Advanced Physics for Brachytherapy 2018

The LBTE



P 





 

pq

ˆ 

) ˆ , ,( Er t

Er

( 



)ˆ, , 

+

=

+



(

−

)

scat q

pr r



p

=

1 4

There is no analytical (closed form) solution of the LBTE.

This is because it is an equation of 6 variables which is integro-differential since the scatter source is:

 







Ed d Er ˆ )ˆ, ,( )ˆ ˆ,  

scat q

Er

E Ers

(

04 )ˆ, ,

 = ,( 

   →



i.e. : I must integrate for the scatter source and this depends on the solution itself at all other (E’,Ω’) points of the phase space…! The only option is to solve the equation numerically (and this is exactly what Acuros and similar algorithms do) by: ➢ separating variables: ➢ solve the LBTE iteratively: • make initial guess for • approximate integration for the scatter source by summation over discrete E, Ω elements • approximate derivatives by finite differences over discrete space elements • correct initial guess, and continue until a convergence criterion is met )ˆ( ) ( )( )ˆ ( ) , ( ) ˆ , ,(  =  =   Y Ef r YE r E r    ) ( r  