Modelling adaptive therapy for tumor using reaction-diffusion equation

Adaptive therapy is an on-and-off treatment depending on the size of tumor which aims to delay the emergence of resistance in cancer treatment. To delay cancer resistance, adaptive therapy will suppress resistant cells growth by allowing a significant number of sensitive cells to remain alive during treatment breaks to compete with resistant cells. It is important to study the characteristics of cancer or factors which will affect the tumor response to adaptive therapy. In this article, we shall model sensitive cells, resistant cells, drug concentration and nutrient concentration by one-dimensional reaction-diffusion equations. It is assumed that the growth of sensitive cells and resistant cells depends on the nutrient concentration. The sensitive cells and resistant cells will compete for resources and we use the Lotka-Volterra model to describe the competition. We study whether the initial fraction of resistant cells and the nutrient concentration will extend the time to progression for adaptive therapy beyond that of the standard of care called continuous therapy. We consider the source of nutrient to be either at both ends of tumor or only at one end of tumor. For a very rare initial resistant cells population, adaptive therapy extends the time to progression the most for both cases of source of nutrient. Tumor progresses slightly faster when the source of nutrient is at both ends of tumor for both adaptive therapy and continuous therapy while the time gained by adaptive therapy is slightly smaller compared to when the source of nutrient is at one end only.