Prospects for model-based dose-calculation in brachytherapy: VCU

work on speed and accuracyJeffrey F. Williamson, Ph.D., FACR

Virginia Commonwealth University, Richmond, VA, USA

Supported in part by NIH Grants R01 CA 46640 and R01 CA 149305, and a grant from Varian Medical Systems

Model-based dose calculation issuesWhat is a “model-based dose-calculation algorithm

(MBDCA)”?What is the clinical rationale for MBDCA in

brachytherapy? VCU’s fast Monte Carlo code for MBDCVCU work on quantitative CT for measurement of

low-energy photon cross sectionsCollaborators

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)

TG-43 dose-calculation algorithm:TG-43 is a table-based source-superposition Model

r

100 cGy/h 50 20 10 7 5 3 1

0.5

Isodoses

Dose distribution in water for single source

Superposition of multiple source doses

TG430K

D(r) G(r)(r) F(r) g( r )S G(r )

Ns

TG43 ii 1

K,iD(r) S (r r )

TG-43 assumptions Patients are composed of 30 cm diameter liquid water spheres Interseed attenuation, tissue composition inhomogeneities,

and applicator shielding have negligable effects

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.

Maughan et al. Med.Phys. 24:1241 (1997)

High Energy: 137Cs or 192Ir Applicator shielding: 5-50% Tissue-air interface: <10%

Low energy: 103Pd or 125I Tissue heterogenities: 5-100% Seed-to-seed attenuation: 5-

10%

What is a “model-based” algorithm?An exact or approximate solution of the underlying radiation transport problem

Cross section data for each tissue voxel and material

Patient Anatomy & segmentations

Source/Applicator geometrySeed locations

4.50

0

0.826 0.612

3.14

0

0.89

0

0.560

1.09

0

0.510

Graphite pelletPb marker

MBDCADose distribution

100100

100

100

0

145145

145

145

45 290

290

290

290

r

en K K( / ), , , d ( ) d

at all ( ,E) for K coh, PE, incoh

What is in the MBDCA Box?

r r

enD( ) (r, ,E) E ( / )( ,E) d dE

Net flux change Attenuation losses

ˆ ˆIn-Scattering: ( ',E') to

s

(

ˆ ˆ ˆ(r, ,E) (r,E) (r, ,E)

ˆ ˆ ˆ ˆ(r, ' ,E' E) (r, ',E')

d ' dE'

Photon sources,E)

ˆS(r, ,E)

MBDCA Box

• An approximate or exact numerical solution of the Boltzmann Transport Equation (BTE)

– ‘first principles’: discrete ordinates or Monte Carlo– ‘Heuristic’: superposition/convolution

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 AcurosTM: 3-8 minute 3D dose calculations:

HDR 192Ir applicator attenuation corrections

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

Zourari Med Phys 649: 2010

Daskalov Med Phys 649: 1999

What BTE solutions are available for MBDCA?

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 N1/2 convergence to exact but statistically imprecise solution long computation times but unbiased solutions

Basic Discrete Event Monte Carlo Algorithm

select distance to next collision

Randomly select Location, direction & energy of primary photon

Select type of collision

Select type of collision

Select Energy and angle of photon leaving collision

Heterogeneity

Photon Collision

Ti capsuleScor ing Bin, V

I-125 Seed

Ag core

Impact on Prostate Dose Delivery

28 prostate 125I seed post-implant CTsCompute Dm,m vs. Dw,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%

Carrier et al., IJROBP 68: 1190 (2007)

Impact of breast anatomy on Pd-103 doses

37 103Pd s/p lumpectomy patients with simulated plans in unirradiated breast

GEANT4 Monte Carlo

0.4

0.6

0.8

1.0

1.2

0.0

0.4

0.8

1.2

1.6

0 5 10 15 20 25 30 35 40

Breast 103Pd Implant D90

Dm,m/ Dw,w: Segmented Breast

Dm,m/ Dw,w: Uniform Breast Mixture

Fraction of Adipose Tissue

Dw,m/Dm.m Uniform Breast TissueDw,m/ Dm,m: Segmented Breast

Dm

,m/ D

w,w

: or A

dipo

se F

ract

ion

Dw

,m/D

m.m

Patient

Segmented Breast

Uniform glandular-

adipose mixture

Homogeneous water

Afsharpour et al., PMB 56: 7045 (2011)

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) 2012Chibani and Williamson: Med Phys 3688: 2005Hedtjarn, Alm-Carlsson, and Williamson: Phys Med Biol 351: 2002

Correlated Sampling concept

Phase space: Precomputed list of single-seed histories transported to seed surface

corrijk het hom

corrcorrijkhet TG43

hom TG43

TG43 TG43

Then Beca

D D (ijk) D (ijk)

D (ijk) D (ijk) DD (ijuse of phase-space source

V

k) D (ijk)

D (ijk) V D (ijk) 0ariance of

corrcorr uncorrijkhet het V D (ijk) V D V D (ijk)

Average dose difference over simulated histories:

If het and het strongly correlated, then

Correlated Sampling PrinciplesWork of Yi Le (Post doc) and Andrew Sampson (Ph.D. student)

homn n n hom

het hetn n n n

hetn

n

n

Sample photon trajectories, assuming

uniform water medium Create heterogeneous geometry

history by

weight recalculation where

r ,E , ,W ,

r ,E , ,W

W

0

0

hom homhomn hom hom

het n

hom n

P , ,W

P , ,

corr het homijk ijk n ijk n

For voxel ijk, tally dose difference:

D D D

Sampson, Ye, and Williamson: Med Phys 2012

0 0 n nwhere probability of selectP , , = i , ,ng

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

Example I: Permanent Seed Implant for Partial Breast Irradiation

High resolution, low energy, 3D breast CT*

156×156×93 grid 0.67 x 0.67 x 0.81 mm3 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) 87 103Pd Theragenics model 200 seeds D90 = 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

Results: Pd-103 Breast Implant

20% difference D90 for MC dose to tissue

Coronal cross sectionPatient prone

Breast Implant: Monte Carlo Dose/TG-43 Dose

Tissue Kerma/TG43: -19.8 ±8.6% in CTV; +22.6 ± 34.5% outside CTV Water Kerma/TG43: +23.0 ± 13.1% in CTV; +100.6 ± 65.4% outside CTV C: 24%-60% by weight & O: 28%- 67% vs. 80% O in water

Monte Dose to Tissue/TG43 Monte Dose to Water/TG43

Monte Dose to Water/Monte Carlo Dose to Tissue

Tissue Kerma/Water Kerma: 53.0 ± 7.2% in CTV; 62.6 ± 22.7.0% outside CTV

Example II: Permanent Seed prostate Implant

78 125I Model 6711 Seeds Prostate Volume 82 cc Planned Dose: V145Gy = 85%, D90 = 130 Gy

Dose calculation 10×15×7.5 cm3 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

Prostate Implant: MC Dose/TG43 DoseMonte Dose to Tissue/TG43 Monte Dose to Water/TG43

Tissue Kerma/TG43: -8.2 ± 9.6% in CTV; -12.6 ± 43.6% outside CTV Water Kerma/TG43: 1.7 ± 2.4% in CTV; -2.0 ± 35.5% outside CTV

Prostate Composition H 9% -11% C 8% - 20% N 2% - 6% O 64%-79%

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

Accuracy of Correlated Sampling Algorithm

corr uncorr

corr uncorrx 2 2

D D

D DNo. SD Z at voxel x

Prostate Breast

Efficiency GainsGrid Size MC Code Time* EGCTV EG20 EG50 EG90

Breast Case

0.67×0.67×0.8 mm3

156×156×93

Uncorr 18.7 min 59.8>99.9%

44.8>99.9%

55.0100%

54.9100%Corr 21.1 sec

Prostate Case

1.0×1.0×1.0 mm3

102×150×72

Uncorr 15.3 min 37.1100%

20.398.2%

26.899.7%

33.6>99.9%Corr 38.6 sec

2.0×2.0×2.0 mm3

51×75×36

Uncorr 1.59 min 44.7100%

23.497.8%

31.899.7%

40.5100%Corr 3.3 sec

3.0×3.0×3.0 mm3

34×50×24Uncorr 30.9 min 41.6

100%22.2

97.7%30.0

99.8%38.1

100%Corr 1.1 sec

*time to achieve average % = 2% within CTV% voxels with efficiency gain > 1

2uncorr uncorr

X% uncorr 902corr corr

(D ) tEG Mean D X% D /100(D ) t

Efficiency Gain vs. HCF

het hom homHCF D D 1 D D

Breast Case Prostate Case

Heterogeneity Correction factor

The RED outline circumscribes the CTV voxels

Efficiency Gain vs. Delivered DoseBreast Case Prostate Case

The RED outline circumscribes the CTV voxels

Percent Standard Deviation vs. Delivered Dose within CTV and Equal CPU Times

Breast Case Prostate Case

21 sec CPU time

0.67×0.67×0.8 mm3 voxels

39 sec CPU time

1×1×1 mm3 voxels

• Correlated

• Uncorrelated• Correlated

• Uncorrelated

Isodose LinesBreast Case 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.

Future Work

2 mm voxels

Hypothesis: Correlated Sampling MC may be able to use coarser voxels than conventional Monte Carlo because ΔD is smoother than Dhet

1 mm voxels

ConclusionsCorrelated 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 upMain downside: Large inhomogeneities de-

correlate parallel histories Max %SD always reduced Weight windows solution under investigation

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

1A. H. Neufeld, Canadian Journal of Research 15B, 132-138 (1937).2B. Brooksby, B. W. et al., PNAS 103 (23), 8828-8833 (2006).3R. A. Geise and A. Palchevsky, Radiology 198 (2), 347-50 (1996).

Impact of compositional uncertainties Impact of randomly distributed calcified

voxels by %weight fraction for 103Pd and 125I prostate seed implants

103Pd 125I

Chibani and Williamson Med Phys 3688: 2005

Materials of interest

* 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), 4115-29 (2006).

“Body”~26 cm x 35 cm

“Head”~ 21 cm

DE process1. Calibrate basis materials2. Scan test material3. Solve BVM for basis coefficients in

each pixel4. Can calculate linear attenuation

coefficient at any energy5. Compare DE estimate to NIST

reference

1 1 1

2 2 2

( , ) ( ) ( ) ( ) ( )

( , ) ( ) ( ) ( ) ( )

x x x

x x x

E w E w E

E w E w E

Ethanol

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 cross-section 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), 4115-29 (2006).

CT Image Reconstruction Problem

X-ray sourcex

Detector array

i ii along ,t

Measured transmission sinogram P( ,t) d(y)

log(transmission)

2D map of tissue attenuation coefficien

Given:

Neets

: ded

(x,y ) c(x)

patient

(x,y)

2P( ,t)

CT Attenuation sinogram

Gantry angleDetectorLocation

Image reconstruction is an inverse problem: Derive image c(x) from projections d(y)

Filtered backprojection (FBP) is exact analytic solution to inverse problem

x Xestimated sinogram

d(y) b(y :c') h(y | x) c'(x)

Scanner PSFBody attenuation

image

Measured Sinogram

y Y

c(x) h(y | x) Filter ln d(y)

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

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)

N

i ix i 1

h(y|x) (E)c (x)

0E

h(y | x) system matrix

g(AM forward

y : c)m

(y)

E (E,y)e

odel

N

i ii 1

(x,

AM object

E) (E)c

model

(x)

J. A. O'Sullivan, and J. Benac, IEEE Trans Med Imaging, 26(3), 283-97 (2007).J. Williamson et al., Med Phys 29: 2404-2418 (2002)

For DECT measuremenc(x)

ts(x), (x)

Alternating Minimization (AM) Objective function:

’: current image estimate d(y): measured data g(y): modelled data from image estimate

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

( ) ( || ) ( )I d g R

J. A. O'Sullivan, and J. Benac, “Alternating minimization algorithms for transmission tomography,” IEEE Trans Med Imaging, 26(3), 283-97 (2007).

AM Algorithm: “Maximum Likelihood” Mean detector response predicted from model

d(y) = noisy measured sinogram

Probability of measuring d(y)

0 i iE x i

g(y :c) (E,y)exp h(y | x) (E)c (x) (y)

d(y)g(y:c)

y

c

g(y : c)P(d : c) e d(y)!

ˆReconstructed Image = c(x) log P(d : c)argmax

An Interesting E-step Property

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

Maximizing log P(d : c) is equivalentto minimizing I[d(y) || g(y : c)]

|| Csiszar's I-divergence, a measure of distance between functions and

( )|| ( ) ln ( ) ( )( )y Y

I m bm b

m yI m b m y m y b yb y

AM Algorithm: M-Step

(k )

(k)(k)0 i ix i

(k) (k)(k)

E'

(k)i i

ˆ ˆq (y,E) I (y,E) exp (E)h(y | x)c (x)

d(y)ˆ ˆp (y,

Energy components of forward projection

Measured and predicted backprojections

E) q (y,E)q (y,E')

ˆb (x) (E)h(y | x)p (y

(k )

(k) (k)

y E

(k)i iy E

(k 1) (k)i ii i

iIterative Update

,E)

ˆ ˆb (x) (E)h(y | x)q (y,E)

1 ˆˆ ˆc (x) c (x) ln b (x) b (x)Z (x)

:

i i(k 1) (k)ˆ ˆGet (k+1)-th estimate c (x) from c (x),

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

Combine Alternating Minimizationand Pose Search

FBP

FBP

AM, no pose search

AM, with pose search

200 AM iterations wit22 ordered subsets

Ryan Murphy, et al.Trans Med Imag 25: 1392 (200

Resolution metric Calculate MTF from each ESF.

Integrate MTF up to a cutoff frequency. Intuitively represents fraction of input

signal recovered after reconstruction

Results here: L = 0.5 mm-1

0

1 ( )L

LA MTF f dfL

Noise matched: [1.09% 0.01%]

Noise-Resolution Tradeoff Curves AM curves below FBP Less noise for same resolution

Strongly edge-preserving (AM-700) NR curve shifts

based on contrast

Dose reduction potential: Ratio of variances

at matched resolution metric

2

2AM

FBP

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)

*J. M. Boone, “The three parameter equivalent spectra as an index of beam quality,” Med Phys 15 (3), 304-10 (1988).

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

Off-axis hardening Transmission measurement only on

central axis (CAX) Harden spectrum off-axis with known BT

geometry

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

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

Mean accuracy: Head phantomPolyenergetic AMFBP

Mean accuracy: Body phantomPolyenergetic AMFBP

Random uncertainty Image noise random uncertainty of DE

cross-section.

AM noise advantage less random cross-section uncertainty.

29% NaClO3 (E=28 keV)

Distribution of errors for PMMA at 28 keV

AM and FBP have same mean AM reduces random error 2- to 4-fold

FBP vs AM Performance: Mapping at 20 keV Uncertainty (1x1x3 mm3 voxels) in image intensity and inplane voxel width needed to estimate at 20 keV with 3% uncertainty

Basis pair (,) Test Mixture  %

140 kVp Sn

Voxel size needed for 3% precision

FBP140 Std

AM140 Sn

Water, 23% CaCl2

18% CaCl2 0.31 % 2.1 mm 1.1 mm

7% CaCl2 0.20 3.4 1.8

29% NaClO3 0.22 3.0 2.3

Teflon 0.16 3.5 1.6

Water, Polystyrene

ETOH 0.09 15.1 7.1

50% ETOH 0.11 8.8 5.6

MEK 0.08 22.4 7.0

PMMA 0.09 8.7 3.8  

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

Virginia Commonwealth University

AM Algorithm: “Maximum Likelihood”• Mean detector response predicted from model

• d(y) = noisy measured sinogram

0E x

g(y : w ,w ) I (y,E)exp h(y | x) w (x) (E) w (x) (E) (y)

d(y)g(y:w)

y

c

g(y :w)P(d:w) e d(y)!

ˆReconstructed Image = w(x) log P(d:w)argmax

P(d:w) Probability of measuring d(y)

Virginia Commonwealth University

An Interesting E-step Property

For non-negative functions m and b, I(mb) is the only discrepancy measure that satisfies Csiszar’s general axioms of formalized inference theory

Maximizing log P(d : c) is equivalentto minimizing I[d(y) || g(y : c)]

|| Csiszar's I-divergence, a measure of distance between functions and

( )|| ( ) ln ( ) ( )( )y Y

I m bm b

m yI m b m y m y b yb y

Virginia Commonwealth University

AM-DE algorithm

j

j

j

j

d (y)2 g (y:c) j

1 2 j 1 yj

(k 1)i

Given: 2 incident spectra, and 2 associated sinograms, Then, data likelihood

I (y,E)d (y), j 1,2

g (y : c)

P(d ,d : c) e d (y)!

is:

Yielding following update ste

ˆ c

p:

(k )

(k )

2i, jj 1(k)

i 2i i, jj 1

b (x)1ˆ(x) c (x) ln ˆZ (x) b (x)

Results: Noiseless Data

• 300 iterations, 22 OS

• 512x512 1 mm pixels,

• Somatom-Plus geometry

• No regularization

Partial Density Images

• nc1 c2

Does AM-DE converge? Is it biased?• Create smaller scale problem

– 61 mm diameter cylinder, 642 pixels– 360 source positions, 92 detectors in array

• Advantages: – examine convergence at large iteration nos.– More advanced performance metrics

1x,x

2

x,x ' yd

Image variance, where is an unbiased estimator of

where

c cˆ Var c(x) F (c)

ln p(d(y) : c

Fisher informa

F(c) = h(y | x)h(y | x ')g(y : c

tion is given

)

by

c(x) c(x ')

AM-DE converges!

• 5,000 iterations • 500,000 iterations

• Ratio: Estimated to true (x,20 keV) at 150,000 AM-DE iterations

Virginia Commonwealth University

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

What is a radiation transport solution?

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 r

enD( ) (r, ,E) E ( / )( ,E) d dE

Net flux change Attenuation losses

ˆ ˆIn-Scattering: ( ',E') to

s

(

ˆ ˆ ˆ(r, ,E) (r,E) (r, ,E)

ˆ ˆ ˆ ˆ(r, ' ,E' E) (r, ',E')

d ' dE'

Photon sources,E)

ˆS(r, ,E)

A numerical solution of the Boltzmann Transport Equation (BTE)

Virginia Commonwealth University

Angular Flux and Fluence

Directions in ,Energies in E,E E

No. photon(b) ICR

s crossinU particle fluence

g sphere(P)Cross-section

: (P)

al area, da

No. photons at r with energy near E and direction near( )Unit

(a) Angular Flux: ( r, ,E)

intensity of photons nAre

eara, Energy and So

with enerl

gid Angle:

y E and dE

eca

irr

tion

Phase r, space ,E