Investigation of the dynamics of chronic lymphocytic leukemia in immuno- and chemotherapeutic treatment: a delayed model

Abstract

The article presents a mathematical model for the development of chronic lymphocytic leukemia, based on the Cauchy problem for a system of ordinary differential equations with delay. The system of equations includes relationships that describe the dynamics of changes in the concentrations of cancer cells, immune cells, and the drug. The interaction between tumor and healthy cells is carried out under the following assumptions: the growth of tumor cells occurs in accordance with the logistic law with a delay, immune cells are produced by the body and die naturally, the relationship between cancer and immune cells has a negative effect on the growth of both populations, the production of cancer cells stimulates growth of immune cells, the action of the chemotherapy drug is subject to the hypothesis of logarithmic destruction, the chemotherapy drug is excreted from the body in accordance with the first order kinetics. For this model, computational experiments were carried out in the absence of treatment, with chemotherapy with cyclophosphamide, and with a combination of chemotherapy with immunotherapy. The influence of the delay parameter on the course of the disease was studied: in the absence of chemotherapeutic treatment, the delay parameter does not affect the dynamics of the process; during chemotherapy or a combination of chemotherapy and immunotherapy, the solutions obtained differ with different options for the delay parameter. The results of the studies allow us to assess the dynamics of the disease and determine the number of courses of chemotherapy and immunotherapy to achieve remission.