# Numerical Analysis of an Osseointegration Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Biological Problem and Its Variational Formulation

**Remark**

**1.**

**Problem P.**Find the density of platelets $c:\overline{\Omega}\times [0,T]\to \mathbb{R}$, the density of osteogenic cells $m:\overline{\Omega}\times [0,T]\to \mathbb{R}$, the concentration of the first growth factor ${s}_{1}:\overline{\Omega}\times [0,T]\to \mathbb{R}$, the concentration of the second growth factor ${s}_{2}:\overline{\Omega}\times [0,T]\to \mathbb{R}$, and the density of osteoblasts $b:\overline{\Omega}\times [0,T]\to \mathbb{R}$ such that,

**Remark**

**2.**

**Problem VP.**Find the density of platelets $c:[0,T]\to E$, the density of osteoblasts $m:[0,T]\to V$, the concentration of the first growth factor ${s}_{1}:[0,T]\to E$, and the concentration of the second growth factor ${s}_{2}:[0,T]\to E$ such that $c\left(0\right)={c}_{0}$, $m\left(0\right)={m}_{0}$, ${s}_{1}\left(0\right)={s}_{{1}_{0}}$, and ${s}_{2}\left(0\right)={s}_{{2}_{0}}$, and, for a.e. $t\in (0,T)$ and for all $u\in V$, $v,\phantom{\rule{0.166667em}{0ex}}w,\phantom{\rule{0.166667em}{0ex}}z\in E$,

**Theorem**

**1.**

## 3. Fully Discrete Approximations and an a Priori Error Analysis

**Problem**${\mathbf{VP}}^{\mathit{h}\mathit{k}}$

**.**Find the discrete density of platelets ${c}^{hk}={\left\{{c}_{n}^{hk}\right\}}_{n=0}^{N}\subset {E}^{h}$, the discrete density of osteoblasts ${m}^{hk}={\left\{{m}_{n}^{hk}\right\}}_{n=0}^{N}\subset {V}^{h}$, the discrete concentration of the first growth factor ${s}_{1}^{hk}={\left\{{s}_{{1}_{n}}^{hk}\right\}}_{n=0}^{N}\subset {E}^{h}$, and the discrete concentration of the second growth factor ${s}_{2}^{hk}={\left\{{s}_{{2}_{n}}^{hk}\right\}}_{n=0}^{N}\subset {E}^{h}$ such that ${c}_{0}^{hk}={c}_{0}^{h}$, ${m}_{0}^{hk}={m}_{0}^{h}$, ${s}_{{1}_{0}}^{hk}={s}_{{1}_{0}}^{h}$, and ${s}_{{2}_{0}}^{hk}={s}_{{2}_{0}}^{h}$, and, for $n=1,\dots ,N$ and for all ${u}^{h}\in {V}^{h}$ and ${v}^{h},\phantom{\rule{0.166667em}{0ex}}{w}^{h},\phantom{\rule{0.166667em}{0ex}}{z}^{h}\in {E}^{h}$,

**Theorem**

**2.**

**Corollary**

**1.**

## 4. Numerical Results

#### 4.1. Numerical Scheme

#### 4.2. Numerical Convergence

#### 4.3. One-Dimensional Examples

#### 4.4. Two-Dimensional Example

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Graph of p defined as a sigmoid (red) and the piecewise form (dashed black), for ${p}_{m}=0.5$.

**Figure 5.**Concentration of platelets along the spatial domain at the final time for the case of ${p}_{m}=0.1$ (blue) and ${p}_{m}=0.5$ (black).

**Figure 6.**Concentration of osteogenic cells (

**left**) and osteoblasts (

**right**) at the final time for the case of ${p}_{m}=0.1$ (blue) and ${p}_{m}=0.5$ (black).

**Figure 7.**Computational domain and mesh for the two-dimensional problem. Due to the symmetry of the problem, only half of the domain was simulated. Interior borders were added to ensure that at the regions of discontinuity for the derivatives of p, only edges were placed. Dimensions in mm.

**Figure 8.**Concentration of platelets at the final time for the case of ${p}_{m}=0.5$ (

**left**) and ${p}_{m}=0.1$ (

**right**).

**Figure 9.**Concentration of osteogenic cells at the final time for the case of ${p}_{m}=0.5$ (

**left**) and ${p}_{m}=0.1$ (

**right**).

**Figure 10.**Concentration of osteoblasts at the final time for the case of ${p}_{m}=0.5$ (

**left**) and ${p}_{m}=0.1$ (

**right**).

h↓$\mathit{k}\to $ | ${2}^{-5}$ | ${2}^{-6}$ | ${2}^{-7}$ | ${2}^{-8}$ | ${2}^{-9}$ | ${2}^{-10}$ | ${2}^{-11}$ | ${2}^{-12}$ | ${2}^{-13}$ |
---|---|---|---|---|---|---|---|---|---|

${2}^{-5}$ | 8.440 | 4.712 | 2.845 | 1.925 | 1.477 | 1.276 | 1.202 | 1.167 | 1.151 |

${2}^{-6}$ | 7.919 | 4.199 | 2.322 | 1.387 | 0.927 | 0.719 | 0.644 | 0.610 | 0.593 |

${2}^{-7}$ | 7.768 | 4.052 | 2.175 | 1.236 | 0.771 | 0.553 | 0.474 | 0.437 | 0.420 |

${2}^{-8}$ | 7.707 | 3.992 | 2.114 | 1.175 | 0.708 | 0.482 | 0.400 | 0.362 | 0.343 |

${2}^{-9}$ | 7.668 | 3.953 | 2.075 | 1.135 | 0.667 | 0.438 | 0.348 | 0.309 | 0.290 |

${2}^{-10}$ | 7.634 | 3.919 | 2.041 | 1.100 | 0.631 | 0.400 | 0.301 | 0.262 | 0.243 |

${2}^{-11}$ | 7.602 | 3.887 | 2.008 | 1.067 | 0.597 | 0.364 | 0.256 | 0.216 | 0.197 |

${2}^{-12}$ | 7.570 | 3.855 | 1.976 | 1.034 | 0.563 | 0.329 | 0.214 | 0.170 | 0.151 |

${2}^{-13}$ | 7.539 | 3.823 | 1.944 | 1.001 | 0.530 | 0.295 | 0.178 | 0.125 | 0.105 |

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**MDPI and ACS Style**

Baldonedo, J.; Fernández, J.R.; Segade, A.
Numerical Analysis of an Osseointegration Model. *Mathematics* **2020**, *8*, 87.
https://doi.org/10.3390/math8010087

**AMA Style**

Baldonedo J, Fernández JR, Segade A.
Numerical Analysis of an Osseointegration Model. *Mathematics*. 2020; 8(1):87.
https://doi.org/10.3390/math8010087

**Chicago/Turabian Style**

Baldonedo, Jacobo, José R. Fernández, and Abraham Segade.
2020. "Numerical Analysis of an Osseointegration Model" *Mathematics* 8, no. 1: 87.
https://doi.org/10.3390/math8010087