Introduction
Cancer is the second leading cause of death in the world [1]. For the past 10 years, remarkable researches have been doing in the diagnosis, treatment, and prevention of cancer [2]. However, despite significant advances in these areas the mortality of cancer is rising [2]. It is predicted 13.2 million deaths due to cancer up to 2030. Hence, cancer modeling and treatment are in the area of interest to many researchers, including biologists, mathematicians and control engineers [3]. Scientists can develop more effective treatment strategies before any time-consuming and expensive clinical trials by using mathematical models and simulations.
A regular schedule should be set for avoiding the side effects of cancer drugs and maintaining the amount of drug use. Various control methods were used to solve this problem. Optimal control approach for drug scheduling is one of the most appropriate approaches. Using this approach not only we can optimize the use of the drugs but also we can help to reduce the effectiveness of cancer cells.
The issue of optimal chemotherapy was initially introduced as an optimal control problem by Swan and Vincent in 1977 [4]. In 2001, Parker and Doyle presented a comprehensive review of mathematical drug delivery [5]. In 2007, pillis et al. developed a mathematical model for the interaction among tumor cells, immune cells, and anticancer drugs [6]. In 2009, DiOnofrio et al. presented an optimal control model to find angiogenesis inhibitors and chemotherapy treatments simultaneously [7]. In 2011, a summary of the optimization methods was presented by shi et al. [8] for chemotherapy. The use of optimal model predictive control was investigated in [9]. In 2013, Moradi et al. made a comparison of the three optimal control methods for cancer treatment [10]. Ghaffari et al. examined combined radiotherapy and chemotherapy in metastatic cancer [11]. Lobato et al. presented an optimal strategy for treating cancer through a multi-objective optimization approach in 2016 [12]. In works [13-15], the adaptive approach for cancer treatment used.
However, nonlinearity in cancer dynamics is one of the challenging problems in cancer control. In [16, 17], the SDRE method is used for optimal chemotherapy due to its flexibility and ease of implimentation. In the presented paper, the pseudo-spectral method is used because it has a high convergence rate and simple implementation structure. This method is easily implemented by solving a nonlinear programming (NLP) problem. Moreover, this method can be demonestrated to general form of nonlinear systems and there is no need to the Lyapunov function to construct the asymptotically stabilizing control [18]. However, these issues are challenging in other methods.
Cancer relapse after a course of treatment is one of the main issues in cancer treatment, which should be considered for finite-duration treatment. This problem was investigated by carefully examining the dynamics of cancer in this paper. It is shown that modifying the dynamics of cancer along with the management of the cells population is necessary. In the proposed treatment strategy, the mixed immunotherapy and chemotherapy is regarded. Immunotherapy is implemented for modifying the dynamics of the cancer-immune system, and then chemotherapy is used for pushing the cell population to the desired level. The target of the immunotherapy is to strengthen the ability of the patient’s immune system.
In this study, we will analyze and expand a model of cancer during the treatment by adding immunotherapy and chemotherapy from a new perspective. Immunotherapy alters the parameters of the system and chemotherapy affects on the states of the system. Physicians recommend different anticancer drugs dose to patients due to side effects of these drugs. It is depend on many factors such as the gender of the patient, age and history of the patient’s diseases. In this paper, the age of the patients is considered in limiting the drug dose by using a fuzzy system.
The structure of the paper is as follows. In the next section, the mathematical model of cancer is presented. The dynamics of the system without any external input is considered, and the stability analysis of the equilibrium points and the bifurcation phenomena are examined. It is found that changes in the parameters of the system can alter the dynamics of the cancer fundamentally. The chemotherapy treatment protocol is derived using the pseudo-spectral method in section 3. A Mamndani fuzzy system is designed to limit the upper bound of the chemo-drug dose in designing chemotherapy protocol based on the age of the patient in section 3. The simulation results have been shown in section 5. In the end, the discussion and conclusion are presented in section 5.
A cancer model including mixed treatment
There are many models for cancer dynamics. A model should consider the cancer cells, immune cells and the effect of external drugs [19]. In this paper the model presented by Kuznetsov et al. [20] is used. This model shows the interaction between tumor cells and immune cells. Despite the simplicity of the model most significant aspects of cancer such as cancer relapse have been shown [13]. The effects of chemotherapy and immunotherapy are imposed in a new approach. Immunotherapy modifies the dynamics of cancer such that after immunotherapy the system’s dynamics has been changed, and chemotherapy affects on the states of the system.
The mathematical model of the system during the mixed treatment is presented as follows: