Sperduto_Khan's Treatment Planning in Radiation Oncology, 5e

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446 SECTION III Treatment Planning: Physics and Dosimetric Principles

The first term in Equation (20.10) represents the velocity times the directional derivative of N e in the direction of Ω . It is known as the streaming term , as it represents the difference in the time derivative between the moving and rest frames, the latter of which also includes the effects of electrons moving past r without any collisions. Upon inserting (20.10) into (20.9), the resulting equa tion becomes ∂ N e _ ∂ t + v . ∇ N e + σv N e = ∬ N e ' { d  2 σ _ d E ′  d Ω ′ } d E ′  d Ω ′ + Q , (20.11) where we have removed the arguments for simplicity. This is the basic form of the transport equation, which is often called the Boltzmann equation because of its simi larity to the expression derived by Boltzmann involving the kinetic theory of gasses. 39 It is more often written in terms of the angular flux, Ψ e , where Ψ e ( r , Ω ,E , t ) = v N e ( r , Ω ,E , t ) : 1 _ v   ∂ Ψ  e _ ∂ t + Ω . ∇ N e + σ Ψ  e = ∬ Ψ  e ' { d  2 σ _ d E ′  d Ω ' } d E ′  d Ω '+ Q , In external beam radiotherapy, the time-independent form of Equation (20.12) is used, since steady state is achieved in a much shorter time than that when the beam is on. 41 Equation (20.12) is an integro-partial-differential equation which can be solved numerically using either stochastic or deterministic methods. Most reports have utilized the latter, employing some form of grid-based numerical method in which phase space is discretized in spatial, angular, and energy coordinates, 40,42,43 although there are some differences in the literature about which techniques are used. Finite difference and finite ele ment methods are used for spatial discretization, and Boman et al. reported using the finiteelement method for all variables. 40 Alternatively, the method of dis creteordinates has been employed for angular discreti zation in the Attila solver, 44,45 and in the subsequent Acuros XB algorithm currently available in the Var ian Eclipse treatment planning system (Varian Assoc, Palo Alto, CA). Energy-dependent coupled photon– electron cross-sectional data are available through CEPXS, which uses the multigroup method to dis cretize the particle energy domain into energy intervals or groups. 46 This class of solvers is commonly known as the discrete ordinates method, although technically the name only refers to the method for numerically dis cretizing in angle. Up to now, we have only discussed electron angu lar density (or angular flux). However, in external beam calculations, collisions involve photons, electrons, and positrons. In principle, Equation (20.12) then becomes a (20.12) Use of the Transport Equations for Photon Beam Calculations

set of coupled equations. For example, excluding positron interactions, we have the following

e ' {

} d E ′  d Ω ′

d  2 σ e γ _ d E ′  d Ω ′

Ω . ∇ Ψ  γ + σ γ Ψ  γ + ∬

γ ' {

} d E ′  d Ω ′ = Q γ

d  2 σ γγ _ d E ′  d Ω ′

+ ∬ Ψ 

γ ' {

} d E ′  d Ω ′

d  2 σ γ e _ d E ′  d Ω ′

Ω . ∇ Ψ  e + σ e Ψ  e + ∬

e ' {

} d E ′ d Ω ′ = Q e (20.13)

d  2 σ ee _ d E ′  d Ω ′

+ ∬ Ψ 

d  2 σ 12 _ d E ′  d Ω ′

where ( Ω ′ , E ′ , Ω , E ) represents the differential crosssection for the creation of particle 2 with energy E , direction Ω , particle 1 of energy E ′ , and direction Ω ′ . Acuros XB Implementation of the Linear BTEs Currently, the only commercial implementation of the linear BTE is the Acuros XB dose calculation algorithm available on the Varian Eclipse treatment planning system. Acuros XB was developed using many of the methods employed with a prototype BTE solver developed at the Los Alamos National Laboratories called Attila, which was co authored by the founders of Transpire, Inc. (Gig Harbor, WA). 47 Transpire, Inc., established a licensing agreement to commercialize Attila for a broad range of applications. Acuros XB has adapted and optimized the methods within Attila for external photon beam calculations. 48 Within the Acuros algorithm, both charged pair production particles are assumed to be electrons, and the contribution of electron-produced Bremsstrahlung within the patient is assumed to be deposited locally. As already mentioned, energy discretization is performed using a multigroup representation of the crosssection. How ever, this is difficult for electrons where the inelastic cross section increases rapidly when energy losses become small. These “soft” interactions would require a very large number of energy bins to accurately describe, which is impractical for an efficient solution. As a result, electron interactions are sep arated into large and small energy losses, the latter of which are described by a continuous slowing-down (CSD) approxi mation. In this case, the angular electron fluence is described by the Boltzmann–Fokker–Planck transport equation: Ω . ∇ Ψ  e + σ e Ψ  e − ∂ S R _ ∂ E Ψ  e + ∬ Ψ  y ' { d  2 σ γ e _ d E ′  d Ω ′ } d E ′  d Ω ′ In this case σ ee represents larger, “catastrophic” inter actions that are represented by standard Boltzmann scattering. 48 Gifford et al. 44 first performed an evaluation of the prototype solver Attila for radiation therapy dose cal culations. Dose calculations performed by Attila were directly compared with those calculated using MC codes MCNPX for a brachytherapy calculation and EGS4 for an + ∬ Ψ  e ' { d  2 σ ee _ d E ′  d Ω ′ } d E ′  d Ω ′ = Q e (20.14)

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