Dynamics of Cancer-Immune System with External Treatment and Optimal Control

Herein, we present a mathematical model of tumour-immune interactions in presence of chemotherapy treatment. The model is governed by a system of delay differential equations with optimal control variables. A discrete time-delay is considered to justify the time-needed for the effector cells to develop a suitable response to the tumour cells. The control variables are included to justify the best treatment strategy with minimum side effects by reducing the production of new tumour cells and keeping the number of normal cells above the average of its carrying capacity. The numerical simulations show that the optimal treatment strategy reduces the load of tumour cells increases the effector cells after few days of therapy. pathogens possess ways to overcome the innate barrier and successfully canalize the host. The adaptive immunity is, however, very specific, it is called into action against pathogens that can evade or overcome innate immune defences. This takes time to develop, but the adaptive immune system ’remembers’ antigens that it has previously encountered and responds immediately the next time they try to infect. There are two types of adaptive immune responses: humoral immunity, mediated by antibodies produced by B lymphocytes, and cell-mediated immunity, mediated by T-lymphocytes [8]. In fact, the innate immune response develops first and occurs on the order of minutes or hours; While adaptive immunity follows, innate and occurs on order of days or weeks. Each has inherit time-delay τ in their developments [9]. The interactions between tumour cells and immune system are very complex and need sophisticated models to describe such interactions [10]. Mathematical models, based on ordinary differential equations, delay differential equations, partial differential equations, have proven to be useful tools in analysing and understanding the IS interactions with viral, bacterial infections and cancerous cells. Several mathematical models have been suggested to describe the interactions of tumour and immune system over time (see e.g. the research papers [11-15]). Most of these papers describe the interactions between tumour cells and immune cells, tumour cells and normal cells alone [16], or consider the interactions of tumour-immune system with chemotherapy treatment [2,3]. In this paper, we provide a mathematical model of tumour-immune interactions in presence of chemotherapy treatment and optimal control variables. The control variables are incorporated to justify the best strategy of treatment and minimize side effects of the external treatment by reducing the production of new tumour cells, while keeping the number of normal cells above the average of its carrying capacity. 1Antigens may be contained within or on bacteria, viruses, other microorganisms, or cancer cells. Citation: Rihan FA, Rihan NF (2016) Dynamics of Cancer-Immune System with External Treatment and Optimal Control. J Cancer Sci Ther 8: 257261. doi: 10.4172/1948-5956.1000423 J Cancer Sci Ther, an open access journal ISSN: 1948-5956 Volume 8(9) 257-261 (2016) 258 Boundedness and Non-Negativity of the Model Solutions To show that the solutions of model (2) are bounded and remain non-negative in the domain of its application for sufficiently large values of time t, we recall the following Lemma: Lemma 1 (Gronwall’s Lemma [19]) Let x, ψ and χ be real continuous functions defined in [a,b], χ ≥ 0 for t ∈ [a,b]. We suppose that on [a,b] we have the inequality


Introduction
Indeed, cancer remains a major cause of death in both developing and developed countries; see the statistics of Health World Organization (WHO) [1]. The term cancer represents numerous diseases that can affect any part of the body. It can similarly be referred to as malignant tumours and neoplasms. The unique procedure that cancer cells adopt includes their ability to multiply rapidly, exceeding their normal boundaries, and accordingly invading the adjacent organs and spreading the disease. Cancer is the most challenging disease to treat, although great research effort dedicated to revealing the relation between the tumour cells and immune system. The desired outcome from cancer treatment should be destruction of all cancer cells in the body, while maintaining the minimum level of healthy cells. Chemotherapy is one of the most highly adopted cancer treatment modalities, however, it was proved to be not the most convenient solution for tumour regression [2,3]. Progress is currently being made in attempting to eliminate tumour cells in the host by using an experimental form of immunotherapy [4,5].
Immune system (IS) is responsible for monitoring substances that be normally present the body. Any new substance in the body that the IS does not recognize raises an alarm, causing the IS to attack it. Substances that cause the IS response are called antigen 1 . Cancer cells are different from normal cells in the body and usually have unusual substances on their outer surfaces that can act as antigens. However, cancer cells look very much like normal cells and are tricky, as they put almost a cape over themselves so that the IS cannot recognize the cancer cell [6].
The immunity (or defence against pathogens or cancer cells) has basically two categories: innate immunity and adaptive immunity [7]. The innate immune system represents a nonspecific (no memory) response to substance to which the body regards as foreign or potentially harmful. The innate system represents the first line of defence and quickly response to certain general signs of infection so that is unable to memorise the same pathogen should the body be exposed to it in the future. In most cases, the innate immunity can suffice to clear pathogens, but sometimes it is insufficient. In fact, pathogens possess ways to overcome the innate barrier and successfully canalize the host. The adaptive immunity is, however, very specific, it is called into action against pathogens that can evade or overcome innate immune defences. This takes time to develop, but the adaptive immune system 'remembers' antigens that it has previously encountered and responds immediately the next time they try to infect. There are two types of adaptive immune responses: humoral immunity, mediated by antibodies produced by B lymphocytes, and cell-mediated immunity, mediated by T-lymphocytes [8]. In fact, the innate immune response develops first and occurs on the order of minutes or hours; While adaptive immunity follows, innate and occurs on order of days or weeks. Each has inherit time-delay τ in their developments [9].
The interactions between tumour cells and immune system are very complex and need sophisticated models to describe such interactions [10]. Mathematical models, based on ordinary differential equations, delay differential equations, partial differential equations, have proven to be useful tools in analysing and understanding the IS interactions with viral, bacterial infections and cancerous cells. Several mathematical models have been suggested to describe the interactions of tumour and immune system over time (see e.g. the research papers [11][12][13][14][15]). Most of these papers describe the interactions between tumour cells and immune cells, tumour cells and normal cells alone [16], or consider the interactions of tumour-immune system with chemotherapy treatment [2,3]. In this paper, we provide a mathematical model of tumour-immune interactions in presence of chemotherapy treatment and optimal control variables. The control variables are incorporated to justify the best strategy of treatment and minimize side effects of the external treatment by reducing the production of new tumour cells, while keeping the number of normal cells above the average of its carrying capacity.

Boundedness and Non-Negativity of the Model Solutions
To show that the solutions of model (2) are bounded and remain non-negative in the domain of its application for sufficiently large values of time t, we recall the following Lemma:

Lemma 1
(Gronwall's Lemma [19]) Let x, ψ and χ be real continuous functions defined in [a,b], χ ≥ 0 for t ∈ [a,b]. We suppose that on [a,b] we have the inequality Therefore, we arrive at the following Proposition:

Proposition 1
Let (E(t), T (t)) be a solution of system (2), then E(t) < M 1 and T (t) < M 2 for all sufficiently large time t, where For the proof, we refer to [9].

Corollary 1
If , T ρ µ η ≥ + then the solutions (E,T) for model (2) are non-negative for any non-negative initial condition. However, if , T ρ µ η < + then there exist non-negative initial conditions such that E(t) becomes negative in a finite time interval.

Model with Chemotherapy and Control
To include external chemotherapy in model (2), we should consider extra two variables namely amount of chemotherapy u(t) and normal cells N (t) with two control variables v(t) and w(t) (Figure 1). We assume the homogeneity of the tumour cells, then the model takes the form The drug kills all types of cells, with different killing rate for each type of cells: ( ) is the fraction cell kill for a given amount of drug, u(t), at the tumour site. The parameters a 1 , a 2 , and a 3 are the three different response coefficients. v(t) represents the amount of dose that is injected into the system, while d 1 is the decay rate of the drug once it is injected. In this case, the quantity we will control directly

Mathematical Models
We first present a simple model that describes the dynamics of tumour cells, T(t), and activated effector cells, E(t), such as cytotoxic T-cells, macrophages, and natural killer cells that are cytotoxic to the tumour cells. We adopt a predator prey formulation of the tumour immunity problem as a battle between immune cells and tumour cells (predators and prey, respectively). The model takes the form with initial conditions: E(0) = E 0 , T (0) = T 0 . Of course, the interaction between the effector and tumour cells leads to a reduction in the size of both populations with different rates, which are expressed by -µE(t)T(t) and -nE(t)T(t), respectively. Because of this interaction, the immune effector cells decrease the population of tumour cells at rate n. While, tumour cells infect some of the effector cells and therefore, the population of uninfected effector cells decreases at the rate µ. If one considers T(t) as prey and E(t) as predator, then F(E,T) is assumed to be Michaelis-Menton form, so that In this term, ρ is the maximum immune response rate and η is the steepness of immune response. The presence of the tumour cells virtually initiates the proliferation of tumour-specific effector cells to reach a saturation level parallel with the increase in the tumour populations. Hence, the recruitment function should be zero in the absence of the tumour cells, whereas it should increase monotonically towards a horizontal asymptote [17]. Here σ represents the normal rate (not increased by the presence of the tumour) of the flow of adult effector cells into the tumour side (region). The source of the immune cells is outside of the system, so it is reasonable to assume a constant influx rate σ. Furthermore, in the absence of any tumour, the cells will die at a rate δ.
To make the mathematical model closer to the reality, one can incorporate a discrete time-lag τ in the model (1) to justify the time required to stimulate the effector cells and develop a suitable response to the tumour cells, after recognizing the tumour cells. The new model with discrete time-lag takes the form This model is called delay differential equations (DDEs), in which we must provide initial functions: E(t) = ψ 1 (t) and E(t) = ψ 2 (t) for all t ∈ [-τ, 0], instead of initial values. In model (2), the presence of tumour cells stimulates the immune response, represented by the positive nonlinear growth term for the immune cells ρE(t-τ )T (t-τ )/(η +T (t-τ )). ρ and η are positive constants, τ ≥ 0 is the time delay that presents the time needed by the immune system to develop a suitable response after recognizing the tumour cells. The saturation term (Michaelis-Menton form) with the E(t) compartment and logistic term with the T(t) compartment are consoled. The presence of the tumour cells virtually initiates the proliferation of tumour-specific effector cells to reach a saturation level parallel with the increase in the tumour populations. Of course, the solution of DDEs model (2) should be bounded and nonnegative [18].
is not u(t), but v(t). The tumour cells and normal cells are modelled by a logistic growth law, with parameters r i representing the growth rate of two types of cells: i=2 identifies the parameter associated with tumour, and i=3 identifies the one associated with the normal tissue. β 1 and β 2 are the reciprocal carrying capacities of tumour cells and host cells respectively. The two terms -c 1 N(t)T (t) and -c 2 N (t)T(t) represent the competition between the tumour and host cells.
Let C = C([−τ, 0], R 4 ) be the Banach space of continuous functions mapping the interval [-τ, 0] into R 4 with the topology of uniform convergence. It is easy to show that there exists a unique solution (E(t), T (t), N (t), u(t)) of system (3) with initial data (E 0 , T 0 , N 0 , u 0 ) ∈ C. For biological reasons, we assume that the initial data of system (3) satisfy E 0 ≥ 0, T 0 ≥ 0, N 0 ≥ 0, u 0 ≥ 0. For τ=0, the model is reduced to ODEs model developed by de Pillis and Radunskaya in [20].
The main objective in developing chemotherapy treatment, in system (3), is to reach either tumour-free steady state or coexisting steady state in which the tumour cells' size is small, while the normal cells' size is closed to its normalized carrying capacity. To keep the patient healthy while killing the tumour, our control problem consists of determining the variables v(t) and w(t) that will maximize the amount of effector cells and minimize the number of tumour cells. We use cost functional of the control with a constraint that to keep normal cells above the average of its capacity. Therefore, our objective is to maximize the functional ( [9]).
Where B u , B w are, respectively, the weight factors that describe the patient's acceptance level of chemotherapy and immunotherapy with a constraint ( , , , , , , ) 0.75 0, We are seeking optimal control pair (v * , w * ) such that where W is, the control set defined by The existence of optimal controls v * (t) and w * (t) for this model is guaranteed by standard results in optimal control theory [21]. Necessary conditions that the controls must satisfy are derived via Pontryagins Maximum Principle [22]. The optimal control problem given by expressions (3)- (7) is equivalent to that of minimizing the Hamiltonian (See the Appendix): and Theorem 1: There exists an optimal pair v*(t) and w*(t) and corresponding solutions E*, T*, N* and u* and that minimizes J(u(t), w(t)) over Ω. The explicit optimal controls are connected to the existence of continuous specific functions λ i for i = 1, 2, 3, 4 satisfying the adjoin system 0 otherwise.
Furthermore, the following properties hold (See the Delay Models with Optimal Control)

Discussion and Conclusion
In this paper, we provided a mathematical model with memory (time-delay) and optimal control variables to describe the dynamics of tumour-immune interactions in presence of chemotherapy treatments. The time-delay has been considered to describe the time needed by the immune system to launch a suitable response after recognizing the non-self-cells or foreign bodies. The numerical approximations of the optimal control problem are carried out using forward and backward Euler's methods. Starting with an initial guess for the value of the controls on the time interval [0, t f ], we solve the state system with control variables (3) using forward Euler's scheme. While, the adjoint system is solved using the solutions of the state system and the transversality conditions (9) backward in time. A Pontryagin-type maximum principle is derived, for optimal control problems with time-lag in the state variable. The control system is subject to a mixed control-state constraint to minimize the cost of treatment, reduce the tumour cells load, and keep the number of normal cells above 75% of its carrying capacity. Figure 2 shows the impact of chemotherapy treatments (with optimal control) when we choose the parameter values in an unstable region (σ =0.2, ρ =0.2, and τ =1.5). The tumour cell population is growing up over time in the absence of chemotherapy, while the presence of treatment helps the immune system to keep the growth of the tumour cells under its control.
The Figure shows that the tumour cells can be eradicated at day 10. The numerical simulations show the rationality of the model presented, which in some degree meets the natural facts.
The theoretical results presented in this paper convey a general insight to biologists and can help to gain better understanding of interaction mechanisms of tumour-immune system. They can be used to evaluate control strategies and applied for real cases in the future research work.

Delay Models with Optimal Control
Mathematical modelling with delay differential equations (DDEs) is widely used for analysis and predictions in epidemiology, immunology, physiology [23][24][25][26][27]. The time delays in these models consider a dependence of the present state of the modelled system on its history. In life, things are rarely so instantaneous; There is usually a propagation delay before the effects are felt. This situation can be modelled using a DDE.
where all the time-lags, τ i , are assumed to be none negative functions of the current time t. Because of these delay terms it is no longer sufficient to supply an initial value, at time t=t 0 , to completely define the problem. Instead, it is necessary to define the history of the state vector, y(t), sufficiently far enough back in time from t 0 to ensure that all the delayed state terms, y(t − τ i ), are always well defined. Thus, it is necessary to supply an initial state profile of the form: It should be noted that ψ(t 0-) need not be the same as y 0 . This immediately introduces the possibility of a discontinuity in the state, y(t).
We mention here that there are many problems in biosciences (such as epidemics, harvesting, chemostat, treatment of diseases, physiological control, vaccination) which can be addressed within an optimal control framework for systems of DDEs [28][29][30]. However, the amount of real experience that exists with optimal control problems (OCPs) is still small. The DDE (12) can be converted into an optimal control problem by adding an m-dimensional control term u(t)