A Mathematical Tumor Model with Immune Resistance and Drug Therapy: An Optimal Control Approach

Abstract

We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four-population model that includes tumor cells, host cells, immune cells, and drug interaction. We analyze the stability of the drug-free equilibria with respect to the immune response in order to look for target basins of attraction. One of our goals was to simulate qualitatively the asynchronous tumor-drug interaction known as “Jeffs phenomenon.” The model we develop is successful in generating this asynchronous response behavior. Our other goal was to identify treatment protocols that could improve standard pulsed chemotherapy regimens. Using optimal control theory with constraints and numerical simulations, we obtain new therapy protocols that we then compare with traditional pulsed periodic treatment. The optimal control generated therapies produce larger oscillations in the tumor population over time. However, by the end of the treatment period, total tumor size is smaller than that achieved through traditional pulsed therapy, and the normal cell population suffers nearly no oscillations.

Evaluating optimal therapy robustness by virtual expansion of a sample population, with a case study in cancer immunotherapy

“…The use of optimal control in the context of the VEPART method would allow a more general optimal protocol to be identified, where "optimal" could be defined in many ways; for instance, one could seek to minimize tumor volume at an endpoint, minimize tumor volume over a time horizon, minimize drug concentration, minimize toxicity, minimize some weighted average the above, etc. (37)(38)(39)(40)(41)(42)(43)(44)(45). Further, this optimization can be performed subject to various types of constraints; for instance, toxicity could be introduced as a constraint on the amount of drug that can be administered (37,46).…”

A mathematical model of immune response to tumor invasion

“…The two terms in the first equation, F N and F L represent the fraction of tumor cells killed in interactions with the two types of immune cells. Traditionally in the literature these competition terms are proportional to the competing populations, (see [2,4,7]). This form is generally justified by considering a cell-kinetic mechanism through which each immune cell has some fixed probability of encountering each tumor cell.…”

“…The mathematical structure of the model is based upon earlier modeling work [2], in which tumor growth, an immune response, and chemotherapy treatment are represented by a system of four differential equations. Our representation also extends other lower-dimensional models, such as that described in [3] in which different cell populations are represented as interacting species.…”

Delay Dynamics of Cancer and Immune Cell Model

“…The weight parameters C, D, E, and F are inserted to allow for discussion of the importance of the minimization strategy as these constants are varied. 6 rate of removal of y and I k 5 , k 7 rate of drugs in removal of y and I γ natural decay rate of chemotherapy τ time of cells in interphase (delay variable)…”

“…This enables a more accurate approach to treatments [18]. With the collaboration among biologists, mathematicians, and medical professionals, mathematical concepts that were more engineering based have been found to be useful in medical applications, [6,7,8,22,9]. The goal of applying optimal control theory to mathematical models representing the interaction between tumor, immune system, and chemotherapy is to determine the ideal mix of treatments that minimizes both tumor mass and negative effects upon the health of the patient.…”