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.
Highlights
The importance of mathematical modeling in dentistry and related fields has steadily increased over the last decades
We here tackle this question by combining the latest state-of-the-art mathematical modeling techniques in the field of multiscale micromechanics, as reviewed in Section 2.1, 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
Where φmpoiclyrHo A and fHpAolyHA are the volume fractions of the micropores and the hydroxyapatite needles; carbonate hydroxyapatite (CHA) and Cmicroφ, respectively, are the fourth-order stiffness tensors of the hydroxyapatite crystals and of the micropores, respectively; θ and φ are the Euler angles quantifying the orientations of the hydroxyapatite crystals; PpcyollyHA(θ, φ) and PpspohlyHA, respectively, are the fourth-order Hill tensors related to cylindrical and spherical inclusions, respectively, embedded in a matrix made up of the microporous hydroxyapatite polycrystal; and I is the fourth-order unit tensor, the components of which are defined via the Kronecker delta δij, namely Iijkl = 1/2(δikδjl + δilδjk)
Summary
The importance of mathematical modeling in dentistry and related fields has steadily increased over the last decades. 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, as reviewed, 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.
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