Multiscale Mathematical Modeling in Dental Tissue Engineering: Toward Computer-Aided Design of a Regenerative System Based on Hydroxyapatite Granules, Focussing on Early and Mid-Term Stiffness Recovery

Abstract

We here explore for the very first time how an advanced multiscale mathematical modeling approach may support the design of a provenly successful tissue engineering concept for mandibular bone. The latter employs double-porous, potentially cracked, single millimeter-sized granules packed into an overall conglomerate-type scaffold material, which is then gradually penetrated and partially replaced by newly grown bone tissue. During this process, the newly developing scaffold-bone compound needs to attain the stiffness of mandibular bone under normal physiological conditions. In this context, the question arises how the compound stiffness is driven by the key design parameters of the tissue engineering system: macroporosity, crack density, as well as scaffold resorption/bone formation rates. We here tackle this question by combining the latest state-of-the-art mathematical modeling techniques in the field of multiscale micromechanics, into an unprecedented suite of highly efficient, semi-analytically defined computation steps resolving several levels of hierarchical organization, from the millimeter- down to the nanometer-scale. This includes several types of homogenization schemes, namely such for porous polycrystals with elongated solid elements, for cracked matrix-inclusion composites, as well as for assemblies of coated spherical compounds. Together with the experimentally known stiffnesses of hydroxyapatite crystals and mandibular bone tissue, the new mathematical model suggests that early stiffness recovery (i.e., within several weeks) requires total avoidance of microcracks in the hydroxyapatite scaffolds, while mid-term stiffness recovery (i.e., within several months) is additionally promoted by provision of small granule sizes, in combination with high bone formation and low scaffold resorption rates.

Strength increase during ceramic biomaterial-induced bone regeneration: a micromechanical study

“…Furthermore, Eqs. (2) and (3) (Scheiner et al 2016), result from the nature of the employed coated inclusionproblem of Hervé and Zaoui (1993), where, in contrast to the classical Eshelby-problem, the stresses and strains are not homogeneous throughout the inclusion.…”

“…For estimating the macroscopic stiffness tensor of the bone-scaffold conglomerate, C congl , the homogenization approach for an nlayered spherical inclusion proposed by Hervé and Zaoui (1993) is specialized for n = 1 (relating to bone tissue), adapted for the case that the stiffness of this layer is transversally isotropic, see (Bertrand and Hellmich 2009), and further combined with a selfconsistent homogenization scheme, in order to account for mutually interacting coated spheres with porous space in-between-in absence of any explicit "matrix phase". This homogenization step is thus based on the composition and morphology of RVE III, as well as on the stiffness tensors of the granule material, C gran , accessible from stiffness homogenization across RVE II, of the bone tissue, C bone , and of the macropores, C macroφ ; the underlying mathematical framework is described at length in (Scheiner et al 2016). …”

“…In vitro studies have shown that the resorption rate may be as high as k res = 0.016 week −1 , depending on the actual composition of the hosting medium (Scheiner et al 2016). …”

“…For the presently studied material, the typical bone growth rate is k growth = (21) induces time-dependencies in the homogenized stiffness expressions relating to the RVEs depicted in Fig. 1, as given in (Scheiner et al 2016).…”

“…For defining the stiffness tensor of newly formed bone tissue, C bone , we consider the orthotropic stiffness tensor determined for mandibular bone by means of ultrasound measurements by Ashman and van Buskirk (1987), and follow then the strategy described in (Bertrand and Hellmich 2009) for converting the ultrasound-based, anisotropic stiffness tensor into the transversally isotropic stiffness tensor related to the bone tissue growing on granules, see also (Scheiner et al 2016) for details, yielding …”