VCU Radiation Oncology

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1 Semi-empirical Dose-Calculation Models in Brachytherapy AAPM 2005 Summer School 25 July 2004 Jeffrey F. Williamson, Ph.D. VCU Radiation Oncology Virginia Commonwealth University

2 Semi-Empirical Dose-Calculation Models Mathematical representation of dose distribution Small number of radionuclide-dependent but source-geometry independent parameters No dose measurements needed to implement for a given source History Isotropic point-source kernel and Sievert integral introduced by sievert, Quimby and Parker in 1920 s Displaced by Monte Carlo- and TLD-based table lookup methods in modern era Accurately represents cesium and iridium dose distributions

3 Model-based Dose Calculation Problem Statement Water-equivalent medium r r D D(r) Ý & Unencapsulated point source of Activity A Isotropic Point Source Air-kerma Strength = S K μgy m 2 /h

4 2004 AAPM Definition of Air-Kerma Strength K & (d) is air-kerma rate in vacuo due to photons > δ ( 5 kev) d L δ S K = K& δ(d) d [ μgy m h = cgy cm h = U] 2 2 ( W) D & air(1 cm) (1 cm) = X(1 & cm) (1 cm) = S K (1 g) S e ( W) = cgy/r g = Fx of KE emitted as photons e K

5 r 1 r 2 Point source emitting P photons Inverse Square Law No.incident photons Φ(r) = Area irradiated Activity P Time = 4πr 2 Dose(r) exposure(r) Φ(r) D(r 1 ) D(r 2 ) = Φ(r 1) Φ(r 2 ) = r 2 r1 2 D& D& med air free space ( ) ( ) ( ) med en en en / = μ ρ μ ρ μ ρ med air air

6 Tissue Attenuation Factor, T(r) 1.40 Tissue Attenuation Factor, T(r) Cs 198 Au 60 Co 169 Yb: Type 8 Seed 125 I: Model Ir: MicroSelectron HDR Describes competition between primary photon and attenuation and scatter buildup T(r ) = 1 ± 0.05 for r < 5 cm when E >200 kev Often derived from 1- D transport calculations Distance (cm) D T(r) = Tissue Attenuation Factor = & D & wat wat (r) in Water (r) in Vacuum

7 Isotropic Point-Source Kernel Assuming air-kerma strength calibration med S K ( μen / ρ) air med 2 D & (r) = T(r) r Assuming contained activity calibration Exposure-rate constant: ( ) Γ = δ x X & (r) r δ A 2 med ( ) ( ) Γδ (W/e) ( μ x en / ρ) air Γδ f x med D & med(r) = A T(r) = A T(r) 2 2 r r

8 TG-43 and Classical Point-kernel Λ D(1 & cm, π / 2) S g(r) L K wat ( μen / ρ) air T(1 cm) G L(1 cm, π /2) D(r, & π / 2) G L(1 cm, π/ 2) T(r) T(1 cm) D(1 & cm, π/ 2) G (r, π/2) L Generally, CPK and modern Monte Carlo-TLD dosimetry agreement within 2%-3% along transverse-axes of 192 Ir and 137 Cs sources

9 TLD-MC vs. Classical For 192 Ir Anctil 1998 Valicenti 1995

10 Y θ' x Δθ θ 1 θ θ 2 y P(r,θ) = P(x,y) Sievert Integral Geometry r r' L X ΔL t ( ) wat air μ t ΔL μen / ρ e Δ D(x,y) & = SK T(x / cos θ)e L 2 (x/cos θ) μ t/cosθ

11 Sievert Filtered Line-Source Integral wat air μ t ( μen / ρ) e 2 θ μ tsec θ θ D(x,y) & = SK e T(x sec θ) dθ Lx 1 μ = effective filtration coefficient (0.021 to mm -1 for Cs-137 and steel ) Best results: treat μ as fitting parameter, not assume μ = μ en L = active length, t = filter thickness Sievert model works well for Cs-137, but >10% errors for lower energy sources off transverse axis

12 Modern 1D Pathlength Algorithms Integrate CDK over 3D source geometry Actual Geometry Cylindrical Core Ceramic Line Source Steel Line Source

13 Modern 3D Pathlength Model with Best Fit CDCS-J 1992 Source 3D geometric Model Best Fit Filtration Coef. RMS error: 0.8% -0.9% to 4.1% range

14 Conventional Sievert Model CDCS-J 1992 Source Ceramic Line Source Model Filtration - μ en RMS error: 3.0% -5.4% to 10.1% range

15 1D Pathlength Model vs. Monte Carlo 3M Cs Tube 0.30 Model 3M 6500 Source: 2 cm Polar profile 0.29 Dose Rate/Air-Kerma Strength (cgy/h per unit U) Monte Carlo Best fit μ en Best Fit μ en } 3D } Geometry Ceramic Line Source Angle (degrees)

16 1D Pathlength Model: HDR 192 Ir %RMS error = 6.9%

17 Scatter Separation Model: HDR 192 Ir Williamson IJROBP 1996 %RMS error = 2.9% Max error = 8.9% D(r, & θ ) = primary dose at (r, θ) + transverse axis scatter dose at (r, π/ 2) D(r, & θ ) = D & (r, θ ) + D & (r, π/2) SPR(r) pri pri

18 Sievert Model: 125 I Seed Best-fit constant filtration coefficients μ Ti = mm -1 μ Resin = 0.00 mm -1 Fractional Error cm 0.5 cm 1.0 cm 2.0 cm 3.0 cm 5.0 cm Polar Angle (degrees)

19 Why is 6711 I-125 Λ lower? Classical ( ) med Λ = μ / ρ T(r ) en 0 air = = TG-43 Λ 85N,95D = % lower! 1985 NIST S K Standard contaminated by 4.5 kev Ti characteristic xrays (10% effect) Conventional T(r) neglects 22 kev Ag characteristic x rays (5% effect)

20 1D pathlength Model Shielded Fletcher-Suit Colpostat Weeks 1998: Best-fit μ for tungsten Solid Line: MCNP Monte Carlo Broken Line: 1D pathlength model

21 Recent Model-based Algorithms Scatter separation Williamson 1996: improves accuracy of 192 Ir anisotropy functions Superposition/Convolution Williamson 1991; Carlsson 2000 Accurate but slow Boltzmann transport equation solutions Discrete ordinates: Daskalov/Williamson Fast Monte Carlo: Le 2005, Chibani 2005 Integral transport: Zhou 2004

22 Current single-source dose superposition algorithm: patients are composed of homogeneous liquid water Cs-137 & Ir-192 sources: 15%- 50% applicator, shielding, bounded scatter-volume effects Pd-103/I-125 interstitial sources: 5%-50% interseed attenuation and tissue inhomogeneity effects 2005 Virginia Commonwealth University

23 Tissue Heterogeneity Effects 78 Model I seeds Single-Energy CT Material Analysis (CTcreate) Z-AXIS (cm) (A) (B) 2D dose ratio D hete /D homo profile A profile B profile C -4 (C) X-AXIS (cm) Virginia Commonwealth University

24 Efficiency Performance LINUX with single Xeon 3.0 GHz processor Geometry Code % σ Time (min) PTRAN_ mm 3 PTAN_CT PTRAN_ mm 3 PTAN_CT PTRAN_ mm 3 PTAN_CT Prostate 2 2 2mm 3 PTAN_CT 2 2 PTRAN_CT vs. PTRAN: 5 times more efficient 2005 Virginia Commonwealth University

25 PTRAN_CT Features Precomputed Phase space source Primary photons tracked to surface of source/applicator with unit weight saved. Used to initialize photon histories during patient simulations Ray tracing Fast parametric ray tracing through voxel grid (Siddon 1985) Preprocessing: identifies voxels that intersect seeds and applicators Analytic ray tracing limited to applicators contiguous to intersected voxels Fast algorithm for merging analytic and voxel path lengths

26 Phase Space Concept Precomputed singlesource histories Effective primary photons for patient simulation

27 PTRAN_CT Ray Tracing Analytic Ray Tracing + Voxel grid Tracing 4 regions and 8 surfaces/ 6711 seed x 78 seeds = Voxel grid Ray Tracing grid = 400K cells integrated Ray Tracing

28 Conclusions Isotropic point-kernel and 1D pathlength models are accurate TG 43 alternatives Interstitial seeds 192 Ir photon energies 137 Cs intracavitary tubes and shielded colpostats including 2D anisotropy 192 Ir anisotropy functions when 1D pathlength + scatter separation is used Semi-empirical model limitations Require careful benchmarking against Monte Carlo to validate for a class of problems Not validated for lower-energy sources More sophisticated models may not be competitive with Monte Carlo dose-calculation engines

29 The Future: Monte Carlo-based Dose-Calculation Subminute Monte Carlo dose calculation for full clinical implant geometries possible Implementation issues More complex applicator/seed localization RTP input will be source/applicator geometry or phase space, not single-source dose distribution Prostate/low energy seed issues How to measure tissue composition? dual-energy CT? CT artifact mitigation» Urethral contrast biases MC» Streaking artifacts from seeds mimic bone

30 Tandem & Colpostat Localization Lerma: Med Phys 29: Weeks CT-compatible applicators with aluminum bodies, afterloading tungsten shields, & Cs-137 tubes Find applicator pose that maximizes coincidence of Monte Carlo applicator geometry with CT contours of applicators