# Modeling the Macrophage-Mediated Inflammation Involved in the Bone Fracture Healing Process

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Biological Background

## 3. Modeling Assumptions

## 4. Model Formulation

## 5. Analysis of the Model

## 6. Numerical Results

#### 6.1. Comparison of Existing Models

#### 6.2. Different Outcomes of the Bone Fracture Healing Process

#### 6.3. Importance of Macrophages during the Bone Fracture Healing Process

#### 6.4. Evolution of the Healing Process for Different Types of Fractures

#### 6.5. Immune-Modulation Therapeutic Treatments of Bone Fractures

#### 6.5.1. Administration of Anti-Inflammatory Drugs at the Beginning of the Healing Process

#### 6.5.2. Cellular Therapeutic Interventions under Immune-Compromised Conditions

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Theorem**

**A1.**

**Proof**

**of**

**Theorem**

**A1.**

**Theorem**

**A2.**

**Proof**

**of**

**Theorem**

**A2.**

**Theorem**

**A3.**

**Proof**

**of**

**Theorem**

**A3**.

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**Figure 1.**Inflammatory, repair, and remodeling phases of the bone fracture healing process. During the inflammatory phase, debris (D) activates the healing process by attracting macrophages ${M}_{0}$ to the injury site, which subsequently activate into their ${M}_{1}$ or ${M}_{2}$ phenotypes. Activated macrophages remove debris and secrete pro- and anti-inflammatory cytokines, such as tumor necrotic factor-$\alpha $ (TNF-$\alpha $) (${c}_{1}$) and interleukin-10 (IL-10) (${c}_{2}$), which regulate the inflammation and the cellular functions. During the repair phase, migrating mesenchymal stem cells (MSCs) up-regulate IL-10 production, proliferate, and differentiate into osteoblasts (${C}_{b}$). Mesenchymal and osteoblast cells synthesize the fibro/cartilage and woven bone, which closes the fracture gap. During the bone remodeling phase, osteoblasts and osteoclasts constantly remove and deposit new bone until the fracture is fully repaired.

**Figure 2.**Flow diagram of the cellular and molecular dynamics during the inflammatory and repair phases of the bone fracture healing process.

**Figure 3.**Comparison of tissues evolution in Model [11] and Model (1)–(10).

**Figure 4.**Cellular and molecular evolution of the resolution of the inflammation in normal conditions.

**Figure 7.**Tissue evolution when macrophages contribute to the healing process (solid line), ${k}_{1},{k}_{2}\ne 0$, and when they do not contribute to the healing process (dashed line), ${k}_{1}={k}_{2}=0$.

**Figure 8.**Tissue evolution when the alternatively activated macrophages, ${M}_{2}$, do not contribute to the healing process (dashed line), ${k}_{2}=0$, and when the classically activated macrophages, ${M}_{1}$, do not contribute to the healing process (dotted line), ${k}_{1}=0$.

**Figure 10.**Tissue evolution in a simple fracture under different initial anti-inflammatory cytokines concentrations, $D\left(0\right)=3\times {10}^{5}$.

**Figure 11.**Tissue evolution in a moderate fracture under different initial anti-inflammatory cytokines concentrations, $D\left(0\right)=2\times {10}^{7}$.

**Figure 12.**Tissue evolution in a moderate fracture under different initial anti-inflammatory cytokine concentrations, $D\left(0\right)=5\times {10}^{7}$.

**Figure 13.**Tissue evolution in a severe fracture under different initial anti-inflammatory cytokines concentrations.

**Figure 14.**Tissue evolution in an advanced age fracture under different initial anti-inflammatory cytokines concentrations.

**Figure 15.**Tissue evolution in a senile osteoporotic fracture under different initial anti-inflammatory cytokines concentrations.

**Figure 16.**Tissue evolution in a severe fracture without therapeutic innervation (solid line) and with ${M}_{0}\left(0\right)$ and ${C}_{m}\left(0\right)$ transplantation (dotted line).

**Figure 17.**Tissue evolution in an aging fracture without therapeutic innervation (solid line) and with MSCs injection (dotted line).

**Figure 18.**Tissue evolution in a senile osteoporotic fracture without therapeutic innervation (solid line) and with MSCs injection (dotted line).

Equilibrium Points | Existence Conditions | Meaning |
---|---|---|

${E}_{0}=(0,0,0,0,0,0,0,0,{m}_{{c}_{0}}^{\ast},{m}_{{b}_{0}}^{\ast})$ | ${m}_{{c}_{0}}^{\ast}\ge 0$, ${m}_{{b}_{0}}^{\ast}\ge 0$ | nonunion |

${E}_{1}=(0,0,0,0,0,0,0,{K}_{lb}(1-{d}_{b}/{k}_{pb}),0,{p}_{bs}/{q}_{bd})$ | ${k}_{pb}>{d}_{b}$ | successful healing |

${E}_{2}=(0,0,0,0,0,{c}_{2}^{\ast},{C}_{m}^{\ast},{C}_{b}^{\ast},{m}_{c}^{\ast},{p}_{bs}/{q}_{bd})$ | ${k}_{pm}>{d}_{m}$ | nonunion or delayed union |

Equilibrium Points | Stability Conditions | Stability |
---|---|---|

${E}_{0}$ | ${k}_{pm}\le {d}_{m}$, ${k}_{pb}\le {d}_{b}$ | ${E}_{0}$ belongs to an attracting local set |

${E}_{0}$, ${E}_{1}$ | ${k}_{pm}\le {d}_{m}$, ${k}_{pb}>{d}_{b}$ | ${E}_{0}$ unstable; ${E}_{1}$ locally stable |

${E}_{0}$, ${E}_{2}$ | ${k}_{pm}>{d}_{m}$, ${k}_{pb}\le {d}_{b}$ | ${E}_{0}$ unstable; ${E}_{2}$ locally stable |

${E}_{0}$, ${E}_{1}$, ${E}_{2}$ | ${k}_{pm}>{d}_{m}$, ${k}_{pb}>{d}_{b}$ | ${E}_{0}$ and ${E}_{1}$ unstable; ${E}_{2}$ locally stable |

Parameter | Description | Range of Values | Reference |
---|---|---|---|

${k}_{{e}_{1}}$ | Engulfing debris rate of ${M}_{1}$ | 3–48/day | [38,41] |

${k}_{{e}_{2}}$ | Engulfing debris rate of ${M}_{2}$ | 3–48/day | [38,41] |

${a}_{ed}$ | Half-saturation of debris | $4.71\times {10}^{6}$ cells/mL | [38] |

${k}_{max}$ | Maximal migration rate | 0.015–0.1/day | [39,45] |

${M}_{max}$ | Maximal macrophages density | $6\times {10}^{5}$–$1\times {10}^{6}$ cells/mL | [27,41] |

${k}_{01}$ | Activation rate of ${M}_{1}$ | 0.55–0.611/day | [29,39] |

${k}_{02}$ | Activation rate of ${M}_{0}$ to ${M}_{2}$ | 0.0843–0.3/day | [29] |

${k}_{12}$ | Transition rate from ${M}_{1}$ to ${M}_{2}$ | 0.083–0.075/day | [29,39] |

${k}_{21}$ | Transition rate from ${M}_{2}$ to ${M}_{1}$ | 0.005–0.05/day | [29] |

${d}_{0}$ | Emigration rate of ${M}_{0}$ | 0.156–0.02/day | [29,39] |

${d}_{1}$ | Emigration rate of ${M}_{1}$ | 0.121–0.2/day | [29,38,39] |

${d}_{2}$ | Emigration rate of ${M}_{2}$ | 0.163–0.2/day | [29,38,39] |

${k}_{0}$ | Secretion rate of ${c}_{1}$ by debris | $5\times {10}^{-7}$–$8.5\times {10}^{-6}$ ng/cells/day | [38] |

${k}_{1}$ | Secretion rate of ${c}_{1}$ by ${M}_{1}$ macrophages | $8.3\times {10}^{-6}$ ng/cells/day | [38] |

${k}_{2}$ | Secretion rate of ${c}_{2}$ by ${M}_{2}$ macrophages | $3.72\times {10}^{-6}$ ng/cells/day | [38] |

${k}_{3}$ | Secretion rate of ${c}_{2}$ by MSCs | $7\times {10}^{-7}$–$8\times {10}^{-6}$ ng/cells/day | [11] |

${d}_{{c}_{1}}$ | Decay rate of ${c}_{1}$ | 12.79–55/day | [29,38] |

${d}_{{c}_{2}}$ | Decay rate of ${c}_{2}$ | 2.5–4.632/day | [29,38] |

${a}_{12}$ | Effectiveness of ${c}_{2}$ inhibition of ${c}_{1}$ synthesis | $0.025$ ng/mL | [29] |

${a}_{22}$ | Effectiveness of ${c}_{2}$ inhibition of ${c}_{2}$ synthesis | $0.1$ ng/mL | [29] |

${a}_{pm}$ | Effectiveness of ${c}_{1}$ inhibition of ${C}_{m}$ proliferation | $3.162$ ng/mL | [11,46] |

${a}_{m{b}_{1}}$ | Effectiveness of ${c}_{1}$ inhibition of ${C}_{m}$ differentiation | $0.1$ ng/mL | [11,47] |

${a}_{01}$ | Half-saturation of ${c}_{1}$ to activate ${M}_{1}$ | $0.01$ ng/mL | [29] |

${a}_{02}$ | Half-saturation of ${c}_{2}$ to activate ${M}_{2}$ | $0.005$ ng/mL | [29] |

${a}_{pb}$ | Effectiveness of ${c}_{1}$ inhibition of ${C}_{b}$ proliferation | 10 ng/mL | [11,48] |

${a}_{p{m}_{1}}$ | Constant enhancement of ${c}_{1}$ to ${C}_{m}$ proliferation | 13 ng/mL | [11,46] |

${k}_{pm}$ | Proliferation rate of ${C}_{m}$ | 0.5/day | [11] |

${d}_{m}$ | Differentiation rate of ${C}_{m}$ | 1/day | [11,30] |

${k}_{pb}$ | Proliferation rate of ${C}_{b}$ | 0.2202/day | [11,30] |

${d}_{b}$ | Differentiation rate of ${C}_{b}$ | 0.15/day | [11,30] |

${p}_{cs}$ | Fibrocartilage synthesis rate | $3\times {10}^{-6}$ g/cells/day | [11,30] |

${q}_{c{d}_{1}}$ | Fibrocartilage degradation rate | $3\times {10}^{-6}$ mL/cells/day | [11,30] |

${q}_{c{d}_{2}}$ | Fibrocartilage degradation rate by osteoclasts | $0.2\times {10}^{-6}$ mL/cells/day | [11,30] |

${p}_{bs}$ | Bone tissue synthesis rate | $5\times {10}^{-8}$ g/cells/day | [11,30] |

${q}_{bd}$ | Bone tissue degradation rate | $5\times {10}^{-8}$ mL/cells/day | [30] |

${K}_{lb}$ | Carrying capacity of ${C}_{b}$ | $1\times {10}^{6}$ cells/mL | [11,30] |

${K}_{lm}$ | Carrying capacity of ${C}_{m}$ | $1\times {10}^{6}$ cells/mL | [11,30] |

$D\left(0\right)$ | Density of necrotic cells | $1\times {10}^{6}$–$2\times {10}^{8}$ cells/mL | [27,38,41] |

${C}_{m}\left(0\right)$ | Initial MSCs density | 1000 cells/mL | [11] |

${M}_{0}\left(0\right)$ | Unactivated macrophage density | 4000 cell/mL | [45] |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Trejo, I.; Kojouharov, H.; Chen-Charpentier, B. Modeling the Macrophage-Mediated Inflammation Involved in the Bone Fracture Healing Process. *Math. Comput. Appl.* **2019**, *24*, 12.
https://doi.org/10.3390/mca24010012

**AMA Style**

Trejo I, Kojouharov H, Chen-Charpentier B. Modeling the Macrophage-Mediated Inflammation Involved in the Bone Fracture Healing Process. *Mathematical and Computational Applications*. 2019; 24(1):12.
https://doi.org/10.3390/mca24010012

**Chicago/Turabian Style**

Trejo, Imelda, Hristo Kojouharov, and Benito Chen-Charpentier. 2019. "Modeling the Macrophage-Mediated Inflammation Involved in the Bone Fracture Healing Process" *Mathematical and Computational Applications* 24, no. 1: 12.
https://doi.org/10.3390/mca24010012