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.
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References
Ivancevic TT, Bottema MJ, Jain LC (2008) A theoretical model of chaotic attractor in tumor growth and metastasis. ar**v: 0807.4272 in Cornell University Library's ar**v.org
Sole R, Goodwin B (2000) Signs of life: how complexity pervades biology. Basic Books, New York, NY
Bar-Yam Y (2011) Concepts: chaos. New England Complex Systems Institute. http://www.necsi.edu/guide/concepts/chaos.html
Guiot C, Degiorgis PG, Delsanto PP, Gabriel P, Deisboeck TS. Does tumor growth follow a universal law? http://arxiv.org/ftp/physics/papers/0303/0303050.pdf
Annibaldi A, Widmann C (2010) Glucose metabolism in cancer cells. PubMed: http://www.ncbi.nlm.nih.gov/pubmed/20473153
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Appendices
Appendix 1: Calculations for DE Model
The derivations for the C/Java model are provided below:
For a unit time change of 1, solve for f new, m new, and c new:
Appendix 2: Sample Source Code
C Programming Language
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define ITERATIONSÂ Â Â Â 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;
 }
}
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Harney, M., Yim, Ws. (2015). Chaotic Attractors in Tumor Growth and Decay: A Differential Equation Model. In: Vlamos, P., Alexiou, A. (eds) GeNeDis 2014. Advances in Experimental Medicine and Biology, vol 820. Springer, Cham. https://doi.org/10.1007/978-3-319-09012-2_13
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DOI: https://doi.org/10.1007/978-3-319-09012-2_13
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