Tortuosity and the Averaging of Microvelocity Fields in Poroelasticity.

The relationship between the macro- and microvelocity fields in a poroelastic representative volume element (RVE) has not being fully investigated. This relationship is considered to be a function of the tortuosity: a quantitative measure of the effect of the deviation of the pore fluid streamlines from straight (not tortuous) paths in fluid-saturated porous media. There are different expressions for tortuosity based on the deviation from straight pores, harmonic wave excitation, or from a kinetic energy loss analysis. The objective of the work presented is to determine the best expression for tortuosity of a multiply interconnected open pore architecture in an anisotropic porous media. The procedures for averaging the pore microvelocity over the RVE of poroelastic media by Coussy and by Biot were reviewed as part of this study, and the significant connection between these two procedures was established. Success was achieved in identifying the Coussy kinetic energy loss in the pore fluid approach as the most attractive expression for the tortuosity of porous media based on pore fluid viscosity, porosity, and the pore architecture. The fabric tensor, a 3D measure of the architecture of pore structure, was introduced in the expression of the tortuosity tensor for anisotropic porous media. Practical considerations for the measurement of the key parameters in the models of Coussy and Biot are discussed. In this study, we used cancellous bone as an example of interconnected pores and as a motivator for this study, but the results achieved are much more general and have a far broader application than just to cancellous bone.

Perspectives on biological growth and remodeling.

The continuum mechanical treatment of biological growth and remodeling has attracted considerable attention over the past fifteen years. Many aspects of these problems are now well-understood, yet there remain areas in need of significant development from the standpoint of experiments, theory, and computation. In this perspective paper we review the state of the field and highlight open questions, challenges, and avenues for further development.

On mechanosensation in bone under microgravity.

Recent research has illuminated the biological response of bone to mechanical loading at the cellular level, but the precise mechanosensory system that signals bone cells to deposit or resorb tissue has not been identified. The purpose of this paper is to describe the current status of this research and to suggest some possible mechanosensory systems by which bone cells might sense environmental loads. The question of whether the mechanosensory system in bone tissue is at the level of the cell or whether it is at the tissue level and involving the cells is considered here. More precisely, the following question is addressed: can an osteocyte or an osteoblast read the gravitational field changes directly (and independent of changes in its environment), or does it detect those changes indirectly from its environment by contact stresses as it must detect other changes in mechanical loading on the surface of the earth? Our strategies for coping with the decay of the musculoskeletal system in long term space flight are somewhat dependent upon the answer to this question.

Dynamic relationships of trabecular bone density, architecture, and strength in a computational model of osteopenia.

A computational model was developed to study the effects of short- and long-term periods of disuse osteopenia and repair to elucidate the interrelationships between bone mass, architecture, and strength. The model is one in which the sequence of structural change events is followed in time. This temporal feature contrasts with studies of real trabecular tissue which are necessarily cross-sectional in nature and do not lend themselves to insights into the dynamic nature of the structural changes with time. In the model it was assumed that the stimulus for bone adaptation to mechanical load is the local mechanical strain rate, according to which the trabecular surfaces are differentially formed and resorbed. The effects of mechanical loading and unloading (disuse) on the cancelous bone properties were studied. The bone mass, architecture, and elastic stiffness were shown to be strongly dependent upon the period of the unloading phase, as well as the period of the reloading phase. Mechanical stiffness is demonstrated computationally to be a multivalued function of bone mass, if architecture is not accounted for. The model shows how the same value of trabecular bone mass can be associated with two or more distinct values of biomechanical stiffness. This result is the first explicit demonstration of how bone mass, architecture, and strength are related under dynamical load-bearing conditions. The results explain the empirical observation that bone mass can account for about 65% of the observed variation in bone strength, but that by incorporating measures of bony architecture into the analysis, the predictability is increased to 94%. The computational model may be used to explore the effects of different loading regimes on mass, architecture, and strength, and potentially for assistance in designing both animal and clinical bone loss studies.

Bone remodeling of diaphyseal surfaces by torsional loads: theoretical predictions.

A theory of surface bone remodeling is extended to include the effects of shearing strains as well as normal strains. It is shown that the surface velocity can only depend upon the square of shearing strains, but that it can be linear as well as quadratic in the normal strains. The theory is applied to predict the surface bone remodeling in the diaphysis of a long bone under combined axial and torsional loading. In the general case the diaphysis of the long bone is modeled as a hollow thin-walled cylinder of arbitrary cross-section and, in a special case, as a right circular thin-walled tube. It is shown here that if a thin-walled right circular cylinder capable of surface remodeling is subjected to an axial compressive load and a twisting torque, then the effect of increasing the torque is the same as the effect of decreasing the axial compressive load, namely the mean radius of the cross-section increases and the wall width thins. Conversely, the effect of reducing the torque is the same as the effect of increasing the axial compressive load, namely the mean radius of the cross-section decreases and the wall width thickens.

On the minimization and maximization of the strain energy density in cortical bone tissue.

Universal minimization and maximization of the strain energy density, while possible in materials with cubic symmetry, is not possible for cortical long bone with its orthotropic material symmetry. Illustrating this point, it is shown that the stress state obtained when an axial load is applied along the long axis of a long bone at the midshaft is a minimizer of the strain energy density, while the stress state obtained when a load is applied perpendicular to the long axis, and perpendicular to the surface, of the mid-diaphysis of a long bone is a maximizer of the strain energy density. Thus, the bone tissue at the midshaft of a long bone is designed by nature so that it has the greatest stiffness in the direction of its long axis and its greatest impact loading resistance in the transverse direction.

Strain or deformation rate dependent finite growth in soft tissues.

The particular portion of the mechanical loading history of a tissue that serves as stimulus for growth or remodeling of that tissue is presently unknown. A kinematic basis for the implementation of strain or rate of deformation as a growth stimulus is presented. It is shown here that a recently proposed continuum theory for finite volumetric growth in soft tissue may be extended to include strain and rate of deformation as growth stimuli. The basis of the presentation is the recognition of two different time scales in the remodeling process, one on the order of seconds associated with the loading, and one on the order of weeks or months associated with the growth.

Deviatoric and hydrostatic mode interaction in hard and soft tissue.

It has been established that many hard and soft tissues have anisotropic material symmetry. It is noted here that the deviatoric and hydrostatic modes interact with each other in a general anisotropic elastic material. In the special case of isotropic, linear elastic, materials these modes are non-interactive. As a consequence of the interaction of these modes encountered in anisotropic materials, the decomposition into hydrostatic and deviatoric modes, and deviatoric mode concepts such as the von Mises effective stress are not appropriate for anisotropic materials in general. The implications of this observation for the presentation of computationally generated stress contours for hard and soft tissues are discussed. It is also pointed out that the mode coupling and mode interaction raise the question of whether anisotropic living tissues respond directly to stress or to some other physical quantity such as strain or strain energy, in view of the recent hypothesis concerning the proliferation and ossification of cartilage.

Imposing thermodynamic restrictions on the elastic constant-fabric tensor relationship.

Zysset and Curnier (1995) rederive the general representation for the fourth rank elasticity tensor in terms of the fabric tensor given originally by Cowin (1985). Zysset and Curnier (1995) then obtain a form of the representation that is sufficiently restricted to satisfy the thermodynamic constraint that the compliance tensor be positive definite. Zysset and Curnier (1995, 1996) argue that their representation has an advantage over that of Cowin (1985) because it has, a priori, been sufficiently restricted to satisfy the thermodynamic constraint that the compliance tensor be positive definite. The purpose of this note is to point out that the limitations of the Zysset and Curnier (1995, 1996) sufficiency representation are quite severe, restricting the flexibility of the model by allowing only five of the nine normally independent orthotropic elastic coefficients to be independent. Potential users should consider employing the Cowin (1985) representation and subsequently imposing the condition that the elasticity tensor be positive definite.

Bone poroelasticity.

Poroelasticity is a well-developed theory for the interaction of fluid and solid phases of a fluid-saturated porous medium. It is widely used in geomechanics and has been applied to bone by many authors in the last 30 years. The purpose of this work is, first, to review the literature related to the application of poroelasticity to the interstitial bone fluid and, second, to describe the specific physical and modeling considerations that establish poroelasticity as an effective and useful model for deformation-driven bone fluid movement in bone tissue. The application of poroelasticity to bone differs from its application to soft tissues in two important ways. First, the deformations of bone are small while those of soft tissues are generally large. Second, the bulk modulus of the mineralized bone matrix is about six times stiffer than that of the fluid in the pores while the bulk moduli of the soft tissue matrix and the pore water are almost the same. Poroelasticity and electrokinetics can be used to explain strain-generated potentials in wet bone. It is noted that strain-generated potentials can be used as an effective tool in the experimental study of local bone fluid flow, and that the knowledge of this technique will contribute to the answers of a number of questions concerning bone mineralization, osteocyte nutrition and the bone mechanosensory system.

Periosteal and endosteal control of bone remodeling under torsional loading.

The shape changes that occur in the mid-diaphysis of a long bone due to adaptive remodeling induced by increasing or decreasing the axial and/or torsional loading of the bone are investigated using a simple model. In this model the mid-diaphysis of a long bone is represented as a hollow thick-walled right-circular cylinder, and different optimal strategies for bone remodeling are considered. It is shown that if such a thick-walled right-circular cylinder capable of surface remodeling is subjected to an axial compressive load and a twisting torque, then the remodeling patterns depend on whether the periosteal surface or the endosteal surface controls the limits of the remodeling process. It is shown that the effect of increasing the torque is always opposite to the effect of increasing the compressive load. Thus, similar remodeling patterns are obtained by increasing one type of loading and decreasing the other. Aside from the restriction of idealized cylindrical geometry, the only assumptions made are that the bone tissue is linearly elastic and that there exists a finite range of remodeling equilibrium stresses. Only those loading situations which maintain the bone in remodeling equilibrium are considered in this work. It follows that the results presented are independent of the specific type of rule governing the temporal evolution of the bone shape, since any such rule applies only in situations where there is active remodeling and, hence, no remodeling equilibrium.

Errors in the orientation of the principal stress axes if bone tissue is modeled as isotropic.

The error in the prediction of the orientation of the principal axes of stress in bone tissue is determined in the case when the tissue is modeled as elastically isotropic rather than as orthotropic, the probable symmetry of bone tissue. Results are two-dimensional and assume the same underlying strain state for both the orthotropic and isotropic cases. The maximum error is 45 degrees, and the typical error is generally significant.

Non-interacting modes for stress, strain and energy in anisotropic hard tissue.

The six non-interacting modes for stress, strain and energy in an orthotropic elastic model of human femoral cortical bone tissue are discussed and illustrated. The stress and strain modes are illustrated using the representation of the stress and strain fields around a circular hole in a flat plate of cortical bone subjected to a uniaxial field of tension as the example. The six modes play a role in the stress analysis of orthotropic elastic materials similar to the roles played by the hydrostatic and deviatoric non-interacting stress, strain and energy modes in isotropic elasticity. The biomechanical significance of the six non-interacting modes for stress, strain and energy in hard tissue is both practical and suggestive. The modes suggest a practical scheme for the representation of stress and strain fields in hard tissue. The existence of the modes suggests physical insights, for example, possible failure mechanisms or adaptation strategies possessed by the hard tissues.

Identification of the elastic symmetry of bone and other materials.

A simplified classification scheme for the elastic symmetries of a solid is applied to the identification of the elastic symmetry of a material by three different methods--visual, stereological and numerical algorithm. Each method is illustrated with an application to bone tissues, but the methods apply to all materials.

Thermodynamic restrictions on the elastic constants of bone.

The thermodynamic restrictions on the elastic coefficients of linear orthotropic elasticity and linear transversely isotropy elasticity are recorded and it is shown that previously reported data for the elastic orthotropic constants of bone satisfy these thermodynamic restrictions.

A model for the excitation of osteocytes by mechanical loading-induced bone fluid shear stresses.

A new experimentally testable hypothesis is advanced for the mechanosensory transduction mechanism by which communicating osteocytes sense the very small in vivo strains in the calcified matrix components of bone. We propose that the osteocytes, although not responsive to substantial fluid pressures, can be stimulated by relatively small fluid shear stresses acting on the membranes of their osteocytic processes. Biot's porous media theory is used to relate the combined axial and bending loads applied to a whole bone to the flow past the osteocytic processes in their canaliculi. In this theory, the bone pores of interest are the proteoglycan filled fluid annuli that surround the osteocytic processes in the canaliculi. We show that previously predicted fluid pore pressure relaxation times were a hundred-fold too short for the lacunar-canalicular porosity because they neglected the fluid drag associated with proteoglycan matrix on the surface membrane of the osteocyte and its cell processes. The recent theory developed in Tsay and Weinbaum [J. Fluid Mech. 226, 125-148 (1991)] for flow through cross-linked fiber filled channels is used to model the flow through this proteoglycan matrix. The predicted pore relaxation time, 1-2 s, closely corresponds to the times measured by Salzstein and Pollack [J. Biomechanics 20, 271-280 (1987)]. Furthermore, using this model, the magnitude of the predicted fluid induced shear stresses, 8-30 dyn cm-2, is shown to be similar to the fluid shear stresses measured in osteoblasts and other cells in which an intracellular Ca2+ shear stress response had been observed. This model is also used, in conjunction with anatomical data and the pore fluid pressure relaxation time data, to show that the spacing between the fibers is approximately 7 nm. The result is consistent with the notion that the canalicular pore space is filled with glycosaminoglycans that are ordered by albumin according to the model of Michel [J. Physiol. 404, 1-29 (1988)]. The new hypothesis is also shown to be consistent with the experiments of McLeod et al. [J. Biomechanics (submitted)] which suggest that high-frequency low-amplitude postural strains can maintain and even increase bone mass.

A case for bone canaliculi as the anatomical site of strain generated potentials.

We address the question of determining the anatomical site that is the source of the experimentally observed strain generated potentials (SGPs) in bone tissue. There are two candidates for the anatomical site that is the SGP source, the collagen-hydroxyapatite porosity and the larger size lacunar-canalicular porosity. In the past it has been argued, on the basis of experimental data and a reasonable model, that the site of the SGPs in bone is the collagen-hydroxyapatite porosity. The theoretically predicted pore radius necessary for the SGPs to reside in this porosity is 16 nm, which is somewhat larger than the pore radii estimated from gas adsorption data where the preponderance of the pores were estimated to be in the range 5-12.5 nm. However, this pore size is significantly larger than the 2 nm size of the small tracer, microperoxidase, which appears to be excluded from the mineralized matrix. In this work a similar model, but one in which the effects of fluid dynamic drag of the cell surface matrix in the bone canaliculi are included, is used to show that it is possible for the generation of SGPs to be associated with the larger size lacunar-canalicular porosity when the hydraulic drag and electrokinetic contribution of the bone fluid passage through the cell coat (glycocalyx) is considered. The consistency of the SGP data with this model is demonstrated. A general boundary condition is introduced to allow for current leakage at the bone surface. The results suggest that the current leakage is small for the in vitro studies in which the strain generated potentials have been measured.

On the dependence of the elasticity and strength of cancellous bone on apparent density.

This paper presents a statistical analysis of the pooled data from a number of previous experiments concerning the dependence of the Young's moduli and strength of cancellous bone tissue upon apparent density. The results show that both the Young's moduli and the strength are proportional to the square of apparent density of the tissue and are therefore proportional to one another. It is shown that the coefficient of proportionality is different for human and bovine tissue. It is concluded that the suggestion of Wolff (Das Gesetz der Transformation der Knochen, Hirschwald, Berlin, 1892) that compact bone tissue is simply more dense cancellous bone tissue is not an accurate statement when only the mechanical properties of these two tissues are considered. It is noted that estimates for the elastic modulus of the individual trabecula of human cancellous bone vary from 1 to 20 GPa and it is suggested that this question needs further study.

Correction formulae for the misalignment of axes in the measurement of the orthotropic elastic constants.

In the experimental determination of the orthotropic elastic constants, one often encounters the situation in which the symmetry axes of the material are not coincident with specimen axes along which the material testing is accomplished. The problem of calculating the compliance coefficients in the symmetry coordinate system from measurements of the compliance coefficients made in an arbitrary, specimen fixed, coordinate system is considered here.

Fluid pressure relaxation depends upon osteonal microstructure: modeling an oscillatory bending experiment.

When bone is mechanically loaded, bone fluid flow induces shear stresses on bone cells that have been proposed to be involved in bone's mechanosensory system. To investigate bone fluid flow and strain-generated potentials, several theoretical models have been proposed to mimic oscillatory four-point bending experiments performed on thin bone specimens. While these previous models assume that the bone fluid relaxes across the specimen thickness, we hypothesize that the bone fluid relaxes primarily through the vascular porosity (osteonal canals) instead and develop a new poroelastic model that integrates the microstructural details of the lacunar-canalicular porosity, osteonal canals, and the osteonal cement lines. Local fluid pressure profiles are obtained from the model, and we find two different fluid relaxation behaviors in the bone specimen, depending on its microstructure: one associated with the connected osteonal canal system, through which bone fluid relaxes to the nearby osteonal canals; and one associated with the thickness of a homogeneous porous bone specimen (approximately 1 mm in our model), through which bone fluid relaxes between the external surfaces of the bone specimen at relatively lower loading frequencies. Our results suggest that in osteonal bone specimens the fluid pressure response to cyclic loading is not sensitive to the permeability of the osteonal cement lines, while it is sensitive to the applied loading frequency. Our results also reveal that the fluid pressure gradients near the osteonal canals (and thus the fluid shear stresses acting on the nearby osteocytes) are significantly amplified at higher loading frequencies.