Mathematical model of immunotherapy response to antibodies as a treatment for cancer

Abstract

Immunotherapy is a significant cancer treatment as it uses the body’s natural immune system to fight cancer. To help boost the immune system, monoclonal antibodies (MABs) are used as they bind to cancer cells helping the immune system recognize these cells. In this paper, we present a mathematical model of nonlinear partial differential equations describing the interaction between the immune cells, MABs, and cancer cells. After nondimensionalizing the model, we analyzed the long-term behavior and found later that it is consistent with the numerical results. Then, we calculated the numerical solutions of the model with different values of the parameters (relative growth rate of cancer cells and the number of immune cells that are removed after killing a cancer cell) to determine the values that help increase the effectiveness of the treatment. We have considered the continuous delivery of antibodies over a certain period of time. These simulations showed that immune cells will eradicate cancer if the number of immune cells that are removed after killing a cancer cell is less than one. However, if each immune cell kills only one cancer cell, then the treatment reduces the cancer to a steady state or almost a steady state. On the other hand, if the relative growth rate of cancer cells is very small and each cancer cell needs more than one immune cell to kill it, then again, we get a steady state for cancer. However, if the relative growth rate is not small, then the cancer will grow after an initial decrease. This study could be implemented into a clinical trial with different delivery protocols of the drug to improve cancer treatment.