Dynamic Behavior of a Tumor Growth Model in Discrete System

IJIRST –International Journal for Innovative Research in Science & Technology| Volume 3 | Issue 02 | July 2016 ISSN (online): 2349-6010

Dynamic Behavior of a Tumor Growth Model in Discrete System A. George Maria Selvam Sacred Heart College, Tirupattur - 635 601, S. India

K. Dhanalakshmi Sacred Heart College, Tirupattur - 635 601, S. India

Abstract In this paper, we consider a discrete model of tumor growth. Interaction of two different types of cells involved in the tumor growth is modeled with difference equations. Existence of steady state is established and local stability analysis is carried. To illustrate our results numerical simulations are also presented.

Keywords: Tumor Growth Model in Discrete System, Tumor Growth Model _______________________________________________________________________________________________________ I.

INTRODUCTION

Cancer is a disease characterized by the uncontrolled growth and spread of abnormal cells. If the spread is not controlled, it can result in death. Cancer arises from one single cell. The transformation from a normal cell into a tumor cell is a multistage process. A proper diagnosis is essential for adequate and effective treatment because every cancer type requires a specific treatment. Every year, cancer claims the lives of more than half a million Americans. About 1,685,210 new cancer cases are expected to be diagnosed in 2016. About 595,690 Americans are expected to die of cancer in 2016, which translates to about 1,630 people per day. With more than 1,300 persons dying of cancer every day, it has become one of the major causes of death in India. There has been close to 5 lakh deaths due to cancer in India in 2014. About 4,91,598 people died in 2014 out of 28,20,179 cases.

Mathematical models are often used to predict progression of cancer and treatment. Tumor growth models are concerned with differential or difference equations to describe the growth of cancer cells. When there is no overlap in population between each generation, discrete models using difference equations are more suitable. The simplest model for growth of a single species can be written as N ( t  1)  f ( N ( t ))

A simple example is N ( t  1)  rN ( t )

This difference equation is linear and sometimes called an exponential or geometric model. The parameter r is the finite growth rate of the population and r is useful in analyzing real population data. Also r is the ratio of the population size at one time to its size one time-unit earlier. Solution is given by r 1  ,  t N ( t )  r N (0)   N (0), r  1  0, r 1 

An extension of the simple model, called the Ricker model includes a reduction of the growth rate for large N ( t )   N (t )   N ( t  1)  N ( t ) exp  r  1    , r  0, K  0 K   

And in non-dimensionalised form One of the most important models in the description of the growth of single species is Gompertz model which can be expressed as x ( t  1)   r x ( t ) log x ( t ).

P. F. Verhulst, a Belgian mathematician studied population models with limitations on resources in the 19th century. The carrying capacity of a population represents the absolute maximum number of individuals in the population, based on the amount of the limiting resource available. P.F.Verhulst asserted that: The growth rate of a population is proportional to the size of the population and to the fraction of the carrying capacity unused by the population. The discrete logistic model (Verhulst model) is N (t )   N ( t  1)  r N ( t ) 1  .

K  

Parameter r is called intrinsic growth rate and K as carrying capacity ( r , K model:

 0) .

The non-dimensional discrete logistic growth

x ( t  1)  r x ( t )(1  x ( t ))

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Dynamic Behavior of a Tumor Growth Model in Discrete System (IJIRST/ Volume 3 / Issue 02/ 070)

Discrete logistic model exhibits greatly different behaviors with only a small change in initial conditions or parameters [12].

Typical dynamics of the logistic growth are shown, see Figure-1.

Fig. 1: Discrete Logistic Growth Model

Lotka and Verhulst also formulated equations describing predator and prey interactions. Assume two species, a prey and a predator, and that the rate of predation is proportional to the size of the prey population and proportional to the size of the predator population. Furthermore: 1) Without the predator, the prey population increases at a rate proportional to the size of the prey population.

2) The prey growth rate is decreased proportional to the rate of predation.

3) Without the prey, the predator decreases at a rate proportional to the size of the predator population (the predator has no alternate food source).

4) The predator growth rate is increased proportional to the rate of predation.

Dynamics of interacting biological species has been studied in the past decades. In predator- prey interactions, the rate of increase of population of one species decreases and the other increases. The Lotka-Volterra model is the simplest model of predator-prey interactions, expressed by the following equations[6, 9]. x   ax  bxy y    cy  dxy

Where x , y are the prey and predator population densities and

a, b, c, d

are positive constants?

II. MODEL DESCRIPTION AND EQUILIBRIUM POINTS Many authors have used mathematical models to describe the interaction among the various components of tumor growth [1,4,10,14]. A tumor is a dynamic nonlinear system in which bad cells grow, spread and eventually overwhelm good cells in the body. Several authors have used the concept of prey-predator type interactions in tumor studies where in general the immune cells play the role of predator and the tumor cells that of prey [5]. Proliferating cells are often presumed to be more mutable.

Mutation is relatively easy to measure in proliferating cells and tissues. Most of the cells are in a quiescent state and only a proportion is in proliferative state. We shall consider two states of cells: proliferative and quiescent. The vast majority of cells, from prokaryotes up to vertebrate organisms, spend most of their time in quiescence. Quiescent cancer stem cells have long been considered to be a source of tumor initiation. Tumor cells multiply more than normal cells and most of the therapies target proliferative cell populations. The discrete time models governed by difference equations are more appropriate when the populations have non overlapping generations. Discrete models can also provide efficient computational models of continuous models for numerical simulations. The maps defined by simple difference equations can lead to rich complicated dynamics [2, 3, 8, 11, 13, 15]. This paper considers the following system of difference equations which describes interactions between two types of cells.

P ( n  1)  rP ( n ) [1  P ( n )]  bP ( n )  cQ ( n ) : proliferative cells, Q ( n  1)  bP ( n )  cQ ( n )  dQ ( n ) : quiescent cells

(1) Where r , b , c , d  0 . Here P ( n ) and Q ( n ) represent the densities with respect to cell size of the proliferating and quiescent classes respectively. The size of the tumor is P ( n )  Q ( n ) . The coefficients b  0 and c  0 represent the transfer from one compartment to the other (depends on growth factors and environmental conditions) and d  0 is the death rate of cells.

An equilibrium point is a point at which variables of a system remain unchanged over time. The system (1) has two equilibrium points E  (0, 0) and 0

2  r 1 b [1  d ] b [ r  1] b [1  d ] E1    ,  2 r [1  c  d ] r [1  c  d ] r [1  c  d ]  r

 . 

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Dynamic Behavior of a Tumor Growth Model in Discrete System (IJIRST/ Volume 3 / Issue 02/ 070)

The existence of trivial equilibrium point E0 is obvious. This equilibrium point implies the complete elimination of tumor.

Biologically, it means that both infected and uninfected tumor cells can be eliminated with time, and complete recovery is possible. The interior point E1 is an positive equilibrium point provided r  1

b (1  d ) (1  c  d )

.

III. STABILITY ANALYSIS In this section, we shall study the equilibrium solutions of model and their local stability. An equilibrium point is locally asymptotically stable if all solutions of the system approaches it as t   . To study the stability behavior of the model, we compute the Jacobian corresponding to each equilibrium point. Jacobian matrix of the system (1) at any point can be written as c  r  2 rP  b  (2) J (P, Q )   

  (c  d ) 

b

We have T race J ( P , Q )  r  2 rP  b  ( c  d ) and D et

J ( P , Q )  ( b  2 rP  r )( c  d )  bc .

For the system (1), we have the following analysis. From (2), Jacobian matrix for E0 is given by r  b J ( E0 )    b

c    (c  d ) 

For E0, we have Trace J ( P , Q )  r  b  c  d and Det J ( P , Q )  bd  r ( c  d ).

The Eigen values of the matrix

J ( E 0 ) are 1, 2 

rcd b 2



1

[( c  b )  ( d  r )] .

2

2

From (2), Jacobian matrix for E1 is given by 2 b (1  d )  2rb (1  c  d ) J ( E1 )    b 

    ( c  d )  c

For E1, we have T race J ( P , Q )  2  r  b  c  d  D et J ( P , Q )  ( r  2)( c  d ) 

2 b (1  d ) (1  c  d )

and

b (1  c  d )[ c  ( c  d )(1  d )]

.

IV. STABILITY OF FIXED POINTS The following lemma [15] is useful in the study of the nature of fixed points.

Lemma 1 p ( )    B   C 2

Let

and 1 ,  2 be the roots of p (  )  0. suppose that p (1)  0 . Then we have

( i ) 1  1 and  2  1 if and only if p (  1)  0 and C  1. ( ii ) 1  1 and  2  1 ( or 1  1 and  2  1) if and only if p (  1)  0 . ( iii ) 1  1 and  2  1 if and only if p (  1)  0 and C  1. ( iv ) 1   1 and  2  1 if and only if p (  1)  0 and B  0, 2 ( v ) 1 and  2 are com plex and 1   2 if and only if B  4 C  0 and C  1.

2

Proposition 2 The fixed point E 0 is a   

r 

Sink if

bd 4( c  d )

r

Source if r

Saddle if

.

bd 4( c  d ) bd 4( c  d )

.

and r 

bd 4( c  d )

.

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441

Dynamic Behavior of a Tumor Growth Model in Discrete System (IJIRST/ Volume 3 / Issue 02/ 070)

Proposition 3 The fixed point E 1 is a Sink if 3  b (1  d )  r  2  (1  c  d )

Source if Saddle if

r  3 r  3

b (1  d ) (1  c  d ) b (1  d )

(1  c  d )

1 (c  d )



and r  2 

b  c  1 d .

(1  c  d )  c  d  1 (c  d )



b  c  1 d .

(1  c  d )  c  d 

.

V. NUMERICAL SIMULATIONS In this section, we will present some numerical examples on the dynamics of tumor growth . Mainly, we present the time plots of the solutions P and Q with phase plane diagrams for the system. Dynamic natures of the system (1) about the equilibrium points under different sets of parameter values are presented. Also the bifurcation diagram indicates the existence of chaos in both prey and predator populations.

Consider the values r = 0.5, b = 0.02,c = 0.01,d = 0.01. It is the trivial equilibrium point. The Eigen values are 1  0.4804 and  2   0.0204

so that 

1,2

1

. Hence the trivial equilibrium point is a Sink. The time plot is presented Figure - 2.

Fig. 2: Stability at for E 0 system (1)

For the values r  2.65, b  0.65, c  0.45, d  2   0.9721 so

that 1,2

 1.

 0.01

it is an interior equilibrium point. The Eigen values are

1  0.1111

and

Hence the interior equilibrium point is a Sink.

Fig. 3: Stability at E1 for system (1)

Studies in nonlinear dynamics aims at identifying qualitative changes in the long-term dynamics predicted by the model.

Bifurcation theory deals with classifying, ordering and studying the regularity in the dynamical changes. Bifurcation diagrams provide information about abrupt changes in the dynamics and the values of parameters at which such changes occur. Also they provide information about the dependence of the dynamics on a certain parameter. Qualitative changes are tied with bifurcation.

A bifurcation point is a point in parameter space where the number of equilibrium points, or their stability properties, or both, change.

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Dynamic Behavior of a Tumor Growth Model in Discrete System (IJIRST/ Volume 3 / Issue 02/ 070)

Fig. 4: Bifurcation for Tumor Growth

This paper, dealt with a 2-dimensional discrete model of interaction between two types of cells. Equilibrium points are computed and stability conditions are obtained. The results are illustrated with suitable sets of parameter values. Numerical simulations are presented to show the dynamical behavior of the system (1).

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