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 therapy and TPSRadiation therapy and TPS
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
Radiation therapy and TPSRadiation therapy and TPS
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
Two mathematical problems in TPSTwo mathematical problems in 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).
Two mathematical problems in TPSTwo mathematical problems in TPS
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?
Overview of different research waysOverview of different research ways
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
Overview of different research waysOverview of different research ways
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
FokkerFokker--Planck equation for 1Planck equation for 1--50 50
MeVMeV electron transportelectron transport
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
FokkerFokker--Planck equation for 1Planck equation for 1--50 50
MeVMeV electron transportelectron transport
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?
FokkerFokker--Planck equation for 1Planck equation for 1--50 50
MeVMeV electron transportelectron transport
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.
FokkerFokker--Planck equation for 1Planck equation for 1--50 50
MeVMeV electron transportelectron transport
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
FokkerFokker--Planck equation for 1Planck equation for 1--50 50
MeVMeV electron transportelectron transport
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
FokkerFokker--Planck equation for 1Planck equation for 1--50 50
MeVMeV electron transportelectron transport
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
FokkerFokker--Planck equation for 1Planck equation for 1--50 50
MeVMeV electron transportelectron transport
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
FokkerFokker--Planck equation for 1Planck equation for 1--50 50
MeVMeV electron transportelectron transport
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
FokkerFokker--Planck equation for 1Planck equation for 1--50 50
MeVMeV electron transportelectron transport
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
FokkerFokker--Planck equation for 1Planck equation for 1--50 50
MeVMeV electron transportelectron transport
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
FokkerFokker--Planck equation for 1Planck equation for 1--50 50
MeVMeV electron transportelectron transport
[ ]
( )
∫
∫
∫
∫
∑ ∑ ∫
∫
−
∞
= −=−
−=
∂
∂
−+
∂
∂−
∂
∂=
′⋅′−
∂
∂
−+
∂
∂−
∂
∂+′⋅′=
′⋅′−
−+
+=
′⋅′−′
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
FokkerFokker--Planck equation for 1Planck equation for 1--50 50
MeVMeV electron transportelectron transport
( )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!
FokkerFokker--Planck equation for 1Planck equation for 1--50 50
MeVMeV electron transportelectron transport
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 equation for 1Planck equation for 1--50 50
MeVMeV electron transportelectron transport
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∂
∂
Broad beam and 2D pencil beam modelsBroad beam and 2D pencil beam models
x
y
z
r
ze
xe
ye
f
q
v,W
Broad beam and 2D pencil beam modelsBroad beam and 2D pencil beam models
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
∂
∂+
∂
∂+
∂
∂−
∂
∂=
∂
∂
Broad beam and 2D pencil beam modelsBroad beam and 2D pencil beam models
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 modelBipartition model
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.
Bipartition modelBipartition model
( ) ( ) ( ) ( )
( ) ( )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
Bipartition modelBipartition model
( ) ( ) ( )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∂
∂−
∂
∂+
∂
∂−
∫
∫
Bipartition modelBipartition model
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )( )
( ) ( ) .,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
Bipartition modelBipartition model
Bipartition modelBipartition 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
Results by Results by SdSd--methodmethod
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
Results by Results by SdSd--methodmethod
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
Results by Results by SdSd--methodmethod
−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
Results by Results by SdSd--methodmethod
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
Results by Results by SdSd--methodmethod
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
Results by Results by SdSd--methodmethod
−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
SD 3.8 cmSD 4.8 cmSD 5.6 cmBipartition 3.72 cmBipartition 4.65 cmBipartition 5.58 cm
Results by Results by SdSd--methodmethod
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
Results by Results by SdSd--methodmethod
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
Results by Results by SdSd--methodmethod
−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
SD 5.2 cmSD 6.5 cmSD 7.8 cmBipartition 5.2 cmBipartition 6.5 cmBipartition 7.8 cm
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!