# 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

- Haas, R.; Polak, C.; Fürhauser, R.; Mailath-Pokorny, G.; Dörtbudak, O.; Watzek, G. A long-term follow-op of 76 Bränemark single-tooth implants. Clin. Oral Implants Res.
**2002**, 13, 38–43. [Google Scholar] [CrossRef] [PubMed] - Soncini, M.; Pietrabissa, R.; Rodriguez y Baena, R. Computational approach for the mechanical reliability of a dental implant. In Computer Methods in Biomechanics and Biomedical Engineering; Middleton, J., Pande, G.N., Jones, M.L., Eds.; Gordon and Breach Science Publishers: London, UK, 2001. [Google Scholar]
- Azcarate-Velázquez, F.; Castillo-Oyagüe, R.; Oliveros-López, L.G.; Torres-Lagares, D.; Martínez-González, Á.J.; Pérez-Velasco, A.; Lynch, C.D.; Gutiérrez-Pérez, J.L.; Serrera-Figallo, M.Á. Influence of bone quality on the mechanical interaction between implant and bone: A finite element analysis. J. Dent.
**2019**, 88, 103–161. [Google Scholar] [CrossRef] [PubMed] - Baggi, L.; Cappelloni, I.; Di Girolamo, M.; Maceri, F.; Vairo, G. The influence of implant diameter and length on stress distribution of osseointegrated implants related to crestal bone geometry: A three-dimensional finite element analysis. J. Prosthet. Dent.
**2008**, 100, 422–431. [Google Scholar] [CrossRef][Green Version] - Cos Juez, F.J.; Sánchez Lasheras, F.; García Nieto, P.G.; Álvarez-Arenal, A. Non-linear numerical analysis of a double-threaded titanium alloy dental implant by FEM. Appl. Math. Comput.
**2008**, 2008, 952–967. [Google Scholar] [CrossRef] - Dorogoy, A.; Haïat, G.; Shemtov-Yona, K.; Rittel, D. Modeling ultrasonic wave propagation in a dental implant—Bone system. J. Mech. Behav. Biomed. Mater.
**2019**, 20, 103. [Google Scholar] [CrossRef] - Farronato, D.; Manfredini, M.; Stevanello, A.; Campana, V.; Azzi, L.; Farronato, M. A Comparative 3D Finite Element Computational Study of Three Connections. Materials
**2019**, 12, 3135. [Google Scholar] [CrossRef][Green Version] - Fernandez, J.W.; Das, R.; Cleary, P.W.; Hunter, P.J.; Thomas, C.D.L.; Clement, J.G. Using smooth particle hydrodynamics to investigate femoral cortical bone remodeling at the Haversian level. Int. J. Numer. Methods Biomed. Eng.
**2013**, 29, 129–143. [Google Scholar] [CrossRef] - Giorgio, I.; Andreaus, U.; Scerrato, D.; Braidotti, P. Modeling of a non-local stimulus for bone remodeling process under cyclic load: Application to a dental implant using a bioresorbable porous material. Math. Mech. Solids
**2017**, 22, 1790–1805. [Google Scholar] [CrossRef][Green Version] - Guan, H.; van Staden, R.C.; Johnson, N.; Loo, Y.-C. Dynamic modeling and simulation of dental implant insertion process—A finite element study. Finite Elem. Anal. Des.
**2011**, 47, 886–897. [Google Scholar] [CrossRef][Green Version] - Hasan, I.; Rahimi, A.; Keilig, L.; Brinkmann, K.T.; Bourauel, C. Computational simulation of internal bone remodeling around dental implants: A sensitivity analysis. Comput. Methods Biomech. Biomed. Eng.
**2012**, 15, 807–814. [Google Scholar] [CrossRef] - He, Y.; Hasan, I.; Keilig, L.; Fischer, D.; Ziegler, L.; Abboud, M.; Wahl, G.; Bourauel, C. Biomechanical characteristics of immediately loaded and osseointegration dental implants inserted into Sika deer antler. Med. Eng. Phys.
**2018**, 59, 8–14. [Google Scholar] [CrossRef] [PubMed] - He, Y.; Hasan, I.; Keilig, L.; Fischer, D.; Ziegler, L.; Abboud, M.; Wahl, G.; Bourauel, C. Numerical investigation of bone remodeling around immediately loaded dental implants using sika deer (Cervus nippon) antlers as implant bed. Comput. Methods Biomech. Biomed. Eng.
**2018**, 21, 359–369. [Google Scholar] [CrossRef] [PubMed] - Hoang, K.C.; Khoo, B.C.; Liu, G.R.; Nguyen, N.C.; Patera, A.T. Rapid identification of material properties of the interface tissue in dental implant systems using reduced basis method. Inverse Probl. Sci. Eng.
**2013**, 21, 1310–1334. [Google Scholar] [CrossRef] - Hou, P.J.; Ou, K.L.; Wang, C.C.; Huang, C.F.; Ruslin, M.; Sugiatno, E.; Yang, T.S.; Chou, H.H. Hybrid micro/nanostructural surface offering improved stress distribution and enhanced osseointegration properties of the biomedical titanium implant. J. Mech. Behav. Biomed. Mater.
**2018**, 79, 173–180. [Google Scholar] [CrossRef] - Joshi, S.; Kumar, S.; Jain, S.; Aggarwal, R.; Choudhary, S.; Reddy, N.K. 3D Finite Element Analysis to Assess the Stress Distribution Pattern in Mandibular Implant-supported Overdenture with Different Bar Heights. J. Contemp. Dent. Pract.
**2019**, 20, 794–800. [Google Scholar] [CrossRef] [PubMed] - Kurniawan, D.; Nor, F.M.; Lee, H.Y.; Lim, J.Y. Finite element analysis of bone-implant biomechanics: Refinement through featuring various osseointegration conditions. Int. J. Oral Maxillofac. Surg.
**2012**, 41, 1090–1196. [Google Scholar] [CrossRef] [PubMed] - Lencioni, K.A.; Noritomi, P.Y.; Macedo, A.P.; Ribeiro, R.F.; Almeida, R.P. Influence of different implants on the biomechanical behavior of tooth-implant fixed partial dentures: A three-dimensional finite element analysis. J. Oral Implantol.
**2019**. [Google Scholar] [CrossRef] - Lima de Andrade, C.; Carvalho, M.A.; Bordin, D.; da Silva, W.J.; del Bel Cury, A.A.; Sotto-Maior, B.S. Biomechanical Behavior of the Dental Implant Macrodesign. Int. J. Oral Maxillofac. Implant.
**2017**, 32, 264–270. [Google Scholar] [CrossRef][Green Version] - Lin, D.; Li, Q.; Li, W.; Duckmanton, N.; Swain, M. Mandibular bone remodeling induced by dental implant. J. Biomech.
**2010**, 43, 287–293. [Google Scholar] [CrossRef] - Murase, K.; Stenlund, P.; Thomsen, P.; Lausmaa, J.; Palmquist, A. Three-dimensional modeling of removal torque and fracture progression around implants. J. Mater. Sci. Mater. Med.
**2018**, 29, 104. [Google Scholar] [CrossRef][Green Version] - Rittel, D.; Dorogoy, A.; Shemtov-Yona, K. Modeling the effect of osseointegration on dental implant pullout and torque removal tests. Clin. Implant Dent. Relat. Res.
**2018**, 20, 683–691. [Google Scholar] [CrossRef] [PubMed] - Sayyedi, A.; Rashidpour, M.; Fayyaz, A.; Ahmadian, N.; Dehghan, M.; Faghani, F.; Fasihg, P.J. Comparison of Stress Distribution in Alveolar Bone with Different Implant Diameters and Vertical Cantilever Length via the Finite Element Method. Long Term Eff. Med. Implant.
**2019**, 29, 37–43. [Google Scholar] [CrossRef] - Sotto-Maior, B.S.; Mercuri, E.G.; Senna, P.M.; Assis, N.M.; Francischone, C.E.; del Bel Cury, A.A. Evaluation of bone remodeling around single dental implants of different lengths: A mechanobiological numerical simulation and validation using clinical data. Comput. Methods Biomech. Biomed. Eng.
**2016**, 19, 699–706. [Google Scholar] [CrossRef] [PubMed] - Wang, C.; Li, Q.; McClean, C.; Fan, Y. Numerical simulation of dental bone remodeling induced by implant- supported fixed partial denture with or without cantilever extension. Int. J. Numer. Method Biomed. Eng.
**2013**, 29, 1134–1147. [Google Scholar] [CrossRef] [PubMed] - Zheng, L.; Yang, J.; Hu, X.; Luo, J. Three dimensional finite element analysis of a novel osteointegrated dental implant designed to reduce stress peak of cortical bone. Acta Bioeng. Biomech.
**2014**, 16, 21–28. [Google Scholar] [PubMed] - Moreo, P.; García-Aznar, J.M.; Doblaré, M. Bone ingrowth on the surface of endosseous implants. Part 1: Mathematical model. J. Theor. Biol.
**2009**, 260, 1–12. [Google Scholar] [CrossRef][Green Version] - Moreo, P.; García-Aznar, J.M.; Doblaré, M. Bone ingrowth on the surface of endosseous implants. Part 2: Theoretical and numerical analysis. J. Theor. Biol.
**2009**, 260, 13–26. [Google Scholar] [CrossRef][Green Version] - Fernández, J.R.; García-Aznar, J.M.; Masid, M. Numerical analysis of an osteoconduction model arising in bone-implant integration. ZAMM Z. Angew. Math. Mech.
**2017**, 97, 1050–1063. [Google Scholar] - Fernández, J.R.; Masid, M. Analysis of a model for the propagation of the ossification front. J. Comput. Appl. Math.
**2017**, 318, 624–633. [Google Scholar] - Lekszycki, T.; dell’Isola, F. A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. Zeit. Ang. Math. Mech.
**2012**, 92, 426–444. [Google Scholar] [CrossRef][Green Version] - Lu, Y.; Lekszycki, T. New description of gradual substitution of graft by bone tissue including biomechanical and structural effects, nutrients supply and consumption. Cont. Mech. Thermod.
**2018**, 30, 995–1009. [Google Scholar] [CrossRef][Green Version] - George, D.; Allena, R.; Rémond, Y. Integrating molecular and cellular kinetics into a coupled continuum mechanobiological stimulus for bone reconstruction. Cont. Mech. Thermod.
**2019**, 31, 725–740. [Google Scholar] [CrossRef][Green Version] - Barbu, V. Optimal Control of Variational Inequalities; Pitman: Boston, MA, USA, 1984. [Google Scholar]
- Brezis, H. Equations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier
**1968**, 18, 115–175. [Google Scholar] [CrossRef][Green Version] - Chau, O.; Fernández, J.R.; Shillor, M.; Sofonea, M. Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion. J. Comput. Appl. Math.
**2003**, 159, 431–465. [Google Scholar] [CrossRef][Green Version] - Ciarlet, P.G. Basic error estimates for elliptic problems. In Handbook of Numerical Analysis; Ciarlet, P.G., Lions, J.L., Eds.; 1993; Volume II, pp. 17–351. [Google Scholar]
- Barboteu, M.; Fernández, J.R.; Hoarau-Mantel, T.V. A class of evolutionary variational inequalities with applications in viscoelasticity. Math. Model. Methods Appl. Sci.
**2005**, 15, 1595–1617. [Google Scholar] [CrossRef] - Campo, M.; Fernández, J.R.; Kuttler, K.L.; Shillor, M.; Viaño, J.M. Numerical analysis and simulations of a dynamic frictionless contact problem with damage. Comput. Methods Appl. Mech. Eng.
**2006**, 196, 476–488. [Google Scholar] [CrossRef] - Alnaes, S.; Blechta, J.; Hake, J.; Johansson, A.; Kehlet, B.; Logg, A.; Richardson, C.; Ring, J.; Rognes, M.E.; Wells, G.N. The FEniCS Project Version 1.5 M. Arch. Numer. Softw.
**2005**, 3. [Google Scholar] [CrossRef] - Logg, A.; Mardal, K.-A.; Wells, G.N. (Eds.) Automated Solution of Differential Equations by the Finite Element Method; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Hecht, F. New development in FreeFem++. J. Numer. Math.
**2012**, 20, 251–265. [Google Scholar] [CrossRef] - Conconi, M.; Parenti-Castelli, V. A sound and efficient measure of joint congruence. Proc. Inst. Mech. Eng. Part H J. Eng. Med.
**2014**, 228, 935–941. [Google Scholar] [CrossRef] - Valigi, M.C.; Logozzo, S. Do Exostoses Correlate with Contact Disfunctions? A Case Study of a Maxillary Exostosis. Lubricants
**2019**, 7, 15. [Google Scholar] [CrossRef][Green Version]

**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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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