Spatiotemporally Optimal Fractionation in Radiotherapy

Abstract

We present a spatiotemporally integrated formulation of the optimal fractionation problem using the standard log-linear-quadratic survival model. Our objective is to choose a fluencemap and a number of fractions so as to maximize the biological effect of tumor dose averaged over its voxels subject to maximum dose, mean dose, and dose-volume constraints for various normal tissues. Constrains are expressed in biologically effective dose equivalents. We propose an efficient convex programming method to approximately solve the resulting computationally difficult model. Through extensive computer simulations on ten head-and-neck and prostate cancer test cases with a broad range of radiobiological parameters, we compare the biological effect on tumor obtained by our integrated approach relative to that from two other models. The first is a traditional IMRT fluence-map optimization model that does not optimize the number of fractions. The second assumes that a fluence-map is available a priori from a traditional IMRT optimization model and then optimizes the number of fractions, thus separating the spatial and temporal components. The improvements in tumor biological effect over IMRT were 9%-52% with average 22%, and 53%-108% with average 69%, for head-and-neck and prostate, respectively. The improvements in tumor biological effect over the spatiotemporally separated model were 15%-45% with average 27%, and 17%-23% with average 21%, for head-and-neck and prostate, respectively. This suggests that integrated optimization of the fluence-map and the number of fractions could improve treatment efficacy as measured within the linear-quadratic framework.

Robust spatiotemporally integrated fractionation in radiotherapy

“…We first briefly recall the nominal spatiotemporally integrated formulation from [18]. The objective in this formulation is to choose the number of sessions and the fluence-map in each session to maximize tumor-BE subject to maximum dose and mean dose constraints on various OAR.…”

“…The objective in this formulation is to choose the number of sessions and the fluence-map in each session to maximize tumor-BE subject to maximum dose and mean dose constraints on various OAR. Dosevolume constraints can also be handled using the approach described in [18]; we omit those here for brevity. The formulation also includes smoothness constraints on the fluence-map to ensure that it is deliverable in practice using IMRT.…”

“…The formulation also includes smoothness constraints on the fluence-map to ensure that it is deliverable in practice using IMRT. Here, we only discuss minimal, crucial ingredients of this formulation; the reader is referred to [18] for an extensive discussion of its derivation, advantages, and limitations.…”

“…Most recently, Saberian et al [18] proposed a spatiotemporally integrated formulation of the fractionation problem, where the number of sessions and the fluence-map were both optimization variables within a single model. The objective was again to maximize the tumor-BE subject to upper bounds on the BED for various OAR.…”

“…We remark as an aside that this reformulation method is more general than and hence applicable to our aforementioned unpublished work on the robust spatiotemporally separated formulation [1]. We numerically quantify the price of robustness for a representative test case for head-and-neck cancer from [18], and also compare the frequency and amount of infeasibility suffered by the nominal and the robust solutions. Our results suggest that the price of robustness is small and the robust solution remains statistically more feasible than the nominal solution even when dose-response parameters vary outside the presumed intervals of uncertainty.…”