Mathematical modeling of COVID-19 fatality trends: Death kinetics law versus infection-to-death delay rule.

The COVID-19 pandemic has world-widely motivated numerous attempts to properly adjust classical epidemiological models, namely those of the SEIR-type, to the spreading characteristics of the novel Corona virus. In this context, the fundamental structure of the differential equations making up the SEIR models has remained largely unaltered-presuming that COVID-19 may be just "another epidemic". We here take an alternative approach, by investigating the relevance of one key ingredient of the SEIR models, namely the death kinetics law. The latter is compared to an alternative approach, which we call infection-to-death delay rule. For that purpose, we check how well these two mathematical formulations are able to represent the publicly available country-specific data on recorded fatalities, across a selection of 57 different nations. Thereby, we consider that the model-governing parameters-namely, the death transmission coefficient for the death kinetics model, as well as the apparent fatality-to-case fraction and the characteristic fatal illness period for the infection-to-death delay rule-are time-invariant. For 55 out of the 57 countries, the infection-to-death delay rule turns out to represent the actual situation significantly more precisely than the classical death kinetics rule. We regard this as an important step towards making SEIR-approaches more fit for the COVID-19 spreading prediction challenge.

"Anatomical simulation" of the biomechanical behavior of the human mandible.

INTRODUCTION: The load-carrying behavior of the human mandible can be described using finite element simulation, enabling investigations about physiological and pathological skeletal adaption. "Anatomical simulation" implies a stepwise approximation towards the anatomical reality. METHOD: The project is structured in three steps. In Step 1, the preprocessing, the simulation model is provided. Step 2 is the numerical computation. Step 3 is dedicated to the interpretation of the results. The requirements of the preprocessing are: a) realization of the organ's individual anatomy, namely its outer shape; b) the tissue's elastic properties, thus its inner consistency; and c) the organ's mechanical loads. For physiological mandibular loading, these are due to muscles, temporomandibular joints, and tooth forces. Meanwhile, the reconstruction of the macroscopic anatomy from computed tomography data is standard. The periodontal ligament is inserted ex post using an approach developed by the authors. The bone is modeled anisotropically and inhomogeneously. By the visualization of the individual fiber course, the muscular force vectors are realized. The mandibular condyle is freely mobile in a kind of simplified joint capsule. For the realization of bite forces, several approaches are available. RESULTS: An extendible software tool is provided, enabling the user - by variable input of muscle and bite forces - to examine the individual patient's biomechanics, eg, the influence of the periodontal ligament, the condition of the temporomandibular joints, atrophic processes, or the biomechanical situation of dental implants. DISCUSSION: By stepwise approximation towards the anatomical reality, the mandibular simulation will be advanced to a valuable tool for diagnosis and prognosis.

Dynamic microvilli sculpt bristles at nanometric scale.

Organisms generate shapes across size scales. Whereas patterning and morphogenesis of macroscopic tissues has been extensively studied, the principles underlying the formation of micrometric and submicrometric structures remain largely enigmatic. Individual cells of polychaete annelids, so-called chaetoblasts, are associated with the generation of chitinous bristles of highly stereotypic geometry. Here we show that bristle formation requires a chitin-producing enzyme specifically expressed in the chaetoblasts. Chaetoblasts exhibit dynamic cell surfaces with stereotypical patterns of actin-rich microvilli. These microvilli can be matched with internal and external structures of bristles reconstructed from serial block-face electron micrographs. Individual chitin teeth are deposited by microvilli in an extension-disassembly cycle resembling a biological 3D printer. Consistently, pharmacological interference with actin dynamics leads to defects in tooth formation. Our study reveals that both material and shape of bristles are encoded by the same cell, and that microvilli play a role in micro- to submicrometric sculpting of biomaterials.

Mineralization-driven bone tissue evolution follows from fluid-to-solid phase transformations in closed thermodynamic systems.

The fundamental mechanisms that govern bone mineralization have been fairly well evidenced by means of experimental research. However, rules for the evolution of the volume and composition of the bone tissue compartments (such as the mineralized collagen fibrils and the extrafibrillar space in between) have not been provided yet. As an original contribution to this open question, we here test whether mineralizing bone tissue can be represented as a thermodynamically closed system, where crystals precipitate from an ionic solution, while the masses of the fibrillar and extrafibrillar bone tissue compartments are preserved. When translating, based on various experimental and theoretical findings, this mass conservation proposition into diffraction-mass density relations, the latter are remarkably well confirmed by independent experimental data from various sources. Resulting shrinkage and composition rules are deemed beneficial for further progress in bone materials science and biomedical engineering.

Fibrillar structure and elasticity of hydrating collagen: a quantitative multiscale approach.

It is well known that hydration of collagenous tissues leads to their swelling, as well as to softening of their elastic behavior. However, it is much less clear which microstructural and micromechanical "rules" are involved in this process. Here, we develop a theoretical approach cast in analytical mathematical formulations, which is experimentally validated by a wealth of independent tests on collagenous tissues, such as X-ray diffraction, vacuum drying, mass measurements, and Brillouin light scattering. The overall emerging picture is the following: air-drying leaves water only in the gap zones between the triple-helical collagen molecules; upon re-hydration, the extrafibrillar space is established at volumes directly proportional to the hydration-induced swelling of the (micro) fibrils, until the maximum equatorial distance between the long collagen molecules is reached. Thereafter, the volume of the fibrils stays constant, and only the extrafibrillar volume continues to grow. At all these hydration stages, the elastic behavior is governed by the same, hydration-invariant mechanical interaction pattern of only two, interpenetrating mechanical phases: transversely isotropic molecular collagen and isotropic water (or empty pores in the vacuum-dried case).

Strength of a cement-based dental material: Early age testing and first micromechanical modeling at mature age.

The compressive strength evolution of 37 centigrade-cured Biodentine, a cement-based dental material, is quantified experimentally by crushing cylindrical specimens with length-to-diameter ratios amounting to 1.84 and 1.34, respectively, at nine different material ages ranging from 1 h to 28 days. After excluding strength values significantly affected by imperfections, formulae developed for concrete are i) adapted for inter- and extrapolation of measured strength values, and ii) used for quantification of the influence of the slenderness of the specimens on the compressive strength. The microscopic origin of the macroscopic uniaxial compressive strength of mature Biodentine is investigated by means of a micromechanics model accounting for lognormal stiffness and strength distributions of two types of calcite-reinforced hydrates. The following results are obtained: The material behavior of Biodentine is non-linear in the first few hours after production. After that, Biodentine behaves virtually linear elastic all the way up to sudden brittle failure. The strength evolution of Biodentine can be well described as the exponential of a function involving the square root of the inverse of the material age. The genuine uniaxial compressive strength evolution can be quantified using a correction formula taken from a standard for testing of concrete, which accounts for length-to-diameter ratios of cylindrical samples deviating from 2. Multiscale modeling suggests that 63% of the overall material volume, occupied by dense calcite-reinforced hydration products, fail virtually simultaneously. This underlines the highly optimized nature of the studied material.

Bone fibrillogenesis and mineralization: quantitative analysis and implications for tissue elasticity.

Data from bone drying, demineralization, and deorganification tests, collected over a time span of more than 80 years, evidence a myriad of different chemical compositions of different bone materials. However, careful analysis of the data, as to extract the chemical concentrations of hydroxyapatite, of water, and of organic material (mainly collagen) in the extracellular bone matrix, reveals an astonishing fact: it appears that there exists a unique bilinear relationship between organic concentration and mineral concentration, across different species, organs, and age groups, from early childhood to old age: During organ growth, the mineral concentration increases linearly with the organic concentration (which increases during fibrillogenesis), while from adulthood on, further increase of the mineral concentration is accompanied by a decrease in organic concentration. These relationships imply unique mass density-concentration laws for fibrillogenesis and mineralization, which - in combination with micromechanical models - deliver 'universal' mass density-elasticity relationships in extracellular bone matrix-valid across different species, organs, and ages. They turn out as quantitative reflections of the well-instrumented interplay of osteoblasts, osteoclasts, osteocytes, and their precursors, controlling, in a fine-tuned fashion, the chemical genesis and continuous transformation of the extracellular bone matrix. Consideration of the aforementioned rules may strongly affect the potential success of tissue engineering strategies, in particular when translating, via micromechanics, the aforementioned growth and mineralization characteristics into tissue-specific elastic properties.

Fiber Rearrangement and Matrix Compression in Soft Tissues: Multiscale Hypoelasticity and Application to Tendon.

It is widely accepted that the nonlinear macroscopic mechanical behavior of soft tissue is governed by fiber straightening and re-orientation. Here, we provide a quantitative assessment of this phenomenon, by means of a continuum micromechanics approach. Given the negligibly small bending stiffness of crimped fibers, the latter are represented through a number of hypoelastic straight fiber phases with different orientations, being embedded into a hypoelastic matrix phase. The corresponding representative volume element (RVE) hosting these phases is subjected to "macroscopic" strain rates, which are downscaled to fiber and matrix strain rates on the one hand, and to fiber spins on the other hand. This gives quantitative access to the fiber decrimping (or straightening) phenomenon under non-affine conditions, i.e. in the case where the fiber orientations cannot be simply linked to the macroscopic strain state. In the case of tendinous tissue, such an RVE relates to the fascicle material with 50 µm characteristic length, made up of crimped collagen bundles and a gel-type matrix in-between. The fascicles themselves act as parallel fibers in a similar matrix at the scale of a tissue-related RVE with 500 µm characteristic length. As evidenced by a sensitivity analysis and confirmed by various mechanical tests, it is the initial crimping angle which drives both the degree of straightening and the shape of the macroscopic stress-strain curve, while the final linear portion of this curve depends almost exclusively on the collagen bundle elasticity. Our model also reveals the mechanical cooperation of the tissue's key microstructural components: while the fibers carry tensile forces, the matrices undergo hydrostatic pressure.

Micromechanics of dental cement paste.

Biodentine is a calcium silicate/calcium carbonate/zirconium dioxide/water-based dental replacement biomaterial, significantly outperforming the stiffness and hardness properties of chemically similar construction cement pastes. We here report the first systematic micromechanical investigation of Biodentine, combining grid nanoindentation with ultrasonic testing and micromechanical modeling. Histograms of nanoindentation-probed hardness and elastic modulus, comprising more than 5700 values each, are very well represented by the superposition of three log-normal distributions (LNDs). Most of the data (74%) belong to the intermediate LND, representing highly dense calcite-reinforced hydration products with on-average more than 60GPa elastic modulus and 3GPa hardness. The remaining data refer, on the one hand, to lower density hydration products, and on the other hand, to single-micron-sized unhydrated clinker and zirconium-dioxide inclusions. Micromechanical homogenization of these three material phases delivers elastic properties of the overall cement paste material, which significantly exceed those probed by more than 300 ultrasonic tests performed in the kHz and MHz regime. This indicates the presence of micro-defects, which slightly weaken the otherwise highly optimized biomaterial system.

Effects of high-urgency ambulance transportation on pro-B-type natriuretic peptide levels in patients with heart failure.

OBJECTIVES: We hypothesized that pro-B-type natriuretic peptide (proBNP) levels in venous blood increase in patients with acute out-of-hospital heart failure because of stress during emergency ambulance transportation, and furthermore, we wanted to show if there is an effect of increasing proBNP levels on hemodynamic parameters. METHODS: Venous proBNP levels, visual analog scale (VAS) for pain and anxiety before and after transportation, heart rate, blood pressure, and transcutaneous Spo(2) were measured every 3 minutes in 32 patients with defined clinical signs of heart failure. RESULTS: ProBNP levels increased significantly (P < .01) during transportation (278.13 +/- 113.20 vs 984.67 +/- 627.33 pg/mL), whereas heart rate and mean blood pressure remained almost stable. There was no significant change in VAS for pain and anxiety (3.79 +/- 3.70 and 2.89 +/- 3.01 vs 2.13 +/- 3.30 and 1.57 +/- 2.78). CONCLUSION: A rapid increase in proBNP levels was shown in patients with acute out-of-hospital heart failure during emergency ambulance transportation but no significant changes in hemodynamic parameters.

"Variances" and "in-variances" in hierarchical porosity and composition, across femoral tissues from cow, horse, ostrich, emu, pig, rabbit, and frog.

It is very well known that bone is a hierarchically organized material produced by bone cells residing in the fluid environments filling (larger) vascular pores and (smaller) lacunar pores. The extracellular space consists of hydroxyapatite crystals, collagen type I molecules, and water with non-collageneous organics. It is less known to which extent the associated quantities (mineral, organic, and water concentrations; vascular, lacunar, and extracellular porosities) vary across species, organs, and ages. We here investigate the aforementioned quantities across femoral shaft tissues from cow, horse, emu, frog, ostrich, pig, and rabbit; by means of light microscopy and dehydration-demineralization tests; thereby revealing interesting invariances: The extracellular volume fractions of organic matter turn out to be similar across all tested non-amphibian tissues; as do the extracellular volume fractions of hydroxyapatite across all tested mammals. Hence, the chemical composition of the femoral extracellular bone matrix is remarkably "invariant" across differently aged mammals; while the water content shows significant variations, as does the partitions of water between the different pore spaces. The latter exhibit strikingly varying morphologies as well. This finding adds to the ample "universal patterns" in the sense of evolutionary developmental biology; and it provides interesting design requirements for the development of novel biomimetic tissue engineering solutions.

Effect of boron incorporation on the bioactivity, structure, and mechanical properties of ordered mesoporous bioactive glasses.

B2O3 doped (0.5-15 mol%) ordered mesoporous bioactive glasses (MBG) with the composition 80% SiO2-15% CaO-5% P2O5 were synthesized via a sol-gel based evaporation-induced self-assembly process using the block-copolymer P123 as a structure directing agent and characterized by biokinetic, mechanical and structural investigations. Nitrogen physisorption isotherms and electron microscopy indicate no detrimental effect of B2O3 on the ordered hexagonal pore structure. Boron incorporation increases both the bulk modulus and hardness of the glasses. In vitro bioactivity tests reveal a rapid initial release of Ca2+ and PO43- ions, followed by formation of hydroxyapatite carbonate within a few hours. Contrary to the tight incorporation of Al in Al-doped MBGs, the rapid release of borate species into simulated-body-fluid suggests loosely bound species localized at the internal surfaces of the mesopores. 29Si, 11B, 31P, and 1H solid state NMR spectroscopy reveal that the majority of the borate is present as anionic BO4/2- species. The need for charge compensation leads to an increase in the average degree of polymerization of the phosphate species for high boron contents. 11B{31P} rotational echo double resonance NMR results reveal the absence of B-O-P linkages. This structural model explains the rapid release of borate and the enhanced dissolution kinetics of the Ca2+ and phosphate species.

Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: experimentally supported micromechanical explanation of bone strength.

There is an ongoing discussion on how bone strength could be explained from its internal structure and composition. Reviewing recent experimental and molecular dynamics studies, we here propose a new vision on bone material failure: mutual ductile sliding of hydroxyapatite mineral crystals along layered water films is followed by rupture of collagen crosslinks. In order to cast this vision into a mathematical form, a multiscale continuum micromechanics theory for upscaling of elastoplastic properties is developed, based on the concept of concentration and influence tensors for eigenstressed microheterogeneous materials. The model reflects bone's hierarchical organization, in terms of representative volume elements for cortical bone, for extravascular and extracellular bone material, for mineralized fibrils and the extrafibrillar space, and for wet collagen. In order to get access to the stress states at the interfaces between crystals, the extrafibrillar mineral is resolved into an infinite amount of cylindrical material phases oriented in all directions in space. The multiscale micromechanics model is shown to be able to satisfactorily predict the strength characteristics of different bones from different species, on the basis of their mineral/collagen content, their intercrystalline, intermolecular, lacunar, and vascular porosities, and the elastic and strength properties of hydroxyapatite and (molecular) collagen.

Mandibular biomechanics after marginal resection: Correspondences of simulated volumetric strain and skeletal resorption.

Serious mandibular diseases such as tumor or osteonecrosis often require segmental or marginal mandibulectomy, the latter with improved outcome thanks to preserved mandibular continuity. Nevertheless, gradual osteolytic and/or osteosclerotic skeletal changes frequently indicate repetitive resections. Based on the fundamental adaptivity of bone to mechanical loads, the question arose whether resection-related anatomical alterations trigger relevant pathological skeletal adaptations. For a clinical case after mandibular box resection due to progressive osteoradionecrosis (ORN), routine biomechanical loading was simulated by finite element method, respecting pathology-related anatomy, tissue properties, and biting capacity. By 3D-visualization of the mandible's pathological development from follow-up-CT's over four years, remarkable correspondences of skeletal resorptions and increased unphysiological strain were revealed. Higher unphysiological load was correlated with more serious and earlier skeletal alterations. Three months post-operatively, serious buccal destruction at the distal resection corner occurred in correspondence with dominant tensile strain. At the resection, elevated strain caused by reduced alveolar height corresponded to skeletal compromise, observed 8-9 months post-operatively. ORN-related lesions, diagnosed before resection, entailed unphysiological strain coinciding with local skeletal alterations. Simulations with "healthy" instead of pathological tissue coefficients induced quantitative improvements of 25-33%, but without fundamental change. These results suggest a decisive contribution of resection-related biomechanical skeletal adaptations to this patient's mandibular decline with hemimandibulectomy about 2.5 years after the first resection. However, mechanical stress concentrations in sharp angles as the distal resection corner and reduced stability due to decreased alveolar height generally bear the danger of pathological biomechanics and severe skeletal adaptations for patients after mandibular box resection.

X-ray physics-based CT-to-composition conversion applied to a tissue engineering scaffold, enabling multiscale simulation of its elastic behavior.

Nowadays, the assessment of the mechanical competence of tissue engineering scaffolds based on computer simulations is a well-accepted technology. Typically, such simulations are performed by means of the Finite Element (FE) method, with the underlying structural model being created based on micro-computed tomography (microCT). Here, this analysis modality is applied to a new, ternary composite, consisting of PHBV, i.e. poly(3-hydroxybutyrate-co-3-hydroxyvalerate), PLGA, i.e. poly(lactic-co-glycolide), as well as of TCP, i.e. tricalcium phosphate hydrate. The studied scaffold structure is made up by fibers of this new composite material, manufactured by means of the rapid prototyping method. The data collected from microCT is utilized for adequately defining the mechanical properties of the FE model. In particular, the three-dimensional field of grey values is interpreted in terms of the underlying field of attenuation coefficients, taking into account the photon energy employed in microCT imaging, eventually allowing for calculation of the three-dimensionally distributed, voxel-specific composition of the studied material. For the sake of keeping the FE simulations as efficient as possible, groups of voxels are combined into one finite element; the grey value of the latter is obtained by volume averaging. Employing a two-step micromechanical homogenization scheme, the experimentally accessible stiffness of the three constituents (PHBV, PLGA, and TCP) is then, finite element by finite element, upscaled to the composition-dependent stiffness of the composite material. The plausibility and adequacy of the FE model is demonstrated by simulating the effects of uniaxial compression on the scaffold structure, in terms of resulting stress and strain fields, highlighting the importance of the fiber junctions (as they are the mechanically most stressed regions), and that neglecting the material heterogeneity would lead to a potentially significant underestimation of stresses and strains. Finally, a comparison is made of the employed analysis modality of microCT data with a previously pursued, simplified analysis strategy, highlighting the conceptual superiority of the former, and pointing out the application limits of the latter.

A mathematical multiscale model of bone remodeling, accounting for pore space-specific mechanosensation.

While bone tissue is a hierarchically organized material, mathematical formulations of bone remodeling are often defined on the level of a millimeter-sized representative volume element (RVE), "smeared" over all types of bone microstructures seen at lower observation scales. Thus, there is no explicit consideration of the fact that the biological cells and biochemical factors driving bone remodeling are actually located in differently sized pore spaces: active osteoblasts and osteoclasts can be found in the vascular pores, whereas the lacunar pores host osteocytes - bone cells originating from former osteoblasts which were then "buried" in newly deposited extracellular bone matrix. We here propose a mathematical description which considers size and shape of the pore spaces where the biological and biochemical events take place. In particular, a previously published systems biology formulation, accounting for biochemical regulatory mechanisms such as the rank-rankl-opg pathway, is cast into a multiscale framework coupled to a poromicromechanical model. The latter gives access to the vascular and lacunar pore pressures arising from macroscopic loading. Extensive experimental data on the biological consequences of this loading strongly suggest that the aforementioned pore pressures, together with the loading frequency, are essential drivers of bone remodeling. The novel approach presented here allows for satisfactory simulation of the evolution of bone tissue under various loading conditions, and for different species; including scenarios such as mechanical dis- and overuse of murine and human bone, or in osteocyte-free bone.

Modal analysis of nanoindentation data, confirming that reduced bone turnover may cause increased tissue mineralization/elasticity.

It is widely believed that the activities of bone cells at the tissue scale not only govern the size of the vascular pore spaces (and hence, the amount of bone tissue available for actually carrying the loads), but also the characteristics of the extracellular bone matrix itself. In this context, increased mechanical stimulation (in mediolateral regions of human femora, as compared to anteroposterior regions) may lead to increased bone turnover, lower bone matrix mineralization, and therefore lower tissue modulus. On the other hand, resorption-only processes (in endosteal versus periosteal regions) may have the opposite effect. A modal analysis of nanoindentation data obtained on femurs from the Melbourne Femur Research Collection (MFRC) indeed confirms that bone is stiffer in endosteal regions compared to periosteal regions (E̅endost = 29.34 ± 0.75 GPa >E̅periost = 24.67 ± 1.63 GPa), most likely due to the aging-related increase in resorption modeling on endosteal surfaces resulting in trabecularization of cortical bone. The results also show that bone is stiffer along the anteroposterior direction compared the mediolateral direction (E̅anteropost = 28.89 ± 1.08 GPa >E̅mediolat = 26.03 ± 2.31 GPa), the former being aligned with the neutral bending axis of the femur and, thus, undergoing more resorption modeling and consequently being more mineralized.

Poromicromechanics reveals that physiological bone strains induce osteocyte-stimulating lacunar pressure.

Mechanical loads which are macroscopically acting onto bony organs, are known to influence the activities of biological cells located in the pore spaces of bone, in particular so the signaling and production processes mediated by osteocytes. The exact mechanisms by which osteocytes are actually able to "feel" the mechanical loading and changes thereof, has been the subject of numerous studies, and, while several hypotheses have been brought forth over time, this topic has remained a matter of debate. Relaxation times reported in a recent experimental study of Gardinier et al. (Bone 46(4):1075-1081, 2010) strongly suggest that the lacunar pores are likely to experience, during typical physiological load cycles, not only fluid transport, but also undrained conditions. The latter entail the buildup of lacunar pore pressures, which we here quantify by means of a thorough multiscale modeling approach. In particular, the proposed model is based on classical poroelasticity theory, and able to account for multiple pore spaces. First, the model reveals distinct nonlinear dependencies of the resulting lacunar (and vascular) pore pressures on the underlying bone composition, highlighting the importance of a rigorous multiscale approach for appropriate computation of the aforementioned pore pressures. Then, the derived equations are evaluated for macroscopic (uniaxial as well as hydrostatic) mechanical loading of physiological magnitude. The resulting model-predicted pore pressures agree very well with the pressures that have been revealed, by means of in vitro studies, to be of adequate magnitude for modulating the responses of biological cells, including osteocytes. This underlines that osteocytes may respond to many types of loading stimuli at the same time, in particular so to fluid flow and hydrostatic pressure.

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

Bone tissue engineering materials must blend in the targeted physiological environment, in terms of both the materials' biocompatibility and mechanical properties. As for the latter, a well-adjusted stiffness ensures that the biomaterial's deformation behavior fits well to the deformation behavior of the surrounding biological tissue, whereas an appropriate strength provides sufficient load-carrying capacity of the biomaterial. Here, a mathematical modeling approach for estimating the macroscopic load that initiates failure of a hierarchically organized, granular, hydroxyapatite-based biomaterial is presented. For this purpose, a micromechanics model is developed for downscaling macroscopically prescribed stress (or strain) states to the level of the needle-shaped hydroxyapatite crystals. Presuming that the biomaterial fails due to the quasi-brittle failure of the most unfavorably stressed hydroxyapatite needle, the downscaled stress tensors are fed into a suitable, Mohr-Coulomb-type failure criterion, based on which the macroscopic failure load is deduced. The change of the biomaterial's composition in response to placing it in physiological solution, caused by growth of new bone tissue on the granules's surfaces, on the one hand, and by resorption of the hydroxyapatite crystals, on the other hand, is taken into account by means of suitable evolution laws. Numerical studies show how the macroscopic load-carrying capacity of the biomaterial is influenced by its design parameters. The presented modeling approach could prove beneficial for the design process of the studied biomaterials (as well as similarly composed biomaterials), particularly in terms of optimizing its mechanical performance.

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.

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.