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Development of trajectory-based techniques for the stereotactic volumetric modulated arc therapy of cranial… Wilson, Byron 2018

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Development of Trajectory-Based Techniques for theStereotactic Volumetric Modulated Arc Therapy ofCranial LesionsbyByron WilsonBSc Physics, University of Toronto, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)December 2018c© Byron Wilson, 2018 The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:  Development of  Trajectory-Based Techniques for the Stereotactic Volumetric Modulated Arc Therapy of Cranial Lesions  submitted by Byron Wilson  in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics  Examining Committee: Ermias Gete, Senior Medical Physicist, BC Cancer Supervisor  Stefan Reinsberg, Physics  Supervisory Committee Member  Cheryl Duzenli, Physics Supervisory Committee Member Arman Rahmim, Radiology University Examiner Septimiu Salcudean, Biomedical Engineering University Examiner   Additional Supervisory Committee Members: Vesna Sossi, Physics Supervisory Committee Member  Supervisory Committee Member  AbstractIntroduction: Stereotactic Radiosurgery is the delivery of a large, highly focusedradiation dose to well defined targets. This thesis explores linac-based in-verse planning algorithms that can be implemented to improve the dosimet-ric and delivery performance of volumetric modulated arc therapy treatmentsfor these indications.Methods: In this work, algorithms for couch-gantry and collimator-gantry trajec-tory optimization were developed. Treatment plans calculated with thesealgorithms were compared dosimetrically to conventional methods used fortreatment planning. Additionally, the clinical feasibility of the methods de-veloped were tested by performing end-to-end patient-specific quality assur-ance on prospective treatments and by developing machine specific qualityassurance for the intra-treatment movement of the couch and collimator.Results: This thesis introduces a robust method for optimizing the trajectory of thecouch by delivering treatments along patient generalized trajectories. Thesetreatments were able to dosimetrically outperform dynamic conformal arcs,and had higher delivery efficiency than multi-arc volumetric modulated arctherapy. Similarly, collimator trajectory optimization was shown to reducethe dose bath when compared with the clinical standard of care. These meth-ods were shown to be safe for delivery using phantom verification studies.Conclusion: This thesis outlines methods for stereotactic radiosurgery which showdosimetric improvement over previous methodology and are clinically fea-sible.iiiLay SummaryLinear accelerators are used to irradiate cancerous tissue in the brain with the hopeof slowing or removing the disease. During radiation treatments, the radiation hasto travel through healthy tissue before it can effect the tumours which can be deepseated within the brain. This work tries to mitigate the exposure of healthy braintissue to unnecessarily high levels of radiation by exploring two methods. Thefirst method optimizes the angles of entrance with the hope of avoiding sensitivehealthy tissue. The second method explores optimizing the rotation of the radiationfield shaping device so that the radiation beam can conform to the tumour as muchas possible. The methods developed were shown to reduce the amount of radiationto the healthy brain tissue while decreasing the delivery time and maintaining thedelivery accuracy.ivPrefaceA version of Chapter 2 has been published in the journal Medical Physics and isbeing republished with permission from Wiley. The original publication can befound here:• Wilson B, Otto K, Gete E. A simple and robust trajectory-based stereotacticradiosurgery treatment. Medical physics. 2017 Jan 1;44(1):240-8.I am the first and corresponding author of this research. I completed this researchunder the supervision of Ermias Gete, who provided research guidance, assistancein the measurements, and clinical expertise. Karl Otto provided technical supportof the codebase which underlies the Varian Eclipse Treatment planning system.This work was featured in:• Ian Randall, TVMAT enhances stereotactic radiosurgery. Medical PhysicsWeb, 2017 Jan 24; http://medicalphysicsweb.org/cws/article/research/67593Additionally, a version of Chapter 3 has also been published in the journalMedical Physics and is being republished with permission from Wiley. The originalpublication can be found here:• Wilson B, Gete E. Machine specific quality assurance procedure for stereo-tactic treatments with dynamic couch rotations. Medical Physics. 2017 Dec1;44(12):6529-37.I am the first and corresponding author of this research. I completed this researchunder the supervision of Ermias Gete, who provided research guidance, designedthe phantom, assisted in the measurements, and co-developed the mathematicalanalysis.vChapter 4 has not yet been published. The work represented in it was com-pleted by myself under the supervision of Ermias Gete. I conducted the numericalsimulations and developed the mathematical models. Gete provided clinical exper-tise which guided the project and assisted in the linac measurements.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Radiotherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Radiotherapy Physics . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Interaction of Photons with Matter . . . . . . . . . . . . . 31.2.2 Linear Accelerators . . . . . . . . . . . . . . . . . . . . . 61.2.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . 111.3 Treatment Planning . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.1 organ at risk (OAR) . . . . . . . . . . . . . . . . . . . . . 151.3.2 Quantitative Analyses of Normal Tissue Effects in the Clinic(QUANTEC) . . . . . . . . . . . . . . . . . . . . . . . . 16vii1.3.3 Treatment Plan Evaluation . . . . . . . . . . . . . . . . . 161.4 Stereotactic Radiosurgery . . . . . . . . . . . . . . . . . . . . . . 191.4.1 Small-Field Dosimetry . . . . . . . . . . . . . . . . . . . 191.4.2 Patient Immobilization . . . . . . . . . . . . . . . . . . . 211.5 Quality Management in Radiation Therapy . . . . . . . . . . . . 231.5.1 The Picket Fence Test . . . . . . . . . . . . . . . . . . . 241.5.2 Coincidence of the Treatment Isocentre . . . . . . . . . . 261.5.3 Patient Specific Quality Assurance . . . . . . . . . . . . . 271.6 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.6.1 Optimization Methodology . . . . . . . . . . . . . . . . . 291.6.2 Cost Functions used in Radiotherapy Planning . . . . . . 351.6.3 Volumetric Modulated Arc Therapy . . . . . . . . . . . . 371.6.4 Treatment Planning for stereotactic radiosurgery (SRS) . . 401.6.5 Couch-Gantry Trajectory-Based Deliveries . . . . . . . . 421.6.6 Collimator-Gantry Based Trajectories . . . . . . . . . . . 431.7 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 452 Couch-Gantry Trajectory-based Stereotactic Radiosurgery Treatments 472.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.2.1 Optimization . . . . . . . . . . . . . . . . . . . . . . . . 492.2.2 Plan Comparisons . . . . . . . . . . . . . . . . . . . . . 532.2.3 Validation of Deliveries . . . . . . . . . . . . . . . . . . 552.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.3.1 Treatment Comparison . . . . . . . . . . . . . . . . . . . 572.3.2 Analysis of Trajectories . . . . . . . . . . . . . . . . . . 602.3.3 Treatment Time Comparison . . . . . . . . . . . . . . . . 632.3.4 Validation of Deliveries . . . . . . . . . . . . . . . . . . 652.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 Machine-Specific Quality Assurance Procedure for Stereotactic Treat-ments with Dynamic Couch Rotations . . . . . . . . . . . . . . . . . 70viii3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.2.1 Set-up and Measurement . . . . . . . . . . . . . . . . . . 723.2.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . 743.2.3 Accuracy of the Procedures . . . . . . . . . . . . . . . . 783.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.3.1 Determination of the couch rotation centre . . . . . . . . 793.3.2 Comparison with the Star-shot Method . . . . . . . . . . 793.3.3 Winston-Lutz Method . . . . . . . . . . . . . . . . . . . 823.3.4 Validation of the Trajectory Logs . . . . . . . . . . . . . 823.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864 Collimator Optimization for VMAT Treatments of Multiple BrainMetastases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.2.1 Patient Selection . . . . . . . . . . . . . . . . . . . . . . 904.2.2 Collimator Angle Optimization Method . . . . . . . . . . 914.2.3 Treatment Plan Cost Function . . . . . . . . . . . . . . . 924.2.4 Direct Aperture VMAT Optimization . . . . . . . . . . . 924.2.5 The Blocking of Fluences Incident on Normal Tissue . . . 934.2.6 Treatment Plan Comparison . . . . . . . . . . . . . . . . 944.2.7 Quality Assurance of Dynamic Collimator Delivery . . . . 944.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.3.1 Collimator Optimization . . . . . . . . . . . . . . . . . . 964.3.2 Treatment Plan Comparison . . . . . . . . . . . . . . . . 974.3.3 Quality Assurance of Dynamic Collimator Delivery . . . . 1024.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.1 The Effect of Couch Rotations in SRS Deliveries . . . . . . . . . 107ix5.2 The Effect of Collimator-Gantry Trajectory Optimization . . . . . 1095.3 Delivery Capabilities of Modern Linacs . . . . . . . . . . . . . . 1105.4 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . 111Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113xList of TablesTable 1.1 QUANTEC dose constraints for single fraction cranial SRS.V12 is the volume that recieves 12Gy and Dmax is the max doseto an organ. Data collected from [9] . . . . . . . . . . . . . . . 16Table 1.2 Summary of the mechanical specifications required for intensitymodulated radiation therapy (IMRT) and volumetric modulatedarc therapy (VMAT) deliveries with a linac. Data was collectedfrom recommendations made by the AAPM task group 142 [48] 25Table 2.1 Patient summary couch trajectory optimization study . . . . . . 54Table 2.2 Maximum velocity model used to estimate delivery time. . . . 55Table 2.3 Beam-on time for competing optimization strategies. . . . . . . 65Table 2.4 Plan Quality Assurance Metrics . . . . . . . . . . . . . . . . . 67Table 2.5 planning target volume (PTV) and normal tissue statistics . . . 69Table 3.1 Glossary of mathematical notation. . . . . . . . . . . . . . . . 74Table 3.2 Accuracy of the developed methods. . . . . . . . . . . . . . . 78Table 3.3 Couch rotation centre offsets with respect to the nominal linacisocentre in the cross-plane, in-plane, and out-of-plane direc-tions (mean ± 2 standard deviations). . . . . . . . . . . . . . . 79Table 3.4 Comparison of film-based with electronic portal imaging device(EPID)-based star-shot measurements. . . . . . . . . . . . . . 81Table 4.1 Summary of Patient Statistics . . . . . . . . . . . . . . . . . . 90xiTable 4.2 Average area of open fluence when optimized with three tech-niques: optimized moving trajectory, optimized static angle,and treatment planner-selected angle. . . . . . . . . . . . . . . 97Table 4.3 Comparison of treatment of dosimetric parameters for four op-timization strategies: algorithm optimized static collimator, al-gorithm optimized trajectory, planner-selected static collimator(PS-Static) and normal tissue blocked. . . . . . . . . . . . . . 98xiiList of FiguresFigure 1.1 A plot of photon attenuation coeficients for water and bone. a)The attenuation coeficients of photons in water (shown here)has very similiar properties to human tissue. At low energies(< 100 keV), the photoelectric effect is dominant, at interme-diate energies (1-10 MeV), Compton scattering is dominant,and at high energies (> 10 MeV), pair production is dominant.b) The attenuation coeficients of photons in bone. At low ener-gies the photoelectric effect is much stronger due to the highereffective atomic number of bone. This phenomenon is respon-sible for the contrast of x-ray scans. At higher energies, Comp-ton and pair production dominate, which depend less on the Zof the material. Therefore dose calculations at these energiesunder the assumption of water equivalence are more accurate.Data for figures (a) and (b) were collected from The NationalInstitute of Standards and Technology database [10, 70] . . . 4Figure 1.2 A linear accelerator (linac) with the gantry in the vertical po-sition and treatment couch (the black carbon-fibre board) isrotated from its home position which is in-line with the gantry.Photons are produced in the gantry and are used to irradiatethe patient who usually lies on the couch. The gantry can berotated± 180 degrees while the couch can be rotated± 90 de-grees provided that collisions are avoided. Image courtesy ofVarian Medical Systems, Inc. All rights reserved. . . . . . . 7xiiiFigure 1.3 a) Cross sectional schematic diagram of a typical linac head.The secondary collimator and multi-leaf collimator (MLC) il-lustrated with a dashed box, is attached to a sliding ring whichallows it to be rotated with respect to the central axis of thebeam. Diagram not to scale. b)External view of the multi-leaf collimator (MLC). The MLC can form customized beamapertures by retracting or extending the individual leaf com-ponents. Image courtesy of Varian Medical Systems, Inc. Allrights reserved. . . . . . . . . . . . . . . . . . . . . . . . . . 9Figure 1.4 The percent depth dose (PDD) curve of a 6MV, 10x10 cm2photon beam measured in a water phantom placed 100 cmfrom the source. The PDD is normalized to 100 at the depth ofmax dose dmax along the central axis. . . . . . . . . . . . . . 10Figure 1.5 Beam profiles for a 6MV 10x10 cm2 photon beam. The blue,red and orange lines correspond to the cross axis beam profilesat depths (d) 1.5, 5 and 10 cm respectively. The dotted linerepresents the geometric field edge defined at the isocentre bythe secondary collimator. . . . . . . . . . . . . . . . . . . . 12Figure 1.6 Level set diagram of the dose deposition kernel of a 6MV pho-ton beam. Doses are normalized to the max dose deliveredby the kernel. A pencil beam of photons (shown as a black ar-row) incident on the water phantom (shown by blue shading) at(0,0). The photon interactions result in high energy electronswhich subsequently deposit their energy in both the forwardand lateral directions. . . . . . . . . . . . . . . . . . . . . . 13Figure 1.7 A typical dose volume histogram for an OAR and PTV is shownin red and blue respectively. Using this illustration, dose statis-tics can be read off the figure axes. A few examples such asV OAR5Gy , the volume of the OAR that receives 5 Gy (≈ 30 %),DPTVmin , the minimum dose of the PTV (≈ 21 Gy), and DOARmax ,the maximum dose of the OAR (≈ 12 Gy) are shown on theplot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18xivFigure 1.8 a) Rigid frame used for SRS deliveries. The frame is affixeddirectly to the skull. b) A positioning box is attached to theframe during imaging and patient set up. It provides fiducialmarkers and visual alignment markers to ease process of align-ing CT information with the delivery isocentre. c-d) Brainlabframe-less mask used in modern day treatments. . . . . . . . 22Figure 1.9 For functions that are convex, there is one minimum which isthe global minimum. The gradient points in the direction offunctional increase, so by searching in the opposite direction,the function minimum can be found. . . . . . . . . . . . . . 29Figure 1.10 This figure shows the level set diagram of a function with in-puts. In this example, the global minimum is outside of thecontained set w ≥ 0. The gradient is calculated (in red), how-ever the solution lies outside of the solution set. This is cor-rected by projecting the solution onto the closest point withinthe solution set. The minimum value that achieves the con-straints lies on the boundary of the constraints (in this exam-ple, the y axis) and the gradient moves along the y axis untilthe minimum is achieved. For these optimizations, the vari-able α has increased importance as the projected gradient canbe much smaller than an optimal step. . . . . . . . . . . . . . 31Figure 1.11 Illustration of dose contribution of fluence subsets of a linacbeam. On the left, the linac beam is subdivided into fluencecontributors. Photons travel through subsection j of the beamand deliver a dose of Di j to voxel i (in blue). Photons fromsubsection j′ can also deliver dose to voxel i, but in a differentamount denoted by Di j′ . Similarly, photons from subsection jalso deliver dose to other voxels such as i′. These dose contri-bution factors build a matrix D which has size bs by v, where bsis the number subsets in the beam times the number of beams,and v is the number of voxels . . . . . . . . . . . . . . . . . 34xvFigure 1.12 Illustration of the progressive sampling algorithm. a) the initialtrajectory is sparsely and discretely sampled as a collection ofcontrol points which are evenly spaced along the gantry trajec-tory. The MLC apertures and monitor unit (MU) of the beamare optimized using stochastic optimization. b) New controlpoints are added in-between two adjacent control points andinitial MLC positions are set to be the linear interpolation ofthe adjacent control points. d) This procedure is continued un-til there are enough control points along the trajectory to ap-proximate a continuous delivery. This figure was reproducedfrom [71] with permission from Wiley Publishing Group. . . . 39Figure 1.13 A beams-eye-view of a typical conformal aperture. The bluelines signify the edges of the MLC leaves, the red blobs arePTVs projected along the axis of the beam, the yellow desig-nates open fluence. The MLC positions are found by takingthe minimum and maximum extent of the PTV contours. . . . 40Figure 2.1 An overview of the Optimization Process. a)A predefined tra-jectory which fully sampled 4pi geometry is fed into the opti-mizer. b) In-house optimization algorithms find the most op-timal MLC and dose rate combinations for a given patient ge-ometry and cost function. c)The sampling of the phase spaceis reduced as much as possible without reducing plan qualitymetrics or cost. d) New MLC sequences are selected using theoptimization algorithms in b. . . . . . . . . . . . . . . . . . 50Figure 2.2 Boxplots of the variables of interest (conformity index, V12and V4) which were statistically significantly different betweenTVMAT and DCA plans. Boxes show mean, quartiles, max-ima, minima and outliers (shown as dots). Variables are nor-malized to mean values of the pooled data sets. TVMAT plansare shown in blue, DCA in orange and VMAT in green. . . . . 58xviFigure 2.3 Dose distribution for patient 3 (right accoustic neuroma (AN))for trajectory volumetric modulated arc therapy (TVMAT) (left)and dynamic conformal arc (DCA) (right). The PTV contour(red) and Brainstem (green) are shown. In addition, dose dis-tributions are shown by yellow (100%), blue (80%), and or-ange (50%) a,b. Transverse slices show the dose distributionslook similar, but with the TVMAT plan able to slightly curtailthe dose away from the brainsteam. c.d. Sagital slices showboth plans were of similar quality. e.f. Frontal slices: bothplans have similar falloff, however the DCA plan comes froma smaller subset of angles so one can see the artefacts of morejagged falloff lines. . . . . . . . . . . . . . . . . . . . . . . 59Figure 2.4 Dose distribution for patient 1 (right AN) for TVMAT (left)and DCA (right). The PTV contour is shown in red. In ad-dition, dose distributions are shown by yellow (100%), blue(80%), and orange (50%) a,b. Transverse slices c.d. Sagitalslices e.f. Frontal slices . . . . . . . . . . . . . . . . . . . . 60Figure 2.5 Dose distribution for patient 4 (right AN) for TVMAT (left)and DCA (right). The PTV contour is shown in red. In ad-dition, dose distributions are shown by yellow (100%), blue(80%), and orange (50%) a,b. Transverse slices c.d. Sagitalslices e.f. Frontal slices . . . . . . . . . . . . . . . . . . . . 61Figure 2.6 Dose distribution for patient 6, met 2 (Multiple Met) for TVMAT(left) and DCA (right). The PTV contour is shown in red. Inaddition, dose distributions are shown by yellow (100%), blue(80%), and orange (50%) a,b. Transverse slices c.d. Sagitalslices e.f. Frontal slices . . . . . . . . . . . . . . . . . . . . 62Figure 2.7 Dose distribution for patient 9, met 1 (Multiple Met) for TVMAT(left) and DCA (right). The PTV contour is shown in red. Inaddition, dose distributions are shown by yellow (100%), blue(80%), and orange (50%) a,b. Transverse slices c.d. Sagitalslices e.f. Frontal slices . . . . . . . . . . . . . . . . . . . . 63xviiFigure 2.8 Analysis of the importance of trajectory on plan quality. Vari-ables are normalized to the values achieved for the eight arcplans. a) The cost function depence on number of partial arcsfor acoustic neuroma plans (patient 1, 2, 3 with data pointsshown in red, green, yellow respectively). The blue line cor-responds to the trend in the TVMAT data and error bars cor-respond to standard deviations from three rounds of TVMAToptimization. b) Optimization of clinical variables of interestfor the AN patients. c) Cost function for 2 Met patients. Itshowed a similiar pattern as the AN patients, however found aminimum at four partial arcs instead of three. d) Optimizationof clinical variables for 2 Met Patients. . . . . . . . . . . . . 64Figure 2.9 a. Dose distribution comparison in coronal plane (film (dot-ted line) vs treatment plan (solid line)) for patient 2. Plan wasscaled to 1/5 of the actual value to have doses in the most ac-curate range for film measurements. b. Vertical profile com-parison for the same treatment. . . . . . . . . . . . . . . . . 66Figure 3.1 Aerial view of the BB phantom mounted to the treatment couchusing the Brainlab couch mount. The phantom consists of fiveBBs (BB1−BB5) affixed to a polystyrene slab. Each BB is lo-cated at different radial distance from the isocentre, with thecentral BB located at the isocentre. The linac is oriented verti-cally with the EPID deployed. . . . . . . . . . . . . . . . . . 73xviiiFigure 3.2 Illustration of the mathematical analysis for BBs 1 and 2. Thenominal linac isocentre is shown as a cross (labelled BB1(θ=0))and the initial BB2 location is shown as a black circle (x2,y2).After rotation of the couch by a given angle, the movement ofBB2 (grey circle) can be represented as a rotation by angle θabout the centre of rotation (cross labelled by ~R0). Addition-ally, a radial scaling S about the centre of rotation accountsfor any out of plane movements. ~dRxy represents the differ-ence between the nominal isocentre and the centre of rotation.Illustration not to scale. . . . . . . . . . . . . . . . . . . . . 76Figure 3.3 Deviation of couch centre of rotation as a function of anglecalculated using the analysis in Equation 3.5. Dataset 1, 2and 3 represent three independent measurements acquired onemonth apart. The solid line represents the data plotted for eachcouch angle (sampled every 10 degrees) while the dotted linerepresents the mean of each dataset, averaged over all of themeasured couch angles. . . . . . . . . . . . . . . . . . . . . 80Figure 3.4 a. The lines represent the calculated EPID based star-shot linessampled every 30 degrees of couch rotation. The circle whichbounds the intersection of these lines is shown. b. Magni-fied version of figure a. The origin (0, 0 mm) represents thenominal linac isocentre. The lines represent calculated star-shot lines, and the intersection points of these lines are shownas circles. The bounding circle is the smallest circle which en-capsulates all of the intersection points. The two crosses repre-sent the centre of the circle and the centroid of the intersectionpoints. c. Raw GafChromic film data collected for traditionalstar-shot analysis. d. Fitted star-shot lines and bounding circlefor the film data. . . . . . . . . . . . . . . . . . . . . . . . . 81xixFigure 3.5 Cross-plane and in-plane deviations obtained from the multi-ple BB method and the Winston Lutz method measured in asingle phantom setup. The dashed lines represent the couchrotation centre deviations (dRx, dRy) obtained from the multi-ple BB measurement, and the solid lines represent the WinstonLutz deviations. Measurements were acquired for two gantrypositions, 0 degrees (above phantom), and 180 degrees (belowphantom). . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 3.6 (a) A plot of the differences between the couch angles recordedby different methodology as the couch rotated through -80→-90 → +90 degrees at its maximum velocity over the courseof 64 seconds (y axis) . The trajectory log couch angle values(θT ) agreed with those recorded in the DICOM header files(θH) to the third significant digit. Additionally, the couch val-ues recorded in the trajectory log and DICOM header (θT andθH respectively) agreed with the calculated values within 0.08degrees. These errors were normally distributed with mean0 and standard deviation 0.025 degrees. Crosses show eightdata points which were excluded from image analysis due tointra-imaging changes in beam intensity. (b) The couch angleposition as a function of delivery time. . . . . . . . . . . . . . 84Figure 4.1 An example of MLC contention issues that may arise in thetreatment of multiple PTVs with a single aperture. PTVs areshown in red, normal tissue is shown in yellow, the MLC isshown in blue and the field jaws are shown in black. (a) Col-limator is rotated to 45 degrees and the MLC aperture is setto conform to the targets. A sizeable amount of normal tissueis being irradiated. (b) The optimal collimator angle occurs at-12 degrees. When the aperture is set to conform to the targets,the normal tissue is efficiently blocked by the MLC. . . . . . 89xxFigure 4.2 MLC aperture used in machine-specific quality assurance ofcollimator rotation. The MLC forms 5 square openings, threeof 1 x 1 cm2 and two of which are 1 x 0.5 cm2. The centralsquare provides the location of the isocentre, while the fartherspaced openings provide an accurate rotational measurement. 95Figure 4.3 A typical area of open fluence level set graph derived frompatient 3. The red line shows the global optimal path lengththrough the graph, while the blue line represents the static an-gle which minimizes the graph (at -15 degrees). . . . . . . . 96Figure 4.4 A DVH comparison for patient 3, which shows the differencein brain dose (shown in green) between the treatment planner-selected collimator angles (plan shown with solid line) andthe trajectory optimized collimator angles (shown as a dashedline). The GTV and PTV doses shown are the summed dosefor all four targets. . . . . . . . . . . . . . . . . . . . . . . . 97Figure 4.5 An example dose distribution comparison between the treat-ment planner-selected plan and the collimator trajectory opti-mized treatment plan. The prescription dose (3150 cGy), 50%(1575 cGy), 1200 cGy and 500 cGy contours are shown in yel-low, orange, white, and blue respectively. a) The clinically de-livered plan has significant dose spillage of the 5 Gy contourinto the brain. b) Using the algorithms presented (both col-limator angle optimization and MLC sequencing algorithms),the low-dose bath was significantly reduced. . . . . . . . . . . 99xxiFigure 4.6 (a) The mean percent reduction in volume recieving x dose(Vx) from the implementation of collimator optimization meth-ods when compared to the treatment planner-selected collima-tor angle. These results are presented after simulated anneal-ing optimization. The optimized collimator angles (red) andcollimator trajectory (blue) are compared with the treatmentplanner-selected collimator angle. Values are averaged for allpatients considered in this study. (b) At the end of simulatedannealing optimization, beam-weights were optimized usinggradient descent. Treatment plans after gradient descent areshown in this graph, and one can see that the difference be-tween the two methodologies disappears while the effect ofcollimator angle optimization still exists. The green line showsthe affect of blocking fluences which do not overlap with PTVtissue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Figure 4.7 Improvement on the optimized cost for patients when com-pared with the clinically selected collimator angles. As can beseen, both strategies, the static algorithm optimized collimatorangle and the optimized collimator trajectory, produced planswith lower cost function evaluations (i.e. higher optimizationcost percent improvement). . . . . . . . . . . . . . . . . . . 102Figure 4.8 DVH comparison between the trajectory optimized collimatorangle (shown as dashed lines) and the normal tissue blockedbeams (shown as solid lines) for patient 3. The brain dose washigher in the trajectory optimized collimator plans. . . . . . . 103Figure 4.9 Comparison of the conformity index (CI) % change producedby the optimization strategies when compared with the CI ofthe treatment planner-selected collimator angle. . . . . . . . 104xxiiFigure 4.10 a)As the collimator rotated for its whole extent [-175 degrees,175 degrees], the angle was measured using the EPID im-ager. Collimator angle was measured from three sources: thedigital imaging and communications in medicine (DICOM)image header (θheader), the trajectory log (θlog), and by analysingthe EPID images (θcalc). b) The first 5 seconds of the QAtest (to the left of the dotted line) involved a clockwise andthen counterclockwise movement of the collimator to providea unique movement to align the trajectory log and EPID im-ages in the time axis. The remainder 60 seconds of the QA testis where the measurements were acquired. . . . . . . . . . . . 105xxiiiGlossaryAAA anisotropic analytical algorithmAAPM American Association of Physicists in MedicineAN accoustic neuromaASTRO American Society for Therapeutic Radiation OncologyBB ball bearingCI conformity indexCP control pointCT computed tomographyCTV clinical target volumeDAO direct aperture optimizationDCA dynamic conformal arcDICOM digital imaging and communications in medicineDR dose rateDVH dose volume histogramEBRT external beam radiation therapyEPID electronic portal imaging devicexxivFMEA failure mode and effects analysisFFF flattening filter freeFO fall offGTV gross tumour volumeHI homogeneity indexICRU International Commission on Radiation Units and MeasurementIMAT intensity modulated arc therapyIMRT intensity modulated radiation therapylinac linear acceleratorMC Monte CarloMLC multi-leaf collimatorMRI magnetic resonance imagingMU monitor unitOAR organ at riskOD optical densityPBC pencil beam convolutionPCA principal component analysisPDD percent depth dosePET positron emission tomographyPTV planning target volumeQA quality assuranceQC quality controlxxvQUANTEC Quantitative Analyses of Normal Tissue Effects in the ClinicRTOG Radiation Therapy Oncology GroupSBRT stereotactic body radiotherapySPECT single photon emission computed tomographySRS stereotactic radiosurgerySSD source skin distanceTVMAT trajectory volumetric modulated arc therapy : VMAT which incorporatestrajectories formed by dynamic movements of the couchVMAT volumetric modulated arc therapyWBRT whole brain radiotherapyxxviAcknowledgementsI would like to first and foremost thank my supervisor, Ermias Gete, for his contin-uous support throughout this project. For every stage of this degree, he went aboveand beyond the expectations of a supervisor and in doing so, has become a valuedfriend and collaborator. In this last half-decade, he has helped me work throughmy weaknesses and taught me many things about medicine and radiation therapy.I hope to one day be as great a teacher and role model as he has been to me.I would like to thank my family for their love and for talking me through manyexistential crisis’s that I encountered while I completed this degree. I would alsolike to thank the friends that I have made in BC and for the many sketch-ballmountain adventures we have shared.I would also like to thank Cheryl Duzenli for cajoling money from the BCCancer agency for this project. Her relentless support of medical physics researchand for the training of the next generation of medical physicists is one of the manythings which make the Vancouver medical physics program world class. She hassaved many lives, not only through her own actions, but by enabling and inspiringthose around her.xxviiChapter 1IntroductionBrain metastases (a secondary malignant growth) affect up to one-third of patientswith cancer [77]. A viable treatment strategy for brain metastasis is stereotacticradiosurgery (SRS) [4] which is the delivery of high intensity, focussed radiationto targets within the brain. However, in radiotherapy, the radiation needs to travelthrough healthy tissue before it can deposit energy in the cancerous tissue. Thiscreates a challenging optimization problem: creating treatments which minimizethe radiation exposure of normal tissue while delivering a sufficient amount ofradiation to control the disease. This thesis focuses on improving SRS treatmentsof brain metastasis.As linear accelerators (linacs) are the most accessible SRS delivery devicesused worldwide, this thesis focuses on linac-based SRS treatments. The introduc-tion will give the reader an overview of photon-based radiotherapy, the physicswhich underpin dose deposition and measurement, biological considerations whentreatment planning, and the optimization methods that are used to create linac-based SRS treatments. Section 1.1 to Section 1.4 (inclusive) provide the readerwith background knowledge of medical physics and can likely be skipped by read-ers with a background in radiation therapy. Section 1.6 provides the reader with aquick introduction to optimization in radiotherapy, and puts the body of this workinto context with previous SRS optimization research. This thesis presents opti-mization techniques that increase the dosimetric quality and delivery efficiency oflinac based SRS treatments. An overview of the work presented in this thesis is1given at the end of this introduction (Section 1.7), after the relevant backgroundconcepts have been introduced.1.1 RadiotherapyRadiotherapy is the delivery of ionizing radiation to tissue with the goal of killingthe diseased subunits of the tissue. The cells within the tissue become diseased byaccumulating mutations in their deoxyribonucleic acid (DNA) code which affecttheir biological function (see [34, 35] for a comprehensive review on cancer). Ra-diation kills cells by causing irreparable DNA damage [66]. The main predictoron the amount of DNA damage to tissue recieving ionizing radiation is the meanenergy absorbed by the medium per unit mass. This quantity is measured in joulesper kg (unit Gray), which for its significance in treatment outcomes in radiotherapy,is also referred to simply as ”dose”.In photon-based external beam radiation therapy (EBRT), radiation dose is de-livered by directing a beam of high energy photons at the treatment site. Thesephotons have typical energies of 4 MeV to 18 MeV [3]. They are delivered witha linac which produces a focused beam of high energy photons which are directedat the target from multiple directions. The control of a medical linac to producethe desired dose distribution in a patient is a complex problem: there are multi-ple entrance angles, intensities, and beam apertures which affect the optimizationof the delivery sequence. Furthermore, there are many considerations which needto be accounted for, such as machine performance, physical dose deposition andradio-biological effects.1.2 Radiotherapy PhysicsThe photons used in radiotherapy do not deliver dose directly, but instead imparttheir energy to electrons which subsequently deposit their energy in the tissue,causing DNA damage. There are four modes of photon interactions in the energyrange used in radiotherapy. These are: coherent scattering, Compton scattering[20], photoelectric effect, and pair/triplet production.21.2.1 Interaction of Photons with MatterAs a photon of a particular energy moves through matter, there is a constant prob-ability of the photon interacting for a given path length. If there are N photonstravelling in a medium, then the number of particles that are removed from theprimary beam as they travel through the medium is given by:∆N =−µN∆x (1.1)where µ is the linear attenuation coefficient, N is the number of photons in thebeam, ∆N is the change in the number of photons in the beam, and ∆x is the pathlength. Letting ∆x tend to the infinitesimal path length dx and solving this equationby the method of separation of variables yields:N = N0e−µx (1.2)where N is the number of photons that has passed through an absorber of thicknessx without interacting. This equation models the intensity of the primary beam as ittravels through a medium. As can be seen, it has the form of an exponential decay.The linear attenuation coefficient µ is dependent on the photon energy andthe medium type and density. It can be calculated by summing over the linearattenuation coefficients for each individual interaction type:µ = σcoh+σinc+ τ+κ (1.3)where the respective attenuation coefficients from different types of interactionsare σcoh (coherent scattering), σinc (compton scattering), τ (photoelectric effect),and κ (pair or triplet production).The mass attenuation coefficient ( µρ ) is defined as the ratio of the linear attenua-tion coefficient to the mass density of the medium. It is more commonly presentedin data tables and plots as it removes the density dependence from the tabulateddata. Mass attenuation coefficients for two of the most common biological materi-als (water and bone) are shown in Figure 1.1.3Figure 1.1: A plot of photon attenuation coeficients for water and bone. a)The attenuation coeficients of photons in water (shown here) has verysimiliar properties to human tissue. At low energies (< 100 keV), thephotoelectric effect is dominant, at intermediate energies (1-10 MeV),Compton scattering is dominant, and at high energies (> 10 MeV), pairproduction is dominant. b) The attenuation coeficients of photons inbone. At low energies the photoelectric effect is much stronger due tothe higher effective atomic number of bone. This phenomenon is re-sponsible for the contrast of x-ray scans. At higher energies, Comptonand pair production dominate, which depend less on the Z of the mate-rial. Therefore dose calculations at these energies under the assumptionof water equivalence are more accurate. Data for figures (a) and (b)were collected from The National Institute of Standards and Technol-ogy database [10, 70]4Coherent ScatteringCoherent scattering is the elastic scattering of photons with molecules (Rayleigh)or the elastic scattering of photons with electrons (Thomson). Neither of theseprocesses impart any energy to the medium, but instead change the direction of thephoton. As can be seen in Figure 1.1, coherent scattering is not a dominant mode ofinteraction at the energies and in the materials of interest in radiotherapy physics.Incoherent ScatteringIncoherent scattering, or Compton scattering, is when a photon inelastically col-lides with a valence electron. The photon imparts some of its energy to the electronand is scattered at an angle. The energy of the scattered photon can be calculatedusing conservation of energy and momentum and depends on the scattering angle:E ′γ =Eγ1+(Eγ/mec2)(1− cosθ) (1.4)where Eγ is the energy of the incoming photon, E ′γ is the energy of the photon afterthe interaction, me is the mass of an electron, c is the speed of light constant, andθ is the scattering angle of the photon. The rest of the energy is imparted to theelectron as kinetic energy:KE ′e = Eγ −E ′γ −Ebinding (1.5)where KE ′e is the kinetic energy of the scattered electron, and Ebinding is the bindingenergy of the electron to the atom. The probability of interaction for a given solidangle is given by the Klein-Nishina [49] cross section which has the form:dσdΩ= α2r2cλ 2λ ′2[λλ ′+λ ′λ− sin2(θ)]/2 (1.6)where α is the fine structure constant, rc is the reduced Compton wavelength of theelectron, λ is the wavelength of the incoming photon and λ ′ is the wavelength ofthe photon after the interaction. This equation dictates that when the incident pho-tons have high energy, most of the outgoing photons and electrons will be forwarddirected.5Photoelectric EffectThe Photoelectric effect is the interaction of a photon with an atom which resultsin the ejection of a bound electron from the atom. This electron absorbs the entireenergy of the incoming photon:KE ′e = Eγ −Ebinding (1.7)where Ebinding is the energy required to eject the electron from the nucleus, andKE ′e is the kinetic energy of the scattered electron after the interaction. The photo-electric cross section depends on the energy of the incident photon and the atomicnumber of the material. It peaks at energies which are in resonance with the bindingenergies of the electrons and has a general trend which decreases as E−3γ . Addi-tionally it is highly dependent on the atomic number (Z) of the materials with ageneral trend of Z3.8.Pair and Triplet ProductionPair production is a process in which the photon interacts with an atom creating anelectron-positron pair. The photon is completely absorbed and the kinetic energyis shared between the outgoing electron and positron:Eγ −2mec2 = KE++KE− (1.8)where mec2 is the rest energy of an electron and KE+,KE− are the kinetic ener-gies of the positron and electron respectively. For a photon to undergo interactionthrough pair production, it must have a minimum energy of 2mec2, the energy re-quired to produce the electron positron pair. Beyond the threshold energy, the crosssection increases rapidly and it is the dominant cross section for high energies. Thecross section also depends linearly on the atomic number of the medium of inter-action.1.2.2 Linear AcceleratorsLinac’s are the most common treatment device for EBRT worldwide [91]. Themost common medical linac is the C-arm linac (an example of which is shown in6Figure 1.2: A linac with the gantry in the vertical position and treatmentcouch (the black carbon-fibre board) is rotated from its home positionwhich is in-line with the gantry. Photons are produced in the gantryand are used to irradiate the patient who usually lies on the couch. Thegantry can be rotated± 180 degrees while the couch can be rotated± 90degrees provided that collisions are avoided. Image courtesy of VarianMedical Systems, Inc. All rights reserved.Figure 1.2). linacs generate a pencil beam of high energy electrons by acceleratingthe electrons in a microwave cavity. Depending on the length of the microwavecavity, the microwave cavity can be placed in-line or at a (typically) 90 degree an-gle with the final beam. If mechanical considerations do not allow the microwavecavity to be in-line with the beam, then the electrons are steered using a bendingmagnet. The electrons are then directed to either a scattering foil to create a broadelectron beam or to a tungsten target to create photons via bremsstrahlung. Modernmedical linacs typically can produce electrons in a range 4 MeV to 20 MeV. How-ever, the majority of EBRT treatments are based on photon beams. Photons areproduced by placing a target in front of the electron beam line. Inside the target,photons are produced from bremsstrahlung radiation, the production of photonsfrom the slowing down of the electrons as they travel through a medium. The spec-tral distribution of photons produced by monoenergetic electrons hitting a target is7given by Kramer’s Law:I(λ ) = K(λλmin−1) 1λ 2(1.9)where I is the irradiance at a wavelength λ , K is a constant, and λmin is the min-imum wavelength possible when all of the kinetic energy of the electron is trans-formed into a single photon. The actual spectrum that is produced by this inter-action deviates from the above formula, as low energy photons are preferentiallyabsorbed by the target and characteristic radiation is preferentially produced by thetarget.Linac Beam CollimationA schematic diagram of a typical linac head is shown in Figure 1.3. High energyelectrons from the microwave cavity are allowed to strike a tungsten target to pro-duce photons. These photons are collimated by the primary collimator so that onlythe photons emitted in the desired direction leave the linac assembly. The photonsare then attenuated by the flattening filter, whose function is to optimize the flu-ence of the beam so that it produces a boxcar function shaped lateral dose profileat 10 cm depth. In modern treatments where a variable aperture collimator can beused to modulate the intensity of the beam, the flattening filter may be removed forachieving higher dose rates.Beam fluences are monitored by the dual-ionization chamber which is locatedbelow the flattening filter. The beam then enters the secondary-tertiary collimatorassembly which can be rotated around the central beam axis to provide an extradegree of freedom. The secondary collimator is comprised of the x-jaw and y-jawswhich define the rectangular edge of the photon beam. The x-jaws and y-jaws arematched with the divergence of the beam and are sufficiently thick so that the beamis practically confined to the opening of the jaws (attenuation of 99.98% for modernVarian linacs). Next, the photons enter the MLC (a form of tertiary collimator)which is placed perpendicular to the axis of the beam and is comprised of 5 -7.5 cm thick tungsten blocks which attenuates approximately 98-99% of the beam(depending on the MLC model)[19]. The MLC leaves are typically between 2.5mm wide to 10 mm wide when their shape is projected to the isocentre correctingfor divergence of the beam. The Varian HD120 MLC used in this dissertation has8Figure 1.3: a) Cross sectional schematic diagram of a typical linac head. Thesecondary collimator and MLC illustrated with a dashed box, is attachedto a sliding ring which allows it to be rotated with respect to the centralaxis of the beam. Diagram not to scale. b)External view of the multi-leafcollimator (MLC). The MLC can form customized beam apertures byretracting or extending the individual leaf components. Image courtesyof Varian Medical Systems, Inc. All rights reserved.9Figure 1.4: The PDD curve of a 6MV, 10x10 cm2 photon beam measured in awater phantom placed 100 cm from the source. The PDD is normalizedto 100 at the depth of max dose dmax along the central axis.64 central leaves (32 leaf pairs) of 2.5 mm width, and 56 outer leaves (28 leaf pairs)of 5 mm width. It can rotate at 15 degrees/second and can linearly move each leafat a max speed between 2.5-5 cm/second [64].Linac Beam CharacteristicsThe human body is very similar in density and composition to water. This makeswater the medium of choice for dosimetric measurements. The dose profile of thecentral axis of a 6 MV 10x10 cm2 photon beam in water is shown in Figure 1.4.As can be seen in the figure, the dose builds to its maximum at a fairly shallowdepth (around 1.5 cm depth). Tumours can be seated close to the surface, but arefrequently at a deeper depth (5-30cm). In the majority of radiotherapy treatmentsituations, a single radiation field will deliver more dose to normal tissue than to10the diseased site since most tumours are located at a depth deeper than the depth ofmaximum dose. It is only by summing the dose contribution from multiple fieldsthat the tumour is made to receive a larger dose than the surrounding tissue. Thisresults in viable control of the disease while reducing the normal tissue toxicities.However, each beam by itself typically delivers more dose to healthy tissue than todiseased tissue, which contribute to a low dose bath to the surrounding tissue.In addition to attenuation of the beam, there is also lateral scatter of the beamwhich spreads the dose beyond the photon beam field edge. This is caused by twoprocesses:1. The lateral scatter of the photons.2. Multiple scatter of the secondary electrons.These factors produce a beam which is spread out in the lateral directions. Profilesproduced by a 6 MV 10x10 cm2 photon beam are shown in Figure 1.51.2.3 Monte Carlo SimulationIn Monte Carlo (MC) simulations [80], the transport of photons are simulated bysampling from stochastic distributions which underlie the interactions of particlesin matter. The dose deposited by these photons is calculated by simulating largenumbers of these interactions. Using this method dose distributions can be calcu-lated on a patient-to-patient basis. MC simulations can also be used to calculatethe dose deposition of a pencil beam of photons (the simulation of a small beam ofphotons incident on a medium). This can be used to approximate the dose from anarbitrary linac aperture using mathematical methods discussed in the next section.Figure 1.6 shows a typical dose deposition pencil beam kernel from a 6 MV beamcalculated using a MC simulation.Pencil Beam ConvolutionThe pencil beam convolution (PBC) algorithm is a model-based method for findingthe dose from photon beams. It works by summing the dose contribution from11Figure 1.5: Beam profiles for a 6MV 10x10 cm2 photon beam. The blue, redand orange lines correspond to the cross axis beam profiles at depths (d)1.5, 5 and 10 cm respectively. The dotted line represents the geometricfield edge defined at the isocentre by the secondary collimator.smaller subsets of the beam. This is accomplished using the following equation:D(x,y,d) =(SSD+dre f )2(SSD+d)2∫ ∫F(x′,y′)K(x− x′,y− y′,d)dx′dy′ (1.10)where source skin distance (SSD) is the distance from the target to the patient skin,dre f is a reference depth used in the calculation of the kernel and intensity, F is flu-ence of the beam (the number of particles passing through a unit area perpendicularto the beam line), and K is the dose deposition kernel (shown in Figure 1.6).The first ratio, (SSD+dre f )2(SSD+d)2 , is known as the inverse square correction factor. Itcorrects for the falloff of the fluence as the radiation source is moved away fromthe dose calculation point. At typical treatment distances, the radiation can be12Figure 1.6: Level set diagram of the dose deposition kernel of a 6MV photonbeam. Doses are normalized to the max dose delivered by the kernel. Apencil beam of photons (shown as a black arrow) incident on the waterphantom (shown by blue shading) at (0,0). The photon interactions re-sult in high energy electrons which subsequently deposit their energy inboth the forward and lateral directions.modelled diverging from a point-source with the particle fluence spread across aspherical surface area. This area depends on the square of the distance from thesource, so the correction factor is a ratio of squared distances.The second part of the equation (the integration) represents a convolution ofthe particle fluence with the dose deposition kernel. The particle fluence repre-sents the number of particles per unit area in the beam cross section. Practically,this method relies on breaking the beam down into subsets. The name ”pencilbeam” comes when the calculation grid size is on the order of mm, for which thelinac beam is modelled as the sum of multiple pencil-sized beam subsets. Thedose deposition kernel models the dose deposited by a each of these beam subsets.13Therefore total dose is the number of particles incident on an x-y coordinate in thebeam crossplane (fluence) multiplied by the dose deposited by the beam subset andsummed in aggregate.As this equation takes the form of a convolution, it can be conducted in thefourier domain for faster calculations, and this forms the basis for time efficienttreatment calculation algorithms [12, 33]. The computational efficiency of thismethod lends itself to iterative optimization problem solving methods which re-quire many dose computations throughout the optimization process.1.3 Treatment PlanningTreatment planning is the process by which a viable EBRT treatment is constructed.The first step in the process is to acquire a computed tomography (CT) scan ofthe patient. CT scans are 3D images produced by x-ray imaging a subject frommultiple directions and then determining the 3D attenuation of the subject usingmathematical methods. This provides anatomical information (as different tissueshave different attenuation coefficients) for planning and dose calculations.The International Commission on Radiation Units and Measurement (ICRU)defines standard ways of reporting doses recieved by tumour and normal tissue vol-umes [41, 67]. The organ at risk (OAR) and tumour target (called the gross tumourvolume (GTV)) are contoured either from the CT images directly or on imagesfrom other modalities, such as magnetic resonance imaging (MRI), positron emis-sion tomography (PET), single photon emission computed tomography (SPECT).Two margins are typically added to the GTV, the first of which is the clinical tar-get volume (CTV) margin which accounts for microscopic spread of the tumourwhich cannot be visualized on the imaging scan. A second margin known as theplanning target volume (PTV) margin is added to account for treatment uncertain-ties such as setup error and patient movement. The PTV margin will depend on thetreatment site, the linacs mechanical accuracy, the accuracy of the patient immobi-lization, and positioning system. A typical GTV to PTV expansion for SRS brainmetastasis is 1-3 mm [45].141.3.1 organ at risk (OAR)An organ is a group of tissues that perform a specific function. Toxicity refers to thedisruption of the biological function of an organ in some way. The goal of treatmentplanning is to create a radiation delivery plan which manages the probability ofnormal tissue toxicity while still delivering enough dose to the diseased tissue tocontrol the disease. The Radiation Therapy Oncology Group (RTOG) defines atoxicity rating quantified on a 5 point scale, with 0 being no symptoms, and 5being death directly related to the radiation effect on the organ [32]. This toxicityrating system is the most commonly used system in North America.The functional subunits within an organ can act with two types of architecturesto achieve the overall function of the organ. The first is a serial architecture whichrequires each subunit of the organ to function for the entire organ to maintain itsfunction. An example of a serial organ is the spinal cord, where if a single subunitof the spinal chord breaks, motor control for a sector of the body can be lost.The radiation dose to toxicity relationship for serial organs will depend on themaximum dose delivered to the organ because in these cases if one part of theorgan is lost, the entire organ fails. The second architecture is a parallel organwhere the organ is made up of subunits which each perform the same function. Ifone subunit fails, the others can still perform the function of the organ. For thesearchitectures, the organ can tolerate large doses to a small subsets of the organ.The radiation dose to toxicity relationship sometimes depends on the mean dosereceived by the organ.Not all organs fit into these two discrete categories, so a mathematical modelis used:gUED = (1NN∑i=1d1ni )n (1.11)where gUED is the generalized uniform equivalent dose, N is the number of voxelsthat make up an organ, d is the dose to a particular voxel, and n is a polynomialfactor (∈ (0,1]) which varies depending on whether the organ behaves in a serialor parallel fashion. When n→ 1, gUED simplifies to the mean dose, and whenn→ 0, the gUED formula simplifies to the max dose formula. More informationon biological modelling in radiotherapy can be found in a review by Marks et al.15Table 1.1: QUANTEC dose constraints for single fraction cranial SRS. V12is the volume that recieves 12Gy and Dmax is the max dose to an organ.Data collected from [9]Critical Structure Constraint Toxicity Rate Toxicity EndpointBrain V12 <5-10 cc <20% Symptomatic necro-sisBrain stem (acoustictumors)Dmax < 12.5 Gy <5% Neuropathy ornecrosisOptic nerve/chiasm Dmax < 12 Gy <10% Optic neuropathySpinal cord (single-fx)Dmax < 13 Gy 1% Myelopathy[61].1.3.2 Quantitative Analyses of Normal Tissue Effects in theClinic (QUANTEC)While the discussion given in the previous section is important for understandingthe basis for clinical practice (and therefore the basis for improvement), there arealso guidelines set in place to assist and guide clinicians in evaluating radiother-apy treatment plans. These guidelines are mainly based on clinical outcomes dataand are enumerated in a publication refered to as Quantitative Analyses of NormalTissue Effects in the Clinic (QUANTEC)[9]. The small subset of these dosimetricconstraints which are applicable for single fraction cranial SRS are shown in Ta-ble 1.1. As can be seen, one of the main considerations in SRS cranial treatmentplanning is to decrease V12Gy of normal brain, the volume of brain that receives 12Gy of radiation, which has been correlated with symptomatic radionecrosis.1.3.3 Treatment Plan EvaluationThere are many quantitative methods by which radiotherapy treatment plans areanalysed. Furthermore, there are different ways to represent the same informationwith little standardization. This can sometimes make the process of treatment planevaluation difficult. This section will focus on the definition of the dose metricswhich are used in this work.16Dose and Volume StatisticsOne of the most commonly used metrics in treatment plan evaluation is the doseand volume statistics. As discussed in Section 1.3.1 and Section 1.3.2, sometimesthe max dose delivered to a structure (Dmax) or the mean dose delivered to a struc-ture (Dmean) is correlated with organ function. Similarly, sometimes toxicity can becorrelated with the volume that receive a certain amount of radiation. These vol-umes are represented with the notation Vx, which corresponds to the total volumewhich receives ”x” radiation or more in Gray.In SRS, the intention is to give the treated volume an ablative dose (a dose sohigh that it kills any tissue recieving the dose), so a treatment plan’s effectiveness ismeasured by the degree to which the prescription dose overlaps and does not extendfrom the treated volume. This property is measured with the quantity known as theconformity index. There are several ways to calculate this metric, but here we willpresent Paddick’s conformity index [73]:CI =(T ∩VP)2T ×VP (1.12)where CI is the conformity index, T is the target volume (which can be the PTVor GTV depending on the application), ∩ is the geometric overlap function, andVP is the volume that receives the prescription dose or more. Mathematically, theconformity index is bounded by 0 and 1. If the conformity index is equal to 1, thenT aligns perfectly with VP (T ∩VP = T = VP). The conformity index can be lessthan 1 for two types of deviations: either under-dosage of T (which would implyT∩VPT < 1), or from dose spillage beyond the target volume (which would implyT∩VPVp< 1) .Another metric used to evaluate treatment plans is the homogeneity index (HI),which measures the dose homogeneity across the PTV. While multiple differentmeasures of homogeneity are used in radiotherapy [44], the formulation used inthis thesis was:HI =DPTVmaxDPTVmin(1.13)where DPTVmax is the max dose to the PTV and DPTVmin is the min dose to the PTV.Recent work has relaxed the goal of homogeneous treatment dose to the PTV in17Figure 1.7: A typical dose volume histogram for an OAR and PTV is shownin red and blue respectively. Using this illustration, dose statistics canbe read off the figure axes. A few examples such as V OAR5Gy , the volumeof the OAR that receives 5 Gy (≈ 30 %), DPTVmin , the minimum dose ofthe PTV (≈ 21 Gy), and DOARmax , the maximum dose of the OAR (≈ 12Gy) are shown on the plot.favour of incorporating dose escalation to subsets of the PTV that are correlated toa higher probability of tumour recurrences [8].dose volume histogram (DVH)The DVH is a widely used tool for evaluating dose distributions. DVHs providevisual representations of the dose statistics. Figure 1.7 shows an illustration of atypical DVH for an OAR and PTV structure. The figure shows how dose volumestatistics, such as V OAR5Gy , DOARmax , and DPTVmin , are visualized. The DVH is versatile asmultiple OAR, PTV, or normal tissues can be plotted and visualized at the sametime (only two are shown in Figure 1.7 for simplicity). Additionally, a summary of18all tissues, sometimes referred to as the ”body” contour, can be plotted to show thedose received by the whole body.1.4 Stereotactic RadiosurgeryCranial SRS is the delivery of highly conformal ablative dose to small, well-definedtargets in the brain. In recent years, linac-based SRS has gained increased rele-vance in the management of cranial lesions due to improvements in machine pre-cision and delivery techniques that allow precise delivery of a highly conformaldose to the target. The high degree of dose conformity with SRS is achieved witheither dynamic conformal arc (DCA) [87] (treatments where the MLC is set to theconform to the PTV and the beam line is rotated around the patient), volumetricmodulated arc therapy (VMAT) [90] (treatments where the MLC, dose rate, andgantry rotation are optimized using mathematical methods and these aspects of thedevice move intra-treatment), or through static beams such as intensity modulatedradiation therapy (IMRT) [25] (a treatment where multiple static beams with vary-ing MLC apertures and dose rates are used to create viable treatments). Typicallythese treatments take between 7 minutes (single isocentre VMAT) to 40 minutes(IMRT). Treatment time mainly depends on how many couch positions are re-quired, while treatment quality depends on how adequately the possible deliveryentrance angles are sampled.1.4.1 Small-Field DosimetrySmall fields provide unique dosimetric challenges in radiotherapy. There are twomain contributors to these challenges. The first of which is high dose gradientsacross the treatment field. Dosimetric measurements of small fields that exhibithigh dose gradients require specialized detectors with a very small active volumeso that readings do not suffer from the partial-volume effect (when readings areaveraged over a volume with inhomogeneous dose distribution).Secondly, lack of electronic equilibrium makes small field dosimetry challeng-ing. Electronic equilibrium is achieved when the electrons leaving a detector’sactive volume are balanced by the same number of electrons of the same kineticenergy entering the volume. However, near the edges of a field, the electrons that19are laterally scattered out of the field edge are not matched by electrons entering theactive volume. Therefore lateral electronic equilibrium does not hold and correc-tions need to be applied for accurate dosimetry. This can be corrected for by usingthe appropriate correction factors [28] or by having a detector with a sufficientlysmall active volume when compared to the field size.Considering these limitations, there are only a handful of measurement deviceswhich are well suited under small field conditions. The two devices used in this the-sis were small volume ionization chambers and radiochromic film. Film providesspatial dose information but are prone to introduced human error. Conversely ion-ization chambers provide accurate point doses, but are not that well suited for dosedistributions. There are also numerous flat panel dosimeters which provide mod-erate spatial information (scale usually in mm to cm). This work relied upon anelectronic portal imaging device (EPID) detector which is made out of amorphoussilicon which can provide 2 mm spatial resolution and relative dose information.Radiochromic FilmRadiochromic film refers to sheets of plastic-like material which is sensitive toradiation and changes colour and hence optical density (OD) when exposed toradiation. The OD is measured as the log ratio of the radiant flux transmitted bythe material (I) divided by the radiant flux received by the material (I0):OD = log(II0) (1.14)The OD-to-dose relationship is non-linear and different for each photon energytransmission measurement. Film is typically scanned with a modern flatbed scan-ner (for example, the Epson 10000XL was used in this study) which provides mea-surements of the absorbance of the film to red green and blue light.Radiochromic film can be used as a form of relative dosimetry: the dose re-sponse curves cannot be derived from first principles and instead need to be ap-proximated by measurements and non-parametric models. The process of calibrat-ing the film involves measuring the dose response by exposing subsections of thefilm to known doses of radiation and measuring the optical density of the irradiatedfilm with a flatbed scanner. The main sources of error which can be introduced with20dose measurement with film are:1. Inconsistency between film batches.2. Different optical properties depending on orientation of the light relative tothe film.3. OD depends on the time since irradiation.4. The scanner has varying sensitivity scan-to-scan.5. The scanner has varying sensitivity laterally intra-scan.If these issues are mitigated, accurate dosimetry with radiochromic film is feasible.For example, the protocol suggested by Fuss et al. [29] quotes that a dosimetricaccuracy of 1% [29] can be achieved. This thesis used GafChromic film (a com-mercial implementation of radiochromic film) using the protocols suggested in themanufacturer’s white-paper [52].1.4.2 Patient ImmobilizationDue to the large, ablative doses delivered to small targets, SRS deliveries requiresmall margins. In order to ensure accurate treatment within these small mar-gins, patient immobilization and intra-treatment imaging are vital. Before intra-treatment imaging was available, patient immobilization was achieved using rigidframes in which an immobilization frame was screwed onto the patients skull by aneurosurgeon (as shown in Figure 1.8a). The rigid frame is attached pre-CT scan-ning and the patient is scanned with the localization box (shown in Figure 1.8b)fastened to the frame to provide fiducial markers for registration.However, the use of rigid frames is time and resource intensive, and is uncom-fortable for patients. Modern treatments use aquaplast moulding which is made of aplastic mesh, which when warmed can form to the patients anatomy and becomesrigid upon cooling. This technique when applied to SRS is known as framelessimmobilization (shown in Figure 1.8c). However, by itself frameless immobiliza-tion does not have the same setup accuracy or reproducibility as the rigid framesbecause they are not attached to patient’s bony structures. This is overcome bysetting up the patient using image guidance and then aligning the patient to the21(a) (b)(c)(d)Figure 1.8: a) Rigid frame used for SRS deliveries. The frame is affixed di-rectly to the skull. b) A positioning box is attached to the frame dur-ing imaging and patient set up. It provides fiducial markers and visualalignment markers to ease process of aligning CT information with thedelivery isocentre. c-d) Brainlab frame-less mask used in modern daytreatments.treatment position with a motorized couch that can precisely move the patient inall six degrees of freedom of motion. Used properly, this technology can provideaccuracy of less than 1 mm [79].221.5 Quality Management in Radiation TherapyA significant consideration in radiation therapy is how to ensure safe and con-sistent treatments on a patient to patient basis. Radiotherapy planning and de-livery is a complex process and there are multiple points of failure that need tobe managed to ensure treatment quality. For this reason, quality management isconducted throughout the radiotherapy planning and delivery process (termed end-to-end quality management). When changes are made to the planning process,methods need to be developed to combat the modes of failure which have beenintroduced.The American Association of Physicists in Medicine (AAPM) has set guide-lines for quality management in the radiotherapy setting. The AAPM report fromtask group 100 [39] (so called TG 100) provides an overview of the application ofrisk analysis methods to ensure treatment quality. It presents various methods suchas failure mode and effects analysis (FMEA), which can be used to find modes offailure which may cause clinically significant events. Once identified, these modesof failure can be mitigated with quality management tools. TG 100 provides rankorder lists of effective tools to use for these applications (listed in descending orderof effectiveness):1. Forcing functions and constraints (such as interlocks, berriers, computerizedentry forms).2. Automation and computerization (such as computerized verification, bar codes,automated monitoring).3. Protocols, standards, and information (such as check-off forms, alarms, es-tablishing protocols).4. Independent double check systems and redundancies (such as redundantmeasurements, independent review, comparison with standards, acceptancetesting).5. Rules and policies (such as establishing a communication line, mandatorypauses, and establishing and performing quality control (QC) and qualityassurance (QA) on hardware and software).236. Education and information (such as training, experience and instruction).The rank order of these methods provides guidelines for developing QA pro-cesses. While some aspects of QA occur closer to the bottom of this list (e.g.education and information) all methods are important in the management of radio-therapy quality. In particular, lower rank methods are implemented in situationswhere the higher order interventions are infeasible or impractical.The AAPM report on Task Group 142 [48] (so called TG 142) set guidelinesfor the quality assurance of a linac. This document sets achievable lower limits onthe accuracy of a linear accelerator for various types of treatments. An abridgedsummary of the mechanical specifications required for SRS VMAT and IMRT de-liveries, and the recommended frequency of quality control is given in Table 1.2.While TG 142 sets guidelines for the mechanical specifications, it does notmake recommendations on how to test for these mechanical specifications. Fur-thermore, as expanded in TG 100, clinical processes are unique to each centre, andquality management should be tailored to the clinical processes which are imple-mented. A rough starting point for the quality management of VMAT treatmentswas presented by Ling et al. [54]. In this work, various methods are introducedfor testing the aspects of the linac enumerated in Table 1.2. Some methods fromthis publication were expanded upon in this work so that the methods developedby Ling et al. could applied to treatments with intra-treatment motion of the couchand collimator.1.5.1 The Picket Fence TestThe picket fence test was developed by Bayouth et al. [6] as a method for ensuringMLC performance for IMRT deliveries (although it is also applicable to VMATdeliveries [54]). In this method, the field jaws are opened and the MLC is set tocreate a 1 mm gap across a 2D dosimeter (film or EPID). This gap is moved acrossthe film in 1.5 cm gaps, with the radiation field turned on when the MLC is static.The irradiation pattern on the 2D dosimeter looks like a picket fence and deviationsin MLC can be identified using visual inspection or computer aided methods.Yu et al. [101] expanded this method to ensure the synchrony of the couch andMLC. In this method, radiochromic film is placed on the treatment couch and a 124Table 1.2: Summary of the mechanical specifications required for IMRT andVMAT deliveries with a linac. Data was collected from recommenda-tions made by the AAPM task group 142 [48]Quality Test AccuracyDailyX-ray output constancy 3 %Laser Localization 1 mmDistance indicator (ODI) 2 mmCollimator size indicator 1 mmMonthlyPhoton beam profile constancy 1 %Light/radiation field coincidence(asymmetric)1 mm or 1 % on a sideJaw poosition indicators 1 mmDose Rate Constancy 2 %Treatment couch position accuracy 1 mm translational, 0.5 degree rotationLocalization lasers <1 mmGantry Collimator angle indicators 1 degreeAnnualX-ray flatness change from baseline 1 %X-ray symmetry change from baseline ± 1 %X-ray output calibration (TG-51) ± 1 %Spot checks for field size dependentoutput factors2 % for field sizes < 4x4 cm2, 1 % >4x4 cm2X-ray beam quality 1 % from baselineX-ray output constancy vs dose rate ± 2 % from baselinex-ray output constancy vs gantry angle ± 1 % from baselineArc Mode (expected MU per degree) ± 1 % from baselineCollimator rotation isocenter ± 1 % from baselineGantry rotation isocenter ± 1 % from baselineCouch rotation isocenter ± 1 % from baselineCoincidence of radiation and mechan-ical isocentre± 1 mm from baseline25mm gap is produced with the MLC. The gap is then scanned across the field whilethe couch moves concurrently at the same speed. Throughout the movement, theradiation field is turned on. This can be used to ensure the simultaneous movementof the couch and collimator.1.5.2 Coincidence of the Treatment IsocentreA medical linac is comprised of many mechanical subsets that act in synergy todeliver a treatment. Each degree of freedom is usually in spherical or cylindricalgeometry with rotations about a single axis. This introduces two forms of errorinto the delivery process: misalignment and wobble of the isocentre. There havebeen various QA methods to measure the sources of error introduced by rotationof the couch, gantry and collimator.Winston-Lutz TestMechanical accuracy of the isocentre rotation was a major hurtle for the use of alinac in SRS treatments. Lutz et al. [58] developed an accurate and robust systemfor measuring the relation of the radiation isocentre with the treatment positionsystem. At the introduction of this technique (and when TG 142 was first authored)the laser localized isocentre was used as the treatment position system. Modernlinacs use an on-board imaging defined isocentre, and the methods developed by[58] has been adapted to these techniques.The method developed by Lutz et al. [58] is now commonly referred to as the”Winston-Lutz Test”. It involves accurately aligning a ball bearing (BB) with thetreatment isocentre, placing a 2D radiation measurement device under the BB, andthen irradiating the phantom with a well defined small field. The location of theBB can be quantified as it is radio-opaque. The 2D dosimeter needs to have enoughspatial resolution to quantify the location of the BB to less than a mm. Initially, filmdosimetry was best suited for this purpose, but modern EPID also provide enoughspatial resolution. From these images, the centre of the BB is found (treatmentisocentre), which is compared to the centre of the radiation field, as defined by thecentre point between the half max of the field edges. The Winston-Lutz test canbe used to measure the mechanical accuracy of the couch, collimator, and gantry.26An up-to-date and more in-depth protocol for the Winston-Lutz test with a modernlinac is provided by Rowshanfarzad et al. [82].Starshot MethodWhile the Winston-Lutz test measures the interaction of the radiation isocentrewith the respective errors of the mechanical degrees of freedom of the linac, thestarshot method tests the mechanical error individually. In this method, the MLC(or sometimes the secondary collimator) is set to form a 1 mm gap along the centralaxis of the field. A 2D dosimeter (previously film, however the EPID can be usedfor the collimator measurement) is irradiated by the linac. Next the degree of free-dom under study is rotated by 30 degrees, and the dosimeter is re-irradiated. This isconsecutively conducted for 180 degrees of rotation (or 6 irradiations). Next a linefitting algorithm is used to fit the strips of irradiation (produced by the 1 mm colli-mated beam). The intersection point between each of the 6 lines is calculated andthe size and centre of the largest circle encapsulating all of the points is reported.This provides a measurement of the true isocentre of rotation for the method understudy. The centre of the circle or centre of mass of the intersection points can becompared to the treatment positioning system defined isocentre.1.5.3 Patient Specific Quality AssuranceThe QA of dose distributions presents a challenging problem: dose distributionspresent 3D data and there are many ways in which the expected dose can differfrom the delivered dose. the goal of analysis is to find gross deviations from clin-ically acceptable treatment plans with one simple to calculate metric. There aretwo main deviations which can occur in radiotherapy: inaccuracy of dose (i.e. thedose is some percentage different than the expected value) and positioning inac-curacies (i.e. the dose distribution is misaligned). Positioning accuracy is veryimportant as un-irradiated tumour tissue will significantly decrease the efficacy ofthe treatment. Similarly, dose inaccuracies can manifest unexpected toxicities forstructures which are close to their limits, and lower the probability of disease-freesurvival if the tumour is under-dosed. Low et al. [57] developed a method that triesto explicitly account for these types of errors, and it is called the gamma pass met-27ric. In this method ideal accuracy specifications, such as the dose value accuracyand positioning accuracy, are specified. In the original publication, an accuracyof 3 %, 3mm in the dose value and position were used respectively. Modern SRStreatment QA uses 2%, 2 mm or 2 %, 1 mm as positional accuracy is of greaterimportance when tight margins are used.Each dose value in the reference distribution is compared to the spatially closedoses of the measured distribution (within δ r mm of the point, where δ r is theaccuracy previously chosen). For each point, the gamma criterion is calculated as :Γ=√(∆rδ r)2+(∆DδD)2 (1.15)where ∆r is the distance between the reference voxel and the measured voxel, δ r isthe specified positional accuracy (i.e. 1 mm), ∆D is the dose difference between thereference voxel and the measured voxel (measured in % of the prescription dose),and δD is the specified dose accuracy. This is conducted for each voxel within δ rof the specified point and amongst these points, the minimum Γ value is quoted. Ifthe Γ value is less than 1, then the voxel passes, otherwise it fails. This analysis isconducted across an entire image and the percentage of voxels which have gammavalue lower than 1 is referred to as the gamma pass rate.1.6 OptimizationOptimization is the process by which a given function is minimized or maximizedfor a set of variable inputs. For the remainder of this introduction, the function tobe minimized (also known as the ”cost” or ”objective” function) will be referred toas f (w), and the variable w refers to a set of input variables. Mathematically, thegoal of optimization is to find the value of w which satisfies:w = argminw[ f (w)] (1.16)Often times there are further constraints on w, for example if w represents beamintensity, then the value of w cannot be negative.28Figure 1.9: For functions that are convex, there is one minimum which is theglobal minimum. The gradient points in the direction of functional in-crease, so by searching in the opposite direction, the function minimumcan be found.1.6.1 Optimization MethodologyGradient DescentGradient descent is a method used in the optimization of convex functions. Ingradient descent, the variables are initialized at some value (typically 0, but if betterestimates of the variables exist, they can be used), and the function minimum isfound by iteratively correcting this guess by moving the solution in the directionof the negative gradient (illustrated in Figure 1.9). This iterative correction of thesolution can be represented as:wt+1 = wt −α∇ f (w) (1.17)29where wt is the variables at iteration t and α is the step size: One example step sizeisα =1max(eig(∇2 f ))(1.18)which denotes the inverse of the maximum eigen value of the Hessian of the costfunction. While this approach guarantees convergence, it usually under-predictsthe magnitude of the step size which results in longer optimization times. A morerobust strategy is an adaptive step size using a backtracking line search method. Inthis method, the stepsize αi (subscript i corresponding to the step size’s dependenceon iteration count) is initialized to a large value, and then it is exponentially de-creased (iteratively multiplied by a number less than 1) until the Armijo-Goldsteincondition [5] is met:f (w−αi∇ f (w))< f (w)−αic||∇ f (w)||2 (1.19)where c is a control parameter set between (0,0.5].Newton-based MethodsNewton-based methods rely on the Hessian of the function to find the perturbation.It relies on fitting a quadratic function to the local gradient and Hessian, and thenminimizing the quadratic. The quadratic fit is accomplished by Taylor expandingthe function about the given iteration’s variables:f (w+∆w) = f (w)+∇ f (w)∆w+∇2 f∆w2 (1.20)This function is quadratic in the variable ∆w and yields the minimized value when∆w = [∇2 f (w)]−1∇ f (w) (1.21)This method is faster than gradient descent when computing the inverse Hessian ofthe function is computationally feasible. For quadratic systems without boundaryconditions, this method converges in a single optimization step.30Figure 1.10: This figure shows the level set diagram of a function with inputs.In this example, the global minimum is outside of the contained set w≥ 0. The gradient is calculated (in red), however the solution liesoutside of the solution set. This is corrected by projecting the solutiononto the closest point within the solution set. The minimum valuethat achieves the constraints lies on the boundary of the constraints (inthis example, the y axis) and the gradient moves along the y axis untilthe minimum is achieved. For these optimizations, the variable α hasincreased importance as the projected gradient can be much smallerthan an optimal step.Constrained OptimizationThere are multiple ways to deal with constrained optimization, the simplest ofwhich is called the projected gradient method. In this method (illustrated in Fig-ure 1.10), the gradient is calculated and then applied to the optimization variable ofthe iteration wi. The solution is then projected onto the set of allowed values of w,and the optimization continues. The projected gradient method does not work for31Newton-based methods. One method to overcome this problem is to add a barrierfunction to the cost, which increases the cost when the optimization is outside thesolution set.Stochastic OptimizationIn the optimization of non-convex functions, non-global minima exist and gradientmethods produce solutions which tend to the local minima that is closest to the ini-tial condition. For computationally simple problems, this feature can be overcomeby sampling multiple initial conditions, allowing the solutions to find the localminima, and then reporting the minimum of the candidate solutions. However, inradiotherapy optimization this strategy is frequently computationally infeasible.A competing strategy to find the global minimum is to use a process knownas simulated annealing [46]. In this method, the function is initialized (w0) andthe variables are optimized using a stochastic iterative process. In each iteration,a perturbation (dw) is sampled from a distribution and applied to the function in-put variable of that loop (wi). The function is then evaluated at wi + dw and thecost at this new variable input is calculated. The change in the function (∆ f ) iscalculated using the formula ∆ f = f (wi)− f (wi + dw). If the function decreasedwhen compared to the previous iteration, then wi+1 = wi + dw, while if the func-tion has increased from the previous iteration, then the perturbation dw may stillbe accepted with a probability given by the Poisson-Boltzman equation.P(i) = exp(− ∆ fkbT (i)) (1.22)where i is the iteration number, P(i) is the probability of accepting the deleteriousperturbation, and T (i) is the ”temperature” of the solution at a particular iteration.This process allows solutions to ”tunnel” out of local minima the same way ashappens to non-globally optimal configurations in potential wells. During opti-mization, temperature is initially high, but as optimization progresses, the temper-ature is decreased so that the solution tends to a stable minimum. This temperaturemodulation is called the annealing schedule.This method is sometimes unsuccessful at overcoming local minima. For meth-ods such as simulated annealing, optimization convergence is slow and warm start-32ing the solution is of increased importance. Warm starting is the concept of choos-ing an initial function input (w0) which is close to the desired minimum. Usuallyw0 can be chosen from previous solutions or from heuristic initialization. In radio-therapy optimization, apertures are warm started using fluence-based optimization(Section 1.6.1) or with apertures defined by a conformal treatment plan.Fluence-Based OptimizationFor fluence-based optimization, the linac beam is subdivided into gridded sections.Photons incident on each of these section will, on average in aggregate, contributedifferent amounts of dose to different voxels within the patient (which may bedesignated as an organ or a target). These dose contributions, denoted as Di j, whereD is the dose deposited to voxel i from subsection j, is illustrated in Figure 1.11.Di j depend solely on the energy spectrum of the beam, and on the geometry of thepatient. While these dose contributions cannot be readily modulated, the fluenceintensity of the beam, w, can be. This is accomplished by changing the shape andintensity of the photon beam using the MLC. The modulated fluence intensity of asubsection can be denoted as w j. There will be bs total fluence weights w j, whereb is the number of beams, and s is the number of subsections in a linac beam. Thenumber of subsections used in this study was 6,400, which represents a 20 cm by20 cm wide beam subdivided into 2.5 mm sections (the width of the MLC). Usingthe above notation, the dose delivered can be calculated as:Dtotal = Dw (1.23)where D is a matrix of dimension bs by v (the number of voxels), and w a columnvector of length bs. One possible cost function is to define the optimal dose thatshould be delivered at each voxel. This can be represented as:C = ||Dw−dcon||2 (1.24)w j ≥ 0 ∀ j (1.25)where C is the objective function and dcon (which is a vector that is is v x 1) arethe optimization constraints which define a dose for each voxel being calculated.33Figure 1.11: Illustration of dose contribution of fluence subsets of a linacbeam. On the left, the linac beam is subdivided into fluence contrib-utors. Photons travel through subsection j of the beam and deliver adose of Di j to voxel i (in blue). Photons from subsection j′ can alsodeliver dose to voxel i, but in a different amount denoted by Di j′ . Sim-ilarly, photons from subsection j also deliver dose to other voxels suchas i′. These dose contribution factors build a matrix D which has sizebs by v, where bs is the number subsets in the beam times the numberof beams, and v is the number of voxelsWhile the matrix D and vector dcon are constants set by geometry and the user re-spectively, w is a variable which can be optimized. This cost function has a welldefined gradient and Hessian and can be optimized with constrained gradient de-scent or a Newton-based method (Section 1.6.1). While this cost is simple from anoptimization standpoint, it does not provide a simple way for practitioners to con-vey their desires to the optimization system. Therefore other cost functions havebeen developed, which are further explained in Section 1.6.2. Optimal fluences34are found by some optimization process, and then these fluences are converted intodeliverable beams using MLC sequencing algorithms.The conceptually simplest MLC sequencing algorithm is a step-and-shoot al-gorithm for which deliverable fluences are considered on a MLC-pair by MLC-pair basis. Each MLC-pair defines a row in the fluence grid (the grid shown inFigure 1.11). The row will define a linear function of optimal fluences. Thesefluence values can be discretized into level sets. These level sets can be deliveredsequentially, delivering the low dose levels first, and then constricting the size ofthe aperture and delivering the higher level sets. The level of discretization willaffect the accuracy of the conversion, however over-discretization of the functionwill require many level sets to be delivered and increases time of delivery and ex-poses the patient to leakage dose as the MLC does not completely block the beam.There are many competing methods to MLC sequencing, but these are beyond thescope of this work as this work mainly focuses on direct aperture optimization.1.6.2 Cost Functions used in Radiotherapy PlanningRadiotherapy cost functions are created to convey the desires of the treatment plan-ner to the optimization software. The simplest possible cost function is one wherethe desired dose of every possible voxel is specified to a particular value (given asEquation 1.24). While this equation is easily understood by optimization software,it is difficult to convey the desired treatment parameters in the defined constraintsDcon. In particular, defining the best possible dose to a particular voxel is a com-plex function of cancer type, target location, OAR location, photon beam energyof the treatment beam, and the capabilities of the optimization software. There hasbeen recent progress in this regards using atlas-based learning methods to producevoxel-based automated dose prescriptions [63]. Previous to this recent progress,other cost functions (explained in the following paragraphs) have been developedto convey the desires of the treatment planner.Another convex function which can be used for dose optimization is the gener-alized p-norm which takes the form:||d||p = (N∑i=1dpi )1p (1.26)35This function is convex and differentiable for p > 1 and is conveniently identicalto the gUED for serial and parallel organs with p= 1n and multiplied by the scalingfactor ( 1N )1p .Another physically meaningful dose criteria is the dose-volume objective [11].These criteria are widely used in radiotherapy and for this reason they are incorpo-rated into some optimization methods. One can express dose-volume objectives ina cost function as:C = ∑organs,targetski∫×(D(v)− c×D0(v))2dv (1.27)In this equation, the dose contribution to each organ is approximated by point cloudrepresentations and calculating the dose to to each point from each beam in thetreatment. Point clouds can be used to approximate dose to OAR, PTV, or normaltissue. The dose delivered to these point clouds can be used to calculate the DVH,which is designated as D(v).However, this cost function does not account for the particular dose effects ondifferent types of tissue. For example, an OAR receiving less dose than the con-strained dose does not deleteriously affect the quality of the treatment. Thereforeno cost should be assigned if the OAR receive less than the constraint dose. ICRUreport 50 [41], and more recently ICRU report 83 [36], provide guidelines for pre-scribing doses in radiotherapy. These guidelines state that for the purposes of find-ing a direct correlation between delivered dose and patient outcome, doses to thePTV should be close to uniform. In particular, ICRU report 50 states that doses tothe PTV should be no less than 95 % of the prescribed dose and no more than 107% of the prescribed dose [41]. This can be achieved in the cost function with twoparameters: one which penalizes under-dosing, and another term which penalizesover-dosing. Typically, there are no further constraints set on the PTV, however,there have been clinical trials which have explored varying the dose inside the PTVguided by biological imaging [8].36These features can be mathematically expressed asC = ∑structuresiwi∫H1× (D(v)− c×D0(v))2dv+∑targetsiwi[∫H2(D(v)−Pmin)2dv−∫H3(Pmax−D(v))2dv](1.28)Where Pmin is the prescribed minimum dose to the target, Pmax is the maximum doseto the target, H1 is the Heaviside function which equals to 1 when D(v)−c×D0(v)is positive and 0 when D(v)− c×D0(v) is negative. H2 and H3 is similarly theHeaviside function, but with D(v)−Pmin and Pmax−D(v) as their respective in-puts. While this function is not convex, it has been shown that local minima aresufficiently close to the global minima such that the they can be used in radiother-apy optimization [97].1.6.3 Volumetric Modulated Arc TherapyVMAT was initially called intensity modulated arc therapy (IMAT) and was pio-neered by Yu [100] as an inverse planning method in which the MLC and gantrydynamically move while the beam is on. A complete history of the development ofVMAT is provided in [15]. At the initial introduction of this method, it was dosi-metrically inferior to static field IMRT. These difficulties are understandable asVMAT is a difficult optimization problem due to the additional degrees of freedomintroduced by the linac gantry rotation. Additionally, these degrees of freedomhave complex constraints due to the continuous movement of the MLC and gantrywhile the beam is on. However, these have since been over-come and VMAT isregarded as dosimetrically equivalent for the treatment of many indications.VMAT trajectories are continuous gantry trajectories when they are delivered.These continuous trajectories can be modelled as static beams (control points (CPs))along the trajectory. For a treatment plan to have accurate dosimetry, CPs need tobe spaced every 1-5 degrees depending on the location and shape of the treatedindication. A set of physical constraints is imposed at each CP due to the physicallimitations of the linac and its components. In particular, as the gantry can be ro-tated at 6 degrees/second, the time interval between two successive control pointscan be as small as 0.2 seconds. Furthermore, the MLC is only able to move be-37tween 2-3 cm/second (depending on model), which imposes a tight constraint onthe allowed MLC positions between successive control points. If MLC ji is the jthMLC position of the ith control point, then this constraint means that:δMLCδ tdt > |MLC ji −MLC ji+1|δMLCδ tdt > |MLC ji −MLC ji−1|(1.29)where δMLCδ t is the MLC velocity and dt is the time between each control point.A typical value for this product in 0.5 cm. Additionally, MLC positions cannotcollide with one another when they move. A safety buffer distance is added whichcan be expressed asb < MLC2n+1i −MLC2ni (1.30)where n is the number of MLC pairs and b is the size of the buffer (e.g. 1 mm).There are two approaches that have found the widest adoption. The first so-lution was the progressive sampling alogorithm [71] which is a direct apertureoptimization (DAO) strategy. Other approaches have incorporated fluence-basedoptimization [7, 14], and both approaches have been found to provide clinicallysuitable dose distributions. The majority of this work is based on the progressiveresolution algorithm which is described in [71].Direct Aperture Optimization VMATAn overview of the progressive sampling algorithm is summarized in Figure 1.12.The gantry arc is initially approximated by a small number of discrete controlpoints. These control points are sufficiently spaced out so that the MLC is able toform any mechanically feasible MLC aperture. These control points have the ini-tialized MLC positions of a conformal treatment plan: the projection of the MLCis set to conform with the beam line projection of the PTV (plus a margin whichdepends on treatment site). A typical conformal aperture is shown in Figure 1.13.The dose deposited by each of the control points’ apertures to the OAR and PTVare calculated using the PBC algorithm. The initial beam weight is set so that themean PTV dose is equal to the prescription dose.38Figure 1.12: Illustration of the progressive sampling algorithm. a) the initialtrajectory is sparsely and discretely sampled as a collection of controlpoints which are evenly spaced along the gantry trajectory. The MLCapertures and MU of the beam are optimized using stochastic optimiza-tion. b) New control points are added in-between two adjacent controlpoints and initial MLC positions are set to be the linear interpolation ofthe adjacent control points. d) This procedure is continued until thereare enough control points along the trajectory to approximate a contin-uous delivery. This figure was reproduced from [71] with permissionfrom Wiley Publishing Group.Next, the MLC positions and MU are subsequently optimized using stochasticoptimization (Section 1.6.1) to meet the dose constraints (cost function) of theparticular plan. This process is continued for a sufficient amount of time until thecost reduction from each optimization loop plateaus around zero.In the next step, CPs are inserted in-between the previously defined CPs as thelinear interpolation of the beam parameters. This includes MLC position, gantryangle, and number of MU delivered. Adding control points in this manner servestwo functions. Firstly, the new control points provide extra freedom for the dosedeposition as the beam can form new MLC apertures and deliver dose from newdirections. Secondly, they increase the accuracy of the dose calculation, which isapproximated by representing the continuous VMAT delivery as an interpolationof a number of static beams defined at each control point. This approximationis less accurate for sparsely spaced control points, but, as the control points arespaced closer and closer together, the approximation sufficiently models the deliv-ered dose.The set of control points formed by the initial CPs and the added CPs areagain perturbed using simulated annealing, however, now the apertures are held39Figure 1.13: A beams-eye-view of a typical conformal aperture. The bluelines signify the edges of the MLC leaves, the red blobs are PTVs pro-jected along the axis of the beam, the yellow designates open fluence.The MLC positions are found by taking the minimum and maximumextent of the PTV contours.to the constraints defined by the mechanical specifications of the MLC (such asin Equation 1.29). The cost is again evaluated between each perturbation, andwhen a minimum is met, further CPs are added. This process is continued until thecontinuous trajectory is accurately approximated. A CP spacing of 2 degrees in thegantry angle is sufficiently accurate for clinical applications.1.6.4 Treatment Planning for stereotactic radiosurgery (SRS)SRS with a linac was pioneered in 1988 by Lutz et al. [58]. At the onset, Lutz et al.realized the importance of multiple entrance angles for these deliveries, and treat-ments were made up of dynamic gantry arcs at various couch angles (around four40non-coplanar arcs per treatment). These treatments were conducted with collima-tion cones, which limited treatments only to small, regularly shaped targets. Withthe introduction of the MLC, the arc technique was improved to include tumors ofirregular shape. This was achieved by dynamically changing the MLC leaf posi-tions so that the beam aperture conforms to the target as the gantry rotates [87].This technique was named as dynamic conformal arc (DCA) and is still used tothis day. While DCA can produce highly conformal treatment plans, it is a forwardplanning method which becomes overcomplicated when treating multiple targets,which is often the case with the treatment of brain metastases.As computer-aided planning became more streamlined, inverse planning meth-ods were developed to utilize the new technology. The first of which was IMRT,however, for this method to have enough entrance positions in the SRS setting,seven or more static fields were typically needed. VMAT treatments allow fluencesfrom multiple angles while having time efficient deliveries and are well suitedfor SRS deliveries. The inverse planning methods developed by various groups[16, 37, 90] allow for complex treatments which treat multiple targets simultane-ously.Single isocentre treatment planning has been shown to be an effective way tocreate highly time-efficient treatments for treating multiple metastases with SRS.Inverse planning methods lend themselves especially well to this technique as highconformity can be achieved through MLC and dose rate modulation (which ensuresthat the amount of healthy tissue which recieves ablative radiation dose is mini-mized). Lau et al. [51] reported comparable clinical outcomes, but with drasticallyreduced treatment times, for patients treated with single isocentre VMAT techniquecontaining one to two arcs when compared to patients treated with conventionalmultiple isocentre treatments. Thomas et al. [90] performed a retrospective treat-ment planning study on patients with multiple metastases previously treated withGamma Knife. They found that their single isocentre VMAT plans with 4 non-coplanar arcs produced dose distributions of comparable quality to other methodsthat used specialized equipment (such as Gamma Knife).411.6.5 Couch-Gantry Trajectory-Based DeliveriesWhile multiple non-coplanar arcs can produce highly conformal dose distributions,they increase the delivery time. The extra time needed to accommodate patientrepositioning for multiple non-coplanar arcs can be mitigated by the use of dy-namic couch motion. This style of treatment was introduced in 1988 by Podgorsakand his colleagues at McGill University [75]. While this method was found to havedosimetric and treatment time saving benefits, it did not gain a wide acceptance asnone of the major linac manufacturers adopted this technology. This method wasalso limited to SRS cones (circular tertiary collimators of varying diameter whichare matched to the divergence of the beam) only as it pre-dated MLCs.With the release of the TrueBeam linac in 2010, Varian introduced a non-clinical Developers Mode [1] that allows trajectory-based delivery. When operatedin Developers Mode, the TrueBeam linac is capable of dynamically moving thegantry, collimator, MLC, jaws, and the couch while the beam is on, allowing com-plex three dimensional trajectory beam delivery. This development has sparkednew interest in the development of an optimization framework for trajectory volu-metric modulated arc therapy (TVMAT) deliveries of cranial [59, 86, 99] and extra-cranial treatments [53, 84, 102] as well as in the development of QA methods forthis mode of delivery [93, 95].Soon after the release of the TrueBeam, Yang et al. [99] devised an optimizationframework to create optimal dynamic couch-gantry trajectories by implementingan algorithm to minimize doses to Organs at Risk (OARs) using an OAR overlapmetric along the delivered trajectory. MacDonald and Thomas [59] and Smythet al. [86] applied this method in cranial SRS and found that the method success-fully improved target dose conformity and lowered doses to OARs when comparedwith standard VMAT plans. However, the trajectories were not necessarily timeoptimal and involved sporadic couch movements, which may be uncomfortable ormay result in intra-fraction motion.The effect of treating with highly co-planar treatments was explored by Nguyenet al. [68] when they developed an unconstrained IMRT optimization process forcranial delivery. In this method (which was named 4pi radiotherapy), fluences wereallowed from candidate beams which were uniformly distributed across the deliv-42erable phase space (couch-gantry angle contributions which would not result incollisions of gantry and patient). The fluences were optimized to produce the glob-ally optimal treatment. Next, fluences directions were removed based on whichdelivered the lowest fluence and the plans were re-optimized. This was continueduntil there were only 20 candidate beams left which was chosen as a trade-off be-tween delivery efficiency and quality. Delivery sequence of these candidate beamswas optimized using a travelling salesman optimization algorithm. This approachwas extended to couch-gantry arc treatments by Langhans et al. [50] who deliveredVMAT treatments which connected the phase space points of the delivered IMRTbeams.Due to the complexity of the TVMAT technique, machine specific QA proto-cols need to be developed prior to its implementation. There has been preliminarywork in this field both within our group and externally. Victoria et al. [93] devel-oped QA methods to tests the synchrony of motion of various components of theTrueBeam linac (couch, gantry, and MLC). However, their work did not includesynchrony tests for dynamic gantry-couch motion including variable dose rate. Toour knowledge, the only reported work in this respect comes from our group.1.6.6 Collimator-Gantry Based TrajectoriesThe collimator and MLC act in synergy to form optimal apertures. However, themechanical specifications of an MLC (defined in Section 1.2.2) set limits on theapertures which the linac can form. Firstly, the MLC provides a rotationally a-symetric forms of collimation as the leaves only protrude into the beam from twodirections. Therefore the shapes which can be produced by the MLC depend on theangle of the collimator. Secondly, the MLC does not completely block the beam.The body of the MLC only blocks approximately 98% of the beam while radiationleakage between parallel leaves (known as inter-leaf leakage 2%) and end leaf gapleakage (the leakage between abutting MLC leaves (1 mm minimum gap size) ) iseven greater. Collimator angle optimization can potentially minimize the effects ofthese mechanical shortcomings.Historically, there has been little conclusive study on the effect of collimatorrotation on the efficiency and quality of radiotherapy delivery. The reason for this43is that the collimator angle will affect MLC sequencing algorithms differently. Ini-tially, when conformal therapy was the standard of care, the collimator angle wouldbe selected using treatment planner experience and a trial-and-error approach. Thisstrategy has continued into modern technology. However, some groups have ex-plored strategies which may out-perform and automate this method.Otto and Clark [72] explored the effect of rotating the collimator at a constantangular velocity intra-treatment during static field IMRT delivery with the hopesof increasing the accuracy and efficiency of fluence map delivery. Milette and Otto[65] tested this method and found that it provided an increase in the accuracy ofthe delivery of concave fluence shapes (such as a shoehorn shape) and a spreadingof the dose contribution from inter-leaf leakage which reduced intensity of the hot-spots due to leakage by a factor of two.Webb [94] explored whether collimator rotation intra-treatment would reducethe frequency and effect of ”parked gaps” (end-to-end leaf leakage) by consider-ing the collimator angle during the optimization process. In this work, a methodwas developed to minimize the number of parked gaps for the delivery of fluenceswhich were convex shapes. This work also explored collimator angle optimizationfor the delivery of multiple concave fluence shapes.Zhang et al. [102] developed methods for the optimization of the collimatorangle to block the spinal cord in stereotactic body radiotherapy (SBRT) treatments.In their work, Zhang et al. [102] found the principal long axis of the spinal cord bydecomposing contours using principal component analysis (PCA), and then align-ing the collimator so that the MLC leaves were perpendicular to the spine, so as tobest protect it from radiation.Yang et al. [99] expanded on the work of Zhang et al. [102] by incorporatingcollimator-couch-gantry trajectories in VMAT deliveries. This work used a heuris-tic OAR-overlap cost function to optimize the couch-gantry trajectory and thenfurther optimized the collimator angle using methods derived from Zhang et al.[102]. The couch trajectory was optimized so that fluences would minimize PTVoverlap with OAR, while the collimator was optimized so that the MLC leaveswould be perpendicular to the long axis of the OAR.Locke and Bush [55] explored the effect of collimator optimization on theprogressive sampling algorithm (described in Section 1.6.3). They developed a44PTV connectivity metric which could be used to initialize DAO-VMAT for com-plex torus-shaped PTVs so that optimization would not get stuck in local minima.Their method also incorporated collimator angle optimization using a graph searchmethod.As single isocentre treatment of multiple metastasis is a relatively new treat-ment modality (pioneered independently by two groups in 2010 by Clark et al. [16]and Hsu et al. [37]) there has been no publications on the effect of intra-treatmentcollimator rotation for VMAT treatments of multiple brain metastasis. However,Wu et al. [98] explored the optimization of static collimator angles for VMAT treat-ments. In this work, a heuristic cost function was developed which was the areaof open fluence for a conformal treatment plan. This function was minimized forstatic collimator angles and treatment plans were optimized using Varian Eclipse.They found a significant decrease in the low dose bath, with the most drastic im-provement in the volume that receives 5 Gy of radiation.MacDonald et al. [60] independently developed a similar cost function to theone explored by Wu et al. [98]. In their formulation, the whitespace was calculatedby adding the open fluence (defined by the jaws) and subtracting the area of over-lap with the MLC, PTV alone and PTV overlapped with OAR. This added extraweighting terms which could be modulated to incorporate various aspects not con-sidered by Wu et al. [98] such as MLC leakage and avoidance of OAR. MacDonaldet al. [60] optimized the collimator trajectory in DCA treatments using a cost-valleytracing algorithm. Using this approach they performed a treatment planning studyof 15 simulated treatment cases which had three and four metastases. They foundthat the dynamic collimator DCA treatments had lower monitor units and dose baththan the VMAT treatments, but failed to find any significant differences betweenthe trajectory-optimized and static-collimator-optimized DCA treatment plans.1.7 Thesis OverviewThis project is aimed at combining the successes of previous treatment modali-ties, to achieve an optimal cranial SRS treatment plan that could be delivered in atime-efficient manner. We explore several algorithms in which the couch and thecollimator are allowed to move dynamically to achieve these goals.45The first strategy that is considered in Chapter 2 is allowing a complete 4pisampling of the allowable phase space by implementing trajectories of the couchand gantry. The optimization strategy that was found to be most efficacious wasa patient-generalized trajectory in which the couch rotates across its full range ofmotion while the gantry delivers partial arcs. This trajectory can be modulated toincrease sampling of the phase space by allowing gantry to sweep more times, orthe sampling can be reduced if the treatment quality is not improved so as to reducetreatment time.This work additionally shows that these methods can be delivered safely byperforming end-to-end patient specific quality assurance for a series of test cases.Delivery verification is further developed in Chapter 3, which presents a machine-specific quality assurance for accurate characterization of the couch rotational ac-curacy.In Chapter 4, this thesis explores the effect of allowing the collimator angleto be a free parameter in the optimization. This is accomplished by implementingan altered version of the heuristic cost function developed by [98], and minimizedwith a constrained Djikstra graph search method. Additionally, the QA methodsdeveloped in Chapter 3 are extended to the collimator.In Chapter 5, the work in the main results of this thesis are briefly summarizedand directions of further inquiry are suggested.46Chapter 2Couch-Gantry Trajectory-basedStereotactic RadiosurgeryTreatments2.1 IntroductionIn cranial SRS, highly conformal ablative radiation is delivered to small, well-defined targets in the brain in a single fraction. For an SRS technique to be success-ful, dose to the target should be highly conformal with rapid dose falloff outside thelesion. Commonly used treatment modalities for these deliveries include special-ized devices such as Gamma Knife and CyberKnife [43], as well as conventionalC-arm linear accelerators. In conventional C-arm linacs, the high degree of targetdose conformality and rapid dose falloff are achieved with multiple beam entranceangles that are typically accomplished with non-coplanar arcs. Two arc techniques,DCA [58, 87] and VMAT [21, 90], are used to achieve these dosimetric objectives.The DCA technique is a forward planning method that becomes increasinglycomplicated to plan and deliver when there are multiple targets, as is typical in thetreatment of brain metastases. Recently, VMAT is becoming increasingly adoptedfor treating multiple metastases with SRS [21, 90] as it uses an inverse planningstrategy. However, non-coplanar VMAT can be cumbersome to plan and deliver47since multiple couch re-positionings are required and the number of arcs and theplanes of inclination of the arcs are not considered during optimization. This worktries to mitigate these shortfalls by the use of simultaneous couch and gantry mo-tion that enables a time-efficient delivery while affording the planning algorithm alarge portion of the couch-gantry phase space.The use of simultaneous couch and gantry motion for SRS was first introducedin 1988 by Podgorsak and his colleagues at McGill University [75]. While theirmethod was found to have dosimetric and treatment time saving benefits, it did notgain a wide acceptance as none of the major linac manufacturers adopted the tech-nology. As linac manufactures opened up the degrees of freedom of the device,there came a renewed interest in this field. The first of which was conducted byYang et al. [99] who devised a method to create dynamic couch-gantry trajectoriesby implementing an algorithm which minimizes beam overlap with OARs. Thiswas accomplished by creating a beam overlap metric for each couch and gantrycombination, and then finding smooth paths through phase space which minimizesthis metric. MacDonald and Thomas [59] and Smyth et al. [86] applied this methodin cranial SRS and found that the method successfully improved target dose con-formality and lowered doses to OARs when compared with standard VMAT plans.However, these trajectories were not necessarily time-efficient or dosimetricallyoptimal as they were predefined before MLC and dose rate modulation. Addition-ally, they involved sporadic couch movements, which may be uncomfortable, andpotentially result in intra-fraction motion.In this work, we propose and validate a method that uses a patient-generalizedtrajectory that approaches 4pi geometry, and thus approximates a fully sampledtrajectory. We also present preliminary analysis on creating time optimal trajec-tories, while still maintaining the treatment plan quality, by developing methodsthat systematically remove portions of the beam trajectory while not significantlycontributing to the dose delivery. These treatments were compared to the dynamicconformal arc method and were dosimetrically validated by delivery on the True-Beam linac via Developer Mode.482.2 Methods2.2.1 OptimizationThe developed optimization protocol has three main components (illustrated in Fig-ure 2.1). First (Figure 2.1a), a patient-generalized trajectory is constructed which iswell suited for cranial indications. Next (Figure 2.1b), this trajectory is fed into anoptimization framework which optimized MLC and dose rate configurations alongthe input trajectory. Finally, the spatial sampling frequency is optimized to ensurea time-efficient delivery (Figure 2.1c-d). These three features combine to produceboth a dosimetrically optimal and time-efficient trajectory. The details will be dis-cussed below.The Couch-Gantry TrajectoryThe central feature of this method is the couch-gantry trajectory in which the couchrotates through 180 degrees while the gantry makes 2-8 partial arc sweeps acrossthe cranium (illustrated in Figure 2.1). As the number of partial arcs increases andthe beams begin to overlap, this trajectory increasingly samples 4pi geometry. Thetrajectory is patient generalized and has a reproducible beam geometry for patientspecific QA. Additionally, while this trajectory allows complete sampling of thephase space, plans are optimized to be patient-specific by variable MLC positionsand dose rates that are calculated to provide maximal OAR sparing and conformityto the target.The couch-gantry trajectory is formed using a trajectory generating function:G =−85cos(N×C)+90 if C < 00 if C = 085cos(N×C)+90 if C > 0(2.1)where G is the gantry angle, C is the couch angle (on the interval [-90,90]), andN is the number of partial gantry arcs (coordinates defined in IEC 61217 [18]).For illustration of this technique, refer to Figure 2.1. When N is set to three, thetrajectory in Figure 2.1 c,d is produced, while when N is set to eight the trajectoryin Figure 2.1 a,b is created. The amplitude of the sinusoid was set to 85, which was49Figure 2.1: An overview of the Optimization Process. a)A predefined tra-jectory which fully sampled 4pi geometry is fed into the optimizer. b)In-house optimization algorithms find the most optimal MLC and doserate combinations for a given patient geometry and cost function. c)Thesampling of the phase space is reduced as much as possible without re-ducing plan quality metrics or cost. d) New MLC sequences are selectedusing the optimization algorithms in b.set lower than 90 because at gantry angle 90, all of the beams would overlap as thegantry would be positioned vertically. In this chapter, one couch gantry trajectorywas centred on each of the lesions.Dose Rate and MLC modulationTreatment plans were optimized using in-house software that was written in MAT-LAB and based on the direct aperture optimization progressive sampling algorithm50described by Otto [71]. The base trajectories were loaded into the optimizer as a setof static control points that designate couch and gantry positions. The optimizer setup the initial condition of optimization by sparsely sampling the trajectory at a re-duced set of control points, which were evenly spaced by 40 degrees. At these con-trol points, the MLC (Varian HD120) was set to conform to the target with a 0 mmmargin. Doses for each control point were calculated using an in-house MATLABimplementation of the pencil beam convolution algorithm. Doses were calculatedonly for critical structures and normal tissue within a 3 cm margin around the PTVso as to reduce computation time. Control point doses were uniformly scaled suchthat the target was covered by the prescription isodose. Once each control pointwas initialized, a scalar cost function (Equation 2.2) was evaluated which relatedthe dose delivered to clinical variables of interest.Next, the initial control points were perturbed stochastically in MLC positionand dose rate (DR). Initially, perturbations were large to ensure the plans avoidedlocal minima, however as optimization progressed, perturbation sizes were linearlydecreased so that minimal cost values could be found. After each perturbation, thecost function was re-evaluated, and if the perturbation was found to reduce the costfunction, it was kept. Otherwise the previous value was retained, and a subsequentperturbation was resampled. As optimization progressed, new control points wereintroduced as linear interpolations of the adjacent control points so as to ensure acontinuous delivery [71]. Additionally, the physical limitations of the device weretaken into consideration for the sampling of the perturbations: only MLC and DRperturbations which could be physically achieved in a continuous gantry-couch arcwere sampled from. For MLC positions, this was defined by the max velocityand for DR, this was the maximum DR given the beam settings. The optimizationwas conducted for 20 minutes with additional control points being added in evenlyspaced increments of 3 couch-gantry degrees.Delivery Time OptimizationDelivery time was optimized by variably sampling the phase space and calculatingcompeting plans. If the plans redundantly sampled the phase space due to the beamtrajectories overlapping, then the treatment time would be increased unnecessarily.51To find the time optimal sampling of the phase space, couch-gantry trajectorieswith varying numbers of partial arcs were constructed. At first, eight partial arcswere used, as this corresponds to a near complete sampling of the phase spaceand gives the treatment planning algorithm a benchmark to compare other plansagainst. Next, the number of partial arcs (N) were varied between eight partial arcs(Figure 2.1a, a near-complete sampling of the phase space) to two partial arcs (aPodgorsak trajectory [75]) in an increment of 0.5 arcs. Each of these plans wereoptimized and then compared. The plan which provided the most sparse samplingof the phase space while producing the same dosimetric result was selected.Cost Function CalculationThe cost function was designed to include the most clinically relevant variables forSRS delivery: doses to OAR, PTV dose conformity, and dose falloff. Doses toOARs were represented as user-assigned dose-volume constraints and relative im-portance factors. The calculation of the cost (C) was conducted using the followingequation:C = [0.5+0.25(CI−CICon)2+0.25(H(FO−FOCon)2]×n∑i=1wi(DV HConi−DV HAchi)2(2.2)where CI is the calculated conformity index; CICon is the optimal conformity index(which was set to 1.0); FO is the calculated dose falloff; FOCon is the constraintfalloff, which was set to 2.0; and wi is the weight for a particular DVH constraintwhich was set to unity for this treatment planning study and judicially changed ifneeded.Doses to OAR and PTV as a function of DVH values were assigned a weight of0.5 of the cost associated with a particular plan, while conformity and dose falloffwere each assigned a weight of 0.25. These weighting parameters (0.5, 0.25, 0.25)were found by manual manipulation to find consistent and high quality optimiza-tions. These calculations used Paddicks conformity index (CI) [73] (Equation 1.12)52and a dose falloff (FO) [74] variable that was defined as:FO =V50%Twhere T is the volume of the target and V50% is the volume that receives 50% ofthe prescription dose.If the achieved DVH, conformity index (CI), or fall off (FO) value was belowthe constraint value, then it was removed from the cost function. This formulationof the cost function allows one to control the relative importance of CI and FOobjectives, while the DVH constraints are of unconstrained magnitude. The CI andFO difference terms represented in equation 2.2 will be of order unity so they eachhave approximately 25% contribution of the total cost. While there are other waysof achieving this goal (ie. adding the conformity index in quadrature, but withvariable weighting terms, or various methods described by [42]), we found thismethod successfully and consistently produced plans of sufficient quality withoutany need of extra variables or complex methods.2.2.2 Plan ComparisonsPatient SelectionTen patients (summarized in Table 2.1) that were previously treated in 2014 withDCA (6 MV beam, planned using iPlan BrainLab AG) at our institution were se-lected for this study. These patients were anonymized and re-planned with theTVMAT method. Patient selection was designed to encapsulate a wide variety ofcases to account for different planning considerations. There was diversity in dis-ease sites with three accoustic neuroma (AN), three single metastasis (met) andfour multiple met cases. Lesion size varied from 0.3 cm3 to 12 cm3 with a medianvolume of 3.4 cm3. Tumor dose varied between 12 Gy to 24 Gy delivered in a sin-gle fraction treatment. The original DCA plans were planned using the iPlan TPS(BrainLAB AG) with one isocentre at the centre of mass of each PTV. The DCAplans were exported to the Varian Eclipse treatment planning system, and dosecalculations were performed using Varian anisotropic analytical algorithm (AAA)without homogeneity correction. Each plan had between three and nine partial arcs53per plan which took approximately 15 to 40 minutes to treat. For comparison pur-poses, plans were also replanned using the ”4 Arc” VMAT geometry presented in[90] with the same optimization methodology (cost function, treatment planningalgorithms) of this study.Table 2.1: Patient summary couch trajectory optimization studyNumber PTV Vol-umeNumberofPrescriptionDosePatient Siteof PTVs (cc) DCAArcs(Gy/Fraction)1 AN (R) 1 3.3 3 12/12 AN (L) 1 5 4 12/13 AN (R) 1 4.6 3 12/14 Met 1 0.9 3 18/15 Met 1 6.4 3 18/16 Met 2 7.7, 12.5 3 15/17 Met 2 4.6, 3.4 5 18/18 Met 2 5.8, 2.2 6 18/19 Met 3 2.6, 2.5,1.79 18/110 Met 3 4.7, 0.4,0.37 15, 24,24/1AN ∼ Acoustic Neuroma, (R) ∼ Right, (L) ∼ Left, Met ∼MetastasisTreatment ComparisonThe primary goal in the development of this method was to reproduce the dosimet-ric results of the DCA method, then judicially try to outperform them in a selectsubset of clinical variables. To do this, the OAR portion of the cost function werebased on the clinically achieved outcomes of the original DCA plans. The OARsDVH of the DCA plans were discretized into a set of constraints which were loadedinto the optimizer. DVH dose constraints for the OARs were scaled by 90% of theclinically achieved values, both to account for minor differences between dose cal-culation engines (our in-house code and Varian AAA) and to ensure an outperfor-mance of the clinical plans. The optimal conformity index, CIOPT , was set to unity,which corresponds to only the PTV receiving the prescription dose. The value for54the most optimal falloff (FOOPT ) was set to two.Treatment Time ComparisonTreatment time was calculated and compared for the acoustic neuroma patients (pa-tient ID 1-3). The delivery was modelled as the linear interpolation of the controlpoints. The assumptions on the gantry mechanical specifications are enumeratedin Table 2.2). For each subset of the delivery (defined by two control points), themost constrained degree of freedom would be identified. For example, if betweenthe first and the second control point the gantry rotates 2 degrees, while 30 MUare delivered and the MLC and couch are static, then the time for gantry rotationwould be 0.33 seconds, while the MU delivery time would be 3 seconds. Thereforethe delivery time would be limited by the dose rate and 3 seconds would be addedto the delivery to account for this segment. This was conducted for all control pointsegments. Two methods were compared: maximal sampling (eight arcs), and theoptimal sampling defined as the trajectory which produced a minimal cost valuewhile having the fewest number of arcs.Table 2.2: Maximum velocity model used to estimate delivery time.Degree of Freedom Max VelocityGantry Rotation 6 degsecDose Rate 600 MUminMLC Leaf Velocity 3 cmsecCouch Rotation 3 degsec2.2.3 Validation of DeliveriesThe optimized plans were exported as static beams to Varian Eclipse, and doseswere calculated using the AAA with dose voxel spacing of 0.1 cm. These controlpoints were spaced every 3 couch-gantry degrees. We tested whether this controlpoint spacing was sufficient by up-sampling the delivery, and comparing the dose55distributions. While control point weights were maintained, plans were renormal-ized so that the minimum dose to the PTV was the prescription dose (plan qualityset to 1). A subset of the patients (patient 1, 2 and 6) were selected for dosemeasurement verification. Due to the fact that Varian has not yet released dynamiccouch motion for clinical deliveries, these measurements were performed in Devel-oper Mode. Prior to delivery, machine commissioning procedures were conductedto ensure the linac was within tolerances for SRS deliveries. Dynamic couch picketfence analysis [101] (Section 1.5.1), as well as an isocentre stability measurement[30] (described in Chapter 3), was conducted. For the dynamic picket fence test,film was placed on the couch, and the couch and collimator were rotated at thesame velocity while the MLC produced the picket fence pattern. Next the film wasreplaced, and the measurement was repeated without movement of the couch orcollimator. The film was compared and the picket fence pattern produced by eachmethod were indistinguishable from each other. Using the stability measurement[30], we found a max couch-isocentre wobble error of 0.4 mm with mean value of0.2 mm.Plans were exported as control points and translated into xml format. Next, thetreatments were delivered on a cube phantom measuring 18.5x18.5x18.5 cm3 forion chamber and film measurements. Ion chamber measurements were performedwith an IBA CC01 chamber with sensitive volume of 0.01 cm3. GaFchromic EBT3film measurements were performed along the sagittal and the coronal planes pass-ing through the isocentre. Gamma analysis [56] was conducted using 2%, 2 mmpassing criterion with a 10% minimum dose threshold.Trajectory log analysis of the delivered plans was conducted and compared tothe trajectory beam parameters of the treatment plans. This was done by compar-ing the expected positions of the beam axes (patient support angle, gantry rotationangle, MLC leaf positions) to the axes position that were recorded in the trajectorylog files during the delivery of the plans. We compared the deliveries at each con-trol point of the delivered plan. Each control point had a cumulative MU, couchangle, and gantry angle (and other beam parameters not compared in this study).The cumulative MU of each CP was found in the trajectory-log time series (± 0.01MU), and the recorded gantry and couch angles at these time points were comparedto the expected values. The root mean squared error (RMSE) was calculated for56these parameters.2.3 Results2.3.1 Treatment ComparisonTreatment Comparison to Dynamic Conformal ArcsOverall the developed TVMAT method was able to produce plans with similaror better dosimetric indices when compared to the DCA plans. Dose distributioncomparisons are shown in Figure 2.4,Figure 2.5, Figure 2.6, and Figure 2.7 whichcorrespond to patients 3,1,6, and 9. One can see that TVMAT produced moreisotropic falloff. Additionally, the prescription isodose conformed more closelyto the PTV. For this particular plan, dose rate modulation was successfully usedto subtly reduce dose to the abutting brainstem structure while not compromisingother planning metrics.An overview of the PTV and normal tissue dose statistics are shown in Ta-ble 2.5. Planning metrics varied widely due to the variation in the location, size,and number of PTVs for each respective patient. When comparing TVMAT to theDCA method using the wilcoxon sign-rank (WSR) test, we found an increase indose conformity from 0.65 to 0.72 (p<0.01), with an average improvement (mean± 2 SE) of 10± 2% . Dose falloff results decreased, but not significantly. TVMATplans had a mean of 3.12 while DCA plans had a mean of 3.27, which amounted toan improvement of 4 ± 2% between the two treatment options. Dose homogeneityindices were similar for both techniques with an average value of 1.23 for TVMATand 1.27 for DCA (% improvement = 3 ± 2 %). If one refers to Table 2.5, one cansee the majority of patients had an improvement in V4 (p<0.05) and V12 (p<0.01)values. This resulted in a relative improvement of 20 ± 10 % for V4Gy and 27 ± 10% for V12Gy. An overview of the significant improvements are shown in Figure 2.2.Doses to OAR varied widely between treatments due to location of the indica-tions with respect to the organs. Therefore it was not possible to find any trends indosimetric values with the sample size used in this study. The DCA plans, VMATand the TVMAT plans all conformed to QUANTEC [9] values. Additionally there57Figure 2.2: Boxplots of the variables of interest (conformity index, V12 andV4) which were statistically significantly different between TVMATand DCA plans. Boxes show mean, quartiles, maxima, minima andoutliers (shown as dots). Variables are normalized to mean values of thepooled data sets. TVMAT plans are shown in blue, DCA in orange andVMAT in green.was a non-significant improvement in the volume weighted mean dose to OAR of13 ± 13 %. Further planning studies with larger patient numbers are required tofind if there is a relationship between this method and reduction of OAR doses.Treatment Comparison: Volumetric Modulated Arc TherapyThe TVMAT technique performed similarly to VMAT when one compares dosi-metric indices of Table 2.5. There were no appreciable differences between anyof the evaluated quantitative values of the two methodology. The differences aresummarized (average % difference ± standard error): conformity (0.7 ± 3 %),homogeniety (0.2 ± 0.7 %) , falloff (2 ± 2 %) , V12 (2 ± 2 %) and V4 (5 ± 3%). This similarity in dosimetric indices was expected due to the fact that the twomethods have very similar beam geometries. This suggests that the main benefitsof this technique over VMAT are only in efficiency of delivery.58Figure 2.3: Dose distribution for patient 3 (right AN) for TVMAT (left) andDCA (right). The PTV contour (red) and Brainstem (green) are shown.In addition, dose distributions are shown by yellow (100%), blue (80%),and orange (50%) a,b. Transverse slices show the dose distributionslook similar, but with the TVMAT plan able to slightly curtail the doseaway from the brainsteam. c.d. Sagital slices show both plans wereof similar quality. e.f. Frontal slices: both plans have similar falloff,however the DCA plan comes from a smaller subset of angles so onecan see the artefacts of more jagged falloff lines.59Figure 2.4: Dose distribution for patient 1 (right AN) for TVMAT (left) andDCA (right). The PTV contour is shown in red. In addition, dose dis-tributions are shown by yellow (100%), blue (80%), and orange (50%)a,b. Transverse slices c.d. Sagital slices e.f. Frontal slices2.3.2 Analysis of TrajectoriesFigure 2.8 shows the impact of the number of partial arcs on the optimized costfor selected number of cases. In all test patients, the optimization algorithm founda cost minimum with fewer than eight partial arcs. This suggests that the eightarc plan adequately samples the phase space and fewer arcs can produce the planswith the same optimized cost values. In the case of the three AN patients, fairly60Figure 2.5: Dose distribution for patient 4 (right AN) for TVMAT (left) andDCA (right). The PTV contour is shown in red. In addition, dose dis-tributions are shown by yellow (100%), blue (80%), and orange (50%)a,b. Transverse slices c.d. Sagital slices e.f. Frontal slicesreproducible results were observed patient to patient. In this subset of patients (Fig-ure 2.8a) the cost showed a minimum value at four partial arcs. Certain variables(shown in Figure 2.8 b) were optimized with even fewer partial arcs. The falloff,conformity, and homogenity were minimized at two partial arcs. This trajectorycorresponds to the one previously studied by Podgorsak [75]. In this trajectory,the gantry and couch both rotate at a constant velocity producing a baseball stitchpattern across the head. The Podgorsak trajectory reportedly produces a spherical61Figure 2.6: Dose distribution for patient 6, met 2 (Multiple Met) for TVMAT(left) and DCA (right). The PTV contour is shown in red. In addition,dose distributions are shown by yellow (100%), blue (80%), and orange(50%) a,b. Transverse slices c.d. Sagital slices e.f. Frontal slicesdose distribution with isotropic falloff [75].While the falloff, conformality, and homogeneity were optimized with fewerpartial arcs, the cost was minimized at four partial arcs for the AN patients. Thesedeliveries had an adequate sampling of the phase space such that they could simul-taneously avoid critical structures, while having enough entrance angles to providefalloff, conformity, and homogeneity.62Figure 2.7: Dose distribution for patient 9, met 1 (Multiple Met) for TVMAT(left) and DCA (right). The PTV contour is shown in red. In addition,dose distributions are shown by yellow (100%), blue (80%), and orange(50%) a,b. Transverse slices c.d. Sagital slices e.f. Frontal slices2.3.3 Treatment Time ComparisonThe three AN patients (patients 1, 2 and 3 chosen arbitrarily) were used to comparethe treatment time of the various methods. As is shown in Figure 2.8, plans forpatients 2 and 3 found the global minimum at only three partial arcs while patient1 needed four partial arcs. These treatments were used as the ”optimized” samplingbenchmark. A summary of the delivery time results are shown in Table 2.3. For63Figure 2.8: Analysis of the importance of trajectory on plan quality. Vari-ables are normalized to the values achieved for the eight arc plans. a)The cost function depence on number of partial arcs for acoustic neu-roma plans (patient 1, 2, 3 with data points shown in red, green, yellowrespectively). The blue line corresponds to the trend in the TVMATdata and error bars correspond to standard deviations from three roundsof TVMAT optimization. b) Optimization of clinical variables of inter-est for the AN patients. c) Cost function for 2 Met patients. It showeda similiar pattern as the AN patients, however found a minimum at fourpartial arcs instead of three. d) Optimization of clinical variables for 2Met Patients.the 12 Gy SRS plans, the fully sampled trajectories (those with eight partial arcs)had an average delivery time of 357 seconds. When the sampling of phase spacewas optimized, we found an average delivery time of 294 seconds. The time todeliver 2 Gy at the maximum dose rate for the fully sampled trajectories was 233seconds, while the optimized trajectories averaged 109 seconds. For large dose64Table 2.3: Beam-on time for competing optimization strategies.Radiosurgery (12 Gy) Radiotherapy (2 Gy)Patient ID Fully Sampled OptimizedSamplingFully Sampled OptimizedSampling1 338 s 289 s 232 s 123 s2 370 s 305 s 233 s 103 s3 363 s 289 s 233 s 102 sMean 357 s 294 s 233 s 109 sper fraction, the dose rate will have a significant effect on treatment time. Linacsoperating in 10 MV flattening filter free (FFF) can achieve a dose rate of 2400MUmin . If the technique developed in this thesis were to be applied with a 10 MVFFF beam, then the relative treatment time reduction would be comparable to whatis observed for the 2 Gy, 600 MUmin treatment. Conversely, the DCA and VMATtreatments took between 720-900 s as they involved multiple high dose arcs (whicheach take approximately 2 minutes, and multiple couch kicks (which also each take2 minutes).2.3.4 Validation of DeliveriesThe results of the dosimetric measurements for patients 1, 2, and 6 (chosen arbitrar-ily) are presented in Table 2.4. Isocentric ion chamber measurements were within2% of measured values for all patients. Uncertainties were mainly attributed touncertainties in small field delivery and the variation of the chamber response withbeam angle. Film measurements provided dose distribution information whichagreed well with the expected values. Sample dose distribution and profile datais shown in Figure 2.9. Dose distributions were compared with gamma analysis(Section 1.5.3) (2%, 2mm passing criterion) and achieved a 98% passing rate onaverage (Table 2.4).For each of the subsequent deliveries, trajectory logs were collected. Therecorded couch and gantry angle were compared with the expected couch andgantry angles. The root mean square deviation of these values were compared.Interestingly, the trajectory log recorded gantry and couch values were an order ofmagnitude closer to their expected values than the machines set tolerances.65(a)(b)Figure 2.9: a. Dose distribution comparison in coronal plane (film (dottedline) vs treatment plan (solid line)) for patient 2. Plan was scaled to1/5 of the actual value to have doses in the most accurate range for filmmeasurements. b. Vertical profile comparison for the same treatment.66Table 2.4: Plan Quality Assurance MetricsPatientIsocentre Dose (cGy) Gamma Analysis Trajectory Log AnalysisAAA IonCham-ber% Dif-ferenceCoronal Sagittal Couch(RMSE)Gantry(RMSE)1 1340 1342 -0.1% 96% 100% 0.046 0.0492 1434 1462 -1.9% 97% 99% 0.052 0.0426.1 1473 1452 1.4% 96% 99% 0.048 0.0506.2 1712 1675 2% 99% 98% 0.041 0.0482.4 DiscussionThe TVMAT technique presented here is an inverse planning method that pro-duces an optimal treatment plan by using MLC and dose rate modulation along apre-defined, over-sampled trajectory. Via dose rate modulation, the optimizationtechnique indirectly determines an optimized beam trajectory by allowing beamdelivery only for optimal beam entrance angles. Our preliminary treatment plan-ning study has shown the dosimetric advantages of TVMAT when compared to theDCA technique due to the increase in conformity, homogeneity while maintainingfalloff and OAR doses.This method attempts to minimize patient discomfort and movement by con-straining the device and treatment plan to have couch rotations in the same directionand, theoretically, have minimal inertial forces acted on the patient. This will bemore comfortable and quicker than multiple static arcs, as the patient will have toundergo shorter treatments with fewer accelerations. However, while the deliveriesin this study tried to limit the accelerations of the couch, the linac control systemallows only for the specification of the location of the linacs degrees of freedomin the form of control points, leaving velocities and accelerations up to the controlof the device. If one wanted to truly limit accelerations felt by the patient, linacmanufacturers would need to release control of these features.The treatment couch-isocentre wobble error can affect the accuracy of theTVMAT delivery technique and should be accurately characterized during com-missioning of this technique. The couch-isocentre wobble error of the TrueBeamlinac on which this study was conducted was less than 0.4 mm. While this error67is relatively small when compared with patient setup error, it should still be con-sidered in the determination of the PTV margin. If couch-isocentre wobble erroris a significant contributor to isocentre localization accuracy, then dynamic couchdeliveries should not be conducted due to the inability to correct for these errors asone possibly could in static couch deliveries.Some of this work is preliminary. Further improvements to the optimizationmethod and a more comprehensive treatment planning study with larger samplesizes are needed to determine the full dosimetric benefits of this technique. Ad-ditional improvements of this method will incorporate single isocentre treatmentplanning for multiple metastases. However, for the single isocentre technique, ac-curate rotational accuracy in patient setup is of paramount importance as smallrotational errors can result in large dosimetric errors when the PTVs are far fromthe isocentre [81]. Preliminary results presented in this paper in trajectory loganalysis suggest that the machine delivery inaccuracies will be insignificant whencompared to patient setup inaccuracies, however the accuracy of these trajectorylogs have not been indepently verified and these results should only be used as aconsistency check. Once implemented, these treatments will allow the treatmentof larger number of targets (more than 3) [69] in a time efficient manner.2.5 ConclusionWe have developed and validated a trajectory-based dose delivery method whichhas dose distribution improvements while having a treatment time of between 3 to8 minutes. Additionally, it has the potential to make the way for a more efficienttreatment planning process while maintaining an accurate delivery on the VarianTruebeam Linac.68Table 2.5: PTV and normal tissue statisticsPatientCI FO HI V12 V4TVMATDCA* VMAT TVMATDCA VMAT TVMATDCA VMAT TVMATDCA*VMAT TVMATDCA*VMAT1 0.80 0.70 0.76 2.8 3.00 2.9 1.19 1.13 1.21 4.03 4.6 4.18 21.7 22.6 222 0.65 0.62 0.72 3.2 3.00 3.1 1.29 1.23 1.25 7.8 8.0 6.9 45.5 38.3 39.73 0.81 0.70 0.78 2.60 2.90 2.66 1.23 1.26 1.27 5.6 6.5 5.86 28.8 31.4 28.84 0.63 0.61 0.69 3.76 4.00 3.55 1.25 1.23 1.22 3 10.0 2.9 17 51.8 14.25 0.76 0.70 0.82 2.60 2.66 2.4 1.21 1.34 1.22 15.5 15.7 14.4 67.2 78.6 61.36.1 0.67 0.69 0.71 3.17 2.70 2.8 1.26 1.26 1.2240.0 42.5 38.8 210.9 205.9 201.16.2 0.77 0.65 0.73 2.50 3.12 2.52 1.20 1.29 1.217.1 0.64 0.50 0.78 3.30 3.01 2.81 1.30 1.42 1.322.3 31.4 20 99.0 155.3 91.67.2 0.80 0.71 0.83 2.70 2.62 2.61 1.20 1.27 1.248.1 0.98 0.91 0.71 2.03 2.10 2.7 1.19 1.33 1.2221.0 20.9 23.37 113.3 147.0 1108.2 0.64 0.64 0.59 3.50 3.25 3.5 1.18 1.27 1.179.1 0.73 0.59 0.71 3.00 4.52 3.32 1.26 1.31 1.259.2 0.71 0.64 0.62 3.45 3.23 3.33 1.22 1.30 1.23 22.1 25.8 22.5 134.9 242.1 1579.3 0.73 0.62 0.70 2.99 3.34 2.94 1.20 1.25 1.1910.1 0.53 0.53 0.57 4.12 4.12 3.7 1.36 1.29 1.2910.2 0.65 0.55 0.68 4.35 4.88 4.00 1.30 1.21 1.27 18.0 19.8 16.6 113.6 165.8 101.110.3 0.55 0.49 0.60 5.13 6.06 4.96 1.22 1.27 1.22CI ∼ Conformity Index, FO ∼ Fall off, HI ∼ Homogeneity index, V12 ∼ The volume that receives 12Gy (in cc). V4 ∼ Thevolume that receives 4Gy (in cc). * shows statistical difference from treatment modalities.69Chapter 3Machine-Specific QualityAssurance Procedure forStereotactic Treatments withDynamic Couch Rotations3.1 IntroductionModern linacs require accurate mechanical specifications to meet the needs ofevolving precision techniques developed for SRS and SBRT. For SRS and SBRTtechniques, the report of AAPM Task Group 142 [47] (TG142) recommends thatthe accuracy of the linac isocentre to be less than 1 mm and the couch rotational ac-curacy to be better than 1 degree. However, modern SRS treatments are becomingincreasingly complex and leverage features that were not considered when TG142was written. One such feature is the incorporation of single isocentre treatments formultiple brain metastases [16, 17, 38, 40, 69, 89]. These treatments have PTVs farfrom the isocentre where positional and rotational inaccuracies of the isocentre willmanifest themselves in greater magnitude. Other technology not considered in theTG142 report include the use of dynamic couch rotations [26, 59, 76, 86, 96, 99]which require the accuracy of the linac to be maintained under rotations of the70couch while the beam is on. Additionally, trajectory logs can be used in the clinicto validate treatments [2, 88], however the couch angle recorded in these logs hasnot been validated in the static or dynamic case. Finally, as the accuracy of imageguidance techniques continue to improve, more accurate couch-based QA meth-ods need to be developed to guarantee that machine precision does not become asignificant obstacle to the quality of treatments.There are several methods developed to measure the linac isocentre. One ofthe most widely accepted methods is the star-shot method [23, 31]. This methodinvolves placing radiosensitive film in the plane of rotation, exposing the film tonarrow fields at different angles and measuring the fields’ overlap. However, thismethod is labour intensive and lacks accuracy as field symmetry is assumed in theanalysis. Another method is the Winston-Lutz (WL) method [58], in which a metalball bearing (BB) is mounted on the treatment couch and precisely aligned withthe linac isocentre. A series of images (film or portal) are taken at various gantry,couch and collimator positions to ascertain the isocentre localization error. Whilethe WL method is a very reliable way of measuring the overall accuracy of thelinac radiation isocentre, it cannot discern couch walkout (isocentre misalignmentdue to couch rotations) from errors due to mechanical misalignments of the gantryand beam collimation system (jaws or MLCs).In this work, we present a method in which the isocentre localization accuracyis measured using a phantom which is comprised of five BBs. Using this phantom,we are able to quantify the linac’s localization error due to couch walkout, quantifythe accuracy of the trajectory logs, and quantify the linac’s ability to maintain theseaccuracies with intra-treatment couch motion. These features will have special rel-evance in the quality assurance of the next generation of SRS treatment techniquesthat involve treatments of multiple targets with a single common isocentre and de-liveries using dynamic couch rotations.3.2 MethodsThe couch isocentre accuracy of a Varian TrueBeam STx Linac (Varian Inc., PaloAlto, California) was evaluated using its EPID and a specialized phantom con-structed for this study. The phantom is a polystyrene slab in which five stainless71steel BBs of 4 mm diameter are placed on the same plane. One BB is placed at thecentre of the polystyrene slab, whilst the remaining four BBs are placed at varyingradial distances and angles in the same plane (radii = 0, 2.8, 4.4, 5.6, and 6.7 cmfor the five respective BBs). This phantom is shown in Figure 3.1. Multiple MVimages of the phantom were acquired at varying couch angles. Algorithms weredeveloped to characterize the couch rotational accuracy from the locations of theBBs identified in the images.3.2.1 Set-up and MeasurementFigure 3.1 shows the phantom setup on the treatment couch with the BrainLabcouch mount (BrainLab, Munich, Germany) which can accommodate 5 degreesof adjustment: tilt, roll, lateral, longitudinal and vertical directions. The phantomwas positioned such that the plane of the BBs was horizontal, while the central BBwas aligned with the room lasers defining the nominal linac isocentre. The gantrywas in the vertical position, and a field size of 20x30 cm2 was used to acquireimages with a 6 MV beam operating at 600 MU/min. The EPID was positioned atthe furthest distance from the source (182 cm) so as to achieve the highest spatialresolution possible.Two treatment modes were considered in this study: static and dynamic de-livery. For the static case, a total of nineteen EPID images were acquired usingthe “High Quality” MV imaging mode. This imaging method had a resolution of1024x768 which translated to an image pixel size of 0.215 mm at the isocentreplane. Images were acquired every 10 degrees as the couch was rotated throughits full range of rotation [-90◦, 90◦]. For the dynamic case, images were acquiredusing the“Continuous” imaging mode in which the MV image readout is synchro-nized with the pauses between beam pulses: the image is read out line by lineuntil the entire image is acquired (at an approximate frequency of 7 Hz). Thismethod has the same resolution and accuracy as the“High Quality” method, if onediscounts pixel blurring introduced by movement of the phantom intra-image ac-quisition. In separate measurements, couch rotation accuracy at varying angularspeeds was measured: max (3 degrees/s), half-max, and static were considered inthis study.72Figure 3.1: Aerial view of the BB phantom mounted to the treatment couchusing the Brainlab couch mount. The phantom consists of five BBs(BB1−BB5) affixed to a polystyrene slab. Each BB is located at differ-ent radial distance from the isocentre, with the central BB located at theisocentre. The linac is oriented vertically with the EPID deployed.73Table 3.1: Glossary of mathematical notation.Symbol DefinitionA The transformation matrix which represents the BBs’movement.θ The calculated couch angle for a given image.~Ro The calculated couch rotation centre in the xy plane.S The radial scaling factor which maximizes the overlap ofthe BBs between rotations.~BBi,θ The x,y coordinate of the ith BB in the reference image fora given angle θ .dRx, dRy and dRz The isocentre misalignment in the cross-plane, in-plane andout-of-plane directions.~L A horizontal line which extends from the nominal isocentre.~LSS A calculated surrogate of the star-shot lines.3.2.2 Data AnalysisThe acquired images were exported in digital imaging and communications inmedicine (DICOM) format and were loaded into in-house MATLAB (The Math-works, Inc. Natick, Massachusetts) analysis software that was developed for thisstudy. The analysis software was used for image segmentation, for localization ofthe BBs, and for calculation of the relevant geometric quantities. The location ofeach BB was identified in the collected imaging sets by first applying a thresholdto the raw EPID images. The centroid of each BB in the thresholded image wasthen calculated and represents the BB location in our analysis.Once BB locations were identified, fiducials were matched to one another us-ing the radius from the centre as a unique identifier. Variables of interest werecalculated using the BB locations as inputs. These variables, as well as all othernotations, are listed in Table 3.1.Determination of the Couch Rotation CentreThe location of the five fiducial markers at each couch angle were compared tothe fiducial locations in the couch zero position. This comparison was conductedusing a MATLAB implemented non-linear least squares optimization (MATLAB’slsqnonlin function) to find a representative transformation matrix (A) which encap-74sulates the rotation and translation information of the fiducials. This transforma-tion matrix was constructed by minimizing the squared error between the fiducialsusing angle conserving scaling, rotations and translations. Mathematically, thesetransformations were represented as: x′y′1= A(S,θ , ~Ro)× xy1 (3.1)A(S,θ , ~Ro) =1 0 Rx0 1 Ry0 0 1×Scosθ −Ssinθ 0Ssinθ Scosθ 00 0 1×1 0 −Rx0 1 −Ry0 0 1 (3.2)where ~Ro = [Rx,Ry,1] represents the couch rotation centre, S represents an an-gle preserving scaling factor, θ represents the calculated angle of couch rotation,~BBi,0 = [x,y,1] represent the cross-plane (x) and in-plane (y) coordinates of the ithBB at couch angle 0, and ~BBi,θ = [x′,y′,1] represent the cross-plane and in-planecoordinates of the ith BB in the image acquired at a couch angle θ . The variablesS, θ , and ~Ro were found such that the following constraint was minimized (anoverview of this analysis is illustrated in Figure 3.2):argminS,θ ,~Ro5∑i=2|| ~BBi,θ −A(S,θ , ~Ro) ~BBi,0||2 (3.3)Once these parameters were found by least squares minimization, they wereused to calculate variables of interest. Using the fact that the BBs lie in a plane,and the photon beam can be modelled as a divergent point source, the vertical offsetwas calculated:dRz = (1−S)DSAD (3.4)where S is the scaling factor, dRz is the isocentre vertical offset error, and DSAD isthe source axis distance, set to 100 cm for this study. The isocentre localizationerror in the xy plane can be calculated by applying the formula :~dRxy = ~Ro− ~BB1(θ=0) (3.5)75Figure 3.2: Illustration of the mathematical analysis for BBs 1 and 2. Thenominal linac isocentre is shown as a cross (labelled BB1(θ=0)) and theinitial BB2 location is shown as a black circle (x2,y2). After rotation ofthe couch by a given angle, the movement of BB2 (grey circle) can berepresented as a rotation by angle θ about the centre of rotation (crosslabelled by ~R0). Additionally, a radial scaling S about the centre of ro-tation accounts for any out of plane movements. ~dRxy represents thedifference between the nominal isocentre and the centre of rotation. Il-lustration not to scale.where ~Ro is the couch isocentre calculated from the above methodology and~BB1(θ=0) is the nominal linac isocentre, defined as the pixel location of the cen-tral BB at couch angle zero.Comparison with Star-shot MethodAdditional analysis was conducted in order to present the couch isocentre walkoutin the same manner as is typically done in “Star-shot” analysis. We defined a linewhich projects horizontally from the isocentre:76~L = ~BB1(θ=0)+[x,0] (3.6)and then transform this line using the matrix A to find the equivalent “Star-shot”lines:~LSS = A~L (3.7)The “Star-shot” lines ( ~LSS) were calculated for couch rotations in 30 degree inter-vals, the points of intersection between each line were determined and the circlewhich encompassed the points was found (see Depuydt et al. [23] for similar anal-ysis). The mean intersection location and the centre and radius of the fitted circlewere compared with film-based star-shot analysis using FilmQA Pro (Ashland Inc.,Covington, Kentucky).Winston-Lutz MethodAn additional measurement was conducted in which both the WL and the multipleBB data were acquired in a single phantom setup. The WL measurement was per-formed by collimating a 1x1 cm2 field with the linac MLC and by acquiring EPIDimages for every 10 degrees of couch rotation. This process was repeated with thegantry above the phantom (0 degrees), and below the phantom (180 degrees). Forthe WL analysis, the location of the central BB was compared to the field centre asdefined by the midpoint of the field border. These deviations were compared to thecouch isocentre misalignment values ( ~dRxy) measured by the multiple BB analysismethod.Validation of The Trajectory LogsThe rotation angle (θ ) extracted by the method discussed in section 3.2.2 provideda method for the validation of the couch angle in the trajectory log. Images wereacquired while the couch moved dynamically and were subsequently exported inDICOM format. The DICOM image format provides each image with a time stampand the expected couch angle in the header of the file. The trajectory log containscouch angle readouts recorded throughout treatment at a sampling rate of 50 Hz,however, it does not contain a time stamp to designate the beginning of data col-77Table 3.2: Accuracy of the developed methods.Measurement (unit) Accuracy (95 % CI)d ~Rxy (mm) 0.07dRz (mm) 0.8θ (degrees) 0.05lection. At the beginning of each measurement, the couch was moved back andforth between positions -80, -90, and -80 degrees to produce a unique movementsignature. The relation between the trajectory logs and the collected images wasestablished by finding the time offset, dt, which minimized absolute difference be-tween the couch angle values from the two datasets for this movement. Using thismethod, the couch angles calculated from the images could be related to the tra-jectory logs. Errors were quantified by subtracting the calculated couch angle ofthe acquired images, from the angle recorded in the trajectory log. This analysiswas conducted for varying couch velocities: max couch velocity (3 degrees/s), halfmaximum velocity, and static deliveries.3.2.3 Accuracy of the ProceduresThe error of the BB localization algorithm was ± half a pixel along each of theimaging axes. The propagation of this source of error was calculated using a boot-strapping approach. Sample data (n = 1000) was created by transforming the setupBB locations about the isocentre by a known angle. These fiducial locations werethen displaced by a normally distributed random error with mean zero and standarddeviation of 0.25 pixels. The methodology of the previous sections was conducted,and the calculated values were compared to the expected values and the error wasreported as the 95% confidence interval of the resultant distributions. The accu-racy of each calculated parameter is presented in Table 3.2. As can be seen, thisphantom is very well suited for quantifying xy offsets (d ~Rxy) as well as rotationalerrors (θ ). Conversely, the least accurate measurement is the dRz, offset which hasan accuracy of 0.8 mm.78Table 3.3: Couch rotation centre offsets with respect to the nominal linacisocentre in the cross-plane, in-plane, and out-of-plane directions (mean± 2 standard deviations).Data Set Cross-Plane (mm) In-Plane (mm) Out-of-plane (mm)(x) (y) (z)Trial 1 0.3 ± 0.2 -0.2 ± 0.3 -0.1 ± 0.6Trial 2 0.3 ± 0.1 -0.7 ± 0.3 0 ± 0.8Trial 3 0.3 ± 0.2 -0.2 ± 0.3 -0.1 ± 0.63.3 Results3.3.1 Determination of the couch rotation centreMeasurements were taken at three distinct time points. The cross-plane (x), in-plane (y), and out-of-plane (z) isocentre position errors (section 3.2.2) were quan-tified for three unique datasets and are plotted against rotation angle in Figure 3.3.Summary statistics such as mean value and standard deviation are summarizedin Table 3.3. As can be seen, the three trials have mean and standard deviationthat overlap and are therefore statistically indistinguishable, with the exceptionof the second trial’s in-plane measurement, which contained an offset of 0.5 mmfrom the other two measurements. As the room lasers were not realigned in themeasurement period, the most reasonable explanation for this result is an inter-measurement variability of phantom setup.3.3.2 Comparison with the Star-shot MethodThe couch star-shot lines were calculated using the method described in Section3.2.2. A plot of the treatment couch star-shot lines (Equation 3.7) for dataset 1 isshown in Figure 3.4. The figure also contains a plot of the smallest circle encom-passing all the intersection points (blue circles) of the star-shot lines. The minimuminscribing circle had a radius of 0.34 mm, while the deviation of the center of thecircle with respect to the nominal linac isocentre had a magnitude of 0.14 mm (0.08mm and 0.12 mm in the cross-plane and in-plane directions respectively). The filmscan, as well as the output analysis, are shown in Figure 3.4 c and d. The star-79Figure 3.3: Deviation of couch centre of rotation as a function of angle cal-culated using the analysis in Equation 3.5. Dataset 1, 2 and 3 representthree independent measurements acquired one month apart. The solidline represents the data plotted for each couch angle (sampled every 10degrees) while the dotted line represents the mean of each dataset, aver-aged over all of the measured couch angles.shot measurements summarized in Table 3.4 are given as the radius of the smallestcircle inscribing all of the points of intersection of the star-shot lines, the distancebetween the linac isocentre and the centroid of the intersection points, the distancebetween the linac isocentre and the centre of the circle and the average rotationalerrors. As can be seen in Table 3.4, the two methods agreed within 0.2 mm forisocentre localization and 0.3 degrees for rotation calculations.80Figure 3.4: a. The lines represent the calculated EPID based star-shot linessampled every 30 degrees of couch rotation. The circle which boundsthe intersection of these lines is shown. b. Magnified version of figurea. The origin (0, 0 mm) represents the nominal linac isocentre. Thelines represent calculated star-shot lines, and the intersection points ofthese lines are shown as circles. The bounding circle is the smallestcircle which encapsulates all of the intersection points. The two crossesrepresent the centre of the circle and the centroid of the intersectionpoints. c. Raw GafChromic film data collected for traditional star-shotanalysis. d. Fitted star-shot lines and bounding circle for the film data.Table 3.4: Comparison of film-based with EPID-based star-shot measure-ments.Analysis Method BB Phantom GafChromic FilmRadius of Circle 0.34 mm 0.5 mmCentroid Distance to Nominal Isocentre 0.22 mm 0.20 mmCentre of Circle Distance to NominalIsocentre0.14 mm 0.2 mmRotation Error 0.05 degrees 0.3 degrees813.3.3 Winston-Lutz MethodThe in-plane and cross-plane deviations obtained from the WL measurements andthe corresponding couch rotation deviations obtained from the multiple BB mea-surements are shown in Figure 3.5. The WL deviations were (mean ± 2 standarddeviations) -0.4 ± 0.3 mm and 0.4 ± 0.3 mm in the cross-plane and in plane di-rections when measurements were taken with the gantry above the phantom whilethey had a measure of 0 ± 0.3 mm and 0± 0.3 mm when the gantry was belowthe phantom (IEC 61217). Conversely, the couch rotation centre deviations frommultiple BB analysis were 0.2 ± 0.1 mm and -0.1 ± 0.3 mm in the cross-planeand in-plane directions for both gantry orientations (IEC 61217). It is interestingto note that unlike the WL measurements, the multiple BB analysis results wereindependent of the gantry orientation and measures only the stability of the couchrotation axis, separate from the mechanical features of the gantry.3.3.4 Validation of the Trajectory LogsData were collected with the couch rotated at its maximum velocity (3 degrees persecond), half maximum velocity, and static. For the couch moving at its maximumvelocity, images were acquired continuously, and 489 data points were collected.There were eight images from which we could not extract BB location informationas the intensity of the beam changed intra-image acquisition. These data points areshown as magenta crosses in Figure 3.6. The couch angle recorded in the imageswere aligned with the trajectory logs in the time domain, and once aligned, themean absolute difference between the trajectory log recorded couch angle withthe angle in the header file was 0.002 degrees. There was no pattern observedbetween rotational error and couch position, and errors were randomly distributedaround zero (Figure 3.6). Figure 3.6 shows the pairwise differences between thetrajectory log (θT ), header file (θH), and calculated (θ ) couch angle for dynamiccouch movement at the couch’s max velocity (3 degrees/s). The difference betweenthe trajectory log and the calculated value was within measurement error of± 0.05degrees in the dynamic case. Similarly, for the static case, the difference betweenthe trajectory log and calculated value was 0.02± 0.04 degrees (mean± 2 standarddeviations). The analysis was repeated for the couch moving at half maximum82Figure 3.5: Cross-plane and in-plane deviations obtained from the multipleBB method and the Winston Lutz method measured in a single phantomsetup. The dashed lines represent the couch rotation centre deviations(dRx, dRy) obtained from the multiple BB measurement, and the solidlines represent the Winston Lutz deviations. Measurements were ac-quired for two gantry positions, 0 degrees (above phantom), and 180degrees (below phantom).83velocity, resulting in a localization error of 0.05± 0.04 degrees. This suggests thatthe linac was able to maintain its rotational accuracy even while the couch movedintra-treatment for varying couch velocities.Figure 3.6: (a) A plot of the differences between the couch angles recordedby different methodology as the couch rotated through -80 → -90 →+90 degrees at its maximum velocity over the course of 64 seconds (yaxis) . The trajectory log couch angle values (θT ) agreed with thoserecorded in the DICOM header files (θH) to the third significant digit.Additionally, the couch values recorded in the trajectory log and DI-COM header (θT and θH respectively) agreed with the calculated valueswithin 0.08 degrees. These errors were normally distributed with mean0 and standard deviation 0.025 degrees. Crosses show eight data pointswhich were excluded from image analysis due to intra-imaging changesin beam intensity. (b) The couch angle position as a function of deliverytime.843.4 DiscussionIn this study, a new EPID-based QA method is proposed for the treatment couch.The method is simple, accurate, and enables the user to access a multitude of com-plementary data with a single measurement. Of particular interest is this method’sability to simultaneously quantify the couch walkout in three dimensions, as wellas the couch rotational accuracy. These tests can be conducted efficiently due toease of set up and analysis. When compared to the traditional film-based star-shottechnique, the method proposed here offers several advantages. First, it is sim-pler in terms of setup and analysis as it does not require film. Second, it is moreaccurate since it does not depend on the accuracy of the field symmetry. Whencomparing this method to the WL method, it provides explicit values of the couchrotational accuracy in three dimensions, and WL measurements can be performedfor the same set-up using the central BB. Additionally, while the work presentedhere is focused on the treatment couch, similar methods can be directly appliedto collimator and gantry rotational measurements. These properties will be of in-creased importance for single isocentre treatment of multiple metastases for whichrotational errors can result in untreated regions of the PTV.Recently, trajectory-based techniques in which there is dynamic motion of thecouch and gantry have been developed for SRS and breast treatments [59, 76, 86,96, 99]. These treatments would require synchronous couch and gantry rotationduring delivery. One way of performing patient-specific quality assurance of suchdeliveries is by comparing the trajectory log data with the trajectory from the treat-ment plan. Before using the trajectory log data for this purpose, it is important toverify its accuracy. The methods to acquire the accuracy of the trajectory log withregards to MLC [2] and gantry rotation [62] have been developed by other groups.In this work, we presented a method for the validation of the couch angles thatare recorded in the trajectory log files. By conducting this method on our centre’sTrueBeam linac, we have demonstrated that the couch angular positions recordedin the trajectory logs are accurate to within 0.05 degrees. Furthermore, these errorsdid not seem to depend on the couch angular velocity. This suggests that the treat-ment couch tested in this work is accurate enough for dynamic couch treatmenttechniques reported recently [53, 59, 76, 86, 96, 99].85Some proposed delivery techniques [53, 59, 76, 86, 99] require accelerations ofthe couch intra-treatment. In our study, when the couch accelerated, the intensityof the beam changed intra-image, resulting in poor quality EPID images. Thethresholding method used for locating the BB positions was not able to reliablyfind all of the BBs in these images, making it impossible to compare the calculatedcouch angle (θ ) with the couch angles from the trajectory log and EPID imageheader. More robust imaging analysis techniques would need to be developed toextend the analysis for cases when the couch does not move at a constant velocity.Additionally, the difference between the couch angle values extracted from thetrajectory log and the EPID image header could be significantly larger in the couchacceleration region (∼0.04 degrees) when compared with those when the couchmoved at constant velocity (<0.01 degrees). However, this difference is still anorder of magnitude smaller than a clinically significant rotational error [81].The mean treatment couch isocentre offset was localized to be 0.3 ± 0.2 mmand -0.2 ± 0.3 mm in the cross-plane and in-plane orientations respectively awayfrom the nominal isocentre. This is due to the nominal isocentre being calibrated tothe isocentre of gantry rotations on this particular machine. For smaller errors, theroom lasers would need to be recalibrated somewhere in-between the mechanicallydefined gantry and couch isocentre.Whilst the measurement of the couch walkout in the z direction is novel, itwas shown to be the least accurate measurement in this work (with accuracy of ±0.8 mm). However, this method may still be capable of demarcating linacs whichwould either pass or fail the TG 142 criteria of 1 mm at the treatment isocentre ifits accuracy is improved. Future work could improve this accuracy by having morepoints farther from the beam central axis (which will decrease the standard error onthe calculated parameters) or by taking image sets at non-coplanar beam geometry.3.5 ConclusionsWe have developed an EPID based quality assurance method for the treatmentcouch which is simple, accurate, and enables the user to access a multitude of com-plementary data with a single measurement. Using this method, we have shownthat the TrueBeam treatment couch that was studied is accurate for both static and86dynamic stereotactic deliveries.87Chapter 4Collimator Optimization forVMAT Treatments of MultipleBrain Metastases4.1 IntroductionVMAT optimization was introduced by Otto [71] as a DAO approach in which thelinac gantry moves intra-treatment. This method was shown to be time-efficientand effective at reproducing dosimetric indices for certain static field IMRT geome-tries [78]. However DAO is a non-convex optimization approach and is more sus-ceptible than IMRT to incorrectly returning local minima after optimization. Onesuch treatment site where this frequently happens is the treatment of multiple brainmetastases with SRS. For these treatments, MLC contention issues (explained inFigure 4.1) provide multiple local minima in the cost function which reduce thefidelity of the optimization[98]. It has been proposed that these shortfalls may bemitigated by incorporating collimator optimization in the form of static collimatoroptimization [98] or dynamic collimator trajectory optimization [55, 60].In a landmark clinical trial by Brown et al. [13], SRS alone was shown to havesimiliar survival outcomes, but better cognitive function when compared with SRSwith whole brain radiotherapy (WBRT) adjuvant therapy. Furthermore, American88(a)(b)Figure 4.1: An example of MLC contention issues that may arise in the treat-ment of multiple PTVs with a single aperture. PTVs are shown in red,normal tissue is shown in yellow, the MLC is shown in blue and the fieldjaws are shown in black. (a) Collimator is rotated to 45 degrees and theMLC aperture is set to conform to the targets. A sizeable amount ofnormal tissue is being irradiated. (b) The optimal collimator angle oc-curs at -12 degrees. When the aperture is set to conform to the targets,the normal tissue is efficiently blocked by the MLC.89Table 4.1: Summary of Patient StatisticsPatient Number Prescription Dose (Gy) Number of Mets Total Volume (cc)1 40/5 7 5.82 40/5 6 3.23 35/5 4 7.34 40/5 4 3.25 35/5 4 5Society for Therapeutic Radiation Oncology (ASTRO) recommends “to not addadjuvant whole brain radiotherapy for stereotactic radiosurgery of limited brainmetastases” [27]. However, SRS without adjuvant therapy increases the probabil-ity of recurrence, and through this, the need for an increased number of salvagetreatments. These salvage treatments can be difficult to plan due to the constraintsset by QUANTEC [9], in particular limits on dose to normal brain, which has beencorrelated with radionecrosis. Hence, it is important that SRS treatments that donot have added adjuvant therapy should reduce the dose bath as much as is possibleto enable future salvage treatments.This work focuses on this goal by incorporating collimator trajectories intoDAO-VMAT treatments. It explores different optimization strategies with the hopesof reducing the dose bath. The strategies explored are (i) treatment planner-selectedstatic collimator, (ii) algorithm-optimized static collimator, and (iii) algorithm-optimized dynamic collimator.4.2 Methods4.2.1 Patient SelectionFive multiple brain metastasis patients who were treated at BC Cancer in 2017were selected for this study. The patients had between 4-7 metastases treated, andan overview is provided in Table 4.1. The patients were treated with a VarianTrueBeam STx linac equipped with a Varian HD120 MLC. The clinical planningprotocol uses two-three VMAT arcs planned using Varian Eclipse, with collimatorangles selected by the treatment planner.904.2.2 Collimator Angle Optimization MethodThe collimator angle was optimized pre-VMAT optimization using a heuristic ob-jective function function:AOpenFluence = A jaw−AMLC (4.1)where AOpenFluence is the are of open fluence; A jaw is the area of the jaw opening;AMLC is the area blocked by the MLC. The area of PTV would be considered as itwas a constant of integration in the shortest path optimization. This function wasthe area of open fluence when the MLC is initialized to conform to the PTV. Foreach MLC pair, the MLC would conform to the maximum extent of the projectionof the PTV onto the MLC plane.The area of open fluence greatly depended on collimator angle as shown inFigure 4.1 which shows two conformal MLC configurations for two possible colli-mator angles. For each possible collimator-gantry combination, the MLC was fit tothe structures and the area of open fluence was calculated. The optimal trajectorythrough collimator-gantry phase space was found using dynamic programming.The method of trajectory optimization was adapted from Locke and Bush [55],with a few modifications to constrain the collimator movement and to force thegantry motion to be on the typical single 360 degree arc trajectory. During the col-limator trajectory optimization process, the collimator rotation velocity was set tobe at maximum 6 degrees per second while the gantry’s velocity was set to 3 de-grees per second. This extra constraint on the collimator velocity (whose hardwarelimit is 15 degrees per second) was set in order to ensure the operational accuracyof the progressive sampling algorithm (see Section 4.2.4 for more information).The shortest path was found between all collimator angles at gantry -180 to all col-limator angles at gantry +180. This was calculated using Dijkstra’s shortest pathalgorithm, where each node of the path were collimator-gantry coordinates (dis-cretized in a grid of 2 degree spacing), and each link between nodes was definedby the the constraints of the linac movement.914.2.3 Treatment Plan Cost FunctionThe main goal of this work was to create a cost function whose minima representeda viable treatment plan, and then ascertain the ability of the optimization strategiesto produce this treatment. We accomplished this by using a simple cost functionthat was based on the clinically delivered plans. We formed the cost function byusing the original clinical plans as a template for the optimizations, with an addedreduction to the dose bath by the expected amount (30% reduction in low dosebath) proposed by implementing collimator optimization [98]. The cost function isgiven by:C = ∑structuresiwi∫H1(D(v)− c(v)D0(v))2dv+∑targetsiwi[∫H2(D(v)−Pmin)2dv+∫H3(Pmax−D(v))2dv](4.2)where Pmin is the prescription minimum dose; Pmax is the prescription maximumdose; D(v) is the DVH achieved in the optimization step; D0(v) is the DVH fromthe original plan; c is the clinical out-performance factor, set to 1 for the highdose region and to 0.7 for doses lower than 12 Gy (decided based on insight from[98]); wi is the weight of each organ, set to the volume of the structure for OARand to 1000 for targets; and H1 is the Heaviside function which equals to 1 whenD(v)− c(v)D0(v) is positive, and 0 when D(v)− c(v)D0(v) is negative. H2 and H3similarly represent the Heaviside function, but with D(v)−Pmin and Pmax−D(v) astheir respective inputs.4.2.4 Direct Aperture VMAT OptimizationSingle, couch zero, 360 degree arc VMAT plans were optimized using an in-houseMATLAB (The Mathworks, Massachusetts) implementation of the progressivesampling method described by Otto [71]. Control points were evenly spaced alongthe pre-defined gantry trajectory (or gantry-collimator trajectory if it was a dy-namic collimator treatment). Initial control points were initialized with conformalapertures and equal MU weighting such that the PTVs received at least the pre-scription dose. While flattening filter free beams have been shown to be equivalent92to flattened beams for VMAT treatments, a 6 MV beam was used in this study.Doses were calculated for each control point using an in-house implementationof the pencil beam convolution algorithm. Doses were calculated for beam aper-tures, and these apertures were optimized using perturbation methods. Initially,MLC leaves were individually perturbed and if the perturbation decreased the costfunction, then the new position was kept. Perturbations were sampled from theuniform random distribution with a width initially set to the full width of avail-able MLC positions (given mechanical constraints), and was linearly decreased asthe optimization progressed. Throughout optimization, additional control pointswere successively added in-between the initial control points along the designatedtrajectory. These control points were initialized as the linear interpolation of theMLC and beam-weights of the neighbouring control points. This sampling wascontinued until the entire trajectory was an accurate approximation of a continuoustrajectory (200 control points per 360 degree arc). As a final optimization step,beam monitor units were optimized using projected gradient descent (constrainedto positive monitor units).4.2.5 The Blocking of Fluences Incident on Normal TissueIn order to measure the dosimetric effect of the MLC contention issues on the op-timized treatment plans, we recalculated the trajectory optimized plans using anidealized MLC model. In this idealized model, the MLC would have no leakageand would be able to completely block the radiation to normal tissue that lies be-tween two PTVs. In this calculation, we recalculated the trajectory optimized planswith the PBC algorithm, however we blocked the open fluence when it did not over-lap with the PTV (plus a 5 mm margin to ensure target coverage was not affected).Once each beam was recalculated, the beam intensities were re-optimized to ensurethat the target dose was sufficiently covered (to account for scatter contribution ofthe blocked fluences).This method provided a measurement of the quality of the treatment plan thatcould be achieved when there were no MLC contention issues. While this a theo-retical limit, the plans created by this method would be dosimetrically equivalentmultiple single isocentre deliveries with a dynamic collimator set to conform to93each of the PTVs for each pass of the delivery. This style of treatment is techni-cally feasible (and therefore provides a good dosimetric treatment goal), however,it is cumbersome to deliver when there are more than three lesions.4.2.6 Treatment Plan ComparisonAll treatment plans were calculated and compared with the same pencil beam con-volution (PBC) calculation algorithm. A randomly selected prospective treatmentplan was recalculated using Varian AAA and the dose distributions were comparedusing (2%, 2 mm) gamma analysis criterion [22] with 20% dose thresholding. ThePBC algorithm had a 98 % pass rate and the algorithm was deemed accurate enoughfor plan-to-plan comparison.Treatment plans were compared on the basis of optimized cost, Vx, the volumethat received x dose or more in Gray, mean brain dose, and Paddicks conformityindex (CI) (elaborated in Equation 1.12).4.2.7 Quality Assurance of Dynamic Collimator DeliveryThe feasibility of dynamic collimator treatments was tested by developing a machine-specific QA technique that measured the collimator rotation intra-treatment withthe EPID. This technique is similar to the method developed for QA of dynamiccouch delivery introduced in Chapter 3. The MLC configuration (illustrated inFigure 4.2) produced a rotationally asymmetric MLC pattern that could be usedto calculate the angle of rotation from EPID images. EPID images were acquiredin “continuous” readout mode for which images are read out line by line at an ap-proximate frequency of 7 Hz. Our centre’s Varian TrueBeam linac (Varian MedicalSystems Inc., Palo Alto, USA), operating in developers mode, delivered a 6 MVbeam at 600 MU/min, while the collimator rotated intra-treatment.Similar to the methods developed in Chapter 3, the collimator angle was ob-tained from three different measurements: the EPID image DICOM header file(θheader), the trajectory log (θlog) and from an EPID based measurement (θcalc).θcalc was calculated by using intensity-based registration between the images col-lected at a given angle and the EPID image collected at collimator angle zero.The registration was conducted by finding the affine transformation that used ro-94Figure 4.2: MLC aperture used in machine-specific quality assurance of col-limator rotation. The MLC forms 5 square openings, three of 1 x 1cm2 and two of which are 1 x 0.5 cm2. The central square provides thelocation of the isocentre, while the farther spaced openings provide anaccurate rotational measurement.tation and translation to best fit the EPID intensity maps. This registration wasconducted using MATLAB’s (The MathWorks Inc., Natick USA) lsqnonlin opti-mization function.The EPID header and trajectory log provided information which was not alignedin the time domain: The EPID image header contained a time stamp, while thetrajectory log collected collimator angles continuously at a 50 Hz sampling rate.These sources of information were aligned by rotating the collimator clockwiseand then counter-clockwise to produce a unique movement, and then aligning thetwo signatures with a time interval dt that minimized the squared error betweenthe data sources. Once aligned in the time domain, the rotational QA data was col-lected. Comparison between angles was conducted for each EPID image, where95Figure 4.3: A typical area of open fluence level set graph derived from patient3. The red line shows the global optimal path length through the graph,while the blue line represents the static angle which minimizes the graph(at -15 degrees).trajectory log values were found by linear interpolation of the time series.4.3 Results4.3.1 Collimator OptimizationThe area of open fluence function varied with both couch angle and collimator an-gle. A typical collimator-gantry cost function level set is shown in Figure 4.3 forpatient 3. This figure shows both the shortest path found using dynamic program-ming as well as the optimal static collimator angle (which occurs at -15 degree). Asan example (for the same patient), the optimal trajectory had a mean open fluenceof 18 cm2, while the algorithm-optimized static and treatment planner-selected96Table 4.2: Average area of open fluence when optimized with three tech-niques: optimized moving trajectory, optimized static angle, and treat-ment planner-selected angle.Patient Optimized Trajec-toryOptimized Static TreatmentPlanner-Selected1 25 cm2 29 cm2 32 cm22 17 cm2 18 cm2 24 cm23 18 cm2 19 cm2 22 cm24 12 cm2 13 cm2 16 cm25 11 cm2 11 cm2 12 cm2Figure 4.4: A DVH comparison for patient 3, which shows the difference inbrain dose (shown in green) between the treatment planner-selected col-limator angles (plan shown with solid line) and the trajectory optimizedcollimator angles (shown as a dashed line). The GTV and PTV dosesshown are the summed dose for all four targets.static angle had a mean open fluence of 19 cm2 and 22 cm2, respectively. A sum-mary of the optimized fluence area for the different patients is given in Table 4.2.4.3.2 Treatment Plan ComparisonA typical DVH comparison between a treatment planner-selected collimator angleand the trajectory optimized collimator angle plan is shown in Figure 4.4. As97Table 4.3: Comparison of treatment of dosimetric parameters for four opti-mization strategies: algorithm optimized static collimator, algorithm op-timized trajectory, planner-selected static collimator (PS-Static) and nor-mal tissue blocked.Patient Number 1 2 3 4 5 MeanCIOptimized Static 0.73 0.84 0.65 0.91 0.90 0.81Trajectory 0.74 0.84 0.76 0.92 0.92 0.84PS-Static 0.72 0.80 0.75 0.85 0.88 0.80Normal Tissue Blocked 0.76 0.83 0.74 0.92 0.91 0.83V5 (cc)Optimized Static 589 152 383 306 141 314Optimized Trajectory 637 154 378 307 141 323PS-Static 830 157 451 358 155 390Normal Tissue Blocked 524 144 295 249 125 267V12 (cc)Optimized Static 174 74 73 89 35 89Optimized Trajectory 166 75 75 88 33 87PS-Static 174 73 88 99 37 94Normal Tissue Blocked 148 72 70 84 34 82Optimized Static 5.6 1.7 4.4 4.1 2.7 3.7Mean Brain Optimized Trajectory 5.7 1.7 4.4 4.1 2.8 3.7Dose(Gy) PS Static 6.5 1.7 4.9 4.5 2.9 4.1Normal Tissue Blocked 4.9 1.6 3.8 3.4 2.2 3.2can be seen, collimator trajectory optimization resulted in a small reduction in thelow-dose bath to the brain. A dose distribution comparison of the same patientwhen compared to the clinically delivered plan is shown in Figure 4.5. While theDVH curve for normal brain does not appear to change significantly, the brain is alarge organ when compared to the PTV and small differences between the curves,correspond to large volumes of unnecessarily irradiated tissue. This is quantifiedin Table 4.3, which shows treatment planning indices between the four methods.As can be seen for patient 3, the methods developed produced a 73 cc reduction inV5 when compared to the planner-selected static collimator method.Both methods, static and dynamic collimator optimization, outperformed theplanner-selected collimator angles in low-dose bath (V5). This is well illustrated inFigure 4.6a, which shows the mean relative improvement in Vx of the two method-ologies when compared to the treatment planner-selected collimator plans. Thesecomparisons were conducted after simulated annealing optimization step and one98(a)(b)Figure 4.5: An example dose distribution comparison between the treatmentplanner-selected plan and the collimator trajectory optimized treatmentplan. The prescription dose (3150 cGy), 50% (1575 cGy), 1200 cGyand 500 cGy contours are shown in yellow, orange, white, and blue re-spectively. a) The clinically delivered plan has significant dose spillageof the 5 Gy contour into the brain. b) Using the algorithms presented(both collimator angle optimization and MLC sequencing algorithms),the low-dose bath was significantly reduced.99can see a noticeable improvement of the collimator trajectory over the optimizedcollimator angle. Treatment beam-weights were then optimized using gradient de-scent with the beam shapes which were produced by the simulated annealing op-timization. When this was conducted, the difference between the two methodolo-gies disappeared, however both methods still outperformed the treatment planner-selected collimator angle (shown in Figure 4.6b). Figure 4.6b also shows the low-dose bath for plans which have the normal tissue dose blocked. This provides agood measure of the dose bath produced by the MLC either from fluence leakagethrough the MLC or from MLC contention issues. One can see that if these is-sues were mitigated, a further 15 percent improvement in low-dose bath could beachieved.Throughout all conducted optimizations, the competing strategies were opti-mized with the same cost function which was derived from the clinically deliveredtreatment plans (as explained in Section 4.2.3). When we compared the treatmentsplans based on this metric, where a low cost value is seen as a ”good” treatmentplan, then we found that both collimator angle optimization, and collimator tra-jectory optimization consistently outperformed or reproduced the quality of thetreatment planner-selected angle (shown in Figure 4.7).The collimator angle optimization methods failed to improve the 7-met treat-ment plan (patient 1) and also made little improvement on the 6-met plan (patient2). For these treatments, due to the large number of PTVs, it was physically impos-sible to find a collimator angle that removes the contention issues. This was mostapparent in patient 1, where treating the patient without the open fluences producedby MLC contention, would have reduced the mean brain dose by 0.7 Gy. A DVHcomparison for this patient, between the fluence blocked plan and the trajectoryoptimized plan is shown in Figure 4.8.These improvements came with modest benefits on the conformity. The confor-mity index (mean± one standard deviation) for the clinically recreated plans (0.80± 0.1) was similar to the optimized static collimator (0.80 ± 0.06) and dynamiccollimator (0.84 ± 0.08) plans. The effect of optimization strategy on conformityindex are summarized in Figure 4.9.100(a)(b)Figure 4.6: (a) The mean percent reduction in volume recieving x dose (Vx)from the implementation of collimator optimization methods whencompared to the treatment planner-selected collimator angle. These re-sults are presented after simulated annealing optimization. The opti-mized collimator angles (red) and collimator trajectory (blue) are com-pared with the treatment planner-selected collimator angle. Values areaveraged for all patients considered in this study. (b) At the end ofsimulated annealing optimization, beam-weights were optimized usinggradient descent. Treatment plans after gradient descent are shown inthis graph, and one can see that the difference between the two method-ologies disappears while the effect of collimator angle optimization stillexists. The green line shows the affect of blocking fluences which donot overlap with PTV tissue.101Figure 4.7: Improvement on the optimized cost for patients when comparedwith the clinically selected collimator angles. As can be seen, bothstrategies, the static algorithm optimized collimator angle and the op-timized collimator trajectory, produced plans with lower cost functionevaluations (i.e. higher optimization cost percent improvement).4.3.3 Quality Assurance of Dynamic Collimator DeliveryA total of 500 images were collected while the collimator rotated dynamically overthe 65 second QA test. Figure 4.10 shows the stability of the collimator rotationalaccuracy throughout this delivery. The three pairwise differences (mean ± 2 stan-dard deviations) between the DICOM header recorded collimator angle (θheader),trajectory log angle (θlog), and calculated collimator angle (θcalc) were 0.04 ±0.06 degrees (θcalc− θheader), 0 ± 0.04 degrees (θlog− θheader), and 0.04 ± 0.06degrees (θcalc−θlog). The maximum lateral displacement introduced by the colli-mator rotation was 0.1 mm, which is not likely to significantly affect the treatmentdosimetry. These results suggest three things. First, the Varian Truebeam linac on102Figure 4.8: DVH comparison between the trajectory optimized collimator an-gle (shown as dashed lines) and the normal tissue blocked beams (shownas solid lines) for patient 3. The brain dose was higher in the trajectoryoptimized collimator plans.which this test was performed can accurately rotate the collimator intra-treatment.Second, the trajectory logs’ recorded collimator angle is accurate, and can be usedin patient-specific QA. Third, machine-specific collimator QA can be deployed ina test that takes less than 2 minutes.4.4 DiscussionDAO-VMAT has been shown to be a clinically viable treatment planning approach.It has been well adapted at many centres and can be used to create clinically equiv-alent plans for many treatment sites. If DAO-VMAT is to remain a standard ofcare, then methods should be implemented to ensure the unnecessary treatment ofnormal tissue is minimized by selecting the best collimator angle. In this work, wereproduced the results of Wu et al. [98] and show that manual selection of collima-tor angles can lead to enhanced MLC contention issues. Additionally, we showedthat an algorithm-based approach for collimator angle selection can produce no-ticeable improvements, most of which comes from optimizing the static collimatorangle. The equivalence between the static and trajectory optimized collimator an-103Figure 4.9: Comparison of the CI % change produced by the optimizationstrategies when compared with the CI of the treatment planner-selectedcollimator angle.gles was most apparent after beam-weight optimization with a gradient method,and we believe this is because the gradient method was able to remove delivereddose from inefficient apertures which result from MLC contention issues.We have shown that our current clinical hardware could safely deliver thesetrajectories. It is still unclear whether the modest improvement of collimator tra-jectory optimized treatments over static optimized collimator angle will exist in afully-developed treatment planning optimization software. We expect that it willhave a diminished importance, as we hope that the clinical software is more ro-bust at avoiding local minima as the treatment planning system vendors have moreresources to invest in perfecting whichever optimization strategy they choose tooffer. Additionally, collimator trajectory delivery will put further burden on clinicsin the form of extra quality assurance. Conversely, an algorithm selected static col-104Figure 4.10: a)As the collimator rotated for its whole extent [-175 degrees,175 degrees], the angle was measured using the EPID imager. Col-limator angle was measured from three sources: the DICOM imageheader (θheader), the trajectory log (θlog), and by analysing the EPIDimages (θcalc). b) The first 5 seconds of the QA test (to the left of thedotted line) involved a clockwise and then counterclockwise move-ment of the collimator to provide a unique movement to align thetrajectory log and EPID images in the time axis. The remainder 60seconds of the QA test is where the measurements were acquired.limator angle would ease the treatment planning process, as it could be automatedif properly integrated into the treatment planning system. It is unclear to us whichof these two methods will best serve the needs of the radiotherapy community.This work showed that DAO-VMAT fails to robustly approach the global min-imum when the cost function contains penalties for low-dose bath to normal tis-sues. This weakness can be mitigated by the use of collimator angle optimizationstrategies. DAO-VMAT with the assistance of collimator optimization, can find105solutions that have lower dose bath when compared to human selected collimatorangles. However, even with the assistance of collimator angle optimization, therewere still unnecessarily opened fluences that contributed to the low-dose bath. Thiscomes from two main sources. First, the progressive sampling algorithm fails toconverge to the optimal solution when there are MLC contention issues: it pro-duces openings that should not exist in an optimal solution. Second, the MLChas small, but non negligible radiation leakage (around 1-2% depending on MLCmodel). The naive solution to this problem is to separate the PTVs into groupsthat do not produce MLC contention issues and then treat the groups separately.However, this approach will produce an increase in the dose from radiation leak-age through the MLC. Finding the optimal strategy under these considerations iscomplex and likely requires further investigation.4.5 ConclusionIn this work, we showed the feasibility of dynamic collimator rotation duringVMAT delivery. However, the majority of treatment improvement can be achievedusing algorithm-based static collimator selection. Future work will test whetherthese results can be reproduced on clinical systems. The improvement made bystatic collimator optimization was already shown by Wu et al. [98] to improve treat-ment planning results on Varian Eclipse. We cannot perform a similar treatmentplanning system comparison of trajectory collimator optimization without accessto the base code of clinical software and incorporating site-specific clinical exper-tise at planning with each method. Future work could compare algorithm selectedcollimator angle DAO-VMAT with fluence-based VMAT on clinical software.106Chapter 5Conclusion5.1 The Effect of Couch Rotations in SRS DeliveriesAt the onset of this work, there were several gaps in knowledge on the effect ofcouch angle optimization on SRS deliveries. In particular, while there were stud-ies that investigated the effect of couch angle optimization on forward planningmethods such as DCA, the effect on inverse planning methods was unknown asVMAT for SRS was still in its infancy. This work found a very modest improve-ment on treatment quality when highly non-coplanar treatments were implementedover single arc VMAT treatments. A single VMAT arc delivers dose from a wideselection of beam directions. Multiple non-coplanar arc VMAT for SRS deliversdose from even more directions and for this reason there is a modest improvementwhen multiple-arc VMAT is used over single arc VMAT.Whether trajectory optimization improves treatment plan quality depends onthe optimization method being used. Many groups have created heuristic algo-rithms predicated on the benefit of avoiding critical OAR. These methods will im-prove dosimetric indices of most forward planning algorithms. However, modernlinacs have the ability to completely shut off the beam using beam weight modula-tion and this feature can be leveraged to remove beams with less optimal directionfrom the beam trajectory. Furthermore, beam weight optimization is a relativelysimple optimization problem as the only constraint on the system is that the beamweights remain positive. Therefore the dosimetric benefit of trajectory optimiza-107tion in forward planning methods can be reproduced by a dense sampling of thephase space during initial optimization and then using beam weight modulation toremove non-optimal beam directions. This is the basis for the methods developedin this work, as well as the methods by Langhans et al. [50] and the 4 pi method[24].There are trade offs between the optimization strategies of the work presentedhere and that of the heuristic couch-gantry optimization presented by other groups(e.g. the work in [59, 86, 99] as a non-exhaustive list). The theoretical analoguebetween the two methods is whether doses to normal tissue should be accountedfor in the cost function, or as a constraint of the optimization. By avoiding certaincouch-gantry angles, the optimization software of other groups is effectively forc-ing a hard constraint on dose from certain directions. There are a few benefits tothis approach enumerated in the following paragraphs.First, it can be difficult to represent the desires of the treatment plan in thecost function, and dose reductions may be missed because they are unknowinglyleft unrepresented in the optimization process. This aspect is very well illustratedin some of the clinical plans that were re-optimized in this thesis. The clinicaltreatment plans used a normal tissue constraint that was not configured optimallyor weighted highly enough. This resulted in unnecessary over-dosage to the normaltissue. These aspects may be overcome by patient specific dose QA, termed asknowledge based planning by some groups [85]. However, these methods are notwidely adopted yet, and may not catch the errors. Furthermore, these methodsare enacted very close to the final delivery and are seen as a reactive (instead of aproactive) approach to quality.Second, enforcing a hard constraint on the trajectory makes the method gener-alizable to any treatment optimization processes. This is important for a multitudeof reasons, the most important of which is in regards to patient safety. The VMAToptimizations used in the clinic are produced by technology that is developed byvendors, and implemented by practitioners. Furthermore, these practitioners donot fully know the inner workings of the optimization software. By implement-ing an approach that is method-generalizable, it can be robust to changes betweensoftware upgrades, under-education of staff and changes of providers. This aspectwas also well represented in the clinically delivered optimizations. The practi-108tioners that optimized the VMAT plans thought that the beam weight optimizationwas well accounted for in the Varian Eclipse optimizations. This functionality wasenabled, and the beam weights varried only between 60% and 100% of maximaloutput instead of the expect 0% to 100%. This misunderstanding is not specific toour clinic, as it is not discussed in prominent publications [90].While there is no dosimetric benefit of optimized couch-gantry trajectories overa dense sampling of the couch-gantry phase space, couch-gantry trajectory opti-mization can create treatments that have faster deliveries. Delivering many non-coplanar arcs can present a burden to patients and clinics. In this work, a methodthat variably samples the delivery phase space in a time efficient manner is pre-sented. The dosimetric delivery quality was gauged for each phase space samplingand the sampling was increased until the delivery quality plateaued. This methodensures that deliveries leverage as many beam angles as is necessary for dosimet-ric performance, while being time efficient. However, this time saving benefit ismodest, and with Varian’s HyperArc framework of arc delivery [83], it may onlyamount to 1-2 minutes of treatment time saving benefit. The importance of whichdepends on staffing and resources of a particular centre. Other groups proposemethods that deliver dose from only the most optimal angles [50], and these meth-ods may reduce the treatment time by a further 30 seconds. However, it should benoted that for SRS deliveries, dose per fraction are high, so dose delivery rate andfluence delivery efficiency of the beam apertures set limits on the delivery time.This work did not considered FFF deliveries even though it is currently the clinicalstandard and will likely have the greatest effect on treatment delivery time. ForSRS dose fractionation, there is no expected clinically relevant improvement ontreatment time over the methods that are currently developed (either in this thesisor by other trajectory optimization groups that use inverse planning).5.2 The Effect of Collimator-Gantry TrajectoryOptimizationThis work explored how collimator rotation affects VMAT deliveries by allowingmore efficient MLC apertures. Efficient and robust treatment optimization is stillan active area of study, and there are still many questions that need to be answered.109DAO-VMAT is particularly well suited for SRS deliveries, as targets are usuallysmall and spherical in geometry. For these shapes, conformal treatments presentan MLC sequence that is very close to the best possible treatment, as they allow asmuch fluence delivered as possible per beam, while naturally producing sphericaldose patterns from the arc trajectory delivery. However, multiple metastases in-crease the complexity of this optimization, which can make DAO-VMAT not idealfor these indications. This work showed that collimator optimization can partiallymitigate the shortcomings of DAO-VMAT by rotating the collimator to minimizeMLC contention issues. Static collimator optimization had similar dosimetric im-provement when compared to the dynamic collimator method, and this is likelydue to dose rate modulation: the beam could be turned off when there were config-urations that had significant contention issues.There are other approaches which could be interesting to test in future work.In particular, fluence-based VMAT may not be hindered by the same shortfalls asDAO-VMAT. Depending on how the optimization algorithm is set up, it may beable to intelligently select which PTVs to deliver dose to when choosing betweencompeting MLC configurations to avoid MLC contention issues. Another strategythat could be explored is delivering dose to only a subset of the PTVs when thereare MLC contention issues, and then delivering dose to the untreated PTVs on asubsequent arc. This strategy is currently commercially implemented [92]. Thereare two short-comings to this strategy, in particular, it increases the treatment timeby a factor of two. Additionally, it may not improve the treatment plan quality:if there is more leakage through the MLC than is produced by MLC contentionissues, then having two arc passes would degrade the treatment quality. Futurework could explore these trade-offs and identify candidate patients which wouldbenefit from this strategy.5.3 Delivery Capabilities of Modern LinacsThis work explored the capabilities of the Varian TrueBeam linac. In short sum-mary, the TrueBeam linac tested in this work was found to be highly capable ofmoving the collimator, gantry, couch, and MLC intra-treatment with a high de-gree of accuracy. In this work, patient specific and machine specific measures110were developed to ensure the delivery accuracy of the linac for dynamic couch andcollimator deliveries. These methods (or process-optimized versions of the samemeasure) would need to be integrated into linac QA procedures if dynamic deliverymethods are to be implemented in the clinic. It is still an open question whetherthe dosimetric and delivery efficiency benefits extolled in this thesis will be offsetby the clinically incurred costs of extra patient-specific and machine-specific QA.5.4 Future DirectionsThere may be future work exploring the interplay between even more aspects of thetreatment, such as moving the isocentre intra-treatment or exploring the interplayof couch-collimator-gantry trajectories. There is still much work to be conductedon collimator-gantry-MLC optimization for certain clinical optimization software.The Varian TrueBeam algorithm fails to robustly reduce the dose bath, and thisshortfall should be rectified. It is still undetermined whether these inadequacieswill exist in fluence-based optimization strategies implemented by Raystation andPinnacle. Future work could explore this question.This work is limited by our current understanding of the biological effect ofradiation on the brain. In this work, we explored methods which would reduce theexposure of healthy brain tissue to radiation. It is important to investigate whetherthis is a suitable goal. Historically, there have been three strategies for the radio-therapy treatment of brain metastasis: WBRT alone, SRS alone, and SRS withadjuvant WBRT. The intention of adding adjuvant WBRT was for the prophylacticreduction of tumour burden. This reduction is measurable: results of a phase IIIclinical trial have shown that SRS alone had a shorter time to intra-cranial failurewhen compared to SRS with WBRT[13]. However, this was coupled with signif-icant decreases in cognitive function and quality of life. Therefore, it is difficultto explore the cost-benefit of delivering radiation to healthy brain tissue. In or-der to find the best trade off, one would need a more in depth understanding ofradiobiology.Finally, there are many non-radiotherapy interventions which can be used totreat patients with brain metastasis. The intervention varies widely based on theprimary tumour site. Future research could provide scientific support for investi-111gating radiation therapy with concurrent, adjuvant or pre-operative chemotherapy.Figuring out the effect of different interventions on radiotherapy would be an in-teresting question. Also, modern interventions are allowing some patients to livelonger. 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