Experimental validation of a mathematical model to describe the drug cytotoxicity of leukemic cells

Chlorambucil (Chl), Melphalan (Mel), and Cytarabine (Cyt) are recognized drugs used in the chemotherapy of patients with advanced Chronic Lymphocytic Leukemia (CLL). The optimal treatment schedule and timing of Chl, Mel, and Cyt administration remains unknown and has traditionally been decided empirically and independently of preclinical in vitro efficacy studies. As a first step toward mathematical prediction of in vivo drug efficacy from in vitro cytotoxicity studies, we used murine A20 leukemic cells as a test case of CLL. We first found that logistic growth best described the proliferation of the cells in vitro. Then, we tested in vitro the cytotoxic efficacy of Chl, Mel, and Cyt against A20 cells. On the basis of these experimental data, we found the parameters for cancer cell death rates that were dependent on the concentration of the respective drugs and developed a mathematical model involving nonlinear ordinary differential equations. For the proposed mathematical model, three equilibrium states were analyzed using the general method of Lyapunov, with only one equilibrium being stable. We obtained a very good symmetry between the experimental results and numerical simulations of the model. Our novel model can be used as a general tool to study the cytotoxic activity of various drugs with different doses and modes of action by appropriate adjustment of the values for the selected parameters.