You may notice some images loading slow across the Open Collections website. Thank you for your patience as we rebuild the cache to make images load faster.

Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Measurement of the arterial input function from radial MR projections Moroz, Jennifer 2019

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata


24-ubc_2020_may_moroz_jennifer.pdf [ 13.29MB ]
JSON: 24-1.0384552.json
JSON-LD: 24-1.0384552-ld.json
RDF/XML (Pretty): 24-1.0384552-rdf.xml
RDF/JSON: 24-1.0384552-rdf.json
Turtle: 24-1.0384552-turtle.txt
N-Triples: 24-1.0384552-rdf-ntriples.txt
Original Record: 24-1.0384552-source.json
Full Text

Full Text

Measurement of the Arterial Input Function from RadialMR ProjectionsbyJennifer MorozB.Sc., University of Alberta, 2007M.Sc., University of Alberta, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)October 2019c© Jennifer Moroz, 2019The following individuals certify that they have read, and recommend to theFaculty of Graduate and Postdoctoral Studies for acceptance, the dissertation enti-tled:Measurement of the Arterial Input Function from Radial MR Projectionssubmitted by Jennifer Moroz in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in PhysicsExamining Committee:Stefan Reinsberg, PhysicsSupervisorPiotr Kozlowski, PhysicsSupervisory Committee MemberCarl Michal, PhysicsSupervisory Committee MemberSan Xiang, PhysicsSupervisory Committee MemberPurang AbolmaesumiUniversity ExaminerRoger TamUniversity ExamineriiAbstractDynamic Contrast-Enhanced MRI (DCE-MRI) data may be used to non-invasivelyinvestigate the health status of tissue. The technique requires that the concentrationof a contrast agent vs. time curve is known in both the tissue of interest and in ablood vessel feeding the tissue - commonly referred to as the arterial input function(AIF). Physiologically relevant parameters are extracted through Pharmacokineticmodeling, though the accuracy is known to be highly sensitive to the quality of theacquired data. It is difficult to get a good measurement of the AIF in pre-clinicalstudies in mice due to their small body size and limited number of vessels of asufficient size. As a result, several groups use a population averaged curve fromthe literature. This curve does not account for inter or intra-individual differences,and impacts the accuracy of the fit parameters.We propose a new projection-based measurement that measures the AIF froma single trajectory in k-space, which provides a temporal resolution equal to therepetition time (TR). This AIF is measured in the mouse tail due to the simplergeometry void of highly enhancing organs nearby. The projection-based AIF isadvantageous as it allows for the acquisition of DCE data, in the tissue of interest,between measurements without affecting the temporal resolution of either data set.We set up a dual coil experimental platform that acquires AIF data at the mouse tailand DCE data at the tumour. Our technique allows for data optimization at bothlocations, without restricting the temporal or spatial resolutions of the AIF or DCEdata. It may be applied to any pre-clinical study using mice or rats.iiiLay SummaryThis thesis presents a technique to measure the concentration of a MRI contrastagent within a blood vessel during a Dynamic Contrast-Enhanced (DCE) MRIstudy in mice. The blood-based concentration (referred to as the arterial inputfunction (AIF)) is measured from a single acquisition, allowing for a higher tem-poral resolution measurement.DCE-MRI data is analyzed quantitatively through modeling. This requires thatthe contrast agent concentration in the tissue of interest (typically a tumour) andin a blood vessel (AIF) that supplies that tissue are known throughout the durationof the scan. It is challenging to accurately measure both concentration-time curvessimultaneously in mice with a high temporal resolution, so most groups use an AIFfrom the literature. We show that our AIF measurement may be performed simul-taneously with a DCE-MRI study without compromising the temporal resolutionsof either, while also improving the model fits. This technique may be applied toany pre-clinical study performed on mice or rats.ivPrefaceThis dissertation is the original intellectual work of Jennifer Moroz. MRI scans onmice, detailed in chapters 4 and 7, were approved by UBCs animal care committee(certificate numbers A09-0943, A13-0053, A16-0105 and A17-0042).MRI coils were designed and constructed by Andrew Yang. This includes boththe strip-line tail coil used for the arterial input function (AIF) measurement, andthe tumour specific surface coil used to acquired the dynamic contrast-enhancedmagnetic resonance imaging (DCE-MRI) data, described in Chapters 4 and 7. Allanimal tail cannulations were performed by Jennifer Baker or Dr. Stefan Reinsberg.The discussion on contrast agents in Chapter 7 was published as a book chapterin Pre-clinical MRI: Methods and Protocols (Moroz, J, Reinsberg, S A, DynamicContrast-Enhanced MRI, Garcı´a Martı´n M., Lo´pez Larrubia P. (eds) PreclinicalMRI. Methods in Molecular Biology, vol 1718. Humana Press, New York, NY,p. 71-87, 2018). Permission to reproduce this information was granted by Springer.Sections 1.1, 1.3, 3.5 and 4.1 were written by me, while sections 1.2, 3.6 and theremainder of section 4 were written by Dr. Stefan Reinsberg. Sections 2 and 3were a completed collaboratively.The projection-based AIF, presented in Chapter 4, was published in MagneticResonance in Medicine (Moroz, J, Wong, C, Yung, A, Kozlowski, P, and Reins-berg, S A, Rapid measurement of arterial input function in mouse tail from pro-jection phases, MRM, Vol 71, p. 238-245, 2014) and modified for a more in-depthdiscussion. Permission to use the figures in this thesis was granted by John Wileyand Sons through their online Copyright Clearance Center Rightslink service.Andrew Yung and Dr. Piotr Kozlowski provided the motivation for this workand assissted with the protocol development. Dr. Stefan Reinsberg supervised thevstudy, assisted with all MRI scanning and made arrangements for the mass spec-troscopy experiment. Clayton Wong designed the pump phantom used to acquiredcolormetric data to validate the projection-based AIF presented in Chapter 4. Heperformed all analysis of the colormetric data. The results from Clayton’s work areshown in Figure 4.5 a) and c). The same phantom was later used in Chapter 6 totest the radial AIF technique.Work from Chapters 4, 5, 6 and 7 were presented at ISMRM conferences.Published abstracts are found in the Proceedings International Society MagneticResonance in Medicine Journal:• Moroz, J, Yung, A, Kozlowski, P, Reinsberg, S A, Estimation of the ArterialInput Function in a Mouse Tail from the Signal Phase of Projection Profiles,Vol 20, p. 239, 2012• Moroz, J, Kozlowski, P, Reinsberg, S A, Determination of Local Tissue En-hancement from Radially Reconstructed Images, Vol 21, p. 3074, 2013• Moroz, J, Yung, A, Kozlowski, P, Reinsberg, S A, Measurement of a hightemporal resolution AIF: extenstion to radial acquisition to compensate forLocal Tissue Enhancement, Vol. 22, p. 526, 2014• Moroz, J, Yung, A, Kozlowski, P, Reinsberg, S A, Interleaved Acquisition ofa Radial Projection Based AIF with a Multi-slice DCE Experiment, Vol. 23,p. 194, 2015Contributions for these abstract are the similar to the publication above. Dr.Stefan Reinsberg and Andrew Yung assisted in writing the pulse program for radialdata acquisition (second and third abstract) and also for the interleaved AIF-DCEexperiment (fourth abstract). Piotr provided feedback on the results and assisted ininterpreting the results. Dr. Stefan Reinsberg assisted with MRI experiments forthe first and fourth abstracts.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Goals of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 22 Magnetic Resonance Imaging Theory . . . . . . . . . . . . . . . . . 52.1 Creation of MR Signal . . . . . . . . . . . . . . . . . . . . . . . 52.2 Concepts and Properties of MR Signal . . . . . . . . . . . . . . . 72.3 Gradient Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 k-Space and Pulse Sequences . . . . . . . . . . . . . . . . . . . . 172.4.1 Cartesian vs. Radial imaging . . . . . . . . . . . . . . . . 262.5 Contrast Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . 27vii3 Dynamic Contrast Enhanced MRI: Theory and Methods . . . . . . 313.1 Angiogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Methods of Imaging Perfusion . . . . . . . . . . . . . . . . . . . 343.3 Methods: DCE-MRI . . . . . . . . . . . . . . . . . . . . . . . . 363.3.1 Considerations for Setting up a DCE-MRI Scan . . . . . . 393.3.2 The Contrast Agent Injection . . . . . . . . . . . . . . . . 403.4 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4.1 Pharmacokinetic Modeling . . . . . . . . . . . . . . . . . 434 High Temporal Resolution AIFMeasurement using the Phase of MRProjections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1 The Arterial Input Function . . . . . . . . . . . . . . . . . . . . . 484.1.1 Requirement for a High Temporal Resolution . . . . . . . 534.1.2 Phase vs Magnitude Derived AIF . . . . . . . . . . . . . 544.2 Alternative Methods . . . . . . . . . . . . . . . . . . . . . . . . 564.2.1 Dual-bolus . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.2 multi-SRT Measurment with Radial Data . . . . . . . . . 574.2.3 Reference Region . . . . . . . . . . . . . . . . . . . . . . 584.3 Projection-Based AIF Measurement . . . . . . . . . . . . . . . . 594.4 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . 594.4.1 Relationship Between Concentration and Signal Phase . . 614.4.2 Validation: Colorometry with a Flow Phantom . . . . . . 624.4.3 In-vivo Measurements . . . . . . . . . . . . . . . . . . . 634.5 Validation of the Phase-Concentration Relationship . . . . . . . . 664.6 Validation with Colorimetry . . . . . . . . . . . . . . . . . . . . 674.7 Projection-Based AIF in-vivo . . . . . . . . . . . . . . . . . . . . 684.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 Radial MR Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.1 Radial MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Improved Temporal Resolution with Compressed Sensing . . . . . 865.3 Methods of Radial Image Reconstruction . . . . . . . . . . . . . 885.3.1 Regridding . . . . . . . . . . . . . . . . . . . . . . . . . 88viii5.3.2 Spatial-Temporal Constrained Reconstruction . . . . . . . 895.3.3 Non-Equidistant Fast Fourier Transform . . . . . . . . . . 895.3.4 Sampling Schemes for Radial Data Collection . . . . . . . 905.4 Comparison of Radial Imaging Techniques . . . . . . . . . . . . 915.4.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 915.4.2 Reconstruction of Radial Images . . . . . . . . . . . . . . 935.4.3 Results: Fully Sampled Radial Images . . . . . . . . . . . 965.4.4 Radial Reconstructions with Fewer Projections . . . . . . 1015.4.5 Recommended Radial Reconstruction Technique . . . . . 1156 Compensation for Local Tissue Enhancement . . . . . . . . . . . . 1186.1 Local Tissue Enhancement . . . . . . . . . . . . . . . . . . . . . 1186.2 Methods: Simulated Tissue Enhancement Study . . . . . . . . . . 1196.2.1 Simulating Local Tissue Enhancement . . . . . . . . . . . 1196.2.2 Quantifying Tissue Enhancement in Radial MRI . . . . . 1246.3 Quantifying Tissue Enhancement from Radial MR Images: Simu-lation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.4 Results: Effects of Local Tissue Enhancement on the Projections . 1386.5 Correcting the Projection-Based AIF for Local Tissue Enhancement 1506.6 Measuring the Radial AIF with Acquired MRI data . . . . . . . . 1576.6.1 AIF Measurement using MRI Data . . . . . . . . . . . . . 1576.6.2 Concluding Remarks . . . . . . . . . . . . . . . . . . . . 1637 Interleaved AIF and DCE Measurement . . . . . . . . . . . . . . . 1657.1 Interleaving a DCE and AIF Measurement . . . . . . . . . . . . . 1657.1.1 Interleaved AIF-DCE Pulse Sequence . . . . . . . . . . . 1687.1.2 Two-coil set-up . . . . . . . . . . . . . . . . . . . . . . . 1697.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . 1717.3 Results from Interleaved Study . . . . . . . . . . . . . . . . . . . 1717.4 Final Thoughts and Directions for Future Study . . . . . . . . . . 1758 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 178Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181ixA Comparing Radial Reconstruction Techniques . . . . . . . . . . . . 206A.1 Shepard’s Method of Interpolation . . . . . . . . . . . . . . . . . 206A.2 Spatial-Temporal Constrained Reconstruction . . . . . . . . . . . 210A.3 Non-Equidistant Fast Fourier Transform . . . . . . . . . . . . . . 211B Radial Acquisition Correction Techniques . . . . . . . . . . . . . . 220B.1 Observed Issues with Radial Sampling . . . . . . . . . . . . . . . 220B.1.1 Eddy Currents . . . . . . . . . . . . . . . . . . . . . . . 221B.1.2 Gradient Timing Delay Correction . . . . . . . . . . . . . 222B.1.3 Trajectory Measurements . . . . . . . . . . . . . . . . . . 223B.2 Attempts at Correcting the Acquired Radial Data . . . . . . . . . 225B.2.1 Post-processing k-Space to Center Echo . . . . . . . . . . 225B.2.2 Effects of NFFT Node Prescription . . . . . . . . . . . . 233B.3 Pre-Acquisition Techniques to Correct Radial Data . . . . . . . . 235B.3.1 Magnetic Field Shimming . . . . . . . . . . . . . . . . . 235C Radial Projection-Based AIF with Imperfect Radial Data: Simula-tion Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245C.1 Methods: Correcting the AIF for Local Tissue Enhancement . . . 245C.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 251C.2.1 The Radial AIF: Centered in Image-space . . . . . . . . . 251C.2.2 AIF Measurement on an Off-Centered Image . . . . . . . 256C.2.3 AIF Measurement with Distortions in k-Space . . . . . . 260C.2.4 Multiple Vessels within the Phantom . . . . . . . . . . . . 266C.2.5 AIF Measurment with a Box Car Injection . . . . . . . . . 276xList of TablesTable 2.1 Nuclear Spin based on atomic weight and nuclear weight . . . 6Table 2.2 Accumulated Phase for Gradient Moment Nulling . . . . . . . 25Table 5.1 mSSIM index for Shepard’s Method of Interpolation . . . . . . 110Table 5.2 mSSIM index for Spatio-Temporal Constrained Reconstruction 110Table 5.3 mSSIM index for the Non-Equidistant Fast Fourier Transform . 111Table 6.1 Average Phase in Tissue Enhancement Image with 55 Projec-tions Average Phase in Simulation 0.85 ± 0.46 rad . . . . . . 133Table 6.2 Mean Percent Difference in the signal Magnitude of the En-hancement Region from the Radial Images and the Simulation:100 x (Shepard’s Method of Interpolation-Simulation) / Pre-Injection Image . . . . . . . . . . . . . . . . . . . . . . . . . 137Table 6.3 Mean Percent Difference in the signal Magnitude of the En-hancement Region from the Radial Images and the Simulation:100 x (STCR-Simulation) / Pre-Injection Image . . . . . . . . 137Table 6.4 Mean Percent Difference in the signal Magnitude from the En-hancement Region of the Radial Images and the Simulation:100 x (NFFT-Simulation) / Pre-Injection Image . . . . . . . . 137Table 6.5 Percent Difference between Snapshot and Dynamic Profiles inthe Enhancement Area (Time points 93.4-225.8 s, Golden anglesampling) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Table 6.6 Average Percent Difference in Profiles for the Vessel VoxelsGolden Angle Sampling (Time points 58.2-225.8 s) . . . . . . 147xiTable 6.7 Average Difference Between the Corrected Radial AIF (Uni-form Sampling) to the Input AIF from time points 108.1-225.8 s 154Table 6.8 Average Difference Between the Corrected Radial AIF (Goldenangle) to the Input AIF from time points 108.1-225.8 s . . . . . 154Table 6.9 Average Ratio of the Corrected Radial AIF (Uniform Sampling)to the Input AIF from time points 108.1-225.8 s . . . . . . . . 156Table 6.10 Average Ratio of the Corrected Radial AIF (Golden angle sam-pling) to the Input AIF from time points 108.1-225.8 s . . . . . 156Table C.1 Ratio of Concentrations Between the Corrected AIF and theSimulated AIF . . . . . . . . . . . . . . . . . . . . . . . . . . 255Table C.2 Ratio of Concentrations Between the Corrected AIF and theSimulated AIF: Image Centered at [112.5 128.5] . . . . . . . . 256Table C.3 Ratio of Concentrations Between the Corrected AIF and theSimulated AIF: Image Centered at [112.5 128.5] . . . . . . . . 259Table C.4 Ratio of Concentrations Between the Corrected AIF and theSimulated AIF: k-Space shifted by [−1.3−1.3] voxels . . . . . 262Table C.5 Ratio of Concentrations Between the Corrected AIF and theSimulated AIF: k-Space shifted by [−2.6−2.6] voxels . . . . . 263Table C.6 Ratio of Concentrations Between the Corrected AIF and theSimulated AIF: k-Space shifted by [−1.3 − 1.3] voxels, andphantom centered at [112.5 128.5] . . . . . . . . . . . . . . . 264Table C.7 Ratio of the Corrected AIF and the Simulated AIF at the PeakUncorrected AIF ratio: 0.954±0.014 . . . . . . . . . . . . . . 271Table C.8 Ratio of the Corrected AIF and the Simulated AIF in the Wash-out Region Uncorrected AIF ratio: 0.783±0.025 . . . . . . . 272Table C.9 Ratio of the Corrected AIF and the Simulated AIF at Late StageUncorrected AIF ratio: 1.50±0.27 . . . . . . . . . . . . . . . 272Table C.10 Ratio of Concentrations Between the Corrected AIF and theSimulated AIF at washout phase . . . . . . . . . . . . . . . . 276Table C.11 Ratio of Concentrations Between the Corrected AIF and theSimulated AIF at Late Stages . . . . . . . . . . . . . . . . . . 277Table C.12 Box Car Injection Summary Statistics . . . . . . . . . . . . . . 277xiiList of FiguresFigure 2.1 Applied gradient fields for spatial localization . . . . . . . . . 12Figure 2.2 Spin Echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Figure 2.3 Gradient Echo . . . . . . . . . . . . . . . . . . . . . . . . . 20Figure 2.4 Flow Compensation . . . . . . . . . . . . . . . . . . . . . . 24Figure 3.1 Vasculature of healthy tissue and with Angiogenesis . . . . . 33Figure 3.2 T1-weighted DCE images pre and post-injection and the Look-Locker T1 map used to calculate the concentration of contrastagent in the tissue of interest . . . . . . . . . . . . . . . . . . 39Figure 3.3 Pictorial representation of the two-compartment model . . . . 45Figure 3.4 The Tofts Model . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 4.1 Schematic of the projection-based AIF measurement . . . . . 60Figure 4.2 Phantom used to validate the linear relationship between signalphase and concentration of Gd-DTPA diluted in saline . . . . 61Figure 4.3 Setup for the injection line with a 25 µl heparin lock, bolusline, and saline flush . . . . . . . . . . . . . . . . . . . . . . 64Figure 4.4 Calibration factor converting a phase difference into a concen-tration of Gd for projections . . . . . . . . . . . . . . . . . . 67Figure 4.5 Signal-time curves for the colorimetry phantom . . . . . . . . 69Figure 4.6 Image-based AIF in the mouse tail . . . . . . . . . . . . . . . 70Figure 4.7 Projection-based AIF in a mouse tail with a temporal resolu-tion of 100 ms . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 4.8 AIF measured in four individual mice and the population average 73xiiiFigure 4.9 Impact of the injection protocol on the expected shape of the AIF 81Figure 4.10 Effects of diffusion of the contrast agent in the injection line . 82Figure 5.1 Three sampling techniques investigated for Radial Imaging . . 91Figure 5.2 Angular gap spacing between neighboring projections whenGolden angle sampling (111.246o spacing) is used . . . . . . 92Figure 5.3 Reference radial magnitude images (233 projections) recon-structed with one of Shepard’s method of interpolation, Spa-tial Temporal Constrained Reconstruction (STCR) or the Non-equidistant Fast Fourier Transform (NFFT) . . . . . . . . . . 98Figure 5.4 Percent difference between the reference Cartesian image andthe radial reconstructions, relative to the Cartesian image . . . 99Figure 5.5 Structural SIMilarity (SSIM) index maps for the reference ra-dial images . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Figure 5.6 Reference radial phase images (233 projections) reconstructedwith one of Shepard’s method of interpolation, Spatial Tempo-ral Constrained Reconstruction (STCR) or the Non-equidistantFast Fourier Transform (NFFT) . . . . . . . . . . . . . . . . 100Figure 5.7 Radial reconstructions with 55 projections . . . . . . . . . . . 103Figure 5.8 Radial images reconstructed with STCR and NFFT using uni-form or Golden angle sampling . . . . . . . . . . . . . . . . 105Figure 5.9 Mean Structural SIMilarity Index (mSSIM) comparing radialimages reconstructed with fewer projections with the referenceimage containing 233 radial projections . . . . . . . . . . . . 109Figure 5.10 Structural SIMilarity Index (SSIM) maps for radial images re-constructed with 55 projections . . . . . . . . . . . . . . . . 113Figure 6.1 Perfusion and extravasation of the contrast agent into the sur-rounding tissue leads to local tissue enhancement . . . . . . . 121Figure 6.2 Radial data sets, particularly with Golden angle sampling, al-low for more flexibility in selecting data for the image recon-struction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Figure 6.3 Schematic for the Static and Dynamic image series . . . . . . 124xivFigure 6.4 Procedure for calculating the error profile for the tissue en-hancement region . . . . . . . . . . . . . . . . . . . . . . . . 126Figure 6.5 Procedure for correcting the AIF for local tissue enhancement 127Figure 6.6 Images of the simulated tissue enhancement from radial im-ages reconstructed with 55 projections at time points 1730 s . 129Figure 6.7 Images of the simulated tissue enhancement from radial im-ages reconstructed with 34 projections at time points 1730 s . 130Figure 6.8 Average phase within the tissue enhancement region from theradial images . . . . . . . . . . . . . . . . . . . . . . . . . . 133Figure 6.9 Percent difference in the signal between the enhancement im-ages from the radial reconstructions (55 projections) and thesimulated enhancement region . . . . . . . . . . . . . . . . . 135Figure 6.10 Average percent difference in the enhancement region for im-ages reconstructed with 55 projections . . . . . . . . . . . . . 136Figure 6.11 Projections of isolated tissue enhancement, from the differenceof post and pre-injection radial images reconstructed with 55projections . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Figure 6.12 Projection profiles of tissue enhancement for the dynamic andsnapshot image series, and the difference between them. Im-ages were reconstructed with 55 projections from uniformlysampled data. . . . . . . . . . . . . . . . . . . . . . . . . . . 142Figure 6.13 Percent difference between the enhancement profiles from theradial images and from the simulated data set, normalized tothe signal of the pre-injection image: 55 Projections . . . . . 145Figure 6.14 Average percent difference in the signal magnitude, within theregion of the vessel, between the enhancement region of the ra-dial images and the expected signal for the pixels correspond-ing to the location of the vessel . . . . . . . . . . . . . . . . . 146Figure 6.15 Average percent difference in the signal magnitude betweenthe enhancement region of the radial images and the expectedsignal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Figure 6.16 Radial projection-based AIFs measured before and after cor-rection for local tissue enhancement with 55 Projections . . . 151xvFigure 6.17 Radial projection-based AIFs measured before and after cor-rection for local tissue enhancement with 34 Projections . . . 152Figure 6.18 Ratio of the radial projection-based AIF (Golden angle sam-pling), after correction for local tissue enhancement, to the in-put curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Figure 6.19 Measured AIF in the tail phantom using the pump phantom . . 159Figure 6.20 Baseline phase of the vessel signal has an angular dependence,independent of sampling method . . . . . . . . . . . . . . . . 160Figure 6.21 Measured AIF in the tail phantom using the pump phantom . . 161Figure 6.22 Sinograms of the acquired projections, background profilesand difference between the two . . . . . . . . . . . . . . . . . 162Figure 6.23 Sinogram of the phase of the vessel signal and a cross-sectionalplot of the data from within the black box . . . . . . . . . . . 162Figure 7.1 Typical locations for tumour implantation and sucessful AIFmeasurement locations in mice . . . . . . . . . . . . . . . . . 166Figure 7.2 Pulse program for the interleaved AIF and DCE measurement 169Figure 7.3 Schematic for the two coil set-up . . . . . . . . . . . . . . . . 170Figure 7.4 Signal magnitude of the DCE-MRI images of the ROI at slice 2 172Figure 7.5 Tofts model fit to the DCE-MRI data from slice 2 . . . . . . . 173Figure 7.6 Magnitude and phase signal of the mouse tail before and afterthe DCE experiment . . . . . . . . . . . . . . . . . . . . . . 174Figure 7.7 Signal from the vessel and local tissue enhancement from theprojections of the post and pre-injection FLASH images . . . 176Figure A.1 Radial magnitude images created with Shepard’s method ofinterpolation with uniform angluar sampling over 180o . . . . 207Figure A.2 Radial magnitude images created with Shepard’s method ofinterpolation and Golden angle sampling . . . . . . . . . . . 208Figure A.3 Radial magnitude images created with Shepard’s method ofinterpolation and randomly sampled data . . . . . . . . . . . 209Figure A.4 Radial magnitude images created with Shepard’s method ofinterpolation and 55 projections . . . . . . . . . . . . . . . . 210xviFigure A.5 STCR magnitude images reconstructed with 233 (reference),144, 89, 55, 34 or 21 uniformly spaced projections . . . . . . 212Figure A.6 STCR magnitude images reconstructed with 233 (reference),144, 89, 55, 34 or 21 projections and Golden angle sampling . 213Figure A.7 STCR magnitude images reconstructed with 233 (reference),144, 89, 55, 34 or 21 randomly selected projections . . . . . . 214Figure A.8 Magnitude images reconstructed using Spatial-Temporal Con-strained Reconstruction (STCR) with 55 projections . . . . . 215Figure A.9 NFFT magnitude images reconstructed with 233 (reference),144, 89, 55, 34 or 21 uniformly spaced projections . . . . . . 216Figure A.10 NFFT magnitude images reconstructed with 233 (reference),144, 89, 55, 34 or 21 projections, and Golden angle sampling . 217Figure A.11 NFFT magnitude images reconstructed with 233 (reference),144, 89, 55, 34 or 21 randomly selected projections . . . . . . 218Figure A.12 Radial images reconstructed with the NFFT and 55 projections 219Figure B.1 Magnitude and Phase of the vessel data (acquired projections- background), and the corresponding radial projection-basedAIFs: k-space centering corrections . . . . . . . . . . . . . . 227Figure B.2 Magnitude and phase of the reconstructed NFFT image of acylindrical phantom . . . . . . . . . . . . . . . . . . . . . . . 228Figure B.3 Effects on the baseline phase after centering the k-space data:First order phase shift has little effect . . . . . . . . . . . . . 229Figure B.4 Effects on the baseline phase after centering the k-space data:First order phase shift has significant effect . . . . . . . . . . 230Figure B.5 Real, imaginary, magnitude and phase signal of the acquiredprojection profiles, the background profiles from the NFFT im-ages and the vessel signal . . . . . . . . . . . . . . . . . . . . 232Figure B.6 Radial projection-based AIF was calculated from an idealizeddata set: Input from projection of NFFT image . . . . . . . . 234Figure B.7 k-Space and projection data with a good shim and poor shim . 236Figure B.8 Pulse Sequence for Radial Projection-based AIF showing trimdefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237xviiFigure B.9 Echo position after introducing small changes in the gradientstrengths in the readout direction . . . . . . . . . . . . . . . . 238Figure B.10 Gradient timing delays for the radial AIF pulse program, forthe three standard slice orientations and Golden angle or uni-form angular sampling . . . . . . . . . . . . . . . . . . . . . 239Figure B.11 Trajectory measurements followed the method by Beaumont . 241Figure B.12 Distance of the measured trajectory from the center of k-space(on the log10 scale) as the readout gradient strength variesfrom its maximum positive value to its maximum negative value 242Figure B.13 Trajectory measurement following the method by Latta . . . . 243Figure B.14 Vessel Signal and phase baseline with the Latta trajectory lo-cations used in the NFFT image reconstruction . . . . . . . . 244Figure C.1 Simulated tail phantom with a single vessel in the top left handcorner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247Figure C.2 Schematic of the radial-based AIF measurement . . . . . . . 249Figure C.3 Schematic for the local tissue enhancement correction . . . . 250Figure C.4 Radial projection-based AIF before and after applying the lo-cal tissue enhancement correction: 144 Projections . . . . . . 252Figure C.5 Radial projection-based AIF before and after applying the lo-cal tissue enhancement correction: 55 Projections . . . . . . . 253Figure C.6 Radial projection-based AIF before and after applying the lo-cal tissue enhancement correction: 34 Projections . . . . . . . 254Figure C.7 Ratio of the corrected AIF to the simulated AIF . . . . . . . . 255Figure C.8 Radial projection-based AIF measured from a phantom off-center in image space . . . . . . . . . . . . . . . . . . . . . . 257Figure C.9 Ratio of the corrected AIF to the simulated AIF: off centre inimage space . . . . . . . . . . . . . . . . . . . . . . . . . . . 258Figure C.10 Tissue enhancement corrected AIFs for a phantom centered inimagespace with those off-center . . . . . . . . . . . . . . . . 259Figure C.11 Radial projection-based AIF measured for a shift in k-spacecenter (shifted by 1.3 voxels in both dimensions from centre) . 261xviiiFigure C.12 Radial projection-based AIF measured for a shift in k-spacecenter (shifted by 2.6 voxels in both dimensions from centre) . 263Figure C.13 Digital phantom with 4 vessels . . . . . . . . . . . . . . . . . 267Figure C.14 Initial AIF measurement in vessel 1 . . . . . . . . . . . . . . 268Figure C.15 Corrected AIF measurement using the radial projection-basedmeasurement . . . . . . . . . . . . . . . . . . . . . . . . . . 269Figure C.16 Concentration ratios of the corrected AIF to the simulated AIF:multi-vessel . . . . . . . . . . . . . . . . . . . . . . . . . . . 270Figure C.17 Corrected AIF using the radial projection-based measurement:multi-vessel, alternating phase . . . . . . . . . . . . . . . . . 274Figure C.18 Concentration ratios of the corrected AIF to the simulated AIFfor a multi-vessel experiment where the phase direction alter-nates between vessels . . . . . . . . . . . . . . . . . . . . . . 275Figure C.19 Radial projetion-based AIFs of an ideal rectangular injectionbolus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278Figure C.20 Concentration ratios of the radial projection-based AIF to thebox car AIF for a multi-vessel experiment . . . . . . . . . . . 280xixGlossaryAIF arterial input functionASL arterial spin labelingCS compressed sensingCT computer-assisted tomographyDCE dynamic contrast-enhancedDCE-MRI dynamic contrast-enhanced magnetic resonance imagingDFT discrete Fourier transformDSC-MRI dynamic susceptibility contrast magnetic resonance imagingEES extravascular extracellular spaceEPI echo planar imagingFFT fast Fourier transform (image space to k-space)FID free induction decayFLASH Fast Low Angle SHotFOV field of viewFT Fourier transformGE gradient echoxxGRE gradient echo pulse sequenceGMN gradient moment nullingIFT inverse Fourier transform (k-space to image space)IDFT inverse discrete Fourier transformIFFT inverse fast Fourier transform (k-space to image space)ICA independent component analysisIR inversion recoveryMR magnetic resonanceMRI magnetic resonance imagingmSSIM mean Structural SIMilarity indexNFFT Non-equidistant Fast Fourier TransformPK Pharmaco-kineticPVE partial volume effectsRF radio-frequencyROI region of interestSR saturation recoverySRT saturation recovery timeSNR signal-to-noise ratioSPGRE spoiled gradient recalled echoSSIM Structural SIMilarity indexSTCR Spatial-Temporal Constrained ReconstructionTE echo timexxiTOI tissue of interestTR repetition timeVFA variable flip anglexxiiAcknowledgmentsThank-you to Stefan Reinsberg and Piort Kozlowksi, my thesis supervisors, forsupporting my work for this degree. Without their support and guidance, this wouldnot have been possible.Thank-you to my family for their support in getting me through this degree.It is no easy feat, but your belief in my ability to achieve my dreams made itmanageable. Another thank-you to my boyfriend, Winston, who looked after methrough all the highs and lows that come with a PhD. You kept me focused throughtriathlon and running races so that I could attach my code during the week.I have been fortunate to have many friends support my academic and athleticdreams over the past 8 years. This includes Tiffany, Brendan, Katherine, Brendan,Shauna, Sarah, as well as everyone at the Fraser street run club and UBC triathlonclub.xxiiiChapter 1IntroductionThe ability to image tissue vasculature non-invasively has applications in identi-fying the presence of a disease (such as cancers), determining the extents/stageof the disease, and monitoring the response to treatment [1, 2]. Positron emis-sion tomography (PET), single photon emission computed tomography (SPECT)and contrast-enhanced computed tomography (CT) have all been used for this pur-pose [3, 4]; however, these methods expose the patient to ionizing radiation [2, 5].Magnetic resonance imaging (MRI) uses time varying magnetic fields to produceimages, and therefore allows a non-invasive technique for imaging soft tissue [6].The field of dynamic contrast-enhanced MRI (DCE-MRI) has grown substan-tially over the last decade to evaluate the health status of tissue non-invasively [7,8]. With its growing popularity in cancer research [9, 10], requirements for datawith concurrent high spatial and temporal resolutions have become apparent [11].DCE-MRI data is analyzed quantitatively through pharmaco-kinetic (PK) model-ing [12]. Most models, however, require that the concentration-time curves areaccurately characterized in the tissue of interest, as well as in a vessel feeding thetissue of interest [1] - which is commonly referred to as an arterial input function(AIF) [13].Individually acquired AIFs are difficult to measure in mice due to their smallbody size [14, 15], rapid heart rate [13, 16] and the limited number of vessels of asufficient diameter for accurate characterization. For these reasons, murine-basedstudies often use a population averaged AIF [17–19] in their analysis. The popula-1tion averaged curve is expected to approximate the true curve, but does not accountfor inter [13] or intra-individual [5, 8] differences. In addition, the population aver-age AIF may only be accurate for a specific injection protocol, contrast agent doseand strain of animal.1.1 Goals of this ThesisOur lab has relied on a mathematical representation of the AIF from Lyng [19] tomodel dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) data.The fitted curve does not have a well defined injection time, so there is a degreeof interpretation when this occurs. A typical DCE-MRI study will have a temporalresolution of a couple seconds. The temporal resolution of the AIF from Lyng, at13 s, is not sufficient for accurate modelling.Limitations in measuring a high temporal resolution AIF in mice have leadgroups to use a population AIF from the literature. Though this curve may be ap-propriate, it does not account for inter or intra-individual differences, nor variationsin the injection protocol. Even if the AIF has a high temporal resolution, the re-sults from modelling may not be specific to the individual. For this reason, it isdesirable to acquire the AIF and DCE data simultaneously. Such studies have beenperformed in mice by Pathak et al. [20] and in rats by McIntyre [21]).The goal of this thesis is to show that a high temporal resolution AIF may be ac-quired in a mouse tail using a projection-based measurement. Since the AIF wouldonly require one line of k-space per measurement, the temporal resolution equalsthe repetition time. A second advantage is our ability to acquire DCE data betweenconsecutive AIF measurements. We show that an interleaved AIF-DCE acquisitionmay be implemented without sacrificing the temporal resolution of either the AIFor DCE data, and may be applied to any mouse or rat experiment.Chapter 2 will introduce magnetic resonance imaging (MRI) and the theoryrelevant for this thesis. This includes the creation of MR signal, how images areconstructed and the physics behind contrast enhancement with a contrast agent.The physics and techniques of DCE-MRI is the subject of Chapter 3. TheChapter opens with a brief discussion of angiogensis - the process in which a tu-mour rapidly develops new blood vessels to supply proliferating cells with nutri-2ents and oxygen. DCE-MRI evaluates the health status of tissue by tracking thedistribution of a contrast agent within a tissue of interest temporally. The primarytechniques of studying perfusion and permeability with MRI are discussed, with aprimary focus on the concepts and methods related to DCE-MRI. The chapter con-cludes with a summary of semi-quantitative and quantitative analysis techniques.We introduce a projection-based arterial input function (AIF) measurement inChapter 4. The measurement is performed with a single line of k-space, and usesthe phase of the MR signal to estimate the concentration of a contrast agent withina vessel (both in a phantom and in-vivo). The Chapter outlines three experiments.The first experimentally measured the phase-concentration conversion factor forour scanner and pulse sequence. The second experiment measured a projection-based AIF within a pump phantom, which was cross-validated using colorimetry.The third measures a projection-based AIF in a mouse tail. The late stage concen-tration was validated with mass spectrometry 20 min post injection in four mice.The permeable nature of capillary walls means that some contrast agent maydiffuse from within the vessel to the surrounding tissue. This could bias the projection-based AIF if the projection is acquired at the same angle for the duration of theexperiment. We propose to extend the measurement to include MR data from mul-tiple angles. Chapter 5 studies three radial reconstruction techniques: interpolationof the radial data onto a Cartesian grid using Shepard’s method of interpolation,Spatial-Temporal Constrained Reconstruction (STCR) and Non-equidistant FastFourier Transform (NFFT). The goal of this chapter is to determine which methodwould be best suited for our application. Since the projection-based AIF is calcu-lated from the difference of two complex signals, both the magnitude and phaseinformation should be preserved in the reconstructed image.Chapter 6 summarizes the results from a simulation study involving radial pro-jections and compensating for contrast perfusion into the surrounding tissue. Thecontrast agent causes an increase in the signal intensity locally, in T1-weightedmagnetic resonance (MR) images, which is referred to as local tissue enhancement.This chapter compares the three radial reconstruction techniques for their ability toreproduce local tissue enhancement within the images. The proposed local tissueenhancement correction uses the projection of these reconstructed images to cor-rect the acquired data. This involves calculating the difference between the pre and3post-injection background signals, and compensating the acquired data with thisinformation. The chapter concludes with the measurement of a projection-basedAIF, after correcting the projections for local tissue enhancement.The interleaved AIF-dynamic contrast-enhanced (DCE) experiment is outlinedin Chapter 7. The chapter opens with a brief description of the pulse sequenceand set-up used for this study. The interleaved experiment was performed both inphantom as well as in-vivo, to validate the technique and evaluate its application ina pre-clinical studies in mice or rats. Recently, most AIFDCE! (AIFDCE!) studieshave avoided the AIF measurement in favour of improved spatial and temporalresolutions at the tissue of interest. Our method could improve the accuracy of themodel fit parameters as the [! ([!)AIF will is specific to the animals physiology atthe time of imaging and the injection protocol used. The use of projections willallow for high temporal resolution measurements for both the [!AIF and tissue ofinterest.4Chapter 2Magnetic Resonance ImagingTheoryX-ray, computer-assisted tomography (CT) and ultrasound were traditionally usedfor imaging internal anatomy [6]. However, these techniques use ionizing radia-tion to produce images, raising concerns for procedures involving repeat imagingsessions [22]. MRI has the advantages that there are no known side-effects fromexposure to the external magnetic or gradient fields, it provides excellent tissuecontrast [23] and places no restrictions on the image orientation [6, 24]. MR imag-ing has been used extensively in clinic to diagnose a broad range of disorders [5].While MRI is based on quantum mechanics, the concepts reduce to classicalmechanics at the macroscopic level [25]. As a result, a majority of MRI theorymay be understood from a classical perspective.2.1 Creation of MR SignalMagnetic resonance is based on the interaction of a proton - possessing a spinand charge - with an external magnetic field [26]. The nuclear spin is an intrinsicproperty of an atom related to its angular momentum. It can take on integer andhalf-integer values, depending on its atomic number (number of protons) and theatomic mass of the nucleus (protons and neutrons) [26]. Table 2.1 summarizes thenuclear spin for atoms of odd and even atomic and nuclear weights.5Table 2.1: Nuclear Spin based on atomic weight and nuclear weightAtomic Number Nuclear Weight Spineven even 0 - no interactionodd even integer spinodd odd half integer spinThe hydrogen atom is used in most MRI applications due to its high relativeabundance (88 M, compared to 80 mM for other atoms like sodium and phos-phorous) in the human body and spin 1/2 [26]. Other common atoms for in-vivoimaging are 13C, 23Na and 31P, which have spins of 12 ,32 and12 respectively [27].When a nucleus with non-zero spin is placed in a static magnetic field, Bo, itprecesses around the field at a constant rate, known as the Larmour frequency:ωo = γBo (2.1)Where ωo is the Larmour frequency (in MHz), γ is the Gyromagnetic ratio (inMHz/T) [28], and Bo is the strength of the external magnetic field. For hydrogen,γ = 42.6 MHz/T [6, 29].The spin creates a constant magnetic moment, which in turn induces a localmagnetic field along the axis of rotation. In the absence of an external magneticfield, the individual magnetic moments are randomly oriented, resulting in a netmagnetization of zero. When placed in an external magnetic field, more mag-netic moments will preferentially align with the main magnetic field, providing anobservable net magnetization along its axis [25]. This interaction between the pro-ton’s magnetic moment and the magnetic field creates magnetic resonance [30].The Zeeman effect states that coupling between a proton and a magnetic field hasquantized values. Since the hydrogen atom has a spin 1/2, there are two uniquestates; spin-up and spin-down.The coupling causes a difference in energy between the two states. A majorityof protons will align parallel to the field (lower energy), resulting in a net magneti-zation described by the Boltzman distribution.6NdownNup= e−∆E/KBT (2.2)Where Ndown is the number of protons aligned anti-parallel to the magneticfield, Nup is the number of protons aligned parallel to the magnetic field, ∆E is theenergy difference between the two states, KB (= 1.381x10−23 J/K) is the Boltzmanconstant, and T is the temperature in K.At room temperature, the difference in protons in each orientation is very small(about 45 out of 10 million at 1.5 T) [30]. However, the proton density in tissue isvery large, resulting in a measurable net magnetization. The vector summation ofall protons provides the net magnetization, denoted as Mo.The next three sections will discuss how we can manipulate the net magnetiza-tion to construct images.2.2 Concepts and Properties of MR SignalThis section cites information from [30], [27] and [31]. Refer to these referencesfor a more in-depth discussion.Signal ExcitationBy convention, the main magnetic field is set along the +Z−axis and the receivercoil oriented such that it can detect a changing magnetic flux in the sample inthe X −Y plane. The net magnetization is tipped into the X −Y -plane with anoscillating radio-frequency (RF) pulse, which is referred to as the B1 pulse. The B1pulse is oriented perpendicular to Bo. The protons in the sample absorb energy fromthe radio-frequency (RF) pulse if the frequency of the RF pulse exactly matches theLarmour frequency [31]. The RF pulse contains a narrow range or bandwidth offrequencies, centered around a central frequency.Since the frequency of the B1 pulse matches the Larmour frequency, the pre-cessing protons will see it as stationary in its frame of reference. We refer to thisas the rotating frame of reference and define its axes with x, y and z (comparedto X, Y and Z for the laboratory frame). Under these conditions, the RF pulseapplies a torque on the magnetization, causing it to rotate towards the x− y-plane.7By convention, the B1 pulse is oriented along the x−axis in the rotating referenceframe [31]. The duration and magnitude of the pulse affects the flip angle, α , atwhich the magnetization tips. The flip angle is determined from [30].α =τ∫0γB1dt (2.3)Where B1 is the strength of the B1 pulse and τ is the duration of the pulse. Theapplication of the B1 pulse forces all individual magnetization vectors to initiallyhave the same phase [32].Once in the x− y-plane, the transverse magnetization (Mxy), continues to pre-cess around the main magnetic field, at a rate of ωo. From Faraday’s law of induc-tion, a changing magnetic flux - such as the spinning magnetization - will induce anelectromotive force in a nearby wire loop. This induces a current in the wire, whichallows the signal to be recorded. The wire loop is referred to as a receiver coil, andmay be designed to optimize the signal-to-noise ratio (SNR) of the anatomy ofinterest.The acquired signal is commonly referred to as the free induction decay (FID).The initial magnitude is characterized by the strength of the net magnetizationimmediately following the B1 pulse, and is a complex summation of all protonswithin the sample. This means that the signal contains multiple frequencies as aresult of variations in the magnetic environment throughout the sample [31]. Thesefrequencies may be extracted using a Fourier transform.MR signal is continuous in nature, but is sampled at discrete points (digitized)with an analog-to-digital converter (ADC) [31]. Sampling at definitive points al-lows for post-processing techniques, such as the fast Fourier transform, to be per-formed. However, sampling by the ADC limits the range of frequencies that may beresolved. If the sampling rate is not sufficient, then higher frequency componentswill be wrapped to a lower frequency, such that fobs = mod( factual, fmax) (wherefobs is the observed aliased frequency, factual is the actual input frequency and fmaxis the maximum frequency that may be resolved by the ADC). This effect is knownas aliasing. To reduce the impacts of aliased signal, the phase-coherent differencesignal between the FID and input RF (frequency and phase) is digitized instead.8This digitized signal is now measured relative to the transmitted frequency, ωT R,and is equivalent to collecting signal in the rotating frame of reference [31].The number of samples taken from the FID and the total sampling time areuser defined. These parameters will impact the maximum frequency that may beaccurately represented, which is referred to as the Nyquist frequency, ωNQ. TheNyquist frequency is defined as [31]:ωNQ =NFID2 ·Ts (2.4)Where NFID is the total number of samples taken from the FID and Ts is thetotal sampling time. Any acquired frequency beyond the Nyquist frequency will beindistinguishable with its mod(ω ,2pi). As a result, the frequencies will be wrappedto the lower frequency. To avoid ghosting artifacts from wrapping, a low pass filtermay be used prior to digitization. This can also improve the SNR of the signal asmost of the higher frequencies are due to noise.Relaxation and the Bloch EquationsThe process of relaxation is well documented ([30], [27], [33], [24]) and is funda-mental to the observed signal and contrast in MRI images. Two processes are ofinterest: the creation of a net magnetization in the direction of the external mag-netic field, and the loss of precessing signal orthogonal to the field.When the B1 pulse is turned off, the protons experience only the main staticmagnetic field, Bo. This means that the individuals spins will eventually return totheir equilibrium state, resulting in a loss of signal in the x−y-plane, and rebuildingof the magnetization along the +z− axis. The time required for both effects isdependent on the tissue type, thus providing tissue contrast in MRI images.The time required to rebuild the net magnetization along the+z−axis is knownas T1 relaxation, or the spin-lattice relaxation time. T1 relaxation is dependent onthe proton’s interaction with its environment. An excited spin will release energyto its environment and return to a lower energy state (with preferential alignmentparallel to the external magnetic field). The governing formula for this process isdefined by the Bloch equation, stating that:9dMzdt=−Mz−MoT1(2.5)Solving this differential equation for Mz, and assuming complete relaxation toregain magnetization Mo (a time interval of 5T1 is recommended for the tissue ofinterest), we get:Mz = Mo(1− exp−t/T1)+Mz(t = 0)exp−t/T1 (2.6)Special cases for this equation are for saturation recovery, in which Mz(T =0) = 0 due to a 90o RF pulse and an inversion recovery experiment, where Mz(t =0) =−Mo. The T1 value for free water is approx. 4 s [34]. As the proton’s environ-ment gets more structured, a proton-lattice interaction is more probable, resultingin shorter T1 for tissues.Signal loss in the x− y-plane results form loss of coherence within Mxy, and isreferred to as T2 relaxation or spin-spin relaxation.The Bloch equation describing the overall T2 relaxation in the rotating frameof reference is:dMxydt=−MxyT2(2.7)Here, Mxy is a 2-dimensional vector representing the transverse magnetizationin the x− y-plane. Solving the equation for Mxy gives us:Mxy = Mxy(0)exp−t/T2 (2.8)Where Mxy(0) is equal to longitudinal magnetization prior to the RF excitationpulse. In cases where the flip angle is not 90o, the transverse equation will includea cos(α) term, where α is the flip angle.There are two processes involved with signal loss in the x− y plane. The firstprocess deals with the energy transfer, ∆E, between neighboring spins in oppos-10ing energy states or from diffusion of the spin to an area with a different Bo. Thisis a non-reversible process, and is governed by the T2 relaxation time constant.The second process is due to magnetic field inhomogeneities. Nearby protons mayexperience a slightly different magnetic environment, and therefore precess at dif-ferent Larmor frequencies. In effect, phase coherence of the protons is lost, assome protons spin faster than the bulk magnetization and others precess slower.This process is static, reversible, and defined by T′2 . The combined relaxation fromreversible and non-reversible effects is defined as T ∗2 , and follows the relation:1T ∗2=1T2+1T ′2(2.9)2.3 Gradient FieldsWhen a radio-frequency (RF) pulse is applied, only the spins that resonate at afrequency within the bandwidth (BW) of the pulse are excited. It is possible tospatially manipulate the magnetic environment experienced by the object, so thatonly a small band of spins resonate at the correct frequencies. This is achievedwith magnetic gradients.The gradient coils adjust the magnetic field strength spatially along the physicalX, Y and Z-axes of the scanner and allow for spatial encoding [33]. They causea linearly varying magnetic field, originating at the isocenter of the MR scanner(see Figure 2.1). The gradient fields are all oriented in the same direction as Bo,such that the resonant frequency of the protons varies in a known (linear) fashion.This information is encoded within the FID signal. It is important to note that thestrength of the gradient fields is much smaller than the external magnetic field,so any field oriented in the X-Y plane is negligible [33]. The gradient fields aremeasured in units of mT/m.The gradient vector may be written as:~G =∂BZ∂XXˆ +∂BZ∂YYˆ +∂BZ∂ZZˆ = GX Xˆ +GY Yˆ +GZZˆ (2.10)Where GX ,Y,Z are the gradient strengths in the X, Y and Z directions respec-tively and ∂BZ/∂X ,Y,Z represents the linear variation in the magnetic field in the11Figure 2.1: Applied gradient fields alter the magnetic field strength linearlyalong the +Z-axis, to allow for spatial localization (exaggerated to showthe effect as ~Bo Gr~r, where~r is the spatial distance from isocenter).The net magnetic field becomes: ~Bnet = ~Bo +Gr~r. There are threephysical gradients that affect the magnetic field strength in the X, Y andZ directions independently. In doing so, the Larmor frequency, whichis dependent on ~Bnet , becomes spatially dependent. The acquired FIDcontains information about all frequencies present in the sample.respective direction. These gradients are generated from three separate coils [33].The gradient fields may be used for slice selection, frequency encoding orphase encoding. This will be the topic of the next section.Slice SelectionMR slice selection is achieved by applying a gradient field, perpendicular to thedesired slice plane, for the duration of the RF excitation pulse. In MRI, the slicesmay be oriented in any direction, allowing for oblique slices [33]. This requires12that two or more of the physical gradient coils are in use. The magnitude of eachis dependent on the angulation of the desired slices relative to the +z-axis [33].The slice-select gradient can introduce phase dispersion of the magnetizationin the slice [33]. This is a combined response of the magnetization to the sliceselect gradient and the shape of the RF pulse. A slice-refocusing, or rephasing,lobe is applied after excitation to compensate for the phase dispersion. This lobehas the opposite polarity of the slice select lobe, and an area identical to the centerof the excitation pulse [33]. For a symmetric RF excitation pulse and slice selectgradient, the area of the rephase lobe is half that of the slice select lobe.The RF excitation pulse has a pre-specified bandwidth (BW). Any spin thatprecesses with a Larmour frequency in that range will be excited. The slice thick-ness, ∆z and the RF pulse bandwidth, ∆ f , are related as:∆z =2pi∆ fγGz(2.11)Where γ is the gyromagnetic ratio and Gz is the strength of the slice selectgradient. Thinner slices are possible by either increasing the strength of the sliceselect gradient or reducing the BW (using a longer excitation pulse).For slices off-set from the magnet isocenter, the RF frequency must be adjustedto match the central Larmor frequency of the slice. The adjustment, δ f can becalculated from:δ f =γGzδ z2pi(2.12)The excitation pulse is often a sinc or Gaussian pulse, which creates a rectan-gular or Gaussian slice profile.Frequency-encodingFrequency encoding, or the read-out, allows for spatial information to be deter-mined from the sample. This is done by applying a linearly varying magnetic fieldacross the sample, orthogonal to the slice direction, which causes the precessionalfrequency to vary linearly [33]. The spatial locations of all spins within the samplemay be determined from the FT of the acquired time-domain signal. Frequency-13encoding may be applied in any direction, though it is typically oriented perpen-dicular to the slice for imaging purposes. By convention, frequency-encoding isalong the x-axis.The frequency-encode gradient waveform typically consists of two lobes [33].The first is a prephasing gradient, and the second is the read-out gradient. Thepurpose of the pre-phase gradient is to prepare the magnetization to form an echoat a later time. The prephase gradient provides a linearly varying magnetic field,causing some isochromats, defined as a cluster of spins resonating at the samefrequency, to precess at a faster rate than others. The net result is a linearly varyingphase accumulation across the sample. When the readout gradient is applied, theisochromats will gradually rephase to produce an echo when the gradient areas ofthe two lobes are equal. The receive coil is on during the readout window so thatthe echo may be acquired.The process of refocusing the spins differs between a spin echo and gradientecho. This will be discussed in greater depth in the next section on pulse sequences.Phase-encodingPhase-encoding provides spatial information about the sample orthogonal to thefrequency-encode direction [33]. This is performed by applying a gradient lobebetween the initial excitation pulse and the readout window. Similar to frequency-encoding, a linear gradient field is applied across the sample, causing isochromatsat different spatial locations to accumulate phase at different rates. The spatialinformation may be determined after application of the Fourier transform. By con-vention, phase-encoding is performed along the y-axis.Phase-encoding localizes signal in a second (or third) orthogonal dimensionthrough spatially-varying phase accumulation. The phase-encode gradient fieldcan be applied concurrently with other gradients, with the exception of the sliceselect and acquisition gradients. To maximize signal consistency between phase-encode steps, many pulse sequences will rewind the phase-encode gradient afterthe readout [33].The phase-encoding gradient is applied N times, where N is the desired num-ber of samples in this direction. The gradient strength may take on multiple values14from the maximum, Gy, to the minimum, −Gy, in equal sized steps (∆y= 2Gy/N).The equal step size allows for a more uniform coverage of k-space so that tech-niques, such as the IFFT, may be applied directly to the data. The value of Gy isdependent on the desired field of view.In 3-D imaging, phase-encoding occurs in two directions; the second of whichis along the slice select direction. This form of imaging greatly enhances the spatialresolution in the slice select direction, but also leads to much longer scan times.Care must be taken to avoid aliasing of the signal into the image. This is not aproblem if the Nyquist criterion is satisfied:∆ky ≤ 1N∆y (2.13)Where ∆ky is the phase encoding step size, ∆y is pixel size, and N is the numberof phase encode samples. The product N∆y is equal to the FOV in the phase-encoding direction.The Fast Fourier TransformMR data is acquired in the frequency domain, which is commonly referred to as k-space. The k-space data is converted to an image through application of the inverseFourier transform (k-space to image space) (IFT). The k-space acquisitions areknown as trajectories, and are composed of a linear combination of signals withinthe slice [6]. Due to the read and phase-encoding gradients, the magnetization ateach location will precess at a different frequency. The IFT extracts the frequencyinformation, and maps it to a location in the image.The Fourier transform (FT), and its corresponding IFT, are defined as [35]:F(ω) =∫ ∞∞f (x)e−iωxdx (2.14)f (x) =12pi∫ ∞∞F(ω)eiωxdω (2.15)Where ω are the frequencies within the k-space signal, x is the spatial location,and i=√−1. These equations are used for continuous signals. MR data, however,is sampled at discrete locations by means of an analog to digital converter [36].15The discrete Fourier transform (DFT) accounts for discrete and periodic signals,mapping the finite set of uniformly spaced sampled onto a uniformly sampled gridin its conjugate space [27]. This is achieved by replacing the integral with a sum-mation at the known sampling positions. The equation for the 1-D DFT and inversediscrete Fourier transform (IDFT) are:Ak =N−1∑j=0a je−i2pik j/N (2.16)a j =N−1∑k=0Akei2pik j/N (2.17)Where Ak is the discretely sampled (complex) data, a j is the image-space data,k is the is the discrete frequencies, j (=0, . . . N-1) is discrete sampling position, andN is the number of samples acquired. The exponential term may be re-written asW jk, where W (= e2pii/N) is the Nth root of unity.The frequency information throughout all of k-space will contribute to eachindividual pixel in the image. The 1-D inverse fast Fourier transform (k-space toimage space) (IFFT) provides the spatial proton density of a projection, while the2-D IFFT produces an image of the sample with proton density information in bothdirections. This is a separable function, so the 2-D (or 3-D) IFT may be applied astwo (or three) 1-D IFTs; one along each matrix dimension [33].The DFT computes an order of N2 iterations per image [35]. A faster techniqueis the fast Fourier transform (image space to k-space) (FFT) (or IFFT), which calcu-lates the DFT through a sequence of algebraic manipulations [36, 37]. This reducesthe number of iterations to NlogN, thus allowing for significant improvements incomputation time [37]. The FFT/IFFT requires that the number of samples is aneven number, though it is most efficient with 2n samples (n=0,1,2,. . .). This is aconsequence of the discrete FFT operating on signal pairs [35] to reduce the num-ber of computations. Acquisitions with a sampling size that is not 2n can takeadvantage of the FFT by zero-filling the matrix to the next power of 2.A property of the discrete FT states that if the signal is discretely sampled inone domain, then it will be periodic in the other [33]. Since the data in k-spaceis discretely sampled, the image-space signal will become as a series of replicates16with a period of N samples. If the acquired data does not satisfy the Nyqist cri-terion, which states that the data sampling rate is at least double the greatest fre-quency found in the signal, then the image-space replicates overlap. This leads toan aliasing artifact.2.4 k-Space and Pulse SequencesThe time-varying signals in MRI may be analyzed by tracking the trajectories in a2-D or 3-D space. This space is the Fourier conjugate of the spatial domain and isreferred to as k-space [33]. The k-space domain can improve our understanding ofpulse sequences as it shows how the MR signal transverses the Fourier domain.MR images are constructed from the k-space trajectories [33]. k-Space data isonly filled when the acquisition window is active, though it is possible to still trans-verse k-space without it. The rate at which we transverse k-space is determinedfrom the gradient strength and the gyromagnetic ratio (|d~k/dt|= |γG|/(2pi)). Thetotal distance covered in k-space is then equal to the area under the gradient wave-form. k-Space may be interpreted as the rate at which a stationary spin accumulatesphase (measured in cycles/meter) under the influence of a gradient[33].The magnitude of the acquired signal is dependent on the repetition time (TR)and the echo time (TE) selected for the scan. The repetition time is defined asthe time interval between successive excitation pulses. Depending on the pulsesequence, TR may be on the order of 10 ms to several seconds [38]. Setting TRto a larger value will allow for more magnetization to rebuild along the +z-axis,and thus improve the SNR. The echo time is defined as the time interval betweenthe center of the 90o excitation RF pulse and the time at which an echo reachesits maximum [39]. This timing is typically set to a value on the order of 1 ms to100 ms [38]. The echo may be achieved using a spin echo or gradient echo, bothof which are described in greater detail in the next sections.k-Space has its maximum intensity at the center, and is symmetric about thispoint. In a majority of applications, the MR signal is acquired as either a spin echoof a gradient echo. This is attractive as data from both sides of k-space is acquired,and the center of k-space can be determined with greater precision. This is a di-rect result of having a non-zero frequency encode gradient on during the readout17window, which affects the precessional frequency (and hence phase accumulation)spatially.Spin EchoSpin echo imaging is popular for T1-weighted applications or for parallel imag-ing techniques due to the improved SNR at the echo [33]. The spin echo involvesthe application of a second, refocusing pulse to rephase the signal. The refocus-ing pulse may be oriented along either the +x-axis or the +y-axis in the rotatingframe [40]. The effect is the same, but the echo will either alternate signs (+y-axis,-y-axis,...) or always be along the +y-axis. The discussion below and in Figure 2.2uses a refocusing pulse along the +x-axis.For maximum echo signal, a 180o pulse is played out at a time of TE/2. Priorto the refocusing pulse, the magnetization will dephase due to magnetic field in-homogeneities. In effect, some spins will experience a higher magnetic field andwill rotate faster than the Larmor frequency, while others will experience a lowermagnetic field and rotate slower than the Larmor frequency. This results in a fan-like pattern in the rotating frame, centered along the +y-axis. The refocusing pulseflips the magnetization across the x-axis, such that the fan-like pattern is now cen-tered along the -y-axis. The spins that were spinning fast continue to spin fasterand gradually approach the -y-axis, while the slower precessing spins continue torotating slower. At the echo time, TE, all spins will be aligned along the -y-axiscreating an echo. The image contrast is provided by the factor e−T E/T2 . The spinecho pulse sequence is summarized in Figure 2.2.The pulse sequence may be designed with the two lobes on either side of therefocusing pulse, or both following the pulse [27]. When the refocusing pulse ispositioned between the two lobes, the gradient lobes will have the same polarity.This design allows for a shorter echo time. In the alternative, the gradient lobeshave different polarity. Though the echo time is longer, the echo is less sensitive toflow [27]. With a simple spin-echo pulse program, only one echo is acquired withina TR. This could result in very long scan times for T2-weighted images due to therequirement to rebuild Mo. Multiple spin echos may be achieved with multiplerefocusing 180o pulses.18Figure 2.2: Pulse program for a spin echo (a). The sequence is characterizedwith an initial 90o pulse, followed by a 1800 pulse at time TE/2. The180o pulse allows the magnetization to rephase, creating an echo at atime of TE. b-e show the response of the magnetization after the ex-citation (90o) and refocusing (180o) pulse. In b, the magnetization istipped into the x-y plane. The spins may precess at different rates dueto magnetic field inhomogeneities, causing them to dephase (c). Afterapplying the 180o refocusing pulse (at TE/2), the magnetization beginsto rephase, forming an echo at TE. The spin echo will have a strongersignal than a gradient echo since it is able to undo phase dephasing dueto magnetic field inhomogenities. The phase-encode gradient is oftenrewound at the end of acquisition, to reset the phase to its initial valueto maintain consistency in the magnetization between repetitions.19Figure 2.3: Pulse program for a gradient echo. In contrast to a spin echo, theecho occurs from a series of read-encode gradient lobes. De-phasingcaused from magnet field inhomogeneities are not refocused, so a gra-dient echo decays faster than a spin echo.Gradient EchoThe gradient echo pulse sequence (GRE) creates the echo though application of twogradient lobes along the same gradient channel. The two lobes are referred to asthe pre-phase and readout gradients, and have opposite polarities. During the pre-phase lobe, the rate of precession of spins within the sample vary linearly. Spins onone side of the sample will accumulate phase much faster than those on the otherside of the sample, resulting in a linearly varying phase distribution across thesample. When the sign of the gradient pulse is inverted, spins continue to precessat the same rate (assuming no magnetic field offsets of inhomogeneities), but in theopposite direction. The GRE forms when the readout gradient area exactly equalsthat of the prephase gradient lobe. The GRE process is summarized in Figure 2.3.20The GRE pulse sequence allows for faster imaging by using a smaller flip anglefor excitation, α . A flip angle less than 90o preserves some longitudinal magneti-zation, so the repetition time of the experiment can be reduced. If α is chosen to bemuch less than 90o, such that sinα ≈α , the longitudinal signal is cosα ≈ 1−α2/2.When compared to a spin echo experiment with similar echo times, a gradientecho will have lower SNR. This is a direct result of signal loss from T ∗2 relaxation(modulated by e−T E/T ∗2 ). Without the extra RF pulse, gradient echos can haveshorter echo times.Multiple gradient echos can be played out in succession within a single repe-tition time. This is the basis for echo planar imaging (EPI). The data acquisitiontransverses k-space back and forth by inverting the gradient area between succes-sive lobes. The maximum number of echos possible is dependent on how rapidlythe signal is lost to T ∗2 relaxation.The sampling locations in k-space are dependent on the net gradient area in thefrequency and phase encode directions. To sample multiple lines within one TR, asmall area phase-encode gradient must be applied between echo acquisitions. Thegradient area is related to the k-space resolution in the phase encoding directionand the desired step size between consecutive k-space lines. These gradients aremade as short as possible to maximize the number of echos that may be acquiredwithin the repetition time.EPI greatly speeds up the total acquisition time by acquiring multiple lines ofk-space with each RF pulse. The total acquisition of all echos is referred to as anecho train, and each acquisition referred to as a shot. For multiple gradient echos,the second half of the readout gradient will act as the prephase gradient for the nextecho, thug allowing for further time savings. The increased speed, however, comesat the cost of geometric distortions from off-resonant spins and Nyquist ghostsresulting from the alternating k-space trajectories.Steady-State MagnetizationThe recovery of longitudinal signal requires a long time. Data collection may besped up using a smaller flip angle for the excitation pulse. This in effect reservesa net longitudinal magnetization (Mo cosθ ) for the next pulse, while still providing21sufficient transverse magnetization for imaging (Mo sinθ ). After multiple excita-tions, the magnitude of the transverse signal approaches a steady state value. Forthe FLASH pulse sequence, the steady state magnetization, Mxy, is [41, 42]:Mxy =Mo(1− e−T R/T1)1− cosθe−T R/T1 sinθe−T E/T ∗2 (2.18)Where TR is the repetition time and θ is the flip angle. The optimal angle ofexcitation, θE is determined from the Ernst equation:θE = cos−1(−T R/T1) (2.19)Where T R is the repetition time of the experiment, and T1 is the longitudinalrelaxation time for the tissue under investigation. With this technique, it is possibleto reduce T R, and reach a steady transverse signal between successive excitationpulses. Often, the phase of the B1 field is adjusted by an integer multiple of aprime-number angle (often 117o) between RF excitations [33] to reduce residualsignal from prior excitations. This process is known as RF spoiling.Flow compensationA gradient echo occurs when the net phase accumulation of spins within a sampleis zero radians [43]. The gradient echo has a maximum magnitude when all spinsare stationary. However, in the case of flow, the magnetic environment experiencedby a moving spin changes between excitation and acquisition. This results in a non-zero net phase accumulation and a loss of signal at the echo or ghosting artifactsif the flow velocity changes between acquisitions [43]. The goal is to refocus bothstationary spins (zeroth order) and those flowing with a constant velocity (first or-der) at the center of the readout. This technique is referred to as flow compensationor gradient moment nulling (GMN) [33].Flow compensation attempts to rephase all spins at the echo, whether they areflowing or not, and is performed individually for each logical gradient axes in-dependently [33]. This is achieved by adding additional gradients, such that thehigher order phase terms can be nulled at the echo [33]. The number of gradientsis dependent on what order we wish to compensate for. For instance, first order22nulling (constant velocity) requires three gradient lobes. Nulling the second orderterm, for constant acceleration, requires four gradient lobes, and so on [33]. Flowcompensation increases the minimum echo time available. Therefore, it is commonto use only first order flow compensation in the clinic [33]. For the simple case ofa constant velocity, flow compensation attempts to solve [43]:φ(x, t) =−γ∫ t2t1G(t ′)[x(t ′)+ vt ′]dt ′ = 0 (2.20)Where φ is the accumulated phase of the transverse magnetization, γ is thegyromagnetic ratio, G(t) is the time course strength of the gradient lobe, v is thevelocity of the flowing spins and t1 and t2 are the times at which the flow compen-sation sequence begins and ends. The echo will form at the end of this sequence.Flow compensation is generally performed in the slice or frequency-encodedirections as the added scan times are much less than for the phase-encode direc-tion [39]. This equation applies for all velocities, as long as it is constant. Higherorder moment nulling (second order for acceleration) is possible with additionalgradient pulses. However, they are rare in clinical imaging due to the longer echotimes [33, 39].A simple case for flow compensation is to use gradients of equal duration,and assume a perfect rectangular pulse (Figure 2.4). The gradients are played outsimultaneously without a break between lobes. The duration of each lobe followsa binomial pattern: the first waveform has amplitude G and duration t, the secondhas amplitude -2G and duration t, and the third has amplitude G and duration t. Forthis series, the echo magnitude reaches its maximum as both the zeroth and firstorder phase accumulations equals 0 at the end of the third pulse.This is only one specific solution to the problem. It is generally true that toperform GMN in the frequency encoding direction for order N, requires a minimumof N+2 gradient lobes with alternating polarity [33]. For shorter echo times, thegradient strength of the first two gradient lobes can be increased. GMN can onlyoccur at one specific time, which is typically chosen at the peak of the echo.In reality, the gradient lobes are trapezoids, rather than rectangles. The gradientmoment nulling should be calculated for each section of the waveform: ramp up,plateau and ramp down. Further, first-order flow compensation is dependent on the23Figure 2.4: Example pulse program for flow compensation. This simple ex-ample has three pulse lobes, with alternating polarity and equal dura-tion. At the time of the echo (3t), both the stationary and flowing spinshave re-phased.reference time (center of the readout window) and the delays between the gradientpulses. To simplify the calculations, the net phase accumulation from all gradientscan be determined as though the gradient starts at time zero (Table 2.2), then applya translation [33].G is the strength of the gradient, τ is the length of the waveform, m0 is the ze-roth order moment and m1 is the first order moment. Translations of the waveformsto a time t ′ = t−∆t requires the corrections [33]:m˜o = mo (2.21)24Table 2.2: Accumulated Phase for Gradient Moment NullingShape G(t=0) G(t=τ) mo m1Ramp up 0 G Gτ2Gτ23Plateau G G Gτ Gτ22Ramp down G 0 Gτ2Gτ26m˜1 = mo∆t+m1 (2.22)Where m˜o and m˜1 represent the zeroth and first order moment for the trans-lated gradient lobes. One interesting property is that the first moment is translationinvariant if and only if mo = 0.Scanning ParametersThe quality of an MRI image can vary dramatically, depending on the imagingparameters, pulse sequence used and other imaging options [39]. This section willbriefly look at some of these parameters and how they affect the MRI image interms of the total scan time, SNR or image contrast.The size of the imaging matrix describes how many read or phase encode sam-ples are taken. The matrix dimensions may be different in the read and phaseencoding directions, depending on the imaging restraints. In general, the read en-code dimension will be larger than the phase encode to reduce the total scan time.Exceptions include images that require an ultra short echo time or when geometricdistortions, susceptibility effects or motion are an issue [39]. The SNR of the im-age is inversely proportional to the square root of the matrix size for a given fieldof view (FOV). Reducing one matrix dimension by a factor of 2 will double thepixel size and lead to an SNR improvement by a factor of 2. The spatial resolutionis determined as the ratio of the FOV to the corresponding matrix size.The FOV defines the spatial extents of the physical image. It may be definedfor a 2-D or 3-D image, with the read encode direction typically referring to thelarger dimension for rectangular images [39]. The SNR of the image is directlyproportional to the FOV for a given matrix size. For instance, doubling the FOV in25one dimension, without chaging the matrix size, will improve the SNR by a factorof 2. However, it is important to select the FOV to cover the region of interest best,as a smaller FOV also provides better spatial resolution. The longer anatomicaldimension is typically chosen to correspond with the read encoding gradient toavoid wrap-around artifacts [39]. To acquire data for an off-center FOV, the RFexcitation pulse is adjusted to match the central Larmor frequency of the slice. TheRF receiver frequency is also adjusted to accommodate this [33].The strength of the gradient fields is calculated from the FOV, matrix size, slicethickness and relative timings of the pulse sequence.2.4.1 Cartesian vs. Radial imagingMRI data may be acquired following any trajectory imaginable. The two mostpopular techniques include Cartesian and radial sampling.The first k-space trajectory used in MRI was projection acquisition by Lauter-bur [44]. MR data is acquired as radial spokes originating at the center of k-spaceand radiating outward. Radial sampling is attractive as all spokes are equally im-portant in the image reconstruction [45], all spokes cross the center of k-space(providing contrast information), motion/flow artifacts are suppressed [46, 47] andreducing the number of projections in the image reconstruction does not affect thespatial resolution - though the SNR does decrease. Radial acquisitions requirelonger scan times to satisfy the Nyquist criterion at the edges of k-space (up to afactor of pi/2 at the edges of k-space for a square FOV). Alternatively, the datamay be under-sampled in the angular direction; though this results in the presenceof streaking artifacts [48].Traditionally, radial MR images were constructed with either filtered back-projection, a technique borrowed from CT [49], or by regridding the data onto aCartesian grid, so that the FFT could be applied [50]. The regridding problem isnot intuitive to solve [51], and requires density compensation prior to appying theIFFT [52, 53]. Several new techniques have emerged recently that allow for imagereconstruction with an under-sampled data set. This will be the focus of Chapter 5.The more popular method is Cartesian sampling, in which the data is acquiredas a series of parallel lines in k-space. Cartesian sampling has been studied more26extensively and holds several advantages over radial sampling. First, the total scantime can be greatly reduced using echo planar imaging (EPI). Though this sam-pling method produces ghosting artifacts, they are well documented and correctionstrategies have been proposed. Generally, imaging artifacts present in Cartesianimaging have known solutions - either preventative or post-processing. Anotherhuge advantage is the ability to apply the FFT directly to the acquired data, thussignificantly reducing the reconstruction times.2.5 Contrast AgentsContrast agents were first used in the 1980s and showed promise in angiographicstudies. Schering was the first company to apply for a patent in 1981 for Gd-DTPA [54].Observed contrast in MRI signal primarily results from changes in the protondensity of tissue or the T1 or T2 relaxation time constants. The contrast agentsgenerally used in DCE-MRI experiments are designed to enhance the contrast be-tween normal and diseased tissue by changing the T1 and T2 relaxation time con-stants [55]. These contrast agents interact with the nuclear magnetic moment ofprotons in nearby tissue, resulting in a change in their phase or orientation withrespect to Bo [5]. The interactions responsible for the changes in T1 and T2 will bediscussed shortly.A majority of contrast agents used in MRI are stable chelates of a paramagneticmetal ion, such as gadolinium, iron or manganese [5, 55, 56]. These ions containunpaired electrons in the outer atomic orbits, which creates strong local magneticfields [5]. Interactions of these fields with the nuclear magnetic moments in tissueselectively induce relaxation. In effect, the effective T1 and T2 relaxation timeconstants are reduced [2].Low molecular weight agents – generally have molecular weights less than1000 Da [4] and contain a variety of gadolinium-based agents (including Gd-DTPA) [3]. Due to their small size, these agents rapidly diffuse through vesselwalls into the extravascular extracellular space [4, 57]. These agents are com-monly used to study angiogenesis in tumours and to monitor the response to an-tiangiogenic therapy [3].27Gadolinium-based contrast agents have been studied most intensively to dateand are commonly used in clinic [58]. Gadolinium ions have seven unpaired elec-trons in their outer orbit, which makes them the most paramagnetic ion [55]. It hasbeen well established that the change in the relaxation rate (inverse of the relaxationtime constants, i.e. 1/Ti where i = 1, 2) is directly proportional to the concentrationof gadolinium within the region [55]. Assuming that the relaxation rates are knownin the presence and absence of gadolinium, the concentration of gadolinium maybe determined from the Solomon-Bloembergen equation [42, 59]:1/Ti = 1/Tio+ ri[Gd] i = 1,2 (2.23)Where Ti is the relaxation time constant in the presence of a contrast agent,Tio is the relaxation time constant in the absence of gadolinium, ri is the relaxivityof the contrast agent, and [Gd] is the concentration of Gadolinium. The relaxivitymay be interpreted as the efficiency at which a paramagnetic ion enhances the re-laxation rate of water protons [55]. It is a function of the magnetic field strengthand the chemical structure of the agent [42]. The above relation has been verifiedin-vitro for T1 and T ∗2 , as well as in-vivo for T1, over a range of concentrations [42].Here, T ∗2 is the transverse relaxation time due to molecular interactions and inho-mogeneities in the magnetic field [5].Changes in the T1 relaxation rate results from the dipole-dipole interaction be-tween the nuclear magnetic moments and the strong magnetic fields created by thecontrast agent [55, 60]. T1 enhancement effects are only observed in the vicinityof the contrast agent since the dipole-dipole interactions are weak. As a result, en-hancement patterns in T1-weighted images define the regions where contrast agentis present. T1 enhancement patterns are most strongly observed in areas wherethere is a uniform distribution of the contrast agent [60]. T1-weighted images willshow an enhanced signal in regions containing the contrast agent [4, 58]. In con-trast, changes in the T ∗2 relaxation rate are due to susceptibility-induced gradientfields surrounding the contrast agent [60]. The induced fields cause long rangemagnetic field inhomogeneities, allowing for T ∗2 shortening further away from thecontrast agent [42]. The effects on the T1 and T ∗2 are complementary, though oneoften dominates over the other [60]. The T1 effect is dominant in areas where the28contrast agent is uniformly distributed due to more close range interactions. While,the T ∗2 effect is dominant when the contrast agent is compartmentalized as this in-creases the induced gradient field [60].Safety concerns and toxicity of Gd-based CAThe unaltered Gd3+ ion is known to be highly toxic in humans as it interferes withthe calcium channels and protein binding sites [61, 62]. The free ions accumulatein the liver, spleen, kidney and bones. Studies have shown that a 50% lethal dose offree Gd in mice is only 0.20 mmol/kg [61]. With these numbers, it is important tochelate the Gd ions with a larger compound that will limit tissue interactions. Gdforms stable chelates with both ethylenediaminetetraacetic acid (EDTA) and di-ethylenetnaminepentaacetic acid (DTPA) [63, 64]. Though dissociation in low pHenvironments is possible, this appears to occur in a very small number of cases [61].The frequency of acute adverse reactions to Gd3+-based contrast agents isabout 0.07-2.4% with doses of 0.1-0.2 mmol/kg. Patients who have had a pre-vious reaction are more likely (eight times greater than the general public) to havea second reaction. The second reaction is often more severe. Patients with aller-gies to other medications or food, or those with asthma, have a greater risk of areaction [62].Acute reactions generally occur within 1 hour of the injection. These rangefrom mild to severe, with most being mild. Late reactions manifest as a skin reac-tion, and often occur between 1 hour to 1 week post injection. Very late reactionsare due to unchelated Gd deposits in the extravascular space. These are most oftenexperienced in patients with renal failure [62]. Use of Gd3+-based contrast agentsis not recommended for patients with renal malfunction, where incomplete excre-tion may be a concern. Studies have shown that a small fraction of these patientsmay have a serious adverse reaction (nephrogenic systemic fibrosis) to the contrastagent [61].A more recent safety concern is disposition of gadolinium within the brain. Astudy by McDonald et al. [65] in 2015 discovered that patients receiving at leastfour gadolinium based contrast agent (GBCA)-enhanced brain MR examinationshad 0.1-58.8 µg gadolinium per gram of neuronal tissue (capillary endothelium and29neuronal interstitium), in a significant dose-dependent manner. These patients allhad normal renal function, and the findings were uncorrelated to age. The depositsshow up as higher signal intensities of non-enhanced T1-weighted images. Depositsare observed in a number of brain structures - particularly in the globus pallidus anddentate nucleus within the brain - as well as in the liver, skin and bone [66].The type on GBCA (linear vs macrocyclic) can impact the quantity of deposits,with greater amounts deposited with linear-type agents [66]. Kang et al. [67] per-formed a study on multiple-sclerosis patients, receiving either nine (high-exposure)or two injections (low exposure) of a linear GBCA (both non-ionic and ionic) in thefirst year and an additional dose in the second year. The results showed a strong cor-relation with the dose, with the high-exposure cohort having enhancement in all re-gions evaluated, while the low-exposure cohort only had an increase in the dentatenucleus. The additional dose in the second year did not appear to affect the signalintensity in either cohort. Signal increases were greater in the high-exposure co-hort when a linear non-ionic contrast agent (such as gadodiamide/Omniscan [66])injection was administered compared to the linear-ionic contrast agents (such asgodopentetate dimeglumine/Magnevist or gadobenate dimeglumine/Multihance [66]).As of 2018, the effects of gadolinium deposits in the brain are still undeter-mined [68] and there is no clear evidence that adverse effects reported after admin-istration of the contrast agent is connected with the deposits [66, 68]. The CanadianAssociation of Radiologist recommends that GBCA administration should be con-sidered with respect to potential risks and benefits and follow the standard dosignguidelines, and that repeat injections should be avoided unless necessary [68]. TheUniversity of British Columbia (UBC) no longer allows the use of Omniscan in theclinic.30Chapter 3Dynamic Contrast EnhancedMRI: Theory and MethodsDCE-MRI is a perfusion-based technique used to evaluate the microvascular struc-ture and function of blood vessels in tissue [7, 8]. Early experiments were per-formed in the mid-1980s, though true perfusion weighting was not realized untiltracer injections and data sampling on the order of seconds was possible. The tech-nique involves the injection of a paramagnetic contrast agent - usually containinga transition element such as gadolinium, manganese or iron - into a peripheral veinand tracking its passage through the capillary bed [3]. By analyzing the biodistri-bution of the tracer, regions with abnormal vasculature are identified [7]. Theseregions are often associated with diseases, such as cancer[4, 69].DCE-MRI operates on the premise that image contrast, caused by the presenceof an injected contrast agent, correlates with the vasculature of various tissues [70,71]. The behavior of the contrast media is monitored through the rapid acquisitionof T1 or T ∗2 -weighted images during the first pass of the contrast agent throughthe tissue [72]. It is expected that any region with highly permeable vasculature orgreater blood flow will enhance rapidly as the contrast agent passes through [4, 42].Depending on the imaging sequence used, this enhancement will reflect an increase(T1-weighted images) or decrease (T ∗2 -weighted images) in signal intensity [60].Based on the observed characteristics, semi-quantitative or quantitative parametersmay be derived to characterize the vasculature [1].31This chapter will summarize the mechanisms for perfusion imaging (angiogen-esis) and briefly discuss three common techniques: DCE-MRI, dynamic suscep-tibility contrast magnetic resonance imaging (DSC-MRI), and arterial spin label-ing (ASL). The focus of the thesis is DCE-MRI, so the discussion will focus on thistechnique in greater depth. The remainder of the chapter will outline Pharmaco-kinetic (PK) modeling and discuss how physiologically relevant parameters canbe derived from the data. But first, it’s important to understand angiogenesis, aphysiological process that allows us to differentiate healthy and abnormal tissue.3.1 AngiogenesisAngiogenesis is the process by which new blood vessels form from a pre-existinghost vasculature [3, 69]. Experiments have shown that tumours cannot grow be-yond a diameter of 2−3 mm from the nearest blood supply due to oxygen diffusionlimits in tissue [1]. Therefore, it is essential that the tumour develops a system ofnew blood vessels that will supply the newly developed cells with oxygen and nu-trients [69]. This blood supply promotes further growth and metastases [1, 3].The vasculature of healthy tissue is organized into a system of arteries, cap-illaries and veins [4, 69]. These vessels are highly efficient in supplying the sur-rounding tissue with essential nutrients [69]. To promote survival of the tissue,these vessels are highly organized and uniformly spaced, such that metabolites canreach all cells through passive diffusion [5]. We can characterize the vasculaturewith parameters that describe the mass blood flow, vessel wall permeability andtissue volume fractions [5].Cancers are known to proliferate rapidly. This requires that new vasculatureforms to supply the new cells with nutrients for survival and further growth [73–75]. Due to the temporal demands of rapid growth, the array of angiogenic ves-sels are disorganized, irregular, fragile, tortuous, have highly permeable walls, andchaotic flow patterns [1, 58]. When compared to healthy tissue, distinct differencesare observed related to blood flow and accumulation in the tissues [4]. Scientistscan exploit these differences to evaluate the health status of the tissue.Figure 3.1 illustrates the difference between the vasculature in normal tissueand in a tumour. Angiogenic vessels exhibit large gaps between endothelial cells,32Figure 3.1: Part a) shows the vascualture of healthy tissue. Blood vessels areorganized and uniformly distributed. Part b) shows the vasculature of atumour resulting from angiogenesis. The blood vessels were rapidly de-veloped, meaning that they are tortuous, disorganized and leaky. Whena contrast agent is present in the blood plasma, it can perfuse into the in-terstitial space of the surrounding tissues more rapidly in a tumour. Thiscauses a diffential enhancement pattern between tumour and healthy tis-sue. DCE-MRI exploits this difference to characterize the health statusof tissues. Figure taken from Emblem et al. [76]. Permission for use hasbeen granted from Nature and from Dr. Kyrre Emblem.within the endothelium and discontinuous basement membranes [3, 69]. As a re-sult, contrast agents perfuse more readily through the vessel walls compared tohealthy tissue [3]. This allows us to characterize tumours.The microvascular density (MVD) (average number of vessels within a smallregion of tissue) is commonly used to assess angiogenesis in tissue [3, 4, 10]. Ithas high correlations with the agressiveness of several cancers [77], patient survivaland risk of metastases [3]. However, the measurement is invasive as a sample ofthe tumour is removed, stained and examined by light microscopy [78]. and it doesnot provide information about blood flow or the hyper-permeability of the vesselwalls. DCE-MRI provides a non-invasive alternative that reveals information aboutthe functionality of the vasculature.333.2 Methods of Imaging PerfusionPerfusion is defined as the delivery of arterial blood to the capillary bed [79]. Per-fusion can be measured from the change in MR signal induced by a contrast agentfrom a series of rapidly acquired MR images [79]. Perfusion imaging generallyfalls into one of three categories of scans: DSC-MRI, DCE-MRI and ASL. Allthree techniques acquire rapid MR images before and after introduction of the con-trast agent. The contrast agent is administered through an injection for DSC-MRIand DCE-MRI, and is referred to as a bolus. In the case of ASL, the contrast agentis magnetically labeled blood.Dynamics Susceptibility Contrast ImagingDynamic Susceptibility Contrast Imaging (DSC-MRI) utilizes an exogenous tracerwhich is injected into a peripheral vein. This technique is used primarily in thebrain and looks at signal intensity losses in T ∗2 -weighted images. The T∗2 effectis much stronger for intravascular tracers, but suffers from quantification issuesif the tracer extravates into the interstitial space as the T ∗2 contrast drops signifi-cantly [79]. For this reason, DSC-MRI is usually restricted to cases where the con-trast agent is compartmentalized. Though most applications are associated withbrain imaging [60], DSC-MRI can be performed anywhere in the body.A blood-pool contrast agent works best as the susceptibility effect extends be-yond the vascular space, resulting in a transient signal drop [60]. Since the bolusinjection will pass through the tissue in a couple seconds, a fast imaging technique- such as EPI or Fast Low Angle SHot (FLASH) - is used. The temporal resolu-tion of the scans is dictated by the transit time of the bolus through the tissue, andtypically a repetition time (TR) of less than 2 s is required [60].Dynamic Contrast-Enhanced MRIDCE-MRI is similar to DSC-MRI since an exogenous tracer is used, but it quan-tifies the change in local T1. DCE-MRI is by far the most common method forstudying perfusion and has applications throughout the body [79]. There has beenwide-spread applications in tumour imaging as the angiogenic vessels allow thecontrast agent to freely perfuse into the surrounding interstitial space [42].34T1 interactions are short range and cause an increase in MR signal in a T1-weighted image. The concentration of agent within the tissue may be deducedfrom the relative change in local T1, in which the change in 1/T1 and concentrationare linearly related [42]. This requires that a high-quality pre-injection T1 map beacquired in addition to the dynamic T1-weighted image series. Popular imagingsequences are EPI or turboFLASH, with a temporal resolutions of a few seconds.DCE-MRI is more sensitive to the leakage of contrast agent into the inter-stitial space [80]. Low molecular weight contrast agents, commonly used in theclinic, readily perfuse from the vasculature into the interstitial space during thefirst pass. This is especially prevalent in cancers. In contrast to DSC-MRI, infor-mation about vessel permeability can be determined from the slow component ofthe concentration-time curve in the tissue of interest [79].Arterial Spin LabelingArterial Spin Labeling (ASL) also measures perfusion. It is a competing techniquewith DCE-MRI, though it is primarily performed to measure blood flow in thewhole brain [81] and quantifies absolute cerebral blood flow [82]. ASL avoids theinjection, and instead uses tissue water as an endogenous tracer [79]. This makesthe technique completely non-invasive [22], and allows for repeat measurements.The technique may be thought of as an inversion recovery experiment, followedby rapid MR imaging of the tissue of interest. A RF pulse excites the arterialblood located upstream of the tissue of interest. This inverted signal (or ’labeledblood’) acts as a temporary contrast agent [22]. The lifetime of the labeled bloodis dependent on the T1 relaxation time of blood, ranging between 1300−1750 msat clinical field strengths.After a delay to allow the blood to reach the tissue of interest (TOI), labeledMR images are acquired which contain a mix of signal from the labeled blood andstatic tissue water [81]. ASL is a differential technique that compares images withand without the labeled blood signal [22]. Any change in tissue magnetization isattributed to perfusion of the excited blood protons into the surrounding tissues.The difference in signal is only a couple percent, so the experiment is performedmultiple times to enhance the SNR [22].353.3 Methods: DCE-MRIThe pioneering experiments on perfusion acquired a single snapshot image of theregion of interest post-injection. Though this provided information about the con-trast agent distribution, it did not yield any functional information about biologicaltissue [7]. As faster imaging techniques were developed, analysis across a seriesof dynamic images became possible.DCE-MRI operates on the premise that image contrast, caused by the presenceof an injected contrast agent, correlates with the vasculature of various tissues [70,71]. Typical contrast agents contain a paramagnetic ion, such as gadolinium (Gd-DTPA, Gd-DOTA) or manganese (MnCl2, MnL1 [83]), though iron oxides can beused as well [75].DCE-MRI data is acquired with a fast imaging technique before, during andafter the rapid administration of a contrast agent (usually gadolinium-based) [57,69, 84]. Each image provides information of the time-resolved distribution of thecontrast agent in the tissue of interest[7, 85]. By analyzing these images, informa-tion regarding tissue physiology and pathology may be extracted. This includesthe size of the extravascular space (EES), vessel wall permeability and the surfacearea of the vessel [5, 42].Three sets of images are acquired in a DCE-MRI experiment [8]. These in-clude the localizer image, pre-contrast T1-weighted images, and rapid T1-weightedimages acquired before, during and after the contrast inject.Localizer ImagesThe localizer images pinpoint the exact location of the region of interest (i.e. tu-mour) and provide anatomical information. The DCE-MRI slices can be positionedand properly aligned from these images.Pre-Contrast T-1 weighted imagesThe pre-contrast images provide a baseline T1 prior to the injection. The T1 relax-ation time constant can be estimated with an inversion recovery (IR) or saturationrecovery (SR) (slow) method, the Look Locker technique or using a variable satu-ration method, such as gradient echo images with variable flip angles [42]. Though36IR or SR techniques are arguably the most precise methods for estimating T1, theyare time consuming [42]. The Look-Locker technique is a faster method than IRand SR, as it acquires T1 maps with a single inversion pulse [86]. The T1 map isoften calculated on a per-pixel basis [5]. The pre-injection T1 times should be ac-quired with high accuracy, as they are used to estimate the post-injection T1 times.IR involves flipping the magnetization by 180o, waiting a variable delay oftime, then applying a 90o pulse to flip the magnetization into the x-y plane. Themagnitude of the signal is then plotted against the delay time. The T1 relaxationtime is calculated by fitting an exponential curve to the MR signal. The measure-ment is typically performed with two or three inversion times [87], though morerepetition times will improve the accuracy at the cost of longer scan times.SR is a faster method of estimating the tissue T1 than IR. The technique in-volves multiple 90o RF pulses at relatively short repetition times, and measuringthe signal intensity for multiple saturation times. A spoiler gradient pulse dephasesthe residual longitudinal magnetization that remains after the excitation pulse. In2017, Wang et al. [88] developed a saturation-inversion-recovery (SIR) sequencethat measures T1 times with sharper T1 resolution than from IR or SR individually.The Look Locker technique provides a fast and efficient method for estimat-ing the T1 map [86]. The technique begins with a 180o inversion pulse, then ap-plies a series of small angle excitation pulses at a variety of well known inversiontimes [89]. Since the angles are small, signal loss along the -z-axis is minimal.The scan is repeated after a period longer than 5T1, such that the magnetization hasfully recovered. From the acquired data, MR images are produced with varyinginversion times. One limitation is that the RF pulses used to acquire the data canaffect the longitudinal magnetization, which results in a faster decay rate, termedT ∗1 -weighting. This weighting is specific to the pulse sequence. The T1-map isdetermined by fitting a smooth curve to the real signal, S (S = A · e−t/T ∗1 ), thenperforming the correction described in Taylor et al. (2015)[89].The variable flip angle (VFA) technique is more widely used as it applies tospoiled gradient recalled echo (SPGRE) scans (also known as FLASH). The T1-weighting is provided by the flip angle and repetition time. Data is acquired formultiple flip angles. T1 is then determined by fitting a non-linear curve to thesignal intensity vs flip angle.37Dynamic Contrast-Enhanced ImagesThe third set of images is the dynamic scans. These are heavily T1-weighted im-ages, and acquired rapidly (every 2−15 s) [69] for at least 5−10 min. However,there is a trade off between acquiring data rapidly and acquiring images with ahigher spatial resolution or tumour coverage [90, 91]. Depending on the goals ofthe study and tumour model used, some groups will sacrifice the temporal resolu-tion to obtain high quality images [92].A majority of DCE-MRI experiments use a spoiled GRE pulse sequence. Com-mon protocols are EPI or turboFLASH, which is a T1-weighted gradient echo (GE)saturation recovery (SR) or inversion recovery (IR) snapshot [4, 42, 69, 79]. Thesesequences acquire data rapidly, provide sufficient temporal and spatial resolutions,and has a good SNR ratio [5]. Typical spatial resolutions are 100−625 µ m in pre-clinical studies in mice (30−40 mm FOV length with 64-256 pixels) [15, 93–98]and 0.5−4.0 mm for in-vivo studies in humans (3.0 T) [99].To maximize T1 contrast in the DCE images, the echo time is set to its minimumvalue (generally 1−2 ms) [79]. Other pulse parameters, such as the repetition timeor flip angle, are then optimized for the desired SNR and temporal resolution. Forthe case of a FLASH experiment, a flip angle of 30−60o [79] is generally used, asit offers a good balance of SNR and temporal resolution.A simple DCE-MRI protocol acquires a single proton density weighted imageprior to the injection, followed by heavily T1-weighted MR images for the rest ofthe experiment [42]. The flip angle is set to a small value for the proton densityimage, and larger for the T1 images. repetition time (TR) is short to achieve goodT1 contrast and to allow for a higher temporal resolution.The concentration of contrast agent within the tissue is estimated from the rela-tive change in MR signal or inverse T1 relaxation time [42]. In the case of a FLASHexperiment, the change in local T1 is determined from a ratio of the signal intensityfrom a dynamic MR image and the proton density image. Figure 3.2 shows the T1-weighted images pre and post-injection, as well as the Look-Locker T1 map. TheLook-Locker map is derived from images with a variety of IR times, and allowsfor the calculation of the concentration of contrast agent in the tissue of interest.38Figure 3.2: T1-weighted DCE images a) pre and b) post-injection and c) theLook-Locker T1 map used to calculate the concentration of contrastagent in the tissue of interest. Tissue enhancement is observed withinthe region outlined in red.3.3.1 Considerations for Setting up a DCE-MRI ScanDepending on the maximum expected concentration of contrast agent in the tissue,the signal intensity or T1 relaxation time may be used for quantification. It is oftenassumed that the change in the signal intensity is linearly related to the concentra-tion of contrast agent in a small region [8, 91], but is only true at low concentra-tions [4]. If higher concentrations are expected, it is advantageous to study changesin the relaxivity (1/T1) of the tissue of interest as this parameter correlates directlywith the concentration of contrast agent [42].The relative difference in T1 is dependent on the native T1 constant pre-contrast(T1o). In effect, the magnitude of the change is greatest in tissues with larger pre-contrast T1 values [84]. The change in the T1 relaxation time constant is convertedinto a concentration with [42]:1/T1 = 1/T1o+[CA] · r1 (3.1)Where T1o and T1 are the pre and post-contrast relaxation time constants, [CA]is the concentration of the contrast agent, r1 is the relaxivity of the contrast agent,defined at a particular Bo strength and temperature.There is a trade off between acquiring data rapidly and acquiring images witha high spatial resolution [90, 91]. Ideally, the DCE data should be acquired with atemporal resolution exceeding the most rapid changes in the tissue [42, 91]. Tissue39coverage and spatial resolution, however, are both sacrificed with high temporalresolution scans [5]. Depending on the goals of the study, some groups chooseto sacrifice the temporal resolution in favour of acquiring higher-spatial resolutionimages to observe heterogeneity in a lesion [92]. Additionally, the temporal reso-lution must be relaxed when the region of interest is large or when a main artery(used for the AIF estimate) is far from the tissue of interest [5, 42]. If the AIF willbe measured, the temporal resolution should be on the order of a second to capturethe rapid contrast kinetics in the blood [42]. Conversely, experiments investigat-ing tumour heterogeneity with a semi-quantitative analysis can relax the temporalresolution in exchange for improved spatial resolution [5].The temporal resolution for DCE-MRI studies are typically on the order ofseconds [8, 69], though some groups have relaxed this requirement and used atemporal resolution of 10-30 seconds for higher quality images [5]. The spatialresolutions are typically 100−625 µm in pre-clinical studies in mice (30−40 mmFOV length with 64-256 pixels) [15, 93–98] and 0.5−4.0 mm for in-vivo studiesin humans (3.0 T) [99].The duration of the scan varies depending on the desired form of analysis.Perfusion weighted scans in human may be acquired in approximately 1 min, whilepermeability weighted scans require approximately 5 min [79]. The scan durationis typically longer in mice [95, 100, 101]. Studies that wish to investigate thetumour heterogeneity are recommended to favor a higher spatial resolution at theexpense of temporal resolution [5].The change in T1 in blood and tissue can vary greatly. It is expected that theT1 of blood may decrease by an order of magnitude at typical clinical doses [102].Conversely, the change in tissue T1 can be much smaller [42]. The chosen pulse se-quence should have good T1 sensitivity and dynamic range to capture both changesaccurately.3.3.2 The Contrast Agent InjectionDCE-MRI investigates signal enhancement in T1-weighted images, induced by anexogenous contrast agent [82]. The bolus of contrast agent is injected into a pe-ripheral vein through a catheter injection. This can be the antecubital in humans,40or the jugular or tail in rodents [5].Typical contrast agents used in DCE-MRI contain a paramagnetic ion, suchas gadolinium (Gd-DTPA, Gd-DOTA) or manganese (MnCl2, MnL1 [83]), thoughiron oxides have been used as well [75]. Paramagnetic ions contain unpaired elec-trons that interact with the protons in tissue, causing a reduction in their T1 andT2 relaxation times [60]. The magnitude of each effect will categorize the contrastagent into T1 or T2 agents, where a T1 agent has a greater relative effect on the T1relaxation time [62]. Most rapid imaging sequences provide greater T1 contrast,and so a T1 agent is often used [5, 103].Conventional contrast agents have concentrations of 0.5− 1.0 M. They arecommonly administered to a dose of 0.1 mM/kg body weight. Typical injectionprotocols in mice involve injecting the bolus 10− 20 s after the start of the scanto allow for pre-injection data to be acquired, and following it with a 20− 30 mLsaline flush [79]. The injection is often performed with a power injector to ensurea reproducible injection between studies [69].The contrast agent only occupies the plasma component of blood [104]. Theconcentration of contrast agent in the plasma space, Cp, is directly related to thatfound in the blood, Cb, by:Cp = Cb/(1−Hct) (3.2)Where Hct is the hematocrit, which describes the fractional volume of redblood cells in blood [105]. Generally, a value between 0.4-0.45 is assumed; but thismay not be valid in advanced cancers, and is lower in the capillaries [104, 106].Following injection, the contrast agent circulates freely throughout the vas-cular plasma space, and diffuses into the interstitial space of surrounding tissuewhere it interacts with the proton spins of the tissue causing a signal increase in T1-weighted images. After several hours, the agent is excreted from the body throughthe kidneys [4]. By analyzing the concentration-time curve in a tissue of interest,perfusion parameters (quantitative or semi-quantitative) can be extracted [104]. Asa result of the contrast dynamics, the tissue signal will increase rapidly initially,peak, then slowly return to its baseline value [72].413.4 Data AnalysisThe analysis of DCE-MRI data is based on the principles of tracer kinetics, whichdescribes how blood is transported through the tissue of interest [79, 80]. Thereare two dominating phenomena that occur simultaneously [107]: the first is therapid perfusion of contrast agent into the microcirculatory network, and the secondis related to the accumulation and slow release of contrast agent from the inter-stitium. DCE-MRI data attempts to extract meaningful parameters related to thehealth status of the tissue. The analysis may be quantitative or semi-quantitative.Semi-quantitative analysis is simple and may be performed quickly. It hasapplications in characterizing tumour growth and tracking its response to treat-ment [5]. Semi-quantitative analysis studies the shape of the AIF [7]. It takes placeover the first pass of contrast agent through the vasculature [104], which describesthe time period beginning with contrast administration and covers a few cardiaccycles [4, 104].The analysis attempts to derive information about the onset time, peak enhance-ment, maximum rate of enhancement, gradient of peak enhancement or washoutand the signal enhancement ratio from these regions [7, 69]. The simplicity of theanalysis, however, comes at the cost of limited physiological information. Thesedescriptors contain information about blood flow, blood volume, contrast agentleakage and ve, though the contribution of each cannot be distinguished [8]. Fur-ther, the reproducibility of the results between trials (within and between institutes)is a major concern [7]. Semi-quantiative parameters have been shown to be depen-dent on the initial conditions [5, 69], and may not accurately reflect the concentra-tion of contrast agent in the tissue of interest [4]. Some groups choose to calculatethe initial area under the enhancement curve (IAUC) [4] as this metric appears tobe more reproducible.Quantitative analysis is preferred by the majority and will be the focus of theremainder of this chapter. Quantitative analysis studies the distribution of contrastagent within the tissue of interest and its elimination from the body, through PKmodeling [5]. The chosen PK model provides physiologically relevant parametersby fitting a mathematical model to the time course signal change within a tissue ofinterest. Most models will include the enhancement characteristics in a blood ves-42sel feeding into the tissue of interest as an input parameter [60]. Theoretically, thederived parameters should be minimally dependent on the injection mode and thepatient’s physiological status [107], therefore allowing for inter and intra-patientcomparisons.To quantify the observed contrast kinetics, the tissue must be divided into wellestablished regions known as compartments [104]. Most models assume that thecontrast agent distributes uniformly throughout the entire compartment for sim-plicity [5]. Depending on the chosen model, the tissue may be represented withthree or four compartments [104], which includes the vascular plasma space, theextravascular extracellular space (EES), the intracellular space, and microscopictissue components such as cell membranes or fibrous tissue. Often compartments3 and 4 are grouped together to simplify the model. These compartments may becharacterized by their fractional volumes, such that:vp+ ve+ vi = 1 and vp = (1−Hct)vb (3.3)Here, vp, ve, vi and vb are the fractional volumes of the vascular plasma space,EES, intracellular space and whole blood space respectively, and Hct is the hema-tocrit.A majority of contrast agents used in DCE-MRI cannot pass into the intracel-lular space due to their size, inertness and non-lipophilicity [104]. For this reason,compartments 3 and 4 are often not considered in the analysis of DCE-MRI data.The resulting model is known as the two compartment model. The next sectionwill discuss two popular models used in DCE-MRI studies.3.4.1 Pharmacokinetic ModelingPK model parameters describe physical properties of the vessels [108], includ-ing blood flow and vessel wall permeability [57]. These may be used to identifythe presence of abnormal vasculature [8], such as that observed in tumours [4].DCE-MRI data is typically characterized with three parameters. The names mayvary, depending on the application of the model being used [57], though ve, Ktransand kep are considered the is the fractional volume of the EES per unit volume of tissue. It is assumed43to be equal to the space filled by the contrast agent [4]. It is approximately 0.2 -0.5 in tumours [57].Ktrans is the volume transfer constant from the plasma to the EES. It determinesthe amplitude of the initial response to injection and is calculated from the absolutevalue of the contrast agent concentration [1]. Ktrans describes the combination ofthe endothelial permeability and surface area product, PS, and blood flow, F, tothe region [1, 57, 69]. Interpretation of Ktrans varies depending on the relativecontributions from blood flow and tissue permeability, though these componentsare difficult to separate [8]. It is well established that tumours have high Ktrans(high permeability and blood flow) [69], while necrotic regions in tumours havesmall Ktrans as they have limited blood flow [4].kep (= Ktrans/ve) is the flux rate constant from the EES to the plasma. It char-acterizes the rate at which the contrast agent returns to the intravascular space. Itis always greater than Ktrans since ve is less than 1. The value is generally onthe order of minutes to hours [4]. kep may be determined from the shape of theconcentration-time curve.These parameters act as probes for monitoring tissue status [109], and haveshown applications in differentiating malignant from benign tumours, tumour stag-ing and monitoring treatment response. The difference can be subtle, therefore itis essential to acquire high quality data. In fact, several studies have demonstratedthat the sensitivity and specificity of DCE-MRI analysis is directly correlated tothe accuracy of the fitted PK model parameters[11, 14, 17, 110–112]. The AIF isknown to have a dramatic impact on parameter accuracy, making it imperative thata high-quality measurement is obtained [11]. It is therefore suggested to measurethe AIF for each experiment [14], including those performed on the same patientmultiple times [108].Two-compartment modelMost pharmacokinetic (PK) models are based on the two compartment model [1,3], in which the vasculature comprises one compartment, and the extracellular ex-travascular space (EES), ve, of the tissue of interest represents the second [110]. Itassumes that contrast agent flows readily between the two compartments [1, 57],44Figure 3.3: Pictorial representation of the two-compartment model. ve is thevolume of the extracellular extravascular space, vp is the plasma volumewithin the vessel, PS is the permeability-surface area product relatingthe rate at which the contrast agent travels between the two regions, andFp is the plasma flow rate.with the rate of diffusive transport dependent on the concentration of agent in thetwo compartments and the permeability of the vessel walls [104]. In addition, themodel assumes that the contrast agent is well mixed within each compartment (i.e.uniform distribution) [5]. A pictorial representation of the two-compartment modelis shown in Figure 3.3 [113].Tofts ModelThe model proposed by Tofts and Kermode [114] in 1991 offers a simple assess-ment of the tissue vasculature, and has become the foundation for more compli-cated PK models [113, 115]. It is important to note that Seymour Kety [116] de-rived similar equations for the exchanges of inert gases at the lungs and tissue,though Tofts and Kermode were the first to apply the concepts to MRI. The Toftsmodel, figure 3.4, has the following functional form:Ct(t) = Ktrans∫ t0Cp(t ′)e−Ktrans(t−t ′)/vedt ′ (3.4)Where Ct is the concentration of contrast agent in the tissue, Cp is the con-45Figure 3.4: The model propsed by Tofts et al. (and various extenstions ofit) is one of the most commonly used PK model for analyzing DCEMRI data. The model requires that the concentration-time curves in thetissue of interest, Ct(t), and blood plasma, Cp(t) or AIF, are known. Theaccuracy of the physiologically relevant perfusion parameters - Ktransand ve - are dependent on the quality of these curves. The AIF needs tobe sampled with a high temporal resolution to capture the rapid contrastchanges in the blood following the injection.centration in the blood plasma, Ktrans is the volume transfer constant relating therate at which the contrast agent perfuses from the vasculature to the tissue, andve is the fractional volume of the EES. These parameters describe physical prop-erties of the vessels [108], and may be used to identify the presence of abnormalvasculature [8], such as that observed in tumours [4].Model Fitting using the AIF and Tissue C-t CurveQuantitative analysis involves fitting a PK model to the concentration-time curves [4,8, 69]. From the fit, physiological parameters, describing the blood flow to the re-gion, and the passage of contrast agent between the plasma space and EES, may bedetermined [69].Typical physiological parameters include the blood flow (perfusion), vesselwall permeability, vessel surface area, intravascular and extravascular extracellu-46lar volume fractions [4, 5]. These parameters are independent of the acquisitionprocedure and only describe tissue properties [8]. Pharmacokinetic (PK) modelsare concerned with contrast agents that readily diffuse across the vessel walls andremain extracellular [57]. Numerous models may be found in the literature.The accuracy of the model fit is highly dependent on the quality of the plasmaconcentration-time curve - often referred to as the AIF. This will be the topic ofthe next chapter.47Chapter 4High Temporal Resolution AIFMeasurement using the Phase ofMR Projections4.1 The Arterial Input FunctionDCE-MRI exams involve the injection of a contrast agent and tracking its distribu-tion within a tissue of interest. Summary parameters, estimated from the images,are known to be dependent on cardiac output, arterial status, injection rate and tis-sue properties [82]. Even though it is possible to control the injection volume andrate, it will not guarantee that the bolus leading into the tissue of interest will havethe same shape. The bolus must first pass through the heart and lungs before beingredistributed around the body, which causes it to disperse as it enters the tissue ofinterest through an artery. If the shape of the incoming bolus is known, informationabout the microvascular structure of the tissue may be derived. Knowledge of thebolus kinetics (maximum concentration, width, etc.) is especially important whenquantitative tissue parameters, such as blood flow or perfusion, are desired.The AIF describes the time-course concentration of a contrast agent in an arterysupplying the tissue of interest [7, 8, 117]. Together with the concentration-timecurve within the tissue of interest (TOI), physiological parameters related to tis-48sue perfusion, vessel wall permeability and the volume of the EES, ve, may beestimated through pharmacokinetic modeling [2]. The AIF is defined in the bloodplasma space, not within the whole blood (equation 3.2). The conversion factor be-tween the two spaces is known as the hematocrit (Hct), with typical values rangingbetween 0.40−0.45 [104].The earliest known measurement of an AIF was in 1991 by Bruce Rosen andcolleagues [118]. In their study, they measured an AIF in the middle cerebralartery of a hypercapnia canine model, using a 1 s single-shot EPI experiment, andcompared it with blood samples taken directly from the femoral artery. The resultsshowed good agreement between the two techniques, which suggested that the AIFcould be measured non-invasively. Later in 1992, Perman et al. developed a dual-FLASH pulse sequence that allowed for simultaneous AIF and DSC measurementsin the neck and brain, respectively [119]. Then in 1996, Fritz-Hansen et al. [102]published a study that confirmed that MRI could be used to non-invasively measurethe AIF in the descending aorta. They used an inversion recovery turboFLASHscan and validated their measurement with direct blood samples.The quality of the AIF can have a significant impact on the accuracy of themodeled parameters [14, 17, 111, 112]. Therefore, the AIF should be sampledwith a sufficient temporal resolution and within an artery supplying the region ofinterest [13, 108]. This can be challenging in situations where a major artery islocated far from the imaging site [10] or when partial-volume effects (PVE) affectthe concentration measurement within small supplying arteries [120]. The vesselselected for the AIF measurement will depend on the goals of the study [82]. Ifbolus dispersion is a concern due to major arterial abnormalities - such as a stenosis- then a smaller vessel closer to the tissue of interest is used. Otherwise, a largervessel further from the site may be the better option to avoid partial volume effects(PVE).The situation is further complicated when imaging small animals, such as mice,due to their small body size[14, 15] and rapid heart rate [13, 16]. In addition,few vessels in the mouse are of sufficient size to measure the AIF with adequatetemporal and spatial resolutions. AIF measurements in mice are often performed inthe left ventricle [13, 15, 19, 94, 121], aorta [100], iliac artery [100] or tail vein [20]as these are the largest vessels.49The AIF is known to vary widely due to variations in the contrast injection,cardiac output and blood supply to the tissue of interest between patients [110].Therefore, it is recommended that the AIF be acquired for each experiment [14,117], including studies performed on the same patient multiple times [108]. It is,however, technically difficult to acquire the AIF when an acceptable vessel is notpresent in the imaging field of view [18].To combat this, multiple groups choose to use a population averaged AIF fromthe literature [17–19] in their analysis. Even though the population averaged AIFis expected to approximate the true curve, it does not account for inter [13] or intra-individual [5] differences. Nor does it reflect the blood flow to the tumour at thetime of the examination [8]. Despite these limitations, the population averaged AIFprovides a reasonable estimate when a high temporal resolution is not possible orwhen a major artery is far from the imaging plane. Care must be taken, however, asthe population averaged AIF may only be accurate for a particular pathology [109],specific injection protocol, contrast agent dose and strain of animal.Two commonly used AIFs in the literature are those proposed by Lyng et al.(mice) [19] and Parker et al. (human) [18].Population Averaged AIF in Mice by Lyng et al.One of the more popular population-averaged AIFs in a mouse was publishedby Heidi Lyng and her team in 1998 [19]. In their study, an amelanotic humanmelanoma xenograft (A-07 or R-18 cell line) was implanted on flanks of femaleBALB/c-nu/nu mice of 8-10 weeks old. A bolus of contrast agent (Gd-DTPA) wasadministered through the tail vein at a constant rate for 5 s duration. The contrastagent had a stock concentration of 0.5 M, and was diluted to 0.03 M with a 0.9%NaCl solution. The injection dose was set to 0.01 ml/g body weight.Imaging was performed with a 1.5 T Signa whole body tomograph. The AIFwas determined from the left ventricle of three separate mice using a T1-weightedspoiled gradient-recalled sequence with T R = 50 ms, T E = 6 ms, flip angle = 80oand temporal resolution of 13 s for a total scan duration of 10 min.Lyng fit a double exponential to the mean concentration-time curve (deter-mined from the relative signal intensity increase), Ca(t), in the left ventricle of50three mice. The functional form is:Ca(t) = Xe−xt +Ye−yt (4.1)Where x, y, X and Y are the fitted constants with values of 2.4 ± 0.9 min−1,0.04 ± 0.02 min−1, 291 ± 111 mM and 98.6 ± 3.4 mM, respectively. They usedthis fitted curve in their analysis rather than the experimental data to minimizenoise fluctuations.Population-Averaged AIF in Humans by Parker et al.Parker et al. provide a widely used population-averaged curve in humans [18]. Inthis study, 23 male patients with advanced cancer and demonstrating abdominal orpelvic masses were scanned four or five times for a total of 113 visits. The first twovisits were to assess reproducibility of the pharmacokinetic parameters, while theremaining three visits (N = 67) were used to calculate the population-averagedAIF.MR images were acquired on a 1.5 T Philips Intera system with a whole-bodycoil for transmission and signal reception. The baseline T1 was determined usingthree axial spoiled gradient echo scans with flip angles of 2o, 10o and 20o, and foursignal averages. The DCE-MRI experiment involved 75 consecutively acquiredaxial volumes with a flip angle of 20o and temporal resolution of 4.97 s. All scanshad 25 slices, T R = 4.0 ms and T E = 0.82 ms.A standard dose (0.1 mmol/kg of body weight) of Omniscan 0.5 mmol/ml (Gd-DTPA-BMA; gadodiamide Nycomed) was injected intravenously through the an-tecubital vein using a power injector at a rate of 3 ml/s. The injection was initiatedat the start of the sixth dynamic scan and was followed with an equal volume salineflush.The AIF was determined in the descending aorta or iliac artery using an auto-mated AIF extraction technique. The user selects the slice for the AIF measure-ment, while the algorithm extracts the signal time-course of every voxel in theslice. The signal intensity is converted to a concentration using equation 3.1, withan assumed contrast agent relaxivity of 4.5 s−1mmol−1. The algorithm then selectscurves that reach a maximum concentration within 10 s of the bolus arrival time51and have a peak concentration in the top 5% of all voxels. The second criteria isexpected to reduce the impacts of PVE.Sixty-seven AIFs were used in the population average. Prior to averaging, allAIFs were manually shifted such that the first-pass peak was aligned at the samepoint. The mean, median and standard deviation were calculated for each timepoint, and fit with the functional form:Cb(t) =2∑n=1Anσn√2pie− (t−Tn)22σ2n +αe−β t1+ e−s(t−τ)(4.2)Where An (0.809 ± 0.044, 0.330 ± 0.040 mmol min), Tn (0.17046 ± 0.00073,0.365 ± 0.028 min) and σn (0.0563 ± 0.0011, 0.132 ± 0.021 min) are the scalingconstants, centers and widths of the nth Gaussian, α (1.050 ± 0.017 mmol) and β(0.1685 ± 0.0056 min−1) are the amplitude and decay constants of the exponential,and s (38.078 ± 16.78 min−1) and τ (0.483 ± 0.015 min) are the width and centerof the sigmoid. The two Gaussian functions represent the first-pass peak and therecirculation peak, while the exponential decay term describes the washout phase.The results showed that the mean and median curves were similar, thus sug-gesting that there were no outliers in the data set. From the standard deviation,there was large variability during the transient first pass, but much less in the slowwashout phase. Variability in the width and peak of the first pass could result fromdifferent doses (determined from the patient mass), heart output and low temporalresolution.Parker et al. performed their analysis on a pixel-by-pixel basis using the ex-tended Kety model. The population-averaged AIF was used as the input curve forthe plasma (Cp = Cb/(1 − Hct)). Repeatability was assessed from the 95%confidence interval of a genuine change in a single individual between the firsttwo visits. Repeatability improved by 41.3% for Ktrans, 41.1% for ve and 22.6%for vp (percent change), when the population-averaged curve was used in place ofthe measured curve. However, this AIF is only valid for tissues in the abdominalregion and could lead to inaccuracies if used in other areas of the body.They concluded that the population-averaged AIF provides greater sensitivityto genuine changes between patients. This is especially important in clinical trialswhich require sensitivity to a physiologically relevant parameter over accuracy.52Since the temporal resolution was a limiting factor, the population-averaged AIFprovided information that may have been missed during the first pass of the bolus.4.1.1 Requirement for a High Temporal ResolutionTo accurately capture the rapid contrast kinetics in the vasculature following in-jection, a high temporal resolution is required [14, 108, 122]. This will captureimportant features, such as the bolus arrival time, the rate at which the bolus ar-rives at the tissue of interest, or the maximum concentration reached.Due to the smaller diameter of capillaries, not all contrast agent molecules willarrive at the same time [107]. The initial upslope of the AIF therefore providesinformation about the tissue perfusion flow. This phenomenon is very fast, andrequires acquisition speeds exceeding 3-5 s per image.Although faster imaging techniques are available, the temporal resolution isstill a concern [18, 19]. More recent measurements of the AIF focus on improv-ing the temporal resolution. Fruytier et al. [100] measured an AIF in the iliacartery of a mouse with a fast gradient echo, providing a temporal resolution of1.19 s. Ragan et al. [15], showed that a compressed sensing approach known ascardiac anatomy-constrained temporally unrestricted sampling (CACTUS) can im-prove the temporal-resolution. They segmented the image into several structures,and updated the images dynamically with two radial projections per measurement.Their AIF, measured in the left ventricle of the mouse heart, had an effective tem-poral resolution of 84 ms.Case Study: Interleaved Measurement of the Signal Intensity Curves inBlood and TissueTaylor et al. [123], developed a method to simultaneously measure the signal intensity-time curves in blood and tissue using a single-angle projection. This provides atemporal resolution of 50 ms for the blood-based measurement, while also allowingfor improved spatial resolution of the tissue of interest (muscle with 0.78 x 1.56 mm2resolution). Their technique alternates acquisitions of a 1-D projection for theblood-based curve and a single phase-encode line at the tissue of interest.The measurements were performed in rats, with an injected dose of 0.1 mmoL/kg53of Gd-DTPA through a tail vein injection. The slice locations for the blood-basedand tissue signal intensity-time curves were in the descending aorta and in mus-cle, respectively. Signal from the aorta was isolated by subtracting the backgroundsignal from the corresponding pixels. The background signal was taken from twonearby pixels in the projection data.The results showed the expected characteristics of an AIF, though the measuredcurve was for signal intensity rather than concentration. The curve was subjectedto a 29-point moving average filter to reduce noise. While their measurementsshowed great potential in capturing the rapid contrast kinetics following the in-jection, T ∗2 effects resulted in significant signal losses, and thus uncertainty in themeasured signal-intensity. Further, the curves need to be converted into a concen-tration before they may be applied for modelling.4.1.2 Phase vs Magnitude Derived AIFThe AIF may be derived from the signal magnitude [18, 19, 21, 124, 125] orfrom the signal phase [16, 126–129]. Traditionally, the AIF was determined fromchanges in the longitudinal (T1) or transverse (T2) relaxation times [19, 21], whichwas then converted to a concentration with an assumed linear relationship [111].Magnitude-based AIFs suffer from signal losses from T ∗2 relaxation effects at thepeak [100, 130], making accurate characterization difficult at high concentrations.Signal truncation becomes a larger problem at higher magnetic field strengths [82].One solution is to measure the AIF in a smaller vessel, where the maximum con-centration is lower, but comes at the cost of greater PVE biases.Most paramagnetic and superparamagnetic contrast agents will alter the bloodsusceptibility or shift its resonance frequency. In effect, it is possible to charac-terize the concentration through phase differences in the MR signal [129]. Recentstudies have looked at measuring the AIF from the signal phase [128]. Phase datais advantages as the signal phase evolves linearly with concentration over a widerange [128, 131], it is expected to have an SNR up to a full order of magnitudegreater than the magnitude data [14, 128], it is less sensitive to partial-volume ef-fects [100], in-flowing blood and the blood hematocrit [82], and it is relativelyinsensitive to T1 and T2 relaxation [126]. However, raw signal phase has a dynamic54range of 2pi radians, meaning that phase wrapping can occur at higher concentra-tions. The larger dynamic range is thought to be the reason for the significantlyhigher SNR potential [82], though phase wrapping becomes problematic when thetemporal resolution is not sufficient to detect the wrap.The phase shift, ∆φ , is dependent on the concentration of contrast agent in thevessel, C, and may be determined as follows [82]∆φ =4piωoχMζCT E3(4.3)Where ωo is the Larmor frequency, χM is the molar susceptibility of the con-trast agent, T E is the echo time and ζ is a factor that reflects the geometricalproperties of the vascular compartment. For the example of an infinite cylinder,ζ = (3cos2 θ − 1)/2, where θ is the angle relative to the static magnetic field. Itwill have a maximum value, of 1/3, when the vessel is oriented parallel to the mainmagnetic field, and disappear completely at an angle of 54.7o. The angular de-pendence must be taken into account when converting the phase difference into aconcentration. This can be done following the method of de Rochefort [129] or bycalculating phase coefficients for the expected vessel alignment [100].Simulating an AIF in a Closed-Loop System by Akbudak et alAkbudak et al. [126] designed a closed-loop system in which varying concentra-tions of a contrast agent could circulate freely, without having to move the phantomduring the scan. Their phantom was motivated to improve the accuracy of estimat-ing the contrast agent concentration through a difference in phase. As noted in theirpaper, magnetic field inhomogeneities can be removed through phase subtractionat two unique time points. But, this assumes that the field gradients, and thus thephantom set-up, are consistent.The phantom includes a long cylindrical tube (for imaging), a mixing reser-voir, an attenuation flask and a variable-rate peristaltic driving pump, all connectedwith transparent Tygon tubing. The cylindrical tube should have a large length todiameter ratio for equation 4.3 to be valid, and be aligned parallel with the mainmagnetic field to maximize the sensitivity of the scan (ζ = 1/3).The results showed an AIF of the expected shape for both a parallel and per-55pendicular orientation, but had a slow upward phase drift which leveled out aftersubtracting the background phase shift. The steady-state concentration was in goodagreement with theory and occurred after five complete passes of the bolus aroundthe system. They tested the linearity of the phase change with concentration forfour echo times (4.66 ms, 12 ms, 18 ms and 26 ms) and found a strong correlation:r = 0.99972 for a plot of ∆φ/T E vs C, with a slope of 2.561 ± 0.0023 deg/m-M/ms and intercept −0.33 ± 0.40 deg/mM. Since the x-intercept of the graph iswithin a standard error of 0, they argued that the finite duration of the RF pulse didnot significantly affect the ∆φ measurements. In addition, they discovered that thephase shift is invariant with T1, T2 and the method in which the echo is sampled(including duration and symmetry). This makes a phase-based AIF measurementmore robust than a magnitude-based approach.4.2 Alternative Methods4.2.1 Dual-bolusThe concentration of the injected bolus must be chosen carefully. A high doseis preferred for improved SNR at the tissue of interest, but also leads to satura-tion effects which make accurate determination of the AIF difficult. Yet, too lowa concentration could mask important signal changes in the tissue of interest, de-spite providing a better estimate of the AIF. Utilizing the benefits of both the lowand high dose injections, Kostler et al. [132] proposed a dual-bolus technique forquantitative multi-slice myocardial perfusion imaging. The technique involves twoconsecutive injections; a low dose bolus for the AIF measurement, followed with ahigher dose bolus for the myocardium measurement for improved signal changes.In their study, Kostler et al. injected Gd-DTPA into the antecubital vein andmeasured the signal changes on a 1.5 T Siemens scanner with a multi-slice, satu-ration recovery trueFISP with TR = 2.6 ms, TE = 1.1 ms and a flip angle of 50o.The two injections were given during two consecutive breath holds with a delay ofless than 1 min. As a proof of principle, they injected boluses of 3 ml, 9 ml and12 ml and compared the AIFs after rescaling for the different doses. The resultsshowed that the up-scaled 3 ml bolus did not match the 12 ml bolus. This con-56firmed that saturation of the blood signal affects the measurement for high doseinjections. They argue that a better method would be to represent the high-dosebolus as a summation of several lower dose boluses, temporally shifted by the du-ration of the injection. This, however, assumes that the system is linear and station-ary. They also discovered that the perfusion values were in better agreement withother studies and had smaller standard deviations when a low-dose AIF was used.Li et al. [133] showed improved results when the temporal resolution of the twoinjection scans were different, provided that the AIF was sampled with a temporalresolution greater than 1 s.The dual-bolus is attractive as the AIF may be measured at the start of eachexperiment, but it doubles the effective imaging time required [134], it assumesphysiological variations are negligible between the two injections [134] and thatthe contrast kinetics are identical for the two injections [111]. The dual-bolustechnique has been primarily applied to cardiac perfusion studies in humans.4.2.2 multi-SRT Measurment with Radial DataAn alternative to the dual-bolus technique is to estimate the AIF from the changein signal magnitude in radially reconstructed images [111, 125]. Since every radialspoke passes through the center of k-space, it is possible to reconstruct image se-ries with varying effective saturation recovery times (eSRT), simply by changingthe number of projections used in the reconstruction. This technique allows for anAIF measurement with a short eSRT, while maintaining high SNR in the tissue ofinterest by using a longer eSRT. As there is only one injection [135], the concern ofdifferent physiological states is removed and the total scan time is dictated by theDCE experiment. Though this technique shows potential for improving DCE-MRIparameter accuracy, Kholmovski noted that the AIF peak concentration was under-estimated in their study.Kim et al. [135] proposed an extension with a multi-saturation recovery time(SRT) method in which the AIF was estimated from three subsets of radially re-constructed images (24 radial spokes each), having different effective saturationrecovery times. They verified that the AIF measured with the multi-SRT tech-nique compared well with the dual-bolus technique in most cases. Since all ra-57dial spokes are acquired in one imaging slice, the supplying blood vessel must belocated within the imaging plane; which, as discussed previously, is not alwayspossible.4.2.3 Reference RegionGradient-echo sequences are known to be sensitive to vessels of all sizes for theAIF measurement [82]. At typical echo times for DSC-MRI (35−45 ms at 1.5 Tor 25− 30 ms for 3 T), there is potential for significant signal losses as a resultof larger amounts of contrast agent in large vessels. As such, the MR signal issusceptible to strong dephasing, which limits our ability to accurately measuringthe AIF. In this situation, measuring the AIF outside the artery could be beneficial.Kovar et al. borrowed a method from positron emission tomography that avoidsthe AIF altogether by comparing the enhancement characteristics of two tissues:the tissue of interest, and a reference tissue with known perfusion values [136].Despite showing promise, the reference region technique did not receive much at-tention until 2005 when Yankeelov et al. investigated its potential uses via simula-tions [109]. The main differences between the two methods are that Kovar tried toestimate the AIF from the reference tissue using the differential form of the Ketyequation, while Yankeelov developed a theory that is completely independent ofthe AIF using the integral form of the Kety equation.The reference region technique uses literature values for Ktrans and ve of thereference tissue, typically skeletal muscle, to estimate the perfusion parametersfor the tissue of interest. Since the AIF is not measured, the experimental timecan be dedicated to obtaining DCE images with greater spatial resolution or SNR.However, the assumption that the AIF is identical for both tissues may not be valid,leading to errors in Ktrans and ve [109]. The simulation results from Yankeelov etal. revealed systematic errors in Ktrans and ve of the tissue of interest that variedlinearly when incorrect values were used for the reference tissue [109]. This canbe an issue if there are differences within a cohort of animals in a study.584.3 Projection-Based AIF MeasurementWe develop a novel method for measuring the AIF in a mouse tail using MR pro-jections and the phase of the MR signal. This method involves the acquisition of asingle 2-D image pre-injection and a series of projections collected before, duringand after the contrast injection. Projection data has a temporal resolution equalto the repetition time, thus offering significant gains in the temporal resolution ofthe AIF, without compromising spatial resolution along the read-encode direction.Since a projection-based AIF is measured rapidly on an individual basis, it wouldbe applicable to DCE-MRI studies performed in small animals.The proposed technique is summarized in Figure 4.1. Data from the vesselis extracted from each projection through a subtraction of the background profile.The background profile is obtained from a projection of the pre-injection 2-D im-age along the direction of phase encode, after removal of the vessel data. Thisis justified as an application of the central-slice theorem as all projections passthrough the center of k-space [137]. The phase of the signal from each projectionis then compared to the average pre-injection value and converted into a concen-tration.The mouse tail was chosen for our analysis because of its simple geometry. Thetail contains four small, isolated point-like vessels in a tissue background void ofcomplicated organs. Since the vessels are located near the surface, the projectionmay be oriented such that the vessels are well separated in the acquired profile toavoid super-position of signals from two. The tail vessels are relatively straight andhave a sufficient diameter for the AIF measurement. The SNR can be improved byincreasing the slice thickness, but the vessels must be properly aligned to PVE.4.4 Experimental MethodsThe experiment was performed in three stages: a phantom-based experiment de-signed to determine the conversion factor between signal phase and concentrationof Gd, a flow-phantom experiment to validate our technique in the presence of flow,and a projection-based AIF measurement performed in-vivo.MRI acquisition took place on a small bore Biospec 70/30 Bruker 7.0 T MRIscanner (Bruker BioSpin Ltd., Etlingen, Germany). A birdcage coil (i.d. = 7.0 cm)59Figure 4.1: Schematic of the projection-based AIF measurement. One 2-Dimage is acquired pre-injection, followed by a series of projections be-fore, during and after contrast injection. This technique may be usedto increase the temporal resolution of the AIF as only one projectionis required per data point. The AIF is determined by first subtractingthe background signal from each projection, then comparing the signalphase to the pre-injection value.and an actively decoupled surface coil designed specifically for the mouse tail(width 7 mm, length 18 mm) were used for signal excitation and reception, re-spectively.The 2-D pre-injection image and the projections were completed as two sep-arate scans using the standard fast low angle shot (FLASH) pulse sequence. Thesettings between both acquisitions were identical, except that the phase-encode gra-dients were set to 0 mT/cm to achieve projections through the center of k-space.The free induction decay (FID) was read into Matlab (The Mathworks, Inc. Natick,MA USA) and centered such that the echo was properly positioned before applyingthe FFT.60Figure 4.2: Phantom used to validate the linear relationship between signalphase and concentration of Gd-DTPA diluted in saline. The phantomconsists of a capillary tube (inner diameter 0.4 mm) placed inside alarger glass tube. The area between the two tubes was filled with tapwater.4.4.1 Relationship Between Concentration and Signal PhaseA calibration phantom was constructed by inserting a capillary tube, with innerdiameter (i.d.) 0.4 mm, inside a larger glass spotting tube (i.d. = 3.7 mm). Theregion between the tubes was filled with tap water [129] to provide additional signalfor magnet shimming and a non-enhancing region to correct for hardware-relatedphase fluctuations [126]. Figure 4.2 depicts the phantom with dimensions.A number of Gd-based solutions, diluted in saline to concentrations between2 and 10 mM, were injected through the capillary tube at three physiologicallyrelevant flow velocities [93]: 1) 0 cm/s, 2) approximately 15 cm/s (1.00 ml/minflow rate) and 3) approximately 30 cm/s (2.00 ml/min flow rate). Gd-DTPA hasa similar relaxivity in saline and tissue fluids [138], so the same calibration factorcan be applied in-vivo.A KD Scientific power injector (model 780220, Holliston, MA, USA) wasused for the injection to reduce variations in injection profile. To investigate theeffects of flow within the capillary tube, the experiment was repeated with flowcompensation along the slice select direction. Under the assumption of plug flowparallel to the main magnetic field, flow compensation was not applied in the reador phase encode directions to keep echo time (TE) as short as possible.FLASH data was acquired with T R = 100 ms, T E = (4.6 or 8.0) ms, flip61angle 30o, FOV 15x15 mm2 and slice thickness 1 mm. The pre-injection 2-D Carte-sian image has a matrix size of 256x256, while the projections are 256x1. The twoecho times were set to the minimum and a value approximately double, and usedto verify if the phase-concentration factor was independent of the echo time. Foreach scan, 20 repetitions were completed to improve the SNR and allow for repro-ducibility of the signal temporally (i.e. phase drift during the experiement). A sec-ond reference phantom was placed in the field of view to track drifts in phase [126].This phantom is sufficiently far from the capillary tube that its signal should not beaffected by susceptibility effects.4.4.2 Validation: Colorometry with a Flow PhantomA colorimetric phantom study was performed to evaluate and cross-validate theprojection-based AIF in the presence of recirculating fluid. The flow phantom wasconstructed, based on the system proposed by Akbudak et al. [126]. It consists ofa peristaltic pump (Minipuls 2, Gilson), a recirculation beaker and three types oftubing: latex (inner diameter 3.2 mm), tygon (inner diameter 3.2 mm) and viton(inner diameter 1.0 mm) tubing. To span the length to and from the centre of thescanner bore to the pump in the adjacent room, 8 m of viton tubing was used.Initially, the system was flushed with tap water to clear out any contrast agentresidue and improve reproducibility between experiments performed within thesame imaging session. The recirculation beaker had an approx. volume of 2 mland served as a mixing site of the injected solution and water.A custom-built colorimeter was used to measure dye concentrations of AlluraRed 40 dye (Kool-Aid, Kraft Foods). This was constructed with a semi-microcuvette, through which fluid could flow through, a light emitting diode (LED)(OVLGC0C6B9, Optek Technology Inc.) and a photodiode (MTD5052N, Mark-tech Optoelectronics). Two operational amplifiers (TL082, Texas Instruments) pro-vided a supplying voltage to the LED and to convert the photocurrent produced bythe photodiode into an output voltage. The voltage was recorded with an oscillo-scope (TDS 3054B, Tektronix) and converted to a concentration of dye followingthe procedure of Sigmann and Wheeler [139]. The system was calibrated usinga set of known solutions of Allura Red 40 dye. A sketch of the flow phantom is62shown in Figure 4.5a.The flow phantom allowed us to temporally control the concentration of con-trast agent circulating around the system. As such, it is possible to validate theprojection-based AIF measurement in the presence of rapid changes in concentra-tion. The flow phantom was filled with water to an initial volume of 24 ml ±1 ml.A 0.8 ml bolus of 0.101 mM Allura Red 40 dye and 10 mM Gd-DTPA was theninjected at a rate of 11 ml/min with the power injector. The bolus was allowedto circulate for approx. 8 min to allow for multiple passes of the bolus throughthe entire system. MRI data was acquired using the standard FLASH experiment,with T R = 100 ms, T E = 3.92 ms, flip angle 30o, 256x256 matrix size, FOV15x15 mm2 and slice thickness 1 mm. The colorimetric data was manually shiftedprior to comparison due to the different sampling locations.4.4.3 In-vivo MeasurementsPrior to any experiments, the injection line was prepared. This included a butter-fly needle, a 25 µl heparin lock (filled with heparinized saline to prevent bloodclots and a pre-mature injection) and PE20 polyethylene tubing (Braintree Scien-tific, Inc.) containing the bolus injection. The length (l) of the injection line wascalculated from the weight of the animal, using l = VCA/(pir2), Where VCA is thedesired injection volume and r = 0.19 mm for PE20 tubing. For this experiment,1.0 M Gd-DTPA was diluted with saline to a final concentration of 30 mM, andinjected to a volume of 5 µl/g body weight (or dose of 0.15 µmol/g body weight).Premature mixing of the solutions in the line was investigated and shown not to bean issue. Fig.4.3 shows the assembly of the injection line.All animal-based scans were approved by the Animal Care Committee at theUniversity of British Columbia. Healthy NOD/SCID immune compromised micewere placed inside a custom-build induction chamber. This chamber has two con-nections for the isoflurane gas, and a v-shaped opening on one side to restrain theanimal while performing the tail vein cannulation. While restraining the mouse, abutterfly needle was inserted into the tail vein as far distal as possible. We chose todo the cannulation prior to anesthesia, as the aesthetic is known to reduce the bloodpressure, which makes the vein more difficult to see. The needle was secured in63Figure 4.3: Setup for the injection line with a 25 µl heparin lock (segmentA), bolus line (segment B), and saline flush (segment C). The heparinlock acts as a buffer between the animal and the contrast-agent volumeto prevent a pre-mature with fast drying glue.The mouse was anesthetized with a mixture of 2% isoflurane and oxygen. Tomaintain a safe body temperature, a heat lamp was shone over the animal duringset-up. Once the respiration rate reduced to approx. 100 beats per minute, themouse was moved to the coil set-up and positioned supine on the animal bed. Asubcutaneous saline injection (approx. 0.5 ml volume) was injected into the looseskin behind the neck to reduce dehydration during the scan. Lacri-lube was appliedto lubricate the eyes. The tail was positioned over the surface coil, straightened andsecured in place with surgical tape (3M Transpore surgical tape). Since the butterflycoil is metal, care was taken to position it as far from the surface coil as possible.The coil was then transported to the scanner where the tail was positioned at themagnet’s isocenter.The animal’s respiration rate and body temperature were monitored duringMR acquisition using MRI-compatible animal monitoring equipment (Small Ani-mal Instruments, Inc., Stony Brook, USA). This includes a fiber-optic temperatureprobe (tip diameter = 1 mm) and a pneumatic pillow placed over the lungs. Thebody temperature was maintained at 37 ± 0.1oC using heated air, and the respira-tion rate between 80-100 beats per minute. Small tweaks to the level of isofluranewere made during the experiment to adhere to these standards.The AIF was measured using an image-based and projection-based approachon separate days and different mice. The image-based measurement served as64a motivation for our newly developed technique and to estimate the concentra-tion long after the bolus injection. A FLASH pilot scan was performed to ensurethat the vessel of interest was aligned along the direction of the main magneticfield [128]. This orientation will maximize the SNR and reduce the presence offringe fields [129]. For this AIF measurement, a standard multi-slice FLASH pro-tocol was used (T R = 100 ms, T E = 6.874 ms, flip angle 90o, 5 slices, FOV15x15 mm2, slice thickness 1 mm) and repeated for eight time points. Each imagehad an acquisition time of 25.6 s and an inter-acquisition time of approx. 11 s,thus providing a temporal resolution of 37 s. A 115 µl bolus of 60 mM Gd-DTPA,diluted with saline, was injected at the start of the fourth image in the series. Thiswas done with a power injector set to 1.00 ml/min. This corresponds to an ap-proximate flow velocity of 15 cm/s in the tubing (PE 20, i.d. = 0.38 mm). Theinjection was preceded with a 25 µl heparin lock and followed with a 40 µl salineflush [111].A projection-based AIF was measured with the acquisition of one pre-injectionimage (256x256 matrix size), followed by a series of projections (256x1 matrixsize, 2560 projections) before, during and after contrast injection (TE = 3.92 ms).This AIF was also measured with a standard multi-slice FLASH (TR = 100 ms,TE = 3.096 ms, flip angle 30o, 5 slices, FOV 15x15 mm2, slice thickness 1 mm),but removed the phase-encoding gradients. To prevent signal loss at the peak ofinjection, a bolus of 30 mM Gd-DTPA, diluted with saline, was injected approxi-mately 25 s into the scan. This provides at least 256 measurements of the baselinephase. Since each projection is one measurement of the AIF, the repetition time(TR) of the scan will dictate the temporal resolution. Projections are much noisierthan MR images, so a long TR and optimized flip angle are desired to maintain asufficient SNR. For the purpose of our study, a repetition time of 100 ms and aflip angle of 30o were used. The long TR will allow for interleaved DCE-MRI dataacquisition between individual AIF measurements in the future.The proposed projection-based method is outlined in Figure 4.1. It involvesthe acquisition of one 2-D Cartesian image before injection, followed by a seriesof projections. The projections were acquired under identical scanning conditions,except that the phase-encode gradients were set to 0 mT/m. From the central-slicetheorem [137], taking the FT of a projection of a 2-D object along one axis is65equivalent to taking the central line of k-space of the 2-D object. Accordingly,a profile of the surrounding tissue was obtained by projecting the image alongthe phase-encode direction, after the vessel data had been removed. Using thebackground profile, we were able to isolate the MR signal from the vessel. TheAIF may be determined by comparing the mean phase for each projection with thepre-injection value. This phase difference is then converted to a concentration ofGd using the calibration factor determined previously.To verify that the concentration at 20 min post-injection is correct, blood sam-ples from four NOD/SCID mice were analyzed for Gd concentration using ICP-MS (inductively coupled plasma mass spectrometry, service provided by MatthewNorman at Exova, Surrey, Canada). These mice were not scanned, but had similarweights to those used for the AIF measurement (range 22.0-30.5 g). The mice wereinjected with a 30 mM bolus of Gd, mixed in saline, to a dose of 5 µl/g weight.The injection was performed manually over a period of approximately 20 s. Themice were euthanized 20 min post injection, at which time, a cardiac puncture wasperformed to extract a large volume of blood (volume collected ranged between200− 500 µl). Samples were brought to a final volume of 10 ml with 1% nitricacid (concentration 0.22 M) to prevent bio-degradation [140]. A control sample,consisting of 50 µl of 30 mM Gd and 9.95 ml of 1% nitric acid for a final concen-tration of 0.15 mM Gd, was also sent for analysis.4.5 Validation of the Phase-Concentration RelationshipThe goal of the calibration experiment was to determine the conversion factor, thatrelates a phase difference to a concentration of Gd, and to investigate the impactflow compensation has on the measurement. The results from this analysis verifiedthat the phase varies linearly over the range of concentrations chosen. In addition,it is independent of the flow velocity and echo time. Phase-concentration curvesfor the projection data, at both echo times, are summarized in Figure 4.4.The slopes of these curves were consistent for all cases studied (three flowvelocities, flow compensation absent/present, two echo times, and images/pro-jections) and had a value of (0.213 ± 0.001) rad/mM/ms. This value is consis-tent with the predicted value of 0.212 rad/mM/ms (∆φ /(∆[C] TE) = 2piγB0χm/3)66Figure 4.4: Calibration factor converting a phase difference into a concentra-tion of Gd for projections. Gd-based solutions, diluted in saline, wereinjected through a capillary tube at three biologically relevant flow ve-locities. The experiment was performed with a standard FLASH pulsesequence, with and without flow compensation.for a long cylinder oriented parallel to B0. γ is the proton gyromagnetic ratio(4.258x107 Hz/T), B0 is the strength of the main magnetic field (7.0 T), and χmis the molar susceptibility of the contrast agent (3.4x10−7 mM−1 for Gd) [141].The phase-concentration relationship holds in the presence and absence of flowcompensation. However, there appears to be a flow-dependent phase shift that issignificantly reduced when flow compensation is used. The extra phase shift be-tween stationary and flowing spins could result from our assumption of plug flow inthe capillary tube. The fluid close to the tube walls may have a slower velocity, andtherefore bias the phase measurement. Blood flow in the arteries and veins is notlikely constant, second order phase compensation would be ideal. But this comesat the expense of longer echo times, which also increases the probability of gettinga phase wrap near the peak. First order compensation brought the phase curvescloser together, and should be sufficient for a projection-based measurement.4.6 Validation with ColorimetryThe colorimetry experiment proved that signal phase is superior to magnitude formeasuring a change in intra-vascular concentration, as shown in Figure 4.5. A67schematic diagram of the flow system is displayed in the inset of Figure 4.5a.Figure 4.5a shows the concentration-time curve for the measurement of the dyeconcentration at different distances downstream from the injection site. Greaterdispersion of the bolus (lower peak height and broader width) can be seen in thecurve corresponding to the measurement made further downstream.Figure 4.5b compares the MR measurement of magnitude and phase during bo-lus passage. The magnitude data does not accurately reproduce the first pass of thebolus, likely a result of signal loss from T ∗2 relaxation. Even though the magnitudedata does show signs of recirculation, it is not to the same extent observed with theMRI phase-based or colorimetric measurement.Figure 4.5c shows the optical and phase-based concentration-times curves su-perimposed on the same graph. Since the location of the colorimetric measurementwas approx. 1 m downstream from the MR coil, it had to be temporally shifted for-ward to align with the MRI curve. The first bolus passage is narrower and higheron MR compared to colorimetry. Subsequent recirculation peaks agree very wellbetween the two modalities. The phase curve had peaks at the same locations tem-porally, but appeared sharper than the cuvette reading. Additional mixing of thedye in the cuvette (located downstream from the MR measurement) could causethis, which would lead to peak dispersion and a lower concentration. peak4.7 Projection-Based AIF in-vivoThe image-based AIF (temporal resolution 37 s) is shown in Figure 4.6. At thisresolution, the shape of the curve is not well characterized, particularly at the peak.Phase data has a dynamic range of 2pi radians, so any phases that exceed this valuewill be reset to a value between 0 and 2pi radians. It is unclear if this has occurred inour measurement. As such the peak concentration could be either 1.46 or 5.78 mM.Additionally, the time at which the bolus arrives at the measurement site is un-known. These ambiguities will cause severe errors in the model fits if the incorrectconcentration is assumed. The lack of temporal information motivated the use ofa MR projection-based approach to measure the AIF. The concentration long afterthe injection is 0.34 mM, which seems reasonable based on other studies[15].A projection-based AIF, having a temporal resolution of 100 ms, is shown in68Figure 4.5: Signal-time curves for the colorimetry phantom. a) shows themeasured colorimetric concentration-time curves from two cuvettes.The inset is a schematic of the flow phantom used for these measure-ments. b) is the average magnitude and phase of the MR signal mea-sured in the tygon tubing. c) compares the phase-based AIF measure-ment and the colorimetric concentration in a simultaneous acquisition.The colorimetric curve has been shifted in time to account for differentlocations of MR coil and cuvette.69Figure 4.6: Image-based AIF in the mouse tail. The AIF was measured inthe vessel indicated by the arrow. At a temporal resolution of 37 s, it isunclear if the measurement at the peak has exceeded the dynamic rangeof 2pi radians and was phase wrapped to a lower angle. By increasingthe temporal resolution, phase wraps will become more obvious. In ad-dition, the details of the curve are not well characterized, and the arrivaltime of the injection is unknown. Increasing the temporal resolutionwill reduce ambiguities in both.Figure 4.7. At this temporal resolution, the details of the curve are better charac-terized throughout the scan, and it is clear when the Gd bolus entered the bloodstream. A double exponential [13, 19], modulated by a sigmoid function [18] to fitthe injection, was fit to the data, having the functional form:C(t) =0.8137 · e−0.0807t +0.33991+4.0415 · e0.9932t (4.4)The double exponential fit, proposed by Lyng et al., is superimposed on theplot [19]. Their AIF was derived from data acquired in the left ventricle of threeseparate mice, and had a temporal resolution of 13 s. The final concentrations70long after injection are comparable between the two techniques, but there is a largediscrepancy at the peak concentration. Lyng determined the concentrations fromthe increase in T1 relaxation rate in a T1-weighted spoiled gradient-recalled (SPGR)experiment, with TR = 50 ms, T E = 6 ms and flip angle = 80o. Their injectionwas 10 µl/g body weight of 30 mM Gd-DTPA diluted in 0.9% NaCl and doneat a constant rate over 5 s. In comparison, our dose was 5 µl/g body weight of30 mM Gd-DTPA, injected over 6− 9 s, depending on the weight of the mouse.With a slower injection, a slightly lower peak concentration is expected. Lavini etal. [117] concluded that effects of flow or T ∗2 decay caused them to mis-measure thepeak concentration. Measuring the AIF with the signal phase should reduce errorsrelated to signal losses from T ∗2 effects. However, the signal from the blood may notrephase fully if the flow velocity is not constant. Higher order flow compensationwould reduce this error, but also leads to longer TE. This could be detrimentalto the measurement at high concentrations the signal decays rapidly due to T ∗2relaxation.Figure 4.8 shows four AIFs measured in four different mice. The injection bo-lus for these mice was 60 mM to see if the higher concentration bolus would affectthe shape of the curve. The AIF was obtained by averaging the phase-time curvesfrom all pixels associated with a vessel (typically 2-6 pixels along the projection).The four curves show similarities in shape and peak height, but also differ fromone another in terms of the rates of enhancement, wash out and final concentration10 min post injection. Superimposed on the figure, is the population averaged AIFfor this cohort (thick black curve). These results support the observation that theAIF varies between individuals.The results from mass spectroscopy showed a blood concentration of 0.170−0.195 mM for the mice of weight 22− 24 g, 0.293 mM for the 30.5 g mouse,and 34.8 mM for the stock solution. These results suggest that the steady-stateGd concentration (20 min post injection) should be approximately 0.2− 0.3 mM,which is consistent with our measured projection-based AIF. Gd clearance in miceis approximately 27 min in mice [5], so we would expect that the concentrationwould gradually decrease throughout the experiment. For comparison, most AIFexperiments were 10:40-38:24 in duration.If all contrast agent were to remain intra-vascular, we would expect a steady-71Figure 4.7: Projection-based AIF in a mouse tail with a temporal resolutionof 100 ms. Our AIF has a functional form as in equation 4.4. The dou-ble exponential proposed by Lyng et al. (temporal resolution 13 s) isshown for comparison [19]. Their curve has a much higher concentra-tion following injection and appears to be shifted temporally relative toour measurement.state concentration of 2.0 ± 0.43 mM Gd. This value was calculated assuminga total blood volume of 6− 8% body weight [142] and a 30 mM Gd-DTPA bolus(with error of 5 mM to account for the measured stock concentration) of volume5 µl/g body weight. The expected concentration is much closer to the observedconcentration at the peak than it is the steady-state value. A possible explanationis the extraction of Gd by the kidney and other tissues immediately following in-jection [4].72Figure 4.8: AIF measured in four individual mice, with an injection of60 mM Gd-DTPA (double strength). The AIFs all have a similar shape,but differ around the peak of the AIF. The population averaged result(solid black line) differs from each of the individual curves.4.8 DiscussionWe acquired an AIF in a mouse tail with a temporal resolution of 100 ms using aprojection-based approach. Our proposed method involves the acquisition of one2-D image before injection, and a series of projections before, during and afterinjection. Scanning parameters (TE, TR, flip angle, etc.) were kept identical forthe image and projection acquisitions. This allowed us to isolate the enhancementin the tail vein from the projection through a subtraction of the background profile.Our measurement was performed in a mouse tail due to the simple geometry andabsence of additional organs that could complicate the measurement. The mousetail contains four vessels along the outer perimeter. Care must be taken duringset-up of the imaging slice to ensure that two vessels will not overlap within the73projection. As such, the slice was often rotated by a couple degrees to ensure thatall vessels could be distinguished in the projection.The AIF was calculated by comparing the average phase between each projec-tion and the pre-injection value, and then converting the phase change into a con-centration with our calibration factor. Despite acquiring data within saline-basedsolutions, there is evidence that the contrast agent affects the signal phase similarlybetween blood and aqueous solutions [127].The AIF may be measured using manual selection of suitable voxels or usingan automated searching algorithm [82]. Though the automated procedure is mucheasier and less time intensive, it could select voxels that do not correlate with avessel [82]. To avoid this issue, the user must confirm that the selected pixels arevalid for the measurement. Manual selection has traditionally been more common.With this method, the user will investigate the concentration-time curves in allvoxels within a desired area, and identify those with the most arterial-like features(early and rapid initial slope, narrow peak and high peak concentration) and goodcontrast-to-noise ratios [82]. This study used manual selection of AIF pixels dueto the small size of the vessels in the image. Most vessels would cover 2-10 pixelstotal. Using an automated algorithm could have introduced PVE biases in the smallvessels that cover few voxels.Concentration-time curves obtained with the signal phase show remarkableagreement with the optical concentration measurements (Figure 4.5c). However,the height and width of the first peak differ between the MR measurement andcolorimetry. This could result from the different measurement volumes betweenMRI and colorimetry. The MRI measurement is made on a 1 mm slice of PE 20tubing (0.011 ml volume), while the colorimetry measurement is made within asemi-micro cuvette with a volume of 3 ml. The cuvette volume is much largerthan the 0.8 ml injection volume, so the maximum concentration of Allura Red 40dye could be underestimated by as much as a factor 3.75. However, this factor isdependent on the flow rate and the injection speed. Since the mixing is not instan-taneous, the peak width of the colorimetric curve increases, its height decreases,and the maximum of the peak is shifted to a later time. Figure 4.5a shows the peakshape in two separate cuvettes within the loop. The first peak for the second cu-vette is broader and has a lower peak concentration, which supports our argument.74All subsequent peaks better resemble one another due to bolus mixing in both thecuvette and recirculation beaker.The AIF proposed by Lyng et al. has become the standard population aver-aged curve for experiments performed in mice. Their curve has a temporal resolu-tion of 13 s and was determined from changes in T1 [19]. A double exponential,C(t) = Xe−xt + Ye−yt , was fit to the post-injection data where X = 5.8 mM,x = 4.4 min−1, Y = 0.7 mM, and y = 0.05 min−1. At their temporal resolution,it is unclear when the bolus injection began, how long it lasted, or if a recirculationpeak is present. It is therefore difficult to temporally align the Lyng population-average AIF to independently acquired DCE data. Furthermore, when fitting theLyng curve to DCE data of higher temporal resolution (< 13 s), concentrations fol-lowing injection may be overestimated, thus leading to errors in pharmacokineticmodel parameters.Attempts to measure murine AIFs have typically been derived from the ob-served change in the signal intensity [17, 112] or tissue T1 [111, 128], whichare then converted into a concentration using an assumed linear relationship [55].However, magnitude-based AIFs suffer from a few limitations. There is evidencethat the signal intensity only varies linearly with concentration over a narrow rangeof concentrations. This non-linearity in the signal intensity results from compet-ing T1 and T ∗2 relaxation effects at high concentrations [111, 127]. In addition, thevalidity of magnitude-based AIFs become questionable when the peak concentra-tions are sufficiently high to cause the signal magnitude to approach the noise floor.Though this affects both magnitude and phase measurements, magnitude-basedmeasurements incur greater errors. This is shown in Figure 4.5b, where the magni-tude data appears to greatly underestimate the peak concentration, while the phasedata provides a better estimate. Losses in MR signal, due to T ∗2 -relaxation [128],may be partially recovered by minimizing the echo time, using a spin-echo se-quence or reducing the concentration of the injected bolus. T1-based measure-ments also require a high-resolution pre-injection T1 map, which will increase thetotal time of the experiment. With the introduction of faster methods, such as theLook-Locker protocol, the T1 map may be collected much faster; but still adds tothe total scan time.Some of these limitations are relaxed when evaluating changes in the signal75phase. Though phase-based measurements are relatively new [126], the number ofstudies using phase have dramatically increased over the last few years [14, 131,141]. Phase is relatively immune to T1 and T2 relaxation times [126], is independentof the blood hematocrit [127], has an increased SNR compared to magnitude data[128] and has an established linear relationship with concentration over a largerrange of concentrations [14].The signal phase can drift slightly through the scan. This may be causedby scanner instabilities, such as drifts in the static field or the transmitter fre-quency [126]. To compensate, a non-enhancing, external reference phantom wasplaced next to the mouse tail such that it was close enough to track small fieldchanges near the point of measurement, yet not close enough for susceptibility is-sues. Analysis of the phase of the reference phantom showed that phase drift wasgenerally negligible for our experiments; but it could be corrected for, if required.Phase measurements have a dynamic range of 2pi radians. At high concen-tration or long echo times, phase wraps could occur when the phase exceeds 2piradians, and is reset to the modulus of the phase and 2pi . Phase wrapping couldlead to uncertainties in the actual concentration if the temporal resolution is insuf-ficient. Fortunately, this issue can be avoided using a faster imaging method toincrease the temporal resolution, or reducing the echo time to a minimum value.Maintaining a sufficient temporal resolution to avoid phase wraps becomes in-creasingly difficult in animals, where the vessel used for the AIF measurement isnot located near the tissue of interest (thus requiring two separate areas to image)and the injection time is short. Our results show that the concentration at the peakfor a 30 mM bolus injection is approximately 1 mM. At this peak concentration,we could increase the echo time to 29.50 ± 0.14 ms before phase-wrapping be-comes an issue at a field strength of 7 T. In general, a longer echo time will allowfor greater phase sensitivity to a change in concentration, but will come at the costof reduced SNR. Increasing the echo time too much could be detrimental to ourmeasurement, as the subtraction of a noisy background profile could introduce ad-ditional sources of noise. It should be noted that even if phase wraps were to occur,the high temporal resolution of our approach is sufficient to detect of these wraps,even during the ’difficult’, rapid enhancement period.Contrary to magnitude-based imaging techniques, flow in blood vessels can not76be suppressed with saturation pulses. Our phantom experiment revealed a velocity-dependent phase shift when flow compensation was not used. This shift appeared tobe consistent for all concentrations as the slopes of the phase-concentration curveswere similar for all cases studied. When flow compensation was applied priorto data acquisition, the phase shift was reduced significantly. As shown in Fig-ure 4.4b, the signal phases at each concentration were nearly identical for flowvelocities of 15 and 30 cm/s. However, there was a small offset in phase betweenthe steady-state and kinetic experiments. This could result from our assumption ofplug flow through the tubing [126] and suggest that first-order phase compensationmay be insufficient in the presence of variable flow velocities, such as observedduring an injection or from the associated increases in heart and respiratory rates.Measuring the AIF in a vein would be advantageous for minimizing ambiguitiesduring pulsitile flow.When measuring the AIF in a vessel, it is important to take into account therelative orientation of the vessel with the main magnetic field. The geometry ofthe vessel or surrounding organs that contains contrast agent must be known foraccurate characterization of the field phase shift. As discussed in the study by deRochefort et al. [129], the individual phase shift effects can be written as a linearcombination from individual organs. In special cases, a simplified model can beassumed. An example is describing the tail vein as an infinite cylinder in our study,which is justified as the length of the vessel is more than four times greater thanthe diameter [127]. More complicated organs may require that a shape factor isestimated. de Rochefort describes how to do this from MRI intensity images.Partial volume effects are consistently a concern when the AIF is measured ina smaller artery or when the spatial resolution is limited due to requirements fora higher temporal resolution or tissue coverage. This can be minimized by ana-lyzing only those pixels where the enhancement kinetics are rapid and follow theexpected shape. However, the signal from the blood vessel and surrounding tissueare complex values, in which the resulting signal will contain both constructive anddestructive contributions [82], thus making it difficult to separate the signal fromeach component.van Osch et al. [127], studied the implications of using a gradient echo se-quence for the AIF measurement and the impacts of PVEs on the concentration77measurement in the internal carotid artery. In their study, they oriented the vesselparallel and perpendicular to the main magnetic field to observe the effects of each.The signal in the extra-vascular compartment is expected to be time-independentfor the parallel orientation, which makes it easier to address PVE issues. van Oschet al. confirmed that the vessel signal follows an inward spiral, where the sig-nal magnitude decreases quadratically with increasing concentration and the phaseincreases linearly with concentration. When the signal is purely from within thevessel, the spiral is centered at the origin. While it is shifted away from the centeras more tissue signal is included. Depending on the magnitude of the shift of thespiral, the AIF may be over or under-estimated, or very distorted [82]. PVE cor-rections were performed using two calibration curves: a R∗2 vs C curve, and a φ vsC curve. The correction was compared with the conventional method of selectivelychoosing voxels with the desired AIF characteristics. The results showed that thecomplex correction - using the calibration curves - did a significantly better jobthan the conventional method did. Our measurement did not address PVEs, butthis could be a future area of study.A major issue with using a metal catheter for the tail vein injection relates tosignificant SNR losses due to magnetic susceptibility effects. The catheter shouldbe placed far from the sensitive region of the receiver coil to minimize this effect.From initial experiments (results not shown), we found minimal distortion and lossof SNR when the butterfly needle was placed outside the sensitive region of thetail surface coil. This complicates the set up as the injection needs to be done asclose to the tip of the tail as possible, and the animal positioned such that the tailcoil is closer to the animals body. An alternative is to use a non-metallic catheter.However, plastic catheters are less stiff and require more effort to use.Bolus delays result from the the AIF and tissue of interest being measured atdifferent locations, and so the AIF is temporally shifted relative to the contrast-uptake in tissue. This issue may be resolved by using a bolus delay-insensitiveimaging sequence, a model that includes a bolus delay term, or the shifting all theconcentration time curves to remove the delay before uptake [82]. Additionally,the use of a 2-D multi-slice imaging procedure for the DCE images means thateach slice is acquired at a different time, and thus have different delay terms. Thisissue is easier to address as the order at which each slice was acquired is known.78Single MR projections are acquired rapidly and have potential for greatly im-proving the temporal resolution of the AIF. However, the data from one projectionis noisier than a projection of an image. The SNR is known to improve by thesquare root of the number of lines used to reconstruct an image. As such, a singleprojection will be eight times noisier than an image acquired with sixty-four phaseencode lines. The SNR can be maximized with a strip-line saddle coil for the tail,using a 90o flip angle and a minimum echo time.Measuring the AIF in small animals is difficult due to the limited number oflarger vessels. Some groups have chosen to measure the AIF in the left ventricleof the heart [13, 15, 143] or in the iliac artery [14, 100]. The left ventricle isattractive for its large size and relatively stationary blood for a short period oftime. But, proper gating is essential for an accurate measurement. This can bedifficult if the heart rate is too rapid. Measuring the AIF in the iliac artery maybe advantageous as it is generally closer to the tissue of interest, but aligning itwith the main magnetic field is challenging [100], and its proximity to a numberof organs will make a projection-based approach more complicated. The tail waschosen for this application as it contains four large, widely spaced vessels (arteriesand veins) and fewer anatomic structures in the background. Since the vessels arerelatively straight and run the length of the tail, it is possible to increase the SNRwith a thicker slice. However, care must be taken during set-up to ensure that thevessel is properly aligned before increasing the slice thickness.Theoretically, the AIF is defined as the tissue response function for an instan-taneous delta function injection. Since the actual injection takes place over a cou-ple seconds, the AIF is determined from the convolution of the injection profileand the tissue response curve [82]. If a power injector is used, the injection pro-file may be assumed as rectangular. Figure 4.9a shows the expected form of theinitial upslope with an assumed rectangular injection pattern with a double expo-nential. The AIF in the figure was measured in a mouse that weighed 24 g. Assuch, the Gd-DTPA injection had a volume of 120 µl, and was injected over a pe-riod of 7.2 s (1.00 ml/min injection rate). To aid our investigation, the duration ofthe bolus injection (red line) and the expected time at which a recirculation peakmay occur (green line) were plotted. The recirculation time was estimated froman assumed blood volume of 6− 8% body weight [142] and a cardiac output of790.73 ± 0.19 ml/min/g [144].Comparing the expected upslope with the measured projection-based AIF re-veals that the enhancement only follows expectations for approx. 3 s. Beyondthis time, the curve levels off rapidly, despite more contrast agent being injected.Possible reasons for the disagreement between the simple model and our obser-vation includes a non-rectangular bolus profile or early contrast perfusion into theextravascular-extracellular space (EES). When two fluids, having different concen-trations of Gd-DTPA, share a boundary, diffusion can occur. Though an early testshowed that mixing of the saline and contrast agent was minimal, it was neverquantified. The duration from setting up the catheter until scanning the animal canbe on the order of 30− 60 min. Even in the presence of very slow diffusion, theboundaries of the contrast injection will be smoothed, meaning that the injectionprofile may not be perfectly rectangular. It is possible that diffusion occurred atboth boundaries. As shown in Figure 4.10, this would lead to a more gradual, s-shaped uptake at the start of the injection and a more rounded shape at the endof the injection with a lower maximum concentration. This is consistent with ourobservations and could result from a lower bolus concentration at the end. Theoverall impact will greatly depend on the rate and extents of diffusion.A second issue with diffusion is the ambiguity of the start of the injection ifthe initial upslope is shallow, which is possible with a trapezoidal injection profile.Early uptake of the contrast agent into the EES would reduce the concentrationof contrast agent in the blood plasma. As a result, the maximum concentrationreached would be lower than expected. The rate of perfusion is known to be de-pendent on the concentration gradient between the blood plasma and tissue. Atthe onset of the injection, the concentration gradient is large, so contrast agent willperfuse at a much faster rate than near the end of the injection when the contrastgradient is smaller as observed. Again, this could cause ambiguity in the locationof the start of the injection. Based on the figure, the injection protocol will have asignificant impact on the early enhancement characteristics of the curve. As such,the AIF should be measured for each injection protocol used. The projection-basedAIF technique will help reduce errors when injections differ between experiments.The injected bolus will flow through the vasculature towards the heart, whereit mixes with blood coming from other areas of the body. This mixing causes80Figure 4.9: Impact of the injection protocol on the expected shape of the AIF.The expected shape for a square injection protocol, with a double expo-nential clearance from the vasculature, is shown in a), while the ob-served upslope from the projection-based AIF is shown in b). Theseresults show that the shape of the AIF differs from expectations whenmeasured in-vivo.81Figure 4.10: Effects of diffusion of the contrast agent in the injection line.A rectangular or trapezoidal injection profile is convolved with a pro-posed double exponential tissue response curve. The expected rectan-gular injection profile shows a steady increase in blood concentrationfor the entire duration of the injection. The trapezoidal injection pro-file, however, produces a rounded shape at the start and end of theinjection, and a lower peak concentration.dispersion of the bolus, such that a second pass of the bolus will have a lower con-centration. The bolus may experience further dispersion after passing through thelungs and left atrium and ventricle. In general, the recirculation peak has a widerwidth than the first pass. This results from recirculation of blood from all over thebody, not just the site of interest. The green line in Figure 4.9b suggests that therecirculation peak may be masked by our injection, and will not be observed in ananimal model. This is consistent with another animal-based study [17].This chapter laid out the frame-work for measuring the AIF from a set of MRprojections. We were successful in measuring the AIF in several mice (temporalresolution of 100 ms), showing that though the shape is consistent between mice,there are subtle differences representing the physiology of the individual at the timeof the scan. One limitation of this technique is the loss of information in a secondspatial dimension. Perfusion of the contrast agent into the surrounding tail tissuecould lead to local tissue enhancement, and alter the shape of the acquired projec-82tion profile. The projection-based AIF assumes that any change in the projectionprofile is solely a result of changing signal within the vessel. Not accounting foraddition siganl changes in the nearby tissue would lead to a bias in the measuredintra-vascular concentration. It is expected that tissue enhancement will occur at aslow rate relative to changes in the blood. Therefore, it may be possible to measurethe degree of tissue enhancement temporally and correct the projections for it. Thisis the focus of the next two chapters.83Chapter 5Radial MR ImagingThe AIF is typically measured through the change in the T1 of blood plasma inMR images after a bolus of contrast agent has been administered [5]. The tem-poral resolution of the AIF measurement is often limited by the time required tocollect data for one image, which is typically on the order of seconds [5]. Thisresolution may not be sufficient if the contrast kinetics in the blood are rapid. Asdiscussed in the previous chapter, the temporal resolution may be accelerated witha projection-based method. Projections are expected to significantly improve thetemporal resolution of AIF measurement as only one line in k-space is needed.The projection-based AIF measurement assumes that all contrast agent remainswithin the vessel. Under this assumption, any observed change in the projectionprofile can be attributed directly to a change within the intra-vascular contrast agentconcentration. However, most vessel walls are permeable to the contrast agent,thereby allowing some to purfuse into the surrounding tissue [74, 75]. The contrastagent will interact with the tissue protons, leading to local concentration-dependentsignal magnitude and phase changes in the tissue[55]. In effect, the shape of themeasured projection profile will be biased. These local changes in MR signal arereferred to as tissue enhancement. Without compensation for tissue enhancement,the AIF measurement may be incorrect.The goal of this chapter is to compare three radial reconstruction methods anddetermine which is best suited for visualizing local tissue enhancement. Radialsampling is attractive since every projection passes through the center of k-space,84and could be used as a single measure for the AIF. Three methods of radial re-construction were applied to the data: 1) Re-gridding the radial data onto a Carte-sian grid, 2) Spatio-Temporal Constrained Reconstruction (STCR) and 3) the Non-Equidistant Fast Fourier Transform (NFFT). In addition, three sampling schemes -uniform, Golden angle and random - were investigated.5.1 Radial MRIMRI data is collected as a series of signal projections in k-space. Two commonmethods of data collection include rectilinear, where series of parallel lines areacquired, or radial sampling in which a set of radial spokes are collected [145].The reconstruction of MRI images from radial data dates back to the early days forMRI [53]. It was considered undesirable due to non-uniform sampling of k-spacedata [48], the presence of streaking artifacts [71], longer scan times [146] and theinability to apply the FFT algorithm directly to the k-space data [46, 147, 148].For these reasons, radial reconstruction has not been studied in great detail untilrecently, when more advanced techniques, such as highly constrained back projec-tion (HYPR) [149] or FOCal Underdetermined System Solver (FOCUSS) [150],proved to construct high quality images. Radial reconstruction is considered advan-tageous over rectilinear reconstruction since all radial spokes are equally importantfor image reconstruction, the spokes always cross the center of k-space, imagequality is not significantly deteriorated with the removal of radial spokes, objectdetails can still be visualized, even with few radial spokes [145], and motion andflow artifacts are suppressed [46]. The last point is attractive for studies involvingblood flow in a mouse tail.The first radial images were reconstructed using filtered back-projection [151],a technique borrowed from CT. Filtered back-projection uses the Inverse Radontransform and operates on the principle of the central slice theorem, which statesthat the 1-D Fourier Transform of a projection is equal to the projection of the2-D FT of the image along a radial spoke passing through the origin [33, 152].The radial spoke and projection are both taken for the same angle. In this respect,full images may be reconstructed from a series of radial projections taken from anumber of different angles spanning pi radians. This technique had limited scope85in MRI, and was later replaced with a gridding method [50, 52].Re-gridding involves interpolating the k-space data onto a Cartesian grid, com-pensating for the variable sampling density, and applying the FFT algorithm [50–52]. Radial reconstruction has gained popularity with the introduction of moreadvanced techniques, include the NFFT [74, 147, 153] and STCR [154, 155].5.2 Improved Temporal Resolution with CompressedSensingcompressed sensing (CS) is a technique that allows MR images to be acquiredrapidly by only acquiring a subset of k-space. It was first introduced by Lustiget al. with their innovative paper in 2007 [156]. A full analysis of CS is beyondthe scope of this thesis, but the main concepts are paramount for our applicationin which CS is applied in radial MRI. The following section briefly overviews thetheory and methodology.CS operates on the premise that any image with a sparse representation can berecovered from randomly undersampled data, provided an appropriate non-linearrecovery scheme is available. The technique was initially motivated by knowl-edge of image compression, in which an image may be represented with less datawithout a noticeable loss in visual quality. The compressed image was insteadrepresented as a vector of sparse coefficients that would hold the important imagedata. This lead to questions about whether it was necessary to collect data over theentire k-space if these images are also compressible. Extending the theory of CS toMRI, Lustig proposed to reconstruct an image from sampled linear combinationsof individual Fourier coefficients or k-space samples.Sparsity means that relatively few voxels (n N) have a non-zero value. Forthe purposes of his article, Lustig focused on images that have sparsity in a fixedmathematical transform domain. Since MRI data is implicitly sparse in k-space,significant reductions in the total scan time is possible by acquiring fewer phase-encode lines. There are three main assumptions regarding the data:The data requires a sparse representation. This includes pixel sparsity in an-giograms, spatial (edges) or temporal finite difference sparsity, sparsity in the im-ages wavelet coefficients or sparsity in k-space.86The data should be randomly sampled, such that it creates incoherent artifactsin the transform domain (appear as additive random noise). Since MRI contrast isfound at the center of k-space, variable sampling density (ie. pseudo-random) willselectively sample data more densely here. Radial sampling is unique in that theartifacts from under-sampling are incoherent streaking artifacts [157], even withuniform angular sampling.A non-linear reconstruction is used to enforce both sparsity of the image rep-resentation and consistency with the acquired data.Under these assumptions, and in the sparse domain, image artifacts becomeincoherent and may be removed with non-linear thresholding. This leaves only thesignificant coefficients, which contain information about the desired image. Thereconstruction is performed by solving the constrained optimization problem:minimize‖ψm+P‖1 s.t. ‖Fum− y‖2 < ε (5.1)Where ψ is the linear operator that transforms from pixel representation into asparse representation, m is the reconstructed image, Fu is the undersampled Fouriertransform, y is the sampled data, ε controls the fidelity of the reconstruction to themeasured data and ‖.‖1,2 represents the mathematical L1,2 norm. P is a penaltyterm which is commonly the total variation constraint (TV).To summarize the steps of the optimization, the randomly sampled, sparse datashows the strong components and incoherent artifacts that appear as additive noise.It is known that the strong components leak energy into the surrounding voxelswhen the Nyquist criteria is not met. This leakage energy is determined by thresh-olding the raw signal and calculating the interference from the remaining signalusing the point spread function. After subtracting off this interference, the interfer-ence level is significantly reduced and previously hidden components are discov-ered. The process is repeated until convergence.In their paper, they compared Cartesian images reconstructed as low resolution,zero-filled with density compensation and the above algorithm. The results showedthat the CS images best reproduced fine details (lost in the low resolution images)and did not suffer from interference artifacts like the zero-filled images did. Theimage quality was good for high acceleration factors, as long as variable density87sampling was used, and if the sampling pattern differed between slices for themulti-slice acquisition. CS is most effective with high contrast images as the strongcoefficients are often sparse. Lustig argues that the worst artifact is loss of lowcontrast features as these may be submerged within the incoherent artifacts.5.3 Methods of Radial Image ReconstructionLocal tissue enhancement may be visualized in MRI images acquired for the du-ration of the DCE study. To maintain the high temporal resolution of our AIFmeasurement, a radial acquisition scheme is applied. In this sense, every projec-tion is used in the estimation of the AIF and in a sliding-window reconstructedimage to assess local enhancement.Three radial reconstruction techniques are compared in this study: 1) Regrid-ding the data onto a Cartesian grid with Shepard’s method of interpolation [158],2) STCR [155] and 3) NFFT [159].5.3.1 RegriddingRadially sampled data may be represented in k-space as a set of ‘spokes’ whichintersect at the center [145]. This data often does not fall onto a Cartesian grid,so it must be first interpolated before applying the FFT [160]. The simplest radialreconstruction technique is re-gridding the radial data onto a Cartesian grid [50].The technique involves the interpolation of the data onto a Cartesian grid, compen-sating for the non-uniform sampling density across k-space [46], and performingthe 2-D FFT [52].Data interpolation may be done with a convolution Kernel [50]. The perfor-mance of the re-gridding method is known to be dependent on the choice of theconvolution Kernel. Other interpolation methods, such as Shepard’s method orDelaunay triangulation may also be used. Shepard’s method involves assigning alldata points to a pixel and calculating a weighted sum for each pixel [158]. Theweighting is defined as the inverse distance from the center of the pixel.885.3.2 Spatial-Temporal Constrained ReconstructionAnother technique - STCR - uses spatial and temporal constraints to reconstructhigh-quality images from sparse k-space data [154, 155]. Images are reconstructedthrough the minimization of a cost function:C = ||WFm−d||22 + α1||√∇tm2 + ε||1 + α2||√∇xm2 + ∇ym2 + ε||1 (5.2)The first term represents the data fidelity which quantifies the error betweenthe estimated solution and acquired data. Here, ||.||2 represents the L2 norm, Wis the binary under-sampling pattern used to obtain the sparse data, F representsthe two-dimensional Fourier Transform, which is applied to each time-frame in thedynamic sequence, m is the estimated image data and d is the acquired k-spacedata. The second and third terms represent the temporal and spatial total variance(TV) constraint terms, and are regularized by the parameters α1 (= 0.04) and α2(= 0.006). Here, ∇t represents the temporal gradient operator, ∇x and ∇y are thegradients of the image in the x and y directions, respectively, ||.||1 represents the L1norm, and ε is a small positive constant used to avoid singularities in the derivativeof the functional. The authors chose the TV constraints to help resolve artifactsfrom under-sampling, while preserving spatial edges and improving the SNR. Thistechnique has been shown to reconstruct high-quality images with as little as 15%of a complete data set (under-sampled in the radial direction) [154]. They defineaccuracy as the ability to successfully resolve fine details with minimal occurrenceof image artifacts.5.3.3 Non-Equidistant Fast Fourier TransformThe third technique is the NFFT [159], which reconstructs an image from datasampled at non-equispaced nodes [161]. The NFFT is an iterative technique thatsolves for the image as an inverse problem [162]. The algorithm estimates an MRimage from the non-uniformly sampled k-space data [153, 161], then comparesits FT domain signal to the acquired k-space data at the known trajectory loca-tions [162]. This comparison involves interpolating the FT data of the image, withtriangulation, at the known input node locations.89Since the FT and interpolation steps are linear, they may be combined into asystems matrix, A. This provides the forward problem:y= Ax (5.3)Where y is the acquired k-space data (under-sampled), and x is the image vec-tor. This problem is best solved iteratively since the image vector may be large,the problem is ill-imposed due to under-sampling and the data vector could be con-taminated with Gaussian noise [162]. The first iteration of the NFFT is equivalentto regridding the data onto a Cartesian grid [160], but without a density compen-sation. Further iterations enhance the accuracy of the reconstruction with the L2norm of the residuum - which is the cost function for the problem [162].φ(x) =12||Ax−y||22 (5.4)The goal is to find a vector x, that minimizes the cost function (x= argminx φ(x)).Block chose a variant of the conjugate gradient method to perform the minimiza-tion. Interested readers can refer to Chapter 5 of his thesis ([162]) for specificdetails.5.3.4 Sampling Schemes for Radial Data CollectionRadial sampling may be performed in numerous ways. The most common modesare incremental sampling, where all projections are equally spaced and collected ineither ascending order or alternating positive and negative angles [163], and Goldenangle sampling [164], where the angular spacing between consecutive projectionsis 111.246o. Random sampling may be used when a compressed sensing algorithmis used. Albeit, this form of sampling may be less efficient than Golden angle incovering k-space, which could introduce additional artifacts. Figure 5.1 providesan example of the distribution of k-space data for the three sampling methods.Golden angle sampling is based on the golden ratio [165], but over 180o sinceparallel opposed projections contain the same MR information. The idea is that theprojections will cover the entirety of k-space quickly and quasi-uniformly. For themost efficient coverage, a Fibonacci number of projections should be used. Under90Figure 5.1: The three sampling techniques investigated. For a better visualcomparison, the figure shows the sampling distribution for 20 or 21 pro-jections. a) shows uniform spacing between the projections. b) is forGolden Angle sampling where the angular spacing between consecu-tive projections is 111.246o. And c) is for random sampling where thechosen angles of acquisition are selected randomly.these conditions, there are two unique gap sizes (Figure 5.2a): Fi−1 larger gapsand Fi−2 smaller gaps, where Fi is the Fibonacci number equal to the number ofprojections used in the reconstruction. If a non-Fibonacci number of projectionsare used, there will be three different gap sizes as shown in Figure 5.2b.5.4 Comparison of Radial Imaging TechniquesThe goal of this study is to determine the radial reconstruction technique that is bestsuited for our application. Suitability is defined as the ability of the technique to re-produce the image with good accuracy (magnitude and phase) and having minimalinterference from image artifacts. Since the projection-based AIF measurementrequires an estimate of the background signal, a projection of the reconstructedimages - perpendicular to the direction of ’read-out’ - is performed. These arecompared to the expected projection profiles to quantify the effect that tissue en-hancement has on the measurement of a projection-based AIF.5.4.1 MethodsTo evaluate the potential of using radial projections to estimate tissue enhance-ment, a cylindrical phantom of similar dimensions to a mouse tail was used. A91Figure 5.2: Angular gap spacing between neighboring projections whenGolden angle sampling (111.246o spacing) is used. If the number ofprojections is a Fibonacci number, F(i), then there are two unique gapssizes: Fi−1 of the larger gaps and Fi−2 of the smaller gaps. But if thenumber of projections is not a Fibonacci number, then there are threeunique gaps sizes. This suggests that a Fibonacci number of projectionsis required for optimal uniformity in k-space if Golden angle samplingis used.92small capillary tube, of inner diameter 0.4 mm, was placed inside a larger glasstube, having internal diameter 3.7 mm. The larger cylinder was 9.2 cm in length,while the capillary tube was slightly longer to provide attachment points on eitherside for the injection line. The space between the tubes was filled with saline to actas tissue. Meanwhile, a solution of Gd-DTPA, diluted with saline to a final con-centation of 5.0 mM, was injected through the capillary tube to represent blood.To avoid additional artifacts due to motion, the Gd-DTPA solution was stationaryfor this experiment.The Cartesian image was acquired with a FLASH experiment on the Bruker 7 TMRI scanner. Signal excitation was done with a volume coil, and signal collectionwith a custom-made stripline surface coil designed for a mouse tail. The scanparameters were TE = 5.00 ms, TR = 100 ms, flip angle = 30o, FOV of 15x15 mm2,matrix size of 256x256. Radial data was acquired with the same scan parameters,but with 233 equi-spaced angles over 360o. The number of angles was chosenbecause it is a Fibonacci number, it is sufficiently small such that the samplingscheme may be repeated in a reasonable amount of time, and has the smallestresidual for Golden angle sampling (determined from the modulus of (1 : 360) ·111.246o with 360o). The flow compensated FLASH experiment was chosen forthis analysis as this will be used to study the mouse tail later.The k-space data was read manually into Matlab, and processed such that theecho occurred at the center of the projection with phase 0 rad. The Cartesian im-age serves as the reference scan with which to compare the fully sampled radialreconstructions.5.4.2 Reconstruction of Radial ImagesRadial images were reconstructed with re-gridding, using Shepard’s method ofinterpolation onto a Cartesian grid, STCR and the NFFT. Since Shepard’s methodof interpolation takes a weighted average of all data points within each pixel, adensity compensation filter was not applied.Reference radial images were constructed with each technique using all 233unique angles. These serve as the gold standard with which to measure the degreeof image degradation when fewer projections were used. This is a better compar-93ison than the Cartesian image as image artifacts and phase distortions observed inthe reference image are likely to show up in the under-sampled images as well. Allreconstructed images were normalized to have a mean value of 1 as this is morestable than using the image maximum in the presence of noise.With a repetition time of 100 ms, the time required to construct a fully sampledimage is 23.3 s. This may be too coarse to accurately characterize the rate of tis-sue enhancement following the injection. Radial images were reconstructed with144, 89, 55, 34 and 21 projections (decreasing Fibonacci numbers) to help iden-tify the optimal temporal resolution for characterizing local tissue enhancement.Three sampling schemes were compared: uniform angular distribution, Golden an-gle [164], and random angular sampling. The angular increment for uniform sam-pling was set to 360o/N, where N = 233,144,89,55,34 and 111.24o for Goldenangle. With random sampling, the projection numbers were randomly sorted, andthe first N projections were selected. This prevented the algorithm from using thesame angle twice and allowed for the pattern to be repeated.For Shepard’s method of interpolation, a pixel was defined as the area within0.5 voxels of the center of a pixel. The value of the pixel was set to the weightedsum of all data points within the pixel area. The weights are the inverse distanceof the sampled data location from the center of the pixel. Any sampled data pointfalling on the Cartesian grid is assumed to be exact and set as the pixel value. Thecenter of k-space was sampled for all projections, so the signal is determined as themean from all projections. The Nyquist criteria is usually not met at the edges ofk-space due to a limited number of projections. As a result, the re-gridding matrixwas zero-filled to prevent ringing artifacts due to truncation of data. The 2-D FFTwas applied to the data matrix to produce the image.Matlab code for the STCR [155, 166] and NFFT [159] techniques are availableonline. The downloaded code was modified for both to allow for Golden angle andrandom angle sampling over 360o. To adjust for angle dependent shifts in the echoposition, the center-of-energy of the first 233 projections was calculated. This issimilar to the center of mass, where the ’energy’ is related to the signal magni-tude. Each projection was individually shifted, such that the echo was properlycentered. A zeroth-order phase correction is applied by multiplying each projec-tion by e−iφecho where φecho is the phase of the center pixel (i.e. the echo). This94approach generally centered the echo better than the method outlined by Anh andCho (1987) [167].All projections were then normalized to have a maximum signal intensity of 1at the echo. A sinogram of the radial k-space data, which is a plot of the projectionsalong the y-axis and the angle of acquisition along the x-axis, showed that theposition of the maximum of each projection followed a sinusoidal curve (amplitudeof 1 pixel). The sinusoidal pattern was not an issue with the image reconstruction.In fact, keeping the pattern in the k-space sinogram provided the most consistentphase between the radial reconstructions and the Cartesian image. Smoothing itout often lead to image artifacts due as the appearance of the echo had a jaggedappearance with angle.Images are compared qualitatively and quantitatively to determine the numberof radial projections required to produce an image of sufficient quality to accuratelyvisualize tissue enhancement. The qualitative investigation involved visually com-paring the Cartesian image with each radially reconstructed image. The images areconsidered sufficient for our application if the signal contrast between importantfeatures (edges, vessel vs. tissue) was correctly represented and no phase artifactswere observed in the object.Quantitatively, images were compared using the Structural SIMilarity index(SSIM) index [168] (MATLAB code available at lcv/s-sim/). The index attempts to automatically predict perceived image quality bylooking at local patterns of pixel intensities and signal dependencies after the im-age has been normalized for luminance and contrast. Image degradation was basedon perceived changes in the structural information within the image.The SSIM compares the reconstructed image to a reference image on threelevels: luminance, contrast, and structure. These are estimated from the mean ofthe signal intensity, the variability of the data via the standard deviation of thesignal and the normalized difference from the mean, ((x− µx)/σx), respectivelyand are relatively independent of one another. The SSIM is calculated as follows:SSIM =M∑j=1[2µxµy+C1µ2x +µ2y +C1]α [2σxσy+C2σ2x +σ2y +C2]β [σxy+C3σxσy+C3]γM(5.5)95Where M is the number of voxels, µx is the average signal of the reconstructedimage, µy is the mean signal of the reference image, σx is the standard deviation ofthe signal in the reconstructed image, σy is the standard deviation of the signal inthe reference image and σxy is the correlation of the two signals from their respec-tive means (x− µx). α , β and γ are weighting factors for the luminance, contrastand structure comparisons. For our analysis, they were all set to 1. Finally, C1,C2 and C3 are small constants to avoid instability when one of the denominatorsapproaches zero. This equation has a maximum value of 1 when the two imagesare identical.The above equation is used to evaluate the global quality of the image. Fora local investigation, the statistics are computed for a smaller square window (eg.8x8 voxels2), which moves pixel-by-pixel over the entire image. This providesmore information about distortions throughout the image from a spatially varyingquality map, thus allowing for a more thorough comparison. The mean StructuralSIMilarity index (mSSIM) is the average of all the local values spanning the image.See [168] for more details.The SSIM index is a full-reference image quality assessment, which means thatthe complete reference image must be known. Our analysis satisfies this condition,as we have a fully sampled Cartesian and Radial images to use as a reference. It isimportant to note that the SSIM only compares the magnitude of images, leavingphase comparisons to subjective evaluation. Comparing the phase maps of theimages may not be resourceful in cases where there is a global phase shift or phasegradient across the image.5.4.3 Results: Fully Sampled Radial ImagesThe reference radial images are displayed in figure 5.3. The signal intensity in thereference radial images using Shepard’s method of interpolation and STCR differfrom the reference Cartesian image. The angle of the signal gradient is greaterthan that observed in the Cartesian image. As a result, a signal intensity cold spotappears in the top right-hand section of the main phantom and a hot spot in thelower left-hand side. Visually, the NFFT technique best resembles the signal in-tensity gradient of the Cartesian image in the main part of the phantom. In all96radial images, the signal intensity of the second, smaller phantom is comparable.Though the edges are blurred in the Shepard’s interpolated and STCR referenceimages. Image artifacts are observed in all reference images (including the Carte-sian image), however they are low intensity and do not appear to affect the signalrepresentation of the phantom.The signal intensity of the capillary tube shows a significant drop-off from thecenter. The signal intensity is comparable between all radial reference images, butlower than that of the Cartesian image (2.27±0.07), even at the center. The signalof the entire capillary tube is 1.48± 0.18 for Shepard’s method of interpolation,1.42± 0.27 for STCR and 1.43± 0.29 with the NFFT algorithm. The signal in-tensities improve at the center of the capillary tube to 1.90± 0.10 for Shepard’smethod of interpolation, 1.98± 0.11 for STCR and 2.02± 0.12 for NFFT. Thismay be attributed to signal smoothing between the high-intensity vessel and thelow-intensity annulus surrounding the vessel.Since all three reference images have the artifact, it is likely a consequenceof the radially acquired data. The k-space sinogram - an image of the echos asa function of the angle of acquisition - shows that the echo is well centered forall angles. The location and intensity of the echo does deviate slightly (followinga sinusoidal curve with period 233 projections), but it’s generally off-set by lessthan 1 pixel from the desired location. It is possible that the MR trajectory missesthe center of k-space for some angles (see Appendix B), which could introduceartifacts into the reconstructed image. The echo was centered with a sub-pixel shiftin k-space and a zeroth-order phase correction prior to image reconstruction. Othermethods of echo centering also produced the artifact. The chosen centering methodproduced the best visual images and the largest mSSIM value.Taking the percent difference between the radial reference images and theCartesian image, relative to the Cartesian image, confirms the above observations(Figure 5.4). The difference with Shepard’s method of interpolation and STCRboth have similar characteristics.The signal intensity difference is greatest on the superior and inferior sides ofthe phantom, with percent differences on the order of 20− 40% and 10− 20%,respectively. The percent difference drops to 0− 10% in the mid-section of thephantom. The difference between the Cartesian reference image and the NFFT97Figure 5.3: Reference radial magnitude images (233 projections) recon-structed with one of Shepard’s method of interpolation, Spatial Tem-poral Constrained Reconstruction (STCR) or the Non-equidistant FastFourier Transform (NFFT). The mSSIM indices are recorded in the ti-tle, which indicates how closely these images compare with referenceCartesian image in terms of luminance, contrast and structure.98Figure 5.4: Percent difference between the reference Cartesian image and theradial reconstructions, relative to the Cartesian image.Figure 5.5: Structural SIMilarity (SSIM) index maps for the reference radialimages. The mean SSIM (mSSIM) indicies for the entire image andonly within the phantom are listed in the title.reference image is more uniform throughout the main phantom, on the order of0− 10% on the left hand side, and increasing to 10− 20% towards the right side.The signal intensity of the phantom is strongest on the left hand side, and graduallydecreases towards the right side. The higher percent error on the right hand sideof the NFFT image could be a direct result of the signal drop-off. The magnitudedifference (Radial - Cartesian) is nearly uniform for the NFFT image, and has hot-spots on the superior and inferior regions with Shepard’s method of interpolationor with the STCR reconstruction. Based on these results, the NFFT would be thebest reconstruction technique.Visually, the NFFT produces the most accurate image, followed by STCR, thenShepard’s method of interpolation. These rankings are consistent with the mSSIM99Figure 5.6: Reference radial phase images (233 projections) reconstructedwith one of Shepard’s method of interpolation, Spatial Temporal Con-strained Reconstruction (STCR) or the Non-equidistant Fast FourierTransform (NFFT). The phase has a similar structure in all images andvaries slightly wihtin the capillary tube.index, with values of 0.720, 0.808 and 0.829 for Shepard’s method of interpolation,STCR and NFFT, respectively. Maps of the SSIM index are shown in Figure 5.5.As shown in the figure, the SSIM index is lowest at the edges of the phantomsand at the image artifacts. This suggests that there may be signal smoothing at theedges or a slight misalignment between the images. It is interesting that the mSSIM100is lower in the phantom than it is across the entire image (exception with the NFFT,in which the region within a radius of 128 voxels was used). The lower intensity ofthe background, void of structural edges, may cause this. Both Shepard’s methodof interpolation and STCR had similar intensity features within the phantom, sothe lower mSSIM of the Shepard’s image is likely related to the increased presenceof image artifacts in the background.The phase images for all three radial reconstructions (Figure 5.6) show similar-ities with the reference Cartesian image. In all images, the phase appears to have aband structure, with cold spots in the upper right-hand and lower left-hand regionsof the phantom. In addition, the phase of the smaller, external phantom appearssimilar in all images. Streaking artifacts, originating from the phantom, are visiblein both the Shepard’s method of interpolation and the STCR techniques.The phase of the capillary tube varies slightly between the four reference im-ages and appears to be consistent across the entire capillary tube. The Cartesianimage has an average phase of -1.17 ± 0.017 rad, while the three radial imageshave average phases of -1.26 ± 0.017 rad for Shepard’s method of interpolation,-1.28 ± 0.026 rad in the STCR image and -1.33 ± 0.018 rad in the NFFT im-age. The subtle differences between the three radial techniques could be due to themethod used for data interpolation and weight compensation. The larger differencecould be related to data centering in k-space, which involves one dimension withthe radial data and in two orthogonal dimensions with the Cartesian images. Cen-tering for the Cartesian image was done with integer shifting, based on the mostprobable location for the maximum.5.4.4 Radial Reconstructions with Fewer ProjectionsWith a temporal resolution of 0.100 s, it would take 23.3 s to acquire data for acomplete radial image. This temporal resolution may be too coarse for a DCE-MRIexperiment, hence it would be advantageous to reconstruct images with fewer ra-dial projections. Accelerated radial images were reconstructed with descendingFibonacci numbers of projections (144, 89, 55, 34 or 21) with all three recon-struction techniques and sampling schemes. Images are considered sufficient if theboundaries of structures are well defined, contrast between neighbouring regions101is preserved and there are minimal image artifacts. Figure 5.7 summarizes the ra-dial images (both magnitude and phase) reconstructed with 55 projections. Thosereconstructed with 144, 89, 34 or 21 projections are reviewed in Appendix A.The magnitude images with 55 projections reveals that uniform and Goldenangle sampling produce similar quality images, and are most comparable with thereference radial image (using all 233 projection angles). Images with random sam-pling are more blurred, have more artifacts, and in the case of NFFT significantlyreduced signal intensity. Images reconstructed with Shepard’s method of interpo-lation or STCR have a hot spot in the lower left-hand quadrant, and a signal coldspot in the upper right-hand quadrant. This is consistent with the reference im-ages, suggesting that the artifact is related to the reconstruction method. The signalintensity of the capillary tube is visibly lower with all three techniques, relativeto the reference image: 82.8 (uniform) and 89.0% (Golden angle) for Shepard’s,75.9 (uniform) and 85.0% (Golden angle) for STCR and 82.3 (uniform) and 93.1%(Golden angle) for NFFT.The phase of the radial images are dependent on the reconstruction methodused. All three reconstruction methods have a distinct streaking pattern in thebackground that radiates out from the main phantom. The patterns are similarbetween Shepard’s method of interpolation and the STCR, while the streaks havegreater clarity with NFFT. With all reconstruction methods, the phase is similarwhen uniform or Golden angle sampling is used, and more blurred for randomsampling. The phase of the capillary tube is uniform for all reconstructions, exceptfor random sampling and NFFT where streaking artifacts affect the entire image. Ingeneral, the average phase is within 4% of the phase of the reference radial images,which is satisfactory. The phase contrast between the smaller external phantomand the background is best with STCR (uniform and Golden angle sampling) andNFFT (all sampling methods). However, there are spatial modulations in the phasedue to background streaking artifacts.Based on these images, the NFFT appears to be the best radial reconstructionmethod, followed by STCR, then Shepard’s method of interpolation. The imagequality is significantly greater with uniform or Golden angle sampling comparedwith random sampling. Figure 5.8 compares the under-sampled radial images re-constructed with STCR or NFFT and uniform or Golden angle sampling.102Figure 5.7: Radial reconstructions with 55 projections. Images were recon-structed with Shepard’s method of interpolation, STCR or NFFT, anduniform, Golden angle or random sampling. All images are on the sameintensity scale.103Qualitative Assessment: Visual AppearanceWith STCR, the boundaries of the main phantom and capillary tube are sharp with89 or 144 projections, and become slightly blurred with 55 projections. Reducingthe number of projections to 34 or 21 projections results in severe blurring. Thesignal contrast between the capillary tube, the surrounding hypo-intense ring andthe main phantom is good with 55 or more projections, though the signal withinthe capillary tube is reduced with fewer projections. The mean signal intensity ofthe capillary tube is within 10.5% of the reference image when at least 89 projec-tions are used in the reconstruction, and drops to 58.2-72.0% with 34 projections.Contrast of the smaller external phantom is good for the images with 89 or 144projections, reasonable with 55 projections, and poor with 34 or 21 projections.Streaking artifacts in the background are low intensity, but become more notice-able with 55 and fewer projections. The randomly sampled images (not show dueto lower quality) followed a similar trend, though the signal intensity within thephantom has a modulating appearance, and the signal drop-off in the capillary tubeis more rapid. With 34 or 21 projections, the shape of the phantom is distorted.The signal phase of the STCR images are similar within the images recon-structed with 55, 89 and 144 projections. Reducing this number to 34 or 21 resultsin smoothing around the edges of the main phantom, and a greater appearance ofstreaking artifacts through the phantom. The average phase of the capillary tubewas consistent in the images with 55-144 projections, and deviated with 34 or 21projections. The phase of all randomly sampled images suffer from steaks through-out the entire image. This could be a result of variable sized gaps in k-space, withsome larger regions void of data. In addition, the average phase of the capillarytube started to deviate from the expected phase (from the reference image) with 55projections and deviated further with fewer projections.NFFT images with at least 89 projections were visually comparable with thereference images when uniform or Golden angle sampling were used. This wasassessed by sharpness of the phantom edges, relative signal contrast between thecapillary tube and the surrounding phantom, and visual appearance of image arti-facts. The images with 55 projections exhibits some signal smoothing, as observedthrough the loss of signal in the hyper-intense regions (left hand side) and loss of104Figure 5.8: Radial images reconstructed with STCR or NFFT and uniform orGolden angle sampling. In general, the image quality between uniformand Golden angle sampling are comparable. Both techniques reproducethe phantom well down to 89 projections, while the image with 55 pro-jections provides a reasonable iamge.105signal contrast between the capillary tube and surrounding phantom. The edgesof these images, however, still appear sharp. Reducing the number of projectionsto 34 or 21 results in more blurring and signal smoothing within the phantom. Inaddition, the lower intensity region between the capillary tube and phantom hasincreased signal relative to the images with 55 projections. The signal intensityof the capillary tube gradually decreases as fewer projections are used in the re-construction. It is within 5.9% of the reference image with 89 projections in thereconstruction, and falls to 54.3-81.8% with 34 projections. The secondary phan-tom is easily distinguished in images with 89 or 144 projections, with reasonablysharp edges. While the signal intensity of the phantom continues to be sufficientlygreater than the background with 55 projections, the edges appear blurred, likelya result of signal smoothing throughout the image. Decreasing the number of pro-jections to 34 or 21 causes further signal smoothing and loss of contrast betweenthe phantom and background due to image artifacts.Again, the images reconstructed with random sampling (not shown) are of in-ferior quality. The edges of the phantom are sharp with 89 or 144 projections, butthere is an observable signal drop-off within the main phantom and streaking arti-facts are present through the phantom. The secondary phantom is distinguishablefrom the background, but is blurred relative to the uniform and Golden angle im-ages. By 55 projections, the signal is significantly lower than the reference andwould not be useful for our application.The image artifacts appear as rippling at the edges of the phantom (high inten-sity regions) and streaks originating from the phantoms. With uniform and Goldenangle sampling, and 89 or 144 projections, only the rippling effect is visually no-ticeable with the current windowing. Streaking artifacts are present, but their in-tensity is low compared to the signal intensity of the phantoms. As we reducethe number or projections to 55, 34 or 21, the streaking artifacts become morenoticeable and with a higher frequency. By 21 projections, the magnitude of thesignal from the artifacts is similar to that of the secondary phantom. With randomsampling, rippling and streaking artifacts are present in all images.The phase of the main phantom is similar to the reference NFFT image downto 55 projections with uniform or Golden angle sampling. Decreasing this to 34 or21 resulted in an increased presence of streaking artifacts throughout the phantom.106The background phase changed from being random Rician noise to clear streaksat 55 projections. The density of the streaks appears to be dependent on the num-ber of projections used, as the frequency dropped in the images with 34 and 21projections. The phase of the smaller reference phantom is comparable in the im-ages with 89 and 144 projections, while streaking artifacts affect the signal with 55and fewer. The average phase of the capillary tube is more stable than with STCRdown to 21 projections, with the maximum deviation being 0.20 rad for the NFFTimage with 34 projections and uniform sampling (compared with 0.55-0.63 radwith STCR). The randomly sampled images suffer from streaking artifacts in allphase images, with the severity dependent on the number of projections used in thereconstruction (worse for fewer projections).The images reconstructed with Shepard’s method of interpolation (not shown)are inferior to STCR and NFFT. Only the images with 89 or 144 projections, anduniform or Golden angle sampling, resembled the reference radial image. As thenumber or projections was reduced to 55 and fewer, the edges of the phantom wasnoticeably blurred, and the signal within appeared smoothed. In addition the con-trast between the capillary tube and the surrounding phantom was greatly reducedas fewer projections were used in the reconstruction. The lower intensity phantomis visible in all images, but suffers from signal losses and blurred edges in the im-ages with 89 and fewer projections. The images with random sampling are againinferior. The shape of the main phantom appears warped in all images with 89 andfewer projections, and signal modulations are observed within the phantom due tostreaking artifacts. Similarly to the other sampling methods, the smaller externalphantom is distinguishable with 89 and 144 projections, but suffers from signallosses and blurred edges with 89 and fewer projections.The image artifacts appear as a curved line, originating at the hyper-intenseregion of the phantom and the lower left-hand corner of the image, and a seriesof ripples on the lower and left hand sides of the phantom. The structure of theartifacts observed in the images with 89 or 144 projections are consistent with thereference image. This suggests that they could be a result of the chosen techniquesfor data interpolation and density compensation. As the number of projections isreduced (34 projections for uniform and Golden angle sampling, or 55 with randomsampling), more irregularly shaped spots are observed in the background. These107have an intensity of approx 30% of the phantom signal.The signal phase is similar between the three sampling methods, and has a dis-tinct structure with four bands originating from the main phantom: two approach-ing each top corner, and two curved features extending towards the lower cornersof the image. As the number of projections is reduced, the edges of the phan-tom become blurred and streaking artifacts are observed within the phase bandsin the background. The phase images become more blurred as fewer projectionsare used in the reconstruction, and streaking artifacts become apparent across thephantom. The average phase of the capillary tube is fairly stable with 55-144 pro-jections (only to 89 projections with random sampling), then deviates with 34 re21 projections. The deviation is slightly less than that seen with STCR.Quantitative assessment: mean Similarity IndexThe mSSIM provides a more quantitative assessment on how closely the under-sampled radial image compares with the fully sampled radial image reconstructedwith the same technique. It is expected that the under-sampled images will havethe same characteristic background artifacts as the reference image. By comparingthe under-sampled images with their respective reference image, these artifacts areaccounted for, and will not further penalize the similarity comparison. The mSSIMwas calculated over two regions: the entire image, and only across the main phan-tom to evaluate if masking would be beneficial for the tissue enhancement correc-tion. The mSSIM values are shown graphically in Figure 5.9, and summarized inTables 5.1, 5.2 and 5.3.As expected, the mSSIM index decreases as fewer projections are used inthe reconstruction, independent of the reconstruction technique or the samplingmethod used. In general, the mSSIM was greater across the main phantom, ascompared with the entire image. This is likely a consequence of background ar-tifacts. Uniform and Golden angle sampling often produced comparable mSSIMresults, and out-performed random sampling. This could result from larger gaps inthe periphery regions of k-space, causing a loss of image contrast and detail.The three tables all confirm that the mSSIM values are generally greater overthe main phantom as compared to the entire image, particularly when 55 or more108Figure 5.9: Mean Structural SIMilarity Index (mSSIM) comparing radial im-ages reconstructed with fewer projections with the reference image con-taining 233 radial projections. The mSSIM for the entire images is in-dicated by the solid line while the dashed line is for phantom only. Thecurves show that the mSSIM index is greatest with uniform and Goldenangle sampling, and lower when random sampling is used.109Table 5.1: mSSIM index for Shepard’s Method of InterpolationEntire ImageNumber of Projections Uniform Golden Angle Random144 0.700 0.706 0.62489 0.648 0.646 0.66255 0.661 0.627 0.58534 0.633 0.586 0.56321 0.563 0.576 0.541Within the Main Phantom OnlyNumber of Projections Uniform Golden Angle Random144 0.758 0.769 0.70589 0.676 0.679 0.73055 0.711 0.658 0.52534 0.655 0.572 0.61221 0.517 0.544 0.385Table 5.2: mSSIM index for Spatio-Temporal Constrained ReconstructionEntire ImageNumber of Projections Uniform Golden Angle Random144 0.841 0.886 0.87789 0.809 0.845 0.81255 0.797 0.800 0.75434 0.673 0.756 0.66221 0.708 0.703 0.637Within Main Phantom OnlyNumber of Projections Uniform Golden Angle Random144 0.899 0.933 0.92089 0.854 0.881 0.82655 0.804 0.820 0.71834 0.655 0.734 0.66421 0.653 0.657 0.622110Table 5.3: mSSIM index for the Non-Equidistant Fast Fourier TransformEntire ImageNumber of Projections Uniform Golden Angle Random144 0.863 0.875 0.73389 0.829 0.829 0.68455 0.748 0.785 0.40034 0.639 0.708 0.44821 0.636 0.638 0.226Within Main Phantom OnlyNumber of Projections Uniform Golden Angle Random144 0.901 0.902 0.79589 0.840 0.842 0.72655 0.764 0.795 0.53734 0.611 0.701 0.55721 0.658 0.633 0.400projections are used in the reconstruction. For this analysis, a threshold of 0.800for the mSSIM provides a good comparison, while a value over 0.900 representsan excellent comparison.With Shepard’s method of interpolation, none of the images meet the criteriafor a good comparison. The mSSIM had a maximum value or 0.700 and 0.706with 144 projections and uniform and Golden angle sampling. Since these imageswere all compared to the reference image with the same reconstruction technique,the lower values could represent loss of resolution or contrast between the differentstructures (main phantom, capillary tube, external phantom). The mSSIM over themain phantom was improved in the images with 89 or 144 projections, similar with34 or 55 projections and worse with 21 projections. Again, no image exceeded thethreshold of 0.800.The mSSIM indices for STCR also show a steady decline as the number ofprojections drops from 144 to 21. This time, the Golden angle sampling schemeperforms best, providing the highest mSSIM values for all acceleration rates tested,followed by uniform sampling then random sampling. All radial images with 89111or 144 projections, as well as the image with Golden angle sampling and 55 pro-jections, meet the criteria of a mSSIM greater than 0.800. The image with uniformsampling and 55 projections is close to 0.800 and could be considered sufficientquality if this form of sampling is desired. When considering the mSSIM withinthe phantom, images with 144 projections have excellent comparability with thereference image, having values of 0.899, 0.933 and 0.920 with uniform, Goldenangle and random sampling. Images with at least 55 projections and uniform orGolden angle sampling, or 89 projections and random sampling all have a mSSIMexceeding 0.800. All of these accelerated images would be considered sufficientfor compensating for local tissue enhancement as they compare well with the ref-erence image (pre-injection with our technique).The mSSIM for the NFFT exceeds 0.800 when uniform or Golden angle sam-pling are used, and 89 or 144 projections. Consistent with the visual analysis,random sampling provides lower mSSIM values. The mSSIM values for the entireimage and across the main phantom were similar when uniform or Golden an-gle sampling were used, and greatly improved for random sampling. Reconstruc-tions with 144 projections and either uniform or Golden angle sampling produceda mSSIM exceeding 0.900, while those with 89 projections exceeded 0.800. Thedrop-off in the mSSIM is significant as fewer projections are used, thus suggest-ing that at least 55 projections should be used in the reconstruction. In addition,uniform or Golden angle sampling is required with the NFFT technique.A more thorough evaluation of the SSIM maps for Shepard’s method of inter-polation shows that the greatest values occur within the capillary tube and at theedges of both phantoms. The maps with 55 projections, and all sampling schemesis shown in the top row of Figure 5.10. This trend was observed with all samplingmethods when at least 55 projections were used in the reconstruction. The regionwithin the two phantoms and in the background appeared as incoherent noise withno obvious structure. These regions cover a majority of the image and will con-tribute more weight to the mSSIM since it’s an un-weighted average. With 34projections, the mSSIM was greatly reduced in the capillary tube from an averageof 0.885±0.007 to 0.702±0.013 (uniform sampling), 0.86±0.02 to 0.61±0.02(Golden angle) and 0.76± 0.02 to 0.55± 0.03 (random). In addition, the ring ofhigh SSIM values at the edge of the phantoms faded, indicating signal smoothing.112Figure 5.10: Structural SIMilarity Index (SSIM) maps for radial images re-constructed with 55 projections. All images are compared with thereference image (233 projections) with the same technique. The aver-aging is done for the entire image with Shepard’s method of interpo-lation and STCR, and only within the circular region 128 voxels fromthe center for NFFT as there is where the image reconstruction is con-strained.These observations are more dramatic with the maps with 21 projections.The second row of Figure 5.10 shows the SSIM maps for images reconstructedwith STCR. The results show that the accelerated images have higher consistencywith the reference image in the background and inside the phantom for all casesstudied. Similar to Shepard’s method of interpolation, the SSIM values were great-113est in the capillary tube and at the edges of the main phantom when at least 55projections are used. The SSIM values at the edges of the secondary phantom arehigh with 144 projections, but decrease rapidly as fewer projections are used. TheSSIM values inside the capillary tube dropped dramatically with 34 projections anduniform (0.52±0.06) or random sampling (0.65±0.03) and approached values of0.53±0.05 (uniform), 0.56±0.03 (Golden angle) and 0.47±0.03 (random) with21 projections. The edges of the main phantom continued to have higher SSIMvalues down to 21 projections, but they were noticeably reduced from the mapswith 55 projections. The SSIM maps show more clearly when image artifacts arepresent. The region inferior of the main phantom experience a rapid drop-off inSSIM values as the number of projections is reduced. By 34 projections, all SSIMvalues are below 0.4 due to streaking artifacts in the background.The third row of Figure 5.10 shows the SSIM maps for NFFT images recon-structed with 55 projections. It is clear that uniform and Golden angle sampling aresuperior to random sampling across the entire image, though the gains are great-est in the background. The SSIM values have a strong dependence on the numberof projections used in the reconstruction, particularly moving down from 89 to 55projections. Generally, the edges of the phantom were sharp in images with 34 ormore projections, as indicated by SSIM values exceeding 0.800. Similar to STCR,the mSSIM in the capillary tube sees a significant drop when the number of projec-tions is reduced from 55 to 34: 0.906±0.008 to 0.52±0.06 for uniform sampling,0.92±0.01 to 0.812±0.02 for Golden angle and 0.866±0.01 to 0.645±0.03 forrandom sampling. The SSIM values within the phantoms and in the backgrounddecrease rapidly from 89 to 55 projections. The SSIM values in the background arerandomly distributed with 144 or 89 projections and uniform or Golden angle sam-pling. With 55 projections, the background had subtle structured artifacts aroundthe phantom. These became more prominent when using 34 or 21 projections in thereconstruction. The maps of all randomly sampled images had structured artifactsin the background, regardless of the number of projections used.1145.4.5 Recommended Radial Reconstruction TechniqueThe criteria for good radial reconstruction technique includes good visual similari-ties (sharp edges, similar signal contrast between distinct regions in the image, andminimal presence of artifacts) and high comparability between images with varyingnumbers of projections. Based on all results, the STCR or NFFT reconstructionsare best suited for a local tissue enhancement correction. For both methods, imagesreconstructed with fewer projections retain structural similarities to the referenceimage. This is important as it provides additional flexibility when correcting forlocal tissue enhancement. The pre-injection image will benefit from using the fulldata set (233 projections for this experiment) to provide the most accurate image.Meanwhile, the temporal resolution of the post-injection images depends on therate at which the contrast agent extravasates into the surrounding tissue. A highertemporal resolution is beneficial for rapid changes, but this comes at the cost ofblurred structural information. The correction involves a comparison of imagesbefore and after the injection, so the comparability of these two images should begood. In addition, image contrast between distinct regions within the image - suchas between the capillary tube and surrounding phantom - must be preserved. This isimportant for an AIF estimation in the mouse tail, as the signal intensities betweenthe vessel and surrounding tissue may be significantly different [169].Uniform and Golden angle sampling consistently produced images of similarquality for all acceleration rates. Generally, images with 89 or 144 projections hadgood quality images, with sharp edges and good contrast between the capillary tubeand the surrounding phantom. Image artifacts were often not an issue with theseacceleration rates. In general, uniform and Golden angle sampling have mSSIMvalues exceeding that of random sampling. This suggests that either uniform orGolden angle sampling should be used. The significantly different values could re-flect the size of gaps in the k-space data, leading to insufficient contrast informationat higher acceleration rates.Re-gridding reconstructions involve a form of data interpolation and weightcompensation. Shepard’s method of interpolation combines both steps by perform-ing a weighted average of data points within the region of a pixel. The accuracyof the data within a particular pixel becomes a function of the data density. The115center of k-space will be most accurate - as noise may be averaged out during theinterpolation step - while the edges of k-space could be sparsely sampled. This canhave an impact on the sharpness of details or edges within the image, and therebyreduce the image quality more rapidly.Zero-filling is a common technique to address the missing data. The zerosshould have little effect closer to the edges of k-space, as the signal intensity isalready near-zero there. But, it could have more dramatic consequences closerto the center of k-space, where the signal gradient between neighbouring voxelscould be large. These sharp edges in k-space could introduce the streaking artifactscommonly seen in radial images, especially with fewer radial projections.Filtering the data with a mean filter (i.e. 3x3 pixel2 area) or a Kaiser Besselfunction can help improve the image quality. The filter has the effect of smoothingthe transition between the acquired data and the zero-filled voxels and also reducesthe number of zeros near the center of k-space. In effect, the presence of imagingartifacts will be less. However, smoothing the data also causes blurring of the edgesof the phantom and further loss of contrast between distinct image features, suchas the capillary tube in the center. Density compensation and filtering are oftenperformed simultaneously in regrdidding techniques. The chosen filter emphasizesthe importance of the high intensity contrast-containing voxels at the center of k-space, and filters more strongly as it approaches the edges of k-space where mostof the voxels have values closer to zero.Van Vaals [170] introduced the key-hole approach in 1993 as a viable solution.Their method updates k-space with variable temporal resolutions, but filling in themissing data - often in the outer regions of k-space - with acquired data from anearlier time. This is advantageous, as the filled data more closely resembles theactual values. The key-hole approach was applied to the image reconstructions inthis study, but did not improve the image quality substantially.In contrast, STCR and NFFT both reconstruct the image as an inverse prob-lem [154, 162], weighted with a cost function to enforce image continuity spa-tially and/or temporally. Iterative techniques are attractive as they account for theunder-sampling pattern and use prior object knowledge to fill in the missing in-formation [162]. The STCR technique reconstructs a batch of images together toenforce temporal regularity between consecutive images. For 200 images, the re-116construction must be broken up into thirteen batches of 16 images, which takesapproximately 15 minutes to complete. The NFFT reconstructs each image sepa-rately, and completes the reconstruction of 200 images on the order of 8 minutes.Since these two images produce similar quality results, the time restriction favorsthe NFFT. Further, the mSSIM values were better over the phantom, rather than theentire image. This suggests that masking of the physical objects is advantageous,especially if image artifacts are observed in the background.117Chapter 6Compensation for Local TissueEnhancement6.1 Local Tissue EnhancementLocal tissue enhancement is a result of contrast agent extravasating from the vas-cular space into the EES. The contrast agent interacts with the protons in the tissue,causing accelerated T1 and T2 relaxation, and thus signal loss and a loss of phasecoherence in T1-weighted images. The projection-based AIF is sensitive to localtissue enhancement as the projection data only contains information along one spa-tial dimension. Acquiring data at other angles, in particular perpendicular to theAIF projections, would be beneficial for visualizing local tissue enhancement.MR projections can be acquired at a number of angles spanning the range of0−360o. This enables the concurrent measurement of a high-temporal resolutionAIF from the projections, and a series of lower temporal-resolution MR images tovisualize tissue enhancement throughout the experiment. The MR projection is acomplex summation of all signal perpendicular to the readout direction. As such,some pixels contain information from the vessel and surrounding tissue. Whentissue enhancement is present, the shape of the projection profile is affected. If thevessel signal were removed from the projections, then the effects of local tissueenhancement may be evaluated quantitatively. This is where constructing radialimages becomes attractive.118Comparing the background profiles (with vessel signal removed), between pro-file i at time t and a pre-injection profile i (same angle), will provide a quantitativemeasure of how the profile shape was affected. Using this information, the projec-tions can be corrected for tissue enhancement prior to extracting the AIF.The goal of this chapter is to demonstrate that local tissue enhancement maybe visualized in radial MR images, and then compensated for, in the post-injectionprojection data. The simulations discussed in this chapter investigates the potentialof three radial reconstruction methods - Shepard’s method of interpolation, STCRand the NFFT - in effectively visualizing, and then correcting for, local tissue en-hancement.6.2 Methods: Simulated Tissue Enhancement StudyThe primary focus of the last chapter was to evaluate three different radial recon-struction techniques and investigate how the number of projections used in thereconstruction and the sampling technique used would affect the image. This studytakes the analysis a step closer to correcting for local tissue enhancement.A simulation study was performed, in which a local tissue enhancement wasadded to a Cartesian image. Projections were calculated from these images throughapplication of the forward Radon transform. From these projections, radial imageswere reconstructed temporally to evaluate the effectiveness of each technique insuccessfully characterizing the tissue enhancement.The purpose of this study is to evaluate our ability to accurately measure localtissue enhancement from radial MR images, and quantify the effectiveness of usingeach reconstruction technique discussed in the previous chapter. Only one vesselwas used in this study for simplicity in the analysis.6.2.1 Simulating Local Tissue EnhancementCapillary walls are permeable, meaning that some contrast agent can extravasatefrom the vascular space into the surrounding tissue (Figure 6.1). A simulation studywas performed, in which extravasation of the contrast agent into the surroundingtissue was added to the Cartesian image of the phantom for 2330 time frames (tem-poral resolution 0.100 s). These images are referred to as the simulated enhance-119ment images. The phase of the vessel signal was based on the mathematical fitof a previously measured projection-based AIF [171]. Gaussian white noise wasapplied to the AIF, such that the SNR of the curve was 40, to make the input curvemore realistic (i.e. presence of small magnetic field inhomogenetities, non-idealgradient waveforms, non-uniform distribution of the contrast agent in the vessel,etc.). Perfusion of contrast agent was seeded at the center of the vessel, and grewoutward spatially described by the function 0.1 · e−5t/230, where t is the time fromthe arrival of the contrast injection and the units are in pixels from the seed position.The shape of the enhancement region was elliptical, in which the radius along thesecond matrix dimension was 0.8 X that of the first dimension. The concentrationof Gd-DTPA in the tissue, CT , was calculated using:CT = Ktrans(AIF ∗ e−Ktranst/ve) (6.1)where Ktrans = 0.1/60s−1 [109, 172], ve = 0.5 [15] and * represents a convolu-tion. The concentration in a given pixel at a radius r from the seed point is a time-shifted version of this curve. The magnitude of enhancement was determined fromthe change in T1, (assuming T1o of 900ms in muscle [6], relaxivity 3.6 (mMs)−1 forGd-DTPA-BMA (Omniscan) at 7T [109, 173–175], and a FLASH experiment witha flip angle of 30o), while the phase was determined directly from the change inconcentration (φ =CT/(0.213 mM/(rad ·ms) ·T E)) with T E = 5 ms [171]. Thisresult describes the signal phase-shift within a vessel aligned parallel to the mainmagnetic field, and will be lower for any other orientation as defined by the geom-etry factor, ζ = (3cos2 θ −1)/2, where θ is the angle between the main magneticfield and the vessel.This study compared the three sampling techniques to evaluate the benefits anddrawbacks of each in a dynamic experiment. Radial images were reconstructedwith various numbers of projections. Winkelmann et al. [165] discovered that themost uniform coverage of k-space with fewer projections, and dynamic acquisi-tions, occurs with Golden angle sampling (angular increment of 111.24o) when aFibonacci number of projections is used. In this study, we define accelerations asreconstructing an image with fewer projections. The complete data set containsa total of 233 projections, while the accelerated images were reconstructed with120Figure 6.1: This figure shows how perfusion and extravasation of the contrastagent into the surrounding tissue leads to local tissue enhancement. Themagnitude images (top row) show varying degrees of local tissue en-hancement at 80 s,130 s,180 s and 230 s after the start of the experiment(the injection took place at 25.6 s). The middle row is the correspondingphase images. The bottom row compares the pre-injection backgroundsignal (black curve) and the projection of the chosen image (pink curve).It is clearly observed that the tissue perfusion affects the shape of theprojection profile in the neighborhood of the vessel. The effects of tis-sue enhancement on the AIF measurement are shown in the plot on theright hand side, where the black curve is the expected AIF, and the pinkcurve shows the measured AIF using the projection-based method.121smaller Fibonacci numbers (144, 89, 55, 34 and 21). We chose 233 projectionsas the full set, as it provided the smallest residual ( mod(233 · 111.246o,360) ≤mod(FN · 111.246o,360), where FN are Fibonacci numbers from 21 to 388) fromGolden angle sampled data.For uniform sampling, the angular increment was set to a constant value be-tween consecutive samples. Since the radon transform only provided angles atmultiples of 360o/233, there may be cases where a projection from the desiredangle was not calculated. In such cases, the projection with the closest angle waschosen. Though the data is not perfectly uniformly sampled, it provides a rea-sonable approximation. The projections for Golden angle sampling were spaced111.246o apart between consecutive samples, while the randomly acquired projec-tions randomly selected one of the 233 projections at each time point, and repeatedthe pattern after 233 samples were collected.Radial images were reconstructed from a subset of all projections using asliding-window approach for a total of 200 images (see Figure 6.2). For this study,the sliding window shift was set to a random number to avoid coherent artifactswithin the image series. Coherent artifacts appeared as angle-dependent phasefluctuations in the calculated projection-based AIF. Having a random window shiftsuppressed the artifact.Radial data was acquired by rotating the image by the desired angle (using bi-cubic interpolation) and projecting the image along the second dimension (equiv-alent to the Phase-encode direction). To address variations in the spatial size ofthe matrix at each angle, the matrix was zero-filled symmetrically to a size of[√2Nread x√2Nread ], where Nread is the number of read-encode points sampled,prior to the rotation. The simulation rotated the image matrix through 233 uniqueangles, determined from mod(n ∗ 111.246,260) with n = 1,2,cot233. This en-forces consistency in the sampled angles between the three sampling methods. A1-D Fourier transform is then applied to convert the projections into k-space data.The image rotation effectively simulates acquisition of data in a radial MRIexperiment. Since the local tissue enhancement can change between sampled timepoints, the selected projections will contain slightly different information about thelocal contrast enhancement. It is expected that if tissue enhancement evolves ata sufficiently slow rate, such that the time required to acquire data for an image122Figure 6.2: Radial data sets, particularly with Golden angle sampling, allowfor more flexibility in selecting data for the image reconstruction. Asliding window reconstruction is a popular method for analyzing dy-namic data sets. This method allows images to be reconstructed tem-porally, by shifting the sampling region between consecutive images.To prevent coherent artifacts between images in the series, we used arandom number generator to determine the size of the fast compared to significant signal changes in the image, than the reconstructedimages should provide a good estimate of tissue enhancement temporally. Thoughthis assumption may not be true in DCE-MRI studies with a highly permeablevasculature.Depending on the rate of local tissue enhancement, sampling data over a finitetime interval could cause temporal blurring of the contrast enhancement in theimage. To quantify errors related to reconstructing images with projections takenat different times, a second data set - referred to as the ’snap-shot’ data - was takento compare with the dynamic experiment. In the snap-shot data set, all projectionswere taken from a single enhancement image, at the mean time of the dynamicimage data. For instance, if an image was reconstructed from projections takenbetween times 10.1 s to 15.5 s, with 0.1 s temporal resolution, then a snap-shotimage was constructed with all projections taken from the image at time 12.8 s.For a fair comparison, the projection angles exactly matched those sampled in the123Figure 6.3: Two situations were analyzed to investigate the effects of tempo-ral blurring in the images. The static case involves taking all projectionsfrom an image at the central time. While the dynamic case involvestaking projections every 0.1 s to more accurately portray a DCE-MRIexperiment. The red lines represent the projections that are sampled ata particular time. The black lines are there to show that the samplingscheme is identical for both the dynamic and snapshot data sets.dynamic case. The 1-D Fourier transform was applied to all projection data priorto image reconstruction. Figure 6.3 illustrates the data sampling for the snap-shotand dynamic experiments.6.2.2 Quantifying Tissue Enhancement in Radial MRIRadial image series were reconstructed for the dynamic and snap-shot data sets forre-gridding with Shepard’s method of interpolation, STCR and NFFT. The imageswere normalized, such that the mean intensity of the pre-injection reference imageswas 1.00. This is more robust than normalizing to the maximum intensity in caseswhere a single pixel could have a higher intensity due to noise or an image artifact.Since characterizing the local tissue enhancement is the focus of this project,the first image in the series was subtracted from all following images after blockingout the signal from the vessel. This provided an image of only the enhancement re-gion and some low-intensity residual imaging artifacts. Artifacts in the background124of the image were removed by applying a mask that encompassed the phantom andcapillary tube.The expected enhancement profiles served as the gold standard. They werecalculated directly from the initial images of the local tissue enhancement, aftersubtracting the baseline image from all in the series. The expected projection pro-file, due to tissue enhancement, was calculated by projecting this difference imagealong the direction perpendicular to the angle of sampling. The change in signalintensity due to enhancement is directly related to the concentration of contrastagent within the tissue. This means that the changes will be small initially, andincrease with time. A quantitative method to compare the enhancement intensitywith the expected values is to look at the difference in signals due to enhancement,relative to the pre-injection image (equation 6.2).Idi f f =IX − IsimulatedIpre−in jection·100 (6.2)Where X represents either the dynamic or snap-shot image series. The units forthis analysis are relative to the signal of the pre-injection image to gauge the errorsassociated with the radial reconstructions. This procedure is outlined in Figure 6.4.The projection-based AIF method from Chapter 4 may be used to measure thesignal in the capillary tube. The ’acquired’ projection data came from the simulatedenhancement image series. Each image was projected along the second matrixdimension (perpendicular to the read encode gradient). The background profilesare calculated from the radial dynamic images, after applying a mask to removethe signal from the vessel.The AIF may be corrected with knowledge of how the local tissue enhance-ment affects the acquired projections. We can get this information by comparingtwo radial images from the dynamic series, after removal of the signal from thecapillary tube: the first image is reconstructed with data prior to the injection, andthe second coming from data acquired after. The comparison involved subtract-ing the pre-injection image from an identically sampled post-injection image. Theremaining non-zero signal in the difference image is related to contrast agent en-hancement.The difference image is then projected vertically (second matrix dimension) to125Figure 6.4: Procedure for calculating the error profile for the tissue enhance-ment region. The radial images are normalized to the mean of the pre-injection image. The tissue enhancement is isolated by subtracting thefirst pre-injection image from all images in the series. Next, the imageof the enhancement is projected along the direction perpendicular to thedesired projection angle. This profile is compared to the expected en-hancement projection using equation 6.2. The units on the differencebetween projection profiles plot is the percentage of the maximum sig-nal of the projection profile.126Figure 6.5: Procedure for correcting the AIF for local tissue enhancement.The correction involves comparing radial images from before and afterthe contrast injection, to isolate any difference between the two situ-ations. The vessel data is zeroed out before projecting the differenceimage along the second dimension. This provides the correction profiledue to local tissue enhancement. Next, the correction profile is sub-tracted from the acquired data and the projection-based AIF is repeated.The units in the difference in profiles is related to the average signalof the phantom in the pre-injection MR images (mean signal intensityis 1.00). The maximum signal in the projections is dependent on thesignal magnitude of the phantom and capillary tube, and the signal in-tensity gradient in the image.127produce a correction profile. This was added to the post-injection projections andhas the effect that it removes the tissue enhancement from the profile and replacesit with the original tissue signal. The projection-based AIF method was repeatedto produce a tissue enhancement corrected AIF. The procedure for the correctionis outlined in Figure Quantifying Tissue Enhancement from Radial MRImages: Simulation Study200 radial images, spanning the 2330 time points, were reconstructed for this anal-ysis with the sliding window method. For a more thorough analysis, and to confirmthat the NFFT provides the more accurate results, all three reconstruction methodswere studied. This includes the three sampling strategies and five acceleration fac-tors. 200 images were selected as a compromise between spanning the entire rangeof data with sufficient resolution and computational time and memory. The threeradial reconstruction techniques were compared based on their ability to accuratelyrecreate the local tissue enhancement. For this analysis, the first image of each se-ries (pre-injection) was subtracted from images in the series. This left an imageof the enhancement pattern and provided insights into potential problematic imageartifacts.Figures 6.6 and 6.7 summarize the local tissue enhancement pattern (magnitudeand phase) from all image series reconstructed with 55 and 34 projections. Thesedata sets were chosen as the image series with 55 generally produced images ofreasonable quality and the greatest drop-off in visual image quality occurred from55 to 34 projections. The image series with 89 or 144 projections were of higherquality (more uniform signal distribution and sharper edges), while the images with21 projections were clearly inferior (significant blurring, loss of contrast betweenstructures and increased presence of artifact). Both figures include the tissue en-hancement images from the three image reconstruction techniques and all threesampling methods. As a reference, the simulated enhancement pattern is plotted inthe top row. The magnitude scale is identical in both figures to make them easierto compare.The results show that both STCR and NFFT reproduce the local tissue en-128Figure 6.6: Images of the simulated tissue enhancement from radial imagesreconstructed with 55 projections at time points 1730 s. The top rowshows the signal magnitude and phase of the simulated enhancement asa reference. The second, third and fourth rows show the signal magni-tude (first, third and fifth column) and phase (second, fourth and sixthcolumn) of the enhancement images for all three radial reconstructionmethods. All images were normalized to the mean signal of the phan-tom in the pre-injection image, and are displayed on the same magnitudescale for an easier comparison.129Figure 6.7: Images of the simulated tissue enhancement from radial imagesreconstructed with 34 projections at time points 1730 s. The top rowshows the signal magnitude and phase of the simulated enhancement,as a reference. The second, third and fourth rows show the signal mag-nitude (first, third and fifth column) and phase (second, fourth and sixthcolumn) of the enhancement images for all three radial reconstructionmethods. All images were normalized to the mean signal of the phan-tom in the pre-injection image, and are displayed on the same magnitudescale for an easier comparison.130hancement well with 55 projections, when uniform or Golden angle sampling isused. For all three radial reconstruction techniques, images reconstructed with uni-form or Golden angle sampling have comparable image quality. Between these twotechniques, NFFT is visually more similar to the simulation. The signal magnitudeof the center of the enhancement region is preserved only with NFFT and uniformor Golden angle sampling. The signal is slightly under-estimated with STCR orShepard’s method of interpolation, with a visible signal variation across the re-gion. With STCR, the signal intensity on the inferior side of the phantom is wellrepresented, while it is underestimated on the superior side. This is consistent in allimages, regardless of the method of sampling used. Uniform sampling producesthe best results, followed by Golden angle sampling, then random sampling. Thesignal for the image with Shepard’s method of interpolation fades more uniformly,but had a hot and cold zones. The outer edge of enhancement is slightly blurred forall three reconstruction techniques, relative to the simulation.When using 89 or 144 projections, the image quality is improved (not shown).Both NFFT and STCR outperform Shepard’s method of interpolation, when uni-form or Golden angle sampling is used. Again, NFFT has a more uniform ap-pearance throughout the entire enhancement region, while STCR shows a slightintensity drop in the upper portion, and Shepard’s method of interpolation contin-ues to have hot and cold spots. The signal of the tissue enhancement region forthe randomly sampled images is visibly inferior to uniform or Golden angle sam-pling for Shepard’s method of interpolation and NFFT. The image quality is betterpreserved with STCR, though uniform or Golden angle sampling still produce bet-ter images. The outer edge of enhancement is sharp with STCR and NFFT, andslightly blurred with Shepard’s method of interpolation, when uniform or Goldenangle sampling is used. The enhancement images with random sampling have thegreatest visual improvement when more projections are used in the reconstruc-tion. While the edges of the randomly sampled images are sharp, the shape of theenhancement region is slightly warped for Shepard’s method of interpolation andwith NFFT. Again, the STCR reconstruction provides a higher quality image withrandom sampling than the other two techniques. In fact, the image quality with allthree sampling methods are similar for STCR.The image quality is visibly degraded after reducing the number of projec-131tions in the reconstruction from 55 to 34. This is especially apparent in the imagesreconstructed with Shepard’s method of interpolation and STCR, both of whichshow a significant drop in the signal intensity of the enhancement region. Whilethe NFFT images continue to have a higher signal intensity, there is some drop-offnear the inner and outer boundaries. In all images with uniform or Golden anglesampling, the edges are blurred. The image quality with random sampling is muchlower than with the other two sampling methods. The images with both Shepard’smethod of interpolation and NFFT had significant signal modulations within thetissue enhancement region and have a warped appearance around the outer bound-ary. However, STCR maintains a better quality image, that is comparable withthe uniform and Golden angle sampled images. These trends are more significantwhen 21 projections are used in the reconstruction.Image artifacts are another concern when reconstructing images with under-sampled data. Streaking artifacts are common for radial under-sampling. The arti-facts in the tissue enhancement images have a slightly different appearance, whichmay be related to the subtraction of two radial images (post minus pre-injection).The artifacts are most often observed in the background. With Shepard’s methodof interpolation, the artifacts become apparent with 89 and fewer projections, whenGolden angle or random sampling is used. The artifacts cause the signal magni-tude to appear jagged, with variable signal intensity in the angular direction, andshifts the phase relative to the uniformly sampled case. With 55 and 89 projections,the artifacts appear as small spots. These have a streaking appearance with 34 or21 projections. Surprisingly, the images with uniform sampling remain artifact freedown to 21 projections. The artifacts are minimal in all images with STCR, regard-less of how the data was sampled. The background artifacts in the NFFT have lowsignal intensity with uniform (34 or more projections) or Golden angle (55 or moreprojections) sampling, and are most easily observed in the signal phase. These ar-tifacts may not impact the tissue enhancement correction due to the low intensity,though a mask may be applied to the image to block them out. The randomly sam-pled NFFT images have an obvious streaking pattern in the background with 55and fewer projections. These artifacts are observed through the entire image andhave a significant effect on the signal of the tissue enhancement.The projection-based AIF measurement compares complex signals from the132Table 6.1: Average Phase in Tissue Enhancement Image with 55 ProjectionsAverage Phase in Simulation 0.85 ± 0.46 radTechnique Uniform sampling Golden angle sampling Random samplingShepard’s 1.15 ± 0.34 rad 1.15 ± 0.44 rad 1.11 ± 0.37 radSTCR 1.140 ± 0.076 rad 1.15 ± 0.22 rad 1.18 ± 0.19 radNFFT 1.121 ± 0.033 rad 1.12 ± 0.20 rad 1.11 ± 0.40 radFigure 6.8: Average phase within the tissue enhancement region from the ra-dial images. The uncertainty bars in the figure represent the standarddeviation of the phase. The average phase is off-set from the simulateddata set (black curve) for all three reconstruction techniques, but areconsistent within the reconstruction technique, independent of how thedata was sampled.background and from the acquired projections. Therefore, the phase data mustalso be conserved as fewer projections are used in the reconstruction. The phaseimages are more visually similar to the simulation down to 55 projections, whenthe data is uniformly or Golden angle sampled. Reducing the projections to 34or 21 does not appear to affect the phase dramatically, but the edges of the tissueenhancement region have a blurred appearance. The phase images with Shepard’smethod of interpolation (Golden angle or random sampling) and NFFT (randomsampling) have visible artifacts through the tissue enhancement region, causinga spotted appearance in the signal phase. Streaking artifacts are present in mostNFFT phase images, though these do not appear to impact the tissue enhancementsignal when at least 55 projections are used in the reconstruction. The averagephase (± standard deviation) within the tissue enhancement region is summarized133in Figure 6.8 and Table 6.1 for reconstructions with 55 projections. The averagephase appears to be consistently off-set from that of the simulation and is indepen-dent of the sampling method used.The phase mis-representation is consistent with all reconstruction methods, in-dicating that the issue could be a result of the Radon transform or how the datawas interpolated onto the Cartesian grid. The phase off-set with 55 projectionsis lowest with NFFT (off-set shift 0.26-0.27 rad), and similar between Shepard’smethod of interpolation and STCR (off-set shift 0.26-0.33 rad). The phase off-setimproves slightly with Shepard’s method of interpolation and STCR (uniform orGolden angle sampling) as more projections are used in the reconstruction. Theaverage phase for the NFFT images does not have an obvious relationship with thenumber of projections used. The observed phase off-sets may not affect the AIFmeasurement, as all radial images will be affected similarly. This will be addressedlater in the chapter.Comparing the enhancement images, through a percent difference, can pro-vide further insights into the effectiveness of each technique in correcting for localtissue enhancement. The enhancement images were compared as a difference be-tween radial and the expected enhancement, and represented as a percent changerelative to the signal of the pre-injection image. This, in effect, provides a morequantitative estimate of the errors introduced by the local tissue enhancement andhow significant these errors are with regards to a radial AIF measurement. Fig-ure 6.9 summarizes the results for the data set with 55 projections.The percent difference is lowest with NFFT, followed by STCR, then Shep-ard’s method of interpolation. The percent difference images with NFFT, and uni-form or Golden angle sampling, has a more uniform appearance than the other tworeconstruction techniques, showing only salt-and-pepper noise. The percent differ-ence increases gradually from 144 to 21 projections, with the most notable changeoccurring from 34 to 21 projections. The percent difference image for STCR iscomparable to NFFT in the lower portion of the tissue enhancement region, butshows a hot spot at the top right hand portion of the enhancement region. This isevident in all images, regardless of the number of projections used or the samplingmethod employed. The percent difference gradually increases as fewer projectionsare used (independent of sampling method used), with the most significant changes134Figure 6.9: Percent difference in the signal between the enhancement imagesfrom the radial reconstructions (55 projections) and the simulated en-hancement region at time point 225.8 s. The two color bars representthe signal magnitude (left bar - columns 1, 3 and 5) and signal phase(right bar - columns 2, 4 and 6). The signal is presented relative to thesignal of the pre-injection image. The results may be interpreted as thepercent error in signal associated with the radial reconstruction. For ref-erence, the mean percent signal intensity of enhancement region risesfrom 3.7% at the first incidence of local tissue enhancement to 66.5% attime point 225.8 s.occurring between 55 and 21 projections. With Shepard’s method of interpolation,a high intensity ring is observed around the outer edge. This is likely a result ofsignal blurring, as observed from the enhancement images. This large differencecould be problematic when compensating for local tissue enhancement, particu-larly if the region overlaps with the vessel signal in the projection. The percentdifference is more uniform closer to the center of the tissue enhancement.The percent difference, of the signal phase, is shown in columns 2, 4 and 6 inFigure 6.9. The phase of the percent difference has a uniform intensity, and seemsto be independent of the sampling method used. The average percent difference135Figure 6.10: Average percent difference in the enhancement region for im-ages reconstructed with 55 projections. The percent difference wasnormalized to the pre-injection image to provide insights into howmuch local tissue enhancement affects the signal intensity. These re-sults suggest that the STCR or NFFT (uniform or Golden angle sam-pling) are the best techniques for compensating for local tissue phase increases slightly when the number of projections is reduced to 55 withNFFT, or to 34 with Shepard’s method of interpolation or STCR.Figure 6.10 summarizes the average percent difference in the tissue enhance-ment region temporally, while Tables 6.2, 6.3 and 6.4 summarize the average per-cent difference (± standard deviation) from time points 108.1-225.8 s. This rangewas selected as the size of the enhancement region was sufficiently strong (size andintensity) to give a good estimate of the differences. As a reference, the averagesignal intensity of the expected tissue enhancement, relative to the pre-injectionimage, increases monotonically from 9.3±4.9 at time point 108.1 s to 66.5±4.4at time point 225.8 s.The results show that the STCR and NFFT perform better than Shepard’smethod of interpolation. The average percent error increases as the enhancementregion grows, a trend consistent between all sampling methods and number of pro-jections. In general, uniform sampling has the lowest errors, followed closely byGolden angle sampling. And the errors increase as fewer projections are used, withdata sets using 89 and 144 projections providing similar results, followed closelyby 55 projections. Reduction from 34 to 21 projections resulted in large jumps136Table 6.2: Mean Percent Difference in the signal Magnitude of the Enhance-ment Region from the Radial Images and the Simulation: 100 x (Shep-ard’s Method of Interpolation-Simulation) / Pre-Injection ImageNumber of Projections Uniform Golden Angle Random144 12.5 ± 2.4 14.0 ± 2.4 15.0 ± 2.189 13.3 ± 3.0 15.8 ± 2.8 16.9 ± 2.755 12.9 ± 2.8 15.5 ± 2.5 16.5 ± 1.934 12.3 ± 2.0 14.4 ± 1.8 18.9 ± 2.221 14.3 ± 1.9 16.0 ± 1.8 21.4 ± 2.3Table 6.3: Mean Percent Difference in the signal Magnitude of the Enhance-ment Region from the Radial Images and the Simulation: 100 x (STCR-Simulation) / Pre-Injection ImageNumber of Projections Uniform Golden Angle Random144 6.53 ± 0.69 7.49 ± 0.86 7.24 ± 0.8589 9.3 ± 1.7 10.2 ± 1.8 9.8 ± 1.555 8.6 ± 1.2 9.2 ± 1.4 9.0 ± 1.234 9.5 ± 1.1 9.1 ± 1.1 9.6 ± 1.021 13.7 ± 1.0 9.3 ± 1.0 10.7 ± 1.0Table 6.4: Mean Percent Difference in the signal Magnitude from the En-hancement Region of the Radial Images and the Simulation: 100 x(NFFT-Simulation) / Pre-Injection ImageNumber of Projections Uniform Golden Angle Random144 4.60 ± 0.26 6.95 ± 0.95 9.8 ± 2.089 9.5 ± 2.3 11.2 ± 2.1 13.4 ± 1.755 9.5 ± 2.6 9.5 ± 2.6 17.0 ± 2.934 9.2 ± 2.3 10.6 ± 1.9 17.88 ± 0.5021 9.1 ± 2.0 10.6 ± 1.3 18.1 ± 1.1in the percent difference, suggesting that the image reconstruction should use atminimum 34 projections, though more is better if the temporal resolution is not aconstraint.137Based on the results of the tissue enhancement images, STCR (any samplingmethod) or the NFFT (uniform or Golden angle sampling) are most appropriatefor a local tissue enhancement correction. The image series with 144 or 89 pro-jections provide the most accurate results, and are the best candidates for the localtissue enhancement correction. Temporal blurring does not seem to have an im-pact on the results as signal changes in the region surround the vessel are slow andsmooth. The next section will evaluate our ability to accurately correct for localtissue enhancement from projections of the radially reconstructed images.6.4 Results: Effects of Local Tissue Enhancement on theProjectionsThe projection-based AIF compares a background signal to the acquired projec-tions. Therefore, it is informative to study the projections of the radial imagesreconstructed with data acquired post-injection (enhancement images). The en-hancement images were summed (complex) along the second dimension to createprojections at each particular time. These projections provide information aboutthe evolution of local tissue enhancement across the experiment. Comparing theseto the projections from the simulation allows us to assess how effective each re-construction method would be in compensating for local tissue enhancement.It is expected that the contrast agent will distribute in the tissue at a slow raterelative to changes in the blood. If true, the lower temporal-resolution of the radialimages - from taking data over a finite time interval - may not compromise theaccuracy of the local tissue enhancement correction. The goal of this study is todetermine the lower limit of the number of projections required to successfullyrecreate the enhancement pattern and provide a reasonable correction to the post-injection projections. Figure 6.11 shows the enhancement profiles (oriented alongthe y-axis) for all radial image series with 55 projections (200 images spanning the2330 sample times).The projections have two distinct regions in which a sharp change in signalis observed: the first at 58.2 s, which corresponds to the injection, and another at108.1 s, when local tissue enhancement is observed in the surrounding tissue. Timepoint 173.0 s is the point at which the signal intensity from the tissue enhancement138Figure 6.11: Projections of isolated tissue enhancement, from the differenceof post and pre-injection radial images reconstructed with 55 projec-tions. The projection profile is oriented along the y-axis and time alongthe x-axis, with points of interested indicated (injection at 58.2 s, tis-sue enhancement observed at 108.1 s). The vessel location is outlinedwith the black box.approaches its asymptotic value. The black box outlines the region associated withthe vessel and is of greatest interest for this study.The tissue enhancement profiles with uniform and Golden angle sampling arevisually superior to those with randomly sampled data. The randomly sampled pro-files suffer from signal blurring and intensity modulations in time. None of theseare of sufficient quality for a tissue enhancement correction. The enhancementprofiles with uniform and Golden angle sampling are comparable with 89 and 144projections. With 55 projections or less, uniform sampling produces enhancementprofiles that vary smoothly across the experiment, while those with Golden anglesampling have slight signal variations between consecutive projections.139Between the three reconstruction techniques (uniform or Golden angle sam-pling), NFFT appears to slightly outperform STCR, while Shepard’s method ofinterpolation is of much lower quality. With Shepard’s method of interpolation,signal intensity modulations are observed in the data sets with 55 and fewer pro-jections, and the signal within the vessel region is lost in the data sets with 34 or21 projections. Only the data set with 144 projections is visually similar with thesimulation. With STCR, data sets with at least 55 projections are comparable withthe simulation, though the data set with Golden angle sampling and 55 projectionshas signal modulations within the region of the vessel. The profiles with 21 or34 projections suffer from blurring and loss of signal within the region of the ves-sel. The enhancement profile from the NFFT images are also comparable with thesimulation with 55 or more projections, though the data sets with 34 projectionsare still of reasonable quality. The data sets with 21 projections are clearly infe-rior, suffering from signal intensity modulations in time, signal blurring and lossof signal within the region of the vessel.There is a sharp increase in the signal difference within the voxels associatedwith the vessel that coincides with the timing of the injection (time = 58.2 s). Localtissue enhancement is not observed this early in the simulation, so all changes in theprojection signal are associated with a change in the vessel signal. The average pre-injection signal intensity of the vessel, within the projection, is 0.130 ± 0.017. Itreaches a maximum intensity of 17.3-17.7 at the center of the vessel or 15.3 ± 2.1over the entire vessel region. The variability is due to the number of voxels thatthe vessel spans in the second dimension. Since the vessel is assumed to be cylin-drical, the center will sum over more voxels than at the edge. For reference, theaverage signal intensity of the pre-injection image in the neighbourhood of thevessel is 79.8 ± 3.5. Recall that these profiles are the calculated from the summa-tion of an enhancement image ((post-injection DCE image at time t - pre-injectionimage)/pre-injection image).The signal magnitude at the center of the vessel reaches a maximum of 7.02-8.62 with Shepard’s Method of interpolation (34-144 projections), 7.10-8.29 withSTCR and 8.14-16.50 with NFFT. The maximum value was achieved with 55 pro-jections (uniform or Golden angle sampling) or with 34 projections (random sam-pling), regardless of the reconstruction technique used. These maxima are much140lower than the expected signal from the simulation, which reaches a peak intensityof 17.3-17.7. The under-estimation in signal intensity may be a result of signalsmoothing. The peak of the AIF in the simulation spans 5 time positions; only twoof which are selected time-positions for image reconstruction. Further, at most 5projections in the reconstruction will contain information of the peak concentra-tion, while the rest were acquired with lower intra-vascular concentrations. Theseresults reveal the importance of using a projection-based AIF measurement imme-diately following the bolus injection. Luckily, tissue enhancement is expected tobe minimal at this point, so the uncorrected AIF may be used.The signal magnitude, for data sets reconstructed with 55-144 projections,tapers off to average values of 6.15-7.01 for Shepard’s method of Interpolation,6.58-6.98 for STCR and 7.14-8.09 for NFFT just before tissue enhancement is ob-served in the surrounding tissue at 108.1 s. The signal in the simulation averages6.927 ± 0.034 over this range. The better agreement between the radial imagesand the simulation further supports the idea that temporal signal blurring may havecontributed to the lower signal intensities within the vessel following injection.Further, these results show that the tissue enhancement correction may still be re-liable in cases where the the contrast agent extravasates into the surrounding tissueearlier in the experiment.The next study investigates whether temporal blurring from data collected overa finite time interval impacts our ability to accurately correct for local tissue en-hancement. Figure 6.12 compares the profiles for a ’snapshot’ and ’dynamic’experiment. Recall that ’snapshot’ means that all projections were taken at onespecific time point, while ’dynamic’ refers to projections taken from a temporalwindow, centered at the time point of the ’snapshot’ image.The simulated profiles are shown in the top row as the gold standard for com-parison. Below it are the projection profiles for the radial reconstructions with 55projections and uniform sampling. The snapshot and dynamic profiles are bothplotted with the same signal intensity scale, while the difference is windowed toa scale set at 1% of the the snapshot and dynamic series. These percentages arevery low since the signal of the pre-injection profile is much greater than that ofthe tissue enhancement.The difference is quite small, but shows greater values in the vicinity of the ves-141Figure 6.12: Projection profiles of tissue enhancement for the dynamic andsnapshot image series, and the difference between them. Images werereconstructed with 55 projections from uniformly sampled data. Thek-space data was either acquired all at one time point (snapshot) orover a finite time interval (dynamic), in which one sample is taken overa temporal window centered at the time point of the snapshot image.The dynamic and snapshot series are windowed identically, while thedifference is set to a window maximum of 1% of the other Dyanmicand snapshot series. The units are relative, as the radial images wereset to have a mean value of 1.0.142sel. The signal within the vessel is on the order of 6.396 ± 0.010 from time points44.7-225.8, while the difference is 0.161 ± 0.013. This equates to differences onthe order of 1.46-2.38% (Shepard’s method of interpolation), 1.28-2.52% (STCR)or 1.41-2.22% (NFFT) of the vessel signal when 34 or more projections are used inthe reconstruction. These errors are relatively small and are not expected to impactthe accuracy of the AIF measurement. At the injection, the errors varies widely(with no clear trends), ranging from 3.6-107.6% in the data sets with 34 or moreprojections. Relying on the projection-based AIF is recommended here.Table 6.5 calculates the average percent difference from these profiles fromtime points 93.4-225.8 s, where local tissue enhancement is strong in the back-ground. The percent difference is relative to the projection of the pre-injectionimage, to better quantify how much of an impact temporal blurring has on thecharacterization of local tissue enhancement. These results are shown for the dataset with Golden angle sampling, though similar values were obtained with uniformand random sampling.Table 6.5: Percent Difference between Snapshot and Dynamic Profiles in theEnhancement Area (Time points 93.4-225.8 s, Golden angle sampling)Number of Projections Shepard’s (%) STCR (%) NFFT (%)144 0.023 ± 0.006 0.030 ± 0.005 0.016 ± 0.00689 0.020 ± 0.008 0.031 ± 0.005 0.012 ± 0.00555 0.022 ± 0.010 0.030 ± 0.005 0.010 ± 0.00534 0.015 ± 0.014 0.027 ± 0.007 0.009 ± 0.00421 0.012 ± 0.009 0.051 ± 0.009 0.008 ± 0.005The average percent difference is comparable between NFFT and STCR, andslightly greater with Shepard’s method of interpolation. They are on the order of0.006-0.051% from time point 93.4 s to the end of the experiment. These val-ues are very low due to the difference being compared to the pre-injection image,which generally has much higher signal than the difference. The percent differenceis greatest around the injection, reaching percent errors as high as 1.60% (range0.97-1.49 for Shepard’s method of Interpolation, 0.80-1.36% for STCR and 0.34-1.60% for NFFT for image reconstructions with 55-144 projections). In general,143the values are greatest with the data sets with 144 projections, and decreases asfewer projections are used. This could be attributed to temporal blurring of signalfrom data taken over a larger temporal window. To put this into perspective, theinjection covers approximately 15 time points (5 to reach the peak concentration,and 10 time points to reach a concentration of 0.655 mM, which is half way be-tween the peak concentration (0.95 mM) and the steady concentration at the end ofthe experiment (0.37 mM)). Only two of these were randomly selected time-pointsfor image reconstruction.These results suggest that temporal blurring, due to taking data from differentdegrees of enhancement (ie. different times), may not be an issue for a smooth,slowly varying enhancement region similar to what was simulated. The percentdifference is small for all data sets studied, including those using 144 projections inthe reconstruction. Reconstructing images with a larger number of projections willimprove the SNR and sharpness of edges of image features, and therefore allowfor a more effective local tissue enhancement correction. When the concentrationgradient is sharp, as is often observed immediately following the injection, fewerprojections may be desired. Local tissue enhancement is expected to be minimalhere, but these images could provide information about motion or dilation of thevessel during and immediately after the injection.A more informative analysis looks at the difference between the profiles fromthe radial images and the simulation. To estimate the magnitude of errors intro-duced by the radial reconstruction, the difference is compared to the average pre-injection image and scaled as a percentage. Figure 6.13 shows the percent dif-ference, between the projection profiles from the radial images and the simulatedprofiles, for all three radial reconstruction techniques with 55 projections and eachsampling scheme. All difference profiles are plotted on the same scale, with amaximum set to 5%.The percent difference profiles are divided into two distinct regions: the firstis the region of the vessel, and the second region is the surrounding tissue on bothsides. Within the vessel, the maximum percent difference occurs at the start of theinjection. This is supported by the previous analysis, in which the signal at thepeak of the AIF suffers from temporal blurring due to data taken before, duringand after the contrast injection.144Figure 6.13: Percent difference between the enhancement profiles from theradial images and from the simulated data set, normalized to the signalof the pre-injection image. 55 Projections were used in the reconstruc-tion. STCR (all sampling methods) and NFFT (uniform or Goldenangle) are superior to Shepard’s method of Interpolation.Once the concentration within the vessel approaches a steady value, the er-ror within the vessel is significantly reduced (except for NFFT with random sam-pling). The average percent difference, when using 55 or more projections andeither STCR (all sampling methods) or the NFFT (uniform or Golden Angle), ison the order of 0.41-0.92%, between time points 58.2 s to 225.8 s. With Shepard’smethod of Interpolation, the average percent difference is low at the center of thevessel and increases at the edges, with average percent differences ranging from1.31-3.33%. Uniform and Golden Angle sampling work best for this technique.The average percent differences, within the vessel and the entire tissue en-hancement, are plotted in Figure 6.14 and Figure 6.15 and summarized in Table 6.6.145Figure 6.14: Average percent difference in the signal magnitude between theenhancement region of the radial images and the expected signal, forthe pixels corresponding to the vessel. The difference is scaled againstthe pre-injection image, so as to estimate the errors introduced at thisstage of the correction. The average percent error consistently shows asharp peak during the contrast agent injection (58.2 s), then approachesa steady value. It is lowest with 144 projections, and increases as fewerprojection are used in the reconstruction.The average is similar between STCR and NFFT, while Shepard’s method of in-terpolation produces larger values (exception NFFT with random sampling).Within the vessel, the average percent difference is lowest with 144 projections,and increases as fewer projections are used. This is consistent for all three recon-struction technique. Uniform and Golden angle sampling often produce similarpercent differences, while random sampling is greater. Based on the figure, recon-146structions with at least 55 projections (any reconstruction technique and uniform orGolden angle sampling) have similar percent differences. The percent differencesincrease more rapidly with 34 projections with uniform sampling than it does withGolden angle or random sampling. Among the three techniques, NFFT (followedclosely by STCR) has the lowest percent differences within the vessel with uniformand Golden angle sampling, and STCR is the best technique for random sampling.Table 6.6: Average Percent Difference in Profiles for the Vessel VoxelsGolden Angle Sampling (Time points 58.2-225.8 s)Number of Projections Shepard’s (%) STCR (%) NFFT (%)144 3.33 ± 0.94 0.73 ± 0.18 0.59 ± 0.1889 1.55 ± 0.25 0.71 ± 0.18 0.74 ± 0.1855 1.70 ± 0.31 0.92 ± 0.23 0.76 ± 0.1634 2.18 ± 0.45 0.89 ± 0.23 0.86 ± 0.2421 3.33 ± 0.93 1.41 ± 0.36 1.54 ± 0.44Extending the analysis to include the surrounding tissue, the average percentdifference is similar with 89 and 144 projections, and increases with fewer pro-jection. This is expected as the signal changes from local tissue enhancement areslow and smoothly varying. In addition, the AIF has reached a near steady valueby this point. Therefore, a coarser temporal resolution can still accurately recreatethe enhancement region. When uniform or Golden angle sampling are used, thepercent errors are comparable with 55-144 projections, and slightly greater with34 projections. The percent errors increase more rapidly when random sampling isused, particularly with the NFFT.The percent differences in the surrounding tissue is generally less than 1.5%for NFFT, averaging 0.86 ± 0.23% (uniform) and 1.01 ± 0.19% (Golden angle)for images with 55 projections. There is a narrow band of higher percent errors justsuperior of the vessel, though the differences rarely exceed 3.5% there. The percentdifferences superior to the vessel are significantly higher with STCR, which maybe related to the signal intensity differences observed in the images. For the datasets with 55 projections, the average percent differences inferior and superior tothe vessel are 0.63 ± 0.12% / 2.69 ± 0.41% (uniform) and 0.70 ± 0.17% /1472.77 ± 0.50% (Golden angle). The importance of these values depend on wherethe vessel is situated. We may have been fortunate that the vessel is located nearthe center of the phantom, so the average percent differences in the proximity ofthe vessel are lower (see above). Shepard’s method of interpolation produces thelargest percent differences, particularly at the outer boundary of enhancement. Toillustrate this, the percent difference increases from 1.49 ± 0.22% (uniform) or1.77 ± 0.24% (Golden angle) near the vessel to 4.26 ± 0.14% / 3.11 ± 0.25%(uniform) or 5.28 ± 0.70% / 3.61 ± 0.48% (Golden angle) at the upper and lowerboundaries, respectively.Since the region within the vessel is of greatest importance, it is imperativeto select a reconstruction technique that minimizes errors in this region. Basedon this figure, it is clear that STCR and NFFT are superior to Shepard’s methodof interpolation, and uniform or Golden angle sampling generally produce lowerpercent errors than random sampling. The largest errors are often found at theboundaries of the vessel, where the signal intensity changes rapidly. This couldbe attributed to signal smoothing due to limited information at the outer regions ofk-space, where image details are stored. It is advisable to use a minimum of 55projections in the local tissue enhancement correction.The optimal number of projections used in the reconstruction is dependent onthe contrast kinetics (i.e. rate of change) and the temporal resolution of the data.There is a trade off between a higher temporal resolution and accurately modelingthe tissue enhancement. In general, if the enhancement curve changes rapidly,it is important to reconstruct images at a high rate to capture key features in thecurve. The best example is at the peak magnitude. Reconstructing images with 233projections could underestimate the degree of enhancement, thus leading to errorsin determining the concentration. Alternatively, image reconstruction with too fewprojections could lead to a greater presence of imaging artifacts and blurring.The temporal resolution of a radial data set is determined by the number of pro-jections used in the reconstruction. If Golden angle sampling is used, it is possibleto retrospectively reconstruct multiple image series with a sliding window recon-struction, each having a different temporal resolution [135, 176]. This is beneficialfor DCE experiments as the trade off between having a high quality image - withhigh spatial resolution - and a high temporal resolution is often a limitation. Image148Figure 6.15: Average percent difference in the signal magnitude between theenhancement region of the radial images and the expected signal. Thedifference is scaled against the pre-injection image. The results showthat STCR and NFFT have the lowest percent errors, which increasesas the number of projections used in the reconstruction is reduced. Forall cases, the average percent error has a sharp peak during the con-trast agent injection (58.2 s), then increases again beyond time point108.1 s, when local tissue enhancement becomes more apparent.149series using a larger number of projections (144, 233, etc.) will provide high con-trast images for visualizing the anatomy, while image series with fewer projections(34 or 55, etc.) can better capture rapid changes between images. By analyzingmultiple data sets in this way, it is possible to extract more information about therate and shape of enhancement.An estimate of the peak concentration in the mouse tail would greatly improveour ability to assess errors in our measurements in-vivo. The simulations involvedadding a local contrast perfusion to an anatomical image and taking one radial pro-jection from each time point. This study shows that a minimum of 55 projectionsare required to produce a satisfactory image and provide a reasonable estimate oflocal tissue enhancement; but this could be greatly dependent on the rate of contrastkinetics in the vasculature.6.5 Correcting the Projection-Based AIF for LocalTissue EnhancementThe final stage of analysis involves measuring the AIF with the projection-basedAIF technique detailed in Chapter 4, then correcting it for local tissue enhancementwith radial images. The radial tissue enhancement correction involves a compari-son of a post-injection image with its sister pre-injection image. Both images usethe same sampling technique (angles of acquisition and number of projections) tominimize reconstruction-related factors, such as artifacts. The difference betweenthese images is then projected along the dimension perpendicular to the angle ofacquisition, after removing the data from the vessels. This provided an estimate ofhow the post-injection profiles were affected by the local tissue enhancement. Bysubtracting the difference profile from the sampled projections, we effectively re-place the tissue enhancement with the original signal pre-injection. The projection-based AIF approach is then applied to get the tissue-enhancement corrected AIF.The tissue enhancement corrected radial AIF’s, with images reconstructed with55 and 34 projections, are shown in Figure 6.16 and Figure 6.17. Both Figures useGolden angle sampling. The sampling pattern is shown in the inset of the Figure.The uncorrected AIF follows the expected trend of the input curve until timepoint 103.1 s, at which point, it diverges due to local tissue enhancement. The150Figure 6.16: Radial projection-based AIFs measured before and after correc-tion for local tissue enhancement. The dashed lines represent the raw,uncorrected AIF, while the solid lines represent the tissue enhance-ment corrected AIFs. In this study, a subset of 55 projections, withGolden angle sampling, were used in the radial reconstruction. Thecorrection, which is outlined in Figure 6.5, was effective in removingthe divergence caused by the tissue enhancement for all three recon-struction methods. The curves with Shepard’s method of interpolationand the NFFT are in good agreement with the input curve, while STCRover-estimates the concentration.151Figure 6.17: Radial projection-based AIFs measured before and after correc-tion for local tissue enhancement. The dashed lines are the uncor-rected AIFs, while the solid lines are the tissue enhancement correctedAIFs. In this figure, 34 projections were used in the reconstruction,with Golden angle sampling. Local tissue enhancement was correctedusing the method outlined in Figure 6.5 and was effective at removingthe divergence for all three radial reconstruction techniques. The re-sults are similar to the case with 55 projections, though the correctedcurves are noisier.152divergence is the greatest with Shepard’s method of interpolation (final concentra-tion of 0.650 mM, or 1.76x greater), then NFFT (final concentration of 0.573 mM,or 1.55x greater), and the lowest with STCR (final concentration of 0.552 mM, or1.49x greater) in the data sets with 55 projections. There are slight differences inthe uncorrected AIF as the sampling scheme changes or when a different number ofprojections are used in the reconstruction. This is likely related to image artifactsthat would carry errors through to the AIF measurement.The tissue enhancement correction was effective in removing the bias for allradial reconstruction techniques. The corrected AIF with Shepard’s method of in-terpolation and the NFFT most closely matched the input AIF, whereas the STCRover-estimated the concentration past the peak by a factor of 1.19-1.34 mM (uni-form sampling), 1.19-1.26 mM (Golden angle) or 1.21-1.41 mM (random sam-pling). Consistent with previous results, the corrected AIF is most comparablewith the simulated curve with uniform or Golden angle sampling, and 55 or moreprojections used in the image reconstruction. With 34 or 21 projections, the cor-rected AIFs are either noisy or over-estimate the concentration when local tissueenhancement is present. The corrected AIFs with random sampling (Shepard’smethod of Interpolation or NFFT) are very noisy with 55 and fewer projections,and would not be effective in modeling DCE-MRI data.The average difference between the tissue enhancement corrected projection-based AIFs and the input curve from time points 108.1 s to the end of the experi-ment are summarized in Tables 6.7 and 6.8.The differences are consistently the lowest when 55, 89 or 144 projectionsare used in the image reconstruction. The average difference between uniformand Golden angle sampling are comparable (within uncertainty), suggesting littledifference between the two sampling methods or all three techniques. Between thethree image reconstruction methods, the NFFT reconstruction most closely agreeswith the input AIF, followed closely by Shepard’s Method of interpolation. Theaverage difference for the STCR reconstruction is consistently greater, by 0.055-0.090 mM. This is a significant amount as the expected intra-vascular concentrationis 0.371 ± 0.010 mM over this time frame.The ratio of the tissue-enhancement corrected projection-based AIF to the inputAIF provides insightful information on the effectiveness of the correction. If the ra-153Table 6.7: Average Difference Between the Corrected Radial AIF (UniformSampling) to the Input AIF from time points 108.1-225.8 sNumber of Projections Shepard’s (mM) STCR (mM) NFFT (mM)144 0.021 ± 0.028 0.088 ± 0.036 -0.002 ± 0.01789 0.004 ± 0.025 0.086 ± 0.022 0.001 ± 0.01655 0.020 ± 0.022 0.073 ± 0.022 0.003 ± 0.01634 0.067 ± 0.026 0.124 ± 0.027 0.056 ± 0.02021 0.031 ± 0.046 0.114 ± 0.046 0.091 ± 0.051Table 6.8: Average Difference Between the Corrected Radial AIF (Goldenangle) to the Input AIF from time points 108.1-225.8 sNumber of Projections Shepard’s (mM) STCR (mM) NFFT (mM)144 0.010 ± 0.034 0.068 ± 0.022 -0.001 ± 0.01889 0.021 ± 0.042 0.083 ± 0.024 0.008 ± 0.01755 0.026 ± 0.039 0.082 ± 0.023 0.012 ± 0.02134 0.058 ± 0.070 0.099 ± 0.030 0.056 ± 0.02021 0.091 ± 0.162 0.101 ± 0.034 0.063 ± 0.085tio is flat beyond the peak of enhancement, then the tissue enhancement correctionwas effective, save a scaling factor. Figure 6.18 shows the ratios for the correctedAIF curves to the input curve, in which Golden angle sampling was used.The ratios are close to 1.00 with Shepard’s method of interpolation and theNFFT, when at least 55 projections are used in the reconstruction. Reducing thisnumber causes the ratio to increase, as a result of over-estimating the intra-vascularconcentration. Since local tissue enhancement resulted in an over-estimation of themeasured concentration post-injection, it is likely that the image quality of the im-ages with 34 and 21 projections was insufficient for accurately accounting for theeffects of local tissue enhancement. The measured AIFs with STCR significantlyover-estimated the concentration, yielding ratios between 1.1-1.3. This is interest-ing as the enhancement images with STCR were most similar to the simulation.The scaling factor for the correction was calculated with the same method for allthree reconstruction techniques.154Figure 6.18: Ratio of the radial projection-based AIF (Golden angle sam-pling), after correction for local tissue enhancement, to the input curve.The results show that STCR overestimates the concentration, whileShepard’s method of interpolation and NFFT are more accurate.The over-estimation in concentration is present for all STCR-corrected AIFs,regardless of the sampling method used. Since the ratio is consistent throughout theexperiment, these curves could be re-scaled if STCR is the preferred reconstructiontechnique. However, the AIFs with NFFT (55 or more projections and uniformor Golden angle sampling) agree with the input curve better. Using the NFFTover STCR would be beneficial, as the scaling factor could introduce additionaluncertainties. Tables 6.9 (uniform sampling) and 6.10 (Golden angle sampling)summarize the ratios of the corrected AIF to the input AIF from time point 108.1 sto the end of the experiment, where local tissue enhancement is observed in theimages.The concentration at the peak is over-estimated post-correction when uniformor Golden angle sampling was used. The degree of over-estimation is related to thenumber of projections used in the reconstruction. In both cases, the over-estimationincreases gradually as the number of projections is reduced from 144 to 34, butnot in a linear fashion. The data sets with random sampling have conflicting re-sults, with most AIFs under-estimating the concentration with Shepard’s Methodof Interpolation, over-estimating with STCR, and having good agreement with theNFFT. In comparison, the simulated peak concentration is 0.929 mM, while the155Table 6.9: Average Ratio of the Corrected Radial AIF (Uniform Sampling) tothe Input AIF from time points 108.1-225.8 sNumber of Projections Shepard’s STCR NFFT144 1.06 ± 0.08 1.25 ± 0.07 1.00 ± 0.0489 1.01 ± 0.06 1.25 ± 0.06 1.01 ± 0.0455 1.06 ± 0.06 1.21 ± 0.06 1.02 ± 0.0434 1.18 ± 0.07 1.34 ± 0.08 1.16 ± 0.0621 1.08 ± 0.12 1.32 ± 0.13 1.25 ± 0.14Table 6.10: Average Ratio of the Corrected Radial AIF (Golden angle sam-pling) to the Input AIF from time points 108.1-225.8 sNumber of Projections Shepard’s STCR NFFT144 1.03 ± 0.09 1.19 ± 0.06 1.00 ± 0.0589 1.06 ± 0.11 1.23 ± 0.06 1.03 ± 0.0555 1.07 ± 0.11 1.23 ± 0.06 1.04 ± 0.0634 1.16 ± 0.19 1.28 ± 0.08 1.16 ± 0.0621 1.25 ± 0.44 1.28 ± 0.09 1.18 ± 0.23uncorrected AIFs have peak concentrations ranging from 0.870-1.175 mM (uni-form), 0.890-1.115 mM (Golden angle) and 0.929-1.148 mM (random). Thoughthese do not all agree with the input AIF, the uncorrected AIF was consistentlycloser to the input AIF at the peak than the tissue enhancement corrected ones.The over-estimation in the corrected curves could be a result of insufficientimage quality to accurately model the tissue enhancement, or from added noiseintroduced by the tissue enhancement correction. The technique involves correct-ing the acquired projections using a projection of a difference of two images. Thecorrection profile will be noisier with images reconstructed with fewer projectionsdue to signal smoothing and a higher probability of reconstruction artifacts.Radially reconstructed images all contain information about the local enhance-ment, and could be used to determine at what stage the correction must be applied.It is recommended to first identify the point at which local tissue enhancement be-comes problematic, and only apply the correction beyond that point. In this study,156local tissue enhancement is observed in the radial images as early as time point108.1 s. Only from this point, will the measurement of the AIF benefit from alocal tissue enhancement correction. Building on this point, if the onset of the lo-cal tissue enhancement is rapid, it is advantageous to use fewer projections in thereconstruction and correct the AIF with a higher temporal resolution despite therisk of lower image quality. But when the rate of enhancement is slower, moreprojections may be used as temporal blurring is less of a concern.The presence of image artifacts could limit our ability to successfully correctthe AIF. This can be seen with the Golden angle data set where the uncorrectedAIF with the NFFT reconstruction does not show significant tissue enhancement,and the corrected curve has an apparent sinusoidal artifact (results not shown). It isimportant that the window shift is set randomly between the reconstructed images.Failure to do so, could result in a coherent oscillating concentration (observed dur-ing study, but not shown here).6.6 Measuring the Radial AIF with Acquired MRI dataThe chapter, to this point, has studied the impact selected radial image reconstruc-tion techniques had on the AIF measurement. The results showed that the NFFTreconstruction was most effective. However, this analysis assumed that all projec-tions were acquired at the same orientation (i.e. 0o) to isolate the errors introducedby the tissue enhancement correction. If a high temporal resolution is desired, itwould be beneficial to measure the AIF directly from the radial data, rather thanalternating AIF and correction profiles. The rest of the chapter, and Appendix C,focuses on measuring the AIF with radial projections.6.6.1 AIF Measurement using MRI DataA cylindrical phantom was imaged with a FLASH protocol on a Bruker 70/30.The data was acquired as radial acquisitions at 233 unique angles, and equi-spacedin the angular direction. Both uniform and Golden angle sampling were investi-gated to see if changing the gradients significantly between measurements had anyimpact on our measurement. The pulse parameter settings were T R = 100 ms,T E = 5ms, flip angle = 30o, 1.5x1.5 cm2 FOV and 256 read-encode samples. The157data is acquired with the stripline surface coil for improved SNR.An injection, of 30 mM Gd-DTPA in saline, was initiated with a peristalticpump (Minipuls 2, Gilson: set to 700 which corresponds to an injection rates of ap-prox. 10 ml/min). The bolus circulated around the system multiple times after pass-ing through a mixing beaker. The temporal resolution of the sampled data is 0.1 s,which is consistent with previous measurements. The measured radial projection-based AIF is shown in Figure 6.19. The curve shows well characterized peaks, andgradually approaches a final steady-state concentration of 0.873 ± 0.082 mM Gd-DTPA. The projection-based AIF is very noisy, so a moving average filter (windowsize of 5) is applied for display purposes.The baseline phase for the first 233 points - defined as the average phase of thesignal from the vessel after it is sorted by angle - has an angular dependence. Thisis clearly observed in Figure 6.20. The pattern is not a simple sinusoid, but doesrepeat every 2pi radians. The baseline pattern appears to be consistent between allangular sampling methods used in the experiment. However, the observed patternseems to be dependent on the set-up, as it changes between scan sessions. Thismay indicate a position-dependent artifact in the placement of the phantom withinthe bore. Care was taken to ensure that the phantom was properly centered prior toimaging, but the phase fluctuations continue to change between sessions.Knowing that the baseline fluctuations are repetitive over 2pi radians, and thatthey are consistent for all scans within a study, a baseline-phase correction couldbe applied. This entails calculating the phase of the first N samples, and using asliding window of size N. After subtracting the baseline-phase from all remainingprojections, the AIF will be smoother, with a significant improvement in the SNR.The baseline-phase corrected AIF is shown in Figure 6.21.Comparing the noise level (standard deviation) of the original and correctedAIFs, the measurement benefits significantly at the pre-injection and late stagetime-points. At the pre-injection stage, the magnitude of the noise, in concentra-tion, reduced from 0.359 mM to 0.049 mM post correction (reduction by a factor of7.4X). Meanwhile, the late stage concentration changed from 0.874± 0.334 mMto 0.873±0.082 mM (An improvement in SNR by a factor of 4).Figure 6.22 was created to help identify and rule out potential sources of thebaseline phase. The figure shows the sinograms (magnitude on the first row, and158Figure 6.19: Measured AIF in the tail phantom using the pump phantom togenerate a series of contrast agent passes. The contrast agent was in-jected into the tubing with a power injector at a rate of 1.000 ml/min,and allowed to circulate for the duration of the experiment. We observea series of peaks that eventally trend towards an equilibrium value. Theinset shows a DCE image of the phantom, which is located 5 mm fromthe location of the AIF slice.phase on the second row) of 233 acquired projections, the estimated backgroundprojections, and the difference between these two images (isolated signal from thevessel). As a reminder, the sinogram is a plot of the radial data as a function of theangle of acquisition. Both sinograms were normalized to have a mean magnitudeof 1.00 before calculating the difference.The sinograms of the acquired data and the background projections show sim-ilarities in magnitude and phase. However, the magnitude of the background has acouple signal hot-spots near the edges of the phantom. These are more easily ob-159Figure 6.20: The baseline phase of the vessel signal, resorted based on theangle of acquisition, shows an angular dependence. The pattern isindependent of the echo time and sampling technique chosen. In ad-dition, it is repetitive after 2pi radians, but does not follow a simplesinusoidal curve.served in the vessel signal sinogram, which is the difference between the acquiredprojections and the background. The hot-spots appear to be well separated from thevessel (narrow band through the middle of the phantom) and are of lower intensity.It is unlikely that these directly contribute to the baseline phase.The signal phase reveals a banding structure, in which the phase in the upperand lower halves of the phantom are close to being conjugates (Figure 6.23). Themeasured phase of the vessel will depend on where it falls within the bandingstructure. In addition, there are a couple angles with a different phase from thedata acquired at a near-by angle (observed as the brighter vertical lines). Theseprojections were acquired at the start of the experiment, so the signal may nothave reached steady-state by this point. Another possibility is a mis-centering of160Figure 6.21: Measured AIF in the tail phantom using the pump phantom togenerate a series of contrast agent passes. A baseline-phase phase cor-rection was applied by subtracting the baseline phase from all remain-ing projections. The correction significantly improved the SNR of themeasrued AIF, confirming that the phase fluctuations have an angualardependence.the echo in k-space by a sub-pixel value. This typically leads to a phase gradientacross the image. The phase of the acquired projections is not symmetric about180o, which could support this claim.When the radial data is read in, it is centered with a global phase shift. Cen-tering each projection individually caused a jagged edge in the k-space sinogram,which in turn produced radial images of lower quality. It is possible that the gra-dient in one dimension is slightly stronger/weaker than expected. In response, thek-space sinogram will be centered appropriately, but have a small shift at someangles (i.e. sinusoidal pattern). Using this data in the radial reconstruction couldcause signal blurring as the center of k-space will be spread over a larger area. Inmore severe cases, the echo may not pass through the center of k-space, which161Figure 6.22: Sinograms of the acquired projections, background profiles anddifference between the two. There appears to be a step-like phase ar-tifact in the difference signal. The phase of the vessel appears to beaffected by the location of the vessel within these two phase bands.Figure 6.23: Sinogram of the phase of the vessel signal and a cross-sectionalplot of the data from within the black box. The phase jumps havea steady phase, transitioning from a phase of −2.13± 0.07 rad in theupper region of the phantom to 2.35±0.05 rad in the lower region. Themeasured phase of the vessel signal appears to vary slightly angularly.This correlates with the positioning of the vessel within each of thesephase bands.162would also impact the signal intensity of the echo.Since the background images are all derived from the same data set, any tem-porally varying change in the acquired projections would be smoothed out. The ra-dial images may have artifacts that consistently occur in the same spatial location,which would impact the background signal differently as each angle. This meansthat the background signal would differ from the acquired data with an angulardependence. Two possible explanations are eddy currents [177] or gradient timingdelays [178]. Appendix B investigates both issues. The results of these studieswere not sufficient for removing the phase baseline in the radial projection-basedAIF.6.6.2 Concluding RemarksThis chapter evaluated three radial reconstruction methods for their effectiveness incorrecting the projection-based AIF for local tissue enhancement. The analysis ofthe enhancement images shows that STCR best reproduced the local enhancementregion, followed by the NFFT. This observation carried through the investigationof the projection profiles, created from a projection of the enhancement imagesperpendicular to the readout direction. In both analyses, Shepard’s method of inter-polation was inferior. The final investigation measured the projection-based AIF,after the sampled projections were corrected for local tissue enhancement. The re-sults showed that Shepard’s method of interpolation and the NFFT technique bestagreed with the input curve, while the concentrations with STCR were consistentlyover-estimated by a factor of 1.19-1.25 when 55 or more projections are used inthe reconstruction.Taking all results into consideration, the NFFT seems to be the best techniquefor correcting for local tissue enhancement, when at least 55 projections are usedin the radial reconstruction. Though it is beneficial to use more projections in theimage reconstruction if local tissue enhancement evolves slowly. In addition, thedata should be sampled with Golden angle sampling, so that radial images may bereconstructed retrospectively at a variety of temporal resolutions.The measurement of the radial projection-based AIF with acquired projectionsrevealed issues in the isolated vessel signal. There appears to be an angular-163dependent phase shift, of which the source remains unknown. Attempts to uncoverthe issue are presented in Appendix B, though the results were inconclusive. Fu-ture work on this project involves further investigation into the source of the phasebanding in the isolated vessel signal. Appendix C explores limitations in the radialprojection-based AIF measurement due to imperfect phantom set-up, off-sets ofthe k-space data, and multiple vessels.The projection-based AIF may be used in current in-vivo AIF-DCE experi-ments, though the best results require that the same angle of acquisition is used.It may be possible to measure a radial AIF in a larger object - such as a rat tail ifspatial resolution is high - but this is beyond the scope of this thesis.164Chapter 7Interleaved AIF and DCEMeasurement7.1 Interleaving a DCE and AIF MeasurementPK model parameters are most specific to a patient and exam if the AIF and DCEdata are acquired simultaneously. This often requires that the blood vessel feedingthe tissue of interest is located close or within the imaging plane [108]. For pre-clinical studies on small animals, this is often difficult to achieve as there are alimited number of vessels of a sufficient size to avoid partial volume effects, andcommon locations for tumour implants are distant from these vessels.Animal-based AIFs have been successfully measured in the left ventricle ofthe mouse heart [15], in the iliac artery [100], and in the mouse tail [20]. Whilethe heart and iliac artery may be closer to a tumour implanted in the mammary fatpad or on the hind flank, these sites would require an image-based measurement.In addition, the estimate within the heart would require a gated scan. This couldfurther impede the temporal resolution of the DCE experiment. The mouse tailprovides a good compromise for being closely located to a tumour implanted onthe hind flank, and simple anatomy to allow a projection-based AIF measurement.The sensitivity and specificity of the DCE-MRI model fit parameters are highlycorrelated with the spatial and temporal resolutions of the acquired data. Great careshould be taken to ensure that the spatial resolution is sufficient to capture tumour165Figure 7.1: Typical locations for implanting tumours in mice (green) andwhere the AIF has been successfully measured in mice (red). There islittle overlap between the two regions, making it difficuilt to acquire anAIF throught the DCE experiment without compromising the temporalresolution.inhomogeneity and to reduce partial volume effects [42], while also attaining asufficiently high temporal resolution to capture the rapid contrast kinetics in thevessel [5, 91]. Satisfying these two conditions is already a challenge when onlyDCE-MRI data is acquired, so adding an image-based AIF to the scan time isundesirable. The projection-based AIF has the advantage that TE can be set muchshorter than TR, thereby allowing us to integrate the AIF measurement into a DCEexperiment with minimal effects on the temporal resolution.Multiple studies have concurrently acquired data for the AIF and DCE exper-iments, though most of them are performed in humans. One animal-based studywas performed by Dominick McIntyre and his colleagues [21], where they per-formed an interleaved AIF and DCE acquisitions in rat tumours. Similar to ourstudy, the AIF was measured in the tail.Case Study: McIntyre et alMcIntyre [21] and colleagues recognized the importance of acquiring the AIF foreach experiment, and performed this study to show that an interleaved AIF andDCE experiment is possible in rats. They looked at both the reproducibility of theAIF measurement in two Wistar Furth rats (no prolactinomas and tail only scans)imaged at 0, 4 and 8 days, as well as the potential for an interleaved AIF-DCE scanwith two separate coils. The interleaved experiment was performed on six rats.These rats had GH3 prolactinomas grown on the flanks.166Data was acquired on a 4.7 T Varian Unity Ivova spectrometer. The experi-ment utilized two coils; a nine-turn solenoid tail coil (length 24 mm, inner diameter8 mm, and oriented with its long axis perpendicular to the main magnetic field) toacquire the AIF and a three-turn solenoid coil wrapped around the tumour (diam-eter 25 mm, length 11 mm and oriented with its long axis vertically) for the DCEdata. The rat was positioned on its side and its tail led through the tail coil. ASPDT PIN diode switch was constructed to alternate between the tumour and tailacquisitions and was remotely controlled by the spectrometer. McIntyre et al. ob-served three distinct vessels in the images of the rat tails (one large artery and twolarge veins). The AIF was measured in any one of these, though the veins producedthe most reliable results.For the reproducibility study, a saturation-recovery gradient-echo pulse se-quence was utilized to avoid signal intensity alterations due to the inflow of unsat-urated spins between the excitation and refocusing pulse. Slice-selective saturationpulses, with a saturation recovery time of 50 ms, were oriented to saturate the fulllength of the tail within the coil volume. A total of 32 scans were obtained priorto injecting a bolus of 0.1 mmol/kg Omniscan to get an accurate measure of thebaseline values. 140 images were acquired after the injection. The results fromthis study showed that the variability of fitted parameters was comparable withinand between rats. This further confirms that the AIF should be taken during eachDCE-MRI scan.DCE-MRI images of a tumour and AIF data at the tail were acquired with a T1-weighted gradient echo pulse sequence with T R = 105 ms, T E = 4 ms and flipangle of 90o (tail) or 50o (tumour). This produced a temporal resolution of 6.72 s.The repetition time was selected to allow for three tumour slices to be imaged,while saturating the signal from the tail to minimize inflow effects. The rat waspositioned such that the tumour was a couple cm beyond the geometric center ofthe magnet. This allowed the tail to lie within the linear region of the gradients.Magnetic field shimming was optimized for the X, Y and Z shims at the tumour,while no shimming was performed on the tail.Their results show that the AIF measurement had superior SNR (standard errorestimates of 2-3%) compared to a measurement in the aorta or vena cava usinga volume coil (standard error estimates of 10%). This confirms that a dual-coil167approach is superior, despite the two coils being off-center. In addition, the AIFmeasurements taken with the interleaved scan were comparable to those from thereproducibility study. This suggests that interleaving the two data sets does notsignificant impact on the reliability of the data.McIntyre calculated Ktrans maps for both a literature AIF (from Rozijn [179])and their individually acquired interleaved AIF. The results show dramatic differ-ences in both the average value and the skewness of the histograms. In addition,the values of ve appear to be more strongly affected. They quote that 48% of pixelslies outside the range of 0 ≤ ve ≤ 1 with Rozijn’s AIF [179], while only 13% lieoutside the range with the interleaved AIF.Our interleaved dual-coil AIF-DCE experiment was inspired by this study.7.1.1 Interleaved AIF-DCE Pulse SequenceThe pulse program used for the interleaved AIF-DCE experiment is shown in Fig-ure 7.2. The pulse program is split into two blocks: one for the AIF measurement,and the second for the DCE experiment at the region of interest (ROI) (often a tu-mour). Within each repetition time, one line of k-space for the AIF and each of theDCE slices is acquired. When setting up the experiment, two slice packs are de-fined. The first slice in the series is always associated with the AIF measurement,while all others are associated with the DCE experiment.The AIF is measured using a flow-compensated FLASH acquisition. The flowcompensation is only applied in the direction of the slice select as the tail is orientedparallel to the main magnetic field. In addition, a radial acquisition is used sothat local tissue enhancement may be assessed throughout the experiment, andcompensated if required.Conversely, the DCE experiment follows the standard protocol used in our lab.This consists of a multi-slice FLASH acquisition with Cartesian sampling. Theinterleaving works by acquiring one line of k-space for the AIF, followed by asingle line of k-space for each DCE image within the repetition time. For a TR of100 ms, up to five DCE slices may be acquired. Interleaving the two experimentsin this way allows us to maintain a high temporal resolution for both the AIF andDCE data. Future studies can use compressed sensing or multi-echo procedures to168Figure 7.2: Pulse program for the interleaved AIF and DCE measurement.During each TR, one line of k-space is acquired for the AIF measure-ment and one line of k-space for each slice in the DCE experiment. Thispattern continues until all data has been collected.acquire the DCE data faster though this could impact the temporal resolution of theAIF if the acquisition window for each slice is increased.7.1.2 Two-coil set-upWith an interleaved experiment, where the AIF and DCE slices are located farapart. The data can be sampled with a single coil at two locations or with twoseparate coils which are optimized for the anatomy of interest. The dual-coil ap-proach is superior as the two coils can have smaller sensitive regions, and thusimprove the SNR of the data. It also for more flexibility for difference sized mice,or locations of the DCE-ROI and catheter for the tail vein injection. However, caremust be taken to ensure that both coils fall within the linear region of the gradientfields [21] and that there is minimal cross talk between the two. Our lab grows169Figure 7.3: Scematic for the Two coil set-up. Surface coils, specific for theAIF and DCE data acquisitions, were used to maximize the SNR of thesignal at each location. The electronic switch allowed us to selectivelychoose which coil was active during the acquisition window, and thusenabling data collection for the AIF and DCE experiments separately.tumours on the hind flank of the mouse, while the AIF is located approximately2− 5 cm away on the average adult mouse. This is within the limits of the linearregion of the volume coil for signal excitation, while the large spacing betweenAIF and DCE measurements benefits from a dual-coil set-up.Having two distinct data collection blocks in the pulse sequence, an electronicswitch was added to select which surface coil is active during the acquisition win-dow. Following the example by McIntyre, a SPDT PIN diode switch was con-structed. The switch operation is based on low and high voltages at the input to thesurface coils, such that one coil will perceive a closed circuit, while the other per-ceives an open circuit. The switch settings are prescribed within the pulse sequenceand is controlled remotely by the scanner. Since one coil is part of an open circuitat any instance, cross talk between these two coils is not a concern. Both surfacecoils are actively decoupled from the volume coil (used for signal excitation).1707.2 Data AcquisitionAn in-vivo measurement was performed on a healthy mouse on a Bruker Biospec7 T MRI scanner. To replicate a DCE experiment, the mouse was set up usingthe same procedure outlined in Chapter 4. This included a tail cannulation and aninjection bolus of 5 ml/g of 30 mM Omniscan. A pre-bolus of 25 ml heparinizedsaline was used to prevent an early injection. The image slice was oriented suchthat no two tail vessels would overlap in the projection.A specialized strip-line coil was used for signal collection at the tail and asingle loop surface coil was used for signal collection of the DCE-data, located atthe kidney. Since the mouse tail tends to be small, we chose to orient the tail coil’slong axis parallel to the main magnetic field to allow for better shimming.In this study, projection data was acquired with an angle of acquisition of 0o forthe AIF measurement, as the phase artifact from the radial data is still unresolved.The DCE slices (N=5) were acquired using the standard FLASH experiment withCartesian sampling. The AIF and DCE data were interleaved, such that one line ofk-space was acquired for each DCE slice between AIF acquisitions. This providedtemporal resolutions of 100 ms for the AIF projections and 12.8 s DCE images(128 phase-encoding lines), respectively. The scan was set-up for 80 repetitions(total time 17:04), with the injection initiated after 60 s.7.3 Results from Interleaved StudyInterleaved AIF-DCE data was acquired with four DCE slices and one AIF slice.The locations of the two slice packs was chosen from a multi-slice FLASH scan ateach location. For the AIF, the selected slice had at least three well defined vesselsand minimal signal from the surrounding tissue. The slice was oriented such thatno two vessels would overlap in the projection profile. The DCE slices coveredthe majority of the kidney volume with 1 mm thick slices, and 2 mm spacing. Thecenter of the two slice packs were separated by 4.6 cm. Images from slice 2 of theDCE pack are shown in Figure 7.4. These images have a temporal resolution of12.8 s and a spatial resolution of 31.2 x 31.2 µm2. The injection took place 60 sinto the experiment, meaning that the first four DCE images are pre-injection, andthe sixth to eightieth are post-injection.171Figure 7.4: Signal magnitude of the DCE-MRI images of the ROI at slice2. The temporal resolution of this experiment is 12.8 s. The bolusof Gd-DTPA was injected 60 s into the experiment, which is duringthe acquisition of the 5th DCE image (data collected between 51.3 s -64.0 s). This figure shows one pre-injection image, the image duringthe injection, and six post-injection time points. The signal throughoutthe animal enhances after injection, though the ROI (outlined with redoval) enhances to the greatest degree.The AIF was measured in each of the four vessels. Only the curves that had thecharacteristic shape of an AIF - initial sharp uptake after the injection, followed bya gradual decrease to a steady concentration - were used in the measurement. Thephase of the external reference phantom tracked phase drift in the imaging plane,and was used to correct the AIF. The vessel located at readout pixels 151-155produced the most AIF-like curves, with pixel 151 selected for the measurement.Other pixels had a large phase shift artifact during the injection, which may haveresulted from movement or a reaction to the injection. The maximum measuredconcentration, for each pixel of the projection corresponding to the vessel, rangedbetween 0.8− 1.7 mM. The selected AIF had a concentration of 0.93 mM at thepeak and a long time concentration around 0.25 mM. This is consistent with theprojection-based AIF discussed in Chapter 4.The ROI for the DCE study was selected by evaluating the concentration-timecurves in locations of significant enhancement over time. The concentration wascalculated from the relative change in T1 (equation 3.1), in which the pre-injection172Figure 7.5: The Tofts model was fit to the DCE-MRI data from slice 2. Theregion of interest (ROI) is outlined in red on the right, while the redcurve on the plot to the left is the concentration of Gd-DTPA within thisarea. The projection-based AIF (blue curve) was used in the model fit-ting (black curve). The Tofts model fit overestimates the concentrationearly, but is considered a reasonable fit.value was measured with a fit to the signal-intensity vs inversion time of a Look-Locker experiment, and the post-injection value calculated from equation 2.18 anda proton density image as an estimate of Mo.The Tofts model was fit to the concentration-time curve from the selected ROI,indicated with the red box on the right side of Figure 7.5. Since this model appearsto fit the data well, and adding a third parameter for the plasma volume (vp) did notimprove the fit, the two-parameter Tofts model would be considered sufficient.The average concentration-time curve for this ROI is the red curve on the left,and the projection-based AIF is in blue. The fit parameters, Ktrans and ve, have val-ues of 0.145 min−1 and 0.269, respectively. These are consistent with the literaturevalues. [98, 109, 180–183].Pre and post injection images of the mouse tail can be instructive in confirmingthat the injection was successful and in assessing local tissue enhancement. Thetail images from the in-vivo study are shown in Figure 7.6. The post-injectionimage ( 36 : 30 after the injection) shows that the blood vessels are all dilated post-173Figure 7.6: Magnitude and phase signal of the mouse tail before and after(approx. 36 : 30 after the injection) the DCE experiment. The injectionwas performed through the superior vessel. These images show that thesuperior vessel is enlarged post-injection, and there are subtle changesin the signal of the surrounding tissue. The phase images shows thatthere was some phase drift during the experiment.injection. Had the AIF projections been acquired radially, the point at which thevessels dilated could be assessed from a sliding window reconstruction. The vesselmask may need to be redefined post-injection to compensate for the enlarged vesselarea or account for any minor shift in position.The tissue surrounding the vessels has changed slightly in the surrounding tis-sue, but it is not clear if this affects the measurement of the AIF as the vessel is onlya few pixels in diameter. If the signal contrast between the vessel and surroundingtissue is low, then the relative contribution of the vessel to the total signal will below. This means that changes in the surrounding tissue could be detrimental to theAIF measurement, even though it does not appear to change significantly betweenpre and post-injection images.174The presence of local tissue enhancement may be assessed from the complexdifference of the post and pre injection images. Figure 7.7 compares the signalmagnitude of the vessels, and from local tissue enhancement. Depending on thelocation, the signal magnitude of the local tissue enhancement varies from 8.7% to178.0% of that from the vessel (50.8% to 76.7% in the vessel used for the AIF).Both are complex signals, so the effect on the signal phase depends on the relativeangle of each. In the best case scenario, the tissue enhancement signal has thesame phase as the vessel, and so the measured AIF is unaffected. The maximumphase difference occurs when the two signals are parallel-opposed, and the tissueenhancement signal is stronger than the vessel.For this study, the AIF projections were all acquired at the same angle, so onlymotion along the projection may be assessed. Movement perpendicular to the ac-quired projection could be a concern if its due to a rotation, as this would changethe background profile of the surrounding tissue. Comparing the FLASH imagesbefore and after the DCE experiment, it appears that the mouse shifted slightly(11.8 µm (2 pixels) to the right and 5.9 µm (1 pixel) down). However, it is un-known when this shift occurred during the experiment. The projection profilesshow a linear shift towards the right by 5-6 pixels during the scan. The movementwas corrected by first identifying the edges of the projection profiles using the So-bel filter. The shift in the edge location, s, relates to the required phase adjustmentin k-space, through the phase term e−i2pixs/256, where x ranges from -128 to 127 inthe readout direction. This correction removed the shift in the readout direction,and produced a more uniform phase across the reference phantom in time.7.4 Final Thoughts and Directions for Future StudyA projection-based AIF is inherently noisy compared to one measured from MRimages. Averaging multiple measurements will improve the SNR, but at the ex-pense of reduced temporal resolution. Taylor et al. [123] averaged 29 sampledtogether to significantly improve the SNR of their measurement. With a movingaverage approach, the temporal resolution (0.050 s) is unaffected, through fine de-tails will be lost. With our technique, averaging as few as 5 time points togetherappeared to be effective in greatly reducing the noise. The contrast changes in175Figure 7.7: Signal from the vessel and local tissue enhancement from the pro-jections of the post and pre-injection FLASH images. The FLASH im-ages were first masked for either the vessel or the surrounding tissue,then the difference (post-pre) was taken. The plot shows the projectionof the difference for the vessel and local tissue enhancement maskedimages. The results show that the relative strength of the tissue en-hancement signal, to that of the vessel, varies. With the vessel chosenfor the AIF, the relative strength is 60.6± 11.1%, suggesting that lo-cal tissue enhancement should be accounted for. The vessel from pixels151-156 was used for the AIF measurement.blood are influenced by the injection rate. With a rate of 1.00 ml/min, the changesoccur on the order of seconds. Therefore averaging 3 or 5 samples with a movingaverage filter should not affect the accuracy of our projection-based AIF.If the DCE data were sampled with a radial technique, a sliding window re-construction may be used to adjust the temporal resolution of the data series. Thesliding window reconstruction is advantageous as it allows for the reconstructionof multiple image series with varying temporal resolutions. There is a trade-offbetween higher temporal resolutions and higher quality images. With fewer pro-jections, more information regarding the contrast kinetics may be extracted. How-176ever, this comes at the cost of larger gaps in k-space, resulting in blurred edges andloss of image contrast. Kholmovski et al. [125] discusses how several image seriesmay be studied to gain further insights.It is expected that only a small fraction of voxels will enhance in the image.Wavelet or independent component analysis (ICA) both have potential in reduc-ing the number of significant variables in the analysis, making interpretation ofthe results more specific to observable changes in the DCE images. Mehrabianet al. [184–186] show that an AIF may be extracted from the complex-signal ofDCE images using ICA. With their approach, the AIF and DCE curves will havethe same temporal resolution. The projection-based AIF measurement allows fora much higher temporal resolution, and has higher potential to capturing the peakconcentration. However ICA could be applied to radially reconstructed images atthe tail to both validate the projection-based AIF measurement and track local tis-sue enhancement more accurately (i.e. ICA could remove noise from the analysis).Multiple groups have studied DCE-MRI data sets with wavelet analysis [176, 187].These would be interesting avenues to explore with future studies.Compressed sensing techniques can significantly reduce scan times as only afraction of the full data set is measured. Fast imaging with Cartesian sampling arewell established based on randomized data collection [156]. Further accelerationsare possible with the techniques such as Grappa [188], where the use of multiplecoils allow for a combination of parallel imaging and compressed sensing. Im-proving the temporal resolution of the DCE data is beyond the scope of this thesis,though it is an area for further investigation.For the best results, we recommend that the tail vein injection is performed witha plastic catheter or is located as distal as possible to the surface coil to minimizesusceptibility artifacts from metallic components in the catheter. Since the diameterof the mouse tail tapers off as we move towards the tip, the plastic catheter is a moreattractive alternative. However, the plastic catheter is more flexible than a needle,which could be more challenging for those less experienced with tail cannulations.177Chapter 8Concluding RemarksThe work detailed in this thesis attempted to improve the temporal resolution ofthe AIF, by measuring it from a series of MR projections. We present a projection-based AIF measurement, in which the AIF is extracted from the phase informationof a single MR projection. This approach has a temporal resolution of 100 ms(the repetition time), and allows for an interleaved AIF-DCE experiment withoutcompromising the temporal resolution of either data set.In Chapter 4, we present the projection-based technique, which measured theAIF from the phase of MR projections. The phase accumulation with concentrationof Gd-DTPA was validated in-vivo to provide a scanner conversion factor for fu-ture experiments. This result ((0.213 ± 0.001) rad/mM/ms) agreed with the the-oretically expected value (0.212 rad/mM/ms) for concentrations ranging between1-10 mM. A projection-based AIF was measured within a tail phantom, concur-rently with a colormetric measurement. The results from this analysis showed thatthe phase data accurately captures changes in intra-vascular concentration. Finally,an AIF was successfully measured in-vivo. The measurement had a temporal res-olution of 100 ms. The long term concentration was validated with a cohort of 4mice, using mass spectrometry.Chapters 5 and 6 studied three radial reconstruction techniques and evaluatedhow they performed with varying numbers of projections and sampling methodswith and without local tissue enhancement present. The results of these chapterssuggest that STCR and the NFFT techniques are superior to Shepard’s method of178interpolation, and that uniform or Golden angle sampling are best. Image serieswith at least 89 projections agreed well with the simulation, though 55 projectionsmay be used for a reasonable estimation of local tissue enhancement. Uniformsampling produced the best results, though Golden angle sampling has the advan-tage of retroactively reconstructing images with different temporal resolutions. TheNFFT technique was selected for further analysis, with Golden angle sampling.Related to this chapter, Appendix C explores potential limitations with theradial projection-based AIF measurement under imperfect data acquisition. Theresults show that the measurement is minimally affected with translations of theobject in image space. The AIF measurement is compromised when fewer pro-jections are used in the tissue-enhancement correction (55 or fewer, in general) orif the k-space data is shifted by a small amount (1.3-2.6 pixel shift in a 256x256image), in which the concentration near the peak was under-estimated, and thetissue-enhancement correction was ineffective at the later stages. The location ofthe vessel within the coils sensitivity zone could also impact the measurement, withvessels closer to the coil being more accurate. The chapter closes with a discussionof an issue that presented with the radially acquired data. The difference betweenthe acquired data and the background profiles (from an NFFT reconstruction) hastwo distinct phase bands. The AIF then has an angular dependence, with the size ofthe effect dependent on the location of the vessel within the phantom. Attempts tocorrect the issue - limiting effects from eddy current (longer TE, varying samplingmethods, etc.), gradient mis-timing measurements or trajectory measurements -were unsuccessful. Resolving the issue continues to be an area for future studies.Chapter 7 detailed the interleaved AIF and DCE measurements with a dual-coilset-up. The phantom experiments verified that the interleaved sequence acquireddata rapidly at the tail and tissue of interest with a temporal resolution of 0.100 s.However, the results showed a phase baseline artefact, that was repeatable within anexperiment, but varied between experiments. Until this issue is resolved, it is rec-ommended to acquire the AIF projections at a single angle. The AIF slice locationwas determined with a multi-slice FLASH experiment. The best location had goodsignal contrast between the vessels and surrounding tissues. An interleaved ac-quisition was successfully applied in-vivo with the single angled projection-basedAIF, providing temporal resolutions of 0.100 s and 12.8 s for the AIF and DCE179data, respectively. Measuring both curves within the same experiment is expectedto improve the accuracy of model fit parameters. This would be beneficial in studiesattempting to differentiate between two or more known populations, and identifytrends between them. Since the projection data is noisy, it is best to measure theAIF in areas with fewer anatomical structures. We chose the mouse tail for ourexperiments for this reason, though other areas are possible. Further the use offlow compensation improved the contrast between the vessel and surrounding tis-sue, providing a more accurate measure of the intra-vascular concentration. Futurestudies could address the issues with the radial data acquisition (AIF) or attempt tospeed up the DCE image acquisition (current temporal resolution of 12.8 s).180Bibliography[1] Peter L Choyke, Andrew J Dwyer, and Michael V Knopp. FunctionalTumor Imaging With Dynamic Contrast- Enhanced Magnetic ResonanceImaging. Journal of Magnetic Resonance Imaging, 17:509–520, 2003.doi:10.1002/jmri.10304.[2] Lindsay W Turnbull. Dynamic contrast-enhanced MRI in the diagnosis andmanagement of breast cancer. NMR in Biomedicine, 22:28–39, 2009.doi:10.1002/nbm.1273.[3] Tristan Barrett, Martin Brechbiel, Marcelino Bernardo, and Peter LChoyke. MRI of Tumor Angiogenesis. journal of magnetic resonanceimaging, 26:235–249, 2007. doi:10.1002/jmri.20991.[4] Anwar R Padhani and Janet E Husband. Review DynamicContrast-enhanced MRI Studies in Oncology with an Emphasis onQuantification, Validation and Human Studies. Clinical Radiology, 56:607–620, 2001. doi:10.1053/crad.2001.0762.[5] Thomas E. Yankeelov and John C. Gore. Dynamic Contrast EnhancedMagnetic Resonance Imaging in Oncology: Theory, Data Acquisition,Analysis, and Examples. Current Medical Imaging Reviews, 3(2):91–107,2009. doi:10.2174/157340507780619179.Dynamic.[6] Donald B Plewes and Walter Kucharczyk. Physics of MRI: a Primer.Journal of Magnetic Resonance Imaging, 35:1038–1054, may 2012. ISSN1522-2586. doi:10.1002/jmri.23642. URL[7] Vanessa N Harry. Novel imaging techniques as response biomarkers incervical cancer. Gynecologic Oncology, 116:253–261, 2010. ISSN0090-8258. doi:10.1016/j.ygyno.2009.11.003. URL[8] J P B O Connor, A Jackson, G J M Parker, and G C Jayson. DCE-MRIbiomarkers in the clinical evaluation of antiangiogenic and vasculardisrupting agents. British journal of cancer, 96:189–195, 2007.doi:10.1038/sj.bjc.6603515.[9] Jelle O Barentsz, Marc Engelbrecht, Gerrit J Jager, J Alfred Witjes, Jeande LaRosette, Boudewijn P.J. van der Sanden, Henk-Jan Huisman, andArend Heerschap. Fast dynamic gadolinium-enhanced MR imaging ofurinary bladder and prostate cancer. Journal of magnetic resonanceimaging, 10:295–304, sep 1999. ISSN 1053-1807. URL[10] Mark A Zahra, Kieren G Hollingsworth, Evis Sala, David J Lomas, andLi T Tan. Dynamic contrast-enhanced MRI as a predictor of tumourresponse to radiotherapy. Lancet Oncology, 8:63–74, 2007.[11] Linda Knutsson, Danielle van Westen, Esben T Petersen, Karin MarkenrothBloch, Stig Holta˚s, Freddy Sta˚hlberg, and Ronnie Wirestam. Absolutequantification of cerebral blood flow: correlation between dynamicsusceptibility contrast MRI and model-free arterial spin labeling. MagneticResonance Imaging, pages 1–7. ISSN 0730-725X.doi:10.1016/j.mri.2009.06.006.[12] Xiangyu Yang, Jiachao Liang, Johannes T Heverhagen, Guang Jia, PetraSchmalbrock, Steffen Sammet, Regina Koch, and Michael V Knopp.Improving the pharmacokinetic parameter measurement in dynamiccontrast-enhanced MRI by use of the arterial input function: theory andclinical application. Magnetic Resonance in Medicine, pages 1448–1456,jun . ISSN 0740-3194. doi:10.1002/mrm.21608.[13] Stephen Pickup, Rong Zhou, and Jerry Glickson. MRI estimation of thearterial input function in mice. Academic Radiology, 10:963–968, sep2003. ISSN 10766332. doi:10.1016/S1076-6332(03)00291-5. URL[14] Anders Garpebring, Ronnie Wirestam, Nils O¨stlund, and Mikael Karlsson.Effects of inflow and radiofrequency spoiling on the arterial input functionin dynamic contrast-enhanced MRI: A combined phantom and simulationstudy. Magnetic Resonance in Medicine, pages 1670–1679, jun . ISSN07403194. doi:10.1002/mrm.22760.[15] Dustin K Ragan, Stephen Y Lai, and James a Bankson. Fast, reproduciblemeasurement of the vascular input function in mice using constrained182reconstruction and cardiac sampling. NMR in biomedicine, 24:373–384,sep 2011. ISSN 1099-1492. doi:10.1002/nbm.1601. URL[16] Emeline Julie Ribot, Eric Thiaudiere, Richard Roulland, Pierre Brugieres,Alain Rahmouni, Pierre Voisin, Jean Michel Franconi, and Sylvain Miraux.Application of MRI phase-difference mapping to assessment of vascularconcentrations of BMS agent in mice. Contrast Media & MolecularImaging, 3:64–71, 2008. ISSN 15554317. doi:10.1002/cmmi.232.[17] Deirdre M McGrath, Daniel P Bradley, Jean L Tessier, Tony Lacey, Chris JTaylor, and Geoffrey J M Parker. Comparison of model-based arterial inputfunctions for dynamic contrast-enhanced MRI in tumor bearing rats.Magnetic Resonance Imaging, pages 1173–1184, may . ISSN 1522-2594.doi:10.1002/mrm.21959.[18] Geoff J M Parker, Caleb Roberts, Andrew Macdonald, Giovanni ABuonaccorsi, Sue Cheung, David L Buckley, Alan Jackson, YvonneWatson, Karen Davies, and Gordon C Jayson. Experimentally-derivedfunctional form for a population-averaged high-temporal-resolution arterialinput function for dynamic contrast-enhanced MRI. Magnetic Resonancein Medicine, 56:993–1000, nov 2006. ISSN 0740-3194.doi:10.1002/mrm.21066. URL[19] Heidi Lyng, Gro A. Dahle, Olav Kaalhus, Arne Skretting, and Einar K.Rofstad. Measurement of Perfusion Rate in Human Melanoma Xenograftsby Contrast-Enhanced Magnetic Resonance Imaging. Magnetic Resonancein Medicine, 40:89–98, 1998. URL[20] Arvind P Pathak, Dmitri Artemov, and Zaver M Bhujwalla. Novel Systemfor Determining Contrast Agent Concentration in Mouse Blood In Vivo.Magnetic Resonance in Medicine, 51:612–615, 2004.doi:10.1002/mrm.10727.[21] Dominick J O McIntyre, Christian Ludwig, Amit Pasan, and John RGriffiths. A method for interleaved acquisition of a vascular input functionfor dynamic contrast-enhanced MRI in experimental rat tumours. NMR inbiomedicine, 17:132–143, may 2004. ISSN 0952-3480.doi:10.1002/nbm.868. URL[22] J Ferre´, E Bannier, H Raoult, G. Mineur, B. Carsin-Nicol, and J.Y. Gauvrit.Arterial spin labeling ( ASL ) perfusion : Techniques and Clinical Use.Diagnostic and Interventional Imaging, pages 1211–1223. ISSN2211-5684. doi:10.1016/j.diii.2013.06.010.[23] O. Arcangioli, L. Arrive, B. Aubert, R. Benacerraf, J.M. Bigot, J. Bittoun,S. Blangy, A. Bonnin, J. Broussin, D. Buthiau, J.N. Buy, A.E. Cabee, J.M.Caille, S. Chagnon, J. F. Chateil, A. Chevrot, C. Coffre, D. Couanet,E. Despres, J. De Treuf, J. Ecoiffier, L. Engelholm, G. Frija, F. Gires,N. Grenier, P. Grenier, C. Guinet, F. Heran, I. Idy-Peretti, J. Jean-Mart,P. Kien, H. Kressel, R. Lavayssiere, J. Leclere, A. Leroy-Willig, L. Malbec,M. Marichez, J. Masselot, E. Meary, Y. Menu, G. Pallardy, R.A. Parienty,J.D. Piekarski, J.C. Roucayrol, E. Schouman-Claeys, C. Segebarth,A. Sibert, E. Somveille, A. Taieb, V. Taviere, D. Vadrot, C. Vallee,D. Vanel, M. Zalcman, and M. Wybier. MRI of the Body. Springer-Verlag,Paris, France, 1 edition, 1989. ISBN 9783642875588.[24] Alfred L. Horowitz. MRI Physics for Radiologist: A Visual Approach.Springer, New York, New York, USA, 3rd edition, 1995. ISBN9780387943725.[25] Lars G Hanson. Is Quantum Mechanics Necessary for UnderstandingMagnetic Resonance? Concepts in Magnetic Resonance Part A, 32A(5):329–340, 2008. ISSN 00178012. doi:10.1002/cmr.a.[26] Brian M Dale and Jeffrey L Duerk. The Use Of Measured K-SpaceTrajectory For Reconstruction Of Radial MRI Data. ProceedingsInternational Society of Magnetic Resonance in Medicine, 10, 2002.[27] Robert W Brown, Yu-Chung N.Cheng, E. Mark Haccke, Michael R.Thompson, and Rames Venkatesan. Magnetic Resonance Imaging. JohnWiley & Sons Inc, Hoboken, New Jersey, second edition, 2014.[28] F. Bloch, W. W. Hansen, and M. Packard. The Nuclear InductionExperiment. Physical Review, 70(7 and 8):474–485, 1946.[29] Brian M Dale, Mark A Brown, and Richard C Semelka. Production of netmagnetization. In MRI Basic Principles and Applications, chapter 1, pages1–9. John Wiley & Sons, Ltd, Chichester, UK, 5th edition, 2015.[30] Muhammed Elmaog˘lu and Azim C¸elik. Fundamentals of MagneticResonance Imaging. In MRI Handbook, chapter 2, pages 7–23. Springer,New York, New York, USA, 2012. ISBN 978-1-4614-1095-9.184doi:10.1007/978-1-4614-1096-6. URL[31] Brian M. Dale, Mark a. Brown, and Richard C. Semelka. Concepts ofmagnetic resonance. In MRI Basic Principles and Applications, volume 0,chapter 2, pages 10–16. John Wiley & Sons, Ltd, Chichester, UK, 5thedition, oct 2015. ISBN 9781119013068. doi:10.1002/9781119013068.URL[32] Stephen Balter. An Introduction Magnetic to the Physics Imaging ofResonance. RadioGraphics, 7(2):371–383, 1987.[33] Matt A Bernstein, Kevin F King, and Xiaohong Joe Zhou. Handbook ofMRI Pulse Sequences. Elsevier Inc, 2004.[34] Stuart Currie, Nigel Hoggard, Ian J Craven, Marios Hadjivassiliou, andIain D Wilkinson. Understanding MRI: basic MR physics for physicians.Postgraduate medical journal, 89:209–223, apr 2013. ISSN 1469-0756.doi:10.1136/postgradmedj-2012-131342. URL[35] Paul Heckbert. Fourier Transforms and the Fast Fourier Transform ( FFT )Algorithm. Computer Graphics, 2:15–463, 1998.[36] J W Cooley and J W Tukey. An Algorithm for Machine Calculation ofComplex Fourier Series, 1965. ISSN 00255718.[37] A. Dutt and V Rokhlin. Fast Fourier Transforms fpr Nonequispaced Data.SIAM Journal Scientific Computing, 14(6):1368–1393, 1993.[38] Robert A Pooley. AAPM / RSNA Physics Tutorial for ResidentsFundamental Physics of MR Imaging 1. RadioGraphics, 25:1087–1099,2005.[39] Muhammed Elmaog˘lu and Azim C¸elik. MR Imaging Parameters andOptions. In MRI Handbook, chapter 5, pages 69–89. Springer, New York,New York, USA, 2012. ISBN 978-1-4614-1095-9.doi:10.1007/978-1-4614-1096-6. URL[40] S. MeiBoom and D. Gill. Modified Spin-Echo Method for MeasuringNuclear Relaxation Times. The Review of Scientific Instruments, 29(8):688–691, 1958.185[41] A Haase, J Frahm, D Matthaei, and K Merboldt. FLASH Imaging . RapidNMR Imaging Using Low Flip-Angle Pulses. Journal of MagneticResonance, 67:258–266, 1986.[42] David L Buckley and Geoffrey J M Parker. Measuring Contrast AgentConcentration in T 1 -Weighted Dynamic Contrast-Enhanced MRI. InLeuven A.L. Baert and Heidelberg K.Sartor, editors, DynamicContrast-Enhanced Magnetic Resonance Imaging in Oncology, chapter 5,pages 69–79. Springer, 2005.[43] E Mark Haacke and Gerald W Lenz. Improving MR image quality in thepresence of motion by using rephasing gradients. American Journal ofRoentgenology, 148:1251–1258, jun 1987. ISSN 0361-803X. URL[44] Paul C Lauterbur. Image Formation by Induced Local Interactions. Nature,242:190, 1973.[45] Jeff L Zhang, Henry Rusinek, Louisa Bokacheva, Qun Chen, Pippa Storey,and Vivian S Lee. Use of cardiac output to improve measurement of inputfunction in quantitative dynamic contrast-enhanced MRI. Journal ofMagnetic Resonance Imaging, pages 656–665, sep . ISSN 1053-1807.doi:10.1002/jmri.21891.[46] M Louis Lauzon and Brian K Rutt. Effects of Polar Sampling in k-Space.Magnetic Resonance in Medicine, 36:940–949, 1996.[47] M I Altbach, T P Trouard, and A F Gmitro. Radial MRI techniques forobtaining motion-insensitive high-resolution images with variable contrast.In 2002 IEEE, pages 125–128, 2002. ISBN 078037584X.[48] Rafael L O’Halloran, Zhifei Wen, James H Holmes, and Sean B Fain.Iterative projection reconstruction of time-resolved images usinghighly-constrained back-projection (HYPR). Magnetic Resonance inMedicine, pages 132–139, jan . ISSN 0740-3194. doi:10.1002/mrm.21439.[49] Eli Benveniste and L. Yaroslavsky. Comparison Of Two-DimensionalRebinning Algorithms For Image Reconstruction From Projections.Technical report, Tel Aviv University, Tel Aviv, 2003. URL{∼}yaro/EliBenveniste/EliBenvenisteSummary.pdf.[50] H Schomberg and J Timmer. The gridding method for image reconstructionby Fourier transformation. IEEE Transactions on Medical Imaging, 14(3):186596–607, jan 1995. ISSN 0278-0062. doi:10.1109/42.414625. URL[51] J D O’Sullivan. A fast sinc function gridding algorithm for fourierinversion in computer tomography. IEEE Transactions on MedicalImaging, 4(4):200–207, jan 1985. ISSN 0278-0062.doi:10.1109/TMI.1985.4307723. URL[52] John I Jackson, Craig H Meyer, Dwight G Nishimura, and AlbertMacovski. Selection of a Convolution Function for Fourier Inversion UsingGridding. IEEE Transactions on Medical Imaging, 10(3):473–478, 1991.[53] Peter M Joseph. Sampling Errors in Projection Reconstruction MRI.Magnetic Resonance in Medicine, 40:460–466, 1998.[54] Muhammed Elmaog˘lu and Azim C¸elik. A Brief History of MagneticResonance Imaging. In MRI Handbook, chapter 1, pages 3–6. Springer,New York, New York, USA, 2012. ISBN 978-1-4614-1095-9.doi:10.1007/978-1-4614-1096-6. URL[55] E´va To´th, Lothar Helm, and Andre´ E Merbach. Relaxivity of MRI ContrastAgents. In Werner Krause, editor, Contrast Agents 1: Magnetic ResonanceImaging, volume 221, pages 61–101. Springer Berline Heidelberg, Berlin,2002.[56] Erez Eyal and Hadassa Degani. Model-based and model-free parametricanalysis of breast dynamic-contrast-enhanced MRI. NMR in biomedicine,22:40–53, jan 2009. ISSN 0952-3480. doi:10.1002/nbm.1221. URL[57] Paul S Tofts, Gunnar Brix, David L Buckley, Jeffrey L Evelhoch, ElizabethHenderson, Michael V Knopp, Henrik B W Larsson, Ting-yim Lee, Nina AMayr, Geoffrey J M Parker, Ruediger E Port, June Taylor, and Robert MWeisskoff. Estimating Kinetic Parameters from DynamicContrast-Enhanced T1-Weighted MRI of a Diffusable Tracer: StandardizedQuantities and Symbols. Journal of Magnetic Resonance Imaging, 10:223–232, 1999.[58] Maya Dadiani, Edna Furman-Haran, and Hadassa Degani. The applicationof NMR in tumor angiogenesis research. Progress in Nuclear MagneticResonance Spectroscopy, 49:27–44, aug 2006. ISSN 00796565.187doi:10.1016/j.pnmrs.2006.06.001. URL[59] Eva To´th, Lothar Helm, and Andre´ Merbach. Relaxivity of Gadolinium(III) Complexes: Theory and Mechanism. In Lothar Helm Andre Merbachand Eva Toth, editors, The Chemistry of Contrast Agents in MedicalMagnetic Resonance Imaging, chapter 2, pages 25–81. John Wiley & Sons,Ltd, New York, New York, USA, second edition, 2013.[60] Fernando Calamante. Quantification of Dynamic Susceptibility ContrastT2* MRI in Oncology. In Leuven A.L. Baert and Heidelberg K.Sartor,editors, Dynamic Contrast-Enhanced Magnetic Resonance Imaging inOncology, chapter 4, pages 53–67. Springer, 2005.[61] Zhuxian Zhou and Zheng-Rong Lu. Gadolinium-Based Contrast Agentsfor MR Cancer Imaging. Wiley Interdisciplinary Reviews. Nanomedicineand Nanobiotechnology, 5(1):1–18, 2013. doi:10.1002/wnan.1198.[62] Brian M Dale, Mark A Brown, and Richard C. Semelka. Contrast Agents.In MRI Basic Principles and Applications, chapter 15, pages 189–195.John Wiley & Sons, Ltd, Chichester, UK, 5th edition, 2015.[63] Hanns-Joachim Weinmann, Robert C Brasch, Wolf-R Press, and George EWesbey. Characteristics of Gadolinium-DTPA Complex: A Potential NMRContrast Agent. American Journal of Roentgenology, 142:619–624, 1984.[64] D. H. Carr, J. Brown, G. M. Bydder, H. J. Weinmann, U. Speck, D. J.Thomas, and I. R. Young. Intravenous Chelated Gadolinium As a ContrastAgent in NMR Imaging of Cerebral Tumours. The Lancet, March 3:484–486, 1984. ISSN 01406736. doi:10.1016/S0140-6736(84)92852-6.[65] Robert J. McDonald, Jennifer S. McDonald, David F. Kallmes, Mark E.Jentoft, David L. Murray, Kent R. Thielen, Eric E. Williamson, andLaurence J. Eckel. Intracranial Gadolinium Deposition afterContrast-enhanced MR Imaging. Radiology, (3):772–782. ISSN0033-8419. doi:10.1148/radiol.15150025.[66] Bang J. Guo, Zhen L. Yang, and Long J. Zhang. Gadolinium Deposition inBrain: Current Scientific Evidence and Future Perspectives. Frontiers inMolecular Neuroscience, 11(September):1–12, 2018.doi:10.3389/fnmol.2018.00335.188[67] H. Kang, M. Hii, M. Le, R. Tam, A. Riddehough, A. Traboulsee, S. Kolind,M. S. Freedman, and D. K.B. Li. Gadolinium deposition in deep brainstructures: Relationship with dose and ionization of lineargadolinium-based contrast agents. American Journal of Neuroradiology,39(9):1597–1603, 2018. ISSN 1936959X. doi:10.3174/ajnr.A5751.[68] Andreu F. Costa, Christian B. van der Pol, Pejman Jabehdar Maralani,Matthew D.F. McInnes, Jason R. Shewchuk, Raman Verma, Casey Hurrell,and Nicola Schieda. Gadolinium Deposition in the Brain: A SystematicReview of Existing Guidelines and Policy Statement Issued by theCanadian Association of Radiologists. Canadian Association ofRadiologists Journal, (4):373–382. ISSN 14882361.doi:10.1016/j.carj.2018.04.002.[69] Baris¸ Tu¨rkbey, David Thomasson, Yuxi Pang, Marcelino Bernardo, andPeter L Choyke. The role of dynamic contrast-enhanced MRI in cancerdiagnosis and treatment. Diagnostic and Interventional Radiology, pages186–192, sep . ISSN 1305-3612. doi:10.4261/1305-3825.DIR.2537-08.1.[70] M. O. Leach, B. Morgan, P. S. Tofts, D. L. Buckley, W. Huang, M. A.Horsfield, T. L. Chenevert, D. J. Collins, A. Jackson, D. Lomas,B. Whitcher, L. Clarke, R. Plummer, I. Judson, R. Jones, R. Alonzi,T. Brunner, D. M. Koh, P. Murphy, J. C. Waterton, G. Parker, M. J. Graves,T. W J Scheenen, T. W. Redpath, M. Orton, G. Karczmar, H. Huisman,J. Barentsz, and A. Padhani. Imaging vascular function for early stageclinical trials using dynamic contrast-enhanced magnetic resonanceimaging. European Radiology, 22:1451–1464, 2012. ISSN 09387994.doi:10.1007/s00330-012-2446-x.[71] A L Martel, R W Chan, E Ramsay, and D B Plewes. Removingundersampling artifacts in DCE-MRI studies using independentcomponents analysis. Magnetic Resonance in Medicine, 59:874–884, apr2008. ISSN 0740-3194. doi:10.1002/mrm.21552. URL[72] Jie Zheng, Kyongtae T Bae, Pamela K Woodard, E Mark Haacke, andDebiao Li. Efficacy of Slow Infusion of Gadolinium Contrast Agent inThree-Dimensional MR Coronary Artery Imaging. Journal of MagneticResonance Imaging, 10(5):800–805, 1999.[73] D Checkley, J J Tessier, J Kendrew, J C Waterton, and S R Wedge. Use ofdynamic contrast-enhanced MRI to evaluate acute treatment with ZD6474 ,189a VEGF signalling inhibitor, in PC-3 prostate tumours. British Journal ofCancer, 89:1889–1895, 2003. doi:10.1038/sj.bjc.6601386.[74] M V Knopp, E Weiss, H P Sinn, J Mattern, H Junkermann, J Radeleff,A Magener, G Brix, S Delorme, I Zuna, and G van Kaick.Pathophysiologic basis of contrast enhancement in breast tumors. Journalof magnetic resonance imaging, 10:260–266, sep 1999. ISSN 1053-1807.URL[75] Tristan Barrett, Hisataka Kobayashi, Martin Brechbiel, and Peter L.Choyke. Macromolecular MRI contrast agents for imaging tumorangiogenesis. European Journal of Radiology, 60:353–366, 2006. ISSN0720048X. doi:10.1016/j.ejrad.2006.06.025.[76] Kyrre E. Emblem, Christian T. Farrar, Elizabeth R. Gerstner, Tracy T.Batchelor, Ronald J. H. Borra, Bruce R. Rosen, A. Gregory Sorensen, andRakesh K. Jain. Vessel calibre—a potential MRI biomarker of tumourresponse in clinical trials. Nature Reviews Clinical Oncology, pages566–584. ISSN 1759-4774. doi:10.1038/nrclinonc.2014.126.[77] Vladimir V. Iakovlev, Manal Gabril, William Dubinski, Andreas Scorilas,Youssef M. Youssef, Hala Faragalla, Kalman Kovacs, Fabio Rotondo,Shereen Metias, Androu Arsanious, Anna Plotkin, Andrew H F Girgis,Catherine J. Streutker, and George M. Yousef. Microvascular density as anindependent predictor of clinical outcome in renal cell carcinoma: Anautomated image analysis study. Laboratory Investigation, (1):46–56.ISSN 00236837. doi:10.1038/labinvest.2011.153.[78] Noel Weidner, Joseph P. Semple, William R. Welch, and Judah Folkman.Tumor Angiogenesis and Metastasis - Correlation in Invasive BreastCarcinoma. The New England Journal of Medicine, 324(1):1–8, 1991.[79] Steven Sourbron. Technical aspects of MR perfusion. European Journal ofRadiology, 76:304–313, 2010. ISSN 0720048X.doi:10.1016/j.ejrad.2010.02.017. URL[80] Tong San Koh, Sotirios Bisdas, Dow Mu Koh, and Choon Hua Thng.Fundamentals of tracer kinetics for dynamic contrast-enhanced MRI.Journal of magnetic resonance imaging, 34:1262–1276, dec 2011. ISSN1522-2586. doi:10.1002/jmri.22795. URL[81] David C Alsop, John A Detre, Xavier Golay, G Matthias, Jeroen Hendrikse,Luis Hernandez-garcia, Hanzhang Lu, Bradley J Macintosh, Laura MParkes, Marion Smits, Matthias J P Van Osch, and Danny J J Wang.Recommended Implementation of Arterial Spin-Labeled Perfusion MRIfor Clinical Applications : A Consensus of the ISMRM Perfusion StudyGroup and the European Consortium for ASL in Dementia. MagneticResonance in Medicine, 00(October 2013), 2014. doi:10.1002/mrm.25197.[82] Fernando Calamante. Arterial input function in perfusion MRI: Acomprehensive review. Progress in Nuclear Magnetic ResonanceSpectroscopy, 74:1–32, oct 2013. ISSN 1873-3301.doi:10.1016/j.pnmrs.2013.04.002. URL[83] Peter Caravan, Christian T Farrar, Luca Frullano, and Ritika Uppal.Influence of molecular parameters and increasing magnetic field strengthon relaxivity of gadolinium- and manganese-based T1 contrast agents.Contrast media & molecular imaging, 4:89–100, 2009. ISSN 1555-4317.doi:10.1002/cmmi.267. URL[84] J A Brookes, T W Redpath, J Gilbert, G Needham, and A D Murray.Measurement of spin-lattice relaxation times with FLASH for dynamicMRI of the breast. The British Journal of Radiology, 69:206–214, 1996.[85] A Jackson, G C Jayson, K L Li, X P Zhu, D R Checkley, J J L Tessier, andJ C Waterton. Reproducibility of quantitative dynamic contrast-enhancedMRI in newly presenting glioma. The British Journal of Radiology, 76:153–162, mar 2003. ISSN 0007-1285. doi:10.1259/bjr/70653746. URL[86] D C Look and D R Locker. Time Saving in Measurement of NMR andEPR Relaxation Times. Review of Scientific Instruments, 41(2):250–251,feb 1970. ISSN 0034-6748. doi:10.1063/1.1684482. URL[87] Yantian Zhang, Matthew O’Donnell, and Paul L Carson. Determination ofSample Time for T1 Measurement. Journal of Magnetic ResonanceImaging, 8:675–681, 1998.[88] Hongzhi Wang, Ming Zhao, Jerome L Ackerman, and Yiqiao Song.Saturation-inversion-recovery : A method for T 1 measurement. Journal of191Magnetic Resonance, 274:137–143, 2017. ISSN 1090-7807.doi:10.1016/j.jmr.2016.11.015. URL[89] Andrew J. Taylor, Michael Salerno, Rohan Dharmakumar, and MichaelJerosch-Herold. T1 Mapping: Basic Techniques and Clinical Applications.JACC: Cardiovascular Imaging, 9(1):67–81, 2016.doi:10.1016/j.jcmg.2015.11.005.[90] M. Keara Boss, N. Muradyan, and D. E. Thrall. DCE-MRI: A review andapplications in veterinary oncology. Veterinary and ComparativeOncology, 11(2):87–100, jun 2011.[91] Jeffrey L Evelhoch. Key factors in the acquisition of contrast kinetic datafor oncology. Journal of magnetic resonance imaging, 10:254–259, sep1999. ISSN 1053-1807. URL[92] Hee Kwon Song and Lawrence Dougherty. Dynamic MRI With ProjectionReconstruction and KWIC Processing for Simultaneous High Spatial andTemporal Resolution. Magnetic Resonance in Medicine, 52:815–824,2004. doi:10.1002/mrm.20237.[93] Elodie Parzy, Sylvain Miraux, Jean-Michel Franconi, and Eric Thiaudie`re.In vivo quantification of blood velocity in mouse carotid and pulmonaryarteries by ECG-triggered 3D time-resolved magnetic resonanceangiography. NMR in biomedicine, 22:532–537, jun 2009. ISSN1099-1492. doi:10.1002/nbm.1365. URL[94] Julien Vautier, Nadine El Tannir El Tayara, Christine Walczak, Joe¨lMispelter, and Andreas Volk. Radial multigradient-echo DCE-MRI for 3DKtrans mapping with individual arterial input function measurement inmouse tumor models. Magnetic Resonance in Medicine, 70:823–828,2013. ISSN 07403194. doi:10.1002/mrm.24504.[95] Jae-Hun Kim, Geun Ho Im, Jehoon Yang, Dongil Choi, Won Jae Lee, andJung Hee Lee. Quantitative dynamic contrast-enhanced MRI for mousemodels using automatic detection of the arterial input function. NMR inBiomedicine, 25:674–684, 2012. ISSN 1099-1492. doi:10.1002/nbm.1784.URL[96] Thomas E Yankeelov, Laura M DeBusk, D Dean Billheimer, Jeffrey J Luci,P Charles Lin, Ronald R Price, and John C Gore. Repeatability of areference region model for analysis of murine DCE-MRI data at 7T.Journal of Magnetic Resonance Imaging, 24:1140–1147, nov 2006. ISSN1053-1807. doi:10.1002/jmri.20729. URL[97] Martin Pellerin, Thomas E Yankeelov, and Martin Lepage. Incorporatingcontrast agent diffusion into the analysis of DCE-MRI data. MagneticResonance in Medicine, 58:1124–1134, dec 2007. ISSN 0740-3194.doi:10.1002/mrm.21400. URL[98] Tormod A M Egeland, Kristine Gulliksrud, Jon-Vidar Gaustad, BeritMathiesen, and Einar K Rofstad. Dynamic contrast-enhanced-MRI oftumor hypoxia. Magnetic resonance in medicine, 67:519–530, feb 2012.ISSN 1522-2594. doi:10.1002/mrm.23014. URL[99] Andrew B Rosenkrantz, Christian Geppert, Robert Grimm, Tobias KBlock, Christian Glielmi, Li Feng, Ricardo Otazo, Justin M Ream,Melanie Moccaldi Romolo, Samir S Taneja, Daniel K Sodickson, andHersh Chandarana. Dynamic contrast-enhanced MRI of the prostate withhigh spatiotemporal resolution using compressed sensing, parallel imaging,and continuous golden-angle radial sampling: preliminary experience.Journal of Magnetic Resonance Imaging, 41:1365–1373, may 2015. ISSN1522-2586. doi:10.1002/jmri.24661. URL[100] A.-C. Fruytier, J. Magat, F. Colliez, B. Jordan, G. Cron, and B. Gallez.Dynamic contrast-enhanced MRI in mice at high field: estimation of thearterial input function can be achieved by phase imaging. Magneticresonance in medicine, 71:544–550, feb 2014. doi:10.1002/mrm.24682.URL[101] Igor Jacobs, Gustav J. Strijkers, Henk M. Keizer, Henk M. Janssen, KlaasNicolay, and Matthias C. Schabel. A novel approach to tracer-kineticmodeling for (macromolecular) dynamic contrast-enhanced MRI.Magnetic Resonance in Medicine, 75:1142–1153, 2016. ISSN 15222594.doi:10.1002/mrm.25704.193[102] Thomas Fritz-hansen, Egill Rostrup, Henrik B W Larsson, LarsSandergaard, Poul Ring, and Ole Henriksen. Measurement of the ArterialConcentration of Gd-DTPA Using MRI : A Step Toward QuantitativePerfusion Imaging. Magnetic Resonance in Medicine, 36:225–231, 1996.[103] Matthias C Schabel and Dennis L Parker. Uncertainty and bias in contrastconcentration measurements using spoiled gradient echo pulse sequences.Physics in Medicine and Biology, 53(9):2345–2373, may 2008. ISSN0031-9155. URL[104] Geoffrey J M Parker and David L Buckley. Tracer Kinetic Modelling for T1 -Weighted DCE-MRI. In Leuven A.L. Baert and Heidelberg K.Sartor,editors, Dynamic Contrast-Enhanced Magnetic Resonance Imaging inOncology, chapter 6, pages 81–92. Springer, 2005.[105] L.J. Liu and M. Schlesinger. MRI contrast agent concentration and tumorinterstitial fluid pressure. Journal of Theoretical Biology, 406:52–60, 2016.[106] Torfinn Taxt, Radovan Jirˇı´k, Cecilie Brekke Rygh, Renate Gru¨ner, MichalBartosˇ, Erling Andersen, Fitz Roy Curry, and Rolf K. Reed. Single-channelblind estimation of arterial input function and tissue impulse response inDCE-MRI. IEEE Transactions on Biomedical Engineering, 59(4):1012–1021, 2012. ISSN 00189294. doi:10.1109/TBME.2011.2182195.[107] C A Cuenod and D Balvay. Perfusion and vascular permeability : Basicconcepts and measurement in DCE-CT and DCE-MRI. Diagnostic andInterventional Imaging, 94:1187–1204, 2013. ISSN 2211-5684.doi:10.1016/j.diii.2013.10.010. URL[108] Mark Rijpkema, Johannes H. A. M. Kaanders, Frank B. M. Joosten,Albert J van der Kogel, and Arend Heerschap. Method for quantitativemapping of dynamic MRI contrast agent uptake in human tumors. Journalof Magnetic Resonance Imaging, 14:457–463, oct 2001. ISSN 1053-1807.URL[109] Thomas E Yankeelov, Jeffrey J Luci, Martin Lepage, Rui Li, Laura Debusk,P Charles Lin, Ronald R Price, and John C Gore. Quantitativepharmacokinetic analysis of DCE-MRI data without an arterial inputfunction: a reference region model. Magnetic Resonance Imaging, 23:519–529, may 2005. ISSN 0730-725X. doi:10.1016/j.mri.2005.02.013.URL[110] Xiaobing Fan and Gregory S Karczmar. A new approach to analysis of theimpulse response function (IRF) in dynamic contrast-enhanced MRI(DCEMRI): a simulation study. Magnetic resonance in medicine, 62:229–239, jul 2009. ISSN 1522-2594. doi:10.1002/mrm.21995. URL[111] Eugene G Kholmovski and Edward V R DiBella. Perfusion MRI withradial acquisition for arterial input function assessment. MagneticResonance Imaging, 57:821–827, may 2007. ISSN 0740-3194.doi:10.1002/mrm.21210. URL[112] Xia Li, E Brian Welch, Lori R Arlinghaus, A Bapsi Chakravarthy, Lei Xu,Jaime Farley, Mary E Loveless, Ingrid A Mayer, Mark C Kelley, Ingrid MMeszoely, Julie A Means-Powell, Vandana G Abramson, Ana M Grau,John C Gore, and Thomas E Yankeelov. A novel AIF tracking method andcomparison of DCE-MRI parameters using individual andpopulation-based AIFs in human breast cancer. Physics in Medicine andBiology, pages 5753–5769, sep . ISSN 1361-6560.doi:10.1088/0031-9155/56/17/018.[113] Steven P. Sourbron and David L. Buckley. Classic models for dynamiccontrast-enhanced MRI. NMR in Biomedicine, 26:1004–1027, 2013. ISSN09523480. doi:10.1002/nbm.2940.[114] Paul S Tofts and Allan G Kermode. Measurement of the blood-brainbarrier permeability and leakage space using dynamic MR imaging. 1.Fundamental concepts. Magnetic Resonance in Medicine, 17(2):357–367,feb 1991. ISSN 0740-3194. URL[115] Rujirutana Srikanchana, David Thomasson, Peter Choyke, and AndrewDwyer. A Comparison of Pharmacokinetic Models of Dynamic ContrastEnhanced MRI. In Proceedings of the 17th IEEE Symposium onComputer-Based Medical Systems, pages 1–6, 2004.[116] Seymour S Kety. The Theory and Applications of the Exchange of InertGas at the Lungs and Tissues. Pharmacological Reviews, 3(1):1–41, 1951.[117] Cristina Lavini and Joost J C Verhoeff. Reproducibility of the gadoliniumconcentration measurements and of the fitting parameters of the vascularinput function in the superior sagittal sinus in a patient population.195Magnetic resonance imaging, pages 1420–1430, dec . ISSN 1873-5894.doi:10.1016/j.mri.2010.06.017.[118] Bruce R Rosen, John W Belliveau, Bradley R Buchbinder, Robert CMcKinstry, Leena M Porkka, David N Kennedy, Michelle S Neuder,C Richard Fisel, Hannu J Aronen, Kenneth K Kwong, Robert M Weisskoff,Mark S Cohen, and Thomas J Brady. Contrast Agents and CerebralHemodynamics. Magnetic Resonance in Medicine, 19(2):285–292, 1991.[119] William H Perman, Mokhtar H Gado, Kenneth B Larson, and Joel SPerlmutter. Simultaneous MR Acquisition of Arterial and Brain SignalTime Curves. Magnetic Resonance in Medicine, 28:74–83, 1992.[120] Adam E Hansen, Henrik Pedersen, Egill Rostrup, and Henrik B W Larsson.Partial volume effect (PVE) on the arterial input function (AIF) inT1-weighted perfusion imaging and limitations of the multiplicativerescaling approach. Magnetic resonance in medicine, 62:1055–1059, oct2009. ISSN 1522-2594. doi:10.1002/mrm.22098. URL[121] Rong Zhou, Stephen Pickup, Thomas E Yankeelov, Charles S Springer Jr,and Jerry D Glickson. Simultaneous Measurement of Arterial InputFunction and Tumor Pharmacokinetics in Mice by Dynamic ContrastEnhanced Imaging : Effects of Transcytolemmal Water Exchange.Magnetic Resonance in Medicine, 52:248–257, 2004.doi:10.1002/mrm.20143.[122] Wei Lin, Junyu Guo, Mark A Rosen, and Hee Kwon Song. Respiratorymotion-compensated radial dynamic contrast-enhanced (DCE)-MRI ofchest and abdominal lesions. Magnetic Resonance in Medicine, 60:1135–1146, nov 2008. ISSN 1522-2594. doi:10.1002/mrm.21740. URL[123] N J Taylor, I J Rowland, S F Tanner, and M Leach. A Rapid InterleavedMethod for Measuring Signal Intensity Curves in Both Blood and Tissueduring Contrast Agent Administration. Magnetic Resonance in Medicine,30:744–749, 1993.[124] Sabine Heiland, Wolfgang Reith, Michael Forsting, and Klaus Sartor. Howdo concentration and dosage of the contrast agent affect the signal changein perfusion-weighted magnetic resonance imaging? A computersimulation. Magnetic Resonance Imaging, 19:813–820, jul 2001. ISSN0730-725X. URL[125] E G Kholmovski, E V Dibella, and A A Samsonov. Perfusion MRI withRadial Sampling : Arterial Input Function and Tissue Enhancement CurveAssessment. In Proceedings International Society of Magnetic Resonancein Medicine, volume 13, page 252, 2005.[126] Erbil Akbudak, Richard E Norberg, and Thomas E Conturo. Contrast-agentphase effects: an experimental system for analysis of susceptibility,concentration, and bolus input function kinetics. Magnetic resonance inmedicine, 38:990–1002, dec 1997. ISSN 0740-3194. URL[127] Matthias J P van Osch, Evert-jan P A Vonken, Max A Viergever, Jeroenvan der Grond, and Chris J G Bakker. Measuring the arterial input functionwith gradient echo sequences. Magnetic Resonance in Medicine, 49:1067–1076, jun 2003. ISSN 0740-3194. doi:10.1002/mrm.10461. URL[128] Melanie S Kotys, Erbil Akbudak, Joanne Markham, and Thomas EConturo. Precision, Signal-to-Noise Ratio, and Dose Optimization ofMagnitude and Phase Arterial Input Functions in Dynamic SusceptibilityContrast MRI. Journal of magnetic resonance imaging, 25:598–611, 2007.doi:10.1002/jmri.20859.[129] Ludovic de Rochefort, Thanh Nguyen, Ryan Brown, Pascal Spincemaille,Grace Choi, Jonathan Weinsaft, Martin R. Prince, and Yi Wang. In vivoquantification of contrast agent concentration using the induced magneticfield for time-resolved arterial input function measurement with MRI.Medical Physics, (12):5328–5339. ISSN 00942405.doi:10.1118/1.3002309.[130] Shiyang Wang, Xiaobing Fan, Milica Medved, Federico D. Pineda,Ambereen Yousuf, Aytekin Oto, and Gregory S. Karczmar. Arterial inputfunctions (AIFs) measured directly from arteries with low and standarddoses of contrast agent, and AIFs derived from reference tissues. MagneticResonance Imaging, 34:197–203, feb 2016. ISSN 0730725X.doi:10.1016/j.mri.2015.10.025. URL[131] Egbert J W Bleeker, Mark A van Buchem, Andrew G Webb, and MatthiasJ P van Osch. Phase-based arterial input function measurements fordynamic susceptibility contrast MRI. Magnetic resonance in medicine, 64:197358–368, aug 2010. ISSN 1522-2594. doi:10.1002/mrm.22420. URL[132] Herbert Ko¨stler, Christian Ritter, Michael Lipp, Meinrad Beer, DietbertHahn, and Jo¨rn Sandstede. Prebolus quantitative MR heart perfusionimaging. Magnetic resonance in medicine, 52:296–299, aug 2004. ISSN0740-3194. doi:10.1002/mrm.20160. URL[133] Ka Loh Li, Giovanni Buonaccorsi, Gerard Thompson, John R. Cain, AmyWatkins, David Russell, Salman Qureshi, D. Gareth Evans, Simon K.Lloyd, Xiaoping Zhu, and Alan Jackson. An improved coverage and spatialresolution-using dual injection dynamic contrast-enhanced (ICE-DICE)MRI: A novel dynamic contrast-enhanced technique for cerebral tumors.Magnetic Resonance in Medicine, 68:452–462, 2012. ISSN 07403194.doi:10.1002/mrm.23252.[134] Jacob U Fluckiger, Matthias C Schabel, and Edward V R DiBella.Constrained estimation of the arterial input function for myocardialperfusion cardiovascular magnetic resonance. Magnetic resonance inmedicine, 66:419–427, aug 2011. ISSN 1522-2594.doi:10.1002/mrm.22809. URL[135] Tae Ho Kim, Nathan A Pack, Liyong Chen, and Edward V R DiBella.Quantification of myocardial perfusion using CMR with a radial dataacquisition: comparison with a dual-bolus method. Journal ofcardiovascular magnetic resonance, 12:45–52, 2010. ISSN 1532-429X.doi:10.1186/1532-429X-12-45. URL[136] David A Kovar, Marta Lewis, and Gregory S Karczmar. A new method forimaging perfusion and contrast extraction fraction: input functions derivedfrom reference tissues. Journal of Magnetic Resonance Imaging, 8(5):1126–1134, 1998. ISSN 1053-1807. URL[137] N Srinivasa, K. R. Ramakrishnan, and K. Rajgopal. Two-DimensionalSpectral Estimation : A Radon Transform Approach. IEEE Journal ofOceanic Engineering, OE-12(1):90–96, 1987.198[138] Kathleen M Donahue, Deborah Burstein, Warren J Manning, and Martha LGray. Studies of Gd-DTPA Relaxivity and Proton Exchange Rates inTissue. Magnetic Resonance in Medicine, 32:66–76, 1994.[139] Samuella B. Sigmann and Dale E. Wheeler. The QuantitativeDetermination of Food Dyes in Powdered Drink Mixes. Journal ofChemical Education, (10):1475–1478, oct . ISSN 0021-9584.doi:10.1021/ed081p1475.[140] David E. Nixon, Mary F Burritt, and Thomas P Moyer. The determinationof mercury in whole blood and urine by inductively coupled plasma massspectrometry. Spectrochimica Acta Part B, 54:1141–1153, 1999.[141] Claire Foottit, Greg O Cron, Matthew J Hogan, Thanh Binh Nguyen, andIan Cameron. Determination of the venous output function from MR signalphase: feasibility for quantitative DCE-MRI in human brain. Magneticresonance in medicine, 63:772–781, mar 2010. ISSN 1522-2594.doi:10.1002/mrm.22253. URL[142] William T Golde, Peter Gollobin, and Luis L Rodriguez. A rapid , simple ,and humane method for submandibular bleeding of mice using a lancet.Lab Animal, 34(9):39–43, 2005.[143] D Stab, J Tran-Gia, C O Ritter, D Hahn, and H Kostler. AIF Determinationfor Quantitative Myocardial Perfusion Imaging Using a Model BasedReconstruction of Radially Acquired Data. In Proceedings InternationalSociety of Magnetic Resonance in Medicine, volume 19, page 1341, 2011.[144] Michael C Kreissl, Hsiao-Ming Wu, David B Stout, Waldemar Ladno,Thomas H Schindler, Xiaoli Zhang, John O Prior, Mayumi L Prins, Arion FChatziioannou, Sung-Cheng Huang, and Heinrich R Schelbert.Noninvasive measurement of cardiovascular function in mice withhigh-temporal-resolution small-animal PET. Journal of Nuclear Medicine,47:974–980, jun 2006. ISSN 0161-5505. URL[145] Shuo Zhang, Kai Tobias Block, and Jens Frahm. Magnetic ResonanceImaging in Real Time: Advances Using Radial FLASH. Journal ofMagnetic Resonance Imaging, 31:101–109, 2010. doi:10.1002/jmri.21987.[146] Rachel W Chan, Elizabeth A Ramsay, Charles H Cunningham, andDonald B Plewes. Temporal stability of adaptive 3D radial MRI using199multidimensional golden means. Magnetic resonance in medicine, 61:354–363, feb 2009. ISSN 1522-2594. doi:10.1002/mrm.21837. URL[147] Potts D., Steidl G., and Tasche M. Fast Fourier transforms fornonequispaced data: A tutorial. In Benedetto J.J. and Ferreira P.J.S.G.,editors, Modern Sampling Theory. Applied and Numerical HarmonicAnalysis, chapter 12, pages 247–270. Birkha¨user, Boston, MA, 2001. ISBN978-1-4612-0143-4. doi: 12.[148] Jeffrey A Fessler and Bradley P Sutton. Nonuniform Fast FourierTransforms Using Min-Max Interpolation. IEEE transactions on signalprocessing, 51(2):560–574, 2003.[149] Charles A Mistretta. Undersampled Radial MR Acquisition and HighlyConstrained Back Projection ( HYPR ) Reconstruction: Potential MedicalImaging Applications in the Post-Nyquist Era. Journal of MagneticResonance Imaging, 29:501–516, 2009. doi:10.1002/jmri.21683.[150] Sungho Tak, Jaeheung Yoo, and Jong Chul. High Resolution ProjectionReconstruction MR Imaging using FOCUSS. In SPIE-IS&T, volume 6498,page 64981A, 2007.[151] Kai Tobias Block, Martin Uecker, and Jens Frahm. Undersampled radialMRI with multiple coils. Iterative image reconstruction using a totalvariation constraint. Magnetic Resonance in Medicine, 57:1086–1098,2007.[152] Avinash C. Kak and Malcolm. Measurement of Projection Data- TheNondiffracting Case. In Robert E. O’Malley Jr., editor, Principles ofComputerized Tomographic Imaging, chapter 4, pages 113–175. SiAM,Philapelphia, PA, first edition, 2001.[153] Leslie Greengard and June-Yub Lee. Accelerating the Nonuniform FastFourier Transform. SIAM Review, 46(3):443–454, 2004.[154] Ganesh Adluru, Ross Whitaker, and Edward V.R. Dibella. Spatio-TemporalConstrained Reconstruction of Sparse Dynamic Contrast Enhanced RadialMRI Data. 2007 4th IEEE International Symposium on BiomedicalImaging: From Nano to Macro, pages 109–112, 2007.doi:10.1109/ISBI.2007.356800. URL[155] Ganesh Adluru, Chris McGann, Peter Speier, Eugene Kholmovski, AkramShaaban, and Edward V R Dibella. Acquisition and Reconstruction ofUndersampled Radial Data for Myocardial Perfusion MRI. Journal ofMagnetic Resonance Imaging, 29(2):466–473, 2009.doi:doi:10.1002/jmri.21585.[156] Michael Lustig, David Donoho, and John M Pauly. Sparse MRI : TheApplication of Compressed Sensing for Rapid MR Imaging. MagneticResonance in Medicine, 58:1182–1195, 2007. doi:10.1002/mrm.21391.[157] K.T. Block and M. Uecker. Simple Method for Adaptive Gradient-DelayCompensation in Radial MRI. In Proceedings International Society ofMagnetic Resonance in Medicine, volume 19, page 2816, 2011.[158] Donald Shepard. A two-dimensional interpolation function forirregularly-spaced data. In Proceedings of the 1968 ACM NationalConference, pages 517–524, New York, New York, USA, 1968. ACMPress. doi:10.1145/800186.810616. URL[159] Jens Keiner, Stefan Kunis, and Daniel Potts. Using NFFT 3 – a softwarelibrary for various nonequispaced fast Fourier transforms. ACMTransactions on Mathematical Software, V(N):1–23, 2008.[160] Tobias Knopp, Stefan Kunis, and Daniel Potts. A note on the iterative MRIreconstruction from nonuniform k-space data. International Journal ofBiomedical Imaging, 2007:24727, jan 2007. ISSN 1687-4188.doi:10.1155/2007/24727. URL[161] Jens Keiner, Stefan Kunis, and Daniel Potts. NFFT 3.0 - Tutorial.Technical report, University of Lubeck, Institute of Mathematics andChemnitz University of Technology, Department of Mathematics,Chemnitz, Germany, 2006. URL[162] Kai Tobias Block. Advanced Methods for Radial Data Sampling inMagnetic Resonance Imaging. PhD thesis, der Georg-August-UniversitatGottingen, 2008.[163] Wei Lin, Feng Huang, Enrico Simonotto, George R Duensing, and ArneReykowski. Off-resonance artifacts correction with convolution in k-space(ORACLE). Magnetic Resonance in Medicine, pages 1547–1555, jun .ISSN 1522-2594. doi:10.1002/mrm.23135.201[164] Stefanie Winkelmann, Tobias Schaeffter, Holger Eggers, and Olaf Doessel.SNR enhancement in radial SSFP imaging using partial k-space averaging.IEEE Transactions on Medical Imaging, 24(2):254–262, feb 2005. ISSN0278-0062. URL[165] Stefanie Winkelmann, Tobias Schaeffter, Thomas Koehler, Holger Eggers,and Olaf Doessel. An optimal radial profile order based on the GoldenRatio for time-resolved MRI. IEEE Transactions on Medical Imaging, 26(1):68–76, jan 2007. ISSN 0278-0062. doi:10.1109/TMI.2006.885337.URL[166] Ganesh Adluru, Suyash P Awate, Tolga Tasdizen, Ross T Whitaker, andEdward V R Dibella. Temporally constrained reconstruction of dynamiccardiac perfusion MRI. Magnetic resonance in medicine, 57:1027–1036,jun 2007. ISSN 0740-3194. doi:10.1002/mrm.21248. URL[167] Z H Ahn, C B, Cho. New Phase Correction Method in NMR ImagingBased on Autocorrelation and Histogram Analysis. IEEE transactions onmedical imaging, MI-6(1):32–36, 1987.[168] Zhou Wang, Alan Conrad Bovik, Hamid Rahim Sheikh, and Eero PSimoncelli. Image Quality Assessment : From Error Visibility to StructuralSimilarity. IEEE Transactions on Image Processing, 13(4):600–612, 2004.[169] Greg J. Stanisz, Ewa E. Odrobina, Joseph Pun, Michael Escaravage,Simon J. Graham, Michael J. Bronskill, and R. Mark Henkelman. T1, T2relaxation and magnetization transfer in tissue at 3T. Magnetic Resonancein Medicine, 54:507–512, 2005. ISSN 07403194. doi:10.1002/mrm.20605.[170] Joop J van Vaals, Marijn E Brummer, W Thomas Dixon, Hans H Tuithof,Hans Engels, Rendon C Nelson, Brigid M Gerety, Judith L Chezmar, andJacques A den Boer. ”Keyhole” method for accelerating imaging ofcontrast agent uptake. Journal of Magnetic Resonance Imaging, pages671–675. ISSN 1053-1807.[171] Jennifer Moroz, Andrew C Yung, Piotr Kozlowski, and Stefan AReinsberg. Estimation of the Arterial Input Function in a Mouse Tail fromthe Signal Phase of Projection Profiles. In Proceedings InternationalSociety of Magnetic Resonance in Medicine, volume 20, page 239, 2012.[172] Patrik Brynolfsson, Jun Yu, Ronnie Wirestam, Mikael Karlsson, andAnders Garpebring. Combining phase and magnitude information for202contrast agent quantification in dynamic contrast-enhanced MRI usingstatistical modeling. Magnetic Resonance in Medicine, 74:1156–1164,2015. ISSN 15222594. doi:10.1002/mrm.25490.[173] Yaqi Shen, Frank L Goerner, Christopher Snyder, John N Morelli, DapengHao, Daoyu Hu, Xiaoming Li, and Val M Runge. T1 Relaxivities ofGadolinium-Based Magnetic Resonance Contrast Agents in Human WholeBlood at 1.5, 3, and 7 T. Investigative Radiology, 50(5):330–338, 2015.ISSN 15360210. doi:10.1097/RLI.0000000000000132.[174] C. Kalavagunta and G.J. Metzger. A field comparison of r1 and r2 *relaxivities of Gd-DTPA in aqueous solution and whole blood : 3T versus7T. Proceedings International Society of Magnetic Resonance in Medicine,18:4990, 2010.[175] Iris M Noebauer-huhmann, Pavol Szomolanyi, Oliver Kraff, and SiegfriedTrattnig. Gadolinium-Based Magnetic Resonance Contrast Agents at 7Tesla. Investigative Radiology, 45(9):554–558, 2010.[176] Rachel W Chan, Elizabeth A Ramsay, Edward Y Cheung, and Donald BPlewes. The influence of radial undersampling schemes on compressedsensing reconstruction in breast MRI. Magnetic resonance in medicine, 67:363–377, feb 2012. ISSN 1522-2594. doi:10.1002/mrm.23008. URL[177] Hao Tan and Craig H Meyer. Estimation of k-Space Trajectories in SpiralMRI. Magnetic Resonance in Medicine, 61:1396–1404, 2009.doi:10.1002/mrm.21813.[178] Dana C. Peters, J. Andrew Derbyshire, and Elliot R. McVeigh. Centeringthe projection reconstruction trajectory: Reducing gradient delay errors.Magnetic Resonance in Medicine, 50:1–6, 2003.[179] T. H. Rozijn, B. P J Van Der Sanden, A. Heerschap, J. H N Creyghton, andW. M M J Bove´e. Influence of the pharmacokinetic model on thequantification of the Gd- DTPA uptake rate in brain tumours using directT1 measurements. Magnetic Resonance Materials in Physics, Biology andMedicine, 6(1):37–43, 1998. ISSN 13528661.doi:10.1016/S1352-8661(98)00008-8.[180] Jonathan M. Bernstein, Jarrod J. Homer, and Catharine M. West. Dynamiccontrast-enhanced magnetic resonance imaging biomarkers in head andneck cancer: Potential to guide treatment? A systematic review. Oral203Oncology, 50:963–970, 2014. ISSN 18790593.doi:10.1016/j.oraloncology.2014.07.011.[181] Chong Duan, Jesper F. Kallehauge, G. Larry Bretthorst, Kari Tanderup,Joseph J.H. Ackerman, and Joel R. Garbow. Are complex DCE-MRImodels supported by clinical data? Magnetic Resonance in Medicine, 77(3):1329–1339, 2017. ISSN 15222594. doi:10.1002/mrm.26189.[182] Andriy Fedorov, Jacob Fluckiger, Gregory D. Ayers, Xia Li, Sandeep N.Gupta, Clare Tempany, Robert Mulkern, Thomas E. Yankeelov, andFiona M. Fennessy. A comparison of two methods for estimatingDCE-MRI parameters via individual and cohort based AIFs in prostatecancer: A step towards practical implementation. Magnetic ResonanceImaging, pages 321–329. ISSN 18735894. doi:10.1016/j.mri.2014.01.004.[183] Xiaoguang Li, Yongshan Zhu, Houyi Kang, Yulong Zhang, Huaping Liang,Sumei Wang, and Weiguo Zhang. Glioma grading by microvascularpermeability parameters derived from dynamic contrast-enhanced MRI andintratumoral susceptibility signal on susceptibility weighted imaging Head& neck imaging. Cancer Imaging, 15(1):1–9, 2015. ISSN 14707330.doi:10.1186/s40644-015-0039-z.[184] Hatef Mehrabian, Chaitanya Chandrana, Ian Pang, Rajiv Chopra, andAnne L Martel. Arterial input function calculation in dynamiccontrast-enhanced MRI: an in vivo validation study using co-registeredcontrast-enhanced ultrasound imaging. European radiology, 22:1735–1747, aug 2012. ISSN 1432-1084. doi:10.1007/s00330-012-2418-1.URL[185] Hatef Mehrabian, Rajiv Chopra, and Anne L Martel. Calculation ofintravascular signal in dynamic contrast enhanced MRI using adaptivecomplex independent component analysis. IEEE Transactions on MedicalImaging, 32(4):699–710, 2013. ISSN 0278-0062.doi:10.1109/TMI.2012.2233747;10.1109/TMI.2012.2233747.[186] Hatef Mehrabian, Masoom A. Haider, and Anne L. Martel. UsingIndependent Compenents Analysis to Calculate Intravascular ContrastAgent Concentration in Prostate Cancer. In IEEE 10th InternationalSymposium on Biomedical Imaging, pages 966–969, 2013. ISBN9781467364553.[187] Robert Marc Lebel, Jesse Jones, Jean-Christophe Ferre, Meng Law, andKrishna S Nayak. Highly accelerated dynamic contrast enhanced imaging.204Magnetic Resonance in Medicine, pages 635–644. ISSN 1522-2594.doi:10.1002/mrm.24710.[188] Mark A Griswold, Peter M Jakob, Robin M Heidemann, Mathias Nittka,Vladimir Jellus, Jianmin Wang, Berthold Kiefer, and Axel Haase.Generalized Autocalibrating Partially Parallel Acquisitions (GRAPPA).Magnetic Resonance in Medicine, 47:1202–1210, 2002.doi:10.1002/mrm.10171.[189] Chao Ma and Xiaohua Jiang. A New Eddy-current Compensation Methodin MRI. PIERS Online, 3(6):874–878, 2007. ISSN 1931-7360.doi:10.2529/PIERS061006031452. URL{&}volume=3{&}number=6{&}page=874.[190] Amir Moussavi, Markus Untenberger, Martin Uecker, and Jens Frahm.Correction of gradient-induced phase errors in radial MRI. MagneticResonance in Medicine, 71:308–312, 2014. ISSN 07403194.doi:10.1002/mrm.24643.[191] Ian C. Atkinson, Aiming Lu, and Keith R. Thulborn. Characterization andcorrection of system delays and eddy currents for MR imaging withultrashort echo-time and time-varying gradients. Magnetic Resonance inMedicine, 62:532–537, 2009. ISSN 07403194. doi:10.1002/mrm.22016.[192] Yantian Zhang, Hoby P Hetherington, Ernest M Stokely, Graeme F Mason,and Donald B Twieg. A novel k-space trajectory measurement technique.Magnetic Resonance in Medicine, 39:999–1004, jun 1998. ISSN0740-3194. URL[193] Peter Latta, Zenon Starcuk Jr., Marco L H Gruwel, Michael H. Weber, andBoguslaw Tomanek. K-space trajectory mapping and its application forultrashort Echo time imaging. Magnetic Resonance Imaging, 36:68–76,2017. ISSN 18735894. doi:10.1016/j.mri.2016.10.012.[194] Marine Beaumont, Laurent Lamalle, Christoph Segebarth, andEmmanuel L Barbier. Improved k -Space Trajectory Measurement WithSignal Shifting. Magnetic resonance in medicine, 58:200–205, 2007.doi:10.1002/mrm.21254.[195] Jennifer Moroz, Piotr Kozlowski, and Stefan A Reinsberg. Determinationof Local Tissue Enhancement from Radially Reconstructed Images. InProceedings International Society of Magnetic Resonance in Medicine,volume 21, page 3074, 2013.205Appendix AComparing RadialReconstruction TechniquesThis appendix shows all of the radial images reconstructed with Shepard’s methodof interpolation, STCR and NFFT, with three sampling methods and five accelera-tion rates.A.1 Shepard’s Method of InterpolationRadial images reconstructed with Shepard’s method of interpolation are presentedin Figures A.1, A.2, and A.3 for reconstructions with 233, 144, 89, 55, 34 or 21projections, and uniform, Golden angle or random sampling.In general, the image quality decreases as fewer projections are used in the re-construction. At least 89 projections are required for the radial image to be visuallycomparable with the reference image (using all 233 radial projections). Reducingthe number of projections to 55 and fewer often causes a loss of contrast betweenthe capillary tube and the rest of the phantom, and blurring of the outer edges ofthe phantom. The smaller external phantom to the right of the main one is visiblein all images with uniform or Golden angle sampling, but only in the images with55 or more projections with random sampling. Blurry, curved streaking artifactsare observed in the background for all images, though the structure is consistentwith the reference image. These artifacts are likely due to the chosen interpolation206Figure A.1: Radial magnitude images created with Shepard’s method of in-terpolation with uniform angluar sampling over 180o. The reconstruc-tions were done with 233 (reference), 144, 89, 55, 34 or 21 projections.Visually, the images with 89 or 144 projections are comparable withthe reference image. As the number of projections is reduced, the im-age becomes blurred and the contrast between the capillary tube andsurrounding region is reduced. The signal intensity gradient across thephantom is preserved down to 34 projections. The mSSIM index, rel-ative to the reference image, gradually decreases from 0.698 with 144projections to 0.563 with 21 projections. Faint artifacts are observed inthe background in all images. These are likely a result of how the datais interpolated onto the Cartesian grid. Additional artifacts are observedin the images with 89 and fewer projections. These appear as blotches.method and density correction. Additional artifacts are observed as the number ofprojections in the reconstruction are reduced.Figure A.4 compared the signal magnitude and phase of the reconstructed im-ages with 55 projections. The magnitude images are similar between the uniform207Figure A.2: Radial magnitude images created with Shepard’s method of in-terpolation and Golden angle sampling (having an angular increment of111.246o). 233 (reference), 144, 89, 55, 34 or 21 projections are usedin the reconstruction. Similar to the uniformly sampled data set, the im-ages with 89 and 144 are visually similar to the reference image. Withfewer projections, the images become blurry and the contrast betweenthe capillary tube and surrounding phantom decreases. The mSSIMvalues, relative to the reference image, are listed in the title, and de-creases from 0.706 with 144 projections to 0.576 with 21 projections.Additional background artifacts are observed in the images with 89 andfewer projections. They are more apparent in the images with 21 or 34projections.208Figure A.3: Radial magnitude images created with Shepard’s method ofinterpolation and randomly sampled data over the angular range of0− 180o. The images were reconsturcted with 233 (reference), 144,89, 55, 34 or 21 projections. Only the image with 144 projections isvisually similar to the reference image. As the number of projectionsdrops from 89 to 21, image contrast between the capillary tube and therest of the phantom decreases rapidly and edges are blurred. The exter-nal phantom to the right of the main one is visible in the images with atleast 55 projections. Beyond this, it blends in with the additional imageartifacts in the background. These artifacts are present in all images,but are more visually apparent as the number of projections is reduced.The mSSIM values listed in the titles decrease from 0.624 with 144projections to 0.541 with 21 projections.209Figure A.4: Radial magnitude images created with Shepard’s method of in-terpolation, 55 projections in the recontruction and uniform, Goldenangle (angular increment 111.246o) or random data sampling. The uni-form and Golden angle sampling schemes produce a simlar images interms of phantom edge sharpness, contrast between the main phantomand capillary tube, and structure of the artifacts. The artifacts appear asblurry curves, not the typical streaking artifacts seen in radial images.The intensity is low, but they are more apparent in the phase image. Therandomly sampled image has a stronger presence of image artifacts andappears more blurred than the other two. This is likely a result of largergaps near the centre of k-space that influence the image contrast.and Golden angle sampled images, but inferior with random sampling (likely dueto larger gaps in near the center of k-space where contrast information is stored).The phase of all images are similar. There are obvious curved artifacts originatingfrom the phantom in the phase image.A.2 Spatial-Temporal Constrained ReconstructionFigures A.5, A.6, and A.7 show the reconstructed images using 233, 144, 89, 55,34 or 21 projections, and using uniform, Golden angle or random sampling.210The STCR reconstructed images showed similarities to the reference image(233 projections) when at least 89 projections are used in the reconstruction foruniform and Golden angle sampling and 144 projections for random sampling. Asthe number of projections is reduced to 55 or fewer, the edges of the phantombecame blurry and contrast between the capillary tube and the main phantom de-grades. The small external phantom to the right is visible in all images. However,it starts to blend in with background image artifacts when 34 or 21 projections areused in the reconstruction for uniform and random sampling. The image set withGolden angle sampling has good visibility of the phantom down to 34 projections.All images have two circular artifacts, the intersect with the edge of the phan-tom. These artifacts have a low magnitude, and seem to only affect the background.As the number of projections is reduced to 55 or less, additional artifacts appear.These artifacts appear as elongated spots that radiate outward from the phantom.Their presence is most noticeable in the images with 34 or 21 projections. Thedata sets with Golden angle or random sampling also show artifacts within themain phantom. These appear to be streaking artifacts, which originate at the cap-illary tube, and are easily observed in the top-right (low intensity) and lower-left(high intensity).Figure A.8 shows the STCR magnitude and phase images reconstructed with 55projections. In general, the images with uniform or Golden angle sampling providesimilar quality, while the randomly sampled image has greater loss of contrastbetween the capillary tube and surrounding vessel and presence of artifacts in thebackground. The phase images are similar. All three sampling methods show radialstreaks from the main phantom. The phase of the smaller phantom differs betweenthe sampling method. However this is more likely a result of the imaging artifactswithin the background.A.3 Non-Equidistant Fast Fourier TransformRadial images reconstructed with the NFFT algorithm are shown in Figures A.9,A.10, and A.11. The reconstructions used 233, 144, 89, 55, 34 or 21 projections,and one of uniform, Golden angle or random sampling.NFFT images with at least 89 projections are visually similar with the reference211Figure A.5: STCR magnitude images reconstructed with 233 (reference),144, 89, 55, 34 or 21 uniformly spaced projections. Visually, the im-ages are comparable with the reference image when at least 89 pro-jections are used in the reconstruction. Signal contrast between thecapillary tube and the main phantom is reduced in all images, relativeto the reference image. However the most dramatic loss of contrast isobserved from 89 to 55 projections. Edge sharpness is good with 55or more projections, then degrades significantly with 34 or 21 projec-tions. The smaller external phantom, on the lower right side, is visiblein all images, though the contrast is best with at lesat 55 projections.The mSSIM index, relative to the reference STCR image, gradually de-creases from 0.841 with 144 projections to 0.708 fwith 21 projections.If the threshold for a clinical-quality image was set to 0.800, then theSTCR reconstruction requires at least 89 projections.212Figure A.6: STCR magnitude images reconstructed with 233 (reference),144, 89, 55, 34 or 21 projections and Golden angle sampling (withan angular increment of 111.246o). Visually, the images are compara-ble with the reference image when at least 89 projections are used inthe reconstruction. As with the uniformly sampled data, the contrastbetween the capillary tube and the rest of the phantom degrades as thenumber of projections drops from 89 to 55 and the edges around themain phantom become blurry. The smaller external phantom is visiblein all images, but is faint in the image with 21 projections. The mSSIMindex gradually reduces from 0.886 with 144 projections to 0.703 with21 projections. Images with 55 or more projections have an SSIM indexexceeding 0.800.213Figure A.7: STCR magnitude images reconstructed with 233 (reference),144, 89, 55, 34 or 21 randomly selected projections over 180o. Theimage with 144 projections is comparable with the reference image,but as the number of projections is reduced, image contrast with thecapillary tube and surrounding phantom is reduced and the edges of thephantom get blurry. Streaking artifacts are observed within the mainphantom with 55 and fewer projections. The external phantom to theright is visible in all images, but with less contrast when 55 or fwereprojections are used. The mSSIM index gradually reduces from 0.877with 144 projections to 0.637 with 21 projections. Only the imageswith 89 and 144 projections have a mSSIM exceeding 0.800, and it hasa sharp drop from 0.754 to 0.661 when the number of projections isreduced from 89 to 55.214Figure A.8: Magnitude images reconstructed using Spatial-Temporal Con-strained Reconstruction (STCR), 55 projections and uniform, Goldenangle (angular increment 111.246o) or random samlping. The uniformand Golden angle sampling schemes both produce an image of compa-rable quality (contrast, presence of artifacts). The randomly sampledimage is more blurred, lower contrast and has a greater presence of ar-tifacts. This likely results from the non-uniform sampling of k-space,resulting in larger gaps that contain image contrast and detail informa-tion.images when uniform or Golden angle sampling were used. This was assessed bysharpness of the phantom edges, relative signal contrast between the capillary tubeand the surrounding phantom, and visual appearance of image artifacts. Althoughthe signal contrast was slightly lower for the image with 89 projections, the restof the image seemed to compare well. This contrast was lost when 55 or fewerprojections were used in the reconstruction. The image quality is poor for theimages with 34 and 21 projections to the point that they are unusable. The edgesof the phantom were significantly blurred and streaking artifacts are seen in the215Figure A.9: NFFT magnitude images reconstructed with 233 (reference),144, 89, 55, 34 or 21 uniformly spaced projections over 180o. Im-ages with 89 and 144 projections are comparable with the referenceradial image, having a mSSIM index of 0.829 and 0.863, respectively.Reducing this number to 55 resulted in loss of contrast between the cap-illary tube and surrounding phantom. With 34 or 21 projections, thereis significant blurring of the phantom edges and significant loss of con-trast between the capillary tube and surrounding phantom. Streakingartifacts are visually apparent in the images with 34 or 21 proejctions.background. Random sampling was clearly inferior for this technique. The imagequality degraded rapidly as fewer projections were used in the reconstruction, withnone of the images resembling the reference NFFT image.The NFFT magnitude images, reconstructed with 55 projections and uniform,Golden angle or random sampling, are shown in Figure A.12. The uniformly sam-pled and Golden angle images are comparable visually, and are of much higherquality than the randomly sampled image. The phase of the image is consistentbetween the three sampling methods, though all three have streaking artifacts and216Figure A.10: NFFT magnitude images reconstructed with 233 (reference),144, 89, 55, 34 or 21 projections, and Golden angle sampling. Imagesreconstructed with at least 55 projections are visually comparable withthe reference image, though the contrast between the capillary tubeand the surronding phantom is lost with 55 projections. Streakingartifacts are observed in the background as early as 55 projections,though they are of low intensity. Images with 34 and 21 projectionare of low quality as the phantom edges are blurry, signal contrast ofthe capillary tube is lost and streaking artifacts are observed in thebackground. The images with at least 89 projections have a mSSIMvalue exceeding 0.800.217Figure A.11: NFFT magnitude images reconstructed with 233 (reference),144, 89, 55, 34 or 21 randomly selected projections over 180o. Thequality of the image degrades as fewer projections are used in thereconstruction. For this sampling method, no image resembles thereference NFFT image: the signal intensity is lower in all images,contrast of the capillary tube and surrounding phantom is lost and theedges of the phantom are blurry. The smaller phantom is observedin images with at least 34 projections, though the contrast with thebackground is low with 34 and 55 projections. Streaking artifacts arepresent in images with 89 and fewer projections, and can be seen inboth the phantom and background. The mSSIM index is 0.733 with144 projections and 0.684 with 89 projections. It drops significantly to0.400 with 55 projections, meaning that this image is not of sufficientquality.218Figure A.12: Radial images reconstructed with the NFFT, 55 projections anduniform, Golden angle (angular increment 111.246o) or random sam-pling. The images with uniform or Golden angle sampling are compa-rable, while the randomly sampled image appears blurry and suffersfrom image artifacts. The phase images look similar for all techniques,though the streaking artifacts in the background are more pronouncedwith random sampling. The phase of the smaller phantom is varied asa result of these artifacts.affect the phase of the smaller side phantom.219Appendix BRadial Acquisition CorrectionTechniquesB.1 Observed Issues with Radial SamplingTheoretically, all radial samples should pass through the center of k-space. This isnot guaranteed as magnetic field inhomogeneities, off-resonance effects, imperfectgradient profiles, scanner timing delays and residual eddy currents are all known tointroduce errors in data positioning. This section will address common issues withradial data sampling and discuss methods to minimize or compensate for them.A simple technique to correct for zeroth and first order phase issues is outlinedin the paper by Ahn and Cho [167]. Their technique is based on the statistical phaseproperties and distributions of the image and provides a more accurate representa-tion of the phase information. Examples include inversion recovery imaging, spec-troscopic imaging, phase-modulated velocity imaging and fast imaging techniquesthat make use of the conjugate symmetry of the FID. Since the projection-basedAIF measurement compared the background profile from an image to an individualprojection, the phase data must be preserved.Phase information may be distorted due to mis-adjustment of the referencephase, delays in the acquisition time, or be introduced by electronic filters. Thecorrection provided by Ahn and Cho [167] involves two parts: first, the first-order phase distortions are addressed, then the zeroth-order phase distortions are220removed. The correction is applied to the MR image, not to the k-space data.The first-order correction factor, ε1, is determined from the phase of the auto-correlation of the image, F(x,y), between neighboring pixels in the direction ofthe read-encode gradient. For instance, if the read-encode is along the x-direction,then the x-directional auto-correlation is used. The image data is then multipliedby the correction factor e−iε1x. The zeroth-order correction is determined from thepeak of the phase histogram of the first-order corrected image. The entire image ismultiplied by the exponential of the zeroth-order conjugate phase.B.1.1 Eddy CurrentsEddy currents are induced electric currents in the conducting structures of the MRscanner when changes in the amplitude of the gradient fields occur [189]. ByLenz’s law, the eddy currents work against the original current and disturb the gra-dient field experienced by the sample [190]. Though they are often more severein permanent magnets than in a superconducting magnet [189], significant artifactsand distortions can degrade the image quality at higher field strengths and whenstronger, rapidly changing gradients are used [191]. The magnitude of the eddycurrents decays in time, resulting in a temporally varying off-resonance effect.They are observed as streaking artifacts in the reconstructed image, originatingfrom the object.Most clinical and research scanners are equipped with active shielding coils,which are designed to minimize fringe fields [189], though further correction is of-ten required. An easily implemented improvement is to add gradient pre-emphasislobes to the gradient waveform. However, these only accommodate a small numberof time constants, so eddy currents can persist at short echo times [191]. Lineareddy current effects present as k-space trajectory distortions (often along the read-encode direction), while Bo eddy currents provide unwanted phase accumulation.Higher order terms and cross-talk have minimal effects on the image and are oftenignored [190].Compensation for eddy currents is essential in under-sampled, non-CartesianFLASH acquisitions at higher field strengths due to the rapidly changing stronggradients. Typical eddy current models represent the system response as a sum of221decaying exponential functions, and convolve this with the time-varying gradientwaveform. Ma et al. [189] suggested a model that compensates for both zeroth-order (spatially invariant) and first-order eddy currents. Their method calculatesthe time derivative of the phase of the echo after a known delay time (range 0-1 s).B.1.2 Gradient Timing Delay CorrectionProjection data is acquired as a series of radial spokes in k-space, consisting of Nrreadout points. Typically, the readout is symmetric, in which the center of k-spaceis acquired as the Nr/2 readout point if Nr is even, or (Nr± 1)/2 readout if Nr isodd. But if the gradient timing is miscalculated, such that the actual start time ofthe gradient readout is different from the requested time, the center of k-space willbe shifted and thus changing the effective echo time. The shift can be differentbetween the three gradient channels, resulting in an angular-dependent shift. Thiswould cause a blurring of the center of k-space and blurring in the reconstructedimage if not corrected.Peters et al. [178] introduced a one-time calibration to apply to future radialscans. Their technique involves acquiring equally spaced radial projections over180o on a homogeneous spherical phantom placed at the scanner isocentre. Thedelays can be determined from the relative shift of the maximum from the expectedlocation. Peters achieved this by applying a Fourier transform to each projection,then determining the shift with the procedure outlined by Ahn and Cho [167]. Theyassume that there are no other sources of linear phase shift due to the object or theimaging procedure. The shift is then converted into a time delay.Peters suggests fitting a sinusoidal curve to the delay vs angle plot. The dataat 0o/180o and 90o will relay information about the two gradient channels. Bycomparing the results from two slice orientations, the delay time may be verified.From that information, the compensatory gradient area can be calculated. This areais then added to the pre and rephasing gradient areas. The compensation areas onlyneed to be measured once for a gradient set. For future use, the areas for each anglecould be placed in a look up table.Peters measured the timing delays in phantom to be as great as 5 µs. However,after adding the compensatory gradient areas, the time delay was reduced to less222than 1 µs, and image artifacts were greatly reduced in both phantom and volunteerdata.B.1.3 Trajectory MeasurementsThe trajectory of MR data is sensitive to imperfections in the gradient amplifierperformance, readout timing and eddy currents induced by the gradient pulse [192,193]. In their presence, the spatial locations of the data points are not consistentwith expectations, which leads to image rotation and artifacts [194]. As scanner de-mands (stronger gradients and faster acquisition times) increase and the trajectorybecomes more complicated - such as spiral imaging - knowing the exact locationwhere the data was collected is essential.Several techniques have been presented to measure the k-space trajectory. Gen-erally, the pulse sequence is altered such that slice selection is in the same directionas the read encode gradient, and often limits the measurement to a single physicalgradient channel at a time for accuracy (to avoid data sampling in areas with lit-tle signal). For instance, if measuring the trajectory along the physical x-gradient,the slice and read encode gradients are both oriented in that direction. This re-quirement limits slice selection to coronal, sagittal or axial orientations. Only onegradient channel is on during the acquisition, which occurs while the read encodegradient is played out. Once the trajectory is known, the information may be usedin the image reconstruction to reduce artifacts (blurring, rotation, intensity varia-tions, etc.).The next two sections will discuss the trajectory measurements introduced byZhang et al., Beaumont et al. and Latta et al.Trajectory Measurement using Phase Differences from Adjacent ImagesZhang et al. proposed a technique that determines the actual k-space trajectoryusing the phase difference between acquired MR signals of adjacent slices alongthe gradient axis of interest. A homogeneous spherical object should be used forthis measurement, as spatially varying susceptibility induced magnetic field inho-mogeneities could introduce errors in the measurement. The results of their studyshowed that the trajectory could be accurately determined and provided significant223improvements in the reconstructed image.Beaumont et al. [194] provide an extension to the technique introduced byZhang et al. [192]. They identified a limitation at high k-space values where thesignal is close to the noise floor. This caused unreliable data for measuring thetrajectory. Beaumont suggested acquiring three trajectories measurements, withtwo offset by a known amount, and averaging them. The pulse sequence is similarto that from Zhang, but includes an added gradient area along the gradient channelof interest prior to the acquisition window. This, in effect, forces the center of k-space to be shifted such that the maximum can cover an area where a signal nullwas present.Most slice profiles are symmetric, so the the critical points where a signal nulloccurs will be symmetric about the center. Beaumont’s technique involves the ac-quisition of three trajectories; one with no shift (0), and two offset measurements(+G and -G). Their results showed artifacts in the trajectory measurements at thenulls when the added gradient was not applied. But after averaging the three sig-nals (-G, 0 and +G), the trajectory was much cleaner and did not present the sameartifacts. Images without correction appeared to be rotated by a few degrees rela-tive to those corrected with the trajectory. They predicted that this was due to anuncontrolled delay between the gradient-waveform and data acquisition. They alsofound that the trajectories appeared compressed slightly, likely due to a mis-settingof the gradient calibration setting or eddy currents.Trajectory Measurement Proposed by Latta el alAn alternative approach is to measure the trajectory with a spin-echo, while utiliz-ing a phase-encode gradient [193]. Their method maps the point-by-point k-spacetrajectory of the examined gradient waveform (X, Y or Z). Slice-selection is doneparallel to the measured gradient axis. After slice-excitation, the phase-encodinggradient, GPE , is applied to introduce spin de-phasing along the chosen gradientdirection prior to the measurement. The phase encoding step is performed in sucha way that the measured k-space trajectory crosses the origin at different times, asobserved through the echo position. The measurement is completed by turning onthe desired gradient waveform, GW , simultaneously with the acquisition window.224The trajectory is reconstructed from a plot of the the applied pre-phased gradientarea, GPE , and the estimated crossing of the k-space origin. This technique is op-timized for short readout gradients, such as radial acquisition. Longer sequences(spiral imaging) would have limited scope.Latta suggests using an adaptive phase encoding gradient such that the trajec-tory has uniform spacing. He argues that the data along the ramp-up and ramp-down sections of the GW waveform are under-sampled if the phase encoding stepsare all equally spaced in area. The adaptive phase encoding gradient is outlinedin their paper. In addition, they used a variable echo time to ensure that all off-resonance effects were compensated for by adjusting the timing between the exci-tation pulse and the refocusing RF pulse.Latta tested their method on phantom and healthy volunteers. The trajectorymeasurements were successfully applied to reconstruct a radial 2-D ultra-shortecho time (UTE) image with significantly improved quality. Prior to correction,the UTE images showed ghosting artifacts. The images were further improved withgradient delay and linear eddy current parametric model corrections. These wereapplied to compensate for missing data at the gradient onset and also reduce thenumber of measurements required to perform the correction. The technique wasalso shown to work well on off-center images of a patient knee offset by 70 mmfrom the scanners isocentre.B.2 Attempts at Correcting the Acquired Radial DataThe radial projection-based AIF measured at the end of Chapter 6 showed an angu-lar dependence on the measured concentration of Gd-DTPA. The organization ofthis Appendix follows the order in which the correction attempts were made. Post-processing corrections were done first, in hopes of reducing the phase baseline to apoint that the AIF may be used for modeling. The focus then shifted to improvingthe quality of the acquired data.B.2.1 Post-processing k-Space to Center EchoHaving identified the angular dependent artifact, we tested three echo correctiontechniques. The first two involved a global shift of all projections by either an in-225teger or a sub-pixel amount. The global integer shift involves finding the pixel atwhich the greatest signal intensity was achieved, and calculating the mean positionacross all angles. Next, all of the projections were shifted by round(Nreadenc/2)−Npixelo f echo. For the sub-pixel shift, the data was up-sampled by a factor of 10,and the position of the echo was determined from the maximum signal intensity.The projections were globally shifted to center the data in k-space, before down-sampling back to its original size. The third technique redefined the locations of thenodes in the NFFT reconstruction so that the maximum in projections k-space co-incided with the center of k-space for the image. To accomplish this, we calculatethe difference in the expected and actual positions of the echo for each projectionand then update the NFFT node positions with that information. Finally, the phaseof the echo was reset to 0 radians. The resulting vessel signal and AIFs are sum-marized in Figure B.1 before and after applying the three correction techniques.The phase baseline of the corrected AIF varies between the three correctiontechniques, but the amplitude of the phase fluctuations are significantly reduced.These results strongly suggest that a k-space correction should be applied to the k-space data prior to proceeding with the projection-based measurement. The smallfluctuations that persist could be a consequence of imperfect centering of the echo.Quantitatively, the phase of the uncorrected AIF covers the range -3.069 to -0.176 rad. It also appears to have a rapid phase jump, which is a consequence ofvessels spatial location within the projection. Applying either the global integer orsub-pixel shift significantly reduced the phase fluctuations (range -1.652 to -1.010rad and -1.576 to - 1.133 rad, respectively). The baseline phase still has the stepin phase, but the difference is a factor of 0.222 and 0.153 of that of the uncor-rected curve. Adjusting the positions of the NFFT nodes significantly improvedthe quality of the baseline as well. However, the phase fluctuations were greaterwith a range from -1.813 to -0.972 rad. From this analysis, it appears that the sub-pixel shift provides the best results, followed by the global integer shift. The phasefluctuations are not desirable, and should be further reduced if possible.The NFFT images from the above analysis are compared in Figure B.2 to gain abetter understanding of what causes the angular phase dependence. The magnitudeimage of the uncorrected data has a hot spot at the center of the phantom, which isnot as dominant after applying a centering correction. Since the background data226Figure B.1: Magnitude and Phase of the vessel data (acquired projections -background), and the average signal phase within the vessel (sorted byangle), for four situations: 1) no pre-processing of the k-space dataof the projections, 2) applying a global integer pixel shift, 3) apply-ing a global sub-integer shift, and 4) adjusting the NFFT node posi-tions based on the expected echo position. The results show that theacquired k-space is not acquired as expected, and requires some post-processing before carrying out the projection based AIF measurement.Of the techniques investigated, a global pixel shift (either sub-pixel orinteger value) performs determined from the summation of many complex signals, regions with highersignal intensity contribute more strongly to the summation. As a result, the phaseof the background projection could be artificially biased towards the phase of thehot spot signal. The hot spot shifted towards the bottom of the phantom when aglobal shift (integer or sub-pixel) was applied to the data. This would affect thebackground signal differently at each angle as the strongest signal is off-center inimage-space. Since the vessel signal is a difference between the acquired data andthe background signal, this angular signal difference could carry a bias through theremainder of the AIF estimation. The NFFT images from the two shifted caseswere similar, which would explain the similar background measurements. Whenthe NFFT nodes were adjusted, the hot spot shrunk and shifted to the lower right-hand side of the phantom. The superior region of the phantom has hypo-intense227Figure B.2: Magnitude and phase of the reconstructed NFFT image of acylindrical phantom. The first column shows the reconstructed imageswhen the raw projection data is used. The remaining three columns cor-respond to one of the following corrections: global integer pixel shift,global sub-pixel shift, or adjusting the locations of the NFFT nodes.Streaking artifacts are observed in all cases, though they are least im-pactful when the global pixel shift is applied (integer or sub-pixel).signal relative to the other three cases. This again will impact the AIF measurementin an angular dependent manner.The second row shows the phase of the NFFT images. Within the phantom,the phase is similar in all cases, although there are slight differences along theperimeter of the phantom. In general, the phase has three hot-spots: two on the left-hand side and a smaller on the lower right-hand side. The phase of the uncorrecteddata set shows a distinct vertical streak. Similarly, the correction with adjustmentsto the NFFT nodes has obvious streaks in the background. These originate fromthe phase hot spots in the phantom. Even though the signal intensity is lower in thebackground, these artifacts could have an impact on the background summationas phase of the complex signal of these voxels all point in the same direction.The phase of the background signal would again have an angular dependence asthe summation will have varying amounts of vectors in the ’hot phase’ and ’coldphase’ setting.228Figure B.3: Effects on the baseline phase after centering the k-space datawith one of two techniques. a) shows the k-space sinogram for thecorrection with a global phase shift and setting the phase of the echoto 0 rad, with the corresponding phase baseline in black (c). b) is thek-space sinogram for the correction following the methods of Ahn andCho, which results in the magenta curve. The k-space sinograms havedistinct differences in appearance, but provide a similar phase baseline.For this experiment, the global phase shift performs slightly better thanthe first order phase shift.Ahn and Cho [167] outline a technique that corrects radial data for zeroth andfirst order phase errors. The effectiveness of the first-order phase shift had conflict-ing results with our data. In some cases, it significantly reduced the phase baseline,while in others it had a limited effect. This illustrates an instability in the radialdata acquisition that needs to be addressed in future studies. Figures B.3 and B.4show data from two experiments performed on different days.Figure B.3 compares k-space and the baseline phase of the projection-basedAIF after centering the data with two techniques. Figure B.3a represents the k-space sinogram after applying a global integer shift to better center the k-spacedata, and then a zeroth order phase correction to set the phase of the echo to 0 rad.229Figure B.4: Effects on the baseline phase after centering the k-space datawith one of two techniques. a) shows the k-space sinogram for thecorrection with a global phase shift and setting the phase of the echoto 0 rad, with the corresponding phase baseline in black (c). b) is thek-space sinogram for the correction following the methods of Ahn andCho, which results in the magenta curve. For this data set, the first or-der phase shift significantly improves the phase-baseline, reducing therange of phase from 1.21 to 0.181 rad.The sinogram has an oscillating pattern with an amplitude of 2 pixels. The resultingphase baseline covers a range of 1.415 to 2.338 rad. Figure B.3b shows the k-space data after a first (shift determined from image-space data) and zeroth orderphase correction [167]. The k-space sinogram no longer had a sinusoidal pattern,though the edges appeared jagged and the amplitude of the echo varied with angle.When put into the NFFT reconstruction, this would put more emphasis on someprojections over others. The phase baseline was noisier than with the global shift,and covered a larger range of 1.180 to 2.349 rad.The data set used in Figure B.4 was significantly improved with the first orderphase correction. Similar to the previous example, the k-space echo was centeredwith either a global phase shift and zeroth order phase correction, or following themethods of Ahn and Cho. The k-space sinogram in Figure B.4a again has an os-230cillating pattern with an amplitude of 2 pixels. The resulting phase baseline (blackcurve) covers a range of 1.21 radians (from -1.217 to 1.208 radians). The k-spacesinogram in Figure B.4b still has an oscillating pattern, but with