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Monte Carlo simulation of x-ray dose distributions for direct aperture optimization of intensity modulated… Bergman, Alanah Mary 2007

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MONTE CARLO SIMULATION OF X-RAY DOSE DISTRIBUTIONS FOR DIRECT APERTURE OPTIMIZATION OF INTENSITY MODULATED TREATMENT FIELDS by ALANAH MARY BERGMAN B.Sc.,McGill University, 1994 M.Sc., McGill University, 1997 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Physics) THE UNIVERSITY OF BRITISH COLUMBIA January, 2007 © ALANAH MARY BERGMAN, 2007 Abstract i i Abstract T h i s thesis investigates methods of reducing radiat ion dose calculat ion errors as applied to a specialized x-ray therapy called intensity modulated radiat ion therapy ( I M R T ) . There are three major areas of investigation. Fi rs t , l imits of the popular 2D penci l beam kernel ( P B K ) dose calcula t ion a lgor i thm are explored. T h e abi l i ty to resolve high dose gradients is par t ly related to the shape of the P B K . Improvements to the spatial resolution can be achieved by modifying the dose kernel shapes already present in the c l in ical treatment planning system. Op t imiza t i on of the P B K shape based on measured-to-calculated test pattern dose comparisons reduces the impact of some l imi ta t ions of this algori thm. However, other l imita t ions remain (e.g. assuming spat ial invariance, no model ing of extra-focal radiat ion, and no model ing of lateral electron t ransport) . These l imita t ions directed this thesis towards the second major investigation - Mon te Ca r lo ( M C ) s imulat ion for I M R T . M C is considered to be the "gold standard" for radia t ion dose calculat ion accuracy. T h i s investigation incorporates M C calculated beamlets of dose deposit ion into a direct aperture opt imiza t ion ( D A O ) algori thm for I M R T inverse p lanning ( M C - D A O ) . The goal is to show that accurate tissue inhomogeneity information and lateral electronic transport information, combined w i t h D A O , w i l l improve the qual i ty /accuracy of the patient treatment plan. M C s imula t ion generates accurate beamlet dose distr ibutions i n t radi t ional ly difficult-to-calculate regions (e.g. air-tissue interfaces or smal l (< 5 c m 2 ) x-ray fields). C o m b i n i n g D A O w i t h M C beamlets reduces the required number of radiat ion units delivered by the linear accelerator by ~30-50%. T h e M C method is cr i t icized for having long s imulat ion times (hours). Th i s can be addressed wi th dis t r ibuted comput ing methods and data filtering ( 'denoising'). T h e th i rd major investigation describes a pract ical implementat ion of the 3D Sav i t zky-Golay digi ta l filter for M C dose 'denoising'. Th i s thesis concludes that M C - b a s e d D A O for I M R T inverse p lanning is c l in ical ly feasible and offers accurate model ing of particle transport and dose deposit ion i n difficult environments where lateral electronic dis-equi l ibr ium exists. Contents i i i Contents A b s t r a c t i i Con ten t s . " i i i L i s t of Tables v i i i L i s t of F igures x Table of A c r o n y m s xxi i i Acknowledgements xxvi 1 I n t roduc t i on to Thesis P ro jec t 1 2 I n t roduc t i on to R a d i a t i o n T h e r a p y Phys ics 6 2.1 His to ry of Rad ia t i on Therapy for Cancer Treatment 6 2.2 Rad i a t i on Bio logy . . . 11 2.2.1 Indirect and Direct A c t i o n Damage to D N A 11 2.2.2 C e l l Surv iva l after Irradiat ion 14 2.2.3 Therapeut ic Ra t io ( T R ) . . . 14 2.3 P h o t o n Interactions w i t h Mat t e r . 16 2.3.1 Rayle igh (coherent) Scattering 18 2.3.2 Photoelectr ic Effect 19 2.3.3 C o m p t o n (Incoherent) Scattering 23 2.3.4 Pa i r P roduc t ion 25 2.3.5 Triple t P roduc t ion 26 2.3.6 Summary . 27 2.4 Charged Par t ic le Interactions wi th Ma t t e r '. 27 2.4.1 Ionizational (Coll isional) Energy Losses 28 Contents iv 2.4.2 Radia t ive (Bremsstrahlung) Energy Losses 30 2.4.3 Elas t ic Scattering 31 2.4.4 Pos i t ron A n n i h i l a t i o n 32 2.4.5 Elec t ron Dosimetry 33 2.5 K E R M A and Absorbed Dose 33 2.6 Rad i a t i on Del ivery 35 2.6.1 Generat ion of Megavoltage X - R a y s - the M e d i c a l Linear Accelerator . . . 38 2.7 Intensity M o d u l a t e d Rad ia t ion Therapy - I M R T . 40 2.7.1 Fluence M o d u l a t i o n Techniques 41 2.7.2 Inverse Treatment P l ann ing 46 2.7.3 The Objective Func t ion . 50 2.7.4 Op t imiza t ion Techniques for M i n i m i z i n g an Object ive (Cost) Func t ion . 56 2.7.5 Techniques for Op t imiza t ion of Rad ia t ion Delivery : Fluence-Based O p -t imiza t ion vs. Direct Aper ture Op t imiza t ion 61 2.7.6 Dosimetr ic Considerations for I M R T 65 3 Dose Calculation Algorithms and Validation Techniques in Radiation Ther-apy 70 3.1 Dose Ca lcu la t ion Algor i thms and Treatment P lann ing Systems ' 70 3.1.1 Character iz ing Rad ia t ion Dose Deposi t ion i n Ma t t e r • 71 3.1.2 Single Penc i l B e a m Convolu t ion A l g o r i t h m E m p l o y i n g 2D Dose Kernels . 77 3.1.3 Superposi t ion / Convolu t ion M e t h o d E m p l o y i n g 3D Point Kernels . . . . 80 3.1.4 Col lapsed Cone Convolu t ion A l g o r i t h m 83 3.1.5 Monte Car lo Simulat ions 84 3.1.6 Tissue Inhomogeneity Corrections 86 3.2 Measurement and Verif icat ion of Rad ia t ion Dose 90 3.2.1 Introduct ion . . . 90 3.2.2 Absolute Dosimeters 93 3.2.3 Relat ive Dosimeters 95 3.3 Compar i son of Rad ia t ion Therapy Dose Dis t r ibut ions 99 3.3.1 Isodose Lines - V i s u a l Inspection 100 3.3.2 Dose Difference 100 Contents v 3.3.3 Distance-to-Agreement 101 3.3.4 Dose Difference / D T A "Pass /Fa i l " Methods 103 3.3.5 T h e G a m m a Factor . 103 4 Improved I M R T Dose D i s t r i bu t i ons using M o d i f i e d P e n c i l B e a m Dose K e r -nels 105 4.1 Prologue 105 4.2 Int roduct ion 106 4.3 Theory 107 4.3.1 Single Penc i l B e a m Dose Kerne l Convolu t ion A l g o r i t h m Us ing 2D Dose Kernels . 107 4.3.2 Dose Ca lcu la t ion Accuracy Assessment - The G a m m a Factor 107 4.4 Mater ia l s and Methods . . *. 108 4.4.1 Measured Reference D a t a 108 4.4.2 Independent Dose Ca lcu la t ion I l l 4.4.3 Dose Kerne l Shape Opt imiza t ion 112 4.4.4 2D C l i n i c a l Fluence M a p Verif icat ion - Example 1 113 4.4.5 2D C l i n i c a l Fluence M a p Verif icat ion - Example 2 . . 113 4.5 Results and Discussion 114 4.5.1 Dose Ke rne l Shape Op t imiza t ion . . . ; 114 4.5.2 5 m m B a r Pa t te rn Fluence 115 4.5.3 2D C l i n i c a l Fluence M a p Dose Verif icat ion - Example 1 116 4.5.4 2D C l i n i c a l Fluence M a p Dose Verification - Example 2 119 4.6 Conclus ion : Modi f ied Penc i l B e a m Dose Kernels 119 5 M o n t e C a r l o S imu la t i on of Pa r t i c l e Transpor t T h r o u g h M a t t e r 122 5.1 Introduct ion to Monte Car lo " 122 5.1.1 S imula t ion Codes - Overview 123 5.1.2 E G S n r c (Electron G a m m a Shower - Na t iona l Research Counci l ) 124 5.1.3 B E A M n r c Code (For radiotherapy source simulation) 126 Contents v i 5.1.4 D O S X Y Z n r c Code (For absorbed dose in matter) 133 5.1.5 Variance Reduc t ion 135 5.1.6 Dis t r ibu ted C o m p u t i n g 137 5.2 Benchmark ing the Monte Car lo System 138 5.2.1 Open F i e l d Verif icat ion . . 139 5.2.2 Absolu te Dose Conversion / Ca l ib ra t ion of V i r t u a l L inac 142 5.2.3 Mul t i - l ea f Col l imator ( M L C ) Modu le for I M R T 145 6 Monte Carlo Input for Direct Aperture Optimization for I M R T 153 6.1 Prologue 153 6.2 Int roduct ion . 154 6.3 Mater ia l s and Methods 157 6.3.1 In i t ia l Treatment Parameters (Fie ld Size) . 159 6.3.2 V i r t u a l Linear Accelerator . . 159 6.3.3 Segmentation of Beamlets from Phase Space 160 6.3.4 Generat ion of Voxe l i zed 'Phan tom 160 6.3.5 Beamlet Dose Dis t r ibu t ion M a t r i x . • 161 6.3.6 Direct Aper tu re Op t imiza t ion ( D A O ) 164 6.3.7 F i n a l Forward Ca lcu la t ion 165 6.3.8 A p p l i c a t i o n of M C - D A O 166 6.3.9 Sources of E r ro r 173 6.4 Results and Discussion 177 6.4.1 P h a n t o m Example - Single F i e l d . 177 6.4.2 P h a n t o m Example - Seven Fields 177 6.4.3 Nasopharynx Recurrence 184 6.4.4 Massive L o w Densi ty Inhomogeneity - L u n g 189 6.5 Conclus ion : M C - D A O 197 7 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions . . . . 199 7.1 Sav i t zky-Golay F i l t e r (Least Squares Fi l ter ) . . . 200 7.1.1 Least Squares A l g o r i t h m for Curve F i t t i n g '. 202 7.1.2 x2 Test 203 • Contents v i i 7.1.3 Adap t ive W i n d o w Sav i tzky-Golay Smoothing A l g o r i t h m 204 7.2 F i l t e red 3D Monte Car lo Dose M a t r i x 205 7.2.1 Reference D a t a Set ( " G o l d Standard") 205 7.2.2 Constant S G Fi l t e r W i n d o w Size / V a r y i n g Uncer ta in ty i n D a t a Set . . . 207 7.2.3 3D W i n d o w Size 212 7.2.4 Sav i tzky-Golay for Summed M u l t i p l e F i e l d Dose Dis t r ibut ions . . . . . . . 213 8 C o n c l u s i o n 218 9 Fu tu re W o r k . 222 B i b l i o g r a p h y 225 / List of Tables v i i i L i s t of T a b l e s 2.1 K and L shell b inding energies (keV) for Various Absorb ing M e d i a 20 3.1 D e p t h of m a x i m u m dose deposition for different photon treatment beams 73 3.2 Advantages/Disadvantages of ion chambers as a radiat ion therapy dosimeter. . . 95 3.3 Advantages/Disadvantages of radiographic film for radiat ion therapy dosimetry. 98 4.1 Sample i r radia t ion parameters for delivery of 5 m m bar pattern (peak-to-peak) fluence map to fi lm plane positioned at depth = 5.0 cm ( S A D 100 cm). I l l 4.2 Cen t ra l Ke rne l Values ( C K V ) for four depths 114 4.3 Propert ies of the masked 2D 7 map dis t r ibut ion (reference data = measured r data). 118 5.1 Argument for calculat ing Dose-to-Water (Dw) versus Dose -To-Medium (Dm). . . 135 5.2 Ion Chamber Measurement vs Monte C a r l o : In i t ia l S l id ing W i n d o w Test . . . 145 5.3 Ion Chamber Measurement vs Monte Car lo : S l id ing W i n d o w Test 147 6.1 Treatment field / op t imiza t ion parameters for A V I D phantom w i t h C-shaped target 167 6.2 D V H planning objectives - P h a n t o m example (C-shape) 167 6.3 Treatment field / opt imizat ion parameters for nasopharynx recurrence example. 169 6.4 , D V H planning objectives - Nasopharynx recurrence example 169 6.5 Treatment field / opt imizat ion parameters for lung tumour example 170 6.6 Dose volume histogram ( D V H ) planning objectives - L u n g tumour example[44, 153, 154] 171 6.7 Doses to structures-of-interest for A V I D phantom example 180 6.8 Ind iv idua l beam weighting (percentage contr ibut ion to dose point w i th in P T V ) . 183 6.9 R a t i o of contributions from the posterior fields versus the anterior fields . . . . 184 List of Tables i x 6.10 Phan tom: Compar i son of to ta l monitor units ( M U ) for V a r i a n Ecl ipse nuence-based plan and M C - D A O plan 184 6.11 Doses to structures of interest - nasopharynx example. Prescr ip t ion Dose = 60Gy. 188 6.12 Ion chamber point dose measurements in cyl indr ica l verification phantom 189 6.13 Nasopharynx Example : Compar ison of monitor unit ( M U ) delivery efficiency for V a r i a n Ecl ipse fluence-based p lan and M C - D A O 189 6.14 Doses to structures of interest for lung example. Prescr ip t ion Dose = 70 Gy. . . 193 6.15 Ion chamber point dose measurements i n A V I D verification phantom for the Ecl ipse fiuence-optimized lung plan u t i l i z ing the penci l beam kernel dose calcu-la t ion 195 6.16 Ion chamber point dose measurements i n A V I D verification phantom for the M C -D A O lung p lan u t i l i z ing Monte Car lo dose calculat ion. 196 6.17 L u n g Example : Compar ison of monitor uni t ( M U ) delivery efficiency for V a r i a n Ecl ipse fiuence-optimized plan and M C - D A O 197 7.1 G A M M A factor analysis (2% / 2 mm) for S G filtered data (coronal plane). " F A I L " = G A M M A value > 1 ' 215 7.2 3D R M S D analysis for raw and S G filtered data 215 List of Figures List of Figures x 2.1 Damage to D N A can occur v i a direct or indirect action. Direct : part icle interacts direct ly w i t h D N A molecule. Indirect: particle ionizes water to produce free radicals which then interact w i t h D N A molecule 12-2.2 Single and double strand breaks causing damage to D N A . If repair mechanisms are unsuccessful, the cell w i l l die 12 2.3 C e l l survival curves depend on the type of radia t ion being used. These two curves compare the cytotoxic effect of single fraction x-ray radia t ion dose to neutron radiat ion dose. (1 G y = 1 J /kg ) A d a p t e d from Johns and C u n n i n g h a m (1983) [42]. : 14 2.4 Hypo the t i ca l tumour Con t ro l P robab i l i ty ( T C P ) curve compared to N o r m a l T i s -sue Compl i ca t ion Probab i l i ty ( N T C P ) curve. Ideally, the two curves are well separated (a). Cl in ica l ly , the difference may not be so great (b) and effort must be made to deliver high doses precisely to the tumour whilst sparing the nearby sensitive healthy organs 15 2.5 Rayle igh (coherent) Interaction, (a) Pho ton (hu) interacts w i t h an a tom but does J not cause an ionizat ion, (b) The a tom as a whole becomes excited and releases excess energy by .emi t t ing a photon. T h e emitted photon has the exact same energy as the incident photon (hy). T h e emitted photon is scattered at an angle very close to the original direction of the incident photon, (c) A t o m returns to ground state 18 r List of Figures x i 2.6 Photoelectr ic Effect (a) Incident photon (hu) interacts w i t h bound atomic elec-t ron (e.g. K shell electron), (b) Elec t ron gains enough energy to be ejected from the a tom w i t h kinet ic energy (KE = hu — En)- A hole in the shell remains and the atom is now ionized, (c) The vacancy is filled by an electron dropping down from a higher energy level (e.g. L shell). The energy difference between the two different shell states (EK — EL) is released from the a tom i n the form of a characteristic x-ray 20 2.7 Auger electron emission (c) competes w i t h characteristic x-r-ays after a vacancy is created by the emission of a photoelectron (b) 21 2.8 P l o t of the mass photoelectric coefficient (r/p) for three absorbing media: water, a luminum, lead. Note the k-edge effect for lead at 88 keV. A d a p t e d from Evans (1955) [46] 22 2.9 C o m p t o n Interaction, (a) Incident photon (hu) interacts w i t h a free a tomic electron, (b) Pho ton transfers energy to electron which is ejected w i t h kinet ic energy (KE). Remain ing photon energy scattered w i t h new energy (hu) 23 2.10 (a) Pa i r P roduc t ion . M i n i m u m photon energy for interaction = 2 m 0 c 2 . (b) Tr ip le t P roduc t ion . M i n i m u m photon energy for interaction = 4 m 0 c 2 26 2.11 Iso-probabil i ty lines define regions of dominant photon interaction type. Absorber atomic number is plotted against incident photon energy. Three major regions are shown: photoelectric effect, compton scatter and pair product ion. A d a p t e d from Evans (1955) [46]. 27 2.12 T h e to ta l mass stopping power for carbon is the sum of the ionizat ion mass s topping power, Sion, and the radiative stopping power, Sra(i- A d a p t e d from Johns and Cunn ingham (1983) [42] 30 2.13 Pos i t ron Ann ih i l a t i on : posi tron loses kinetic energy and comes almost to rest where it interacts w i t h a stat ionary electron. The electron / posi t ron convert their masses completely into energy v i a E = rnc2. The energy takes form of two 511 k e V photons (m0c2) which are emitted at a relative angle of 180° (or very close) ^ 32 2.14 Energy spectrum for a 6 M V x-ray photon beam simulated by the author using B E A M n r c Monte Car lo code. Note the presence of the characteristic 511 k e V annihi la t ion spike. 33 V List of Figures x i i 2.15 T h e relationship between K E R M A (kinetic energy released i n medium) and dose. 35 2.16 Schematic of medical linear accelerator. A d a p t e d from K a r z m a r k et al.[1973] [37] 38 2.17 Schematic of medical linear accelerator treatment head.( l ) electron gun, (2) s tanding waveguide, (3) bending magnet, (4) electron beam steering control , (5) target, (6) carousel of scattering foils and flattening filters, (7) ion chamber, (8) asymmetric co l l imat ing jaws, (9) beam shaping device (multileaf col l imator) . . 40 2.18 (a) open beam fluence for a square field, (b) simple fluence modula t ion , (c) C o m -plex fluence modula t ion. Insets = horizontal profile through mid-image 41 2.19 Wedged shaped dose dis t r ibut ion i n tissue-equivalent plastic caused by wedged shaped attenuator 42 2.20 Pho to of a multi- leaf coll imator looking up from the patients' perspective. Not ice the tw in banks of 0.5 cm - 1 cm wide tungsten leaves 44 2.21 Schematic of V a r i a n 120 leaf M L C . (a) Beam' s -Eye-View of field shaping capa-b i l i ty of M L C (b) end-on cut-away demonstrating tongue-and-groove design (c) aspect view i l lus t ra t ing rounded leaf end shape 44 2.22 F i r s t Row: Three M L C - d e f i n e d apertures delivered sequentially. Second Row: T h e corresponding fluence maps delivered by s t he M L C . Last column = sum of the three apertures. T h i r d Row: Profile representations of the fluence maps. . . 45 2.23 T imel ine for two different fluence map delivery modes for the mult i leaf col l imator (a) Stat ic M L C ( s M L C ) and (b) D y n a m i c M L C ( d M L C ) 46 2.24 Dose Volume Histograms for two treatment plans. D(Rx) indicates prescript ion dose, (a) A poor p lan showing inhomogeneous tumour ( P T V ) coverage and high organ-at-risk ( O A R ) doses, (b) Improved plan w i t h good tumour dose uniformity and reduced O A R dose 48 2.25 (a) Open field fluence (b) Fluence map divided into beamlets of open intensity (c) Beamlet weights are opt imized to obtain a fluence map 49 2.26 Soft constraints give 'negotiable' upper and lower dose l imi ts on d is t r ibut ion of dose wi th in structure. For the target, ideally 100% of the prescript ion dose covers 100% of the target volume and 0% of the volume exceeds the prescript ion dose. . 52 2.27 Dose-volume constraints minimize the number of voxels that fall under the D V H curve, bounded by Drnax and D* (shaded area) [29]. D* is located at the intersec-t ion between the volume constraint, Vmax and the D V H curve 54 List of Figures x i i i 2.28 Effect of pr ior i ty setting on opt imizat ion of dose-volume histograms, (a) L o w pr ior i ty on enforcing m a x i m u m dose hard constraint, (b) H i g h priority. 54 2.29 P l o t of objective (cost) function value vs. suggested solution, (a) Gradient based op t imiza t ion method may converge on a local m i n i m u m which is not ideal, (b) Stochastic op t imiza t ion methods find preferred global m i n i m u m 56 2.30 Simulated annealing opt imiza t ion process for direct aperture op t imiza t ion (Fields, apertures and M L C properties are randomly selected 57 2.31 E x a m p l e of a temperature cooling curve for simulated annealing opt imiza t ion . . 59 2.32 F l o w diagram for fiuence-based opt imizat ion 61 2.33 Compare optimal (dashed line) and actual (solid line) D V H curves for the fiuence-based optimization method. Note the decrease in the P T V mean dose and increase i n O A R m a x i m u m dose i n the final d is t r ibut ion 63 2.34 F l o w diagram for direct aperture opt imizat ion 63 2.35 (a) M L C diagram indicat ing locat ion of dose profile (vertical solid line), (b) E x a m p l e of dose profile demonstrat ing interleaf transmission 65 2.36 E n d - o n diagram of M L C showing tongue and groove design. Leaves slide in to /ou t of the page, (a) A b u t t i n g leaves, (b) single protruding leaf 66 2.37 Tongue-and-Groove effect (arrows showing yellow underdosage lines) visible on this coronal plane of a seven field I M R T plan 66 2.38 E n d - o n diagram of M L C w i t h profiles, (a) T w o adjacent leaves are opened to create a uniform fluence profile, (b) The same open field is delivered i n two segments. T h e summat ion of the two segments does not equal the open profile i n (a). A tongue-and-groove error results. 67 3.1 D i a g r a m depict ing measurement conditions for determining a percentage depth dose 72 3.2 (a) 2D contour plot of f i lm measured P D D for open 10x10 c m 2 6 M V photon field, (b) Profile along depth axis of same data 73 3.3 D i a g r a m depict ing measurement geometry for T i s sue -Ai r -Ra t io measurements. . 74 3.4 D i a g r a m depict ing measurement conditions to obta in a T M R ratio. 75 3.5 (a) A modula ted photon fluence treatment field, (b) Segmentation into 0.25 x 0.5 c m 2 beamlets : 77 List of Figures x i v 3.6 I D dose deposit ion kernel derived from ion chamber measured' commissioning data. D a t a exported from E c l i p s e ® v 6 . 5 T P S (depth = 5.0 cm) 78 3.7 T h e photon fluence is corrected for off-axis intensity variat ion and then convolved w i t h a 2D depth-specific penci l beam dose deposit ion kernel. The final result is a dose plane. . 78 3.8 (a) P h o t o n incident on phantom interacts at a point. T h e tota l energy re-• leased (kinetic energy transferred to electron plus scatter photon energy) is called T E R M A . (b) Poin t dose kernel describes dose dis t r ibut ion about the interact ion point 81 3.9 Col lapsed cone convolution calculates the 3D deposited energy wi th in a series of cones emanating from the photon interaction point. T h e doses w i t h i n the cone are collapsed onto the central ray of each cone projection 83 3.10 P h o t o n beam incident on (a) uniform water phantom, (b) water phantom w i t h embedded air inhomogeneity, thickness (da). Note increase i n lateral scatter of photon i n air w i l l reduce scatter dose to point P. .- 86 3.11 P h o t o n beam incident on phantom containing an inhomogeneity. T h e B a t h o Power L a w correction factor considers the locat ion of the point of interest, P relative to the inhomogeneity 88 3.12 Schematic of a point dose kernel i n (a) water, and (b) air. Note the increased spread of the dose deposit ion i n the low density mater ial 90 3.13 Examples of different point-dose dosimeters: parallel plate ion chamber, cy l ind r i -cal ion chamber and diode 91 3.14 Examples of different tissue phantoms, (a) Rounded square rexalite AVID phan-t o m ( M D X , Vancouver, Canada) , (b) anthropomorphic solid water phantom, (c) square slabs of solid water plastic ' 92 3.15 D i a g r a m of typica l farmer-type cyl indr ica l ion chamber. Incident photon ionizes air (33.85 e V / i o n pair) and the ion / e~ charge is collected at inner and outer electrodes 93 3.16 T y p i c a l characteristic ( H and D) curve for radiographic film. Note the low re-sponding toe region and the saturating shoulder region. Dosimetry measurements w i t h film should fall w i t h i n the linear response region of the film 97 List of Figures x v 3.17 F i l m response curves to increasing dose for two different f i lm types manufactured by K o d a k ( E D R and X V ) 99 3.18 (a) 2D Isodose overlay showing good qualitative agreement (b) Poor agreement. T h e isodoses exhibi t a geometric shift in the y-axis 100 3.19 T w o examples of dose discrepancy (a) Isodoses are spat ial ly shifted. T h e dose difference method is sensitive to smal l shifts in the high gradient region, (b) Isodoses are shifted i n dose value. The distance-to-agreement method is sensitive to smal l dose differences i n low gradient regions 101 3.20 Dose comparison methods, (a) F i l m measured I M R T dose plane (b) Ca lcu la ted dose plane (c) Percentage Dose difference map, (d) distance-to-agreement map, (e) B i n a r y map : percentage dose difference A N D distance-to-agreement cr i ter ia (2% / 2 mm) fail (white), (f) G A M M A map w i t h 2% / 2 m m pass/fai l cri teria. . 102 4.1 (a) 2D ideal fluence pattern comprised of ten 5 m m wide bars of uni t intensity (white), (b) actual fluence pat tern obtained when the ideal fluence is modified by the dosimetric properties of the multi- leaf col l imator beam shaping device (transmission, dosimetric leaf gap). The leaf travel direction is from left-to-right, (c) profile through ideal fluence map. Peak-to-valley ratio = 1.0. (d) profile through actual fluence map. Peak-to-valley ratio = 0.85 109 4.2 Frequency domain profiles comparing spatial frequency components of the Fourier transform of a c l in ica l fluence map (solid line) to the Fourier transform of a 5 m m R E C T function ( F T ( R E C T ) = S I N C ) (dashed line) 110 4.3 I D Dose deposit ion kernel derived from ion chamber measured commissioning data. D a t a exported from Eclipse v6.5 T P S (depth = 5.0 cm) 112 4.4 C y l i n d r i c a l l y symmetric I D dose kernel for depth = 5.0 cm. Solid line: i n i t i a l ion chamber-derived kernel from T P S . Dashed lines: several potential central kernel values (radius = 0 cm) sampled dur ing the opt imiza t ion process 114 4.5 Dose kernel profiles for depth = 5.0 cm. 2D kernel area has been normal ized to 1.0. Solid line: i n i t i a l ion chamber-derived kernel from T P S . Dashed line: modified kernel shape. 115 List of Figures x v i 4.6 5 m m bar pat tern (peak-to-valley) dose profiles. D e p t h = 5.0 cm. Solid line: measured film doses. Dashed line: calculated doses. Dose kernel generated 1 from: (a) ion chamber commissioning data, (b) film commissioning da ta [13], (c) modified dose kernel 116 4.7 (a) C l i n i c a l fluence map. (b) Dose dis t r ibut ion measured w i t h film (depth = 5.0cm). L ine indicates locat ion of profile data 117 4.8 I M R T dose profiles (2.5 m m inferior, to isocentre). Depth=5.0 cm. Solid line: film measured dose data. Dashed line: calculated data, (a) ion chamber dose kernel (b) modified dose kernel 117 4.9 2D gamma map comparing measured data to (a) ion chamber kernel and (b) modified kernel calculated doses. . . -. i 118 4.10 His togram dis t r ibut ion of gamma values for calculated vs. measured dose com- x parison 118 4.11 C l i n i c a l example #2. Depth=5.0 cm. ( a ) I M R T dose dis t r ibut ion. (b),(c): X - a x i s , and (d),(e): Y - a x i s dose profiles at point (-0.4 cm,4.3 cm) relative to isocentre. Solid line: f i lm measured data. Dashed line: calculated data. (b),(d): ion chamber dose kernel (c),(e): modified dose kernel • . . . . 120 5.1 Linear accelerator construction model for V a r i a n 2 1 E X unit - photon mode (ren-dered i n B E A M n r c ) . Yel low dashed lines indicate simulated photon trajectories. 127 5.2 Phase Space planes are captured above and below the secondary jaws, (a) loca-t ion of phase plane above secondary jaws on linac model , (b) locat ion of phase space plane below jaws, (c) X - Y scatter plot of particle d is t r ibut ion above jaws, (d) Phase space dis t r ibut ion below jaws set to produce 10 x 10 c m 2 field at isocentre (100cm) : . . 130 5.3 Examples of information contained wi th in the phase space file (6 M V x-ray beam col l imated to 10 x 10 c m 2 at 100 cm.) (a) Pho ton energy spectrum. Note the 511 k e V peak originating from posi tron annihi la t ion events, (b) Relat ive photon planar fluence as a function of radial distance from the central axis, (b) Relat ive photon energy fluence 131 List of Figures x v i i 5.4 Schematic of M L C leaf segmentation into two regions. V i r g i n i a Commonwea l th Univers i ty M L C model calculates thickness of leaf mater ial traversed by an inc i -dent photon and calculates probabi l i ty of a single compton scatter. 132 5.5 Percentage depth dose curves for three different fields sizes: 3 x 3, 10 x 10 and 40 x 40 c m 2 . Sol id line: ion chamber measured data. Dot ted Line : Monte Ca r lo simulated data 140 5.6 Off-axis profile curves for three different depths i n phantom (1.5 cm, 10.0 cm and 30.0 cm). Different field sizes shown: (a) 3 x 3 c m 2 , (b) 10 x 10 c m 2 and (c) 40 x 40 c m 2 . Sol id line: ion chamber measured data. Do t t ed Line : Mon te C a r l o s imulated data, (d) O A R comparison for three different field sizes at depth 1.5 cm. 141 5.7 M C model of the V a r i a n C L 2 1 E X linear accelerator head (a) Group A - tar-get, p r imary coll imator, monitor chamber, mylar mirror , and Phasespace-A. (b) G r o u p B - monitor chamber, mylar mirror , secondary co l l imat ion (jaws) and Phasespace-B ^ . . i 143 5.8 (a) Dose dis t r ibut ion for 10x10 c m 2 6 M V x-ray field incident on a cyl inder phantom. Dep th to isocentre = 13.5 cm. 100 M U delivered, (b) P D D pro-file. Errorbars: Monte Car lo absolute dose data. Sol id line: treatment p lanning system data (Var ian E c l i p s e ® P B K algorithm) 144 5.9 M L C leaves define an open window. T h e window is s l id across the open beam deposi t ing dose i n an integrating ion chamber. W i n d o w widths = 1, 5, 10, and 90 m m ' 146 • 5.10 Rela t ionship between Monte Car lo dose calculat ion error (compared to measure- \ ment) and the Physical Leaf Offset setting i n V C U particle transport code. Linear regression fit to data .• . . 147 5.11 2D film plane of bar pattern M L C delivery. 6 M V , S A D 100, depth = 12.8 cm. (a) Peak-to-valley bar widths: 30 m m , 20 m m , 10 m m (b) B a r widths: 7.0 m m , 5.0 m m , 3.5 m m 148 5.12 Compar i son of 2D plane for an M L C bar pattern (a) film measured doses, (b) Penc i l B e a m calculated doses, (c) Monte Car lo simulated doses. 6 M V photon beam, S A D 100 cm, depth = 12.8 cm. B a r widths = 7.0, 5.0 and 3 .5 'mm 149 List of Figures x v i i i 5.13 2D fi lm plane of bar pattern ( > l c m bar width) M L C delivery. 6 M V , S A D 100 cm, depth = 12.8 cm. B a r widths = 30 m m , 20 m m , 10 m m (a),(b), and (c) respectively. Profile locations are shown as a dashed horizontal line. Profiles comparing film measurement (black solid line), P B K convolut ion (blue dashed line), and Monte Car lo s imulat ion (red dashed line w i t h error bars) drawn through (d) 30 m m , (e) 20 m m , and (f) 10 m m bar pat tern widths. ' 150 5.14 2D f i lm plane of bar pat tern (< l c m bar width) M L C delivery. 6 M V , S A D 100 cm, depth = 12.8 cm. Peak-to-valley bar widths = 7.0 m m , 5.0 m m , 3.5 m m (a),(b), and (c) respectively. Profiles comparing film measurement (black solid l ine), P B K convolution (blue dashed line), and Monte Ca r lo s imulat ion (red dashed line w i t h error bars) drawn through (d) 7.0 m m , (e)"5.0 m m , and (f) 3.5 m m bar pat tern widths. 151 5.15 (a) Dose d is t r ibut ion resulting from delivering a bar pat tern (< l c m bar width) fluence. L ine indicates locat ion of profile data perpendicular to leaf travel direc-t ion, (b) Profile data across the M L C leaves showing penci l beam kernel ( P B K ) a lgor i thm (blue dashed line) agreement w i th measured (black solid line) interleaf leakage. Note that the P B K algori thm does not model interleaf leakage at a l l . (c) Monte carlo s imulat ion profile (red dashed line w i t h error bars) compared to f i lm measured profile (black solid line) 152 6.1 Process d iagram w i t h approximate t imings for M C - D A O technique. A l l B E A M -n r c / D O S X Y Z n r c simulations performed on 30 A M S Opteron 2100 processors. . 158 6.2 Objects extraneous to the patient body contour that are not existent dur ing the actual treatment must be masked prior to undergoing Monte Car lo s imulat ion. (a) Before the masking, (b) After masking 161 6.3 Dose d is t r ibut ion from a single beamlet. S O L I D L I N E : P T V (a) A x i a l view, (b) Sagi t ta l v iew 162 6.4 Open-field phase space is segmented into 2.5 x 5.0 m m 2 beamlets. E a c h beamlet, <pj, contributes dose to a voxel, i, w i th in a structure-of-interest. T h i s da ta is stored i n a beamlet dose dis t r ibut ion mat r ix 162 6.5 (a) T h e beamlet dose contr ibut ion to the patient is calculated, (b) T h e doses only to the structure of interest (in this case, the P T V ) are extracted 163 List of Figures x i x 6.6 Water equivalent A V I D phantom w i t h air cavity, (a) Structure set w i t h C-shaped P T V and spinal cord-like O A R . (b) F i e l d placement 167 6.7 Ante r io r bias beam arrangement for nasopharynx recurrence example 168 6.8. F ive field lung tumour example. F i e l d arrangement shown i n the top left corner.-T h e G T V (smaller blue sphere) is the gross tumour volume ( tumour visual ized on the C T image). The P T V (planning target volume) is created by adding a uniform 10 m m margin about the G T V 171 6.9 Open field percentage depth dose through 5.0 cm air cavity. D O T T E D L I N E : P B K algor i thm (without inhomogeneity correction), D A S H E D L I N E : P B K (with modified Ba tho correction), S O L I D L I N E : Monte Car lo 177 6.10 3D representation of the 95% prescription isodose surface (light green) encom-passing the C-shaped target (red). T h e air cavity surface rendering is not shown for c lar i ty - box drawing indicates relative posit ion, (a) P B K algor i thm opt imiza-t ion / P B K final forward calculat ion, (b) Same p lan from (a) re-calculated using M C simulat ion, (c) M C - D A O plan w i t h 6 apertures per field 178 6.11 T h e opt imized D V H (before final calculation) and final D V H (after final dose calculation) are compared for the (a) fluence-based opt imiza t ion and, (b) D A O plans. ( D A S H E D L I N E ) = opt imized D V H . ( S O L I D L I N E ) = final D V H 179 6.12 F i l m Ca l ib ra t i on Curves are constructed by correlating known P D D doses (for 50, 100 and 200 M U ) to the scanned film image pixel intensity 181 6.13 Corona l dose plane M C - D A O plan: (a) film measurement ( L E F T ) and Monte Car lo ( R I G H T ) . Ar rows show tongue-and-groove effect. Profiles (b) parallel and (c) perpendicular to leaf motion. S O L I D : M C dose. D A S H E D : F i l m measured dose, (d) Cropped film plane (e) Corresponding dose-difference ( D D ) / distance-to-agreement ( D T A ) map[119]. Da rk = D T A or D D passed. W h i t e = failed. (Pass /Fa i l : D D <3% max dose / D T A <3 mm) 182 6.14 Nasopharynx Recurrence : M C - D A O Opt imiza t ion / M C - F o r w a r d Ca lcu la t ion . T H I C K W H I T E S O L I D L I N E : P T V contour. W H I T E D O T T E D L I N E : 95% isodose 185 6.15 Nasopharynx Recurrence : M C - D A O plan wi th colorwash dose representation. . 185 List of Figures xx 6.16 Mon te Car lo reveals 9.0% under-dosage of P T V (mean dose) for this I M R T plan. S O L I D L I N E : P B K opt imized / P B K forward calculat ion. D A S H E D L I N E : P B K opt imized / M C forward calculat ion 186 6.17 T h e optimized DVH (before final calculat ion) ' and final D V H (after final dose calculation) are compared for the (a) fiuence-based opt imiza t ion and, (b) D A O plans. ( D A S H E D L I N E ) = opt imized D V H . ( S O L I D L I N E ) = final D V H . . . . . 187 6.18 L u n g : (a) Ecl ipse TPS-generated I M R T plan w i t h P B K forward calculat ion, (b) save plan as (a), but w i th M C forward calculat ion. Note massive underdosage revealed by M C , 190 6.19 L u n g : M C - D A O 2D axial dose dis t r ibut ion 190 6.20 L u n g D V H : T h e Eclipse T P S generates a fluence-optimized treatment p lan based on P B K calculated beamlet dose distributions. T h e plan then undergoes a for-ward calculat ion using two methods: 1) P B K convolution (solid line) and, 2) Monte Car lo (dashed line). These D V H s demonstrate a systematic dose calcula-t ion error 192 6.21 L u n g D V H : The Eclipse T P S w i l l generate em ideal fluence-optimized treatment p lan based on P B K calculated beamlet dose distr ibutions. T h i s ideal Ecl ipse p lan D V H (dashed line) is compared to the M C - D A O generated D V H (solid line). 192 6.22 T h e optimized lung plan D V H (before final calculation) and final D V H (after final dose calculation) are compared for the (a) fiuence-based opt imiza t ion and, (b) D A O plans. ( D A S H E D L I N E ) = opt imized D V H . ( S O L I D L I N E ) = final D V H . 194 6.23 F i l m verification for Ecl ipse plan. P l a n generated w i t h Ecl ipse P B K T P S i n lung and transferred to a uniform phantom. A coronal f i lm plane is compared to Mon te Ca r lo simulations 195 6.24 F i l m verification for M C - D A O plan. D A O plan generated from M C da ta is trans-ferred to a uniform phantom. A coronal film plane at isocentre is compared to Monte Car lo simulations 196 7.1 (a) I D moving average square filter, b) triangle, c) Gaussian 200 7.2 Square wave filter (moving average) (a) original I D signal, (b) filter w i t h constant ampl i tude moved across the signal calculat ing the average of values under the curve (c) F i n a l smoothed filter (dashed line) overlaid on original signal (solid l ine). 201 L i s t of Figures x x i 7.3 Seven F i e l d I M R T reference plan w i t h 'C ' -shaped high dose region. M e a n uncer-ta inty i n high-dose region : ~1 .0%. (a) axia l dose plane, (b) coronal dose plane, (c) axia l contour plot, (d) coronal contour plot. Contour lines: 0.5, 1.0, 1.5, 1.8, 2.0, 2.2 G y 206 7.4 Dose d is t r ibut ion at coronal plane of 7 field I M R T plan. Monte Car lo s imula t ion performed w i t h 300, 50, 25 and 10 mi l l ion histories per field, (a) dose plane w i t h locat ion of profile, (b) dose profile comparison through raw data, (c) uncertainty profile at same locat ion 208 7.5 Isodose contours through coronal plane for seven field I M R T example. Unfutered da ta is shown i n a l l images, (a) reference data set w i t h 1% average uncertainty i n the high dose region. T h e following contours show the reference dataset as an underlaying solid black contour, (b) 2% uncertainty plan (dashed),(c) 2.5% uncertainty plan (dashed), and (d) 3.5% uncertainty plan (dashed) 209 7.6 Isodose contours. Sol id contour underlay = reference d a t a ( l % uncertainty), (a) raw dose dis t r ibut ion : 2.0% uncertainty, (b) S G filter applied to(a). (c) raw dose d is t r ibut ion : 2.5% uncertainty, (d) S G filter applied to (c). (e) raw dose d is t r ibut ion : 3.5% uncertainty, (f) S G filter applied to (e). 210 7.7 G a m m a M a p results comparing raw dose distr ibutions of vary ing uncertainty to reference dose plane (1% uncertainty), (a) 2% uncertainty dose d is t r ibut ion (b) 2.5% uncertainty, (c) 3.5% uncertainty. G a m m a cri ter ia = 2 m m / 2% ( R x dose) 211 7.8 G a m m a M a p examples demonstrat ing improvement i n appearance after applica-t ion of Sav i t zky-Golay (SG) filter, (a) raw dose d is t r ibut ion for the 2.5% uncer-tainty case (25 mi l l ion histories / field), (b) After S G filter has been applied to each field and the result summed, to create the final dose dis t r ibut ion. G a m m a cr i ter ia = 2 m m / 2% ( R x dose) 212 7.9 Isodose contours comparing the order i n which the 3D S G filter is applied. L E F T column: S G filter applied to each I M R T fields prior to summing final dose dis t r i -but ion. R I G H T column: R a w field doses are summed and the S G filter is applied once to the result. (a),(b) 5.3% uncertainty/f ield, (c),(d) 6.6% uncertainty/f ie ld , (e),(f) 9.3% uncertainty/f ield. 214 7.10 Effectiveness of the Sav i tzky-Golay filter in terms of order of appl icat ion and in i t i a l raw data uncertainty (millions of histories simulated per field) 215 List of Figures x x i i 7.11 3D Root M e a n Square Difference data comparing to the reference dose dis t r ibu-t ion ( 1 % uncertainty sum plan). The effect of increasing the number of histories s imulated and the order in which the Sav i tzky-Golay filter is applied is shown. . 216 9.1 Effect of flattening filter on (a) photon fluence, and (b) percentage depth dose ( P D D ) for a 6 M V x-ray medical linear accelerator treatment beam, (dashed red c line — flattening filter removed) / solid blue line = flattening filter present) . . . 224 Table of Acronyms x x i i i Table of Acronyms B E V Beam's Eye V i e w CF Cos t Func t ion - also - Correct ion Factor C K V Cen t ra l Kerne l Value CRT Conformal Rad ia t i on Therapy C T C o m p u t e d Tomography C T V C l i n i c a l Tumour Volume DAO Direct Aper ture Op t imiza t ion DD Dose Difference 1 dMLC D y n a m i c M u l t i l e f Col l imator DTA Distance to Agreement D V H Dose Volume His togram EDW Enhanced D y n a m i c Wedge EGSnrc E lec t ron G a m m a Shower - Na t iona l Research Counc i l EqTAR Equivalent T i s sue -Ai r R a t i o EUD Equivalent Un i fo rm Dose F F T Fast Fourier Transform -F W H M F u l l - W i d t h H a l f - M a x i m u m G T V Gross Tumour Volume H U Hounsfield U n i t ICRU International Commiss ion on Rad ia t ion Uni t s IMRT Intensity M o d u l a t e d Rad ia t ion Therapy K E K i n e t i c Energy K E R M A K i n e t i c Energy Released in Mat t e r Table of Acronyms x x i v k e V Ki lo-e lec t ron Voltage (describes monoenergetic photon beams) k V p Ki lovol tage Peak potential (across x-ray tube - describes polyenergetic photon beams) L E T Linear Energy Transfer M C Monte Ca r lo M C - D A O Monte Ca r lo Direct Aper ture Op t imiza t ion M L C Mul t i l e a f Co l l ima to r M R I Magne t ic Resonance Imaging M U M o n i t o r U n i t M e V Mega-electron Voltage (describes monoenergetic photon beams) M V Megavoltage peak accelerating potential (across linear accelerator waveguide - describes polyenergetic photon beams) N T C P N o r m a l Tissue Compl ica t ion Probab i l i ty O A R Organ at R i s k - also - Off -Axis -Rat io P B K Penc i l B e a m Kerne l P D D Percentage D e p t h Dose P E T Pos i t ron Emiss ion Tomography P T V P l a n n i n g Target Vo lume R M S D R o o t M e a n Square Distance R T A R R a t i o of T i s sue -Ai r Ra t io S A D Source- to-Axis Distance S G Sav i t zky-Golay s M L C Stat ic Mul t i l ea f Col l imator S P E C T Single Pho ton Emiss ion Computed Tomography S S D Source-to-Surface Distance T A R T i s sue -Ai r Ra t io T C P Tumour Con t ro l P robab i l i ty T E R M A To ta l Energy Released in Mat t e r T L D Thermoluminescent Dosimeter T M R Tissue M a x i m u m R a t i o Table of Acronyms x x v T P R Tissue P h a n t o m Ra t io T P S Treatment P l a n n i n g System V C U V i r g i n i a Commonweal th Univers i ty Acknowledgements x x v i Acknowledgements I would like to acknowledge my co-supervisors, D r . C h e r y l Duzenl i , D r . K a r l O t to , and D r . Tony Popescu at the Vancouver Cancer Centre ( V C C ) for their support of this thesis and for being the ones that have to ask the difficult (but necessary) questions. C h e r y l keeps me th ink ing of the big picture and has always reminded me that I a m my harshest cr i t ic which has really helped w i t h some of those 'bad ' days that popped up every now and again. K a r l has been a colleague for seventeen years now and his thoughtful and (sometimes painfully) honest insight into m y radia t ion therapy research has been invaluable. His dai ly sense of off-beat humour really has been appreciated. Tony's insatiable enthusiasm for the field of Monte Car lo research has been greatly mot iva t ing and inspir ing. I would also like to thank the other members of my thesis committee, D r . A l e x M a c K a y and Dr . Vesna Sossi. T h e y forced me to th ink outside the boundaries of radia t ion therapy and try to look for connections between other fields of research. T h e y also are very good at asking those tough questions! I a m deeply grateful to fellow students Mar ie-Pier re Mi le t t e and K a r l B u s h for provid ing me w i t h the boosts of momentum that I needed to keep pushing forward w i t h this thesis project. Mar ie -Pier re ' s direct aperture opt imizat ion M a t l a b code and K a r l Bush ' s phase space segmen-ta t ion scripts and C code provided a cr i t ica l springboard for this Monte Carlo-based inverse p lanning project. T h e y have also been very patient w i th my many questions and discussions. M y fellow office-mates are Peter Petr ic and Tony Mestrovic . They offered encouragement, com-ments on papers, practice presentations and impor tant ly the desperately needed comic relief that one must have to survive this long endeavor. V i c t o r i a Cancer Centre physicists, D r . Wayne B e c k h a m and D r . Sergei Zavgorodni have provided resources, codes, great ideas and a regular forum for meeting and discussing the various Monte Car lo research and cl in ica l implementat ion projects w i t h i n the agency. I thank my family who offered uncondit ional support for this crazy undertaking. T h e T h o m -son family have also been so encouraging every step of the way which really made me feel great. I th ink that they are a l l breathing a b ig sigh of relief now! F ina l ly , Ian Thomson patiently suffered through my erratic swings between frustration and elation. T h i s P h D project has been the demanding chi ld of our family for the last four and a half years and I absolutely would not be able to finish this wi thout his calming, loving support (...and his m a d C/C++ programming skil ls) . The poor guy didn ' t know what he was signing up for when he started dat ing a P h D student. In my opinion, al l spouses/partners should get at medal on graduation day! Note: T h i s doctoral project has been supported financially by the Michae l S m i t h Foundat ion for Hea l th Research - Trainee A w a r d , the Univers i ty of B r i t i s h C o l u m b i a - Univers i ty Graduate Fellowship, and the B C Cancer Agency. Thei r contributions are acknowledged w i t h much appreciat ion. Chapter 1. Introduction to Thesis Project 1 Chapter 1 Introduction to Thesis Project Ionizing radia t ion therapy is commonly used to treat human disease, i n part icular carcinoma, or cancer. There are benefits to s imulat ing a patient's treatment prior to the actual appl icat ion of the therapeutic radiat ion beam. P r io r knowledge of the dose deposit ion bo th i n the cancerous treatment target area and the surrounding normal tissue w i l l allow doctors to determine an op t ima l radia t ion therapy plan that w i l l destroy the disease while sparing healthy tissue or avoiding side-effects due to treating radiat ion sensitive tissues (e.g. optic nerve or spinal cord damage). A n in t roduct ion to effects of radiat ion on biological systems and the identification of therapeutic goals is discussed i n Section 2.2 Intensity M o d u l a t e d Rad ia t ion Therapy ( I M R T ) is a sophisticated x-ray photon radia t ion delivery technique capable of generating rapidly changing regions of h igh and low photon flu-ence (and thus dose) across a plane perpendicular to the beam axis. Smal l , mult i- leaf col l imator ( M L C ) defined beam segments are used to deliver the photon beam. The combinat ion of pho-ton fluence modula t ion w i t h several treatment beam angles produces three dimensional dose dis tr ibut ions that conform to very complex tumour targets while sparing nearby cr i t ica l struc-tures or healthy tissue. I M R T planning is d ivided into two major components: 1) op t imiza t ion of the treatment parameters using an inverse treatment planning technique, and 2) final dose calculat ion. A detailed in t roduct ion to I M R T theory is presented i n Section 2.7. I M R T is often used i n the treatment of head and neck cancer. T h e tumours i n this site are often i n close proximi ty to sensitive structures such as the parot id glands and bra in stem / spinal cord. Treatment beams may be forced to cross significant regions of tissue inhomogeneity such as sinus air cavities. M a n y I M R T planning systems use the concept of op t imiz ing the weight of smal l subfields or beamlets of radia t ion such that the total contr ibut ion of a l l the beamlets produces the desired dose dis t r ibut ion. If the relationship between a beamlet 's properties and Chapter 1. Introduction to Thesis Project 2 the dose to the patient for a l l fields can be established, the opt imiza t ion process can quickly manipulate this da ta as part of the inverse planning procedure[l , 2]. It has been shown that smal l fields applied to inhomogeneous materials challenge t rad i t iona l dose calculat ion algorithms that do not model electronic disequilibrium(e.g. single penci l beam kernel ( P B K ) convolution) [3, 4, 5, 6]. Electronic disequi l ibr ium occurs when the electron fluence entering a calculat ion region does not equal the electron fluence exi t ing the region. If the dose calculat ion accuracy is in question, bo th the I M R T opt imiza t ion process and the final forward dose calculat ion could be compromised. In effect, the opt imizat ion process is mis led by these dose errors because i t cannot recognize problem areas (regions of electronic disequi l ibr ium). T h i s eventually leads to an opt imiza t ion convergence error [7]. A n outcome of this error is that plans that appear to adequately cover the planning target volume ( P T V ) may be revealed as having an under or over-dosage once the plan is re-calculated using an accurate dose calculat ion a lgor i thm or s imulat ion (e.g. M o n t e Car lo simulation) [8, 9]. T h i s thesis explores the effect of dose calculat ion accuracy on these different aspects of I M R T p lanning and treatment. There are three major sections. F i r s t , the l imita t ions of a. popular 2D single pencil beam kernel ( P B K ) convolut ion a lgor i thm for forward dose calculations are explored (Chapter 4) [10]. Th i s a lgor i thm is implemented in some commercia l treatment planning systems ( T P S ) . The algori thm starts by subdiv id ing an open, irregularly shaped radiat ion field into a finite number of near-zero-area penci l beams. Associa ted w i t h each penci l beam is a dose deposit ion kernel that describes the physical d is t r i -bu t ion of absorbed dose about the narrow beam axis in a water-equivalent mater ia l (phantom). T h e to ta l dose to a point of interest i n phantom is the superposit ion of the dose contributions f rom. a l l the penci l beams i n the field (mathematically, this can be represented by a convo-lut ion) . T h e shape .of the dose kernels depends on the beam energy spectrum and thus are depth-dependent. T h i s dose calculat ion algori thm is described i n detai l i n Section 3.1. T h e abi l i ty of this a lgor i thm to resolve areas of high dose gradients is par t ly related to the shape of the penci l beam dose deposition kernel. The shape of the kernel is derived from a series of in-phantom measured beam data (profiles and percentage depth doses) [11, 12]. B y generating dose kernels based on measurements in phantom, beam spectrum (including scatter) characteris-Chapter 1. Introduction to Thesis Project 3 tics at depth are automat ical ly incorporated. For I M R T applications, i t has been demonstrated that higher resolution dosimeters (e.g. f i lm, diodes) are necessary to accurately capture the high dose gradient regions and to generate appropriate dose kernels for I M R T [ 1 3 , 14]. T h e sharper dose kernels w i l l improve dose calculat ion accuracy for the complex I M R T fluence map deliv-eries that produce high dose gradient regions. However, discrepancies between measured and calculated doses for I M R T fields s t i l l exist. Chapter 4 demonstrates that further improvements to the dose calculat ion accuracy can be achieved by iteratively modifying the shape/structure of the penci l beam dose kernel. Op t imiza t ion of the P B K shape based on measured-to-calculated test pat tern dose comparisons reduces the impact of some l imitat ions of this a lgor i thm. T h e results of this s tudy have been published i n the peer reviewed journal , Medical Physics (Bergman et aL,Me<i.P/tys.,31(12):3279-87,2004)[10]. There are, however, other remaining l imi ta t ions (e.g. assuming a spat ial ly invariant dose kernel[15, 16], the lack of model ing of extra-focal radia t ion from the linear accelerator[17], the lack of model ing of lateral electron transport[5, 18], and the use of a coarse kernel sampling grid (2.5 mm)) . These l imita t ions suggest that I M R T may require a more accurate dose calculat ion method. Th i s realization leads into the second major area of investigation of this theses (Chapter 6). Mon te Car lo s imulat ion is a general stochastic stat ist ical sampl ing technique used to solve mathemat ica l problems. In this applicat ion, Monte Car lo calls upon the fundamental physics of radia t ion interactions to simulate particle (photon, neutron, or charged particle) transport and ul t imate ly dose deposit ion i n different media. T h i s s imulat ion method is widely recognized as the most accurate dose calculat ion engine for radiat ion therapy planning par t icular ly i n regions of electronic disequi l ibr ium (e.g. interfaces between l o w / h i g h density materials and smal l field sizes used w i t h high beam energies) [5, 18, 19, 20]. A detailed description of Monte Ca r lo theory is presented i n Chapter 5. Trad i t iona l I M R T opt imiza t ion methods involve changing the weights or intensities of the fluence pixels i n a 2D pixelated map to obtain an optimal fluence or intensity map[21, 22] (See Section 2.7.4 for a discussion on opt imiza t ion methods). After performing this fluence-based optimization, a leaf sequencing a lgor i thm is applied to determine the M L C leaf motions that w i l l deliver the desired distribution[23, 24, 25]. The leaf sequencer incorporates M L C dosimetric Chapter 1. Introduction to Thesis Project 4 and geometric properties such as transmission and leaf-motion l imitat ions. Since the optimal fluence map is not physical ly deliverable by the M L C , the closest i t can come to this ideal s i tuat ion is called the actual or deliverable fluence map [26]. Several groups have introduced Mon te Car lo simulations into this conventional fiuence-based I M R T opt imiza t ion process[8, 27]. De Gersem et a/.(2001) and Shepard et a/.(2002) bo th proposed that some M L C properties can be direct ly incorporated into the opt imizat ion process thus e l iminat ing the leaf sequencing step[28, 29]. Shepard et al. label their technique direct aperture optmization ( D A O ) . T h e advantage of this method is that the resulting M L C apertures are inherently deliverable and are generally more efficient than the t radi t ional fluence/intensity map opt imiza t ion i n that fewer treatment segments and radiat ion units (called monitor units ( M U ) ) are required. D A O is discussed further i n Section 2.7.5. It has been suggested that an ideal inverse planning system would include the mult i- leaf col l imator i n the op t imiza t ion process (e.g. D A O ) and employ an accurate dose calculat ion method (e.g. Monte Car lo simulations) [30]. Shepard et al. have used an in-house Monte Carlo-style point-kernel superposit ion a lgor i thm in their D A O process [31, 32]. Incorporat ing Mon te Ca r lo calculations into the I M R T opt imiza t ion process w i l l ensure that photon/e lect ron transport w i t h i n or adjacent to smal l air cavities are properly modeled. T h e op t imiza t ion can now at tempt to compensate for the dose perturbat ion caused by the inhomogeneity. In addi t ion, Mon te Ca r lo can provide valuable information about the dose contr ibut ion from extra-focal sources (e.g. flattening filter). Th i s is often neglected by some t rad i t iona l dose calculat ion algori thms. T h i s second section of the thesis uses Monte Car lo simulations (based on the E G S n r c par-ticle transport code) to calculate the entire beamlet dose deposit ion ma t r ix under full scatter conditions. T h e beamlet doses provide input to a direct aperture op t imiza t ion inverse p lanning program. T h i s technique is referred to in the thesis as MC-DAO (See Chapter 6). Incorporat ing Mon te C a r l o calculations into the I M R T opt imiza t ion process w i l l ensure that dose i n regions w i t h i n or adjacent to smal l air cavities is properly modeled and any dose perturbations caused by the inhomogeneity are recorded. T h e goal of merging M C and D A O is to assess if improved tissue inhomogeneity and electronic equi l ibr ium information combined w i t h the D A O technique Chapter 1. I n t r o d u c t i o n t o Thesis P r o j e c t 5 w i l l improve the accuracy and delivery efficiency of the opt imized treatment p lan. T h e results of this s tudy have been published i n Medical P h y s i c s (Bergman et al.,Med.Phys.,33(10):3Q6Q-79,2006)[33]. T h e M C method is cri t icized for having a relatively long s imulat ion t ime (on the order of hours). T h i s is somewhat of an artificial l imi ta t ion as increasing computer speed and paral lel comput ing w i l l reduce calculat ion times. Several authors have proposed strategies that w i l l reduce the number of particles required to complete a s imulat ion while main ta in ing acceptable s tat is t ical uncertainty. These are called v a r i a n c e r e d u c t i o n s t r a t e g i e s (see Section 5.1.5). A n -other approach is to simulate fewer particles and then apply a d igi ta l smoothing filter to the 'noisy' data. T h i s technique is discussed i n the th i rd section (Chapter 7). T h i s chapter describes a prac t ica l implementat ion of the 3D Sav i tzky-Golay digi ta l filter. It offers d ig i ta l smoothing of da ta possessing higher uncertainties associated w i t h each dose value wi thout sacrificing dose accuracy / resolution. T h i s thesis concludes by stat ing that the 2D P B K convolution a lgor i thm has major l imi t a -tions for I M R T treatment planning despite efforts to improve the spatial resolution of the dose deposit ion kernel. Monte Carlo-based D A O for I M R T is feasible and offers exquisite model ing of environments where electronic disequi l ibr ium exists. The s imulat ion times are improved by implement ing a large computer cluster environment i n conjunction w i t h a 3D dig i ta l Savi tzky-Golay smoothing filter. Chapter 2. Introduction to Radiation Therapy Physics 6 Chapter 2 Introduction to Radiation Therapy Physics 2.1 History of Radiation Therapy for Cancer Treatment Rad ia t i on oncology describes the treatment of main ly malignant (but sometimes benign) disease w i t h particle and photon ionizing radiat ion sources. X - r a y photons, gamma ray photons and electrons are most commonly used for radiat ion treatment. W i t h i n months of the discovery of x-rays by W . C . Roentgen i n 1895 in Germany, E . H Grubbe at tempted to cure a breast cancer by del ivering eighteen one-hour x-ray sessions to the disease site[34]. It quickly became apparent that there was a need to quantify the qual i ty of the radia t ion beam, (energy) and the amount of radia t ion delivered to the patient. A chromoradiometer was invented by G . Holzknecht i n 1902 which correlated the photochemical colour change of a sod ium carbonate and potassium chloride solution wi th the amount of radia t ion dose absorbed[35, 36]. He proposed a " H " scale of dosimetry which was used for nearly th i r ty years. For example, the value of " 3 H " corresponded to a m i l d skin reaction. T h e discovery of r ad ium i n 1898 by Mar i e and Pierre Cur ie introduced another source of radia t ion that could be used for treatment. Radium-226 has a number of decay products, each i n t u r n are also radioactive. T h e half-life is 1602 years. A l p h a particles, be ta particles and gamma rays are generated dur ing the decay chain, and it is the gamma rays that are used for therapy. W h e n rad ium decay is i n equi l ibr ium w i t h it 's decay products (radon, lead, polonium, and bismuth) , a complicated energy spectrum is generated. However, two m a i n lines of energy are often quoted: 0.55 and 1.65 M e V . R a d i u m tubes quickly became a popular therapy format as they could be inserted into cavities or surgically implanted. In 1909, G . Forssell i n S tockho lm Chapter 2. Introduction to Radiation Therapy Physics 7 in i t ia ted a large-scale radiat ion treatment program using a technique of implan t ing r ad ium tubes. It is this physician that coined the term "brachytherapy" (or near / close therapy), a te rm s t i l l commonly used today. T h e field of radia t ion therapy was originally embraced by dermatologists for the treatment of superficial disease. T h e implantable rad ium tubes were later used to treat fair ly accessible internal tumours (e.g. gynecological applications). T h e next level i n radia t ion delivery was "roentgentherapy", also known as "teletherapy" (far therapy) or external beam therapy. T h i s modal i ty would be used to treat deep seated tumours. The problem w i t h the t rad i t iona l x-ray tubes used for dermatology applications was that the energy was i n the 50 - 60 k V p range and not very penetrating. (Note: T h e term kVp refers to kilovoltage peak potential ( x l 0 3 e V ) across the x-ray tube. It is used to describe photon beams having a spectrum of energies.) Innovations in x-ray tube design increased the m a x i m u m energy of the x-ray photons to ~ 200 keV. (Note: keV represents the term kiloelectron volt and is used when referring to monoenergetic photon energies.) These beams were called supervoltage x-rays, later to be known as orthovoltage x-rays. T h e international unit of x-ray exposure, the roentgen (R) was introduced i n 1926. It was a measure of the ionizat ion per mass of air and had units of coulombs per k i logram (C/kg). "Telecurietherapy" using rad ium sources to create an external treatment beam was explored. R a d i u m was l imi ted i n availabil i ty and was expensive. There was also a risk that the radioactive radon gas daughter product could leak out of the treatment unit and pose a health hazard. Because the. amounts of rad ium were generally quite smal l (1 - 4 g), short source-to-surface distances were required to mainta in reasonable treatment times. It is for these reasons that this technique was inferior to orthovoltage x-ray beam therapy and was eventually abandoned i n the 1930 - 1940's. A r o u n d this t ime, the concept of fractionated radiat ion therapy was introduced. T h e goal was to increase the therapeutic ratio of the x-ray treatments (increase tumour k i l l whi le sparing healthy tissue): In 1934, Couta rd presented his c l in ica l findings on fractionation. He proposed that the tota l dose of radiat ion is divided into mult iple smaller doses al lowing for the recovery of normal tissue, the repopulat ion of tumour cells, the reoxygenation of tumour cells to increase radiosensitivity, and the reassortment of tumour cells w i t h i n the cell cycle. T h e first 6 0 C o (cobalt) treatment unit was unveiled i n 1950. It generated a M e V photon Chapter 2. Introduction to Radiation Therapy Physics 8 beam (1.173 M e V and 1.332 M e V ) , had a higher act ivi ty and was less expensive than rad ium. (Note: M e V is defined as mega-electron volt ( x l O 6 eV) and is used to describe monoenergetic photon beams.) Generally, a single average value of 1.25 M e V is quoted as the energy. The half-life is 5.27 years. The radioactive source was created by bombarding a 5 9 C o target w i t h thermal neutrons to create the unstable 6 0 C o isotope. Th i s isotope decays to 6 0 N i v i a gamma photon and beta particle emission. T h e first c l inical 6 0 C o radiotherapy unit i n N o r t h A m e r i c a was instal led at the Saskatchewan Univers i ty Hospi ta l i n Saskatoon, Canada i n 1951 w i t h the support of physicist H . E . Johns. Simultaneously, a group at the V i c t o r i a Hosp i t a l i n London , Ontar io was also working on a similar cobalt therapy system. T h e Saskatoon unit was up and running on October 23, 1951, only a couple of days before the L o n d o n unit . However, the L o n d o n hospi tal d id treat the first patient (Oct. 27th, 1951), followed by Saskatoon (Nov 8th, 1951). Sadly, the first London patient died a few weeks later. The first Saskatoon patient completed treatment successfully and l ived for several decades after. The Saskatoon 6 0 C o therapy unit treated patients for 21 years and was finally decommissioned i n 1972. Linear accelerators and betatrons also became available for radia t ion therapy i n N o r t h A m e r -ica s tar t ing ~1955. D u r i n g W o r l d War II, much research effort was directed towards the de-velopment of powerful radar systems. The magnetron and klys t ron microwave power source technology serendipitously lent itself well to medical linear accelerator design [37]. Electrons could now be accelerated on demand well into the megavoltage energy range (4 - 25 M V ) before smashing into the x-ray target (tungsten and copper). (Note: M V is defined as megavoltage and is used to describe the accelerating potential of the linear accelerator. It refers to polyenergetic photon beams.) The bremsstrahlung x-rays generated from the coll is ion were used to treat the patient. T h e target could also be removed and a scattering foil inserted to facilitate par t i -cle therapy using the electron beam. M o d e r n medical linear accelerators can generate photon beams up to ~25 M V . O n l y six years after the discovery of the neutron by J .Chadwick (1932), therapy beams were being tested on different disease sites. Neut ron therapy beams are obtained by accelerating protons or deuterons and col l iding them w i t h a be ry l l ium target. T h e protons/deuterpns are typ ica l ly i n the 16 - 100 M e V range and the resulting neutron beam energy is ~ 8 - 40 M e V . Chapter 2. Introduction to Radiation Therapy Physics 9 Neut ron therapy was not available i n N o r t h A m e r i c a unt i l the early 1970's. S tar t ing i n 1954, protons (and deuterium) were used as a treatment moda l i ty for deeply seated tumours. T h e y had desirable dose deposition patterns such as dense ioniza t ion at the end of the penetration distance, cumulat ing i n a Bragg peak. T h e y also exhibi t low lateral beam spread i n tissue. T h e first large scale, dedicated proton therapy cl inic opened i n 1990 at L o m a L i n d a Univers i ty i n Cal i fornia . B e a m energies can vary between 50 - 300 M e V . T h e only proton treatment facility i n Canada is located at T R I U M F ( T R I Univers i ty Meson Fac i l i ty ) , Univers i ty of B r i t i s h Co lumbia , Vancouver, B C and is used exclusively to treat ocular disease (e.g. choroidal melanoma). B o r o n neutron >capture ( B N C ) therapy was suggested as a way to treat b ra in tumours back i n the 1950 - 1960's. Boron-10 is irradiated w i t h thermal neutrons and the coll is ion produces an unstable a tom that disintegrates into an alpha particle and a recoiling l i th ium-7 nucleus. T h e relat ively heavy particles leave a short but destructive path of ionizat ion, essentially deposit ing a high dose i n a very localized area. A s a treatment, B N C therapy was not par t icular ly successful i n the 1950-1960's main ly due to the lack of tumour-selective boron-containing drugs and access to neutron beam sources. Improved boron delivery mechanisms revived this technique i n the 1990's Negative pi-meson (pion) therapy was proposed in 1961 and cl in ica l trials were in i t ia ted i n 1975 at T R I U M F . The trials ended i n the m i d 1990's and no benefit over conventional photon therapy was demonstrated for the two disease sites (prostate and gl ioblastoma bra in tumours) . T h e p ion program was discontinued. In 1975, at the Conference Generale des Poids et Mesures, the (SCI) internat ional uni t for absorbed dose (J/kg) was named the gray (Gy) in honour of L . H . Gray, a B r i t i s h physicist who studied the effects of radiat ion on biological systems. For the past 20 years, the ma in focus i n radiat ion therapy has been on improved deliv-ery of these megavoltage x-ray photon beams to the patient. M o r e control of the co l l imat ion of the photon beam was achieved w i t h asymmetric secondary jaw (collimator) mot ion (early 1990's). A l l four edges of the rectangular beam could be adjusted independently. Poured lead or cerrobend alloy was used to create blocks that would tai lor the rectangular beam to the Chapter 2. Introduction to Radiation Therapy Physics 10 tumour shape and protect nearby sensitive structures. T h i s was a t ime intensive process and required l i f t ing of the heavy blocks by the radiat ion therapists. A multi- leaf col l imator ( M L C ) was developed to eliminate the need for poured blocks. T h i s device was comprised of t w i n banks of typ ica l ly ~fif ty-two 1 cm wide tungsten leaves that could be moved independently to create complex beam por ta l shapes. T h e original mult i leaf col l imator concept was patented i n 1959 and it 's use was described by Takahashi i n 1965[38]. However, i t wasn't' un t i l the early 1990's that these devices were availably commercially. Three dimensional patient anatomical information was integrated into the radiat ion therapy planning process in the 1970-1980's w i t h the development of the computed tomography ( C T ) x-ray scanner[39]. Rad ia t i on beams could now be shaped to the projection of the tumour resulting i n a conformal 3D dose d is t r ibut ion ( 3 D - C R T ) . N o w that precise, complex radiat ion beam shapes could be delivered to the patient, the concept of inverse treatment planning based on opt imiz ing complex photon fluence delivery maps was introduced by Brahme (1982) [40]. He published a solution for a simple case having rotat ional symmetry i n 1988[22]. F r o m there, opt imiza t ion schemes for inverse p lanning and techniques for the delivery of these opt imized fluence maps using M L C - b a s e d intensity modula ted radia t ion therapy became a major focus of research in medical physics and radia t ion oncology. Mos t radia t ion therapy facilities today now have the technology to plan and deliver h igh doses to the tumour while sparing nearby sensitive structures. Current ly, the move from the rea lm of 3D treatments to 4D treatments is occurring w i t h the in t roduct ion of adaptive radiation therapy. T h i s involves t racking and treating moving targets (e.g. lung tumours) , or t reat ing only dur ing one phase of the mot ion cycle (e.g. treat only on exhale). Other precision radiat ion therapy delivery modali t ies are currently being developed and commercial ized. These include tomotherapy (the linear accelerator (linac) rotates continuously around the patient while delivering dose i n a modulated fan beam while the patient moves through the beam) and the cyberknife (linac on a robotic a rm can deliver 100's of radia t ion beams i n a non-coplanar manner). Several l ight ion particle beams (such as he l ium (1957), neon (1977) and carbon (1994)) have become available as a treatment option[41]. The potential for biological damage for these • a Chapter 2. Introduction to Radiation Therapy Physics 11 h i g h e r - L E T (linear energy transfer) radiat ion beams are less affected by tissue oxygenation and variations i n the cell cycle and D N A repair. Generally, their availabil i ty is l imi t ed to large research facilities w i t h sophisticated cyclotrons that can generate the desired particle beams. Imprecision in c l in ica l target volume definition is currently a l i m i t i n g factor i n precision ra-d ia t ion therapy treatment. Funct ional and molecular imaging modali t ies ( P E T , M R I , S P E C T ) are currently i n the process of being integrated into the c l in ica l radiat ion therapy environment to better localize the biological extent of malignant tumour sites. 2.2 Radiation Biology Oncology describes the study of cancer. Oncogenes are modified normal genes w i t h i n a cell that increase the probabi l i ty that a malignancy w i l l develop. Cancer is a malignant disease state where the specific cells w i t h i n the body lose their abi l i ty to regulate cell d ivis ion. T h i s results in an uncontrolled explosion of tissue growth, often to a point of destroying a functional organ or sloughing malignant cells that then spread throughout the body. The use of radia t ion to k i l l tumour cells has been documented since 1895. T h e exact mechanism of cell death due to radiat ion is an active area of research today. T h e general v iew supports the theory that most cell damage occurs when radiat ion induces single or double-strand breaks i n the nuclear D N A helix structure (sugar / phosphate backbone) or damage to one of the base molecules (cytosine, thymine, adenine, or guanine). Damage to the proteins i n the cell membrane is another possible mechanism that would disable a cell . Damage could affect the cell membrane permeabil i ty to certain molecules (e.g. electrolytes). D N A damage is s t i l l considered to be the pr imary cause of cytotoxicity. 2.2.1 Indirect and Direct Action Damage to D N A T h e mechanism for damage is either by a direct interaction between the energetic part icle and the D N A molecules or by an indirect interaction v i a the- product ion of highly reactive free-radicals that then attack the D N A molecules (see Figure 2.1). Damage to the D N A might involve a single strand break, a double strand break or a modif icat ion to the base pairs (Figure Chapter 2. Introduction to Radiation Therapy Physics 12 2.2). A complete, double strand break can lead to irreversible damage to the D N A ' s abi l i ty to reproduce, eventually leading to cell death. A single strand break has a chance of being repaired by the D N A . However, i f the repair is incomplete, or contains an error i n the protein sequences, cell death or muta t ion may occur. hv hv Figure 2.1: Damage to D N A can occur v i a direct or indirect action. Direct : part icle interacts direct ly w i t h D N A molecule. Indirect: particle ionizes water to produce free radicals which then interact w i t h D N A molecule. Single strand break Double strand break Figure 2.2: Single and double strand breaks causing damage to D N A . If repair mechanisms are unsuccessful, the cell w i l l die. Free radicals are created when the radiat ion causes hydrolysis of water. T h i s is a multi-step process and can take one of two routes: ionizat ion or excitat ion. T h e first route starts w i t h the absorpt ion of enough energy by the neutral water molecule to cause it to become ionized (see E q u a t i o n 2.1)[42]. T h e free electron becomes temporar i ly t rapped between water molecules (~ 1 ms) and is referred to as an aqueous electron (eaq). H20 + hv —i H20+ + e~ —• H20+ + e" (2.1) T h e posi t ively charged water molecule then breaks up into a bare proton and a hydroxy l free radical , OH (Equat ion 2.2). The OH radical has an unpaired electron and is looking for • \ Chapter 2. Introduction to Radiation Therapy Physics 13 another electron to pair wi th . It is highly reactive and has a lifespan of 1 /v,s. H20+ —> OH + H+ (2.2) T h e second route leading to the hydrolysis of water involves having the water molecule become excited, but not ionized after absorbing energy from the incident part icle (Equa t ion 2,3). H20 + hv —> H20* —> H • + OH (2.3) T h e excited water molecule then s imply disintegrates into two electrically neutral radicals, hydroxy l (-OH) and hydrogen (H-). For low linear energy transfer ( L E T ) radiations (i.e. electrons, photons), the p r imary process for D N A damage is indirect action. For high L E T radiat ion (i.e. a lpha particles, protons, ions), the frequent mechanism for damage is direct action[43]. The level of D N A damage can be modified by several biological factors. Different stages i n the cell cycle have different sensi t ivi ty to radia t ion effects [43]. The reason for this sensitivity is s t i l l an active area of research and is not wel l understood. T h e late G 2 (gap period when cell prepares to divide prior to mitosis) and M (mitot ic cell division) phases are the most sensitive. It has been suggested that the density of the D N A mater ia l may be a factor. In the late G 2 and M phase, the D N A has completed repl icat ion and the chromosome mater ial condenses. T h e concentration of sulfhydryl compounds also varies through the cell cycle. These compounds are known to be "radioprotectors"^ and were of significant interest to the Un i t ed States A r m y .during the C o l d War . The presence of oxygen w i l l also enhance the amount of indirect D N A damage causing cell death because the formation of hydroperoxy radicals which have a longer lifespan and can travel further to reach a D N A interact ion site (see Equa t ion 2.4). H- + 02 —> H02 (2.4) Chapter 2. Introduction to Radiation Therapy Physics 14 2.2.2 Cell Survival after Irradiation After i r radia t ing a popula t ion of cells to a given dose, many w i l l die. However, some cells may survive because damaged D N A has an inherent mechanism for self-repair. T h e survival response of a group of cells to radiat ion dose is described as the surviving fraction. T h e surv iv ing fraction can be plot ted against dose to generate a cell survival curve (Figure 2.3). T h e shape of the cell survival curve w i l l depend on the type of cell being irradiated, the type of radia t ion being used (photon or particle). F r o m Figure 2.3, it is clear that the neutron beam is much more effective at cell k i l l i ng than the x-ray beam - even at lower doses. 10° 10"' C O "•g:io'z LL Iio"3 3 tf> io-4 Iff6 0 5 10 15 20 Gy Figure 2.3: C e l l survival curves depend on the type of radia t ion being used. These two curves compare the cytotoxic effect of single fraction x-ray radiat ion dose, to neutron radi -a t ion dose. (1 G y = 1 J /kg ) A d a p t e d from Johns and Cunn ingham (1983) [42]. 2.2.3 Therapeutic Ratio (TR) The goal of radia t ion therapy is to exploit the smal l differences between tumour and normal tissue response to radiat ion. Preferably, the tumour is more sensitive to radia t ion than the healthy tissue. If this is the case, it should be easy to deliver a tumourc ida l dose to the cancer while min imiz ing side-effects (damage) to the surrounding, normal tissue (radiat ion dose tolerance da ta for different tissues summarized by E m a m i et al. (1991)[44]). However, it is not unusual to have a s i tuat ion where the normal tissues are more sensitive than the tumour . If this is the case, it becomes cr i t ica l that the high doses of radia t ion are contained w i t h i n the Chapter 2. Introduction to Radiation Therapy Physics planning target volume ( P T V ) and low doses are delivered to the normal tissues. T o achieve this condi t ion, precision delivery of complex radiat ion beams is required. T w o types of probabi l i ty curves are commonly referred to in radiat ion therapy. F i r s t is the tumour control probability (TCP) curve. O n first approximation, this curve could be considered to be the inverse of the cell survival curve discussed i n the previous section. It describes the probabi l i ty of suppressing a cancer recurrence at different doses (see Figure 2.4). T h e second curve is the normal tissue complication probability. Th i s curve is specific to different healthy tissues and organs. It describes the chance of encountering a negative side-effect of the radia t ion treatment dose. Ideally, these curves are well separated w i t h the N T C P curve to the right and the T C P curve to the left. In reality, these curves may be very close together (Figure 2.4(b)) and if the healthy organ is in close proximi ty to the tumour (e.g. tumour adjacent to optic nerve or spinal cord), t reat ing the patient effectively without causing injury may be a fine line. T h e te rm therapeutic ratio is used to describe the relative posi t ioning of the T C P and N T C P curves[45]. Quanti tat ively, the therapeutic ratio has been defined a number of different ways, but i n general, it is defined as a ratio of two cl inical endpoints. For a given dose of radia t ion, ! ) , the numerator is often the probabi l i ty that the cancer w i l l be controlled. T h e denominator is Dose -> Dose -> (a) . ' (b) Figure 2.4: Hypothe t ica l tumour Con t ro l P robab i l i ty ( T C P ) curve compared to N o r m a l Tissue Compl i ca t ion Probab i l i ty ( N T C P ) curve. Ideally, the two curves are well separated (a). Cl in ica l ly , the difference may not be so great (b) and effort must be made to deliver high doses precisely to the tumour whilst sparing the nearby sensitive healthy organs. Chapter 2. Introduction to Radiation]Therapy Physics 16 often the probabi l i ty that some undesirable side-effect w i l l be experienced by the patient (e.g. i r r i ta ted skin , loss of saliva, radiat ion pnumonit is i n the lung, rectal bleeding, cataracts) (see Equa t ion 2.5). M a n y different endpoints have been used i n determining this factor. E m p l o y i n g highly conformal radiat ion treatment techniques such as 3D-conformal radia t ion therapy (3D-C R T ) or intensity modulated radiat ion therapy ( I M R T ) or particle therapy (e.g. protons), w i l l allow for the safe treatment of patients that may be straddl ing the line between tumour control and unacceptable side effects. N o r m a l Tissue Compl ica t ion Probab i l i ty (D) 2.3 Photon Interactions with Matter Photons incident on matter have a probabi l i ty of interacting w i t h the atoms contained wi th in . T h e probabi l i ty depends on the type and density of the absorbing material , and the energy of the incident photons. T h i s probabi l i ty per unit length of absorbing mater ia l is described by a quant i ty called the linear attenuation coefficient, p. For monoenergetic x-rays, the at tenuation is described by an exponential function. The number of incident x-rays expected to penetrate through a thickness, x, of absorbing material without interacting is described by E q u a t i o n 2.6 Where • N0 = i n i t i a l number of photons incident on the medium • N = number of photons t ransmit ted through the medium Re-arranging Equa t ion 2.6 to form Equa t ion 2.7, one can see that since the numerator is a fraction and thus unitless, the units for the linear attenuation coefficient is i n 1/distance, or TR = Tumour Cont ro l P robab i l i t y (D) (2.5) [42]. (2.6) 1/crn. Chapter 2. Introduction to Radiation Therapy Physics 17 P = (2.7) x Often, the physical density (p) dependence of this factor is divided out to create the mass at tenuation coefficient (^). Th i s is done to reveal only the dependence on the material 's a tomic composi t ion. These units are quoted i n cm2/g. There are several different types of interactions between the photon and the a tom that may occur. O n l y the interactions relevant to radiat ion therapy photon energies (1 - 25 M e V ) are listed below. E a c h interaction is associated w i t h a specific linear attenuation coefficient (shown i n brackets): • Ray le igh Scattering (acoherent) • Photoelectr ic Effect ( T ) • C o m p t o n Scattering (o-incoherent) • Pa i r P roduc t i on (K) • Tr ip le t P roduc t ion («; t) T h e to ta l linear attenuation coefficient, p, is s imply the sum of the ind iv idua l interact ion at tenuation coefficients (Equat ion 2.8). T h e first interaction, Rayleigh scattering, is an elastic interaction involving the photon and the entire atomic electron. The photoelectric effect occurs when the photon transfers a l l of its energy to the bound electron and that energy is high enough to eject it from its a tomic energy orbi ta l . A cascade of energy level transfers between the remaining orbi ta l electrons results in the emission of another photon. Compton scattering is an inelastic transfer of energy from the photon to a free atomic electron resulting i n a scattered photon and a free electron. T h e last ' two types of interactions (pair and triplet production) represent a beautiful example of energy to mass conversion. T h e photon transforms into an electron-positron pair i n the electric field of an P ^coherent "T" T -f" Oincolierent ~f~ ~f* Chapter 2. Introduction to Radiation Therapy Physics 18 atomic nucleus (pair product ion) , or i n more rarer cases, the electric field of a bound electron (triplet product ion) . A detailed description of each interaction is provided in the following sections of this chapter. 2.3.1 Rayleigh (coherent) Scattering Rayle igh scattering (also called coherent scattering) describes the elastic interaction between an incident photon and bound atomic electrons (see Figure 2.5). T h e photon transfers its energy to the entire a tom put t ing it into an excited state. Th i s exci ta t ion energy is not enough to ionize the atom. T h e electron releases its energy i n the form of a photon having the exact same energy as the incident photon (elastic interaction). The incident photon is often described as being elastically scattered. The Rayle igh scattering contr ibut ion to the total photon interaction cross-section is generally very smal l for c l inical ly relevant ( 1 - 2 5 M e V ) energies and absorbers (e.g. tissue). T h e percentage contr ibut ion to the to ta l interaction cross-section is ~ 0 . 0 2 % for 2 M e V photons i n carbon. It is more prevalent for low photon energies (< 100 keV) incident on high atomic number materials. F igure 2.5: Rayle igh (coherent) Interaction, (a) Pho ton (hu) interacts w i th an a tom but does not cause an ionizat ion, (b) T h e a tom as a whole becomes excited and releases excess energy by emit t ing a photon. The emitted photon has the exact same energy as the incident photon (hu). The emitted photon is scattered at an angle very close to the original direction of the incident photon, (c) A t o m returns to ground state. Chapter 2. Introduction to Radiation Therapy Physics 19 2.3.2 Photoelectric Effect T h e photoelectric effect, is characterized by a complete transfer of energy (hi/) from an inci -dent x-ray photon to a bound orbi ta l electron. Interactions wi th " K " shell electrons (having the strongest b inding energy) have a higher interaction cross-section (so long as the incident photon has at least this m i n i m u m energy). Once the electron b inding energy (EB) has been overcome, the remaining energy is converted into kinetic energy (KE). T h e now free electron (photoelectron) has a kinet ic energy described by Equa t ion 2.9. W i t h the departure of the bound electron, a hole is formed i n the vacated energy orbi ta l (often K shell). T h e a tom is now i n an excited state. There are two main mechanisms for releasing the excess energy and returning to the ground state: the emission of characteristic radia t ion or an Auger electron. Characteristic Radiation A n electron from a higher orbi ta l (L or M shell) w i l l drop into this hole, returning the a tom to a stable state. D u r i n g the t ransi t ion from higher to lower energy, an x-ray energy photon is emit ted (Figure 2.6). The energy of the emitted photon is exactly equal to the differential between the two energy orbitals (see Equa t ion 2.10). here EK and EL are the b inding energies of the K - and L- shells respectively. T h e difference between shell levels is unique and discreet for every element, thus the term characteristic radia t ion is used. T h e most common interaction that generates characteristic photons occurs when a K-she l l electron is ejected and replaced by a M or L shell electron. T h e major K-she l l energies for several common atoms are shown i n Table 2.1. KE = hv - EH (2.9) hv' = E K - E L (2.10) Chapter 2. Introduction to Radiation Therapy Physics 20 KE = hv - E K (a) (b) (c) Figure 2.6: Photoelectric Effect (a) Incident photon (hv) interacts with bound atomic electron (e.g. K shell electron), (b) Electron gains enough energy to be ejected from the atom with kinetic energy (KE = hv — EK)- A hole in the shell remains and the atom is now ionized, (c) The vacancy is filled by an electron dropping down from a higher energy level (e.g. L shell). The energy difference between the two different shell states (EK — EL) is released from the atom in the form of a characteristic x-ray. Table 2.1: K and L shell binding energies (keV) for Various Absorbing Media H C 0 A l Pb Atomic Number (Z) 1 8 6 13 82 K-Shell (keV) 0.0136 0.283 0.531 1.559 88.001 L-Shell (keV) 0.087 15.870 Auger Electrons Characteristic radiation is not the only mechanism for an excited atom to return to a stable state after undergoing a photoelectric interaction. There is a second spontaneous process in which an atom with an electron vacancy in the K shell readjusts itself by ejecting one or more electrons instead of an x-ray photon (see Figure 2.7). The electrons ejected in this process are called Auger electrons (named for Pierre-Victor Auger who discovered this behaviour in 1925). Much like how the x-ray photons are emitted with a discreet energy, Auger electrons have a discreet kinetic energy. The energy of an Auger electron is described by Equation 2.11. KEA = E X - E 2 - E B A (2.11) Where KEA = Kinetic Energy possessed by Auger electron Chapter 2. Introduction to Radiation Therapy Physics 21 • Ei = B i n d i n g energy of in i t i a l photoelectron released (K-shell) • E2 = B i n d i n g energy of higher orbi ta l electron that fills vacancy left by photoelectron (L or M shell). • EBA — B i n d i n g energy of orbi ta l where Auger electron is ejected K E = hv - E K ^ * K E ' = EM - E K - E L '»,**•& <?-'*V* "^ -**c-.,. .1 (a) (b) (c) Figure 2.7: Auger electron emission (c) competes w i t h characteristic x-rays after a vacancy is created by the emission of a photoelectron (b). Energy and Atomic Number Dependance - Photoelectric Effect Generally, the dependence of the photoelectric mass attenuation coefficient, r / p , on the absorb-ing med ium composi t ion (atomic number (Z)) and incident photon energy (E) is derived from experimental results and theoretical interpolation. For the materials and energies encountered i n radia t ion therapy, the following relationship is accepted (see Equa t ion 2.12). where A = atomic mass. T h e var ia t ion of the mass photoelectric coefficient, r / p , w i th photon energy is shown i n F igure 2.8 for three absorbing media (water, a luminum and lead). T h e 1/E3 dependance on incident photon energy translates into a rapid decrease in the importance of the photoelectric effect when delivering therapeutic radiat ion doses to patients Chapter 2. Introduction to Radiation Therapy Physics 22 Photon Energy (MeV). Figure 2.8: P lo t of the mass photoelectric coefficient ( r /p ) for three absorbing media: water, a luminum, lead. Note the k-edge effect for lead at 88 keV. A d a p t e d from Evans (1955) [46] (1 - 25 M e V ) . For energies (> 100 keV) the attenuation co-efficient i n water is negligible. It should be noted that al though the photoelectric effect may not be important for megavoltage radia t ion therapy, it plays a significant role i n x-ray imaging where the photon energies are i n the 25 k e V (mammography) to 130 k e V ( C T ) range. Photoelectr ic interactions are more l ikely to occur when the energy of the incident photon is just s l ight ly larger than the energy required to free a bound atomic shell electron (usually the K shell). T h i s can result i n a dramatic discontinuity i n the mass attenuation coefficient (see plot for lead i n Figure 2.8). T h i s effect is called the "K-edge effect". T h e dependance on atomic number, Z , varies sl ightly depending on which end of the atomic scale the absorbing med ium falls into. For low Z materials (e.g. water, air, carbon), the dependance is ~ Z 3 . Since p / p does have a 1/A dependance and Z / A is essentially a constant (~ 1/2), the atomic number dependance is essentially oc Z 2 . T h i s strong dependance on atomic number provides the strong contrast differences between tissue (Zeff ~ 7.5) and bone ( Z e / / ~ 12.31) i n diagnostic x-ray imaging. Chapter 2. Introduction to Radiation Therapy Physics 23 2.3.3 Compton (Incoherent) Scattering C o m p t o n scattering occurs when an incident photon transfers some of its energy to a "free" electron (free : b inding energy -C incident photon energy). The incident photon and the tar-get electron are scattered away at an energy and angle that conserves the to ta l energy and momentum of the system (Figure 2.9). T h e scattered photon has a reduced energy relative to the incident photon thus the terms inelastic or incoherent scattering are also used to describe this interaction. T h e change i n energy of the photon is visualized as a lengthening in the wavelength, A A (Equa t ion 2.13). Xf — Xi = A A = —— (1 — cos 9) (2.13) where • h = P lanck ' s constant = 6.626 x 1 0 ~ 3 4 J s • m0 = rest mass of electron • c = speed of light (3.0 x 10 8 m / s F igure 2.9: C o m p t o n Interaction, (a) Incident photon (hv) interacts w i t h a free atomic electron. (b) P h o t o n transfers energy to electron which is ejected w i t h kinetic energy (KE). Remain ing photon energy scattered w i t h new energy (hv). Chapter 2. Introduction to Radiation Therapy Physics 24 • 0 = scatter angle of the photon T h e electron gains kinetic energy and moves away from the interaction site. It is sometimes referred to as a Gompton electron. The conservation of energy and momentum equations can also be used to derive the energy, E, and scattering angle, <f>, of the C o m p t o n electron (Equat ions 2.14 and 2.15). „ , ail — cosO) ,n . E = hva-—y— 2.14 l + a(l-cos6) ( ' cos(/)= (1 + a) — - • (2.15) where a = hu0/m0c2 A t lower (non relativistic) energies, the classical Thomson formula (Equat ion 2.16) is used to calculate the interaction cross-section for an inelastic (Compton) scatter event between an incident photon and electron [42]. ^Thomson = ^ ( ^ \ = 6-65 X 1 ( T 2 9 171* (2.16) 3 \m0c2J where • e c = the permi t t iv i ty of free space = 8.854 x 10^ 1 2 . farads/meter • k = 8.9875 x 10° N m 2 / C 2 (from Coulombs Law) T h e quant i ty ke2/(m0c2) = 2.8 x 1 0 ~ 1 5 m is known as the classical (or Compton) electron radius, rQ. A t higher energies, relat ivist ic quantum mechanics effects come into play and the classical T h o m p s o n scatter equation is no longer val id . Sp in and magnetic moment corrections were int roduced by K l e i n and Ni sh ina (Equations 2.17 and 2.18). C h a p t e r 2. I n t r o d u c t i o n t o R a d i a t i o n Therapy P h y s i c s 25 where F ••( 1 V 7 n « 2 ( i - ^ g ) 2 \ f 2 1 « * K N \ 1 + a ( l - cosO) J { [1 + a ( l - cos0)]{l + c o s 2 6 ) J 1 ' T h e C o m p t o n mass attenuation coefficient has the following dependencies (see E q u a t i o n 2.19) 0incoherent Z \ , . oc -7—j= (2.19) Note that Z / A i s essentially a constant, therefore, the C o m p t o n mass attenuation coefficient, a / p , has no dependance on atomic number. C o m p t o n scattering is the dominant interact ion i n the c l in ica l ly relevant energy range ( 1 - 2 5 M e V ) . Since there is no atomic number dependance, images produced w i t h megavoltage energies have very poor contrast between tissue and bone compared to diagnostic imaging energies. 2.3.4 Pair Production T h i s is the dominant interaction process for high-energy photons (> 10 M e V ) . It is a unique interaction i n that it exemplifies Einstein 's theory of the direct conversion of mass to energy and vice versa. T h i s interaction can be represented by the famous equation E = m c 2 . Pa i r p roduct ion occurs when a megavoltage x-ray photon interacts w i t h the strong electric field of an a tomic nucleus. The photon decays into a particle-antiparticle pair (in this case an electron-posi t ron pair) whilst conserving both energy and momentum (see F igure 2.10(a)). To conserve energy, the m i n i m u m energy that a photon must have to undergo this interact ion is equal to the rest mass of the electron-positron pair ( 2 m 0 c 2 = 1.02 M e V ) . If the incident photon has an energy greater than 1.02 M e V , the newly formed electron-positron pair is endowed w i t h kinet ic energy (see Equa t ion 2.20). E K = h u - 2 m 0 c 2 (2.20) T h e pair product ion mass attenuation coefficient is represented by n / p . T h e dependence of Chapter 2. Introduction to Radiation Therapy Physics 26 e e o + (a) (b) Figure 2.10: (a) Pair Production. Minimum photon energy for interaction = 2m0c2. (b) Triplet Production. Minimum photon energy for interaction = 4m G c 2 . this coefficient on the atomic number of the absorbing medium and the energy of the incident photon is described in Equation 2.21. Again, since Z/A is a constant, the pair production mass attenuation coefficient really varies with Z. This interaction becomes significant for high energies and high atomic mass materials. Triplet production is similar to pair production in that a photon interacts with an electric field. The photon energy converts completely into mass to form an electron - positron pair. The source of the electric field in pair production is the nucleus. The source of the electric field for triplet production is a bound atomic electron (see Figure 2.10(b)). If the electron receives enough energy to overcome its binding energy to the nucleus, it is set free and joins the electron-positron pair to form a triplet of charged leptons. Conservation of energy and momentum laws can show that the incident photon threshold for triplet production is 4m 0 c 2 . The probability of this interaction occurring is quite small relative to pair production (triplet:pair ~ 0.01). Often the cross-section for triplet production (2.21) 2.3.5 Triplet Production Chapter 2. Introduction to Radiation Therapy Physics 27 is combined w i t h pair product ion (K) when quoting an attenuation coefficient. 2.3.6 Summary T h e probabi l i ty of a given interaction occurring clearly depends on the energy of the incident photon, and often the atomic number of the absorbing material . F igure 2.11 identifies regions where a certain interaction is dominant. Iso-probabili ty lines depict equal importance between the C o m p t o n / Photoelectr ic effect and the Compton / Pa i r P roduc t ion effect. ,100 90 80 70 I 60 < ° 50 z o E o <C 40 30 20 10 Photoelectric Effect, Dominant, Compton Effect Dominant 10 Photon Energy (MeV) Pair Production Dominant 10 10' Figure 2.11: Iso-probabili ty lines define regions of dominant photon interact ion type. Absorber atomic number is plotted against incident photon energy. Three major regions are shown: photoelectric effect, compton scatter and pair product ion. A d a p t e d from Evans (1955) [46]. 2.4 Charged Particle Interactions with Matter Energetic charged particles interact w i t h the electric field of nearby atoms i n matter. T h e inter-actions relevant for radia t ion therapy are divided into three ma in classifications: ionizat ional , radiative, and elastic scattering interactions. Ionizational (also called collisional) interactions involve a transfer of the electron's kinetic energy to the a tom causing an exci ta t ion or ionizat ion. Chapter 2. Introduction to Radiation Therapy Physics 28 Radia t ive interactions occur when an electron is influenced by the atomic nucleus' electric field and the loss of kinetic energy takes on the form of a radiated photon (bremsstrahlung). Elas-t ic interactions w i t h the atomic nucleus results in the electron becoming deflected v i a coulomb scattering wi thout any loss i n kinetic energy. It is important to note that local absorbed dose is solely caused by atomic ionizations (col-lisions). T h e photon produced dur ing a radiative (bremsstrahlung) interaction w i l l t ravel away from the charged particle pa th and may undergo a photon interaction elsewhere. Meanwhi le , the electron is deflected sl ightly from its original course. The elastic scattering interact ion i m -parts no energy to the surrounding medium and s imply changes the direct ion of the electron travel. 2.4.1 Ionizational (Collisional) Energy Losses A s a charged particle traverses an absorbing medium, smal l discreet exchanges of energy occur along the pa th of travel. T h e kinetic energy of the charged' particle gradually decreases un t i l it comes to rest. The amount of kinetic energy transferred to the med ium per unit thickness of travel (MeV/cm) is called the stopping power (see Equa t ion 2.22). dE Stopping Power =-j— (2.22) T h e mass s topping power is often quoted and is represented by Sion (MeV cm2/g). T h i s value is obtained by d iv id ing the stopping power by the mater ial density (Equa t ion 2.23). Mass Stopping Power = Sion ~ - \ —— ) (2.23) p \ dx J Consider ing a non-relativistic, heavy, charged particle to start, it can be shown using the classical laws of conservation of energy and momentum that the energy transfer to an electron in a m e d i u m is inversely propor t ional to the kinetic energy ( K E ) of the incident part icle (Equa t ion 2.24) [42]. A £ ( 6 ) = £ ! r | ^ i L ( 2 , 4 ) Chapter 2. Introduction to Radiation Therapy Physics 29 where • b = impact parameter (min imum radius before direct impact w i t h electron occurs) • z = positive charge • r0 = classical electron radius • rn0 — rest mass of electron • M = mass of incident heavy charged particle • KE = kinet ic energy = \Mv2 (v = velocity of incident particle) A s the charged particle slows down, the energy transfer increases at a rate of 1/v2. T h e energy transfer is also proport ional to the square of the incident particle 's charge, z. T h e to ta l energy loss of a traveling charged particle can be obtained by integrating the various energy transfers occurring w i t h i n a cyl indr ica l shell defined by the m i n i m u m and m a x i m u m impact parameter rad i i , brnin/brnax. T h e impact parameter, b, describes the m i n i m u m distance that two charged particles can approach before a direct hi t on the electron occurs. In most radia t ion therapy applications, the charged particles interact ing w i t h the med ium are electrons (although heavier particles such as protons, he l ium ions and carbon ions have been studied). Electrons are smal l and light and are capable of achieving high velocities. Rela t iv i s t ic effects must be accounted for i n the mass ionizat ion stopping power, Sion. T h e b ind ing energy of atomic electrons to the nucleus w i l l also affect the incident particle 's abi l i ty to ionize the target atom. A factor called the mean ionizational energy, I, was also introduced. F ina l ly , a density or screening effect correction, 5, must be considered as distant electron interactions may be modified by the electric field of closer electrons. Bethe considered all of the above issues and derived an equation for ionizat ional losses per distance by the incident charged particle (Equa t ion 2.25) [42]. Slon = 2nr2N( mnc P2 ' KE2(KE + 2rn0c2) KE2/8 — (2KE In z + m0c2)m0c2ln2 2m0c2P (KE + m0c2)2 + 1 - p2 - 5 (2.25) Chapter 2. Introduction to Radiation Therapy Physics 30 where • Ne = number of electrons per gram of absorber • P = v/c ratio T h e var ia t ion of the ionizat ional mass stopping power w i t h electron energy for carbon is shown i n Figure 2.12. tot ion rad rad 10"? 10"1 10° . 101 102 Energy (MeV) Figure 2.12: T h e to ta l mass stopping power for carbon is the sum of the ionizat ion mass stopping power, Sion, and the radiative stopping power, Srad- A d a p t e d from Johns and Cunn ingham (1983) [42]. 2.4.2 Radiative (Bremsstrahlung) Energy Losses Radia t ive losses by the charged particle is called bremsstrahlung, G e r m a n for braking (decel-erating) radiation. T h i s phenomenon occurs when a charged particle experiences a change i n veloci ty (acceleration). A charge experiencing an acceleration w i l l radiate energy i n the form of a photon. A n acceleration may occur when an electron interacts w i t h the electric field of an atomic nucleus (and to a much lesser extent, the electric field of another electron). T h e proba-b i l i ty of producing a bremsstrahlung photon increases as the electron's p rox imi ty to the atomic nucleus decreases. The deflection (change of velocity vector) of the charged part icle depends on the charge of the nucleus as well as the charge and the mass of the incident part icle (usually an Chapter 2. \ Introduction to Radiation Therapy Physics 31 electron). T h e radiated energy is proport ional to the square of the acceleration (see Equa t ion 2.26). Radiated Energy - • . 2 / Force \ 2 / zZe2 time where , ^ ' -.2 ( r y I zZe2\ oc (acceleration) oc ( ^ - ^ — ) oc —^J (2.26) • ^ = atomic number of incident particle • Z = a tomic number of nucleus • e = elemental charge (electron charge) • M =.mass of incident particle For an electron, the radiative mass stopping power was. 'derived by Heit ler (1954) and is presented below (Equat ion 2.27). c ir2NeZKE ' 2(KE + m0c2) 1 l U 2 Q (2.27) T h e values for Sra<i as they vary w i t h electron energy are shown i n Figure 2.12 for carbon. 2.4.3 Elastic Scattering Elas t i c scattering occurs when the energy of the incident charged particle is conserved and only the direct ion of travel is changed by the coulomb potential of the target. W h e n the charged particle is an electron and i t is deflected by the coulomb force from an atomic nucleus, i t is referred to as Rutherford elastic scattering. Electron-electron scattering is also called Moiler scattering. Electron-posi t ron scattering was described by Bhabha and thus takes his name-sake. In a radiat ion therapy context, elastic scattering only serves to distr ibute the electrons throughout the absorbing medium without the deposition of energy (dose). Chapter 2. Introduction to Radiation Therapy Physics 32 2.4.4 Positron Annihilation Pa i r p roduct ion is one of several possible photon interactions w i t h matter for megavoltage st-ray energies and is discussed in Section 2.3.4. The posi tron that is created dur ing the process undergoes a series of elastic and inelastic collisions and gradually loses energy to the surrounding medium. W h e n the posi tron comes to rest (or near rest), it w i l l collide w i t h a nearby electron and bo th particles undergo a complete conversion from mass into energy v i a posi t ron annihi la t ion (See Figure 2.13). T h e rest mass of the electron (and the positron) is 511 k e V . To conserve energy, the total photon energy after the interaction must be 2 x 511 keV. However, the L a w of Conservat ion of M o m e n t u m must also be met. Assuming that the posi tron and electron are at rest when the annihi la t ion occurs, the total momentum before the interaction is zero. To main ta in this zero momentum, the annihi la t ion process creates two anti-parallel photons, each carry ing away 511 keV. Often, the positron or the electron has some smal l residual kinet ic energy at the moment when the annihi la t ion occurs. To conserve momentum, the two anti-parallel photons may not be exactly aligned at 180°. A smal l angle between the two photons w i l l exist. T h e photon energy spectrum for an x-ray beam w i l l exhibit a characteristic annihi la t ion peak at an energy of 511 k e V (see Figure 2.14). ^ hv, = m 0 c 2 Figure 2.13: Pos i t ron Ann ih i l a t i on : posi tron loses kinetic energy and comes almost to rest where it interacts w i t h a stat ionary electron. The electron / positron convert their masses completely into energy v i a E = mc2. The energy takes form of two 511 k e V photons ( m 0 c 2 ) which are emitted at a relative angle of 180° (or very close). Chapter 2. Introduction to Radiation Therapy Physics 33 120 100 80 u> c CD *~< u 60 > t5 CU CC 40 20 °0 1 2 3 4 5 6 Energy (MeV) Figure 2.14: Energy spectrum for a 6 M V x-ray .photon beam simulated by the author using B E A M n r c Monte Car lo code. Note the presence of the characteristic 511 k e V annihi la t ion spike. 2.4.5 Electron Dosimetry Elec t ron dosimetry requires knowledge of the rate of dose deposit ion i n matter by the charged particle as it traverses the material , leaving a path of atomic ionizat ion and exci tat ion. T h e inelastic exchange of energy w i t h the medium is described previously i n Section 2.4.1. Often a continuous slowing down approximation ( C S D A ) is used to calculate the dose deposit ion to matter. T h i s method follows an electron pa th i n a piece-wise fashion. A n in i t i a l s topping power is obtained from tables of data and the amount of energy deposited to the med ium over a smal l interval calculated. T h e energy loss is subtracted from the in i t i a l electron energy to obta in a new, lower electron energy. Th i s new energy is then used to look up a new stopping power value. T h e process is repeated over contiguous intervals un t i l the entire electron energy has been deposited into the absorbing medium. 2.5 KERMA and Absorbed Dose Megavoltage x-ray (and electron) beams are produced i n a linear accelerator and used to treat various human diseases, main ly cancer. Defining a concept of dose of radiat ion, much like a Chapter 2. Introduction to Radiation Therapy Physics 34 dose of prescript ion drugs, is essential. Different types of cancers w i l l respond (i.e. be kil led) differently to different amounts of radiat ion dose. Similar ly , healthy tissue has a l imi t on how much dose i t can receive before permanent radiat ion injury" occurs. Before the quanti ty of dose can be defined, a more fundamental concept, called K E R M A , must be discussed. F i r s t consider an x-ray photon beam incident on matter, for example, water. There is a finite probabi l i ty that one of the photon interaction events discussed i n Section 2.3 w i l l occur. If some of the in i t i a l photon energy is transferred to the kinetic energy of an electron, the energy at that interaction point is called K E R M A (kinetic energy released i n matter) . T h e units are i n energy transferred (by the uncharged particles) per unit mass of matter (5Etr/Sm). T h e p r imary x-ray photons are not direct ly ionizing, however they generate a cascade of secondary photons and electrons i n the medium. It is the charged particles (electrons and positrons) that deposit energy direct ly into matter. The amount of energy absorbed at a given point i n matter is called, dose. A g a i n , the units are in energy absorbed per unit mass, (SEab/5m). A special SI unit has been defined to describe the general concept of energy per unit mass. It called the gray (Gy) , named after L . H . Gray, a B r i t i s h physicist who worked ma in ly on the effects of radia t ion on biological systems. He is also involved i n establishing the Bragg- Gray Cavity Theory, used today to convert air ion chamber measurements acquired i n media to an actual dose to the media. T h e term gray can be applied to both K E R M A and absorbed dose which can sometimes cause confusion. T h e gray is defined i n Equa t ion 2.28.' \Gray = l3-^ (2.28) kg T h e relationship between K E R M A and Dose is i l lustrated i n Figure 2.15. Note the dose bui ld-up region of electronic disequil ibr ium. Photons that interact near the surface of the med ium w i l l produce secondary electrons that w i l l travel away from the point of interaction (downstream) to deliver dose. Since not as. many secondary electrons are generated i n the air outside the medium, the amount of energy entering the region close to the surface of the m e d i u m is less than the amount of energy leaving the region. T h i s is what is referred to as electronic Chapter 2. Introduction to Radiation Therapy Physics 35 disequi l ibr ium. A t a depth equal to the mean interaction distance for the photon, a condi t ion is reached where the number of electrons entering a region of interest equals the electrons leaving that same region. Elec t ronic equi l ib r ium has been established. Even though equi l ib r ium exists, i n Figure 2.15, K E R M A w i l l always be slightly less than dose. Th i s is because the dose deposited at a point of interest originated from the K E R M A deposited slightly upstream. K E R M A is affected by the at tenuation of the photon beam only and does not experience the same "bui ld-up region" as Dose. ! electronic equilibrium 4 I I 1 L Depth: in Medium Figure 2.15: The relationship between K E R M A (kinetic energy released in medium) and dose. 2.6 Radiation Delivery E a c h type of radia t ion source and energy, whether i t .be a photon or a particle w i l l produce a d is t inc t ly characteristic pat tern of dose deposition i n tissue. T h e type of radia t ion used and the mechanism for delivery depends on the disease type, the staging of the disease and its physical locat ion. Del ivery of the radiat ion to the disease site can be achieved v i a several different ways Chapter 2. Introduction to Radiation Therapy Physics 36 1. Ex t e rna l B e a m Rad ia t ion - this technique involves projecting a beam of radia t ion at the patient from an external source. It is sometimes referred to as teletherapy (therapy from a distance). T h i s beam is commonly comprised of either x-ray photons, h igh energy gamma photons, electrons, or protons. T h e radiat ion must penetrate the skin and under ly ing tissue to reach the diseased area (tumour). Often several intersecting beams are used to focus the radiat ion damage onto the tumour. Ex te rna l source beams are created one of two ways: either passively w i t h a radioactive isotope (beam essentially always 'on ' but thick lead shielding is either open or closed), or actively w i t h a medical linear accelerator (beam is only generated when power to the unit is applied). T h e original teletherapy units were comprised of a powerful, yet slowly decaying Cobalt-60 radioactive isotope (two gamma energies at 1.17 and 1.33 M e V and a half-life of 5.3 years). Today, most modern radia t ion therapy departments now rely on high-energy x-ray generation using a medical part icle linear accelerator (linac). Details of the linac construction and function are described i n Section 2.6.1. 2. Encapsula ted (sealed) Sources - These radioactive sources are placed direct ly inside the treatment area ( tumour). T h i s technique is often referred to as brachytherapy (near therapy) or interst i t ia l (between cells) therapy. The benefit of this type of therapy is that the radia t ion source is inside the tumour - you don't have to pass through healthy tissue to reach the disease as is done w i t h external beam therapy. The dose fall-off distance (from high dose to low dose) is much shorter when the treatment area is very close to the source due to the properties of the inverse square law for diverging radia t ion sources. T h i s advantage means that very large doses can be deposited local ly into the tumour wi thout r isking damage to near-by healthy or radiat ion sensitive organs. The sources are either temporar i ly placed into posi t ion using a catheter and radioactive beads or a pellet, or they can be permanently implanted w i t h a min imal ly invasive surgery technique. T h e characteristic commonly used to describe the radioactive source strength is air kerrna strength, Sk, which is measured i n units of U, where 1 U = 1 / i - G y - m 2 / h . A i r kerma strength per activity, Sk/A, is also quoted ( / i G y - m 2 / h / C i ) . Brachytherapy is generally Chapter 2. Introduction to Radiation Therapy Physics 37 div ided into two categories - high dose rate and low dose rate. H i g h dose rate treatments often utilizes an Iridium-192 (air kerma rate per act ivi ty: 109.1 / i G y - m 2 / h / G B q [ 4 7 ] = 4.04 p G y - m 2 / h / m C i ) pellet on the end of a wire (e.g. Nucle t ron m i c r o s e l e c t r o n ® ( V e e n e n d a a l , T h e Netherlands) remote afterloader). Treatment sites include esophageous, bronchus (lung), breast and treatment times are "typically of the order of minutes. Cesium-137 stackable pellets have also been used, main ly for gynecologic treatments (air kerma rate per act iv i ty : 82.1 p G y - m 2 / h / G B q [ 4 7 ] = 3.04 p G y - m 2 / h / m C i ) and the treatment times are i n the order of hours (e.g. selectron remote afterloader). L o w dose rate treatments employ permanent implantable seeds, often used in low grade prostate treatments. U p to one hundred, 1 x 5 m m seeds are used. C o m m o n sources are iodine-125 (air kerma rate per ac t iv i ty = 1.27 p G y - m 2 / h / m C i [ 4 8 ] ) , and pal ladium-103 (1.29 p G y - m 2 / h / m C i [ 4 8 ] ) . T h e permanent 1-125 seed implants for prostate cancer have a half-life (time to decay to half the original act ivi ty) of 59.4 days. A l t h o u g h the sources are never removed, a negligible external dose rate is measured around the pelvic area after a couple of months. Gold-198 (2.06 / i G y - m 2 / h / m C i [ 4 8 ] ) is used to create surgical ly placed and removed eye plaque treatments for intraocular tumours. T h e treatment t ime is on the order of several days. 3. Unsealed (unencapsulated) sources - Th i s type of radiat ion delivery usually involves the ingestion or injection of a soluble radioactive material , usually found in the form of a l iqu id . It is not contained i n a pellet or capsule as described-in the previous paragraph. The most common appl icat ion is the delivery of iodine-131 for the treatment of bo th benign and malignant thyroid conditions. Iodine natural ly seeks the thyro id gland ensuring that the majori ty of the radiat ion is concentrated i n the treatment area rather than spreading uniformly throughout the healthy parts of the body. The iodine that is not taken up by the thyro id is excreted through the kidneys and into the urine. Other unsealed sources include phosphorous-32 "(targets bone marrow), strontium-89 (bones), and y t t r ium-90 (knee joint) . S t r ic t handl ing procedures are required to contain the radioactive solutions. Chapter 2. Introduction to Radiation Therapy Physics 38 2.6.1 Generation of Megavoltage X-Rays - the Medical Linear Accelerator T h e medica l linear accelerator (linac) is designed specifically to accelerate electrons to velocities close to the speed of light. T h e acceleration is achieved by injecting low energy electrons into a h igh frequency electromagnetic waveguide that has an oscil lat ing E M field t imed to push the negatively charged electrons through the guide much like r id ing a wave (see Figure 2.16). Electron Gun Pulsed klystron or | Modulator ^ III Magnetron \ Linac •* V head Figure 2.16: Schematic of medical linear accelerator. A d a p t e d from K a r z m a r k et al.[1973][37] M o s t linacs operate i n the "S" band, or at a frequency of 2856 M H z . T h e corresponding wavelength, A, according to the equation, A = c/v, (where c = the speed of light 3 x l 0 1 0 cm/s , and v = 2856 x 10 6 / s ) , is 10.5 cm. The waveguide can be designed to operate i n a traveling wave mode or a standing wave mode. The electrons undergo a short bunching phase where groups or pulses of electrons are emit ted by an electron gun and undergo an in i t i a l acceleration. T h e y are then introduced into the high energy waveguide and gain energy as they travel down the length of the unit un t i l they reach megavoltage levels. T h e exact energy of the electrons w i l l depend on the design of the waveguide and the amount of microwave power flooding the guide at the t ime of acceleration. The microwave energy is generated by a klystron or a magnetron which is carefully coupled to the accelerating waveguide in terms of design and t iming . In Chapter 2. Introduction to Radiation Therapy Physics 39 s tanding waveguides, a circulator unit is often added between the microwave power source and the waveguide to prevent power reflections from the waveguide from interfering w i t h the k lys t ron or magnetron. T h e synchronici ty between the arr ival of the electron bunches and the microwave power is cr i t ica l . A pulse modulator controls both the k lys t ron/magnet ron and the electron gun t iming . T h e accelerated electron beam exits the waveguide and i n most gantry-based linacs, forced into a 90° or even 270° tu rn where it then collides wi th a th in "transmission" target. T h e size of the electron beam is less than 1 m m i n diameter. The target composi t ion is usual ly a mix ture of tungsten and copper. The rapid deceleration of the electron beam i n the target generates bremsstrahlung radiat ion. A t megavoltage energies, the bremsstrahlung fluence is forward peaked and emanates as a point source from the target. T h e x-rays now exit the evac-uated target region and are projected towards the patient. Between the target and the patient are several beam modifiers, namely the tungsten pr imary coll imator , the copper flattening filter (used to flatten the profile of the forward peaked beam), a transparent kapton ion chamber (to moni tor the dose delivered), secondary tungsten col l imat ing jaws w i t h independent mot ion (to create rectangular shapes), and finally an optional tert iary col l imator (e.g. tungsten mult i leaf col l imator (see Chapter 2.7.1 for description)). A schematic is shown i n Figure 2.17. Linear accelerators must be calibrated such that a given "beam on" t ime delivers a known amount .of radia t ion dose to the patient for a given set of reference conditions (e.g. depth in tissue, field size, beam energy). Linear accelerator "beam on" t ime is measured i n units called monitor units (MU) and is related to the amount of radia t ion dose measured by the monitor ion chamber located i n the treatment head (component (7) in Figure 2.17). A t the Vancouver Cancer Centre, the amount of dose absorbed i n the monitor chamber per moni tor uni t ( M U ) is adjusted by the physicist such that 1 M U w i l l deliver 1 c G y of dose to water (tissue-equivalent material) at the depth of m a x i m u m dose absorption, drnax (1.5 c m for a 6 M V photon 'beam). T h e field size is set to 10 x 10 c m 2 . Chapter 2. Introduction to Radiation Therapy Physics 4 0 Figure 2.17: Schematic of medical linear accelerator treatment head.( l ) electron gun, (2) stand-ing waveguide, (3) bending magnet, (4) electron beam steering control , (5) target, (6) carousel of scattering foils and flattening filters, (7) ion chamber, (8) asymmet-ric co l l imat ing jaws, (9) beam shaping device (multileaf coll imator) 2.7 Intensity Modulated Radiation Therapy - IMRT Intensity M o d u l a t e d Rad ia t i on Therapy ( I M R T ) is a radia t ion treatment p lanning and delivery technique used to generate highly conformal x-ray radiat ion doses to complex-shaped tumours while sparing nearby sensitive tissues (e.g. eyes, spinal cord, or rectum). The conformal shapes are created by modula t ing the photon fluence across a plane perpendicular to the beam axis. T h e probabi l i ty of tumour control increases w i t h increasing absorbed dose to the cancer cells. Ideally, the prescript ion dose would be increased unt i l a l l the cancer cells are eradicated. T h e l i m i t i n g factor in this p lan is often the tolerance l imi ts of nearby healthy organs. For example, the benefit of increasing the tumour control probabi l i ty to a vertebral cancer may be lost if the patient becomes paralyzed due to radiat ion damage to the spinal cord. I M R T is a technique that offers an oppor tuni ty to achieve excellent high-dose conformity to the target while sparing healthy sensitive organs. A fluence map refers to the 2D dis t r ibut ion of x-ray photon fluence perpendicular to the beam direct ion. T h e photon fluence is incident on the treatment volume and deposits dose Chapter 2. Introduction to Radiation Therapy Physics 11 in three dimensions. In open beams, the fluence is quite uniform (solid grey area of Figure 2.18(a)). However, the fluence can be modulated to at ta in fine control over the dose deposition. For example, the beam can be attenuated i n a non-uniform manner to produce a wedge-shaped d is t r ibut ion (see Figure 2.18(b)). Even more complex fluence map distr ibutions can be obtained by overlapping many mult i leaf col l imator (MLC)-def ined segments (Figure 2.18(c)). 10 20 30 40 50 60 70 (a) 30 JO 50 60 70 80 90 103 (b) 10 20 30 40 S3 60 70 80 90 (c) Figure 2.18: (a) open beam fluence for a square field, (b) simple fluence modula t ion, (c) Complex fluence modula t ion . Insets = horizontal profile through mid-image. 2.7.1 Fluence Modulation Techniques Fluence modula t ion across the open radiat ion beam can be achieved using several different methods ranging from very simple (solid brass wedged shaped beam attenuators) to very com-Chapter 2. Introduction to Radiation Therapy Physics 12 Figure 2.19: Wedged shaped dose dis t r ibut ion i n tissue-equivalent plastic caused by wedged shaped attenuator plex (120 independently motor-driven tungsten leaves). T h e term I M R T generally applies to the more complex method implementing the multi leaf coll imator, but technically any method of spat ia l ly manipula t ing the photon fluence intensity could be labeled as intensity modulated radiation therapy. Physical (Hard) Wedges Phys i ca l (hard) wedges are often made of brass and are mounted onto the exit port of the x-ray photon beam. Several different geometries are often available generating a range of wedge angles (e.g. 15°, 30°, 45° or 60°) . The photon beam is attenuated preferentially i n the thicker end of the wedge. The dose dis t r ibut ion to the patient w i l l also be a wedge shape (see Figure 2.19). These attenuators are useful for generating uniform dose distr ibutions when the patients' surface of incidence is non-uniform, or wedge-shaped (e.g. tangential breast geometries). Dynamic Wedges T h e cumbersome implementat ion of the heavy brass wedges motivated research into alternative methods to achieve the same type of dose modula t ion. B y moving one of the sol id tungsten sec-ondary jaws into or out of the treatment field while the beam is on, a wedge shaped dose profile can be achieved. Factors such as the desired wedge angle, the open field size, and the prescrip-Chapter 2. Introduction to Radiation Therapy Physics 43 t ion dose w i l l affect the characteristics of the jaw motion. Tables can be compiled that .allow for a very fast ' look-up ' of the treatment requirements and from there a computer controlled jaw mot ion file can be generated. T h i s technique is sometimes referred to as enhanced dynamic wedging ( E D W ) . T h e E D W technique w i l l remove the requirement for the technologists to lift heavy physical wedges, but two treatment fields w i l l s t i l l be required if two different wedge angle directions are desired. E D W s are used quite commonly as a simple missing tissue compensator. To achieve more complex photon fluence modulat ion, a more sophisticated delivery mechanism is required. Physical Compensators Phys ica l compensators are s imilar to wedges in that the photon beam is preferentially absorbed depending on the thickness of the attenuator material . These offer more complex 2D photon fluence modula t ion . The compensators can be made by mi l l ing styrofoam i n two dimensions and then fi l l ing the mi l led cavities w i t h attenuating steel shot. These compensators are effective, but the construct ion is labor-intensive. Multileaf Collimation T h e term I M R T is generally synonymous w i t h mult i leaf coll imator ( M L C ) fluence modula t ion . T h e mult i leaf col l imator is an indispensable beam shaping device (see Figure 2.20). There are two banks of tungsten leaves (up to s ix ty leaves per bank) . In the V a r i a n 120 leaf model , the central 40 leaves have a w i d t h of 0.5 cm. There are 10 leaves flanking each side of the central group and they have a w id th of 1.0 cm. A series of independent motors control the leaf mot ion into and out of the photon field (see Figure 2.21). T h e leaves are 6.0 cm thick i n a direct ion paral lel to the beam axis. The full potential of the M L C is exploited not as a general beam shaping device, but as a tool for delivering complex I M R T fluence distr ibutions. Chapter 2. Introduction to Radiation Therapy Physics 11 Figure 2.21: Schematic of Var i an 120 loaf M L C . (a) Beam' s -Eye-View of field shaping capabi l i ty of M L C (b) end-on cut-away demonstrat ing tongue-and-groove design (c) aspect view i l lus t ra t ing rounded leaf end shape. Chapter 2. Introduction to Radiation Therapy Physics 45 Figure 2.22: F i rs t Row: Three M L C - d e f i n c d apertures delivered sequentially. Second Row: T h e corresponding fluence maps delivered by the M L C . Last column = sum of the three apertures. T h i r d Row: Profile representations of the fluence maps. It is possible to modulate the photon fluence of a treatment beam by summing together the cont r ibut ion of smaller M L C - d e f i n e d apertures (see Figure 2.22). There are two modes of operation that are available that allow delivery of a complex photon fluence dis t r ibut ion: static ( sMLC)[23 , 25] and dynamic ( d M L C ) [ 4 9 , 50]. Stat ic delivery is the simpler of the two concepts and generally more intui t ive (this is the example shown i n Figure 2.22. A desired photon fluence dis t r ibut ion is divided up into a number of smaller, contiguous, unmodula ted aperture shapes. T h e radiat ion beam is delivered i n an aperture-by-aperture fashion in that the beam is off while the M L C leaves are i n t ransi t ion between two different aperture shapes (see Figure 2.23(a)). Once the desired aperture shape has been achieved, the beam w i l l t u rn on for the calculated t ime (monitor units) . Depending on the number of apertures to deliver, the l inac may have to be started and stopped numerous times while delivering a single treatment field. T h e superposit ion of a l l the aperture fluence contributions is equivalent to the overall desired photon fluence. Chapter 2. Introduction to Radiation Therapy Physics 46 BEAM ON • = - = BEAM ON •• BEAM OFF , ^ | _ _ a H _ _ f l L _ _ f i L _ BEAM OFF . MLC I -I • ' MLC POSITION I ! ! EEd! S J POSITION Aperture Aperture Aperture Aperture Moving Aperture #1 #2 #3 #4 T I M I = 4 : > TIME » » (a) (b) Figure 2.23: T imel ine for two different fluence map delivery modes for the mult i leaf col l imator (a) Stat ic M L C ( s M L C ) and (b) D y n a m i c M L C ( d M L C ) . .' D y n a m i c delivery involves moving the M L C leaves while the beam is on (Figure 2.23(b)). T h e linear accelerator is started and stopped only once per treatment field. To calculate the M L C leaf motions that w i l l generate a desired photon fluence is much more complicated. Dose-rate and leaf mot ion velocities must be considered. Precise M L C mot ion control is imperat ive to ensure that the complex dose delivery is delivered accurately. 2.7.2 Inverse Treatment Planning Inverse treatment p lanning was developed to take full advantage of the fine control over dose delivery offered by M L C - I M R T . Inverse planning is the opposite of the conventional iterative forward p lanning method. Forward planning involves placing a treatment beam on the v i r tua l patient, calculat ing the doses, and assessing the resultant d is t r ibut ion for c l in ica l suitabil i ty. T h i s process requires manual interaction and needs to be repeated as many times as needed to obta in an op t imal plan. Conversely, inverse planning requires the user to pre-define some desired dose d is t r ibu t ion constraints such as the dose to the target or the m a x i m u m dose allowed to nearby cr i t ica l structures. T h i s may sound like a problem that could be solved analyt ica l ly as a system of equations w i t h a l imi ted number of unknowns (fluence p ixe l weights). However, given the number of beams and fluence pixels involved, and the addi t ional cr i ter ia that no negative fluence pixels are allowed (we can only deliver, dose to the patient, not remove i t ) , an analy t ica l solut ion is not generally feasible. Instead, opt imiza t ion methods are employed. T h e goal is to minimize an objective or cost function describing the difference between the desired dose d is t r ibut ion and the optimized dose dis t r ibut ion (see Section 2.7.3). There are different Chapter 2. Introduction to Radiation Therapy Physics 47 methods available for min imiz ing the objective function and they are discussed in Section 2.7.4. It is impor tant to realize that the term "'optimization" can be used to describe two different steps i n the I M R T inverse p lanning process. Fi rs t , op t imiza t ion methods are used to min imize the objective function (see Section 2.7.4). Second, the physical radia t ion beam delivery to the patient must be determined for each treatment field. T h i s can be done by either op t imiz ing the intensity of the photon beamlets that make up a fluence map (the fluence map is defined at the beginning of Section 2.7), or by directly opt imiz ing physical radia t ion beam apertures defined by the mult i- leaf col l imator ( M L C ) . A comparison of these fluence-based optimization and direct aperture optimization methods is presented i n Section 2.7.5. For bo th of the methods discussed, the radia t ion delivery opt imiza t ion is performed on a l l treatment beams simultaneously. Dose Volume Histograms (DVH) Cumula t ive Dose Volume Histograms ( D V H s ) are often used to quant i ta t ively assess the qual i ty of a 3 D dose d is t r ibut ion w i t h respect to meeting certain c l in ical dosimetric goals. For example, it is desirable that 100% of the treatment dose prescribed by the radia t ion oncologist covers 100% of the cancerous tumour. . Similar ly , i t is desirable that no dose (or 0% of the tumour prescript ion dose) be deposited i n sensitive / cr i t ical organs or non-involved healthy tissue. Cumula t ive D V H s represent the volume, V (often quoted as a percentage), of a specific structure that receives a dose that is at least equal to or greater than a specific dose of interest, D. T h e result is displayed as a 2D D V H curve. Figure 2.24 illustrates cumulat ive D V H curves for two sample treatment plans. In Figure 2.24(a), the dose to the planning target volume (tumour) is not very uniform (shallow D V H slope). Th i s non-homogenous dis t r ibut ion may lead to non-uniform tumour cell k i l l i ng increasing the risk of a cancer recurrence. In the same diagram, the dose to certain regions of the cr i t ica l structures (organs-at-risk) meets and even exceeds the prescript ion dose. T h i s high dose may result in damage to sensitive structures leading to unacceptable side-effects of the treatment. Examples of.notable sensitive structures include the spinal cord, lung, rectum or lens of the eye. In Figure 2.24(b), the tumour receives a more homogenous dose. T h i s is represented by a steep D V H curve and indicates that close to 100% of the tumour volume is covered by the prescription dose and very l i t t le over-dosing is present. Chapter 2. Introduction to Radiation Therapy Physics 48 Dose D(Rx) Dose D(Rx) (a) (b) Figure 2.24: Dose Volume Histograms for two treatment plans. D(Rx) indicates prescript ion dose, (a) A poor p lan showing inhomogeneous tumour ( P T V ) coverage and high organ-at-risk ( O A R ) doses, (b) Improved plan w i t h good tumour dose uniformity and reduced O A R dose. T h e dose to the sensitive structure is also minimized . T h e treatment p lan represented by Figure 2.24(b) is considered to be a superior treatment compared to the plan represented i n Figure 2.24(a). Beamlets of Photon Intensity Inverse p lanning for I M R T begins by d iv id ing the beam's-eye-view ( B E V ) of the p lanning tar-get volume ( P T V ) into a 2D grid .of radiat ion fluence pixels, or beamlets. A beamlet is the dose d is t r ibut ion result ing from one beam element. A relationship between a given beamlet of radia t ion and the dose deposited to the target and/or nearby, healthy, sensitive structures can be calculated. In a fluence-based opt imizat ion concept, the weight of every beamlet for every treatment field is simultaneously opt imized to achieve the prescription dose to the P T V while m i n i m i z i n g doses to the sensitive organs (Figure 2.25). T h e opt imized map of beamlet fluences is delivered to the patient using the sophisticated M L C beam shaping device. The dimensions of each beamlet element i n the B E V grid is arbitrary, but bounded. Generally, i n a direction perpendicular to leaf travel, the m i n i m u m pixe l size is the m i n i m u m leaf w i d t h (often 0.5 cm) and the m a x i m u m size is the largest open area defined by the M L C bank (40.0 cm). In the direction parallel to leaf mot ion, the m i n i m u m / m a x i m u m Chapter 2. Introduction to Radiation Therapy Physics 19 (a) (b) (c) Figure 2.25: (a) Open field fluence (b) Fluence map divided into beamlets of open intensity (c) Beamlet weights are opt imized to obtain a fluence map. p ixe l size is equal to the m i n i m u m / m a x i m u m allowable gap between opposing leaves (~0.05 c m and 15.0 c m respectively for Var i an 120 leaf M L C ) . A typica l m i n i m u m beamlet dimension is 0.50 x 0.25 c m 2 . Novel methods have been explored to reduce the m i n i m u m beamlet size. O t t o et al.[51] demonstrated that by rotat ing the M L C through 180°, a circular beamlet of diameter 0.25 m m is achievable . Bortfeld et al. [52] suggest a t ranslat ion of the treatment table by a distance equal to one half of the leaf w i d t h to improve sampling i n a direction perpendicular to leaf travel (leaf w i d t h direction). F i n d i n g the op t imal beamlet dimensions that w i l l balance delivery efficiency w i t h excellent dose conformity has been studied by other authors. Generally, dose conformity and cr i t ica l s tructure sparing improves as the beamlet dimensions decrease[31]. Bortfeld et al. [52] used a general sampl ing theory and the theory of linear systems to determine the ideal beamlet w i d t h (perpendicular to leaf travel) that w i l l allow the radiat ion delivery system to approach its inherent spat ia l resolution l imi t . The beamlet w id th should be a factor of 1.7 less than the beam penumbra w i d t h (penumbra = dose fall-off distance between the 80% and 20% isodose level = A j /80%-20%)- For a 6 M V beam and relatively smal l fields, this beamlet dimension would be: 2.5 m m - r 1.7 = 1.5 m m . The beamlet w id th should be equal to the sampling size (in this case, 1.5 m m ) . W i t h the exception of some specialized stereotactic radiosurgery m i c r o - M L C (having a leaf w i d t h of 1.7 - 3 mm)[53], most M L C s w i l l not meet this cr i ter ia (leaf w i d t h 5 - 1 0 mm) . Decreasing the beamlet size offers improved "fine t iming" of the dose deposition, but this comes w i t h a cost. A s the beamlet d is t r ibut ion becomes finer, the fluence map becomes increasingly more complex to deliver - often resulting i n less efficient beam deliveries that translate into longer Chapter 2. Introduction to Radiation Therapy Physics 50 treatment times, h igh monitor unit delivery and increased M L C photon transmission (leakage). Other considerations for selecting beamlet sizes is the resolution of the dose calculat ion gr id , the finite x-ray source size and scatter / electron transport[54]. Implementing beamlets that are smaller than the dose calculat ion grid spacing or the spatial resolution capabi l i ty of the dose spread kernel offers no benefit. 2.7.3 The Objective Function T h e goal of I M R T opt imiza t ion is to assemble al l the metrics that can describe a treatment plan's deviat ion from a desired qual i ty or objective, generate a function that reduces them a l l down to a single number, and then minimize that number by modifying the treatment plan. T h e equation that can generate this single unified metric is called an objective function, (also known as a cost function or penalty function). T h e objective function does not have any physical meaning i n itself, but the value can be used as a relative amount to determine whether or not changes to the plan improves or deteriorates the quali ty of treatment. M o s t objective functions fall into one of three categories: dose-based, dose-volume based and biological. T h e dose and dose-volume based objective functions deal w i t h physical dose deposit ion information. T h e term physical refers to quantities that can be measured, such as dose and volume. For example, the objective function may include information about the number of dose points that meet some prescript ion cri teria. Similar ly , i t may include information about what volume of a cr i t ica l structure receives more than the tolerance dose. Another important physical constraint to consider dur ing the I M R T inverse planning process is the requirement that the radia t ion beamlet weights must be non-negative. Negative weights may theoretically offer an exact solut ion to the inverse problem in I M R T , but are not physical ly deliverable on a real linear accelerator. A n increasingly popular type of objective function is based on biological response of certain tissues to radia t ion dose deposition. Th i s technique includes tumour control probabi l i ty ( T C P ) and normal tissue complicat ion probabi l i ty ( N T C P ) models. T h e information is not measured direct ly but is derived from tissue response models to radiat ion. Chapter 2. Introduction to Radiation Therapy Physics 51 Dose-based Mean Square Deviation Objective Function One of the most basic objective functions has the form of a least squares equation. Equa t ion 2.29 applies an objective or cost measure to a single structure w i t h i n the patient[55]. Other structures may be added l inearly w i t h ind iv idua l weighting or importance factors, ws (Equa t ion 2.30). Npts Npts • Cost Function = Fobj(x) = £ (A - dPf = (a* ' x ~ dp? (2-29) »=i i=i where • Di is the to ta l dose to a point or voxel, i, from al l beamlet contributions • dp is the prescribed or target dose • a; is the dose contr ibut ion to voxel, i, from beamlet, j. (ai = {%}) • x = the ma t r ix of beamlet weights, (x = {XJ}) • Npts = number of points (voxels) in structure - :  ( In the target, the goal is to minimize the square of the difference between the calculated dose, Di, and the prescribed dose, dp. Th i s form assesses the sui tabi l i ty of the proposed treatment p lan using a point-by-point assessment. There are two problems w i t h this method. F i r s t , large organs w i l l dominate the opt imiza t ion because of sheer volume of data. T h i s can be resolved by normal iz ing the structure weighting factor, wa, to the number of voxels (or volume) i n the structure of interest, Ns (Equat ion 2.30). ^ W = E ? " f ( A ^ ) 2 (2-30) 3 = 1 J V S 1=1 where A ^ s t r = number of structures Secondly, this equation only has a mechanism to ensure that the dose i n the structure of interest does not deviate from the single desired dose. Some authors have modified the basic Chapter 2. Introduction to Radiation Therapy Physics 52 form of the objective function to allow separately defined penalties for over-dosing vs. under-dosing. T h i s type of objective function is often associated w i t h what are called soft constraints. These are m a x i m u m and m i n i m u m dose l imits that may by violated, but a penalty w i l l apply. In practice, a m i n i m u m and m a x i m u m dose constraint (Dmin and Drnax) are specified and each constraint has an ind iv idua l weighting (Equat ion 2.31 and Figure 2.26). ( Nstr „..min Mpts Nst,r ,...rnax v1" £ - f r - E ( A - Dmm)2H(Dmm - A ) + £ £ ( A - A ^ ) 2 ^ ( A - Dmax) s=l i = \ s=l " /V« 1=1 (2.31) where, I 1 A D > 0 H(AD) = I y 0 AD < 0 j T h i s objective function works well for the target where a uniform dose is desired, but for sensitive structures, the goal is to reduce the dose to the structure as much as possible. Equa t ion 2.30 does not t ry to minimize this dose. It can only a im the plan towards the tolerance dose, dp. Linear forms of this objective function exist and include parameters such as mean / m a x dose to target or cr i t ica l structure. In reality, not a l l constraints are achievable thus compromises must be accepted. 100 90 X -80 £ 70 1 60 o * 50 OJ CO c 10 • 20 10 n 20 40 60 60 100 Dose (Gy) Figure 2.26: Soft constraints give 'negotiable' upper and lower dose l imi ts on d is t r ibu t ion of dose w i t h i n structure. For the target, ideally 100% of the prescript ion dose covers 100% of the target volume and 0% of the volume exceeds the prescript ion dose. Chapter 2. Introduction to Radiation Therapy Physics 53 Dose-Volume Objective Function T h e concept of a dose-volume min imiza t ion scheme rather than a point-by-point dose comparison scheme has also proven to be very useful, par t icular ly for sensitive structures. Dose-volume constraints require that the user define a m a x i m u m dose (Drnax) and volume (Vmax) constraint to a structure-of-interest. For example, in Figure 2.27 the orange star is a dose constraint point. T h e goal is to irradiate.no more than 35% of lung to a dose that exceeds 20 Gy[56, 57, 58, 29, 55]. E a c h dose-volume constraint is associated wi th a weight or pr ior i ty which can be adjusted indiv idual ly . T h e goal is to minimize the volume of voxels exceeding a m a x i m u m dose for a given structure. T h i s is different from the dose-based objective described above which favors a uni form dose dis t r ibut ion for a l l points wi th in a structure. For this method volumes of voxels are min imized , not just the square of the difference between ind iv idua l points. T h e general form of the equation for this dose-volume objective function (F^J) is shown below (see E q u a t i o n 2.32 and Figure 2.27),. where • Dmax = dose specified i n dose-volume constraint number, v • D* = dose where horizontal line at height Vmax intersects w i t h D V H curve • Nfoc — dose volume constraint for structure, str A n extension of this type of objective function allows the user to assign hard constraints. H a r d constraints define a m a x i m u m dose which must not be exceeded, wi thout compromise. For example, a hard constraint would be applied to the spinal cord where above 45 G y permanent paralysis is possible. Because every dose-volume constraint is associated w i t h a pr ior i ty or importance, one can assign a very high prior i ty to ensuring that a m a x i m u m dose is not exceeded. ) (2.32) v=l i Y s i = l and Chapter 2. Introduction to Radiation Therapy Physics 54 ao 100 Dose (Gy) Figure 2.27: Dose-volume constraints minimize the number of voxels that fall under the D V H curve, bounded by Drnax and D* (shaded area) [29]. D* is located at the intersection between the volume constraint, Vmax and the D V H curve. F igure 2.28(a) demonstrates an example where meeting the m a x i m u m dose l imi t is set to some moderate value. Figure 2.28(b) assigns a high pr ior i ty to meeting the m a x i m u m dose l imi t . 00 90-80 70 60-> 50r 1 40: I 30 20 10 0-0 Dose (Gy) (a) Figure 2.28: Effect of pr ior i ty setting on opt imizat ion of dose-volume histograms, (a) L o w pr ior i ty on enforcing m a x i m u m dose hard constraint, (b) H i g h priority. T h e work presented i n this thesis is based on opt imiz ing the treatment p lan using a combina-t ion of dose and dose-volume constraints. T h e target dose d is t r ibut ion is assessed by summing up the square of the difference between the prescribed and current dose over a l l the target points. Dose-volume constraints are applied to the cr i t ica l (sensitive) structures. Chapter 2. Introduction to Radiation Therapy Physics 55 Biological Response Objective Function Authors have suggested that I M R T opt imizat ion should be based on the biological effects of dose to specific tissues[59, 60, 61]. Radiobiological models can be used to predict the tumour control probabi l i ty ( T C P ) and the normal tissue complicat ion probabi l i ty ( N T C P ) . New models also account for the dependence of the tumour / organ's response to par t ia l ly i r rad ia t ing the volume and inhomogeneity in the delivered dose dis t r ibut ion. The goal of a biological objective function is to maximize the T C P while min imiz ing the N T C P . Au tho r s have described an alternative biological constraint called the equivalent uniform dose (EUD)[62 , 63]. E U D describes the effect of the spat ial d is t r ibut ion of radia t ion dose on tumour cell k i l l . If a prescription dose is delivered non-uniformly (i.e. there are regions of under-dosage and /or over-dosage), the cell k i l l is not cis efficient compared to a uniform dose delivery. Non-uni form dose distr ibutions require a higher prescription dose to achieve the same amount of cell k i l l as uniform dose distributions. T h e advantage of this index is that i t can convert a physical dose dis t r ibut ion into a biological objective without completely defining complex T C P and N T C P functions. It does not predict biological response directly, however i t can be used to maximize T C P and minimize N T C P . W h e n u t i l i z ing the E U D , several assumptions are made [64]: • the major effect on the T C P is the m i n i m u m dose to the target - only a smal l weighting is given to hot spots (regions of m a x i m u m dose) • the N T C P for serially functioning normal organs (e.g. spinal cord) is main ly affected by the m a x i m u m dose - only a smal l weighting is given to cold spots (regions of under-dosage) • the N T C P for paral lel functioning normal organs (e.g. lungs, kidneys) is main ly affected by the mean dose A l t h o u g h biological objective functions can often generate superior treatment plans com-pared to physical objective functions, the tumour response and normal tissue damage models (e.g. dose-volume relationships) that are the cornerstone of this method are not completely characterized and are s t i l l debated by clinicians. Chapter 2. Introduction to Radiation Therapy Physics 56 500 450 400 5350 :300 )250 U-200 <j150 100 50 0 Starting Point #1 20 40 60 Suggested Solution # 80 100 40 60 Suggested Solution # 100 (a) (b) F igure 2.29: P l o t of objective (cost) function value vs. suggested solution, (a) Gradient based opt imiza t ion method may converge on a local m i n i m u m which is not ideal, (b) Stochastic opt imizat ion methods find preferred global m in imum. 2.7.4 Optimization Techniques for Minimizing an Objective (Cost) Function T h e goal of inverse treatment planning is to minimize an objective function, examples of which are discussed i n Section 2.7.3. Generally, the I M R T inverse p lanning problem cannot be solved analy t ica l ly as there are l imitat ions on many parameters, such as the requirement for non-negative beamlet weights (only dose deposition allowed, not dose extraction) or ' there may be just too many variables (thousands) to solve for mathematical ly in an efficient manner. Instead, op t imiza t ion methods are employed. Generally, objective function opt imiza t ion (minimizat ion) techniques fall into two major categories: deterministic and stochastic (random). Determinis t ic op t imiza t ion moves "downhi l l" from a start ing point (Figure 2.29(a)) and adjusts parameters according to some algori thm. The gradient method is an example. Stochastic methods adjust parameters i n a more random fashion (Figure 2.29(b)). Simulated annealing is an example of this class of op t imiza t ion method. A desirable opt imiza t ion technique w i l l quickly converge on the global m i n i m u m of the solution space rapidly wi thout becoming t rapped in a loca l m i n i m u m . Different techniques have various advantages and disadvantages and they are described briefly below. Chapter 2. Introduction to Radiation Therapy Physics 57 Simulated Annealing (stochastic optimization) Simulated annealing falls under the category of stochastic techniques. The name originated due to parallels w i t h solid state physics and the behaviour of metals as they cool i n temperature and form a lowest energy state in a crystall ine form[65]. T h i s method is used extensively i n this thesis thus more detai l is dedicated to this topic i n this section. 1) S E L E C T F I E L D 2) S E L E C T A P E R T U R E 3) S E L E C T P R O P E R T Y 4) S E L E C T M A G N I T U D E of C H A N G E L E A F P O S I T I O N DIRECTION MAGNITUDE APERTURE WEIGHT D I R E C T I O N M A G N I T I T J E F igure 2.30: Simulated annealing opt imiza t ion process for direct aperture op t imiza t ion (Fields, apertures and M L C properties are randomly selected. F igure 2.30 illustrates the opt imiza t ion process for the direct aperture op t imiza t ion process. F i rs t , a beam direct ion (one of the pre-determined beam angles) is randomly selected. T h e op t imiza t ion must then randomly choose whether the aperture weight or the aperture shape of the identified beam w i l l be modified. • If the aperture shape is selected, one of the 40 central M L C leaves (5 m m width) is randomly activated. A change i n leaf posi t ion is calculated. T h e new leaf posi t ion is accepted only if it falls w i th in the allowable "solution space". Th i s solution space includes only those M L C leaf positions that are physical ly deliverable Chapter 2. Introduction to Radiation Therapy Physics 58 • If the aperture weight is selected, a new weight is selected by the opt imiza t ion . Negative aperture weights are forbidden. T h e m a x i m u m amount that the leaf posit ion or the aperture weight is allowed to change (step size) is determined by Equa t ion 2.33. The variable, R s t e p , describes the rate of decrease for the allowable size of any changes. Initially, large changes (steps) w i t h i n the solut ion space is possible. A s the number of successful iterations, nsucc, increases, smaller and smaller step sizes w i l l be allowed, thus shr inking the solution space. X(nsucc) = X o (2.33) ( n s u c c + l ) « s t c p where • XQ = i n i t i a l step size • nSucc = number of successful iterations • RsteP = rate at which the step size of any suggested changes decreases. Un l ik e gradient based methods that can only move "downhi l l " , s imulated annealing w i l l accept solutions that worsen the objective (cost) function value despite the fact that the ul t imate goal is to min imize this function. T h i s feature gives the opt imiza t ion a chance to escape from any potent ia l loca l m i n i m a and converge on the global min imum. The probabil i ty, P, that a worse solut ion (increase i n the objective (cost) function) is accepted is governed by the Met ropol i s cr i ter ion having a form described i n Equa t ion 2.34[66]. Note the use of a temperature property the op t imiza t ion equations. Th i s term is adopted from simulated annealing concepts used in solid state physics and describes the 'energy' of the system. The idea is that as the temperature decreases ( 'cools'), the magnitude of allowable changes also decreases. T h e chance that a worse objective (cost) function value is accepted by the opt imizat ion also decreases. ACF P — g T(nsucc) (2.34) Chapter 2. Introduction to Radiation Therapy Physics 59 where • ACF = the deterioration i n the cost (objective) function value (increase) . • T(nsucc) = current "temperature" of the opt imiza t ion T h e reduct ion in temperature over t ime is called the cooling schedule and can be described by any number of functions-(e.g. linear, quadratic, etc.). A common function is some form of an inverse power (See Equa t ion 2.35 and Figure 2.31). T0 T{nsucc) = —-j- (2.35) (1 + nsucc)Rr where • T(nsucc) = temperature after nsucc successful iterations • T0 = the in i t i a l temperature of the system • RT = rate of temperature cooling 100 90 80 70 ) l . , , , 1 0 20 40 60 80 100 iteration Figure 2.31: Example of a temperature cooling curve for simulated annealing opt imiza t ion . F r o m Equa t ion 2.34, at h igh temperatures (early on i n the opt imizat ion) , the value of P is close to one thus solutions that increase the objective function are allowed more frequently. A s the op t imiza t ion progresses, solutions that result i n a worse objective (cost) function value are not as readily accepted. Similar ly , solutions that produce a large increase in the objective (cost) Chapter 2. Introduction to Radiation Therapy Physics 60 function are accepted w i t h a lower probabi l i ty than smal l increases. T h e selection of the in i t i a l "temperature" value and the rate of temperature cooling is quite important . T h e temperature should be large enough to allow the opt imiza t ion to escape from local m in ima . However, i f the temperature is too high, the search space w i l l be completely random and the 'opt imizat ion may take a very long t ime to converge on a final solution. Simulated annealing is slower than the gradient method, but i t has an inherent mechanism for escaping local min ima . T h e applicat ion of this technique i n a radiotherapy context was proposed i n the late 1980's[67, 68, 69] G r a d i e n t M e t h o d (determinis t ic op t imiza t ion) T h e gradient method is a deterministic technique that sets the first derivative (gradient) of the objective function, F0Uj(x) , to zero (see Equat ions 2.36 and 2.37) and then attempts to solve for the unknown parameter, x (beamlet weights) [55]. N p o i n t s ^ p o i n t s minimize F0bj(x) = j\2 ( A — dp) 2 = 2\2 ( a i ' x — °W 2 (2.36) i=l i=l N p o i n t s V F o 6 i ( x ) = 2 £ ( a i - x - d p ) - a , = 0 (2.37) i=l - where • a; = a-ij = dose deposit ion from a beamlet j , to dose point, i • x = Xj = weighting factor of beamlet, j • • dp = prescription or target dose Gradient methods are fast, but prone to terminat ing in a local m i n i m a dur ing the opt imiza t ion process (Figure 2.29(a)). T h e best solution w i l l lie i n the global m i n i m u m of the objective function, but this method may never find that locat ion. Fortunately, simple objective functions w i l l avoid this condi t ion and generally the local m i n i m a are not far from the global m i n i m u m solution. Chapter 2. Introduction to Radiation Therapy Physics 61 Other Objective Function Minimization Techniques Other op t imiza t ion methods that have been employed for objective function min imiza t ion in -clude: conjugate gradient method, linear programming[70], genetic algorithms[71], mixed integer programming[72], i terative filtered back projection[73], and the random search method[74]. 2.7.5 Techniques for Optimization of Radiation Delivery : Fluence-Based Optimization vs. Direct Aperture Optimization T w o leading methods have emerged for opt imiz ing I M R T radiat ion delivery. T h e first, and possibly the most common at present, is a fluence-based op t imiza t ion method. The second is called direct aperture op t imiza t ion ( D A O ) . B o t h methods are compared i n this section. Fluence-based Optimization T h i s technique is implemented in many treatment p lanning systems ( T P S ) for I M R T and a pic tor ia l flow-chart is shown i n Figure 2.32. (a) ( b ) (c) (d) (e) Figure 2.32: F low diagram for fluence-based opt imizat ion . T h e goal is to establish a relationship between the locat ion of a fluence map pixe l (or beamlet) and the dose to the patient or phantom (Figure 2.32(a)). T h e weights of the beamlets are then opt imized by defining an objective function and min imiz ing the output of that function using one of the methods described i n Section 2.7.4. A n ideal or optimal fluence map is generated (Figure 2.32(b)) and the corresponding optimal dose-volume histogram ( D V H ) representing the Chapter 2. Introduction to Radiation Therapy Physics 62 dose delivered by this fluence map is shown i n Figure 2.32(c). T h e fluence map must now be translated.into a series of mult i leaf coll imator control files v i a a leaf motion calculator program (Figure 2.32(d)). T h e M L C is characterized by leaf mot ion l imi ts , M L C transmission, interleaf leakage and rounded (vs straight) leaf ends. The leaf mot ion calculator w i l l a t tempt to take into account only some of these properties, namely transmission, leaf mot ion l imi ts , and leaf t ip shape. Often the ideal fluence map is not physically deliverable so a closest match is offered by the leaf mot ion calculator. Th i s modified fluence map is often referred to as the actual or deliverable fluence map. The final dose dis t r ibut ion is calculated from this modified map result ing i n a dose dis t r ibut ion that may deviate from the opt imized doses obtained from the beamlet weighting step (Figure 2.32(e)). T h e advantage of this method is that the leaf mot ion calculator step can generate M L C control files that can be delivered either i n a step-and-shoot fashion ( s M L C ) or dynamica l ly ( d M L C ) . T h e ma in disadvantage is that the final or actual fluence map may contain compromises compared to the optimal fluence map because no M L C dosimetric or geometric properties / l imi ta t ions are included dur ing the opt imizat ion step. A common side-effect of not considering these properties is that the final c l in ical D V H may not agree w i t h the optimized D V H which was specified by the radiat ion oncologist. The P T V coverage may not be as uniform as expected or the organ-at-risk ( O A R ) doses may be higher than that predicted by the op t imiza t ion (see Figure 2.33). Direct Aperture Optimization Direc t aperture opt imiza t ion has been a fairly recent development i n I M R T [ 2 8 , 29]. A l l M L C leaf mot ion constraints are incorporated into the opt imizat ion itself and the M L C apertures are opt imized direct ly (Figure 2.34). M L C transmission is also included i n the opt imiza t ion . T h i s is in contrast to the fiuence-based method which optimizes beamlet weights w i th no consideration for the geometric or dosimetric l imitat ions of the M L C . The D A O approach does not require a separate leaf-sequencing step as the final deliverable M L C control files natura l ly fall out of the opt imiza t ion . In this step-and-shoot approach, only the beam gantry angles and the number of apertures Chapter 2. Introduction to Radiation Therapy Physics 63 Dose[%] Figure 2.33: Compare optimal (dashed line) and actual (solid line) D V H curves for the fluence-based optimization method. Note the decrease i n the P T V mean dose and increase in O A R m a x i m u m dose in the final dis t r ibut ion. ( a ) ( b ) Figure 2.34: F low diagram for direct aperture opt imiza t ion per field need to be specified by the user. The in i t i a l conditions for the D A O opt imiza t ion is such that a l l apertures are set to the beam's-eye-view ( B E V ) projection of the p lanning target volume ( P T V ) (plus a smal l margin (e.g. 0.5 cm)). A t each i terat ion the op t imiza t ion must randomly select (1) which of the n treatment fields to optimize, (2) which of the m apertures w i t h i n that field to optimize, (3) whether the aperture weight or the aperture shape w i l l be adjusted, and (4) the amount by which the weight or shape of the aperture w i l l change. T h e ind iv idua l M L C leaf positions (defining the aperture shape) and the aperture weights are op t imized using simulated annealing[65] to minimize a dose-volume objective function[29]. A t ( C ) (d) Chapter 2. Introduction to Radiation Therapy Physics 64 the start of the opt imizat ion, large changes to the M L C leaf positions and aperture weights are allowed. However, as the opt imiza t ion continues, the allowable adjustments to the M L C . shapes and weights decrease. T h i s is due to the implementat ion of a "cooling schedule" in the op t imiza t ion (see Section 2.7.4 for a description of simulated annealing and the cooling schedule). T h e dose-volume constraints (e.g. P T V coverage, max dose to organ-at-risk) are specified by the user. T h e number of voxels that are non-compliant w i th these dose constraints is min imized dur ing the opt imizat ion . The beamlet dose contr ibut ion to a structure-of-interest is updated by subtract ing or adding the dosimetric effect of the new M L C leaf posi t ion or aperture weight. T h e dosimetric properties included i n the aperture shape opt imiza t ion are the M L C transmission and leaf mot ion l imitat ions. Leaf-t ip and tongue-and-groove effect are not included. After the M L C aperture shapes and weighting are opt imized, each field (wi th a l l its aper-tures) is wr i t t en into a M L C control file that can be immediately delivered by the linear accel-erator. T h e tota l number of monitor units associated w i t h each treatment field is determined by summing the aperture weights associated w i t h the beam. It is important to note that there is no user input into the final shape of the apertures, except to define l imi ts where the opt imizer may explore. These aperture shapes are not "pre-selected". T h e user may specify the number of apertures to use, a m i n i m u m number of radiat ion units (called monitor units ( M U ) ) per aperture (thus ensuring a m i n i m u m beam weight) or a m i n i m u m open field area (defined as a percentage of the "open" P T V B E V aperture). A systematic s tudy of simple and complex p lanning targets by J iang et a/. [75] indicates that for most c l in ica l situations, a good dose dis-t r ibu t ion for a D A O plan can be achieved w i t h only five apertures. Complex cases may require up to nine apertures. In the Shepard et al. [29] paper, three to five apertures per field were tested. Since the user has control over the number of apertures (and thus fluence intensity levels), the D A O treatment plans tend to require fewer monitor units to deliver compared to the fiuence-based method (20 - 50% fewer MU)[76] . T h i s reduction i n M U could have large, positive impl icat ions for room shielding design and secondary cancer risk due to whole body exposure to leakage radiat ion from the linear accelerator[77]. Another benefit of this method is that the final D V H s tend to show better agreement w i t h the optimized D V H . T h i s property Chapter 2. Introduction to Radiation Therapy Physics 65 of D A O is explored in detai l in Chapter 6 of this thesis. A disadvantage of this method is that it tends to be sensitive to tongue-and-groove effect dose artifacts. Th is issue is discussed in the next section (Section 2.7.6). A l though , M L C leaf mot ion constraints and transmission are included i n the opt imiza t ion of the M L C apertures, not a l l characteristics are considered. Inter-leaf leakage, tongue-and-groove, and the effect of the rounded-leaf t ip construct ion are not included i n the opt imiza t ion as implemented by the original authors [29]. 2.7.6 Dosimetric Considerations for I M R T M L C Transmission A l t h o u g h the tungsten M L C leaves are 6 c m thick in a direct ion parallel to the beam axis, they are not perfect attenuators. The M L C w i l l t ransmit anywhere from 1 - 2% of the incident photon beam, which contributes significantly to the overall dose delivered to the patient. Transmission effects must be taken into consideration when converting an optimal fluence map to a deliverable fluence map (see Section 2.7.5). Note that i n a direction perpendicular to the leaf travel (across the leaves), the transmission is not uniform. There tends to be interleaf leakage which is larger than the intraleaf leakage (see Figure 2.35). Generally, an average value is quoted. Figure 2.35: (a) M L C diagram indicat ing locat ion of dose profile (vertical solid l ine), (b) E x -ample of dose profile demonstrat ing interleaf transmission. Chapter 2. Introduction to Radiation Therapy Physics 66 The Tongue-and-Groove Effect A tongue-and-groove design is implemented in the Var i an M i l l e n n i u m 120 leaf muli t leaf col l i -mator ( M L C ) . T h i s design reduces the amount of leakage radiat ion between adjacent leaves (see Figure 2.36). c < << < > > ? r y • Jtongue •I" (a) (b) Figure 2.36: End-on d iagram of M L C showing tongue and groove design. Leaves slide in to /ou t of the page, (a) A b u t t i n g leaves, (b) single protruding leaf. • — ^ --20 « 60 80 100 120 140 160 180 Figure 2.37: Tongue-and-Groove effect (arrows showing yellow underdosage lines) visible on this coronal plane of a seven field I M R T plan. T h e tongue-and-groove effect is a dosimetric artifact visual ized as a line of underdosage oriented paral lel to the leaf motion, and located between adjacent leaves of the M L C (See Figure 2.37). It occurs when ind iv idua l M L C leaves are allowed to protrude into the x-ray Chapter 2. Introduction to Radiation Therapy Physics 67 field for a significant amount of t ime during beam delivery. It is par t icular ly noticeable when adjacent leaves are allowed to protrude i n an alternating fashion. Figure 2.38(a) shows the open field fluence profile for two open M L C leaves. The same open field can be delivered i n two segments (Figure 2.38(b)). However,, the intensity profiles from the two half-apertures do not add up to create a uniform intensity, fluence as shown i n (a). A smal l gap of under dosage equal i n w i d t h to the w i d t h of the tongue (or groove) is now present. T h i s occurs because that region is always covered by either the tongue or the groove and never experiences a completely open fluence intensity. (a) (b) Tongue-and-groove artifact T Jin h h Figure 2.38: E n d - o n diagram of M L C w i t h profiles, (a) T w o adjacent leaves are opened to create a uniform fluence profile, (b) The same open field is delivered i n two segments. The summat ion of the two segments does not equal the open profile i n (a). A tongue-and-groove error results. Some authors have reported lines of underdosage of 10 - 15% for a single field delivery [78, 79]. Deng et al. suggest that the effect should become blurred out for mult iple fields (>5) because the effect may not always appear i n the same plane for every field[79]. In addi t ion, patient set-up uncertainty contr ibut ion should also reduce the effect. The tongue-and-groove problem can be min imized by careful M L C leaf sequencing[80, 81], rotat ing the l inac col l imator such that the M L C leaf travel is 90° relative to the direction of gantry rotat ion, or even ro ta t ing the Chapter 2. Introduction to Radiation Therapy Physics 68 col l imator dur ing beam delivery[51]. L e a f - T i p Shape The V a r i a n 120 leaf mult i leaf coll imator can create apertures of varying field sizes, up to 4 0 x 4 0 c m 2 . Ideally, the ends of the leaves w i l l match the divergence of the x-ray field perfectly toN reduce penumbra effects. Since the angle of these leaf-ends w i l l have to vary depending on the field size, V a r i a n has chosen a compromise by manufacturing their 120 leaf M L C w i t h rounded leaf ends (see Figure 2.21). T h i s w i l l ensure that a constant penumbra is generated for any leaf posi t ion. T h e leaf end radius on this model is 80 m m . Since the ends do transmit some radia t ion and can never be direct ly abut t ing when in opposit ion, dosimetric corrections are required when creating M L C leaf sequences for delivering I M R T plans. Essentially, a smal l gap is introduced into the M L C leaf sequence to ensure that the measured dosimetry matches the planned values. T h i s gap is referred to as the dosimetric leaf gap. T o t a l B o d y Dose D u e to Scat ter and Leakage R a d i a t i o n It is well established that radia t ion exposure can k i l l malignant tumour cells and is the basis for the entire field of radia t ion therapy. It is also well established that radia t ion can induce cancer i n humans. T h e goal of therapy is to focus the radiat ion beams such that a large dose is absorbed by the tumour, but none is absorbed by the surrounding healthy tissue. P rac t i ca l ly this is not realistic as some i r radia t ion of the healthy tissue is required to access the tumour. Regardless of the radia t ion delivery method ( I M R T or simple conformal fields w i t h poured blocks), the risk of inducing a cancer i n healthy tissue located wi th in the treatment area, twenty or so years i n the future, is often considered acceptable when compared to the benefit of k i l l i ng the immediate threat of the active malignancy. •* However, there is a secondary threat to healthy areas i n the body.' T h i s threat arises from whole body exposures to the out-of-field scatter radiat ion and leakage from the linear accelerator treatment head. Th i s whole body dose w i l l be proport ional to the amount of t ime that the radia t ion beam is on. The shift from 3D-conformal radiat ion therapy to I M R T has realized a 3 - 6 x increase i n the number of monitor units required to deliver the same dose to the Chapter 2. Introduction to Radiation Therapy Physics 69 patient. T h i s would imp ly that the risk of inducing a secondary cancer due to leakage and scatter increases w i t h the use' of I M R T . However, the problem is not simple as I M R T has the added benefit of improving the conformality of the dose dis t r ibut ion, w i t h the effect of reducing the amount of healthy tissue exposed to high radiat ion dose. T h i s issue is discussed i n detai l by H a l l and W u u (2003) and the reader is referred to their publication[77]. Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 70 v. Chapter 3 Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 3.1 Dose Calculation Algorithms and Treatment Planning Systems It is essential to know what dose a patient w i l l receive dur ing treatment prior to tu rn ing on the megavoltage x-ray beam. Rad ia t ion treatment planning in a cl in ical setting is only possible if the radia t ion dose to biological materials for patient-specific geometries can be predicted. There are two aspects to this problem: 1. Charac ter iz ing the radiat ion beam behaviour i n a uniform water mater ia l (penetration, lateral dis t r ibut ion) , and 2. A p p l y i n g the beam characterization to patient-specific geometries by correcting for surface contours and even tissue inhomogeneities. Before the integration of 3D computed tomography ( C T ) imaging techniques into the radia-t ion therapy cl inic , treatment planning was essentially a 2D process. The radia t ion oncologists only had orthogonal projection images (e.g. diagnostic x-ray films) and I D mechanical ly as-sisted (or even hand-drawn) body contours to t ry and locate the tumour and nearby sensitive structures. Before computers, hand calculations were used to translate tables of dose deposi-t ion characteristics of the x-ray beam into a dose to the patient. T h e abi l i ty to correct for dose \ Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 71 perturbations caused by tissue inhomogeneities (lung being an exception) was very rudimentary. A radia t ion therapy beam's dose deposition characteristics can be calculated to any point i n the phantom/pat ient using the following two parameters : 1. B e a m Penetrat ion (Paral lel to Beam Axis ) - defined by the Percentage Depth Dose (PDD), the Tissue-Air-Ratio (TAR), or the Tissue-Phantom-Ratio (TPR). These values help ob-ta in a measure of the dose deposition along the central axis of the beam for a given field size and beam energy. 2. B e a m Profile (Perpendicular to Beam Axi s ) - the radiat ion beam intensity is not perfectly uniform across the open por t ion of the radiat ion field. A n Off-Axis-Ratio (OAR) is used to move lateral ly away from the central axis. It is defined for a specific depth and field size. Because bo th of these parameters are affected by beam energy, field size, and distance of the phan tom away from the source, tables and formulas for dose corrections (relative to a set of reference conditions) are required. A brief description of useful beam characteristics and their behaviour is provided i n the next section. O n l y those factors that are relevant to this thesis are discussed. It is clear that many factors w i l l have to be accounted for to ensure an accurate dose calculat ion. It is important to note that a l l of these dose calculat ion parameters are acquired assuming the condi t ion of electronic equi l ibr ium has been met (electrons entering the point of interest equals the electrons leaving). Regions of disequi l ibr ium, for example the bui ld-up region, are difficult to model and generally are a source of large errors. T h i s is a problem that plagues the most sophisticated of treatment planning dose calculat ion algorithms. 3.1.1 Characterizing Radiation Dose Deposition in Matter Percentage Depth Dose T h e Percentage D e p t h Dose ( P D D ) is a measure of the penetration properties of a radia t ion beam. Referring to Figure 3.1, the P D D is defined by Equa t ion 3.1. Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 72 PDD(d,SSDJsdmax,E) D(d, SSD, fsdrnax, E) D(dmax, SSD, fSdrnax, E) x 100% (3.1) where • d = depth of interest • dmax = depth of m a x i m u m dose deposition • SSD = source-to-surface distance • fsdrnax = held size at depth of m a x i m u m dose • E = energy of radiat ion beam • D{d, SSD, fsdmax, E) = dose to the point of interest • D(dmax, SSD, fsdrnax, E) = dose to reference conditions (at depth of m a x i m u m dose) A 2 D depth dose plane and the corresponding P D D profile for a 6 M V , 10 x 10 c m 2 photon beam are shown i n Figure 3.2. Note i n Figure 3.2(b) the following features: the build-up region (before ver t ical dashed line), the depth of m a x i m u m dose deposition, dmax, and the attenuation Figure 3.1: D i a g r a m depict ing measurement conditions for determining a percentage depth dose. Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 73 region (after d m a x ) . In the bui ld-up region electronic equi l ib r ium has not been established, thus there is more energy leaving the volume of interest than entering. T h e bui ld-up region continues un t i l it reaches a depth that equals the range of the secondary electrons l iberated by interactions w i t h the incident pr imary photons. For a 6 0 C o beam, this depth is 0.5 cm. The depth where electronic equi l ibr ium is established corresponds to dmax, the depth of m a x i m u m dose deposit ion. A table of dmax values for different photon beams is shown in Table 3.1. Beyond dmax, the dose begins to fall due to attenuation of the photon beam as it travels deeper into matter. X(cm) cm (a) (b) Figure 3.2: (a) 2D contour plot of film measured P D D for open 10x10 c m 2 6 M V photon field, (b) Profile along depth axis of same data. Table 3.1: D e p t h of m a x i m u m dose deposition for different photon treatment beams. Pho ton Energy depth of m a x i m u m dose (dmax) V61Cs (0.662 M e V ) 0.1 cm 6QCo (1.25 M e V ) 0.5 cm 6 M V 1.5 c m 10 M V 2.5 c m 18 M V 3.3 c m 25 M V 5.0 cm Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 74 Tissue-Air-Ratio and Peak Scatter Factor T h e T i s sue -Ai r -Ra t io ( T A R ) is useful for calculat ing dose at different depths i n phantom or for different field sizes, especially when the S S D value is non-standard (not 80 or 100 cm). A special case is defined where the depth in phantom is equal to the depth of m a x i m u m dose deposition, dmax (see P D D above). For this si tuation, the term peak-scatter factor (PSF) is used. T h e geometry for measuring the T A R is shown in Figure 3.3. T h e ratio is defined in Equa t ion 3.2. TAR(d,fs,E) = (3.2) Dair(fs,E) where SAD = source-to-axis distance which is kept constant (100 c m for most linear acceler-ators, 80 c m for some 6 0 C o units). Note that there is no source-to-surface distance dependance for this rat io. Johns et a/.(1958) showed that this is true for geometries where the S S D is > 50 cm[82]. Generally, T A R decreases w i t h depth and to a lesser extent, field size. ) T h e denominator i n Equa t ion 3.2 is the dose delivered to a smal l mass of tissue i n air. T h i s value is difficult to measure for photon beams having an energy > 6 0 C o as electronic equ i l ib r ium cannot be established. T h e T A R and P S F parameters are thus replaced by the Tissue-Phantom-1 SAD SAD fs (a) (b) Figure 3.3: D iag ram depict ing measurement geometry for T i s sue -Ai r -Ra t io measurements. Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 75 R a t i o ( T P R ) and the T i s sue -Max imum-Ra t io ( T M R ) respectively for these higher energies (see next section). Tissue-Phantom-Ratio and Tissue-Maximum-Ratio T h e Tissue-Phantom-Rat io ( T P R ) compares the dose at depth i n phantom to the dose at a reference depth. The measurement point is always kept at the isocentre of the treatment uni t (source-to-axis distance = 100 cm). Effectively, the phantom is shifted to accommodate measurement of this ratio. T h e geometry for measurement is shown i n F igure 3.4. and the corresponding ratio presented in Equa t ion 3.3. TPR(d,fs,E) = D(SAD, d, fs, E) (3-3) D(SAD,dref,fs,E) T h e special case, where dref = dmax is called the Tissue-Maximum-Ratio (TMR). T h e T P R and T M R are commonly used dosimetry data. T h e T P R decreases w i t h depth and field size. I * 1 i + — • — * * f S D S A D water (a) f s D S A D max water (b) Figure 3.4: D iag ram depict ing measurement condit ions,to obta in a T M R ratio. Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 76 Discussion A l l of the above factors assume that the density of mater ial i n which the dose is being cal-culated is uniform. Electronic equi l ibr ium is also assumed. In reality, a patient is composed of muscle, fat, bones, lung, and air. Post-calculat ion correction factors can be appl ied to t ry and account for these heterogeneities. Tissue inhomogeneity corrections are discussed i n Sec-t ion 3.1.6. In addi t ion, most physical beam characterization measurements are performed i n a tissue-equivalent (often water or "solid water" plastic) square phantom. A real patient has irregular surface contours. Ca lcu la t ing 2D or 3D doses by hand would pose a real challenge due to the changing depth of tissue above the isocentre plane. Correc t ion factors are available, but again, are cumbersome to implement. M u c h research has been dedicated to the topic of producing fast, accurate, 3D dose calculat ion algorithms. A couple of common algori thms i m -plemented i n commercial treatment planning computer systems are described i n the sections below. In general, the dose calculat ion algorithms discussed i n this thesis can be d iv ided into three categories: 1. I D point dose calculations (previously discussed i n Section 3.1.1) 2. Kernel-based convolut ion algorithms (e.g. pencil-beam kernel model , point-kernel super-posi t ion/convolut ion) 3. Direc t s imulat ion of photon/electron interaction and transport from fundamental physics (e.g. Mon te Car lo) A t t en t i on is drawn to the. fact that the first two algorithms are models of the dose deposit ion i n matter. T h e only true dose calculat ion method based on the fundamental physics of photon and electron interactions i n matter is Monte Car lo s imulat ion. T h i s method is a major focus of this thesis and is described in detai l in Chapter 5. Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 77 3.1.2 Single Pencil Beam Convolution Algorithm Employing 2D Dose Kernels T h e single pencil beam kernel ( P B K ) convolution algori thm falls into the kernel-based con-volution category of dose calculat ion algorithms. It is the calculat ion algori thm used by the commercia l treatment p lanning system at the Vancouver Cancer Centre ( E c l i p s e ® v . 6 . 5 ) so more deta i l w i l l be provided compared to some of the other dose calculat ion a lgor i thm descriptions. T h i s a lgor i thm starts by subdiv id ing the photon radiat ion field into a finite number of of smal l "beamlets", for example 0.25 x 0.50 m m 2 (see Figure 3.5). The 3D project ion of the smal l radia t ion beam emanating from each beamlet locat ion is called a "pencil beam". A 2D penci l beam dose kernel attempts to model the spatial d is t r ibut ion of energy deposit ion (dose) about one of these penci l beams of photons at depth i n water[83, 84] (see Figure 3.6). Since the shape of this kernel depends on the photon energy spectrum and the produc t ion /d i s t r ibu t ion of secondary electrons i n the water, it is depth-specific. F igure 3.5: (a) A modula ted photon fluence treatment field, (b) Segmentation into 0.25 x 0.5 c m 2 beamlets. T h e dose d is t r ibut ion is defined in a plane perpendicular to the beam axis. T h e tota l dose to a point of interest i n phantom is the superposit ion of the dose contributions from al l the penci l beams i n the field. If spat ial invariance of the pencil beam dose kernel is assumed [85, 83], the dose d is t r ibut ion across an irregularly shaped field at a given depth can be calculated by convolving a modified photon fluence field, F'(x',y'), w i t h the depth dependant pencil beam kernel, K(x, y, d) . T h e modified photon field fluence is obtained by mul t ip ly ing the idealized Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 78 photon fluence F(x', y') by a depth dependant pr imary beam off-axis intensity profile (accounts for presence of flattening filter), Pini(x', y', d) (Equat ion 3.4 and Figure 3.7)[12, 17]. (SSD + d A2 r00 r°° D(x,y,d;SSD) = { J+ ^ J ^F{x')y')Pmt{x\y\d)K{x-x'\y -y',d)dx''dy' (3.4) Where , • SSD = Source-to-Surface-Distance (SSD) • d = depth of dose calculat ion • dref = reference depth defined by user radial distance from centre (mm) Figure 3.6: I D dose deposit ion kernel derived from ion chamber measured commissioning data. D a t a exported from E c l i p s e ® v 6 . 5 T P S (depth = 5.0 cm). raw fluence Intensity map pencil beam 2 D dose plane (at depth) dose kernel (at depth) (at depth) Figure 3.7: The photon fluence is corrected for off-axis intensity var ia t ion and then convolved w i t h a 2D depth-specific penci l beam dose deposit ion kernel. T h e final result is a dose plane. Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 79 T h e penci l beam dose kernels are defined at several discreet depths (e.g. 1.5, 5.0, 10.0, 20.0, and 35.0 cm). Ca lcu la t ion points located at intermediate depths use interpolated values[86]. Generating the Pencil Beam Dose Kernel Penc i l beam kernels for radiat ion dose calculations were original ly constructed from Monte Car lo simulation[84]. However, the kernel properties could only be simulated if detailed knowl-edge about the beam spectrum at depth in phantom was known. Ahnesjo et al. [87] proposed a semianalyt ic method for generating pencil beam dose kernels based on a weighted sum of precalculated Monte Ca r lo monoenergetic kernels. The weighting was derived from beam spec-t r u m information obtained from in-phantom measurements. Some authors quote an analyt ic form represented by a double exponential (Equat ion 3.5) [88] or even a-double/ t r iple Gaussian [89]. Ae~ar + Be-* Kn- -2 (3-5) C h u i et al.[11] and Storchi et al.[12] demonstrated that the pencil beam dose kernel can be derived from measured commissioning data (e.g. jaw-defined P D D ' s and profiles). The energy spectrum at depth is affected by the amount of pr imary beam attenuation and the amount of phantom scatter. The penumbra region of the jaw-defined open beam field profiles is used to determine the shape of the dose spread kernel. The measured da ta in this h igh gradient region w i l l be affected by the sampling rate and the sensitive volume of the dosimeter used [90, 91]. Larger volume dosimeters w i l l cause spatial averaging (smoothing) of the measured da ta producing broader (larger full w i d t h half maximum) dose kernels compared to those derived from higher resolution commissioning measurements (e.g. film) [13]. Advantages / Disadvantages T h e single penci l beam dose calculat ion algori thm has been implemented in many commercia l treatment p lanning systems. For most c l in ical situations, the a lgor i thm is fast and accurate and simple to commission. However, there are several factors that can affect the dose calculat ion Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 80 accuracy of this method. (1) Q u a l i t y of Input D a t a for K e r n e l Gene ra t ion - Since the dose kernel is generated from penumbra measurements in water, the quali ty of the -measured data is important . T h e high-gradient region from which the dose kernel parameters are extracted must be measured w i t h a high-resolution detector (i.e. film or diode or small-volume ion chamber). G o o d qual i ty input da ta does not guarantee that the dose kernel w i l l model the dose deposit ion perfectly. T h i s issue is addressed i n Chapter 4 of the thesis. (2) Spa t i a l Invariance of Dose K e r n e l - The concept of a spat ial ly invariant dose de-posi t ion kernel is a key property of any convolution-based calculat ion a lgor i thm. However, the dose kernel is not t ru ly spatial ly invariant due to off-axis energy spectrum differences (beam energy softening) and kernel t i l t ing (beam divergence) effects. T h e issue of spat ia l ly varying dose deposit ion kernels has been studied[15, 16] and even accounted for[92] by other authors. (3) M o d e l i n g Tissue Inhomogeneit ies - The most serious disadvantage of the single penci l beam dose calculat ion algori thm is its inabi l i ty to model changes i n lateral electron scatter i n materials other than water. M a n y algorithms start by calculat ing the dose to water and then apply a post-calculation, I D path-length correction for regions of inhomogeneity. A summary of various tissue inhomogeneity methods is provided i n Section 3.1.6. 3.1.3 Superposition / Convolution Method Employing 3 D Point Kernels T h i s is another kernel-based convolution a lgori thm. The convolut ion now takes place between 3D point kernels (instead of the 2D penci l beam kernels described above i n Section 3.1.2) and a T E R M A (total energy released i n material) factor. It is also refereed to as a superposi-tion/convolution a lgor i thm (see Equa t ion 3.6[93]). (3.6) where • T ( r ' ) = T E R M A at photon interaction site r' Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 81 • p ( r ' ) = density at r ' • H(r — r', pave) = energy' deposit ion kernel value for a displacement, r — r' i n a med ium of density pave • Pave = average density between r' and r T h i s method models the pr imary dose deposition and the first (and subsequent mult iple) scatter events. There are three basic steps (see Figure 3.8): • 1) Calcula te the to ta l energy released ( T E R M A ) to the med ium by the pr imary photons (see Equa t ion 3.7). The term T E R M A encompasses the kinet ic energy gained by the sec-ondary electrons plus the energy of the scattered (e.g. compton or photoelectric) p r imary photon. • 2) Transpor t the energy deposit ion about the pr imary photon interaction site (described by a 3D point dose deposit ion kernel) • 3) Determine dose to medium by superimposing the kernel at a l l p r imary interact ion sites. (a) (b) Figure 3.8: (a) P h o t o n incident on phantom interacts at a point. T h e tota l energy released (kinetic energy transferred to electron plus scatter photon energy) is called T E R M A . (b) Point dose kernel describes dose dis t r ibut ion about the interaction point < Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 82 In i t 's most basic form the calculat ion of T E R M A can be represented by Equa t ion 3.7[32], T E R M A = $ x ^ (3.7) P where • ^ ' = mass attenuation coefficient for the medium of interest and at the photon energy of interest • — photon energy fluence i n the calculat ion voxel of interest T h e point dose kernel can be separated into two components: 1) the pr imary dose kernel, and 2) the first (and multiple) scatter dose kernel. Monte Car lo methods have been used to calculate the pr imary and the secondary (and higher order) scatter kernels. T h e y can also be derived analyt ica l ly using K l e i n - N i s h i n a cross-sections for C o m p t o n interactions. T h i s method was clearly of some interest to the medical physics communi ty s tar t ing i n 1984 as several papers and conference presentations (Ahnesjo, Boyer and M o k , C h u i and M o h a n , and Mack ie and Scrimger) exploring this method appeared a l l w i th in the same year[94, 95, 96, 97, 98]. A major advantage of this method is that dose inhomogeneity effects can be incorporated direct ly into the dose kernel on a voxel by voxel basis (see p(r') factor in E q u a t i o n 3.6). T h e dose deposit ion dis t r ibut ion surrounding an interaction point depends on the type of med ium in which the interaction occurred. The dose kernel for uniform water can be scaled l inearly by the density of the medium at the voxel of interest (interaction site) prior to performing the superposit ion. T h i s a lgor i thm is superior to the post-calculation correction-based methods. If 3D spat ia l invariance of the dose kernel is assumed, fast Fourier transform ( F F T ) methods can be appl ied to the convolution[99]. However, this method inhibi ts the voxel-by-voxel scaling of the dose kernel w i t h density. Even i n a uniform material , the dose kernels are not spat ia l ly invariant due to beam spectrum variations and the t i l t ing of the dose kernel due to beam divergence. T h e assumption that the dose kernel is spatial ly invariant would l imi t the accuracy of this method. Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 83 Advantages / Disadvantages T h e superposit ion convolution method is an accurate dose calculat ion a lgor i thm, and even performs well in inhomogeneous media. However, its main l imi ta t ion is calculat ion speed. 3.1.4 Collapsed Cone Convolution Algorithm T h e collapsed cone convolution algori thm is a modificat ion of the superposi t ion/convolut ion point kernel-based a lgor i thm described i n Section 3.1.3. A point dose kernel describes the 3D dis t r ibu t ion of pr imary and scatter dose about a point of interaction. Imagine a discreet number of lines projecting from the in i t i a l interaction point and each line is the axis of a narrow cone (Figure 3.9). W h e n a l l of the cones of dose are added together the complete 3D dose kernel I Pwater | Figure 3.9: Col lapsed cone convolution calculates the 3D deposited energy w i t h i n a series of cones emanating from the photon interaction point. The doses w i t h i n the cone are collapsed onto the central ray of each cone projection. energy is recovered. Now, if the total energy contained wi th in each cone is collapsed onto its central axis, you have a series of rays emanating from the pr imary interaction point . A g a i n , the to ta l energy contained wi th in a l l of the rays w i l l add up to the energy contained w i t h i n the or iginal 3D dose deposit ion kernel. T h i s technique was described by Ahnesjo i n 1989 and has been implemented commercial ly w i t h much success [100]. The form of the kernel is s imilar to E q u a t i o n 3.5, but an addi t ional dependance on polar angle is added (see Equa t ion 3.8). Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 84 k(r,9) = (3 .8) Advantages / Disadvantages T h e major advantage to this method is that the inhomogeneity corrections can be implemented direct ly into the a lgor i thm (much like the standard superposi t ion/convolut ion method) but the calculat ion speed is greatly increased. 3 . 1 . 5 Monte Carlo Simulations Monte Ca r lo is a stochastic stat ist ical sampling technique used to solve mathemat ica l problems. It w i l l r andomly sample probabi l i ty distr ibutions that represent the fundamental interactions of photons and electrons in matter. Indiv idual incident photons (or electrons) are followed through the med ium and the dose deposit ion patterns recorded. The type of interaction that occurs is selected at random (although weighted by a probabi l i ty dis t r ibut ion) . For example, an x-ray photon at an energy typica l ly used for radiat ion therapy treatments (e.g. 6 M V ) has a certain probabi l i ty of either undergoing: • i) no interaction, • ii) photoelectric effect event. • i i i) compton scattering event, or • iv) pair -product ion event A n y progeny particles that may arise from the interactions between the photons (or electrons) w i t h atoms (e.g. compton electrons, bremsstrahlung etc.) are followed and scored. T h e analysis of one photon (or electron) journey (called a history) in itself does not contain much information about the macroscopic properties of the environment under study, just like how one photon of visible l ight w i l l not reveal the composi t ion of a photograph. However, when mil l ions of photons and their subsequent interactions are allowed to accumulate, they do start to reveal the b ig picture. In a medical physics context, this big picture may include informat ion such as Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 85 types and frequencies of certain interactions, geographic and spectral d is t r ibut ion of particles as they pass through different media, sources of scattered radiat ion, and ul t imate ly dose deposited w i t h i n a patient or phantom. Generally, Monte Car lo simulations i n radiat ion therapy medical physics fall under three major categories: 1. S imula t ion of radiat ion beam (electron or photon beam) from a medical linear accelerator (energy spectra, geographic d is t r ibut ion of particles and sources of scatter) 2. S imula t ion of dose in target media such as human tissue or tissue-equivalent phantoms (dose, scattering properties, spectral changes) 3. S imula t ion of different dosimeters (dose deposition properties, detector response and test-ing of new design concepts) After much development and rigorous verification, Monte Car lo is considered to be the gold standard for dose calculat ion accuracy. Tissue inhomogeneities are inherently modeled in this system and no correction factors are required. A s this method contributes significantly to this thesis, an entire chapter is dedicated to the topic (see Chapter 5). Advantages / Disadvantages Monte Car lo simulations are currently the gold standard for dose calculat ion accuracy especially i n difficult treatment / anatomical geometries (e.g. wi th in /near tissue inhomogeneities and regions of lateral electronic disequil ibrium). Since this is a random sampling technique, a large number of interactions (histories) must be simulated to achieve stat is t ical significance (i.e. mil l ions to bil l ions of ind iv idua l particle interactions). Because of this requirement, the largest disadvantage of Monte Car lo to date is the slower calculat ion speed (hours). In general, the longer the s imulat ion t ime, the lower the statist ical uncertainty i n the data. Clever s tat is t ical variance techniques can reduce the number of simulations required to acquire low-noise data[101, 102]. Cluster (also called parallel) comput ing can reduce the overall calculat ion t ime. F ina l ly , d ig i ta l denoising filters can be applied i n an attempt to smooth stat is t ical ly noisy data[103, 104]. T h i s last opt ion is discussed i n detail in Chapter 7. Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 86 3.1.6 Tissue Inhomogeneity Corrections M o s t dose calculat ion algorithms start w i t h the assumption that an irradiated patient or phan-t o m is comprised of a uniform volume of water. Water is often used to represent a patient as it is a simple mater ial that has an atomic number and radiological properties nearly ident ical to human muscle tissue (Zwater ~ 7.51 , Zmuscie ~ 7.64). A real patient i s of course much more complex than this simple assumption. A human has bones, air cavities and lungs embedded wi th in . These tissue inhomogeneities w i l l affect the dose deposited i n the patient thus a uni -form water representation for dose calculations is no longer appropriate. M a n y dose calculat ion algori thms w i l l calculate dose to water, then attempt to apply a post-calculat ion correction to regions affected by the presence of tissue inhomogeneities. Post-calculat ion corrections are difficult to apply rigorously as the dose to a point of interest is a combinat ion of three dimen-sional contributions originat ing from the forward pr imary beam as well as forward, side and back scattered photons (see Figure 3.10). A comparison summary of several of the methods described below is available from W o n g et al[105] and duPlessis et al. [106]. (a) (b) Figure 3.10: Pho ton beam incident on (a) uniform water phantom, (b) water phan tom w i t h embedded air inhomogeneity, thickness ( d a ) . Note increase i n lateral scatter of photon i n air w i l l reduce scatter dose to point P . Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 87 R a t i o of T i s s u e - A i r - R a t i o ( R T A R ) C o r r e c t i o n M o s t dosimetric ratios and characterizations are quoted assuming the mater ial being irradiated is uniform water. However, when tissue inhomogeneities are present, an effective or equivalent water depth needs to be quoted. T h i s is s imply.obta ined by scaling the depth of the inhomo-geneity by the physical density. T h e simplest correction method would be to take a ratio of tissue-air-ratios ( T A R ) (see Section 3.1.1 for definition of T A R ) . Equa t ion 3.9 and Figure 3.11 describes this ratio [42]. - TAR(deff,fsd) °* " TAR(dw,fsd) ^ where • dw = depth i n uniform water • deff = effective depth i n water = d'w + (dapa) • fsd = field size of photon beam at point P at depth d T h e problem wi th this approach is that there is no information about the locat ion of the air cavi ty relative to the point of interest (e.g. is the point of interest close to the ai r /water interface?) and the lateral extent of the air cavity is not considered. Power L a w (modif ied Ba tho ) C o r r e c t i o n Batho (1964) [107], w i t h corrections by Sontag (1977) [108], at tempted to include the locat ion of the point of interest below the inhomogeneity i n the correction factor. L ike the R T A R method, the B a t h o correction uses a ratio of tissue-air-ratios, but the T A R values are now raised to an exponent. T h e exponent is based on density differences between /the two types of at tenuating media, for example, water and air (see Equa t ion 3.10 and Figure 3.11). TAR(zufsdy^ C F ~ TAR(z2,fsdy^  ( 3 ' 1 0 ) A l t h o u g h the posi t ion of the dose calculat ion point relative to the tissue inhomogeneity along the beam axis is considered, the lateral extent of. the inhomogeneity is not. Essentially, Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 88 the inhomogeneity is considered to be an infinite slab. Points located adjacent to an finite w i d t h inhomogeneity w i l l not be corrected for changes i n lateral phantom scatter. T h e (modified) Ba tho correction is currently implemented i n the commercial treatment p lanning system used at the Vancouver Cancer Centre ( E c l i p s e ® v . 6 . 5 , V a r i a n M e d i a l Systems, Pa lo A l t o , C A ) and is wor th some extra attention. A l t h o u g h for many c l in ica l situations this correction works satisfactorily, there are sometimes environments where lateral electronic dise-qu i l i b r ium becomes a factor. T h e Ba tho correction s imply does not acknowledge this condi t ion. T h e average range of a 6 M e V electron (R50) in water is approximately 3 cm. T h e Ba tho cor-rect ion performs well for larger (> 6 x 6 c m 2 ) photon fields where electronic equi l ib r ium exists at the centre of, the field. In air, the electron range is larger than 3 cm and for higher energy photon (or electron beams) the range is further increased.. W h e n the range of the secondary electron meets or exceeds the photon field size, lateral electronic disequi l ibr ium is said to exist. Dose calculat ion errors up to 14.5% for a. 5 x 5 c m 2 field and 32% for a 2 x 2 c m 2 field size i n lung have been reported[109]. P i V' [P~2 T Figure 3.11: Pho ton beam incident on phantom containing an inhomogeneity. T h e Ba tho Power L a w correction factor considers the locat ion of the point of interest, P relative to the inhomogeneity. Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 89 Equiva len t T i s s u e - A i r - R a t i o ( E q T A R ) C o r r e c t i o n T h e equivalent tissue-air ratio method ( E q T A R ) was proposed by Sontag and C u n n i n g h a m i n 1978 to t ry and incorporate the finite shape of the inhomogeneity into the dose correction factor[110]. The correction factor takes into account the change i n pr imary photon fluence by u t i l i z ing the effective depth (deff), much like in the R T A R method. However, an addi t ional step attempts to account for changes i n the scattered photon fluence by modifying the equivalent radius field size component (reff) used to look up the T A R values. T h e effective radius, reff, is determined by the surrounding tissue density. It is obtained by mul t ip ly ing the original field size radius at depth, rd, by the weighted average of surrounding voxel densities (see E q u a t i o n 3.11). . ' reff = rx >-V^. "•-,>./• A,v.A M i l ) £i,j,k wi,j,k T h i s method was applied specifically to C T density data which was a technology previously not available to the physicist. The E q T A R correction factor is shown i n Equa t ion 3.12. ni? _ TAR(deff,reff) ' /a- I O N C F - TAR(d,rd) ( 3 - 1 2 ) T h i s method has been shown to be quite accurate (albeit slow) for situations where lateral electronic equi l ib r ium exists. The E q T A R method w i l l underestimate the correction factor required for smal l (< 5 x 5 c m 2 ) fields incident on low density inhomogeneities. S u p e r p o s i t i o n / C o n v o l u t i o n and Fast Four ie r Transform ( F F T ) M e t h o d s T h i s 3D dose calculat ion a lgor i thm is described i n Section 3.1.3. T h e advantage of this method is that the effect of the tissue inhomogeneity correction is incorporated direct ly into the algor i thm instead of being applied as a post-calculation after-thought. T h i s dose kernel can be separated into two components: 1) the pr imary dose kernel, and 2) the first (and mult iple) scatter dose kernel. T h i s a lgor i thm deals w i t h tissue inhomogeneities by scaling the point dose kernel for uniform water linearly. T h e scaling factor is s imply the relative density of the med ium at the voxel of interest (interaction site). Figure 3.12 shows the effect of a low density mater ia l on the Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 9 0 Pwater I (a) (b) Figure 3.12: Schematic of a point dose kernel in (a) water, and (b) air. Note the increased spread of the dose deposition in the low density material. scale of the point dose kernel relative to water. This method offers superior dose calculation accuracy in tissue inhomogeneities compared to the Batho or E q T A R methods. Other Methods Other corrections are available, namely: effective SSD [111], isodose shift[112] differential scat-ter air ratio (DSAR)[113], and the delta volume method[114]. The reader is referred to the corresponding papers for more details. 3.2 Measurement and Verification of Radiation Dose 3.2.1 Introduction A n important aspect of treatment planning and delivery is the verification that the planned or calculated doses are indeed that which are actually delivered to the patient. A good verification process will ensure that the treatment planning tools are correctly predicting the patient dose and that the delivery system is operating within specifications. Ideally, some kind of measure-ment tool would measure 3D distribution of absolute absorbed dose within the patient in real time during delivery. Unfortunately, no such system currently exists and to invasively insert Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 91 a dosimeter inside a real patient is generally infeasible. Some smal l detectors have been used to measure intra-cavity doses, but they are isolated point doses and not a 3D representation. Some examples of point dose dosimeters are shown i n Figure 3.13. Figure 3.13: Examples of different point-dose dosimeters: paral lel plate ion chamber, cy l indr ica l ion chamber and diode Since the dosimetry w i t h i n a real patient cannot currently be direct ly measured, the next best opt ion is to measure doses to an inanimate patient surrogate, such as a solid block of tissue-equivalent plastic or a tank of water. Th i s surrogate is called a phantom and i t is designed to accept different dosimetry systems (e.g. ion chambers, diodes, film or T L D s ) . The plastic or water phantom has radiographic properties (photon absorption and scattering) that are equivalent to tissue (muscle). Conveniently, human tissue is well represented by pure water. Often a phantom w i l l be referred to as being "water equivalent". Examples of tissue phantoms are shown i n Figure 3.14. A radia t ion dosimeter should be able to measure dose (absorbed energy per unit mass) i n media. There are only a few methods that can obtain this information direct ly (calorimetry, chemical dosimetry, ion chamber), so indirect methods are often employed. Quanti t ies such as the amount of ionizat ion i n air or chemical composi t ion changes can be translated into dose either by apply ing an absolute conversion (based on fundamental physics effects) or by construct ing a relative dose response cal ibrat ion curve. Dosimeters that can relate a measurement direct ly to dose v i a fundamental physics calculations is called an absolute dosimeter (e.g. ion chamber). Dosimeters that require a cal ibrat ion curve (known dose to a point is assigned a uni t of measure on the dosimeter) are called relative dosimeters (e.g. film). Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 92 Figure 3.14: Examples of different tissue phantoms, (a) Rounded square rexalite AVID phan-tom ( M D X , Vancouver, Canada) , (b) anthropomorphic solid water phantom, (c) square slabs of solid water plastic A n ideal measurement device for patient verification would have the following properties: • provide accurate absolute dosimetry • be reproducible • provide 3D data • provide in-vivo da ta • have high spatial resolution • be insensitive to beam energy • be insensitive to the type of radiat ion (e.g. photon vs electron) • be independent in response to dose or dose rate • have a linear response to absorbed dose • have a large measurement range (span low doses to high doses wi thout saturation) • have a direct ional ly independent response Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 93 • be resistant to radiat ion damage • be easy to use and read-out • be reusable v • be inexpensive T o date, there is no single dosimeter that can meet this t a l l l ist of desirable properties. A review of several different available dosimeters is provided below. 3.2.2 Absolute Dosimeters In this section an emphasis w i l l be made on ion chambers as an absolute dosimeter as it is the p r imary device used i n this thesis. Other dosimeters are mentioned, but details are not supplied. Ion iza t ion C h a m b e r M o s t medical physics departments have some form of ion chamber available for photon dosimetry measurements. It is considered to be the workhorse of point dosimetry systems. T h e ion chamber is also an absolute dosimeter. T h i s is because the quanti ty measured wi th in the chamber can be related to dose v i a well understood physical principles. The ion chamber is essentially an air cavi ty bounded by two electrodes having a potential difference (See Figure 3.15 for a schematic of a cy l indr ica l ion chamber). outer electrode (0 V) Incident photon (graphite or conducting plastic) creates ion / e" pair \ inner electrode (+/- volts) (aluminum or steel) air cavity Figure 3.15: D iag ram of typ ica l farmer-type cyl indr ica l ion chamber. Incident photon ionizes air (33.85 e V / i o n pair) and the ion / e~ charge is collected at inner and outer electrodes. Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 94 Opposi te ly charged particles (electrons and positive ions) are produced when electrons cross-ing the cavity transfer 'enough energy to the gas (air) to cause an ionizat ion. These positive and negative charged ionizat ion products separate and follow the electric field lines to opposite electrodes. One of the electrodes, the collecting electrode, is monitored by an electrometer and provides a readout of the charge collected. Generally, the dose to a point in water or phantom mater ial is desired. T h e physics describing this process is called the Bragg-Gray cavity theory[42]. The determinat ion of dose to phan tom from an air cavi ty measurement can be divided into three major steps: 1. calculate dose to gas (air) based on the amount of ionizat ion products collected 2. determine dose to the chamber wal l (bounding the air cavity) 3. convert this wal l dose to a point dose in phantom T h e intermediary mater ial separating the phantom from the chamber air cavi ty is the cham-ber wal l . T h e construct ion of an ion chamber is such that the air cavi ty is bound by some k i n d of wa l l mater ia l (often plastic or graphite). T h e chamber wal l provides a boundary or volume w i t h i n which the ionizat ion products are collected, and it provides a s t ructural substrate to at tach the electrodes. If the wal l is th in and made of a mater ia l having radiologic properties s imilar to that of the medium, one can assume that the photon spectrum in the phantom and i n the chamber wal l are the same. If the electron spectrum is not perturbed by the presence of the air cavi ty or the wal l , the dose to medium, Dmea can be obtained from the charge collected from an ioniza t ion chamber by the following equations (Equat ion 3.13 and 3.14) (3.13) (3.14) where Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 95 • W = average absorbed energy (electron volt: eV) required to produce one ion pair i n air (33.85 eV) • Q = amount of charge of one sign (coulombs) produced dur ing i r radia t ion (C) • m = mass of air (gas) w i t h i n collecting volume (kg) • §™as = r a t i o of S ^ 1 and mass stopping powers for the wal l and gas materials respectively, averaged over a l l p r imary photon energies and subsequent secondary electron energies (stopping power units = M e V ^ c m 2 ) ( \ rned fiabsorbad \ _ r a t i 0 0 f mass attenuation coefficients for the medium and the ion chamber /' / wall wal l (mass attenuation coefficient units = ^ ) A common design is the cyl indr ica l ion chamber (see Figure 3.13). Other designs include a paral lel plate geometry (Figure 3.13) or even spherical geometry. The energy response of most ion chambers is relatively constant. Table 3.2: Advantages/Disadvantages of ion chambers as a radiat ion therapy dosimeter. ' Advantages Disadvantages energy independent OD 'point ' dosimeter (or pseudo-ID if scanned linearly) inexpensive easily integrated into phantom integrating reusable absolute dosimeter Other Other absolute dosimeters include calorimetry and chemical dosimetry (Fricke solutions). T h e y have characteristic advantages and disadvantages but are generally not used c l in ica l ly on a routine basis. 3.2.3 Relative Dosimeters Radiographic f i lm is the pr imary modal i ty for measuring 2D dose distr ibutions i n this thesis. Other dosimeters are also available, but they are not discussed i n detail . Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 96 R a d i o g r a p h i c F i l m Radiographic film is by far the most common 2D close verification method. It is also one of the oldest radia t ion detectors, employed by W . Rontgen to further his experiments w i t h the newly discovered x-ray. F i l m is often used as a 2D dosimeter i n a c l in ica l setting. It is useful for verifying single x-ray treatmentMields or can be used as an integrating dosimeter to measure the cumulat ive dose dis t r ibut ion delivered by a complex multi-f ield treatment plan. If desired, many 2D film planes can be sandwiched between phantom mater ial to generate a quasi-3D dose d is t r ibut ion . X - r a y film is comprised of a th in , flexible p las t ic /polymer substrate (e.g. cellulose triacetate or polyester) onto bo th sides of which a photosensitive emulsion is applied. T h e emulsion is comprised of a heterogeneous m i x of silver halide crystals (e.g. AgBr, AgCl, Agl) and gelatin. T h e gelatin is quite permeable to the wet processing chemistry that develops the image, making it a good choice to b ind the photosensitive emulsion. Commercial ly , the silver halide crystals come i n several different architectures: cubic, tabular and amorphous. W h e n an x-ray photon is incident on the radiographic film, the following chemical reaction occurs (see Equat ions 3.15 and 3.16). Note that AgBr is used i n this example. T h e accumulat ion of elemental silver makes a latent image. The image is revealed when the film is subjected to wet, chemical processing. A n y silver ions that d id not react to produce an elemental form are washed away. The silver is black, thus areas of the film that received the most radia t ion w i l l appear dark (this is why bones appear white on a diagnostic x-ray). F i l m dosimetry can be performed if a relationship between dose and fi lm darkening (optical density) can be established. Since film is a relative dosimeter, a cal ibrat ion curve is required. F i l m emulsion batch characteristics and the film processing environment may vary from day to day. For this reason, a cal ibrat ion curve must be acquired prior to every film dosimetry session. 7 + Br~ Br + e (3.15) e~ + Ag+ Ag (3.16) Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 97 T h i s can be quite t ime consuming. F i l m does have a linear response to dose w i t h i n a very specific window of radia t ion exposure. The characteristic curve describes the film opt ical density (darkening) response to log relative exposure (Figure 3.16). 3.5: Log Relative Exposure Figure 3.16: T y p i c a l characteristic ( H and D) curve for radiographic film. Note the low re-sponding toe region and the saturat ing shoulder region. Dos imet ry measurements w i t h film should fall w i th in the linear response region of the film. Note that at low relative exposures, the film may show very l i t t le dose response (film dark-ening). T h i s is called the "toe" of the curve. A t high exposures, the film response saturates (the "shoulder" of the curve). It is important to match the film sensit ivity to the appl icat ion. F i l m manufacturers w i l l offer several different types of film emulsions to ensure that dose range to be delivered can be properly captured by the product. Since silver is a major component of the radiosensitive emulsion, the high atomic number (ZAI — 47) means that i t has a high cross-section for the photoelectric effect at lower ( sub-MeV) photon energies (see Section 2.3.2). Radiographic film is ~ 2 0 x more sensitive to 40 k e V x-rays than 1 M e V x-rays[115]. Scattered photons from wi th in the phantom or from the treatment linear accelerator components have a lower energy component. F i l m w i l l show a dose over-response to these photons. A common region where film demonstrates this problem is i n the penumbra just outside the x-ray field edges. It has also been suggested that film w i l l over-respond for some smal l intensity modulated radiat ion therapy ( I M R T ) beam segments due to the many overlapping penumbras generated by the multi- leaf col l imator ( M L C ) . Es thappen et Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 98 al. [116] provide an overview of the work done by herself and other authors to characterize the energy response of radiographic film i n an I M R T context. Despite these shortcomings, film is s t i l l commonly used and the spat ial resolution is incom-parable. T h e physical l imi t would be due to the grain size which is ~ 1 fim. Table 3.3 reviews i the benefits and drawbacks of using radiographic film as.a dosimeter. Table 3.3: Advantages/Disadvantages of radiographic film for radia t ion therapy dosimetry. Advantages Disadvantages 2D dosimeter 2D dosimeter (not true 3D) flexible (like a sheet of paper) relative dosimeter relatively inexpensive energy dependant (esp. at lower energies) easily integrated into phantom requires wet processing integrating not reusable white light sensitive.. . In a c l in ica l context, the verification of a patient treatment may involve i r radia t ing a film w i t h mul t ip le field angles and doses. It is important that the film characteristics match the doses being delivered. T h e dose range of interest should fall on the linear part of the characteristic curve. T y p i c a l doses to the fi lm plane after a l l fields have been summed together is 150 - 250 cGy. For this type of composite film verification, K o d a k E D R (extended dose range) film is often used (see Figure 3.17). T h e linear range is from ~0-350 c G y i n this example. For single-field or lower dose applications,, the K o d a k X V film w i t h a linear range of ~0-50 c G y would be a more appropriate choice (Figure 3.17). Other Other relative dosimeters include radiochromic fi lm, gel dosimetry, thermoluminescent dosime-•tery, s i l icon diode detectors, M O S F E T s , d iamond detectors and scint i l la t ion systems. Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 99 Dose Delivered to Film: (cGy) Figure 3.17: F i l m response curves to increasing dose for two different f i lm types manufactured by K o d a k ( E D R and X V ) . 3 . 3 Comparison of Radiation Therapy Dose Distributions . In a c l in ica l setting, dose verification of a patient's treatment p lan prior to therapy is an essential part of a comprehensive qual i ty assurance program. Several tools, are available that provide information on how well two different dose distributions agree. Often the comparison is made between planned (calculated) doses and measured doses. Some of the tools are quali tat ive and many are quantitat ive. W h e n the planned dose d is t r ibut ion and the measured dose d is t r ibut ion are found to be not quite a perfect match, "pass / fai l" cri teria need to be defined to assist the physicist and c l in ic ian w i t h their decision of whether or not a patient should be treated w i t h the proposed plan[117]. T h e tools listed below al l have strengths and weaknesses which means that a combinat ion of tests are usually employed to judge the qual i ty of agreement between two dose distr ibutions. In a pract ical setting, quantitative "pass / fai l" cr i ter ia is only part of a bigger picture. Regions of poor agreement may not be located i n an area of c l in ica l significance (e.g. not in /near the tumour or sensitive healthy structures). F r o m a c l in ica l perspective, the Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 100 information provided by the dose comparison information must be integrated w i t h professional judgement. For the purpose of this thesis, only the information generated by these comparison tools w i l l be considered. 3.3.1 Isodose Lines - Visual Inspection T w o dimensional dose distr ibutions are typica l ly compared i n a c l in ica l setting by visual inspec-t ion of isodose lines. W h e n verifying calculated doses to measured doses, i t is expected that if good agreement has been achieved, the isodose lines should overlay perfectly (Figure 3.18). In reali ty this is not always the case. A quick, quali tat ive assessment can be done using this method, but the "pass/fail" outcome of this type of analysis is highly subjective. (a) . (b) Figure 3.18: (a) 2D Isodose overlay showing good quali tat ive agreement (b) Poor agreement. T h e isodoses exhibi t a geometric shift in the y-axis. 3.3.2 Dose Difference A dose difference map is a more quantitative way to compare two dose distr ibutions. Essentially, one d is t r ibut ion is subtracted from the other, point-by-point, and the differences are represented either as absolute values or as percentage differences from a reference condi t ion. T h i s type of analysis is most useful for comparing regions where the dose gradients are quite low. In the simple example of a one dimensional open held dose profile, i t is clear that the difference map is very sensitive to smal l shifts i n isodose line positions in high-gradient regions (e.g. field edges). Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 101 Figure 3.19(a) demonstrates the large dose differences encountered when smal l spat ial shifts i n the isodose curve are present. 110 100 90 3" 80 g 7 0 O O 60 « S 5 0 § 40 CD 0- 30 20 '• 10 DTA- small Dose Difference = very large /T -15 -10 -5 0 5 Profile x-Position (cm) 10 110 100 90 3- 80 S 7 0 o Q 60 cp s 50 I 40 CD 0- 30 20 10 0t Dose Difference = small .__„„* ..^ DTA = very large -• --20 -15 -10 -5 0 5 Profile x-Position (cm),. 10 15 .(a) (b) F igure 3.19: T w o examples of dose discrepancy (a) Isodoses are spat ia l ly shifted. T h e dose dif-ference method is sensitive to smal l shifts i n the high gradient region, (b) Isodoses are shifted i n dose value. The distance-to-agreement method is sensitive to smal l dose differences i n low gradient, regions. In Figure 3.20(a), a c l in ical I M R T dose dis t r ibut ion measured w i t h f i lm is shown. Figure 3.20(b) shows the calculated dis t r ibut ion. Figure 3.20(c) shows a map of the percentage dose difference as calculated i n Equa t ion 3.17. Note that regions that exceed + / - 5 % have been highlighted. V a n D y k et al . (1993) suggested that a 3% dose difference would be considered to be c l in ica l ly acceptable[117]. „ ^ ^ (calculated dose — measured dose) „ „„IV. %DD = - — , , , , x 100% calculated dose (3.17) 3.3.3 Distance-to-Agreement A n alternative to the dose difference map is the distance-to-agreement ( D T A ) map. T h i s map documents how far from a point of interest one must travel before the same dose value is encountered i n the comparison dose dis t r ibut ion (see Figure 3.19(b)). Note that i n low gradient regions, large D T A values may be encountered even though the dose difference is very smal l . It rap id ly becomes clear that both types of analysis are essential i n order to comprehensively Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 102 (b) (0 20 40 60 30 100 120 140 160 (d) * I t B 1 ' ' f • : t a * 1 #• 4 40 60 80 100 120 (f) F igure 3.20: Dose comparison methods, (a) F i l m measured I M R T dose plane (b) Calcu la ted dose plane (c) Percentage Dose difference map, (d) distance-to-agreement map, (e) B i n a r y map : percentage dose difference A N D distance-to-agreement cr i ter ia (2% / 2 mm) fail (white), (f) G A M M A map w i t h 2% / 2 m m pass/fai l cri teria. Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 103 assess the "goodness" of agreement between different dose distr ibutions. Distance-to-agreement mapping is performed on the measured and calculated dose dis t r i -butions shown i n Figures 3.20(a) and (b). T h e result is shown i n Figure 3.20(d). Note that the color scale is set to highlight regions i n the map that exceed 2 m m . T h i s is defined as the "pass" cr i ter ia for this test. It has been suggested by other authors that a < 3 m m distance to agreement is c l in ica l ly acceptable as a "pass" value for most treatment plans[117]. 3.3.4 Dose Difference / D T A "Pass/Fail" Methods Cer t a in comparison techniques are more meaningful i n either high-dose gradient areas or low-dose gradient regions, but often, not both. T h i s issue led authors such as Cheng et al. (1996), Harms et al. (1998), to develop a composite index that incorporates bo th comparison methods (dose subtract ion and D T A ) and highlights regions that fail a set of comparison criteria[118, 119]. Generally, the two types of dose comparisons are assessed and regions where bo th tests fail the pass fail cr i ter ia are highlighted as problem areas. The percentage dose difference and distance-to-agreement results from Figures 3.20(c) and (d) are compared and a composite map is produced i n Figure 3.20(e). In this map, a "1" (black) represents a pixel where either the dose-difference or the D T A map passed a 2% / 2 m m test criteria. A "0" (white) represents a pixel where bo th tests failed. A l t h o u g h this type of analysis does highlight pass / fail regions, there is not a lot of quan-t i ta t ive da ta that describes by how much the test failed. T h i s is addressed i n the next section. 3.3.5 The Gamma Factor L o w et al.[120] further developed the composite analysis concept into a more quanti tat ive method for comparing measured data to calculated data by in t roducing the gamma factor, 7 . T h e gamma factor offers a metric that is essentially a combinat ion of the dose difference and the distance-to-agreement comparison results. The user specifies an acceptabil i ty cr i ter ion for. each test. T h e advantage of this method is that the influence, of the ind iv idua l tests vary spa-t ia l ly across high and low dose gradient regions. T h e overall gamma factor provides a consistent Chapter 3. Dose Calculation Algorithms and Validation Techniques in Radiation Therapy 104 metric to describe the qual i ty of agreement between two dose distr ibutions. T h e acceptabil i ty cr i ter ia suggested by V a n D y k et a/.(1993)[117] is a 3% dose difference and 3 m m distance-to-agreement. T h e gamma-factor analysis can be applied to a I D profile, or to a n-dimensional dose dose dis t r ibut ion. / In the 2D example( the j(rm) values are incrementally bui l t up, by stepping through the reference d is t r ibut ion pixel-by-pixel . (Equations 3.18, 3.19)[120]. r ( r m , r c ) = r2(rm,rc) 62{rm,rc) + A n 2 — 'M \ Ad2M AD j(rm) = minimum{T(rrn)rc)} for all rc (3.19) where • r = distance between measured ( r m ) and calculated (r c ) data points • AdM = m a x i m u m distance-to-agreement considered acceptable (often set to 3mm[117]) • S(rm, rc) = dose difference between measured data at rm and calculated data at rc • ADM = m a x i m u m percentage dose difference acceptable (often set to 3% of the dose normal iza t ion value[117]) In the following cl in ical example, a gamma map is calculated using a tighter "pass" "fai l" cr i ter ia than that suggested by V a n D y k et al . ( 3% / 3 m m ). In Figure 3.20(f), the acceptance cr i ter ia used is 2% / 2 m m . A n advantage to the G a m m a M a p analysis is that the degree of agreement is incorporated into the map data. In addi t ion, it can be extended to n-dimensions. Chapter 4. Improved IMRT'Dose Distributions using Modified Pencil Beam Dose Kernels 105 C h a p t e r 4 I m p r o v e d I M R T D o s e D i s t r i b u t i o n s u s i n g M o d i f i e d P e n c i l B e a m D o s e C o n v o l u t i o n K e r n e l s 4.1 Prologue For the single penci l beam dose kernel convolution algori thm, the abi l i ty to resolve areas of h igh dose structure is par t ly related to the shape of the penci l beam dose kernel (similar to how a photon detector's point spread function relates to imaging resolution). Improvements i n dose calculat ion accuracy have been reported when the treatment p lanning system ( T P S ) is re-commissioned using.high-resolution measurement data as input . T h i s study proposes to improve the dose calculat ion accuracy for I M R T planning by modifying cl inical dose kernel shapes already present i n the T P S , thus avoiding the need to re-acquire higher resolution commissioning data. T h e in-house opt imiza t ion program minimizes a objective-function based on a 2D composite dose subtract ion / distance-to-agreement (gamma) analysis. T h e final modified kernel shapes are re-introduced into the treatment planning system and improvements to the dose calculat ion accuracy for complex I M R T dose distr ibutions evaluated. T h e central kernel value (radius = 0 cm) has the largest effect on the dose calculat ion resolution and is the focus of this study. The da ta presented in this chapter has been published i n the peer-reviewed journal , Medical Physics (Bergman et ai, "The use of modified single penci l beam dose kernels to improve I M R T dose calculat ion accuracy" ,Med.P%s.,31(12):3279-3287, 2004)[10]. Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 106 4.2 Introduction T h e single penci l beam dose kernel convolution algori thm is implemented i n some commer-cial treatment p lanning systems ( T P S ) , for example, V a r i a n E c l i p s e ® v . 6 . 5 (OS System B u i l d 7.1.31, V a r i a n Inc., Pa lo A l t o , CA)[86] . Th i s a lgori thm starts by subdiv id ing the open photon radia t ion field into a finite number of of smal l "beamlets" of radiat ion, for example 0.25 x 0.50 m m 2 . T h e 3D projection of the smal l radiat ion beam emanating from each beamlet locat ion is called a "pencil beam". Associated w i t h each penci l beam is a dose kernel that describes the physical d is t r ibut ion of absorbed dose about the narrow beam axis i n a water-equivalent mater ia l (phantom). T h e total dose to a point of interest in phantom is the superposi t ion of the dose contributions from al l the penci l beams in the field. T h i s can be represented by a convolut ion of the dose kernel w i t h the pr imary fluence at depth. The shape of the dose kernels depends on the beam energy spectrum and thus are depth-dependent. T h e penci l beam kernels are derived from a series of in-phantom measured beam da ta (pro-files and percentage depth doses) [11, 12]. B y generating dose kernels based on measurements i n phantom, beam spectrum (including scatter) characteristics at depth are automat ical ly incor-porated. T h e in-phantom T P S commissioning measurements are t radi t ional ly acquired w i t h an ion chamber (e.g. Wellhofer IC-10, cavity radius = 3 mm). For I M R T applications, it has been demonstrated that higher resolution dosimeters (e.g. film, diodes) are necessary to generate sharper dose kernels[13, 14]. The sharper dose kernels w i l l improve dose calculat ion accuracy for the complex I M R T fluence map deliveries that produce high dose gradient regions. However, discrepancies between measured and calculated doses for I M R T fields s t i l l exist. Th i s s tudy w i l l demonstrate that further improvements to the dose calculat ion accuracy can be achieved by i terat ively modifying the shape/structure of the pencil beam dose kernel. Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 107 4.3 Theory 4.3.1 Single Pencil Beam Dose Kernel Convolution Algorithm Using 2D Dose Kernels T h i s dose calculat ion a lgor i thm is described i n Section 3.1.2. To summarize, a penci l beam dose kernel at tempts to describe the 2D spatial d is t r ibut ion of energy deposit ion (dose) about a smal l , but finite penci l beam of photons incident on matter[83, 84]. T h e shape of this kernel depends on the photon energy spectrum at the depth of interest and the p roduc t ion /d i s t r ibu t ion of secondary electrons i n the mater ial of interest (usually tissue-equivalent). T h e dose d is t r ibut ion is defined i n a plane perpendicular to the-beam axis. If spat ia l i n -variance of the penci l beam dose kernel is assumed, the dose d is t r ibut ion across an irregularly shaped field at a given depth can be calculated by convolving a modified photon fluence. field, F'(x',y'), w i t h the pencil beam kernel, K(x,y,d) (see Equa t ion 3.4 and Figure 3.7)[12, 17]. T h e penci l beam dose kernels are defined at several discreet depths (e.g. 1.5, 5.0, 10.0, 20.0, and 35.0 cm). Ca lcu la t ion points located at intermediate depths use interpolated values[86]. T h e concept of a spat ial ly invariant dose deposit ion kernel is a key property of a convolut ion-based calculat ion algori thm. However, the dose kernel is not t ru ly spat ia l ly invariant due to energy spectrum differences (beam softening) and kernel t i l t ing (beam divergence) across the calculat ion plane. In this study, no specific corrections are.made to account for the use of a spat ia l ly invariant dose kernel. 4.3.2 Dose Calculation Accuracy Assessment - The Gamma Factor Several methods for comparing reference (measured) doses to calculated doses are available and include quali tat ive isodose overlays, quantitative dose subtract ion methods, distance-to-agreement ( D T A ) methods[117], or some combinat ion of the above[119]. Ce r t a in comparison techniques are more meaningful in either high-dose gradient areas or low-dose gradient regions, but often, not both . T h e differences between these two close dis t r ibut ion comparison methods are discussed i n Section 3.3. Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 108 L o w et al. [120, 121] extended the composite test concept and developed a more quanti tat ive method for compar ing measured data to calculated data by int roducing the gamma factor, 7 . T h e gamma factor offers a metric that is spatial ly independent. It is essentially a combinat ion of the dose difference and the distance-to-agreement comparison results. T h e user specifies an acceptabi l i ty cri ter ion for each test; i n this study, a 3% dose difference and 3 m m distance-to-agreement is used (as suggested by V a n D y k et al. (1993)[117]). The gamma-factor analysis can be applied to a I D profile, or to a n-dimensional gamma map. A 2D gamma map is used i n this s tudy to compare dose planes calculated from modified and unmodified dose kernels. 4.4 M a t e r i a l s a n d M e t h o d s 4.4.1 Measured Reference Data The dose kernel opt imizat ion process independently calculates the dose for a test pat tern flu-ence and i terat ively modifies the dose convolution kernel un t i l the calculated doses match the measured 2D dose dis t r ibut ion. T h e measured dose dis t r ibut ion data is considered to be the 'gold s tandard ' to which T P S dose calculat ion accuracy can be compared. A complex, 2D measured dose dis t r ibut ion is used to establish an ideal 'a impoint ' for the op t imiza t ion of the penci l beam dose kernel shapes. Dose information for bo th the kernel op t imiza t ion process and the final dose calculat ion verification are performed w i t h radiographic film. Parameters used to model the mult i leaf coll imator dosimetric properties (transmission / leaf gap) are kept constant in order to isolate the effect of the kernel modification. 2D Reference F luence M a p s A n ideal 2D fluence map is created consisting of a high resolution bar pattern. T h e pat tern is comprised of ten bars of alternating unit and zero photon fluence intensity values. (Figure 4.1(a)). T h e peak-to-valley distance is 5 m m and each bar is 100 m m long. A peak-to-valley rat io, (peak value — valley value)/(peak value) is defined to describe the .fluence map test pat tern intensity variat ion. In Figure 4.1(c), the peak-to-valley ratio is 1.0. Chapter 4 . Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 109 (c) (d) Figure 4.1: (a) 2D ideal fluence pat tern comprised of ten 5 m m wide bars of uni t intensity (white), (b) actual fluence pattern obtained when the ideal fluence is modified by the dosimetric properties of the multi- leaf col l imator beam shaping device (trans-mission, dosimetric leaf gap). The leaf travel direct ion is from left-to-right, (c) profile through ideal fluence map. Peak-to-valley ratio = 1.0. (d) profile through actual fluence map. Peak-to-valley ratio = 0.85. T h e ideal fluence map is impor ted into the Var i an Ecl ipse v6.5 T P S and applied to a square homogenous phantom. Before calculat ing the dose to the phantom, the T P S accounts for the dosimetric properties / l imitat ions of the field shaping device (e.g. mult i leaf col l imator ( M L C ) transmission, leaf gap properties) and generates an actual (or deliverable) fluence map (Figure 4.1 (b)). T h e leaf transmission is modeled as the average of the intra-leaf and inter-leaf t ransmission values. In this example, the leaf transmission reduces the peak-to-valley ratio to 0.85 (Figure 4.1 (d)). T h e 5 m m bar pat tern (peak-to-valley) fluence map was chosen because it is sensitive to changes/modifications to the dose calculat ion kernel. It also contains spatial frequencies that meet or exceed the spat ial frequencies encountered in typ ica l c l in ical I M R T fluence dis tr ibut ions Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 110 (Figure 4.2). Fourier transform ( F T ) analysis has been previously applied to several interesting radia t ion therapy applications[52, 54]. In this study, Fourier transforms are used to compare the spat ial frequencies of a cl inical fluence map to those of a 5 m m R E C T function (representing the 5 m m bar pattern). T h e discrete Fourier transform has been computed using a fast Fourier transform algor i thm as implemented i n the M A T L A B (Mathworks inc., Nat ick , M A ) Image Processing Toolbox. T h e complex modulus (magnitude) is shown. Ana ly t i ca l ly , the F T of a I D R E C T function of w i d t h a, results i n a S I N C function w i t h first zero crossing appearing at frequency 1/a [122]. A s shown i n Figure 4.2, the F T of the 5 m m R E C T pat tern encompasses the spat ia l frequencies contained in the cl inical map. cycles/mm Figure 4.2: Frequency domain profiles comparing spatial frequency components of the Fourier transform of a cl inical fluence map (solid line) to the Fourier transform of a 5 m m R E C T function ( F T ( R E C T ) = S I N C ) (dashed line). Doses are calculated at four different depths (1.5, 5.0, 10.0, 20.0 cm) using the 6 M V V a r i a n 2 1 E X photon pencil beam kernels stored in. the T P S and the actual fluence map that is exported to the independent dose calculat ion algori thm (described in section 4.4.2) T h e 2D dose distr ibutions are measured using K o d a k E D R 2 radiographic film (Eas tman K o d a k Inc., Rochester, N Y ) loaded into a light-tight solid water (water-equivalent plastic) cassette. T h e cassette is designed such that the film is sandwiched between 2 cm slabs of solid water. T h e cassette is positioned perpendicular to the beam axis, at 100 c m S A D , and placed at a specified depth i n solid water. A m i n i m u m of 20.0 cm of solid water mater ia l is placed below the film plane to provide adequate backscatter. T h e fluence maps are delivered by a V a r i a n C l inac 2 1 E X medical linear accelerator using a sl iding window dynamic M L C technique. Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 111 Sample dosimetric parameters for the 5.0 cm depth bar pattern fluence map are shown i n Table 4.1. , . Table 4.1: Sample i r radia t ion parameters for delivery of 5 \mm bar pat tern (peak-to-peak) flu-ence map to film plane positioned a t depth = 5.0 cm ( S A D 100 cm). • D e p t h Prescribed Dose Calcula ted Delivered Dose Ra te (cm) ( G y to Isocentre) Mon i to r Uni t s ( M U ) ( M U / m i n ) 5.0 2.25 3859 400 A s radiographic film is a relative dosimeter, a cal ibrat ion curve must be constructed to convert film opt ical density into absorbed dose. T h i s is achieved by acquiring two 5 x 5 c m 2 , 100 cm S S D percentage depth dose v film images exposed w i t h different moni tor uni t ( M U ) settings (to capture the relevant dynamic dose range of the film). T h e resulting film opt ical density is correlated w i t h ion chamber measured data acquired under similar i r radia t ion conditions. A l l films are developed en masse in a K o d a k 3000RA film processor operating a s tandard cycle ( I l l s drop-t ime for 35 cm film length) and developer temperature of 33 .5°C. T h e automatic replenishment pumps are disabled while processing the smal l batch of films. The processed films are digi t ized w i t h a V i d a r V X R - 1 6 Dos imet ryPro film scanner (Vida r Systems Corp . , Herndon, V A ) using 71 p ixe l s / inch resolution (0.0358 cm/p ixe l ) and 16 bit depth settings. T h e isocentre posi t ion on each fluence film image is identified by a set of fiducial markers inherent i n the solid water film cassette. 4.4.2 Independent Dose Calculation T h e single penci l beam dose kernel convolution a lgor i thm is reconstructed i n M A T L A B ( M a t h -works Inc., Nat ick , M A ) by mul t ip ly ing a 2D fluence map, F(x',y') (includes effects due to M L C transmission, m i n i m u m leaf gap etc.) by the 2D pr imary beam off-axis intensity profile, Pint(x',y', d), and convolving the result w i t h a 2D pencil beam dose kernel, K(x,y,d) (refer to section 3.1.2 for theory). The Ecl ipse T P S implements an inherent 2.5 m m calculat ion gr id which is maintained i n the independent dose calculat ion. \ Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 112 _ 4 . 4 . 3 Dose Kernel Shape Optimization T h e dose kernel shapes used by the T P S calculat ion a lgor i thm are opt imized using an in-house-program. T h e modified kernels are re-inserted into the T P S for assessment. T h e procedure is described below. E x t r a c t i n g Dose K e r n e l s f rom T P S T h e I D dose kernel profiles are exported from the T P S as an A S C I I text file and read into the in-house calculat ion and opt imizat ion program (Figure 4.3). These kernels are defined at several depths (e.g. 1.5, 5.0, 10.0, 20.0, and 35.0 cm). The exported kernel has data-points located every 2.5 m m along a I D radial line emanating from the or igin (radius = 0 cm). The kernel is assumed to be cyl indr ica l ly symmetric and is converted to a 2D dose kernel ma t r ix having a grid-point spacing of 2.5 m m by linear interpolat ion. The effect of modifying the dose kernel shape can now be immediately assessed i n a manner independent from the T P S . radial distance from centre (mm) Figure 4.3: I D Dose deposit ion kernel derived from ion chamber measured commissioning data. D a t a exported from Eclipse v6.5 T P S (depth = 5.0 cm). . Dose K e r n e l Shape O p t i m i z a t i o n In this study, only the central kernel value ( C K V ) (radius r = 0 cm) is modified. The de-sired kernel shape is obtained by using the opt imizat ion tools offered in M A T L A B , namely the Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 113 fminbnd function. T h i s function is based on the Golden Section search (a method of locat ing a m i n i m u m along a I D , bracketed, continuous function and parabolic interpolation) [123]. It i terat ively modifies the C K V and attempts to minimize the value of a user-specified objective function. T h e objective function is defined to be the sum of a l l p ixel values i n a 2D gamma fac-tor map ( S T ) representing a calculated dose-to-measured dose comparison [120]. T h e pass/fai l cr i ter ia for the gamma analysis is set to: dose difference = 3% of prescript ion dose and, distance-to-agreement = 3 m m . 4.4.4 2D Clinical Fluence Map Verification - Example 1 Fol lowing the opt imiza t ion process described i n section 4.4.3, the modified kernel is tested on a c l in ica l I M R T field. A single 6 M V photon fluence map from a seven-field c l in ica l I M R T p lan is delivered to K o d a k E D R 2 radiographic film located perpendicular to the beam axis i n a solid water phantom. The film is placed at a depth of 5.0 cm, S A D 100 cm. T h e moni tor units delivered to the fi lm is 1315 M U (based on a 2.0 G y single-field normal iza t ion dose at isocentre). T h e doses delivered to the film plane are calculated by the T P S using bo th the or iginal ion chamber-derived kernels and the modified kernels. The agreement between the calculated data and the measured data is assessed using a gamma factor map analysis technique and I D profile comparisons. 4.4.5 2D Clinical Fluence Map Verification - Example 2 A dynamic M L C control file is imported into the T P S and the doses calculated using both the ion chamber measurement-derived kernel and the modified kernel. Profiles through the m a x i m u m peak of the calculated dose dis t r ibut ion are compared to film measurement. Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 114 4.5 Results and Discussion 4.5.1 Dose Kernel Shape Optimization Dose kernels that are derived from ion chamber measured commissioning da ta w i l l be referred to as ion chamber kernels. Kernels derived v i a the iterative op t imiza t ion process w i l l be referred to as modified kernels. T h e dose calculat ion a lgor i thm implements a piecewise linear kernel shape (datapoints every 2.5 m m ) . T h e iterative opt imiza t ion process samples many different potent ial central kernel values using the cl in ical ly available ion chamber kernels as an in i t i a l condi t ion (Figure 4.4). 10 •sr 9 0 i L — . 1 . . •• , 1 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 Distance from Interaction.Point (cm) Figure 4.4: C y l i n d r i c a l l y symmetr ic I D dose kernel for depth = 5.0 cm. Solid line: i n i t i a l ion chamber-derived kernel from T P S . Dashed, lines: several potent ial central kernel values (radius = 0 cm) sampled dur ing the opt imiza t ion process. For clarity, only radius = 0 to radius = 2.5 cm are plot ted and only one half of the I D kernel is shown (it is cyl indr ica l ly symmetric) . The central kernel values predicted by the opt i -miza t ion process are listed i n Table 4.2. P r io r to performing the dose calculat ion convolut ion, the kernel 2D area is normalized to 1.0. Table 4.2: Cent ra l Ke rne l Values ( C K V ) for four depths. D e p t h i n phantom Ion Chamber C K V F i l m Der ived C K V [13] Modi f ied C K V (cm) (from Eclipse v6.5) (from Eclipse v6.5) 1.5 1.6061 - 9.1403 5.0 1.7410 3.8119 6.9769 10.0 1.6415 - 5.0976 20.0 1.2784 - 2.3773 Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 115 T h e in i t i a l 6 M V ion chamber kernel shape is compared to the final modified kernel shape for dep th = 5.0 c m i n Figure 4.5. 0.4, •— r-^ —, •' 0.35-§ 0.25 / \ T3 -10 -5 0 5 10 mm Figure 4.5: Dose kernel profiles for depth = 5.0 cm. 2D kernel area has been normal ized to 1.0. Solid line: i n i t i a l ion chamber-derived kernel from T P S . Dashed line: modified kernel shape. 4.5.2 5 mm Bar Pattern Fluence Dose profiles calculated by the T P S using the unmodified ion chamber kernels are compared to measured doses (film) for a 5 m m bar pattern (peak-to-valley) fluence map i n Figure 4.6(a). T h e dose d is t r ibut ion is normalized to 100% at the average peak-to-valley height. T h e ion chamber kernels are unable to resolve the correct dose in the peaks and valleys result ing i n dose differences of ± 1 7 % between measured and calculated data. The discrepancy between calculated and measured data outside the field edge (arrow : Figure 4.6(a)) is a docu-mented problem inherent to the pencil beam dose calculat ion algori thm. The issue is related to the lack of model ing of the extra-focal scattered radiat ion (e.g. from the jaws / flattening filter) [17]. T h e modified close kernel shapes are impor ted into the T P S , the 5 m m bar pat tern doses are re-calculated and the result compared to measured (film) data (Figure 4.6(c)). Note the improved performance w i t h the implementat ion of the modified dose kernels. O t to et al. (2002) generated a dose kernel based on high-resolution commissioning data (film) for one depth (5.0 cm) [13]. In their study, the performance of the dose calculat ion algori thm using bo th the film input da ta and the ion chamber input data to generate the dose kernels are compared. T h e Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 116 (a) 160 140 g 1 2 0 g 100 o ' •I 80 J5 at OH 60 40 20 160 140 ,120 ; 100 ! '80 ! 60 40 20 Measured Data — Modified Kernel Measured Data —• Film Kernel ft fi A A v 9 0 -10 -8 (b) -6 -4 -2 0 2 4 6 8 10 cm -6 -4 -2 0 2 4 6 8 10 Figure 4.6: 5 m m bar pat tern (peak-to-valley) dose profiles. D e p t h = 5.0 cm. Solid line: measured film doses. Dashed line: calculated doses. Dose kernel generated from: (a) ion chamber commissioning data, (b) film commissioning data [13], (c) modified dose kernel. calculated doses based on the film input data is- compared to the measured delivered dose in Figure 4.6(b)). Despite the use of a high resolution dosimeter to commission the T P S , the 5 m m bar pat tern s t i l l exhibits a discrepancy between calculated and measured da ta of ± 8 % . 4.5.3 2D Clinical Fluence Map Dose Verification - Example 1 T h e performance of the dose calculat ion algori thm using the modified kernels are compared to the ion chamber kernels currently implemented in the T P S . Figure 4.7 illustrates a 2D measured dose d is t r ibut ion (b) resulting from a. single I M R T fluence map (a). T h e dose d is t r ibut ion is masked to highlight the information located wi th in the treatment field edges. Figures 4.8 (a),(b) compare I D calculated and measured dose profiles through a point 2.5 Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 117 -5 -2.5 0 2.5 5 7.5 -7.5 -6 -4.5 -3 -1.5 0 1.5 3 4.5 6 -7.5 cm cm (a) (b) Figure 4.7: (a) C l i n i c a l fluence map. (b) Dose dis t r ibut ion measured w i t h film (depth = 5.0cm). L ine indicates locat ion of profile data. (a) (b) Figure 4.8: I M R T dose profiles (2.5 m m inferior, to isocentre). Dcpth=5.0 cm. Solid line: film measured dose data. Dashed line: calculated data, (a) ion chamber dose kernel (b) modified dose kernel. m m inferior to the isocentre (to avoid the M L C leaf leakage line) at a depth of 5.0 cm. The profile is i n a direct ion parallel to the M L C leaf motion. The modified kernel improves the agreement between the measured data and the calculated data by increasing the height and depth of the dose peaks and valleys. A 2D gamma factor ( 7 ) analysis is shown i n Figure 4.9. T h e comparison and analysis is performed wi th in the field edges. His tograms can provide a more quantitative index for comparing the agreement between the calculated doses and the measured doses for both dose kernels. Figure 4.10 illustrates the dis t r ibu t ion of the gamma values in the 2D dis t r ibut ion for the ion chamber kernel and the modified kernel calculated doses. The dis t r ibut ion of gamma factor values favor the modified Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 118 ION CHAMBER DOSE KERNEL (a) MODIFIED DOSE KERNEL (b) Figure 4.9: 2D gamma map comparing measured data to (a) ion chamber kernel and (b) mod-ified kernel calculated doses. 8000 7000 6000 _ i LU X 5000 rx O 4000 a. LU CO 3000 s Z 2000 1000 0 •P Ion Chamber Kernel ; Modified Kernel I 2 3 4 5 6 GAMMA FACTOR BINS Figure 4.10: His togram dis t r ibut ion of gamma values for calculated vs. measured dose compar-ison. kernel calculated da ta i n that the histogram dis t r ibut ion is shifted towards the lower (more desirable) gamma values. Ideally, the sum of a l l gamma values w i t h i n the field would equal zero (perfect agreement between the measured and calculated data). T h e sum of a l l gamma, J2l{iij)i 3 5 w e U ^ the mean and max values for the 2D masked distr ibutions is presented i n Table 4.3. Table 4.3: Propert ies of the masked 2D 7 map dis t r ibut ion (reference data = measured data). Ion Chamber Kerne l Modif ied Kerne l 5287 4638 mean 0.68 0.57 max 7 ( ' i , j) 5.9 4.2 There are several factors (in addi t ion to the accuracy of the dose kernel shape) that contribute Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 119 to the agreement between a measured I M R T dose dis t r ibut ion and a calculated dose d is t r ibut ion . For example, the single penci l beam dose kernel convolution algori thm does not model the effects of M L C inter-leaf leakage and the tongue-and-groove effect. In this study, these factors were kept constant i n an effort to isolate the effect of the dose kernel shape. V 4.5.4 2D Clinical Fluence Map Dose Verification - Example 2 A second c l in ica l I M R T dose dis t r ibut ion is shown i n Figure 4.11(a). The x- and y- profiles ( indicated as dashed lines on Figure 4.11(a)) are plot ted in Figure 4.11(b),(c). T h e modified penci l beam kernel calculat ion demonstrated a closer agreement to measured data. T h e original ion chamber-derived penci l beam kernel was unable to resolve the dose peak, par t icular ly along the x-prohie. 4.6 Conclusion : Modified Pencil Beam Dose Kernels T h e penci l beam kernels that are t radi t ional ly derived from ion chamber measurements are increasingly found to be unsatisfactory for high-resolution I M R T dose calculations. T h e use of smal l volume detectors (e.g. diodes, film) during commissioning w i l l improve the calculat ion accuracy by reducing spat ial averaging of the input data. However, the use of a smal l volume detector may not completely resolve the problem. Other l imitat ions include the assumption of a spat ia l ly invariant dose kernel, use of a coarse kernel sampling gr id (2.5 mm) , lack of model-ing the extra-focal radia t ion contr ibut ion, and lack of model ing the lateral electron transport in inhomogeneous materials. T h i s study, proposes a technique that i terat ively optimizes pre-exist ing dose kernels to improve dose calculations for I M R T (as implemented i n the V a r i a n E c l i p s e ® v 6 . 5 / O S System B u i l d 7.1.31). For the 5 m m bar test pattern, the modified dose ker-nels reduced the discrepancy between the measured and calculated doses in the peak and valley region from ± 1 7 % (for the ion chamber-derived kernels) to less than ± 2 % . Bet ter agreement w i t h measured data is achieved when using modified dose kernels even when compared to the high resolution, film derived dose kernels (film kernels demonstrate ± 8 % discrepancy between measured and calculated doses i n the peak and valley regions). A more modest improvement Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 120 in the calculated dose accuracy for two cl inical ly relevant single-held I M R T fluence maps is observed. Modi f i ca t ion of the kernel shape based on measured-to-calculated test pat tern dose agreement w i l l reduce the impact of some l imitat ions of the penci l beam convolut ion dose a l -gor i thm i n terms of dose gradient resolution. However, the penci l beam a lgor i thm does have ION C H A M B E R Kernel M O D I F I E D Kernel (b) (c) 70 60 50 <?40 I-20 10 I j j (ci) (e) F igure 4.11: C l i n i c a l example #2. Depth=5.0 cm. ( a ) I M R T dose dis t r ibut ion. (b),(c): X - a x i s , and (d),(e): Y - a x i s dose profiles at point (-0.4 cm,4.3 cm) relative to isocentre. Solid line: f i lm measured data. Dashed line: calculated data. (b),(d): ion chamber dose kernel (c),(e): modified dose kernel. Chapter 4. Improved IMRT Dose Distributions using Modified Pencil Beam Dose Kernels 121 other l imi ta t ions unrelated to its resolving capability. T h i s a lgor i thm assumes that the entire pa t ien t /phan tom i n which the dose is calculated is comprised entirely of water. Realis t ical ly, a human is comprised of not only water-like tissues, but of bone, air and lung as well . T h e penci l beam convolut ion implements an equivalent pathlength, Ba tho inhomogeneity correction (see Section 3.1.6 for a description). Th i s method has some serious dose calculat ion accuracy l imi ta t ions w i t h i n and adjacent to certain non-water tissue types (e.g. lung, bone). T h e prob-lem, is related to the absence of any k ind of model ing of lateral electronic scattering. T h i s issue applies to smal l radia t ion fields as well (< 4 x 4 c m 2 ) or any s i tuat ion where lateral electronic d isequi l ibr ium may exist. T h e combinat ion of these different disadvantages (dose kernel resolution issues i n high--gradient regions and dose errors in regions of tissue inhomogeneity) provided a very strong indica t ion that a completely different dose calculat ion algori thm for I M R T was required. T h e decision was made to move away from the single pencil beam dose kernel convolut ion a lgor i thm and embrace instead, Monte Car lo methods for the s imulat ion of photon/electron transport in tissue. T h e rest of this thesis is dedicated to explorations of how the advantages of Mon te Car lo can be exploi ted to improve the quali ty of intensity modulated radiat ion treatments. Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 122 C h a p t e r 5 M o n t e C a r l o S i m u l a t i o n o f P a r t i c l e T r a n s p o r t T h r o u g h M a t t e r 5.1 Introduction to Monte Carlo Monte Ca r lo is a stochastic statist ical sampling technique used to solve mathemat ica l problems. T h i s method describes a process where the end result depends on a series of input variables that are taken from w i t h i n a fixed range and selected to fit a probabi l i ty d is t r ibut ion function. T h e variables are essentially selected at random, much like the games of chance i n the casinos of Mon te Car lo , Monaco . For the example of photon interaction model ing applications, the type of photon interaction about to occur is selected at random (although weighted by a probabi l i ty dis t r ibut ion) . For example, an x-ray photon at an energy typical ly used for radia t ion therapy treatments (e.g. 6 M V ) has a certain probabi l i ty of either undergoing: • i) no interaction, • ii) photoelectric effect event, • i i i) compton scattering event, or • iv) pair -product ion event Mon te Ca r lo w i l l track the history of every daughter particle or photon generated after the in i t i a l interaction. T h e analysis of one photon (or electron)'s journey or history i n itself does not contain much information about the macroscopic properties of the environment under study, just like how one photon of visible light w i l l not reveal the composi t ion of a photograph. However, when mil l ions of photons and their subsequent interactions are allowed to accumulate, Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 123 they do start to reveal the big picture. In a medical physics context, this b ig picture may include information such as types and frequencies of certain interactions, geographic and spectral d is t r ibut ion of particles as they pass through different media, sources of scattered radiat ion, and u l t imate ly dose deposited w i th in a patient or phantom. Generally, Monte Ca r lo simulations i n radia t ion therapy medical physics fall under three major categories: 1. S imula t ion of radiat ion beam (electron or photon beam) from a medical linear accelerator (energy spectra, geographic dis t r ibut ion of particles and sources of scatter) 2. S imula t ion of dose in target media such as human tissue or tissue-equivalent phantoms (dose, scattering properties, spectral changes) 3. S imula t ion of different dosimeters (dose deposit ion properties, detector response and test-ing of new design concepts) 5 .1 .1 Simulation Codes - Overview Several Monte Car lo radiat ion transport code-bases are available for radia t ion therapy applica-tions. These include: • E G S n r c (Electron G a m m a Shower) - Na t iona l Research C o u n c i l of C a n a d a / Stanford Linear Accelerator Centre • M C N P (Monte Car lo N-Par t ic le) - Los Alamos Na t iona l Laboratory, N M • E T R A N (Electron Transport) -Nat iona l Institute of Standards and Technology, M D • P E R E G R I N E - Lawrence Livermore Nat iona l Laboratory, C A 1 • D P M (Dose P l ann ing Method) - Univers i ty of Mich igan , M I • P E N E L O P E - Nuclear Energy Agency, Organisat ion for Economic Co-operat ion and De-velopment, F rance /Spa in /Argen t ina • G E A N T 4 - Centre Europeen de l a Recherche Nucleair ( C E R N ) , Switzer land Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 124 • V M C + + (Voxel based Monte Carlo) - Nat iona l Research Counc i l of C a n a d a ( N R C ) These codes have differences i n the particle transport calculations, and performance compar-isons for bo th photon and electron radiotherapy applications are available i n the literature[124, 125, 126]. In this report, the E G S n r c system w i l l be ut i l ized. A description of this par t icular part icle t ransport code follows. 5.1.2 EGSnrc (Electron Gamma Shower - National Research Council) H i s t o r y E G S is a powerful publ ic Monte Car lo program to simulate electron and photon interactions w i t h matter. It was first developed in the 1970's at S L A C (Stanford Linear Accelerator Centre) . E G S stands for electron gamma shower and is apt ly named to describe the many daughter particles and photons that are produced dur ing an electromagnetic cascade. In the m i d - 1980's, the software was developed into a more user-friendly package that was made available to scientists at no cost[127]. It is copyrighted by Stanford Univers i ty and the Government of C a n a d a (Nat ional Research Counc i l ) . D e s c r i p t i o n E G S is able to simulate the radiat ion transport of particles (+ or -) or photons through prac-t ica l ly any med ium element, compound or mixture. (Note: The media and their properties are contained i n a stand-alone subroutine called P E G S 4 ) . The energy range that can be handled is quite broad (~1 k e V to several thousand G e V ) . The interactions that can be simulated are: P h o t o n Interact ions: • i) Ray le igh (coherent) scattering • ii) Photoelectr ic effect i • i i i) C o m p t o n (incoherent) scattering Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 125 • iv) Pa i r product ion E l e c t r o n Interact ions: • i) M0ller (electron-electron) and B h a b h a (electron-positron) scattering • ii) Mol ie re mult iple scattering (small angle version of the screened Rutherford cross section w i t h screening effects assessed using the Fermi-Thomas atomic model.) • i i i) Bremsst rahlung photon product ion • iv) Pos i t ron annihi la t ion in flight and at rest • v) Col l i s iona l energy losses dur ing charged particle transport E l e c t r o n Shower Gene ra t ion T h e electron shower describes a process where an in i t i a l electron (+ or -) produces a cascade of secondary particles and photons. A s an electron traverses matter, i t w i l l undergo a process of continuous energy loss. Th i s energy loss may be due to radiative processes (bremsstrahlung photon generation) or coll isional processes (excitation or ionizat ion of local atoms). For ra-diative processes, the bremsstrahlung interaction w i l l result i n the or iginal electron (possessing a lower energy than before the interaction) plus a bremsstrahlung photon. Depending on the d is t r ibut ion of energy between the electron and the photon, the electron may continue to ion-ize or excite local atoms, or it may deposit the rest of its energy locally. T h e bremsstrahlung photon may go on to interact w i t h matter, producing daughter electrons and photons v i a the processes listed above. For coll isional processes, some of the.neighbouring atoms may absorb enough energy from the passing electron and become ionized. T h i s interaction w i l l produce a secondary electron (c5-ray) that may create its own path of exci ta t ion and ionizat ion, given enough energy. It is these cascading interactions that generates the electron gamma shower. Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 126 5.1.3 B E A M n r c Code (For radiotherapy source simulation) In response to a need to model radiat ion therapy sources for applications to 3D treatment planning, a col laborat ion between Canada 's Na t iona l Research C o u n c i l and the Univers i ty of Wiscons in was ini t ia ted. T h e project was called O M E G A (Ot tawa Mad i son E lec t ron G a m m a Algo r i t hm) and the B E A M n r c source s imulat ion code was published i n 1995 as part of this undertaking[128]. T h e B E A M n r c code accepts user input i n the form of radia t ion sources and linear accelerator physical components. T h e component geometry, composi t ion and relative positions are carefully constructed v i a a graphic user interface. To construct a medica l linear accelerator used to generate high energy x-ray photons, the following components are required (see Figure 5.1): • i) target and pr imary coll imator • ii) flattening filter • i i i) radia t ion monitor (ion chamber) • iv) mylar mirror • v) secondary col l imat ing jaws ( X and Y opposing directions) • vi) exit window w i t h reticule (optional) A d d i t i o n a l components can be added if required: • i) ter t iary collimators (e'.g. blocks, stereotactic radiosurgery coll imators, M L C ) • ii) compensators (e.g. wedges) E a c h component is created from a selection of pre-defined geometries, or component mod-ules ( C M ) . These modules are oriented in a plane perpendicular to the beam axis and can be customized based on the specifications of the part icular linear accelerator being modeled. Some typ ica l component modules are listed below: Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 127 target + ——— primary collimator flattening filter ion chamber—— mylar mirror • secondary jaws phase space — Figure 5.1: Linear accelerator construction model for V a r i a n 2 1 E X unit - photon mode (rendered i n B E A M n r c ) . Yel low dashed lines indicate simulated photon trajectories. • i) S L A B S (square boundaries of user-defined dimensions, can be layered and stacked - e.g. W / C u electron target) • i i ) C O N E S T A K (truncated cones of user defined radi i , can be layered and stacked e.g. p r imary coll imator) • i i i) F L A T F I L T (application of the C O N E S T A K used to facilitate design of conical flat-tening filters for photon beams) • iv) C H A M B E R (layers having cyl indr ica l symmetry used to facilitate design of moni tor ionizat ion chambers) • v) J A W S (models pairs of opposing surfaces or blocks - e.g. X - or Y - secondary jaws) Chapter 5: Monte Carlo Simulation of Particle Transport Through Matter 128 L i n e a r Acce le ra to r (linac) S imu la t i on T h e linear accelerator to be simulated is a V a r i a n Cl inac 2 1 E X located at the Vancouver Cancer Centre (see Figure 5.1). A 6 M V x-ray spectrum is generated using B E A M n r c by specifying an incident electron energy of .6 M e V (the electron energy is physical ly determined from the bending magnet current indicator i n the linac cabinet). The shape of the electron beam intensity d is t r ibut ion incident on the tungsten target (spot size) is also used to characterize the resulting x-ray spectrum. A Gaussian shape describes the electron beam spread and the fulh-width / half-m a x i m u m of this Gaussian is set to 0.75 m m . The energy and Gauss ian F W H M of the electron beam is adjusted dur ing the benchmarking phase of a M C simulat ion such that the result ing s imulated dose d is t r ibut ion to water matches the physical ly measured doses (ion chamber or diode). T h e rectangular radia t ion field size is defined by the secondary coll imators (jaw). T h e field size is specified at a distance 100 cm from the target (at the isocentre of most medical linear accelerators). A l l component properties i n the linear accelerator head, (target, p r imary col l imator , flattening filter, monitor chamber, mirror , jaws etc.), are obtained from manufacturer specification documentat ion. T h e B E A M n r c software is capable of t racking the primary, secondary and later generations of photons and electrons emerging from the target. T h e information that is stored includes photon/e lec t ron trajectories (direction, energy, locat ion of point of last interaction etc.) and the amount of energy deposited i n locations of interest (e.g. moni tor ion chamber). T h e photon/par t ic le data is scored at a plane perpendicular to the x-ray beam pr imary axis at a user-specified distance from the target. T h i s plane is called a phase space (see Figure 5.1). Phase Space Fi les and Scor ing Planes Scoring planes or phase spaces are v i r tua l particle counting planes located perpendicular to the beam axis at user-specified distances from the radiat ion source. T h e information recorded at each scoring plane includes the traversing particles' posi t ion (x-y location), energy, direct ion, charge (+ / - / 0), weight and point-of-origin. Scoring planes are often located at the end ) Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 129 of a chain of components that w i l l not change (e.g. after the mylar mirror , but before the secondary adjustable jaws). A scoring plane is always located at the very end of the entire component chain (e.g. after the tert iary coll imator - M L C , stereotactic, blocks). F igure 5.2 indicates two common locations for phase spaces. Phase space files can either be re-used by the B E A M n r c code (e.g. for changing tert iary coll imator settings), or it can be directed at a v i r t ua l patient or phantom for the purpose of calculat ing absorbed doses. Recording a particle 's his tory or point of origin can be a very useful tool . A t a given phase space, i t is possible to ascertain the dose contributions from different components of the linear accelerator (e.g. scatter from the jaws v.s. flattening filter). It is also possible to discover if a given particle has ever generated a bremsstrahlung photon or a secondary electron dur ing its travels and where that event took place. Energy spectra (Figure 5.3(a)), particle fluence vs. radia l posi t ion (Figure 5,.3(b)), and energy fluence vs. posit ion (Figure 5.3(c)) plots are also very useful when analyzing the properties of the radiat ion beam. P a r t i c l e Transpor t T h r o u g h Ter t i a ry C o l l i m a t o r s ( M L C ) T h e B E A M n r c code is capable of model ing expl ic i t ly the M L C composi t ion and geometry for the V a r i a n 120 leaf model used in this thesis. T h e V A R M L C static M L C component module was or iginal ly developed by K a p u r and M a and models the air gaps between leaves, the tongue-in-groove mechanism by which adjacent leaves slide against each other, and the d r iv ing screws at the top and bo t tom of each leaf used to open and close the leaves[129]. D y n a m i c M L C modes are accommodated by a new component module called D Y N V M L C , developed by Hea th and Seuntjens[130]. A n alternative to explici t Monte Car lo model ing was developed by K e a l l and Siebers at the V i r g i n i a Commonweal th University[131, 132] and is the method used for al l s imulations pre-sented i n this thesis. T h i s method transports the open-field phase space through the static or dynamic M L C very rapidly by making some simplifying assumptions about the M L C geometry and the physical interactions between the incident photons/electrons and the tungsten mate-r ia l . T h e weight of an incident phase space particle (photon) is recalculated based on photon at tenuation and first C o m p t o n scatter (scatter photon energy and scatter angle) properties of Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 130 -101 ' 1 1 1 -10' ' — ' 1 -10 -5 0 5 10 -10 -5 0 5 10 cm cm (c) (d) Figure 5.2: Phase Space planes are captured above and below the secondary jaws, (a) locat ion of phase plane above secondary jaws on linac model , (b) locat ion of phase space plane below jaws, (c) X - Y scatter plot of particle d is t r ibut ion above jaws, (d) Phase space d is t r ibut ion below jaws set to produce 10 x 10 c m 2 field at isocentre (100cm). Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 131 120 100 .•S- 80 60 •CC 40 20 511 keV peak from positron annihilation 2 3 4 Energy (MeV) (a) -2 0 2 Radial distance (cm) (b) -2 0 2 Radial distance (cm) •(c) Figure 5.3: Examples of information contained w i t h i n the phase space file (6 M V x-ray beam col l imated to 10 x 10 c m 2 at 100 cm.) (a) Pho ton energy spectrum. Note the 511 k e V peak originat ing from posi tron annihi la t ion events, (b) Relat ive photon planar fluence as a function of radial distance from the central axis, (b) Relat ive photon energy fluence. Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 132 tungsten for a given energy. The complex design of the M L C is segmented into simplified geometric regions which add together to provide pathlength information through the at tenuating tungsten mater ia l (see Figure 5.4). The probabi l i ty of a C o m p t o n scatter w i t h i n a sub-region depends on the tota l thickness, L, of M L C mater ial that the photon must traverse. If a C o m p t o n event is invoked, the locat ion of the event at thickness,t, is randomly sampled from 0 to L assuming exponential attenuation. T h e incident photon distance, t, and the C o m p t o n scatter photon distance, t' is calculated. E lec t ron transport (scattering and bremsstrahlung production) is ignored and this code should be used wi th caution when examining skin and near surface dosimetry as electron contaminat ion does contribute to this region. Pa i r p roduct ion events are also ignored. Leaf t ip and tongue-and-groove effects are effectively modeled even w i t h the simplif ied geometry. If the photon enters the M L C at X-pos i t i on between Xa and Xt2, a leaf-tip thickness calculator program is invoked. T h e thickness of at tenuating mater ia l depends on the entry and exit points of the photon along the arc of the leaf t ip . T h i s particle transport model is implemented despite the simplifying assumptions listed above. T h e reason is that it is fast (only requir ing a few minutes per field), especially when compared to a full Mon te Ca r lo s imulat ion. Benchmark ing tests (see Section 5.2.3) demonstrate that this model can accurately model the mult i - leaf col l imator , even for dynamic beam deliveries (beam on while the M L C moves). T h e process of integrating the complex dynamic M L C patterns into the particle transport Region 1 Region 2 no interaction compton scatter Figure 5.4: Schematic of M L C leaf segmentation into two regions. V i r g i n i a Commonwea l th Univers i ty M L C model calculates thickness of leaf mater ial traversed by an incident photon and calculates probabi l i ty of a single compton scatter. Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 133 code starts by reading the leaf t ip positions i n from the l inac M L C control file. T h e leaf positions i n the file are a projection to 100 cm from the target. These projections are demagnified back up to the level of the M L C at 45 cm from the target. The delivery index associated w i t h each M L C control point establishes a relationship between the leaf posi t ion and the monitor units j delivered. A phase space located below the secondary jaws, but above the M L C (at 45 cm), is now ready to be transported through the M L C model . T h e exi t ing phase space is modula ted by weighting the incident phase space according to the probabi l i ty that a leaf (or leaf t ip) is block-ing the passage of a given particle. The probabi l i ty function is obtained from the file containing the M L C leaf positions and the delivery index. Th i s method works for bo th step-and-shoot and dynamic M L C delivery modes. 5.1.4 DOSXYZnrc Code (For absorbed dose in matter) Simi la r to the B E A M n r c code for s imulat ing photon/electron interactions w i t h i n the various components of a medical linear accelerator, the D O S X Y Z n r c code is a EGSnrc -based Monte Ca r lo s imula t ion package. The intent is to calculate absorbed radiat ion doses w i t h i n a 3D voxel-based phantom containing I C R U (International Commiss ion on Rad ia t i on Uni ts ) defined media and their interaction cross-sections. The phantom can be manual ly constructed or gen-erated from computed tomography electron density data. Rad ia t ion beams are directed at the phantom by defining the distance between the v i r tua l isocentre of the l inac, the locat ion of the phase-space containing the photon/electron properties, and the angle of incidence relative to the pa t i en t /phan tom axis. T h e radiat ion beam data may be generated from published spectra or from pre-simulated phase-space files as described i n Section 5.1.3. T h e same photon inter-act ion probabi l i ty distr ibutions used for the linear accelerator s imulat ion are applied to the pa t ien t /phan tom dose deposit ion simulat ion. Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 134 R e p o r t i n g D o s e - t o - M e d i u m (Dm), or D o s e - t o - W a t e r - i n - M e d i u m (Dw) Tradi t ional ly , dose calculat ion algorithms were developed to calculate the dose to water, Dw. T h i s is reasonable given that 50 - 65% of the human body is comprised of this medium. Treat-ment machine dosimetry cal ibrat ion protocols (e.g. T G - 5 1 , A A P M ) specify a reference dose to water (e.g. 1 monitor unit ( M U ) = 1 c G y of dose to dmax). T h e commissioning of most commercia l treatment p lanning systems ( T P S ) requires extensive radiat ion dose measurements (profiles, depth doses, output factors) to be acquired i n water or a water-equivalent medium. T h e data is ut i l ized by the dose calculat ion algori thm specific to the T P S . T h i s methodology opens up the issue of dealing wi th dose perturbations caused by tissues in the body that are not radiological ly equivalent to water (e.g. lung, air, cort ical bone). T h e effect of.these relatively l o w / h i g h density media on the dose calculat ion a lgor i thm and how to correct the calculated doses to compensate for this effect is described i n Section 3.1.6. Mon te Ca r lo simulations calculate dose to the medium (Dm) specified w i t h i n each voxel , not the dose-to-water-in-medium (Dw). The common default media specified i n the s imula t ion are tissue, bone, air and lung, however any number of media may be used. To convert from Dm to Dw a rat io of stopping powers must be applied s A" a t ° r • For soft tissues, the difference in the value for dose-to-water and dose-to-medium is ~1%[133]. In cort ical bone, this difference may reach 11.7% for beam energies in the range of 4 - 2 4 M V . The effect of calculat ing Dm vs. Dw on c l in ica l head/neck and prostate plan D V H s has been recently reported by Dogan et al. [134]. The mean differences (Dw-Dm) i n the 98% P T V volume dose coverage for the head/neck plans and the prostate plans are 1.6% and 0.0% respectively. The mean differences between the cr i t i a l structure mean doses for the head/neck plans (cord) and prostate plans (rectum) are 1.4% and 0.4% respectively. T h e largest dose differences between the Dm plans and the Dw plans occurs in the bony femoral heads for the prostate examples (Dw plans report a mean dose 5.1% higher than the Dm plans). It seems that for every argument that the dose-to-medium (Dm) should be converted to dose-to-water-in-medium (Dw), a counter-argument exists[135]. A summary of arguments for and against the use of Dm over Dw is given i n Table 5.1. T h i s is a very current and controversial topic. The Monte Car lo doses reported i n this thesis are dose-to-medium. Chapter -5.. Monte Carlo Simulation of Particle Transport Through Matter 135 Table 5.1: Argument for calculat ing Dose-to-Water (Dw) versus Dose -To-Med ium (Dri Dose-to-Water (Du Dose-to-Medium (Dm) Consistent w i t h A A P M T G - 5 1 cal ibrat ion protocols Current dose prescriptions based on experience w i t h Dw M a n y biological indices ( T C P , N T C P ) based on Dw values. Represents true dose to tissue Conversion to Dw introduces uncertainty -defeats purpose of using Mon te Ca r lo C l i n i c a l impact of using Dm over Dw not significant D.m doses not ideal for comparison w i t h other T P S ' Dw main ly a historical appl icat ion motivated by a lack of accurate heterogeneity corrections 5.1.5 Variance Reduction Monte Ca r lo is inherently a statistics-based method. The ul t imate goal is to be able to rapid ly simulate doses w i t h h igh resolution (small voxel sizes) and low uncertainty. H i g h levels of s tat is t ical noise w i l l produce undesirable results when dealing w i t h radia t ion therapy treatment planning. Deasy identified several effects of Monte Car lo noise i n this context including[104]: 1. uncertainty in the est imation of relevant dosimetric and radiobiological indices (e.g. D V H , T C P , N T C P , max or m i n dose)[136, 137, 138, 139] 2. degradation of isodose plots (noisy and broken lines) assessed by the radia t ion oncologist 3. uncertainty i n the beamlet doses used for inverse treatment p lanning result ing i n an op-t imiza t ion convergence error[140, 141] T o avoid these issues, the most obvious way to improve statistics i n a voxel is to increase the number of photons/electron histories simulated. The uncertainty (standard deviat ion, o) in. a voxel is inversely proport ional to the square root of the number, N, of histories or simulations that accumulates i n the voxel ( a oc 1/y/N ). T h i s can be done at the expense of computa t ion t ime which is proport ional to N. Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 136 Al ternat ively , the number of histories per voxel w i l l increase as the x3, where x is the length of one size of the symmetr ic voxel. Therefore the uncertainty is proport ional to 1 /x 3 / 2 . T h e disadvantage of increasing the voxel size, is a loss in spatial resolution. Several vvariance reduction techniques are available w i th in the E G S n r c code that can increase the speed of the simulations whilst mainta ining good statistics. For photons, these techniques include, i) bremsstrahlung spl i t t ing and ii) photon forcing. For charged particles, techniques include, i) Russ ian roulette and ii) range rejection[142, 102, 143]. • i) B r e m s s t r a h l u n g S p l i t t i n g : this technique increases the number of bremsstrahlung photons created at a bremsstrahlung electron interaction site. E a c h ' spl i t ' photon w i l l have a weight inversely proport ional to the total number of photons generated. • i i ) R u s s i a n Rou le t t e : is applied to charged particles and is typica l ly used i n association w i t h bremsstrahlung spl i t t ing. The number of post-bremsstrahlung-interaction electrons followed by the Monte Car lo software is reduced to the number that would have been followed had the 'bremsstrahlung spl i t t ing ' opt ion not been implemented. Electrons are e l iminated by assigning a random weight to the 'e lec t ion and compar ing to a ' survival threshold ' weight. Electrons exceeding this threshold are el iminated. Surv iv ing electrons increase i n weight un t i l one remains. • i i i ) P h o t o n Forc ing : photons are forced to interact w i t h i n a specified component module. Th i s is usually applied in regions where there are sparse interactions due to low density or t h in geometric cross-section. Th i s technique is useful for investigating the source of contaminant electrons i n a photon beam when s imulat ing a linear accelerator head. T h e number of forced interactions for.a given photon can be specified. • iv) Range Re jec t ion : this technique is applied to charged particle transport . T h e range of a given particle i n a medium is projected based on the particle energy and the restricted s topping power of the medium. If calculations indicate that the particle w i l l not escape from the current region (assuming coll isional losses only) , the t racking of that particle 's his tory is discontinued. It is assumed that if any bremsstrahlung photons are generated by the particle 's interaction w i t h matter, these photons w i l l not escape the region. Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 137 T h e last method discussed here is the use of 3D digi ta l filters that smooth arid denoise Monte Car lo da ta obtained w i t h less than opt imal stat ist ical uncertainty (> 2%). T h e objective is to decrease Monte Car lo calculat ion times by requiring fewer histories to be simulated. T h e addi t ion of a denoising filter. reduces the statist ical uncertainty of the data to a useable level. Var ious filters have been explored and most operate on random noise only. These filters include Gauss ian smoothing filters[104], adaptive noise reduction tool (ANRT)[144] , i terative reduction of noise ( I R O N ) [145] and the Savi tzky-Golay (local least-squares) filter[104, 103]. Kawrakow's local ly adaptive version of the Savi tzky-Golay filter[103] has been implemented i n this thesis and is discussed i n detail i n Chapter 7. 5.1.6 Distributed Computing T h e nature of Monte Ca r lo s imulat ion lends itself very well to dis t r ibuted (or parallel) com-put ing. T h e user w i l l specify how many particles (histories) to run and each particle is tracked indiv idual ly . Typica l ly , a complete linear accelerator photon beam s imulat ion w i l l require ~1,000,000,000 histories for good statistics (many become attenuated in the flattening filter or are scattered out of the pr imary beam). A t a s imulat ion speed of 0.0005 s /h is tory on an A M D Opteron 2100 2.8 G H z processor, 1,000,000,000 histories through an entire l inac would take 139 hours to simulate. These large number of histories can be broken up into smaller groups (e.g. 10 computers calculate 100,000,000 histories each over 13.9 hours). T h e cont r ibut ion from each computer can be summed together to achieve the results for the entire s imulat ion. Th i s k i n d of dis t r ibuted comput ing requires a dedicated cluster of computer processors a l l being coordinated by a central scheduling system. T h e majori ty of this thesis has been simulated on three P e n t i u m 4 computers ( x 2 processors each) operating on a L i n u x operating system plat-form. E a c h of the 6 processors operates at a speed of 3.2 G H z . T h e "Condor" queueing software determines which processors are available to receive a chunk of data for s imulat ion. T i m e gains can also be achieved by reusing data that does not change from simulat ion-to-simulat ion. For example, i n a l inac head, the only moving components are the secondary asymmetric jaws and the ter t iary coll imators (e.g. mult i leaf col l imator) . A phase space can be recorded just above the jaws and used as a s tar t ing point for any future simulations (i.e. jaw size changes). Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 138 5.2 Benchmarking the Monte Carlo System For Mon te Car lo simulations to be useful in a radiat ion therapy context, the entire v i r tua l system must be benchmarked (verified) against measured data obtained from the actual linear accelerator being modeled. The ul t imate goal is to obtain accurate dose deposit ion information inside a patient or patient-like mater ial (phantom). A l l Mon te Car lo simulations described i n this thesis were generated by the author. The measured data used to benchmark the Monte Ca r lo simulations is a combinat ion of archived measurement data acquired by c l in ica l medical physicists plus supplemental ion chamber and film data acquired by myself. T h e linear accelerator (linac) simulated i n this thesis is a V a r i a n 2 1 E X model (Palo A l t o , C A ) . A l t h o u g h the linac is capable of creating a 6 and a 18 M V x-ray photon beam plus a 4, 6, 9, 12, or 15 M e V electron beam, only the 6 M V photon x-ray beam is modeled for this project. The v i r t ua l accelerator is constructed manual ly from ind iv idua l , contiguous components along the beam pa th (see Section 5.1.3). T h e exact geometry and mater ial composi t ion of each component is specified by the manufacturer. The v i r tua l x-ray beam is generated by direct ing a megavoltage (6 - 25 M e V ) electron beam onto a tungsten target. B o t h the energy and the Gauss ian spread of the incident electron beam can be adjusted by the user. Set t ing the electron beam energy and Gauss ian spread parameters to obtain a perfect match w i t h measurements can be a very painstaking, i terative process. The E G S n r c code w i l l track the electron beam interactions w i th in the target and the components along the beam path to generate the characteristic properties (spectrum and fluence / energy distr ibution) of the x-ray beam (see Figure 5.3 for examples). Defining the properties of the incident electron beam is a cr i t ical step i n matching a Monte Ca r lo s imulated linear accelerator to the real c l in ical accelerator being used to treat patients. T h e B E A M n r c code offers some options to describe the incident electron beam (parallel beam, point source, circular beam w i t h Gaussian dis t r ibut ion etc.). T h e electron beam used in this thesis is defined i n B E A M n r c as having an energy of exactly 6.0 M e V and a circular, Gauss ian intensity d is t r ibut ion . T h e ful l -width hal f -maximum ( F W H M ) of the electron beam is set to 0.75 m m . T h e dosimetric properties of the resulting v i r tua l x-ray beam (percentage depth doses ( P D D ) Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 139 and off-axis rat io ( O A R ) dose profiles) are compared to ion chamber measurements i n water (for a review of the P D D and O A R concept, see Section 3.1.1). These measurements involve acquir-ing dose profiles bo th i n a direction parallel to the beam ( P D D ) and perpendicular to the beam ( O A R ) i n a water phantom using a IC10 0.14cc waterproof ionizat ion chamber (Scanditronix-Wellhofer, Uppsa la , Sweden - see Figure 3.13). The measurements used to benchmark the V a r i a n 2 1 E X v i r t ua l linear accelerator are obtained from archived data acquired by medical physics staff dur ing c l in ica l commissioning of the real linear accelerator at the Vancouver Cancer Centre ( V C C ) i n 2004 ( V C C linac name = Unit 7). 5.2.1 Open Field Verification Measured dose profiles i n water are obtained from cl inical commissioning data for a 6 M V photon beam col l imated to produce three different field sizes ( 3 x 3 c m 2 , 10 x 10 c m 2 , and 40 x 40 c m 2 ) . T h e source-to-surface-distance (SSD) is set to 100 cm and the field sizes are defined at 100 cm. T h e v i r tua l Monte Car lo linear accelerator also generated photon beams for the same field sizes. T h e phase space from these beams are directed onto a uniform water phantom (having the same set-up geometry as the measured data) and the dose d is t r ibut ion simulated using D O S X Y Z n r c . A diagram of the set-up geometry is shown i n Figure 3.1. Figure 5.5 compares measured and Monte Car lo dose profiles i n a direct ion paral lel to the beam axis (percentage depth dose - P D D ) for three different field sizes (3, 10 and 40 x 40 c m 2 ) . T h e da ta is normalized to 100% on the central axis at a depth of m a x i m u m dose deposit ion (depth = 1.5 cm for a 6 M V photon beam). A good match along this axis confirms that the incident electron energy on the tungsten target specified i n B E A M n r c is correct. F igure 5.6(a),(b) and (c) compares measured and Monte Car lo cross-beam profiles for three field sizes (3 x 3, 10 x 10, and 40 x 40 c m 2 ) . The profiles for three depths (1.5, 10.0 and 30.0 cm) are compared. D u r i n g the Monte Car lo s imulat ion, a v i r tua l electron beam strikes a v i r t ua l x-ray target. If the Gaussian w i d t h of the electron beam has been correctly specified, good agreement between the Monte Car lo calculat ion and measurement i n the penumbra region (field edges) w i l l be achieved. Agreement w i th in error is observed for the 3 x 3 c m 2 , 10 x 10 c m 2 , and 40 x 40 c m 2 fields for a l l depths. Chapter. 5. Monte Carlo Simulation of Particle Transport Through Matter 140 10 0 0 3 x 3 cnrf 10 x 10 c m 2 40 x 40 c m ' 10 Depth (cm) 15 20 Figure 5.5: Percentage depth dose curves for three different fields sizes: 3 x 3 , 1 0 x 1 0 and 40 x 40 c m 2 . Sol id line: ion chamber measured data. Dot ted Line : Monte Car lo s imulated data. Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 141 -3 -2 -1 0. 1 2 3 Off-Axis. Distance(cm) 1,5 cm •aS^at^EJiv .. . • 100 cm-» • 30.0 cm -5 -2.5 0 2.6 5 Off-Axis Distance (cm) (a) (b) -20 -10 0 10 20 Off-Axis.Distance (cm) — 70 & 60 • c ; a 40 4) 0 . 30 20 10 f -T40x40cm'; "1.0x10cm; 3x3cm* -36 -30 -26 -20 -15 -10 -5 0'. 5 10 15 20 25 ,30 35 Off-Axis Distance(cm) (c) (d) Figure 5.6: Off-axis profile curves for three different depths in phantom (1.5 cm, 10.0 cm and 30.0 cm). Different field sizes shown: (a) 3 x 3 c m 2 , (b) 10 x 10 c m 2 and (c) 40 x 40 c m 2 . Sol id line: ion chamber measured data. Do t t ed Line : Monte Ca r lo s imulated data, (d) O A R comparison for three different field sizes at depth 1.5 cm. Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 142 5.2.2 Absolute Dose Conversion / Calibration of Virtual Linac T h e M C simulat ion of the linear accelerator is divided into two steps (Figures 5.7(a) and (b)). T h i s is to facilitate absolute dose calculations using a technique described by Popescu et al.(2005) [146]. T h i s paper addressed the question: "How do you calibrate a v i r tua l linear accelerator?". T h e components involved i n the two steps of the linac s imulat ion are as follows: 1. P r i m a r y B e a m and F o r w a r d Scat ter C o n t r i b u t i o n to M o n i t o r C h a m b e r : (Figure 5.7(a)) T A R G E T , P R I M A R Y C O L L I M A T O R , F L A T T E N I N G F I L T E R , M O N I T O R C H A M -B E R and M Y L A R M I R R O R A phase space plane located perpendicular to the beam axis (Phasespace-A) is located a t . the end of this chain, just above the secondary col l imator (movable x / y jaws). T h e phase space records the charge, energy, x-y location, and direct ion cosines of any particle that passes through i t . Note that since the secondary jaws have not been included i n this s imulat ion, the monitor chamber records no backscatter dose contr ibut ion from this component. O n l y the forward dose produced by particles or iginat ing from the target is recorded. 2. Backsca t t e r C o n t r i b u t i o n to M o n i t o r C h a m b e r : (Figure 5.7(b)) M O N I T O R C H A M B E R , M I R R O R and S E C O N D A R Y C O L L I M A T O R S PhaseSpace-A is used as input to a second s imulat ion that now includes the adjustable secondary coll imator back up to the monitor chamber. The secondary col l imator jaw sizes are set to match the cl inical field sizes. Phasespace-B is located at, a distance of 45 cm from the electron target (at the level of the multi- leaf col l imator) . T h e backscatter dose contr ibut ion to the monitor chamber is recorded and provides important information for absolute dose calculations. Essentially, the effect of different jaw settings can be assessed by compar ing the backscatter components alone. Note that Phasespace-A needs only to be calculated once for al l patients and fields (using the 6 M V photon beam) as there are no adjustable component settings. Phasespace-B is generated Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 143 "—target -—primary collimator flattening filter —-monitor chamber •mylar mirror _ PHASE SPACE/ * monitor chamber -—mylar mirror ^secondary jaws • • PHASE SPACEB (b) (a) F igure 5.7: M C model of the Var i an C L 2 1 E X linear accelerator head (a) Group A - target, p r imary coll imator , monitor chamber, mylar mirror , and Phasespace-A. (b) G r o u p B - monitor chamber, mylar mirror , secondary col l imat ion (jaws) and Phasespace-B. for every secondary col l imator (jaw) setting. T h e ul t imate goal is to determine the amount of dose absorbed i n the monitor chamber per moni tor unit such that 1 M U w i l l deliver 1 c G y to the patient at dr:mx for a 10 x 10 c m 2 field size. A B E A M n r c s imulat ion is performed for the 10 x 10 c m 2 field. T h i s beam is projected onto a uniform phantom and the Monte Car lo dose is simulated using D O S X Y Z n r c . T h e cal ibra t ion geometry for the v i r tua l l inac is S A D = 100 c m and depth = dmax. T h e raw dose from D O S X Y Z n r c is in units of dose absorbed per particle incident on the target. If the cal ibra t ion condi t ion is enforced (1 M U = 1 c G y at dmax), Popescu et a/.[146] showed that this ca l ibra t ion can be achieved for 8.129 x 1 0 1 3 ± 1.0% electrons incident on the target and a tota l dose of 20.87 c G y dfc 1.0% i n the monitor chamber for a 6 M V beam. T h e conversion factor from dose per particle incident on the target (Gy/particle) to dose (Gy) for a given field size is achieved by applying the following equation [146]: D xyz,abs = D (Df—d + DT(10 x 10)) D ^ a b s Tl/Z (Df—d + DT(fs)) T~)c.al xyz U (5.1) Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 144 where • DXyz = relative dose to voxel of interest (dose p e r p a r t i c l e i n c i d e n t on target (Gy/particle)) • U = moni tor units delivered (1 M U = 1 c G y at d r n a x ) • u^°J"ward — forward dose to monitor chamber from components above the jaws • D ^ , f l c k (10 x 10) = backscatter dose to monitor chamber for a 10 x 10 c m 2 field size (reference condit ion) • Drflck(fs) = backscatter dose to monitor chamber for a given field size of interest • DcxayZabs = absolute dose to phantom (reference condi t ion — 1 c G y at d m a x ) • Dxyz = relative dose to phantom under cal ibrat ion conditions (dose p e r p a r t i c l e i n c i d e n t on target) T h i s ca l ibra t ion is applied to a simple open field, 10 x 10 c m 2 geometry (see Figure 5.8). T h e isocentre is at a depth of 10.0 c m and the number of monitor units ( M U ) delivered is 100. F igure 5.8: (a) Dose d is t r ibut ion for 10 x 10 c m 2 6 M V x-ray field incident on a cylinder phantom. D e p t h to isocentre = 13.5 cm. 100 M U delivered, (b) P D D profile. Errorbars: Monte Ca r lo absolute dose data. Sol id line: treatment p lanning system da ta (Var ian E c l i p s e ® P B K algori thm). Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 145 5.2.3 Multi-leaf Collimator (MLC) Module for I M R T T h e V a r i a n 120-leaf Mul t i - l ea f Col l imator ( M L C ) is a complicated component to simulate, es-pecial ly since the tungsten leaves can be moving while the beam is turned on. E a c h bank has 60 tungsten leaves. T h e central 20 cm of the M L C is comprised of 0.5 c m wide (perpendicular to leaf motion)leaves. T h e outer 20 leaves have a w id th of 1 cm. T h e V i r g i n i a Commonwea l th Univers i ty ( V C U ) has developed code (writ ten in the C programming language) that w i l l trans-port an open field phase space file through the dynamic M L C by generating probabi l i ty maps of a par t icular M L C leaf's posit ion. T h e probabi l i ty map is based on the data contained w i t h i n the s tandard M L C control file. A discussion of the V C U method is found i n Section5.1.3. S l i d i n g W i n d o w Test w i t h Ion C h a m b e r Ver i f i ca t ion A 6 M V dynamic s l iding window test is used to benchmark the M L C output. T h e M L C leaves are open to a set w id th , essentially leaving a slit of open radiat ion beam. T h e unblocked area is then s l id across the active beam. T h e dose to a P T W farmer-type cy l indr ica l ion chamber (0.6 c m 3 active volume) is integrated as the sl iding window moves across the field (see Figure 5.9). The ion chamber is located at a depth of 5.0 cm i n a solid water phantom. Four different slit widths were delivered: 1 m m , 5 m m , 10 m m , and 90 m m . T h e measured and simulated absolute doses are compared in Table 5.2. Table 5.2: Ion Chamber Measurement vs Monte Car lo : In i t ia l S l id ing W i n d o w Test W i d t h of Sl id ing W i n d o w (mm) Ion Chamber Dose (cGy) Monte Car lo Dose (cGy) % Difference 1 4.87 ± 0.10 4.51 ± 0.05 7.66 5 9.18 ± 0.18 8.82 ± 0.09 3.97 10 15.17 ± 0.30 14.87 ± 0.18 1.99 90 52.09 ± 1.04 52.15 ± 0.52 -0.12 Clearly, there is a problem w i t h the Monte Car lo s imulat ion of these dynamic fields. A closer look at the particle transport C code shared by the V i r g i n i a Commonweal th University[132] revealed that there is an adjustable parameter labeled physical leaf gap. In the unaltered code, this variable is set to 0.008 cm. It is not surprising that the transport code needs to be cal ibrated to match a specific l inac. E v e n on the actual c l in ical linacs themselves, smal l variations i n the Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 146 S A D 100cm /, sliding window' width integrating ion chamber,.. solid water; Figure 5.9: M L C leaves define an open window. The window is s l id across the open beam deposit ing dose i n an integrating ion chamber. W i n d o w widths = 1, 5, 10, and 90 m m . M L C cal ibra t ion means that the identical, M L C control file delivered by bo th machines w i l l not produce the same measured dose dis t r ibut ion. T h e physical leaf offset parameter was changed systematical ly and the open field phase space re-transported through the M L C control file for the 1 m m sl id ing gap example. The dose calculated to a depth of 12.8 cm on the central 'axis of the beam is recorded for each offset value. A plot of the offset value versus dose is shown i n Figure 5.10. A linear relationship is observed. F r o m this data, it was determined that the particle transport code physical leaf offset value should be set to 0.0175 cm (0% difference between Monte Car lo calculated'dose and measured dose for 1 m m sliding window test). The variable s l iding window test is repeated w i t h the new leaf offset and the Monte Car lo simulated doses compared to the measured doses i n Table 5.3. Chapter 5.. Monte Carlo Simulation of Particle Transport Through Matter 147 0.04 0.06 leaf offset ( cm) Figure 5.10: Relat ionship between Monte Car lo dose calculat ion error (compared to measure-ment) and the Physical Leaf Offset setting i n V C U particle transport code. Linear regression fit to data. Table 5.3: Ion Chamber Measurement vs Monte Car lo : S l id ing W i n d o w Test W i d t h of Sl id ing W i n d o w Ion Chamber Dose Monte Car lo Dose % Difference (mm) . (cGy) (cGy) .1 4.87 ± 0.10 4.87 ± 0.05 0.00 5 9.18 ± 0.18 9.25 ± 0.11 -0.71 10 . 15.17 ± 0.30 15.18 ± 0.17 -0.09 90 52.09 ± 1.04 52.53 ± 0.94 -0.84 B a r P a t t e r n Test w i t h 2D F i l m Ver i f i ca t ion A second benchmarking test verifies two M L C control files, each containing three bar patterns of varying peak-to-peak distances (see Figure 5.11). The first file contains bar patterns having peak-to-valley distances of 30 m m , 20 m m , and 10 m m (Figure 5.11(a)). The second file contains bar patterns w i t h 7.0 m m , 5.0 m m , and 3.5 m m peak-to-valley distances (Figure 5.11(b)). These M L C files are delivered on the cl inical V a r i a n 2 1 E X linear accelerator using a ' 6 M V photon beam. A 2D coronal film plane perpendicular to the beam axis is posit ioned at a depth of 12.8 cm i n an A V I D ( M D X , Vancouver, B C ) solid water phantom. The film was cal ibrated using three P D D cal ibra t ion films (100 cm S A D , 12.8 cm depth) each corresponding to a different M U setting (50 Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 148 I 1 1 I I 1 f l f i f i i W inutility 1 1 1 1 1 • I B l W Wr mm -80-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70 80 mm (a) -80-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70 mm 00 Figure 5.11: 2D film plane of bar pat tern M L C delivery. 6 M V , S A D 100, depth = 12.8 cm. (a) Peak-to-valley bar widths: 30 m m , 20 m m , 10 m m (b) B a r widths: 7.0 m m , 5.0 m m , 3.5 m m . M U , 100 M U , 200 M U ) . The h i m is processed on a K o d a k 3000 R A film processor on standard cycle (developer temperature = 34 .5°C) . T h e dose to phantom from the same two M L C control files are simulated using Monte Car lo (6 M V photon beam). T h e relative doses are converted to absolute dose using the technique described i n Section 5.2.2. The phantom voxel size is 2.5 m m 3 . For comparison, the dose to phan tom is also calculated using the single penci l beam kernel (as implemented in the Var i an E c l i p s e ® v . 6 . 5 treatment planning software). A comparison of the film measured, P B K calcu-lated and Monte Ca r lo s imulated dose distr ibutions for one bar pattern delivery is shown i n F igure 5.12. T h e bar peak-to-valley widths shown i n this example are 7.0 m m , 5.0 m m , and 3.5 m m . Profiles through the dose planes are compared i n Figures 5.13 and 5.14. T h e V C U M L C particle transport code w i l l model interleaf leakage which many non-Monte Ca r lo methods are unable to do. Figure 5.15(a) shows the locat ion of a line profile acquired across the M L C s , i n a direct ion perpendicular to the leaf motion. Accompany ing this image is the line profile da ta comparing the film measured profile to the penci l beam kernel ( P B K ) calculated profile (Figure 5.15(b)). The P B K algori thm has no mechanism for model ing the M L C interleaf leakage. A n average leaf transmission is used to calculate dose. Figure 5.15(c) compares the film measured profile to M C simulat ion. T h e M C simulat ion was able to reproduce the peaks-and-valleys indicat ing M L C interleaf leakage lines. Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 149 Figure 5.12: Comparison of 2D plane for an M L C bar pattern (a) film measured doses, (b) Pencil Beam calculated doses, (c) Monte Carlo simulated doses. 6 M V photon beam, S A D 100 cm, depth = 12.8 cm. Bar widths = 7.0, 5.0 and 3.5 mm. Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 150 3% ,p a a n i C PO 30 fJNLJ -80-70-60-50-4O-30-20-10 0 10 20 30 40 50 60 70 80 mm -80-70-60-50-40-3a20-10 0 10 20 30 40 50 60 70 80 mm -80-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70 80 (a) (b) (c) (d) (e) (f) F igure 5.13: 2D film plane of bar pattern ( > lcm bar width) M L C delivery. 6 M V , S A D 100 cm, depth = 12.8 cm. B a r widths = 30 m m , 20 m m , 10 m m (a),(b), and (c) respectively. Profile locations are shown as a dashed horizontal line. Profiles comparing film measurement (black solid line), P B K convolution (blue dashed line), and Monte Car lo s imulat ion (red dashed line w i t h error bars) drawn through (d) 30 m m , (e) 20 m m , and (f) 10 m m bar pattern widths. Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 151 • f e - 4 0 -30 -20 "10 T 1 0 2 0 H I "40 : " i f - f e " 4 4 1 4 - 1 0 0 1 0 20 30 40 K> -fe - 4 0 - 3 0 - 2 0 - 1 0 0 1 0 20 " 30 40 50 mm mm mm (d) (e) (f) F igure 5.14: 2D fi lm plane of bar pattern (< 1cm bar width) M L C delivery. 6 M V , S A D 100 cm, depth = 12.8 cm. Peak-to-valley bar widths = 7.0 m m , 5.0 m m , 3.5 m m (a),(b), and (c) respectively. Profiles comparing film measurement (black solid l ine), P B K convolut ion (blue dashed line), and Monte Car lo s imulat ion (red dashed line w i t h error bars) drawn through (d) 7.0 m m , (e) 5.0 m m , and (f) 3.5 m m bar pat tern widths. Chapter 5. Monte Carlo Simulation of Particle Transport Through Matter 152 (a) Film measured Monte Carlo (b) (c) Figure 5.15: (a) Dose d is t r ibut ion resulting from delivering a bar pat tern (< lcm bar width) fluence. L ine indicates locat ion of profile data perpendicular to leaf travel direc-t ion, (b) Profile data across the M L C leaves showing penci l beam kernel ( P B K ) a lgor i thm (blue dashed line) agreement w i th measured (black solid line) interleaf leakage. Note that the P B K algori thm does not model interleaf leakage at a l l . (c) Monte carlo s imulat ion profile (red dashed line w i t h error bars) compared to film measured profile (black solid line). Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 153 Chapter 6 Monte Carlo Input for Direct Aperture Optimization for IMRT 6.1 Prologue T h i s work introduces an EGSnrc-based Monte Car lo ( M C ) beamlet dose d is t r ibut ion ma t r ix into a direct aperture opt imiza t ion ( D A O ) a lgor i thm for I M R T inverse planning. T h e technique is referred to as Monte Car lo - Direct Aper ture Op t imiza t ion ( M C - D A O ) . T h e goal is to assess if the combinat ion of accurate Monte Car lo tissue inhomogeneity model ing and D A O inverse p lanning w i l l improve the dose accuracy and treatment efficiency for treatment planning. Several authors have shown that the presence of smal l fields and /or inhomogeneous materials i n I M R T treatment fields can cause dose calculat ion errors for algorithms that are unable to accurately model electronic disequilibrium[3, 4, 5]. Th i s issue may also affect the I M R T opt imiza t ion process because the dose calculat ion algori thm may not properly model difficult geometries such as targets close to low-density regions (lung, air etc.). A c l in ica l linear accelerator head is simulated using B E A M n r c ( N R C , Canada) . A novel in -house a lgor i thm wr i t ten in C code subdivides the resulting phase space into 2.5 x 5.0 m m 2 beam-lets. E a c h beamlet is projected onto a patient-specific phantom. T h e beamlet dose contr ibut ion to each voxel i n a structure-of-interest is calculated using D O S X Y Z n r c . T h e mult i leaf col l imator ( M L C ) leaf positions are l inked to the locat ion of the beamlet dose distr ibutions. T h e M L C shapes are opt imized using direct aperture opt imizat ion ( D A O ) . A final Monte Car lo calculat ion w i t h M L C model ing is used to compute the final dose dis t r ibut ion. Monte Car lo s imula t ion can generate accurate beamlet dose distr ibutions for t radi t ional ly difficult-to-calculate geometries, par t icular ly for smal l fields crossing regions of tissue inhomogeneity. T h e in t roduct ion of D A O Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 154 results i n an addi t ional improvement by increasing the treatment delivery efficiency. For the examples presented i n this s tudy the reduction i n the to ta l number of monitor units to deliver is ~ 3 0 - 50% compared to fluence-based opt imizat ion methods. A large por t ion of this chapter has been publ ished i n the peer-reviewed journal , Medical Physics (Bergman et al., "Direct aperture op t imiza t ion for I M R T using Monte Car lo generated beamlets",MeciP/iys. ,33(10):3666-3679, 2006) [33]. 6.2 Introduction T h e nature of intensity modulated radiat ion therapy ( I M R T ) op t imiza t ion is such that smal l beam segments defined by the mult i leaf coll imator ( M L C ) are used to generate photon fluence dis tr ibut ions. The goal is to conform volumes of high-dose deposit ion to complex planning targets while sparing healthy tissue. These targets may be i n close p rox imi ty to sensitive structures such as the parot id glands, brain stem, spinal cord and / or opt ica l apparatus. Treatment beams may be forced to cross regions of tissue inhomogeneity such as lung, or sinus air cavities found i n the head and neck region. T h e presence of smal l treatment fields and/or inhomogeneous materials has been found to cause dose calculat ion errors for algorithms that are unable to model lateral electronic disequi l ibr ium [3, 4, 5, 147]. M a n y I M R T models use the concept of opt imiz ing the weight of smal l subfields or beamlets of radia t ion such that the to ta l contr ibut ion of a l l the beamlets produces the desired dose dis t r ibut ion. It has been shown that smal l fields applied to inhomogeneous materials challenge t rad i t iona l dose calculat ion algorithms (e.g. single penci l beam kernel ( P B K ) convolution). If the dose calculat ion accuracy is i n question, it is expected that the I M R T opt imiza t ion process would also be compromised. It is important to dist inguish the difference between a dose calculat ion (or systematic) error and a convergence error. Systematic errors are due to inaccuracies in the dose calculat ion a lgor i thm compared to a gold standard (e.g. measurement or Monte Ca r lo simulations). T h e presence of such systematic dose errors leads to a convergence error, where the opt imiza t ion process converges at a subopt imal solution. [7, 148]. Convergence errors may cause plans to appear as though the planning target volume ( P T V ) is covered Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 155 adequately dur ing the opt imiza t ion step. However, these opt imized plans may be revealed as having a P T V under-dosage (or other dose coverage / sparing issues) once the final forward calculat ion is performed using an accurate dose a lgor i thm (e.g. Monte Car lo) [8, 9]. Mon te Ca r lo is a general stochastic statist ical sampling technique used to solve mathemat i -cal problems. In this applicat ion, Monte Car lo calls upon the fundamental physics of radia t ion interactions to simulate particle (photon, neutron, or charged particle) transport and ul t imate ly dose deposit ion i n different media. Th i s s imulat ion method is widely recognized as the most accurate dose calculat ion engine for radiat ion therapy planning par t icular ly i n regions of elec-tronic d isequi l ibr ium (e.g. interfaces between low/h igh density materials and smal l field sizes used w i t h h igh beam energies) [5, 18]. Incorporat ing Monte Car lo calculations into the I M R T opt imiza t ion process w i l l ensure that photon/e lect ron transport w i th in or adjacent to smal l air cavities are properly modeled such that an attempt can be made to compensate for the dose per turbat ion caused by the inhomogeneity. In addi t ion, Monte Car lo can provide valuable information about the dose contr ibut ion from extra-focal sources (e.g. flattening filter). T h i s scatter is not modeled i n several common dose calculat ion algorithms implemented in commercial treatment systems[17]. Monte , Ca r lo calculations also have the advantage of accurate model ing of beam modifiers, such as the mult i leaf coll imators ( M L C ) or wedges. The resultant scattered photon and electron doses, M L C transmission and leakage between M L C coll imator leaves w i l l contribute to the dose i n bo th the high-dose region and the beam penumbra. Trad i t iona l opt imiza t ion methods involve breaking down an open 2D field into independent beamlets (fluence elements) and then changing the weights or intensities of these beamlets to obta in an optimal fluence or intensity map[21, 22]. T h i s technique is commonly referred to as fluence-based optimization. If the relationship between a beamlet 's intensity and the dose to the patient / phantom for a given field can be established, the opt imiza t ion process can quickly manipulate this data as part of the inverse p lanning process [1, 2]. In this study, the large ma t r ix that describes the dose contr ibut ion from every beamlet from every field to the patient / phan tom is called a beamlet dose distribution matrix (also referred to i n the li terature as dose deposition coefficients). Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 156 After performing this fluence-based opt imizat ion, a leaf sequencing a lgor i thm is applied to determine the M L C leaf motions that w i l l deliver the desired distribution[24, 25]. T h e leaf sequencer incorporates M L C dosimetric and geometric properties such as transmission and leaf-mot ion l imita t ions . Since the optimal fluence map may not be physical ly deliverable by the M L C , the closest i t can come to this ideal s i tuat ion is called the actual or deliverable fluence map[26]. Several groups have introduced Monte Car lo simulations into this conventional fluence-based I M R T op t imiza t ion process[8, 27]. D e Gersem et a/.(2001)[28] and Shepard et a/.(2002)[29] both propose that some M L C prop-erties can be direct ly incorporated into the opt imiza t ion process thus e l iminat ing the leaf se-quencing step. Shepard et al. label their technique direct aperture optimization ( D A O ) . T h e advantage of this method is that the resulting M L C apertures are inherently deliverable and are generally more efficient than the t radi t ional fluence/intensity map opt imiza t ion i n that fewer treatment segments and monitor units ( M U ) are required. The efficiency gains are a result of several factors. F i r s t , the user can specify a m i n i m u m allowable open aperture area and moni tor uni t delivery (beam weighting) thus el iminat ing small , inefficient beamlets from the solut ion space. Second, the l imi ted number of apertures are in i t ia l ized as large open segments conforming to the B E V of the P T V . It has been suggested that an ideal inverse planning system would include the mult i leaf col-l imator i n the opt imiza t ion process and employ accurate dose model ing (especially for difficult treatment geometries) [30]. Shepard et al. use an in-house Monte Carlo-style point-kernel super-posi t ion a lgor i thm i n their photon D A O process [31, 32]. A form of direct aperture op t imiza t ion can also be applied to electron I M R T planning. A l - Y a h y a et ai.[149] calculate pre-defined geo-metr ic "fieldlet" doses ("fieldlet" is a term the authors use to describe smal l electron sub-fields of the order of several square centimetres) using Monte Car lo data. T h e fieldlet selection and electron energy are then opt imized ( E M R T - energy modulated radiat ion therapy). T h e technique proposed i n this s tudy implements a novel method for segmenting a uniform phase space output from a B E A M n r c linear accelerator s imulat ion into ind iv idua l phase space beamlets[150]. These ind iv idua l beamlets of phase space are projected onto a patient-specific phan tom and the dose deposit ion dis t r ibut ion is simulated using D O S X Y Z n r c . T h e relationship Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 157 between a beamlet locat ion and the resulting dose to a structure-of-interest is recorded i n the form of a large dose matr ix . Changes to the beamlet dose contr ibut ion are updated by assessing the effect of varying mult i leaf coll imator apertures on the beamlet weights. Geometr ic l im i t a -tions i n M L C mot ion plus transmission are considered dur ing the opt imiza t ion . A simulated annealing a lgor i thm provides the backbone for this opt imiza t ion method. T h i s style of inverse p lanning is referred to i n this thesis as M C - D A O . 6.3 Mater ia l s and Methods Briefly, the steps involved i n generating a M C - D A O plan are listed below (also see numbering on Figure 6.1): 1. Es tab l i sh the treatment held geometry (number of fields, gantry angles etc.). 2. Simulate the 6 M V x-ray open field phase space for each field size using B E A M n r c . 3. Segment the open phase space into 0.25 x 0.50 c m 2 beamlets 4. Calcula te the beamlet dose contr ibut ion to the phantom ( D O S X Y Z n r c ) and generate a beamlet-to-dose deposit ion mat r ix 5. Introduce a series of in i t ia l ly open mult i leaf coll imator apertures and calculate the effect on the beamlet dose deposit ion weighting (e.g. effect of M L C transmission). T h e number of apertures per field are user-defined. 6. Op t imize the M L C aperture shapes /. weights using simulated annealing to min imize dose-volume constraints 7. Perform a final Monte Car lo forward dose calculat ion on the opt imized M L C apertures. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 158 ONCE X # Fields (3) § 4 5* min / fld beamlets (c code) 15* min / fld DOSXYZnrc ONCE (5J/«6) X # Fields (7) 2-25 min DAO (MATLAB) 30* min / fld DOSXYZnrc * 30 x S u n F i re 2100 O p t e r o n P r o c e s s o r s (2.8 MHz) Figure 6.1: Process d iagram wi th approximate t imings for M C - D A O technique. A l l B E A M -n r c / D O S X Y Z n r c simulations performed on 30 A M S Opteron 2100 processors. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 159 6.3.1 Initial Treatment Parameters (Field Size) T h e 6 M V photon treatment beam arrangement and in i t i a l field sizes must be determined prior to the opt imiza t ion . The field isocentre is placed at the middle of the planning target volume ( P T V ) . A l l treatment p lan parameters and the structure set (organ contours) are exported in D I C O M format from the treatment planning computer. A n in-house M A T L A B software tool determines the field sizes to set based on the beam's-eye-view ( B E V ) projection of the target plus a uni form 0.5 cm margin. 6.3.2 Virtual Linear Accelerator T h e B E A M n r c ( N R C , Ot tawa, Canada) [143, 128] Monte Car lo software is used to simulate a 6 M V x-ray photon beam output from a V a r i a n C L 2 1 E X (Var ian M e d i c a l Systems, Pa lo A l t o , C A ) medical linear accelerator. B E A M n r c is an appl icat ion of the E G S n r c particle transport code[127]. T h e main components i n this c l in ica l accelerator are the target, p r imary (stationary) col l imator , moni tor chamber, mirror , and secondary col l imat ing jaws (adjustable). T h e ter t iary col l imator model ( M L C ) is dealt w i t h separately from the B E A M n r c code and is described in section 6.3.7. T h e photon cut-off energy ( P C U T ) is set to 10 k e V and the electron cut-off energy ( E C U T ) is set to 700 keV. T h e M C simulat ion of the linear accelerator is divided into two steps (Figure 5.7) to facilitate absolute dose calculations using a technique described by Popescu et al.(2005) [146]. Note that Phasespaee-A needs only to be calculated once for a l l patients and fields (using the 6 M V photon- beam) as there are no adjustable component settings. Phasespace-B is generated for every secondary col l imator (jaw) setting. T h e M C cluster at our facility is currently comprised of three Pen t ium 4, 3 . 2 G H z dual pro-cessor computers. A Monte Car lo job submission can thus be div ided into 6 paral lel processes. T h e t ime to calculate 250 mi l l ion histories for Phasespace-B for a single field is ~11 minutes. ( N O T E : at the t ime of wr i t ing this thesis, the lab was making a t ransi t ion from the older three P e n t i u m 4 processor cluster to a new high performance cluster of 30 A M S 2100 Opteron single core processors (2.8 M H z each). T h e comput ing speed is now 4 - 5 x faster.) Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 160 6.3.3 Segmentation of Beamlets from Phase Space T h e phase space plane located below the secondary jaws of the linear accelerator (Phasespace-B ) is d iv ided into 2.5 x 5.0 m m 2 (at 100 cm S A D ) beamlets using in-house software developed by Popescu and B u s h (2005) [150]. The software modifies the B E A M n r c 23-bit latch-bit identi-fication tag, t radi t ional ly used to identify from which component w i t h i n the l inac head a given particle interacts. There are 23 independent la tch bits thus up to 23 components can be iden-tified. In this part icular applicat ion, the latch bi t concept is modified to identify a geographic region w i t h i n the phase space rather than a component w i t h i n the linac head. A s there can be many more than 23 beamlets w i t h i n a typica l treatment field (^200 - 800), a creative solut ion must be found. T h e latch-bit modificat ion software (writ ten i n C code) converts the 23 inde-pendent la tch bi t tags into one single 23-bit binary number. T h e number of beamlets that a phase space can be div ided into has now increased from 23 to 2 2 3 . T h e open-field phase space is converted into n i nd iv idua l phase space files - one per beamlet. E a c h beamlet can now be ind iv idua l ly projected onto a patient / phantom and the dose contr ibut ion calculated. 6.3.4 Generation of Voxelized Phantom A voxelized phantom is generated from D I C O M format C T slice data. F i r s t , the D I C O M C T slice da ta is processed using pre-existing in-house software (wri t ten i n C code) such that C T artifacts and a l l objects (eg. C T couch) that are extraneous to the body contour are removed (i.e. Hounsfield units set to air). A new masked C T image is created. Figure 6.2 illustrates this step. T h e masking program can work on any contoured structure ( P T V , cord, parotids etc.) and is useful for isolat ing structures of interest for other applications (e.g. extract ing dose information for a part icular structure when generating D V H curves). T h e removal of extraneous-to-body objects is done to ensure that the upcoming M C dose s imula t ion does not at tempt to calculate dose i n regions of mater ial that are non-existent dur ing treatment. T h e C T Hounsfield units (HU) are converted to density and are then segmented into a smal l subset of four specific tissue types (air, lung, tissue bone) using ctcreate ( N R C , Ot tawa , Canada) . T h e H U thresholds are: air (< 50 H U ) , lung (< 300 H U ) , tissue (< 1125 H U ) , bone Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 161 (a) (b) F igure 6.2: Objects extraneous to the patient body contour that are not existent dur ing the actual treatment must be masked prior to undergoing Monte Ca r lo s imulat ion, (a) Before the masking, (b) After masking. (< 3000 H U ) . T h e voxel size used i n this s tudy is 2.5 x 2.5 x 2.5 m m 3 . 6.3.5 Beamlet Dose Distribution Matrix E a c h beamlet is ind iv idua l ly projected onto a patient-specific phantom at the correct gantry angle and the doses are calculated using D O S X Y Z n r c ( N R C , Ot tawa, Canada) [151] (Figure 6.3). T h i s process is repeated for every treatment field. The ul t imate goal is to bu i ld a dose deposit ion ma t r ix that describes the relationship between a beamlet locat ion and the dose to a structure-of-interest (see Figure 6.4). For a given field, the to ta l dose, DSi, to a voxel, i, w i th in a structure-of-interest, s is given by Equa t ion 6.1. icamlets Dsl= ":r'\i (6-1) 3 In vector notat ion, this is described as : £ y = a j - X &i = {%•}, * = j = ^-•••Nbeamlets (6.2) Where is the dose contr ibut ion to voxel, i, from beamlet, j, having a weight, Xj. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 162 pixel • • l l l l B I I I B B i a i l ! • • • • • • • • • • • • • I - • • • • • • • • • • • • • • • • • • • • • • • • • I • • • • • • • • • • • • • I • • • • • • • • • • • • • I • • • • • • • • • • • • • I Open Field Phase Space Figure 6.4: Open-field phase space is segmented into 2.5 x 5.0 m m 2 beamlets. E a c h beamlet, 4>j, contributes dose to a voxel, i, w i th in a structure-of-interest. T h i s da ta is stored in a beamlet dose dis t r ibut ion matr ix . C h a p t e r 6. M o n t e Carlo Input for D i r e c t A p e r t u r e O p t i m i z a t i o n for IMRT 163 (a) (b) F igure 6.5: (a) T h e beamlet dose contr ibut ion to the patient is calculated, (b) T h e doses only to the structure of interest (in this case, the P T V ) are extracted Note that D O S X Y Z n r c calculates the dose-to-medium, not the dose-to-water. If the dose-to-water is desired, a rat io of stopping powers would have to be applied[133]. T h e raw D O S X Y Z n r c doses are i n relative units, quoted in gray per electron incident on the target. E a c h field dose is converted to absolute dose (gray) [146]. T h i s method requires the in i t i a l B E A M n r c phase space generation to be d iv ided into two steps as described in section 6.3.2. To convert to absolute dose, the number of moni tor units assigned to each field must be known. However, the opt imized moni tor units ( M U ) cannot be determined unt i l the op t imiza t ion process is complete. T o avoid this conundrum, an in i t ia l iza t ion value of 100 M U is arbi t rar i ly assigned to each field. T h e final M U settings are later obtained by appl icat ion of the opt imized weighting factor. T h e ul t imate goal is to bu i ld a dose deposit ion mat r ix that describes the relationship between a treatment field beamlet and the absolute dose to a voxel w i th in a structure-of-interest. To isolate this information, a subset of structure doses are 'cut out ' from the to ta l beamlet dose d is t r ibut ion and the da ta stored (see Figure 6.5). Note that a l l beamlet dose distr ibutions are calculated under full scatter conditions. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 164 6 . 3 . 6 Direct Aperture Optimization (DAO) Unl ike t rad i t iona l fluence-based techniques, direct aperture opt imizat ion ( D A O ) integrates dosi-metr ic and geometric (i.e. leaf motion) properties of the mult i leaf col l imator ( M L C ) direct ly into the op t imiza t ion process. The in-house software used i n this study[76] is an implementat ion of the work described by Shepard et al. (2002) [29]. In this step-and-shoot approach, only the beam gantry angles and the number of apertures per field need to be specified by the user. T h e in i t i a l condit ions for the D A O opt imiza t ion is such that a l l apertures are set to the beam's-eye-view ( B E V ) projection of the P T V (plus a 0.5 cm margin). A t each i terat ion the opt imizer must randomly select (1) which of the n treatment fields to optimize, (2) which of the m aper-tures w i t h i n that field to optimize, (3) whether the aperture weight or the aperture shape w i l l be adjusted, and (4) the amount by which the weight or shape of the aperture w i l l change. The i nd iv idua l M L C leaf positions (defining the aperture shape) and the aperture weights are opt imized using simulated annealing[65] to minimize a dose-volume objective function[29]. A t the start of the opt imizat ion, large changes to the M L C leaf positions and aperture weights are allowed. However, as the opt imizat ion continues, the allowable adjustments to the M L C shapes and weights decrease. Th i s is due to the implementat ion of a cooling schedule in the opt imiza t ion . T h e dose-volume constraints (e.g. P T V coverage, max dose to organ-at-risk) are specified by the user. T h e number of voxels that are non-compliant w i t h these dose constraints is min imized dur ing the opt imizat ion . The beamlet dose contr ibut ion to a structure-of-interest is updated by subtract ing or adding the dosimetric effect of the new M L C leaf posi t ion or aperture weight. The dosimetric properties included i n the aperture shape opt imiza t ion are the M L C transmission and leaf mot ion l imitat ions. Leaf-tip and tongue-and-groove effect are not included. After the M L C aperture shapes and weighting are opt imized, each field (wi th a l l its aper-tures) is wr i t ten into a M L C control file. The to ta l number of moni tor units associated w i t h each treatment field is determined by summing the aperture weights associated w i t h the beam. It is impor tant to note that there is no user input into the final shape of the apertures, except to define l imi t s where the optimizer may explore. These aperture shapes are not "pre-selected". Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 165 However, the user may specify a m i n i m u m number of M U per aperture (thus ensuring a min i -m u m beam weight) or a m i n i m u m open field area (defined as a percentage of the "open" P T V B E V aperture). A systematic study of simple and complex planning targets by J iang et al. (2005) indicate that a good dose dis t r ibut ion for a D A O plan can be achieved w i t h only five apertures for most c l in ica l si tuations. Complex cases may require up to nine apertures[75]. In the Shepard et al.(2002) paper, three to five apertures per field were tested. 6.3.7 Final Forward Calculation T h e M C - D A O plans undergo a full forward calculat ion using Monte Ca r lo simulations. T h e M C simulations have been benchmarked against open-field ion chamber measured percentage depth doses and profiles for field sizes of 3 x 3 c m 2 to 40 x 40 c m 2 and agreement is. w i t h i n 1.5%. T h e mult i leaf col l imator is introduced using phase space transport code provided by the V i r g i n i a Commonweal th Univers i ty research group [132]. In this model , simplified geometric regions are used and it is assumed that' the photons only undergo at tenuation and a single C o m p t o n scatter interaction. Pa i r product ion and electron interactions w i t h i n the M L C are not considered. The mult i leaf coll imator s imulat ion has been benchmarked against ion chamber measurements for the s l iding window test of varying sizes ( 1 - 9 0 m m gap) and the agreement is <1%. F i l m verification for a sample I M R T field was also performed w i t h acceptable agreement. W i t h the six-processor computer cluster currently i n use, the dose d is t r ibut ion from each field takes ~ 2 hours. T h e dose output of the M C simulat ion is in units of gray per particle incident on the electron target. These relative doses are converted to absolute doses by apply ing the monitor unit settings generated by the optimization[146]. For the V a r i a n 2 1 E X linear accelerator, i t is assumed that the M L C shapes do not impact the dose to the monitor chamber Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 166 6.3.8 Application of M C - D A O P h a n t o m w i t h Inhomogenei ty - Single F i e l d A 25.6 x 25.6 x 28.0 c m 3 water-equivalent AVID treatment verification phantom ( M D X M e d i -cal , Vancouver , Canada) is imaged at 130 k V p on a Picker PQ5000 x-ray C T scanner. T h e slice thickness is 3.0 m m and the axia l resolution is 0.68 m m . T h e D I C O M images are converted into a Mon te Ca r lo voxel phantom (voxel size = 2.5 x 2.5 x 2.5 m m 3 ) . A 5.0 c m thick air cavity is introduced into the phantom and 'C ' -shaped P T V w i t h embedded spinal cord-like organ-at-risk ( O A R ) contour is posit ioned start ing 2.0 m m below the air cavity (see Figure 6.6(a)). There is 6 m m between the concave surface of the P T V and the O A R . A single 6 M V , 6.0 x 7.0 c m 2 open photon field from a V a r i a n 2 1 E X is s imulated using B E A M n r c . T h e resulting phase space is directed at the anterior face of the v i r t ua l phantom and the doses calculated using D O S X Y Z n r c ( S A D = 100 cm, depth to isocentre = 12.8 cm). A percentage depth dose obtained from Monte Car lo is compared to the doses obtained from the single penci l beam algori thm implementing the modified Ba tho correction [152]. The non-corrected dose d is t r ibut ion (no inhomogeneity) is also calculated for comparison. P h a n t o m w i t h Inhomogenei ty - Seven Fie lds Seven coplanar fields having equally spaced gantry angles are placed around the A VID phantom (Figure 6.6(b)). T h e open fields are segmented into beamlets and the Monte Car lo dose depo-si t ion ma t r ix is constructed. The calculat ion t ime per beamlet to simulate 700,000 histories on a 102 x 110 x 111 voxel phantom is ~ 8 2 s. T h e uncertainty associated w i t h each beamlet is ~ 2 . 9 % . T h e tota l calculat ion t ime for this example is 9.2 hours (82 s/beamlet x 2436 beamlets total) . Note that i n this example, the beamlet doses were calculated i n series. T h e code has since been modified to take advantage of a new 30-processor comput ing cluster ( A M S Opteron 2100 series, single core). The beamlets are now calculated in parallel which decreases the tota l ca lcula t ion t ime by ~ 5 x . Six apertures per field are specified i n the D A O software. A summary of some of the planning parameters is presented i n Table 6.1. T h e p lanning objectives are listed i n Table 6.2. T h e opt imized plan produces a series of M L C Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 167 Field 1 (a) (b) F igure 6.6: Water equivalent A V I D phantom w i t h air cavity, (a) Structure set w i t h C-shaped P T V and spinal cord-like O A R . (b) F i e l d placement. Table 6.1: Treatment field / opt imizat ion parameters for A V I D phantom w i t h C-shaped target. Example Number B e a m Number D A O T y p i c a l Fie lds Arrangement Aper tures F i e l d Size 1 7 equally spaced angles 6 7.5 x 7.5 cm2 Table 6.2: D V H planning objectives - P h a n t o m example (C-shape). P T V > 95% volume covered by 95% R x dose < 5% volume exceeds 110% R x dose O A R < 5% volume exceeds 60% R x dose B O D Y < 5% volume exceeds 110% R x dose control files and monitor units. The opt imizat ion t ime is ~ 2 0 minutes. T h i s p lan undergoes a final forward Monte Ca r lo s imulat ion using D O S X Y Z n r c to determine dose to the phantom. The calculat ion t ime for seven fields (250,000,000 histories per field) is ~ 2 hours. T h e uncertainty i n the h igh dose region is < 2%. A comparison p lan is generated using the V a r i a n E c l i p s e ® v . 6 . 5 treatment p lanning system ( T P S ) implemented i n our cl inic. The T P S pairs a fluence-based opt imiza t ion method w i t h the single penci l beam kernel ( P B K ) dose calculat ion algori thm. The modified B a t h o heterogene-i ty correction is employed for the pre-opt imizat ion (calculation of dose deposit ion coefficients) and the final dose calculat ion stage. The treatment efficiency of the two methods ( T P S vs Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 168 M C - D A O ) are compared by stat ing the total number of monitor units required to deliver one fraction. Deficiencies i n the T P S final dose calculat ion are explored by repeating the forward dose calcula t ion using M C simulat ion. T h e seven-field M C - D A O plan undergoes fi lm verification on a Var i an 21 E X linear accel-erator using K o d a k E D R 2 film embedded in a film-specific insert for the AVID phantom. A coronal film plane is i rradiated at isocentre (depth = 12.8 cm). A cal ibrat ion film is i rradiated i n the same phantom by acquiring an axial percentage depth dose plane. The fi lm is processed at 34 .5°C in a K o d a k 3000 R A processor operating on the standard cycle (111 s for 35 cm length). T h e cal ibra t ion film opt ical density is assigned a dose by comparing to ion chamber measured P D D da ta (corrected for the 84.5 c m S S D AVID phantom geometry). T h e M C - D A O p lan coronal plane can now be converted to dose and compared to the M C simulat ion result. C o m p l e x Pa t ien t A r c h i t e c t u r e P l u s A i r C a v i t y - N a s o p h a r y n x Recur rence A patient example w i t h recurrent nasopharynx cancer is first planned on the c l in ica l fiuence-based T P S . T h e p lan is comprised of seven, 6 M V photon beams distr ibuted w i t h an anterior bias (Figure 6.7). T h e optimal D V H obtained from the raw opt imiza t ion is compared to the actual or final Figure 6.7: Anter ior bias beam arrangement for nasopharynx recurrence example Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 169 D V H . T h e final D V H is obtained by apply ing the leaf mot ion calculator to the optimal fluence and then performing the final forward calculat ion. T h e T P S utilizes a P B K dose calculat ion a lgor i thm for bo th the opt imiza t ion step and the final forward calculat ion. For comparison, the T P S p lan is also simulated using Monte Car lo for the final calculat ion step. A second I M R T plan is generated using M C - D A O . T h e same seven field beam arrangement as shown i n Figure 6.7 is used. The open field is d ivided into beamlets and the calculat ion t ime per beamlet to simulate 1.2 mi l l ion histories on a 69 x 88 x 61 voxel phantom is ~130 s (on a Intel P e n t i u m 4, 3.2 M H z processor). T h e uncertainty associated w i t h each beamlet is ~2 .9%. T h e to ta l calculat ion t ime for this example is 6.9 hours (130 s/beamlet x 1148 beamlets total) . A g a i n note that i n this example, the beamlets were calculated i n series, not i n parallel . Ten apertures per field are specified for the direct aperture opt imizat ion. The M L C control files and monitor units are generated by D A O w i t h an opt imizat ion t ime of ~ 1 5 minutes. A final forward calculat ion is performed using Monte Car lo . The opt imized M C - D A O D V H is compared to the 1 final M C forward calculated D V H . A summary of the treatment field and opt imiza t ion parameters for this nasopharynx recurrence patient is presented i n Table 6.3. Table 6.3: Treatment field / op t imiza t ion parameters for nasopharynx recurrence example. E x a m p l e Number Fields B e a m Arrangement Number D A O T y p i c a l Apertures F i e l d Size 1 7 anterior bias 10 5.5 x 3.5 cm2 T h e p lanning objectives for this example are determined by a radia t ion oncologist and are l isted i n Table 6.4. In the case where it is not possible to meet a l l p lanning objectives, the oncologist w i l l pr iori t ize the p lanning objectives to guide the planning. Table 6.4: D V H planning objectives - Nasopharynx recurrence example. P T V >95% volume covered by 95% R x dose < 5% volume exceeds 110% R x dose brainstem minimize volume exceeding 20 G y dose L t / R t Tempora l Lobes minimize volume exceeding 15 G y dose C h i a s m minimize volume exceeding 15 G y dose B O D Y < 5% volume exceeds 110% R x dose T h e treatment delivery efficiency (number of M U per treatment) for the M C - D A O p lan is compared to the current I M R T planning method implemented i n our cl inic . Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 170 B o t h the fiuence-based plan and the M C - D A O plan are transferred onto a 27 cm diameter cy l indr ica l verification phantom. A n ion chamber measurement is acquired at the center of the phantom (depth = 13.5 cm). T h e ion chamber used is a Farmer-type P T W - F r e i b u r g type 30001. T h i s P M M A - g r a p h i t e wal l / a luminum electrode chamber has a sensitive volume of 0.6 c m 3 (3.05 m m radius, 25.9 m m length). M a s s i v e L o w Dens i ty Inhomogenei ty - L u n g One of the most difficult p lanning situations (from a dose calculat ion accuracy perspective) is a soli tary lung mass embedded w i t h i n the low density lung region. The fluence map delivery is not par t icular ly complex as the. tumour is not abut t ing any sensitive organs (except for the unaffected, healthy lung surrounding i t ) , however the t ransi t ion from lung to tumour to lung is dosimetr ical ly challenging. Th i s site is par t icular ly difficult to plan and deliver i n a c l in ical sett ing as the tumour is also moving due to the patient breathing motion. Tumour t racking and mot ion compensation techniques are a major field of study and not explored i n this thesis. T h i s thesis does emphasize, however, the issue of dose calculat ion accuracy. T h e assumption is thus made that the target (lung tumour) is stationary i n this example. L i k e the previous nasopharynx example, an I M R T plan is generated w i t h the V a r i a n Ecl ipse T P S which employs a fiuence-based opt imizat ion and a single penci l beam kernel ( P B K ) convolu-t ion dose calculat ion algori thm. Tissue inhomogeneity corrections are applied post-convolution using the modified Ba tho method. F ive non-equidistant 6 M V fields are applied w i t h a left side bias to avoid the healthy right lung and spinal cord (Figure 6.8). A summary of the treatment field and opt imiza t ion parameters for this lung example is presented i n Table 6.5. Table 6.5: Treatment field / opt imizat ion parameters for lung tumour example. Example Number B e a m Number D A O T y p i c a l Fie lds Arrangement Apertures F i e l d Size 2 5 left bias 3 6.0 x 5.0 c m 2 T h e p lanning objectives are listed in Table 6.6. Note that much of this P T V (planning target volume) contour encompasses a por t ion of the low-density, lung material . It is expected that meeting the treatment planning objectives would be very difficult for this structure. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 171 Table 6.6: Dose volume histogram ( D V H ) planning objectives - L u n g tumour example[44, 153, 154], P T V >95% volume covered by 95% R x * dose < 5% volume exceeds 105% R x dose G T V >98% volume covered by 98% R x dose < 5% volume exceeds 102% R x dose L t L u n g < 20% volume exceeds 20 G y mean lung dose < 15 G y R t L u n g < 15% volume exceeds 10 G y B O D Y < 5% volume exceeds 110% R x dose * R x = 70 G y / 35 fractions T h e optimal dose volume histogram ( D V H ) obtained from the raw opt imiza t ion is compared to the actual or final D V H for the Ecl ipse T P S plan (calculated wi th the P B K algori thm). To compare dose calculat ion algorithms, the Ecl ipse generated plan is also simulated using Monte Ca r lo for the final calculat ion step. Figure 6.8: F ive field lung tumour example. F i e l d arrangement shown i n the top left corner. T h e G T V (smaller blue sphere) is the gross tumour volume ( tumour visual ized on the C T image). The P T V (planning target volume) is created by adding a uniform 10 m m margin about the G T V . Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 172 A second I M R T p lan is generated using M C - D A O . T h e same five field beam arrangement as shown i n Figure 6.8 is used. The open field is divided into beamlets and the calculat ion t ime per beamlet to simulate ~900,000 histories on this 138 x 79 x 72 voxel phan tom is ~140 s (on an Intel Pen t ium 4, 3.2 M H z processor). T h e beamlet uncertainty is ~ 3 . 0 % . T h e tota l beamlet dose calculat ion t ime for this example is 7.8.hours (again, the calculat ion t ime has since been reduced by a factor of ~ 5 x w i t h the instal la t ion of the new 3 0 - A M S Opteron cluster and the implementat ion of parallel beamlet dose calculations). Since this is a geometrically simple example, only three apertures per field are specified for the direct aperture opt imiza t ion . The M L C control files and monitor units are generated by D A O w i t h an op t imiza t ion t ime of ~ 2 0 minutes. T h e opt imiza t ion t ime was quite long because the two large lung organs were included i n the op t imiza t ion process. The voxel density is 1 voxel / 2.5 m m 3 result ing in ~120,000 voxels per lung. A n increase i n the opt imizat ion speed could be realized by reducing the calculat ion voxel density i n the lung (1 voxel / 5.0 m m 3 w i l l decrease the number of calculat ion voxels by a factor of 8). A final forward calculat ion of the resulting D A O plan is performed using Mon te Car lo ( M C - D A O ) . The opt imized M C - D A O D V H is compared to the final M C forward calculated D V H . ' T h e treatment delivery efficiency (number of M U per treatment) for the M C - D A O p lan is compared to the current I M R T planning method implemented in our cl inic (Eclipse fluence-based opt imiza t ion) . B o t h the fluence-based plan and the M C - D A O plan are transferred onto a 25.6 x 25.6 x 28.0 c m 3 water-equivalent A V I D treatment verification phantom. A n ion chamber measurement and coronal film plane is acquired at isocentre located at the centre of the phantom (depth = .12.8 cm). T h e ion chamber used is a cy l indr ica l IC10 Wellhofer model (Scandi t ronix Wellhofer A G , Germany) . T h i s C-552 plastic wal l / C-552 plastic electrode chamber has a sensitive volume of 0.147 c m 3 (3.0 m m radius, 6.0 m m length). T h e coronal plane verification was performed using K o d a k E D R 2 radiographic film processed i n a K o d a k 3000 R A processor (std cycle, 33 .5°C) . Three percentage depth dose cal ibrat ion films (75 M U , 150 M U and 250 M U ) were acquired so that the known doses can be correlated w i t h the scanned film p ixe l intensity. A S A D (100 cm) set-up was used and the field size was 5 x 5 c m 2 . Th i s process generates a ca l ibra t ion curve. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 173 6.3.9 Sources of Error There are many sources of error i n radiat ion therapy planning, dose calculat ion and x-ray beam delivery. It is important to identify and minimize these errors to ensure that the treatment goals for these real, human patients are achieved in a consistent and accurate manner. Ideally, the radia t ion dose of the tumour w i l l not vary by more than 5% of the prescript ion dose. Sources of error associated w i t h I M R T treatments i n general can be div ided into four major categories: 1. Ident i f ica t ion of diseased tissues and nearby heal thy organs 2. C a l c u l a t i o n of dose depos i t ion i n tissue and t reatment p l a n o p t i m i z a t i o n 3. R a d i a t i o n dose del ivery on l inear accelerator 4. Pa t i en t setup and i m m o b i l i z a t i o n for t reatment M a n y of these sources of error are shared wi th the treatment p lanning methods currently i n use c l in ica l ly (e.g. identification of the tumour, radia t ion dose delivery and patient set-up). A l t h o u g h there are dose calculat ion and p lan opt imiza t ion errors specific to the M C -D A O method, the goal of this, thesis is to show that M C - D A O offers an improvement over the t rad i t iona l p lanning/ca lcu la t ion methods currently i n use. T u m o u r L o c a l i z a t i o n Before one can even attempt to treat a tumour, one must first be able to find i t . Before the integration of 3D imaging modali t ies into the field of radiat ion oncology, doctors relied on 2D planar x-rays, 2D planar nuclear medicine scans (e.g. gamma camera) and physical palpat ion of the mass (if accessible) to locate the tumour. Generally, treatment fields were quite large and encompassing to ensure that no part of the tumour was missed, however large volumes of healthy tissue were also treated i n the process. The in t roduct ion of 3D C T x-ray imaging provided a detailed look into the interior of the human body. T h e tumor could be identified slice-by-slice to form a 3D representation and the radiat ion beams were designed to conform to the tumour shape. However, i n C T imaging, the boundary between diseased tissue and healthy tissue may not always be distinct. There are many tumours that are just not visual ized wel l w i t h Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 174 x-ray imaging (e.g. subtle bra in tumours embedded i n bra in matter or microscopic extensions of cancerous cells). Tumour specific contrast agents may help in some cases. Soft tissue imaging w i t h magnetic resonance ( M R I ) and metabolic imaging wi th posi t ron emission tomography ( P E T ) offer complimentary modalit ies for visual iz ing the cancerous tissue. However, these alternative imaging modali t ies are used i n specific situations and are not the s tandard of care i n Canada . C T imaging, however, is standard. Depending on the locat ion and type of tumour, a marg in of 'error' is typ ica l ly added to the visible tumour region to account for macroscopic extension of the disease. Th i s margin is usually a few mill imetres. Dose C a l c u l a t i o n E r r o r s Dose calcula t ion errors occur when the predicted dose does not match the actual dose delivered to the patient. The magnitude of this error w i l l vary depending on the calculat ion model used, the qual i ty of the patient C T imaging data, the homogeneity of the tissues that are being treated (e.g. soft tissue only vs. a combinat ion of soft tissue, bone and a i r / lung) , and the s ize /complexi ty of the radiat ion fields being delivered. Mos t treatment p lanning systems currently available can achieve agreement w i t h measured (in phantom) doses w i t h i n 2% error for uniform, water-like tissues. However, large errors associated w i t h smal l fields and the presence of low-density inhomogeneities (e.g. air / lung) of the order of up to 20%-70% have been reported[106]. A l t h o u g h Monte Ca r lo is considered to be the "gold standard" for dose calculat ion accuracy, it is not free from sources of error. T h e error can be divided into two types: Systematic Error or Statistical Error. Systematic error includes uncertainty associated wi th : • interact ion cross-section data • the random number sampling process • approximations introduced by the Monte Car lo software for particle transport • inaccuracies i n the s imulat ion geometry Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 175 • errors associated w i t h converting C T image 'Hounsfield ' data into physical density and tissue types W h e n s imulat ing the x-ray photon beam emerging from the linear accelerator, detailed knowledge of the accelerator component geometry and material composi t ion must be known. Errors i n this information w i l l affect the spatial and energy dis t r ibut ion of the v i r t ua l x-ray beam. Similar ly , when calculat ing doses to the patient or some other material , knowledge of the composi t ion of the patient is also required. Th i s information is obtained by assigning mater ia l types and physical density to the voxels i n the C T image based on the Hounsfield number data contained wi th in . If ' the cal ibrat ion between Hounsfield number and physical density is not accurate, the density or even the tissue type w i l l be mis-labeled. Radiologic properties of each of these materials are obtained from look-up tables stored w i t h i n the Mon te Ca r lo system (called ' P E G S ' data). A typ ica l C T representation of the patient has voxel size dimensions <1 x 1 x 5 m m 3 . Mon te Ca r lo phantoms typica l ly use a voxel size of 2.5 x 2.5 x 2.5 m m 3 . T h e C T voxels are thus re-sampled using averaging methods to match the desired Monte Car lo phan tom voxel size. T h i s may lead to spatial averaging errors. For example, i f a larger voxel contains some tissue-like voxels and some air-like voxels, the average voxel value may mistakenly appear as some intermediate tissue, such as lung. The reason that larger voxels sizes are used i n Monte Ca r lo is because smal l voxels w i l l require more simulations i n order to achieve an acceptable s tat is t ical uncertainty which means longer s imulat ion times and larger memory requirements. T h e 2.5 m m 3 voxel is selected i n an attempt to balance s imulat ion t ime, memory requirements and good spat ial resolution. Statistical Error is related to the number of particles simulated. Increasing the number of s imulated photon/electron interactions decreases the stat ist ical error. T h e error is propor t ional to 1/yfN, where N = the number of simulations. Increasing the voxel size w i l l also decrease the s tat is t ical error per voxel. For a cubic voxel, the error is propor t ional x 3 , where x = the length of the voxel along one dimension. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 176 Trea tment P l a n O p t i m i z a t i o n E r r o r s T h e M C - D A O method incorporates some properties of the M L C direct ly into the opt imiza t ion ( M L C leaf transmission and leaf mot ion l imitat ions) , but this is not a complete list. However, for the final dose calculat ion (which occurs after the opt imizat ion step) a l l properties are included in the calculat ion. Th i s incomplete list of M L C properties dur ing the op t imiza t ion w i l l lead to what is called a convergence error; the final calculated doses do not match exactly the opt imized doses. R a d i a t i o n Dose D e l i v e r y E r r o r s A linear accelerator w i t h a M L C col l imat ion system delivers the radiat ion dose. T h e treatment unit must be cal ibrated by a physicist to ensure that the amount of radiat ion we p lan to deliver matches the radia t ion we actually deliver. T h e monitor chamber is tuned such that we can define 1 M U (monitor unit) to equal 1 c G y of dose at a depth of m a x i m u m dose (dmax) i n water for a 10 x 10 c m 2 field size. Th i s cal ibrat ion is performed w i t h an ion chamber collected to an electrometer. Errors in the ion chamber measurement w i l l contribute to errors in the linear accelerator cal ibrat ion. Generally, these types of errors are less than 1%. In addi t ion, strict performance specifications must be imposed on the mult i leaf col l imator . Intensive quali ty assurance is performed to ensure that the the M L C leaves are reaching the expected positions w i t h i n a fraction of a mil l imetre. Pa t i en t S e t - U p / I m m o b i l i z a t i o n E r r o r s E v e n w i t h the most perfect dose calculat ion, p lan opt imiza t ion and beam delivery scheme, a large source of error is related to patient set-up and immobi l iza t ion . Rad ia t i on beam local iza-t ion marks are usually tat tooed onto the skin of the patient. These skin marks are not often ideal because the interior organs of the body can move relative to the skin. T h e most diffi-cult treatment site is the lung as smal l tumours embedded i n the lung volume w i l l experience breathing mot ion. To allow for motion/set-up errors, the cl inical target volume is increased in size by a margin ( typical ly 5 - 1 0 m m for the prostate, and even more for the lung.). Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 177 6.4 Results and Discussion 6.4.1 Phantom Example - Single Field A depth dose curve drawn through a single field dose d is t r ibut ion illustrates the problem en-countered when target volumes are located direct ly beyond an air cavity. T h e Monte Car lo s imula t ion models the per turbat ion due to the air cavity, namely the drop in dose w i t h i n the air cavi ty and the dose re-buildup region located at the air-tissue interface. T h e locat ion of the P T V i n this example is such that the re-buildup region can cause an under-dosage at the side of the target proximal to the x-ray source (see Figure 6.9). F igure 6.9: O p e n field percentage depth dose through 5.0 cm air cavity. D O T T E D L I N E : P B K a lgor i thm (without inhomogeneity correction), D A S H E D L I N E : P B K (wi th modified Ba tho correction), S O L I D L I N E : Monte Car lo . 6.4.2 Phantom Example - Seven Fields Dose D i s t r i b u t i o n : 7 F i e l d P h a n t o m T h e problem encountered when the tissue inhomogeneity is not adequately modeled dur ing the I M R T op t imiza t ion phase is i l lustrated in the following phantom example. In Figure 6.10(a) the Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 178 3D representation of the C-shaped P T V (red) is shown w i t h a 95% isodose surface (light green) encompassing > 99% of the target area. Th i s p lan was opt imized and the forward calculat ion performed using the P B K algori thm. W h e n the same plan parameters are simulated w i t h Monte Car lo , a problem w i t h the P T V coverage close to the air-phantom interface is revealed (Figure 6.10(b)). In the region prox imal to the air cavity, there is systematic c l ipp ing of the P T V by the 95% isodose surface. T h e 99% volume coverage or iginal ly predicted by the P B K dose calculat ion algori thm now drops to 96%. A l t h o u g h this s t i l l meets the 95% dose / 95% volume P T V acceptance criteria, the systematic under-dosage to this anatomical region may not be agreeable to the cl inician. T o address this problem, Monte Car lo simulated beamlet dose da ta is used for the opt imiza-Figure 6.10: 3D representation of the 95% prescription isodose surface (light green) encompass-ing the C-shaped target (red). The air cavity surface rendering is not shown for c lar i ty - box drawing indicates relative posit ion, (a) P B K a lgor i thm opt imiza t ion / P B K final forward calculat ion, (b) Same plan from (a) re-calculated using M C simulat ion, (c) M C - D A O plan w i t h 6 apertures per held. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 179 t ion step. T h e opt imized p lan then undergoes a final dose calculat ion again using Mon te Car lo . T h i s is referred to as the MC-DAO plan. Since the Monte Car lo beamlet da ta accurately models the effect of the air cavity, the opt imizat ion can attempt to provide adequate coverage in the difficult region close to interface (Figure 6.10(c)). The l imi t ing factor for obtaining a good p lan is dictated by what is physical ly achievable by the optimizer i n light of the bu i lup /bu i ld -down dose regions adjacent to the air cavity. Dose V o l u m e His tog rams - 7 F i e l d P h a n t o m T h e dose volume histograms associated w i t h Figures 6.10(a) and (b) are shown i n F igure 6.11(a). A summary of the relevant doses is shown in Table 6.7. The mean P T V dose is 3% higher for the P B K opt imized / M C calculated plan, but the 95% isodose coverage is compromised. In addi t ion, the mean O A R dose calculated w i t h M C is 5% higher than that predicted by the P B K algor i thm. A summary of relevant doses are shown i n Table 6.7. % Dose. % Dose (a) (b) F igure 6.11: The opt imized D V H (before final calculation) and final D V H (after final dose cal-culation) are compared for the (a) fiuence-based opt imiza t ion and, (b) D A O plans. ( D A S H E D L I N E ) = opt imized D V H . ( S O L I D L I N E ) = final D V H . Figure 6.11(b) compares the M C - D A O plan to the P B K o p t i m i z e d / M C calculated p lan (also shown as the dashed line in Figure 6.11(a)). T h e M C - D A O p lan has the 95% isodose covering 95% of the P T V which meets the planning objectives. T h e systematic under-dosage Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 180 Table 6.7: Doses to structures-pf-interest for A V I D phantom example. P B K op t imized / P B K op t imized / M C - D A O P B K Forward Calcu la t ion M C Forward Ca lcu la t ion (6 a p e r t u r e s / F L D ) P T V % V o l > 95% R x Dose 99% 96% 95% M e a n Dose 102% 105% 102% M i n Dose 83% 78% . 75% M a x Point Dose 110% 114% 116% % V o l > 110% 0.0% 3.7% 1.5% O A R M e a n Dose 31% 36% . 36% M a x Poin t Dose 57% 76% 67% % V o l >60% R x Dose 0.0% 6.2% 3.6% B O D Y Hot Spot Dose(0.5 c m 3 Vol . ) 106% 110% 109% V o l . Receiv ing 95% Dose 28 cc 36 cc 24 cc V o l . Receiving 50% Dose 338 cc 293 cc 362 cc demonstrated i n Figure 6.10(b) has now been resolved. In the Introduct ion, the concepts of systematic error and convergence error were described. Figures 6.11(a) and (b) exemplify the difference between the two types of error. In Figure 6.11(a), the same p lan undergoes a forward calculat ion using two different dose calculat ion methods ( P B K convolution and Monte Car lo) . Assuming that the Monte Ca r lo plan is the "gold s tandard", the P B K plan D V H exhibits a systematic error. In Figure 6.11(b), the same final forward calculat ion is used (Monte Car lo) , but the algorithms used to opt imize the p lan are different. T h e P B K opt imized D V H demonstrates a convergence error. It is also interesting to note that even w i t h repeated optimizations, the fluence-based plan consistently produces a r ing (or annulus) of 95% isodose. These plans met the P T V coverage cri teria, but they also increase the dose to the 'normal ' tissue volume. The volume of normal tissue that receives >95% or more of the prescription dose for the P B K opt imized / M C forward calculated p lan is 36 cc. A method commonly used to control the presence/location of high-dose volumes i n normal tissues involves defining a dummy 'hot spot (max imum dose) reduct ion ' contour that can be included in the re-optimization. Th i s would have to be done to improve this fluence-based plan. T h e M C - D A O p lan does not have this issue and repeated opt imizat ions for this example produced the desired 'C ' -shaped 95% isodose coverage. The volume of normal tissue receiving Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 181 >95% of the prescript ion dose for the M C - D A O plan is 24 cc. Plan Verification : 7 Field Phantom To verify the M C simulated plan doses, three cal ibrat ion curves are first acquired to correlate a known dose to phantom to a scanned film image pixel intensity (see Figure 6.12). A cal ibrated measured coronal film plane is then compared to the Monte Car lo v i r tua l dose plane i n Figure 6.13(a). Dose profiles i n a direction parallel and perpendicular to the M L C x 1 04 Calibration Curve (PDD) 2.5 — 1 5 > S 1 CL U 0 50 100 150 200 25 Dose (cGy) 50 MU 100 MU 200 MU Figure 6.12: F i l m Ca l ib ra t i on Curves are constructed by correlating known P D D doses (for 50, 100 and 200 M U ) to the scanned film image pixel intensity. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 182 (d) (e) Figure 6.13: Corona l dose plane M C - D A O plan: (a) film measurement ( L E F T ) and Monte Car lo ( R I G H T ) . Arrows show tongue-and-groove effect. Profiles (b) paral lel and (c) perpendicular to leaf motion. S O L I D : M C dose. D A S H E D : F i l m measured dose, (d) Cropped film plane (e) Corresponding dose-difference ( D D ) / distance-to-agreement ( D T A ) map[119]. D a r k = D T A or D D passed. W h i t e = failed. (Pass /Fa i l : D D <3% max dose / D T A <3 mm) Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 183 leaf travel are compared i n Figures 6.13(b),(c). T h e Monte Car lo calculated doses and the film measurement show excellent agreement. A 2D map of "Pas s /Fa i l " cr i ter ia based on a combinat ion of percentage dose difference and distance-to-agreement tests[119] for the cropped image i n F igure 6.13(d) is shown i n Figure 6.13(e). The blue area shows where the comparison between the film measured doses and the Monte Car lo doses passed either the Dose Difference (< 3%) test or the distance to agreement test (< 3 mm) . O n l y 1.8% of the da ta points failed this test. T h i s par t icular p lan exhibits some visible tongue and groove effect (arrows i n F igure 6.13(a)). T h i s is an artifact of delivering D A O apertures that contain adjacent leaves that protrude i n an al ternat ing fashion (see Section 2.7.6). The tongue and groove effect can be min imized by using a 90° col l imator angle (leaf mot ion is perpendicular to the plane of gantry rotation) instead of the 0° angle used i n this example. Us ing rotat ing apertures w i l l further reduce this effect[51, 76]. Since the M C - D A O method is able to accurately model the effect of the air cavi ty on the dose dis t r ibut ion, it can now attempt to compensate for these effects. T h i s is reflected in the choice of beam weighting across the seven fields. The percentage contr ibut ion from each field to a point w i t h i n the P T V is recorded for bo th the Ecl ipse P B K plan (wi th P B K forward calculation) and the M C - D A O plan (see Table 6.8). The percentage doses are normalized to give a to ta l of 100% at the point of interest. Table 6.8: Ind iv idua l beam weighting (percentage contr ibut ion to dose point w i t h i n P T V ) F i e l d F i e l d F i e ld F i e ld F i e l d F i e l d F i e l d T O T A L 1 2 3 4 5 6 7 Ecl ipse P B K (%) 4.1 19.8 17.6 14.6 10.1 16.1 17.7 * 100 * M C - D A O (%) . 3.1 14.8 17.8 16.8 18.5 19.8 9.2 * 100 * Fie lds 1, 2 and 7 a l l pass through the air inhomogeneity and w i l l be referred to as the anterior fields (see Figure 6.6(b)). Fields 3, 4, 5, and 6 a l l are directed posterior to the air cavi ty and are referred to as posterior fields. The ratio between posterior contr ibut ion and anterior contr ibut ion can be calculated (see Table 6.9). T h e M C - D A O plan has shifted the beam weighting to the posterior fields. The posterior beams do not experience dose re-buildup issues as the anterior fields do when re-entering the phantom after the air cavity. It should be Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 184 noted that t he posterior beam direction is not completely ideal either. Beams exi t ing phantom mater ia l and entering a low density mater ial do experience a slight bui ld-down region of dose just prior to the air cavity. U l t ima te ly physics w i l l l imi t the coverage of a P T V located adjacent to a significant air cavity. Table 6.9: Ra t io of contributions from the posterior fields versus the anterior fields Eclipse P B K 1.4 M C - D A O 2.7 Trea tment Efficiency : 7 F i e l d P h a n t o m T h e tota l number of monitor u n i t s , ( M U ) required to deliver the fluence-based p lan (Eclipse T P S ) is compared to the number of monitor units required to deliver the 6 aperture/f ield M C -D A O plan. The M U per field are listed i n Table 6.10. In this example, the M C - D A O p lan is 32% 'more efficient to deliver i n terms of the number of monitor units. Table 6.10: Phan tom: Compar i son of to ta l monitor units ( M U ) for V a r i a n Ecl ipse fluence-based plan and M C - D A O plan. F i e l d 1 F i e l d . 2 F i e l d 3 F i e l d 4 F i e l d 5 F i e l d 6 F i e l d 7 T O T A L Fluence Op t . M C - D A O 207 155 244 87 167 163 258 180 236 186 200 172 207 97 * 1519 * * 1040 * 6.4.3 Nasopharynx Recurrence Dose D i s t r i b u t i o n : N a s o p h a r y n x A 2 D axia l slice for the final M C - D A O forward calculated (Monte Car lo simulated) dose dis-t r ibu t ion is shown i n Figure 6.14. T h e M C - D A O generated an acceptable p lan that provides adequate coverage to the P T V (98% of the P T V is covered by the 95% isodose). A colorwash representation i n a l l three anatomical planes is shown i n Figure 6.15. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 185 Figure 6.15: Nasopharynx Recurrence M C - D A O plan w i t h colorwash dose representation. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 186 Dose V o l u m e His tog rams : N a s o p h a r y n x Ecl ipse f iuence-opt imized p lan - P B K vs. M C forward ca lcu la t ion Figure 6.16 demon-strates how Monte Ca r lo s imulat ion can reveal,,problems w i t h the P B K calculated doses for this small-field (< 6 x 6 c m 2 ) case. For this difficult nasopharynx recurrence example, the P B K - o p t i m i z e d and P B K - f o r w a r d calculated plan demonstrates fair coverage of the planning target volume (91% P T V volume covered by the 95% isodose surface).. However, when the same plan is calculated using Monte Car lo simulat ion, an under-dosage is revealed (16% P T V volume covered by the 95% isodose surface). The mean dose to the P T V for the M C forward calculated p lan is 9% lower than that predicted by the P B K algori thm. The issue appears to be associated w i t h the performance of the dose calculat ion algori thm. Figure 6.16: Monte Ca r lo reveals 9.0% under-dosage of P T V (mean dose) for this I M R T plan. S O L I D L I N E : P B K opt imized / P B K forward calculat ion. D A S H E D L I N E : P B K opt imized / M C forward calculat ion. M C - D A O p lan - M C forward ca lcu la t ion M C - D A O can generate a plan for this patient example that has dosimetric properties com-parable to that approved by the radiat ion oncologist on the original T P S (see Table 6.11). T h e difference is that the verification measurements now match the M C - D A O planned doses (see next section for p lan verification). Monte Car lo models the smal l field doses accurately. Dose (%) Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 187 O p t i m i z e d vs . F i n a l D V H T h e advantage of inc luding the M L C properties in the I M R T opt imiza t ion is demonstrated by compar ing the D V H obtained from the opt imizat ion step to the final D V H obtained after forward .calculation (Figures 6.17(a) and (b)). "T" Figure 6.17: T h e optimized D V H (before final calculation) and final D V H (after final dose calculation) are compared for the (a) fluence-based opt imiza t ion and, (b) D A O plans. ( D A S H E D L I N E ) = opt imized D V H . ( S O L I D L I N E ) = final D V H . For the fluence-based method, there is a discrepancy between the optimal D V H and the final D V H . T h i s is most notable for the P T V . T h e optimal D V H does not include M L C properties such as transmission. Since the D A O approach already includes M L C transmission and physical leaf mot ion con-straints dur ing the opt imizat ion , the optimal D V H is a much more faithful representation of the final D V H (after final calculation). However, the M C - D A O does not include a l l M L C properties. T h e leaf t ip shape and inter-leaf leakage is not included in the opt imizat ion . T h i s may be the reason for the smal l remaining deviat ion between the final and opt imized M C - D A O planning target volume D V H . Chapter 6. Monte Carlo Input for Direct 'Aperture Optimization for IMRT 188 Table 6.11: Doses to structures of interest - nasopharynx example. Prescr ip t ion Dose = 60Gy. T P S 'Accepted ' Dose M C - D A O P l a n (10 a p e r t u r e s / F L D ) P T V % V o l > 95% R x Dose 91% 98% M e a n Dose 100% 101% M i n Point Dose 87% 82% M a x Point Dose 111% 111% % V o l > 110% R x dose 0.0% 0.0% brainstem % V o l > 20 G y 1.7% 1.1% M e a n Dose 5.2 G y 7.1 G y M a x Point Dose 37 G y 29 G y L t . T e m p . Lobe % V o l > 15 G y dose 27% 22% M e a n Dose 9.8 G y 8.5 G y , M a x Point Dose 54 G y 54 G y R t . T e m p . Lobe % V o l > 15 G y dose . 43% 38% M e a n Dose 16 G y 14 G y M a x Point Dose 65 G y .65 G y C h i a s m % V o l > 15 G y dose 29% 34% M e a n Dose 12 G y 6.7 G y M a x Point Dose 38 G y 20 G y B O D Y H o t Spot Isodose(0.5 c m 3 Vol . ) 98% 100% % V o l > 110% R x Dose 0.0% 0.0% P l a n Ver i f i ca t ion : N a s o p h a r y n x Point dose verification measurements are performed on a uniform cylinder phantom w i t h a 0.6cc Farmer-type ion chamber insert. A n error of 1.2% based on reproducibi l i ty and posi t ioning only is assumed. T h i s verification test reveals a 9.0% discrepancy between the P B K calculated doses and measurement. Bouchard et al. (2004) caution that ion chamber measurements may exhibi t a large uncertainty (up to 10% for a single field) when applied to fields exhibi t ing a large fluence per turbat ion (e.g. I M R T fields) and an ion chamber I M R T correction factor may be required[155]. However, film verification results and micro-ion chamber (0.01 cc) measurements confirm this large discrepancy between measurement and the T P S doses. In addi t ion, the Monte Car lo calculated plan shows very good agreement w i t h the ion chamber data (0.2% difference) (see Table 6.12). T h e P B K algori thm is failing to model these relatively smal l field sizes (~5.5 x 3.5 c m 2 ) . In this example the dose discrepancy problem is not the air cavity because the same calculated-measured dose discrepancy was demonstrated on a uniform cy l indr ica l verification phantom. I Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 189 Table 6.12: Ion chamber point dose measurements i n cy l indr ica l verification phantom. Ca lcu la t ion M e t h o d Dose (cGy) Percent Difference from Measurement ((Planned - Measured) /Measured)* 100% Ion Chamber Measurement 146.96 ± 1 . 8 0.0% , P B K Convolu t ion 160.20 . +9.0% Mon te Car lo S imula t ion 147.23 ± 0.11 +0.2% Trea tment Efficiency : N a s o p h a r y n x T h e to ta l number of moni tor units ( M U ) required to deliver the fiuence-based p lan is compared to the number of moni tor units required to deliver the M C - D A O plan. The M U per field are l isted i n Table 6.13. Table 6.13: Nasopharynx Example : Compar ison of monitor unit ( M U ) delivery efficiency for V a r i a n Ecl ipse fiuence-based plan and M C - D A O F i e l d F i e l d F i e l d F i e l d F i e l d F i e l d F i e l d T O T A L 1 2 3 4 5 6 7 Fluence Opt . 126 208 137 97 119 79 93 * 859 * M C - D A O 93 74 90 92 82 58 82 * 571 * T h e M C - D A O p lan requires 33% fewer monitor units ( M U ) for this example. Note that even w i t h the 859 moni tor units quoted for the fluence-optimized P B K plan, the ion chamber verification measurements indicate that there is s t i l l a 9.0% under-dosage of the P T V . To even at tempt to resolve this discrepancy, the number of monitor units would have to be increased beyond 859 M U . 6.4.4 Massive Low Density Inhomogeneity - Lung Dose D i s t r i b u t i o n : L u n g A 2D ax ia l d is t r ibut ion for the Ecl ipse plan forward calculated using two methods ( P B K and M C ) is shown i n Figure 6.18. The M C plan reveals a significant underdosage of the P T V and G T V that is not predicted by the P B K dose calculat ion a lgor i thm (see Figure 6.18(b) and Table 6.14). T h e P B K does not model the lateral electronic disequi l ibr ium environment encountered in the lung. The effect is comparable to the A V I D phantom, example (wi th air inhomogeneity) described i n Sections 6.4.1 and 6.4.2. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 190 Figure 6.18: L u n g : (a) Ecl ipse TPS-generated I M R T plan w i t h P B K forward calculat ion, (b) save plan as (a), but w i t h M C forward calculat ion. Note massive underdosage revealed by M C . Figure 6.19: L u n g : M C - D A O 2D axia l dose dis t r ibut ion. T h e M C - D A O axia l dose dis t r ibut ion is shown in Figure 6.19. A l t h o u g h the coverage of the P T V by the 95% isodose is improved compared to Figure 6.18(b), it is physical ly quite difficult to achieve the goal of 95% volume coverage by the 95% isodose surface to the portions of the P T V encompassing the low density lung mater ial without over-dosing the solid tumour region ( G T V ) or the surrounding healthy tissue. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 191 Dose V o l u m e His tograms : L u n g Ec l ip se f iuence-opt imized p lan - P B K vs. M C forward ca lcu la t ion T h e D V H corresponding to Figures 6.18(a) and (b) are plot ted i n Figure 6.20. T h e only difference between the two plans is the type of final forward calculat ion used ( P B K vs M C ) . T h e p lanning objective is to cover at least 95% of the P T V volume w i t h the 95% isodose surface (See Table 6.6 for planning objectives). For the Eclipse fiuence-optimized plan w i t h P B K dose calculat ion, 95% of the P T V volume is covered by the 94% isodose surface (see Table 6.14). However, when the p lan is recalculated using Monte Car lo , a large underdosage of the P T V is revealed (95% of the P T V volume is covered by only 67% of the prescript ion dose and the mean dose drops by 17%). Th i s is due to the inabi l i ty of the P B K / M o d i f i e d B a t h o correction calculat ion to model the extent of lateral scatter of electrons i n the low density mater ia l (lung) dur ing the opt imiza t ion (and forward calculation) steps. The opt imiza t ion overestimates the dose to the P T V thus no attempt is made to boost the dose to this difficult region. A summary of several D V H plan characteristics for the Ecl ipse ( P B K forward calculated), the Ecl ipse ( M C forward calculated) and the M C - D A O plans is presented i n Table 6.14. T h e heart and spinal cord dose are not reported as they are essentially far enough from the treatment site to be considered as "uninvolved" structures. M C - D A O p l a n - M C forward ca lcu la t ion T h e D V H s generated by the idealized Ecl ipse fluence-optimization ( P B K calculat ion algo-r i thm) are compared to the M C - D A O plan D V H s in Figure 6.21. T h e M C - D A O p lan is able to achieve the target 95% prescription dose coverage of 95% of the P T V volume. T h i s is a major improvement over the 67% dose offered by the Ecl ipse plan (wi th M C forward calculat ion). T h e M C - D A O is unable to achieve the same dose homogeneity across the P T V as the idealized Ecl ipse P B K plan. T h i s is visualized as a shallower slope on the D V H curve. T h e dose homo-geneity is fundamentally l imi ted by the physics of particle transport as the photon beam crosses the lung- tumour- lung interfaces. A single 6 M V photon beam w i l l experience a re-build up region of ~1 .5 c m as it enters the solid tumour from the low density lung mater ial . In addi t ion, a smal l bu i ld-down region exists where the beam exits the tumour and re-enters the lung. T h e Chapter '6. Monte Carlo Input for Direct Aperture Optimization for IMRT 192 100 90 80 E 60 3 50 o > § 40 CD 0_ 30 20 10 0 : "Tx i A: • % .1 ^ 1 • % \ 1 -.* • 4. . : -~- - - •* •• •-. • a. -a • & PTV V • : ** •  -ft: : 1 | * g. j ^ . It » It-1* ;:* Iv \ * \ \ *. LJ""*1^^ \ 1 20 40 60 80 Percent Dose (%) too 120 Figure 6.20: L u n g D V H : The Eclipse T P S generates a fiuence-optimized treatment p lan based on P B K calculated beamlet dose distributions. The p lan then undergoes a forward calculat ion using two methods: 1) P B K convolut ion (solid line) and, 2) Monte Car lo (dashed line). These D V H s demonstrate a systematic dose calculat ion error. F igure 6.21: L u n g D V H : The Eclipse T P S w i l l generate an ideal fiuence-optimized treatment p lan based on P B K calculated beamlet dose distr ibutions. T h i s ideal Ecl ipse p lan D V H (dashed line) is compared to the M C - D A O generated D V H (solid l ine). Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 193 Table 6.14: Doses to structures of interest for lung example. Prescr ip t ion Dose = 70 Gy . P l a n Eclipse T P S Ecl ipse T P S M C - D A O Cr i t e r i a P l a n P l a n P l a n ( P B K For .Calc . ) ( M C For .Ca l . ) (3 apertures) P T V % Dose Covering > 95% V o l . 95% 67% 95% M e a n Dose 98% 81% 101% M i n Point Dose 78% 45% 50% M a x Dose (> 0.5 c m 3 Vol . ) 100% 93% 109% % V o l > 105% R x dose 0% 0% 13% G T V % Dose Cover ing > 95% V o l . 97% 84% 99% M e a n Dose 99% 89% 102% % V o l > 105% R x dose 0% 0% 2% L t L u n g % V o l > 20 G y 13% 12% 15% M e a n Dose 6.7 G y 8.4 G y 11.2 G y M a x Dose (> 0.5 cm3 Vol . ) 70.0 G y * 65.1 G y * 76.3 G y * R t L u n g % V o l > 10 G y dose 0.6% 0.2% 10% M e a n Dose 0.8 G y 1.0 G y 2.8 G y M a x Point Dose (> 0.5 c m 3 Vol . ) 20.2 G y 18.3 G y 23.6 G y B O D Y Hot Spot Isodose(0.5 c m 3 Vol . ) 100% 93% 109% % V o l > 105% R x Dose 0% 0% 0% Note: these max point doses fall w i th in the P T V increased P T V isodose coverage demands a compromise and it takes the form of increased dose to the ipsi lateral and contralateral lung dose, plus the overall 'hot-spot ' (maximum) radiat ion dose. T h e mean dose delivered to the left lung by the idealized Ecl ipse P B K plan and the M C - D A O p lan is 6.7 G y and 11.2 Gy , respectively. The mean dose to the right lung is 0.8 G y and 2.8 Gy , respectively. T h e global 'hot-spot' dose increased from 100% for the Ecl ipse P B K plan to 109% for the M C - D A O plan. Despite this increase i n dose to the structures-at-risk, the planning objectives defined in Table 6.6 are s t i l l met by the M C - D A O plan. O p t i m i z e d vs. F i n a l D V H T h e D V H obtained from the optimization step is compared to the final D V H (obtained after forward calculation) for both the Ecl ipse I M R T plan and the M C - D A O plan i n Figures 6.22(a) and (b), respectively. For the fiuence-based method (Figure 6.22(a)), there is a large shift (~5%) between the optimal D V H and the final D V H , par t icular ly for the P T V structure. T h e D A O method demonstrates closer agreement between the optimal and final P T V D V H (see Figure 6.22(b)). However, some discrepancy may be observed because of the fact that the op t imiza t ion does Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 194 Percent.Dose (%) Percentage.Dose (.%) (a) (b) Figure 6.22: T h e optimized lung plan D V H (before final calculation) and final D V H (after final dose calculation) are compared for the (a) fiuence-based opt imiza t ion and, (b) D A O plans. ( D A S H E D L I N E ) = opt imized D V H . ( S O L I D L I N E ) = final D V H . N O T include M L C leaf t ip and interleaf leakage properties. It was noted that left lung D V H curves do show a difference between the opt imized and final dose dis t r ibut ion. T h e reason for this is s t i l l under investigation. P l a n Ver i f i ca t ion : L u n g T h e dose d is t r ibut ion predicted by the Ecl ipse fluence-optimization and the M C - d i r e c t aperture op t imiza t ion are verified i n the 26.6 x 26.6 x 28.0 c m 3 A V I D phantom using bo th an IC10 0.14 cc cy l indr ica l ion chamber and K o d a k E D R 2 film located at isocentre. T h e absolute point dose measured at the isocentre is compared to the predicted (calculated) dose for the Ecl ipse fiuence-based p lan u t i l i z ing the single pencil beam kernel calculat ion a lgor i thm and M C . The results are summarized in Table 6.15. T h e measured doses are w i t h i n 1% agreement w i t h the Ecl ipse P B K calculated dose and the M C dose. T h i s agreement indicates that the P B K does not have an issue calculat ing doses to a uniform phantom for this combinat ion of field sizes and M L C shapes. A l so , the agreement verifies the M C simulated absolute doses as being correct. A n y problems w i t h the dose dis t r ibut ion on the cl inical lung patient geometry w i l l be due to tissue inhomogeneity effects. Two-dimensional coronal dose distr ibutions are also measured w i t h film ( K o d a k E D R 2 ) and Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 195 compared to the P B K calculated dose and Monte Car lo simulated dose for the Eclipse-generated I M R T plan. F i r s t a dose-to-pixel intensity cal ibrat ion curve is constructed. T h e coronal film plane for the Ecl ipse plan is then calibrated and compared to the Monte Ca r lo result (see F igure 6.23). The profile comparison indicates that good agreement is achieved on the uniform phantom. T h e large underdosage of the c l in ical patient lung P T V (Figure 6.20), as revealed by Mon te Car lo , is completely due to the lung inhomogeneity. Table 6.15: Ion chamber point dose measurements in A V I D verification phantom for the Ecl ipse fiuence-optimized lung plan u t i l iz ing the penci l beam kernel dose calculat ion. Dose (cGy) Percent Difference from Measurement Ecl ipse I M R T P l a n ((Planned - Measured) /Measured)*100% Ion Chamber Measurement 140.1 ± 2.8 c G y * P B K Convolu t ion 138.7 -1.0% Monte Ca r lo S imula t ion 141.3 ± 1.5 c G y +0.8% T h e M C - D A O p lan also undergoes ion chamber verification and film verification i n a uniform A V I D phantom. T h e results shown i n Table 6.16 and Figure 6.24. T h e M C measured dose to the centre of the phantom (isocentre) is w i t h i n 1% of the ion chamber verification measurement. I Me Open Export Info Results Dose (cGy) Figure 6.23: F i l m verification for Ecl ipse plan. P l a n generated w i t h Ecl ipse P B K T P S i n lung and transferred to a uniform phantom. A coronal film plane is compared to Monte Ca r lo simulations. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 196 50 75 Oose (cG 125 225 -80 -60 Max Display Dose (cOy i mean 1 0614 w*x -1 872 Dcrse-153 8634 cGy txxti siza * 0 5mm matrix sue * V0.5 K 170 6 «*nA2 : x«-»mm y».5mm Do5?.152 9925cOy pixel sie « 0.5mm matrix sfcte* 1705 X 1705mmA2 Fife name ' nHHHHH Coronal_Lung_DA02.iif Dose s c a m g i Ciientation. c o r o n a f i l e name : MCDAO2_NoG_008t« D o t s S c a i i n j t Figure 6.24: F i l m verification for M C - D A O plan. D A O plan generated from M C data is trans-ferred to a uniform phantom. A coronal film plane at isocentre is compared to Monte Ca r lo simulations. Table 6.16: Ion chamber point dose measurements i n A V I D verification phantom for the M C -D A O lung plan u t i l i z ing Monte Car lo dose calculat ion. Dose (cGy) Percent Difference from Measurement M C - D A O P l a n ((Planned - Measured) /Measured)* 100% Ion Chamber Measurement 142.8 ± 2.9 c G y Monte C a r l o S imula t ion 142.2 ± 1.6 c G y -0.4% The 2 D fi lm profile comparison (measured vs. Monte Car lo) indicates that good agreement i n two dimensions has been achieved. Note the annulus structure in the M C - D A O p lan coronal plane. T h i s is an indica t ion that the M C - D A O is a t tempting to compensate for the poor dose deposi t ion i n the annulus region of the P T V containing the lung material . Trea tment Eff iciency : L u n g T h e to ta l number of moni tor units ( M U ) required to deliver the fluence-based p lan is compared to the number of moni tor units required to deliver the M C - D A O plan. T h e M U per field are l isted in Table 6.17. T h e M C - D A O plan requires 51% fewer monitor units to deliver an acceptable plan. Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 197 Table 6.17: L u n g Example:- Compar ison of monitor uni t ( M U ) delivery efficiency for V a r i a n Ecl ipse fluence-optimized plan and M C - D A O F i e l d F i e l d F i e l d F i e l d F i e l d T O T A L 1 2 3 4 5 Fluence Opt . 199 103 145 201 154 * 802 * M C - D A O 77 79 105 61 73 * 395 * 6.5 Conclusion : MC-DAO A technique is presented for calculat ing Monte Car lo beamlet da ta using E G S n r c particle trans-port code for direct aperture opt imizat ion I M R T inverse p lanning ( M C - D A O ) . For t rad i t iona l ly difficult-to-calculate treatment geometries (small fields and/or large tissue inhomogeneities), the M C - D A O technique offers a clear benefit. The loss of lateral electronic equi l ib r ium i n low density materials is now properly modeled (phantom w i t h air cavity and lung examples). T h e addi t ion of direct aperture opt imiza t ion results i n a reduction i n the to ta l number of monitor units by ~30-50% for the three examples presented (phantom, nasopharynx, lung). T h e opti-mized D V H s generated by the M C - D A O software are already a fairly faithful representation of the final M C forward calculated doses as there is no addi t ional leaf sequencing step required. T h e M C calculated beamlets originate from an open'phase space that has been art if icial ly segmented into 2.5 x 5 m m 2 bins. M L C characteristics are not accounted for dur ing this stage. T h e D A O does incorporate M L C leaf transmission and mot ion l imita t ions but interleaf leakage and leaf t ip shape are not included. Th i s warrants further investigation. A l t h o u g h the agreement between the opt imized D V H s and the final forward calculated D V H s for the M C - D A O method are an improvement over the fiuence-based opt imiza t ion method, some discrepancy does s t i l l remain. T h i s is most l ikely due to the remaining sources of convergence error introduced dur ing the beamlet generation / opt imiza t ion stage as described above. Despite these remaining l imitat ions, M C - D A O technique w i l l be useful for p lanning small-field I M R T cases for P T V s located wi th in or adjacent to tissue inhomogeneities. T h e planning t ime from start to finish for a single patient is currently 3 - 4 days. T h i s t ime w i l l decrease as the entire process becomes more streamlined (the various programs can be scripted to provide a continuous t ransi t ion from one step to the next). It is estimated that a full Mon te Ca r lo D A O Chapter 6. Monte Carlo Input for Direct Aperture Optimization for IMRT 198 p lan inc luding the hnal dose calculat ion can be generated i n less than two days. A l l of the Monte Car lo calculations can be left to run unattended overnight. It would not be feasible, nor would it be a benefit to apply this technique to every patient treated at this cl inic. However, a site that par t icular ly benefits from this technique is the lung as it is t radi t ional ly one of the most difficult areas to calculate dose to accurately. Dose escalation studies have shown that the 5-year cancer control rate can be improved by increasing the treatment dose to the tumour (e.g. 5-year control rate was 12%, 35%, and 49% for 6369, 7484, and 92103 Gy, respectively)[156]. However, the l i m i t i n g factor preventing the delivery of these increased doses is severe morb id i ty rates to the lung organ (radiation-induced pneumonitis and fibrosis) [157]. The M C - D A O method can benefit the patient because the radiat ion dose can now be escalated unt i l the lung dose tolerance l imi t is met. T h i s can only be done because accurate lung dose information is now available. Overa l l , this technique w i l l offer a smal l number of patients (5 - 10%) a potent ial ly large benefit. Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 199 C h a p t e r 7 3D S a v i t z k y - G o l a y D i g i t a l F i l t e r f o r M o n t e C a r l o D o s e D i s t r i b u t i o n s D i g i t a l filters have been used extensively i n image and signal processing applications for several decades. T h e y are useful for contrast enhancement, smoothing, noise reduction and feature extract ion, just to name a few applications. D i g i t a l smoothing filters replace a raw data point, Pi, w i t h some form of local averaging or smoothing of surrounding elements, ps.mooth. In one dimension, the influence or weight of each of the surrounding elements on the data point of interest can be described by a function, fn, for example, a square, triangle, exponential or po lynomia l (see Equa t ion 7.1 and Figure 7.1). vTootH = £ f n p i + n ( 7 . 1 ) n=-nL where • riL = number of da ta points to the left of posi t ion i • TIR = number of da ta points to the right of posit ion i • riL + TIR = smoothing window size. If the form of / „ is a square wave, for example, the filter is called a moving window average (Figure 7.1(a)). For this type of filter, a data point is replaced by the mean value of the neighbouring elements w i t h i n a well defined window. The window is then moved over by one data element and the process repeated (Figures 7.2(a) and (b)). The coefficients that describe the elements of a moving window average are constant. Generally, this form of filter is not desirable for data containing high gradients (e.g. peaks or valleys) as the filter w i l l reduce the Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 200 > £Z Figure 7.1: (a) I D moving average square filter, b). tr iangle, c) Gaussian height of the peak and increase the w i d t h (see Figure 7.2(c)). More complex filters can be described by a higher order polynomial , often a quadratic w i t h the coefficients determined v i a a least squares fitting method (as described i n the following section). Generally, for a smoothing filter to work well , it is assumed that the data is bo th relatively slowly vary ing and corrupted by random noise. The data points must also be equally spaced. If the filter has invariant coefficients, convolution methods employing fast Fourier transforms may be applied. 7.1 Savitzky-Golay Filter (Least Squares Filter) A I D least squares po lynomia l fit method (see Section 7.1.1) was.reported by A . Savi tzky and M . J . E . Go lay i n 1964 and is often referred to as a Sav i tzky-Golay filter ( S G filter)[158]. Sav i tzky and Golay ' s goal was to remove random noise i n one dimensional spectro-photometer data for analyt ica l chemistry applications. T h i s group explored different filter options, such as the moving ' average or R C (exponential) filters. These filters d id reduce the noise in the data, but at the cost of a degradation i n the peak intensity. The least squares method fits the local Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 201 (c) Figure 7.2: Square wave filter (moving average)(a) original I D signal, (b) filter w i t h constant ampli tude moved across the signal calculat ing the average of values under the curve (c) F i n a l smoothed filter (dashed line) overlaid on original signal (solid line). da ta to a po lynomia l curve by min imiz ing a least squares objective function. T h e output value (ordinate) is obtained by re-introducing the input data (abscissa) value back into the equation for the curve. T h e coefficients of the curve are selected so as to minimize the square of the difference between the smoothed and raw data over al l data points. The filter coefficients w i l l vary as it moves across the data set as they are influenced by the element values and their associated uncertainty w i th in the smoothing window. T h e usefulness of a 3D generalization of this filter for Monte Car lo dose calculat ion appl i -cations was explored by Kawrakow i n 2002[103]. He furthered the development of this filter by adding a local ly adaptive smoothing window size (instead of a fixed window used by Savi tzky Chapter 7. 3D Savitzky-Golay Digital Filter for-Monte Carlo Dose Distributions 202 and Golay ) . The appropriate size for each dimension of the 3D smoothing window is queried based on a x 2 test (see Section 7.1.2). If the smoothed data w i t h i n the 3D window and the original raw data (wi thin the same 3D filter window) pass the x2 test, then the window size and the smoothed point at the voxel of interest is accepted. The result is that for low-gradient areas of a dose ma t r ix (e.g. open area of a square field), a larger smoothing window can be used. T h i s is an advantage because more da ta points are made available to create a more accurate least squares curve fit to the raw data. For high-gradient regions (e.g. field edges), a large smoothing window may degrade the slope of a steep dose fall-off. The window size can be reduced un t i l the x2 test passes the defined cri teria. There may even be situations where a da ta point w i l l undergo no smoothing at al l . The performance of the Kawrakow implementat ion of the adaptive Sav i t zky-Golay 3d denoising filter compared to other available 3D denoising methods, such as the content adaptive median hybrid fi,lter[159], wavelet threshold denoising[160], anisotropic dif-fusion[161], and iterative methods[145] was presented by E l N a q a (2003)[162]. Overa l l , the 7 x 7 x 7 voxel adaptive Sav i tzky-Golay filter clearly performed the best as assessed by reductions i n the mean square-error, the V a n D y k pass / fail cr i ter ia (using 2% / 2 mm) [117], and the smoothness of isodose lines. 7.1.1 Least Squares Algorithm for Curve Fitting The least mean squares ( L M S ) fit determines the ideal coefficients for a curve that fits a given set of data (Equa t ion 7.2) by min imiz ing the square of the difference between a desired and an actual signal (Equat ion 7.3) [163]. Assuming that a function, y(x) can be described by coefficients, a,\ ... o « , such that: y(x) =y(x;al...aM) (7.2) T h e n the appropriate coefficients are found when : N minimize over ai ... aM ^2[Vi — y{xi] a i - - - a i w ) ] 2 (7-3) A s an example, a smoothed dose point i n a 3D dose matr ix , D^ may be calculated from Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 203 a l inear combinat ion of the surrounding raw dose points, Di'j'k', where each raw dose point is scaled by a coefficient, aijk^j'k1 (see Equa t ion 7.4). i + i i j k , R j + j i j k , R k+kijk>R D i j k = Y Y Y a i j k , i ' j ' k ' Di'j'k' (7.4) • i ' = i - i i j k , L j ' = j - j i j i , L k' = k - k i j i i L 7.1.2 x2 Test T h e x2 test, or "goodness-of-fit" test provides a quantitative comparison between observed and expected results and estimates how likely the match is. x2 = E ^ H r ^ (7-5) Where • N = number of data points • Xi = measured da ta point • u, = mean of a l l da ta points, and • a = s tandard deviat ion of the data. It is expected that x2 w i l l be approximately equal to N, the number of da ta points. If x2 is much larger than expected, something is wrong w i t h the data comparison. In the appl icat ion of smoothing 3D dose matrices, if the smoothed data point has a value that is not likely, given the surrounding environment, this test w i l l fail. It is for this reason that the x2 test lends itself well as a " P A S S / F A I L " assessment cri ter ia for the adaptive smoothing window technique. H i g h gradient regions w i l l fail for large window sizes. The smoothing window is then "adaptively" reduced (asymmetrically, one dimension at a time) un t i l the smoothed data point becomes a "l ikely" value, thus passing the x2 test. A t any given t ime, this filter is comparing a smoothed set of da ta points to the raw data points. The x2 test cr i ter ia is often set such that x2/N must be < 1 to pass. T h e comparison between two 3D dose d is t r ibut ion matrices is shown i n E q u a t i o n 7.6. Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 204 Xijk _ . y** fPjjkji >j>k) Dy-yk1 J < ]_ (7 6) Nijk rip . j f = j — J L k'=k — kL ^ AD-i' JI k'. where • ' i , j , /c = locat ion of the central da ta point i n the 3D mat r ix • ih / ift, = left / right window w i d t h i n x-dimension • 3L I 3R = left / right window wid th i n y-dimension • kL j ku = left / right window w i d t h i n z-dimension • Pijk{i',f> k') — fit dose at posi t ion i',j', k' w i th in smoothing window • Di'j'k' = unsmoothed dose at posi t ion i',f, kl w i th in smoothing window • ADi'j'fi' — uncertainty i n voxel at posit ion i',j', k! w i th in smoothing window. • Nijk = number of elements i n the smoothing window • np = number of free parameters i n the nth order po lynomia l fit 7.1.3 Adaptive Window Savitzky-Golay Smoothing Algorithm K a w r a w k o w introduced the adaptive window concept into the t rad i t iona l Sav i t zky-Golay (least squares) filter for 3D Monte Car lo dose distr ibutions[l03]. The smoothed dose data at points (i',f, k') centred at voxel k), can be described not by a linear combinat ion of the surround-ing 'dose voxels as i n Equa t ion 7.4, but by a polynomial , Pijk(i', j', k')-.oi the form shown i n E q u a t i o n 7.7. n n—X n-X—n Piitfj', *') = £ £ E fcUi' - o v - WW - W (7-7) A=0 M=0 ^=0 where n is the power of the po lynomia l fit. A couple of assumptions are then made to simplify the problem of determining not only the coefficients of the po lynomia l , but also the uncertainty on the smoothed data point . Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 205 1. T h e stat is t ical uncertainties w i th in the 3D window of interest are s imilar such that A D ? . f c « <AD2> 2. A l t h o u g h any order of po lynomia l can be used, a 2nd order is assumed (n •— 2) 3. T h e 3D smoothing window is positioned symmetr ica l ly about the voxel of interest k). Solving for the coefficients, bx^, is not t r iv ia l . Mathematics-dedicated software (Maple) was used by T . Popescu to derive the equations describing each coefficient. T h e Sav i t zky -Golay filter implemented here was original ly started as a MAT LAB appl icat ion at the V i c t o r i a Cancer Centre. It was brought to our lab, debugged, converted to the C p rogramming language and opt imized for speed. A n example of the 6QOO coefficient for a 2nd order po lynomia l is shown i n Equa t ion 7.8[103]. -\ " i rij r l k &000.= A+i'j+j'.fc+fc' i' =—rii j' =—Tij /c' =—nfc where i _ ^ 5(P-9l) i>.i>.k>  A9i~^ l=i',j',k ni(nL + 1) 9i = 3 ' (7-• Nijk = to ta l number of voxels i n 3D window ((2rij + l ) (2n j + l)(2n'fc + 1)) • rij, rij, r%k = number of voxels in window to the left /r ight of central voxel 7.2 Filtered 3D Monte Carlo Dose Matrix 7.2.1 Reference Data Set ("Gold Standard") A seven-field I M R T plan is generated to conform a high dose region to a'difficult art if icial ' C -shaped target. The p lan undergoes Monte Car lo s imulat ion w i t h 300 mi l l i on histories per field (see F igure 7.3(a),(b) for axia l and coronal dose planes). The corresponding dose contour plots are also shown i n Figures 7.3(c) and (d). The uncertainty in the high dose region per field (ADi) is ~ 2 . 5 % . W h e n the seven fields are added together, the dose uncertainty per field (in G y ) adds i n quadrature per s tandard error propagation theory (Equat ion 7.10). T h e uncertainty to a Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 206 given voxel i n the sum plan, ADsum, is expected to be ~ 1 % . It is important to note that for an overall uncertainty of <1% in a sum plan, each field does not have to have <1% uncertainty. If N D s u m = ^ f t (7-9) then A D L m = E ( A A ) (7.10) (c) (d) Figure 7.3: Seven F i e l d I M R T reference plan w i t h 'C ' -shaped high dose region. M e a n uncer-ta inty i n high-dose region : ~1 .0%. (a) axia l dose plane, (b) coronal dose plane, (c) ax ia l contour plot, (d) coronal contour plot. Contour lines: 0.5, 1.0, 1.5, 1.8, 2.0, 2.2 Gy . Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 207 T h e reference p lan employing 300 mi l l ion histories per field is considered to be the "gold s tandard" for a l l subsequent comparisons. T h e average uncertainty i n this p lan is 7.2.2 Constant SG Filter Window Size / Varying Uncertainty in Data Set A single I M R T p lan is re-simulated w i t h decreasing sets of histories (photon simulations) per field, namely 50 mi l l i on (2% uncertainty), 25 mi l l ion (2.5% uncertainty) and 10 m i l l i o n (3.5% uncertainty) , and the data is de-noised using a 3D Sav i tzky-Golay filter ( 5 x 5 x 5 voxel max window). The goal is to assess if there is a m a x i m u m uncertainty level above which the Sav i t zky-Golay filter fails to be a useful smoothing filter. Figure 7.4(a) shows the locat ion of a line profile acquired through the axial dose plane. Figure 7.4(b) shows the dose profiles through al l four sets of da ta (300, 50, 25, and 10 mi l l ion histories) and (d) is a zoom image of the peaks. T h e associated uncertainty profiles are also plot ted i n Figure 7.4(c). It is clear from these plots that as number of histories simulated, TV, increases, the uncertainty, a, is reduced (according to standard a oc 1/y/N theory) and the agreement w i t h the reference da ta improves. , T h e coronal plane isodose contours are shown i n Figure 7.5. In this figure, the isodose contours for vary ing levels of overall dose uncertainty (2%, 2.5% and 3.5%) are compared to the reference image (1% error). A 5 x 5 x 5 voxel 3D Sav i tzky-Golay filter is applied to the coronal plane of the summed field dose d is t r ibut ion for the 3 distr ibutions shown in Figure 7.5(b),(c) and (d). T h e effect of the 3D S G filter on the : 1) qualitative isodose agreement, 2) root mean square difference ( R M S D ) , and 3) number of pixels that fail the gamma cri terion for these distr ibutions is examined. Isodose C o m p a r i s o n - S G filter appl ied to 7 -Fie ld S u m P l a n The isodoses obtained from the 3D S G filter applied to the summed (7 fields) dose d is t r ibut ion are shown i n Figure 7.6. Note that the S G filter is a 3D filter, but the isodose planes shown in the thesis are 2D planar subsets of the 3D volume. Visua l ly , the filter is very effective and producing a match between the 2% uncertainty dose d is t r ibut ion and the reference dose Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 208 dis t r ibu t ion (Figure 7.6(a),(b)). Improvement i n agreement for the filtered 2.5% uncertainty d is t r ibut ion is also demonstrated (Figure 7.6(c),(d)). However, the dose d is t r ibut ion w i t h 3.5% uncertainty d id not show a visual improvement in isodose lines (Figure 7.6(e),(f)). Th i s result indicates that there may be a l imi t to the effectiveness of this filter and that a m i n i m u m uncertainty value i n the raw dose d is t r ibu t ion is required. -84.9 -83.6 -82.4 -81.1 -79.9 -78.6 -77.4 -76.1 -74.9 cm (a) O ^_ t— LU c CD CL 300 M 50 M 25 M 10 M «-> - 2 - 1 0 1 2 3 cm (c) F igure 7.4: Dose d is t r ibut ion at coronal plane of 7 field I M R T plan. Monte Car lo s imulat ion performed w i t h 300, 50, 25 and 10 mi l l ion histories per field, (a) dose plane wi th locat ion of profile, (b) dose profile comparison through raw data, (c) uncertainty profile at same locat ion. Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 209 40 45 50 55 60 65 70 75 40 45 50 55 60 65 70 2.5% Unc. (25 Million histories / field) 3.5% Unc. (10 Million histories / field) (c) (d) Figure 7.5: Isodose contours through coronal plane for seven field I M R T example. Unfi l tered da ta is shown i n a l l images, (a) reference data set w i t h 1% average uncertainty i n the high dose region. T h e following contours show the reference dataset as an under-laying solid black contour, (b) 2% uncertainty plan (dashed),(c) 2.5% uncertainty p lan (dashed), and (d) 3.5% uncertainty plan (dashed). Chapter 7. 3 D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 210 Raw (Unfiltered) Doses 3D Savitzky-Golay Filter Applied '40 45 50 65 70 75 (a) (b) 70 . 75 70 75 (c) (d) (e) (f) F igure 7.6: Isodose contours. Sol id contour underlay = reference d a t a ( l % uncertainty), (a) raw dose d is t r ibut ion : 2.0% uncertainty, (b) S G filter applied to(a). (c) raw dose d is t r ibut ion : 2.5% uncertainty, (d) S G filter applied to (c). (e) raw dose d is t r ibut ion : 3.5% uncertainty, (f) S G filter applied to (e). Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 211 G a m m a M a p C o m p a r i s o n A G a m m a map comparison between the S G filtered doses and the reference data set is performed using 2 m m / 2% pass/fai l cr i ter ia (See Chapter 3.3.5 for description of the G a m m a factor). The effect of the field uncertainty on the 2D G a m m a map result (comparing raw da ta doses to reference dose (1% uncertainty)) is shown i n Figure 7.7. A 2D G a m m a dis t r ibut ion is shown, but a 3D extension of the G a m m a factor calculat ion could easily be applied (a nice feature for compar ing 3D dose distr ibutions) . T h e percentage of pixels that fail the G a m m a cr i ter ia (2 m m / 2% ( R x dose)) are shown in Table 7.1. (a) (b) (c) Figure 7.7: G a m m a M a p results comparing raw dose distr ibutions of varying uncertainty to reference dose plane (1% uncertainty), (a) 2% uncertainty dose d is t r ibu t ion (b) 2.5% uncertainty, (c) 3.5% uncertainty. G a m m a cri ter ia = 2 m m / 2% ( R x dose) T h e 3D Sav i t zky-Golay filter is applied to each field dose d is t r ibut ion prior to performing the final summat ion . The improvement i n the G a m m a map appearance is shown i n Figure 7.8. T h e percentage of pixels that fail the G a m m a cri ter ia (2 m m / 2% ( R x dose)) for these images is shown i n Table 7.1. R o o t M e a n Square Difference ( R M S D ) T h e R M S D is a commonly used quantitative measure of the average squared error between a set of parameters. T h e squared term renders i t sensitive to differences between two values (if the squared error is smal l then the actual difference w i l l be smal l as well). A 3D R M S D analysis is performed on the S G filtered dose distr ibutions (2%; 2.5% and 3.5% uncertainty compared. 1.0% uncertainty). The R M S D is defined i n Equa t ion 7.11. T h e effect Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 212 (a) (b) Figure 7.8: G a m m a M a p examples demonstrat ing improvement i n appearance after appl ica t ion of Sav i t zky-Golay (SG) filter, (a) raw dose d is t r ibut ion for the 2.5% uncertainty case (25 mi l l i on histories / field), (b) After S G filter has been applied to each field • and the result summed to create the final dose dis t r ibut ion. G a m m a cri ter ia = 2 m m / 2% ( R x dose) of the Sav i t zky-Golay filter on raw dose distr ibutions of varying uncertainty is shown i n Table 7.2 and Figure 7.11. E g i T,&1(DsG(i,3,k)-Dref(i,j,k))2 ( 7 n ) 7.2.3 3D Window Size In general, the larger the m a x i m u m 3 D window size set, the larger the number of voxels are used to fit the po lynomia l smoothing filter and the better the fit. T h e adaptive part of the adaptive window Savitzky-Golay filter indicates that the 3 D window is adjusted unt i l the x2 test cr i ter ia is met. For distr ibutions containing low dose gradients it may be possible to use a maximum, window size of 10 or greater voxels in a l l three dimensions. I M R T distr ibutions however tend to contain many high dose gradients. Even if the m a x i m u m window size is in i t i a l ly set to a very large value, i t w i l l be forced to shrink and shrink unt i l the x2 test cr i ter ia is met. For the example shown i n this section, a typical ' f inal ' window size is more i n the range of 5 x 5 x 5 voxels. There is really no point in setting the m a x i m u m in i t i a l window size to much larger than this value. The S G filtering process is slowed down immensely w i t h increasing 3 D window size. It is therefore advantageous to use a window size that w i l l ensure a good po lynomia l fit RMSD = \ Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 213 given the data it is t ry ing to smooth, while not wasting t ime on shr inking an overly-large search space. T h e Kawrawkow paper showed results from the appl icat ion of a 7 x 7 x 7 voxel 3D filter size. T h e I M R T examples tested i n this thesis confirm that this is a good window size, and even suggest that a 5 x 5 x 5 voxel may be sufficient for many cl inical cases. 7.2.4 Savitzky-Golay for Summed Multiple Field Dose Distributions For mul t ip le fields, the S G filter can be introduced i n two different places along the da ta pro-cessing chain. Each ind iv idua l treatment field may undergo S G filtration prior to being summed together to obtain the final dose dis t r ibut ion. Th i s would require n applications of the filter depending on the number, n , of treatment fields involved. Al ternat ively , the unfiltered doses can be summed together and the filter applied only to the final dose dis t r ibut ion. F igure 7.9 compares isodoses for the three dose distr ibutions being analyzed i n this section (2%, 2.5% and 3.5% uncertainty i n the seven-field sum plan). In the left-hand column, the raw dose d is t r ibut ion from the seven I M R T fields were summed, then the S G filter applied. In the r ight-hand column, the raw doses were filtered ind iv idua l ly and then the result ing seven fields summed together. The uncertainty on the ind iv idua l fields i n these examples translate as follows (defining seven fields / sum plan): • 1.0% uncertainty i n sum plan 2.6% per field • 2.0% uncertainty in sum plan 5.3% per field • 2.5% uncertainty i n sum plan 6.6% per field • 3.5% uncertainty i n sum plan 9.3% per field A G a m m a map was generated comparing the part icular S G method (Sum-Then-Fi l te r or F i l t e r -Then-Sum) to the reference data. The pass/fai l cr i ter ia set is 2 m m / d % (of R x dose) distance-to-agreement and dose difference. The number of pixels that failed the G a m m a test are recorded for each coronal dose comparison and the da ta stored i n Table 7-1 and plot ted i n F igure 7.10. Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 214 Field Data Summed, Then SG Filter Applied SG Filter Applied to Each Field, Then Summed (c) (d) (e) (f) Figure 7.9: Isodose contours comparing the order in which the 3D SG filter is applied. L E F T column: SG filter applied to each I M R T fields prior to summing final dose distribu-tion. R I G H T column: Raw field doses are summed and the SG filter is applied once to the result. (a),(b) 5.3% uncertainty/field, (c),(d) 6.6% uncertainty/field, (e),(f) 9.3% uncertainty/field. Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 215 f Table 7.1: G A M M A factor analysis (2% / 2 mm) for S G filtered data (coronal plane). " F A I L " = G A M M A value > 1 Number of R a w D a t a S u m T h e n A p p l y A p p l y S G Fi l t e r Histories S G filter T h e n S u m (% Pixels Fail) (% Pixe ls Fail) (% Pixe ls Fail) 50 m i l l i o n (2% Unc . sum plan) 4.9 4.7 3.0 25 m i l l i o n (2.5%' Unc . sum plan) 10.6 10.0 6.7 10 m i l l i o n (3.5%.Unc. sum plan) 21.7 17.8 17.1 A.30 •4-* w <D 120 CD § 1 0 b_ a> 5 o CU Figure 7.10: Effectiveness of the Sav i tzky-Golay filter i n terms of order of appl ica t ion and in i t i a l raw data uncertainty (millions of histories simulated per field). Note that the general (and expected) trend indicates that as the number of histories increases (and the uncertainty decreases), the number of pixels fail ing the gamma map test decreases. It is also expected that the appl icat ion of the S G filter w i l l improve the agreement w i t h the reference da ta (300 mi l l ion histories) compared to the raw data. Table 7.2: 3D R M S D analysis for raw and S G filtered data Number of R a w D a t a S u m T h e n A p p l y A p p l y S G F i l t e r Histories S G filter T h e n S u m (Gy) (Gy) (Gy) 50 m i l l i o n (2% Unc . sum plan) 0.018 0.017 0.015 25 m i l l i o n (2.5% Unc . sum plan) 0.025 0.023 0.021 10 m i l l i o n (3.5% Unc . sum plan) 0.039 0.034 0.031 -+- Raw Data "-+-• Sum Field Dose, Then SG Filter . ••+•- SG Filter Each Field, Then Sum 20 40 60 Number of Histories (Millions) Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 216 >f0.045 Q 0:04 W •| 0.035 8 0.03 G CD £ 0:025 i i ' a 0.02 CD i §0.015 CT w c 0.01 CD CD 2 0.005 o o n — i — Raw Data Sum Field Doses, Then SG Filter -+-• SG Filter Each Field, Then Sum ^ ^^^^ - v.. >^^_ *"*»>... ,%^ >n& " -+ - -10 20 30 40 50 Number of Histories (Millions) 60 Figure 7.11: 3D Roo t M e a n Square Difference data comparing to the reference dose dis t r ibu-t ion (1% uncertainty sum plan). The effect of increasing the number of histories s imulated and the order i n which the Sav i tzky-Golay filter is applied is shown. A s w i t h the appl icat ion of .most filters to de-noise or ' improve' raw data, caut ion must be employed. Monte Ca r lo is a statistics-based dose s imulat ion method. S imula t ing w i t h a large number of histories w i l l decrease the uncertainty i n the dose to a given voxel. However, the dose to the voxel falls on a Gaussian probabi l i ty dis t r ibut ion. Even though it is most l ikely that the dose is w i t h i n two sigma (2cr) of error, there is a non-negligible chance that the data point may lie i n the far t a i l of the probabi l i ty distribution,, fall ing beyond the predicted uncertainty-range and producing an outlier i n dose. W h e n applying the S G filter, it is possible that this outlier may cause the x2 to fail consistently unt i l a point is reached where the da ta point i s s imply not smoothed at a l l . Kawrakow (2002) showed that this may lead to a co ld /ho t voxel i n the smoothed (and raw) dose distribution[103]. The influence of an outlier on the to ta l dose d is t r ibut ion is decreased i f the dis t r ibut ion is broken down, into sub-sections of data first, the filter then applied to each piece, and the result re-summed together[103]. Rad i a t i on therapy treatment p lanning lends itself very well to this approach as we are often s imula t ing independent radia t ion fields, then summing the result to obtain a final dose dis t r ibut ion. It would make sense to apply the S G filter to each field separately then sum the filtered doses to obta in the final J Chapter 7. 3D Savitzky-Golay Digital Filter for Monte Carlo Dose Distributions 217 dose d is t r ibut ion . The disadvantage of this method is calculat ion efficiency, as the filter must now be applied n times for each of the n fields, as compared to one appl icat ion of the filter on the summed raw doses. It is for this reason that the size of the m a x i m u m 3D window size be carefully selected. The larger the in i t i a l window, the longer the denoising filter w i l l take to traverse the 3D dose mat r ix . A p p l y i n g the S G filter once, to the summed dose d is t r ibu t ion is not necessarily discouraged if t ime is an issue, but a caveat of caut ion should be applied. If the comput ing t ime is not a factor, the recommended method would be to filter the ind iv idua l fields first. Chapter 8. Conclusion 218 Chapter 8 Conclusion T h e original goal of this thesis was to explore the deficiencies of the popular single penci l beam a lgor i thm and at tempt to improve the spatial resolution abi l i ty of the dose deposit ion kernel for high dose gradient regions (as encountered in intensity modulated radia t ion therapy ( I M R T ) x-ray photon treatments). The penci l beam kernels that are t radi t ional ly derived from ion chamber measurements are increasingly found to be unsatisfactory for high-resolution I M R T dose calculations. The use of smal l volume detectors (e.g. diodes, film) dur ing commissioning w i l l improve the calculat ion accuracy by reducing spatial averaging of the input data. However, the use of a smal l volume detector may not completely resolve the problem. Other l imi ta t ions include the assumption of a spat ial ly invariant dose kernel, use of a coarse kernel sampl ing grid (2.5 mm) , lack of model ing the extra-focal radiat ion contr ibut ion, and lack of model ing the lateral electron transport i n inhomogeneous materials. The first part of this thesis proposes a technique that i terat ively optimizes pre-existing dose kernels to improve dose calculations for I M R T (as implemented i n the Var i an E c l i p s e ® v 6 . 5 / O S , System B u i l d 7.1.31). T h e modified dose kernels significantly improve the calculated dose accuracy for a 5 m m bar (peak-to-valley) test pat tern when compared to the original ion chamber-derived kernels. Bet ter agreement w i t h measured data is achieved when using modified dose kernels even when compared to the high resolution, film-derived dose kernels. A more modest improvement i n the calculated dose accuracy for two cl inical ly relevant single-field I M R T fluence maps is observed. Modi f i ca t ion of the kernel shape based on measured-to-calculated test pat tern dose agree-ment w i l l reduce the impact of some l imitat ions of the penci l beam convolut ion dose a lgor i thm. However, several l imita t ions w i t h this a lgor i thm remain - issues unrelated to the spat ia l reso-lu t ion capabi l i ty or spat ial invariance property of the dose deposit ion kernel. A n increasingly impor tant disadvantage of this a lgor i thm is related to the lack of model ing of lateral electronic Chapter 8. Conclusion 219 scatter i n different media, par t icular ly low density materials such as lung and air cavities (e.g. sinus). M o s t dose calculat ion algorithms rely on the presence of lateral electronic equ i l ib r ium w i t h i n the medium. T h a t is why the build-up region near the surface of a patient or phan tom has t rad i t iona l ly been a difficult area to calculate. Other situations may arise where lateral electronic equi l ibr ium is not established. Smal l treatment fields, where the half -width of the beam is less than the range of the secondary electrons set in mot ion by the photon beam (1.5 cm for 6 M V beam), experience significant calculat ion errors. Regions wi th in . and adjacent to low density materials (e.g. lung or air) can also experience a loss i n electronic equ i l ib r ium due to the increased scatter range of the electrons. The single penci l beam algor i thm relies on I D pathlength corrections to correct the dose beyond a non-water equivalent mater ial . T h e lateral dispersion of the secondary electrons are not accounted for. It is for these reasons that the bulk of this thesis moved its focus away from the single penci l beam algor i thm and towards Monte Car lo simulations. Monte Car lo provides accurate model ing of dose deposit ion i n many difficult geometries, inc luding wi th in , adjacent and beyond tissue inhomogeneities and smal l fields. Including Monte Car lo simulations i n bo th the inverse treatment p lanning step (beamlet optimizat ion) and the final forward dose calculat ion w i l l ensure that the treatment p lan w i l l be opt imized using correct dose information and the final dose d i s t r ibu t ion to the patient can be presented to the radiat ion oncologist w i t h confidence that the dose d is t r ibut ion is accurate. A method is presented for calculat ing Monte Car lo beamlet data using E G S n r c part icle trans-port code for direct aperture opt imizat ion I M R T inverse p lanning ( M C - D A O ) . For t rad i t iona l ly difficult-to-calculate treatment geometries (small fields and/or large tissue inhomogeneities), the M C - D A O technique offers a clear benefit. The loss of lateral electronic equi l ibr ium i n low density materials is now properly modeled. T h e addi t ion of direct aperture op t imiza t ion results in a reduct ion i n the to ta l number of monitor units by ~ 3 0 - 50% for the examples presented. T h e opt imized D V H s generated by the M C - D A O software are already a faithful representation of the final M C forward calculated doses as there is no addi t ional leaf sequencing step required. T h e M C calculated beamlets originate from an open phase space that has been art if icial ly segmented into 2.5 x 5 m m 2 bins. M L C characteristics are not accounted for dur ing this stage. Chapter 8. Conclusion 220 T h e D A O does incorporate M L C leaf transmission and mot ion l imita t ions but interleaf leakage and leaf t ip shape are not included. T h i s warrants further investigation. A l t h o u g h the agreement between the opt imized D V H s and the final forward calculated D V H s for the M C - D A O method are an improvement over the fiuence-based opt imiza t ion method, some discrepancy does s t i l l remain. T h i s is most l ikely due to the remaining sources of convergence error in t roduced dur ing the beamlet generation / op t imiza t ion stage as described above. A common cr i t ic i sm of the Monte Car lo method is the t radi t ional ly long calcula t ion times required to simulate the hundreds of mil l ions of particles. T h i s l imi t a t ion is rap id ly becoming a non-issue w i t h increased computer processing speeds and the implementat ion of large networks capable of dis t r ibuted computing. D i g i t a l filters such as the Sav i tzky-Golay filter can be used to denoise Mon te Car lo distr ibutions. T h e advantage of this method is that long simulations can be terminated sooner. The increased stat ist ical noise penalty associated w i t h early s imulat ion terminat ion can be smoothed. Explorat ions into the use of this filter i n an I M R T context results i n the recommendation that the sum dose fields have a to ta l uncertainty level of ~ 2.5 — 3.0% for the filter to be effective. For the seven field example, this translates into a mean uncertainty per field of ~ 6.5 — 7.5%. It is preferable to have ind iv idua l fields filtered prior to the summat ion that results i n the final dose dis t r ibut ion. T h i s is to maximize the effectiveness of this filter and to increase the probabi l i ty of smoothing large stat ist ical fluctuations that may occur i n i nd iv idua l radia t ion fields. T h e M C - D A O technique w i l l be useful for planning small-field I M R T cases for P T V s located w i t h i n or adjacent to tissue inhomogeneities. The planning t ime from start to finish for a single patient is currently 3 - 4 days. T h i s t ime w i l l decrease as the entire process becomes more streamlined (the various programs can be scripted to provide a continuous t ransi t ion from one step to the next) . It is estimated that a full Monte Car lo D A O plan inc luding the final dose . calculat ion can be generated i n less than two days. A l l of the Monte Car lo calculations can be left to run unattended overnight. It would not be feasible, nor would it necessarily be a benefit to apply this technique to every patient treated at this cl inic. However, as mentioned i n Chapter 6, a site that par t icular ly benefits from this technique is the lung as it is t radi t ional ly one of the most difficult areas to calculate dose to accurately. Dose escalation studies have Chapter 8. Conclusion 221 shown that the 5-year cancer control rate can be improved by increasing the treatment dose to the tumour (e.g. 5-year control rate was 12%, 35%, and 49% for 6369, 7484, and .92103 Gy, respectively) [156]. However, the limiting factor preventing the delivery of these increased doses is severe morbidity rates to the lung organ (radiation-induced pneumonitis and fibrosis) [157]. The MC-DAO method can benefit the patient because the radiation dose can now be escalated until the lung dose tolerance limit is met. This can only be done because accurate lung dose information is now available. Overall, this technique will offer a small number of patients (5 -10%) a potentially large benefit. f Chapter 9. Future Work 222 Chapter 9 Future Work T h e M C - D A O method presented here addresses several of the issues plaguing t rad i t iona l I M R T op t imiza t ion and delivery, namely: • systematic errors introduced by an inadequate dose calculat ion a lgor i thm in regions of tissue inhomogeneity and smal l fields • convergence errors dur ing the opt imiza t ion for the same reason stated above • the discrepancy between the opt imized and final D V H curves as observed for the fluence-based method because M L C properties have not been accounted for • inefficient treatment field delivery (large number of monitor units required to achieve the dose to the target). It is important to note that al though some of the M L C properties are included i n direct aperture op t imiza t ion (leaf mot ion l imita t ions and transmission), this is not a complete list. A full Mon te Ca r lo s imulat ion of the M L C , or an excellent model such as the Siebers et al. V C U transport code[132], w i l l include the leaf t ip shape and interleaf leakage. It may be possible to implement only the V C U particle transport code to calculate the effect of the changing M L C leaf positions on the beamlet dose dis t r ibut ion weighting. T h e V C U code may be fast enough to iterate the full open-field phase space through the new M L C aperture. A more efficient technique would be to only transport those particles that interact w i t h the ind iv idua l M L C leaf of interest and update just a subset of beamlet dose dis t r ibut ion weightings instead of the entire collect ion of open-field beamlets. Tradi t ional ly , the role of the flattening filter was to ensure that the forward peaked bremsstrahlung intensity d is t r ibut ion was modulated such that a uniform, rather than peaked, dose profile would Chapter 9. Future Work 223 be generated i n the patient at a depth of 10 cm. Th i s modula t ion simplified patient dosime-t ry for simple open and wedged beams significantly. For I M R T , the benefit of a flat profile is negated by the fact that the field is now deliberately modulated to generate complex flu-ence distr ibutions. Authors have recently proposed that perhaps it is t ime to eliminate the flattening filter as it serves only to attenuate the photon beam and requires longer beam-on times to achieve a prescription dose to the patient[164, 165, 166]. The treatment delivery ef-ficiency could potential ly be doubled w i t h the removal of this component. It is also a source of extra-focal scatter radia t ion which is not properly modeled i n many commercia l dose cal-cula t ion algorithms. Tomotherapy technology does take advantage of this design modif icat ion and does not include a flattening filter when generating the fan-beam of treatment x-rays. In a t rad i t iona l gantry-based linear accelerator, the overall x-ray beam spectrum, the energy fluence and the scattering characteristics w i l l change dramat ical ly w i t h the removal of this component. For example, the photon fluence at the beam central axis is 2.7 x higher w i t h the flattening filter removed (see Figure 9.1). The reduction i n extra-focal scatter radia t ion also reduces the to ta l whole-body scatter dose received by the patient, possibly decreasing the risk of secondary malignancies to parts of the body not direct ly involved w i t h the tumour being targeted. The removal of the flattening filter results i n more lower energy photons entering the treatment field that would have been otherwise attenuated out of the beam. T h e result is a reduction i n the beam hardening of the x-ray beam (mean beam energy is reduced). T h e dose deposit ion characteristics i n water (phantom) are different than when the filter is present. T h e beam has a sl ight ly lower penetrat ing power thus the dose fall-off is expected to increase (see Figure 9.1(b) for a percentage-depth-dose curve comparison). Some investigators have started to explore this opt ion and i t is an open area of research. Combin ing the modified linear accelerator w i t h the M C - D A O treatment p lanning approach could garner further increases i n external beam delivery efficiency. T h e denoising abi l i ty of the Sav i tzky-Golay filter has been shown to be useful when applied to dose matrices associated w i t h I M R T treatment fields. The filter is applied after the M C - D A O process has been completed. A l t h o u g h not explored i n this thesis, the Sav i t zky-Golay filter may offer a benefit when applied early on i n the M C - D A O process, namely, to the calcula t ion of the Chapter 9. Future Work 224 (a) (b) F igure 9.1: Effect of flattening filter on (a) photon fluence. and (b) percentage depth dose ( P D D ) for a 6 M V x-ray medical linear accelerator treatment beam, (dashed red line = flattening filter removed) / solid blue line — flattening filter present) thousands of ind iv idua l Monte Car lo beamlet (pencil beam) doses. T h e speed of the beamlet dose calculat ion process may be increased i f fewer photon/electron interactions per beamlet are simulated. One may also consider that for a fixed number of simulations / beamlet the op t imiza t ion convergence error introduced by the stat ist ical uncertainty of the beamlets may be reduced by in t roducing the Sav i tzky-Golay filter. ^ F ina l ly , the V a r i a n 2 1 E X medical accelerator has been benchmarked for a 6 M V x-ray photon beam. Higher energies are available c l inical ly (10 or 18 M V ) and i t would be beneficial to benchmark these beams as well. 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