Selecting Radiotherapy Dose Distributions by Means of Constrained Optimization Problems

Abstract

The main steps in planning radiotherapy consist in selecting for any patient diagnosed with a solid tumor (i) a prescribed radiation dose on the tumor, (ii) bounds on the radiation side effects on nearby organs at risk and (iii) a fractionation scheme specifying the number and frequency of therapeutic sessions during treatment. The goal of any radiotherapy treatment is to deliver on the tumor a radiation dose as close as possible to that selected in (i), while at the same time conforming to the constraints prescribed in (ii). To this day, considerable uncertainties remain concerning the best manner in which such issues should be addressed. In particular, the choice of a prescription radiation dose is mostly based on clinical experience accumulated on the particular type of tumor considered, without any direct reference to quantitative radiobiological assessment. Interestingly, mathematical models for the effect of radiation on biological matter have existed for quite some time, and are widely acknowledged by clinicians. However, the difficulty to obtain accurate in vivo measurements of the radiobiological parameters involved has severely restricted their direct application in current clinical practice.

In this work, we first propose a mathematical model to select radiation dose distributions as solutions (minimizers) of suitable variational problems, under the assumption that key radiobiological parameters for tumors and organs at risk involved are known. Second, by analyzing the dependence of such solutions on the parameters involved, we then discuss the manner in which the use of those minimizers can improve current decision-making processes to select clinical dosimetries when (as is generally the case) only partial information on model radiosensitivity parameters is available. A comparison of the proposed radiation dose distributions with those actually delivered in a number of clinical cases strongly suggests that solutions of our mathematical model can be instrumental in deriving good quality tests to select radiotherapy treatment plans in rather general situations.

Appendix

We provide here the main ingredients in the proofs of Theorems 2.1 and 2.2 in Sect. 2.

Proof of Theorem 2.1

It is readily seen that the functional given in (7) is lower semicontinuous (l.s.c.) on the space W ^1,∞(Ω) endowed with the uniform convergence. Existence of at least one minimizer follows from the fact that the associated functional

$$ J^{*}(D) = J(D) + I_{K}(D), $$

(17)

where K is as in the statement of part (a) in the Theorem 2.1 and I _K (D)=0 when D∈K,I _K (D)=+∞ otherwise, is also l.s.c. on W ^1,∞(Ω) with respect to the uniform convergence, since K is compact for that convergence. Then a minimizer of (17) (hence for the problem under consideration consisting of minimizing (7) under constraints (8)–(10)) exists by classical results (cf. Buttazzo 1989, Buttazzo et al. 1998).

On the other hand, a direct computation (similar to that performed in Alfonso et al. (2012) for a related problem) shows that the integrand in (7) is convex when inequalities (11) and (12) are satisfied. This in turn implies the convexity of J(D) in (7), whereupon uniqueness follows. □

Proof of Theorem 2.2

It is quite similar to that of Theorem 2.1. In particular, the functional

$$ J^{*}(D) = J(D) + I_{\overline{K}}(D) $$

(18)

is l.s.c. on the space W ^1,∞(Ω) endowed with the uniform convergence, since $\overline{K}$ is compact for that convergence. This yields the existence of minimizers of (18), and hence for the problem under consideration consisting of minimizing (7) under constraints (8)–(10) and (13), (14). In its turn, uniqueness follows exactly as in Theorem 2.1. □