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^{1}Department of Mathematical Sciences, College of Science, United Arab Emirates University, UAE

^{2}Faculty of Pharmacy, Clinical Program, Cairo University, Kasr El-Aini St., Cairo 11562, Egypt

- *Corresponding Author:
- Rihan FA

Department of Mathematical Sciences

College of Science, United Arab Emirates University

Al Ain, 15551, UAE

**Tel:**+971-3-76 73 333

**E-mail:**[email protected]

**Received date:** September 09, 2016; **Accepted date:** October 27, 2016; **Published date:** October 29, 2016

**Citation: **Rihan FA, Rihan NF (2016) Dynamics of Cancer-Immune System with External Treatment and Optimal Control. J Cancer Sci Ther 8:257-261. doi:10.4172/1948-5956.1000423

**Copyright:** © 2016 Rihan FA, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Cancer Science & Therapy

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.

Chemotherapy; Immune system; Mathematical modeling; Optimal control; Time-delay; Tumour

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 [11-15]). 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.

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

(1)

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

(2)

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].

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

Then

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 then the solutions (E,T) for model (2) are non-negative for any non-negative initial condition. However, if, 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

(3)

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 is not u(t), but v(t). The tumour cells and normal cells are modelled
by a logistic growth law, with parameters ri 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], R4) be the Banach space of continuous functions
mapping the interval [-τ, 0] into R4 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, u0 ≥ 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]).

(4)

Where B_{u}, B_{w} are, respectively, the weight factors that describe the
patient’s acceptance level of chemotherapy and immunotherapy with
a constraint

(5)

We are seeking optimal control pair (v*, w*) such that

(6)

where W is, the control set defined by

*W = {(v,w) : (v,w*) piecewise continuous, such that

(7)

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):

(8)

and γ ≥ 0 with γ(t)k(t) = 0, Where

A standard application of Pontryagins Maximum Principle leads to the following result:

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

(9)

with transversality conditions

(10)

Furthermore, the following properties hold

(11)

(See the Delay Models with Optimal Control)

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.

Mathematical modelling with delay differential equations (DDEs) is widely used for analysis and predictions in epidemiology, immunology, physiology [23-27]. 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.

(12)

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:

(13)

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 [28-30]. 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)

(14)

and a suitable objective functional (measure): J0(u)

Maximize

(15)

and subject to control constraint a ≤ u(t) ≤ b, and state constant
y(t) ≤ c, where a and b are the lower and upper bounds. The integrand,
L(:) is called the Lagrangian of objective functional which is continuous
in [0,t_{f}]. Additional equality or inequality constraint(s) can be imposed
in terms of J_{i}(u).

Pontryagin’s Maximum Principle [22] gives necessary conditions that the control and the state need to satisfy, and introduces an adjoint function to affix to the differential equation to the objective functional. The necessary conditions needed to solve the optimal control problem are derived from the so called ’Hamiltonian’ H which is given by the equation

(16)

Here, λ^{T} (t) is a vector of costate variables of the state variables y(t),
which is the solution of the equation

(17)

Where

Given the nonlinear Hamiltonian (16) in the controls v and w, the process of solving the optimal control problem is to solve the state system (14) together with the adjoint equation (17) and the following conditions:

(18)

This work is generously supported by NRF Project (UAE). The authors would like to thank the reviewers for their constructive comments which improved the manuscript.

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