On the Transverse Diffusion of Beams of Photons in Radiation Therapy

Abstract

Typical external radiotherapy treatments consist in emitting beams of energetic photons targeting the tumor cells. Those photons are transported through the medium and interact with it. Such interactions affect the motion of the photons but they are typically weakly deflected which is not well modeled by standard numerical methods. The present work deals with the transport of photons in water. The motion of those particles is modeled by an entropy-based moment model, i.e., the $M_1$ model. The main difficulty when constructing numerical approaches for photon beam modeling emerges from the significant difference of magnitude between the diffusion effects in the forward and transverse directions. A numerical method for the $M_1$ equations is proposed with a special focus on the numerical diffusion effects.

Appendix : Computation of Bounds on the Eigenvalues of the Jacobian of the Flux

Using rotational invariance and a normalization (see [11] for details), the closure (4) can also be rewritten under the form

$$\begin{aligned} \psi ^2= & {} \psi ^0 \left[ \frac{1-\chi }{2} Id + \frac{3\chi -1}{2} \frac{\psi ^1 \otimes \psi ^1}{|\psi ^1|^2} \right] , \end{aligned}$$

(17)

where the Eddington factor $\chi$ is a scalar function of the scalar $|\psi ^1|/\psi ^0$.

Consider that $\psi ^1$ is colinear to $e_1$ (otherwise just use a rotation to work in such a reference frame). Using the form (17) of the closure, the fluxes $\mathbf {F_n}$ in the direction $n\in S^2$ read

$$\begin{aligned} \varvec{\Psi }= & {} \left( \psi ^0, \ \psi ^1 \right) , \\ \forall n \in S^2, \qquad \mathbf {F_n}(\varvec{\Psi })= & {} \left( \psi ^1.n, \ \frac{\psi ^0}{2} \left[ (1-\chi )n + (3\chi -1) \frac{ (\psi ^1.n) \psi ^1 }{|\psi ^1|^2} \right] \right) , \end{aligned}$$

Chose a reference frame such that $\psi ^1 = \psi _1^1 e_1$ with $\psi _1^1 \ge 0$. In this reference frame, the spectrum of the Jacobian of the flux $\mathbf {F_1}$ along the direction $n=e_1$ (direction of the beam) and along $n=e_2$ (direction normal to the beam) read

$$\begin{aligned} Sp\left( \partial _{\varvec{\Psi }}\left( \mathbf {F_1}(\varvec{\Psi })\right) \right)= & {} \left( \frac{3\chi -1}{2N_1^1}, \frac{\chi ' \pm \sqrt{\chi '^2 + 4 (\chi -N_1^1\chi ')}}{2} \right) ,\\ Sp\left( \partial _{\varvec{\Psi }}\left( \mathbf {F_2}(\varvec{\Psi }) \right) \right)= & {} \left( 0, \ \pm \sqrt{\frac{1-\chi + N_1^1\chi ' - \frac{3\chi -1}{2N_1^1} \chi ' }{2}} \right) . \end{aligned}$$

Now, in order to come back to the computations in any reference frame, one can simply use a rotation R such that $R \psi ^1 = \psi _1^1 e_1$. One has

$$\begin{aligned} \begin{array}{rcl} \forall n\in S^2, \qquad \partial _{\varvec{\Psi }} \left( \mathbf {F_n}(\varvec{\Psi }) \right) &{}=&{} \partial _{(\psi ^0,R\psi _1^1e_1)} \left( \mathbf {F_n}(\psi ^0,R\psi _1^1e_1) \right) \\ &{}=&{} R_2 \partial _{\varvec{\Psi }} \mathbf {F_{R^Tn}}\left( \psi ^0,|\psi ^1|e_1\right) R_2^T, \end{array} \qquad R_2 = \left( \begin{array}{cccc} 1 &{} 0_{\mathbb {R}^3} \\ 0_{\mathbb {R}^3} &{} R \end{array} \right) . \end{aligned}$$

The spectrum of such a matrix can be bounded using the previous computations

$$\begin{aligned} \forall n\in S^2,&\quad&Sp\left( \partial _{\varvec{\Psi }} \left( \mathbf {F_n}(\varvec{\Psi }) \right) \right) \subset [ b_-, b_+ ], \end{aligned}$$

(18a)

$$\begin{aligned}&b_-(N^1,n) = (1-\theta ) \min S_t(|N^1|) + \theta \min S_n(|N^1|), \end{aligned}$$

(18b)

$$\begin{aligned}&b_+(N^1,n) = (1-\theta ) \max S_t(|N^1|) + \theta \max S_n(|N^1|), \qquad \theta = \frac{N^1.n}{|N^1|}. \end{aligned}$$

The exact bounds $b_-$ and $b_+$ of $Sp\left( \partial _{\varvec{\Psi }} \mathbf {F_n}(\varvec{\Psi }) \right)$ could be computed analytically as the eigenvalues of the $4\times 4$ matrix $\partial _\Psi \left( \mathbf {F_n}(\Psi )\right)$. However, using such analytical formulae may introduce errors at the numerical level that may be non-negligible. Computing the bounds in (18) is easier, and they are sufficient for the present applications.