Permeability study of vertebral cancellous bone using micro-computational fluid dynamics

Understanding of cancellous bone permeability is lacking despite its importance in designing tissue engineering scaffolds for bone regeneration and orthopaedic surgery that relies on infiltration of bone cement into porous cancellous bone. We employed micro-computational fluid dynamics to investigate permeability for 37 cancellous bone specimens, eliminating stringent technical requirements of bench-top testing. Microarchitectural parameters were also determined for the specimens and correlated, using uni-variate and multi-variate regression analyses, against permeability. We determined that bone surface density, trabecular pattern factor, structure model index and trabecular number are other possible predictors of permeability (with R values of 0.47, 0.44, 0.40 and 0.33), in addition to the commonly used porosity parameter (R value of 0.38). Pooling these parameters and performing multi-variate linear regression analysis improved yield the R-value of 0.50, indicating that porosity alone is a poor predictor of cancellous bone permeability and, therefore, other parameters should be included for a better and improved linear model.     

Permeability study of vertebral cancellous bone using micro-computational fluid dynamics

Understanding of cancellous bone permeability is lacking despite its importance in designing tissue engineering scaffolds for bone regeneration and orthopaedic surgery that relies on infiltration of bone cement into porous cancellous bone. We employed micro-computational fluid dynamics to investigate permeability for 37 cancellous bone specimens, eliminating stringent technical requirements of bench-top testing. Microarchitectural parameters were also determined for the specimens and correlated, using uni-variate and multi-variate regression analyses, against permeability. We determined that bone surface density, trabecular pattern factor, structure model index and trabecular number are other possible predictors of permeability (with R values of 0.47, 0.44, 0.40 and 0.33), in addition to the commonly used porosity parameter (R value of 0.38). Pooling these parameters and performing multi-variate linear regression analysis improved yield the R-value of 0.50, indicating that porosity alone is a poor predictor of cancellous bone permeability and, therefore, other parameters should be included for a better and improved linear model.     

Experimental and theoretical investigation of directional permeability of human vertebral cancellous bone for cement infiltration

The use of acrylic polymers in infiltrating the porous bone structure is an emerging procedure for the augmentation of osteoporotic vertebrae. Although this procedure is employed frequently, it is performed based on empirical knowledge, and therefore, does not take into consideration the porosity-dependent permeability of human vertebral cancellous bone. The purpose of this study was to: (a). experimentally and theoretically investigate interdependence of the vertebral cancellous bone permeability and porosity, and (b). examine if the bone permeability of spinal cancellous bone can be predicted using bone mineral density measurements. If these relations can be established, they can be useful in optimizing the injection conditions for predicable cement infiltration. To determine the porosity-dependent and directional permeability, 34 bone cores-20 samples in the superior-inferior (SI) direction and 14 in the anterior-posterior (AP) direction-were cut from 20 lumbar vertebrae and infiltrated with silicone oil with a viscosity matching that of PMMA. The permeability of the cores was determined based on Darcy's law. The mean permeability of SI and AP cores was 4.45+/-1.72 x 10(-8) and 3.44+/-1.26 x 10(-8)m(2), respectively. An interesting finding of this study was that the permeability of the AP cores was approximately 78% of that of SI cores, though the porosity of the SI and AP cores taken from the same vertebra was approximately equal. In addition, we provided a theoretical model for the porosity-dependent permeability that accurately described non-linear dependency of the bone permeability and porosity in both directions. Although the relation of the bone permeability and porosity was established, bone mineral density was a weak predictor of the bone permeability. The experimental and theoretical results of this study can be used to understand polymer flow in cement infiltration procedures.     

Permeability analysis of scaffolds for bone tissue engineering

Porous artificial bone substitutes, especially bone scaffolds coupled with osteobiologics, have been developed as an alternative to the traditional bone grafts. The bone scaffold should have a set of properties to provide mechanical support and simultaneously promote tissue regeneration. Among these properties, scaffold permeability is a determinant factor as it plays a major role in the ability for cells to penetrate the porous media and for nutrients to diffuse. Thus, the aim of this work is to characterize the permeability of the scaffold microstructure, using both computational and experimental methods. Computationally, permeability was estimated by homogenization methods applied to the problem of a fluid flow through a porous media. These homogenized permeability properties are compared with those obtained experimentally. For this purpose a simple experimental setup was used to test scaffolds built using Solid Free Form techniques. The obtained results show a linear correlation between the computational and the experimental permeability. Also, this study showed that permeability encompasses the influence of both porosity and pore size on mass transport, thus indicating its importance as a design parameter. This work indicates that the mathematical approach used to determine permeability may be useful as a scaffold design tool.     

Predicting the permeability of trabecular bone by micro-computed tomography and finite element modeling

The intrinsic permeability of bone plays an important role in the transport of nutrients and minerals within the tissue, and affects the mechanical stimuli that are related to the fate of the stem cells. The objective of this study was to establish a method to assess trabecular bone permeability using experimental and finite element (FE) modeling approaches based on micro computed tomography (µCT) images. Human cadaveric tibia cube specimens (N=23) were scanned with µCT. The permeability was measured experimentally using a custom-developed constant-head permeameter, and computationally by a poroelastic formulation to simulate the fluid flow within the discretized bone matrix and pore phase. The average of the experimentally measured permeability was 4.84×10⁻¹⁰ m² with a standard deviation of 3.70×10⁻¹⁰ m². A regression model of the µCT determined that the maximum bone area to total area ratio (maxBA/TA) for all slices that are perpendicular to the direction of fluid flow explained 84% of the variability of the natural logarithm of the experimentally measured permeability. The 2D measure of maxBA/TA performed better than 3D measures in general, although some parameters were reasonably well associated with permeability such as bone volume ratio (BV/TV, r=−0.71), the bone surface/bone volume (BS/BV, r=0.73), and the trabecular thickness (TbTh, r=−0.71). The correlation between the permeability predicted with FE models and experimentally measured permeability was reasonable (r=0.69), but the FE approach did not accurately represent the wide variability of permeability measured experimentally. The results of this study suggest that the changes in the trabecular bone microarchitecture have an exponential relationship with permeability, and the use of µCT-based 2D measurement of maxBA/TA performs well at predicting permeability, thus providing a convenient approach to measure this important aspect affecting biomechanical functions in the tissue.     

Age-related changes in mouse bone permeability

The determination of lacunar-canalicular permeability is essential for understanding local fluid flow in bone, which may indicate how bone senses changes in the mechanical environment to regulate mechano-adaptation. The estimates of lacunar-canalicular permeability found in the literature vary by up to eight orders of magnitude, and age-related permeability changes have not been measured in non-osteonal mouse bone. The objective of this study is to use a poroelastic approach based on nanoindentation data to characterize lacunar-canalicular permeability in murine bone as a function of age. Nine wild type C57BL/6 mice of different ages (2, 7 and 12 months) were used. Three tibiae from each age group were embedded in epoxy resin, cut in half and indented in the longitudinal direction in the mid-cortex using two spherical fluid indenter tips (R=238 μm and 500 μm). Results suggest that the lacunar-canalicular intrinsic permeability of mouse bone decreases from 2 to 7 months, with no significant changes from 7 to 12 months. The large indenter tip imposed larger contact sizes and sampled larger ranges of permeabilities, particularly for the old bone. This age-related difference in the distribution was not seen for indents with the smaller radius tip. We conclude that the small tip effectively measured lacunar-canalicular permeability, while larger tip indents were influenced by vascular permeability. Exploring the age-related changes in permeability of bone measured by nanoindentation will lead to a better understanding of the role of fluid flow in mechano-transduction. This understanding may help indicate alterations in bone adaptation and remodeling.     

Experimental and numerical identification of cortical bone permeability

Bone is a complex system, and could be modeled as a poroelastic media. The aim of this paper is to identify the macroscopic value of the cortical bone permeability coefficient. A simple experimental method was designed in order to determine the permeability coefficient. Two bone samples taken from different ox femurs were filled with water, to place them under internal pressure. The measurements gave both the fluid flow through the lateral surfaces and the internal pressure. The originality of this work is the coupling between an experimental process and a structural computation performed with a finite element method. The mean cortical bone permeability coefficient identified was about k=1.1x10(-13)m(2). This value tends to confirm other values found in the literature, obtained by different methods and often at macroscopic scale. It confirms also the domination of vascular permeability (Haversian and Volkmann's canals).     

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.     

A mixed boundary representation to simulate the displacement of a biofluid by a biomaterial in porous media

Characterization of the biomaterial flow through porous bone is crucial for the success of the bone augmentation process in vertebroplasty. The biofluid, biomaterial, and local morphological bone characteristics determine the final shape of the filling, which is important both for the post-treatment mechanical loading and the risk of intraoperative extraosseous leakage. We have developed a computational model that describes the flow of biomaterials in porous bone structures by considering the material porosity, the region-dependent intrinsic permeability of the porous structure, the rheological properties of the biomaterial, and the boundary conditions of the filling process. To simulate the process of the substitution of a biofluid (bone marrow) by a biomaterial (bone cement), we developed a hybrid formulation to describe the evolution of the fluid boundary and properties and coupled it to a modified version of Darcy's law. The apparent rheological properties are derived from a fluid-fluid interface tracking algorithm and a mixed boundary representation. The region- specific intrinsic permeability of the bone is governed by an empirical relationship resulting from a fitting process of experimental data. In a first step, we verified the model by studying the displacement process in spherical domains, where the spreading pattern is known in advance. The mixed boundary model demonstrated, as expected, that the determinants of the spreading pattern are the local intrinsic permeability of the porous matrix and the ratio of the viscosity of the fluids that are contributing to the displacement process. The simulations also illustrate the sensitivity of the mixed boundary representation to anisotropic permeability, which is related to the directional dependent microstructural properties of the porous medium. Furthermore, we compared the nonlinear finite element model to different published experimental studies and found a moderate to good agreement (R(2)=0.9895 for a one-dimensional bone core infiltration test and a 10.94-16.92% relative error for a three-dimensional spreading pattern study, respectively) between computational and experimental results.     

Estimation of bone permeability using accurate microstructural measurements

While interstitial fluid flow is necessary for the viability of osteocytes, it is also believed to play a role in bone's mechanosensory system by shearing bone cell membranes or causing cytoskeleton deformation and thus activating biochemical responses that lead to the process of bone adaptation. However, the fluid flow properties that regulate bone's adaptive response are poorly understood. In this paper, we present an analytical approach to determine the degree of anisotropy of the permeability of the lacunar-canalicular porosity in bone. First, we estimate the total number of canaliculi emanating from each osteocyte lacuna based on published measurements from parallel-fibered shaft bones of several species (chick, rabbit, bovine, horse, dog, and human). Next, we determine the local three-dimensional permeability of the lacunar-canalicular porosity for these species using recent microstructural measurements and adapting a previously developed model. Results demonstrated that the number of canaliculi per osteocyte lacuna ranged from 41 for human to 115 for horse. Permeability coefficients were found to be different in three local principal directions, indicating local orthotropic symmetry of bone permeability in parallel-fibered cortical bone for all species examined. For the range of parameters investigated, the local lacunar-canalicular permeability varied more than three orders of magnitude, with the osteocyte lacunar shape and size along with the 3-D canalicular distribution determining the degree of anisotropy of the local permeability. This two-step theoretical approach to determine the degree of anisotropy of the permeability of the lacunar-canalicular porosity will be useful for accurate quantification of interstitial fluid movement in bone.     

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.     

Finite element modelling of polymethylmethacrylate flow through cancellous bone

A mathematical model based on the Finite Element Method is developed to simulate the non-linear flow of acrylic bone cement through cancellous bone. The cancellous bone bed is modelled as a bed of parallel capillaries filled with equal spaced toroidal trabeculae. By manipulating the relative size of the torus and the capillary, the flow within bone of varying porosity is simulated. An apparent permeability based on the volume weighted average viscosity and Darcy's law is developed to describe the flow of the acrylic through the cancellous bone bed. The model predicts a cancellous bone permeability of 5.6 x 10(-9)-8.3 x 10(-9) m2 for linear flow. The non-linear behavior of the acrylic cement results in an increase of apparent permeability when compared to the permeability computed for linear flow. Estimates of penetration are achieved by running the model in a quasi-steady state fashion with pressure applied over a fixed time increment. Close agreement is shown between model predictions of penetration depth and experimental results available in the literature.     

Poroelastic analysis of interstitial fluid flow in a single lamellar trabecula subjected to cyclic loading

Trabecula, an anatomical unit of the cancellous bone, is a porous material that consists of a lamellar bone matrix and interstitial fluid in a lacuno-canalicular porosity. The flow of interstitial fluid caused by deformation of the bone matrix is believed to initiate a mechanical response in osteocytes for bone remodeling. In order to clarify the effect of the lamellar structure of the bone matrix—i.e., variations in material properties—on the fluid flow stimuli to osteocytes embedded in trabeculae, we investigated the mechanical behavior of an individual trabecula subjected to cyclic loading based on poroelasticity. We focused on variations in the trabecular permeability and developed an analytical solution containing both transient and steady-state responses for interstitial fluid pressure in a single trabecular model represented by a multilayered two-dimensional poroelastic slab. Based on the obtained solution, we calculated the pressure and seepage velocity of the interstitial fluid in lacuno-canalicular porosity, within the single trabecula, under various permeability distributions. Poroelastic analysis showed that a heterogeneous distribution of permeability produces remarkable variations in the fluid pressure and seepage velocity in the cross section of the individual trabecula, and suggests that fluid flow stimuli to osteocytes are mostly governed by the value of permeability in the neighborhood of the trabecular surfaces if there is no difference in the average permeability in a single trabecula.     

HISTOCHEMICAL DEMONSTRATION OF DEHYDROGENASE ACTIVITY IN THE CELLS OF NORMAL HUMAN BLOOD AND BONE MARROW

Endogenous and succinic dehydrogenase activity was demonstrated in the living cells of normal human blood and bone marrow using a buffered nitro BT-succinate incubating solution. With this technique dehydrogenase activity was localized primarily in the granular leukocytes and the sites of enzymatic activity appeared to be non-mitochondrial. The addition of a non-ionic surface active agent to the incubating solution resulted in marked differences in the cellular and intracellular localization of dehydrogenase activity. With this method it was possible to demonstrate dehydrogenase activity in the mitochondria of most of the formed elements of the blood and bone marrow, including developing granulocytes and erythroid cells, agranulocytes, and blood platelets. Mature erythrocytes also exhibited a minimal dehydrogenase reaction with this procedure. This investigation indicated that in order adequately to demonstrate and evaluate dehydrogenase activity in the cells of the blood and bone marrow it was necessary to have increased cellular and mitochondrial permeability, as well as partially viable cells with an intact dehydrogenase system.     

Probing the tissue to subcellular level structure underlying bone's molecular sieving function

Transport of fluorescent probes between 300 and 2,000,000 Da was studied in mechanically loaded and unloaded ulnae of skeletally mature rats to characterize the permeability of the pericellular space of the lacunocanalicular system (LCS), and the microporosity of the bony matrix. The mineral matrix porosity allowed for penetration of the 300 Da probe but impeded transport of larger probes. The pericellular space of the LCS was permeable up to 10 kDa; above 10 kDa, diffusion was ineffective for transport through the pericellular space. Convective transport via load-induced fluid flow increased penetration of all probes up to 70 kDa. Above this threshold, probes were excluded from bone, both with and without loading. This exploratory study suggests that bone acts as a molecular sieve and that mechanical loading modulates transport of solutes through the pericellular space that links osteocytes deep within the tissue to the blood supply and to osteoblasts and osteoclasts on bone forming and resorbing surfaces. This provides support for the postulate of transport modulated bone remodeling in which osteocytes are influenced by and modulate the local permeability of their surroundings as a means for survival (Knothe Tate et al. 1998, [28]) and has profound implications for osteocyte viability and intercellular communication in bone.     

Multiscale modeling of bone tissue with surface and permeability control

Natural biological materials usually present a hierarchical arrangement with various structural levels. The biomechanical behavior of the complex hierarchical structure of bone is investigated with models that address the various levels corresponding to different scales. Models that simulate the bone remodeling process concurrently at different scales are in development. We present a multiscale model for bone tissue adaptation that considers the two top levels, whole bone and trabecular architecture. The bone density distribution is calculated at the macroscale (whole bone) level, and the trabecular structure at the microscale level takes into account its mechanical properties as well as surface density and permeability. The bone remodeling process is thus formulated as a material distribution problem at both scales. At the local level, the biologically driven information of surface density and permeability characterizes the trabecular structure. The model is tested by a three-dimensional simulation of bone tissue adaptation for the human femur. The density distribution of the model shows good agreement with the actual bone density distribution. Permeability at the microstructural level assures interconnectivity of pores, which mimics the interconnectivity of trabecular bone essential for vascularization and transport of nutrients. The importance of this multiscale model relays on the flexibility to control the morphometric parameters that characterize the trabecular structure. Therefore, the presented model can be a valuable tool to define bone quality, to assist with diagnosis of osteoporosis, and to support the development of bone substitutes.     

A note on the units of membrane permeability to water

In the usual experimental technique, the membrane permeability to water in the units of cm/sec is not measured directly. The quantity which is usually obtained may be considered to be a measure of a coefficient of penetrability having the dimensions time/length. The purpose of this paper is to show that by introducing the bulk modulus of water, an expression is obtained from which it is possible to calculate the water permeability in the units of cm/sec from the values in conventional units.     

Interstitial fluid flow in the osteon with spatial gradients of mechanical properties: a finite element study

Bone remodelling is the process that maintains bone structure and strength through adaptation of bone tissue mechanical properties to applied loads. Bone can be modelled as a porous deformable material whose pores are filled with cells, organic material and interstitial fluid. Fluid flow is believed to play a role in the mechanotransduction of signals for bone remodelling. In this work, an osteon, the elementary unit of cortical bone, is idealized as a hollow cylinder made of a deformable porous matrix saturated with an interstitial fluid. We use Biot’s poroelasticity theory to model the mechanical behaviour of bone tissue taking into account transverse isotropic mechanical properties. A finite element poroelastic model is developed in the COMSOL Multiphysics software. Elasticity equations and Darcy’s law are implemented in this software; they are coupled through the introduction of an interaction term to obtain poroelasticity equations. Using numerical simulations, the investigation of the effect of spatial gradients of permeability or Poisson’s ratio is performed. Results are discussed for their implication on fluid flow in osteons: (i) a permeability gradient affects more the fluid pressure than the velocity profile; (ii) focusing on the fluid flow, the key element of loading is the strain rate; (iii) a Poisson’s ratio gradient affects both fluid pressure and fluid velocity. The influence of textural and mechanical properties of bone on mechanotransduction signals for bone remodelling is also discussed.     

Basophilic line of the articular cartilage in normal and various pathological states

Epiphyses of long tubular bones in the man and animals of various age, as well as experimental material of the adjuvant arthritis, with special reference to the basal part of the articular cartilage have been studied by means of histological, histochemical and histometrical methods. The structural-chemical organization of the basophilic line (tidemark) of the articular cartilage ensures its barrier role and participation in regulating selective permeability. Reconstruction of the tidemark in the process of physiological ageing and in cases of the articular pathology is aimed to preserve its integrity and in this way a complete differentiation of the noncalcified and calcified structures is secured. Disturbance of the basophilic line results in changes of the articular selective permeability, in invasion of vessels and structural elements of the bone marrow, and in development of profound distrophic and destructive changes of the cartilage--in deforming artrosis. Deflations in the structural-chemical organization of the tidemark indicate certain disturbances in the state of the system articular cartilage--subchondral bone. These data can be of prognostic importance.     

Analysis of the source of heterogeneity in the osmotic response of plant membrane vesicles

Plasma membrane vesicles have been widely employed to understand the biophysics of water movements, especially when active aquaporins are present. In general, water permeability coefficients in these preparations outcome from the analysis of the osmotic response of the vesicles by means of light scattering. As from now, this is possible by following a theoretical approach that assumes that scattered light follows a single exponential function and that this behavior is the consequence of vesicle volume changes due to an osmotic challenge. However, some experimental data do not necessarily fit to single exponentials but to double ones. It is argued that the observed double exponential behavior has two possible causes: different vesicle population in terms of permeability or in terms of size distribution. As classical models cannot identify this source of heterogeneity, a mathematical modeling approach was developed based on phenomenological equations of water transport. In the three comparative models presented here, it was assumed that water moves according to an osmotic mechanism across the vesicles, and there is no solute movement across them. Interestingly, when tested in a well described plasma membrane vesicle preparation, the application of these models indicates that the source of heterogeneity in the osmotic response is vesicles having different permeability, clearly discarding the variable size effect. In conclusion, the mathematical approach presented here allows to identify the source of heterogeneity; this information being of particular interest, especially when studying gating mechanisms triggered in water channel activity.