Efficient modeling of chemotherapy regimens using mixed-integer linear programming

In this article, we focus on determining a minimum-cost treatment program aimed at maintaining the size of a cancerous tumor at a level that allows the patient to live comfortably. At each predetermined point in a treatment horizon, the patient either receives drug treatment or does not. In the first case, the tumor shrinks and its size is multiplied by a constant factor lower than 1; in the second, it grows following an exponential or Gompertz growth law. We first demonstrate that a simple heuristic solution provides an optimal treatment program. We then show that the Gompertz function can be described, like the exponential function, by a simple recurrence relation that does not explicitly depend on time. Thanks to the characteristics of the logarithmic function, this property allows us to formulate the problem as a mixed-integer linear program. This result makes it possible to solve the problem very efficiently using one of the many solvers available to handle this type of program, and above all to consider and solve several extensions to the problem. In particular, we show how to determine which of the optimal equivalent solutions to the initial problem are the most relevant. We also show how to measure the effect of a marginal increase in the treatment budget on patient quality of life. Numerous computational experiments are presented to illustrate these issues.