# A Bi-Geometric Fractional Model for the Treatment of Cancer Using Radiotherapy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Non-Newtonian Calculi and Fractional Operators

#### Fractional Operators on Non-Newtonian Calculus

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

## 3. Mathematical Model

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

## 4. Numerical Solution

**Example**

**1.**

**Proposition**

**4.**

**Proof.**

#### Computer Program and Determining Constants

^{−3}for cancerous and healthy cells, respectively. The associated ${\alpha}_{i}$ in the bi-geometric analogue can be calculated as ${\alpha}_{1}=\mathrm{exp}\left(9.7041\times {10}^{-4}\right)$ and ${\alpha}_{2}=\mathrm{exp}\left(0.3396\right)$. In the absence of radiation, cancer wins, resulting in the following conditions [4]:

^{3}multiplied by a billion.

## 5. Discussion

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

clear; clc; % Defining the constants for the model a1=exp(0.00097041); a2=exp(0.3396); b1=exp(0.0433); b2=exp(0.2385); k1=exp(1); k2=exp(1); e=exp(0.0008); q=exp(0.45); L=exp(0.26); u0=exp(0.284); v0=exp(0.284); h=0.01; mu=0.8; w=exp(35);X=u0; Y=v0; Rn=u0; Sn=v0; Z=u0; N=v0; Kn=u0; Jn=v0; % Defining the time variable as a vector A=zeros(1,100); B=zeros(1,100); C=zeros(1,100); D=zeros(1,100); E=zeros(1,100); F=zeros(1,100); for m=1:100 A(1,m)=exp(m*0.1); D(1,m)=m*0.1; end for s=1:100 t=A(1,s); t0=D(1,s); % Defining the partition n=round(log(t)/h); n0=round(t0/h); Un=u0; Vn=v0; Om=u0; Lm=v0; % Finding the predictor-corrector by applying 3 times radiotherapy for i=0:n-1 if (1<=exp(i*h)) && (exp(i*h)<=L) Un=exp(((h^mu)/(gamma(mu+1)))*(((n-i)^mu)-((n-i-1)^mu))*(a1*(1-Un/k1)-(b1*Vn)-(e*q))*exp(i*h)); Vn=exp(((h^mu)/(gamma(mu+1)))*(((n-i)^mu)-((n-i-1)^mu))*(a2*(1-Vn/k2)-(b2*Un)-q)*exp(i*h)); else Un=exp(((h^mu)/(gamma(mu+1)))*(((n-i)^mu)-((n-i-1)^mu))*(a1*(1-Un/k1)-(b1*Vn))*exp(i*h)); Vn=exp(((h^mu)/(gamma(mu+1)))*(((n-i)^mu)-((n-i-1)^mu))*(a2*(1-Vn/k2)-(b2*Un))*exp(i*h)); end X=X*Un;Y=Y*Vn; end Un=X; Vn=Y; % the predictor value is reserved to Un and Vn for k=0:n0-1 if (0<=k*h) && (k*h<=L) Om=((h^mu)/(gamma(mu+1)))*(((n-k)^mu)-((n-k-1)^mu))*(a1*(1-Om/k1)*Om-(b1*Om*Lm)-(e*q)*Om); Lm=((h^mu)/(gamma(mu+1)))*(((n-k)^mu)-((n-k-1)^mu))*(a2*(1-Lm/k2)*Lm-(b2*Om*Lm)-q*Lm); else Om=((h^mu)/(gamma(mu+1)))*(((n-k)^mu)-((n-k-1)^mu))*(a1*(1-Om/k1)*Om-(b1*Om*Lm)); Lm=((h^mu)/(gamma(mu+1)))*(((n-k)^mu)-((n-k-1)^mu))*(a2*(1-Lm/k2)*Lm-(b2*Om*Lm)); end Z=Z+Om;N=N+Lm; end Om=Z; Lm=N; % Finding the value of healthy cells population Xn and cancerous cells population Yn Xn=u0; Yn=v0; for j=0:(n-1) if (1<=exp(j*h)) && (exp(j*h)<=L) Xn=exp(((h^mu)/(gamma(mu+2)))*(((n-j-1)^(mu+1))-(2*(n-j)^(mu+1))+((n-j+1)^(mu+1)))*(a1*(1-Xn/k1)-(b1*Yn)- (e*q))*exp(j*h)); Yn=exp(((h^mu)/(gamma(mu+2)))*(((n-j-1)^(mu+1))-(2*(n-j)^(mu+1))+((n-j+1)^(mu+1)))*(a2*(1-Yn/k2)-(b2*Xn)- q)*exp(j*h)); else Xn=exp(((h^mu)/(gamma(mu+2)))*(((n-j-1)^(mu+1))-(2*(n-j)^(mu+1))+((n-j+1)^(mu+1)))*(a1*(1-Xn/k1)-(b1*Yn))*exp(j*h)); Yn=exp(((h^mu)/(gamma(mu+1)))*(((n-j-1)^(mu+1))-(2*(n-j)^(mu+1))+((n-j+1)^(mu+1)))*(a2*(1-Yn/k2)-(b2*Xn))*exp(j*h)); end Rn=Rn*Xn;Sn=Sn*Yn; end if (1<=exp(n*h)) && (exp(n*h)<=L) Rn=Rn*exp((h^mu)/(gamma(mu+2))*((n-1)^(mu+1)+(n^mu)*(mu-n+1))*(a1*(1-u0/k1)-b1*v0)- (e*q))*exp((h^mu)/(gamma(mu+2))*(a1*(1-Un/k1)-(b1*Vn)-(e*q))*exp(n*h)); Sn=Sn*exp((h^mu)/(gamma(mu+2))*((n-1)^(mu+1)+(n^mu)*(mu-n+1))*(a2*(1-v0/k1)-b2*u0)-q)*exp((h^mu)/(gamma(mu+2))*(a2*(1-Vn/k2)-(b1*Un)-q)*exp(n*h)); else Rn=Rn*exp(h^mu)/(gamma(mu+2))*((n-1)^(mu+1)+(n^mu)*(mu-n+1))*(a1*(1-u0/k1)-b1*v0)*exp((h^mu)/(gamma(mu+2))*(a1*(1-Un/k1)-(b1*Vn))*exp(n*h)); Sn=Sn*exp((h^mu)/(gamma(mu+2))*((n-1)^(mu+1)+(n^mu)*(mu-n+1))*(a2*(1-v0/k1)-b2*u0))*exp((h^mu)/(gamma(mu+2))*(a2*(1-Vn/k2)-(b1*Un))*exp(n*h)); end Zn=u0; Nn=v0; for a=0:(n0-1) if (0<=a*h) && (a*h<=L) Zn=((h^mu)/(gamma(mu+2)))*(((n-a-1)^(mu+1))-(2*(n-a)^(mu+1))+((n-a+1)^(mu+1)))*(a1*(1-Zn/k1)*Zn-(b1*Nn*Zn)- (e*q*Zn)); Nn=((h^mu)/(gamma(mu+2)))*(((n-a-1)^(mu+1))-(2*(n-a)^(mu+1))+((n-a+1)^(mu+1)))*(a2*(1-Nn/k2)*Nn-(b2*Nn*Zn)- q*Nn); else Zn=((h^mu)/(gamma(mu+2)))*(((n-a-1)^(mu+1))-(2*(n-a)^(mu+1))+((n-a+1)^(mu+1)))*(a1*(1-Zn/k1)*Zn-(b1*Nn*Zn)); Nn=((h^mu)/(gamma(mu+2)))*(((n-a-1)^(mu+1))-(2*(n-a)^(mu+1))+((n-a+1)^(mu+1)))*(a2*(1-Nn/k2)*Nn-(b2*Nn*Zn)); end Kn=Kn+Zn;Jn=Jn+Nn; end if (0<=n0*h) && (n0*h<=L) Kn=Kn+((h^mu)/(gamma(mu+2))*((n0-1)^(mu+1)+((n0)^mu)*(mu-n0+1))*(a1*(1-u0/k1)*u0-b1*v0*u0)-(e*q))+((h^mu)/(gamma(mu+2))*(a1*(1-Om/k1)*Om-(b1*Om*Lm)-(e*q))); Jn=Jn+((h^mu)/(gamma(mu+2))*((n0-1)^(mu+1)+((n0)^mu)*(mu-n0+1))*(a2*(1-v0/k1)*v0-b2*v0*u0)-q)+((h^mu)/(gamma(mu+2))*(a2*(1-Om/k2)*Om-(b1*Om*Lm)-q)); else Kn=Kn+((h^mu)/(gamma(mu+2))*((n0-1)^(mu+1)+((n0)^mu)*(mu-n0+1))*(a1*(1-u0/k1)*u0-b1*v0*u0))+((h^mu)/(gamma(mu+2))*(a1*(1-Om/k1)*Om-(b1*Om*Lm))); Jn=Jn+((h^mu)/(gamma(mu+2))*((n0-1)^(mu+1)+((n0)^mu)*(mu-n0+1))*(a2*(1-v0/k1)*v0-b2*v0*u0))+((h^mu)/(gamma(mu+2))*(a2*(1-Om/k2)*Om-(b1*Om*Lm))); end E(1,s)=(Kn); F(1,s)=(Jn); B(1,s)=(Rn); C(1,s)=(Sn); end plot(A,B); plot(A,C); plot(B,C);

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**Figure 1.**The results sample for phase diagram and cancerous cell populations according to the given constants in Appendix A: (

**a**) The phase diagram; (

**b**) Cancerous cell population.

Type of Calculus | Derivative (μ = 1) | Integral (μ = −1) |

Newtonian | ${f}^{\prime}\left(x\right)$ | ${}_{a}{}^{}I{}_{x}\left(f\right)\left(x\right)={{\displaystyle \int}}_{a}^{x}f\left(s\right)ds$ |

Bi-Geometric | ${f}^{*}\left(x\right)=\mathrm{exp}\left(\frac{x\left(f\left(x\right)\right){}^{\prime}}{f\left(x\right)}\right)$ | ${}_{a}{}^{\mathrm{e}\mathrm{x}\mathrm{p}\left(x\right)}I{}_{x}\left(f\right)\left(x\right)=\mathrm{exp}\left({{\displaystyle \int}}_{a}^{x}ln\left(f\left(s\right)\right)\frac{ds}{s}\right)$ |

${t}^{p}$-calculus | ${f}^{\left(p\right)}\left(x\right)=\frac{{\left(f\left(x\right)\right)}^{q}{\left(f\left(x\right){}^{\prime}\right)}^{p}}{{x}^{q}},\frac{1}{p}+\frac{1}{q}=1$ | ${}_{a}{}^{{x}^{p}}I{}_{x}\left(f\right)\left(x\right)={\left(\frac{1}{p}{{\displaystyle \int}}_{a}^{x}{f}^{\frac{1}{p}}\left(s\right){s}^{\frac{1}{q}}ds\right)}^{p}$ |

Bi-Positive Calculus | ${f}^{\left(\mathcal{l}\right)}\left(x\right)=ln\left(f{\left(x\right)}^{\prime}\right)+f\left(x\right)-x$ | ${}_{a}{}^{ln\left(x\right)}I{}_{x}\left(f\right)\left(x\right)=ln\left({{\displaystyle \int}}_{a}^{x}\mathrm{e}\mathrm{x}\mathrm{p}\left(sf\left(s\right)\right)ds\right)$ |

$\alpha $-Calculus | ${}_{}{}^{\alpha \left(x\right)}D\left(f\right)\left(x\right)=\alpha \left(\frac{{\alpha}^{-1}\left(f\left(x\right)\right){}^{\prime}}{{\alpha}^{-1}\left(x\right){}^{\prime}}\right)$ | ${}_{a}{}^{\alpha \left(x\right)}I{}_{x}\left(f\right)\left(x\right)=\alpha \left({{\displaystyle \int}}_{a}^{x}{\alpha}^{-1}\left(f\left(s\right)\right){\alpha}^{-1}\left(s\right){}^{\prime}ds\right)$ |

Type of Calculus | Fractional Integral (μ) | |

Newtonian | ${}_{a}{}^{}I{}_{x}^{\mu}\left(f\right)\left(x\right)=\frac{1}{\Gamma \left(\mu \right)}{\int}_{a}^{x}{\left(x-s\right)}^{\mu -1}f\left(s\right)ds$ | |

Bi-Geometric | ${}_{a}{}^{\mathrm{e}\mathrm{x}\mathrm{p}}I{}_{x}^{\mu}\left(f\right)\left(x\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{1}{\Gamma \left(\mu \right)}{\int}_{a}^{x}{\left(ln\left(\frac{x}{s}\right)\right)}^{\mu -1}ln\left(f\left(s\right)\right)\frac{ds}{s}\right)$ | |

${t}^{p}$-calculus | ${}_{a}{}^{{x}^{p}}I{}_{x}^{\mu}\left(f\right)\left(x\right)={\left(\frac{1}{p\Gamma \left(\mu \right)}{{\displaystyle \int}}_{a}^{x}{\left({x}^{\frac{1}{p}}-{s}^{\frac{1}{p}}\right)}^{\mu -1}{f}^{\frac{1}{p}}\left(s\right){s}^{\frac{1}{q}}ds\right)}^{p}$ | |

Bi-Positive Calculus | ${}_{a}{}^{ln}I{}_{x}^{\mu}\left(f\right)\left(x\right)=ln\left(\frac{1}{\Gamma \left(\mu \right)}{{\displaystyle \int}}_{a}^{x}{\left({e}^{x}-{e}^{s}\right)}^{\mu -1}\mathrm{e}\mathrm{x}\mathrm{p}\left(f\left(s\right)+s\right)ds\right)$ | |

$\alpha $-Calculus | ${}_{a}{}^{}I{}_{x}^{\mu}\left(f\right)\left(x\right)=\alpha \left(\frac{1}{\Gamma \left(\mu \right)}{{\displaystyle \int}}_{a}^{x}{\left({\alpha}^{-1}\left(x\right)-{\alpha}^{-1}\left(s\right)\right)}^{\mu -1}{\alpha}^{-1}\left(f\left(s\right)\right){\alpha}^{-1}\left(s\right)\right)$ | |

Type of Calculus | Caputo Fractional Derivative (−μ) | |

Newtonian | ${}_{a}{}^{C}D{}_{x}^{\mu}\left(f\right)\left(x\right)=\frac{1}{\Gamma \left(n-\mu \right)}{{\displaystyle \int}}_{a}^{x}{\left(x-s\right)}^{n-\mu -1}{f}^{\left(n\right)}\left(s\right)ds$ | |

Bi-Geometric | ${}_{a}{}^{\mathrm{e}\mathrm{x}\mathrm{p}}D{}_{x}^{\mu}\left(f\right)\left(x\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{1}{\Gamma \left(n-\mu \right)}{{\displaystyle \int}}_{a}^{x}{\left(ln\left(\frac{x}{s}\right)\right)}^{n-\mu -1}{\left(s\frac{d}{ds}\right)}^{n}ln\left(f\left(s\right)\right)\frac{ds}{s}\right)$ | |

${t}^{p}$-calculus | ${}_{a}{}^{{x}^{p}}D{}_{x}^{\mu}\left(f\right)\left(x\right)={\left(\frac{1}{\Gamma \left(n-\mu \right)}{{\displaystyle \int}}_{a}^{x}{\left({x}^{\frac{1}{p}}-{s}^{\frac{1}{p}}\right)}^{n-\mu -1}{\left(\frac{1}{p}{s}^{\frac{1}{q}}\frac{d}{ds}\right)}^{n}{f}^{\frac{1}{p}}\left(s\right){s}^{\frac{1}{q}}ds\right)}^{p}$ | |

Bi-Positive Calculus | ${}_{a}{}^{ln}D{}_{x}^{\mu}\left(f\right)\left(x\right)=ln\left(\frac{1}{\Gamma \left(n-\mu \right)}{{\displaystyle \int}}_{a}^{x}{\left({e}^{x}-{e}^{s}\right)}^{n-\mu -1}{\left({e}^{s}\frac{d}{ds}\right)}^{n}\mathrm{e}\mathrm{x}\mathrm{p}\left(f\left(s\right)\right){e}^{s}ds\right)$ | |

$\alpha $-Calculus | ${}_{a}{}^{}D{}_{x}^{\mu}\left(f\right)\left(x\right)=\alpha \left(\frac{1}{\Gamma \left(n-\mu \right)}{{\displaystyle \int}}_{a}^{x}{\left({\alpha}^{-1}\left(x\right)-{\alpha}^{-1}\left(s\right)\right)}^{n-\mu -1}{\left(\frac{1}{{\alpha}^{-1}\left(s\right){}^{\prime}}\frac{d}{ds}\right)}^{n}{\alpha}^{-1}\left(f\left(s\right)\right){\alpha}^{-1}{\left(s\right)}^{\prime}ds\right)$ |

**Table 2.**Patient data: tumor Type (Squamous Cell Carcinoma, Adenocarcinoma), therapy (Radiotherapy).

Patient 1 | Patient 2 | |
---|---|---|

Tumor | SCC | SCC |

Staging | T1bN_{0}M_{0} | T1bN_{0}M_{0} |

therapy | RT | RT |

Initial and final population | 24.1 × 10^{9}–3.59 × 10^{9} | 17.4 × 10^{9}–8.61 × 10^{9} |

Final population by model | 3.26 × 10^{9} | 8.16 × 10^{9} |

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**MDPI and ACS Style**

Momenzadeh, M.; Obi, O.A.; Hincal, E.
A Bi-Geometric Fractional Model for the Treatment of Cancer Using Radiotherapy. *Fractal Fract.* **2022**, *6*, 287.
https://doi.org/10.3390/fractalfract6060287

**AMA Style**

Momenzadeh M, Obi OA, Hincal E.
A Bi-Geometric Fractional Model for the Treatment of Cancer Using Radiotherapy. *Fractal and Fractional*. 2022; 6(6):287.
https://doi.org/10.3390/fractalfract6060287

**Chicago/Turabian Style**

Momenzadeh, Mohammad, Olivia Ada Obi, and Evren Hincal.
2022. "A Bi-Geometric Fractional Model for the Treatment of Cancer Using Radiotherapy" *Fractal and Fractional* 6, no. 6: 287.
https://doi.org/10.3390/fractalfract6060287