Comparing the Streamline
Diffusion Method and Bipartition
Model for Electron Transport
Jiping Xin
Department of Mathematical Science,
Chalmers University of Technology
Content
 A brief introduction of the background
 Comparing Sd-method and bipartition model
for electron transport
 Future work
Background





Radiation therapy and TPS
Two mathematical problems in TPS
Transport theory and conservative law
Equations
Overview of different research ways
Radiation therapy and TPS
Radiation therapy is the
medical use of ionizing
radiation as part of cancer
treatment to control
malignant cells.
From the website of Medical Radiation
Physics, KI
Pencil beam
Radiation therapy and TPS
In radiotherapy, Treatment
Planning is the process in
which a team consisting of
radiation oncologists, medical
radiation physicists and
dosimetrists plan the
appropriate external beam
radiotherapy treatment
technique for a patient with
cancer.
From Phoenix TPS
Two mathematical problems in TPS
 Radiation transport simulations: This process
involves selecting the appropriate beam type
(electron or photon), energy (e.g. 6MV, 12MeV)
and arrangements.
Our problem! General, flexible, accurate, efficient
algorithm!
 Optimization: The more formal optimization
process is typically referred to as forward planning
and inverse planning inference to intensity
modulated radiation therapy (IMRT).
Two mathematical problems in TPS
GMMC, Dep. of Math. Sci., CTH:
Research project: Cancer treatment through the
IMRT technique: modeling and biological
optimization
 Models and methods for light ion-beam transport
 Biological models and optimization for IMRT
planning
Questions
How to describe this kind of physical
phenomena?
Transport theory and conservative law
Transport theory:
1. Transport theory is based on nuclear physics, quantum
physics, statistical physics, etc.
2. Gas dynamic, neutron transport (nuclear weapon after
WWII), astrophysics, plasma, medical physics
3. Transport theory (discontinuous field) is more
microscopic compared with fluid dynamic or heat transfer
(continuous field) and more macroscopic compared with
quantum mechanics or nuclear physics, it describes a
group of particles. Very important!!!
4. We study charged particle transport theory!
Transport theory and conservative
law
The particle phase space density: Boltzmann
u  ( ,  )
•Cross sections!
•Particle phase space density!
•6 variables!
Photon: Compton scattering, pair
production, photon-electron
Electron: elastic scattering, inelastic
scattering, bremsstrahlung
Equations (Models)
•Transport equation (Boltzmann equation)
u  f  


  f (r, u' , E' ) (E' , E, u'u)du' dE' f (r, u, E)    (E, E' , u'u)du' dE'  S
•Fokker-Planck equation (approximation)
0
4
s
0
4
s

1 2f   fS 2fR
2 f
u  f  T ( E ) 
(1   )



S
2
2 
2
  1      E
E
 
•Fermi equation (approximation)
 2 f  2 f 
f
f
f  f S 1  2 f R
x
y


 T ( E) 2 
S
2
2
z
x
 y  E 2 E
   x   y 
•Bipartition model
Our final goal is to solve the 6
dimensional pencil beam equation!!!
Questions
 Which equation we should begin with? (M)
 The approximation (Fermi and Fokker-Planck) is accurate?
In which kinds of cases they are accurate? (M)
 Which particle we should begin with? (photon - popular,
electron - complex, ion - hot topic, proton) (M)
 How to solve the equation? Analytical method (FT),
numerical method (FEM), stochastic method (MC). (S)
 Obviously it will be difficulty to solve the 6 dimensional
equations directly, then how to simplify the equations? (S)
broad beam model and 2D pencil beam model
 How to go back to 6D model from simplified equations?
Overview of different research ways
Sichuan University
Zhengming Luo’s group
Karolinska Institute
Anders Brahme’s group
Ion:
Fermi, FT, 6
Proton, Ion:
Fermi, FT, 6
Photon:
TE, CL, 6
Electron:
BiM, FT, FDM, 3
Hybrid PBM, 6
Chalmers
Mohammad Asadzadeh
Fermi and FP, 3, FEM
University of Kuopio
Jouko Tervo’s group
TP 4 FEM
TP 6 FEM (failed)
EGSnrc
Monte Carlo
Overview of different research ways
 Electron transport (more complex than ion)
 1-50 MeV (Fokker-Planck approximation is
accurate)
 Fokker-Planck equation (PDE)
 Broad beam model (3 dimensions)
 Finite element method (Sd-method)
Comparing Sd-method and bipartition
model for electron transport
 Fokker-Planck equation for 1-50 MeV
electron transport
 Broad beam and 2D pencil beam models
 Bipartition model
 Results by Sd-method
Fokker-Planck equation for 1-50
MeV electron transport
Transport equation
u  f ( r , u , E ) 
N AD
 f (r , u, E )  f (r , u, E ) N ( E , u  u )du

4

A
N A D E0
N AD E





f (r , u, E ) r ( E , E  E )dE 
f (r , u, E ) r ( E , E  E )dE 


E
0
A
A
N D E0
 A Z   f (r , u, E ) M ( E , E   E ) u  u   ( E , E   E )dudE 
E
4
A
N AD E

Z   f (r , u, E ) M ( E , E  E ) u  u   ( E , E  E )dudE 
E / 2 4
A
Fokker-Planck equation for 1-50
MeV electron transport
The screened Rutherford cross section:
r02 Z ( Z  1)( m0c 2 ) 2 ( E  m0c 2 ) 2
 N ( E,  ) 
E 2 ( E  2m0c 2 ) 2
1  Z 1/ 3 
 

4 121.25 
2

1
 Z
1
1  2 Z  1 


 



2
3/ 2
137
2
137
1






1



2

1


2







2
2 2

 Z  E  m0c
1.13  3.76

2
137

 E E  2m0c


E ( E  2m0 c 2 )
E  m0 c 2


2

m0c 2

2
 E E  2m0c
 

Fokker-Planck equation for 1-50
MeV electron transport
Bremsstrahlung cross section
4r02 Z 2
 r ( E, T ) 
137T
 E  T 2 2 E  T  1   1

 E T



 ln Z  
 

1 
2

E
3 E  4
3
 6E


m0c 2T
  100
E ( E  T ) Z 1/ 3
1    a1e a   b1eb 
2
   c1e c2  d1e  d 2
2
Moller cross section

r02 m0c 2 E  m0c 2
 M E , T  
E E  2m0c 2



2
1
T2


2

2 E  m0c 2 m0c 2 T 
T2
E2
T
 

1 

  
2
2 2
2 2
E
E

T


E

T
E  m0c  
E  m0c






Questions
How to derive Fokker-Planck equation from
transport equation and check the accuracy?
Fokker-Planck equation for 1-50
MeV electron transport
The classical contributions about the Fokker-Planck
approximation are summarized by Chandrasekhar in [28] and
Rosenbluth in [32]. [29,30,31] are studying the case of the
linear particle transport. These works give a heuristic
derivation of the Fokker-Planck operator. In [33] Pomraning
gave a formalized derivation of the Fokker-Planck operator as
an asymptotic limit of the integral scattering operator where a
peaked scattering kernel is a necessary but not sufficient
condition in the asymptotic treatment.
Fokker-Planck equation for 1-50
MeV electron transport
Laplace’s method for Laplace integrals
b
I ( x)   f (t )e
a
x ( t )
dt
If  (t ) has a global maximum at t  c with a  c  b and if
f (c)  0, then it is only the neighbourh ood of t  c that
contribute s to the full asymptotic expansion of I ( x) as x  
Fokker-Planck equation for 1-50
MeV electron transport
Step 1. we may approximat e I ( x) by I ( x;  ) where
 c  f (t )e x ( t ) dt , if a  c  b,
c 
 a 
I ( x;  )   f (t )e x (t ) dt , if c  a,
a
 b
x ( t )

f
(
t
)
e
dt , if c  b.

 b 
This approximat ion is truth since the changing limits
of integratio n only introduces exponentia lly small errors.
Fokker-Planck equation for 1-50
MeV electron transport
Step 2. Now   0 can be chosen small enough so that
it is valid to replace  (t ) by the first few terms in its
Taylor or asymptotic series expansion.
Step 3. Having substitute d the approximat ions for 
and f indicated above, we now extend the endpoints
of integratio n to infinity, in order to evaluate the resulting
integrals (again thi s only introduces exponentia lly small
errors).
Fokker-Planck equation for 1-50
MeV electron transport
Surface harmonic expansion and Legendre
polynomial expansion (very important part compared
with the heuristic derivation for the angular
integration, vector)
 2n  1 

anm f nm (r , E )Ynm (u )

n  0 m   n  4 

f (r , u, E )  
n
 2k  1 
 sk ( E , E ) Pk (u  u )
k  0  4 

n

 2n  1 







anmYnm (u ) 

0 4  s ( E , E, u  u) f (r, u , E )du dE  
2

n 0 m   n 

 s ( E , E , u  u )   


0
1
f nm ( E )  Pn ( ) s ( E , E ,  )ddE 
1
Fokker-Planck equation for 1-50
MeV electron transport
Example: the screened Rutherford cross section
It is not accurate for low energy electron
transport, so we choose 1-50 MeV.
N AD
 f (r , u, E )  f (r , u, E ) N ( E, u  u)du

4

A

n
1
N AD
 2n  1 



f
(
r
,
u
,
E
)

(
E
,
u

u
)
d
u

a
f
(
r
,
E
)
Y
(
u
)
Pn ( ) N ( E ,  )d

 nm nm


N
nm


4


1
A
2 
n 0 m   n 
1

1
Pn ( ) N ( E,  )d
Fokker-Planck equation for 1-50
MeV electron transport
Step 1:
1
I   Pn ( ) N ( E ,  )d
1

1
1

  Pn ( ) N ( E ,  )d   Pn ( ) N ( E ,  )d : I1  I 2
Choose   (  n ,1). Here  n is the largest positive root
of the Legendre polynomial Pn ( ).
I1
I2
 lim
a 0 


1
 N ( E ,  )d
1 a
Pn ( )   N ( E ,  )d

 0  I  I2
Fokker-Planck equation for 1-50
MeV electron transport
Step 2: I   P (1)  P (1)(  1) ( E,  )d : I 
1
2
n


n

N
2

I1 :  Pn (1)  Pn (1)(  1)  N ( E,  )d
Step 3:
I1
 lim
I 2 a0
(1)
1
 P (1)  P
 P (1)  P

1
1 a

(1)
n
n
n
n
(1)

(1)(  1) 
(1)(  1)  N ( E ,  )d
(1)
1

N
( E ,  )d
 0  I  I 2  I 2  I1  I 2

I   Pn (1)  Pn (1)(  1)  N ( E,  )d
1
(1)
Fokker-Planck equation for 1-50
MeV electron transport
N AD
 f (r , u, E )  f (r , u, E ) N ( E , u  u )du

4

A
1
N A D  n  2n  1 
(1)


anm f nm (r , E )Ynm (u ) 1 Pn (1)  Pn (1)(  1)  N ( E ,  )d


A n 0 m   n  2 
N D
 A  f (r , u , E ) N ( E , u  u )du 
A 4
 
N AD

1
2 
2
f

f (r , u , E ) N ( E , u   u )du   T1 ( E )
(1   )

2
2 

4

A
 (1   )  
 

N AD
f (r , u , E ) N ( E , u  u )du 

4

A
 

1
2 
2
f
 T1 ( E )
(1   )

2
2 
 (1   )  
 

N AD 1
T1 ( E ) 
  (  1) N ( E ,  )d
1
A

Fokker-Planck equation for 1-50
MeV electron transport
The screened Rutherford cross section
with corrections for low energy electron
transport
r02 Z 2 (m0c 2 ) 2 ( E  m0c 2 ) 2
1
 N ( E,  ) 
E 2 ( E  2m0c 2 ) 2
1    2 2
The advantage of bipartiton model
for large angle!
Fokker-Planck equation for 1-50
MeV electron transport
Bremsstrahlung and inelastic scattering are similar
with elastic scattering, we won’t repeat them again.
Then we have:
N A D E0
N AD E
S1 ( E ) f  2 R1 ( E ) f
f (r , u, E ) r ( E, E   E )dE  
f (r , u, E ) r ( E , E  E )dE  

A E
A 0
E
E 2
S1 ( E ) 
N AD E
 r ( E , T )TdE 

0
A
R1 ( E ) 
N AD E
 r ( E , T )T 2 dE 

2A 0
N A D E0
Z   f (r , u, E ) M ( E , E   E ) u  u   ( E , E   E )dudE 
E
4
A
N D E
 A Z   f (r , u, E ) M ( E , E  E ) u  u   ( E , E  E )dudE 
E / 2 4
A
 

1
2 
S 2 ( E ) f  2 R2 ( E ) f
2
f 
 T2 ( E )
(1   )


2
2 




(
1


)



E
E 2


Fokker-Planck equation for 1-50
MeV electron transport
Fokker-Planck equation

1 2f   fS 2fR
2 f
u  f  T ( E ) 
(1   )



2
2 
2




1





E

E


T ( E )  T1 ( E )  T2 ( E )
S ( E )  S1 ( E )  S 2 ( E )
R( E )  R1 ( E )  R2 ( E )
Broad beam and 2D pencil beam models
Broad beam and 2D pencil beam models
We could consider it as a monoenergetic and
monodirectional plane source embedded in
an infinite homogeneous medium. We
should emphasize that the emission
direction of the source is paralled to x-axis.
So by the symmetry f(r,u.E) is independent
of y, z and phi, and we may simplify the
Fokker Planck equation to obtain the broad
beam equation:
 
f ( x,  , E )
 

2
2
S ( E ) f ( x,  , E )  R( E ) f ( x,  , E )

 T (E)
(1   )
f ( x,  , E )  
x
 


E
E2

Broad beam and 2D pencil beam models
The 2D pencil beam model has been used
in Fermi-Eyges theory in [30,40]. We may
view it as a projection of 3Dpencil beam
model on yz-plane. Note that the emission
direction of the source is now paralled to
the y-axis. Because if we use the same
emission direction as the broad beam
model, then only mu will not be sufficient
to characterize the particle phase density
and we should still keep phi. But if we use
y-axis as the emission direction, we may
neglect mu and just keep phi. Finally we
assume that 2PBM is independent of
energy. Then we simplify the FokkerPlanck equation to get:
f ( y, z,  )
f ( y, z,  )
 2 f ( y, z ,  )
cos 
 sin  
 T (E)
y
z
2
Mohammad: [19,20,21,22,25,27,26]
Results by Sd-method
Results by Sd-method
Results by Sd-method
Results by Sd-method
Results by Sd-method
Results by Sd-method
Results by Sd-method
Results by Sd-method
Results by Sd-method