comparing the streamline diffusion method and bipartition ...€¦ · comparing the streamline...

Comparing the Streamline Comparing the Streamline

Diffusion Method and Bipartition Diffusion Method and Bipartition

Model for Electron TransportModel for Electron Transport

JipingJiping XinXin

Department of Mathematical Science, Department of Mathematical Science,

Chalmers University of TechnologyChalmers University of Technology

ContentContent

A brief introduction of the backgroundA brief introduction of the background

Comparing Comparing SdSd--method and bipartition model method and bipartition model

for electron transportfor electron transport

Future workFuture work

BackgroundBackground

Radiation therapy and TPSRadiation therapy and TPS

Two mathematical problems in TPSTwo mathematical problems in TPS

Transport theory and conservative lawTransport theory and conservative law

EquationsEquations

Overview of different research waysOverview of different research ways


Radiation therapyRadiation therapy is the is the

medical use of ionizing medical use of ionizing

radiation as part of cancer radiation as part of cancer

treatment to controltreatment to control

malignant cells. malignant cells.

From the website of Medical Radiation

Physics, KI

Pencil beam


In radiotherapy, In radiotherapy, Treatment Treatment

PlanningPlanning is the process in is the process in

which a team consisting of which a team consisting of

radiation oncologists, medical radiation oncologists, medical

radiation physicists and radiation physicists and

dosimetristsdosimetrists plan the plan the

appropriate external beam appropriate external beam

radiotherapy treatment radiotherapy treatment

technique for a patient with technique for a patient with

cancer. cancer.

From Phoenix TPS


Radiation transport simulations:Radiation transport simulations: This process This process involves selecting the appropriate beam type involves selecting the appropriate beam type (electron or photon), energy (e.g. 6MV, 12MeV) (electron or photon), energy (e.g. 6MV, 12MeV) and arrangements. and arrangements.

Our problem!Our problem! General, flexible, accurate, efficient General, flexible, accurate, efficient algorithm! algorithm!

Optimization:Optimization: The more formal optimization The more formal optimization process is typically referred to as forward planning process is typically referred to as forward planning and inverse planning inference to intensity and inverse planning inference to intensity modulated radiation therapy (IMRT).modulated radiation therapy (IMRT).


GMMC, Dep. of Math. Sci., CTH:GMMC, Dep. of Math. Sci., CTH:

Research project: Cancer treatment through the Research project: Cancer treatment through the

IMRT technique: modeling and biological IMRT technique: modeling and biological

optimization optimization

Models and methods for light ionModels and methods for light ion--beam transportbeam transport

Biological models and optimization for IMRT Biological models and optimization for IMRT

planning planning

QuestionsQuestions

How to describe this kind of physical How to describe this kind of physical

phenomena? phenomena?

Transport theory and conservative lawTransport theory and conservative law

Transport theoryTransport theory::

1.1. Transport theory is based on nuclear physics, quantum Transport theory is based on nuclear physics, quantum physics, statistical physics, etc. physics, statistical physics, etc.

2.2. Gas dynamic, neutron transport (nuclear weapon after Gas dynamic, neutron transport (nuclear weapon after WWII), astrophysics, plasma, medical physicsWWII), astrophysics, plasma, medical physics

3.3. Transport theory (Transport theory (discontinuous fielddiscontinuous field) is more ) is more microscopicmicroscopic compared with fluid dynamic or heat transfer compared with fluid dynamic or heat transfer ((continuous fieldcontinuous field) and more ) and more macroscopicmacroscopic compared with compared with quantum mechanics or nuclear physics, it describes a quantum mechanics or nuclear physics, it describes a group of particles. group of particles. Very important!!!Very important!!!

4.4. We study charged particle transport theory!We study charged particle transport theory!

Transport theory and conservative Transport theory and conservative

lawlaw

The particle phase space density: BoltzmannThe particle phase space density: Boltzmann

),,( Eurf ),,( zyxr = ),( φθ=u

•Cross sections!•Particle phase space density!•6 variables!

Photon: Compton scattering, pair

production, photon-electron

Electron: elastic scattering, inelastic

scattering, bremsstrahlung

Equations (Models)Equations (Models)

SE

Rf

E

SfffETfu +

∂++

−+−=∇⋅

2

2

2

2

2

2

1

1)1()(

∂

∂

∂

φ∂

∂

µµ∂

∂µ

µ∂

∂

Sff

ETE

Rf

E

Sf

y

f

x

f

z

f

yx

yx +

++

∂+=

∂++

2

2

2

2

2

2

)(2

1

θ∂

∂

θ∂

∂∂

∂

∂

∂θ

∂

∂θ

∂

∂

•Bipartition model

•Fermi equation (approximation)

•Fokker-Planck equation (approximation)

•Transport equation (Boltzmann equation)

SdEduuuEEEurfdEduuuEEEurffu ss +⋅−⋅=∇⋅ ∫ ∫∫ ∫∞∞

0 40 4'')',',(),,('')',,'()',',(

ππσσ

Our final goal is to solve the 6 dimensional pencil beam equation!!!

QuestionsQuestions

Which equation we should begin with? (M)Which equation we should begin with? (M)

The approximation (Fermi and FokkerThe approximation (Fermi and Fokker--Planck) is accurate? Planck) is accurate? In which cases they are accurate? (M)In which cases they are accurate? (M)

Which particle we should begin with? (photon Which particle we should begin with? (photon -- popular, popular, electron electron -- complex, ion complex, ion -- hot topic, proton) (M)hot topic, proton) (M)

How to solve the equation? Analytical method (FT), How to solve the equation? Analytical method (FT), numerical method (FEM), stochastic method (MC). (S)numerical method (FEM), stochastic method (MC). (S)

Obviously it will be difficult to solve the 6 dimensional Obviously it will be difficult to solve the 6 dimensional equations directly, then how to simplify the equations? (S) equations directly, then how to simplify the equations? (S) broad beam model and 2D pencil beam modelbroad beam model and 2D pencil beam model

How to go back to 6D model from simplified equations?How to go back to 6D model from simplified equations?


Sichuan University

Zhengming Luo’s group

Proton, Ion:

Fermi, FT, 6

Photon:

TE, CL, 6

Electron:BiM, FT, FDM, 3Hybrid PBM, 6

Karolinska Institute

Anders Brahme’s group

Ion:

Fermi, FT, 6

University of Kuopio

Jouko Tervo’s group

TP 4 FEM

TP 6 FEM (failed)

Chalmers

Mohammad Asadzadeh

Fermi and FP, 3, FEMEGSnrc

Monte Carlo


Electron transport (more complex than ion)Electron transport (more complex than ion)

11--50 50 MeVMeV (Fokker(Fokker--Planck approximation is Planck approximation is

accurate)accurate)

FokkerFokker--Planck equation (PDE)Planck equation (PDE)

Broad beam model (3 dimensions)Broad beam model (3 dimensions)

Finite element method (Finite element method (SdSd--method)method)

Comparing Comparing SdSd--method and bipartition method and bipartition

model for electron transportmodel for electron transport

FokkerFokker--Planck equation for 1Planck equation for 1--50 50 MeVMeV

electron transportelectron transport

Broad beam and 2D pencil beam modelsBroad beam and 2D pencil beam models

Bipartition modelBipartition model

Results by Results by SdSd--methodmethod

FokkerFokker--Planck equation for 1Planck equation for 1--50 50

MeVMeV electron transportelectron transport

[ ]

[ ]

[ ] EdudEEEuuEEEEurfZA

DN

EdudEEEuuEEEEurfZA

DN

EdEEEEurfA

DNEdEEEEurf

A

DN

uduuEEurfEurfA

DNEurfu

E

EM

A

E

EM

A

E

rA

E

Er

A

NA

′′′−−⋅′′−−

′′−′′−⋅′−′′′′+

′′−−′−′′′+

′⋅′−′=∇⋅

∫ ∫

∫ ∫

∫∫

∫

2/ 4

4

0

4

),(),(),,(

),(),(),,(

),(),,(),(),,(

),(),,(),,(),,(

0

0

π

π

π

ϕδσ

ϕδσ

σσ

σ

Transport equation



The screened Rutherford cross section:The screened Rutherford cross section:

( ) ( )

−

+−

−+

+−+

++=

ξ

πββ

ξ

πβ

ηξξσ

1

1

1372

1

1

1

137221

1

)2(

)())(1(),( 2

2/3222

0

2

22

0

22

0

2

0 ZZ

cmEE

cmEcmZZrEN

( )( )

( )( )2

0

22

0

2

0

22

0

223/1

2213776.313.1

25.1214

1

cmEE

cm

cmEE

cmEZZ

+

+

+

+

=η

2

0

2

0 )2(

cmE

cmEE

+

+=β



BremsstrahlungBremsstrahlung cross sectioncross section

( ) ( ) ( )

∆

−+

−

−−

−+= γ

γφσ

E

TEZ

E

TE

E

TE

T

ZrTEr

6ln

3

1

43

21

137

4),( 1

2

222

0

3/1

2

0

)(100

ZTEE

Tcm

−=γ ( ) γγγφ 22

111

baebea

−− += ( ) γγγ 22

11

dcedec

−− +=∆

MollerMoller cross sectioncross section

( ) ( )( ) ( ) ( )

( )( )

−+

+−

++

−+

+

+=

TE

T

cmE

cmcmE

E

T

cmE

E

TE

T

TcmEE

cmEcmrTEM 22

0

2

0

2

0

2

22

0

2

2

2

22

0

22

0

2

0

2

0 21

1

2,σ

QuestionsQuestions

How to derive FokkerHow to derive Fokker--Planck equation from Planck equation from

transport equation and check the accuracy?transport equation and check the accuracy?



The classical contributions about the FokkerThe classical contributions about the Fokker--Planck Planck

approximation are summarized by Chandrasekhar in [28] and approximation are summarized by Chandrasekhar in [28] and

RosenbluthRosenbluth in [32]. [29,30,31] are studying the case of the in [32]. [29,30,31] are studying the case of the

linear particle transport. These works give linear particle transport. These works give a heuristic a heuristic

derivationderivation of the Fokkerof the Fokker--Planck operator. In [33] Planck operator. In [33] PomraningPomraning

gave a formalized derivation of the Fokkergave a formalized derivation of the Fokker--Planck operator as Planck operator as

an asymptotic limit of the integral scattering operator where a an asymptotic limit of the integral scattering operator where a

peaked scattering kernel is a necessary but not sufficient peaked scattering kernel is a necessary but not sufficient

condition in the asymptotic treatment. condition in the asymptotic treatment.



LaplaceLaplace’’s method for Laplace integralss method for Laplace integrals

∫=b

a

txdtetfxI

)()()( φ

+∞→

=≠

≤≤=

xxI

ctcf

bcactt

as )( ofexpansion asymptotic full the toscontribute

that of oodneighbourh only the isit then ,0)(

if and with at maximum global a has )( If φ



errors. smalllly exponentia introducesonly n integratio of

limits changing thesinceion approximat justified a is This

. if ,)(

, if ,)(

, if ,)(

);(

where);(by )( eapproximatmay we1. Step

)(

)(

)(

=

=

<<

=

∫

∫

∫

−

+

+

−

b

b

tx

a

a

tx

c

c

tx

bcdtetf

acdtetf

bcadtetf

xI

xIxI

ε

φ

ε φ

ε

ε

φ

ε

ε



expansion. series asymptoticor Taylor

itsin termsfewfirst by the )( replace to validisit

thatsoenough smallchosen becan 0 Now 2. Step

tφ

ε >

errors). smalllly exponentia introducesonly s(again thi

integrals resulting theevaluate order toin infinity, n tointegratio of

endpoints theextend now we),0)( (cont. above defined being

with and for ionsapproximat thedsubstitute Having 3. Step

≠cf

fφ



Surface harmonic expansion and Legendre Surface harmonic expansion and Legendre

polynomial expansion polynomial expansion (important part no used in(important part no used in

the heuristic derivation for the angular integration)the heuristic derivation for the angular integration)

∑ ∑∞

= −=

+=

0

)(),(4

12),,(

n

n

nm

nmnmnm uYErfan

Eurfπ

)(),(4

12),,(

0

uuPEEk

uuEE ksk

k

s ⋅′′

+=⋅′′ ∑

∞

=

σπ

σ

∫ ∫

∑ ∑∫ ∫∞

−

∞

= −=

∞

′′′

⋅

+=′′′′⋅′′

0

1

1

00 4

),,()()(

)(2

12),,(),,(

EddEEPEf

uYan

EdudEurfuuEE

snnm

n

n

nm

nmnms

ξξσξ

σπ



Example: the screened Rutherford cross section Example: the screened Rutherford cross section

for elastic scattering for elastic scattering

[ ]∫ ′⋅′−′π

σ4

),(),,(),,( uduuEEurfEurfA

DNN

A

∑ ∑ ∫∫∞

= −=−

+=′⋅′′

0

1

14),()()(),(

2

12),(),,(

n

n

nm

NnnmnmnmNA dEPuYErfa

nuduuEEurf

A

DNξξσξσ

π

∫−1

1),()( ξξσξ dEP Nn



Step 1:Step 1:

21

1

1

1

1

:),()(),()(

),()(

IIdEPdEP

dEPI

NnNn

Nn

+=+=

=

∫∫

∫

−

−

ε

εξξσξξξσξ

ξξσξ

).( polynomial Legendre theof

root positivelargest theis Here ,1). ( Choose

ξ

ξξε

n

nn

P

∈

21

1

02

1 0

),()(

),(lim II

dEP

dE

I

Ia

Nn

N

a≈⇒=≤

∫

∫−

−

+→

ε

ε

ξξσε

ξξσ



Step 2:Step 2: ( ) 2

1 )1(

2 :),()1)(1()1( IdEPPI Nnn′=−+≈ ∫ε ξξσξ

( )∫− −+=′ε

ξξσξ1

)1(

1 ),()1)(1()1(: dEPPI NnnStep 3:Step 3:

( )( ) 21221 )1(

1

)1(

02

1 0),()1)(1()1(

),()1)(1()1(lim IIIII

dEPP

dEPP

I

Ia

Nnn

Nnn

a′+′≈′≈≈⇒=

−+

−+=

′

′

∫

∫−−

+→

ε

ε

ξξσξ

ξξσξ

( )∫− −+≈1

1

)1(),()1)(1()1( ξξσξ dEPPI Nnn



[ ]

( )

∫

∫

∫

∫

∑ ∑ ∫

∫

−

∞

= −=−

−=

∂

∂

−+

∂

∂−

∂

∂=

′⋅′−

∂

∂

−+

∂

∂−

∂

∂+′⋅′=

′⋅′−

−+

+=

′⋅′−′

1

11

2

2

2

2

1

4

2

2

2

2

14

4

0

1

1

)1(

4

),()1()(

)1(

1)1()(

),(),,(

)1(

1)1()(),(),,(

),(),,(

),()1)(1()1()(),(2

12

),(),,(),,(

ξξσξπ

φµµµ

µ

σ

φµµµ

µσ

σ

ξξσξ

σ

π

π

π

π

dEA

DNET

fET

uduuEEurfA

DN

fETuduuEEurfA

DN

uduuEEurfA

DN

dEPPuYErfan

A

DN

uduuEEurfEurfA

DN

NA

NA

NA

NA

n

n

nm

NnnnmnmnmA

NA



( )222

0

2

22

0

22

0

22

0

21

1

)2(

)()(),(

ηξξσ

+−+

+=

cmEE

cmEcmZrEN

The screened Rutherford cross section without corrections for low energy electron transport

This is the advantage of bipartitonmodel for large angle!



BremsstrahlungBremsstrahlung and inelastic scattering are similar and inelastic scattering are similar

with elastic scattering, we wonwith elastic scattering, we won’’t repeat them again. t repeat them again.

Then we have:Then we have:

2

1

2

1

0

)()(),(),,(),(),,(

0

E

fER

E

fESEdEEEEurf

A

DNEdEEEEurf

A

DN E

rA

E

Er

A

∂

∂+

∂

∂=′′−−′−′′′ ∫∫ σσ

∫∫ ′=′=E

rA

E

rA EdTTE

A

DNERETdTE

A

DNES

0

2

10

1 ),(2

)( ),()( σσ

[ ]

[ ]

2

2

2

2

2

2

2

2

2

2/ 4

4

)()(

)1(

1)1()(

),(),(),,(

),(),(),,(0

E

fER

E

fESfET

EdudEEEuuEEEEurfZA

DN

EdudEEEuuEEEEurfZA

DN

E

EM

A

E

EM

A

∂

∂+

∂

∂+

∂

∂

−+

∂

∂−

∂

∂=

′′′−−⋅′′−−

′′−′′−⋅′−′′′′

∫ ∫

∫ ∫

φµµµ

µ

ϕδσ

ϕδσ

π

π



FokkerFokker--Planck equationPlanck equation

)()()(

)()()(

)()()(

1

1)1()(

21

21

21

2

2

2

2

2

2

ERERER

ESESES

ETETET

E

Rf

E

SfffETfu

+=

+=

+=

∂++

−+−=∇⋅

∂

∂

∂

φ∂

∂

µµ∂

∂µ

µ∂

∂


x

y

z

r

ze

xe

ye

f

q

v,W


We could consider it as a We could consider it as a monoenergeticmonoenergetic and and

monodirectionalmonodirectional plane source embedded in plane source embedded in

an infinite homogeneous medium. We an infinite homogeneous medium. We

should emphasize that the emission should emphasize that the emission

direction of the source is direction of the source is paralledparalled to xto x--axis. axis.

So by the symmetry So by the symmetry f(r,u.Ef(r,u.E) is independent ) is independent

of y, z and phi, and we may simplify the of y, z and phi, and we may simplify the

Fokker Planck equation to obtain the broad Fokker Planck equation to obtain the broad

beam equation:beam equation:

x

y

z

[ ] [ ]),,()(),,()(),,()1()(),,(

2

22 ExfER

EExfES

EExfET

x

Exfµµµ

µµ

µ

µµ

∂

∂+

∂

∂+

∂

∂−

∂

∂=

∂

∂


The 2D pencil beam model has been used The 2D pencil beam model has been used

in in FermiFermi--EygesEyges theorytheory in [30,40]. We may in [30,40]. We may

view it as a projection of 3Dpencil beam view it as a projection of 3Dpencil beam

model on model on yzyz--plane. Note that the emission plane. Note that the emission

direction of the source is now direction of the source is now paralledparalled to to

the ythe y--axis. axis. Because if we use the same Because if we use the same

emission direction as the broad beam emission direction as the broad beam

model, then only mu will not be sufficient model, then only mu will not be sufficient

to characterize the particle phase density to characterize the particle phase density

and we should still keep phi.and we should still keep phi. But if we use But if we use

yy--axis as the emission direction, we may axis as the emission direction, we may

neglect mu and just keep phi. Finally we neglect mu and just keep phi. Finally we

assume that 2PBM is independent of assume that 2PBM is independent of

energy. Then we simplify the Fokkerenergy. Then we simplify the Fokker--

Planck equation to get:Planck equation to get:

x

y

z

A

B

2

2 ),,()(

),,(sin

),,(cos

φ∂

φ∂φφ

φφ

zyfET

z

zyf

y

zyf=

∂

∂+

∂

∂

Mohammad: [19,20,21,22,25,27,26]


Bipartition model was presented in 1967 by Luo in [3]. The main ideaof this model

is to separate the beam into a diffusion group and a straight forward group and

deal with them separately. [4,7] are the two most important papers about

bipartition model for electron transport. In [4] Luo used bipartition model for 20

keV-1 MeV electron transport which takes into account both elastic and inelastic

scatterings. In [7] Luo extended bipartition model into the energy range 1-50 MeV,

considered the influence of the energy-loss straggling, secondary-electron

production, and bremsstrahlung. The applications of bipartition model for

inhomogeneous problems and ion transport are discussed in [5,6]. The history and

development of the transport theory of charged particles and bipartition model are

summarized in [8,9]. Bipartiton model could also be combined with Fermi-Eyges

theory to produce the hybrid electron pencil beam model for 3D problems.


( ) ( ) ( ) ( )

( ) ( )ExSExC

EdEEEA

DNExfff

Ex

ff

E

f

E

Er

Arcc

,,,,

,,,2

1 0

2

2

µµ

σµϕµρ

++

′−′′′=+Ω∂

∂−

∂

∂+

∂

∂− ∫

Lewis transport equation (broad beam model,

only Fokker-Planck approximation for energy,

the asymptotic approximation for screened

Rutherford cross section not accurate)

( ) ( ) ( )[ ] ( )∫ ′⋅′−′=π

σµµµ4

,,,,,,, uduuEA

DNExfExfExC MF

Af


( ) ( ) ( )ExfExfExf ds ,,,,,, µµµ +=

( ) ( )

( ) ( )[ ] ( ) ( )ExSSuduuEA

DNExfExf

ffEx

ff

E

dMFA

ss

srscs

sc

,,,,,,,

2

1

4

2

2

µσµµ

ϕµρ

π

+−′⋅′−′

=+Ω∂

∂−

∂

∂+

∂

∂−

∫

( ) ( ) ( ) ( )

( ) ( )[ ] ( ) ( )ExSSuduuEA

DNExfExf

EdEEEA

DNExff

Ex

ff

E

dMFA

dd

E

Er

Asdt

ddt

,,,,,,,

,,,2

1

4

2

20

µσµµ

σµµρ

π

++′⋅′−′+

′−′′′=Ω∂

∂−

∂

∂+

∂

∂−

∫

∫


( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( )( )

( ) ( ) .,1,0 ,,,,,

11 ,

4

12

,4

12,,

,4

12,,

,4

12,,

0

1

00

0

0

0

miExSExC

Eddx

Ed

Rt

tt

adtxAP

nC

ExSpl

ExS

ExNpl

Exf

ExApl

Exf

idif

E

En

nnnf

m

l

lld

l

lld

l

lls

sL==

′

′−=

+

+−=

+=

+=

+=

∫∑

∑

∑

∑

−∞

=

=

∞

=

∞

=

µµ

αµ

π

µπ

µ

µπ

µ

µπ

µ

Kernel for

bipartition

model


( ) ( )[ ] ( )[ ] ( )

( ) ( ) ( ) mlEExExAD

ExAExAEx

AExAEL

E

l

ml

lll

lrlcl

al

≤−+=

+Ω∂

∂−

∂

∂+∆

∂

∂−

′

∞

+=′′′∑ ,,

,,2

1,,

0

1

2

2

δδϕ

ϕµ

( ) ( )[ ] ( )[ ] ( )

( ) ( ) ( ) mlEExExA

ExAExAEx

AExAEL

E

ll

lrlcl

al

>−+−=

+Ω∂

∂−

∂

∂+∆

∂

∂−

,,

,,2

1,,

0

2

2

δδϕ

ϕµ

( ) ( )∫ +

−+

+′−′′′+−∂

Ω∂=

∂

∂

++

∂

∂

+

++

∂

∂−

0

,,2

1

1212

1

2

2

11

E

Elrl

All

lt

lllt

SEdEEEExAA

DNN

E

N

x

N

l

l

x

N

l

l

E

N

εσϕ

ρ

FT

FD


0 1 2 3 4 5 6 7 8 9 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

MeV

Re

lative

En

erg

y D

istr

ibu

tio

n

SD 2 cmSD 2.5 cmSD 3 cmBipartition 2 cmBipartition 2.5 cmBipartition 3 cm


0 2 4 6 8 10 12−0.2

0

0.2

0.4

0.6

0.8

1

1.2

MeV

Rela

tive E

nerg

y D

istr

ibution

SD 2 cmSD 2.5 cmSD 3 cmMC 1.8−2 cmMC 2.4−2.6 cmMC 2.8−3 cm


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

µ

Rela

tive A

ngle

Dis

trib

ution

SD 2 cmSD 2.5 cmSD 3 cm


0 2 4 6 8 10 12 14 16 18 20−0.2

0

0.2

0.4

0.6

0.8

1

1.2

MeV

Rela

tive E

nerg

y D

istr

ibution

SD 3.8 cmSD 4.8 cmSD 5.6 cmBipartition 3.72 cmBipartition 4.65 cmBipartition 5.58 cm


0 5 10 15 20 25−0.2

0

0.2

0.4

0.6

0.8

1

1.2

MeV

Rela

tive E

nerg

y D

istr

ibution

SD 3.8 cmSD 4.8 cmSD 5.6 cmMC 3.6−3.9 cmMC 4.5−4.8 cmMC 5.4−5.7 cm


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

µ

Re

lative

An

gle

Dis

trib

utio

n



0 5 10 15 20 25 30−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

MeV

Rela

tive E

nerg

y D

istr

ibution

SD 5.2 cm SD 6.5 cmSD 7.8 cmBipartition 5.2 cmBipartition 6.5 cmBipartition 7.8 cm


0 5 10 15 20 25 30 35−0.2

0

0.2

0.4

0.6

0.8

1

1.2

MeV

Rela

tive E

nerg

y D

istr

ibution

SD 5.2 cmSD 6.5 cmSD 7.8 cmMC 5.2−5.6 cmMC 6.4−6.8 cmMC 7.6−8 cm


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

µ

Re

lative

An

gle

Dis

trib

utio

n


Future work

Energy deposition

3D pencil beam model

Inhomogeneous medium and irregular geometry

Ion transport

More finite element methods

Error estimates

People in this field

C. BORGERS (Tufts University)

E. W. Larsen (Univ. of Michigan, Ann Arbor)

G. C. Pomraning (UCLA)

Lawrence H. Lanzl (University of Chicago, the Manhattan

Project )

David Jette (Dep. of Med. Phy., Rush-Presbyterian-St

Luke's Medical Center, Chicago, Illinois)

Anders Brahme (Karolinska Institute)

Zhengming Luo (Sichuan University)

Mohammad Asadzadeh (CTH)

Thank you very much

for your listening!

comparing the streamline diffusion method and bipartition ...€¦ · comparing the streamline...

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