Abstract
In this paper, an antitumour model for characterising radiotherapy and immunotherapy processes at different fixed times is proposed. The global attractiveness of the positive periodic solution for each corresponding subsystem is proved with the integral inequality technique. Then, based on the differentiability of the solutions with respect to the initial values, the eigenvalues of the Jacobian matrix at a fixed point corresponding to the tumour-free periodic solution are determined, resulting in a sufficient condition for local stability. The solutions to the ordinary differential equations are compared, the threshold condition for the global attractiveness of the tumour-free periodic solution is provided in terms of an indicator , and the permanence of a system with at least one tumour-present periodic solution is investigated. Furthermore, the effects of the death rate, effector cell injection dosage, therapeutic period, and effector cell activation rate on indicator are determined through numerical simulations, and the results indicate that radioimmunotherapy is more effective than either radiotherapy or immunotherapy alone.
1. Introduction
Cancer is a major public health issue and the leading cause of death worldwide. According to the World Health Organization (WHO), there were nearly million cancer-related deaths in [1]. The global cancer burden is expected to rise to nearly million cases and million deaths by 2030 [2]. Although numerous effective medical treatments against cancer have been developed, cancer treatment remains a challenging problem in neoteric medicine [3]. Host cells, or normal cells, should be kept above their minimum level throughout the entire body during cancer remission. As a result, modern techniques, such as surgery, chemotherapy, and radiotherapy, fail to destroy cancerous cells due to a lack of effective treatment strategies. In addition, chemotherapy harms cells in the bone marrow (myelosuppression), hair follicles (alopecia), and digestive tract (mucositis) under normal conditions. Therefore, chemotherapy depletes the immune system of the patient, leading to dangerous infections. Therefore, many patients suffer from the adverse effects of the treatment in addition to therapeutic resistance and cancer recurrence.
Novel therapeutic strategies have been investigated, and immunotherapy has been recently approved for the treatment of various types of cancer [4]. Immunotherapy includes the use of antigen- and nonantigen-specific substances, such as cytokines, as well as adoptive cellular immunotherapy (ACI) [3]. Cytokines, such as IL-2 and IFN- , are soluble proteins that mediate cell-to-cell communication [5]. During ACI, tissue cells are cultured to enhance and expand the immune system. ACI can be administered in two ways: (i) lymphokine-activated killer (LAK) cell therapy, in which cells are extracted from patients and cultured in vitro with high concentrations of IL-2 in peripheral blood leukocytes before being injected back into the cancer site; and (ii) tumour-infiltrating lymphocyte (TIL) therapy, in which cells are extracted from lymphocytes recovered from the patient with cancer and incubated with high concentrations of IL-2 before being injected back into the patient. The use of ACI slows or stops the spread of cancer cells to other parts of the body and helps the immune system become more effective by eliminating cancer cells.
Various mathematical models have been studied for cancer treatments with virotherapy, radiotherapy, chemotherapy, and immunotherapy [6–11]. Based on the inhibition model, Piantadosi model, and autostimulation model, Antonov et al. investigated impulsive tumour growth models to describe medical interventions during cancer treatments [12]. Sigal et al. modelled the effects of immunotherapy, specifically dendritic cell vaccines and T cell adoptive therapy, on tumour growth with and without chemotherapy [13]. The model demonstrated that chemotherapy increases tumorigenicity, whereas CSC-targeted immunotherapy tumorigenicity. Pratap proposed a model that describes the nonlinear dynamics between tumour cells, immune cells, and three forms of therapy: chemotherapy, immunotherapy, and radiotherapy [14]. The model was used to develop optimized combination therapy plans using optimal control theory. Feng and Navaratna demonstrated that the initial ratio between regulatory T cells and effector T cells impacts the tumour recurrence time and that the effectiveness of IL-2 use may reverse the immunotherapy outcome [15].
Dong et al. investigated the role of helper T cells in the tumour immune system and proposed the following model [16]:
where , and represent the populations of tumour cells (TCs), effector cells (ECs), and helper T cells (HTCs), respectively. The first equation describes the rate of change in the TC population. Here, the logistic growth term was chosen, where is the maximal growth rate of the TC population, and is the carrying capacity of the TC biological environment. The second equation describes the rate of change in the EC population. ECs have an average lifespan of . is the EC stimulation rate by EC-lysed TC debris. is the EC activation rate by the HTCs. The third equation describes the rate of change in the HTC population. is the birth rate of the HTCs produced in the bone marrow. HTCs have an average lifespan of . is the HTC stimulation rate in the presence of identified tumour antigens. To address the lack of biostability, Talkington et al. assumed that and introduced saturation into the tumour interactions [17]: where is the birth rate of the ECs, and and are half-saturation constants.
As discussed above, radiotherapy is usually used in cancer treatment because it permanently damages the DNA of tumour cells, destroying these cells [18, 19]. While nearby healthy tissue cells can suffer temporary damage from this radiation, these cells can repair the DNA damage and continue to grow normally. Numerous studies have shown that radioimmunotherapy is more effective for inhibiting tumour growth than radiotherapy [4, 20]. Thus, compared to the continuous system models mentioned above, we introduce pulsed ACI and radiotherapy into system (2) and analyse the effect of the combined treatment [7, 21–23]. Our novel system is formulated as follows:
where , , and denote the death rates of the ECs, HTCs, and TCs due to radiotherapy at time , respectively. Here, , , and are the therapeutic period. represents the dosage of infusing the ECs with antitumour activity at time .
In this article, we study the effects of impulsive perturbations on the tumour-free solution of model (3) and the threshold values of its stability conditions. In addition, the mathematical criteria for the permanence of system (3) are investigated. Numerical simulations were carried out to validate our analytical results.
The article is organized as follows. In Section 2, for convenience, we present some definitions and lemmas. In Section 3, the local stability and global attractiveness of the tumour-free periodic solution are studied by means of the linearized Floquet stability and comparison techniques. Several additional technical computations that were used to establish the results presented in this section are deferred to see appendix. In Section 4, it is shown that once the threshold condition is satisfied, as well as certain other conditions, system (3) is permanent, with at least one tumour-present periodic solution. Numerical simulations that confirm our theoretical findings are discussed in Section 5 and Figures 1 and 2. Finally, a discussion of the theoretical and numerical results is provided.
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2. Preliminaries
In this section, we introduce some definitions and preliminary lemmas that are useful for establishing our results.
Definition 1 (see [24]). System (3) is said to be permanent if there are constants (independent of the initial values) and a finite time such that, for all solutions, with all initial values , and hold for all . Here, may depend on the initial value .
Similar to Lemma 1 in [21], we obtain that the solution of is
Thus, it follows that for , with and . Then, the following lemma is valid.
Lemma 2. is a positively invariant region for system (3).
Let ; then, system (3) can be reduced to the following system:
According to (5), we can obtain that
It is clear that
Let
then, we have
Thus, when (15) is valid, system (5) has a unique positive periodic solution, which can be formulated as follows: where
Hence, we can obtain the following conclusion.
Lemma 3. System (5) has a unique positive periodic solution if and only if and, for every solution of (5), it follows that
Proof. It is easy to prove that .
For an arbitrary , we choose an that is sufficiently small such that
where the first inequality in (17) is valid based on (15), and
Without loss of generality, assume that
for .
When and , it follows from (5) and (19) that
which implies that
Integrating (21) from or gives
It should be noted that
Then, it follows from (22) and (23) that
which implies that
Then, according to (25) and (17), there exists an such that when , it holds that
Hence, when , where , it follows from (22), (26), and (17) that
Since is arbitrary, we conclude that .
This completes the proof.
Similarly, we arrive at the following conclusion.
Lemma 4. For every solution of system (3), there exist three positive constants and such that for sufficiently large , provided that where with
Proof. On the basis of (29), we choose an that is sufficiently small; then,
where is defined in (36).
According to Lemma 2 and (3), there exists a such that when , it holds that
Then, consider the following system:
for . Similar to Lemma 3 it follows from (32) that
where is a solution of (34), and
with
and
with
Based on Lemma 2 and (3), it holds that for ; thus,
for , which implies that for . Thus, for arbitrary , there exists a such that when , it holds that
This completes the proof.
3. The Stability of the Tumour-Free Periodic Solution
Let denote the solution of the first three equations of (3) for initial data and , as follows:
Additionally, we can define the mappings as follows:
and the map as
Theorem 5. (i)The tumour-free periodic solution of system (3) is locally asymptotically stable provided that (ii)The tumour-free periodic solution of system (3) is globally attractive provided that
Proof. (1) According to (A.12), (A.13), (15), and (45), the three eigenvalues of the Jacobian matrix of map at point are
which implies that the tumour-free periodic solution is locally stable [25].
(2) Considering (46), we choose an such that
Let denote the solution of (5). Then, according to Lemma 2 and (3), we have ; thus,
which implies that [26]. Then, according to Lemmas 3 and 4, there exists a such that
for .
Then, for , we have
which implies that
Furthermore, we have that
It follows from (48) and (53) that
Moreover, it follows from (3) that
Based on (54) and (55), we have .
Similar to (50), we can prove that for arbitrary , there exists a such that
for . In addition, we can choose an that is sufficiently small such that
where and are defined in (60).
Based on the fact that , there exists a such that for . Then, the following system is considered:
For , we obtain the following positive periodic solution:
with
Similar to Lemma 3, it follows from the first inequality of (57) that , where is a solution of (58). Thus, there exists a such that
When , it follows from (59), (12), and (57) that
Therefore, based on (61), (62), and (56), we can infer that
Since is arbitrary, we conclude that .
To prove that , we choose an such that
where the first inequality results from (15), and the expressions of and are defined in (67).
Based on the fact that , there exists a such that for . Then, the following system is considered:
For , we obtain the following positive periodic solution:
with
According to (66), (11), and (64), when , it holds that
Similar to Lemma 3, we can prove that , where is a solution of (65). Thus, there exists a such that