Chaotic Attractors in Tumor Growth and Decay: A Differential Equation Model

Abstract

Tumorigenesis can be modeled as a system of chaotic nonlinear differential equations. A simulation of the system is realized by converting the differential equations to difference equations. The results of the simulation show that an increase in glucose in the presence of low oxygen levels decreases tumor growth.

Appendices

Appendix 1: Calculations for DE Model

The derivations for the C/Java model are provided below:

$$ {\displaystyle \int}\frac{\mathrm{d}n}{\mathrm{d}t}\mathrm{d}t=0 $$

$$ \dot{f}=\frac{f_{\mathrm{new}}-{f}_{\mathrm{old}}}{t}=\alpha \eta \left(m-f\right) $$

$$ \dot{m}=\frac{m_{\mathrm{new}}-{m}_{\mathrm{old}}}{t} = \beta \kappa n+f\left(\gamma -c\right)-m $$

$$ \dot{c}=\frac{c_{\mathrm{new}}-{c}_{\mathrm{old}}}{t}=\nu fm-\omega n-\delta \phi c $$

For a unit time change of 1, solve for f _new, m _new, and c _new:

$$ {f}_{\mathrm{new}}=\left(1-\alpha \eta \right){f}_{\mathrm{old}}+\alpha \eta {m}_{\mathrm{old}} $$

$$ {m}_{\mathrm{new}}=\beta \kappa n+{f}_{\mathrm{old}}\left(\gamma -c\right) $$

$$ {c}_{\mathrm{new}}=\left(1-\delta \phi \right){c}_{\mathrm{old}}+\nu {f}_{\mathrm{old}}{m}_{\mathrm{old}}-\omega n $$

Appendix 2: Sample Source Code

C Programming Language

#include <math.h>

#include <stdio.h>

#include <stdlib.h>

#define ITERATIONS    900

int main ()

{

inti;

FILE *F1;

floatn_old = 0.50; // tumor cell density 50

floatf_old = 0.0; // matrix–metalloproteinases concentration

floatf_new = 0.0;

floatm_old = 0.0; // matrix-degradative enzyme concentration

floatm_new = 0.0;

floatc_old = 0.0; // Oxygen concentration

floatc_new = 0.0;

float alpha = 0.06; // tumor cell volume

float beta = 0.05; // glucose level

float gamma = 0.265; // number of tumor cells 26.5

float delta = 0.40; // diffusion from the surface 40

floatdn = 0.0005;

float dm = 0.0005;

float dc = 0.5;

float rho = 0.01;

float eta = 0.50; //was 50

float kappa = 1.0;

float sigma = 0;

float nu = 0.5;

float omega = 0.57;

float phi = 0.025;

F1 = fopen("chaotumor.txt","w");

for (i = 0; i< ITERATIONS; i++)

    {

        f_new = alpha*eta*(m_old - f_old) + f_old;

        m_new = beta*kappa*n_old - f_old*c_old + gamma*f_old;

        c_new = nu*f_old*m_old - omega*n_old - delta*phi*c_old + c_old;

        f_old = f_new;

        m_old = m_new;

        c_old = c_new;

        fprintf (F1,"%f",m_new);

        fprintf (F1," %f\n",f_new);

        fprintf (F1," %f\n",c_new);

    }

fclose(F1);

}

Java Programming Language

importjava.util.Scanner;

import java.io.*;

importjava.lang.Math.*;

importjava.util.Random;

importjava.util.*;

public class ChaosTumor

{

 // initialize system parameters

static double alpha = 0.06; //tumor cell volume

static double beta = 0.05; // glucose level

static double gamma = 0.265; // number of tumor cells, 26.5

static double delta = 0.40; // diffusion from surface, 40

 // initalize system constants

static double n = 50;

static double dn = 0.0005;

static double dm = 0.0005;

static double dc = 0.5;

static double rho = 0.01;

static double eta = 50;

static double kappa = 1.0;

static double sigma = 0;

static double nu = 0.5;

static double omega = 0.57;

static double phi = 0.025;

public static void main (String[] args)

 {

 // initialize parameters

double f; // matrix-metalloproteinases concentration

double m; // matrix-degradative enzyme concentration

double c; // oxygen concentration

 // ask user input for growth, capacity, and initial population

 Scanner scan = new Scanner (System.in);

System.out.println("Chaos in Tumor Growth Dynamics Study");

System.out.println("*****************************");

System.out.print("Enter MM concentration, f: ");

 f = scan.nextDouble();

System.out.print("Enter MDE concentration, m: ");

 m = scan.nextDouble();

System.out.print("Enter the oxygen concentration, c: ");

 c = scan.nextDouble();

System.out.println("*****************************");

 // open file

try

 {

    // open output file

 File outFile = new File("ChaosTumor.txt");

BufferedWriter writer = new BufferedWriter(new FileWriter(outFile));

 // print out the inital condition

    writer.write("Initial conditions :"+m+" "+f+" "+c);

System.out.println("Initial population: "+m+" "+f+" "+c);

    writer.newLine();

    double [] temp = new double[3];

    for(inti=1; i<51; i++)

    {

        temp = compute(m,f,c);

        f = temp[0];

        m = temp[1];

        c = temp[2];

        // remove negative values

        if(m<0)

            m=0;

        if(f<0)

            f=0;

        if(c<0)

            c=0;

        writer.write("Iteration "+i+": "+m+" "+f+" "+c);

        writer.newLine();

        System.out.println("Iteration "+i+": "+m+" "+f+" "+c);

    }

    // close output file

writer.close();

 }

catch (IOException e)

 {

System.err.println(e);

System.exit(1);

 }

 // close file

 } // end main

 /**

 * method: compute(float m, float f, float c)

 * computes an array containing [m,f,c] at t+1

 * @param: float m, float f, float c

 * @precondition: none

 * @postcondition: none

 * @return: returns m, f, and c concentrations at t+1

 **/

public static double[] compute(double m, double f, double c)

 {

    double[] array = new double[3];

array[0] = (1-alpha*eta)*f + alpha*eta*m;

    array[1] = beta*kappa*n +(gamma-c)*f;

    array[2] = (1-delta*phi)*c + nu*f*m - omega*n;

return array;

 }

}