# Dynamical Simulation of Effective Stem Cell Transplantation for Modulation of Microglia Responses in Stroke Treatment

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Representation of the SDRPA Model

## 3. The SDRPA Model’s Equilibrium Points

- $\frac{dS}{dt}=0\iff $$$\begin{array}{c}\hfill [\sigma -{\alpha}_{5}{M}_{a}-{\alpha}_{6}{M}_{p}-{\gamma}_{s}]S=0,\end{array}$$
- $\frac{dD}{dt}=0\iff $$$\begin{array}{c}\hfill [({\delta}_{2}-{r}_{1}){M}_{p}-{r}_{2}{M}_{a}]D=0,\end{array}$$
- $\frac{d{M}_{r}}{dt}=0\iff $$$\begin{array}{c}\hfill {\alpha}_{0}-({\alpha}_{1}+{\alpha}_{2}+{\gamma}_{0}){M}_{r}=0,\end{array}$$
- $\frac{d{M}_{p}}{dt}=0\iff $$$\begin{array}{c}\hfill {\alpha}_{1}{M}_{r}-({\alpha}_{4}S+{\delta}_{1}D+{\alpha}_{3}+{\gamma}_{1}){M}_{p}=0,\end{array}$$
- $\frac{d{M}_{a}}{dt}=0\iff $$$\begin{array}{c}\hfill {\alpha}_{2}{M}_{r}+({\alpha}_{5}S-{\gamma}_{2}){M}_{a}+({\alpha}_{3}+{\alpha}_{4}S){M}_{p}=0.\end{array}$$

**Proposition**

**1.**

- ${r}_{1}<{\delta}_{2}$
- ${\gamma}_{s}<\sigma $
- $\frac{{\alpha}_{0}{\alpha}_{2}({r}_{2}{\alpha}_{6}+{\alpha}_{5}({\delta}_{2}-{r}_{1}))}{z\omega}<{\gamma}_{2}(-{r}_{1}+{\delta}_{2})-{r}_{2}{\alpha}_{3}$
- ${r}_{2}{\alpha}_{3}<{\gamma}_{2}({\delta}_{2}-{r}_{1}))$
- ${r}_{2}({\alpha}_{1}{\alpha}_{3}+{\alpha}_{2}({\alpha}_{3}+{\gamma}_{1}))<{\gamma}_{2}{\alpha}_{1}({\delta}_{2}-{r}_{1})$

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 4. The Equilibrium Points’ Stability of the SDRPA Model

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Remark**

**1.**

- Theorem 1 indicates that the damage, D, can penetrate the SDRPA model, if ${\lambda}_{2}>0$.
- Theorem 2 indicates that the damage, $D>0$, penetrated the brain.
- Theorem 3 indicates that stem cell transplantation, $S>0$, modulates the inflammatory environment in a stroke, $D>0$.
- The SDRPA model is considered stable when the immunomodulation from transplanted stem cells can be one of the mechanisms of post-stroke recovery.

## 5. Numerical Results and Analysis

#### 5.1. Determination of Parameters

#### 5.2. Comparison of Experimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Effectiveness of stem cell transplantation on immune cells’ phenotype behavior and dead brain cells within 72 h. Where ${\alpha}_{4}=0.5314,{\alpha}_{5}=0.3468$ and ${\alpha}_{6}=0.1419$.

**Table 1.**Parameters values for transplanted stem cells–damaged brain cells–resting microglia–pro-inflammation microglia–anti-inflammation microglia (SDRPA) model.

Parameters | Values | Descriptions | Sources |
---|---|---|---|

${S}^{*}$ | 1 | SCs initial concentration | [31] |

${D}^{*}$ | 0.4 | damage initial concentration | [4] |

${M}_{r}^{*}$ | 1 | resting microglia initial concentration | [4] |

${M}_{p}^{*}$ | 0.1415 | pro-inflammation initial concentration | [4] |

${M}_{a}^{*}$ | 0.02 | anti-inflammation initial concentration | [4] |

$\sigma $ | 0.69 | the reproduction rate of stem cells | [42] |

${\alpha}_{0}$ | 0.38 | the resting microglia source | [4] |

${\alpha}_{1}$ | 0.12 | activation rate of ${M}_{r}$ into ${M}_{p}$ | [4] |

${\alpha}_{2}$ | 0.017 | activation rate of ${M}_{r}$ into ${M}_{a}$ | [4] |

${\alpha}_{3}$ | 0.11 | the rate transference from ${M}_{p}$ to ${M}_{a}$ | [4] |

${\delta}_{1}$ | 0.2854 | the cytotoxic effects due to ${M}_{p}$ | [5] |

${\delta}_{2}$ | 0.1 | the death rate of brain cells due to ${M}_{p}$ | [5] |

${\gamma}_{s}$ | 0.1 | the natural death rate of S | [42] |

${\gamma}_{0}$ | 0.003 | the natural death rate of ${M}_{r}$ | [4] |

${\gamma}_{1}$ | 0.06 | the natural death rate of ${M}_{p}$ | [4] |

${\gamma}_{2}$ | 0.05 | the natural death rate of ${M}_{a}$ | [4] |

${r}_{1}$ | 0.05 | the decay rate of concentration of the D by ${M}_{p}$ | [4] |

${r}_{2}$ | 0.0125 | the decay rate of concentration of the D by ${M}_{a}$ | [4] |

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Alqarni, A.J.; Rambely, A.S.; Hashim, I.
Dynamical Simulation of Effective Stem Cell Transplantation for Modulation of Microglia Responses in Stroke Treatment. *Symmetry* **2021**, *13*, 404.
https://doi.org/10.3390/sym13030404

**AMA Style**

Alqarni AJ, Rambely AS, Hashim I.
Dynamical Simulation of Effective Stem Cell Transplantation for Modulation of Microglia Responses in Stroke Treatment. *Symmetry*. 2021; 13(3):404.
https://doi.org/10.3390/sym13030404

**Chicago/Turabian Style**

Alqarni, Awatif Jahman, Azmin Sham Rambely, and Ishak Hashim.
2021. "Dynamical Simulation of Effective Stem Cell Transplantation for Modulation of Microglia Responses in Stroke Treatment" *Symmetry* 13, no. 3: 404.
https://doi.org/10.3390/sym13030404