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: