# Mathematical Modeling and Analysis of Tumor Chemotherapy

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## Abstract

**:**

## 1. Introduction

## 2. The Model

#### 2.1. Mathematical Model

- Both immune effector cells and chemotherapy decrease the tumor population.
- The population of effector cells decreases due to the degradation process, consumption when killing tumor cells, and the effect of chemotherapy.
- Chemotherapy drugs can affect tumor cells and immune effector cells through a mass-action mechanism.
- A higher constant input of the drug dose can result in both higher tumor and immune effector cell depletion.

#### 2.2. The Reduced Model

## 3. Dynamics

**Lemma**

**1.**

**Proof.**

#### 3.1. Equilibria of Dimensionless Model

#### 3.2. Stability of Equilibrium States

#### 3.2.1. Dead Equilibrium State

#### 3.2.2. Tumor-Free Equilibrium State

#### 3.2.3. Tumor-Present Equilibrium State

#### 3.2.4. Coexisting Equilibrium State

## 4. Parameter Sensitivity Analysis

## 5. Numerical Simulations

#### 5.1. Numerical Simulations of the Equilibrium States

#### 5.2. Simulations Using Different Immune Strengths

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Model assumptions about the detailed interactions between tumor cells, the immune system, and chemotherapy.

**Figure 2.**Sensitivity analysis of model (3) by changing the values of four parameters. (

**a**) Parameter p. (

**b**) Parameter k. (

**c**) Parameter s. (

**d**) Parameter f.

**Figure 3.**Numerical simulations of the model for four equilibrium states. (

**a**). Dead equilibrium state ${E}_{0}$. (

**b**) Tumor–free equilibrium state ${E}_{1}$. (

**c**) Tumor–present equilibrium state ${E}_{2}$. (

**d**) Coexisting equilibrium state ${E}_{3}$.

**Figure 4.**The numbers of tumor cells and immune cells under chemotherapy with different immune intensities. (

**a**) Simulation using a strong immune system strength with $p=20$. (

**b**) Simulation using a weak immune system strength with $p=10$.

**Table 1.**Description and values of the parameters in model (1).

Parameters | Units | Description | Value | Reference |
---|---|---|---|---|

a | day${}^{-1}$ | Growth rate of NK cells | none | none |

b | cell${}^{-1}$ | Inverse of NK cells capacity | $3.17\times {10}^{-6}$ | fitting |

c | day${}^{-1}$ | Growth rate of tumor | $5.14\times {10}^{-1}$ | [33] |

d | cell${}^{-1}$ | Inverse of tumor capacity | $1.02\times {10}^{-9}$ | [33] |

r | cell${}^{-1}$ day${}^{-1}$ | Activation rate of CTLs | $1.1\times {10}^{-7}$ | [34,35] |

$\mu $ | day${}^{-1}$ | CTL death rate | $2.0\times {10}^{-2}$ | [34] |

${\alpha}_{1}$ | cell${}^{-1}$ day${}^{-1}$ | NK cell death rate | $1.0\times {10}^{-7}$ | [33] |

${\alpha}_{2}$ | cell${}^{-1}$ day${}^{-1}$ | Tumor death rate of NK | $6.41\times {10}^{-11}$ | [33,36] |

${\beta}_{1}$ | cell${}^{-1}$ day${}^{-1}$ | CTL death rate | $3.42\times {10}^{-10}$ | [37] |

${\beta}_{2}$ | cell${}^{-1}$ day${}^{-1}$ | Rate of CTL-induced tumor death | $3.5\times {10}^{-7}$ | fitting |

v | dose | Influx of drug | none | none |

$\omega $ | day${}^{-1}$ | Drug decay rate | $9\times {10}^{-1}$ | [12] |

${k}_{N},{k}_{L}$ | day${}^{-1}$ | Immune cell killed by drug | $6\times {10}^{-1}$ | [12] |

${k}_{T}$ | day${}^{-1}$ | Tumor cell killed by drug | $8\times {10}^{-1}$ | [12] |

**Table 2.**Endemic equilibrium states of model (3) and the corresponding parameter ranges.

No. | p | k | f | Equilibrium Points |
---|---|---|---|---|

1 | $2.92\times {10}^{4}$ | $(5.23\times {10}^{-3},$ +$\infty )$ | $({f}_{1}^{\ast},$ +$\infty )$ | ${E}_{3}({N}_{1}^{\ast},{L}_{1}^{\ast},{T}_{1}^{\ast},\frac{s}{f})$ |

2 | $(1.23\times {10}^{4},\phantom{\rule{3.33333pt}{0ex}}2.92\times {10}^{4})$ | $(0,\phantom{\rule{3.33333pt}{0ex}}\frac{1}{1.8\times {10}^{-2}p})$ | $(0,\phantom{\rule{3.33333pt}{0ex}}{f}_{3}^{\ast})$ | ${E}_{3}({N}_{2}^{\ast},{L}_{2}^{\ast},{T}_{2}^{\ast},\frac{s}{f})$ |

3 | $(0,\phantom{\rule{3.33333pt}{0ex}}5.06\times {10}^{3})$ | $(0,\phantom{\rule{3.33333pt}{0ex}}{k}_{1}^{\ast})$ | $({f}_{2}^{\ast},$ +$\infty )$ | ${E}_{3}({N}_{3}^{\ast},{L}_{3}^{\ast},{T}_{3}^{\ast},\frac{s}{f})$ |

No. | p | k | f | Equilibrium Points |
---|---|---|---|---|

1 | $(0,+\infty )$ | $(0,+\infty )$ | $(0,+\infty )$ | ${E}_{0}(0,0,0,\frac{s}{f})$ |

2 | $(\frac{s}{f},+\infty )$ | $(0,+\infty )$ | $(0,+\infty )$ | ${E}_{1}(\frac{p-\frac{s}{f}}{1.8\times {10}^{-2}p},0,0,\frac{s}{f})$ |

3 | $(0,+\infty )$ | $(0,+\infty )$ | $(\frac{ks}{25.7},+\infty )$ | ${E}_{2}(0,0,1.91\times {10}^{2}(25.7-\frac{ks}{f}),\frac{s}{f})$ |

4 | $2.92\times {10}^{4}$ | $(5.23\times {10}^{-3},+\infty )$ | $({f}_{1}^{\ast},+\infty )$ | ${E}_{3}({N}_{1}^{\ast},{L}_{1}^{\ast},{T}_{1}^{\ast},\frac{s}{f})$ |

5 | $(1.23\times {10}^{4},2.92\times {10}^{4})$ | $(0,\frac{1}{1.8\times {10}^{-2}p})$ | $(0,{f}_{3}^{\ast})$ | ${E}_{3}({N}_{2}^{\ast},{L}_{2}^{\ast},{T}_{2}^{\ast},\frac{s}{f})$ |

6 | $(0,5.06\times {10}^{3})$ | $(0,{k}_{1}^{\ast})$ | $({f}_{2}^{\ast},+\infty )$ | ${E}_{3}({N}_{3}^{\ast},{L}_{3}^{\ast},{T}_{3}^{\ast},\frac{s}{f})$ |

No. | p | k | s | f | Equilibrium Points |
---|---|---|---|---|---|

1 | 10 | $0.4$ | 10 | $0.1$ | ${E}_{0}$ |

2 | 10 | 10 | 1 | $0.1$ | ${E}_{1}$ |

3 | 20 | $0.1$ | 10 | $0.2$ | ${E}_{2}$ |

4 | 20 | $0.01$ | 10 | $0.6$ | ${E}_{3}$ |

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**MDPI and ACS Style**

Song, G.; Liang, G.; Tian, T.; Zhang, X.
Mathematical Modeling and Analysis of Tumor Chemotherapy. *Symmetry* **2022**, *14*, 704.
https://doi.org/10.3390/sym14040704

**AMA Style**

Song G, Liang G, Tian T, Zhang X.
Mathematical Modeling and Analysis of Tumor Chemotherapy. *Symmetry*. 2022; 14(4):704.
https://doi.org/10.3390/sym14040704

**Chicago/Turabian Style**

Song, Ge, Guizhen Liang, Tianhai Tian, and Xinan Zhang.
2022. "Mathematical Modeling and Analysis of Tumor Chemotherapy" *Symmetry* 14, no. 4: 704.
https://doi.org/10.3390/sym14040704