Model for tumour growth with treatment by continuous and pulsed chemotherapy

Abstract

In this work we investigate a mathematical model describing tumour growth under a treatment by chemotherapy that incorporates time-delay related to the conversion from resting to hunting cells. We study the model using values for the parameters according to experimental results and vary some parameters relevant to the treatment of cancer. We find that our model exhibits a dynamical behaviour associated with the suppression of cancer cells, when either continuous or pulsed chemotherapy is applied according to clinical protocols, for a large range of relevant parameters. When the chemotherapy is successful, the predation coefficient of the chemotherapic agent acting on cancer cells varies with the infusion rate of chemotherapy according to an inverse relation. Finally, our model was able to reproduce the experimental results obtained by Michor and collaborators [Nature 435 (2005) 1267] about the exponential decline of cancer cells when patients are treated with the drug glivec.

Introduction

Cancer is the name given to a cluster of more than 100 diseases that presents a common characteristic, the disorderly growth of cells that invade tissues and organs (Anderson et al., 2001, Brú et al., 2003). These cells may spread to other parts of the body rapidly forming tumours (Baserga, 1965).

An important mechanism of body defence against a disease caused by a virus, bacteria or tumour is the destruction of infected cells or tumours by actived cytotoxic T-lymphocytes (CTL) cells also known as hunter lymphocytes. CTL are able to kill cells or to induce a programmed cell death (apoptosis). The biological activation process occurs efficiently when the CTL receive impulses generated by T-helper cells (T_H). The stimuli occur through the release of cytokines. This phenomenon is not instantaneous; besides the time elapsed to convert resting T-lymphocytes in CTL, there is also a natural delay of the cytological process (Wodarz et al., 1998, Iarosz et al., 2011). Banerjee and Sarkar (2008) studied the dynamical behaviour of tumour and immune cells using delay differential equations. They observed the existence of oscillations in tumour cells when a time delay was considered in the growth of T-cells.

A possible way to stop the growing of cancer cells is chemotherapy. That is, the treatment with a drug or combination of drugs through some protocol. There are many experimental and theoretical studies about the effects of the chemotherapy on the cells. Moreover, mathematical models have been considered to simulate the growth of cancer cells (Liu et al., 2012), as well as, tumour-immune interactions with chemotherapy (De Pillis et al., 2007).

In this paper we investigate a mathematical model for the growth of tumours that not only take into consideration the time delay character of the lymphocytes dynamics, but also the effect of the chemotherapy. We extend the model of Banerjee and Sarkar (2008) by adding the chemotherapy, and by considering some clinically plausible protocols. Firstly, a continuous chemotherapy is analysed. Secondly, the traditional or pulsed chemotherapy protocol is analysed, in which the drug is administered periodically. According to experimental protocols, we have used both a constant amplitude (Ahn and Park, 2011) and an oscillatory amplitude (Kuebler et al., 2007) for the continuous infusion rate of chemotherapy (Pinho et al., 2002).

One of our main results is to show that there are a large range of relevant parameters that lead to a successful chemotherapy. In a successful chemotherapy, the predation coefficient of the chemotherapic agent acting on the cancer cells and the infusion rate of the chemotherapy are inversely related. For the continuous chemotherapy, we have ensured the stability of the non-cancer state (i.e., a successful chemotherapy) by calculating the Lyapunov exponents of the non-cancer solution. Finally, our model was able to reproduce the experimental results obtained by Michor et al. (2005) about the exponential decline of cancer cells when patients are treated with the drug glivec.