Stephen C. Cowin

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. Biots 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 continuous wave technique for the measurement of the elastic properties of cortical bone

A continuous wave technique is described for measuring the nine independent orthotropic elastic coefficients from a single cubic specimen. The side dimensions of this cubic specimen are on the order of 5 mm. Because of the small size of the specimen, the spatial resolution of material inhomogeneity using this technique is quite good. Although it is possible to apply this technique to any elastic material such as woods or metals, the elastic properties of human and canine cortical femora are presented here. The orthotropic elastic coefficients and the variation of these coefficients are presented as a function of anatomical position.

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 Youngs moduli and strength of cancellous bone tissue upon apparent density. The results show that both the Youngs 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.

LINEAR ELASTIC MATERIALS WITH VOIDS

A linear theory of elastic materials with voids is presented. This theory differs significantly from classical linear elasticity in that the volume fraction corresponding to the void volume is taken as an independent kinematical variable. Following a discussion of the basic equations, boundary-value problems are formulated, and uniqueness and weak stability are established for the mixed problem. Then, several applications of the theory are considered, including the response to homogeneous deformations, pure bending of a beam, and small-amplitude acoustic waves. In each of these applications, the change in void volume induced by the deformation is determined. In the final section of the paper, the relationship between the theory presented and the effective moduli approach for porous materials is discussed.In the two year period between the submission of this manuscript and the receipt of the page proof, there have been some extensions of the results reported here. In the context of the theory described, the classical pressure vessel problems and the problem of the stress distribution around a circular hole in a field have uniaxial tension have been solved [19,22]. The solution given in the present paper for the pure bending of a beam when the rate effect of the theory is absent is extended to case when the rate effect is present in [21]. The various implications of the rate effect in the void volume deformation are pursued all the subsequent works [19,20,21,22].

Bone remodeling I: theory of adaptive elasticity

A thermomechanical continuum theory involving a chemical reaction and mass transfer between two constituents is developed here as a model for bone remodeling. Bone remodeling is a collective term for the continual processes of growth, reinforcement and resorbtion which occur in living bone. The resulting theory describes an elastic material which adapts its structure to applied loading.ZusammenfassungEine Thermo-mechanische kontinuum Theorie als Modell für die Knochenrekonstrucktion wird entwickelt, die eine chemische Reaktion und einen Massentransport zwischen zwei Komponenten behandelt. Knochenrekonstruktion ist ein Sammelbegriff für die kontinuierlichen Prozesse des Wachsens, der Verstärkung und des Abbaus wie sie im lebenden Knochen auftreten. Die Theorie beschreibt ein elastisches Material, das sich in der Form der Belastung anpasst.

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 Fullers 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.

Mechanotransduction and flow across the endothelial glycocalyx

In this inaugural paper, we shall provide an overview of the endothelial surface layer or glycocalyx in several roles: as a transport barrier, as a porous hydrodynamic interface in the motion of red and white cells in microvessels, and as a mechanotransducer of fluid shearing stresses to the actin cortical cytoskeleton of the endothelial cell. These functions will be examined from a new perspective, the quasiperiodic ultrastructural model proposed in Squire et al. [Squire, J. M., Chew, M., Nneji, G., Neal, C., Barry, J. & Michel, C. (2001) J. Struct. Biol. 136, 239–255] for the 3D organization of the endothelial surface layer and its linkage to the submembranous scaffold. We shall show that the core proteins in the bush-like structures comprising the matrix have a flexural rigidity, EI, that is sufficiently stiff to serve as a molecular filter for plasma proteins and as an exquisitely designed transducer of fluid shearing stresses. However, EI is inadequate to prevent the buckling of these protein structures during the intermittent motion of red cells or the penetration of white cell microvilli. In these cellular interactions, the viscous draining resistance of the matrix is essential for preventing adhesive molecular interactions between proteins in the endothelial membrane and circulating cellular components.

Candidates for the Mechanosensory System in Bone

Some potential mechanisms by which bone cells sense mechanical loads are described and hypotheses concerning the functioning of these mechanisms are explored. It is well known that bone tissue adapts its structure to its mechanical load environment. 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.

A model for strain amplification in the actin cytoskeleton of osteocytes due to fluid drag on pericellular matrix

A model is presented that provides a resolution to a fundamental paradox in bone physiology, namely, that the strains applied to whole bone (i.e., tissue level strains) are much smaller (0.04-0.3 percent) than the strains (1-10 percent) that are necessary to cause bone signaling in deformed cell cultures (Rubin and Lanyon, J. Bone Joint Surg. 66A (1984) 397-410; Fritton et al., J. Biomech. 33 (2000) 317-325). The effect of fluid drag forces on the pericellular matrix (PM), its coupling to the intracellular actin cytoskeleton (IAC) and the strain amplification that results from this coupling are examined for the first time. The model leads to two predictions, which could fundamentally change existing views. First, for the loading range 1-20MPa and frequency range 1-20Hz, it is, indeed, possible to produce cellular level strains in bone that are up to 100 fold greater than normal tissue level strains (0.04-0.3 percent). Thus, the strain in the cell process membrane due to the loading can be of the same order as the in vitro strains measured in cell culture studies where intracellular biochemical responses are observed for cells on stretched elastic substrates. Second, it demonstrates that in any cellular system, where cells are subject to fluid flow and tethered to more rigid supporting structures, the tensile forces on the cell due to the drag forces on the tethering fibers may be many times greater than the fluid shear force on the cell membrane.

The fabric dependence of the orthotropic elastic constants of cancellous bone.

It has been proposed that the orthotropic elastic constants of cancellous bone depend upon a tensorial measure of anisotropy called fabric as well as the tissues structural density. Cowin (1985, Mechanics Mater, 4, 137-147; 1986, J. biomech. Engng 108, 83-88) developed explicit relationships for the elastic constant, structural density and fabric relationship. In this study the orthotropic elastic moduli, structural density, and fabric components were measured for 11 cancellous bone specimens from five bovine femora and for 75 specimens from three human proximal tibiae and fitted to these relationships using a least squares analysis. The relationships explained between 72 and 94% of the variance in the elastic constants. The relationships between the elastic constants and squared or cubed power functions of structural density had better predictive value over the entire distribution of the data than did expressions with linear functions of structural density.