Estimation of the poroelastic parameters of cortical bone

Prediction of the time course of callus stiffness as a function of mechanical parameters in experimental rat fracture healing studies--a numerical study

Numerous experimental fracture healing studies are performed on rats, in which different experimental, mechanical parameters are applied, thereby prohibiting direct comparison between each other. Numerical fracture healing simulation models are able to predict courses of fracture healing and offer support for pre-planning animal experiments and for post-hoc comparison between outcomes of different in vivo studies. The aims of this study are to adapt a pre-existing fracture healing simulation algorithm for sheep and humans to the rat, to corroborate it using the data of numerous different rat experiments, and to provide healing predictions for future rat experiments. First, material properties of different tissue types involved were adjusted by comparing experimentally measured callus stiffness to respective simulated values obtained in three finite element (FE) models. This yielded values for Young's moduli of cortical bone, woven bone, cartilage, and connective tissue of 15,750 MPa, 1,000 MPa, 5 MPa, and 1 MPa, respectively. Next, thresholds in the underlying mechanoregulatory tissue differentiation rules were calibrated by modifying model parameters so that predicted fracture callus stiffness matched experimental data from a study that used rigid and flexible fixators. This resulted in strain thresholds at higher magnitudes than in models for sheep and humans. The resulting numerical model was then used to simulate numerous fracture healing scenarios from literature, showing a considerable mismatch in only 6 of 21 cases. Based on this corroborated model, a fit curve function was derived which predicts the increase of callus stiffness dependent on bodyweight, fixation stiffness, and fracture gap size. By mathematically predicting the time course of the healing process prior to the animal studies, the data presented in this work provides support for planning new fracture healing experiments in rats. Furthermore, it allows one to transfer and compare new in vivo findings to previously performed studies with differing mechanical parameters.

Interstitial fluid flow in canaliculi as a mechanical stimulus for cancellous bone remodeling: in silico validation

Cancellous bone has a dynamic 3-dimensional architecture of trabeculae, the arrangement of which is continually reorganized via bone remodeling to adapt to the mechanical environment. Osteocytes are currently believed to be the major mechanosensory cells and to regulate osteoclastic bone resorption and osteoblastic bone formation in response to mechanical stimuli. We previously developed a mathematical model of trabecular bone remodeling incorporating the possible mechanisms of cellular mechanosensing and intercellular communication in which we assumed that interstitial fluid flow activates the osteocytes to regulate bone remodeling. While the proposed model has been validated by the simulation of remodeling of a single trabecula, it remains unclear whether it can successfully represent in silico the functional adaptation of cancellous bone with its multiple trabeculae. In the present study, we demonstrated the response of cancellous bone morphology to uniaxial or bending loads using a combination of our remodeling model with the voxel finite element method. In this simulation, cancellous bone with randomly arranged trabeculae remodeled to form a well-organized architecture oriented parallel to the direction of loading, in agreement with the previous simulation results and experimental findings. These results suggested that our mathematical model for trabecular bone remodeling enables us to predict the reorganization of cancellous bone architecture from cellular activities. Furthermore, our remodeling model can represent the phenomenological law of bone transformation toward a locally uniform state of stress or strain at the trabecular level.

Dynamic permeability of the lacunar-canalicular system in human cortical bone

A new method for the experimental determination of the permeability of a small sample of a fluid-saturated hierarchically structured porous material is described and applied to the determination of the lacunar-canalicular permeability [Formula: see text] in bone. The interest in the permeability of the lacunar-canalicular pore system (LCS) is due to the fact that the LCS is considered to be the site of bone mechanotransduction due to the loading-driven fluid flow over cellular structures. The permeability of this space has been estimated to be anywhere from [Formula: see text] to [Formula: see text]. However, the vascular pore system and LCS are intertwined, rendering the permeability of the much smaller-dimensioned LCS challenging to measure. In this study, we report a combined experimental and analytical approach that allowed the accurate determination of the [Formula: see text] to be on the order of [Formula: see text] for human osteonal bone. It was found that the [Formula: see text] has a linear dependence on loading frequency, decreasing at a rate of [Formula: see text]/Hz from 1 to 100 Hz, and using the proposed model, the porosity alone was able to explain 86 % of the [Formula: see text] variability.

Prediction of fracture healing under axial loading, shear loading and bending is possible using distortional and dilatational strains as determining mechanical stimuli

Numerical models of secondary fracture healing are based on mechanoregulatory algorithms that use distortional strain alone or in combination with either dilatational strain or fluid velocity as determining stimuli for tissue differentiation and development. Comparison of these algorithms has previously suggested that healing processes under torsional rotational loading can only be properly simulated by considering fluid velocity and deviatoric strain as the regulatory stimuli. We hypothesize that sufficient calibration on uncertain input parameters will enhance our existing model, which uses distortional and dilatational strains as determining stimuli, to properly simulate fracture healing under various loading conditions including also torsional rotation. Therefore, we minimized the difference between numerically simulated and experimentally measured courses of interfragmentary movements of two axial compressive cases and two shear load cases (torsional and translational) by varying several input parameter values within their predefined bounds. The calibrated model was then qualitatively evaluated on the ability to predict physiological changes of spatial and temporal tissue distributions, based on respective in vivo data. Finally, we corroborated the model on five additional axial compressive and one asymmetrical bending load case. We conclude that our model, using distortional and dilatational strains as determining stimuli, is able to simulate fracture-healing processes not only under axial compression and torsional rotation but also under translational shear and asymmetrical bending loading conditions.

Advances in assessment of bone porosity, permeability and interstitial fluid flow

This contribution reviews recent research performed to assess the porosity and permeability of bone tissue with the objective of understanding interstitial fluid movement. Bone tissue mechanotransduction is considered to occur due to the passage of interstitial pore fluid adjacent to dendritic cell structures in the lacunar-canalicular porosity. The movement of interstitial fluid is also necessary for the nutrition of osteocytes. This review will focus on four topics related to improved assessment of bone interstitial fluid flow. First, the advantages and limitations of imaging technologies to visualize bone porosities and architecture at several length scales are summarized. Second, recent efforts to measure the vascular porosity and lacunar-canalicular microarchitecture are discussed. Third, studies associated with the measurement and estimation of the fluid pressure and permeability in the vascular and lacunar-canalicular domains are summarized. Fourth, the development of recent models to represent the interchange of fluids between the bone porosities is described.

Interstitial fluid flow within bone canaliculi and electro-chemo-mechanical features of the canalicular milieu: a multi-parametric sensitivity analysis

Canalicular fluid flow is acknowledged to play a major role in bone functioning, allowing bone cells' metabolism and activity and providing an efficient way for cell-to-cell communication. Bone canaliculi are small canals running through the bone solid matrix, hosting osteocyte's dendrites, and saturated by an interstitial fluid rich in ions. Because of the small size of these canals (few hundred nanometers in diameter), fluid flow is coupled with electrochemical phenomena. In our previous works, we developed a multi-scale model accounting for coupled hydraulic and chemical transport in the canalicular network. Unfortunately, most of the physical and geometrical information required by the model is hardly accessible by nowadays experimental techniques. The goal of this study was to numerically assess the influence of the physical and material parameters involved in the canalicular fluid flow. The focus was set on the electro-chemo-mechanical features of the canalicular milieu, hopefully covering any in vivo scenario. Two main results were obtained. First, the most relevant parameters affecting the canalicular fluid flow were identified and their effects quantified. Second, these findings were given a larger scope to cover also scenarios not considered in this study. Therefore, this study gives insight into the potential interactions between electrochemistry and mechanics in bone and provides the rational for further theoretical and experimental investigations.

Silicate bioceramics induce angiogenesis during bone regeneration

The capacity to induce rapid vascular ingrowth during new bone formation is an important feature of biomaterials that are to be used for bone regeneration. Akermanite, a Ca-, Mg- and Si-containing bioceramic, has been demonstrated to be osteoinductive and to promote bone repair. This study further demonstrates the ability of akermanite to promote angiogenesis and investigates the mechanism of this behavior. The akermanite ion extract predominantly caused Si-ion-stimulated proliferation of human aortic endothelial cells. The Si ion in the extract was the most important component for the effect and the most effective concentration was found to be 0.6-2 μg ml(-1). In this range of Si ion concentration, the stimulating effect of the ceramic ion extract was demonstrated by the morphology of cells at the primary, interim and late stages during in vitro angiogenesis using ECMatrix™. The akermanite ion extract up-regulated the expression of genes encoding the receptors of proangiogenic cytokines and also increased the expression level of genes encoding the proangiogenic downstream cytokines, such as nitric oxide synthase and nitric oxide synthesis. Akermanite implanted in rabbit femoral condyle model promoted neovascularization after 8 and 16 weeks of implantation, which further confirmed its stimulation effect on angiogenesis in vivo. These results indicate that akermanite ceramic, an appropriate Si ion concentration source, could induce angiogenesis through increasing gene expression of proangiogenic cytokine receptors and up-regulated downstream signaling. To our knowledge, akermanite ceramic is the first Si-containing ceramic demonstrated to be capable of inducing angiogenesis during bone regeneration.

Analytical basis for the determination of the lacunar-canalicular permeability of bone using cyclic loading

An analytical model for the determination of the permeability in the lacunar-canalicular porosity of bone using cyclic loading is described in this contribution. The objective of the analysis presented is to relate the lacunar-canalicular permeability to a particular phase angle that is measurable when the bone is subjected to infinitesimal cyclic strain. The phase angle of interest is the lag angle between the applied strain and the resultant stress. Cyclic strain causes the interstitial fluid to move. This movement is essential for the viability of osteocytes and is believed to play a major role in the bone mechanotransduction mechanism. However, certain bone fluid flow properties, notably the permeability of the lacunar-canalicular porosity, are still not accurately determined. In this paper, formulas for the phase angle as a function of permeability for infinitesimal cyclic strain are presented and mathematical expressions for the storage modulus, loss modulus, and loss tangent are obtained. An accurate determination of the PLC permeability will improve our ability to understand mechanotransduction and mechanosensory mechanisms, which are fundamental to the understanding of how to treat osteoporosis, how to cope with microgravity in long-term manned space flights, and how to increase the longevity of prostheses that are implanted in bone tissue.

Combined in vivo/in silico study of mechanobiological mechanisms during endochondral ossification in bone healing

Mechanobiological theories have been introduced to illustrate the interaction between biology and the local mechanical environment during bone healing. Although several theories have been proposed, a quantitative validation using histomorphometric data is still missing. In this study, in vivo histological data based on an ovine animal experiment was quantified and used to validate bone healing simulations focussing on the endochondral ossification process. The bone formation at different callus regions (periosteal and endosteal bone at the medial and lateral side) was analyzed from in vivo data and quantitatively compared with in silico results. A histomorphometric difference was found in medial and lateral hard callus formation 3 weeks after osteotomy. However, the same amount of new bone was formed on both sides between week 3 and 6. Using a parametric approach, distinct ranges for mechanical strain levels regulating tissue formation were found, for which the in silico prediction agrees with the in vivo endochondral ossification both in pattern and quantity. According to this finding, a strain range of 1 to 8% seems to be conducive for cartilage formation while bone formation may be facilitated by strains up to 4%. This study demonstrates the potential of a thorough validation of in silico results for gaining a better understanding of mechanobiological mechanisms during bone healing.

Numerical model of bone remodeling sensitive to loading frequency through a poroelastic behavior and internal fluid movements

In this article, a phenomenological numerical model of bone remodeling is proposed. This model is based on the poroelasticity theory in order to take into account the effects of fluid movements in bone adaptation. Moreover, the proposed remodeling law is based on the classical 'Stanford' law, enriched in order to take into account the loading frequency, through fluid movements. This coupling is materialized by a quadratic function of Darcy velocity. The numerical model is carried out, using a finite element method, and calibrated using experimental results at macroscopic level, from the literature. First results concern cyclic loadings on a mouse ulna, at different frequencies between 1 Hz and 30 Hz, for a force amplitude of 1.5 N and 2 N. Experimental results exhibit a sensitivity to the loading frequency, with privileged frequency for bone remodeling between 5 Hz and 10 Hz, for the force amplitude of 2 N. For the force amplitude of 1.5 N, no privileged frequencies for bone remodeling are highlighted. This tendency is reproduced by the proposed numerical computations. The model is identified on a single case (one frequency and one force amplitude) and validated on the other ones. The second experimental validation deals with a different loading regime, an internal fluid pressure at 20 Hz on a turkey ulna. The same framework is applied, and the numerical and experimental data are still matching in terms of gain in bone mass density.

Effects of loading frequency on the functional adaptation of trabeculae predicted by bone remodeling simulation

The process of bone remodeling is regulated by metabolic activities of many bone cells. While osteoclasts and osteoblasts are responsible for bone resorption and formation, respectively, activities of these cells are believed to be controlled by a mechanosensory system of osteocytes embedded in the extracellular bone matrix. Several experimental and theoretical studies have suggested that the strain-derived interstitial fluid flow in lacuno-canalicular porosity serves as the prime mover for bone remodeling. Previously, we constructed a mathematical model for trabecular bone remodeling that interconnects the microscopic cellular activities with the macroscopic morphological changes in trabeculae through the mechanical hierarchy. This model assumes that fluid-induced shear stress acting on osteocyte processes is a driving force for bone remodeling. The validity of this model has been demonstrated with a remodeling simulation using a two-dimensional trabecular model. In this study, to investigate the effects of loading frequency, which is thought to be a significant mechanical factor in bone remodeling, we simulated morphological changes of a three-dimensional single trabecula under cyclic uniaxial loading with various frequencies. The results of the simulation show the trabecula reoriented to the loading direction with the progress of bone remodeling. Furthermore, as the imposed loading frequency increased, the diameter of the trabecula in the equilibrium state was enlarged by remodeling. These results indicate that our simulation model can successfully evaluate the relationship between loading frequency and trabecular bone remodeling.

Improvement of the shear fixation stability of intramedullary nailing

BACKGROUND

The healing outcome of long bone fractures is strongly influenced by the mechanical environment. High interfragmentary movement at the fracture site is detrimental to the fracture healing process. Long bone fractures stabilized with thin intramedullary nails commonly used for unreamed intramedullary nailing might be very flexible in shear direction and therefore critical for the fracture healing outcome. The aims of this study were to simulate the shear interfragmentary movement during gait for a human tibia treated with intramedullary nailing and to investigate if this movement could be lowered by implant design modifications.

METHODS

The shear movement was calculated with a 3D finite element model based on computer tomograph images of a cadaver bone-implant complex of a transverse tibia fracture treated with a Stryker T2 Standard Tibial Nail. This model was validated through in vitro test results under pure shear, axial, bending and torsional loading.

FINDINGS

High shear movements of approximately 4mm were calculated during gait. These shear movements could be reduced by approximately 30% either by implant modifications or the use of a 1mm thicker nail. Combining the implant modifications with a 1mm thicker nail, the shear movements could be reduced by 54%.

INTERPRETATION

The increase of the fixation stiffness by using an implant material with a high Young's modulus in combination with an angle-stable nail-screw fixation helps to reduce the shear movement during gait and possibly to lower the risk of a prolonged healing time with unreamed intramedullary nailing.

Wavelet decomposition of transmitted ultrasound wave through a 1-D muscle-bone system

In the attempt for using ultrasound as a diagnostic device for osteoporosis, several authors have described the result of the in vitro experiment in which ultrasound is passed through a cancellous bone specimen placed in a water tank. However, in the in vivo setting, a patient's cancellous bone is surrounded by cortical and muscle layers. This paper considers in the one-dimensional case (1) what effect the cortical bone segments surrounding the cancellous segment would have on the received signal and (2) what the received signal would be when a source and receiver are placed on opposite sides of a structure consisting of a cancellous segment surrounded by cortical and muscle layers. Mathematically this is accomplished by representing the received signal as a sum of wavelets which go through different reflection-transmission histories at the muscle-cortical bone and cortical-cancellous bone interfaces. The muscle and cortical bone are modeled as elastic materials and the cancellous bone as a poroelastic material described by the Biot-Johnson-Koplik-Dashen model. The approach presented here permits the assessment of which possible paths of transmission and reflection through the cortical-cancellous or muscle-cortical-cancellous complex will result in significant contributions to the received waveform. This piece of information can be useful for solving the inverse problem of non-destructive assessment of material properties of bone. Our methodology can be generalized to three-dimensional parallelly layered structure by first applying Fourier transform in the directions perpendicular to the transverse direction.

Trabecular bone remodelling simulation considering osteocytic response to fluid-induced shear stress

In bone functional adaptation by remodelling, osteocytes in the lacuno-canalicular system are believed to play important roles in the mechanosensory system. Under dynamic loading, bone matrix deformation generates an interstitial fluid flow in the lacuno-canalicular system; this flow induces shear stress on the osteocytic process membrane that is known to stimulate the osteocytes. In this sense, the osteocytes behave as mechanosensors and deliver mechanical information to neighbouring cells through the intercellular communication network. In this study, bone remodelling is assumed to be regulated by the mechanical signals collected by the osteocytes. From the viewpoint of multi-scale biomechanics, we propose a mathematical model of trabecular bone remodelling that takes into account the osteocytic mechanosensory network system. Based on this model, a computational simulation of trabecular bone remodelling was conducted for a single trabecula under cyclic uniaxial loading, demonstrating functional adaptation to the applied mechanical loading as a load-bearing construct.

In situ permeability measurement of the mammalian lacunar-canalicular system

Bone is capable of adapting its mass and structure under mechanical cues. Bone cells respond to various mechanical stimuli including substrate strain, fluid pressure, and fluid flow (shear stress) in vitro. Although tissue-level strains are well documented experimentally, microfluidic parameters around bone cells are quantified mainly through theoretical modeling. A key model parameter, the Darcy permeability of the bone lacunar-canalicular system (LCS), is difficult to measure using traditional methods due to the co-existence of the larger vascular and smaller LCS porosities. In this paper, we developed a novel method to measure the LCS permeability by rapid compaction of intact mammalian bones and recording the intramedullary pressure (IMP). Six canine metacarpals were subjected to three step compression tests with peak loads of 50, 100, or 200lbs, while the IMP was simultaneously recorded using a catheter pressure transducer. The loading ramp time was chosen to be ~2ms, which was long enough to allow pressure equilibrium to be established between the marrow cavity and the vascular pores, but short enough to observe the LCS fluid flowing into and out of the vascular pores. This loading scheme permitted us to differentiate the contribution of the two intermingled porosities to the IMP responses. The time constant of the IMP pressurization and relaxation due to the LCS was found to be 8.1+/-3.6s (n=18). The mid-shaft cortex of the metacarpals mainly consisted of osteons with an average radial thickness of 65+/-27microm, which served as the characteristic distance for the LCS fluid to relax. The LCS permeability was obtained via poroelastic analysis to be 2.8+/-1.8x10(-)(23)m(2), which was smaller than previous theoretical predictions (order of 10(-)(19) to 10(-)(22)m(2)), but within the range of previous experimentally based estimations (order of 10(-)(22) to 10(-)(25)m(2)). Our results also show that osteoblasts and osteocytes experience hydraulic pressures that differ by three orders of magnitude under physiological compressive strains. These estimates of the in vivo mechanical environments may be used to design in vitro models for elucidating the cellular and molecular mechanisms of bone adaptation and pathological bone loss.

Experimental determination of the permeability in the lacunar-canalicular porosity of bone

Permeability of the mineralized bone tissue is a critical element in understanding fluid flow occurring in the lacunar-canalicular porosity (PLC) compartment of bone and its role in bone nutrition and mechanotransduction. However, the estimation of bone permeability at the tissue level is affected by the influence of the vascular porosity in macroscopic samples containing several osteons. In this communication, both analytical and experimental approaches are proposed to estimate the lacunar-canalicular permeability in a single osteon. Data from an experimental stress-relaxation test in a single osteon are used to derive the PLC permeability by curve fitting to theoretical results from a compressible transverse isotropic poroelastic model of a porous annular disk under a ramp loading history (2007, "Compressible and Incompressible Constituents in Anisotropic Poroelasticity: The Problem of Unconfined Compression of a Disk," J. Mech. Phys. Solids, 55, pp. 161-193; 2008, "The Unconfined Compression of a Poroelastic Annular Cylindrical Disk," Mech. Mater., 40(6), pp. 507-523). The PLC tissue intrinsic permeability in the radial direction of the osteon was found to be dependent on the strain rate used and within the range of O(10(-24))-O(10(-25)). The reported values of PLC permeability are in reasonable agreement with previously reported values derived using finite element analysis (FEA) and nanoindentation approaches.

Hierarchical poroelasticity: movement of interstitial fluid between porosity levels in bones

The governing equations for the theory of poroelastic materials with hierarchical pore space architecture and compressible constituents undergoing small deformations are developed. These equations are applied to the problem of determining the exchange of pore fluid between the vascular porosity (PV) and the lacunar-canalicular porosity (PLC) in bone tissue due to cyclic mechanical loading and blood pressure oscillations. The result is basic to the understanding of interstitial flow in bone tissue that, in turn, is basic to understanding of nutrient transport from the vasculature to the bone cells buried in the bone tissue and to the process of mechanotransduction by these cells. A formula for the volume of fluid that moves between the PLC and PV in a cyclic loading is obtained as a function of the cyclic mechanical loading and blood pressure oscillations. Formulas for the oscillating fluid pore pressure in both the PLC and the PV are obtained as functions of the two driving forces, the cyclic mechanical straining and the blood pressure, both with specified amplitude and frequency. The results of this study also suggest a PV permeability greater than 10(-9) m(2) and perhaps a little lower than 10(-8) m(2). Previous estimates of this permeability have been as small as 10(-14) m(2).

Simulation of the nutrient supply in fracture healing

The healing process for bone fractures is sensitive to mechanical stability and blood supply at the fracture site. Most currently available mechanobiological algorithms of bone healing are based solely on mechanical stimuli, while the explicit analysis of revascularization and its influences on the healing process have not been thoroughly investigated in the literature. In this paper, revascularization was described by two separate processes: angiogenesis and nutrition supply. The mathematical models for angiogenesis and nutrition supply have been proposed and integrated into an existing fuzzy algorithm of fracture healing. The computational algorithm of fracture healing, consisting of stress analysis, analyses of angiogenesis and nutrient supply, and tissue differentiation, has been tested on and compared with animal experimental results published previously. The simulation results showed that, for a small and medium-sized fracture gap, the nutrient supply is sufficient for bone healing, for a large fracture gap, non-union may be induced either by deficient nutrient supply or inadequate mechanical conditions. The comparisons with experimental results demonstrated that the improved computational algorithm is able to simulate a broad spectrum of fracture healing cases and to predict and explain delayed unions and non-union induced by large gap sizes and different mechanical conditions. The new algorithm will allow the simulation of more realistic clinical fracture healing cases with various fracture gaps and geometries and may be helpful to optimise implants and methods for fracture fixation.

Sensitivity of tissue differentiation and bone healing predictions to tissue properties

Computational models are employed as tools to investigate possible mechano-regulation pathways for tissue differentiation and bone healing. However, current models do not account for the uncertainty in input parameters, and often include assumptions about parameter values that are not yet established. The aim was to clarify the importance of the assumed tissue material properties in a computational model of tissue differentiation during bone healing. An established mechano-biological model was employed together with a statistical approach. The model included an adaptive 2D finite element model of a fractured long bone. Four outcome criteria were quantified: (1) ability to predict sequential healing events, (2) amount of bone formation at specific time points, (3) total time until healing, and (4) mechanical stability at specific time points. Statistical analysis based on fractional factorial designs first involved a screening experiment to identify the most significant tissue material properties. These seven properties were studied further with response surface methodology in a three-level Box-Behnken design. Generally, the sequential events were not significantly influenced by any properties, whereas rate-dependent outcome criteria and mechanical stability were significantly influenced by Young's modulus and permeability. Poisson's ratio and porosity had minor effects. The amount of bone formation at early, mid and late phases of healing, the time until complete healing and the mechanical stability were all mostly dependent on three material properties; permeability of granulation tissue, Young's modulus of cartilage and permeability of immature bone. The consistency between effects of the most influential parameters was high. To increase accuracy and predictive capacity of computational models of bone healing, the most influential tissue mechanical properties should be accurately quantified.

Fluid and Solute Transport in Bone: Flow-Induced Mechanotransduction

Much recent evidence suggests that bone cells sense their mechanical environment via interstitial fluid flow. In this review, we summarize theoretical and experimental approaches to quantify fluid and solute transport in bone, starting with the early investigations of fluid shear stress applied to bone cells. The pathways of bone interstitial fluid and solute movement are high-lighted based on recent theoretical models, as well as a new generation of tracer experiments that have clarified and refined the structure and function of the osteocyte pericellular matrix. Then we trace how the fluid-flow models for mechanotransduction have evolved as new ultrastructural features of the osteocyte lacunar-canalicular porosity have been identified and how more recent in vitro fluid-flow and cell-stretch experiments have helped elucidate at the molecular level the possible pathways for cellular excitation in bone.

Modeling fluorescence recovery after photobleaching in loaded bone: potential applications in measuring fluid and solute transport in the osteocytic lacunar-canalicular system

Solute transport through the bone lacunar-canalicular system is essential for osteocyte viability and function, and it can be measured using fluorescence recovery after photobleaching (FRAP). The mathematical model developed here aims to analyze solute transport during FRAP in mechanically loaded bone. Combining both whole bone-level poroelasticity and cellular-level solute transport, we found that load-induced solute transport during FRAP is characterized by an exponential recovery rate, which is determined by the dimensionless Strouhal (St) number that characterizes the oscillation effects over the mean flows, and that significant transport occurs only for St values below a threshold, when the solute stroke displacement exceeds the distance between the source and sink (the canalicular length). This threshold mechanism explains the general flow behaviors such as increasing transport with increasing magnitude and decreasing frequency. Mechanical loading is predicted to enhance transport of all tracers relative to diffusion, with the greatest enhancement for medium-sized tracers and less enhancement for small and large tracers. This study provides guidelines for future FRAP experiments, based on which the model can be used to quantify bone permeability, solute-matrix interaction, and flow velocities. These studies should provide insights into bone adaptation and metabolism, and help to treat various bone diseases and conditions.

Determining the most important cellular characteristics for fracture healing using design of experiments methods

Computational models are employed as tools to investigate possible mechanoregulation pathways for tissue differentiation and bone healing. However, current models do not account for the uncertainty in input parameters, and often include assumptions about parameter values that are not yet established. The objective of this study was to determine the most important cellular characteristics of a mechanoregulatory model describing both cell phenotype-specific and mechanobiological processes that are active during bone healing using a statistical approach. The computational model included an adaptive two-dimensional finite element model of a fractured long bone. Three different outcome criteria were quantified: (1) ability to predict sequential healing events, (2) amount of bone formation at early, mid and late stages of healing and (3) the total time until complete healing. For the statistical analysis, first a resolution IV fractional factorial design (L(64)) was used to identify the most significant factors. Thereafter, a three-level Taguchi orthogonal array (L(27)) was employed to study the curvature (non-linearity) of the 10 identified most important parameters. The results show that the ability of the model to predict the sequences of normal fracture healing was predominantly influenced by the rate of matrix production of bone, followed by cartilage degradation (replacement). The amount of bone formation at early stages was solely dependent on matrix production of bone and the proliferation rate of osteoblasts. However, the amount of bone formation at mid and late phases had the rate of matrix production of cartilage as the most influential parameter. The time to complete healing was primarily dependent on the rate of cartilage degradation during endochondral ossification, followed by the rate of cartilage formation. The analyses of the curvature revealed a linear response for parameters related to bone, where higher rates of formation were more beneficial to healing. In contrast, parameters related to fibrous tissue and cartilage showed optimum levels. Some fibrous connective tissue- and cartilage formation was beneficial to bone healing, but too much of either tissue delayed bone formation. The identified significant parameters and processes are further confirmed by in vivo animal experiments in the literature. This study illustrates the potential of design of experiments methods for evaluating computational mechanobiological model parameters and suggests that further experiments should preferably focus at establishing values of parameters related to cartilage formation and degradation.

Influence of location, fluid flow direction, and tissue maturity on the macroscopic permeability of vertebral end plates

STUDY DESIGN

We implemented a pilot study in a growing animal model. The macroscopic permeability of the vertebral endplates was measured. The influence of location, tissue maturity, and fluid flow direction was quantified.

OBJECTIVE

We hypothesized that the macroscopic permeability of vertebral endplate may decrease with maturity of the vertebral segment.

SUMMARY OF BACKGROUND DATA

The alternation of loading induced by the diurnal cycle generates convective flux into the vertebral segment with the dominant flow path through the vertebral endplates. The alteration of mass transport at the disc-vertebrae interface may interrupt the mechanobiologic balance, and have an effect such as degenerative changes or scoliosis.

METHODS

A previously validated method for measuring permeability, based on the relaxation pressure caused by a transient-flow rate was used. Three specimens were extracted from each L1 to L5 endplate. Seventy-one specimens were frozen, and 64 were stored fresh in a standard culture media. A microscopic analysis completed the biomechanical analysis.

RESULTS

At 2, 4, and 6 months, the mean permeability (10(-14) m/N x s, flow-in/flow-out) of the central zone was respectively: 1.23/1.66, 1.03/1.29, and 0.792/1.00. Laterally, it was 1.03/1.19, 1.094/1.001, and 0.765/0.863. For all groups, cartilage endplate and growth plate were both thinner in the center of the plate. Weak differences of the vascular network were detected, except for a small increase of vascular density in the central zone.

CONCLUSION

The results from this animal study showed that the central zone of the vertebral endplate was more permeable than the periphery and the flow-out permeability was up to 35% greater than the flow-in permeability. Increase of permeability with decrease of cartilage thickness was noticed within the same age group. We also found a statistically significant decrease of the macroscopic permeability correlated with the tissue maturity.

A measurement technique to evaluate the macroscopic permeability of the vertebral end-plate

The remodelling response of vertebral segments in idiopathic scoliosis or disc degeneration, shows a modification of the disc hydration. The investigation of mass transport between the disc and the vertebral body is relevant to understanding the normal and pathological behaviour of the spine. The measurement method we adopted, to derive the macroscopic permeability of the vertebral end-plate, used the relaxation pressure due to a transient-flow rate into the biological structure. Our approach mimicked the in vivo conditions of mass transfer between the disc and the vertebral bodies. It minimized the flow-induced matrix compaction and it allowed the boundary conditions of the specimen to be controlled. We proposed a specific theoretical method and an associated device according to poroelastic theory. A preliminary evaluation with a controlled porous medium, preceded a pilot study in an animal model (lumbar L(4)-L(5) segment of a 4 months old female pig). The macroscopic permeability of multilayered cylindrical specimens including the trabecular bone, subchondral bone and cartilage end-plate was measured, using a 'cartilage-towards-bone' fluid flow direction. A histologic evaluation completed the biomechanical approach. Results showed that the central zone was more permeable than the periphery and this concurred with qualitative studies from the literature.

Poromechanics of compressible charged porous media using the theory of mixtures

Osmotic, electrostatic, and/or hydrational swellings are essential mechanisms in the deformation behavior of porous media, such as biological tissues, synthetic hydrogels, and clay-rich rocks. Present theories are restricted to incompressible constituents. This assumption typically fails for bone, in which electrokinetic effects are closely coupled to deformation. An electrochemomechanical formulation of quasistatic finite deformation of compressible charged porous media is derived from the theory of mixtures. The model consists of a compressible charged porous solid saturated with a compressible ionic solution. Four constituents following different kinematic paths are identified: a charged solid and three streaming constituents carrying either a positive, negative, or no electrical charge, which are the cations, anions, and fluid, respectively. The finite deformation model is reduced to infinitesimal theory. In the limiting case without ionic effects, the presented model is consistent with Blot's theory. Viscous drag compression is computed under closed circuit and open circuit conditions. Viscous drag compression is shown to be independent of the storage modulus. A compressible version of the electrochemomechanical theory is formulated. Using material parameter values for bone, the theory predicts a substantial influence of density changes on a viscous drag compression simulation. In the context of quasistatic deformations, conflicts between poromechanics and mixture theory are only semantic in nature.

The effect of recovery time and test conditions on viscoelastic measures of tensile damage in cortical bone

Stiffness degradation and strength degradation are often measured to monitor and characterize the effects of damage accumulation in bone. Based on evidence that these properties could be affected by not only damage magnitude but also test conditions, the present study investigated the effect of hold condition and recovery time on measures of tensile damage. Machined human femoral cortical bone specimens were subjected to tensile tests consisting of a pre-damage diagnostic loading cycle, a damage loading cycle and post-damage cycle. Controlled variables were recovery time (1, 10, and 100 min) and hold condition (zero load or zero strain) after the damage cycle. Damage measures were calculated as the ratio of each post-damage cycle to the pre-damage value for loading modulus, secant modulus, unloading modulus, stress relaxation and strain (stress) recovery at 1 min post-diagnostic time. The damage cycle caused reductions in all measures, and some measures varied with recovery time and hold condition. Apparent modulus degradation for both hold conditions decreased with recovery time. Stress relaxation was unaffected by recovery time for both hold conditions. Zero-strain hold conditions resulted in lower values for degradation of modulus and change of relaxation. Stress or strain recovery after the damage cycle was evident through 100 min, but 90% of the recovery occurred within 10 min. The results demonstrate that choice of test conditions can influence the apparent magnitude of damage effects. They also indicate that 10 min recovery time was sufficient to stabilize most measures of the damage state.

An estimate of anisotropic poroelastic constants of an osteon

The anisotropic poroelastic constants of an osteon are estimated by micromechanical analysis. Two extreme cases are examined, the drained and the undrained elastic constants. The drained elastic constants are the porous medium's effective elastic constants when the fluid in the pores easily escapes and the pore fluid can sustain no pore pressure. The undrained elastic constants are the porous medium's effective elastic constants when the medium is fully saturated with pore fluid and the fluid cannot escape. The drained and undrained elastic constants at the lacunar and canalicular porosity tissue levels are estimated by using an effective moduli model consisting of the periodic distribution of ellipsoidal cavities. These estimated anisotropic poroelastic constants provide a database for the development of an accurate anisotropic poroelastic model of an osteon.

Predicting fracture of the femoral neck

A prediction of the probability of safe loading of the femoral neck, based on queueing theory, is presented. The following methods have been applied: (I) criterion of bone fracture was formulated, taking into consideration the complex state of stress-strain in the porosity zones of the bone; (II) tensile stresses around pores in the stretched zone of the bone were evaluated; (III) the influence of random events of the critical regimes of loading was modeled. The evaluation of the probability of safe loading of bones was obtained based on the levels of the tensile stresses, Young's moduli and ultimate tensile stresses which are affected by the increase in bone porosity and the distribution of the pores. Examples of analysis involving typical mechanical properties of bone in areas of vascular and lacunar-canalicular porosity are demonstrated. The ranges of initial average values of effective Young's moduli and ultimate tensile strengths were taken as 15.8-17.5GPa and 83-95MPa, respectively. The present analysis discovers the existence of three levels of safe loading: (1) a relatively safe level of the nominal tensile stresses (smaller than (2.8-3.2)MPa) where the probability of safe loading is of the order of 0.95 for the bone porosity which is less than 0.15; (2) an intermediate level of safety where the nominal tensile stresses are below (4.2-4.8)MPa and the probability of safe loading is 0.89 or higher, for the same level of bone porosity; (3) a critical level of safe loading where the nominal tensile stresses are about (8.3-9.5)MPa; they lead to sharp drop of probabilities of safe loading to 0.85-0.8 if the porosity is about 0.10 and to probabilities of 0.41-0.4 if the porosity is about 0.15.

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.