Comparison of patient-specific computational models vs. clinical follow-up, for adjacent segment disc degeneration and bone remodelling after spinal fusion.

Spinal fusion is a standard surgical treatment for patients suffering from low back pain attributed to disc degeneration. However, results are somewhat variable and unpredictable. With fusion the kinematic behaviour of the spine is altered. Fusion and/or stabilizing implants carrying considerable load and prevent rotation of the fused segments. Associated with these changes, a risk for accelerated disc degeneration at the adjacent levels to fusion has been demonstrated. However, there is yet no method to predict the effect of fusion surgery on the adjacent tissue levels, i.e. bone and disc. The aim of this study was to develop a coupled and patient-specific mechanoregulated model to predict disc generation and changes in bone density after spinal fusion and to validate the results relative to patient follow-up data. To do so, a multiscale disc mechanoregulation adaptation framework was developed and coupled with a previously developed bone remodelling algorithm. This made it possible to determine extra cellular matrix changes in the intervertebral disc and bone density changes simultaneously based on changes in loading due to fusion surgery. It was shown that for 10 cases the predicted change in bone density and degeneration grade conforms reasonable well to clinical follow-up data. This approach helps us to understand the effect of surgical intervention on the adjacent tissue remodelling. Thereby, providing the first insight for a spine surgeon as to which patient could potentially be treated successfully by spinal fusion and in which patient has a high risk for adjacent tissue changes.

Osteochondral resurfacing implantation angle is more important than implant material stiffness.

Osteochondral resurfacing implants are a promising treatment for focal cartilage defects. Several implant-factors may affect the clinical outcome of this treatment, such as the implant material stiffness and the accuracy of implant placement, known to be challenging. In general, softer implants are expected to be more accommodating for implant misalignment than stiffer implants, and motion is expected to increase effects from implant misalignment and stiffness. 3D finite element models of cartilage/cartilage contact were employed in which implantation angle (0°, 5°, 10°) and implant material stiffness (E = 5 MPa, 100 MPa, 2 GPa) were varied. A creep loading (0.6 MPa) was simulated, followed by a sliding motion. Creep loading resulted in low maximum collagen strains of 2.5% in the intact case compared to 11.7% with an empty defect. Implants mostly positively affected collagen strains, deviatoric strains, and hydrostatic pressures in the adjacent cartilage, but these effects were superior for correct alignment (0°). The main effect of implant misalignment was bulging of opposing cartilage tissue into the gap caused by the misalignment. This increased collagen strains and hydrostatic pressures. Deviatoric strains were increased adjacent to the gap. Subsequent sliding initially increased strains for a stiff, misaligned implant, but generally sliding decreased strains. In conclusion, implants can decrease the detrimental effect of defects, but correct implant alignment is crucial, more than implant material stiffness. Implant misalignment causes a gap, causing potentially damaging cartilage deformation during prolonged loading, for example, standing, even for soft implants. Mild motion may positively affect the cartilage. © 2018 The Authors. Journal of Orthopaedic Research® published by Wiley Periodicals, Inc. on behalf of Orthopaedic Research Society. J Orthop Res 36:2911-2922, 2018.

Mechanical behavior of a soft hydrogel reinforced with three-dimensional printed microfibre scaffolds.

Reinforcing hydrogels with micro-fibre scaffolds obtained by a Melt-Electrospinning Writing (MEW) process has demonstrated great promise for developing tissue engineered (TE) constructs with mechanical properties compatible to native tissues. However, the mechanical performance and reinforcement mechanism of the micro-fibre reinforced hydrogels is not yet fully understood. In this study, FE models, implementing material properties measured experimentally, were used to explore the reinforcement mechanism of fibre-hydrogel composites. First, a continuum FE model based on idealized scaffold geometry was used to capture reinforcement effects related to the suppression of lateral gel expansion by the scaffold, while a second micro-FE model based on micro-CT images of the real construct geometry during compaction captured the effects of load transfer through the scaffold interconnections. Results demonstrate that the reinforcement mechanism at higher scaffold volume fractions was dominated by the load carrying-ability of the fibre scaffold interconnections, which was much higher than expected based on testing scaffolds alone because the hydrogel provides resistance against buckling of the scaffold. We propose that the theoretical understanding presented in this work will assist the design of more effective composite constructs with potential applications in a wide range of TE conditions.

Influence of the temporal deposition of extracellular matrix on the mechanical properties of tissue-engineered cartilage.

Enhancement of the load-bearing capacity of tissue-engineered (TE) cartilage is expected to improve the clinical outcome of implantations. Generally, cartilage TE studies aim to increase the total extracellular matrix (ECM) content to improve implant mechanical properties. Besides the ECM content, however, temporal variations in deposition rate of ECM components during culture may also have an effect. Using a computational approach, the present study aims to quantify possible effects of temporal variations in the deposition of glycosaminoglycan (GAG) at given collagen synthesis rates on the mechanical stiffness of cartilage TE constructs. Maturation of a cylindrical cartilage TE construct over 42 days of culture was simulated using a composition-based finite element model that accounted for the transient deposition of GAG and collagen. Results showed an effect of GAG deposition rate on the swelling behavior and the collagen network strain, which resulted in significant changes in the compressive stiffness of cartilage TE constructs. When collagen deposition was first allowed in the constructs while the GAG deposition was delayed for the first 2 or 4 weeks, the collagen more effectively restricted tissue swelling later during the culture. Consequently, while the ultimate amount of ECM at day 42 was identical between the constructs, those with delayed GAG deposition contained elevated internal osmotic swelling pressure (up to 48%). This increased the compressive stiffness (up to 60%) of cartilage TE constructs at day 42. These findings clarify similar, yet unexplained, experimental observations. By providing further insights into mechanical effects inside cartilage TE constructs, these analyses are expected to help in designing culture regimes for engineering TE cartilage with improved load-bearing properties.

The effects of matrix inhomogeneities on the cellular mechanical environment in tissue-engineered cartilage: an in silico investigation.

Mechanical stimulation during cartilage tissue-engineering enhances extracellular matrix (ECM) synthesis and thereby improves the mechanical properties of tissue engineered (TE) cartilage. Generally, these mechanical stimuli are of a fixed magnitude. However, as a result of ECM synthesis and spatial variations thereof at both the macroscopic and microscopic scales, the internal mechanical conditions in the constructs change with time. Consequently, the physical signals in the environment of the cells will vary spatially and temporally, even though macroscopically the same loading is applied to the construct. The purpose of the present study was to numerically quantify such effects and thereby reveal the importance of adjusting loading regimes during cartilage tissue-engineering. A validated nonlinear fiber-reinforced poroviscoelastic swelling cartilage model that can accommodate for effects of collagen reinforcement and swelling by proteoglycans was used. At the microscopic scale, ECM was gradually varied from localized in the pericellular area, toward equally distributed throughout the surrounding interterritorial matrix. At the macroscopic tissue scale, ECM was gradually varied from predominantly localized in the periphery of the TE construct toward homogeneously distributed. Both concentration of ECM in the pericellular area and concentration of ECM in the periphery of a construct alter the physical signals up to an order of magnitude compared to those at the onset of the culture. Of particular interest, is the effect of elevated osmotic swelling pressure in the pericellular area, which shields not only the cells from receiving external mechanical compression, but also directly induces tension on the cells. Based on the present computational simulations, it is therefore, proposed that cartilage TE studies should consider ECM distribution as an important factor when developing loading protocols for cartilage culturing process. For instance, the level of mechanical compression should gradually increase to sufficiently deform chondrocytes over time, in case there is matrix accumulation in the pericellular area.

How preconditioning affects the measurement of poro-viscoelastic mechanical properties in biological tissues.

It is known that initial loading curves of soft biological tissues are substantially different from subsequent loadings. The later loading curves are generally used for assessing the mechanical properties of a tissue, and the first loading cycles, referred to as preconditioning, are omitted. However, slow viscoelastic phenomena related to fluid flow or collagen viscoelasticity are initiated during these first preconditioning loading cycles and may persist during the actual data collection. When these data are subsequently used for fitting of material properties, the viscoelastic phenomena that occurred during the initial cycles are not accounted for. The aim of the present study is to explore whether the above phenomena are significant for articular cartilage, by evaluating the effect of such time-dependent phenomena by means of computational modeling. Results show that under indentation, collagen viscoelasticity dominates the time-dependent behavior. Under UC, fluid-dependent effects are more important. Interestingly, viscoelastic and poroelastic effects may act in opposite directions and may cancel each other out in a stress-strain curve. Therefore, equilibrium may be apparent in a stress-strain relationship, even though internally the tissue is not in equilibrium. Also, the time-dependent effects of viscoelasticity and poroelasticity may reinforce each other, resulting in a sustained effect that lasts longer than suggested by their individual effects. Finally, the results illustrate that data collected from a mechanical test may depend on the preconditioning protocol. In conclusion, preconditioning influences the mechanical response of articular cartilage significantly and therefore cannot be neglected when determining the mechanical properties. To determine the full viscoelastic and poroelastic properties of articular cartilage requires fitting to both preconditioning and post-preconditioned loading cycles.

A review of the combination of experimental measurements and fibril-reinforced modeling for investigation of articular cartilage and chondrocyte response to loading.

The function of articular cartilage depends on its structure and composition, sensitively impaired in disease (e.g. osteoarthritis, OA). Responses of chondrocytes to tissue loading are modulated by the structure. Altered cell responses as an effect of OA may regulate cartilage mechanotransduction and cell biosynthesis. To be able to evaluate cell responses and factors affecting the onset and progression of OA, local tissue and cell stresses and strains in cartilage need to be characterized. This is extremely challenging with the presently available experimental techniques and therefore computational modeling is required. Modern models of articular cartilage are inhomogeneous and anisotropic, and they include many aspects of the real tissue structure and composition. In this paper, we provide an overview of the computational applications that have been developed for modeling the mechanics of articular cartilage at the tissue and cellular level. We concentrate on the use of fibril-reinforced models of cartilage. Furthermore, we introduce practical considerations for modeling applications, including also experimental tests that can be combined with the modeling approach. At the end, we discuss the prospects for patient-specific models when aiming to use finite element modeling analysis and evaluation of articular cartilage function, cellular responses, failure points, OA progression, and rehabilitation.

The effect of tissue-engineered cartilage biomechanical and biochemical properties on its post-implantation mechanical behavior.

The insufficient load-bearing capacity of today's tissue-engineered (TE) cartilage limits its clinical application. Focus has been on engineering cartilage with enhanced mechanical stiffness by reproducing native biochemical compositions. More recently, depth dependency of the biochemical content and the collagen network architecture has gained interest. However, it is unknown whether the mechanical performance of TE cartilage would benefit more from higher content of biochemical compositions or from achieving an appropriate collagen organization. Furthermore, the relative synthesis rate of collagen and proteoglycans during the TE process may affect implant performance. Such insights would assist tissue engineers to focus on those aspects that are most important. The aim of the present study is therefore to elucidate the relative importance of implant ground substance stiffness, collagen content, and collagen architecture of the implant, as well as the synthesis rate of the biochemical constituents for the post-implantation mechanical behavior of the implant. We approach this by computing the post-implantation mechanical conditions using a composition-based fibril-reinforced poro-viscoelastic swelling model of the medial tibia plateau. Results show that adverse implant composition and ultrastructure may lead to post-implantation excessive mechanical loads, with collagen orientation being the most critical variable. In addition, we predict that a faster synthesis rate of proteoglycans compared to that of collagen during TE culture may result in excessive loads on collagen fibers post-implantation. This indicates that even with similar final contents, constructs may behave differently depending on their development. Considering these aspects may help to engineer TE cartilage implants with improved survival rates.

Influence of tissue- and cell-scale extracellular matrix distribution on the mechanical properties of tissue-engineered cartilage.

The insufficient load-bearing capacity of today's tissue- engineered (TE) cartilage limits its clinical application. Generally, cartilage TE studies aim to increase the extracellular matrix (ECM) content, as this is thought to determine the load-bearing properties of the cartilage. However, there are apparent inconsistencies in the literature regarding the correlation between ECM content and mechanical properties of TE constructs. In addition to the amount of ECM, the spatial inhomogeneities in ECM distribution at the tissue scale as well as at the cell scale may affect the mechanical properties of TE cartilage. The relative importance of such structural inhomogeneities on mechanical behavior of TE cartilage is unknown. The aim of the present study was, therefore, to theoretically elucidate the influence of these inhomogeneities on the mechanical behavior of chondrocyte-agarose TE constructs. A validated non-linear fiber-reinforced poro-elastic swelling cartilage model that can accommodate for effects of collagen reinforcement and swelling by proteoglycans was used. At the tissue scale, ECM was gradually varied from predominantly localized in the periphery of the TE construct toward an ECM-rich inner core. The effect of these inhomogeneities in relation to the total amount of ECM was also evaluated. At the cell scale, ECM was gradually varied from localized in the pericellular area, toward equally distributed throughout the interterritorial area. Results from the tissue-scale model indicated that localization of ECM in either the construct periphery or in the inner core may reduce construct stiffness compared with that of constructs with homogeneous ECM. Such effects are more significant at high ECM amounts. At the cell scale, localization of ECM around the cells significantly reduced the overall stiffness, even at low ECM amounts. The compressive stiffness gradually increased when ECM distribution became more homogeneous and the osmotic swelling pressure in the interterritorial area increased. We conclude that for the same amount of ECM content in TE cartilage constructs, superior mechanical properties can be achieved with more homogeneous ECM distribution at both tissue and cell scale. Inhomogeneities at the cell scale are more important than those at the tissue scale.

Design of next generation total disk replacements.

To improve the treatments for low back pain, new designs of total disk replacement have been proposed. The question is how well these designs can act as a functional replacement of the intervertebral disk. Four finite element models were made, for four different design concepts, to determine how well they can mimic the physiological intervertebral disk mechanical function. The four designs were a homogenous elastomer, a multi-stiffness elastomer, an elastomer with fiber jacket, and a hydrogel with fiber jacket. The best material properties of the four models were determined by optimizing the model behavior to match the behavior of the intervertebral disk in flexion-extension, axial rotation, and lateral bending. It was shown that neither a homogeneous elastomer nor a multi-stiffness elastomer could mimic the non-linear behavior within the physiological range of motion. Including a fiber jacket around an elastomer allowed for physiological motion in all degrees of freedom. Replacing the elastomer by a hydrogel yielded similar good behavior. Mimicking the non-linear behavior of the intervertebral disk, in the physiological range of motion is essential in maintaining and restoring spinal motion and in protecting surrounding tissues like the facet joints or adjacent segments. This was accomplished with designs mimicking the function of the annulus fibrosus.

Composition of the pericellular matrix modulates the deformation behaviour of chondrocytes in articular cartilage under static loading.

The aim was to assess the role of the composition changes in the pericellular matrix (PCM) for the chondrocyte deformation. For that, a three-dimensional finite element model with depth-dependent collagen density, fluid fraction, fixed charge density and collagen architecture, including parallel planes representing the split-lines, was created to model the extracellular matrix (ECM). The PCM was constructed similarly as the ECM, but the collagen fibrils were oriented parallel to the chondrocyte surfaces. The chondrocytes were modelled as poroelastic with swelling properties. Deformation behaviour of the cells was studied under 15% static compression. Due to the depth-dependent structure and composition of cartilage, axial cell strains were highly depth-dependent. An increase in the collagen content and fluid fraction in the PCMs increased the lateral cell strains, while an increase in the fixed charge density induced an inverse behaviour. Axial cell strains were only slightly affected by the changes in PCM composition. We conclude that the PCM composition plays a significant role in the deformation behaviour of chondrocytes, possibly modulating cartilage development, adaptation and degeneration. The development of cartilage repair materials could benefit from this information.

Remodeling of fracture callus in mice is consistent with mechanical loading and bone remodeling theory.

During the remodeling phase of fracture healing in mice, the callus gradually transforms into a double cortex, which thereafter merges into one cortex. In large animals, a double cortex normally does not form. We investigated whether these patterns of remodeling of the fracture callus in mice can be explained by mechanical loading. Morphologies of fractures after 21, 28, and 42 days of healing were determined from an in vivo mid-diaphyseal femoral osteotomy healing experiment in mice. Bone density distributions from microCT at 21 days were converted into adaptive finite element models. To assess the effect of loading mode on bone remodeling, a well-established remodeling algorithm was used to examine the effect of axial force or bending moment on bone structure. All simulations predicted that under axial loading, the callus remodeled to form a single cortex. When a bending moment was applied, dual concentric cortices developed in all simulations, corresponding well to the progression of remodeling observed experimentally and resulting in quantitatively comparable callus areas of woven and lamellar bone. Effects of biological differences between species or other reasons cannot be excluded, but this study demonstrates how a difference in loading mode could explain the differences between the remodeling phase in small rodents and larger mammals.

A poroviscoelastic description of fibrin gels.

The mechanical induction of specific cell phenotypes can only be properly controlled if the local stimuli applied to the cells are known as a function of the external applied loads. Finite element analysis of the cell carriers would be one method to calculate these local conditions. Furthermore, the constitutive model of the construct material should be able to describe mechanical events known to be responsible for cell stimulation, such as interstitial fluid flow. The aim of this study was to define a biphasic constitutive model for fibrin, a natural hydrogel often used for tissue engineering but not yet thoroughly characterized. Large strain poroelastic and poroviscoelastic constitutive equations were implemented into a finite element model of a fibrin gel. The parameter values for both formulations were found by either analytically solving equivalent low strain equations, or by optimizing directly the large strain equations based on experimental stress relaxation data. No poroelastic parameters that satisfactorily described the fibrin carrier behaviour could be found, suggesting that network viscoelasticity and fluid-flow time-dependent behaviour must be separately accounted for. It was demonstrated that fibrin can be described as a poroviscoelastic material, but a large strain characterization of the parameter values was necessary. The analytical resolution of the low strain poroviscoelastic equations was, however, accurate enough to serve as a reliable initial condition for further optimization of the parameter values with the large strain formulation.

Stress-relaxation of human patellar articular cartilage in unconfined compression: prediction of mechanical response by tissue composition and structure.

Mechanical properties of articular cartilage are controlled by tissue composition and structure. Cartilage function is sensitively altered during tissue degeneration, in osteoarthritis (OA). However, mechanical properties of the tissue cannot be determined non-invasively. In the present study, we evaluate the feasibility to predict, without mechanical testing, the stress-relaxation response of human articular cartilage under unconfined compression. This is carried out by combining microscopic and biochemical analyses with composition-based mathematical modeling. Cartilage samples from five cadaver patellae were mechanically tested under unconfined compression. Depth-dependent collagen content and fibril orientation, as well as proteoglycan and water content were derived by combining Fourier transform infrared imaging, biochemical analyses and polarized light microscopy. Finite element models were constructed for each sample in unconfined compression geometry. First, composition-based fibril-reinforced poroviscoelastic swelling models, including composition and structure obtained from microscopical and biochemical analyses were fitted to experimental stress-relaxation responses of three samples. Subsequently, optimized values of model constants, as well as compositional and structural parameters were implemented in the models of two additional samples to validate the optimization. Theoretical stress-relaxation curves agreed with the experimental tests (R=0.95-0.99). Using the optimized values of mechanical parameters, as well as composition and structure of additional samples, we were able to predict their mechanical behavior in unconfined compression, without mechanical testing (R=0.98). Our results suggest that specific information on tissue composition and structure might enable assessment of cartilage mechanics without mechanical testing.

Importance of collagen orientation and depth-dependent fixed charge densities of cartilage on mechanical behavior of chondrocytes.

The collagen network and proteoglycan matrix of articular cartilage are thought to play an important role in controlling the stresses and strains in and around chondrocytes, in regulating the biosynthesis of the solid matrix, and consequently in maintaining the health of diarthrodial joints. Understanding the detailed effects of the mechanical environment of chondrocytes on cell behavior is therefore essential for the study of the development, adaptation, and degeneration of articular cartilage. Recent progress in macroscopic models has improved our understanding of depth-dependent properties of cartilage. However, none of the previous works considered the effect of realistic collagen orientation or depth-dependent negative charges in microscopic models of chondrocyte mechanics. The aim of this study was to investigate the effects of the collagen network and fixed charge densities of cartilage on the mechanical environment of the chondrocytes in a depth-dependent manner. We developed an anisotropic, inhomogeneous, microstructural fibril-reinforced finite element model of articular cartilage for application in unconfined compression. The model consisted of the extracellular matrix and chondrocytes located in the superficial, middle, and deep zones. Chondrocytes were surrounded by a pericellular matrix and were assumed spherical prior to tissue swelling and load application. Material properties of the chondrocytes, pericellular matrix, and extracellular matrix were obtained from the literature. The loading protocol included a free swelling step followed by a stress-relaxation step. Results from traditional isotropic and transversely isotropic biphasic models were used for comparison with predictions from the current model. In the superficial zone, cell shapes changed from rounded to elliptic after free swelling. The stresses and strains as well as fluid flow in cells were greatly affected by the modulus of the collagen network. The fixed charge density of the chondrocytes, pericellular matrix, and extracellular matrix primarily affected the aspect ratios (height/width) and the solid matrix stresses of cells. The mechanical responses of the cells were strongly location and time dependent. The current model highlights that the collagen orientation and the depth-dependent negative fixed charge densities of articular cartilage have a great effect in modulating the mechanical environment in the vicinity of chondrocytes, and it provides an important improvement over earlier models in describing the possible pathways from loading of articular cartilage to the mechanical and biological responses of chondrocytes.

Are disc pressure, stress, and osmolarity affected by intra- and extrafibrillar fluid exchange?

Because extrafibrillar water content dictates extrafibrillar osmolarity, we aimed to determine the influence of intra- and extrafibrillar fluid exchange on intradiscal pressures and stresses. As experimental results showed that extrafibrillar osmolarity affects intervertebral disc cell gene expression and crack propagation, quantification of the effects of changes in intra- and extrafibrillar fluid exchange is physiologically relevant. Therefore, our 3D osmoviscoelastic finite element (FE) model of the intervertebral disc was extended to include the intra- and extrafibrillar water differentiation. Two simulations were performed, one without intrafibrillar fluid and one with intrafibrillar fluid fraction as a function of the extrafibrillar osmotic pressure. The intrafibrillar fluid fraction as a function of the extrafibrillar osmotic pressure was exponentially fitted to human data and implemented into the model. Because of the low collagen content in the nucleus, no noticeable differences in intradiscal pressure estimation were observed. However, values of extrafibrillar osmolarity, hydrostatic pressure, and the total tissue stress calculated for the annulus were clearly different. Stresses, hydrostatic pressure, and osmolarity were underestimated when the intrafibrillar water value was neglected. As the loading increased, the discrepancies increased. In conclusion, the distribution of pressure and osmolarity in the disc is affected by intra- and extrafibrillar water exchange.

Characterization of articular cartilage by combining microscopic analysis with a fibril-reinforced finite-element model.

Load-bearing characteristics of articular cartilage are impaired during tissue degeneration. Quantitative microscopy enables in vitro investigation of cartilage structure but determination of tissue functional properties necessitates experimental mechanical testing. The fibril-reinforced poroviscoelastic (FRPVE) model has been used successfully for estimation of cartilage mechanical properties. The model includes realistic collagen network architecture, as shown by microscopic imaging techniques. The aim of the present study was to investigate the relationships between the cartilage proteoglycan (PG) and collagen content as assessed by quantitative microscopic findings, and model-based mechanical parameters of the tissue. Site-specific variation of the collagen network moduli, PG matrix modulus and permeability was analyzed. Cylindrical cartilage samples (n=22) were harvested from various sites of the bovine knee and shoulder joints. Collagen orientation, as quantitated by polarized light microscopy, was incorporated into the finite-element model. Stepwise stress-relaxation experiments in unconfined compression were conducted for the samples, and sample-specific models were fitted to the experimental data in order to determine values of the model parameters. For comparison, Fourier transform infrared imaging and digital densitometry were used for the determination of collagen and PG content in the same samples, respectively. The initial and strain-dependent fibril network moduli as well as the initial permeability correlated significantly with the tissue collagen content. The equilibrium Young's modulus of the nonfibrillar matrix and the strain dependency of permeability were significantly associated with the tissue PG content. The present study demonstrates that modern quantitative microscopic methods in combination with the FRPVE model are feasible methods to characterize the structure-function relationships of articular cartilage.

Bone regeneration during distraction osteogenesis: mechano-regulation by shear strain and fluid velocity.

Corroboration of mechano-regulation algorithms is difficult, partly because repeatable experimental outcomes under a controlled mechanical environment are necessary, but rarely available. In distraction osteogenesis (DO), a controlled displacement is used to regenerate large volumes of new bone, with predictable and reproducible outcomes, allowing to computationally study the potential mechanisms that stimulate bone formation. We hypothesized that mechano-regulation by octahedral shear strain and fluid velocity can predict the spatial and temporal tissue distributions seen during experimental DO. Variations in predicted tissue distributions due to alterations in distraction rate and frequency could then also be studied. An in vivo ovine tibia experiment evaluating bone-segment transport (distraction, 1 mm/day) over an intramedullary nail was used for comparison. A 2D axisymmetric finite element model, with a geometry originating from the experimental data, was created and included into a previously developed model of tissue differentiation. Cells migrated and proliferated into the callus, differentiating into fibroblasts, chondrocytes or osteoblasts, dependent on the biophysical stimuli. Matrix production was modelled with an osmotic swelling model to allow tissues to grow at individual rates. The temporal and spatial tissue distributions predicted by the computational model agreed well with those seen experimentally. In addition, it was observed that decreased distraction rate (0.5 mm/d vs. 0.25 mm/d) increased the overall time needed for complete bone regeneration, whereas increased distraction frequency (0.5 mm/12 h vs. 0.25 mm/6 h) stimulated faster bone regeneration, as found in experimental findings by others. Thus, the algorithm regulated by octahedral shear strain and fluid velocity was able to predict the bone regeneration patterns dependent on distraction rate and frequency during DO.

Osmoviscoelastic finite element model of the intervertebral disc.

Intervertebral discs have a primarily mechanical role in transmitting loads through the spine. The disc is subjected to a combination of elastic, viscous and osmotic forces; previous 3D models of the disc have typically neglected osmotic forces. The fibril-reinforced poroviscoelastic swelling model, which our group has recently developed, is used to compute the interplay of osmotic, viscous and elastic forces in an intervertebral disc under axial compressive load. The unloaded 3D finite element mesh equilibrates in a physiological solution, and exhibits an intradiscal pressure of about 0.2 MPa. Before and after axial loading the numerically simulated hydrostatic pressure compares well with the experimental ranges measured. Loading the disc decreased the height of the disc and results in an outward bulging of the outer annulus. Fiber stresses were highest on the most outward bulging on the posterior-lateral side. The osmotic forces resulted in tensile hoop stresses, which were higher than typical values in a non-osmotic disc. The computed axial stress profiles reproduced the main features of the stress profiles, in particular the characteristic posterior and anterior stress which were observed experimentally.