Blood and interstitial flow in the hierarchical pore space architecture of bone tissue.

There are two main types of fluid in bone tissue, blood and interstitial fluid. The chemical composition of these fluids varies with time and location in bone. Blood arrives through the arterial system containing oxygen and other nutrients and the blood components depart via the venous system containing less oxygen and reduced nutrition. Within the bone, as within other tissues, substances pass from the blood through the arterial walls into the interstitial fluid. The movement of the interstitial fluid carries these substances to the cells within the bone and, at the same time, carries off the waste materials from the cells. Bone tissue would not live without these fluid movements. The development of a model for poroelastic materials with hierarchical pore space architecture for the description of blood flow and interstitial fluid flow in living bone tissue is reviewed. The model is applied to the problem of determining the exchange of pore fluid between the vascular porosity and the lacunar-canalicular porosity in bone tissue due to cyclic mechanical loading and blood pressure. These results are 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.

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.

Microarchitecture and bone quality in the human calcaneus: local variations of fabric anisotropy.

The local variability of microarchitecture of human trabecular calcaneus bone is investigated using high-resolution micro-computed tomography (µCT) scanning. The fabric tensor is employed as the measure of the microarchitecture of the pore structure of a porous medium. It is hypothesized that a fabric tensor-dependent poroelastic ultrasound approach will more effectively predict the data variance than will porosity alone. The specific aims of the present study are as follows: (1) to quantify the morphology and local anisotropy of the calcaneus microarchitecture with respect to anatomical directions; (2) to determine the interdependence, or lack thereof, of microarchitecture parameters, fabric, and volumetric bone mineral density (vBMD); and (3) to determine the relative ability of vBMD and fabric measurements in evaluating the variance in ultrasound wave velocity measurements along orthogonal directions in the human calcaneus. Our results show that the microarchitecture in the analyzed regions of human calcanei is anisotropic, with a preferred alignment along the posterior-anterior direction. Strong correlation was found between most scalar architectural parameters and vBMD. However, no statistical correlation was found between vBMD and the fabric components, the measures of the pore microstructure orientation. Therefore, among the parameters usually considered for cancellous bone (ie, classic histomorphometric parameters such as porosity, trabecular thickness, number and separation), only fabric components explain the data variance that cannot be explained by vBMD, a global mass measurement, which lacks the sensitivity and selectivity to distinguish osteoporotic from healthy subjects because it is insensitive to directional changes in bone architecture. This study demonstrates that a multidirectional, fabric-dependent poroelastic ultrasound approach has the capability of characterizing anisotropic bone properties (bone quality) beyond bone mass, and could help to better understand anisotropic changes in bone architecture using ultrasound.

Quantitative model for predicting lymph formation and muscle compressibility in skeletal muscle during contraction and stretch.

Skeletal muscle is widely perceived as nearly incompressible despite the fact that blood and lymphatic vessels within the endomysial and perimysial spaces undergo significant changes in diameter and length during stretch and contraction. These fluid shifts between fascicle and interstitial compartments have proved extremely difficult to measure. In this paper, we propose a theoretical framework based on a space-filling hexagonal fascicle array to provide predictions of the displacement of blood and lymph into and out of the muscle's endomysium and perimysium during stretch and contraction. We also use this model to quantify the distribution of blood and initial lymphatic (IL) vessels within a fascicle and its perimysial space using data for the rat spinotrapezius muscle. On average, there are 11 muscle fibers, 0.4 arteriole/venule pairs, and 0.2 IL vessels per fascicle. The model predicts that the blood volume in the endomysial space increases 24% and decreases 22% for a 20% contraction and stretch, respectively. However, these significant changes in blood volume in the endomysium produce a change of only ∼2% in fascicle cross-sectional area. In contrast, the entire muscle deviates from isovolumetry by 7% and 6% for a 20% contraction and stretch, respectively, largely attributable to the significantly larger blood volume changes that occur in the perimysial space. This suggests that arcade blood vessels in the perimysial space provide the primary pumping action required for the filling and emptying of ILs during muscular contraction and stretch.

Role of structural anisotropy of biological tissues in poroelastic wave propagation.

Ultrasound waves have a broad range of clinical applications as a non-destructive testing approach in imaging and in the diagnoses of medical conditions. Generally, biological tissues are modeled as an homogenized equivalent medium with an apparent density through which a single wave propagates. Only the first wave arriving at the ultrasound probe is used for the measurement of the speed of sound. However, the existence of a second wave in tissues such as cancellous bone has been reported and its existence is an unequivocal signature of Biot type poroelastic media. To account for the fact that ultrasound is sensitive to microarchitecture as well as density, a fabric-dependent anisotropic poroelastic ultrasound (PEU) propagation theory was recently developed. Key to this development was the inclusion of the fabric tensor - a quantitative stereological measure of the degree of structural anisotropy of bone - into the linear poroelasticity theory. In the present study, this framework is extended to the propagation of waves in several soft and hard tissues. It was found that collagen fibers in soft tissues and the mineralized matrix in hard tissues are responsible for the anisotropy of the solid tissue constituent through the fabric tensor in the model.

Mixture theory-based poroelasticity as a model of interstitial tissue growth.

This contribution presents an alternative approach to mixture theory-based poroelasticity by transferring some poroelastic concepts developed by Maurice Biot to mixture theory. These concepts are a larger RVE and the subRVE-RVE velocity average tensor, which Biot called the micro-macro velocity average tensor. This velocity average tensor is assumed here to depend upon the pore structure fabric. The formulation of mixture theory presented is directed toward the modeling of interstitial growth, that is to say changing mass and changing density of an organism. Traditional mixture theory considers constituents to be open systems, but the entire mixture is a closed system. In this development the mixture is also considered to be an open system as an alternative method of modeling growth. Growth is slow and accelerations are neglected in the applications. The velocity of a solid constituent is employed as the main reference velocity in preference to the mean velocity concept from the original formulation of mixture theory. The standard development of statements of the conservation principles and entropy inequality employed in mixture theory are modified to account for these kinematic changes and to allow for supplies of mass, momentum and energy to each constituent and to the mixture as a whole. The objective is to establish a basis for the development of constitutive equations for growth of tissues.

A symmetry invariant formulation of the relationship between the elasticity tensor and the fabric tensor.

The fabric tensor is employed as a quantitative stereological measure of the structural anisotropy in the pore architecture of a porous medium. Earlier work showed that the fabric tensor can be used additionally to the porosity to describe the anisotropy in the elastic constants of the porous medium. This contribution presents a reformulation of the relationship between fabric tensor and anisotropic elastic constants that is approximation free and symmetry-invariant. From specific data on the elastic constants and the fabric, the parameters in the reformulated relationship can be evaluated individually and efficiently using a simplified method that works independent of the material symmetry. The well-behavedness of the parameters and the accuracy of the method was analyzed using the Mori-Tanaka model for aligned ellipsoidal inclusions and using Buckminster Fuller's octet-truss lattice. Application of the method to a cancellous bone data set revealed that employing the fabric tensor allowed explaining 75-90% of the total variance. An implementation of the proposed methods was made publicly available.

Fabric dependence of quasi-waves in anisotropic porous media.

Assessment of bone loss and osteoporosis by ultrasound systems is based on the speed of sound and broadband ultrasound attenuation of a single wave. However, the existence of a second wave in cancellous bone has been reported and its existence is an unequivocal signature of poroelastic media. To account for the fact that ultrasound is sensitive to microarchitecture as well as bone mineral density (BMD), a fabric-dependent anisotropic poroelastic wave propagation theory was recently developed for pure wave modes propagating along a plane of symmetry in an anisotropic medium. Key to this development was the inclusion of the fabric tensor--a quantitative stereological measure of the degree of structural anisotropy of bone--into the linear poroelasticity theory. In the present study, this framework is extended to the propagation of mixed wave modes along an arbitrary direction in anisotropic porous media called quasi-waves. It was found that differences between phase and group velocities are due to the anisotropy of the bone microarchitecture, and that the experimental wave velocities are more accurately predicted by the poroelastic model when the fabric tensor variable is taken into account. This poroelastic wave propagation theory represents an alternative for bone quality assessment beyond BMD.

The specific growth rates of tissues: a review and a re-evaluation.

The first objective of this review and re-evaluation is to present a brief history of efforts to mathematically model the growth of tissues. The second objective is to place this historical material in a current perspective where it may be of help in future research. The overall objective is to look backward in order to see ways forward. It is noted that two distinct methods of imaging or modeling the growth of an organism were inspired over 70 years ago by Thompson's (1915, "XXVII Morphology and Mathematics," Trans. - R. Soc. Edinbrgh, 50, pp. 857-895; 1942, On Growth and Form, Cambridge University Press, Cambridge, UK) method of coordinate transformations to study the growth and form of organisms. One is based on the solid mechanics concept of the deformation of an object, and the other is based on the fluid mechanics concept of the velocity field of a fluid. The solid mechanics model is called the distributed continuous growth (DCG) model by Skalak (1981, "Growth as a Finite Displacement Field," Proceedings of the IUTAM Symposium on Finite Elasticity, D. E. Carlson and R. T. Shield, eds., Nijhoff, The Hague, pp. 348-355) and Skalak et al. (1982, "Analytical Description of Growth," J. Theor. Biol., 94, pp. 555-577), and the fluid mechanics model is called the graphical growth velocity field representation (GVFR) by Cowin (2010, "Continuum Kinematical Modeling of Mass Increasing Biological Growth," Int. J. Eng. Sci., 48, pp. 1137-1145). The GVFR is a minimum or simple model based only on the assumption that a velocity field may be used effectively to illustrate experimental results concerning the temporal evolution of the size and shape of the organism that reveals the centers of growth and growth gradients first described by Huxley (1924, "Constant Differential Growth-Ratios and Their Significance," Nature (London), 114, pp. 895-896; 1972, Problems of Relative Growth, 2nd ed., L. MacVeagh, ed., Dover, New York). It is the method with an independent future that some earlier writers considered as an aspect of the DCG model. The development of the DCG hypothesis and the mixture theory models into models for the predicted growth of an organism is taking longer because these models are complicated and the development and refinement of the basic concepts are slower.

Fabric dependence of wave propagation in anisotropic porous media.

Current diagnosis of bone loss and osteoporosis is based on the measurement of the bone mineral density (BMD) or the apparent mass density. Unfortunately, in most clinical ultrasound densitometers: 1) measurements are often performed in a single anatomical direction, 2) only the first wave arriving to the ultrasound probe is characterized, and 3) the analysis of bone status is based on empirical relationships between measurable quantities such as speed of sound (SOS) and broadband ultrasound attenuation (BUA) and the density of the porous medium. However, the existence of a second wave in cancellous bone has been reported, which is an unequivocal signature of poroelastic media, as predicted by Biot's poroelastic wave propagation theory. In this paper, the governing equations for wave motion in the linear theory of anisotropic poroelastic materials are developed and extended to include the dependence of the constitutive relations upon fabric-a quantitative stereological measure of the degree of structural anisotropy in the pore architecture of a porous medium. This fabric-dependent anisotropic poroelastic approach is a theoretical framework to describe the microarchitectural-dependent relationship between measurable wave properties and the elastic constants of trabecular bone, and thus represents an alternative for bone quality assessment beyond BMD alone.

Fabric dependence of bone ultrasound.

Current diagnosis of bone loss and osteoporosis is based on the measurement of the Bone Mineral Density (BMD) or the apparent mass density. Unfortunately, in most clinical ultrasound densitometers: 1) measurements are often performed in a single anatomical direction, 2) only the first wave arriving to the ultrasound probe is characterized, and 3) the analysis of bone status is based on empirical relationships between measurable quantities such as Speed of Sound (SOS) and Broadband Ultrasound Attenuation (BUA) and the density of the porous medium. However, the existence of a second wave in cancellous bone has been reported, which is an unequivocal signature of poroelastic media, as predicted by Biot's poroelastic wave propagation theory. A fabric-dependent anisotropic poroelastic approach is empolyed as a theoretical framework to describe the microarchitectural-dependent relationship between measurable wave properties and the elastic constants of trabecular bone, and thus represents an alternative for bone quality assessment beyond BMD alone.

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).

The estimated elastic constants for a single bone osteonal lamella.

Micromechanical estimates of the elastic constants for a single bone osteonal lamella and its substructures are reported. These estimates of elastic constants are accomplished at three distinct and organized hierarchical levels, that of a mineralized collagen fibril, a collagen fiber, and a single lamella. The smallest collagen structure is the collagen fibril whose diameter is the order of 20 nm. The next structural level is the collagen fiber with a diameter of the order of 80 nm. A lamella is a laminate structure, composed of multiple collagen fibers with embedded minerals and consists of several laminates. The thickness of one laminate in the lamella is approximately 130 nm. All collagen fibers in a laminate in the lamella are oriented in one direction. However, the laminates rotate relative to the adjacent laminates. In this work, all collagen fibers in a lamella are assumed to be aligned in the longitudinal direction. This kind of bone with all collagen fibers aligned in one direction is called a parallel fibered bone. The effective elastic constants for a parallel fibered bone are estimated by assuming periodic substructures. These results provide a database for estimating the anisotropic poroelastic constants of an osteon and also provide a database for building mathematical or computational models in bone micromechanics, such as bone damage mechanics and bone poroelasticity.

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.

The significance of bone microstructure in mechanotransduction.

Recent developments in modeling the relationship between bone microstructure and mechanotransduction are reviewed. The focus is on the relationship between the bone microstructure and the mechanosensation mechanism by which osteocytes sense the bone fluid motion propelled by the mechanical loading of the whole bone.

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.

Analysis of avian bone response to mechanical loading. Part two: Development of a computational connected cellular network to study bone intercellular communication.

Mechanical loading-induced signals are hypothesized to be transmitted and integrated by connected bone cells before reaching the bone surfaces where adaptation occurs. A computational connected cellular network (CCCN) model is developed to explore how bone cells perceive and transmit the signals through intercellular communication. This is part two of a two-part study in which a CCCN is developed to study the intercellular communication within a grid of bone cells. The excitation signal was computed as the loading-induced bone fluid shear stress in part one. Experimentally determined bone adaptation responses (Gross et al. in J Bone Miner Res 12:982-988, 1997 and Judex et al. in J Bone Miner Res 12:1737-1745, 1997) are correlated with the fluid shear stress by the CCCN, which adjusts cell sensitivities (loading and signal thresholds) and connection weights. Intercellular communication patterns extracted by the CCCN indicate the cell population responsible for perceiving the loading-induced signal, and loading threshold is shown to play an important role in regulating the bone response.

Analysis of avian bone response to mechanical loading-Part one: Distribution of bone fluid shear stress induced by bending and axial loading.

Mechanical loading-induced signals are hypothesized to be transmitted and integrated by a bone-connected cellular network (CCN) before reaching the bone surfaces where adaptation occurs. Our objective is to establish a computational model to explore how bone cells transmit the signals through intercellular communication. In this first part of the study the bone fluid shear stress acting on every bone cell in a CCN is acquired as the excitation signal for the computational model. Bending and axial loading-induced fluid shear stress is computed in transverse sections of avian long bones for two adaptation experiments (Gross et al. in J Bone Miner Res 12:982-988, 1997 and Judex et al. in J Bone Miner Res 12:1737-1745, 1997). The computed fluid shear stress is found to be correlated with the radial strain gradient but not with bone formation. These results suggest that the radial strain gradient is the driving force for bone fluid flow in the radially distributed lacunar-canalicular system and that bone formation is not linearly related to the loading-induced local stimulus.

Tensile and compressive stress yield criteria for cancellous bone.

Experimentally based isotropic tensile and compressive strain yield criteria for cancellous bone are shown to imply tensile and compressive stress yield criteria, respectively. In particular, it is deduced that the directional variation of uniaxial yield stress is proportional to the one of Young's modulus. The obtained tensile and compressive stress yield criteria, in turn, imply information about the total stress yield criterion for cancellous bone.