Modeling bone regeneration around endosseous implants

14 Chapter 2. Linear stability analysis2.1 IntroductionA stability analysis is performed for a biological model for peri-implantosseointegration, which was proposed in Moreo et al. [67]. This model allowsto take into account an implant surface microtopography. The resultspresented in Moreo et al. [67] were in agreement with experiments. Theauthors reported, that model can predict contact and distance osteogenesismodes of bone formation [24].From the numerical simulations, which were performed for a differentgeometry of the healing site, it was found that the system of equations,proposed in Moreo et al. [67], is characterized by appearance of a wave-likeprofile in the solution for a certain range of parameters. This profile wasinitially recognized in the solution of the model equations for the 1D domainof length 2.5mm (Figure 2.1b). This domain was chosen for the simulationsof bone formation near the cylindrical implant, located within the bonechamber, used in the experiments by Duyck et al. [29], Vandamme et al.[92, 93, 94]. The authors reported that in experiments, new bone was formedat all distances from the host bone, and integration of bone and implant wasachieved. That wave-like profile has not been noticed by Moreo et al. [67],since for the geometry used in his simulations, in which the distance fromhost bone to implant was 0.6mm, only a part of ’wave’ is visible in thesolution (Figure 2.1a), and a wave-like profile is not distinguishable. Forlarger domains, more ’waves’ appear in the solution. The solution for thedomain of length 5mm is shown in Figure 2.1c.s242(a)01.5 2x1050(b)1 2 3x86420(c)2 4 6xm2101.5 2x64201 2 3x4202 4 6xDay 300 Day 500 Day 1000Figure 2.1: Osteogenic cell m and growth factor 2 s 2 distributions at differenttime moments, obtained from the numerical solution of model equations,defined in Moreo et al. [67], for 1D axisymmetric domain with length (a)L = 0.6 mm and (b) L = 5 mmThe conditions, under which a wave-like profile appears, are studied in