Optimal Control of Antigen Specific Antibody Interactions for Cancer Immunotherapy

Abstract

In the history of cancer treatment, the immunotherapy is considered to be the most promising treatment approach. The idea behind this breakthrough is to stimulate the patient’s own immune system to recognize the cancer cells and destroy them. In this therapy, the antibodies are known to be powerful medications to activate the immune system in different ways. They circulate throughout the body until they discover a substance that body recognize as alien i.e. antigen and bind to them. Similarly, cancer cells often have molecules on their surface known as tumor-associated antigens. The researchers can design many clones of the antibody that only target a certain antigen type such as one found on tumors or cancer cells. Then, these are used as an effective drug for treating cancer. Thus, the antigen specific antibody interactions play a vital role in cancer immunotherapy. In this study, we propose a dynamic model to represent the population of antigens and antibodies in cancer patients; in particular we focus on the antigen-specific-antibody interactions to elicit an immune response that leads to the death of cancer cells. We formulate a terminal control problem where the schedule and doses of these antibodies are considered as control variables. The objective functional has been formulated as a measure of antigen population at the end of the treatment period. Pontryagin minimum principle (PMP) has been used to obtain the optimal control policies. For illustration, a series of numerical results is presented showing the effectiveness of immune therapy for cancer treatment corresponding to the different scenarios, choices of parameters and treatment periods. The results indicate that the control doses are followed by the emergence of antigen population. This approach would be potentially applicable to determine and prescribe the optimal doses and schedules for cancer patients.