Imputing radiobiological parameters of the linear-quadratic dose-response model from a radiotherapy fractionation plan.

The objective in cancer radiotherapy is to maximize tumor-kill while limiting toxic effects of radiation dose on nearby organs-at-risk (OAR). Given a fixed number of treatment sessions, planners thus face the problem of finding a dosing sequence that achieves this goal. This is called the fractionation problem, and has received steady attention over a long history in the clinical literature. Mathematical formulations of the resulting optimization problem utilize the linear-quadratic (LQ) framework to characterize radiation dose-response of tumors and OAR. This yields a nonconvex quadratically constrained quadratic program. The optimal dosing plan in this forward problem crucially depends on the parameters of the LQ model. Unfortunately, these parameters are difficult to estimate via in vitro or in vivo studies, and as such, their values are unknown to treatment planners. The clinical literature is thus replete with debates about what parameter values will make specific dosing plans effective. This paper formulates this as an inverse optimization problem. The LQ dose-response parameters appear in the objective function, the left hand side, and the right hand side of the forward problem, and none of the existing generic methods can provide an exact solution of the inverse problem. This paper exploits the structure of the problem and identifies all possible parameter values that render the given dosing plan optimal, in closed-form. This closed-form formula is applied to dosing-plans from three clinical studies published within the last two years.

Optimal modality selection in external beam radiotherapy.

The goal in external beam radiotherapy (EBRT) for cancer is to maximize damage to the tumour while limiting toxic effects on the organs-at-risk. EBRT can be delivered via different modalities such as photons, protons and neutrons. The choice of an optimal modality depends on the anatomy of the irradiated area and the relative physical and biological properties of the modalities under consideration. There is no single universally dominant modality. We present the first-ever mathematical formulation of the optimal modality selection problem. We show that this problem can be tackled by solving the Karush-Kuhn-Tucker conditions of optimality, which reduce to an analytically tractable quartic equation. We perform numerical experiments to gain insights into the effect of biological and physical properties on the choice of an optimal modality or combination of modalities.

Adaptive treatment-length optimization in spatiobiologically integrated radiotherapy.

Recent theoretical research on spatiobiologically integrated radiotherapy has focused on optimization models that adapt fluence-maps to the evolution of tumor state, for example, cell densities, as observed in quantitative functional images acquired over the treatment course. We propose an optimization model that adapts the length of the treatment course as well as the fluence-maps to such imaged tumor state. Specifically, after observing the tumor cell densities at the beginning of a session, the treatment planner solves a group of convex optimization problems to determine an optimal number of remaining treatment sessions, and a corresponding optimal fluence-map for each of these sessions. The objective is to minimize the total number of tumor cells remaining (TNTCR) at the end of this proposed treatment course, subject to upper limits on the biologically effective dose delivered to the organs-at-risk. This fluence-map is administered in future sessions until the next image is available, and then the number of sessions and the fluence-map are re-optimized based on the latest cell density information. We demonstrate via computer simulations on five head-and-neck test cases that such adaptive treatment-length and fluence-map planning reduces the TNTCR and increases the biological effect on the tumor while employing shorter treatment courses, as compared to only adapting fluence-maps and using a pre-determined treatment course length based on one-size-fits-all guidelines.

A theoretical stochastic control framework for adapting radiotherapy to hypoxia.

Hypoxia, that is, insufficient oxygen partial pressure, is a known cause of reduced radiosensitivity in solid tumors, and especially in head-and-neck tumors. It is thus believed to adversely affect the outcome of fractionated radiotherapy. Oxygen partial pressure varies spatially and temporally over the treatment course and exhibits inter-patient and intra-tumor variation. Emerging advances in non-invasive functional imaging offer the future possibility of adapting radiotherapy plans to this uncertain spatiotemporal evolution of hypoxia over the treatment course. We study the potential benefits of such adaptive planning via a theoretical stochastic control framework using computer-simulated evolution of hypoxia on computer-generated test cases in head-and-neck cancer. The exact solution of the resulting control problem is computationally intractable. We develop an approximation algorithm, called certainty equivalent control, that calls for the solution of a sequence of convex programs over the treatment course; dose-volume constraints are handled using a simple constraint generation method. These convex programs are solved using an interior point algorithm with a logarithmic barrier via Newton's method and backtracking line search. Convexity of various formulations in this paper is guaranteed by a sufficient condition on radiobiological tumor-response parameters. This condition is expected to hold for head-and-neck tumors and for other similarly responding tumors where the linear dose-response parameter is larger than the quadratic dose-response parameter. We perform numerical experiments on four test cases by using a first-order vector autoregressive process with exponential and rational-quadratic covariance functions from the spatiotemporal statistics literature to simulate the evolution of hypoxia. Our results suggest that dynamic planning could lead to a considerable improvement in the number of tumor cells remaining at the end of the treatment course. Through these simulations, we also gain insights into when and why dynamic planning is likely to yield the largest benefits.

Optimal fractionation in radiotherapy with multiple normal tissues.

The goal in radiotherapy is to maximize the biological effect (BE) of radiation on the tumour while limiting its toxic effects on healthy anatomies. Treatment is administered over several sessions to give the normal tissue time to recover as it has better damage-repair capabilities than tumour cells. This is termed fractionation. A key problem in radiotherapy involves finding an optimal number of treatment sessions (fractions) and the corresponding dosing schedule. A major limitation of existing mathematically rigorous work on this problem is that it includes only a single normal tissue. Since essentially no anatomical region of interest includes only one normal tissue, these models may incorrectly identify the optimal number of fractions and the corresponding dosing schedule. We present a formulation of the optimal fractionation problem that includes multiple normal tissues. Our model can tackle any combination of maximum dose, mean dose and dose-volume type constraints for serial and parallel normal tissues as this is characteristic of most treatment protocols. We also allow for a spatially heterogeneous dose distribution within each normal tissue. Furthermore, we do not a priori assume that the doses are invariant across fractions. Finally, our model uses a spatially optimized treatment plan as input and hence can be seamlessly combined with any treatment planning system. Our formulation is a mixed-integer, non-convex, quadratically constrained quadratic programming problem. In order to simplify this computationally challenging problem without loss of optimality, we establish sufficient conditions under which equal-dosage or single-dosage fractionation is optimal. Based on the prevalent estimates of tumour and normal tissue model parameters, these conditions are expected to hold in many types of commonly studied tumours, such as those similar to head-and-neck and prostate cancers. This motivates a simple reformulation of our problem that leads to a closed-form formula for the dose per fraction. We then establish that the tumour-BE is quasiconcave in the number of fractions; this ultimately helps in identifying the optimal number of fractions. We perform extensive numerical experiments using 10 head-and-neck and prostate test cases to uncover several clinically relevant insights.

A Markov decision process approach to multi-category patient scheduling in a diagnostic facility.

OBJECTIVES: To develop a mathematical model for multi-category patient scheduling decisions in computed tomography (CT), and to investigate associated tradeoffs from economic and operational perspectives. METHODS: We modeled this decision-problem as a finite-horizon Markov decision process (MDP) with expected net CT revenue as the performance metric. The performance of optimal policies was compared with five heuristics using data from an urban hospital. In addition to net revenue, other patient-throughput and service-quality metrics were also used in this comparative analysis. RESULTS: The optimal policy had a threshold structure in the two-scanner case - it prioritized one type of patient when the queue-length for that type exceeded a threshold. The net revenue gap between the optimal policy and the heuristics ranged from 5% to 12%. This gap was 4% higher in the more congested, single-scanner system than in the two-scanner system. The performance of the net revenue maximizing policy was similar to the heuristics, when compared with respect to the alternative performance metrics in the two-scanner case. Under the optimal policy, the average number of patients that were not scanned by the end of the day, and the average patient waiting-time, were both nearly 80% smaller in the two-scanner case than in the single-scanner case. The net revenue gap between the optimal policy and the priority-based heuristics was nearly 2% smaller as compared to the first-come-first-served and random selection schemes. Net revenue was most sensitive to inpatient (IP) penalty costs in the single-scanner system, whereas to IP and outpatient revenues in the two-scanner case. CONCLUSIONS: The performance of the optimal policy is competitive with the operational and economic metrics considered in this paper. Such a policy can be implemented relatively easily and could be tested in practice in the future. The priority-based heuristics are next-best to the optimal policy and are much easier to implement.