Mathematical Modeling, Analysis, and Simulation of Tumor Dynamics with Drug Interventions

Abstract

Over the last few decades, there have been significant developments in theoretical, experimental, and clinical approaches to understand the dynamics of cancer cells and their interactions with the immune system. These have led to the development of important methods for cancer therapy including virotherapy, immunotherapy, chemotherapy, targeted drug therapy, and many others. Along with this, there have also been some developments on analytical and computational models to help provide insights into clinical observations. This work develops a new mathematical model that combines important interactions between tumor cells and cells in the immune systems including natural killer cells, dendritic cells, and cytotoxic CD8+ T cells combined with drug delivery to these cell sites. These interactions are described via a system of ordinary differential equations that are solved numerically. A stability analysis of this model is also performed to determine conditions for tumor-free equilibrium to be stable. We also study the influence of proliferation rates and drug interventions in the dynamics of all the cells involved. Another contribution is the development of a novel parameter estimation methodology to determine optimal parameters in the model that can reproduce a given dataset. Our results seem to suggest that the model employed is a robust candidate for studying the dynamics of tumor cells and it helps to provide the dynamic interactions between the tumor cells, immune system, and drug-response systems.

Mathematical Modelling for the Role of CD4+T Cells in Tumor-Immune Interactions

“…It describes regulation and suppression of CD8 + T cell activity, which occurs when there are very high levels of activated CD8 + T cells without responsiveness to cytokines present in the system [40,42,59,60]. e activating effect of cytokines on CD8 + T cells is represented in equation (3) by the term (p i I/(g i + I))L [40,42,61]. CD4 + T cell population is represented by equation (4) which has a constant decaying rate μ 1 .…”

Mathematical Modelling of the Dynamics of Tumor Growth and its Optimal Control

“…where f (·) and g(·) are proliferation, competition and inhibition functions of the cells and K [·] , z(M )[·] are the killing cells term arising out of the effect of chemotherapy drug thus if K [·] = 0, there is no effect of drug. The term z(M ) = 1 − e −M represents the effectiveness of chemotherapy at some specific phases of cell cycles (Chang et al , 2003;Unni & Seshaiyer, 2019).…”

“…The drug intervention terms d 4 M and d 5 I in 26 and 27 represent the amount of chemotherapy and immunotherapy drugs in some interval of time. It ay be noted that the chemotherapy and immunotherapy drugs are eliminated from the body over time at a rate proportional to its concentration (Unni & Seshaiyer, 2019).…”

“…All the parameters in this section are estimated in cells −1 day −1 unit. As in (Unni & Seshaiyer, 2019), the effects of the existence of dendritic cells, and chemotherapy drugs, immunotherapy drugs etc.…”

Computational Mathematical Model Based on Lyapunov Function for the Hormonal Storage Control

“…Models and computer simulations, using real data, are capable to generate insights. They have the potential to predict normal and abnormal behaviors of a cell, organs, and other components of living systems [11,13,15,39]. The work of Cortez et al [13] used the Lotka-Volterra equations to verify the dynamics of storage-synthesis control in a cell.…”