Model for tumour growth with treatment by continuous and pulsed chemotherapy
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 (TH). 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.
Section snippets
The model
We extend a mathematical model proposed by Sarkar and Banerjee (2005) including the chemotherapic agent. The model is based on the predator-prey system. The T-lymphocyte is the predator, while the tumour cell is the prey that is being attacked. The predators can be in a hunting or a resting state. The resting cells do not kill tumour cells, but they can become hunters. The activation occurs not only due to cytokines released by macrophages that absorb tumour cells, but also by direct contact
Continuous chemotherapy
In this section we consider the continuous application of chemotherapy, without pause or interruption. That is, the value of the Δ is constant in time.
Pulsed chemotherapy
Often chemotherapy treatments are carried out in cycles. The repeated application of drugs for a short time is a typical protocol for chemotherapy, called pulsed chemotherapy (De Pillis and Radunskaya, 2003). For example, in this protocol, one may use the drug doxorubicin combined with other drugs to treat some types of cancer. The chemotherapy with these drugs is given through cycles of treatment according to the type of cancer (Shulman et al., 2012).
In the following, we will consider two
Conclusions
We propose a delay differential equations model for the evolution of cancer under the attack of both the immune system and chemotherapy. The novelty in this model is the introduction of the chemotherapy and the adjustment of parameters according to recent experimental evidence. We considered some types of protocols aiming at the cancer suppression.
We studied a continuous administration of drugs. The solutions of the system are stable, presenting a limit cycle behaviour. We identified domains of
Acknowledgements
This study was partially supported by the following Brazilian Government Agencies: CNPq, CAPES, FAPESP and Fundação Araucária. HPR, MSB, CG acknowledge the RSE-NSFC (443570/NNS/INT-61111130122). K. C. Iarosz acknowledges CAPES Foundation, Ministry of Education of Brazil, Brasília - Processo n. 1965/12-3.
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