Research Article - Onkologia i Radioterapia ( 2020) Volume 14, Issue 4

# Inspection of the global behaviour of a radiogenic cancer treatment nonlinear model

Kaushik Dehingia^{1}

^{*}, Bhagya Jyoti Nath

^{2}, Hemanta Kumar Sarmah

^{1}and Anusmita Das

^{1}

^{1}Department of Mathematics, Guwahati University, Assam, India

^{2}Department of Mathematics, Barnagar College, Sorbhog, Assam, India

**Kaushik Dehingia, Department of Mathematics, Guwahati University, Assam, India,**

^{*}Corresponding Author:**Email:**kaushikdehingia17@gmail.com

**Received: **03-Jul-2020
**Accepted: **20-Jul-2020
**Published:**
30-Jul-2020

## Abstract

Neoplastic diseases are liable for 12% of deaths around the world. The researchers of several disciplines tried to observe the dynamics of tumour growth with the response to other cell population of the physical body. They need the used ever-new technique to regulate the tumour growth and its eradication from the physical body. Clinically, there are many treatment strategies like surgery, chemotherapy, radiotherapy, gene therapy, immunotherapy and combination of this etc. to beat the disease. During this paper, we analysed a model for the effect of radiogenic therapy within the treatment of cancer. Local stability analysis at various equilibrium points has discussed during this paper. For greater interest, we have shown that the tumour free equilibrium point is globally asymptotically stable. We even have done numerical verification of our mathematical result.

#### Keywords

Cancer, radiogenic therapy, positivity, bounded-ness, local stability, global stability

#### Introduction

A multi-faceted disease, which involves heterogeneous interaction between abnormal cells and surrounding environment of it, called Cancer [1]. When the body is not capable to control the growth of abnormal cell and when the body cell lose their ability to die, occur cancer [2]. Surgery, chemotherapy, radiotherapy, hormone therapy, and immunotherapy, either in isolation or in the combination of two or more of these are prominently used in cancer treatment [3]. With the utilization of radioisotopes, radiation is highly effective to eradicate tumours [4]. Chemotherapy is the treatment in which mostly used chemically designed drugs. Today, the researchers heavily busy with immunotherapy because it contains lesser side effects than other treatments. Immunotherapy increasing the effectiveness of the immune system to counter cancer cells by emphasizing the body's own natural ability to battle cancer. It encompasses the use of cytokines with ACI, engineering genetic antibody called (mAbs), and IL-2 therapy to improve or restore immune system function. However, chemo-immunotherapy enriches tumour recession and antitumor immune response [1]. Speciously vaccines, monoclonal antibodies, lymphocytes and cytokines are the key tools for tumour immunology [5]. The harshness of disease, treatment procedure and asset of patients’ immune system are the key factors on which cancer treatment depends [6-9].

Gene therapy may use when no other therapies which are mentioned above treated well to the patient’s body. By introducing a well-designed gene into the cells of a patient gene therapy correct an inborn inaccuracy of metabolism, alter or repair an acquired genetic abnormality, and provide a new function to a cell [10]. Recently, the research has proven that with the lesser effect on normal tissues a combination of gene and radiotherapy called radiogenic therapy may perform well in cancer curability [4,11,12]. Also, it has clinically demonstrated that the use of radiogenic therapy has many potential reimbursements with different toxicity profiles of both two therapies [13]. Radiations counter abnormal cells and increase the proliferation rate of the body’s immune cells and greater the anti-tumour response [11]. At the same time, gene therapy also kills tumour cells by inserting toxin genes under the control of a tumour-specific promoter and blocks the mechanisms by which tumours evade immunological destruction [2].

Mathematical modelling is increasing rapidly to predict the effectiveness of specific treatment. Modelling of any biological phenomena links the experimental result to computational results [14,15]. Mathematical models played an active role in the research of disease dynamics and are becoming instrumental in planning treatment strategies and give deep insight into understanding complex biological processes [16]. Modelling of cancer is usually divided into two phases, the one is to deal with the concept of the system without treatment, and another one is treatments in the evocative form [1]. Treatments with new features are evolved in nowadays, and it becomes a broad research area for any kind of researchers. In this study, we have proposed a radiogenic therapy cancer model by modifying the Tsygvintsev et al. gene therapy model [12] and based on radiotherapy model [9] with some assumptions. Further, we have analysed the dynamics of the model using the basics of mathematics.

**Some mathematical preliminaries**

Suppose that *y=f(y*) is a vector form of a set of nonlinear ODE.
Suppose that *y ^{*}* is an equilibrium point. Let the Jacobian matrix
corresponding to the equilibrium point

*y*be as

^{*}*J*.

^{*}**Lemma:** An equilibrium point *y ^{*}*of the differential equation
y=f(y) is stable if all the eigenvalues of

*J*, the Jacobian evaluated at

^{*}*y*, have negative real parts. The equilibrium point is unstable if at least one of the eigenvalues has a positive real part [17].

^{*}**Definition: **A Lyapunov function W(Y) is a function consisting
of state variables which has a minimum at an equilibrium point
and which has no local minima [17].

**Lemma:** Let Ω be a region of phase space containing the
equilibrium point *y ^{*}*. Let

*W: Ω→R*be a continuous and differentiable function. W is a positive definite function for the point

*y*if it satisfies the following two conditions.

^{*}(i)*W(y ^{*}) = 0* and

(ii) *W(y)>0 for y∈Ω −{y ^{*}}* [17]

**Lemma:** Let *y ^{*}* be an equilibrium point of the differential
equation

*y=f(y)*and let

*W*be a positive definite function for this point. The equilibrium point is globally asymptotically stable i.e., the solutions tend to this point for initial conditions in the neighbourhood of

*y*if

^{*}*W (y) <0 for all y ∈ Ω -{y*[17].

^{*}}**The model**

In this study, we formulate our model that primarily based on
Tsygvintsev et al. [12] gene therapy model. The model deals
with effector cell population* E(t)*, tumour or cancerous cell
population *T(t)*, gene therapy *u _{1}* (one kind immunotherapy)
and a single dose of radiotherapy

*γ*.

The effector cell: To model effector cells *E(t)* at any time
t>0, our first assumption is that the NK cells, Interleukin-2,
LAK, Lymphocytes, T helper cells all belong to effector cells
population. Next, we supposed that during gene therapy, the
immune system strongly respond to the tumour site and it is
represented by the term *cT*. The effector cells proliferate itself
in its site which is represented by a Michaelis-Menten fashion , which is a decreasing function of effector cells. *–dE* is
the natural decay of effector cells. The external immunotherapy
source to the effector site is represented by the term *u _{1}*, which
is time-dependent. From these assumptions, the dynamics
of effector cell population can be expressed by the differential
equation

The tumour cell: Tumour cells *T(t) *proliferate logistically which
can be modelled by* rT(1-bT)*, at any time* t>0 *with parameters
and *b* assigning the cancer growth rate and cancer cell capacity
respectively. The term used for the high cancer clearance
during gene therapy with the effector cells negative response.
The single-dose radiation *γ* kills only the tumour direct contact
with tumour site [9]. Keeping in these, the dynamics of tumour
cells represented by following differential equation.

The overall model: With the above discussion, we may finalize our model, which is

**Assumptions:**

1. The effector cells *E(t)* proliferate by itself with a decreasing
function of Michaelis-Menten form

2. The effector cells *E(t)* arise with direct contact of tumour *T(t) *Both the two therapies can have the ability to clear the tumour
at the same time at a different level

3. Gene therapy clears the tumour combined with effector cells *E(t)*

4. Radiation γ supply directly to the tumour site such that it kills
tumour cells *T(t) *only

**Parameters**

**Positivity and bounded-ness**

Since the right-hand side of the system *(3)* is completely
continuous and locally Lipschitzian on C, the solution *(E(t),
T(t))* of the model with initial conditions *E(0) ≥ 0, T(0) ≥ 0* exists and is unique on* (ζ, 0), where 0 < ζ ≤ ±∞* [18]. From
system with initial conditions* E(0) ≥ 0, T(0) ≥ 0*, we have from
the second equation of the system.

Again

Since,

Therefore, we can drop these non-negative quantities, hence we get

Hence, all the solution of the system with initial conditions* E(0)
≥ 0 *and *T(0) ≥ 0* exists in the interval *(0,1)* and* E (t) ≥ 0* and *T(t) ≥ 0 *for all* t ≥ 0*

From the second equation of system, it follows that

From the standard Kamke comparison theory [19], we get

Now, from the first equation of the system (3), it follows that

*[Since T (t) bounded, Let sup T(t)=T _{s}*

*Taking, cT _{s} + u_{1}=K*

*[By assumption and parameter values this is constant]*

Hence,

The generalised Gronwall Lemma [20] gives *E (t)<M _{1} where M_{1}* is uniformly bounded.

Therefore,* T (t)* of the system subject to initial conditions* E (0)
≥ 0, T (0) ≥ 0* is bounded but *E (t) *may be bounded under some
conditions among parameters and the bound of *T (t) *for *t>0.*

**Local stability analysis at equilibrium points**

In this section, we will study the existence and stability analysis of the system (3) at various equilibrium points. The equilibrium points of the system (3) are

2. Cancer infected equilibrium

For stability analysis linearizing the system (3) to obtain Jacobian as follows:

Local stability at each equilibrium point is checked by using
Lemma 2.1 and the parameter **Table 1** and has the following
results.

The coexistence of both effector and tumour cell: With using
the value of the parameters from **Table 1** and the radiation dose *γ=0.1* the systems has two biologically feasible equilibrium
points out of which are cancer-free shows saddle
point behaviour with the unstable result and cancer infected
equilibrium point shows inward spiral
behaviour with locally asymptotically stable nature. Hence,
at this point, both effector and tumour compete with each
other in presence of radiogenic therapy. That means both cells
population goes to damped oscillation behaviour about zero
instead of asymptotes to zero (**Figure 1a**).

**Figure 1:** Radiogenic therapy; (a) Radiation dose γ=0.1; (b) Radiation dose *γ=0.195*; (c) Immunotherapy term u1 =53, cancer clearance term a=5 and
Radiation dose *γ=0.1*,

The radiogenic therapy may cut out: Employing the values
presented in **Table 1** as well as radiation dose *γ=0.195* number of stationary points is only one which
is cancer-free. This stationary point is asymptotically stable
and the system shows nodal sink nature at this equilibrium.
That means the effector cell population incorporates with the
treatment given can suppress the tumour growth to zero with
time increase (see **Figure 1b**). However, because of the higher
dose of radiation, which gives a negative impact on effector
cells, the effector cell population also goes down rapidly with
the increase of time. Which is not the aim of the treatment and
the treatment fails in this case.

Parameters | Meaning | Values | interval | Source |
---|---|---|---|---|

c |
cancer antigenicity | 0.05 (1/time) | [10^{-3},0.5] |
[12] |

u_{1} |
Immunotherapy term | 1 (cell/time) | [10^{-2},10^{2}] |
[12] |

p |
proliferation rate of E |
0.1245 (1/time) | 0.1245 | [12] |

f |
half-saturation for E proliferation term |
10^{-3}, (cells) |
[10^{-5},1] |
[12] |

d |
The half-life of eﬀector cells E |
0.03 (1/time) | 0.03 | [12] |

R |
cancer growth rate | 0.18 (1/time) | [10^{-1},2] |
[12] |

b |
cancer cell capacity | 10^{-9}(1/cells) |
10^{-9} |
[12] |

a |
cancer clearance term | 1 (1/cells) | [10^{-2},10^{2}] |
[12] |

g |
Half-saturation, for cancer clearance | 10^{5} (cells) |
10^{5} |
[12] |

**Table 1.** The parameter value for the model

The radiogenic therapy succeeds: Through the values presented
in **Table 1** as well as the immuno-therapy dose *u _{1} =53*, cancer
clearance term

*a=5*and the radiation dose γ=0.1, there exists only one equilibrium point . That means, at this, the body may be in a healthy stage as because the system behaves asymptotically stable with nodal sink nature. The tumour cells asymptote to zero rather than oscillating about zero (

**Figure 1c**) i.e., after a sufficient amount of time and a suitable amount of radiation and immuno-therapy dose the tumour cells are vanished by the use of treatment. This shows the combined effect of radiogenic therapy.

The initial values used in these simulations are *E(0)=T(0)=1000.*

#### Global Stability

Linear stability analysis shows how a system behaves in a neighbourhood of an equilibrium point. It does not, however, describe anything about what happens further away from equilibrium. In this section, we will show the point is globally stable by constructing a Lyapunov function, which described the total eradication of the tumour.

Define a Lyapunov functional of the model as

Differentiating on both sides with respect to time, we get

Where

Here,

And

∴*W <0*, because the matrix A is positive definite.

The equilibrium point satisfies all the conditions of
Lyapunov stability theorem i.e. tumour-free equilibrium point to be globally asymptotically stable [Lemma 2.3]. Hence,
the equilibrium point is globally asymptotically
stable on the domain (where is denote the nonnegative
octant of R^{2}) if the Lyapunov *W<0* at this point
(**Figures 2 and 3**).

**Figure 2:** Time series solution for various initial values for* E(t) and T(t)*

**Figure 3:** This figure depicts the point *P ^{*}_{1} (1770,0)* is globally asymptotically stable

#### Conclusion

Cancer research has undergone radical changes in the last
few decades. The main subject of this paper to investigate the
effectiveness of radiogenic therapy. Here, we have observed
that if we treat tumour site with sufficient amount of immunotherapy
drugs and radiotherapy dose under some circumstances,
cancer may be eradicated from the body. We further investigated
for the total eradication of cancer by checking global stability at
tumour free equilibrium. From our observation, the trajectory
of the system goes towards the point for various initial conditions (**Figure 2**) which described the global stability
of the system (**Figure 3**). Hence, we may conclude that cancer
may eradicate from the body with the use of radiogenic therapy
by choosing a suitable amount of drug dose. This research gives
small amounts of help to the oncologists in practice the cancer
treatment. In future, we will analyse the model with pulsed
radiation and optimal continuous radiation therapy using delay
differential equation for a better realistic scenario.

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