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Three dimensional ultrasound elasticity imaging Abeysekera, Jeffrey Michael 2016

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Three Dimensional Ultrasound Elasticity ImagingbyJeffrey Michael AbeysekeraB.A.Sc., The University of British Columbia, 2008M.A.Sc., The University of British Columbia, 2010a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Mechanical Engineering)The University of British Columbia(Vancouver)April 2016c© Jeffrey Michael Abeysekera, 2016AbstractChanges in tissue elasticity are correlated with certain pathological changes, suchas localized stiffening of malignant tumours or diffuse stiffening of liver fibrosis orplacenta dysfunction. Elastography is a field of medical imaging that characterizesthe mechanical properties of tissue, such as elasticity and viscosity. The elastog-raphy process involves deforming the tissue, measuring the tissue motion using animaging technique such as ultrasound or magnetic resonance imaging (mri), andsolving the equations of motion. Ultrasound is well suited for elastography, how-ever, it presents challenges such as anisotropic measurement accuracy and provid-ing two dimensional (2d) measurements rather than three dimensional (3d). Thisthesis focuses on overcoming some of these limitations by improving upon meth-ods of imaging absolute elasticity using 3d ultrasound. In this thesis, techniquesare developed for 3d ultrasound acquired from transducers fitted with a motor tosweep the image plane, however many of the techniques can be applied to otherforms of 3d acquisition such as matrix arrays. First, a flexible framework for 3dultrasound elastography system is developed. The system allows for comparisonand in depth analysis of errors in current state of the art 3d ultrasound shear waveabsolute vibro-elastography (swave). The swave system is then used to measurethe viscoelastic properties of placentas, which could be clinically valuable in diag-nosing preeclampsia and fetal growth restriction. A novel 3d ultrasound calibrationtechnique is developed which estimates the transducer motor parameters for accu-rate determination of location and orientation of every data sample, as well as forenabling position tracking of a 3d ultrasound transducer so multiple volumes can becombined. Another calibration technique using assumedmotor parameters is devel-oped, and an improvement to an existing N-wire method is presented. The swaveiiresearch system is extended to measure shear wave motion vectors with a new ac-quisition scheme to create synchronous volumes of ultrasound data. Regularizationbased on tissue incompressibility is used to reduce noise in the motion measure-ments. Lastly, multiple ultrasound volumes from different angles are combined formeasurement of the full motion vector, and demonstrating accurate reconstructionsof elasticity are feasible using the techniques developed in this thesis.iiiPrefaceThis thesis is primarily based on several manuscripts and conference proceedings,resulting from collaboration of multiple researchers. All publications have beenmodified to make the thesis coherent.Thework in Chapter 2was the result of the collaboration ofmultiple researchers.The author’s contribution was designing and building the ultrasound motor con-troller, helping determine the program structure, developing the ultrasound acquisi-tion and GUI modules, evaluating the phasor measurement accuracy and sensitivitythrough simulation, and writing the chapter. Julio Lobo helped determine the pro-gram structure, tested different experimental configurations, and collected the ex-perimental repeatability measurements. Dr. Hani Eskandari helped determine theprogram structure and helped developed the software. Dr. Reza Zahiri and WeiqiWang were the primary developers of the radio frequency (rf) motion tracking C++and CUDA code. Dr. Ali Baghani and Nabil Lathiff were the primary developersof the phasor fitting and local frequency estimation (lfe) C++ and CUDA code.Dr. Robert Rohling and Dr. Tim Salcudean developed the main research ideas be-hind the project and provided helpful guidance.The work in Chapter 3 included a study of human placenta (H15-00974) per-formed under written informed consent (see Appendix B) after approval by the UBCChildren’s and Women’s Research Ethics Board. The research was the result of thecollaboration of multiple researchers. The author’s contribution was in helping de-sign the experimental protocol, helping collect experimental data, processing andanalysing the data, and writing the chapter. Mehran Pesteie helped design the exper-imental protocol and helped collect experimental data. Manyou Ma helped collectexperimental data. Ashton Ellis and Tamsin Tarling obtained consent from the pa-ivtients. Dr. Denise Pugash provided radiology support at BC Women’s Hospital.Support for the elastography system was led by Tim Salcudean and Julio Lobo.Dr. Jeff Terry coordinated placenta handling at BCWomen’s Hospital and providedinformation on placenta anatomy and pathology. Dr. Robert Rohling developed themain research ideas behind the project, helped design the experimental protocol,and provided useful suggestions.A version of Chapter 4 has been published inJ. M. Abeysekera, M. Najafi, R. Rohling, and S. E. Salcudean, “Cal-ibration for position tracking of swept motor 3-D ultrasound”, Ultra-sound in medicine and biology, vol. 40, no. 6, pp. 1356–1371, 2014The author’s contribution in that paper was developing the idea, implementing themethods, verifying the methods through numerical simulations and experiments,and writing the manuscript. Dr. Mohammad Najafi provided the code for his previ-ously published two dimensional (2d) calibration algorithm and the computer-aideddesign (cad) file for the calibration phantom. Dr. Robert Rohling and Dr. Tim Sal-cudean assistedwith their suggestions and contributed toward editing themanuscript.A portion of Chapter 5 was orally presented and published in the conferenceproceedings inJ. Abeysekera, R. Rohling, and S. Salcudean, “Vibro-elastography:Absolute elasticity from motorized 3D ultrasound measurements ofharmonic motion vectors”, in IEEE International Ultrasonics Sympo-sium, 2015, pp. 1–4In addition, another portion of Chapter 5 has also been submitted for publication asJ. M. Abeysekera, M. Honarvar, R. Rohling, and S. E. Salcudean, “3Dultrasound shear wave absolute vibro-elastography (SWAVE) from fullvector motion field measurements”, submittedThe author’s contribution was developing the idea, implementing the methods, ver-ifying the methods through experiments, and writing the manuscript. Dr. Moham-mad Honarvar reprocessed the experimental measurements using his own inversionvalgorithm. Dr. Robert Rohling and Dr. Tim Salcudean assisted with their sugges-tions and contributed toward editing the manuscript.A portion of Chapter 6 was orally presented and included in the conferenceproceedings inJ. M. Abeysekera, M. Honarvar, S. E. Salcudean, and R. Rohling,“Combining axial meaures to estimate 3D motion over an ultrasoundvolume”, in Twelfth International Tissue Elasticity Conference, 2013,p. 101The author’s contribution was developing the idea, implementing the methods, ver-ifying the methods through simulation, and writing the manuscript. Dr. Moham-mad Honarvar processed the simulation results using his own inversion algorithm.Dr. Robert Rohling and Dr. Tim Salcudean assisted with their suggestions and con-tributed toward editing the manuscript.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Ultrasound Background . . . . . . . . . . . . . . . . . . . . . . . 81.4.1 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.2 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.3 Transducers . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.4 Beamforming . . . . . . . . . . . . . . . . . . . . . . . . 131.4.5 Image Formation . . . . . . . . . . . . . . . . . . . . . . 181.4.6 Doppler . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.5 Elastography Background . . . . . . . . . . . . . . . . . . . . . . 29vii1.5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.5.2 Additional Modelling Considerations . . . . . . . . . . . 331.5.3 Motion Measurement . . . . . . . . . . . . . . . . . . . . 351.5.4 Elasticity Estimation . . . . . . . . . . . . . . . . . . . . 421.6 Ultrasound Spatial Calibration Background . . . . . . . . . . . . 591.6.1 Calibration Methods . . . . . . . . . . . . . . . . . . . . 601.6.2 Position Tracking . . . . . . . . . . . . . . . . . . . . . . 632 Development and Analysis of a 3D System for Absolute ElasticityMeasurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.2.1 System Overview . . . . . . . . . . . . . . . . . . . . . . 722.2.2 Software Design . . . . . . . . . . . . . . . . . . . . . . 742.2.3 Ultrasound Interface . . . . . . . . . . . . . . . . . . . . 742.2.4 Motor Control . . . . . . . . . . . . . . . . . . . . . . . 762.2.5 Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . 782.2.6 Sampling Sequences . . . . . . . . . . . . . . . . . . . . 792.2.7 Motion Estimation . . . . . . . . . . . . . . . . . . . . . 812.2.8 Phasor Fitting . . . . . . . . . . . . . . . . . . . . . . . . 812.2.9 Phasor Compensation . . . . . . . . . . . . . . . . . . . 832.2.10 Synchronization . . . . . . . . . . . . . . . . . . . . . . 852.2.11 Elasticity Estimation . . . . . . . . . . . . . . . . . . . . 862.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.4.1 Motor Attenuation . . . . . . . . . . . . . . . . . . . . . 912.4.2 Phasor Synchronization . . . . . . . . . . . . . . . . . . 922.4.3 Elasticity Repeatability . . . . . . . . . . . . . . . . . . . 932.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.5.1 Phasor Fitting Results . . . . . . . . . . . . . . . . . . . 952.5.2 Motor Attenuation Results . . . . . . . . . . . . . . . . . 1002.5.3 Phasor Synchronization Results . . . . . . . . . . . . . . 1002.5.4 Elasticity Repeatability Results . . . . . . . . . . . . . . . 107viii2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123 Viscoelastic Characterization ofExVivoPlacenta Tissue Using SWAVE1143.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1143.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.2.1 Rheological Modelling . . . . . . . . . . . . . . . . . . . 1193.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1233.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344 Spatial Calibration of Swept 3D Ultrasound . . . . . . . . . . . . . . 1364.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.2.1 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . 1414.2.2 N-wire Calibration (N2D, N3D, and NFull3D ) . . . . . . . . . 1434.2.3 Planar Calibration (P3D) . . . . . . . . . . . . . . . . . . 1474.2.4 Wedge Calibration on 2D Slices and then Fitting (W2D→3D) 1514.2.5 Validation of the Calibration Methods . . . . . . . . . . . 1544.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1564.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615 Measurement of the Full Shear Wave Motion Vector in SWAVE . . . 1635.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1635.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1695.2.1 Synchronization . . . . . . . . . . . . . . . . . . . . . . 1695.2.2 Displacement Estimation . . . . . . . . . . . . . . . . . . 1695.2.3 Displacement Regularization . . . . . . . . . . . . . . . . 1725.2.4 Elasticity Reconstruction from lfe . . . . . . . . . . . . . 1745.2.5 Elasticity Reconstruction from shear-FEM . . . . . . . . . 1755.2.6 Phantom Experiment . . . . . . . . . . . . . . . . . . . . 1765.2.7 Reconstruction Performance Evaluation . . . . . . . . . . 1795.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180ix5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1885.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956 Combining Multiple Views for Increased Accuracy of the Full ShearWave Motion Vector in SWAVE . . . . . . . . . . . . . . . . . . . . . 1976.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1976.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1996.2.1 Finite Element Simulation . . . . . . . . . . . . . . . . . 1996.2.2 Simulating Displacement Measurements . . . . . . . . . . 2006.2.3 Solving for 3D Displacement . . . . . . . . . . . . . . . . 2046.2.4 Simulating Calibration Error . . . . . . . . . . . . . . . . 2066.2.5 Elasticity Inversion . . . . . . . . . . . . . . . . . . . . . 2086.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2106.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2116.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2167 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2177.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2177.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2207.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2227.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226A Index Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264B Placenta Study Consent Form . . . . . . . . . . . . . . . . . . . . . 266C N-wire Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273D Logarithm and Exponential Maps of Dual Quaternions . . . . . . . 277E Solving for Geodesic Direction on the Quaternion Manifold . . . . . 279F Simulation Test of Rotation Fitting Algorithm . . . . . . . . . . . . 284xG Linear Least Squares Displacement Gradient . . . . . . . . . . . . . 287xiList of TablesTable 1.1 Commercially available ultrasound elastography systems. . . . 3Table 1.2 Technical specifications of common position tracking systems. 66Table 3.1 Rheological model parameters and root mean square (rms) errorcorresponding to the fits to the shear wave speed dispersion forsix placenta samples . . . . . . . . . . . . . . . . . . . . . . . 130Table 4.1 Standard deviations of the three translation degree(s) of freedom(dof) (mm) and three rotation dof (degrees) after 10 trials usinga single data set for calibration. . . . . . . . . . . . . . . . . . 157Table 5.1 Manufacturer specifications for the CIRS Model 049 elasticityquality assurance phantom. The background and measured in-clusion (stiffest of four inclusions) are tabulated. . . . . . . . . 177Table 5.2 Performancemeasures of the reconstructed elasticity values basedon the measured and regularized displacements. . . . . . . . . 187Table 6.1 Parameter values used for simulating jitter error in the displace-ment measurements with Equation 6.2. . . . . . . . . . . . . . 204xiiList of FiguresFigure 1.1 A block diagram of an ultrasound system. . . . . . . . . . . . 9Figure 1.2 The components of an ultrasound transducer. . . . . . . . . . 12Figure 1.3 The layout of a linear array and the array coordinates. . . . . . 13Figure 1.4 Ultrasound transmit focusing is achieved using variable timedelays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 1.5 Ultrasound transmit focusing is achieved using variable timedelays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 1.6 Example of apodization weighting functions applied to the el-ements of an array. . . . . . . . . . . . . . . . . . . . . . . . 16Figure 1.7 Simulated response from a point scatterer located at the trans-ducer’s focus for the different apodization weightings shown inFigure 1.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Figure 1.8 Block diagram of in-phase and quadrature (i/q) processing. . . 18Figure 1.9 The i/q demodulation process. . . . . . . . . . . . . . . . . . 19Figure 1.10 Example of an rf signal and its corresponding envelope. . . . 20Figure 1.11 Example of blood flow through the carotid artery as depictedby Pulsed Wave Doppler. . . . . . . . . . . . . . . . . . . . . 23Figure 1.12 Example images of blood flow through the carotid artery asdepicted by Colour Flow and Power Doppler imaging modes. . 25Figure 1.13 Example of an rf signal before and after tissue compression. . 35Figure 1.14 Triangulation configurations for determining three dimensional(3d) marker positions with optical tracking systems. . . . . . . 64Figure 1.15 Markers for position tracking systems. . . . . . . . . . . . . . 65xiiiFigure 2.1 Block diagram of the elasticity measurement system. . . . . . 73Figure 2.2 A photograph of the motor control box connected to the Ultra-sonix scanner and a motorized ultrasound transducer. . . . . . 77Figure 2.3 Flowchart describing the synchronization between the ultra-sound acquisition and the excitation in the shear wave absolutevibro-elastography (swave) system. . . . . . . . . . . . . . . 85Figure 2.4 Simulated excitation signals for analysis of phasor fitting. . . . 89Figure 2.5 Front view of the CIRS 049 elastography quality assurance phan-tom with the experimental scanning regions indicated. . . . . 93Figure 2.6 Diagram of the different exciter locations used to test the re-peatability of the elasticitymeasurements as viewed from abovethe phantom. . . . . . . . . . . . . . . . . . . . . . . . . . . 94Figure 2.7 Example phasor fitting result to simulated noisy displacementmeasurements. . . . . . . . . . . . . . . . . . . . . . . . . . 95Figure 2.8 Error in the least squares fit of a simulated 200 Hz cosine exci-tation at 10 µm amplitude with varying number of displacementsamples, displacement jitter error, and frame rate error. . . . . 96Figure 2.9 A demonstration of a phasor wave artefact observed in exper-imental bandpass sampled phasor images. . . . . . . . . . . . 98Figure 2.10 A profile across the width of an image containing the phasorwave artefact. . . . . . . . . . . . . . . . . . . . . . . . . . . 99Figure 2.11 Measurements comparing the ultrasound signal strength withand without the motor driver between the transducer and thescanner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Figure 2.12 Results of phase compensation measured with the Porta andTexo ultrasound interfaces. . . . . . . . . . . . . . . . . . . . 102Figure 2.13 Results of phase compensation measured with the BK ultra-sound interface. . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure 2.14 Estimated amplitude, noise, and time shifts to minimize thesquared error between the first phasor image collected and eachsubsequent compensated phasor image. . . . . . . . . . . . . 103xivFigure 2.15 Results of phase compensation using a trigger to synchronizethe start of each phasor image, using the Texo ultrasound inter-face, with the excitation. . . . . . . . . . . . . . . . . . . . . 104Figure 2.16 Example elasticity images of the soft and stiff inclusions mea-sured in the CIRS 049 elastography quality assurance phantom. 105Figure 2.17 Repeated measurements of the mean elasticity in a stiff and softinclusion, in a moderately stiff background with different mo-tion sampling techniques. . . . . . . . . . . . . . . . . . . . . 106Figure 2.18 Repeated measurements of the mean elasticity in a stiff and softinclusion, in a moderately stiff background with different ex-citer positions. . . . . . . . . . . . . . . . . . . . . . . . . . 108Figure 3.1 The experimental apparatus for placenta swave measurements. 117Figure 3.2 A mechanical schematic of the (a) Voigt, (b) Maxwell, and (c)Zener rheological models. . . . . . . . . . . . . . . . . . . . 121Figure 3.3 The 2d image plane from the centre of the volumetric ultra-sound sweep for placenta Sample 5. . . . . . . . . . . . . . . 124Figure 3.4 Mean Young’s modulus measurements as a function of fre-quency for the six placenta samples. . . . . . . . . . . . . . . 125Figure 3.5 The mean amplitude of the measured phasor for each placentasample over all excitation frequencies in the region where themean elasticity was calculated. . . . . . . . . . . . . . . . . . 126Figure 3.6 The shear wave speed dispersion relations found using theVoigt,Maxwell, and Zener models for each of the placenta samples. . 128Figure 3.7 The brightness mode (b-mode) and Young’s modulus imageplanes from the centre of the volumetric ultrasound sweep fromthree different regions of placenta Sample 4. . . . . . . . . . . 129Figure 3.8 Mean Young’s modulus measurements as a function of fre-quency for placenta Sample 4, repeated at three separate lo-cations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Figure 3.9 The range ofmeanYoung’smoduli reported in the elastographyliterature for normal placenta. . . . . . . . . . . . . . . . . . 132xvFigure 4.1 The coordinate systems used in the calibration procedure. . . . 140Figure 4.2 The 3d transducer, encased in a holder with infrared light emit-ting diode (ired) markers. . . . . . . . . . . . . . . . . . . . 142Figure 4.3 The phantom used for the planar and wedge calibration methods.143Figure 4.4 The top view of one row of the N-wire phantom and the inter-secting scan plane. . . . . . . . . . . . . . . . . . . . . . . . 144Figure 4.5 A 3D view of the three N-wire rows. . . . . . . . . . . . . . . 145Figure 4.6 A top view of the sameN-wire phantom and scanning geometryas presented in Figure 4.5. . . . . . . . . . . . . . . . . . . . 146Figure 4.7 A rendering of the phantom used for the planar and wedge cal-ibration methods. . . . . . . . . . . . . . . . . . . . . . . . . 148Figure 4.8 The translation and rotation parameter values plotted for tenindependent tests depicting the variation in the solutions. . . . 157Figure 4.9 Point reconstruction error for the calibration techniques. . . . 158Figure 5.1 A schematic of the synchronization between themechanical ex-citation and the volumetric ultrasound acquisition. At a givenmotor position, each line in the 2d image plane is triggered se-quentially at a common excitation phase which is then repeatedfor a number of phase offsets. After all phase offsets are col-lected, the motor is stepped to the next position and the processis repeated. Once the collection is finished for all motor posi-tions, the data are reorganized into volumes that were triggeredat a common phase. . . . . . . . . . . . . . . . . . . . . . . . 170Figure 5.2 Flowchart of the synchronization between the ultrasound ac-quisition of each beam line and the excitation signal at multiplephase offsets (extended from Figure 2.3). . . . . . . . . . . . 176Figure 5.3 A b-mode image generated from the envelope of one of the rfimages. The hypoechoic circle in the bottom left shows a crosssection of the stiff inclusion. . . . . . . . . . . . . . . . . . . 177xviFigure 5.4 From left to right, slices through the volume of the displace-ments measured in the x, y, and z directions. (a)–(c) The dis-placements estimated using speckle tracking alone, and (d)–(f)the displacements after post-processing the displacement mea-surements with divergence regularization. The x, y, and z di-rections roughly correspond to the lateral, axial, and elevationaltransducer coordinates. . . . . . . . . . . . . . . . . . . . . . 181Figure 5.5 The estimated Young’s modulus volumes using the lfe andshear-FEM inversion algorithms. . . . . . . . . . . . . . . . . 182Figure 5.6 Profiles of the reconstructed elasticity values passing throughthe centre of the inclusion in the x, y, and z-directions. . . . . 183Figure 5.7 Boxplots describing the distribution of Young’s modulus val-ues in the inclusion and in the background. . . . . . . . . . . 185Figure 5.8 Plot of the real part of the displacement component along thex-direction, across the x-dimension of the volume, for differentvalues of the regularization parameter, α. The balance betweenthe divergence constraint and the measurement fit is plotted forthe same values of α. . . . . . . . . . . . . . . . . . . . . . . 186Figure 6.1 The view geometry is described as a rotation about an axis pass-ing through the vertex of a tetrahedron and perpendicular to theopposite face of the tetrahedron. . . . . . . . . . . . . . . . . 201Figure 6.2 Geometry of the mutiple view angles with respect to the phantom.202Figure 6.3 The ultrasound axialmeasurement direction over the entire sweptvolume with an outline of the simulated phantom provided forreference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203Figure 6.4 The real part of the complex displacement phasors over the vol-ume shown in three orthogonal cross-sectional slices. . . . . . 211Figure 6.5 The estimated elasticity and difference in elasticity from groundtruth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212Figure 6.6 The ultrasound axialmeasurement direction over the entire sweptvolume with an outline of the simulated phantom provided forreference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213xviiFigure C.1 By observation of the inter-point distances for one N-wire ata time, the transducer orientation can be constrained to rotateabout an axis. . . . . . . . . . . . . . . . . . . . . . . . . . . 274Figure C.2 By observation of the inter-point distances for two N-wires ata time, the transducer orientation can be fully constrained. . . 275Figure C.3 Observing the inter-point distances for all three N-wires at thesame time allows for only one coplanar solution. . . . . . . . 276Figure F.1 Plot of the mean and standard deviation of the best fit geodesicerrormetric after taking the natural logarithm of the set to trans-form the distribution closer to normal. . . . . . . . . . . . . . 285Figure F.2 A visual demonstration of fitting the rotation samples with axesindicating the rotation orientation. . . . . . . . . . . . . . . . 286xviiiGlossary1d one dimensional2d two dimensional3d three dimensionaladc analog-to-digital converterapi application programming interfacearf acoustic radiation forceb-mode ultrasound brightness mode imagebi-rads Breast Imaging Reporting and Data System, a tool for breast cancer as-sessment and categorizationcad computer-aided design, used for manufacturingcnr contrast-to-noise ratioct computed tomographycte contrast-transfer efficiencydac digital-to-analog converterdct discrete cosine transformdof degree(s) of freedomxixfem finite element methodfft fast Fourier transformgui graphical user interfacei/q in-phase and quadrature, baseband ultrasound signalsired infrared light emitting diodeiugr intrauterine growth restrictionlfe local frequency estimationmri magnetic resonance imagingmre magnetic resonance imaging elastographyoct optical coherence tomographypca principal component analysispet positron emission tomographypga principal geodesic analysisprf pulse repetition frequencypsf point spread functionpzt lead-zirconate-titanate, a piezoelectric materialrf radio frequency, unprocessed ultrasound signalsroi region of interestrms root mean squaresduv shear wave dispersion ultrasound vibrometryslam simultaneous localization and mappingsmurf spatially modulated ultrasound radiation forcexxspect single-photon emission computed tomographysnr signal-to-noise ratioswave shear wave absolute vibro-elastography, a method to provide absolute mea-surements of tissue elasticty used and developed in this thesistcp/ip Transmission Control Protocol/Internet Protocoltgc time gain compensationtrus transrectal ultrasoundxxiAcknowledgmentsI would like to thank my supervisors, Professors Robert Rohling and Tim Salcud-ean, for their valuable guidance and support during the course of this thesis. I amgrateful for the many members of the Robotics and Control Lab for their collabo-rations, insights, and friendship. I would like to acknowledge the funding that sup-ported my research during the course of my thesis from the National Science andEngineering Council, the Canadian Institutes of Health Research, the C. A. LaszloChair in Biomedical Engineering, and the UBC Four Year Fellowship scholarship.Finally, this work would not have been possible without the incredible support frommy family and friends, who provided the motivation and encouragement needed toachieve my dreams.xxiiChapter 1Introduction1.1 MotivationIt is often desirable to gain insight into the inner workings of the human body. Sev-eral techniques have been developed to visualize internal anatomical structures orpathology frommeasurements of tissue properties. One of themost commonly usedmedical imaging techniques projects X-rays from one side of the body through theother to create a two dimensional (2d) image of X-ray absorption. The X-ray sourceand detector may also be rotated around the body to create a cross-sectional slicewhich forms the basis of computed tomography (ct) imaging. The slice locationsmay be translated to create a three dimensional (3d) ct image. In positron emis-sion tomography (pet) and single-photon emission computed tomography (spect)imaging, a gamma ray detector is rotated around the body to measure rays pro-duced by an injected radioisotope designed to trace specific biochemical reactions.magnetic resonance imaging (mri) measures the decay rate of proton spins as theyrelax from an excited state to a lower energy state aligned with a strong magneticfield. Ultrasound imaging measures high frequency acoustic waves reflected backto a transducer caused by reflection of the waves by scatterers and interfaces ofdifferent acoustic impedance [263].Elastography is a branch of medical imaging techniques that provides measure-ments and images of the mechanical properties of tissue. Most commonly, elastog-raphy infers the elastic properties (i.e. Young’s modulus or shear modulus) of tissue.1The elastic properties can vary over several orders of magnitude between differenttypes of tissue and between normal and diseased tissues [115]. Both relative andabsolute measurements of tissue elasticity have proven useful in several clinical ap-plications such as improving breast cancer classification with Breast Imaging Re-porting and Data System (bi-rads) [101, 147], targeting prostate cancer for biopsyand focal therapy [239, 282], and liver fibrosis assessment and staging [53, 145],among others. Quantifying the absolute elasticity values may be beneficial in char-acterizing tissue types, determining the different stages of a disease, or monitoringthe progress of a treatment [86].The assessment of tissue elasticity in identifying disease is familiar in medicine.Digital palpation is used in physical examinations to locate stiff tumours of thebreast and prostate. A related “mechanical imaging” technique measures the pres-sure at the tissue surface in order to quantify the sensation felt by the fingertips dur-ing palpation [94]. Instead, elastography determines the elastic properties of thetissue by applying a force to the tissue, measuring the resulting internal motion ofthe tissue, and solving the equations of motion derived from a continuummechanicsmodel of the tissue behaviour relating the measured motion to the elastic properties.Several approaches have been developed for each of these three steps. For provid-ing measurements of internal tissue motion, the majority of elastography systemsrely on ultrasound, with some systems using mri, and a small number using opticalcoherence tomography (oct) and X-ray. A number of ultrasound based systems,described in Table 1.1, are currently commercially available from major manufac-turers such as Siemens Healthcare (Erlangen, Germany), Philips Healthcare (Best,Netherlands), GE Healthcare (Little Chalfont, United Kingdom), Echosens (Paris,France), SuperSonic Imagine (Aix-en-Provence, France), Hitachi Medical (Tokyo,Japan), BKMedical (Herlev, Denmark), Toshiba (Otawara, Japan), andWuxi HiskyMedical Technology (Beijing, China).Standard medical ultrasound imaging has gained popularity due to its low cost,portability, relatively high frame-rate, ease of use, and safety. Ultrasound uses com-pressional waves created by short pressure pulses on the surface of the tissue withcentre frequencies in the 1 MHz to 20 MHz range. The pressure wave propagatesinto the tissue and the resulting echoes are recorded. An ultrasound transducerconverts between voltages and pressures using piezoelectric crystals. The crys-2Table 1.1: Commercially available ultrasound elastography systems.System Typea AbsoluteMeasurementMeasurementLocationIntended ClinicalApplicationSiemenseSie Touch Strain No 2d Image Liver, Breast,ThyroidVTi Amplitude No 2d Image Liver, Breast,ThyroidVTq Transient Yes Pointb Liver, Breast,ThyroidVTIQ Transient Yes 2d Image Liver, Breast,ThyroidPhilipsElastography Strain No 2d Image BreastElastPQ Transient Yes Pointb LiverGEElastography Strain No 2d Image BreastShear WaveElastography Transient Yes 2d Image Liver, BreastEchosensFibroScan Transient Yes Pointc LiverSuperSonic ImagineShearWaveElastography Transient Yes 2d/3d Image Liver, BreastHitachiReal-time TissueElastography Strain No 2d ImageBreast, ThyroidProstate, PancreasBKElastography Strain No 2d Image Breast, ProstateToshibaElastography Strain No 2d Image BreastShear WaveElastography Transient Yes 2d Image LiverWuxi HiskyFibroTouch Transient Yes Pointc Livera The different types of elastography are described in detail in Section 1.5.4.b Average over small 2d region of interest (roi).c Linear regression along one dimensional (1d) line.3tals are grouped in an aperture and electronically pulsed in a timed sequence tocreate a transmit pressure field. Inhomogeneities in the density and compressibil-ity of the tissue cause the wave to scatter. The returning pressure waves (echoes)are measured over an aperture, delayed, summed, and digitally sampled. The re-sulting collection of echo signal lines is processed to create a standard brightnessmode (b-mode) image [16, 69].Due to the complex scattering pattern of ultrasound waves in biological tissuesand the highly anisotropic beam pattern, some ultrasound images may be difficultto interpret. 3d ultrasound offers a number of advantages over conventional 2d ul-trasound. One clear advantage is that the 3d spatial relationship between structuresin the imaging volume is already available in the volume image, in comparison tomentally visualizing the 3d structures from a series of 2d cross-sections. From the3d image, different 2d imaging planes can be “resliced” from the volume to provide2d image orientations that would otherwise be impossible to acquire. Alternatively,surface or volume rendered models can be created which may reveal pathology thatmay be difficult to see in conventional ultrasound images [219]. Imaging in 3d canprovide more information on the geometry of anatomical structures and an accu-rate measure of their volume. 3d ultrasound can also help in needle localizationand guidance during biopsy [85].Use of 3d ultrasound can also benefit elastography. The above mentioned ad-vantages for 3d ultrasound in terms of easing the interpretation of the geometricalrelationships in the data and improving the accuracy of geometrical measures stillapply [36]. Further, since elastography involves solving the equations of motionderived from a continuum mechanics model, measurement of the variation of themotion over space can help reduce errors when a 2d planar model does not ac-curately describe the experimental conditions [96]. The most accurate approachwould include measurement of motion in all three directions over the volumetricregion of interest (roi) [343].Since ultrasound is a highly anisotropic imaging modality, the motion mea-surements with ultrasound are most accurately computed in the axial direction, theprimary direction of ultrasound wave propagation. In the other two directions, lat-eral and elevational, motion measurements are typically an order of magnitude lessaccurate due to lack of phase information [42], wider extent of the point spread4function [167], and larger sampling intervals [187]. In contrast, magnetic reso-nance imaging elastography (mre) is able to measure motion accurately in all threedirections, which helps to reduce errors in the elasticity measurements.One approach to improve ultrasoundmotion measurements is to move the trans-ducer to different view angles so the accurate axial motion measurements from sev-eral directions can be combined [4, 32, 211]. In general this requires a spatial cal-ibration to determine the translation and rotation between each view. The problemof ultrasound spatial calibration has been studied extensively [143, 206], howeverlimited research has investigated 3d ultrasound calibration at accuracies requiredfor elastography applications.In summary, elastography provides a means of measuring the elastic proper-ties of tissues. Contrast in elastic properties is correlated with clinically relevantpathology. Elastography requires a method of measuring internal tissue motion,with most systems using ultrasound. The use of 3d ultrasound and quantitativeabsolute elastography techniques may help improve visualization and specificationof physiological abnormalities. Improvements in 3d ultrasound spatial calibrationmay enable more sophisticated methods of measuring 3d tissue motion vectors.1.2 HypothesisThe overall hypothesis is that a 3d approach to measurement can be used with ul-trasound elastography to provide accurate measurements of tissue elasticity. Toexamine this hypothesis, the following objectives are set:• Develop a modular 3d elastography research platform that allows for flexibleuse of different techniques and equipment for comparison and optimization.• Test the 3d elastography research platform through simulations and experi-ments on tissue mimicking phantoms and ex vivo tissue.• Develop highly accurate ultrasound calibration techniques that can be appliedto 3d elastography.• Develop a method for measuring the full shear wave motion vector field overa volume with 3d ultrasound and test on a tissue mimicking phantom.5• Investigate the use of spatial calibration for combining multiple ultrasoundvolumes from different views to determine the full shear wave motion vectorfield over a volume through simulation.1.3 Thesis OutlineThe rest of this chapter is devoted to a detailed description of the background in thefields of ultrasound, elastography, and ultrasound spatial calibration. This is meantto provide sufficient information for the non-specialist to understand the rest of thisthesis and to provide a more thorough review of relevant literature.Chapter 2 describes the design, development, and testing of a 3d elastogra-phy research platform. The details of the separate components of the system andhow they operate are described. Different methods of sampling tissue motion aretested through simulation and experiments. It is shown that single frequency andmulti-frequency tissue motion can be recovered with acceptable accuracy using asub-Nyquist sampling rate, assuming consistent tissue motion, with reasonable as-sumptions of errors due to measurement jitter and uncertain frame rate, using arange in the number of temporal samples acquired. An artefact from using the sub-Nyquist sampling approach, appearing as a wave with a similar spatial frequencyto the waves induced by the applied excitation, is discovered and explained. It isshown that the artefact is less prevalent when using a higher temporal sampling rate.Measurements on a tissue mimicking phantom demonstrate the repeatability of theelasticity measurements for repeated acquisitions and different excitation locations.In Chapter 3 the 3d elastography research system of Chapter 2 is applied tohealthy ex vivo human placentas. Measurement of multi-frequency shear wavesover a band of 60 Hz to 200 Hz is demonstrated and the feasibility of estimatingplacenta elasticity is shown. Viscoelastic parameters based on Voigt, Maxwell, andZener rheological models are estimated, showing a strong viscous dispersion.In Chapter 4 a novel spatial calibration technique for 3d ultrasound is described.The method makes fewer assumptions about the motor geometry than previous cal-ibration techniques, and thus has the potential to reduce errors introduced throughinaccurate scan conversion of ultrasound data from transducer to Cartesian coordi-nates. The method is based on calibrating individual image slices across the volume6using an accurate 2D calibration technique. A best fit to a subset of slices is per-formed to decrease data collection time compared to calibrating all of the slicesmaking up the volume, and reduce the influence of random errors in individual cal-ibrations. The method is compared to the widely used N-wire calibration technique.A novel extension of the N-wire calibration is developed to incorporate more datapoints that are typically discarded and contrasted with the traditional N-wire tech-nique. Finally, another novel 3d ultrasound calibration technique is developed thatuses assumed motor geometry and planar calibration features. Experiments showthat the proposed multi-slice technique produces the smallest point reconstructionerror; at 0.82 mm, the method is the first sub-millimetre 3d ultrasound calibrationtechnique reported.In Chapter 5 a solution for measuring dynamic motion on the order of hundredsof cycles per second using 3d ultrasound is proposed based on synchronizing theultrasound acquisition with the mechanical exciter producing the tissue motion. Anovel regularization algorithm is developed using a physical constraint of tissue in-compressibility to help reduce noise in the motion measurements. The elasticitydistribution of a tissue mimicking phantom containing a stiff sphere is estimatedusing two algorithms; a local frequency estimator and a finite element solution.The best elasticity estimate reaches within 1 kPa of the reference Young’s modu-lus value provided by the phantom manufacturer at the centre of the stiff sphere.Mean elasticity measurements are compared to previous mre estimates, and areshown to be within the variance of current elastography methods. It is shown thatusing displacements after regularization results in more than a 2 dB improvementin contrast-to-noise ratio (cnr) ratios of the elasticity images using either algorithmcompared to displacements before regularization.In Chapter 6 the potential of combining multiple ultrasound volumes from dif-ferent angles for measurement of the full motion vector over a volume is investi-gated. A method is presented to solve for 3D motion vectors from three volumes ofaxial motion measurements, and the technique is verified through simulation. Theeffects of scan conversion, interpolation, and calibration are modelled to study howthey affect the displacement and elasticity estimates. It is shown that the proposedmulti-view approach can reliably solve for dynamic 3D displacement vectors suit-able for elasticity estimation. The error in elasticity increases by less than 8 % for a7typical level of calibration error (< 1 mm and 2◦) compared to elasticity measuredwith no calibration error.In Chapter 7 the thesis is concluded by relating the results of the precedingchapters to each other and placing the results in the context of the current state of thefield of elastography. The strengths and weaknesses of the research are discussedand recommendations for future research directions are provided.Seven appendices are included at the end of the manuscript. Appendix A pro-vides some useful information on index notationwhich is used to describe the elastictheory underlying elastography. In Appendix B a copy of the consent form used inthe placenta study described in Chapter 3 is provided. In Appendix C the difficultyin determining the location of the edge points in the popular N-wire spatial cal-ibration phantom is demonstrated using geometrical examples, and the particularsolution proposed in Chapter 4 is shown to demonstrate how that design leads tothe ability to determine the edge points. In Appendix D the exponential and log-arithmic mapping to and from the dual quaternion manifold is explained, which isused in the rotation fitting and interpolation used in Chapter 4. In Appendix E themethod for determining the direction of the best fit geodesic on the quaternion man-ifold used in Chapter 4 is described, it is proven that the distance metric used for thegeodesic fitting is independent of the choice of coordinate system, and the fittingalgorithm is compared in more detail to alternative approaches. In Appendix F asimulation is described that evaluates the rotation fitting algorithm from Chapter 4.In Appendix G a robust method for computing displacement gradients using linearleast squares, which is used in the elasticity estimation algorithm in Chapter 6, isdescribed.1.4 Ultrasound BackgroundThis section describes the physical processes, signal processing, and methods ofmedical ultrasound imaging. A block diagram providing an overview of an ultra-sound imaging system is shown in Figure 1.1.An ultrasound transducer is applied to the body and is connected to an ultra-sound scanner which controls the transducer, and interprets and processes the trans-ducer measurements. A transducer creates acoustic pulses at the surface of the body8SoundWavesMux andTx/Rx SwitchesBodyTransducerDisplayCableScanConversionEnvelope Detectionand Log CompressionI/QDemodulationDopplerProcessingRxBeamformerTxDelaysADCAmpTGCUltrasoundScanner ScanSequencerFigure 1.1: A block diagram of an ultrasound system. The scan sequencercontrols the transmit (Tx) and receive (Rx) timing. The beamformedreceived data is then processed for b-mode/Doppler display.which transmit waves that travel into the underlying tissue. Echoes are produced bythe tissue and are measured by the transducer over time. The distance or depth ofthe reflecting echoes are resolved, creating a spatial map of the acoustic propertiesof tissue.The depth, d, is determined from the time it takes for the ultrasound pulse totravel to the target and backt =2dc, (1.1)where c is the speed of the ultrasound wave and the factor of 2 accounts for thetravel of the transmitted wave to the target and the travel of the reflection back tothe transducer, assuming a straight line of travel. Usually a constant speed of soundof 1540 m/s is assumed which corresponds to an average value for tissues typicallyimaged with ultrasound. Excluding fat, tissues that are usually imaged by ultra-9sound are within 5 % of this value [140]. Some scanners use a different assumedspeed of sound using presets depending on the tissue type selected by the user, orwill allow the user to set the value directly. This will correct for errors in depthestimation from Equation 1.1, as well as errors in focusing which are discussed inmore detail in Section 1.4.4.Echoes of the ultrasound waves are scattered or reflected by the tissue, wherescattering refers to the interaction between waves and particles smaller than thewavelength, and reflection refers to the interaction between the waves and objectslarger than the wavelength. The echoes are caused by a combination of changes inthe tissue density and compressibility.1.4.1 ReflectionTissue structures with features larger than the wavelength, such as interfaces be-tween organs, adipose tissue, cysts, and bone create prominent reflections. Thereflections are often described using the characteristic acoustic impedance [161]Z = ρc, (1.2)where ρ is the mass density and c is the speed of the sound waves travelling inthe tissue. It is generally assumed that the propagation of an ultrasound wave in thebody can be modelled by a longitudinal wave propagating in a fluid under isentropicconditions [73], resulting in a propagation speed ofc2 =∂p∂ρ=Kρ, (1.3)where p is the pressure and K is the Bulk modulus.At a boundary between two objects with impedances Z1 and Z2, the ratio of thereflected pressure of an incident wave normal to the interface is described by thereflection coefficient [69]R =Z2 − Z1Z2 + Z1. (1.4)The received ultrasound signal is closely approximated as a convolution be-tween the spatial distribution of the reflection coefficients of the tissue and the point10spread function (psf) of the imaging system, with the assumptions of linear wavepropagation and weak scattering [208]. Typically the reflection coefficient is smallfor soft tissues, meaning most of the energy of the wave is transmitted through theinterface. This is important as it allows echoes to return from structures deeper thanthe first few reflecting interfaces. Large differences in acoustic impedance, whichresult in large reflections, can cause a “shadowing” artefact, displaying a dark regionbelow the reflecting object. This is also the reason for using an acoustic coupling gelbetween a transducer and the skin as there is a large impedance mismatch betweenair and the body.1.4.2 ScatteringInhomogeneities in acoustic impedance of tissue on scales smaller than the wave-length result in scattered echoes. The scattered waves interfere constructively anddestructively depending on the relative phases of the waves. This leads to the famil-iar granular appearance of ultrasound images, also known as speckle. It has beenshown that despite its appearance as tissue texture, speckle is not a direct depic-tion of the underlying tissue structure, but instead carries information dependent onthe tissue, transducer, and point spread function [327]. However, the speckle pat-tern is deterministic, not random, and can be reproduced given the same imagingconditions [306].1.4.3 TransducersThe transducer is made up of several components, namely piezoelectric elements,a backing layer, a matching layer, and an acoustic lens as shown in Figure 1.2. Thepiezoelectric elements, commonly composed of lead-zirconate-titanate (pzt), areconnected to electrodes leading from the cable attached to the scanner and are usedto create the ultrasound waves by applying high voltage pulses (usually 50 V to100 V), as well as to detect echoes from the body when the reflected waves applypressure on the transducer face. The backing layer is used to damp the vibrations ofthe elements which helps to reduce the number of cycles in the pulse, and thus in-crease the bandwidth. Thematching layer reduces the change in acoustic impedancebetween the transducer and the body which would otherwise cause most of the wave11Cable to scannerBackingPiezoelectric element(s)Electrical connection to electrodesMatching layer(s)CasingAcoustic lensFigure 1.2: The components of an ultrasound transducer.to reflect before transmitting into the body as explained in Section 1.4.1. The acous-tic lens functions to focus the acoustic beam in the elevational direction. The depthof the elevational focus is usually fixed.While an ultrasound transducer could be made using a single piezoelectric ele-ment and the components mentioned in the previous paragraph, only the anatomicalinformation along a single line would be measured. Most modern ultrasound trans-ducers are composed of several piezoelectric elements arranged in a row, calledan array. A common configuration is the linear array in which the elements areplaced along a straight line, creating rectangular shaped images. Phased arrays usefewer closely packed elements to create a smaller footprint, which allows accessto small windows of the body such as between the ribs, and sweep the ultrasoundbeam across a large range of angles creating a fan shaped image (explained in Sec-tion 1.4.4). Curvilinear arrays place the elements along a curved surface, creatingan annular sector shaped image which can be useful for imaging large structuressuch as organs in the abdomen. The layout of a linear array and the coordinate12Axial (depth)Lateral (width)Elevation(thickness)PitchxyzFigure 1.3: The layout of a linear array and the array coordinates. The axialdirection corresponds to the 0◦ direction of sound propagation and thusthe depth, the lateral direction (also referred to as azimuth) correspondsto the width of the image, and the elevational direction the thickness ofthe image. The spacing between the centre of the elements is called thepitch.system of the array used throughout this thesis is shown in Figure 1.3.The voltage signal measured by the piezoelectric elements is referred to as anradio frequency (rf) signal. When referring to the voltage from a single elementor channel, this would be called prebeamformed rf data. Most ultrasound systemshowever do not save the channel data, but rather use hardware to combine data fromseveral elements in a process called beamforming.1.4.4 BeamformingBeamforming is the process of combining the acoustic energy from a group of ele-ments, or an aperture, to create a beam in which the majority of the acoustic energyis along the beam direction. The postbeamformed rf signal is used to create a linein a b-mode image. By changing the location or direction of the beam, themultitudeof lines can be combined to form an image.The beam is typically focused by combining the approximately spherical emis-sions from each element and controlling the pulse timing of each element so that the13Time delaysShifted transmit pulsesWave frontsFocal zoneElementsFigure 1.4: Ultrasound transmit focusing is achieved using variable time de-lays.waves converge to a focal zone as shown in Figure 1.4. The pulse timing may alsobe adjusted to steer the beam to different angles. The delays for each element arecalculated based on geometry assuming straight line wave propagation as shown inFigure 1.5. The time for the wave emitted by the element located at x to reach thefocal point at r at a steering angle of θ ist(x) =1c(√r2 + x2 − 2r x sin θ − r), (1.5)where c is the speed of sound. For a given aperture size, the maximum propagationtime can be calculated using Equation 1.5 and used to bias the time delays for eachlocation along the array so the element that is furthest from the focal point will havea delay of zero [12]. Thus the time delay for each element isτ(x) = tmax − t(x). (1.6)14dmaxrFocal pointyxθFigure 1.5: Ultrasound transmit focusing is achieved using variable time de-lays.Simple receive beamforming essentially performs the reverse of the transmitfocusing. The signals across the aperture are aligned in time, using the principle ofcoherence where coherent signals will add to create a large signal, and incoherentsignals will tend to cancel and appear as random low level signal [306]. The delaysare computed using geometry as in transmission. However, unlike transmission,which can only use one set of fixed delays to create a physical wave pattern, thereceive delays can be changed dynamically. In dynamic receive focusing the delaysare adjusted as a function of depth so the entire image is focused. Usually a constantf-number (ratio of the focal distance to the aperture size) is desired, so the numberof elements used in the summation is gradually increased with increasing depth.The received signal can contain energy from directions other than the mainbeam axis. The off-axis directions containing energy are referred to as side andgrating lobes. Apodization, which weights each element in transmit and receive,can help to reduce the amplitude of these lobes with the trade-off of increasing the15Figure 1.6: Example of apodization weighting functions applied to the ele-ments of an array.width of the main lobe. Common apodiziation functions are box, Hanning, andBlackman which are shown in Figure 1.6. To demonstrate the effect of apodiza-tion, the reflection from a point scatterer was simulated using Field II [148], a soft-ware for simulating the spatial impulse responses of ultrasound transducers, for theapodizations mentioned. The magnitude of the received signal at the axial coordi-nate of the focus point plotted across the width of the transducer shows the effecton the beam in Figure 1.7. The −6 dB width of the main lobe has increased fromabout 0.5 mm to 0.8 mm, but the presence of the side lobes and grating lobes hasbeen greatly reduced.The beamforming delays described in this section are based on the geometry ofthe elements and the focus. Heterogeneities in the speed of sound cause a changein the geometric time differences to reach to focus because the waves from each16Figure 1.7: Simulated response from a point scatterer located at the trans-ducer’s focus for the different apodization weightings shown in Figure1.6.element travel through different paths. The heterogeneities also cause refraction,bending the waves in different directions. These aberrations cause an increase inbeam width and decrease in amplitude [316].An important consideration in ultrasound image formation is the attenuationof the waves due to viscous and scattering effects. Attenuation is depth and fre-quency dependent and is usually on the order of 1 dB/(cm MHz) [140]. Becausethe amount of energy lost is a function of how far the waves have travelled, a com-pensation based on the depth of the received signal is commonly applied in a processcalled time gain compensation (tgc). The tgc is increased gradually with depth toefficiently use the dynamic range of the analog-to-digital converter (adc) hardwareand to achieve uniform brightness along the depth.17LPFLPFr (t)I (t)Q(t)cos(ωt)− sin(ωt)Figure 1.8: Block diagram of i/q processing.The resolution of the beamformed ultrasound image is anisotropic and spatiallyvariant. The axial resolution is proportional to the transmit frequency and the frac-tional bandwidth of the system. The lateral and elevational resolution are also pro-portional to the transmit frequency, and are directly proportional to the f-number(in the elevational direction the f-number is fixed). The axial resolution is usuallyhigher than the lateral and elevational resolution by factors of 3–8 and 10, respec-tively [17].The frequency dependence of resolution and attenuation lead to a fundamentaltrade-off: short duration, high frequency pulses allow clear separation of fine detailsin the image; long duration, low frequency pulses allowwaves to penetrate deep intothe body without images becoming dominated by noise.1.4.5 Image FormationThe b-mode image formation process uses beamformed rf data and applies i/qdemodulation, envelope detection, log compression, and scan conversion processesbefore final display. These processes are described in this section.The beamformed rf data is an amplitude modulated signal. To retrieve thetissue reflectivity information the carrier frequency is removed in a process called18(a) (b)(c) (d)Figure 1.9: The i/q demodulation process starts with beamformed rf datashown (a) as a function of time and (b) frequency. (c) The spectrum isshifted to the left after down mixing and (d) after low pass filtering thesignal only contains a band of energy that was originally centred aroundthe positive centre frequency of the rf signal.19Figure 1.10: Example of an rf signal and its corresponding envelope.i/q demodulation, shown schematically in Figure 1.8. First, the rf data is downmixed which shifts the frequency spectrum down. This is commonly achieved bymultiplying the rf signal with cosine and sinusoid signals at the centre frequency,or generating the quadrature component by sampling the rf signal with a delayof 1/(4ω) (see [296] for a description of sampling strategies). This can also beaccomplished digitally by multiplying with a complex exponentialr (t)e−jωt = r (t) · (cos(ωt) − jsin(ωt)) = I (t) + jQ(t), (1.7)where r (t) is the rf signal, I (t) andQ(t) are the in-phase and quadrature i/q compo-nents, and ω is the centre frequency of the rf signal. After down mixing, the signalis low pass filtered to remove the negative frequency spectrum of the real rf signaland noise outside of the desired bandwidth. The spectrum of the signal during thesteps of the demodulation process is shown in Figure 1.9. For more efficient storageand processing, the i/q data is usually downsampled because the sampling rate canbe reduced to twice the cut-off frequency of the low pass filter without introducingaliasing. In addition to the b-mode image formation described below, the i/q datais also used for Doppler motion detection as described in Section 1.4.6.The amplitude of the i/q data, or the envelope, is given bye(t) =√I2(t) +Q2(t). (1.8)Figure 1.10 shows an rf signal and the corresponding envelope. Log compression20follows envelope detection to ensure that the image has an appropriate dynamicrange for the display and for human visual perception.To create a geometrically accurate image for display, the compressed envelopedata must be mapped from transducer coordinates to Cartesian coordinates in a pro-cess called scan conversion. This can be accomplished by creating a regular gridin transducer coordinates for the envelope data points, a regular grid in Cartesiancoordinates for the display, computing the location of the Cartesian grid in trans-ducer coordinates using the beam geometry, and finally using bilinear interpolationto estimate the envelope for the Cartesian grid points.1.4.6 DopplerIn diagnostic ultrasound, Doppler is used to detect and measure blood flow. Firstdescribed by Christian Doppler in 1842 in a study of the colour of light from stars, itexplains how the frequency of a wave is perceived to change relative to movement.The premise of the Doppler effect is that the apparent frequency of a detected wavewill change when the source or receiver is moving.The Doppler equation for ultrasound waves transmitted and received by a sta-tionary transducer and interacting with a moving scatterer can be derived as follows.Consider a scattering particle moving towards a transducer transmitting a wave witha centre frequency of f0. The scatterer is acting as a moving receiver, increasing therate at which it receives the transmitted sound waves, causing an apparent increasein observed frequencyfS = f0c + v cos θc, (1.9)where fS is the frequency experienced by the scatterer, v is the velocity of the scat-terer, c is the speed of sound in tissue, and θ is the angle between the sound beamand the scatterer. Now part of the sound is echoed by the scatterer back towards thetransducer. The scatterer is acting as a moving transmitter, causing an increase inthe frequency emitted by the scattererfR = fScc − v cos θ . (1.10)Substituting Equation 1.9 into Equation 1.10 and defining fR = f0 + fD , where fD21is the Doppler frequency shift measured by the transducer, we obtainf0 + fD = f0c + v cos θc − v cos θ . (1.11)Considering the flow velocity observed in clinical Doppler examinations is typically0.1 % of the value of the speed of sound in tissue, Equation 1.11 can be simplifiedtofD = 2 f0v cos θc. (1.12)Clearly in a real blood vessel in the body the received echo signal will not befrom a single scatterer moving at a single velocity, but from a collection of scatterersat different velocities. This results in a spectrum of measured frequency shifts.A number of blood flow measurement techniques based on the Doppler princi-ple have been developed for ultrasound scanners. The different techniques and theiradvantages and disadvantages are discussed in the following sections.Continuous Wave and Pulsed WaveContinuousWaveDopplermode on ultrasound scanners continuously transmit soundbeam and receive the echoes from the beam. A specialized transducer containing atransmit element and receive element may be used, or an array with separate aper-tures dedicated to each function. The detected echoes at the receiver undergo i/qdemodulation to shift the spectrum to the base band and remove the high frequencycomponents. The remaining signal contains the spectrum of Doppler shift frequen-cies, fD , in the region of the overlapping transmit and receive beams. In addition tothe signal due to blood motion, there are contributions from other structures such asthe vessel wall which typically have larger echo amplitudes and could overwhelmthe blood motion signals. These other structures typically move slower than theblood so a high-pass filter, sometimes referred to as a “wall filter” or a “clutter”filter, can remove these signals from the spectrum [140]. This unfortunately alsoremoves the signal from slow moving blood.The resulting Doppler shift spectrum for physiological flows is typically in thekHz range, and can be applied to a loudspeaker for audible feedback. Stereo speak-ers may be used to differentiate between positive and negative flow. The signal is22Figure 1.11: Example of blood flow through the carotid artery as depicted byPulsed Wave Doppler. The horizontal teal lines indicate the gate loca-tion, and the angled teal lines indicate the estimated flow angle whichwas manually set to approximately parallel to the vessel direction. Thewaveform under the b-mode image shows the estimated velocity as afunction of time. In this case, a purely positive flow is measured (flowtowards the transducer), with a peak velocity of approximately 60 cm/s.also often processed by a fast Fourier transform (fft) and plotted over time for visualfeedback. A broad spectrum often indicates abnormal or turbulent flow. Quantifica-tion of the velocity based on the fft is dependent on the relative angle between thebeam and the flow which can be difficult to correct for in Continuous Wave mode,however the ratio between the peak-systolic and end-diastolic values is independentof the angle and can be useful for characterizing the flow [169].The continuous operation of the transmit and receive operations removes thedepth ranging ability of pulse echo ultrasound. This makes it difficult to localizeblood flow measurements from Continuous Wave Doppler. Localization may beimportant for separating flow information from different vessels or to quantify dif-ferent regions (e.g. turbulent and laminar regions) inside a single vessel. Pulsed23Wave Doppler provides this feature by only providing Doppler shift measurementsfrom a small sample volume or “gate.” The transducer transmits a pulse and thereceive signal is ignored until enough time has elapsed for echoes from the start ofthe gate to return and then is ignored again after the amount of time to reach theend of the gate’s depth. Because the location of the gate is known, the angle of thevessel can be estimated in the region of the gate and corrected for. The signal pro-cessing as well as the audio and visual feedback are similar to Continuous Wave.An example of a Pulsed Wave Doppler image is shown in Figure 1.11.The pulse repetition frequency (prf) is the effective sampling rate of the PulsedWave Doppler signal, which limits the frequency range and thus the maximum ve-locity that can be measured according to the Nyquist sampling theorem|v | ≤ c fPRF4 f0 cos θ. (1.13)For a deeply placed gate, the waiting time for the echoes to return from the gatelocation can be long, which could severely limit the prf. Since the echoes beforethe start of the gate do not need to be recorded, a high prf may be achieved bytransmitting additional pulses during the period of time before the first pulse returnsfrom the gate [276]. The disadvantage of the high prf approach is that if there isany motion between the transducer and the gate there will be some ambiguity aboutwhere the motion originated.The number of samples, N , for the fft computation also puts a restriction on theminimum velocity that can be measured with the PulsedWave mode, given by [280]|v | ≥ c fPRF2N f0 cos θ. (1.14)Continuous Wave and Pulsed Wave may be combined in a duplex fashion withconventional b-mode imaging to provide both Doppler measurements and informa-tion about the anatomical structures.Colour FlowColour Flow mode displays measurements of the mean velocity along the beam di-rection from a number of gates in a region of interest as a 2d image overlaid on24(a)(b)Figure 1.12: Example images of blood flow through the carotid artery as de-picted by (a) Colour Flow and (b) Power Doppler imaging modes.Rapid transitions from yellow to light blue in the Colour Flow imagelikely indicate aliasing.25a b-mode image as shown in Figure 1.12. The colour mapping typically presentsflow towards the transducer as red and flow away from the transducer as blue. Thegating process is analogous to the Pulsed Doppler gating, except Colour flow willcontain multiple different gates along a beam line. The same beam line is transmit-ted repeatedly at a prf similar to Pulsed Doppler until a defined number of lineshave been acquired, called an ensemble. Adjacent lines over the roi are acquiredsequentially after an ensemble for each line is collected.Due to the limited number of ensembles causing variability in the fft outputand the computational cost of the fft operation on each gate across the roi, differentestimators were developed for measuring the mean frequency. Almost all modernscanners use a phase-based estimator [102]. The most common technique is basedon the autocorrelation technique introduced in [158]. The technique relates themean frequency computed from the power spectrumω¯ =∞´−∞ωP (ω) dω∞´−∞P (ω) dω(1.15)to the autocorrelation function using the relationship expressed by the Wiener-Khintchine theorem [59],Γ (τ) =∞ˆ−∞P (ω) ejωτdω. (1.16)Using Equation 1.15, Equation 1.16, and the derivative of Equation 1.16, it can beshownjω¯ =Γ˙ (0)Γ (0). (1.17)Representing the autocorrelation asΓ (τ) = |Γ (τ) | ejφ(τ), (1.18)the mean Doppler frequency shift is the derivative of the phase of the autocorrela-26tion function which can be approximated asω¯ = φ˙ (0) ≈ φ (TPRF) − φ (0)TPRF=φ (TPRF)TPRF, (1.19)where TPRF is the period between pulses.Using the definition of the discrete autocorrelation at a single lag (TPRF), if thedemodulated received signal is represented as a function of ensemble number i,I (i) + jQ (i), then the autocorrelation over an ensemble length of N isΓ (TPRF) =N−1∑i=1[I (i) + jQ (i)] [I (i − 1) − jQ (i − 1)] , (1.20)and the phaseφ (TPRF) = tan−1*....,N−1∑i=1Q (i) I (i − 1) − I (i)Q (i − 1)N−1∑i=1I (i) I (i − 1) +Q (i)Q (i − 1)+////-. (1.21)Note that the i/q signals used in Colour Flow processors are often integrated overthe depth of a gate, which reduces the samples along the gate to a single sample foreach pulse in the ensemble.The presence of the arctangent function in Equation 1.21 limits the phase es-timation to the interval (−pi, pi], which puts a restriction on the maximum velocitythat can be measured|v¯ | ≤ c ω¯2ω0 cos θ= cpi2ω0TPRF cos θ= cfPRF4 f0 cos θ, (1.22)which is equivalent to the Nyquist limit from Equation 1.13.The variance of the spectrum can also be computed with the autocorrelationfunctionσ2 ≈ 2T2PRF(1 − |Γ (TPRF) |Γ (0)), (1.23)but is less commonly displayed.Since the flow direction typically varies across the 2d image, angle correction27is often not applied to accurately estimate the actual velocity value. Instead, ColourFlow is usually used to identify interesting flow regionswhich can then be quantifiedin detail in Pulsed Wave mode [261].Significant accuracy improvements can be made by estimating the centre fre-quency of the backscattered signal rather than assuming that the centre frequencyremains constant and is equal to the centre frequency of the transmitted pulse. Ad-ditionally, the estimation of both the Doppler shift and centre frequency shift canbe made for every sample along the depth of the gate, instead of integrating thei/q signals. Representing the i/q signals as a function of ensemble number i andgate sample k, the velocity estimate incorporating these two ideas can be obtainedfrom [194]v cos θ =c2TPRF·tan−1*..,M∑k=1N−1∑i=1Q(i,k)I (i−1,k)−I (i,k)Q(i−1,k)M∑k=1N−1∑i=1I (i,k)I (i−1,k)+Q(i,k)Q(i−1,k)+//-ω0*..,1 + tan−1 *..,M−1∑k=1N∑i=1Q(i,k)I (i,k−1)−I (i,k)Q(i,k−1)M−1∑k=1N∑i=1I (i,k)I (i,k−1)+Q(i,k)Q(i,k−1)+//- /2pi+//-(1.24)where M is the number of samples contained along the depth of the gate.Power DopplerPower Doppler images display the integrated power spectrum of the Doppler signalinstead of its mean Doppler frequency shift. While this does not provide quantita-tive measurements of velocity magnitude or direction, it is useful for detecting thepresence of small flow. The images are computed from the zero lag of the autocor-relation function, which is clear from setting τ = 0 in Equation 1.16,Γ (0) =∞ˆ−∞P (ω) dω. (1.25)The detection sequence is the same as used in Colour Flow imaging up to the pointof computing the autocorrelation. The value of the Power Doppler signal is relatedto the number and size of the scattering particles, or blood cells, which is non-28linearly dependent on the blood flow velocity, shear rates, and hematocrit level [49,277].Compared to Colour Flow imaging, Power Doppler imaging provides severaladvantages. One critical difference is the noise appears differently. In Colour Flowimaging, the noise as a random process acts to introduce random phase angles intothe measurement which may then appear as a random hue. The noise has low powercompared to the desired Doppler signal, therefore, in a Power Doppler image, thenoise appears over a small range of hues which can easily be visually ignored. Thiseffectively allows the use of a larger gain without the Power Doppler image becom-ing dominated by noise. The greater sensitivity is useful for detecting small vesselsand vessels with low flow velocity, however . Power Doppler also has greater sensi-tivity when flow within the sample volume occurs in multiple directions, which forColour Flowwill create frequency shifts that tend to cancel each other out when cal-culating the mean shift, while the Power Doppler integral will add the signal fromeach shift. Another difference between Colour Flow and Power Doppler modes isthat the mean frequency of the Doppler signal will change as a function of the angle,which changes the velocity measurement for Colour Flow, while the energy of thesignal does not depend on the angle and therefore does not affect the Power Dopplermeasurement. The last major advantage is that Power Doppler does not alias likeColour Flow, as the integral of the spectrum is the same whether the phase wrapsaround or not [198].The primary disadvantage of Power Doppler compared to Colour Flow is thatit does not provide measurements of direction or speed. The greater sensitivity ofPower Doppler also makes it more susceptible to motion artefacts such as contrac-tion of the heart organ shift during breathing [213].1.5 Elastography BackgroundA brief overview of Elastography was provided earlier in this chapter in 1.1. Thatoverview covered the basic concepts, clinical applications, available commercialsystems, and some of the current limitations. This section will go into more de-tail about the governing principles and the various techniques used to infer tissueelasticity.291.5.1 TheoryElastography relies on the field of continuum mechanics. The following sectiondescribes how the motion of the material is related to the forces acting upon thematerial and its material properties. This section introduces some new notationthat will be used throughout this thesis unless otherwise specified. Normal fontsdenote scalar parameters while vectors are shown in bold lowercase and matricesin bold uppercase. In index notation, the Einstein summation convention is usedwhere repeated indices are summed over. A brief description of index notation isprovided in Appendix A.There are two common descriptions of material motion; Lagrangian and Eule-rian. The Lagrangian description studies the motion of a particle that is specified byits initial position. The Eulerian description studies a particle that occupies a speci-fied region of space. In elastography the two descriptions are considered equivalentas the spatial fluctuations in displacement, stress, and strain, are much larger thanthe amplitudes of the displacements [232].A fundamental law governing the mechanics of motion is the conservation oflinear momentum over a volume V with surface S∂∂t˚Vρ∂u∂tdV =˚VfdV +¨ST (n) dS, (1.26)where ρ is the mass density, u is the displacement, f is the body force per unitvolume, and T (n) is the traction on the surface with outward unit normal n.The tractions acting on the surface of the volume can be described using thestress tensorTi = σ jin j . (1.27)The conservation of angular momentum implies that the stress tensor is symmetric,reducing the number of independent stress components from nine to six.Using Equation 1.27 and applyingGauss’s divergence theorem to Equation 1.26,the equation of motion is obtainedρu¨i = f i + σi j, j . (1.28)30The small deformations observed in elastography can be described in terms ofthe infinitesimal strain tensor i j =12(ui, j + u j,i). (1.29)In a linear elastic material, the stress and strain are related through a generalizationof Hooke’s lawσi j = Ci jklkl, (1.30)where C is the fourth-order stiffness tensor. Due to symmetries in σ,  , and C, thenumber of independent components of C is reduced from 81 to 21. It is the goal ofelastography to identify the components of C. To simplify the problem, the tissueis often modelled as an isotropic material, resulting in two independent materialparametersCi jkl = λδi jδkl + µ(δikδ jl + δilδ jk), (1.31)where λ and µ are the Lamé parameters, and δi j is the Kronecker delta (1 for i = jand 0 otherwise). Using Equation 1.31, Hooke’s law for isotropic materials can bewritten asσi j = λkkδi j + 2µ i j . (1.32)Young’s modulus, the ratio of tensile stress to extensional strain, and Poisson’sratio, the ratio of transverse strain to axial strain, are related to the Lamé parametersasE =µ (3λ + 2µ)λ + µ, (1.33)andν =λ2 (λ + µ). (1.34)Conversely,µ =E2 (1 + ν), (1.35)andλ =νE(1 + ν) (1 − 2ν) . (1.36)The adiabatic compressibility of tissues is determined primarily by the molecu-31lar content of the tissue and intermolecular interactions [287]. Most tissues imagedby ultrasound are composed of 70 % to 80 % water. Consequently the hydrationof soft tissues leads to near incompressible behaviour. The implications of incom-pressibility on the mechanical properties are Poisson’s ratio is almost 0.5 (ν ≈ 0.5),through Equation 1.35 Young’s modulus is almost three times the shear modulus(E ≈ 3µ), and through Equation 1.36 λ  µ.Combining the equation of motion described in Equation 1.28 with the defini-tions of the strain tensor in Equation 1.29 and Hooke’s law in Equation 1.32 resultsin a description of the elastic response in terms of displacementsρu¨i = f i +[µ(ui, j + u j,i)], j+(λuk,k),i . (1.37)In a homogeneous region, where λ and µ are constant in space, Equation 1.37can be written asρu¨i = f i + µui, j j + (λ + µ) u j, ji . (1.38)In the case of dynamic motion, it is often convenient to represent the motion interms of propagatingwaves. According toHelmholtz’s decomposition theorem, anysquare integrable vector field in three dimensions can be resolved into the sum ofa divergence-free field and a curl-free field [161]. The divergence-free componentis often termed the distortion wave, transverse wave, or shear wave and consists ofthe componentu¨i =µρui, j j = c2sui, j j, (1.39)where cs is the shear wave speed. The curl-free component is often termed the di-latation wave, longitudinal wave, or compression wave is consists of the componentu¨i =λ + 2µρui, j j = c2pui, j j, (1.40)where cp is the compressionwave speed [164]. For nearly incompressiblematerials,such as soft tissues, the compression wave is orders of magnitude faster than theshear wave. The shear wave is typically the component of interest in elastographyas it is both easier to measure due to its slower speed, and because its properties are32more directly related to the shear modulus (the compression wave includes the firstLamé parameter).1.5.2 Additional Modelling ConsiderationsThe previous section provided theory for purely elastic materials because most elas-tography techniques measure quantities only related to linear elasticity. While inmany cases this description is adequate [286], in general soft tissue behaviour ismore complex and accounting for these differences can in some situations provideadditional information and amore accurate fit to themeasurements [173]. Examplesof more complicated phenomena include viscoelasticity, anisotropy, and nonlinear-ity.ViscoelasticityMechanical testing of soft tissue has demonstrated viscoelastic behaviour in whichthere is a time or frequency dependence for stress and strain [111]. Research onmeasuring viscosity using elastography techniques has suggested clinical relevance.For example, measurements of the dynamic modulus (complex shear modulus)demonstrated a power law behaviour as a function of excitation frequency in breastlesions, with malignant lesions showing a tendency for larger power law exponents,likely related to high vascularity and more liquid-like behaviour [293]. In addition,the creep behaviour of malignant breast tumours is thought to be related to the pHand collagen density of the tumour micro-environment [146].In modelling viscoelastic solids, it is generally assumed that the elastic restor-ing forces are proportional to the displacement and the dissipative forces are pro-portional to the velocity. It is necessary to choose a tissue model to determine theviscoelastic parameters. Common models include Maxwell (spring and damper inseries), Voigt (spring and damper in parallel), and Zener (spring and damper in se-ries and in parallel with another spring) [111]. The appropriate model can dependon the material and the excitation conditions. For example, measurements on exvivo bovine muscle and phantoms of agar gels using a transient planar shear waveexcitation showed a Voigt model fit the data better than a Maxwell model [54].In contrast, a different study of in vivo brain and liver tissues undergoing multi-33frequency sinusoidal excitation found that a Maxwell model provided a better fitcompared to a Voigt model, while a Zener model provided the best fit [162].AnisotropyIn soft tissues with highly organized structure, for example the alignment of musclefibres along particular directions, the mechanics of the material will also dependon the direction of the organization. Measurements on human skeletal muscle haveshown shear modulus in the direction parallel to the muscle fibres is greater thanthat measured perpendicular to the fibres [119]. Myocardial fibre orientation canbe mapped by determining the direction in which a shear wave propagates with thegreatest velocity, and could be used to detect myofiber disarray associated with car-diac pathologies such as postinfarction myocardial remodelling and hypertrophiccardiomyopathy [181]. White matter tracts in the brain have been observed toact as waveguides for propagating waves and can be modelled using a transverseisotropic model, which has potential for improving accuracy when monitoring clin-ical changes in the brain [273]. Elastography measurements of the breast in two pa-tients with fibroadenoma and invasive ductal carcinoma found regions of elevatedfibre organization corresponding to the two masses, however differentiation was notpossible [291].NonlinearityThe generalized Hooke’s law in Equation 1.30 assumes a linear relationship be-tween stress and strain. However, this assumption only holds over a small rangeof strains for most soft tissues. For example, the stress-strain relationship is some-times approximated by three regions; two quasi-linear regions with a relatively lowelastic modulus at low stress and a relatively high modulus at high stress, and aregion between where there is a constant change in the gradient [89]. The actualstress-strain relationship in real tissue is likely more complex.There is significant diagnostic potential in differentiating tumours from mea-surements of nonlinear elastic properties. For example, compression tests of ex vivoprostate tissue samples showed a twofold increase in elastic modulus for canceroustissue between pre-compression levels of 2 % and 4 %, while there was a negligible34Figure 1.13: Example of an rf signal before and after tissue compression.The data were collected from an Ultrasonix SonixTouch scanner witha L14-5/38 linear array transducer while applying freehand compres-sion with the transducer to a homogeneous polyvinyl chloride tissuemimicking phantom.change over the same range for normal tissue and benign prostatic hyperplasia [171].Another study examining ex vivo breast tissue compared five hyperelastic modelsand found almost all of the parameters corresponding to pathological tissues werebetween two times to two orders of magnitude larger than those from normal tissue,suggesting nonlinear elastography could distinguish between normal and patholog-ical tissues with further potential to classify cancers, especially invasive lobularcarcinoma [228]. An in vivo imaging study of ten patients with benign and malig-nant masses demonstrated the potential of nonlinear elastography in distinguishingbetween the two types, where no statistically significant difference was found be-tween shear moduli for fibroadenomas and invasive ductal carcinomas, however asignificant difference was found in the exponential stiffening parameter of a reducedVeronda-Westmann model describing the nonlinearity of the tissues [116].1.5.3 Motion MeasurementElastography uses measurements of tissue motion to infer the mechanical proper-ties of tissues using the theory described in Section 1.5.1 and Section 1.5.2. Thefollowing section describes the measurement of tissue motion using medical ultra-sound.35Speckle TrackingAs mentioned in Section 1.4.2, small scale acoustic inhomogeneities create specklepatterns. If it is assumed that a group of scatterers produces a unique speckle pat-tern, and the motion of the tissue results in corresponding motion of this pattern,then motion can be measured by matching the speckle pattern over successive sig-nal acquisitions [130]. This assumption holds up well for uniform translation [43],or for small deformations (stretching or compressing) [207], but does not hold forgeneral motion (e.g. for rotations greater than a few degrees [156]. For the motionconditions usually encountered in elastography it has proven adequate.Figure 1.13 shows an rf signal of tissue before and after undergoing static com-pression using the probe face to indent the tissue. The post-compression signalclearly has a similar shape to the pre-compression signal. Looking at only a smallregion of the signal, the change between the two signals can be modelled as a timeshift, or temporal translation. Dividing echo signals into small regions, or blocks,and determining the translation between blocks from images captured at differenttimes by finding the best match is the basis of speckle tracking tissue motion mea-surement. Blocks are selected to be large enough to adequately capture the specklepattern, while small enough such that the change in the block between frames ap-pears as mainly a shift, instead of a stretch which would change the shape of thesignal and make it difficult to match [74].One method of determining the translation of a block between images is todefine a pattern matching function which is maximized/minimized by varying thetranslation. The most commonly used method is maximizing the normalized cross-correlation, which with the common assumption that the rf is zero mean is com-puted between two signals s1 and s2 bymaximizeuR(u) =B/2∑i=−B/2s1(i) · s2(i + u)√B/2∑i=−B/2s1(i)2√B/2∑i=−B/2s2(i + u)2(1.41)where i is the sample index, u is the integer sample translation, and B is the numberof samples in a block. The normalized cross-correlation has proven to be accu-36rate and robust, however it can be computationally expensive, creating difficulty intracking tissue motion at rates similar to the ultrasound frame rate. Also, the cycli-cal nature of the rf signals can lead to finding false matches, called “peak-hopping”errors due to the highest correlation coefficient occurring at a secondary peak in thecorrelation function [252]. Applying a spatial filter post-processing step can help toreduce the random peak-hopping errors [270]. Starting the search with large blocksand iteratively using smaller blocks (coarse-to-fine) is another approach to reducethe peak-hopping error with a small increase in computational load [62]. Both thespeed and peak-hopping problems can be mitigated by bracketing the search re-gion using the previously computed translation from the neighbouring block [345].Speed can also be improved by using different matching functions with reducedcomputational complexity, such as minimizing the sum of absolute differences orsum of squared differences between signal blocks, where squared differences pro-vide similar motion estimation precision to normalized correlation, while absolutedifferences generally performs with less precision [326].So far this section has discussed estimating motion to the nearest digital sam-ple. For common system parameters of 40 MHz sampling rate and 1540 m/s speedof sound, the spacing between samples computed using Equation 1.1 is 19.25 µm.Often a finer precision is desired so interpolation is used. One option is to inter-polate between the samples of the rf signals, however this becomes increasinglycomputationally expensive for finer precision [253]. Alternatively, the peak of thecorrelation function can be estimated using discrete computations of the correla-tion coefficient, either exactly using sinc interpolation, or more commonly usingparabolic or cosine interpolation between the three largest samples. Parabolic andcosine interpolation methods are popular because they are computationally inex-pensive, however they introduce cyclical bias errors [57].A popular alternative to cross-correlation peak searching is phase-based track-ing. If the rf signal, s(t), is represented analyticallysˆ(t) = s(t) − js˘(t), (1.42)where s˘(t) is the one dimensional (1d) Hilbert transform of s(t), then the magnitudeof the complex cross-correlation between two acquisitions will contain the same37maximum as the rf cross-correlation function, and the phase of the complex cross-correlation will cross zero at the location of the maximum [196]. Alternatively, ifbaseband i/q signals are used, the phase of the complex cross-correlation is propor-tional to the displacement at the location of the maximum [226]. The phase-basedmethods attempt to find the shift that produces a phase angle of zero (for analyticsignals), or alternatively finds the phase angle between the two signals (for base-band signals). This second description is reminiscent of the phase-based trackingin Doppler Colour Flow, but in this case only two images are used instead of theseveral measurements making up a Colour Flow ensemble. Removing TPRF fromEquation 1.24 produces a measurement of displacement along the axial beam di-rection. Phase-based speckle tracking which operates on i/q data produces a mathe-matically equivalent result using analytic rf signals, and further, is mathematicallyequivalent to cross-correlation speckle tracking if it is assumed that the phase varieslinearly and that motion produces only shifts of the rf signal [194]. For typicalsignal-to-noise ratio (snr) in elastography imaging scenarios (≈ 30 dB) and smalldisplacements (<25 µm) the phase-based Doppler algorithms can provide similarperformance to normalized cross-correlation speckle tracking with the advantageof approximately an order of magnitude increase in computation speed [254]. Onechallenge of using the phase-based tracking method for elastography is aliasingwhich will occur for any displacement exceeding (cpi)/(2ω0) which is typicallyaround 0.25 mm to 1.0 mm. Aliasing can be reduced by using the neighbouringblock’s displacement as an initial guess in an iterative Newton search for the rootof the cross-correlation phase [251].While most speckle tracking methods assume that the speckle pattern does notchange due to motion, under large deformations it may be necessary to remove thedistortions in the pattern. One approach leads to a model that shows the referenceimage is a filtered version of a motion corrected image, under the assumptions of alinear image formationmodel and that acoustic impedance is not affected bymotion.The filter is dependent on the motion, and corrections for the speckle changes canbe applied by solving for an affine transform that minimizes the difference betweenthe filtered image and the reference image [202]. Companding is another approachwhich warps the reference image by first applying a coarse 2d shift and stretch, thena fine scale 2d shifting, before a final pass of correlation based motion tracking38along the axial direction [60].A different approach to speckle tracking is to track specific features detectedfrom the speckle patterns. Features of the rf signal such as the zero crossings canbe detected using linear interpolation [298], or the peaks using the zero crossings ofa wavelet transformed rf signal [95]. Instead of providing the motion measurementfor a block of the image, feature tracking methods provide a measurement for everyfeature location. This generally leads to a higher spatial density of measurements,although the spacing may be nonuniform. The feature tracking algorithms are lesssusceptible to signal decorrelation due to stretching, however the algorithms can bemore sensitive to signal noise.A continuous representation of the rf data, for example by fitting a piece-wisecubic spline, allows for an analytical solution to the best match of the pattern match-ing function. Blocks of discrete data from the tracked frame can be compared tothe continuous reference frame, and the sum squared error between the two can beminimized by finding where the derivative is zero [325]. This method provides lowbias and variance in the motion estimates, however finding the roots of the fifthorder polynomial representing the analytical derivative of the sum squared erroris generally more computationally demanding compared to other speckle trackingtechniques. The size of the signal block used to compute the squared error can bereduced all the way down to a single sample, providing high resolution and com-parable accuracy and precision to typical block sizes (on the order of 1 mm), withthe trade-off of greater sensitivity to signal noise [346]. Representing both the ref-erence and tracked signals using splines improves accuracy and precision but iscomputationally more expensive [253].Speckle tracking methods can be extended to measuring 2d (adding lateral) and3d (adding elevational) displacement vectors by extending the pattern matchingsearch to those directions, with the exception of phase-based methods because theoscillations at the carrier frequency of the pulsed wave normally only occur alongthe beam (axial) direction, which means that phase changes cannot be observed inthe non-axial directions. The best match (usually the peak of the cross-correlationfunction) can be found independently in each direction [347], iteratively in eachdirection [105], or interpolated by fitting a 2d/3d function to the correlation co-efficients [67, 349]. The accuracy and precision of the estimates in the lateral and39elevational directions are poor compared to the axial direction, which is in large partcaused by a larger sample spacing (typical element pitch is on the order of 300 µmcompared to digital sample spacing of 19.25 µm), poorer beam resolution in thenon-axial directions [195], and a typical lack of phase information in the non-axialdirections [42]. Additional rf lines in the lateral and elevational directions can beinterpolated to improve the sample spacing [165]. The resolution in the non-axialdirections can be improved using a synthetic aperture beamforming approach, how-ever this requires a system that can save pre-beamformed received rf data for eachelement [167]. Phase information can be introduced into the non-axial directionsto improve tracking accuracy. One approach introduced phase information in thelateral direction synthetically by removing either the negative or positive half ofthe lateral spectrum [66]. Another approach that several groups have investigatedis modifying the apodization weightings used in beamforming to introduce oscilla-tions in the other directions [15, 149, 186, 302].A different approach for estimating 2d and 3dmotion vectors is to combine ax-ial measurements from overlapping beams from different directions, often referredto as angular compounding. This takes advantage of the superior measurementaccuracy of the axial speckle tracking and uses known geometry of the beams toreconstruct the components of the motion vector mathematically. As only measure-ments along the beam are used, this technique can also be used with conventionalDoppler based velocity tracking and is termed Vector Doppler. The different axialbeams can be generated electronically by introducing beamforming delays and us-ing multiple transmit and receive frames at different angles [125, 268], or a singletransmit and multiple receive angles computed digitally for a faster sampling ratebut poorer snr [170, 310]. Instead of electronic beamforming methods, multipleangular frames can be created by moving a transducer. For example, a phased ar-ray or curvilinear array transducer can be mechanically translated laterally, whereeach beam line in a conventional image can be rearranged to create an angled im-age from each translated location [313]. Another example is rotating a transduceraround the roi [211]. Multiple axial beam angles can also be acquired using mul-tiple transducers directed towards the same roi [4, 33, 90]. An advantage of themoving transducer and multiple transducer techniques over electronic beam steer-ing is that larger inter-beam angles, and thus greater measurement accuracy, can be40achieved. A disadvantage is that the moving and multiple transducer techniques re-quire calibration to spatially align the measurement from each transducer location.The geometry of electronic beam steering methods lends itself well to imaging su-perficial tissue, such as the carotid arteries, but not for deeper tissue such as liver.The opposite is generally true for moving and multiple transducer methods.Vibration DopplerConventional Doppler, as described in Section 1.4.6, measures the average velocityof a steady and slowly varying flow, and is not suitable for measuring vibrating tis-sue. While Colour Flow algorithms can be modified to measure the displacementthrough each gate or block between individual frames as described in the previoussection, this requires access to the unprocessed i/q data. It is still possible however toestimate steady state vibrations using the output of the Doppler processing (i.e. theDoppler spectrum). A vibrating tissue scatterer modulates the Doppler power spec-trum, with harmonics spaced proportional to the vibrating frequency and weightedby Bessel functions of the first kind [133]. The modulation parameter of the Besselfunctions is related to the vibration amplitude asβ =2ξω0c0, (1.43)where ξ is the vibration amplitude along the ultrasound beam axis, ω0 is the centrefrequency of the ultrasound pulse, and c0 is the propagation speed of the ultrasoundpulse.Estimating the modulation parameter, β, leads to an estimate of the vibrationamplitude. There are two common methods to do this; one based on estimating thestandard deviation of the power spectrum, and the other based on ratios of spectralharmonics. The standard deviation of the Doppler power spectrum is related to themodulation parameter through [144]β =√2σωe, (1.44)where ωe is the frequency of the vibration.The amplitude ratio of adjacent Bessel bands in the Doppler power spectrum41are related to the modulation parameter through [339]Ai+1Ai= Ji+1(β)Ji (β) , (1.45)where Ai is the amplitude of the ith spectral band, and Ji is the ith order Besselfunction. The ratio can be computed as a function of β beforehand, and then usedto estimate β during the experiment.The phase of the vibration can be estimated directly from the phase of theDoppler spectra [339].1.5.4 Elasticity EstimationMethods for generating images depicting tissue elasticity have been in developmentfor over 20 years, resulting in a diverse set of techniques. The methods can beroughly categorized by the type of tissue excitation applied (static/dynamic) andthe type of measurement provided (relative/absolute).StrainUnder conditions of purely uniaxial compression, from Equation 1.32 and Equa-tion 1.33, Hooke’s law reduces toE =σ1111. (1.46)Thus, the problem of determining tissue elasticity is reduced to finding the ratio ofstress to strain in the direction of compression.Strain can be computed from ultrasound based motion measurements (see Sec-tion 1.5.3) using the slope from a least-squares line fit over a small spatial win-dow [155]. However, it is often difficult to measure the stress distribution in thetissue. A common assumption is a uniform axial stress field. Using a large com-pressor area can help mitigate errors in this assumption. Since the magnitude ofthe stress is usually not known, only a relative estimate of the elasticity is possible.Commonly the strain is displayed instead of the relative Young’s modulus [233].Instead of using an extra apparatus for applying compression, it is common touse the ultrasound transducer face to apply quasi-static tissue compression [123]. In42this situation, the pressure is generally higher near the transducer face compared todeep tissue, resulting in a “hardening” artefact which increases apparent elasticitywith increasing depth [234].The relation between strain and stress given in Equation 1.46 applies to thecase of uniaxial compression, but achieving this type of loading inside the body isdifficult in practice. More generally, the relation between axial strain and stress is11 =1E(σ11 − νσ22 − νσ33) . (1.47)Poisson’s ratio is usually assumed to be approximately 0.5 because most soft tis-sues are nearly incompressible, leaving the three stresses and Young’s modulus asunknown. It is possible to attempt to estimate the three components of stress usingan analytical model and knowledge of the boundary conditions and compressor ge-ometry [256]. However, the stress concentrations caused by elastic heterogeneitiesare difficult to predict because the underlying Young’s modulus distribution is un-known, and hence the 1d model of Equation 1.46 is usually assumed for strainelastography.The displacement estimation used for computing strain images produces sev-eral artefacts. For example, horizontal “zebra” stripes, caused by cyclical bias er-rors introduced by sub-sample interpolation in correlation based speckle tracking,and smaller “worm” artefacts caused by large signal overlaps in speckle trackingcreating correlated noise patterns [235]. The mechanical compression of the tissuealso causes compression of the backscattered signal, meaning the block matchingcan only match in a certain region of the block, where the location of the matchwill be biased toward the higher amplitude region of the block [58]. This causeserrors in strain computation because the distance between displacement estimatesis assumed to be the distance between block centres. Methods to help mitigate thislocation error include reducing the dynamic range of the echo signals [11], scalingand shifting the pre-compression signal to improve coherence during block match-ing [60], or correcting for the location of the estimates [188].In addition to inferring relative Young’s modulus distributions, strain imaginghas other clinical uses, such as using shear strain to assess mobility breast lesionsto help differentiate between malignant and benign lesions [314], or in measuring43poroelastic behaviour [271]. Creep tests using strain imaging have been used to esti-mate time constants associated with the extracellular environment of breast lesions,where a shorter time constant has been demonstrated for malignant masses [265].Another class of elastography methods using strain measurements is directedtoward finding a relative distribution of the tissue elasticity based on solving par-tial differential equations. Starting with Equation 1.28 under conditions of staticcompression and no body forces, the equation of equilibrium is obtainedσi j, j = 0. (1.48)Combining compatibility with Hooke’s law as in Equation 1.32, equations relating,strain, spatial derivatives of strain, and spatial derivatives of shear modulus canbe obtained. One approach solves the equations under plane strain conditions tofind the spatial distribution of the relative shear modulus over 2d using finite differ-ences [294]. A different approach solves the equations under plane stress conditionsfor the gradient of the relative shear modulus in 2d and calculates the logarithm ofthe relative shear modulus by integrating the gradient over a path using the relation-ship between the logarithm and its derivative [305]∂ log(g(x)g(x0))∂x=1g(x)∂g(x)∂x, (1.49)where g(x0) represents a reference shear modulus value at location x0. Both ofthese approaches suffer from requiring third order derivatives of noisy displacementmeasurements.An alternative approach that reduces sensitivity to derivatives is using a weakformulation of the equations of motion and elasticity in a finite element method(fem). The elasticity can be determined using an iterative approach to minimize afunctional summing the squared difference between the measured tissue displace-ments and the displacements computed by the fem using the current estimate ofelasticity [154]. While an initial guess is required for the elasticity, a spatially uni-form value can be used and does not affect solution [86]. The elasticity is updatedfor each iteration by computing the gradient of the functional with respect to theunknown elasticity parameters (e.g. the Young’s modulus for every finite element),44resulting in a Jacobian matrix. The columns of the Jacobian matrix, which corre-spond to the gradient of the functional with respect to a single elasticity parameter,can be computed by solving the fem forward problem where the forcing function ischanged to the product between the current fem computed displacement field andthe partial derivative of the stiffness matrix with respect to the elasticity parameter.This means the Jacobian takes N forward fem solutions, where N is the numberof elements, which is computationally demanding. Instead of the computing theJacobian, the gradient can be obtained more efficiently using the adjoint methodwhich only requires two forward problem solves per iteration independent of thenumber of elasticity parameters [225]. Usually the iterative fem approaches ig-nore the lateral tissue displacement component because of the poorer displacementmeasurement accuracy in this direction and the sensitivity of the inverse problem toerrors in displacement measurement. Regularization can be used to help solve theequations to compensate for missing data and noise. Typically a plane strain modelis assumed to transform the elasticity problem to 2d to accommodate displacementmeasurements obtained using 2d ultrasound. The fem problem can also be solveddirectly by rearranging the equations in terms of the elastic parameters, since thediscretized system of equations is linear with respect to the elastic parameters [354].A slightly different use of fem in an iterative approach has been proposed toobtain a relative modulus distribution based on Equation 1.47 using strain mea-surements along the axial direction [255]. The method uses an initial guess for themodulus distribution and uses a 3d fem to estimate stresses. The modulus is thenupdated at every pixel location using the estimated stresses and the axial strain mea-surements, and the process is repeated until the modulus converges. The methodrequires an assumption for the Poisson’s ratio which is typically assumed a constant0.495.There has been some work on creating absolute rather than relative measure-ments of Young’s modulus by also measuring the stress distribution over the areaof contact between the ultrasound transducer and the tissue. Pressure sensors canbe mounted around the outside of the transducer and used to improve estimating 3dstress field [344]. Similarly, a thin film pressure sensor can be placed directly underthe transducer with a small reduction in ultrasound signal quality [318]. Instead ofusing pressure sensors, a thin compliant layer with known elasticity can be placed45between the transducer and the tissue. The strains can be measured in the compliantlayer in the same way as is done in tissue, but since the elasticity is known the stressmay be computed. This may be accomplished using Equation 1.46 and assumingconstant stress along each ultrasound beam line [201], or using a 3d fem modelof the compliant layer with some assumptions about the underlying tissue [160].While these methods allow for quantification of elasticity values and to correct fornonuniform pressure at the surface of the tissue, the methods cannot reliably predictthe stress concentrations caused by inclusions.AmplitudeSimilar to strain imaging, amplitude based elastography displays relative differ-ences in tissue stiffness based on displacement measurements. These methods usea dynamic excitation source as opposed to a static source in strain imaging, and themaximum amplitude of the displacement over the measurement time is displayed ateach location. An increase in tissue stiffness results in a decrease in displacementamplitude. For example, an inhomogeneity with an area 0.5 % of the surroundingtissue with a Young’s modulus three times greater than the surrounding tissue canbe detected by a 20 % drop in shear wave vibration amplitude [113].“Sonoelasticity” imaging, an amplitude based elastography method, appliessteady state vibrations over a band of about 20 Hz to 1000 Hz to the tissue. Theamplitude of the wave pattern is measured using the vibration Doppler techniqueof estimating the power spectrum standard deviation as described in Section 1.5.3.Sonoelasticity methods typically create shear waves in the tissue by applying vibra-tions on the surface that radiate like a point source such as using an acoustic hornwith a tapered cone [183] or a metal rod acting like a piston [312]. Standing wavesin the vibration field produce modal patterns with areas of increased or decreaseddisplacement which interferes with the detection of elastic inhomogeneities. Toreduce this effect, multiple excitation frequencies can be applied simultaneouslyto excite a greater number of eigenmodes, which will tend to smooth the overalldisplacement field [312].An alternative to using vibrating sources to produce shear waves in tissue isto use the acoustic radiation force (arf) to create impulsive excitations. High in-46tensity acoustic beams are focused in the tissue using the same ultrasound trans-ducer as used for b-mode imaging and displacement tracking, creating localizeddisplacements on the order of tens of microns over a volume approximately 1 mm3in size [222]. Tissue displacements are typically tracked using correlation basedtechniques [254]. The advantages of arf techniques are that the nonuniform stressand modal artefacts seen in the stain and sonoelasticity methods respectively areeliminated, the force can be applied anywhere the imaging system can focus, andno extra equipment is required for applying the tissue excitation. The disadvantagesare risks of tissue heating, limiting the number of pushing pulses, and the require-ments on the ultrasound scanner being able to produce the high voltages and beamsequences to create the push pulses.Transfer FunctionConsidering the tissue as a linear dynamic system, transfer functions of the tissuemotion can be used to describe its mechanical response as a function of frequency.The tissue motion can be described by [319]Md2u(t)dt2+ Bdu(t)dt+Ku(t) = f(t), (1.50)where u(t) and f(t) are vectors of tissue displacement and force as a function oftime, and M, B, and K are matrices representing the mass, damping, and stiffnessof the tissue. The matrices are dependent on the mechanical model, but do nothave to be specified if model parameters are not desired. The Fourier transform ofEquation 1.50 yields the transfer function at a specified location in the tissueH (ω) =u(ω)f (ω)=1−ω2m + jωb + k . (1.51)From Equation 1.51 it is clear that at ω = 0 the transfer function contains informa-tion only about the compliance (inverse of stiffness) of the tissue. Similar to strainimaging, the force is usually not known so a relative estimate of compliance can beobtained by replacing the force with a reference displacementf = krur . (1.52)47If the reference stiffness is set to unity, then the relative compliance between thereference and the ith displacement location is [281]H ir (0) =  ui (0)ur (0) . (1.53)The transfer function can be computed from the power spectral densities ofcorrelation speckle tracking measurementsH ir (ω) =Pir (ω)Pii (ω), (1.54)where Pir (ω) is the cross-spectral density between the reference and ith displace-ment, and Pii (ω) the spectral density of ui.Usually it can be assumed that over a small band of low frequencies that themagnitude of the transfer function is flat and therefore the stiffness estimate canbe averaged over this band to reduce errors due to measurement noise. A measureof confidence can be used to weight the transfer function over the low frequencyband. The coherence function indicates what portion of the input energy from urat frequency ω appears at the output in ui at the same frequency [281]Cir (ω) =Pir (ω)2Pii (ω)Prr (ω). (1.55)The coherence function covers the interval [0, 1], where a value close to 1 indicatesa linear system with high snr. The final weighted intensity value for display can becomputed via [210]Ii =ω2´ω1Cir (ω) · H ir (ω) dω(ω2 − ω1)ω2´ω1Cir (ω)dω, (1.56)where ω2 and ω1 are the upper and lower bounds of the frequency band. Typicallythe excitation is band-pass filtered white noise (ranging from 1 Hz to 20 Hz), appliedeither via the transducer or through a separate shaker. Compared to strain imaging,transfer function images have demonstrated lower variance and greater accuracy48which may be due to including more temporal and frequency information in theestimation [319].The magnitude and phase of the transfer functions over a broader frequencyrange can also be used to estimate the viscoelastic parameters of amaterial model [97].This is typically done using a mechanical vibration source. An alternative methodis to use the arf to generate a step response in the tissue. The temporal deriva-tive of the step response is the impulse response, and the Fourier transform of theimpulse response is the transfer function between the applied force and tissue dis-placement [110].Transient Shear WavesAs mentioned in Section 1.5.1, the behaviour of shear waves is directly related totissue elasticity. Transient techniques use a short duration, impulse-like excitationto generate a shear wave. The propagation of the wave is measured using speckletracking techniques over time as the wave travels through the roi, and the speedof the wave is inferred from the motion measurements. From Equation 1.39, theshear wave speed is proportional to the square root of the shear modulus. Typicalshear wave speeds in soft tissue are between 1 m/s to 5 m/s, so the motion must besampled at a relatively high rate to capture the evolution of the wave over time. Thepropagation speed of the ultrasound pulse is about three orders of magnitude greaterthan the shear wave speed, so it is fast enough to sample the motion, however thesequential nature of forming a 2d image results in a frame rate which is too slow tosample the shear wave. Instead thewave can be tracked by using rapid interrogationswith a single beam line, repeated with different laterally spaced beam lines acrossthe roi [223]. Alternatively, full channel parallel receive beamforming hardwarecan be used which can sample the entire roi using a planar transmit wave [310].Mechanical excitation can be used to generate the shear waves, such as usinga single element transducer as a piston [283]. The shear wave speed is estimatedusing linear regression on the phase of the wave as a function of depth, and can berelated to the shear modulus through Equation 1.39. The downside of the regressiontechnique is that spatial information is lost (i.e. elastic inhomogeneities whichwouldresult in different shear wave speeds along the imaging depth are not captured). This49approach is the basis for the commercial FibroScan and FibroTouch systems usedfor liver fibrosis examination.A similar approach is used for 2d imaging, where the shear wave is generatedby applying a short pulse using the face of a linear array or two rods on either sideof the array in the elevational direction, which creates a shear wave that propa-gates primarily along the beam axis and does not significantly diffract in the eleva-tional plane [32, 284]. The 2d approach retains spatial information by estimatingthe shear modulus algebraically using a simplified Helmholtz equation, based onEquation 1.39, using only the axial component of motion and ignoring out of planederivatives. The simplified Helmholtz equation can be transformed to the frequencydomain, either to restrict the inversion to a specific band where the snr is high, orto examine the complex wave field for viscoelastic parameter estimation [54].Transient shear waves can also be created using the arf which creates a point-like source that radiates in an approximately cylindrical shape [288]. The shearmodulus can be estimated using the same simplified Helmholtz algebraic inver-sion [221], however due to the sensitivity to noise when taking second-order deriva-tives in the algebraic Helmholtz approach, time-of-flight methods have become apopular alternative. The time-of-flight methods assume lateral propagation of thearf induced shear wave, and measure the time for the peak displacement to occur atlaterally spaced locations (adjacent beam lines) [241]. It is also possible to track theshear wave propagation along the pushing beam axis, however it is more challeng-ing as it can be difficult to track the motion along this path, and the time-of-flighthas a depth dependence and is affected by the spatial distribution of the arf, lead-ing to greater variance in the elasticity measurements [329]. Another method forgenerating transient shear waves is to apply the arf push at different depths in rapidsuccession (faster than the speed of the shear waves). The constructive interferenceof the shear waves from each source creates a planar shaped wave front with greateramplitude, termed the “Mach cone” [34].In most arf techniques, an image is constructed from repeated interrogations toenable imaging a large field of view and to average measurements to increase snr.A single focal arf excitation can enable measurement over an area approximately2 cm × 2 cm [223], with the axial extent limited by the arf depth of focus and thelateral extent limited by shear wave attenuation. A “Mach cone” type of excitation50can enable a larger measurement region, however repeated interrogations are stillusually required as measurement along the push beam is unreliable because thereis no lateral wave front to track and it is still desirable to improve snr throughaveraging [311].In the transient arf techniques described thus far, each interrogation consistsof a single push beam and several tracking beams spaced laterally around the pushlocation to measure the time-of-flight. It is also possible to measure the shear wavetime-of-flight by using a single tracking beam at a fixed location and sequentiallymoving the push beam laterally around the tracking location. This method has theadvantage of cancelling out position dependent speckle bias in the motion trackingbecause the motion is always biased in the same way when the tracking beam isfixed. The disadvantage is that several pushes need to be applied which generallyincreases the acquisition time [134].In an effort to increase acquisition speed for transient arf techniques, a push-ing strategy that combines unfocussed arf beams from multiple small spaced sub-apertures is used to generate multiple shear waves throughout the roi, a techniquetermed comb-push [295]. The resulting waves interfere in a complicated patternmaking shear wave speed estimation more difficult, however the tissue motion canbe directionally filtered to examine waves only propagating from left-to-right andfrom right-to-left. The wave speed can be easily detected from each of these filteredimages and averaged to create a 2d elasticity image from a single arf interrogation.In a technique specifically focused on measuring the frequency dependenceor viscoelastic properties of tissue, shear wave dispersion ultrasound vibrometry(sduv) applies a narrow-band oscillatory arf by modulating the acoustic ampli-tude. Similar to time-of-flight measurements, the shear wave is interrogated at dif-ferent laterally spaced locations, but speed is calculated using the known excitationfrequency and by measuring the phase change of the wave over the lateral distance(i.e. the spatial frequency) [63]. This is typically repeated for different excitationfrequencies to measure the shear wave dispersion.The sduv technique uses excitation at a known temporal frequency and mea-surements of the spatial frequency of the shear wave to determine the shear wavespeed. It is also possible to do the opposite by using an excitation of known spatialfrequency and measurements of the temporal frequency, a technique termed spa-51tially modulated ultrasound radiation force (smurf) [203]. In smurf, the arf ismodulated in the lateral direction, x, taking the formF (x) = (1 + cos (kx)) ψ (x) , (1.57)where ψ (x) is a smoothly varying, positive envelope function, such as a Gaussian.The modulation can be created by weighting the transmit apodization with a pair oflaterally spaced Gaussians, intersecting two unfocussed (planar) beams at obliqueangles to each other, or by rapidly firingmultiple push beams laterally. The resultingaxial velocity, which can be measured using speckle tracking techniques, takes theformu˙y (x, t) = (1 + cos (kx) cos (ωt)) ψ (x) . (1.58)After finding the temporal frequency, ω, from the axial velocity at any trackinglocation, the shear wave speed can be calculated at the focus of the applied forceusingcs =ωk. (1.59)By varying the spatial frequency of the force, several temporal frequencies can beexamined to observe the dispersion in a similar manner to sduv.Steady-State Shear WavesSteady-state elastography techniques use a continuous harmonic excitation to gener-ate shear waves in tissue. Instead of directly measuring the shear wave propagationspeed as in transient techniques, steady-state techniques measure the shear wave-length or wave number (inverse of wavelength) to relate motion measurements toelastic properties. The wave speed is related to the wave number as shown in Equa-tion 1.59. From Equation 1.39, the wave number is inversely proportional to thesquare root of the shear modulus.The tissue is typically excited using vibrations applied to the tissue surface atfrequencies ranging from 10 Hz to 1000 Hz [138]. The cyclical nature of the exci-tation has made it popular amongst mre techniques as the acquisition can be syn-chronized with different phases of the excitation [214, 278]. For ultrasound elas-tography, tissue motion may be sampled using a sequence similar to Colour Flow52Doppler and compensated using the known acquisition delays [21], or captured us-ing full parallel receive hardware [205].One class of elasticity estimation methods using steady state shear waves is al-gebraic inversion, which directly solves the partial differential equations describingelastic wave motion for the elastic parameters. One approach is to measure andmodel the full wave field, however this approach can be sensitive to tissue incom-pressibility since second derivatives of displacement are multiplied by a relativelylarge value (see Equation 1.38) [290]. If it is assumed that the tissue is fully incom-pressible, then the motion field will only contain shear waves and the shear moduluscan be recovered by measuring only one component of the motion vector over a vol-ume using Equation 1.39 [231]. Instead of ignoring compression waves, they can beremoved completely by applying the curl operator to the displacement field beforeinversion, however this requires measurement of the full 3d vector field over a vol-ume and results in taking third order spatial derivatives of measured displacementfor the solution [292].Similar to the strain methods, the equations of motion can also be describedusing a weak formulation and discretized using the fem. Again, a functional min-imizing the squared error between the measured displacements and the displace-ments produced by the fem for the current elasticity estimate can be iterativelysolved [321]. The adjoint method can also be applied to conditions with steadystate harmonic excitation to reduce the computational load of the gradient calcu-lation, as was demonstrated for a 2d plane stress condition [352]. The equationsof motion can also include viscoelastic parameters, however the solution of theseparameters can be more sensitive to noise [96].Instead of solving an optimization problem iteratively, the fem problem canalso be solved directly by rearranging the equations in terms of the elastic param-eters. The equations can be rearranged to directly solve for both Lamé parameters,however this approach can be sensitive to noise for tissue that is nearly incompress-ible [274]. Instead, the tissue can be assumed to be nearly incompressible witha guess for Poisson’s ratio which leaves Young’s modulus as the only unknown(optionally also including viscosity) [99]. Another approach to dealing with near53incompressibility is to introduce a pressure termp = λuk,k, (1.60)where λ is the first Lamé parameter and pressure is treated as an unknown quan-tity [243]. This pressure term can be substituted into Equation 1.37 and discretizedusing the fem, avoiding the homogeneity assumption, which has been shown tointroduce artefacts into the elasticity estimation. Sparsity inversion has been pro-posed to help condition the problem with the additional unknowns introduced byincluding the pressure term [135]. The pressure term can be eliminated to reducethe number of unknowns by applying the curl operator. The curl can be applied di-rectly to Equation 1.37 before being discretized, again avoiding the assumption oflocal elastic homogeneity [136]. These methods generally require measurement ofthe full motion vector over a volume. Alternatively, the compression and couplingterms can be removed, while still avoiding the homogeneity assumption, resultingin independent equations relying on only one component of themotion vector [137].A different approach to determine the elasticity is to extract the wavelength fromthe measured displacement data. The wave length can be determined by measuringthe length between peaks or troughs of the waves [20, 180, 339], fitting a sinusoidalong a profile of the displacement data [88, 337], or by computing the gradient ofthe phase of the wave along a direction [184, 199]. The drawback of these methodsis that they assume that the shear wave is propagating locally as a plane wave andthey require an estimation of the propagation direction.The local frequency estimation (lfe) algorithm estimates the wave number ofthe measured displacement without the assumption of plane wave propagation. Asmentioned previously, under the assumptions of incompressibility and homogene-ity, the tissue motion can be described with the Helmholtz equation taking the formof Equation 1.39. Oliphant et al. [230] showed that the Helmholtz equation can besolved using the ratio of two filters that are related through spatial differentiation.The lfe algorithm makes use of a special case of these pairs of filters: lognormalquadrature filters defined as the product between radial and directional components.A series of filter pairs with different centre frequencies and bandwidths are used toobtain a wideband estimate of the wave number [163, 214]. The lfe approach may54use only a single component measured over a volume or a 2d plane if out-of-planespatial derivatives are assumed to be negligible. If all components of the motionvector are measured, lfe can be applied to each direction and the results can beaveraged [199]. A disadvantage of the lfe approach is that the spatial resolution ofthe estimate is dependent on the centre frequency and bandwidth of the filter pairthat best match the wave number.While the lfe algorithm naturally contains directional filtering though its fil-ter definition, the techniques based on a plane wave assumption can be improved byadding a directional filtering preprocessing step which helps to reduce the complex-ity of the wave field from interfering wave fronts. Further, low-pass and high-passfilters can be applied to reduce noise in the displacements and the effects of thecompression wave, respectively [200].Instead of using a directional filter to remove the effects of interfering wavepatterns, a model can be designed that incorporates waves travelling in different di-rections. This approach is used in the Travelling Wave Expansion technique, wherethe general solution to a Helmholtz decomposition of the displacement field is rep-resented as an integral over a unit sphere [22]f (x) =‹Sa (φ, θ) ejk (n(φ,θ) ·x) dφ dθ, (1.61)where this function represents waves with amplitude a and wave number k travel-ling in the direction n. The problem is solved by choosing values for a and k thatminimize the error between the model and the measured data over small regionsassumed to have constant mechanical properties.In another technique, the centroid of a thresholded spatial Fourier transformof the displacement field is used to determine the principal wave number in a re-gion [204]. The advantage of this method is that it performs well in the presence ofnoise in the displacement measurements compared to lfe. The disadvantages arethat location information is lost so the entire region must be assumed homogeneous,and the region must be large enough to contain at least one wavelength.The above techniques make use of both the magnitude and phase of the wavemotion. When only the amplitude of the waves is available, such as when measur-55ing motion using the vibration Doppler method described in Section 1.5.3, specialwave patterns proportional to the shear wavelength or speed can be generating us-ing two vibration sources. When two sources are placed on either side of a roi anddriven at the same frequency and amplitude, the technique is referred to as the staticinterference approach, and the amplitude of resulting interference pattern has a spa-tial period equal to half the shear wavelength, assuming plane waves are emitted bythe vibration sources. When the two sources are driven with a small frequency dif-ference, one atω and the other atω+∆ω, the pattern moves slowly towards the lowfrequency source at a speed proportional to the shear wave speedcpattern ≈ ∆ω2ω cs . (1.62)Because a small frequency difference is used, the observed speed is less than thetrue shear wave speed, resulting in the term crawling waves, and can be measuredwithout the special techniques used for transient shear wave measurement [338].Time ReversalTime reversal elastography measures tissue motion excited by a short transient ex-citation to obtain information about the impulse response of the tissue at a givenlocation. The technique makes use of principles used in Transfer Function, Tran-sient Shear Wave, and Steady-State Shear Wave elastography methods. The timereversal technique relies on two assumptions, namely time invariance and spatialreciprocity. Time invariance is valid if there are negligible viscous losses, as u(t)and u(−t) are both solutions to the elastic wave equation in Equation 1.37. Spa-tial reciprocity implies that the excitation source and measurement receiver can beinterchanged without a change in measurement, which again is valid for losslessmedia. While these assumptions may not hold completely in soft tissues, the errorsintroduced by their violation have proven to be small enough to allow estimation ofthe elastic properties.In an acoustic time reversal experiment, first a short excitation pulse is sentthrough the medium. The scattered waves are recorded, time reversed, and re-applied to the medium from a secondary transducer surrounding all or part of themedium. The opposite scattering propagation occurs through the medium, acting56to refocus the waves toward the original excitation source [103]. A similar approachis used for time reversal elastography, with an additional use of spatial reciprocityto use the same excitation source in both steps to create a focusing of the elasticwave at the measurement location.The time reversal technique can be applied using either the active approach orthe correlation based approach. The active approach is most similar to the acoustictime reversal process just described. In the active approach, first an exciter emitsone to two cycles of a sinusoidal signal, usually in the band 60 Hz to 150 Hz. Thetissue motion resulting from the excitation is recorded at a single point, xo, for alength of time, T , usually on the order of 500 ms. The displacement along the axialdirection measured by ultrasound, ui, can be described in terms of the excitationand the impulse response through convolutionui (xo, t) = fn (xs, t) ⊗ hni (xs, xo, t) , (1.63)where fn is the excitation applied along direction n at location xs, hni is the impulseresponse between the excitation andmeasurement, and ⊗ represents the convolutionoperation. Next, the exciter is programmed with a time reversed recording of themeasured motion from the last step. The axial displacement after time reversal isuTRi (x, t) = ui (xo,T − t) ⊗ hni (xs, x, t) , (1.64)which can be described in terms of the excitation using Equation 1.63uTRi (x, t) = fn (xs,T − t) ⊗ hni (xs, xo,T − t) ⊗ hni (xs, x, t) . (1.65)Using the principle of reciprocity, if the excitation was applied at the measurementlocation, xo, along the axial direction, the resulting displacement at the location ofthe exciter, xs, along the excitation direction would be the same as the measureddisplacementui (xo, t) = un (xs, t) = f i (xo, t) ⊗ hin (xo, xs, t) . (1.66)Since the exciter was programmed to use the time reversed measured displacement,57Equation 1.66 can be substituted into Equation 1.64, resulting inuTRi (x, t) = f i (xo,T − t) ⊗ hin (xo, xs,T − t) ⊗ hni (xs, x, t) . (1.67)The tissue motion at the same measurement location, xo, will now observe a focus-ing of energy at time T . The information about the mechanical properties definingthe resulting wave motion is contained in the impulse responses hin and hni, wherethe observed field is filtered by the excitation [30].The disadvantages of the active approach are that a fully programmable exciteris required and the entire experiment must be repeated for every point in the roi.The correlation based approach takes advantage of signal processing to effectivelyachieve time reversal focusing without needing to re-apply the excitation. Similarto the first step of the active approach, a short duration excitation is applied to thetissue surface and the resulting displacement is measured over a period of time. Thetemporal cross-correlation is computedR (xo, x, t) = ui (xo,T − t) ⊗ ui (x, t) . (1.68)Using Equation 1.63 and Equation 1.66, the cross-correlation can be expressed interms of the excitation and impulse responseR (xo, x, t) = f i (xo,T − t) ⊗ hin (xo, xs,T − t) ⊗ fn (xs, t) ⊗ hni (xs, x, t) . (1.69)Noting that the first, second, and fourth terms make up Equation 1.67,R (xo, x, t) = uTRi (x, t) ⊗ fn (xs, t) , (1.70)demonstrating that the cross-correlation field is the same as the time reversal field,uTRi , filtered by the excitation [31].The cross-correlation can be post-processed in the frequency domain to opti-mize the information obtained. Taking the Fourier transform of the cross-correlationresults in the cross-spectral density between xo and x. All of the frequency contentin the signal can receive equal weighting by dividing by the magnitude of the cross-spectral density, resulting in58P (xo, x, t)|P (xo, x, t) | = ej(φ(x,ω)−φ(xo,ω))ej2ωT , (1.71)where φ is the phase of the impulse response. The constant phase offset can beremoved and a band pass filter applied to limit the signals to the bandwidth of theexcitation.Due to the complexity of the impulse response, or elastodynamic Green’s func-tion, it is difficult to obtain a direct inversion for the shear wave properties from themix of both near and far field compression and shear wave terms [29]. However,two simple methods based on wave length and wave speed estimation have provenuseful. The most widely used to date is the wave length based estimator, probablydue to its robustness to low snr, where it has been confirmed through simulationand experiment that the−6 dB width of the displacement along the axial direction atthe focus time is approximately equal to the shear wave length [55]. The wave speedcan be determined using the phase as a function of x, divided by the frequency ω,which provides the time of flight. The slope of a linear regression to the time offlight provides the speed [31].The time reversal technique works best in highly reflective and reverberant me-dia, as the reverberation acts to increase the effective aperture, increasing the abilityto focus waves [104]. Alternatively, multiple uncorrelated random source can alsoachieve the desired diffuse field [29]. Excitation can be applied through random fin-ger taps, or through passive physiological processes such as arterial pulses to excitethe thyroid and brain, or a combination of arterial pulses, heart beats, and musclecontractions to excite the liver [56, 112, 355]. However, the passive techniques typ-ically result in low displacement snr in ultrasound which can limit the reliabilityof the results.1.6 Ultrasound Spatial Calibration BackgroundUltrasound spatial calibration describes the problem of determining the positionand orientation of ultrasound images with respect to a sensor attached to the ultra-sound transducer, which, in turn, is used to track the movement of the transducerwith a position tracking system. Most commonly this is used for “freehand 3d ul-trasound”, which tracks the movement of a 2d ultrasound image so that multiple59images can be reconstructed into a volume. The same general principles applywhen it is desired to track a 3d ultrasound transducer to collect multiple volumesfrom different positions and orientations. Other applications include tracking nee-dle trajectory in ultrasound-guided needle insertion procedures [218], or ultrasoundguided registration of surgical tools to preoperative ct [340].The position tracking system provides measurements of the sensor mounted onthe ultrasound transducer, rather than the ultrasound image itself. Since the locationof the image origin and the orientation of the beam lines is not known relative tothe transducer casing, the calibration generally cannot be performed through directmeasurements of the position of the sensor with respect to the transducer casing.Instead, ultrasound spatial calibration techniques have been developed based onimaging an object of known geometry, also referred to as a calibration phantom. Thefollowing sections provide some background on the various calibration techniquesas well as the position tracking systems used with them.1.6.1 Calibration MethodsMany calibration techniques have been developed involving different phantom de-signs and algorithms. Most phantoms are constructed using a highly acousticallyreflective material such as metal, glass, nylon, or rubber, and are placed in a waterbath to enable transmission of the ultrasound energy between the transducer andthe phantom. Calibration algorithms may be iterative optimization techniques orclosed-form solutions, and can vary in the number of unknown parameters. Iter-ative methods can be sensitive to initial estimates and can result in sub-optimalsolutions located in local minima [93]. Iterative methods also typically requiremore measurements to achieve the same level accuracy as closed-form methods.Reducing the number of unknown parameters also reduces the number of requiredmeasurements, and can improve accuracy because the parameters are often coupledand errors in estimation can propagate through the solution.All calibration methods are affected by the ability to localize image features.The resolution of the ultrasound system, which is both anisotropic and spatiallyvariant causes distortion of the imaged shapes. Variations in speed of sound causefurther problems in inaccurate depth estimation and ultrasound beam refraction.60Another related problem is that the ultrasound image is often modelled as a plane,however the beam has a variable thickness. An example that illustrates the problemis locating a point target which can persistently appear in the ultrasound image asthe transducer is moved in the elevational direction over several millimetres.One of the simplest phantom designs is a fixed point target, which can be con-structed with a small spherical bead [182], the crossing of two wires [82], or avirtual point located at the centre of a sphere [47]. The point target is scanned fromseveral viewing angles and positions by moving the transducer. The point targetwhen segmented from the ultrasound image and transformed to either the trackeror phantom coordinates should remain in a fixed position. Therefore, the unknowncalibration transformation can be found using an iterative optimization algorithmto minimize the spread of the transformed point locations. To help reduce the ele-vational beam thickness problem in accurately locating the point, a solution using aspecialized miniature ultrasound transducer as a phantom that can detect when themain transducer beams reach the phantom and actively send a pulse to locate thispoint for the main transducer has been proposed [121].Another simple phantom design is the planar phantom [262, 275]. Similar tofixed point target, the tracked transducer is placed in several different positions. Theline feature detected where the planar phantom intersects the imaging plane whentransformed to the tracker or phantom coordinate system when using the correctcalibration matrix should be in the same plane. The planar method is attractive be-cause it enables automatic segmentation, and unlike the point target, there is no needto carefully align the transducer with the phantom. A solution for the elevationalbeam thickness problem for planar phantoms has been developed where a trans-ducer is mounted on a moving phantom consisting of a metal bar, with its top edgecentred with the axis of a wheel [262]. The wheel is placed on a flat surface andcan be translated or rotated to collect multiple images, while the metal bar remainsat a fixed height above the flat surface, creating a virtual plane that is detected withultrasound that is always aligned with the beam. Another planar based techniquethat reduces the effect of beam thickness is using virtual planes created by pairs ofparallel wires [40, 77]. Planar phantoms using a pre-calibration step to determinethe location of the plane in tracker coordinates using a calibrated stylus has beenused to reduce the number of unknowns in the calibration problem [27], or to obtain61a closed-form formulation [217].Another method of spatial calibration that does not require knowledge of thephantom location in the tracker coordinate system is the hand-eye calibration prob-lem [35, 39, 41, 68, 176]. In this approach, the goal is to solve the matrix equationAX = XB, (1.72)where A is the rigid transformation between two ultrasound images, B is the rigidtransformation between the sensor measurements for the two images, and X is theunknown calibration transformation between the ultrasound image and the sensor.The matrix B is easily computed by chaining together two measurements from theposition tracking system, and A can be determined either by registering the ultra-sound images directly to one another, or by registering each image to the phantomand chaining the two transformations together. In the case that A is found by regis-tering images to each other, it is possible to achieve calibration without knowledgeof the phantom geometry.Another straightforward approach to calibration is to generate a set of pointswith known correspondence between the ultrasound and tracker coordinates. Thiscan be accomplished by imaging the tip of a tracked stylus [212]. A similar cali-bration approach is to use the stylus to locate point features on a phantom which isimaged, resulting in the same corresponding point data [129, 168, 249]. The solu-tion can be obtained iteratively, through geometry, or using a closed-form solution.A proposed solution to the elevational beam thickness problem is to use two conesthat meet at a point along the stylus axis a known distance away from the styluspivot point. The decreasing radii of the cones help to guide the ultrasound plane tothe intersection point [142].A popular phantom that uses known geometry in the tracker coordinate sys-tem and corresponding points is the N-wire (or Z-wire) phantom [65, 190, 238].The N-wire phantom is scanned approximately perpendicular to the plane of thethree wires which results in three collinear lines appearing in the ultrasound im-age. Given knowledge of the end points of the wires in the tracker coordinatesthrough a pre-calibration using a calibrated stylus, the centre point along the diag-onal of the N-shape can be determined in tracker coordinates using the geometrical62lengths between the three points measured in the ultrasound image. This creates acorresponding point between the ultrasound image and tracker coordinate systemsfor each N-shape in each image. Similar to the tracked stylus calibration technique,when at least three non-collinear points are identified the calibration transformationcan be solved either iteratively or in closed-form.A few iterative techniques for more generic wire phantoms have been developedas well, such as using an optimization algorithm to maximize the correlation be-tween image intensities and a geometrical model of the wire pattern [259]. Anotherexample is using the iterative closest point algorithm to determine the calibrationmatrix to match segmented points from ultrasound to a random collection of wiretargets [334]. Use of the virtual centre point of hollow tubes rather than wires hasbeen proposed to help improve automatic segmentation and line localization [61].1.6.2 Position TrackingPosition tracking systems determine the position and orientation of a sensormountedon an ultrasound transducer and in some cases a calibration phantom as well. Aftercalibration, the ultrasound image pixels are mapped to the 3d coordinate systemof the tracking system using sensor readings during image acquisition. Optical orelectromagentic based tracking systems are commonly used. The following sec-tions describe the two types of systems and Table 1.2 summarizes the technicalspecifications of commonly used systems.Optical SystemsOptical tracking systems use cameras to detect markers attached to the object(s) ofinterest. The position of the markers is determined using triangulation techniques,and the geometry of the marker placement can be used to determine the orienta-tion of the object. A common camera configuration is two 2d cameras which areoriented to view the same scene (i.e. stereo vision). These cameras can be video-based such as the MicronTracker (Claron Technology, Toronto, Ontario, Canada),or infrared-based such as the Polaris (NDI, Waterloo, Ontario, Canada). Anotherapproach is to use three 1d image sensors, such as the infrared-based Certus (NDI).Figure 1.14 demonstrates the triangulation process for both configurations (for more63bfPpleftpright(a)PC1C2C3(b)Figure 1.14: Triangulation configurations for determining 3d marker posi-tions with optical tracking systems. (a) Stereo vision uses the geometryof two calibrated cameras that have lens centres separated by a knownbaseline, b, and known focal distances, f , to determine the 3d positionof P based on perspective projections onto the left and right cameras,ple f t and pright . (b) A combination of three 1d cameras determinesthe 3d position of P as the intersection between three planes definedby the camera sensor readings (C1−3) and the calibrated relative geom-etry between the cameras.64Figure 1.15: Markers for position tracking systems. From left to right, a videocontrast X-point, a retroreflective infrared sphere, an active ired, and aelectromagnetic sensor coil. Ruler shown for reference scale.65Table 1.2: Technical specifications of common position tracking systems. Accuracy is expressed as root mean squarederrors. Fields for which data were not available from the manufacturer are listed as not reported (NR).Tracking System Accuracy [mm] MeasurementRate [Hz]MaximumDistance [mm]MinimumDistance [mm] Resolution [mm]OpticalNDI OptotrakCertus 0.14600N+1.31 7000 1500 0.010NDI Polaris 0.25 60 2400 950 NRClaronMicronTracker 0.35 20 2000 400 0.015ElectromagneticAscension 3DGuidance 1.4 20–2552 250–780 3 NR 0.5NDI Aurora 0.7 40 660 50 NR66details see [335] for stereo cameras and [197] for 1d cameras). Some commonlyusedmarkers are shown in Figure 1.15. Markers can be passive, such as intersectinghigh-contrast regions creating an X-point marker or retroreflective infrared spher-ical markers, or active such as an ired. The advantages of active markers are thatthey are less dependent upon lighting conditions affecting reflection and a uniquegeometry is not required to identify a group of markers representing a single trackedobject. The disadvantages of active markers are that usually some timemultiplexingmust be used to distinguish different markers and a wired connection to the track-ing system is usually required. There is no statistically significant difference in theaccuracies of passive and active markers [336].Electromagnetic SystemsElectromagnetic tracking systems track the position of sensor coils in an electro-magnetic field generated by a source transmitter. The systems use the principle ofelectromagnetic induction to measure the induced currents in the sensor coils dueto changes created by the source transmitter. If the position of the moving sensoris described with respect to the transmitting source in spherical co-ordinates withdistance ρ, azimuth angle α, and elevational angle β, and the orientation of the sen-sor is described using the angles ψ, φ, and θ, then the output current of the sensorcoils, y, is related to the current of the source coils, x, through [266]y = M2piρ3RφRθRψR−1α R−1β SRβRαx (1.73)where M is the magnetic moment of the coil, R is a rotation matrix for a givensubscripted angle, andS =1 0 00 −12 00 0 −12, (1.74)accounts for the scaling factor between radial and tangential magnetic coupling foraligned source and sensor coils. Assuming that there are only small changes inposition and orientation between acquisitions, Equation 1.73 can be linearized andsolved using three independent transmission vectors [266].67A line-of-sight is not required between the transmitting source and the trackedsensor coils, which is the primary advantage of electromagnetic systems over op-tical systems. The primary disadvantage of electromagnetic tracking systems isa sensitivity to environmental factors that alter the magnetic field. For example,power supplies and other electronics generate their own magnetic fields that in-terfere with the field created by the source transmitter. Ferromagnetic materialssuch as iron and steel affect the homogeneity of the transmitted magnetic field. Fortransmitters using alternating current, the presence of conductive material such ascopper or aluminum will induce eddy currents causing field distortions. Anotherdisadvantage is the systems usually have a smaller working volume compared to op-tical systems. Most electromagnetic systems share the drawback of active opticalmarkers in that they require a wired connection between the tracked object and themeasurement system. The Calypso system (Calypso Medical Technologies, Seat-tle, WA, USA) provides wireless tracking by measuring the resonance response ofthree tuned transponders with a planar sensor array that triangulates the transponderlocations [24].68Chapter 2Development and Analysis of a 3DSystem for Absolute ElasticityMeasurement2.1 IntroductionMany pathological conditions are associated with a change in the mechanical prop-erties of tissue, such as increased elasticity in focal cancerous lesions [114] and ar-terial plaques [79], or diffuse diseases such as liver cirrhosis [108] or autoimmunethyroiditis [297]. Elastography is an imaging tool that can depict these changes tohelp identify, diagnose, monitor, and guide treatment for these conditions. Elas-tography uses an elastic model to relate the mechanical properties of tissue to itsmotion in response to a force. The tissue motion is measured inside the body us-ing an imaging modality to track changes over time, most commonly with eitherultrasound or mri.As mentioned in Chapter 1, current commercial ultrasound elastography sys-tems that produce images of the spatial variation in elastic modulus are based onstrain or transient arf techniques. Strain can be implemented with simple hard-ware and can acquire images continuously in real-time, however strain only pro-vides relative measures of tissue elasticity. The arf based approaches require a69cool-down time to limit tissue and transducer heating to safe levels [240], whichcan limit the acquisition rate and ability to perform continuous acquisition. A fur-ther disadvantage is that arf methods are typically only offered on higher cost ul-trasound scanners which have the hardware to provide the necessary power for thepush beam [205]. There is also a limit on the depth penetration for measuring thearf induced transient shear wave. For example, the depth limit on a Siemens Acu-son S2000 system is 5.5 cm [84].Quantifying absolute measures of the elastic modulus, rather than relative mea-sures of elasticity, can reduce dependence on operator skill and improve repro-ducibility [8, 52], and can be useful in determining the different stages of a dis-ease [307]. In static or quasi-static low frequency deformation (<20 Hz) such as instrain imaging, with the force boundary conditions unknown, the absolute valuesof the elastic modulus cannot be obtained. Dynamic excitation produces inertialforces inside the tissue and enables the calculation of the absolute values of elastic-ity. While the arf technique provides a convenient way of applying dynamic exci-tation integrated with the imaging system, external mechanical harmonic vibrationscan also be used for absolute elastography without the previously mentioned draw-backs of arf [214, 290, 339]. Of course, such methods also have disadvantages tobe explored later.Measurement of dynamic tissue motion is challenging because acquisition ratesmust be sufficient to capture the full spectral content of the motion, and acquisitionover a roi should describe a common motion state (i.e. the tissue deformation atan instant in time should be captured). Under the assumption of consistent andrepeatable cyclical tissue motion, mre approaches solve this problem by using syn-chronization between a mechanical exciter and the mri scanning sequence [290].For ultrasound measurement of transient arf propagation, a small imaging regioncan be used to achieve a high effective temporal sampling rate, and repeated inter-rogations can be used to scan a larger roi [222]. Some ultrasound scanners canreceive data in parallel across all channels, allowing for ultrafast acquisition on theorder of 1000 Hz using planar transmit beams [34].Conventional ultrasound measures 2d image planes, however tissue deforma-tion is 3d in general. Measurement in 3d allows for a more accurate description oftissue mechanics, which is one of the main advantages of mre over conventional70ultrasound elastography [87]. Volumes of ultrasound data can be acquired usingmotorized swept ultrasound transducers, or more sophisticated matrix array trans-ducers. Measurements of motion over a volume can improve the precision of shearwave time-of-flight measurements [331], reduce shear wave diffraction bias [343],and enable characterization of anisotropic behaviour [330].Transient arf elastography techniques have become well-known with manycommercial and research systems developed for a wide variety of clinical appli-cations. These techniques are limited in measurement depth and fundamental com-promises of tissue heating, transducer heating, and the size of the roi. mre systemshave the advantage of measuring the full 3d displacement vector over a volume ofinterest, using steady state, low frequency vibrations that can penetrate deep intotissue. However, mre has disadvantages in long acquisition time (usually severalminutes), and high cost of equipment relative to ultrasound. It is worth exploringsteady state excitation with 3d ultrasound to investigate the possible advantages anddetermine the limitations of this type of elastography system.Previously a 3d ultrasound elastography system was developed at UBC basedon repetitive acquisition of small sectors for a high effective frame rate of tissuemotion [23]. An alternative bandpass motion sampling technique that is compatiblewith conventional ultrasound b-mode sequences was developed and implemented in2d with an Ultrasonix scanner (Richmond, BC, Canada) [98]. The latter techniquewas also adapted to work with a BK scanner (Herlev, Denmark) with a transrectalultrasound (trus) transducer mounted on a motorized cradle for 3d imaging [191].The three systems described in the previous paragraph were developed for thespecific configuration used in the application. It is desirable to create a generic re-search platform that can flexibly adapt to different hardware and approaches. Thiswill allow for comparison between the different approaches, as well as selectionof the optimal tools for a given application. The focus of the work presented inthis chapter is to present a unified framework that combines the different tech-niques developed at UBC. The system has been termed shear wave absolute vibro-elastography (swave), as it uses mechanical vibrations to generate shear waveswithin the tissue to quantify absolute measures of elasticity. Details of the configu-ration options that have currently been developed are described, and an analysis ofthe reliability of the system under expected experimental conditions are quantified71through simulation and experiment on a tissue mimicking phantom.Specifically, a modular software project is developed that implements the pre-viously developed 3d sector based Ultrasonix system, the 3d bandpass based BKsystem, and the 2d bandpass based Ultrasonix system. The flexibility of the systemis demonstrated by adapting the system for the first implementation of a 3d bandpassbased Ultrasonix system. A new motor control hardware unit is described whichcan be used to interface with any swept motor 3d ultrasound transducer withoutmodification. A new triggering system is described which is used to synchronizeultrasound acquisition with mechanical excitation. Methods for detecting complexmotion phasors from time harmonic displacement measurements are described andsimulations are used to investigate the sensitivity of the phasor detection in the pres-ence of errors in frame rate and displacement measurement, as a function of thenumber of temporal measurements. The ability to synchronize measurements oftissue motion over the roi is tested experimentally for various configurations. Therepeatability of elasticity measurements is investigated using experiments compar-ing the measurements from bandpass based and sector based motion measurement,and from different excitation geometry.2.2 Methods2.2.1 System OverviewThe main components of the system are the ultrasound interface to the scannerand transducer, the motor control of the ultrasound transducer to collect volumetricdata, the excitation control of the function generator and vibration shaker, and theelastography processing sub-system that comprises motion tracking and elasticityestimation. The components are controlled by a user through a graphical user inter-face (gui) on the scanner which also provides real-time display of b-mode, tissuemotion, and elasticity images, as well as data saving for offline use. Presets de-signed for specific hardware and measurement applications can by loaded by thegui to initialize the different components to appropriate settings. The componentsand their connections are shown in the block diagram in Figure 2.1.The ultrasound component is used to interrogate the tissue over a period of72GUIUltrasoundInterfaceMotor ControlExcitationControlFile Input & OutputUltrasoundScannerUltrasoundTransducerBodySteppingCommandsVoltagesSettingsFunctionGeneratorDriverShakerWaveformVoltagesAmplifiedVoltagesVibrationsSettingsTriggerRFor I/Q SyncAcousticPulsesEchoesRFor I/QMotionEstimationDerivePhasorsEnvelope DetectionPhaseCompensationElasticityEstimationScan ConversionDisplay RFor I/QDisplacementTime SeriesCorrected PhasorElasticityDisplacement PhasorFigure 2.1: Block diagram of the elasticity measurement system. In this diagram the ellipses represent physical ob-jects and the rectangles represent programming modules, or classes. The elastography processing sub-system iscontained in the dashed box.73time while steady-state multi-frequency vibrations are applied to the tissue surfaceby the excitation component. The ultrasound image is swept over a volume using acustomized motor control component. The tissue motion caused by the excitationis estimated along the axial direction using correlation-based displacement trackingfrom the collected ultrasound data. The temporal displacements are converted intoa phasor representation in the frequency domain and elasticity is estimated basedon the shear wavelength over the volume.2.2.2 Software DesignThe rectangular components shown in Figure 2.1 correspond to different classes orcollections of classes separated to achieve a specific task in the elasticity measure-ment process. Each component consists of a base class which defines a commoninterface and typically contains multiple derived classes for different hardware orsoftware implementations. Configuration of software projects for different compo-nent combinations is controlled through CMake (Kitware Inc., Clifton Park, NY,USA), a cross-platform open-source build system. Allowing CMake to include andexclude certain components provides the advantage that a given system only re-quires the dependencies for the desired combination of components, rather thanneeding to install the dependencies for every combination. Because each compo-nent uses a common interface, different configurations can easily be swapped andnew hardware can implemented without affecting the other components of the sys-tem.Projects were developed with Microsoft Visual C++ (Redmond, WA, USA).Processes for motion tracking, elasticity estimation, envelope detection, and scanconversion were accelerated using the CUDA application programming interface(api) to work with an NVIDIA 580 GTX GPU (Santa Clara, CA, USA) or greater.The Ultrasonix scanner wasmodified to include this GPU and a larger power supply,while the BK system used an external PC with the GPU installed.2.2.3 Ultrasound InterfaceThe primary role of the ultrasound interface is to communicate with the scannerto obtain rf or i/q ultrasound data as it becomes available and to record the time74when the data is acquired. When possible, it also enables controlling acquisitionsettings and timing. The interface has specific implementations to work with Ul-trasonix and BK scanners in real-time. The Ultrasonix systems are primarily usedwith hand-held swept motor 3d ultrasound transducers which can be used to scanvarious organs such as the breast, liver, kidney, or thyroid. A longer term goal withthe Ultrasonix systems is to implement the system with the Sonix Embrace auto-mated breast scanner. The BK scanner is used primarily with a trus transducerwith a sagittal array, mounted in a cradle driven by an external stepper motor foracquiring 3d ultrasound volumes of the prostate.The Ultrasonix interface actually has three implementations based on differ-ent apis offered and runs directly on the scanner. It is designed to work with thecurrent generation scanner, the SonixTouch, as well as the older MDP and RP mod-els. The first Ultrasonix interface is based on the Porta api which provides accessto the imaging “engine backend.” It uses C-style callbacks to transfer rf data di-rectly from the “cineloop” buffer and provides various functions for controlling thescanner. The advantages of the Porta api are a lightweight interface for communi-cating with the scanner and access to the clinical exam b-mode images which arecomputed using a proprietary algorithm and cannot be reproduced exactly from therf signals. The disadvantage of the Porta api is a lack of fine level control of thescanner, such as sequence parameters. The second Ultrasonix interface is basedon the Texo api, which solves the limitation of Porta by providing access to thesequencing and beamforming parameters of the scanner. Similar to Porta, it pro-vides callbacks and functions to communicate with the scanner. However, the Texoapi only provides rf data. While this is the data of interest for elastography as aninput for motion estimation, it is difficult to use it in clinical settings because theb-mode is computed from the rf by the research software and has not been ap-proved for clinical use. The third Ultrasonix interface is based on the Ulterius apiwhich uses Transmission Control Protocol/Internet Protocol (tcp/ip) to stream rfdata from the Ultrasonix clinical exam software to a client application. The ad-vantages of the Ulterius api are that the familiar clinical interface is available foruser to change ultrasound imaging settings, as well as the clinical exam b-modeas with Porta. The disadvantages are shared with Porta, with additional problemswith dropped frames due to tcp/ip packet loss and increased CPU usage because75the clinical exam software. Acquisition timing information for all three Ultrasonixinterfaces is determined using a frame counter and interrogating the scanner framerate or frame period.The BK interface is designed to work with the Pro Focus 2202 scanner equippedwith a research interface to stream i/q data to an external PC through a DALSAXcelera-CL PX4 frame grabber card (Teledyne DALSA, Waterloo, ON, Canada).Settings may be controlled via the Grabbie api which communicates over tcp/ip,however in practice the settings are often set manually on the scanner by the user.Timing information for each frame is provided by time stamps from the frame grab-ber.The system is capable of working with only 2d images (no motorized sweep),but 3d is preferred for greater accuracy.2.2.4 Motor ControlIn a conventional 3d ultrasound scan, the scan plane is swept over the roi in theelevation direction in slow steady motion. The sweeping rate is limited by the 2dframe rate of the ultrasound system which in turn is limited by the amount of timefor the ultrasound waves to reach the imaging depth and back and the number ofbeam lines across the 2d frame. The motor can either briefly pause at each motorposition while a 2d frame is acquired, or can move continuously but slowly enoughthat the distortion caused by acquiring a frame while moving is negligible. Thisresults in a volumetric sample on the order of one volume per second.In the development of the 3d ultrasound elastography research system, it wasdesired to gain greater control over the motion of the scan plane during imaging.Specifically, for measuring dynamic motion on the order of 20 Hz to 300 Hz, a rea-sonable sampling rate and number of samples can be achieved by stepping themotorto a position, collecting several frames (for motion measurements), and then step-ping motor to the next position in the sweep. The designed motor control box pro-vides motor commands to the transducer without any modification to the transduceror scanner. The motor control box is placed between the transducer and scanneras shown in Figure 2.2. The motor control box passes ultrasound signals straightthrough but intercepts motor signals from the scanner and provides its own motor76Figure 2.2: A photograph of the motor control box connected to the Ultra-sonix scanner and a motorized ultrasound transducer.commands as directed by the software running on the PC. The motor control boxcontains the same receptacle as the scanner for inserting the transducer. On theother end of the motor control box, the same type of plug as the transducer is usedto connect the box to the scanner. From the scanner’s perspective, the transducerappears as if it is normally connected.The ultrasound signals are routed directly from the transducer receptacle to thescanner plug on a 12-layer PCB. Finger stock gaskets are used to couple the trans-ducer shield to the motor driver shield and the scanner chassis shield to limit noiseentering the system. A stepper motor driver circuit is placed on the same PCB as77is connected to the transducer motor pins via the transducer receptacle. Any motorsignals from the scanner via the plug are left floating. The stepper motor circuit re-ceives step and direction signals from two pins of the parallel port of the on-boardscanner PC which can be controlled from the developed software. An external 12 VDC supply provides power to the motor driver circuit.2.2.5 ExcitationSome previous elastography methods have applied vibrations to tissue using an ex-citer oriented in a transverse direction to the tissue surface in order to generate shearwaves [38, 46, 113, 173]. Conceptually this is the most straight forward way to gen-erate shear waves, as the applied deformation is a shear at the surface. For simplegeometry, the resulting waves have a planar wave pattern, allowing for some simpli-fying assumptions in the elasticity inversion process. However, the wave penetra-tion using transverse surface vibrations can be poor, limiting the available imagingregion, especially for larger depths. Alternatively, longitudinal vibrations appliednormal to the tissue surface can be applied which generally provide better wavepenetration into the body [290, 322, 339, 353]. These longitudinal vibrations willstill generate shear waves, since an incident elastic wave at the interface of two me-dia can result in the reflection and refraction of both compression and shear waves,regardless of the type of incident wave [164]. Further, the types of waves generatedby the excitation source are dependent on geometry. For example, a finite sizedrectangular strip or circular disk radiates both compression and shear waves [209].To achieve deep tissue penetration, the swave system typically applies longitudinalvibration to the surface of the tissue with a small circular disk at the end of a rod,vibrating normally to the surface of the medium. Mode conversion results in shearwaves at depth.Excitation signals are defined using a data array in software with each entry inthe array corresponding to a voltage at a specific time. The signals can be arbi-trary, but are typically sinusoidal. Multi-frequency signals are created by addingmultiple sinusoids together. The excitation frequencies are usually picked within aband of 20 Hz to 400 Hz, which balance a trade-off of greater wave penetration atlower frequencies and finer elasticity resolution with higher frequencies. The sig-78nal generating hardware is programmed with the excitation signal arrays over USB.The current system supports three hardware signal generators: Agilent 33220AandU2761A function generators (Santa Clara, CA, USA), as well as the Data Trans-lation DT9812 digital-to-analog converter (dac) (Marlboro, MA, USA).The output signal power of the signal generators is amplified before being sentto the vibration shaker. Any amplifier with sufficient bandwidth and wattage can beused.2.2.6 Sampling SequencesAs mentioned in the previous section, the swave system requires measurement oftissue motion within a band of 20 Hz to 400 Hz. The swave system has been de-signed to be able to switch between two ultrasound sequences capable of measuringthis motion. The first sequence, called sector-based scanning, divides the width ofimage into small sectors which are repetitively scanned at a high effective samplingrate [21]. The second sequence, called bandpass sampling, uses a conventionalb-mode sequence of single continuous scanning, and uses knowledge of the ultra-sound frame rate and the excitation frequencies to reconstruct the amplitude andphase of the motion while sampling at a rate lower than the Nyquist rate [98].The ultrasound acquisition rate is determined by the time of flight of the ultra-sound pulse to reach the imaging depth and reflect back to the transducer (Equa-tion 1.1), and the number of beam lines across the width of the image. In a conven-tional b-mode sequence, the beam lines are acquired one-by-one starting with thefirst line at one end of the array, and ending with the last line at the opposite end ofthe array. This constitutes a single frame of data. For a typical image with 128 beamlines, a depth of 5 cm, and a speed of sound of 1540 m/s, the maximum possibleframe rate is 120 Hz and therefore tissue motion measurements will be aliased.The sector-based scanningmethod divides the image into sectors, typically con-taining between 1 and 16 beam lines, and changes the sequence in which the beamlines are acquired such that each sector is repeated immediately for the desired num-ber of temporal motion samples to increase the sampling rate over the sector region.For example, if an image consisting of 128 beam lines is divided into sectors of 8beam lines each, the effective sampling rate of each sector is 16 times greater than79when acquiring entire image frames sequentially. However, the total acquisitiontime remains the same as the same total number of beam lines are acquired in thesequence. After motion detection, the phase of the motion measured within eachsector is aligned across the image using knowledge of the acquisition delays, whichis described in more detail in Section 2.2.9.In the bandpass samplingmethod, because the sampling rate and the frequencycontent of the excitation signal are known, the amplitude and phase of the tissuemotion can be reconstructed from the displacement time series without having tosatisfy the Nyquist sampling rate requirement of a sampling rate greater than twicethe highest frequency component in the signal. Given the excitation frequency andsampling rate, the aliased motion signal will appear at a known frequency in themeasurable frequency band. The sampling requirement can be restated for a real-valued band-limited time-domain signal as requiring the sampling rate to be greaterthan twice the positive spectral bandwidth of the signal. This will result in a replicaof the signal spectrum in the measurable baseband as long as the sampling fre-quency, fs, satisfies2 f0 + Bm + 1≤ fs ≤ 2 f0 − Bm , (2.1)where f0 is the centre frequency of the signal, B is the positive frequency band-width, and m is an integer corresponding to the minimum number of spectral half-shifts required to project the original signal to the baseband. While the excitationis typically a single frequency signal, or a combination of widely spaced single fre-quency signals, a bandwidth of about 10 Hz around each frequency component isusually assumed to account for the effect of finite-time windowing and distortion.The aliased signal will have the same amplitude as the true signal, and the phase ofthe signal will either match the true signal or be 180◦ out of phase depending onthe known number of spectral half shifts due to aliasing. The spectral shifting ofthe spectrum to the baseband frequencies causes a decrease in snr proportional tothe number of shiftsSNR =PS(m + 1)PN, (2.2)where PS and PN are the spectral power densities of the signal and white noise,respectively. For multi-frequency excitation, one further note about the bandpasssampling method is the sampling frequency should be chosen carefully to avoid80overlap between the aliased spectral bands.2.2.7 Motion EstimationTissue motion is estimated between sequences of rf data provided by the ultrasoundinterface. In the cases when i/q data are provided by the ultrasound interface in-stead, the rf signals are reconstructed by reversing the demodulation process. Therf data is divided into overlapping blocks of several data samples and the maximumnormalized cross-correlation is found along the beam axis for each block. Neigh-bouring blocks which have already been computed are used to bracket the searchregion to improve speed and reduce errors [345]. Displacement shifts at sub-sampleresolution are obtained by estimating the cross-correlation peak using cosine inter-polation on the discrete normalized correlation coefficients [57]. The output of themotion estimation block is the displacement component along the beam axis for ev-ery block location and every acquisition of a given line. Every block represents thetissue motion at the block’s location over the sampling time. Subsequent analysisis carried out in the frequency domain as described in the following sections.2.2.8 Phasor FittingTo provide input data for the inverse elasticity algorithm the speckle tracking mea-surements which describe the displacement between ultrasound frames in time needto be converted into displacement phasors at each excitation frequency. Given Nsspeckle tracking measurements, the displacement in the time domain at a givenlocation can be described byu (tk ) =N f∑m=1Am cos (ωmtk + θm) , (2.3)where tk represents a discrete time sample, k ∈ 1 . . . Ns, Nf is the number of ex-citation frequencies used, ωm is the excitation frequency, and Am and θm are theamplitude and phase at a given excitation frequency. The displacements can also81be described using the complex exponential byu (tk ) =N f∑m=1<{Amejθmejωmtk}. (2.4)Replacing the complex exponentials with their trigonometric representations andthen taking the real part results inu (tk ) =N f∑m=1Am cos (θm)︸         ︷︷         ︸xrcos (ωmtk ) −Am sin (θm)︸          ︷︷          ︸xisin (ωmtk ) . (2.5)The two identified scalars, xr and xi, are calculated to find the best fit phasor to thespeckle tracking measurements.The values of xr and xi are identified for each frequency by expressing Equa-tion 2.5 in matrix form,Mx = u, whereM =cos (ω1t1) sin (ω1t1) . . . cos(ωNf t1)sin(ωNf t1)1..................cos(ω1tNs)sin(ω1tNs). . . cos(ωNf tNs)sin(ωNf tNs)1, (2.6)u =u (t1)...u(tNs), (2.7)x =xr1xi1...xrNfxiNfC, (2.8)where C accounts for a DC offset in the measurements. The solution is found in theleast squares sensex = (MᵀM)−1 Mᵀu. (2.9)82The phasor at a given frequency isU (ωm) = Amejθm (2.10)= Am cos (θm) + jAm sin (θm) (2.11)= xr + j (−xi) . (2.12)To provide feedback on the goodness of the fit, a quality factor is computedusing the ratio of the signal energy of the fit and the summation of the energy of thefit and the difference between the fit and the original data. This is expressed as| |M0x| | 2| |M0x| | 2 + | |Mx − u| | 2 , (2.13)whereM0 is the same asM except the last column is set to zeros to remove the DCcomponent from the fit.2.2.9 Phasor CompensationIt is the nature of ultrasound acquisition that leads to the sampling of each displace-ment signal to occur at different times dependent on spatial location. The finitepropagation speed of the ultrasound pulses means that points located deeper withinthe tissue are sampled at a later time relative to the shallower depths. Typicallyultrasound scanners create a 2d image by sequentially scanning beam lines, mean-ing that each line is also separated in time. To use spatial information about themeasured motions and relate the information to tissue mechanics, the displacementmeasurements should be synchronous. For periodicmotions, under the assumptionsthat the transducer does not move relative to the tissue and the ultrasound speed ofsound is constant through the roi, the time delays can be compensated in the fre-quency domain as described by Baghani et al. [21]. The technique corrects for theacquisition delays by multiplying the phasors at each spatial location by a complexexponential to shift the phase of the phasor proportional to the time difference fromthe first measured phasor.Ucomp(a, b, c) = Umeas(a, b, c)e−j2pi fe∆t (2.14)83whereUmeas(a, b, p) is the phasor fitted to the temporal displacements at beam linea and depth b in the pth 2d image plane, Ucomp(a, b, p) is the phasor after com-pensation, fe is the excitation frequency, and ∆t is the acquisition time differencebetween the phasor at (a, b, p) and the phasor at the origin. The time difference canbe decomposed into intraline, interline, intersector, and interplane compensations.The intraline compensation accounts for the time delay between acquiring ultra-sound echo signals along the axial or depth dimension, which is the time for theultrasound pulse to reach the phasor at depth b and return to the transducer, wherethe wave travels at a speed of c, resulting in∆tintraline =2bc. (2.15)The interline compensation accounts for the delay between the acquisition of eachbeam line. Given the pulse repetition period, TPRF, which is typically the time forthe ultrasound pulse to reach the maximum depth in the image and back, 2d/c, thenthe time delay between the ath line and the first is∆tinterline = TPRF (a − 1) . (2.16)The interplane compensation accounts for the delay between the acquisition of sub-sequent 2d phasor image planes, and is typically computed using the differencebetween the time stamp of the first ultrasound rf frame used in the first phasorcomputation, and the time stamp of the first ultrasound rf frame used for the pthphasor image plane.If a sector-based high frame rate sequence is used to sample the tissue motion,then the interline compensation would be applied independently to each individualsector using the first line in the sector as the reference, and a fourth compensationwould be applied to account for the delay between each sector acquisition. For Llines in a sector, each scanned K times, the delay between the first line in the firstsector and the sth sector is∆tsector = LKTPRF (s − 1) . (2.17)84Ready SignalExciter Output Trigger Synchronization CircuitReady & Trigger?Ultrasound Trigger InAcquire DataYesNoFigure 2.3: Flowchart describing the synchronization between the ultrasoundacquisition and the excitation in the swave system.2.2.10 SynchronizationSynchronization between the ultrasound acquisition and the excitation signal canbe used as a complement or replacement for the phasor compensation described inthe previous section. One option is to trigger the start of the acquisition of a definednumber of ultrasound frames for computing a phasor image, with each trigger oc-curring at a common excitation phase. This would remove the need for interplanecompensation, and might be more reliable than frame time stamps over long acqui-sition periods. Another option is to trigger the acquisition of each ultrasound frame.In this case, triggers would need to shift to different phases over the excitation todetect tissue motion between ultrasound frames. Synchronizing each ultrasoundframe also removes the need for interplane compensation, while providing finercontrol over the exact motion phases that are measured rather than measuring mo-85tion phases depending on the ultrasound frame rate. Finally, each ultrasound beamline can be triggered, again while shifting the phase of the trigger for subsequent ac-quisitions to observe motion. This final synchronization scheme removes the needfor interline compensation, and can enable tracking of all components of the motionvector as discussed in more detail in Chapter 5.The implementation of the synchronization for the Ultrasonix systems is de-scribed by the flowchart in Figure 2.3. A custom synchronization circuit is used tooutput a trigger pulse to the ultrasound machine which collects the specified ultra-sound data (one beam line, one frame, or a defined number of frames) in its standardsequence after receiving the trigger. The synchronization circuit takes two inputs.One input is a ready signal from the ultrasound machine which allows the programto indicate when it would like to acquire a beam line which is realized through apulse sent through the parallel port of the ultrasound machine. Once the synchro-nization circuit has received the ready signal, it will wait for the function generatorto output a trigger pulse, which happens when the function generator reaches theprogrammed phase. The synchronization circuit then outputs a pulse which is sentto the ultrasound machine, and resets itself to wait for the next ready signal. Thesynchronization circuit is realized using two flip-flop chips. The ultrasound scannertransmits and receives as soon as the trigger pulse is received from the synchroniza-tion circuit. The process is repeated until all of the desired data are collected.2.2.11 Elasticity EstimationThe linear elastic wave equation relating the displacement in the frequency domainto the mechanical properties of the tissue is given by Equation 1.37, which for har-monic excitation at a frequency of ω is written in matrix form as∇ ·[µ(∇d + (∇d)T)+ λ (∇ · d) I]= −ρω2d, (2.18)where λ and µ are the Lamé parameters, ρ is the density, and d is the tissue dis-placement. The density of most soft tissue is close to the density of water, so ρ isusually assumed to be uniformly 1000 kg/m3. Tissues imaged in elastography areoften considered nearly incompressible (∇ · d ≈ 0), which implies the Poisson’s ra-tio, ν, is approximately 0.5 and the Young’s modulus is related to the shear modulus86throughE = 2µ (1 + ν) (2.19)≈ 3µ. (2.20)To simplify the problem, the shear modulus is usually assumed to be locallyconstant (i.e. homogeneity assumption) meaning that the spatial derivatives of theshear modulus are negligible. These assumptions allow Equation 2.18 to be sim-plified to a Helmholtz equationµ∇2dγ = −ρω2dγ, (2.21)which is independently satisfied by each orthogonal displacement component γ.The shear modulus can be determined using a ratio of the output of two filters oper-ating on the displacement [230]. This can be accomplished over a broadband rangeof elastic wave spatial frequencies with a collection of lognormal quadrature filters,defined as a product between radial and directional components [163]qi (k) = ri (k)∑pnp (k) , (2.22)where k is the spatial frequency variable, k = ‖k‖ is the wave number, and p is thenumber of directions. The radial component is defined asri (k) = e− 12 ln 2 ln2(kki), (2.23)where ki is the centre frequency of the filter. In this work, the filter banks consistof 11 frequencies spaced an octave apart, with the highest frequency correspond-ing to a wavelength of two pixels. For a 200 Hz excitation frequency with 1 mmpixel spacing, this results in the filter banks containing a centre frequency rangecorresponding to Young’s modulus values of approximately 0.5 kPa to 500 MPa.The directional component of the filters have a cos2 dependence in the half-space of the selected direction, and zero in the other half-space, with selected di-rections corresponding to the positive and negative x, y, and z axes [200]. This87filter design helps to separate interfering waves due to reflection and refraction sothey can be processed separately and combined.Let s represent the result of the convolution between a quadrature filter and thedisplacement; then, the frequency estimate for a pair of filters iskˆi = <(si+1si) √kiki+1. (2.24)The final local frequency estimate is a weighted summation of the estimates fromthe pairs of filtersk¯ =∑N−1i=1< (si) kˆi∑N−1i=1< (si) . (2.25)Once the local spatial frequency has been determined, the Young’s modulus canbe estimatedE = 3ρω2k¯2. (2.26)This method has been used formre inversion and has been termed lfe. In the swavesystem, axial displacement phasor image planes are computed at each excitationfrequency for each motor position in the 3d sweep of the ultrasound transducer andscan converted to a 3d Cartesian grid before applying the lfe inversion algorithmto estimate elasticity over the volume. If the elasticity can be considered to beindependent of frequency, for example if the tissue is not significantly viscous, or ifthe excitation frequencies are similar enough that dispersive effects are negligible,then the elasticity can be averaged over each excitation frequency. This can helpreduce artefacts caused by low amplitude nodes in the tissue motion modal pattern,as each frequency should produce a different pattern [312].2.3 SimulationsSimulation was used to study the errors in the phasor calculation for common usecases and expected errors in the system. Synthetic speckle tracking measurementswere generated for a single location in space by sampling a synthetic excitationsignal at a given frame rate for a given number of desiredmeasurements, then addinga random error to the displacement value representing jitter error.The phasor calculation was first tested using a single frequency cosine signal.880 0.1 0.2−10010Time (s)Amplitude(µm)(a)0 0.1 0.2−20−1001020Time (s)Amplitude(µm)(b)Figure 2.4: Simulated excitation signals for analysis of phasor fitting. (a) Sin-gle frequency excitation signal consisting of a cosine at 200 Hz with a10 µm amplitude and 0◦ phase. (b) Multi-frequency excitation signalcreated from the summation of the signal (a) and a second cosine sig-nal at 240 Hz with a 8 µm amplitude and 30◦ phase. The true signal isshown in black and the signal in the base band is shown in grey. Thecircles represent 20 displacement samples acquired at a frame rate of70 Hz.89The signal had a frequency of 200 Hz, an amplitude of 10 µm, and a phase of 0◦.This is a typical excitation usedwith the elastography system. The chosen frequencyis low enough to provide adequate penetration into the tissue without substantial at-tenuation, and high enough such that the measured elastic wavelengths fit within theregion of interest and allow for identification of small scale variations in elasticity.The amplitude is at a level typically observed in vivo and is measured accuratelywith speckle tracking. The amplitude is small enough to limit observable nonlineareffects in the Young’s modulus.The number of samples acquired is limited by the amount of time to performthe measurement and the number of scan lines that can be programmed in the ultra-sound sequencer. The time to perform the measurement is ideally as small as possi-ble to limit possible artefacts from patient motion as well as to provide results faster.The scan lines are programmed in the ultrasound sequencer to acquire ultrasoundimages for a given number of repetitions to provide the number of desired speckletracking measurements. The limit is typically somewhere between 1000 and 2000lines. The transducers on the Ultrasonix machines have 128 elements. Althoughthe sequence does not necessarily have to program 128 lines, reducing the numberof lines also reduces either the field of view or the lateral spatial spacing of thedisplacement samples. For most practical sequences the number of speckle mea-surements is limited to be between 10 and 30 samples. This study looked betweenthat range in increments of 5 samples.A nominal frame rate of 70 Hz was used to sample the speckle tracking mea-surements. The maximum frame rate possible is limited by the time required forthe ultrasound acoustic pulse to reach the desired imaging depth and reflect back tothe transducer surface as described by Equation 1.1, and the number of beam linesin a frame. For a conventional ultrasound sequence with depths between 6 cm and15 cm and 128 beam lines across the image, the maximum frame rate is somewherebetween 30 Hz and 100 Hz. The frame rate is chosen to be below this limit to ensurethat no frames are dropped, but close to this limit to ensure a short acquisition time.After an ultrasound sequence is programmed on the machine, the software re-quests the scanner to report the frame rate it is using. There is potentially a smalldifference between the frame rate reported by the scanner and the actual frame rate.The frame period is provided by the Ultrasonix apis at a resolution of 1 µs. The90time stamps from the frame grabber used with the BK scanners are provided at aresolution of 1 µs as well. At a frame rate of 70 Hz, an error in the frame period of1 µs would result in a error in the frame rate of 0.05 dHz. Even though the resolu-tion of the frame period is 1 µs, the error could potentially be larger. When settingthe frame rate on the Ultrasonix scanners, the maximum resolution is 1 dHz. Theerror in the actual frame rate is likely much less than this, but it serves as an upperbound for the expected error. This study investigated the frame rate error between0 dHz and 0.2 dHz, in increments of 0.05 dHz.The jitter error is expected to be under 4 µm for speckle tracking under typicalconditions [254]. The jitter was modelled as normally distributed with standarddeviations starting from 0 µm and incremented by 1 µm to 4 µm. The jitter was ran-domly sampled for each displacement measurement and added to the displacementvalue. The error was randomly sampled 1000 times for each test case and the resultswere averaged over the 1000 trials.The simulation compared the phasor fit to the synthetic speckle tracking mea-surements to the known true signal. The difference between the phasor amplitudeand phase were computed, as well as the quality factor of the fit as computed byEquation 2.13.2.4 Experiments2.4.1 Motor AttenuationPlacing the motor control box between the transducer and the scanner poses therisk of attenuating the ultrasound signals by passing them through two additionalconnectors and a few centimetres of copper trace (approximately 3 cm to 7 cm long,0.2 mm wide, and 36 µm thick traces). The attenuation could reduce snr and limitthe depth of penetration of the ultrasound. To test whether the motor control boxcauses significant attenuation, a 4DL14-5/38 motorized ultrasound transducer wasused to image a CIRS 040 general purpose multi-tissue phantom. The phantomconsists of Zerdine, a proprietary elastic polymer that mimics the mechanical andacoustic properties of tissue. The phantom was measured by the manufacturer tohave a speed of sound of 1541 m/s and attenuation of 0.63 dB/(cm MHz). The91phantom contains multiple nylon monofilament wire targets which appear as brightpoint reflectors in ultrasound. A group of wires aligned along a vertical column andspaced by 1 cm was imaged to examine the amplitude of the ultrasound signal as afunction of depth. Plots of the amplitude were used to compare the signal strengthwith and without the motor control box.2.4.2 Phasor SynchronizationWhile the phasor calculation simulation investigated the ability of the fitting algo-rithm to recover a phasor when the frame rate is unknown, it did not investigate theerrors introduced by the phase compensation when an incorrect frame rate is used.To study how this might affect the phasor measurement, a transducer is fixed over atissue mimicking phantom and 2d image frames are collected over a period roughly40–60 seconds long while steady-state harmonic motion is applied to the surfaceof the phantom. Phasors are computed using every 20 consecutive image framesand compensated according to Equation 2.14. Each computed phasor image in theset should be the same after compensation, so the effect of an incorrect frame ratecan be observed by computing the root mean square (rms) error between the firstphasor image and each subsequent phasor imageURMS(i) =√√√ Nc∑a=1Nr∑b=1(Ui+1(a, b) −U1(a, b))2NcNr, (2.27)where Uγ is the γth phasor image, Nc and Nr are the number of columns and rowsin the roi, i ∈ 1 . . . Np − 1, and Np is the number of phasor images.The study was done with the Porta and Texo Ultrasonix ultrasound interfacesusing a 4DL14/38 transducer on a CIRS 049 elastography phantom, and the BKframe grabber interface using a 8848 trus transducer on a CIRS 066 prostate elas-tography phantom.An estimate of any phase shift present in the phasor images due to systemic orconstant timing errors is quantified using an optimization to minimize the squareddifference between the first phasor image,U1, and the k th phasor image,Uk , wherethe k th phasor image is shifted by phase θ, amplitude weighted by A, and adjusted9235Stiff InclusionROI Soft InclusionROIØ 2030Figure 2.5: Front view of the CIRS 049 elastography quality assurance phan-tom with the experimental scanning regions indicated (not to scale).for the DC component of noise ηminimizeθ,A,ηU1 − AUkejθ − η2 . (2.28)The experiment was also repeated using the trigger described in Section 2.2.10with the Texo ultrasound interface to synchronize each phasor image with the exci-tation (i.e. every 20th ultrasound frame was triggered). If there is a constant error inthe frame time stamps, then the synchronization should limit the error in the phasormeasurements.2.4.3 Elasticity RepeatabilityAn important aspect of the system is the ability to reliably provide the same mea-surement of absolute elasticity over repeated tests. To test the repeatability of themeasurements, five volumes were acquired sequentially using an Ultrasonix Sonix-Touch scanner with a 4DL14-5/38 motorized linear transducer. The experimentwas repeated using both the Porta ultrasound interface with the bandpass samplingmethod and Texo ultrasound interface with the sector-based scanningmethod. Notethat the bandpass samplingmethod can be implemented on either the Porta or Texo93UltrasoundTransducerFootprintExciterLocations132LateralElevationalFigure 2.6: Diagram of the different exciter locations used to test the repeata-bility of the elasticity measurements as viewed from above the phantom.The positions are shown for measuring the stiff inclusion. The exciterpositions were mirrored about the vertical when measuring the soft in-clusion to keep them on the surface of the phantom.interfaces, while the sector-based scanning method can only be implemented withthe Texo interface at this time. For clinical studies, the Porta interface would bepreferred as it can stream the clinically approved b-mode image from the scanner’sexam software.The transducer was placed over two regions of a CIRS 049 elastography qualityassurance phantom to capture data for the stiffest (62 kPa) and the softest (6 kPa)spheres in the phantom, as shown in Figure 2.5. The mean elasticity within theinclusions and in the surrounding background (29 kPa) was computed and com-pared for each volume. To study the effect of different boundary conditions, themeasurements were repeated using three different exciter placements as shown inFigure 2.6.Ultrasound images were acquired to a depth of 5 cm using a centre frequencyof 5 MHz for the transmit pulses. The phasors were fit using 25 displacement mea-surements at each location in the volume. The Porta interface, using the bandpass94Figure 2.7: Example phasor fitting result to simulated noisy displacementmeasurements.samplingmethod, acquired each 2d ultrasound image, consisting of 128 beam lines,at a rate of 72 Hz. The Texo interface, using the sector-based scanning method, di-vided each 2d ultrasound image into 8 adjacent sectors of 16 beam lines, and eachsector was acquired at a rate of 625 Hz. The 2d images were swept into a thin 3dvolume using 9 motor steps in 0.45◦ increments. Excitation was applied at 200 Hz.2.5 Results2.5.1 Phasor Fitting ResultsAn example of a phasor fitting result is shown in Figure 2.7. The fitted signal closelymatches the true simulated signal despite the presence of displacement measure-ment errors.The first row of Figure 2.8 shows how the error in the magnitude of the phasor95102030 02400.511.5Samples Jitter (µm)PhasorMagnitudeError(µm)00.10.2 02400.511.5∆Frame Rate (dHz)Jitter (µm)00.10.2 10203000.511.5∆Frame Rate (dHz)Samples0.0µm0.5µm1.0µm1.5µm102030 0240246810Samples Jitter (µm)PhaseAngleError(◦)00.10.2 0240246810∆Frame Rate (dHz)Jitter (µm)00.10.2 1020300246810∆Frame Rate (dHz)Samples0◦2◦4◦6◦8◦102030 0240.80.91Samples Jitter (µm)QualityFactor00.10.2 0240.80.91∆Frame Rate (dHz)Jitter (µm)00.10.2 1020300.80.91∆Frame Rate (dHz)Samples0.800.850.900.951.00Figure 2.8: Error in the least squares fit of a simulated 200 Hz cosine excita-tion at 10 µm amplitude with varying number of displacement samples,displacement jitter error, and frame rate error. The first row shows thedifference between the fitted phasor magnitude and the excitation refer-ence, the second row the difference between the recovered phasor angleand the excitation reference, and the third row the quality factor of thefit as defined in Equation 2.13. The first column shows results with noerror in the frame rate. The second column displays the fitting resultsusing 20 displacement samples. The third column has a fixed jitter errorof 2 µm.96changes with the number of displacement samples, the displacement jitter error,and the frame rate error. The results show that for a fixed level of jitter error (2 µm),the magnitude error varies from 3.9 % with 30 samples and no frame rate error,to 6.7 % with 10 samples and 0.2 dHz frame rate error. For a fixed number of 20samples, the magnitude error increases approximately linearly with jitter error from0 to 9.9 %, while the frame rate error contributes to less than 0.4 % change in theerror varying from 0 dHz to 0.2 dHz. Looking at the error in estimating the pha-sor magnitude varying only the jitter error and number of samples with no framerate error, the magnitude error again increases approximately linearly with jitter er-ror, while decreasing the number of samples increases the slope of the error lineexponentially.In the second row of Figure 2.8 the phase angle error is analysed. The phaseangle error increases dramatically at large jitter error levels, especially with fewerdisplacement samples. The phase angle error also generally increases with greatererror in the frame rate. At a level of 0.2 dHz frame rate error, the phase angle erroractually worsens when more samples are collected, from 4.3◦ at 10 samples to 4.8◦at 30 samples. This is explained by the longer amount of time required to collectthe samples, allowing the error in frame rate to accumulate.In the third row of Figure 2.8 it can be seen that the quality factor drops off as thejitter error in the displacements increases. The quality factor does not seem to bestrongly affected by the level of error in the frame rate or the number of displacementsamples collected.A second test was conducted for multiple frequencies by adding a second cosinesignal at 240 Hz with an amplitude of 8 µm and phase angle of 30◦ to the previouslydescribed 200 Hz cosine signal. The signal is shown in Figure 2.4 (b). Of the 125different cases using different combinations of displacement samples, displacementjitter error, and frame rate error, the recovered 200 Hz signal was almost identicalto the single frequency case. The largest differences in magnitude and phase were0.015 µm and 0.18◦.Experimental measurements applying the bandpass sampling technique showeda wave pattern polarized in the horizontal direction of the image, regardless of theexcitation position. In fact, the pattern can be observed without any excitation forceand will be referred to as the phasor wave artefact. An example is shown in Fig-97Width [mm]0 10 20 30Depth [mm]0102030405060(a)Width [mm]0 10 20 30Depth [mm]0102030405060(b)Width [mm]0 10 20 30Depth [mm]0102030405060[7m]-5-4-3-2-1012345Width [mm]0 10 20 30Depth [mm]0102030405060(c)Width [mm]0 10 20 30Depth [mm]0102030405060(d)Figure 2.9: A demonstration of a phasor wave artefact observed in experi-mental bandpass sampled phasor images. The temporal displacementsmeasured from (a) a frame early in the acquisition and (b) a frame col-lected later in time show small shifts across the image. The fitted phasor(c) before applying interline compensation shows a small uniform ampli-tude as a result from the temporal displacements, and (d) after applyinginterline compensation shows a wave pattern across the horizontal di-mension.98Width [mm]0 10 20 30 40Amplitude [7m]-1-0.500.51Figure 2.10: A profile across the width of an image containing the phasorwave artefact.ure 2.9. The pattern is caused by fluctuations in the displacement tracking and theinterline compensation. The temporal displacement frames can contain random dis-placement shifts on the order of a micron either towards or away from the transduceras shown in Figure 2.9 (a) and Figure 2.9 (b). This noise can cover a wide band,and can therefore appear at one of the excitation frequency components, shown asa mostly uniform displacement across the frame in the real part of the fitted phasorin Figure 2.9 (c). This inaccuracy on its own should have little effect on the elas-ticity measurements because of its low magnitude and low spatial frequency whichis removed during the lfe processing. However, the interline compensation, whichis supposed to correct for the phase shift of the measured excitation wave patterncaused by sequential line acquisition, when applied to this uniform shift introducesa phase change across the lateral direction. Figure 2.10 shows a profile of the wavepattern across the width of the image. Measuring the distance from crest-to-crestand trough-to-trough gives an estimated wave length of approximately 13 mm. The99significance of this length on the measurements and how it can be predicted is dis-cussed in Section 2.6.2.5.2 Motor Attenuation ResultsThe results from comparing the ultrasound signal strength with and without themotor control box placed between the transducer and the scanner are shown in Fig-ure 2.11. The b-mode shows the vertical column of strong point reflectors used forthe comparison. From top-to-bottom, the brightness increases as the focus depth isreached, then gradually decreases as the signal is absorbed by the phantom mate-rial. Looking at just the rf envelope amplitude along the beam line intersecting thevertical column, it is evident that the peak signal amplitude from each point targetwith the motor control box inserted is within 2 % of the peak signal amplitude whenconnecting the transducer directly to the scanner.2.5.3 Phasor Synchronization ResultsThe results from measuring the same repeated motion over a period of 40–60 sec-onds using the Ultrasonix Porta and Texo interfaces on a CIRS 049 elastographyquality assurance phantom are plotted in Figure 2.12. The vibration was appliedat 200 Hz and was measured at a frame rate of 72 Hz, measuring a peak phasoramplitude of 4 µm. The rms error when considering only the absolute part of thephasor remains small in magnitude and flat over time, indicating that the same waveamplitudes are being detected. There is a 0.41 µm jump over about 0.5 seconds forthe absolute phasor at around 35 seconds in the Porta dataset. A 0.17 µm jump isalso visible at around 42 seconds in the Texo dataset. These jumps were possiblycaused by someone walking near the apparatus during the acquisitions, or an ac-cidental bump of the ultrasound scanner, transducer clamp, or phantom. The realand imaginary parts of the phasor show a gradual increase in rms error over time,indicating that the phase of the measurements is slowly drifting and causing a dif-ference compared to the first detected phasor. There is a sudden increase of 0.59 µmand 0.44 µm in the error for the real and imaginary components for the Porta mea-surements at around 25 seconds. The Porta timing is provided by a frame counterand the frame rate, and the sudden increase occurs when the frame counter reaches100(a)(b)Figure 2.11: Measurements comparing the ultrasound signal strength withand without the motor driver between the transducer and the scanner.(a) The b-mode image with a yellow dashed line indicating the line in(b) which is used to plot the signal in both cases.101(a) (b)Plane Time Stamp [s]0 20 40 60Error [7m]00.20.40.60.81AbsReIm(c)(d) (e)Plane Time Stamp [s]0 20 40 60Error [7m]00.050.10.150.20.250.3(f)Figure 2.12: Results of phase compensation measured with the (top) Portaand (bottom) Texo ultrasound interfaces showing from left-to-right theabsolute magnitude of the phasor, the real part of the phasor, and a plotof the phasor rms error over time for the absolute, real, and imaginarycomponents of the phasor. The magenta box indicates the region thatthe rms is calculated over.102(a) (b)Plane Time Stamp [s]0 20 40 60Error [7m]01020304050 AbsReIm(c)Figure 2.13: Results of phase compensationmeasuredwith the BK ultrasoundinterface showing from left-to-right the absolute magnitude of the pha-sor, the real part of the phasor, and a plot of the phasor rms error overtime for the absolute, real, and imaginary components of the phasor.The magenta box indicates the region that the rms is calculated over.Plane Number0 100 200Estimated Amplitude00.20.40.60.81(a)Plane Number0 100 200Estimated Noise [7m]00.20.40.60.81(b)Plane Number0 100 200Estimated Compensation [7s]-800-600-400-2000200 Linear Fit(c)Figure 2.14: Estimated amplitude, noise, and time shifts to minimize thesquared error between the first phasor image collected and each sub-sequent compensated phasor image, corresponding to the Porta ultra-sound interface measurements.103(a) (b)Plane Time Stamp [s]0 20 40 60Error [7m]00.050.10.150.20.250.3AbsReIm(c)Figure 2.15: Results of phase compensation using a trigger to synchronize thestart of each phasor image, using the Texo ultrasound interface, with theexcitation. From left-to-right are the absolute magnitude of the phasor,the real part of the phasor, and a plot of the phasor rms error over timefor the absolute, real, and imaginary components of the phasor. Themagenta box indicates the region that the rms is calculated over.its maximum allowable value (experimentally determined to be 16000) and startscounting up again from zero. Since the maximum counter number is fixed it can bedetected and corrected for.The test was also applied to the BK ultrasound interface, instead using a CIRS066 prostate elastography phantom, resulting in the measurements plotted in Fig-ure 2.13. In this case, the vibration was applied at 165 Hz and was measured at aframe rate of 43 Hz, measuring a peak phasor amplitude of 78 µm. The roi in thephasor images is smaller than in the previous results because the waves are onlymeasured properly within the ellipse shaped prostate mimicking region. Similar tothe results obtained with the Ultrasonix interfaces, the rms error of the absolutepart of the phasor remains flat over time, while the rms error in the real and imagi-nary parts increases with time. The magnitude of the error appears to be larger forthe BK measurements, but this might be partially explained by the larger phasoramplitudes.The larger phase drift in the BK measurements is confirmed in the results ob-tained by applying the optimization in Equation 2.28 to estimate the phase shift104(a) (b)[kPa]510152025303540Figure 2.16: Example elasticity images of the (a) soft and (b) stiff inclusionsmeasured in the CIRS 049 elastography quality assurance phantom us-ing the Porta ultrasound interface on an Ultrasonix scanner. These im-ages are taken from the centre 2d image slice of the swept volume.The dotted circles correspond to the inclusion location. Excitation wasapplied at 200 Hz.between phasor images. For the Porta interface, the estimated phase shift for eachplane to correct for the drift of the phasor images over time was 3.03 µs, after cor-recting for the discontinuity in the real and imaginary parts of the phasor due tothe frame counter roll-over. The Texo interface had an estimated shift of 2.07 µs,and the BK interface 30.2 µs per phasor image. In all cases, the fit to Equation 2.28produced amplitude weightings consistently close to unity, noise at a fraction of amicron, and time delays that increased in magnitude linearly for each phasor image,even in the BK case despite the curving in the rms error. An example of the fit forthe Porta interface is shown in Figure 2.14.The rms error over time when triggering the ultrasound acquisition with theexciter is shown in Figure 2.15. In this case, the error remains flat over time for allof the absolute, real, and imaginary parts of the phasor.105BandpasssamplingSector-basedscanningYoung's Modulus [kPa]0510152025303540InclusionBackground(a)BandpasssamplingSector-basedscanningYoung's Modulus [kPa]0510152025303540InclusionBackground(b)Figure 2.17: Repeated measurements of the mean elasticity in a (a) stiff and(b) soft inclusion, in a moderately stiff background with different mo-tion sampling techniques. Excitation was applied at 200 Hz.1062.5.4 Elasticity Repeatability ResultsFigure 2.16 displays example elasticity images of the soft and stiff inclusions inthe CIRS 049 elastography quality assurance phantom. The displayed images cor-respond to the centre image slice in one of the volumetric acquisitions using thebandpass sampling technique with the Porta ultrasound interface. The location ofthe inclusions was determined through manual segmentation of the b-mode ultra-sound images, and are indicated with dotted circles in Figure 2.16. As expected,the estimated elasticity values are lower within the soft inclusion region, and higherin the stiff inclusion region, relative to the background. Some variation in the esti-mated elasticity is observed in regions expected to be homogeneous. This variationis likely caused by regions of low displacement amplitude, or nodes in the wavefield, which result in poor estimation of derivatives and corrupt the elasticity mea-surements [136, 199].The following repeatability results averaged the Young’s modulus values withinthe segmented region to obtain a measure for the inclusion elasticity, and outsideof the segmented region for a measure of the background elasticity. The measure-ments of the mean elasticity of the stiff and soft CIRS inclusions and the back-ground from five separate volumetric acquisitions using either bandpass samplingor sector-based scanning are shown in Figure 2.17. The measurements of the back-ground elasticity in the region near the stiff inclusion are consistent, all near 17 kPawith a range of less than 0.5 kPa. The measurements of the background in the re-gion near the soft inclusion are also consistent but are slightly larger in magnitudeat around 19 kPa, with a range of less than 1 kPa. Measurements of the stiff inclu-sion elasticity show the greatest variation in these results with the a range of 4 kPaaround amean of about 31 kPa. The soft inclusionmeasurements show repeatabilitysimilar to the background, with an average stiffness of about 9.5 kPa and a range ofvalues spanning less than 1 kPa. For all of these measurements there appears to beno substantial difference between the bandpass sampling or sector-based scanningmethods.The repeatability of mean elasticity measurements as a function of exciter po-sition are shown in Figure 2.18. For the measurements around the stiff inclusion,exciter positions 1 and 2 demonstrate similar results to the previous measurements107Exciter Position1 2 3Young's Modulus [kPa]0510152025303540InclusionBackground(a)Exciter Position1 2 3Young's Modulus [kPa]0510152025303540InclusionBackground(b)Figure 2.18: Repeated measurements of the mean elasticity in a (a) stiff and(b) soft inclusion, in a moderately stiff background collected usingsector-based scanning at three exciter positions. Excitation was ap-plied at 200 Hz.108both in terms of mean elasticity and spread and in both the inclusion and back-ground. At exciter position 3, both the inclusion and background show increasesin mean elasticity by approximately 7 kPa and 2.5 kPa, respectively. The measure-ments however have a similar spread compared to the other exciter positions. Themeasurements around the soft inclusion show a similar trend, with increased meanelasticity of approximately 0.5 kPa and 2 kPa for the inclusion and background, re-spectively, again with a similar spread for repeated volume acquisitions.2.6 DiscussionAn important aspect of the system is the elasticity image frame rate. The totalcomputational time for motion tracking and elasticity inversion using a GPU-basedimplementation is 71 ms [23]. The frame rate of the elasticity images is limitedby the rate of ultrasound data collection rather than computation time for typicalimaging settings. For example, an acquisition with an imaging depth of 6 cm and128 laterally spaced beam lines, using 20 ultrasound frames to estimate the mo-tion phasor, the 2d elasticity image could be updated at a rate of 5 Hz. Elasticitymeasurements have shown improvements from 3d volumes with as few as 5 to 7elevationally spaced planes, with a volumetric elasticity update every one to twoseconds [23]. To obtain a full volumetric sweep, using a 4DL14-5/38 motorizedlinear array transducer with a default motor step of 0.45◦, a total of 61 planes isrequired which reduces the volumetric acquisition time to around 10–20 seconds.In all cases, the acquisition can be repeated continuously for an indefinite time,providing updated elasticity measurements during a procedure.Care must be taken in selecting multi-frequency signals so that they do not over-lap in the base band. For example, excitation frequencies of 200 Hz and 270 Hzsampled at a frame rate of 70 Hz both appear at 10 Hz in the base band. In thiscase the phasor calculation is no longer unique and therefore they cannot be reli-ably found. The solution to the least squares fit is no longer unique because thematrixM defined in Equation 2.6 becomes rank deficient. Observing that the timesamples, tk , are taken at multiples of the sampling rate, andω1fs(mod pi) =ω2fs(mod pi), (2.29)109it is clear thatcos (ω1tk ) = cos (ω2tk ) (2.30)andsin (ω1tk ) = sin (ω2tk ) . (2.31)The phasor fitting results in this chapter were presented by fitting a signal atthe excitation frequency, however the procedure can also be accomplished usingthe base band frequencies instead. For the test cases mentioned in this section, thedifference between the results was close to the machine precision, indicating nodifference between the two methods.In the phasor fitting results, it was revealed in Figure 2.9 that small randomshifts in the displacement estimates from experimental measurements resulted in aphasor wave artefact across the lateral dimension of the phasor image. The waveappearance is caused by the interline compensation which is used to shift the phaseof tissue motion collected at different times. Ignoring any motion induced by theexcitation, from Equation 2.14 the wave appears across the lateral coordinate as aplane waveAe−j2pi feTPRFl+θ, (2.32)where TPRF is the time delay between acquiring adjacent beam lines, l is the beamline number, A is the amplitude of the phasor fitted to the random displacementshifts appearing in the band at the excitation frequency, fe, and θ is a random phaseresulting from the phasor fit. For a linear array transducer, this will appear in Carte-sian coordinates as a horizontally polarized plane wave. The elasticity measuredby the lfe algorithm can be predicted by computing the wave number from Equa-tion 2.32k =dφdx=feTPRF∆x, (2.33)where ∆x is the spacing between beam lines. For curvilinear transducers, the wavewill no longer appear purely as a plane wave in Cartesian coordinates, but will bestretched with increasing depth, which would show as an increase in elasticity withdepth. For the example in Figure 2.9, the data was acquired with a linear array,the excitation frequency was 210 Hz, the interline time was 110 µs, and the linespacing was 0.3 mm, resulting in a wave number of 0.077 mm−1 which corresponds110to a wave length of approximately 13 mm which matches the wave length estimatedfrom Figure 2.10. The elasticity, as computed from Equation 2.26 is 22 kPa, whichis within the range of soft tissue elasticity. This makes it difficult to filter the phasorwave artefact from the phasor images because the spatial frequency is within theband of the expected shear wavemeasurements, unlike the compression wave whichcan be filtered due to its low spatial frequency away from the measurements. It maybe possible to apply a filter before applying the interline compensation. Furtherinvestigation into whether this could affect the measured shear wave field would bean interesting area of future research.The phasor synchronization results show that the wave pattern can slowly driftout of phase over time if it is not synchronized with the exciter. The fact that therms error does not increase over time for the absolute part of the phasor, but doesfor the real and imaginary parts indicate that it is a timing issue. This is furtherconfirmed by the elimination of the increase in error with time when the acquisitionis triggered by the excitation module. The underlying cause is likely either an errorin estimating the frame time stamps, or clock skew caused by different clocks usedfor the ultrasound scanner and excitation modules. The latter might be solved bydriving both by a master clock. If the master clock is sufficient for removing thephase drift error, it may become the preferred method in most use cases comparedto the triggered synchronization proposed in this work, since there would never be alag introduced by waiting for a trigger pulse to start the next ultrasound acquisition.The elasticity measurements of the CIRS 049 elastography phantom are slightlydifferent from the values reported by the manufacturer. For soft inclusion there isan overestimation of the reported elasticity, and for the stiff inclusion and the back-ground there is an underestimation of the reported elasticity. This is consistent withprevious measurements by our group at UBC using both ultrasound and mri basedelastography methods, with a variety of different inversion algorithms [22, 135–137]. The reasons for this discrepancy could be the difference in temperature, ex-citation frequency, or changes of the material properties of the phantom over time.The measurements done by CIRS were performed in a temperature controlled en-vironment at 22 ◦C, using quasi-static compression, and at the time of manufactureapproximately 10 years before the measurements were collected for this work. Inthis work, the temperature was at the ambient room temperature of the laboratory,111and was not controlled or measured. The frequency dependence of the elastic mod-ulus of the phantom is not known, however there could be substantial differencebetween 0 Hz and 200 Hz.The elasticity measurements at different exciter locations show a systemic in-crease in elasticity values for the third exciter location. This location is the further-est from the transducer. The increase can be explained as a diffraction bias, wherethe wave motion is no longer primarily along the axial direction, and therefore theprojected motion vector results in an underestimation of the wave number. Thedecrease in wave amplitude due to the greater wave attenuation over the longer dis-tance also likely contributes to greater error in the elasticity measurements for thethird location.2.7 ConclusionThis chapter has described the design and analysis of a flexible research based elas-tography platform. The implementation details for several different ultrasound in-terfaces has been described. The system is generic and can be adapted to any ul-trasound scanner that allows access to beamformed rf or i/q echo signals (usingthe bandpass sampling method). Simulations demonstrated that a conventionalb-mode pulse sequence can be used to measure steady state shear wave motion,and the phasor motion estimation is robust to speckle tracking jitter and errors inestimating the frame time stamps or frame rate. A customized ultrasound motordriver was developed that can interface with any motorized ultrasound transducerwithoutmodification, which provides greater control over the ultrasound acquisitionwithout reducing ultrasound signal quality. Experimental measurements showed agradual shift in motion phase over long acquisitions. For most cases, the acquisi-tion time should be short enough to safely ignore the error, but for large volumetricacquisitions, or for combining multiple measurements over a long period of time, asynchronization between the excitation and ultrasound scanner using a trigger caneliminate the problem. The elasticity estimates are repeatable, but have a slightdependence on excitation conditions.The following chapter describes some initial measurements obtained by apply-ing the system to ex vivo placenta tissue. The system is also being used concurrently112by other researchers at UBC for studies of liver, kidney, and prostate elasticity.113Chapter 3Viscoelastic Characterization ofEx Vivo Placenta Tissue UsingSWAVE3.1 IntroductionThe placenta is the organ connecting a developing fetus to the uterine wall, and is re-sponsible for the exchange of oxygen, nutrients, and waste between the mother andfetus. The fetal blood circulates through the umbilical cord to the placenta where itbranches into vessels and eventually a network of branching chorionic villi. Placen-tal abnormalities, such as preeclampsia and intrauterine growth restriction (iugr),can have long-term impacts on both maternal and fetal health. Preeclampsia isa progressive hypertensive disorder of pregnancy, and is associated with approxi-mately one third of severe obstetric morbidity [332]. iugr is the pathologic restric-tion of fetal growth due to adverse genetic or environmental factors, characterizedby an estimated fetal weight below the 10th percentile. The incidence of iugr isestimated to be 5 % to 7 % of all pregnancies and results in an increased risk ofperinatal, childhood, and adult morbidities [48]. The etiologies of both conditionsare associated with placental factors such as poor uteroplacental perfusion, abnor-mal villous structure, and placental infarction [132, 285].114Due to the vascular nature of the placenta, medical Doppler ultrasound servesas a natural tool to assess placental function. The pulsatility index of the uterineartery, which is a calculated as the difference between peak systolic and end dias-tolic flow divided by the mean velocity, is the most predictive Doppler measurementfor diagnosis of preeclampsia or iugr. The presence of a prediastolic notch in thevelocity waveform provides further predictive power [72]. However, the sensitiv-ity of Doppler ranges from 20 % to 60 %, with a positive predictive values of 6 %to 40 %, and therefore Doppler assessment alone has limited value as a screeningtest [26].Histology indicates that abnormal placenta microstructure is significantly dif-ferent from normal placenta, and is suggested as a cause of significant differencesin elastic and viscous properties in mechanical testing [179]. Elastography offersa non-invasive method of characterizing changes in mechanical properties in vivo.Recently, a few studies have investigated applying elastography to the placenta inboth ex vivo and in vivo settings [51, 185]. Initial studies have shown promisein identifying placental abnormalities. For example, Kılıç et al. [159] estimatedelasticity using arf induced shear wave speed measurements and found a signifi-cant difference in placental elasticity between women diagnosed with preeclampsiaand healthy controls for gestational ages between 28–35 weeks. Similarly, signif-icant increases in elasticity have been found for measurements during the secondtrimester of pregnancy using shear wave speed [70], and relative stiffness inferredfrom strain [71]. A significant increase in shear wave speed for pregnancies diag-nosed with either fetal growth restriction or pregnancy induced hypertension hasbeen observed [229]. Sugitani et al. [301] found a significant difference betweennormal and growth restricted placental shear wave speed, but did not find a signifi-cant difference for pregnancy induced hypertension.Elastography can also determine the viscous properties of tissue by examiningthe frequency dependence of the shear waves [63, 81, 162]. Measurement of theviscous properties of placenta tissue is limited, though a dependence on the defor-mation rate has been observed in mechanical testing [333]. A recent elastographystudy of the placenta fit a power law rheological model to account for shear wavedispersion over a band of 20 Hz to 80 Hz [51].Most elastography studies of the placenta to date have been based on arf tran-115sient shear wave speed measurements at a single point on anterior placentas. Theapplication of arf to other placenta locations might be difficult due to depth lim-itations. For example, one study was unable to study placentas located at depthsgreater than 8 cm [70]. The swave system uses longitudinal vibration which gen-erates shear waves deep into the body. The placement of the excitation source canalso be selected by the operator, so it could be placed in a location optimized forthe placenta location without affecting the placement of the ultrasound transducer.Elasticity is computed quickly over a large volumetric roi which can provide addi-tional information about the spatial variability of placenta elasticity.The purpose of this study is to obtain initial placenta data using the swavesystem and to select appropriate experimental factors and swave parameters. Theswave system is used to measure the Young’s modulus of six placenta samples overa band of 60 Hz to 200 Hz. The frequency dependence is observed and the viscosityis quantified by fitting rheological models to the data. This is meant as a proof-of-concept to demonstrate the swave system and start work on better understanding theplacenta, and not meant as a final diagnostic tool for a specific medical condition.3.2 Methodsswave measurements were applied to six healthy intact placenta samples. The pla-centas were obtained after full-term delivery, and were measured approximately4 hours after delivery on average, with a maximum delay of 10 hours after deliv-ery. This study (H15-00974) was performed under written informed consent afterapproval by the UBC Children’s andWomen’s Research Ethics Board. The consentform used for this study is provided in Appendix B. Study data were collected andmanaged using REDCap electronic data capture tools hosted at BC Women’s andChildren’s Hospital [128].Samples were stored immediately after delivery in a refrigerator at 4 ◦C, andwere heated to 37 ◦C to approximate in vivo temperature just before swave mea-surements by submerging the samples in a constant temperature water bath (Cole-Parmer, Montreal, Quebec, Canada). The maternal side of the placenta was placedon over an acoustic absorbing pad to reduce reverberation artefacts, and both theplacenta and pad were placed on top of a steel plate and wrapped loosely in a mesh116Figure 3.1: The experimental apparatus for placenta swave measurements.The placenta (1) is submerged in the water bath after being placed ona acoustically absorbent pad and a steel plate and wrapped loosely in amesh bag. The ultrasound transducer (2) is submerged in the water bathand is imaging the fetal side of the placenta. The transducer is placed ina holder and positioned using a flexible arm. The disk at the end of theexciter (3) makes contact with the fetal side of the placenta to generateshear waves.bag to keep the placenta submerged and stationary in the water bath. The exciterused to generate shear waves in the placenta was partially submerged into the waterbath to make contact with the fetal side of the placenta. The exciter consisted of a3 cm diameter circular steel plate which was placed in contact with the placenta onone end and at the other attached to a 10 cm long steel shaft threaded into a voicecoil actuator (LDS V203, Brüel & Kjær, Nærum, Denmark) which was held sta-tionary above the water bath using a 3-prong clamp (Fisher Scientific, Waltham,MA, USA). A 4DL14-5/38 motorized swept volume ultrasound transducer (Ver-mon, Tours, France) was placed in a custom made 3d printed case which was usedto attach the transducer to a flexible positioning arm (CIVCO Medical Solutions,Kalona, IA, USA). The transducer face was submerged in the water bath and wasfixed by the positioning arm with light contact on the fetal side of the placenta orwith a small gap of less than a few millimetres. This placement was used to avoidpre-compression effects which could potentially increase the observed elasticity.The experimental apparatus is shown in Figure 3.1.The ultrasound transducer was connected to a SonixTouch ultrasound machine117(Ultrasonix Medical Corp., Richmond, BC, Canada). The swave system was con-figured with the Porta ultrasound interface, the Data Translation excitation gener-ator, and the custom stepper motor driver for Ultrasonix machines to control thevolume acquisition, all described in Chapter 2. The ultrasound scanner transmit-ted pulses centred at 5 MHz, which corresponds to the recommended abdominaltransducer frequency for obstetric examinations by the American Institute of Ul-trasound in Medicine [14]. The thickness of a full term placenta is expected to bebetween 1.5 cm to 2.5 cm [157], so the scanner was set to a depth of 3 cm to cap-ture the full thickness of all placentas and allow for a small water gap between thetransducer and placenta. Monochromatic harmonic excitation was used to generateshear waves in the placentas. A single ultrasound volume was collected for each ofsix excitation frequencies at 60 Hz, 80 Hz, 90 Hz, 100 Hz, 120 Hz, and 200 Hz. Theexcitation frequencies were chosen to cover a wide range to examine the frequencydependence of the elasticity measurements, to line up nicely within the basebandso they could easily be differentiated, and to create at least one wavelength acrossthe depth of the placenta sample to ensure lfe algorithm reached the correct esti-mate because it requires a transition zone of half a wavelength into a region [199].The scanner collected 2d rf image frames at a rate of 72 Hz, so the excitation fre-quencies were placed at −12 Hz, 8 Hz, 18 Hz, 28 Hz, −24 Hz, and −16 Hz in thebaseband. Using measurements of placenta elasticity from the literature [229],the average healthy placenta has a Young’s modulus of approximately 5 kPa, sothe chosen excitation frequencies result in expected wavelengths of 2.2 cm, 1.6 cm,1.4 cm, 1.3 cm, 1.1 cm, and 0.65 cm. Based on the simulations from Chapter 2,a total of 25 ultrasound rf image frames were collected at each motor position inthe volume for measuring tissue displacement and accurately fitting phasors at eachexcitation frequency. The motor was stepped to 10 positions, subtending an angleof approximately 4.5◦. This relatively thin motor sweep decreased the acquisitiontime, which helped to limit the phase drift error described in Chapter 2, while stillcovering enough distance in the elevation direction to fit at least a quarter of a shearwavelength based on estimates of placenta elasticity from the literature and the ex-citation frequencies chosen in this study. The thin volume approach has producedaccurate results on phantoms in a previous study [23]. Shear wave motion was esti-mated along the axial direction of the transducer, and the Young’s modulus at each118excitation frequency was computed from the scan converted motion phasors usingthe lfe algorithm and Equation 2.26.To ensure quality measurements, the phasor amplitude was monitored afterstarting the excitation at 60 Hz to check that the amplitude adequately penetratedthe roi and had a magnitude between 5 µm to 50 µm which is in a range accuratelytracked by the cross-correlation speckle tracking algorithm. The accuracy of thespeckle tracking was monitored by observing images of the normalized correlationcoefficient and ensuring that the value was greater than 0.95 over the majority of theroi. The phasor fit to the displacement measurements was monitored by ensuringthe quality factor, as computed by Equation 2.13, was greater than 0.8, averaged overthe entire image. Real-time display of the b-mode, phasor amplitude, and Young’smodulus images during acquisition also provided quality control. Data were savedfor offline processing.For each placenta sample the images were examined and an annular sector roiwas manually selected containing placenta tissue (avoiding regions in the ultra-sound volumes that imaged outside of the placenta) with good wave amplitude (try-ing to avoid nodes in the wave pattern). This roi generally covered the majorityof the image volume. The mean and standard deviation of the Young’s moduluswas computed for each excitation frequency on each placenta sample. To help gaininsight into the frequency dependence of the results, viscoelastic models were fit tothe measurements as described in the following section.3.2.1 Rheological ModellingThe viscoelastic properties of tissue can be determined by examining the frequencydependence or dispersive behaviour of the shear waves. The frequency dependencecan be modelled by replacing the shear modulus with the complex shear modulusµ∗ = µ + jµ′. (3.1)The shear wave number also becomes complexk∗ =√ρω2µ∗ . (3.2)119The shear wave speed can be obtained from the real part of the shear wave num-ber [227]cs =ω< (k∗) . (3.3)As mentioned in Section 1.5.4, the lfe algorithm provides measurements of thewave number, so the swave system can easily convert its output to shear wave speedusing Equation 3.3. For this work, the mean wave number over the roi is used.Using Equation 3.1 and Equation 3.2, the shear wave speed can be related to thecomplex shear modulus usingcs =√√√√ 2 (< (µ∗)2 + = (µ∗)2)ρ(< (µ∗) +√< (µ∗)2 + = (µ∗)2) . (3.4)The complex shear modulus can be related to the viscoelastic properties us-ing rheological models. Commonly used models for biological soft tissues includeVoigt, Maxwell, and Zener [111]. A mechanical schematic representation of themodels is shown in Figure 3.2. To derive the complex shear modulus for thesemodels, it is convenient to use mechanical impedance, which measures how mucha structure resists motion. It is usually expressed as a ratio of force to velocity, butin this chapter in order to relate impedance to continuum mechanics, shear stressreplaces force and shear strain rate replaces velocity, which in the frequency domainresults inZ =σ (ω)jω (ω). (3.5)Since the shear modulus is the ratio of shear stress and shear strain, the shear mod-ulus can be expressed in terms of impedance asµ∗ = jωZ . (3.6)This makes it simple to determine the shear modulus from the schematic repre-sentations in Figure 3.2, and then obtain the shear wave speed from Equation 3.4.The total impedance of components in parallel is the summation of the componentimpedances. For components in series, the inverse total impedance is the sum of120η µ (a)µ η (b)µ2 η µ1 (c)Figure 3.2: A mechanical schematic of the (a) Voigt, (b) Maxwell, and (c)Zener rheological models.121the inverses of the component impedances. The impedance of a spring elementparametrized by µ isZ =µjω, (3.7)and for a dashpot parametrized by η isZ = η. (3.8)For the Voigt model, this results in a shear modulus ofµ∗ = µ + jωη, (3.9)and a shear wave speed ofcs =√2(µ2 + ω2η2)ρ(µ +√µ2 + ω2η2) , (3.10)where µ and η refer to the spring and dashpot parameters in Figure 3.2a.For the Maxwell model, the shear modulus isµ∗ =jωµηµ + jωη, (3.11)and the shear wave speedcs =√√ 2µρ(1 +√1 + µ2ω2η2) . (3.12)For the Zener model, the shear modulus isµ∗ =µ1µ2 + jω (µ1η + µ2η)µ2 + jωη, (3.13)122and the shear wave speedcs =√√√ 2 (µ21µ22 + ω2η2 (µ1 + µ2)2)ρ(µ1µ22 + ω2η2 (µ1 + µ2) +√(µ21µ22 + ω2η2 (µ1 + µ2)2) (µ22 + ω2η2)) .(3.14)To estimate the parameters in each of the models, a least squares fit over thesix excitation frequencies is found between the model and the measured shear wavespeed from swaveminimizeχ6∑i=1(cs(ωi, χ) − cˆs (ωi))2subject to χa ≥ 0, a = 1, . . . , n,(3.15)where χ is a vector of n rheological parameters, and cˆs (ωi) is the measured shearwave speed from swave at excitation frequency ωi. The lower bound constraintwas applied to avoid non-physical solutions for the rheological parameters. Thegoodness of the fit is evaluated using the rms errorcRMS =√√166∑i=1(cs(ωi, χ) − cˆs (ωi))2. (3.16)3.3 ResultsAn example of the waves measured inside of a placenta sample is displayed in Fig-ure 3.3. The b-mode image shows mostly uniform speckle through the imagingregion, with some bright reflections near the centre of the image likely caused bya collagenous matrix encasing damaged villi (discussed further later). The acous-tically absorbent pad is visible at the bottom of the image at a small angle to thehorizontal and a dark region in the top left of the image is created from waves pass-ing through water where the exciter has depressed the tissue away from the face ofthe ultrasound transducer. The real part of the complex motion phasor shows thewave pattern at one phase of the excitation. The waves appear to be propagatingroughly from left to right. The wavelength appears to decrease with increasing ex-123(a) (b) (c)[7m]-10-50510(d) (e)(f) (g)Figure 3.3: The 2d image plane from the centre of the volumetric ultrasoundsweep for placenta Sample 5. (a) The b-mode image shows the placenta,with bright reflections near the centre of the image. The correspondingreal part of the complex motion phasor is shown at the excitation fre-quencies (b) 60 Hz, (c) 80 Hz, (d) 90 Hz, (e) 100 Hz, (f) 120 Hz, and (g)200 Hz. All images have a depth of 3 cm and width of 3.8 cm.124Excitation Frequency [Hz]60 80 90 100 120 200Young's Modulus [kPa]0510152025303540455055Sample 1Sample 2Sample 3Sample 4Sample 5Sample 6Figure 3.4: Mean Young’s modulus measurements as a function of frequencyfor the six placenta samples (standard deviation shown as bars). Theplot markers are spread a small amount about each excitation frequencyto improve visualization—this does not correspond to actual variationsin the physically applied excitation frequency.citation frequency. The region containing the damaged villi seems to change thewave pattern slightly which is especially visible at higher frequencies.Mean Young’s modulus measurements, as computed by the lfe algorithm andEquation 2.26, as a function of excitation frequency for the six placenta samples areplotted in Figure 3.4. All samples show a general trend of increasing elasticity withincreased excitation frequency as expected because Equation 2.26 does not includea separate viscous term. The mean elasticity values between samples are similarat each excitation frequency, with greater spread occurring at higher frequencies.The variance of the elasticity within a sample also increases with frequency. Thismay be partially explained by the decrease in excitation amplitude, as shown inFigure 3.5, possibly leading to a poorer snr at high frequencies.125Excitation Frequency [Hz]60 80 90 100 120 200Phasor Amplitude [7m]05101520253035Figure 3.5: The mean amplitude of the measured phasor for each placentasample over all excitation frequencies in the region where the mean elas-ticity was calculated. The solid line connects the median of the meansfor each frequency, indicating that the general trend is decreasing phasoramplitude with frequency.Figure 3.6 shows the fits to the shear wave speed dispersion using the Voigt,Maxwell, and Zener models. The viscoelastic parameters which correspond to thefitted models for each sample and the rms error in the fits are provided in Table 3.1.All of the models provide a similar fit to the measurements, with the Voigt andZener models providing an almost identical curve. For this data set the Voigt modelprovides the best fit in terms of rms error.Although most of each placenta appears as uniformly textured tissue in b-modeimages, some placentas did exhibit some heterogeneities which were also investi-gated. Measurements were repeated in three different locations on placenta Sample4. The b-mode and Young’s modulus (computed at 120 Hz) images for the differ-ent regions are shown in Figure 3.7. The elasticity in the first measurement region126Frequency [Hz]60 80 90100 120 200Shear Wave Speed [m/s]0.511.522.533.5MeasurementsVoigt FitMaxwell FitZener Fit(a)Frequency [Hz]60 80 90100 120 200Shear Wave Speed [m/s]0.511.522.533.5MeasurementsVoigt FitMaxwell FitZener Fit(b)Frequency [Hz]60 80 90100 120 200Shear Wave Speed [m/s]0.511.522.533.5MeasurementsVoigt FitMaxwell FitZener Fit(c)Frequency [Hz]60 80 90100 120 200Shear Wave Speed [m/s]0.511.522.533.5MeasurementsVoigt FitMaxwell FitZener Fit(d)(continued)is mostly uniform with increasing elasticity with increasing depth. Computing themean Young’s modulus over the first 3 mm and last 3 mm depth quantifies the in-crease from 6.5 kPa to 16.4 kPa. The second measurement region contains a darkcircular region near the centre of both b-mode and elasticity images, likely cor-responding to a blood clot. The third measurement region contains some brightreflections near the bottom centre of the b-mode image, again likely caused by dam-aged villi as in Figure 3.3, and an increase in elasticity seems to correspond to thisarea. The mean and standard deviation of the elasticity for each of these regions127Frequency [Hz]60 80 90100 120 200Shear Wave Speed [m/s]0.511.522.533.5MeasurementsVoigt FitMaxwell FitZener Fit(e)Frequency [Hz]60 80 90100 120 200Shear Wave Speed [m/s]0.511.522.533.5MeasurementsVoigt FitMaxwell FitZener Fit(f)Figure 3.6: The shear wave speed dispersion relations found using the Voigt,Maxwell, and Zener models for each of the placenta samples ((a)–(f)correspond to samples 1–6).over the full range of excitation frequencies is shown in Figure 3.8. For computingthe statistics, the second region did not include the blood clot, but the third regiondid contain the damaged villi. Visually the mean elasticity measured in the first tworegions appears very similar, especially for frequencies ≤ 120 Hz, however thereis a statistically significant difference between the two regions over all frequencies,using the Student’s t-test for comparison (p < 0.001). The third region shows anincrease in elasticity compared to the other two regions over all frequencies whichis likely due to the presence of the damaged villi.3.4 DiscussionThe elasticity values measured in this work can be compared with previous placentaelastography studies. Figure 3.9 summarizes the range of mean Young’s modulifrom several studies. When necessary, measurements were converted from shearwave speed to Young’s modulus usingE = 3c2s ρ, (3.17)128(a) (b)[kPa]051015202530(c) (d)(e) (f)Figure 3.7: The b-mode (first column) andYoung’smodulus (second column)image planes from the centre of the volumetric ultrasound sweep fromthree different regions of placenta Sample 4. The Young’s modulus im-ages shown here were computed from waves excited at 120 Hz.129Table 3.1: Rheological model parameters and rms error corresponding to thefits to the shear wave speed dispersion shown in Figure 3.6 for six placentasamples. The final column shows the inter-sample mean of each quantity(standard deviation in brackets).Model Sample Mean1 2 3 4 5 6Voigtµ (kPa) 0.369 0.733 0.690 0.451 0.295 0.607 0.524 (0.179)η (Pa s) 13.6 21.4 22.7 14.7 11.0 22.1 17.6 (5.05)cRMS (m/s) 0.164 0.301 0.099 0.396 0.070 0.291 0.220 (0.129)Maxwellµ (MPa) 7.64 9.15 5.64 19.8 3.82 13.6 9.95 (5.89)η (Pa s) 11.4 17.4 18.7 12.0 9.19 18.3 14.5 (4.12)cRMS (m/s) 0.189 0.320 0.144 0.429 0.098 0.333 0.252 (0.128)Zenerµ1 (kPa) 0.367 0.731 0.692 0.455 0.296 0.609 0.525 (0.179)µ2 (kPa) 413 486 456 498 326 489 445 (65.9)η (Pa s) 13.7 21.4 22.7 14.7 11.0 22.1 17.6 (5.07)cRMS (m/s) 0.164 0.301 0.102 0.397 0.071 0.293 0.221 (0.129)where ρ was assumed to be 1000 kg/m3, and the tissue was implicitly assumed tobe completely incompressible. The low frequency measurements from this workcorrespond to the low end of the previously reported range, and some of the highfrequency measurements extend beyond the high end of the previously reportedrange. Only the study by Callé et al. [51] used mechanical excitation, howeverit used transient rather than steady-state excitation, and the reported speeds wereextracted from the spectrum at 50 Hz. The other studies used arf excitation, how-ever the spectral content of the excitation for the commercial systems used is notreported.The repeated measurement of elasticity for a single placenta at different loca-tions showed similar mean elasticity results but with a significant difference. Previ-130Excitation Frequency [Hz]60 80 90 100 120 200Young's Modulus [kPa]0510152025303540455055Location 1Location 2Location 3Figure 3.8: Mean Young’s modulus measurements as a function of frequencyfor placenta Sample 4, repeated at three separate locations (standard de-viation shown as bars). The plot markers are spread a small amountabout each excitation frequency to improve visualization—this does notcorrespond to actual variations in the physically applied excitation fre-quency.ous studies have compared mean elasticity measurements from different regions ofthe placenta with one finding differences [301], one not finding differences [185],and one finding difference for abnormal placenta tissue but no difference for healthyplacenta tissue [159]. The results in this work are not directly comparable becausethe difference in mean elasticity is compared within a single placenta sample, ratherthan the differences between groups of mean elasticity as in the literature. The re-sults here add additional insights that the particular structures in the roi can affectthe elasticity measurements.Kılıç et al. [159] measured a larger maximum elastic modulus on the maternalsurface of the placenta than on the fetal surface. The maternal side correspondsto the deeper region in our measurements, so the observation of increased elastic-131Young's Modulus [kPa]0 5 10 15 20 25 30 35Sugitani et al . (2013)Ohmaru et al . (2015)Li et al . (2012)Kiliç et al . (2015)Cimsit et al . (2015)Callé et al . (2015)200 Hz120 Hz100 Hz90 Hz80 Hz60 HzThis study (2016)Figure 3.9: The range of mean Young’s moduli reported in the elastographyliterature for normal placenta, with the range of mean Young’s modulifrom this study at each frequency provided for comparison. The valueswere extracted from Callé et al. [51], Cimsit et al. [70], Kılıç et al. [159],Li et al. [185], Ohmaru et al. [229], and Sugitani et al. [301]. The Calléet al. [51] work includes two transient excitation methods, the topmostcorresponding to arf induced shear waves, and the lower a mechanicalexcitation.ity with depth in Figure 3.7 seems to match the previous measurements. Furtherresearch into the biophysical basis for differences in elasticity between the chori-onic plate, intervillous space, and basal plate would be helpful for explaining theseobservations. Based on our current understanding, our best explanation is the in-creased stiffness is caused by a collagenous layer separating the fetal and maternaltissues. The placenta is organized into cotyledons that each correspond to a singlemajor fetal vascular unit. At term, the three dimensional organization of placentalcotyledons resembles an egg carton. The lateral and deep margins of each cotyle-don are defined by a transition from fetal placental tissue to maternal decidual tissueunder and between each cotyledon. At the border of this transition the placenta ex-udes a contiguous layer of collagenous material which effective separates the fetaland maternal solid tissues. Because this layer is solid and contiguous, this likelyrepresents areas of increased stiffness at the base of the placenta and intermittently132where the lateral aspect of bordering cotyledons are included in the scanned area.When placental villi are damaged, which occurs in all placentas including clini-cally normal placentas, a similar collagenous matrix is produced which encases thedamaged villi and frequently concatenates adjacent villi. This process is a likely ex-planation for focal areas of increased stiffness and density seen away for the basaland lateral cotyledonmargins. Occasionally these areas, particularly along the basalaspect in placentas over 40 week gestational age, may calcify and this would con-tribute to increased echogenicity and stiffness (e.g. bright spots in Figure 3.3(a) andFigure 3.7(e)).The placenta elasticity measurements demonstrated a strong frequency depen-dence. This agrees with previous mechanical testing that concluded that the me-chanical response of placenta tissue is dependent on strain rate [333]. It is difficultto compare the results to previous elastography studies of the placenta as they havenot reported viscosity values. Comparing the mean viscosity of 17.6 Pa s using theVoigt model for the placenta measurements from this work to other tissues reportedin the literature indicates that the placenta is highly viscous. For example, mea-surements of breast tissue have reported a much lower mean viscosity at 2.4 Pa s formalignant cancer, 2.1 Pa s for fibroadenoma, and 0.55 Pa s for surrounding breasttissue [292]. The mean viscosity for healthy brain and liver tissue is also lower at6.7 Pa s and 5.5 Pa s, respectively [162]. The viscosity of the vastus medialis musclein the thigh when actively contracting is closer to but still smaller than the placentaat 12 Pa s [80].There are several limitations in the estimation of viscoelastic properties in thiswork. These limitations include lower tissue displacement snr at higher frequen-cies, simplifications in the lfe inversion algorithm for determining the shear wavenumber, lack of consideration for fitting attenuation, lack of spatial information,and simplifications in rheological modelling. The tissue displacement snr affectsthe viscoelastic parameter estimation in a similar way to the Young’s modulus esti-mation, where the low amplitude of the waves at higher frequencies leads to greateruncertainty in the lfe output. The lfe inversion algorithm also includes simplifica-tions which do not account for any dispersive effects, which likely leads to inaccu-racies in the shear wave speed estimation. A more sophisticated inversion methodthat can directly obtain the complex shear modulus or complex shear wave number133directly would likely be more accurate. One possible approach to find the imagi-nary part of the complex wave number is to measure the phase angle between theshear wave and the applied force [323].The lack of spatial information in the viscoelastic results is due to using anaveraged wave number over a roi for the wave speed calculation. This was done tosimplify the fitting operation from applying it to many thousands of voxels for eachplacenta sample to applying it just once per sample. This also made interpretingthe results simpler and easier to compare between rheological models and differentplacenta samples. However, just as the spatial distribution of Young’s moduluscan contain interesting information as demonstrated in Figure 3.7, the change inviscosity over space could provide some useful diagnostic information.The rheological modelling was simplified by ignoring inertial terms. Elastogra-phy studies typically do not model inertial effects as the fits to experimental resultsappear to be adequate without the need for adding an additional parameter. Whileinertial effects could play a role, the strong effect of viscosity likely produces ahighly overdamped second order system [97, 246].The different excitation frequencies applied in this work were applied in a se-quential manner, with an independent volumetric ultrasound displacement mea-surement acquired for each frequency. This allowed complete isolation of each fre-quency component to help interpret the quality of acquisition of each componentand their influence on the system. For the ex vivo setting this was an acceptablecompromise, but for in vivo applications, the multiple acquisitions may not be ableto be matched sequentially due to motion of the mother or fetus between excitationfrequencies. A multi-frequency excitation approach, which combines the excita-tion signals of each desired frequency, is the recommended approach for in vivomeasurements in order to reduce acquisition time. This efficacy of the simultane-ous multi-frequency approach has previously been demonstrated for in vivo prostatetissue [191].3.5 ConclusionThe feasibility of applying the swave technique to placenta tissue has been demon-strated. Measurement of six healthy placenta samples produced similar elasticity134measurements over a band of excitation frequencies of 60 Hz to 200 Hz, with valuesagreeing with previous elastography measurements in the literature.This is the first study to apply Voigt, Maxwell, and Zener rheological modelsto elastography measurements of placental tissue. Results indicate that all modelsprovide a similar fit, with the Voigt model providing the best fit by a slim marginbased on rms error. The results indicate the placenta is a very viscous organ, whichcorroborates with its highly vascular structure.Future work will compare elastic and viscoelastic measurements of placentasfrom normal pregnancies to those diagnosed with preeclampsia or fetal growth re-striction, to determine if there are differences that may be used to help detect dis-ease. Applying the swave system in vivo should include a study optimizing theexciter design and placement for sufficient generation of shear waves within theplacenta, but it would otherwise be straightforward to measure the placenta vis-coelastic behaviour during pregnancy with this system. Increased accuracy in elas-ticity inversion could be obtained by measuring the full 3d displacement vector.The following chapters describe methods for improving the accuracy in measuringthe full 3d displacement vector for swave.135Chapter 4Spatial Calibration of Swept 3DUltrasound4.1 IntroductionUltrasound has gained popularity because it is low cost, portable, safe, and has ahigh frame-rate. While the majority of diagnostic ultrasound procedures use 2dimaging, 3d imaging can provide significant advantages [85, 219]. For example,different 2d imaging planes may be “re-sliced” from the 3d ultrasound volume toprovide 2d image orientations that would be otherwise impossible to acquire, suchas parallel to the skin surface. Additionally, surface or volume rendered models canbe generated which may reveal pathology that is more difficult to discern in con-ventional ultrasound images. Imaging in 3d can provide more information on thegeometry of anatomical structures and more accurate measures such as volume. 3dultrasound can also help in needle localization and guidance during biopsy [28]. Inthis thesis, 2d imaging refers to planar cross-sectioned images produced by trans-ducers such as linear, curvilinear, or phased arrays, and 3d imaging refers to ultra-sound volumes produced from swept motor transducers.3d ultrasound data can be especially beneficial in elastography. While inducinglow frequency vibration of the tissue, 3d ultrasound can observe the waves prop-agating in all 3 dimensions over a volume [260]. 3d measurements can reduceoverestimation of elasticity calculations by up to 60 % compared to 2d measure-136ments [23].Typically a swept motor transducer relies on a model of the motor geometry tocalculate the location of 2d image slices with respect to another. In elastography,the effect of uncertainty or error in the parameters, such as sweep angle, sweep stepsize, scale of the voxels, and other geometry used by the ultrasound machine to re-construct each sweep of image slices into a Cartesian volume, has not been studiedin detail. While most of these errors are generally not considered large enough topose a problem with 3d diagnostic ultrasound imaging, errors in the parameterscould be expected to contribute to elastography errors in two major ways. The firstis by incorrect geometrical placement of image data, which for swave techniqueswould affect the apparent shear wavelength, resulting in misestimation of elastic-ity. The other contribution from errors in the parameters is errors in the assumeddirection of motion measurement. Previous work has demonstrated that assuminga purely vertical measurement direction for motion that is actually measured in acylindrical coordinate system leads to underestimation of elasticity [137]. Someform of calibration to solve for these parameters for an individual transducer mayhelp reduce these errors.A further extension of applying 3d ultrasound to elastography is the possibil-ity to combine multiple intersecting volumes tilted at different angles which wouldallow for accurate measurement of the full 3dmotion vector, rather than the 1d com-ponent of the motion vector that is typically measured with ultrasound [4]. A posi-tion tracking sensor can be attached to a swept motor transducer to allow tracking ofmultiple 3d ultrasound volumes. Using the position tracking sensor measurements,these volumes can be transformed to a global coordinate system and combined intoa composite volume. 3d ultrasound spatial tracking of this nature has other clinicalapplications including intra-operative visualization [237], ablation guidance [19],and spatial compounding [172], among others.Generally, position tracking systems measure the pose of a sensor mounted onthe ultrasound transducer with respect to a fixed base; therefore, it is necessary tocalibrate the pose of the image with respect to the sensor to enable these trackingtechniques. Typically, this calibration is accomplished by collecting ultrasound im-ages of objects, called phantoms, with known geometry and calculating the rotationand translation parameters of the calibration that best match the image features to137the geometry. Phantoms are typically constructed from spherical beads [182], nylonwires [262], and planar surfaces [275].Ultrasound calibration remains a critical factor in the overall system accuracy ofmany ultrasound guided procedures [189]. The 2d ultrasound calibration problemhas been studied extensively, with reviews ofmany of the techniques published [143,206]. However, it is generally not straightforward to extend 2d techniques to makeuse of the extra data available in 3d ultrasound volumes. Various aspects of the3d ultrasound calibration problem have been investigated. One study used a 2dtechnique to calibrate a 2d transducermounted on amotor to replicate the action of a3d transducer [45]; however, in commercially available swept motor 3d transducers,the array is enclosed in the transducer housing, so tracking the array location asthe motor is swept is infeasible. Tracking both the transducer and phantom hasshown greater accuracy compared to registering ultrasound volumes to each other,as in hand-eye techniques [35]. Point phantoms have been used for 3d calibration,and are easier to use in 3d compared to 2d because aligning the point phantomprecisely with the scan plane is not as difficult with many regularly spaced slicesin the volume [175, 259]. For wire phantoms, the intensities of the wire reflectionsover the volume are iteratively matched to a reference model of the straight linesmatching the phantom geometry [35, 258]. A few techniques have also used planarfeatures, and are solved by either using intensity information to iteratively match areference, similar to the wire methods [177], or matching the zero component ofthe reference model of the plane to points on the plane in the volume [27, 258].As previously mentioned, a swept motor transducer relies on several geomet-rical parameters to calculate a volume based on the location of a set of 2d imageslices. Reliance on these parameters is considered to contribute to spatial calibra-tion error [258]. It is a problem during calibration acquisition, since each volumeobserving the phantom will incorrectly locate the phantom’s features. It furtherbecomes a problem after calibration when tracking the transducer, as not only arethe calibration transforms incorrect, leading to placing the volumes in the wrongposition and orientation in the global coordinate system, but each volume in thecollection is distorted with respect to the local transducer coordinates as well. Pre-vious work has studied the distortions caused by errors in the sweep parametersfor an untracked swept motor transducer, where highly accurate measures of organ138volume were desired, and a method was presented for estimating the parametersassuming accurate alignment of the transducer with two specially designed phan-toms [315].The work presented in this chapter aims to reduce errors caused by uncertaintyin the motor parameters in tracked 3d ultrasound. This should help to reduce errorsin 3d elastography which relies upon these parameters, as well as provide improve-ment to 3d ultrasound spatial calibration which can be applied to advanced elas-tography methods and several other clinical application. Since recent techniquesdeveloped for 2d calibration demonstrate very high accuracy, and, assuming motorpositioning is repeatable such that each image slice in a swept volume can be con-sidered a 2d imaging source, a novel technique is proposed that calibrates multipleindividual 2d image slices along the sweep, instead of scan converting the slicesto a Cartesian volume and then performing a single 3d calibration. For example,a study of a closed-form 2d calibration using differential measurements from in-clined wedges reported a point reconstruction error under 0.3 mm [215]. Whileit is difficult to compare calibration accuracy across the literature, sub-millimetrepoint reconstruction accuracy has not been previously demonstrated in 3d calibra-tion techniques.Ideally, 2d calibrationwould be performed for each slice in the volume to achieveultimate accuracy. However, it is impractical and unnecessary to calibrate everyslice. Instead, a best fit path through calibration solutions calculated for a subsetof the images slices along the sweep is used. The fit will be used to determine thecalibration transforms between the calibrated slices. The fit can also reduce the in-fluence of error in individual calibrations, analogous to a least-squares model fittingto noisy linear data to reduce error of individual measurements.In particular, a method for fitting calibrations is presented which separately fitsthe translation and rotation parameters. The translation parameters are found byfitting an arc to the origin of each calibration. The rotation parameters are found bysolving for the best fit geodesic on the quaternion manifold, with each calibration’srotation component representing one quaternion sample on the manifold. The fit-ting method presented in this work is general and would function to extend any 2dcalibration technique to a 3d swept motor transducer. The 2d differential wedgetechnique is used in this work due to its high accuracy [215]. Since the technique139Figure 4.1: The coordinate systems used in the calibration procedure.uses multiple 2d slices of the wedge phantom over a 3d volume, this technique islabelled “W2D→3D”.The proposed multi-slice calibration approach is evaluated by comparing it toseveral single-calibration (slice or reconstructed volume) techniques. One approachis to use a 2d technique on a single slice in the volume, such as the well known N-wire (or Z-wire) [75, 238], and rely on the motor parameters to locate all otherpoints in the volume relative to the calibrated slice (“N2D”). Since this calibrationtechnique ignores data at all locations along the sweep except one slice, there is po-tential for a lever arm effect to magnify errors when the motor is swept far from thecalibrated slice. To reduce the lever arm effect, a novel technique is developed toextend the N-wire concept to a reconstructed volume (3d) by first scan convertingthe segmented points into a 3d point cloud before solving for the calibration param-eters (“N3D”). A possible improvement to the general N-wire calibration conceptis also presented by solving for the outside edge points of the N-wire, in additionto the centre point used in previous works (“NFull3D ”), explained in detail later. TheN-wire technique provides the advantage of a closed form solution compared to aniterative optimization, but has the disadvantage of providing sparse point featurescompared to feature-rich planes [262]. For a final comparison, a novel closed formplanar-fitting calibration technique is presented that relies on scan converted 3d data(“P3D”). Details of these methods are provided below.1404.2 MethodsThis section describes the calibration problem and the five techniques used in thischapter to solve it. An illustration of the components and coordinate systems for thecalibration is shown in Figure 4.1. The goal of calibration is to determine the fixed,rigid rotation and translation relating the image coordinate system, I, and the coor-dinate system of the position sensor on the transducer holder,H . Here the positionsensor consists of markers on the holder that are tracked by a fixed optical camerasystem, O, which also tracks markers mounted on the phantom body,M. Often itis convenient to define the coordinate system of the phantom not atM, but withrespect to a geometrical feature that appears in the images. This coordinate systemis called P, and the fixed relationship between P andM is determined separatelya priori with tracked stylus measurements (using an optical tracking system with0.02 mm mean error as explained later in Section 4.4).4.2.1 Data AcquisitionAll images were acquired on a SonixTouch ultrasound machine (Ultrasonix Med-ical Corp., Richmond, BC, Canada), equipped with a 4DL14-5 linear array sweptmotor transducer, operating at a transmit frequency of 10 MHz and a depth set-ting of 45 mm. The transducer and phantoms were submersed in a solution of 9 %by volume glycerol combined with distilled water to create a medium for the ul-trasound waves to travel through at 1540 m/s to match the ultrasound machine’ssettings [224].Based on data from the transducer manufacturer (Vermon, Tours, France), theradius of the motor is 81 mm and has an angular travel of 0.45◦ between slices.Both parameters were used for 3d scan conversion. To gain complete control overthe transducer’s stepper motor, an intermediate circuit was constructed and placedbetween the transducer and the ultrasound machine (see Section 2.2.4). The circuitpassed the ultrasound signals directly through the connectors, but provided its ownstepper motor signals controlled by the parallel port on the SonixTouch.The transducer and phantoms used for calibration had ired markers mountedto their bodies which were optically tracked by an Optotrak Certus motion capturesystem (Northern Digital Inc., Waterloo, Ontario, Canada). This provided both141Figure 4.2: The 3d transducer, encased in a holder with ired markers.position and orientation of coordinate systems defined using the markers on thetransducer and phantom bodies with respect to the Optotrak coordinate system. Tofind the relationship between the iredmarkers on the phantom body and the imagefeatures on the phantom, anOptotrak 4Marker Digitizing Probe stylus with a known1.5 mm radius ball tip was used to locate points inM with known correspondencein P.The transducer was placed inside a holder that was precisely manufactured tomatch its outer dimensions with an Objet30 3d printer (Objet Inc., Billerica, MA,USA) and held in place using constraining pins pushing against the top of the trans-ducer. The ired markers were press fit into small circular bore holes in the holder,142Figure 4.3: The phantom used for the planar and wedge calibration methods.which was reinforced with a stainless steel frame to reduce the possibility of theholder flexing and changing the markers’ positions. The holder was mounted on agoniometer, a linear translation stage, and a rotary stage to allow fine adjustmentsof the transducer pose, before being attached to a positioning arm (model 811-002;CIVCO Medical Solutions, Kalona, IA, USA) to fix the assembly in space. Thephantoms were also attached to a positioning arm (model 244; Manfrotto, Cassola,Italy) to keep the phantom fixed while recording data. The holder and one of thecalibration phantoms are pictured in Figure 4.2 and Figure 4.3.4.2.2 N-wire Calibration (N2D, N3D, and NFull3D )In N-wire calibration, the geometry of the N shape is used to calculate the positionof the centre point, ~f , based on measurements from an ultrasound image. Once143Figure 4.4: The top view of one row of the N-wire phantom and the intersect-ing scan plane.at least three non-collinear centre points have been found in both coordinate sys-tems, the transform that best matches the points provides the calibration solution.Multiple centre points can be obtained by acquiring an image with more than oneN-wire shape visible, or acquiring additional images after transducer motion. Bothtechniques are used in this work to obtain an over-constrained set of points, with aphantom containing three N-wire shapes, and a collection of 10 ultrasound volumesfrom different positions and orientations.The phantom was constructed with 0.3 mm nylon wire interwoven through sixholes (~h1−6) in a N shape, as shown in Figure 4.4. In total the phantom containedthree rows of N-wires, each spaced by 5 mm. The top row geometry had 30 mmbetween ~h1 and ~h3, 50 mm between ~h1 and ~h4, and 20 mm horizontally between ~h2and ~h5. The middle row had the same dimensions, but the N shape was mirrored.144Figure 4.5: A 3d view of the three N-wire rows. The ultrasound scan plane,shown in grey, intersects each N-wire at three points, shown by the cir-cles. A dashed line is shown connecting the intersection points in thescan plane. Based on the same inter-point distances, the lines could bemirrored to alternate orientations, shown as dash-dot lines.The bottom row had the same orientation as the top, but ~h3, ~h5, and ~h6 were shifted5 mm to the left. The holes were precisely located by manufacturing a rectangularblock with the 3d printer (0.028 mm resolution and 0.1 mm accuracy, as stated bythe manufacturer).The geometry of the N-wire was defined by the points ~a, ~b, ~c, and ~d whichwere identified in the phantom markers’ coordinate system,M, with a stylus andthe Optotrak system. The stylus measured the hole locations, ~h1...6, and used theknown geometry to find the corners of the N shape. The corner ~a was locatedsome scalar distance ξ along the direction(~h1 − ~h4)from the point ~h1. The scalardistance was determined using the triangle formed byξ(~h1 − ~h4)+ ξ(~h5 − ~h2)= ~h2 − ~h1. (4.1)145Figure 4.6: A top view of the same N-wire phantom and scanning geometryas presented in Figure 4.5. The N-shapes are omitted in this view, butthe intersection points are shown as hollow circles along the ultrasoundscan plane, shown in black. The dashed lines from Figure 4.5 all liewithin the scan plane and are occluded by the black line. Consideringonly one N-wire row at a time, the inter-point distances could be createdby the scan plane being aligned along the dash-dot line for the top (pink),middle (green), or bottom (blue) rows. Noting equal depth between eachdash-dot line into the page from this view, it would be impossible for thescan plane to possibly pass through all three of the dash-dot lines at thesame time, or any combination of dash-dot and dashed (occluded by theblack) lines.The other corner points were found in a similar manner.The nine points in each slice in the ultrasound volume were automatically seg-mented by performing morphological opening operations to reduce speckle andthen searching for parallel lines of three collinear dots [65]. Using the segmentedpoints, the location of the centre point in the phantom marker coordinates was cal-culatedM ~f = M~a +‖I~e − I ~f ‖‖I~e − I~g‖(M ~d − M~a) , (4.2)where M ~f and I ~f are located in the marker and image coordinate systems, respec-tively. The N3D technique used all of the segmented points, while the N2D only usedthe points segmented in the centre slice. For each calibration technique, 10 volumeswere acquired.The collection of points described inM can be transformed to H using Op-totrak measurements. The calibration transform can be found by matching corre-sponding points between I andH . The translation vector is the difference betweenthe centroid of the two sets of points, and the quaternion describing the rotation is146found by solving an eigenvalue problem [139].The description thus far has only demonstrated how to estimate the location ofthe centre point along the angled segment of the N-wire, as used previously in theliterature, ignoring the points on the parallel outside wires (~e and ~g in Figure 4.4).Including the outside points could improve the corresponding point transformationestimation. The reason that these points were ignored in previous works might bethe fact that the location of the outside points in the phantom’s coordinate systemis not unique when based only on the lengths from the ultrasound image used tofind the centre point. In fact there are two possible locations for the outside points,mirrored about a line perpendicular to outside wires and passing through the centrepoint as shown in Figure 4.5 and Figure 4.6. Considering only one N-wire at a time,the ultrasound scan plane could pass through either the dashed line or the dash-dotline in Figure 4.5 and still produce the same inter-point distances in the image.However, it is possible to determine the correct line when considering all threeN-wires. Considering the three interior points in Figure 4.5, and all the possibleexterior points (intersections of dashed and dash-dot lines with exterior wires), thereare 23 = 8 possible sets of nine points that could lie on the ultrasound plane. Ingeneral, only one of these sets lies in a plane. In Figure 4.6, it is clear that a planecould not pass through any set of three lines that contains any of the dash-dot lines,leaving the combination of the three dashed lines as the only possibility in thisexample. To select the correct set of points in this work, a plane is fitted for eachcase, and the plane with the smallest mean squared distance to the points is selected.Further discussion on the ambiguity of transducer pose based on N-wire lengthsmeasured in ultrasound is explored in Appendix C.4.2.3 Planar Calibration (P3D)The method of planar calibration presented here is similar to a technique developedfor simultaneous localization and mapping (slam) [247]. The slam technique wasapplied to a laser range finding sensor, mounted on amobile robot, which obtained acloud of points to be segmented into planar surfaces. The planes were then matchedafter a change in position or orientation of the sensor to estimate the motion of thesensor. Our method does not match two sets of segmented data. Rather, it matches147Figure 4.7: A rendering of the phantom used for the planar and wedge calibra-tion methods, excluding the ired marker attachment. The phantom in-cludes a grid of hemispherical indents for the stylus to measure the rigidtransformation between the phantom coordinate system and the mountedired markers.planes segmented in ultrasound, I, to a mathematical model of the planes describedin the phantom coordinate system, P. Only one reconstructed ultrasound volumeis required. However, collecting additional volumes after transducer motion canserve to reduce errors in calibration calculations via averaging. Our method uses aphantom designed for ultrasound calibration, where the angles for the planes werechosen to balance image quality and reliable feature extraction, while providing an-gles that reduce the sensitivity of the calibration to measurement errors [215]. Thecalibration solution is found in two steps. First, the rotation is found by aligning thenormal vectors of the planes segmented in the ultrasound volumes to the mathemat-ical model. Next, the translation is found using the rotation solved in the first step,and by satisfying the planar equation in both the phantom and image coordinatesystems.Previous studies using planar features in 3d swept ultrasound have not attemptedto detect planes using 3d data, instead using 2d data to create clouds of pointsdefined by maximum intensity gradients [177], Canny edge detection [259], and2d Hough transforms [27]. These studies have used iterative methods to solve the148calibration problem. The technique presented here uses total least squares to findthe best fit planes to 3d data and has a closed form solution. Based on research on2d calibration, both aspects tend to provide more accurate solutions [216, 262].The phantom was manufactured with the 3d printer, containing five planar sur-faces as pictured in Figure 4.3. The phantom also contains a 6 × 5 grid of hemi-spherical indents, spaced 10 mm apart in each direction, pictured in Figure 4.7. Thelocation of the indents with respect to the planar surfaces is known to within the tol-erance of the 3d printer, and the radius of the hemispheres matches the radius ofthe stylus tip. The points are reliably located in both the phantom, P, and marker,M, coordinate systems with a stylus. The rigid transformation between P andMis determined with a corresponding point algorithm [139].The lines produced by the phantom surfaces are segmented in each slice in thesweep by finding the maximum intensity along each scan line. The user identifieslocal regions where each unique plane is located in the image. Best fit planes are fitto the collection of maximum intensity points using principal component analysis(pca) to minimize the orthogonal distances of the points to the plane [153].A plane can be described by the equation~n · ~x = d, (4.3)where ~n is the unit normal of the plane and the distance to the origin is d. To find therotation from the ultrasound volume to the phantom, the dot product is maximizedbetween the normal vectors detected from the ultrasound volume and the normalvectors of the planes known from the phantom’s manufactured geometry describedin the phantom coordinate system (i.e. the corresponding normal vectors are madeas close to parallel as possible). Using unit quaternions to represent rotation, thealgorithm finds the unit quaternion PI q that maximizes:k∑i=1(PI qIni PI q∗) · Pni, (4.4)where Ini is the ith normal vector in the ultrasound volume coordinate system,written as a purely imaginary quaternion[0, ~nTi]T, q∗ is the conjugate of q, and k is149the number of planes in the volume. This equation is essentially the same as findingthe rotation in the corresponding point problem with quaternions. The difference isusing normal direction vectors instead of position location vectors. The quaternionthat maximizes Equation 4.4 can be determined by finding the eigenvector corre-sponding to the maximum eigenvalue of a 4 × 4 symmetric matrix composed ofsums of products of the elements of the normal vectors [139].The next step is to find the translation vector from the ultrasound volume to thephantom. Consider transforming a point from one of the planes in the ultrasoundvolume coordinate system to the phantom coordinate systemP ~p = PI RI ~p + PI~t, (4.5)where PI R is the rotation found from Equation 4.4, but expressed as a 3×3 rotationmatrix to simplify the notation, and PI~t is the translation to solve for between thetwo coordinate systems. Using Equation 4.3, the translation vector is related to theplane parameters viaPd = P~n · P ~p= P~nT(PI RI ~p + PI~t)=(PI RI~n)T PI RI ~p + P~nT PI~t= I~nT I ~p + P~nT PI~t= Id + P~nT PI~t . (4.6)Given three or more planes in the ultrasound volume, PI~t can be solved for in theleast squares sensePI~t =(AT A)−1AT ~d, (4.7)whereA =P~nT1...P~nTk, (4.8)150and~d =Pd1 − Id1...Pdk − Idk. (4.9)The five planes in the phantom used in this experiment already provide redun-dant information with a single ultrasound volume. To match the other techniques,ten volumes in total were acquired.4.2.4 Wedge Calibration on 2D Slices and then Fitting (W2D→3D)The methods described thus far rely upon knowing the geometry of the transducerand motor precisely to determine where each of the 2d image slices in the ultra-sound volume is located with respect to another to scan convert the slices into a3d volume before identifying phantom features. To reduce errors introduced from3d scan conversion, 2d calibrations are performed at multiple slice locations andthe best fit path through the 2d calibrations is found to link them together. Any 2dcalibration technique may be applied in this framework. For this work, a recentlydeveloped highly accurate technique is chosen which uses slope measurements ofwedge features from the phantom in Figure 4.3 [215]. Calibrations on five slicesevenly distributed across the motor sweep are used to fit the 3d calibration path.The trade-off of improving the best fit path with more calibrated slices is the timerequired to collect data. The 2d wedge calibration typically requires at least tenunique images and poses per slice for a reliable result. Collecting data for eachslice would require 10 × 64 = 640 unique images and poses compared to the 50chosen to be collected here. Using the information from the 50 poses, the cali-bration transforms at intermediate positions are found by interpolating along thebest fit path. This method does not rely on the motor geometry, however in-planescan conversion is assumed to be correct (i.e. the piezoelectric element spacing isknown). In addition, it is assumed that the array travels along a circular arc and therelative angular positions produced by the stepper motor are accurate and repeat-able. The rest of this subsection describes the method of interpolating between 2dcalibrations.The translation and rotation parts of the calibration interpolation are considered151separately. The translation is found using the best fit circle to the individual trans-lation samples from each slice calibration along the arc. In this work, circle fittingis performed using a closed form algebraic fit, optimized to eliminate bias in theestimation [10].The rotation is found using the best fit geodesic on the unit quaternion manifold.The advantages of parametrizing the rotation using quaternions are that they are in-tuitively and compactly described by a rotation axis and angle, and it is simple toremove computational drift and restore a valid rotation through vector normaliza-tion. The quaternion manifold is a 3-sphere (S3) sitting in four dimensional space.A number of researchers have focused on the problem of interpolating rotations,where the resulting path must pass through the set of control points. Examplesinclude spherical linear interpolation [289], acceleration minimizing cubics [245],and spherical splines [50]. The problem of finding a best fit path through regres-sion on the manifold is far less studied. One example attempts to find a geodesic onS3 using the analogous slope and intercept for linear regression in Euclidean spacewith an optimization algorithm minimizing distances to the rotation samples [106].The method presented here is similar, however the mean of the rotation samples isused to locate the geodesic on the manifold, and only the direction of the geodesicis optimized instead of solving for both quantities simultaneously. The method hereis only one such method of fitting the calibration methods. Other approaches arediscussed in Appendix E.In Euclidean space, R3, a best fit line can be parametrized by a point on the lineand a direction vector. The line passes through the centroid, or mean, of the samplepoints, and the sum of the orthogonal distances between the samples and the line isminimized. This line can be written as~x = ~xm + t~v, (4.10)where ~xm is themean point,~v is the direction of the line, and t is the scalar parameterof the line.This same procedure can be extended to finding a best fit geodesic on a quater-nion manifold. The direction of the line in Equation 4.10 can be interpreted as lyingin the tangent space at the mean point. On a quaternion manifold, some mapping is152required to pull the tangent back to the manifold. This is called the exponential map,which also has an inverse mapping from the manifold to the tangent space calledthe logarithmic map. The correspondence between the two can be seen as follows.A quaternion, q, describing a rotation around the normalized axis of rotation, ~u, byan angle, θ, can be written as [13]q = cos(12θ)+ sin(12θ)~u= e12 θ~u . (4.11)Taking the logarithm of Equation 4.11 results in~v =12θ~u. (4.12)The logarithmic and exponential maps are explained in more detail in Appendix D.The + operator in Equation 4.10 is replaced by the Lie group operator, which forunit quaternions is multiplication. Therefore, the best fit geodesic on the quater-nion manifold can be written using the mean quaternion, qm of the set of samplequaternions, and a tangent vector at the mean quaternion, ~vq = qm exp(t~v). (4.13)The mean quaternion, is found using spherical averages by first calculating aEuclidean average and projecting back to the 3-sphere, then iteratively mapping thesample points to the tangent space at the current estimate of the mean, calculatingthe Euclidean average in the tangent space, then mapping back to the sphere by theexponential map until the iterations converge to a single point on the sphere [50].The tangent vector at the mean quaternion, ~v, is found by solving a gradientdescent minimization problem, described in Appendix E. A simulation verifyingthe efficacy of the algorithm is provided in Appendix F.The interpolated rotation path is bi-invariant, which means that the result isindependent of the choice of coordinate system for the transducer holder (left-invariance), H , as well as the image slices (right-invariance), I. See Appendix Efor a demonstration of the left- and right-invariance of the distance metric used153for optimization. If all of the quaternion sample points were left multiplied by anarbitrary quaternion, a, then the resulting path would beq = aqm exp(t~v). (4.14)Right multiplication of the quaternions would result in the pathq = qm exp(t~v)a. (4.15)In other words, if any of the axes of our co-ordinate systems were redefined,the rotation describing the redefinition would be the same as the rotation betweenthe original and new solution paths. This is an intuitive and desirable property, butit is not guaranteed in general for paths on the manifold. If bi-invariance was notsatisfied, the path would depend upon the definitions of the co-ordinate systemswhich can be defined arbitrarily.4.2.5 Validation of the Calibration MethodsValidation of the calibration methods was performed by assessing repeatability andaccuracy. Repeatability refers to how similar calibration solutions are to each othergiven different input data—repeating calibrations with different images should ide-ally still give the same transform between I and H because it is fixed. Accuracyrefers to how close the calibration solution is to the true transform between I andH . Since this is unknown, accuracy cannot be measured directly. Instead, accuracyis inferred by reconstructing physical quantities with known location or dimension.To evaluate the repeatability, the standard deviation of the six degrees of free-dom can be calculated after calibration with multiple independent data sets [2, 39].In this work, since 10 data sets were acquired for each technique, and each techniquecan be solved using only one data set, the standard deviation for each technique iscalculated for 10 independent trials. In the case ofW2D→3D, the standard deviationis calculated after applying fits to 10 randomly selected unique combinations of the10 calibrations at each of the five calibration locations.Two point reconstruction tests were used to evaluate the calibration accuracy.Point reconstruction tests calculate the difference between points with known lo-154cations and segmented points transformed to the same coordinate system using acalibration solution. In both of the tests, all 10 data sets were used to solve for asingle least squares solution for each method.The first point reconstruction test used a tracked stylus as the object to be im-aged [258]. The stylus and transducer were both moved between each of 25 volumeacquisitions. The ball tip was manually segmented by finding the slice in eachvolume with the strongest comet-tail reverberation, centring a line along the rever-beration, and selecting the top of the bright intensity response of the tip [7]. Thebright response corresponds to the top of the sphere [122]; the centre of the spherewas estimated by adding a distance equal to the radius along the direction of thescan line. The point reconstruction error is calculated between the stylus tip, asmeasured by the Optotrak system, and the tip segmented in the images and thentransformed into the Optotrak coordinates using the calibration solutionEs (v) = ‖O~x − OHTv HI T I~x‖, (4.16)where the ~x is the stylus tip, T describes the rigid transformation operation, error iscalculated for the vth volume acquired of the stylus tip.The second point reconstruction test used the N-wire phantom [215]. A singlevolume of the phantom was acquired, distinct from the 10 used in the N-wire cal-ibrations. Only the centre points of each N shape were used for evaluation with atotal of 78 centre points segmented in the volume. The centre points are known inM from geometry as described in the Methods section. These points can be trans-formed to H using the Optotrak measurements. The same centre points in I canbe transformed to H using the calibration solution. The error is the magnitude ofthe difference between these two sets of pointsEn (i, v) = ‖OHT−1v OMTv M~xi − HI T I~xi ‖, (4.17)where the error is calculated for i ∈ [1 . . . 3] centre points in each volume.155t x [mm]16182022242628303234N 2DN 3DNFull3D P 3DW 2D!3D(a), [deg]-104-102-100-98-96-94-92-90-88-86-84-82-80-78N 2DN 3DNFull3D P 3DW 2D!3D(b)t y [mm]3234363840424446485052N 2DN 3DNFull3D P 3DW 2D!3D(c)- [deg]-20-18-16-14-12-10-8-6-4-20246N 2DN 3DNFull3D P 3DW 2D!3D(d)(continued)4.3 ResultsThe calibration results from each of the 10 data sets for all of the calibrationmethodsare plotted in Figure 4.8. The results are separated into six dof; translation fromthe image to the holder (tx , ty , tz) and Euler angles defined using the fixed imagecoordinate axes and applying first a rotation by γ around the x-axis, followed bya rotation by β around the y-axis, and finally a rotation by α around the z-axis.It should be noted that these results cannot be used to assess the accuracy of the156t z [mm]-98-96-94-92-90-88-86-84-82-80-78N 2DN 3DNFull3D P 3DW 2D!3D(e). [deg]-64-62-60-58-56-54-52-50-48-46-44-42-40-38N 2DN 3DNFull3D P 3DW 2D!3D(f)Figure 4.8: The translation and rotation parameter values plotted for ten in-dependent tests (•), depicting the variation in the solutions. Note, N2Dresults are converted to the same coordinates as the other methods (i.e.the first slice of the sweep) using the assumed motor geometry. The pa-rameters corresponding to a least squares solution () using all of thedata sets are also plotted.Table 4.1: Standard deviations of the three translation dof (mm) and threerotation dof (degrees) after 10 trials using a single data set for calibration.N2D N3D NFull3D P3D W2D→3Dtx 3.05 1.09 0.89 0.73 0.63ty 2.72 1.35 1.00 0.80 0.44tz 5.55 1.31 1.09 0.66 0.92α 6.38 0.94 1.09 0.55 0.58β 7.13 1.73 1.57 0.68 0.89γ 7.96 1.16 1.08 0.51 1.54157Point Reconstruction Error [mm]01234N 2DN 3DNFull3D P 3DW 2D!3DEnEsFigure 4.9: Point reconstruction error for the calibration techniques, using theN-wire phantom as validation (En) and the stylus (Es). The bars indicatethe mean and the error bands indicate the standard deviation.calibration methods, as the true calibration solution is unknown, however it doesprovide a measure of the calibration repeatability. A calibration method is morerepeatable when the solutions are more closely spaced. Based on Figure 4.8, themost repeatable methods are P3D andW2D→3D. This is quantified later in the resultssection using standard deviation. The least squares solution using all of the 10 datasets for each calibration method is also plotted. While a few points appear to beoutliers, the outliers do not correspond to each other across the six dof (e.g. anoutlier in tx does not correspond to an outlier in ty). The least squares solution isclearly not a mean of the individual solutions in each dof, and does not necessarily158even lie within the spread of the individual solutions in each dof, because the leastsquares solution fits all six degrees of freedom at once.The standard deviations of the six dof for each of the calibration techniques arepresented in Table 4.1. A smaller standard deviation indicates that the solutions arerepeatable with a different input of transducer poses and images. This is desirableas it demonstrates that the method is not sensitive to the exact motion of the trans-ducer used to collect the images as well as the image quality. The most repeatablecalibration based on the standard deviation results isW2D→3D for tx and ty and P3Dfor the remaining dof.The mean and standard deviation of the point reconstruction error as calcu-lated using both the N-wire phantom and stylus tip as validation tools are presentedin Figure 4.9. These results serve as an indirect measure of calibration accuracy.Based on the mean error, W2D→3D is significantly the most accurate technique inboth the En and Es tests, where significance is found using the Student’s t-test (p <0.01). All of the techniques are significantly more accurate than N2D.4.4 DiscussionThe stylus used in this study served two purposes; to measure the coordinate sys-tems for the phantoms, and to provide points measured in the tracker’s coordinatesystem for validation in the point reconstruction test. To understand how the sty-lus contributed to the errors in this study, a study of how well the stylus measuresknown distances, as well as how much the stylus location varies at a measurementpoint was done. The columns of the stylus grid on the slope phantom are 50 mmin total length. Averaging over the five columns, the mean and standard deviationof the absolute error in the distance measurement using the stylus were 0.02 mmand 0.03 mm, respectively. To measure how much the stylus location varies, thestandard deviation of the stylus location at each of the 30 grid points was averaged,resulting in a value of 0.11 mm. These results suggest stylus error has little effecton calibration or the point reconstruction test.Comparing the performance of the calibration methods studied in this chapterto previous results produced by other research groups should be done with cautiondue to differences in materials, equipment, and acquisition protocol. In addition,159not all methods in the literature evaluate the calibrations using the same defini-tions of point reconstruction error. One study matched five tracked ultrasound vol-umes to a model of a phantom containing two egg-shaped 3d features to calibrate,and simulating optical tracking errors found a mean point reconstruction error of2.0 mm [176]. A study using only a single volume and four point fiducials for cali-bration measured rms point reconstruction errors of 5.1 mm and 5.5 mm, over twocross-wire point measurements repeated 10 times each and located in tracker co-ordinates using a stylus [258]. A study comparing three calibration methods (IXI-wire, cube, and stylus) found rms point reconstruction errors over 10 stylus vali-dation points of 2.15 mm, 4.91 mm, and 2.36 mm, using one, one, and five trackedultrasound volumes, respectively [259]. Another study comparing separable andcombined hand-eye calibrations (the phantom is not tracked) as well as a trackedphantom approach, all with a cross-wire phantom, calculated error between pointssets generated over a 503 mm3 volume spaced by 5 mm in each direction, and foundmean errors of 5.9 mm, 3.5 mm, and 3.3 mm, respectively [35]. To the best of ourknowledge, W2D→3D is the first 3d ultrasound calibration to demonstrate potentialfor sub-millimetre point reconstruction accuracy with a mean of 0.82 mm with thestylus validation, and 0.97 mm with the N-wire validation. To compare to the re-sults in the literature reported as rms values, the rms error for theW2D→3D methodwith the stylus validation was 0.93 mm, and 1.03 mm with the N-wire validation.Including the outside points for the full N-wire calibration (NFull3D ) did not pro-vide a significant improvement in accuracy compared to the traditional centre pointN-wire calibration (N3D). However, NFull3D did provide a small improvement in re-peatability. It is important to remember that these results used multiple volumetricsweeps over the phantom, resulting in thousands of corresponding points used forcalibration. In cases where the number of points may be more limited, for examplein the case of 2d calibration, the full N-wire technique may prove to be of greaterpractical significance. Using a single 2d image for calibration, including the out-side points reduced the standard deviation of the six dof (tx , ty , tz , α, β, γ) overten independent trials by 29 %, 11 %, 10 %, 28 %, 8 %, and 22 %, respectively.The N-wire configuration used in this experiment enabled unique determina-tion of the edge points for NFull3D . The edge points cannot be determined for allpossible N-wire configurations. For example, using only two rows of the config-160uration presented in this chapter allows the possibility of the plane tilting to passthrough two of the alternate lines shown in Figure 4.6. However, finding the com-mon plane to the three row configuration presented in this chapter is not the onlymethod of determining the edge points. One possible alternative is to place twoN-wires side-by-side. In this configuration, the alternate line for one of the N-wireswould produce incorrect inter-point distances in the second N-wire. The trade-offfor the side-by-side configuration is a lower sensitivity because of a shallower anglefor the diagonal segments to fit the width of the image.4.5 ConclusionThe process of scan conversion to convert swept ultrasound data into a Cartesianvolume relies on estimates of motor parameters which can introduce errors intothe calibration process. To overcome the reliance on motor parameters, a novelapproach was proposed in which multiple 2d calibrations are performed and a bestfit path through them is found (W2D→3D). The proposed method can be used withany 2d calibration technique, however a wedge phantom was chosen due to its highaccuracy in 2d. Four other techniques were developed to compare against the fittingmethod. Three were based on extending the 2d N-wire calibration technique into3d. The fourth (P3D) used planar features for calibration, and is the first planartechnique to have a closed form solution for 3d swept ultrasound.The repeatability of the calibrations was evaluated using the standard devia-tion of the six dof after multiple trials on independent input data, and it was foundP3D was the most repeatable. Overall accuracy was evaluated using two point re-construction tests andW2D→3D demonstrated the highest accuracy in both tests. Theaccuracy and repeatability of the N-wire phantommethods were improved by using3d data compared to sweeping from a 2d calibration.The fitted calibration approach,W2D→3D, was able to obtain sub-millimetre ac-curacy. Based on previous work, sub-millimetre tracking accuracy is required inelastography applications to realize the benefits of multiple ultrasound views fortissue motion measurements compared to beam steering methods [4]. This shouldenable use of tracked swept motor ultrasound in elastography to improve 3d tissuemotion estimates. Future work will investigate the benefits of using tracked 3d ul-161trasound to measure tissue motion and calculate elasticity. In Chapter 5 the fittedcalibration parameters are used in the scan conversion process to convert 3d speckletracking displacement measurements from transducer coordinates to Cartesian co-ordinates. In Chapter 6 three ultrasound volumes are combined to compute 3ddisplacement measurements and elasticity, with calibration errors simulated basedon the errors measured in this chapter.162Chapter 5Measurement of the Full ShearWave Motion Vector in SWAVE5.1 IntroductionElastography measures the mechanical properties of tissue and tissue elasticity (e.g.Young’s modulus) is the most common quantity inferred from these measurements.Determining tissue elasticity is clinically relevant because changes in tissue elastic-ity are correlated with certain pathological changes, such as stiff cancerous tumourssurrounded by soft normal tissue [236]. Elastography has found clinical applica-tions in improving breast cancer bi-rads classification [101, 147], targeting prostatecancer for biopsy and focal therapy [239, 282], and liver fibrosis assessment andstaging [53, 145], among others.Tissue elasticity is determined by applying a mechanical excitation to the tissue,measuring the resulting tissue displacements, typically with ultrasound or mri, andreconstructing the elasticity distribution from the measured displacements by ap-plying the laws of continuum mechanics [25, 308]. The mechanical excitation canbe quasi-static, where the strain computed from the displacements provides infor-mation on the relative elasticity distribution, or dynamic, where the inertial forcesin the tissue allow for calculation of absolute values of elasticity [135]. The swavetechnique uses dynamic multi-frequency mechanical excitation to deform the tissueand estimate absolute elasticity values. Quantifying the absolute elasticity values163may be beneficial in characterizing tissue types, determining the different stages ofa disease, or monitoring the progress of a treatment [86].One of the main advantages of mre over conventional ultrasound elastogra-phy is that displacement can be measured accurately in all three directions overa volume [87]. Elasticity reconstructions based on the full 3d displacement fieldmeasurements are more accurate than 2d or 1d approaches [299]. Conventionalultrasound measures 2d image planes. Volumes of ultrasound data can be collectedwith a swept ultrasound transducer that contains a stepper motor to sweep the 2dimage plane over a 3d field of view. Volumes may also be acquired with matrixarray transducers that contain multiple rows of elements [220].Motion measurement, or speckle tracking, is typically measured in ultrasoundby dividing the echo signals into small regions or blocks and matching the blocksbetween successive acquisitions, for example by maximizing the correlation coeffi-cient between blocks [254, 326, 345]. Previous studies have demonstrated tracking3dmotion components over ultrasound volumes by extending blockmatchingmeth-ods to 3d [67, 105, 317]. Most of these studies measure quasi-static compressionmotion to generate strain images. In comparison, dynamic motion is a challenge aseach ultrasound beam line must be collected at a common motion state to allow forblock matching in 3d.In most conventional ultrasound scanners, beam lines are scanned individuallyfrom one side of the array to the other. For performing 3d measurements of dy-namic motion, this presents two problems. First, the frame rate is limited by theacoustic propagation speed of the ultrasonic pulses and the number of beam linesin a frame. Second, within each frame as each subsequent scan line is acquired thetissue scatterers will continue to move to different locations, making speckle patternmatching across lines unreliable.As a result of the first problem, in most imaging scenarios the motion must belimited to under 50 Hz based on the Nyquist sampling criterion. For band-limitedperiodic motion, the magnitude and phase of the displacement can be recoveredfrom the undersampled measurements in the baseband using knowledge of the ex-citation and sampling frequencies [98]. The frame rate can be increased by receiv-ing a collection of lines, usually four, beamformed in parallel within the transmittedbeamwidth with minimal changes to conventional hardware, which has been used to164measure motion in arf imaging [76]. In a 3d cardiac strain imaging study, a matrixarray was triggered to collect sub-volumes synchronized with an ECG, resulting ina high frame rate relative to the heart rate [193]. For measuring periodic motion,the effective sampling rate can be increased using a different sequence, similar tocolor Doppler, acquiring a small sector of lines repeatedly before moving to thenext sector. After measuring displacement in one direction along the lines, the de-lays between each line and each sector can be compensated using the known delaysbetween acquisitions [21].The second problem with measuring 3d dynamic motion, that of tissue scat-terers being in different locations when each scan line is acquired, results in errorsin the speckle tracking process. The displacement measured from block matchingis the average motion of the scatterers of the block, ignoring biases towards highersignal amplitudes [58]. If an entire block was acquired at the same time, most of thescatterers would be expected to generally move in the same direction because theshear wavelength is typically much larger than the block size. If a 3d block containslines acquired at different times, then the shear wave will have travelled betweenacquisitions and the scatterers will no longer generally move in the same directionacross the block. In addition, the change in speckle pattern during the acquisitionof a frame caused by scatterer motion may cause biases because the speckle patternmay match over a larger volume. A solution to the problem is to synchronize the ex-citation and ultrasound acquisition. One study triggered the start of a single elementultrasound transducer acquisition with the start of harmonic tone burst excitations,reaching an effective sampling rate of 10 kHz, and repeated the procedure at dif-ferent locations to create a 2d image [91]. Similarly, a single element transducerwas used to visualize heart contraction by triggering the start of acquisition fromthe R-wave peak of an ECG and recording the signal over several cardiac cycles be-fore moving the transducer laterally and repeating to create a 2d image [250]. Thesame concept has been used with a linear array by triggering the start of one beamline with the start of a mechanical excitation, where additional beam lines werecollected subsequently in the same manner by electronically changing the positionof the beam centre with the linear array and repeating the mechanical excitationto measure both harmonic and transient motions [131]. In a study measuring theshear wave propagation from a vibrating needle, the start of acquisition of a small165sector of lines was triggered by the actuator controller and, after a period of record-ing time, the procedure was repeated to collect additional sectors until the entireimage was collected [83]. A method that programs the ultrasound sequencer to usea line repetition frequency that is an integer multiple of the excitation frequency,and acquires one line per excitation period can achieve synchronization without atrigger [350]. All of these methods ensure that, across the image, the motion wasmeasured at the same relative temporal sample points and could potentially be usedto match speckle patterns across lines for multi-dimensional motion measurement.However, these studies only measured one component of the motion vector.Combining ultrasound planar transmit (the pulse is applied to all array elementsat the same time) with hardware that receives and samples echoes on each elementchannel individually in parallel results in very high frame rates by collecting an en-tire image in one transmit and receive event [309]. The lack of transmit focusingresults in a lower acoustic energy density and larger errors in displacement mea-surements, however coded pulses can be used to compensate for the loss in acousticenergy [320]. Because all beam lines are active at the same time, they all measurethe motion at the same time and there is no need for additional synchronizationwithin an image. Extending this concept to a matrix array might allow synchronousacquisition of a volume [264], however a swept ultrasound transducer would re-quire synchronization between each motor position in the sweep. While not usinga packaged swept ultrasound transducer, a similar concept has been used to collectvolumes of ultrasound data with linear arrays mounted on stepper motors eitherconfigured as two perpendicular arrays intersecting at a line [33], or with a singlearray rotated around the circumference of a phantom [211], using plane wave trans-mission and parallel receive hardware to quickly measure several phases of motionwith synchronization to the excitation applied at each motor position. These workswere able to obtain measurements of the full vector shear wave motion field over avolume, the same goal as this work, but used 1d displacement measurements fromdifferent angles to reconstruct the motion vector.In mre, the start of the motion sensitive gradient used to measure tissue dis-placements is triggered by the mechanical actuator at several different excitationphase offsets for each motion encoding direction [199, 290]. Our work on ultra-sound follows a similar approach by triggering each beam line with a function gen-166erator applying harmonic excitation. For successive frames, the phase of the excita-tion signal is shifted such that each frame measures a different phase of the motion.This approach creates volumes of beamformed ultrasound data at common motionstates. The frame rate or effective temporal sampling rate is no longer an issue be-cause the time to acquire a single line is almost negligible compared to the periodof excitation, and the temporal samples of the excitation signal are not related to theline repetition rate, but are chosen by selecting the trigger offsets.It should be noted that the motion measurements with ultrasound are most ac-curately computed in the axial direction, or the direction of ultrasound wave prop-agation. The other two directions, lateral and elevational, are typically an orderof magnitude less accurate due to lack of phase information [42], wider extent ofthe point spread function [167], and larger sampling intervals [187]. It is desir-able to mitigate the measurement errors in the other two directions, as elasticityreconstruction methods are sensitive to noise [231]. The motion measurements canbe improved using special beamforming strategies such as introducing phase in-formation in other directions through transverse oscillations [186], improving theresolution of the point spread function using synthetic aperture techniques [167], orelectronically steering the ultrasound beam to combine axial measures in differentdirections [267]. All require changes to standard b-mode imaging.Alternatively the displacement measurements can be processed to reduce er-rors, for example through spatial filtering with a median filter [270], however sucha filter does not have a meaningful relationship to the physics of the motion. Apply-ing a physical constraint on the motion using reasonable assumptions can be usedas an alternative to spatial filtering. A common physical constraint on the motionto reduce displacement estimation error is continuity. For example, during blockmatching the 2d displacement in a local neighbourhood was estimated by simul-taneously minimizing the speckle decorrelation and a continuity penalty based onsums of the derivative of the motion in each direction [151]. Other examples ofenforcing continuity on the displacement include using a cost function minimizingthe squared difference in the displacement measurement in adjacent samples in theaxial and lateral directions [272], or in a local neighbourhood [248], while minimiz-ing the sum squared or absolute difference between images. Another regularizationpenalty based on the strain energy in a deformable mesh has also been used [341].167For vector measurements, regularization based on the divergence and curl ofthe vector field has been proposed as a method to include every derivative-basedquantity that has a direct physical interpretation [18]. In the case of elastography, acommon assumption of tissue incompressibility can be enforced by minimizing thedivergence. In mre it is common to process the measured displacements to obtaina divergence-free field, for example by applying the curl operator [292], perform-ing Helmholtz-Hodge decomposition [291], or applying a high pass filter to removecontributions from compressional waves and a low pass filter to remove noise [162,200]. For quasi-static ultrasound elastography, an algorithm for reducing the vari-ance in lateral displacement estimates based on an integral of the axial strain and anaverage of the lateral displacement speckle tracking measurements in a plane strainincompressible condition demonstrated signal-to-noise improvements in the lateraldisplacement of a factor greater than 10 [195]. Under similar experimental condi-tions, a recent study applied a penalty on the divergence of the 2d displacement toiteratively reduce noise in both the axial and lateral speckle tracking measurementsby up to a factor of 17 [120]. Displacement vectors in 3d after static compressionwere estimated by simultaneously minimizing the squared intensity differences be-tween two volumes, and penalties on the smoothness and divergence of the dis-placement, approximated using a finite element mesh [269]. A similar approachhas been used as a post processing step on mre displacement data, finding a newdisplacement field that minimizes the difference between the new field and the mea-surements as well as the new field’s smoothness and divergence [243]. Our workattempts to reduce noise in all three components of the displacement by applyinga regularization based on the divergence, with no assumption on the deformationcondition, and is solved directly based on a least squares formulation.In summary, there is a need for full vector motion field measurement with stan-dard 3d ultrasound to obtain absolutemeasures of elasticity. The goal of this chapteris to develop a novel method that measures harmonic tissue motion vectors over avolume with a standard swept b-mode ultrasound transducer, applies a novel regu-larization to the motion to improvemeasurements, and produces elasticity estimatescomparable to mre. The method is validated on measurements of a tissue mimick-ing phantom, and the results are compared to a previous mre study on the samephantom. Elasticity is estimated using both the lfe method and a state-of-the-art168fem based inversion algorithm to evaluate the effect of the inversion model on theelasticity estimates. Several metrics are used to compare the inversion algorithmsand the difference between using the measured displacements and regularized dis-placements as inputs to the algorithms. This work also includes a study of how theregularization parameter affects the displacement.5.2 Methods5.2.1 SynchronizationAnalogous tomre, a steady-state mechanical excitation is applied externally to gen-erate shear waves in the tissue. Beamformed ultrasound rf data are acquired overa volume of tissue by collecting 2d frames of standard delay and summed channeldata at a given motor position for a period of time before stepping the motor to thenext position in the volumetric sweep. To enable displacement speckle tracking of3d blocks of rf data shifted over a 3d volume, volumes of rf data at common mo-tion states are acquired by synchronizing each ultrasound beam line in a 2d framewith the mechanical excitation. Once a 2d frame is collected, the exciter is repro-grammed to synchronize at a phase shifted point along its cycle, and another 2dframe is collected. When the motor is stepped to the new position, the exciter isreprogrammed to the first synchronization phase used in the previous motor posi-tion. This is shown in Figure 5.1 and the pseudo-code provided in Algorithm 5.1.It is assumed that the motion within the tissue is consistent across all cycles of theexcitation. Once the entire collection is complete the rf lines are rearranged intovolumes that were triggered at common phases. Displacements are then estimatedby comparing the volumes to each other, resulting in a complete volume of dis-placement vectors for each common excitation phase.5.2.2 Displacement EstimationOnce the volumes of rf data at several excitation phases are collected, 3d speckletracking and scan conversion is used to find the 3d displacement vectors over thevolumes, as done in previous work on static 3d displacement tracking [260]. Blocksof rf data approximately 1.2 mm × 1.5 mm × 2.25◦ in the axial, lateral, and eleva-169Figure 5.1: A schematic of the synchronization between the mechanical ex-citation and the volumetric ultrasound acquisition. At a given motorposition, each line in the 2d image plane is triggered sequentially at acommon excitation phase which is then repeated for a number of phaseoffsets. After all phase offsets are collected, the motor is stepped to thenext position and the process is repeated. Once the collection is finishedfor all motor positions, the data are reorganized into volumes that weretriggered at a common phase.170Algorithm 5.1 Ultrasound pulse sequence to synchronize with excitation.Start exciterfor φ← −Φ to + Φ do . for all motor anglesfor θ ← 0 to Θ do . for all phase offsetsfor l ← 0 to L do . for all beam linesSend ready signalWait for next triggerAcquire line l at phase θ and motor angle φend forend forend fortional directions of the transducer were used for speckle tracking, and were over-lapped by spacing the blocks by approximately 0.3 mm × 0.3 mm × 0.45◦. Sub-sample displacement shifts in transducer coordinates are estimated by maximizingthe normalized cross-correlation in 3d. First, a coarse estimate of the axial shift isestimated using a 1d search for the maximum correlation along an rf line, bracketedby the previous block’s estimated shift to improve speed and reduce peak hoppingerrors [345]. Next, the correlation is computed at the neighbouring sample pointsin the axial, lateral, and elevational directions, resulting in a 3 × 3 × 3 volume ofcorrelation coefficients. A 23 coefficient 3d polynomial of the formg (u, v,w) = a0 + a1u + a2v + a3w + a4uv + a5uw+ a6vw + a7u2 + a8v2 + a9w2 + a10u2v+ a11u2w + a12v2u + a13v2w + a14w2u+ a15w2v + a16uvw + a17u2v2 + a18u2w2+ a19v2w2 + a20u2vw + a21v2uw+ a22w2uv (5.1)is fit to the correlation coefficients, and themaximum of the polynomial is computedusingNewton’smethod, providing the sub-sample shifts in each direction [349]. Allcorrelations are computed using the volume triggered at zero phase as a reference.The displacements in Cartesian coordinates are calculated by using scan con-version to compute the location of the blocks both before a displacement shift and171after a displacement shift and taking the difference between these points. Usuallythe scan conversion parameters can be obtained through the transducer manufac-turer. To achieve greater accuracy, in this work the scan conversion parameterswere determined using theW2D→3D calibration technique described in Chapter 4.Using a least squares fit, the multiple volumes of displacement measurements atdifferent excitation phases are converted to a single volume (for each measurementdirection) of complex phasors in the form Aejθ (see Section 2.2.8).5.2.3 Displacement RegularizationAs mentioned, the displacement measurements are often precise in the axial direc-tion, but contain an order of magnitude larger variance in the lateral and elevationaldirections. Generally, elasticity estimation algorithms are sensitive to this varianceso it is desirable to find a way to smooth the displacement field. While a spatial filtercould be used, a regularization of the displacements that uses a priori informationmay be more appropriate. One possible constraint is assuming tissue incompress-ibility, which is applicable to most soft tissues due to the high volume fraction ofwater, provided that the water does not diffuse out of the local tissue region duringmechanical excitation. The incompressibility constraint can be expressed mathe-matically as minimizing the divergence of the displacements. A displacement field,d, that balances a fit to the displacement measurements, m, with minimizing thefield’s divergence can be found viaminimized‖d −m‖2 + α2 ‖∇ · d‖2 , (5.2)where α is a regularization parameter. The displacement measurements in Carte-sian coordinates in Equation 5.2 have been arranged in a single vectorm =mx[0]my[0]mz[0]...mz[N − 1], (5.3)where mγ[i] is the measurement at the ith pixel in the γ direction, and N is the172total number of pixels, resulting in m containing 3N entries. The pixel orderingis organized to first traverse the x direction, then the y direction, and finally the zdirection. The divergence operator is approximated using finite differences∇ · d = ∂dx∂x+∂dy∂y+∂dz∂z(5.4)≈ Ld (5.5)≈ 1h(Lx + Ly + Lz)d, (5.6)where h is the pixel spacing (assumed uniform in each direction) and Lx , Ly , andLz are sparse arrays with difference kernels oflx = [ 1 0 0 −1 ] , (5.7)ly =[ 1 01×3Nx+1 −1 ] , (5.8)andlz =[ 1 01×3NxNy+2 −1 ] , (5.9)starting at the first, second, and third columns of the arrays, respectively, and shift-ing by three columns for every row of the arrays. The kernels perform a simplefinite difference between adjacent pixels in the x, y, and z directions, with the zerosin the centre of the kernels accounting for the three components in the displacementvector and the number of pixels in each direction of the volume, denoted Nx , Ny ,and Nz . Differences at the boundary are taken to be zero.The minimization problem in Equation 5.2 can be solved by taking the deriva-tive with respect to d and setting it equal to 0, resulting ind =(I + α2LTL)−1 m, (5.10)where I is a 3N × 3N identity matrix. This is equivalent to the least squares solutionto [ IαL]d =[m0], (5.11)where 0 is a 3N × 1 vector of zeros. The solution to Equation 5.11 can be computed173directly using matrix factorization [127]. The uncertainty in the measurements canbe accounted for by applying the weightingW =1/σx1/σy1/σz. . ., (5.12)where σγ is the standard deviation of the measurement in the γ direction. Thestandard deviation values used in this work are taken from a planar motion testin a previous study [260]. The least squares problem in Equation 5.11 with theweighting applied becomes [ WαL]d =[Wm0]. (5.13)The choice of the regularization parameter, α, affects how much influence thedivergence penalty is given. A small value will provide little influence and the resultwill be very close to the displacementmeasurements and a value that is too largewillproduce a result that does not resemble the displacement measurements. Severalvalues of α between 10−10 to 106 are tested and based on the results presented laterin this paper, the elasticity is reconstructed using a value of 102.5.2.4 Elasticity Reconstruction from lfeAs described in Chapter 2 in Section 2.2.11, under the assumptions that the elas-tic properties of the tissue are linear, isotropic, lossless, and locally constant, andthe tissue can be considered incompressible, the motion can be described with anindependent Helmholtz equation in each directionµ∇2dγ = −ρω2dγ, (5.14)where λ and µ are the Lamé parameters, ρ is the density, ω is the frequency ofexcitation, and γ is the chosen direction. The lfe inversion method uses the outputof the ratio of pairs of filters from a filter bank that spans a large range of spatial fre-quencies corresponding to the range of possible shear wave numbers. In Chapter 2,174only axial displacement measurements were collected, so Equation 5.14 was solvedusing only the axial component. With all three components of the displacement vec-tor measured in this chapter, the final lfe reconstructions are averages of separatereconstructions calculated from the x, y, and z displacement data sets [199].5.2.5 Elasticity Reconstruction from shear-FEMIn the derivation of the Helmholtz equation in Equation 5.14 for the lfe reconstruc-tion, one of the assumptions is that the spatial derivatives of the shear modulus arenegligible (homogeneity assumption). For heterogeneous media, this assumptionbreaks down at the boundary of the two materials. In lfe, the spatial support ofthe filters is proportional to the wavelength, and thus creates a transition zone inthe results where the elasticity gradually changes between the two materials with aresolution of one-half of the wavelength [199].It is also possible to formulate the inverse problem without this assumption.Starting from Equation 2.18 and removing all coupling and compression terms re-sults in∇ ·(µ∇dγ)= −ρω2dγ . (5.15)This model for the tissue motion is discretized using the fem, and has been termedthe shear-FEM method [137]. The fem implementation uses linear shape functionsfor the displacement and test functions, and piece-wise constant shape functions forthe shear modulus. The equations are arranged in terms of shear modulus so thesystem of equations can be solved directlyKµ = f, (5.16)where K and f are formed using any one of the displacement components. Sinceall three displacement components are measured, the equations can be formed foreach and then stacked and solved simultaneously [138]KxKyKz µ =fxfyfz . (5.17)175Ready SignalExciter Output Trigger at Phase θ Synchronization CircuitReady & Trigger?Ultrasound Trigger InAcquire LineYesNoFigure 5.2: Flowchart of the synchronization between the ultrasound acquisi-tion of each beam line and the excitation signal at multiple phase offsets(extended from Figure 2.3).The inverse problem is stabilized using sparsity regularization, choosing a trun-cated discrete cosine transform to approximate the solution [135]. The volume ismeshed using eight-node hexahedral elements, with a length of 1.5 mm which isnear the optimal element size to elastic wavelength ratio which balances the trade-off in increasing inertial forces with decreasing derivative accuracy [138].5.2.6 Phantom ExperimentTo test the methods, ultrasound data were captured from an elastography qual-ity assurance phantom (Model 049, CIRS Inc, Norfold, VA, USA). This phantomwas chosen because the manufacturer provides reference elasticity values based onquasi-static compression tests on batch samples, andmremeasurements of the same176Figure 5.3: A b-mode image generated from the envelope of one of the rfimages. The hypoechoic circle in the bottom left shows a cross sectionof the stiff inclusion.Table 5.1: Manufacturer specifications for the CIRS Model 049 elasticityquality assurance phantom. The background and measured inclusion(stiffest of four inclusions) are tabulated.Background InclusionSpeed of Sound [m/s] 1545 1541Attenuation [dB/(cm MHz)] 0.5 0.53Elasticity [kPa] 29 62177phantom have been performed in a previous study [22]. The CIRS phantom is com-posed of an elastic tissue mimicking material called Zerdine, and contains two sets(10 mm and 20 mm diameter) of four spherical inclusions constructed with elastici-ties ranging from soft to stiff. Images of the stiffest 20 mm diameter inclusion of thephantom were acquired. The phantom specifications measured by the manufacturerare provided in Table 5.1.The images were acquired with a 4DL14-5 motorized linear array ultrasoundtransducer connected to an SonixTouch scanner (Ultrasonix Medical Corp., Rich-mond, BC, Canada). Each beam line was synchronized with an exciter (33220A,Agilent, Santa Clara, CA, USA) applying 200 Hz vibrations before reprogrammingthe exciter to a total of 25 phases distributed evenly across the excitation period.After collecting one 2d image frame for each exciter phase, the transducer motorwas stepped to the next position in the sweep. The sweep comprised a total of 31motor positions, subtending an angle of approximately 14◦.The synchronization between the ultrasound acquisition and the excitation isshown inmore detail in Figure 5.2, which is a specialized version of the general syn-chronization scheme for the swave system in Figure 2.3. As shown in the flowchart,a custom synchronization circuit is used to output a trigger signal to the ultrasoundmachine which is programmed to collect one beam line in its standard sequenceafter receiving the trigger. The ultrasound machine transmits and receives a beamline as soon as the trigger pulse is received from the synchronization circuit. Theprocess is repeated until all of the data are collected as described in Algorithm 5.1.The location of the stiff inclusion was determined using its known geometryprovided by themanufacturer and comparing it to features in corresponding b-modeimages. The inclusion appeared as a slightly hypoechoic circle in individual b-modeslices through the volume as shown in Figure 5.3. The edges of the inclusion weremanually segmented in several slices, creating a cloud of points over the volume.The centre of a 10 mm radius sphere was calculated by minimizing the distancebetween the edge of the sphere and the point cloud. To reduce the influence of theboundary artefacts, the background region was cropped 1 cm from the edges of thevolume.1785.2.7 Reconstruction Performance EvaluationThe rms error can be used to compare the performance of the two elasticity recon-struction techniques, lfe and shear-FEM, using both the regularized displacementsand unprocessed displacements as inputs. It is defined asERMS =√√1NN∑i=1(E − E∗)2, (5.18)where E and E∗ are the Young’s modulus values at each voxel in the volume esti-mated by the reconstruction technique and provided by the manufacturer, respec-tively, and N is the number of voxels.It is also of interest to compare the elasticity reconstructions from the measuredmotion from 3d ultrasound in this work to a previous study using mre measure-ments. Using rms error in this case would be difficult because it would containregistration errors from aligning the two data sets. Instead the mean value in theinclusion and background regions are compared and reported as a percentage, re-sulting in the mre error metricEUS→MR =|EUSE − EMRE |EMRE× 100%, (5.19)where EUSE and EMRE are the mean Young’s modulus values in a region computedfrom ultrasound elastography and mre measurements, respectively.The elastographic cnr is commonly used to quantify the detectability of an in-clusions or inhomogeneity [37]. In this work it is computed from the reconstructedYoung’s modulus valuesCNR = 20 log10*..,2(Einc − Ebkg)2σ2inc + σ2bkg+//- , (5.20)where Einc and Ebkg are the mean moduli, and σ2inc and σ2bkgthe variance of themoduli in the inclusion and background regions, respectively.The elastographic snr can be used as a measure of how much variation or noiseis present in a homogeneous region [324]. In this work it is computed from the179reconstructed Young’s modulus values in regions that should have a constant elas-ticity valueSNR = 20 log10(Eσ), (5.21)where E and σ are the mean and standard deviation of the reconstructed Young’smodulus values in the roi. The snr is computed for both the inclusion and back-ground regions.The contrast-transfer efficiency (cte) describes the efficiency with which levelsof true modulus contrast are converted into levels of contrast in the reconstructedelasticity image. It is defined as the ratio of the observed contrast to the true con-trast [257]. Expressed in decibels, it is given byCTE = 20 log10(EincEbkg)− 20 log10 *,E∗incE∗bkg+- , (5.22)where E∗inc and E∗bkgare the Young’s modulus values for the inclusion and back-ground, respectively, provided by the manufacturer.5.3 ResultsThe motion measured over the ultrasound volume in the x, y, and z directions arepresented in Figure 5.4. The figure displays the real part of the displacement pha-sor for both the speckle tracking measurements and the regularized displacement.The regularized displacements show a reduction in variance, particularly in the xand z directions, as these measurements are more influenced by the poorer speckletracking ability in the lateral and elevational directions, respectively.The volumes of reconstructed elasticity using the lfe and shear-FEM methodsfrom both displacement measurements and regularized displacement are shown inFigure 5.5. All of the elasticity images show a stiff inclusion in a homogeneousbackground as expected. The location and size of the inclusion appears to agreewith the location estimated from segmenting the inclusion in b-mode images.Profiles of the elasticity images through the centre of the inclusion in the x, y,and z-directions are shown in Figure 5.6, using the b-mode segmentation to definethe inclusion centre. For the lfe reconstruction using 1d displacement, the peak180(a) (b) (c)(d) (e) (f)Figure 5.4: From left to right, slices through the volume of the displacementsmeasured in the x, y, and z directions. (a)–(c) The displacements esti-mated using speckle tracking alone, and (d)–(f) the displacements afterpost-processing the displacement measurements with divergence regu-larization. The x, y, and z directions roughly correspond to the lateral,axial, and elevational transducer coordinates.elasticity reaches close to the manufacturer’s specification in each direction, withthe peak occurring a few millimetres off centre. The lfe reconstruction using 3ddisplacement slightly underestimates manufacturer’s specification of the inclusionelasticity, but the peak of the profiles appears more centred. The lfe reconstructionusing the regularized 3d displacement produces more of an underestimation, but theprofiles appear to maintain a more consistent elasticity value within the inclusion.For the shear-FEM reconstruction with 1d displacement the inclusion elasticity ismostly underestimated, however a large peak appears off centre along the x direc-tion. The reconstruction with 3d displacement has a similar underestimation of181(a) (b) (c)(d) (e) (f)Figure 5.5: The estimated Young’s modulus volumes using the (top row) lfeand (bottom row) shear-FEM inversion algorithms. The first columnuses only the displacement component along the y direction, the secondcolumn uses all three displacement components, and the third columnuses all three displacement components after applying regularization.The white dotted outline indicates the intersection between the edgesof the spherical stiff inclusion and the displayed cross-sectional planes.The x, y, and z directions roughly correspond to the lateral, axial, andelevational transducer coordinates.182Distance [mm]-15 -10 -5 0 5 10 15Young's Modulus [kPa]010203040506070true x y z(a)Distance [mm]-15 -10 -5 0 5 10 15Young's Modulus [kPa]010203040506070(b)Distance [mm]-15 -10 -5 0 5 10 15Young's Modulus [kPa]010203040506070(c)Distance [mm]-15 -10 -5 0 5 10 15Young's Modulus [kPa]010203040506070true x y z(d)Distance [mm]-15 -10 -5 0 5 10 15Young's Modulus [kPa]010203040506070(e)Distance [mm]-15 -10 -5 0 5 10 15Young's Modulus [kPa]010203040506070(f)Figure 5.6: Profiles of the reconstructed elasticity values passing through thecentre of the inclusion in the x, y, and z-directions using the (top row)lfe and (bottom row) shear-FEM inversion algorithms. The first columnuses only the displacement component along the y direction, the secondcolumn uses all three displacement components, and the third columnuses all three displacement components after applying regularization.The “true” elasticity profile based on the manufacturer’s specificationsis shown as the solid line. The distance along the abscissae is measuredfrom the centre of the inclusion.the inclusion elasticity, but without the large peak. The regularized displacementsproduce a smooth bell shaped profile with a peak located in the centre of the inclu-sion that slightly overestimates the expected elasticity. A boundary artefact causesthe estimated elasticity values to rise rapidly toward the edges of the lfe plots inthe z-direction (bright edges in Figure 5.6(a), Figure 5.6(b), and Figure 5.6(c)).Similarly, the values decrease to zero for the shear-FEM plots near the edges (darkedges in Figure 5.6(d), Figure 5.6(e), and Figure 5.6(f)). The boundary effect ap-pears larger for the shear-FEM and for this reason is also visible in the profile across183the x-direction.The distribution of the elasticity values within the inclusion and outside of theinclusion are shown for all of the methods in Figure 5.7. In all cases the medianvalues underestimate the manufacturer’s specification. For the lfe reconstructions,the range of elasticity values decreases when changing the input from 1d to 3d dis-placements, and from 3d displacements to regularized 3d displacements. For theshear-FEM reconstructions, the range of elasticity values decreases when changingthe input from 1d to 3d displacements, but increases from 3d displacements to reg-ularized 3d displacements. Decreasing the variance is desirable since the regionsrepresented in these box plots are expected to have a single homogeneous elasticity.However, a decrease in variance should not come at the cost of decreased accuracyin the estimate. This balance is quantified in the performance measure presentednext.The performance metrics for the lfe and shear-FEM methods using unpro-cessed 1d and 3d displacements, as well as regularized 3d displacements as inputsfor estimating the elasticity are summarized in Table 5.2. In terms of rms error, theshear-FEM reconstruction using 1d displacements provided the best performance,while lfe performed better in both 3d displacement cases. Using previous mremeasurements on the same phantom as a reference [22], the percent error for themean inclusion value was lowest (6.79 %) for the shear-FEM method using the 1ddisplacements, closely followed by the shear-FEM reconstruction using the regu-larized 3d displacements (8.44 %). For the background, lfe with the regularized3d displacements produced the lowest error (9.63 %). The reconstructions showat least a 2 dB improvement in cnr from unprocessed 3d displacements to regu-larized displacements with both methods. Similarly, a 2 dB improvement in snr isobserved in the inclusion and 3 dB in the background for lfe from unprocessed 3ddisplacements to regularized displacements, however there is drop in snr for theshear-FEM method which seems to be caused by the larger variance as observedin Figure 5.7. The cte shows that the shear-FEM method using the regularized3d displacements does the best job of displaying the true contrast of the phantom(−1.38 dB away from zero).The effect of the regularization parameter, α, on the displacements is exam-ined in Figure 5.8. Profiles of the real part of the displacement component along184LFE shear-FEM LFE shear-FEM LFE shear-FEMYoung's Modulus [kPa]01020304050607080 Regularized3DMeasured3DMeasured1D(a)LFE shear-FEM LFE shear-FEM LFE shear-FEMYoung's Modulus [kPa]01020304050607080 Regularized3DMeasured3DMeasured1D(b)Figure 5.7: Boxplots describing the distribution of Young’s modulus in the (a)inclusion and (b) background. Each plot is divided into three sections,with the leftmost based on 1d displacement measurements, the middlebased on 3d displacement measurements, and the rightmost based on 3ddisplacements after divergence regularization.1854030x [mm]20100-10010dx [7m]10210610 -410 -610 -810 -1010 -20,(a)|| Wd - Wm ||10 -10 10 -5 100 105|| Ld ||10 -1510 -1010 -51001051010,  = 10 -10,  = 10 -8,  = 10 -6,  = 10 -4,  = 10 -2,  = 10 2,  = 10 6(b)Figure 5.8: (a) Plot of the real part of the displacement component along thex-direction at y = 10 mm and z = 15 mm across the x-dimension of thevolume for different values of the regularization parameter, α. (b) Thebalance between the divergence constraint and the measurement fit isplotted for the same values of α.186Table 5.2: Performance measures of the reconstructed elasticity values based on the measured and regularized displace-ments. The 1d displacement measurements correspond to the y component of motion. Some measures are specificto the stiff inclusion (inc) or soft background (bkg) regions.ERMS [kPa] EUS→MRinc [%] EUS→MRbkg [%] cnr [dB] snrinc [dB] snrbkg [dB] cte [dB]Measured 1d Displacementlfe 14.4 16.5 10.9 12.7 12.4 16.0 -2.45shear-FEM 13.8 6.79 10.4 12.9 13.3 15.7 -2.74Measured 3d Displacementlfe 14.5 16.6 11.0 12.4 12.4 14.8 -2.46shear-FEM 19.5 32.3 29.1 6.54 11.1 14.1 -2.98Regularized Displacementlfe 14.7 21.2 9.63 14.6 14.4 17.8 -3.06shear-FEM 16.8 8.44 24.7 8.63 9.30 9.53 -1.38187the x-direction are shown in Figure 5.8(a) for several values of α, at the locationy = 10 mm and z = 15 mm in the volume. The regularization initially shows littlechange in the displacement profile for increasing α, then starts to produce smooth-ing before dominating the solution and no longer fitting the measurements. Thereis a visible change between the profiles at α = 10−4 and 10−6. There appears tobe little change in the profiles between 10−2 and 102, indicating that increasing theregularization is not producing any further improvement. At very large values of α,the regularization term dominates and a field with no displacement and thus zerodivergence is found. A common method to visualize the balance between fitting themeasurements and fitting the regularization term is to plot the norm of the residualfor each term against each other for different values of regularization [127]. This isdone in Figure 5.8(b) using the terms from Equation 5.13 with the divergence norm,‖Ld‖, along the ordinate, and the weighted difference between the solution and themeasurements, ‖Wd −Wm‖, along the abscissa. This plot confirms the findingsfrom Figure 5.8(a), with small values of α not affecting the divergence residual andshowing the measurement residual increases but remains small (measured in mi-crons and summed across the entire volume in three displacement directions). Thepoint at α = 10−4 shows a change in the divergence residual, corresponding to thestart of visual smoothing in Figure 5.8(a). The divergence residual continues to de-crease as α increases without change in the measurement residual until a very largeα is used, corresponding to the zero displacement solution.5.4 DiscussionCompared to the values of elasticity reported by the phantom manufacturer, themean and median values of all reconstructions underestimate the inclusion (62 kPa)and the background (29 kPa). This discrepancy was also observed in Chapter 2. Aprevious mre study on the same phantom describes possible reasons for the dis-crepancy between the manufacturer’s values and the elasticity estimates, such asdifferences in temperature, excitation frequency, or change in properties due to age-ing [22]. The same reasons also apply to discrepancies between the elasticity esti-mates from the mre study and this study, with differences such as segmentation ofthe inclusion also contributing.188The lfe elasticity estimates based on the 1d displacement measurements aresimilar to the measurements reported in Chapter 2. The mean Young’s moduluscomputed in the inclusion is 36 kPa compared to 31 kPa in Chapter 2, and in thebackground is 23 kPa compared to 17 kPa in Chapter 2. The data in this chapterwere acquired by triggering every beam line in the sequence, while there was notriggering at all in the Chapter 2 data used for the elasticity comparison, so phasorshad to be compensated based on estimated timing. The volume sweep in this chap-ter is also more than three times larger than the sweep used in Chapter 2. Anotherimportant difference is that while only a single component of displacement wasused in this chapter, the tracking was done in 3d. Tracking in 3d improves accuracyby reducing signal decorrelation because the blocks of rf data can shift in morethan one direction to get the best match, and by improving snr by increasing theamount of rf data in each block by extending it along each dimension [141]. They component of the displacement used in the 1d elasticity estimation in this chap-ter also corrects for geometrical errors caused by different motor angles creating adifferent angle for the axial measurements across the volume.The magnitude of the EUS→MR error reported in Table 5.2, comparing the in-versions from the ultrasound data collected in this work to mremeasurements froma previous work, appears large in some cases. While closer agreement is desir-able, the field of elastography has been challenged by the inability to obtain thesame quantitative elasticity measurements between different methods. For exam-ple, using the samemremeasured motion field with the lfe inversion algorithm andan algebraic inversion algorithm produced of a difference in mean Young’s modu-lus of 24.1 % [199]. A study comparing mre measurements and ultrasound combshaped arf elastography found a 15 % difference in modulus [295]. Another studycomparing different commercial ultrasound elastography methods found an error of31.5 % comparing the 25th and 75th percentiles from modulus measurements usinga Philips system [124]. The same study found the percent error between Philips andSiemens median modulus measurements was 12.4 %. The study also detected a de-pendence on transducer type and measurement depth. Despite these differences, fora given modality, algorithm, and measurement protocol, valuable clinical data areobtained which can be used for liver fibrosis staging [53] and breast cancer bi-radsclassification [101], among other applications.189Using the regularized 3d displacements rather than the unprocessed 3d dis-placement measurements as an input to the shear-FEM algorithm improves the re-sult in every metric except for snr. It does not provide as clear a benefit to the lfealgorithm, where the cnr, snr and EUS→MR error in the background region all im-prove, but the rms error, cte, and EUS→MR error in the inclusion all worsen. Thesame can be said for the lfe algorithm when comparing the 1d displacement tothe regularized 3d displacement. The shear-FEM inversion using 1d displacementsoutperforms the regularized 3d displacements in all metrics except the cte. Whilea better result is expected using 3d motion measurements, this indicates that thereare likely still errors in the x and z displacement components that are corrupting thefem solution. This is further backed up by the inability of a full fem based inversionwith coupled displacement terms to produce a reasonable elasticity estimate (resultsnot reported). Ideally, all three components would have the best (axial) accuracy,which is the goal of Chapter 6.The method of regularizing the displacement field by minimizing the field’sdivergence presented in this work operates on a phasor representation of the dis-placements. It is also possible to apply the smoothing operation on each volume ofdisplacements, representing a measurement at different phase or time samples, andthen derive the phasor representation of the smoothed displacement measurements.Based on the data collected in this work, there is no difference in the resulting pha-sor between the two methods up to about machine precision. The choice to applythe smoothing on the phasor representation instead of the raw displacement mea-surements was made to reduce the computation time since the operation only needsto be applied once instead of 25 times (number of phase offsets).The regularization parameter, α in Equation 5.13, was chosenmanually by start-ing with a small value and slowly increasing it until a visually smooth displacementfield was produced. The L-curve criterion can be used as an automated approach tochoosing the regularization parameter [127]. Figure 5.8(b) is a plot of the residuals,however it does not display an L-shape for the values of α examined, suggesting itmay not be appropriate for picking the parameter in this case. In another work usingdivergence regularization on the displacement field, it is suggested that minimizingthe divergence term should not be considered a true regularization term, but a con-straint being enforced by a penalty, and it should be chosen as the largest value after190which no improvement is observed in the displacement [269]. This is essentiallythe manual approach used in this work.Based on the displacement phasors shown in Figure 5.4, the regularization ap-pears to have the greatest smoothing effect on the x-component of the displacement.The x-component is based on the lateral speckle tracking measurements, while they- and z-components have spatially varying dependencies on the axial and eleva-tional speckle tracking measurements. Since the axial component is measured mostaccurately, the y- and z-components in Cartesian coordinates are expected to bemore accurate than the x-component. This is observed in a previous translationalmotion tracking experiment of volumetric displacement components [260]. Thevariances measured in that experiment are also used to weight the different compo-nents during regularization as described in Equation 5.13, meaning the measure-ments of the x-component contribute less to the final regularized solution of thedisplacement vector field (the y- and z-components contribute 50 and 2.5 timesmore to the solution, respectively).The displacement measurement weighting matrix as defined in Equation 5.12assumes spatially uniform variance in each of the vector components. However,since the point spread function in ultrasound is spatially variant, better performancemight be observed using a spatially varying weighting [304]. Instead of using ex-perimental translation motion tests to estimate the variances, which depend on thetransducer and imaging system characteristics, theoretical lower bounds on the esti-mation accuracy could be used [166], eliminating the need to repeat the experimentsfor different imaging scenarios.The synchronization between the mechanical excitation and the acquisition ofeach beam line increases the total acquisition time compared to standard imagingwithout synchronization. In standard ultrasound b-mode imaging, the minimumtime required to acquire a beam line is determined by the depth that the ultrasonicpulse needs to travel to and back and the speed at which the pulse travels. For exam-ple, typical values of 5 cm depth and 1540 m/s speed of sound results in a time ofapproximately 65 µs. The synchronization used in this work triggered a beam lineonce every excitation period, which at 200 Hz is 5000 µs. For an acquisition with31 motor positions, 25 frames in each position, and 128 beam lines in each frame,the total acquisition time is 496 s with synchronization, compared to 6.45 s without191synchronization. This is a worst case comparison as these timings do not accountfor any pauses for motor movement, which is not a significant factor for the synchro-nized case because of the long time between beam line triggers, but would increasethe time for the case without synchronization. It is difficult to directly compare thetiming to mre since the acquisition is dependent on a number of factors such as themri sequence, echo times, volume dimensions and resolution, excitation frequency,and number of phase offset measurements. The mre study used for comparison inthis work reported an acquisition time of 300 s, however the volume was thin andwide compared to the ultrasound volume used in this work, and the number of phaseoffsets was 8 compared to 25 [22].The acquisition time could be improved by designing a new synchronizationsequence. For example, when a beam line is triggered, it could be repeatedly firedwith known delays between firing to collect all desired phase offsets before one ex-citation period is complete, and then the next line could be triggered at the start ofthe next period. This would reduce the time in the above example by a factor of 25(number of frames in each motor position) for a total acquisition time of approxi-mately 19.8 s. This would be similar to the work sequence and synchronization usedin previous works of dynamic motion measurement [83, 91, 131, 250]. The spacingbetween the phase offsets as determined by the line repetition time could also be setsuch that the acquisition of subsequent lines is frequency-locked to the excitationfrequency, eliminating the need to trigger between each line [350], though someform of synchronization would be required between motor positions to collect syn-chronous volumes. The disadvantage of this method is it requires reprogrammingof the pulse sequence which is possible on Ultrasonix machines, but not all others.In the method used in this work, the only requirement is being able to trigger eachtransmission; there is no modification to the standard ultrasound sequence. If fullchannel parallel receive hardware is available such that all 128 lines could be re-ceived at the same instant, in combination with a planar transmit as in [309], entireframes could be acquired in 65 µs and the trigger at the start of an excitation periodwould only be necessary for each motor position, resulting in an acquisition timeof approximately 0.16 s, neglecting pauses for motor movement. It might also bepossible to eliminate the need for synchronization if tissue motion is constrainedto primarily the axial direction by applying compensation to the rf signals for the192delay in acquiring subsequent beam lines, similar to previous work applying com-pensation for transducer motion [283].Methods for the acquisition of the full 3d vector shear wave motion over a vol-ume using ultrasound have been described before using a different approach [33,211]. In these works a plane wave transmit and full channel parallel receive hard-ware was used to measure the motion at a high rate. These works use a linear arraymounted on a stepper motor to obtain a volume which is similar to a swept motorultrasound transducer but with the motor not integrated into the casing and cabling.Synchronization between the exciter and pulse sequence is applied at each motorposition. Elasticity is estimated by applying the curl operator to the displacementfield and solving an Helmholtz equation algebraically which is similar in conceptto the lfe inversion used in this work. The main difference between the work de-scribed in this paper and these previous works is the previous works use 1d dis-placement tracking from multiple angles, obtained through both beamforming andphysical repositioning of the ultrasound array, and combine these 1dmeasures intoa 3d vector. This paper uses 3d displacement tracking to obtain the 3d vector di-rectly. The advantages of 3d tracking over 1d are greater measurement accuracydue to the ability to shift in multiple directions to reduce signal decorrelation andthe option to use signal blocks that span all three directions which should improvesnr, and the ability to use a single view angle for the full 3d displacement vectorwhich reduces system complexity and acquisition time. Combining multiple trans-ducer view angles has the advantage of using axial displacement measures whichare more accurate than lateral and elevational measures and therefore the regular-ization used in this work to reduce noise is not as important [4, 5]. However, thenumber of organs whichmultiple transducers may be applied is limited by geometryand the need to surround the organ from different transducer viewing positions. It isalso important that the multiple transducers do not change the mechanical boundaryconditions (e.g. changing the areas of transducer contact with the patient) duringacquisition, otherwise the elastic wave field will change and it would be impossi-ble to combine the measurements into a 3d vector. For these reasons, a specializedbreast scanner using a water bath between the transducer and tissue has been usedin the previous studies [33, 211].Longer acquisition times, such as those caused by the synchronization used in193this work, are more susceptible to errors caused by patient or transducer motion.Some 3d ultrasound procedures might be able to restrict some undesired motionssuch as an automated breast scan [351], or trus imaging of the prostate [281], sosuch errors could be minimized.The measurement of tissue motion in this work relies upon the assumption thatconsistent reproducible motion is produced, and thus a snapshot of the motion canbe measured over a region at different acquisition times but identical mechanicalexciter states. In a clinical setting, patient motion (e.g. through breath or heartbeat)breaks this assumption and would introduce error into the measurements. As men-tioned in the previous paragraph, the long acquisition time for the method proposedin this work is more susceptible to these errors, but certain clinical applications,such as automated breast scanning, where motion is restricted may be able to re-duce the magnitude of these errors. mre is susceptible to these same problemsbut has been used in vivo [162, 290, 342]. Several ultrasound based tissue motionmeasurement methods also use this assumption, for example for ex vivo liver [83]and in vivo cardiac [250] measurements. Based on the success of tissue motionsampling in these previous studies, it seems that error introduced by patient motioncan be managed to acceptable levels. There remains the question of whether theproposed method can measure consistent reproducible motion in an ideal settingwhen the patient does not contribute any additional motion. Forced harmonic vi-brations approach steady state in exponential time for a nearly incompressible 3delastic body [152]. To quantify the time to reach steady state, a simplified analysisof 1d damped forced vibration can be used as an approximation of tissue motion.The solution to the differential equation governing a mass-spring-damper vibrationexcited by a harmonic forcing function results in a transient term that oscillates atthe damped natural frequency of the system and decays exponentially, and a steady-state term that oscillates at the forcing frequency. The exponentially decaying termtakes the forme−η2m t = e−tτ , (5.23)where η and m are the local lumped damping and mass parameters, τ is the timeconstant of the exponential decay, and t is the elapsed time. A rough guideline is toignore transient effects when t/τ ≥ 5 (the exponential decay is less than 0.67 % of194its original amplitude) [300]. To estimate the time constant, following the method-ology of Turgay et al. [319], the ratio of viscous damping to the local lumped massis estimated to be 4000 Pa s/kg. This means that transient effects can be ignoredafter 2.5 ms have elapsed. In this work care was taken to only start sampling ul-trasound data a few seconds after starting the vibration source so a consistent andreproducible steady-state motion should be expected in the phantom experiments.Another potential change that could disrupt the assumption of consistent repro-ducible motion is thermal effects. Repeated acoustic pulses from the ultrasoundtransducer may increase tissue temperature, changing the elastic properties of thetissue and the speckle pattern used formotion tracking. Previous research has shownthese thermal effects to be small, with a temperature change under 2 ◦C and a shiftof under 5 % of a sample in the speckle pattern (under one micron) [78].This study used monochromatic mechanical excitation. Multi-frequency vibra-tions have been measured in mre [162], so the synchronization presented in thiswork should allow for ultrasound measurement of multi-frequency displacementvector components over a volume. The multi-frequency data can mitigate measure-ment noise by changing locations of small displacement [242], and increasing thenumber of the equations to improve the conditioning of the inverse problem [136],or can also be used for spectral analysis [81, 97].5.5 ConclusionThis study has demonstrated measurement of the full 3d harmonic motion vectorover a volume by adding a synchronization step to the tracking steps and addingdivergence-based regularization. Using a physical constraint consistent with softtissues, divergence regularization post-processing creates a smoother field of dis-placement vectors. The regularized displacements produce a more accurate volumeof elasticity estimates using the shear-FEM inversion method compared to usingthe unprocessed 3d displacement measurements. The accuracy does not seem toimprove with lfe inversion using the regularized displacements, however improve-ments in cnr and snr are observed. It is difficult to determine which inversionmethod is superior, however the profile of shear-FEM shows a symmetric and con-sistent profile in each direction with a peak at the centre of the inclusion that is195within 1 kPa of the manufacturer’s specification. Results similar to mre are gener-ated using either method.Future work will investigate the use of more sophisticated elasticity estima-tion algorithms, measurements on tissue instead of phantommaterial, and improve-ments in acquisition speed. Investigating repeatability and sensitivity to differentboundary conditions and excitation modes are also of interest for particular clinicalapplications.196Chapter 6Combining Multiple Views forIncreased Accuracy of the FullShear Wave Motion Vector inSWAVE6.1 IntroductionImages of tissue elasticity are formed by measuring tissue motion in response to amechanical stimulus. This field of medical imaging, called elastography, is com-monly based on ultrasound measurements of tissue motion because ultrasound isubiquitous, non-invasive, safe, inexpensive, and portable.High quality measurements of the full 3d displacement vector over a volume arenecessary to accurately solve the inverse problem for elastic parameters [96, 299].Unfortunately, in the directions orthogonal to the ultrasound beam axis, the lack ofphase information, poorer resolution, and larger sample spacing leads to over anorder of magnitude increase in displacement measurement uncertainty comparedto measurements along the beam axis [42, 167, 187]. The measurement noise inthese directions can cause errors so large in the inversion process that replacingthe measurements with a false assumption of no motion in the orthogonal direc-197tions can lead to more accurate estimates of the elastic parameters [99]. In Chap-ter 5, amethod formeasuring 3d displacement vectors using cross-correlation basedspeckle tracking on synchronized ultrasound acquisition with a motorized 3d trans-ducer was proposed, and a technique for reducing noise in the displacement mea-surements using regularization based on tissue incompressibility was implemented.However, the two inversion methods used in that study decoupled the displacementcomponents in the equations of motion, because inversion based on a fully coupled3d motion model was not able to properly estimate elasticity.One way to obtain multiple displacement components is to vary the ultrasoundbeam angle to obtain different axial measurements. Angular compounding is a tech-nique which electronically steers the ultrasound beam axis to different angles, andcombines the displacement measurements from each of these angles to computethe displacement vector components [267, 348]. The angular compounding tech-nique improves displacement estimates by using multiple accurate axial displace-ment measurements of the projected displacement vector, however the steering an-gle is usually limited to angles up to 15◦ to maintain acceptable image quality, andeven in special cases the angle is typically limited to 45◦ at most [126]. The max-imum imaging depth becomes limited with increasing steering angles because thevector components can only be estimated in the region where the beams from eachangle overlap.An alternative approach is to use multiple transducer views with overlappingbeams crossing at different angles. This has been successfully used in a customizedDoppler instrument formeasuring the blood velocity vector at a point in space [109].A more recent work used multiple volumes collected by a hand held matrix arrayto compute 3d velocity vectors in the heart [118]. There has been some researchinterest in using this approach in elastography to improve displacement measure-ment accuracy in different directions. Some studies have used a transducer ori-ented orthogonal to the applied compression so that accurate axial displacementmeasurements could be used to measure the strains transverse to the primary exci-tation direction [60, 303]. Another study used two coplanar and orthogonal lineararray transducers to measure 2d strains over a plane using axial displacement mea-surements from each transducer to obtain symmetric accuracy in the measurementdirections [150]. In my master’s research [4], I compared 2d displacement and198strain measurement accuracy using 2d correlation speckle tracking, angular com-pounding, and two coplanar linear arrays with relative beam angles varied between40◦ to 90◦ and investigated errors introduced by inaccurate calibration between thetwo transducers.A combination approach of multiple transducer views and angular compound-ing has been used in an elastography study measuring shear waves using two or-thogonal linear arrays, oriented such that their image planes intersect along a line,mounted on stepper motors to measure 3d displacements over a 3d grid [33]. Asimilar approach is used by Muller et al. [211], using a single linear array mountedon a rotating table to circle the roi from multiple view directions.In this chapter it is proposed to combine axial measurements from different mo-torized 3d ultrasound transducers scanning the same volume from different angles,or, alternatively, a single tracked motorized 3d ultrasound transducer sequentiallyscanning a volume from different angles. It is the focus of this chapter to solve for 3dmotion vectors from three volumes of axial motion measurements, and to study theeffects of scan conversion, interpolation, and calibration on the displacement andelasticity estimates. Practical issues of the possibility of applying the techniqueclinically for breast cancer imaging are examined by simulating feasible scanninggeometry and realistic calibration errors based on the results of Chapter 4.6.2 Methods6.2.1 Finite Element SimulationA 3d fem simulation was used to investigate the effectiveness of the proposedmultiple view method in a controlled and repeatable environment. The fem solu-tion provided synthetic displacement data to generate simulated ultrasound speckletracking measurements and a ground truth for comparing the elasticity results. A40 × 40 × 40 mm3 cube with a 5 mm radius stiff sphere was modelled using ANSYSfem software (Cecil Township, PA, USA). The simulated phantomwasmeshed with10-node tetrahedral elements (SOLID187) with a 0.5 mm nominal side length. Thestiff sphere was modelled with a Young’s modulus of 62 kPa, with the rest of thecube modelled with a background Young’s modulus of 29 kPa. All elements were199modelled with a density of 1000 kg/m3 and a Poisson’s ratio of 0.495 to simulatesoft tissue. A 300 Hz harmonic excitation with an amplitude of 0.6 mm was appliedto the top surface of the cube in the normal direction of the top surface (longitudinalvibration in the y direction). The bottom surface of the cube was constrained in itsnormal direction (y), and constrained in the x and z directions along their respectivelines of symmetry. All other sides of the cube were left free to deform.6.2.2 Simulating Displacement MeasurementsUltrasound volumes were modelled using the geometry of a 4DL14-5/38 transducer(Vermon, Tours, France). Each volume contained 61 2d image slices swept over anarc in 0.45◦ increments with a radius of 81 mm between the centre of rotation andthe start of the image. Each 2d image was 38.1 mm wide and 60 mm deep.In total three ultrasound volume views were modelled to measure different com-ponents of the phantom motion vector using axial measurements from each ul-trasound volume. To obtain the highest quality vector displacements from com-bined ultrasound axial displacement measurements, three orthogonal views shouldbe used such that each axial measurement corresponds to one component of thevector. The volumetric sweep the 3d motorized ultrasound transducer changes thedirection of axial measurement as a function of motor position, making it difficultto ensure the views are orthogonal over the full volume. Instead, it was decided tomake the central 2d image plane of the motor sweep from each view orthogonal,while the other image planes would deviate slightly from the ideal measurementgeometry. To make the method easy to adapt to experimental and clinical settings,it was desired to design the geometry such that the transducer views all generallypointed downwards so that they could all be placed on the top surface of a waterbath, or flipped and placed under a prone patient. This geometry can be achievedby orienting the axial direction of the centre image plane at an angle of 35.4◦ down-ward from the horizontal, and rotating the transducer about a vertical axis in 120◦increments to obtain the three views. The geometry can be described as rotatingaround the top face of a tetrahedron as depicted in Figure 6.1.With respect to the fem model, the first transducer view was placed such thatthe central 2d image plane had a 5 mm gap between the transducer face and the200Rotation axisRotation path View 1View 2 View 3Figure 6.1: The view geometry is described as a rotation about an axis (dashedgrey line) passing through a vertex of a tetrahedron and perpendicularto the opposite face of the tetrahedron (shown here as the top face). Theviews follow a 120◦ rotation along a circular path (grey dashed circle)on the top face of the tetrahedron, with each view centred on one of thetetrahedron edges and directed to the centre of the opposite edge (blackarrows and dashed lines). The three views intersect at a common pointand are orthogonal.phantom, and the start of the image corresponded to the same height as the top ofthe phantom. The second and third transducer views were obtained by rotating thefirst view about the centre of the cube. The axes of the central image planes for eachof the three views around the phantom are shown in Figure 6.2. The geometry ofthe sweep of the first view with respect to the phantom is shown in Figure 6.3.Time-varying axial displacement measurements were generated by projectingthe fem solution onto the axial directional cosines of each image slice. Let u(x, t)represent the displacements computed by ANSYS at position x at time t, and nγ (x)represent the axial directional cosines of the γth view, then the simulated axial dis-201z-20020x-20 0 20(a)-200z20200x-2002040y(b)x-20 0 20y02040(c)y02040z-20020(d)Figure 6.2: Geometry of the mutiple view angles with respect to the phantom,showing (a) top, (c) front, (d) side, and (b) isometric views of the sim-ulation. The black lines indicate the edges of the phantom and the red,green, and blue lines indicate the x, y, and z axes of the centre plane ofthe volume for each view, respectively.placement is computed as a dot producta(x, t) = nγ (x) · u(x, t) + η(x, t), (6.1)where η represents jitter error in the ultrasound speckle tracking process. The jittererror in the motion measurements was simulated by adding zero mean Gaussiannoise with variance equal to the Cramér-Rao lower bound for partially decorrelated202y [mm]-20-1001020304050x [mm]-30 -20 -10 0 10 20 30Figure 6.3: The ultrasound axial measurement direction over the entire sweptvolume (blue arrows) with an outline of the simulated phantom providedfor reference (black square).speckle [328]σ =c2·√32pi f 30T(B3 + 12B) *, 1R2(1 +1SNR2)2− 1+-, (6.2)where c is the ultrasound pulse speed, f0 is the centre frequency of the ultrasoundpulse, T is the displacement block length (measured in terms of echo time), B is thefractional bandwidth of the ultrasound system, R is the normalized cross-correlationcoefficient, and snr is the ultrasound signal-to-noise ratio based on signal corrup-tion due to electronic noise. The parameter values used in this study are shownin Table 6.1 and were chosen to match the 4DL14-5/38 transducer and expectedexperimental measurement conditions.203Table 6.1: Parameter values used for simulating jitter error in the displace-ment measurements with Equation 6.2.c 1540 m/sf0 10 MHzT 1.3 µsB 0.7R 0.98snr 40 dB6.2.3 Solving for 3D DisplacementThe axial displacements from each volume were mapped to a common grid, cov-ering the extent of the overlapping region of the volumes, using 3d linear interpo-lation. The grid was regularly spaced, with a resolution of 1 mm in each directionchosen to achieve balance for the inversion algorithm in the element size trade-offin increasing inertial forces with decreasing derivative accuracy [138]. The axialdirectional cosines of each volumewas also mapped to the common grid, and a leastsquares solution, similar to beam steering methods [313], was used to calculate thedisplacement vectors in terms of the fem coordinate system. The relationship be-tween the unknown displacement, d, and the axial measurements is modelled asVd = a, (6.3)204whereV =nᵀ1[0] 01×3 . . . 01×301×3 nᵀ1[1]....... . .01×3 nᵀ1[N − 1]nᵀ2[0] 01×3 . . . 01×301×3 nᵀ2[1]....... . .01×3 nᵀ2[N − 1]nᵀ3[0] 01×3 . . . 01×301×3 nᵀ3[1]....... . .01×3 nᵀ3[N − 1], (6.4)nγ[i] is the axial directional cosine vector for the γth view at the ith grid location,N is the total number of grid locations,d =dx[0]dy[0]dz[0]...dz[N − 1], (6.5)dx , dy , and dz are the x, y, and z components of the displacement vector at each205grid location,a =a1[0]a1[1]...a1[N − 1]a2[0]...a2[N − 1]a3[0]...a3[N − 1], (6.6)and aγ is the axial measurement for the γth view at each grid location. The leastsquares solution for d is computed asd = (VᵀV)−1 Vᵀa. (6.7)6.2.4 Simulating Calibration ErrorThe process of mapping each volume of data to a common grid requires knowledgeof the spatial transformations between each of the volumes. This requires a cali-bration procedure, which could consist of determining the transformation betweena transducer and a tracking sensor as described in Chapter 4, or could consist ofa registration to align ultrasound volumes scanning the same target from differentpositions. In either case, some residual error in the translation and rotation param-eters is expected. This will contribute to errors both in spatially aligning the datafrom each volume, and in describing the measurement directions accurately in themodel as described by Equation 6.3.Conceptually, the calibration error in this work models the situation where thedata are measured at the nominal locations and directions as described in Sec-tion 6.2.2, but the calibration provides the wrong measurement location and direc-tion. Before calibration error is introduced, the axial displacement data are given byEquation 6.1 at spatial location x0. If the calibration error is described by a trans-lation, sΛ, and a rotation, RΘ, then the axial displacement measurements from the206γth view are mapped to a new locationx = RᵀγRΘRγx0 + RγsΛ, (6.8)where Rγ describes the rotation from phantom coordinates to the central imageplane of the γth view. The application of Rγ in Equation 6.8 is to ensure the cali-bration error is the same for each of the three views in the local view coordinates.In addition to mapping the data to a new location, the axial directional cosines aremodifiedmγ[i] = RᵀγRΘRγnγ[i], (6.9)where the matrix V in Equation 6.4 is now constructed using the modified direc-tional cosines, mγ[i].The calibration errors are composed of a magnitude and a direction. For thetranslation, this is just a scalar distance, Λ, multiplied by a unit vectorsΛ = ΛvˆΛ. (6.10)For the rotation, Rodrigues’ rotation formula is used to obtain the rotation matrixfrom the rotation axis unit vector and rotation angle magnitudeRΘ = I + sin (Θ) [vˆΘ]× + (1 − cos (Θ)) [vˆΘ]2× , (6.11)where I is a 3 × 3 identity matrix and [vˆΘ]× is a skew symmetric cross productmatrix constructed from the components of the rotation axis unit vector[vˆΘ]× =0 −vˆΘ[2] vˆΘ[1]vˆΘ[2] 0 −vˆΘ[0]−vˆΘ[1] vˆΘ[0] 0. (6.12)To investigate the error in estimating elasticity as a function of calibration error,Λ and Θ were tested at several values. The magnitude of calibration errors can bedifficult to estimate because there often is no gold standard to compare a calibrationresult against. In Chapter 4, calibration error was quantified by calculating the thepoint-to-point distance in mapping the tip of a tracked stylus to a global coordinate207system using the calibration solution and position tracking measurements. Basedon the results of Chapter 4, the best calibration method,W2D→3D had an error reach-ing up to 1.0 mm (95 % confidence interval using the Student’s t-distribution). Itis also of interest to investigate the effect of larger calibration errors, so based onthe results of Chapter 4, and neglecting the N2D method as a naive approach for 3dcalibration, the worst calibration errors were observed with N3D, reaching 2.7 mm(95 % confidence interval). Unfortunately this calibration error metric cannot sep-arate errors in the the translation and rotation parameters, so instead the magnitudeof the translation error was assumed to vary up to 2 mm, and the rotation parame-ters caused a position error on the order of 0.7 mm to 2.7 mm, which correspondsto approximately 1◦ to 3◦ for at an imaging depth of 60 mm. For the simulations, Λwas tested at values of 0 mm, 0.5 mm, 1 mm, 1.5 mm, and 2 mm, and Θ at 0◦, 1◦,2◦, and 3◦. At each value of Λ and Θ, 50 random trials were simulated, each witha unique translation direction, vˆΛ, and rotation axis, vˆΘ. These vectors were sam-pled uniformly over all directions using the direction defined by uniform randomquaternions [174].6.2.5 Elasticity InversionEstimates of Young’s modulus were therefore obtained using a recently developeddirect (rather than iterative) inversion algorithm based on a dynamic displacement-pressure fem, or mixed-fem, formulation with sparsity regularization [135]. Themixed-fem technique uses a full volumetric model of 3d elastic deformation with-out assumptions about elastic parameter homogeneity, or material incompressibil-ity. As a result of the more realistic model, this inversion algorithm has demon-strated high quality results when all three displacement components are accuratelymeasured, as in mre, with both phantom and patient data [279]. Since ultrasoundbased displacement measurements using the multiple view approach should be ofsimilar accuracy, this inversion algorithm is well suited for this work.Using the governing equation of motion for a linear isotropic elastic materialundergoing a time harmonic excitation (Equation 1.37), and defining the hydrostaticpressure (as in Equation 1.60) as an additional unknown, the tissue mechanics are208described by∇ ·[µ(∇d + (∇d)T)+ pI]= −ρω2d, (6.13)where µ is the shear modulus, p the pressure, d the displacement, ρ the density, andω the frequency of excitation. The density of most soft tissue is close to the densityof water, so ρ is assumed to be uniformly 1000 kg/m3.For the fem formulation, Equation 6.13 is written in weak form and discretizedusing 8-node hexahedral elements, with piece-wise constant shape functions for theshear modulus and pressure and linear shape functions for the displacement. Theinverse problem can be written in matrix form as[A C] µp = f. (6.14)Sparsity regularization is used to ensure that the problem is full rank, to improvethe condition number of the matrix, and to reduce the sensitivity of the solution tonoise. The shear modulus and the pressure are assumed to have a sparse represen-tation in the discrete cosine transform (dct) domain, and can be represented usingthe lower frequency part of the domain. In this work, the dct domain is truncated toselect 85 % and 90 % of the coefficients for the shear modulus and pressure, respec-tively. Let Tµ and Tp represent the inverse truncated dct transforms for the shearmodulus and pressure respectively, then the regularized inverse problem becomes(ATµ)ᵀ (ATµ) (ATµ)ᵀ (CTp)(CTp)ᵀ (ATµ) (CTp)ᵀ (CTp)µ˜p˜ =(ATµ)ᵀ f(CTp)ᵀ f , (6.15)where µ˜ and p˜ are approximated dct transform domain shear modulus and pressure.After solving Equation 6.15, the shear modulus is obtained viaµ = Tµ µ˜. (6.16)In this work, the shear modulus values calculated by the inversion algorithm areconverted to Young’s modulus assuming fully incompressible tissue, resulting in asimple scalar relation E ≈ 3µ.To decrease the computation time, the roi is divided into several 13 × 13 × 13209sub-domains and the solution is computed independently in each sub-domain. Thetotal computation time using Matlab (Mathworks, Natick, MA, USA) running ona 3 GHz Intel Core2 Duo PC for a full elasticity volume was approximately twominutes on average.The inversion process is sensitive to errors introduced by taking derivatives ofnoisy displacement measurements. A linear least squares fit in overlapping 3 ×3 × 3 regions is used as a robust method to calculate the spatial gradient of thedisplacements used in the inversion. This is described inmore detail in Appendix G.6.3 ResultsThe real part of the complex phasor motion field is shown in Figure 6.4 for boththe ANSYS solution and the displacement computed from three simulated axialdisplacement volumes. All three components are recovered from the combinationof axial displacements and the wave pattern closely resembles the ANSYS solution.The common overlapping volume of the three views is slightly smaller than fullphantom cube, so some regions close to the edges do not contain displacement inthe volumes computed from axial measurements.Examples of estimated elasticity volumes and the difference between the es-timates and the ground truth are shown in Figure 6.5. In all cases a stiff spheri-cal inclusion is visible in a soft background. The difference images highlight thelargest errors near the border between the background and inclusion. The inclusionappears to generally be underestimated, while the background appears to generallybe overestimated. There is a band of underestimated background values around theboundaries of the roi which seems to grow in spatial extent with increased simu-lated calibration error.The rms error in the estimated elasticity volumes is depicted in Figure 6.6. Therms error is averaged over 50 randomized trials at each of the calibration errorsettings to create one point on the plot. The rms error for the elasticity estimatedbased on the ANSYS displacements is 3.50 kPa. Adding the steps of projecting thedisplacements onto the axial directions, adding speckle tracking noise at a 40 dBsnr, interpolating displacements to a common grid, and computing the orthogonaldisplacement components increases the mean rms error in the elasticity estimates210(a) (b) (c)(d) (e) (f)Figure 6.4: The real part of the complex displacement phasors over the vol-ume shown in three orthogonal cross-sectional slices. The displace-ments in the x, y, and z directions are depicted from left to right, withthe top row displaying the fem solution computed by ANSYS, and thebottom row showing an example of the displacements computed fromEquation 6.7 using three simulated axial measurements (with no cali-bration error applied).to 6.22 kPa (with no calibration error applied). As expected, the error increaseswith increasing calibration error in translation and rotation, both in isolation andcombined. For low calibration errors (< 1 mm and 2◦), the elasticity error increasesslowly (< 0.5 kPa) as calibration error increases. The last value on the 3◦ curve at2 mm is beyond the range of the plot as the error increases greatly to 494 kPa.6.4 DiscussionA significant source of the error in the elasticity estimates is caused by several typesof errors in the displacement measurements. Neglecting calibration and speckle211(a) (b) (c) (d)(e) (f) (g) (h)Figure 6.5: The (top row) estimated elasticity and (bottom row) difference inelasticity from ground truth. The estimated elasticity volumes are shownusing input displacement (a) computed by ANSYS, and computed fromthree simulated views with (b) no calibration error, (c) 1 mm and 1◦ cal-ibration error, and (d) 2 mm and 3◦ calibration error. The differencebetween the estimated elasticity and ground truth are shown in the sameorder as the estimated elasticity. Positive values indicate overestimationand negative underestimation in the difference images.tracking noise, the displacements already accumulate errors from the three inter-polation steps: (1) from the irregular tetrahedral fem mesh in ANSYS to a regulargrid in Matlab using the ANSYS element shape functions; (2) from the regular gridcomputed in (1) to the irregularly spaced ultrasound measurement points (i.e. thecentres of each pixel after scan conversion from transducer to Cartesian coordinates)using 3d linear interpolation; and (3) from the ultrasound measurement points tothe common regular grid for the overlapping measurements from each view using3d linear interpolation. Due to the motor rotation to create the swept ultrasoundvolume, the intersecting beam angles are not always orthogonal, so after addingspeckle tracking noise a spatially dependent projection error is created. This maypartially explain why the elasticity estimates are poorer towards the edges of the212$  [mm]0 0.5 1 1.5 2ERMS [kPa]0510152025#  = 0°#  = 1°#  = 2°#  = 3°ANSYSFigure 6.6: The mean rms error in the estimated elasticity computed over theentire volume for all 50 trial randomized realizations of error, at eachcalibration error setting ranging from 0 mm to 2 mm in translation and0◦ to 3◦ in rotation.volume. For low calibration error levels, the elasticity error seems to be dominatedby displacement snr, however as the calibration error increases the elasticity errorcan grow by orders of magnitude.Part of the elasticity error is also caused by differences between the inversemodel and the forward model, as evidenced by the error produced when the dis-placements from the ANSYS solution are used directly as the input for the inver-sion algorithm. These discrepancies include differences in the assumptions used toderive the equations of motion, approximations used to discretize continuous quan-tities, mesh and element definitions, node placement and resolution, and algorithmsused to compute derivatives, integrals, and solve equations.The results presented in Figure 6.5 demonstrate the elasticity error is higher inregions where the elasticity suddenly changes from the soft background to the stiff213inclusion. This error is partially caused by the mesh, where some elements spanthe interface of regions of different elasticity. Since an element can only represent asingle value of elasticity, it will contain somemix between the two elasticities withinit. An optimized mesh adapted for the specific spatial distribution of elasticity,which could be approximated from a quasi-static strain image, would reduce theseerrors near the interface of the soft and stiff elasticity regions [117].Inaccurate elasticity estimation, especially at the boundaries between regions ofdifferent elasticity, is also caused by the sparsity regularization, where the truncateddct causes ringing artefacts near step changes in elasticity [135]. Using a differentsparsity promoting transform, or a more intelligent method of choosing dct coeffi-cients could help reduce the ringing artefacts. For example, if a quasi-static strainimage is available it can be used as an approximation of the spatial distribution ofthe shear modulus for selecting the sparsity pattern [138].For consistency, the same mesh was used for the elasticity inversion of the AN-SYS solution was also used for all of the simulated ultrasound displacement mea-surements. This resulted in some of the nodes falling outside of the overlappingregion of the volumes. This might explain why the elasticity estimates for the sim-ulated ultrasoundmeasurements have a larger error towards the edges of the overlap-ping region. A special mesh designed for the irregular geometry of the overlappingregion might help to eliminate the edge error.The error in ultrasound speckle tracking displacement measurements was mod-elled as voxel-wise additive Gaussian noise. However, the quality of displacementmeasurements in real ultrasound signals is spatially dependent. Attenuation of theultrasound signal with depth causes the snr of the ultrasound signal and thus thedisplacement measurement quality to decrease with depth. The psf of the ultra-sound system also varies across the image plane, with the finest resolution locatednear the transmit focus depth and poorer resolution in the shallow and deep regions.The psf also is poorer toward the lateral edges of the image as the beam cannot beproperly focussed without array elements extending beyond the width of the image.Finally, displacement estimates are also more strongly influenced by high amplitudeechoes in the rf signal.The calibration error model used in this work kept the measured data constantand introduced an error in the measurement location and direction. An alternative214way to model the calibration error would be to fix the location and direction (i.e. thecalibration says there is perfect alignment), while the measured data is actually ac-quired at a different location at a different projection angle. Neither approach mod-els the experimental conditions closer than the other, however there could be somedifference in simulated results. The approach used in this work changes the mea-surement direction matrix in Equation 6.4, and the alternative approach does notmodify the measurement direction matrix but changes the measured displacements(a in Equation 6.1), while both approaches model misregistration of the volumes asin Equation 6.8.This study neglected errors caused by refraction of the ultrasound beam whencrossing between regions with different propagation speeds, and improper estima-tion of echo depth due to difference in the speed of sound used by the scanner andthe true propagation speed of the echoes. The nature of these errors is similar to thecalibration error that was modelled, however the errors would be spatially variant,depending on the underlying tissue composition and structure. The magnitude ofthe errors would likely be similar to the calibration error modelled in this work.For example, in the breast with a subcutaneous layer of fat followed by glandulartissue, the ultrasound speed can be expected to be 1480 m/s and 1580 m/s, respec-tively [92]. For an incident angle of 10◦ between the ultrasound beam and the tran-sition layer between the fat and glandular tissue, the change in beam angle is 0.68◦,and at 45◦ incidence the change is 1.7◦. A previous study modelling the combinedmisregistration caused by refraction and speed of sound depth misplacement in 2dfor orthogonal coplanar linear arrays estimated less than 1.0 mm error for breasttissue for imaging depths up to 30 mm [4].An implicit assumption in this work is that displacement measurements canbe acquired synchronously across all three volumes. The techniques described inChapter 2 should be sufficient, with either accurate time stamp information for eachultrasound acquisition combined with phase compensation, or a trigger between theexcitation and ultrasound scanner.A long term goal of this research direction is to apply this multi-view approachclinically as a 3d elastography breast cancer system. An effort was made to de-sign the transducer and view geometry such that it could be applied to the breast,either through direct skin contact or through a water bath similar to other similar215approaches [32, 211]. The elastic boundary conditions should not change betweenview acquisitions, so for direct skin contact this system would probably be best re-alized using multiple transducers rigidly fixed together or a customized transducerwith multiple view directions [4]. For a water bath system, a single transducer couldbe used instead and rotated to each view position.6.5 ConclusionThis work simulated acquiring volumetric ultrasound axial motion measurementsof harmonically excited tissue from three different view points which are combinedto solve for the full motion vector. The modelled geometry could feasibly be used ina clinical breast scanner for cancer imaging. Realistic calibration errors in spatiallyaligning the three volumes were studied. The proposed three view method can reli-ably solve for high quality 3d displacement vectors suitable for elasticity estimationbased on an full 3d model of tissue mechanics.216Chapter 7Conclusion7.1 SummaryIn this chapter, a full analysis of the research and its conclusions is presented inlight of current research in the field. The contributions of the thesis are delineatedand the limitations of the approaches undertaken in this thesis are described. Thechapter is concluded with a description of possible future research directions.In Chapter 2 the design and implementation of a 3d ultrasound elastographysystem was described. This system realizes a generalized modular system based onseveral previously developed specialized systems [23, 98, 191], and introduced thefirst implementation of a 3d band pass based system with an Ultrasonix ultrasoundscanner. Simulations demonstrated the robustness of estimating motion amplitudeand phase as a function of the number of samples used, the uncertainty in samplingrate, and errors in ultrasound speckle tracking based displacement measurements.b-mode images collected on a quality assurance phantom showed no substantialdegradation of snr by introducing extra hardware for precise 3d ultrasound mo-tor control. An analysis of experimental measurements of motion showed a gradualdrift in motion phase as well as a false wave pattern appearing frommotion compen-sation, and methods for correcting for these errors were presented. Experimentalmeasurements of absolute elasticity values were collected on a CIRS 049 elastog-raphy phantom. Consistent elasticity measurements were produced from multipleacquisitions and different excitation conditions. The elasticity values were in agree-217ment with previous measurements of the phantom [22, 135–137].In Chapter 3 the 3d ultrasound elastography system developed in Chapter 2was used to quantify the elasticity and viscosity of healthy ex vivo placenta tissue.Measurements were collected from six placenta samples using vibrations over aband of 60 Hz to 200 Hz. Mean measurements of Young’s moduli matched previ-ously reportedmeasurements of healthy placenta tissue in the literature [51, 70, 159,185, 229, 301]. This chapter presented the first elastography research to fit Voigt,Maxwell, and Zener rheologicalmodels to fit the placenta dispersionmeasurements.The Voigt model provided the best fit to the measurements, agreeing with conclu-sions from elastographymeasurements of bovinemuscle [54] and porcine liver [64].The strong viscous behaviour measured in this chapter matches observations fromconventional mechanical testing of the placenta [333].In Chapter 4, a novel calibration method, W2D→3D, is described for simultane-ously calibrating the motor parameters of a 3d ultrasound transducer and determin-ing the rigid transform from the ultrasound data to a tracked sensor. Previous 3dultrasound spatial calibration research has assumed that the motor parameters areknown a priori, however it has been suggested that this may contribute to errors inthe calibration result [258], and may be a reason that, prior to the results presentedin Chapter 4, no 3d ultrasound calibration technique has demonstrated calibrationerror under 1 mm. The novel method uses accurate 2d ultrasound calibration at mul-tiple locations along the motor sweep to fit the path of the ultrasound image. Thetechnique is general and can make use of the diverse set of 2d calibration methodsdescribed in the literature. Chapter 4 also presents a novel 3d ultrasound calibrationmethod that uses assumed motor parameters and uses planar features and a closedform solution. Finally, a method for extending the widely used N-wire technique toadd additional data points from the usually ignored edges of the N-shapes. Overall,theW2D→3D method performed best. It improves the accuracy of 3d scan conversionin general, which can help improve the accuracy of any 3d ultrasound elastographytechnique using a swept motor ultrasound transducer, as well as any elastographytechnique that uses a tracked 3d ultrasound transducer, such as the multiple viewmethod described in Chapter 6.In Chapter 5 a novel method for measuring dynamic shear waves with volumet-ric ultrasound using a synchronization scheme was described. The method used the218W2D→3D calibration from Chapter 4 for scan conversion to Cartesian coordinates.This chapter reported the first results of measuring the full 3d displacement vec-tor over a volume using a swept motor 3d ultrasound transducer for shear wavesgenerated using excitation frequencies on the order of hundreds of cycles per sec-ond. A novel method of regularizing the displacement measurements by apply-ing an incompressibility constraint based on tissue motion physics was developedwhich reduced noise in the displacement measurements without distorting the over-all wave pattern. Elasticity measurements on a CIRS 049 elastography phantomwere calculated using two previously developed inversion algorithms (lfe [200],and shear-FEM [137]) and 3d measurements (both regularized and unprocessed)are contrasted with 1d measurements. The elasticity estimated using the shear-FEM algorithm with 3d regularized displacement measurements provided the mostaccurate depiction of the true elasticity contrast and expected elasticity profile.In Chapter 6 a method for measuring the full 3d displacement vector over a vol-ume was investigated using the axial displacement measurement from three approx-imately orthogonal ultrasound views, as a potentially more accurate measurementmethod compared to Chapter 5. A simulation of a practical geometry for a clinicalbreast scanning system was used to evaluate the proposed approach, with calibra-tion errors added to study the effect of misalignment of the views. Based on thelevel of calibration errors found in Chapter 4, accurate estimates of elasticity canbe obtained of a stiff sphere located in a soft background in a simulated phantom.Previous work on using multiple views to improve ultrasound motion measurementdemonstrates promise for applying this approach clinically [33, 118, 211].At this point, it can be stated that the primary objectives of the thesis have beenachieved. In particular,• A modular 3d elastography research platform was developed and used tocompare and optimize parameter selection for different techniques and equip-ment.• The 3d elastography research platform was tested through simulations andexperiments on tissue mimicking phantoms and ex vivo tissue.• Two highly accurate ultrasound calibration techniques were developed that219can be applied to 3d elastography.• Amethod for measuring the full shear wavemotion vector field over a volumewith 3d ultrasound was developed and tested on a tissue mimicking phantom.• A method for combining multiple ultrasound volumes from different viewsto determine the full shear wave motion vector field over a volume was inves-tigated through simulation incorporating realistic modelling of spatial cali-bration.In summary, the hypothesis of this thesis, that a 3d approach to measurementcan be used with ultrasound elastography to provide accurate measurements oftissue elasticity, is confirmed through numerical simulations and experiments ontissue-mimicking phantoms and ex vivo tissue.7.2 ContributionsThe contributions of this thesis are summarized as follows:• A modular 3d ultrasound elastography research platform, swave, enablingeasy implementation of different algorithms and use of different hardware.The systems allows for comparison of different elastography approaches andcan help in the development of new elastography techniques.• Extending swave and implementing a technique for measuring the viscoelas-tic parameters in the first application of swave to placenta tissue. The tech-nique uses longitudinal mechanical vibration and measures the wave patternwithin the placenta with a 3d ultrasound transducer using the band pass mo-tion sampling technique. Elasticity estimates are consistent with previouselastography measurements of placenta reported in the literature.• A novel technique to calibrate a 3d swept motor ultrasound transducer with-out knowledge of the transducer’s motor parameters (calledW2D→3D in Chap-ter 4). The technique fits a rigid transformation path through multiple 2d ul-trasound calibration solutions obtained along the motor path. The techniquecan use the solution from any 2d calibration method, and is implementedwith the closed form multi-wedge method [215].220• A novel technique to calibrate a 3d swept motor ultrasound transducer usingclosed-form solution based on planar image features (called P3D in Chap-ter 4). The solution uses the plane parameters fit to the segmented lines ineach 2d image slice along the motor sweep and calculates the pose of theultrasound volume with respect to the phantom’s coordinate system. Theclosed-form solution, unlike iterative solutions, is fast to compute, not sub-ject to sub-optimal local minima, and is not sensitive to initial estimates forthe calibration parameters.• A novel extension of the N-wire ultrasound calibration technique to solve forthe location of the two edge points of each N-shape instead of ignoring themfor the registration between the ultrasound and phantom points (called NFull3Din Chapter 4). The conventional approach only uses the centre point from thediagonal of the N-shape because its location in phantom coordinates can becomputed using only ratios from the ultrasound image, and in general therecan be ambiguity in determining the edge points in phantom coordinates. Thenew approach uses a specific placement of multiple N-shapes to uniquelyidentify the edge points by constraining all of the points from a given imageto be located on a plane.• A novel technique to measure dynamic shear wave vectors with 3d ultrasoundfor swave. The approach synchronizes the acquisition of each beam line withthe mechanical excitation that produces the shear waves. The trigger from theexciter to the ultrasound scanner is shifted in phase relative to the excitationsignal such that multiple motion states are observed in the ultrasound data.The displacement between synchronous ultrasound volumes is measured us-ing 3d cross-correlationwith polynomial fitting for sub-sample precision, andscan conversion to convert from transducer to Cartesian displacements.• A novel technique to apply regularization to 3d displacement measurementsof soft tissue motion for swave. The technique applies a divergence penaltythat encourages solutions with incompressible tissue motion which is ex-pected in most elastography imaging scenarios. The solution can be com-puted directly using matrix inversion and has demonstrated a qualitative re-221duction in displacement measurement noise for 3d shear wavemeasurements.• A code base to solve for the three components of a motion vector given threeoverlapping volumes of motion measurements and the relative rigid transfor-mations between each volume.7.3 LimitationsThe results reported in this thesis are very promising, however the proposed tech-niques suffer from some limitations which will pose challenges for adoption of thetechniques in clinical practice.The methods in this thesis assume that the transducer is stationary during ac-quisition. For the experiments in this thesis, the transducer was always fixed usinga clamp. In a clinical setting, this would require the radiologist to hold their handsteady for a few seconds while a volume is acquired. For the 3d vector motion mea-surement described in Chapter 5, where the acquisition time is significantly longer,the transducer would likely need to be fixed.Even with a perfectly fixed transducer, the in vivo setting poses new challenges.Patient movement, breathing, and cardiac motion can all cause unwanted motionsin the measured displacement field. In the worst case, large motions can move thetracked scatterers outside of the field of view, making it impossible to continuouslymeasure tissue displacements. For smaller motions, these can disrupt the excitationwave field and destroy the repeatability of the excitation which is assumed by thephase compensation algorithm.Another practical consideration is the application of excitation to the tissue.The design of excitation devices for generating waves in deeply situated organs isa challenging task. Attenuation also constrains the range of excitation frequenciesthat can penetrate the organ of interest. The ideal excitation frequencies are alsorelated to the elasticity of the tissue, as this affects the wavelength and resultingresolution of the elasticity images. The ability to measure waves in the liver, createdby vibrations from a small disk applied to the surface of the body, with 3d ultrasoundhas been demonstrated up to depths of 15 cm for excitation frequencies up to 60 Hz,and to depths of 8 cm for excitation frequencies up to 100 Hz [23].222For elasticity inversion, in this thesis the tissue has been modelled as linear andisotropic, with a constant density equal to the density of water. Aside from thedispersion analysis of the placenta measurements in Chapter 3, this thesis also as-sumes that the tissue is perfectly elastic with no viscous losses. The assumptionof linearity is reasonable because the magnitude of the deformations is on the or-der of microns of displacement. It should be noted that real tissue is expected tobehave in a nonlinear fashion for increasing levels of strain. This could affect therepeatability of elasticity measurements as the level of pre-compression could differbetween measurements. While some tissues, such as liver, are close to isotropic onthe macro scale, other tissues such as muscle can have different elastic propertiesin different directions, and the methods developed in this thesis may not properlycharacterize these tissues. Most soft tissues imaged with ultrasound elastographyhave a high water content and have a density close to 1000 kg/m3, any deviationfrom this density will cause errors in the elasticity measurement.7.4 Future WorkThe swave system could be enhanced by including other inversion algorithms, suchas a GPU accelerated fem inversion. The swave system could also be extended toinclude angular compounding, similar to the 2d system described by Zahiri-Azaret al. [348], which would provide 2d displacement vectors over a volume, allowingfor more sophisticated inversion algorithms to be used. Another extension to theswave system could be to work with parallel received pre-beamformed data, for ex-ample by using the SonixDAQ add-on to the Ultrasonix systems. This would enableshorter acquisition times and could be used to measure 2d displacement vectors us-ing the transverse oscillation technique [186]. The use of matrix array transducerswith the swave system would also reduce the acquisition time by eliminating theneed to move a motor in the transducer, and could improve image quality with thegreater ability to focus acoustic beams. Matrix arrays may also allow for 3d angularcompounding for measurement of the full motion vector.Further ex vivo measurements of placenta tissue, especially from pregnanciesdiagnosed with preeclampsia and iugr would determine if the swave system iscapable of differentiating between normal and abnormal placenta tissue. The vis-223coelastic model fitting could be extended to produce high resolution spatial mapsof viscous properties over a volume. A careful comparison between viscoelasticparameters, b-mode features, and histology would be useful in understanding thebiophysical basis for changes in the viscoelastic properties, and could help explainany observed changes in abnormal placenta tissue. If the swave system proves to bea useful tool for diagnosing placenta abnormalities, an in vivo study of the optimalexcitation design and placement would be valuable for creating a clinically viableproduct.The calibration methods developed in this thesis can be extended to work withcurvilinear transducers. The wedge phantom may need to be modified to optimizethe image quality for the different geometry of a curvilinear transducer. It wouldbe interesting to extend the W2D→3D method to also solve for the radius of curva-ture of the array and the element pitch and study if solving for these parameterscan reduce calibration errors as solving for the 3d motor parameters. A study ofthe calibration error as a function of the number of 2d calibration solutions (eachat a unique motor position) used in the 3d calibration fitting procedure would beof interest for determining the optimal number of calibrations to perform. Inte-grating the calibration algorithms into a powerful open-source toolkit, such as thePLUS toolkit [178], would make it easier for other researchers to use and would bebeneficial by enabling the software to provide real-time feedback on the calibrationquality and full control and synchronization of the spatial tracker and ultrasoundscanner from a single interface.The measurement of the full 3d shear wave motion vector could be tested withdifferent modulus contrast (e.g. with a soft inclusion) to further investigate its per-formance in comparison to 1d tracking and mre measurements with various in-version algorithms. With a goal of ultimately applying the technique in vivo, oneof the faster acquisition schemes described in Chapter 5 should be implemented.To further decrease the acquisition time, the number of phase offsets might be re-duced. A study of the displacement tracking error as a function of the number ofacquired phase offsets would be valuable for determining the minimum number ofoffsets needed for a desired accuracy. Measurement of multi-frequency excitationis a natural extension of the work, which should also help improve the accuracy ofthe elasticity inversion.224A dedicated apparatus for multiple view ultrasound acquisition should be de-veloped for use in ex vivo and in vivo settings. The apparatus could be realized withmultiple 3d transducers fixed using a manufactured holder. A two view implemen-tation providing 2d displacement over a volume could also be studied as it wouldprovide advantages in acquisition rate and cost. The two view implementation coulduse a curl-based inversion such as the c-FEM [136]. 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