Prospects for model-based dosecalculation. work on speed and accuracy

Transcription

1 Prospects for model-based dosecalculation in brachytherapy: VCU work on speed and accuracy Jeffrey F. Williamson, Ph.D., FACR Virginia Commonwealth University, Richmond, VA, USA Supported in part by NIH Grants R01 CA and R01 CA , and a grant from Varian Medical Systems

2 Model-based dose calculation issues What is a model-based dose-calculation algorithm (MBDCA)? What is the clinical rationale for MBDCA in brachytherapy? VCU s fast Monte Carlo code for MBDC VCU work on quantitative CT for measurement of low-energy photon cross sections Collaborators Washinton University EE: Jody O Sullivan, Dave Politte U Pitt (Radiol): Bruce Whiting VCU: J. Williamson, A. Sampson (GS 5), J. Evans (Ex GS) Yi Le (ex PD), Y. Yu (GS 4), D. Han (GS 1)

3 TG-43 dose-calculation algorithm: TG-43 is a table-based source-superposition Model D(r) G(r) TG43(r) F(r) g(r ) S G(r ) K 0 r Isodoses 100 cgy/h Dose distribution in water for single source TG-43 assumptions N s D(r) S K,i TG43(r r i ) i1 Superposition of multiple source doses Patients are composed of 30 cm diameter liquid water spheres Interseed attenuation, tissue composition inhomogeneities, and applicator shielding have negligable effects

4 Why Model-based Dose Calculation? TG-43 is correction-based algorithm Patient modeled as fixed-size, uniform water phantom Neglects tissue heterogeneities, seed-to-seed attenuation, applicator shielding effects, and tissue-air interfaces. High Energy: 137 Cs or 192 Ir Applicator shielding: 5-50% Tissue-air interface: <10% Low energy: 103 Pd or 125 I Tissue heterogenities: 5-100% Seed-to-seed attenuation: 5-10% Maughan et al. Med.Phys. 24:1241 (1997)

5 What is a model-based algorithm? An exact or approximate solution of the underlying radiation transport problem Patient Anatomy & segmentations MBDCA Cross section data for each tissue voxel and material ( en / ),, K,d K ( ) d at all ( r,e) for K coh, 0 Dose distribution PE, incoh Source/Applicator geometry Seed locations Pb marker Graphite pellet

6 What is in the MBDCA Box? ˆ (r, ˆ,E) (r,e) (r, ˆ,E) Net flux change Attenuation losses s(r, ˆ' ˆ,E' E) (r, ˆ',E')d ˆ' de' S(r, ˆ,E) Photon sources In-Scattering: ( ˆ ',E') to ( ˆ,E) D( r) (r,,e) E ( / )( r,e) dde MBDCA Box An approximate or exact numerical solution of the Boltzmann Transport Equation (BTE) first principles : discrete ordinates or Monte Carlo Heuristic : superposition/convolution en

7 What BTE solutions are available for MBDCA? Deterministic transport solutions: Discrete Ordinates Method (DOM) Systematically discretize (re) phase space and iteratively solve BTE as coupled set of difference equations Rapidly converging, statistically precise solutions but no guarantee of freedom from systematic error Example: Varian Acuros TM : 3-8 minute 3D dose calculations: HDR 192 Ir applicator attenuation corrections Zourari Med Phys 649: 2010 Example of DOM ray-effect artifact in HDR source dose distribution due to angular space discretization DOM Tetrahedral and rectangular meshes for describing patient and HDR source geometries Daskalov Med Phys 649: 1999

8 What BTE solutions are available for MBDCA? Basic Discrete Event Monte Carlo Algorithm Randomly select Location, direction & energy of primary photon Photon Collision select distance to next collision Select type of collision Select type of collision Ti capsule Ag core I-125 Seed Scoring Bin, V Heterogeneity Select Energy and angle of photon leaving collision Monte Carlo method Randomly construct large number of photon tracks or histories Outcome of every photon interaction is randomly selected from basic cross section data Dose = average energy deposited/detector mass per history Exact, approximation-free MC solutions of BTE are feasible Slow N 1/2 convergence to exact but statistically imprecise solution long computation times but unbiased solutions

9 Impact on Prostate Dose Delivery Carrier et al., IJROBP 68: 1190 (2007) 28 prostate 125 I seed post-implant CTs Compute D m,m vs. D w,w (TG-43) using ICRP-23 (1975) composition. Average effects: Overall: -6.9±2.0% Tissue Composition: -2.6±0.4% Interseed Attenuation: -4.0±1.7%

10 Impact of breast anatomy on Pd-103 doses Breast 103 Pd Implant D 90 Dm,m/ Dw,w: or Adipose Fraction Dw,m/Dm.m Uniform Breast Tissue Dw,m/ Dm,m: Segmented Breast Dm,m/ Dw,w: Segmented Breast Dm,m/ Dw,w: Uniform Breast Mixture Fraction of Adipose Tissue Patient Dw,m/Dm.m Segmented Breast Uniform glandularadipose mixture Homogeneous water Pd s/p lumpectomy patients with simulated plans in unirradiated breast Afsharpour et al., PMB 56: 7045 (2011) GEANT4 Monte Carlo

11 Novel model-based Dose-Calculation algorithms: VCU work Super-fast Monte Carlo using correlated sampling Work of Yi Le (Post doc) and Andrew Sampson (Med. Phys. Ph.D. student) Sampson, Ye, and Williamson: Med Phys (In Press) 2012 Chibani and Williamson: Med Phys 3688: 2005 Hedtjarn, Alm-Carlsson, and Williamson: Phys Med Biol 351: 2002

12 Then Correlated Sampling concept Average dose difference over simulated histories: corr ijk het hom D D (ijk) D (ijk) corr het D (ijk) D (ijk) D TG43 TG43 Phase space: Precomputed list of single-seed histories transported to seed surface hom TG43 corr ijk Because of phase-space source D (ijk) D (ijk) Variance of D (ijk) V D (ijk) 0 If het and het strongly correlated, then corr corr uncorr het ijk het V D (ijk) V D V D (ijk) TG43

13 Correlated Sampling Principles Work of Yi Le (Post doc) and Andrew Sampson (Ph.D. student) Sample photon trajectories, hom For voxel ijk, tally dose difference: corr het hom ijk ijk n ijk n D D D n r,e, n n n,w hom, assuming uniform water medium Create heterogeneous geometry het n n n n history r,e,,w by weight recalculation where hom hom P het 0 hom het,, n Wn Wn hom hom P hom 0,, n where P,, = probability of selecti ng,, 0 n 0 n het n Sampson, Ye, and Williamson: Med Phys 2012

14 Other correlated and Uncorrelated Code Features Chibani and Williamson: Med Phys 2005 Geometry Siddon voxel-grid ray tracing integrated with combinatorial geometry ray tracing Voxel indexing and phase-space Voxel-by-voxel cross-section table and density assignment, e.g., by EGS CTcreate Transport and scoring Efficient expected-value tracklength estimator Simplified tissue collision model: KN + PE (PTRAN_Correl only) Seed positions and contours from VariSeed Output to Pinnacle or in-house DVH software

15 Example I: Permanent Seed Implant for Partial Breast Irradiation High resolution, low energy, 3D breast CT * grid 0.67 x 0.67 x 0.81 mm 3 voxels Tissue segmentation: skin, adipose, and glandular 1 and 2 Tissue composition: Woodard and White 1986 Simulated lumpectomy cavity with permanent implant Spherical cavity (7 cc) with 1 cm CTV expansion (44 cc) Pd Theragenics model 200 seeds D 90 = 118 Gy planned by VariSeed using 2D TG-43 Protocol VariSeed optimized followed by manual adjustments Code implementation Based upon extensively benchmarked PTRAN code family Fortran 90: Intel Fortran compiler 10.0/ O2 optimization Executed on single 3.2 GHz processor of AMD Hexacore chip in Linux environment * Breast CT exam provided by Dr. John Boone, UC Davis

16 Results: Pd-103 Breast Implant Coronal cross section Patient prone 20% difference D 90 for MC dose to tissue

17 Breast Implant: Monte Carlo Dose/TG-43 Dose Monte Dose to Tissue/TG43 Monte Dose to Water/TG43 Tissue Kerma/TG43: ±8.6% in CTV; ± 34.5% outside CTV Water Kerma/TG43: ± 13.1% in CTV; ± 65.4% outside CTV C: 24%-60% by weight & O: 28%- 67% vs. 80% O in water

18 Monte Dose to Water/Monte Carlo Dose to Tissue Tissue Kerma/Water Kerma: 53.0 ± 7.2% in CTV; 62.6 ± % outside CTV

19 Example II: Permanent Seed prostate Implant I Model 6711 Seeds Prostate Volume 82 cc Planned Dose: V 145Gy = 85%, D 90 = 130 Gy Dose calculation cm 3 ROI with variable grid size (0.5 mm to 2 mm voxels) Day 30 post-implant CT exam with contoured prostate, bladder, urethra, and rectum Tissue and density assignments made through DOSXYZnrc code package ctcreate using a ramp function of 55 materials

20 Prostate Implant: MC Dose/TG43 Dose Monte Dose to Tissue/TG43 Prostate Composition H 9% -11% C 8% - 20% N 2% - 6% O 64%-79% Monte Dose to Water/TG43 Tissue Kerma/TG43: -8.2 ± 9.6% in CTV; ± 43.6% outside CTV Water Kerma/TG43: 1.7 ± 2.4% in CTV; -2.0 ± 35.5% outside CTV

21 Monte Dose to Water/Monte Carlo Dose to Tissue Tissue Kerma/Water Kerma: 12.1 ± 12.0% in CTV; 24.3 ± 43.7% outside CTV

22 Accuracy of Correlated Sampling Algorithm Prostate Breast D corr Duncorr No. SD Z x at voxel x 2 2 D corr Duncorr

23 Efficiency Gains Grid Size MC Code Time * EG CTV EG 20 EG 50 EG 90 Breast Case mm Uncorr 18.7 min 59.8 Corr 21.1 sec >99.9% 44.8 >99.9% % % Prostate Case mm Uncorr 15.3 min 37.1 Corr 38.6 sec 100% % % 33.6 >99.9% mm Uncorr 1.59 min 44.7 Corr 3.3 sec 100% % % % mm Uncorr 30.9 min 41.6 Corr 1.1 sec 100% % % % 2 (D uncorr ) tuncorr EGX% Mean Duncorr X% D 2 90 /100 (D corr ) tcorr * time to achieve average % = 2% within CTV % voxels with efficiency gain > 1

24 Efficiency Gain vs. HCF Breast Case Prostate Case The RED outline circumscribes the CTV voxels Heterogeneity Correction factor HCF Dhet Dhom 1 D Dhom

25 Efficiency Gain vs. Delivered Dose Breast Case Prostate Case The RED outline circumscribes the CTV voxels

26 Percent Standard Deviation vs. Delivered Dose within CTV and Equal CPU Times Breast Case Prostate Case Correlated Uncorrelated Correlated Uncorrelated 21 sec CPU time mm 3 voxels 39 sec CPU time mm 3 voxels

27 Breast Case Isodose Lines Prostate Case Isodose curves for low low-uncertainty uncorrelated Monte Carlo (dashed lines) and correlated Monte Carlo (solid). 39 s for prostate and 21 s for breast.

28 Future Work 1 mm voxels 2 mm voxels Hypothesis: Correlated Sampling MC may be able to use coarser voxels than conventional Monte Carlo because ΔD is smoother than D het

29 Conclusions Correlated Monte Carlo gives accurate and precise patient-specific 3D doses in a clinically feasible time. Accuracy: Solution matches un-correlated MC within statistical fluctuations Precision: Mean 2% std: < 3 min within CTV on 1 mm voxel grid Mean 2% std: < 20 sec within CTV on 2 mm voxel grid Parallel processing: another factor of 10 speed up Main downside: Large inhomogeneities decorrelate parallel histories Max %SD always reduced Weight windows solution under investigation

30 Inadequate knowledge of tissue composition ICRP and ICRU bulk tissue compositions Based on sparse measurements from the 1930 s to 60 s e.g. water content of prostate (82.5%) single specimen of 14 year old boy from 1935! 1 Substantial tissue composition variability e.g. water content of adipose tissue: 23% to 78% 2 Patient-specific distribution of tissue types e.g. breast glandularity: 16% to 68% 3 Need non-invasive method: x-ray CT In low energy range, cross sections can be not be described with fewer than two parameters. single-energy CT not an option Dual-energy CT is logical choice 1 A. H. Neufeld, Canadian Journal of Research 15B, (1937). 2 B. Brooksby, B. W. et al., PNAS 103 (23), (2006). 3 R. A. Geise and A. Palchevsky, Radiology 198 (2), (1996).

31 Impact of compositional uncertainties Impact of randomly distributed calcified voxels by %weight fraction for 103 Pd and 125 I prostate seed implants 103 Pd 125 I Chibani and Williamson Med Phys 3688: 2005

32 Materials of interest Head ~ 21 cm Body ~26 cm x 35 cm * J. F. Williamson, S. Li, S. Devic, B. R. Whiting, and F. A. Lerma, On two-parameter models of photon cross sections: application to dual-energy CT imaging, Med Phys 33 (11), (2006).

33 DE process ( x, E ) w ( x) ( E ) w ( x) ( E ) ( x, E ) w ( x) ( E ) w ( x) ( E ) Calibrate basis materials 2. Scan test material 3. Solve BVM for basis coefficients in each pixel 4. Can calculate linear attenuation coefficient at any energy 5. Compare DE estimate to NIST reference Ethanol

34 DECT limitation DECT cross-section estimation is highly sensitive to input CT image errors. 1 Random (noise) Systematic (artifacts) Essential research motivation: SIR may better support quantitative DECT photon crosssection estimation 1 J. F. Williamson, S. Li, S. Devic, B. R. Whiting, and F. A. Lerma, On two-parameter models of photon cross sections: application to dual-energy CT imaging, Med Phys 33 (11), (2006).

35 CT Image Reconstruction Problem Detector array P( 2,t) (x,y) Given: Measured transmission sinogram P(,t) d(y) i along,t log(transmission) i i patient X-ray source x Needed: 2D map of tissue attenuation coefficients (x,y) c(x)

36 CT Attenuation sinogram Detector Location Gantry angle

37 Image reconstruction is an inverse problem: Derive image c(x) from projections d(y) Body attenuation Scanner PSF image d(y) b(y :c') h(y x) c'(x) xx estimated sinogram Filtered backprojection (FBP) is exact analytic solution to inverse problem yy Measured Sinogram c(x) h(y x) Filter ln d(y)

38 Statistical Image Reconstruction FBP: data incompleteness, inconsistency, noise, nonlinearity = ARTIFACTS! Poses image reconstruction as an optimization problem Find the image most likely to have generated the measured data Assumes measurements are randomly distributed per Poisson or Gaussian Minimizes image noise Physically realistic forward model used to calculate expected data means from image estimate Eliminates model mismatch artifacts (streaking, cupping, etc) Image iteratively refined: maximize fit between measured and modeled data

39 Alternating Minimization (AM) Object model: Assume voxel x composed of N materials Forward model: Incident x-ray energy spectrum, 0 (E,y) Represents an implicit beam-hardening correction Scatter estimate, (y) AM object model N (x, E) (E)c (x) i1 For DECT measurements c(x) (x), (x) i i AM forward model g( y:c) (y) E E 0 (E,y)e h(y x) system matrix h(y x) N x i1 i(e)c i(x) J. A. O'Sullivan, and J. Benac, IEEE Trans Med Imaging, 26(3), (2007). J. Williamson et al., Med Phys 29: (2002)

40 Alternating Minimization (AM) Objective function: : current image estimate d(y): measured data g(y): modelled data from image estimate ( ) Id ( g) R( ) I(d g): data-mismatch term Minimizing I-divergence is equivalent to maximizing Poisson log-likelihood R( ): penalty function to smooth noise : strength of penalty function J. A. O'Sullivan, and J. Benac, Alternating minimization algorithms for transmission tomography, IEEE Trans Med Imaging, 26(3), (2007).

41 AM Algorithm: Maximum Likelihood Mean detector response predicted from model g(y:c) 0(E,y)exp h(y x) i(e)c i(x) (y) E x i d(y) = noisy measured sinogram P(d : c) y Probability of d(y) measuring d(y) e g(y:c) g(y : c) d(y)! Reconstructed Image = c(x) ˆ argmax log P(d : c) c

42 An Interesting E-step Property Maximizing logp(d : c) is equivalent to minimizing I[d(y) g(y : c)] I m b Csiszar's I-divergence, a measure of For non-negative functions m and b, I(m b) is the only discrepancy measure that satisfies Csiszar s general axioms of formalized inference theory distance between functions m and m( y) I m b m( y) ln m( y) b( y) yy b( y) b

43 AM Algorithm: M-Step Energy components of forward projection (k) (k) i (k) 0 x i i ˆ i q ˆ (y,e) I (y,e) exp (E)h(y x)c (x) (k) (k) d(y) p ˆ (y, E) q ˆ (y,e) (k) ˆq (y,e') i E' Measured and predicted backprojections (k) b (x) (E)h(y x)p ˆ (y,e) (k) i y y Iterative Update: E E i (k) b ˆ (x) (E)h(y x)q ˆ (y,e) (k1) (k) Get (k+1)-th estimate c ˆ (x) from c ˆ (x), i 1 (k) (k) ˆ (k1) (k) ˆ i ˆ c (x) c i (x) ln b i (x) b i (x) Z(x) i i

44 Iterative E-M reconstruction: FBP alternative Can incorporate realistic detector model into FP operation that includes nonlinear detector behavior More robust: will generate best image in presence of incomplete or inconsistent data Can constrain image formation process using a priori knowledge, e.g., known shape, composition of metal rods Drawback: very computationally intensive

45 Combine Alternating Minimization and Pose Search FBP FBP AM, no pose search 200 AM iterations wit 22 ordered subsets AM, with pose search Ryan Murphy, et al. Trans Med Imag 25: 1392 (200

46 Calculate MTF from each ESF. Integrate MTF up to a cutoff frequency. Resolution metric Intuitively represents fraction of input signal recovered after reconstruction AL 1 L L 0 MTF( f ) df Results here: L = 0.5 mm -1 Noise matched: [1.09% 0.01%]

47 Noise-Resolution Tradeoff Curves AM curves below FBP Less noise for same resolution Strongly edgepreserving (AM-700) NR curve shifts based on contrast Dose reduction potential: Ratio of variances at matched resolution metric 2 AM 2 FBP

48 X-ray energy spectrum estimation Al and Cu filters for transmission measurements No beam-hardening correction applied to data 2-parameter Birch-Marshall (BM) model for spectrum* kvp + inherent Al filtration (mmal) Minimize difference between measurement and model 90, 120, and 140 kvp beams All fit with less than 1.35% RMSE Fit kvp within 1 kev of nominal *J. M. Boone, The three parameter equivalent spectra as an index of beam quality, Med Phys 15 (3), (1988).

49 Off-axis hardening Transmission measurement only on central axis (CAX) Harden spectrum off-axis with known BT geometry

50 CAX scatter measurement 6.25 mm (1/4 ) Pb interrupts primary beam Can separate scatter and primary signals First order scatter correction: Assume constant scatter for all detector positions and gantry angles

51 Accuracy Comparisons FBP vs. Statistical Iterative Reconstruction (AM algorithm) Hypothesize AM will perform better than FBP Increase spectral separation between 2 scans Hypothesize DECT problem will be better conditioned 0.5 mm of Tin

52 Mean accuracy: Head phantom FBP Polyenergetic AM

53 Mean accuracy: Body phantom FBP Polyenergetic AM

54 Random uncertainty 29% NaClO 3 (E=28 kev) Image noise random uncertainty of DE cross-section. AM noise advantage less random crosssection uncertainty.

55 Distribution of errors for PMMA at 28 kev AM and FBP have same mean AM reduces random error 2- to 4-fold

56 FBP vs AM Performance: Mapping at 20 kev Uncertainty (1x1x3 mm 3 voxels) in image intensity and inplane voxel width needed to estimate at 20 kev with 3% uncertainty Basis pair (,) Water, 23% CaCl 2 Test Mixture % 140 kvp Sn Voxel size needed for 3% precision FBP AM 140 Std 140 Sn 18% CaCl % 2.1 mm 1.1 mm 7% CaCl % NaClO Teflon ETOH Water, Polystyrene 50% ETOH MEK PMMA

57 Conclusions Tissue and applicator inhomogeneity corrections are large Development of fast MBDCAs (MC or DOM) is nearly complete No conceptual/engineering barrier to implementing MBDCA for high energy sources and possibly prostate seed Modifying clinical physics practice is major effort Non-invasive mapping of cross sections is major unsolved problem low-energy BTx MBDCA DECT i i i b t i l d l t t

58 AM Algorithm: Maximum Likelihood Mean detector response predicted from model g(y : d(y) w,w = ) noisy I 0(y,E)exp measured h(y sinogram x) w (x) (E) w (x) (E) (y) E x P(d : w) P(d : w) y Probability of measuring d(y) e g(y:w) g(y : w) d(y)! d(y) Reconstructed Image = w(x) ˆ argmax log P(d : w) c Virginia Commonwealth University

59 An Interesting E-step Property Maximizing logp(d : c) is equivalent to minimizing I[d(y) g(y : c)] I m b Csiszar's I-divergence, a measure of distance between functions m and m( y) I m b m( y) ln m( y) b( y) yy b( y) For non-negative functions m and b, I(mb) is the only discrepancy measure that satisfies Csiszar s general axioms of formalized inference theory b Virginia Commonwealth University

60 AM-DE algorithm Given: 2 incident spectra, I(y,E) and 2 associated sinograms, Then, data likelihood P(d,d : c) (k1) i d(y), j 1,2 j is: 2 g(y:c) j j 1 2 e j1 y Yielding following update step: ĉ 1 (x) c ˆ (x) ln (k) j1 i 2 j g(y:c) d(y)! (k) (k) Z(x) i b (x) 2 j1 b ˆ i,j i,j j (x) d(y) j Virginia Commonwealth University

61 Results: Noiseless Data 300 iterations, 22 OS 512x512 1 mm pixels, Somatom-Plus geometry No regularization

62 Partial Density Images c 1 c 2 n

63 Does AM-DE converge? Is it biased? Create smaller scale problem 61 mm diameter cylinder, 64 2 pixels 360 source positions, 92 detectors in array Advantages: examine convergence at large iteration nos. More advanced performance metrics Image variance, where ĉ is an unbiased estimator of c Var c(x) ˆ 1 F (c) where Fisher information is given by x,x 2 ln p(d(y) : c F(c) = h(y x)h(y x')g(y :c) x,x' y c(x) c(x') d

64 AM-DE converges! 5,000 iterations 500,000 iterations Ratio: Estimated to true (x,20 kev) at 150,000 AM-DE iterations

65 AM Dual Energy Cross-Section Estimation Problem is to accelerate convergence rate so that AM-DE multi-component reconstructions are feasible Assess various regularization schemes using small-scale test problems Invertible Fisher Information matrices Use Fessler s extension of Cramer-Rao bound to biased (regularized) estimations Hypothesis: Condition number (ratio of max to min variance matrix eigenvalue) is measure of estimator stability Not invertible: insufficient data or ill posed Virginia Commonwealth University

66 What is a radiation transport solution? A numerical solution of the Boltzmann Transport Equation (BTE) ˆ (r, ˆ,E) (r,e) (r, ˆ,E) Net flux change Attenuation losses s(r, ˆ' ˆ,E' E) (r, ˆ',E')d ˆ' de' S(r, ˆ,E) Photon sources In-Scattering: ( ˆ ',E') to ( ˆ,E) giving the angular flux, (re), the density of photons as a function of position, direction and energy. From (re), any dosimetry quantity, dose, can be calculated. r en r D( ) (r,,e) E ( / )(,E) d de Virginia Commonwealth University

67 Angular Flux and Fluence Directions in, Energies in E,E E (b) ICRU particle fluence: (P) No. photons crossing sphere (P) Cross-sectional area, da (a) Angular Flux: ( r,,e) No. photons at r with energy near E and direction near () Unit Area, Energy and Solid Angle: a E intensity of photons near r with energy E and direction Phase space r,,e Virginia Commonwealth University