Global dynamics of reaction-diffusion oncolytic M1 virotherapy with immune response
Introduction
Cancer is a group of diseases caused by the growth and spread of malignant tumor cells. If it is not controlled, it can lead to death. According to The International Agency for Research on Cancer (IARC), approximately 18.1 million new cancer cases and 9.6 million deaths were estimated in 2018 [2]. Many therapies have been used to treat cancer like radiotherapy and chemotherapy [3], [4]. However, these therapies are not selective and can invade tumor cells as well as normal cells. Thus, they may cause many side effects including fatigue and hair loss [4]. Oncolytic virotherapy is an experimental cancer treatment which has occupied a large area in medical and mathematical research in the last few years [5], [6]. It uses selective oncolytic viruses which are engineered to target tumor cells but stay away from normal cells. Oncolytic viruses infect and reproduce inside tumor cells resulting in cell death. After the death of a tumor cell, a large number of new viruses are released that are able to spread and kill other tumor cells [6]. Therefore, the ability of oncolytic virotherapy to completely remove the tumor depends on the efficacy of the therapy.
Many oncolytic viruses have been developed and used in clinical trials, and great results have been achieved [3], [7], [8], [9], [10]. Nevertheless, there are many challenges to this type of treatment which may reduce its efficacy and need more investigations [7], [11], [12]. One of these challenges is the immune responses against the tumor cells, which can limit the replication of the oncolytic viruses and thus cause a decrease in the number of viruses. One approach to solve this issue is to design selective viruses which can rapidly spread and kill tumor cells [9], [11], [12]. However, the relation between oncolytic virotherapy and tumor-specific immune responses is very complex and it is an active area of research. Sometimes, immune responses are stimulated to support the oncolytic virotherapy and clear the tumor [3], [11], [13].
Mathematical models have been used to help understand the complicated dynamics of oncolytic virotherapy in order to design better and more effective treatments for cancer. Some of these models are similar to HBV and HIV viral infection models [14], [15], [16], [17], [18], [19], [20]. For example, Wang et al. [7] extended the basic oncolytic virotherapy model [21] and studied the effect of the virus burst size, which represents the number of new viruses formed inside a tumor cell, on viral therapy. They found that the virus burst size is an element that should be taken into account when developing new viruses for viral therapy. Okamoto et al. [11] introduced a model to test the ability of selective and non-selective oncolytic viruses to eliminate tumor and improve treatments. Malinzi et al. [5] suggested a model to study the effect of combining oncolytic virotherapy with chemotherapy. Their model reflects the relations between the tumor cells, immune responses, oncolytic virotherapy and chemotherapy. They showed that virotherapy can enhance chemotherapy if the correct optimal dosage is used. Kim et al. [22] studied the interactions between tumor cells, cytotoxic T lymphocyte (CTL) immune response and virotherapy through using a mathematical model and investigated the conditions for the existence of a Hopf bifurcation. Ratajczyk et al. [6] analyzed a five-dimensional model to explore the effect of using oncolytic virotherapy with TNF- α inhibitors. TNF- α can kill tumor cells and limit the production of viruses. They showed that the inhibition of TNF- α can increase the efficacy of oncolytic virotherapy. However, spatial effects were not considered in the aforementioned models.
The diffusion of oncolytic viruses within the tumor may play a critical role in the delivery and success of the treatment [12], [13]. Many mathematical models have been improved to include the spatial and temporal distributions of viruses and cells. For instance, Tao and Guo [12] considered the movements of viruses and immune cells and studied a nonlinear system of partial differential equations with a moving boundary. They found out that the immune response against tumor cells may restrict the replication of oncolytic viruses and weaken the efficacy of virotherapy. Wang et al. [10] introduced a reaction-diffusion model and assumed that the normal cells, tumor cells and free virus diffuse in a bounded domain. They suggested different treatment strategies for oncolytic virotherapy. In each case, they determined the optimal dosage required to completely remove the tumor depending on gene mutations in the cell. Malinzi et al. [4] obtained traveling wave solutions of a reaction-diffusion system and studied the effect of virotherapy on the concentration of tumor cells in the presence of CTL immune response.
Lin et al. [23] identified a naturally occurring alphavirus M1 as a selective oncolytic virus targeting cancer cells that lack Zinc-finger antiviral protein (ZAP). The virus showed high efficiency in killing tumor cells without harming normal cells. To understand the role of this virus in oncolytic virotherapy, Wang et al. [1] formulated an ordinary differential equation model with competition between normal cells and tumor cells on a limited nutrient source. They investigated the effect of using the oncolytic M1 virus on the growth of tumor and normal cells, and they determined the minimum virotherapy dosage required to effectively eliminate the tumor. Their model takes the formwhere H(t), N(t), Y(t) and V(t) represent the concentrations of nutrient, normal cells, tumor cells and free M1 virus, respectively. The parameter κ represents the nutrient recruitment rate, and µ represents the minimum effective dosage of M1 virus. The normal and tumor cells consume the nutrient at rates β1HN and β2HY, respectively. The growth rate of normal cells as a result of consuming the nutrient is given by α1β1HN, while the growth rate of tumor cells is given by α2β2HY. The virus infects and kills tumor cells at rate β3YV, and it replicates at rate α3β3YV. The parameter d is the washout constant rate of nutrient and bacteria. The parameters η1, η2 and η3 are the natural death rate constants of normal cells, tumor cells and M1 virus, respectively.
In this paper, we extend model (1) by including the effect of CTL immune response on the oncolytic virotherapy. We assume that all model components undergo diffusion which gives a system of partial differential equations. We investigate the non-negativity and boundedness of the solution of the proposed model. In addition, we analyze all equilibrium points of the extended model, which was not accomplished in (1) as they only focused on the tumor-free equilibrium point. The paper is organized as follows. In Section 2, we give the details of the model under study. In Section 3, we investigate the non-negativity and boundedness of solutions. Moreover, we find all possible equilibrium points and determine the threshold conditions required for their existence. In Section 4, we prove the global stability and local instability of the corresponding equilibrium points. In Section 5, we carry out some numerical simulations to confirm the theoretical results of the previous sections. The conclusion is stated in Section 6.
Section snippets
A reaction-diffusion oncolytic M1 virotherapy model
In this section, we extend the model studied in [1] to include spatial effects and an immune response. The immune cells can be considered as cytotoxic T-lymphocytes (CTLs) which destroy tumor cells. The model is given by the following system of partial differential equations:
Basic properties
In this section, we show that the solutions of system (2), (3), (4) exist, non-negative and bounded. Also, we find the possible equilibrium points of (2) and the threshold conditions which govern the existence of these points. Theorem 1 Assume that . Then, model (2) has a unique, non-negative and bounded solution defined on for any given initial data satisfying (3). Proof Let be the set of all bounded and uniformly continuous functions from to and .
Global properties
In this section, we prove the global asymptotic stability of the six equilibrium points E0, E1, E2, E3, E4 and E5. Also, we discuss the instability of these points. Theorem 3 The competition-free equilibrium E0 is (a) globally asymptotically stable if and R1 ≤ 1, and (b) unstable if or R1 > 1. Proof Define a Lyapunov functionalThen, we get
Numerical simulations
In this section, we perform some numerical simulations in order to validate the theoretical results of the previous sections. For this purpose, we choose the spatial domain as with a step size . We consider the simulations on a time interval from to with a step size . We take the parameter values as follows: and . The rest of the parameters are taken as free parameters. Some of the parameter
Conclusion
In this paper, we studied the dynamics of a diffusive oncolytic virotherapy model with CTL immune response. The purpose was to study the effect of the CTL immune response against tumor cells on the efficacy of oncolytic M1 viral therapy. We found that the model has six possible equilibrium points which are stable under the following conditions:
- (a)
The competition-free equilibrium E0 exists and is globally asymptotically stable if and R1 ≤ 1. These threshold conditions determine when both the
Acknowledgment
This project was funded by the research and development office (RDO) at the Ministry of Education, Kingdom of Saudi Arabia. Under grant no. (HIQI-34-2019). The authors also, acknowledge with thanks research and development office at King Abdulaziz University (RDO-KAU) for technical support.
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