# Optimal Control Analysis of a Mathematical Model for Breast Cancer

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Formulation

## 3. Model Analysis

#### 3.1. Boundedness and Positivity of Solutions

**Theorem**

**1.**

**Proof:**

#### 3.2. The Equilibrium Points of System (5)

#### 3.3. The Reproductive Number and Tumor-Free Equilibrium Point

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 3.4. Co-Existing Equilibrium Points

**Theorem**

**5.**

**Proof.**

## 4. Uncertainty and Sensitivity Analysis

## 5. Analysis of Optimal Control

#### Existence of an Optimal Control

**Theorem**

**6.**

- $f$ is not empty;
- The admissible control set $U$ is closed and convex;
- Each right hand side of the state system is continuous, is bounded above by the sum of the bounded control and the state, and can be written as a linear function of $\overline{{u}_{i}^{*}(t)}$ with coefficients depending on time and the state;
- The integrand of ${J}_{1}\overline{({u}_{i}^{*}})$ is convex on $U$ and is bounded below by $-{c}_{2}+{c}_{1}{\overline{u}}^{2}$ with ${c}_{1}>0$.

**Proof.**

- Strategy 1: Anti-cancer drug treatment control on tumor cells (control ${u}_{1}(t)$ only);
- Strategy 2: Ketogenic diet control on excess estrogen and tumor cells (control ${u}_{2}(t)$ only);
- Strategy 3: Anti-cancer drug and ketogenic diet treatment combined control on tumor cells growth and excess estrogen (controls ${u}_{1}(t)$ and ${u}_{2}(t)$).

**Theorem**

**7.**

**Proof.**

## 6. Numerical Simulations and Discussion

#### Effects of Control on the System (9)

## 7. Conclusions

- The conditions of stability of the tumor-free equilibrium (TFE) was established and the system is only local asymptotically stable (LAS) if a certain threshold quantity, known as the reproductive number, is less than unity (${R}_{0}<1$). It implies that the number of tumor cells in the body will be brought to zero if proper treatments and a ketogenic diet that can force make the threshold to a value less than unity are monitored.
- An individual has the chance of developing breast cancer depending on the level of the immune system (s), the efficacy of the anti-cancer drug (k) and the rate at which the ketogenic diet (d) is being taken to fight tumor cells. We also found out that the presence of excess estrogen in system makes it unstable, as depicted in Figure 7. This implies that any additional estrogen quantity introduced into the body through the birth control, and hormone replacement therapy (HRT) enhances the rate of tumor formation. Thus, the development of breast cancer is certain.
- The transition from normal cells class to tumor cells class plays a crucial role in breast cancer dynamics $({\lambda}_{1})$. More tumor is formed if the DNA is damaged or altered as a result of excess estrogen, which reduces the number of normal cells being produced by red blood cells.

_{0}is properly analyzed. In addition, moderation is conceivable if the planning of intercessions is sufficiently quick and if the arrangement includes the utilization of more than one therapy procedure. No therapy (ketogenic diet and anti-cancer drug) is possible, unless minimal resources are accessible.

## 8. Further Research

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**PRCCs of homogeneous model parameters with the tumor cells as the baseline variable. All parameter values were varied in 25% of their baseline values in Table 1. The most sensitive parameters are shown to be $P-values$ of ${\alpha}_{1},g,{\mu}_{1},{\gamma}_{3}$ and $\omega $ are less than 0.01.

**Figure 2.**PRCCs of homogeneous model parameters with the tumor cells as the baseline variable. All parameter values were varied in 25% of their baseline values in Table 1. The most sensitive parameters are shown to be $P-values$ of $s,{\gamma}_{2},{\mu}_{3}$ and $\rho $ are less than 0.01.

**Figure 3.**The variation of proportion of Tumor cell population for different values of $d$ with other parameters fixed.

**Figure 4.**The variation of proportion of Estrogen level population for different values of $k$ with other parameters fixed.

**Figure 5.**The variation of proportion of Tumor cell population for different values of $k$ with other parameters fixed.

**Figure 6.**The variation of proportion of Immune booster population for different values of $\beta $ with other parameters fixed.

**Figure 9.**The variation of proportion of Normal cell population for different values of ${\lambda}_{1}$ with other parameters fixed.

**Figure 10.**Simulation result of the model (9), showing normal cell population against time with and without control.

**Figure 11.**Simulation result of the model (9), showing tumor cell population against time with and without control.

**Figure 12.**Simulation result of the model (9), showing estrogen level against time with and without control.

**Figure 13.**Simulation result of the model (9), showing immune response against time with and without control.

Parameter | Symbol | Value | Unit | Refs |
---|---|---|---|---|

Per capita growth rate of normal cells | ${\alpha}_{1}$ | 0.70 | day^{−1} | [12] |

Per capita growth rate of tumor cells | ${\alpha}_{2}$ | 0.514 | day^{−1} | [5] |

Natural death rate of normal cells | ${\mu}_{1}$ | 0.00003 | day^{−1} | Assumed |

Natural death rate of tumor cells | ${\mu}_{2}$ | 0.01 | day^{−1} | [7] |

Rate of inhibition of normal cells | ${\varphi}_{1}$ | 6 × 10^{−8} | day^{−1} | [1] |

Tumor cells death rate due to immune response | ${\gamma}_{2}$ | 3 × 10^{−6} | day^{−1} | [12] |

Interaction coefficient rate with immune response | ${\gamma}_{3}$ | 1 × 10^{−7} | day^{−1} | [5] |

Source rate of immune cells | $s$ | 1.3 × 10^{4} | day^{−1} | [12] |

Source rate of estrogen | $\u03f5$ | 1.3 × 10^{4} | day^{−1} | est |

Immune threshold rate | $\omega $ | 3 × 10^{5} | day^{−1} | [5] |

Immune response rate | $\rho $ | 0.20 | day^{−1} | [13] |

Natural death rate of immune cells | ${\mu}_{3}$ | 0.29 | day^{−1} | [5] |

Efficacy of anti-cancer drug | $k$ | 0–1 | day^{−1} | Assumed |

Supplement for immune booster | $\beta $ | 0.01 | day^{−1} | est |

Tumor formation rate as a result of DNA damage by excess estrogen | ${\lambda}_{1}$ | 0.20 | (Pg/mL)^{−1}day^{−1} | est |

Immune suppression rate due to excess estrogen | ${\lambda}_{3}$ | 0.002 | day^{−1} | est |

Assume constant of value of decay factor | $g$ | 0.1 | day^{−1} | est |

Natural death rate of estrogen | ${\mu}_{4}$ | 0.97 | day^{−1} | [19] |

Death rate due to ketogenic diet | ${\mu}_{5}$ | 2.0 | day^{−1} | est |

Constant rate of ketogenic diet | $d$ | 0.5 | day^{−1} | est |

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**MDPI and ACS Style**

Oke, S.I.; Matadi, M.B.; Xulu, S.S. Optimal Control Analysis of a Mathematical Model for Breast Cancer. *Math. Comput. Appl.* **2018**, *23*, 21.
https://doi.org/10.3390/mca23020021

**AMA Style**

Oke SI, Matadi MB, Xulu SS. Optimal Control Analysis of a Mathematical Model for Breast Cancer. *Mathematical and Computational Applications*. 2018; 23(2):21.
https://doi.org/10.3390/mca23020021

**Chicago/Turabian Style**

Oke, Segun Isaac, Maba Boniface Matadi, and Sibusiso Southwell Xulu. 2018. "Optimal Control Analysis of a Mathematical Model for Breast Cancer" *Mathematical and Computational Applications* 23, no. 2: 21.
https://doi.org/10.3390/mca23020021