On the relationship between the orthotropic Young's moduli and fabric.
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Animals↗
Biomedical subjects
S C Cowin
Publications and source records attributed to S C Cowin.
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.
Biomechanical Phenomena↗
An evolutionary Wolff's law for trabecular architecture.
A continuum model is proposed to describe the temporal evolution of both the density changes and the reorientation of the trabecular architecture given the applied stress state in the bone and certain material parameters of the bone. The data upon which the proposed model is to be based consist of experimentally determined remodeling rate coefficients and quantitative stereological and anisotropic elastic constant measurements of cancellous bone. The model shows that the system of differential equations governing the temporal changes in architecture is necessarily nonlinear. This nonlinearity is fundamental in that it stems from the fact that, during remodeling, the relationship between stress and strain is changing as the stress and strain variables themselves are changing. In order to preserve the remodeling property of the model, terms that are of the order strain times the changes in density and/or microstructural properties must be retained. If these terms were dropped, there would be no feedback mechanism for architectural adaptation and no adaptation of the trabecular architecture. There is, therefore, no linearized version of the model of the temporal evolution of trabecular architecture. An application of the model is illustrated by an example problem in which the temporal evolution of homogeneous trabecular architecture is predicted. A limitation of the proposed continuum model is the length scale below which it cannot be applied. The model cannot be applied in regions of cancellous bone where the trabecular bone architecture is relatively inhomogeneous or at a bone-implant interface.
Bone Density↗
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.
Biomechanical Phenomena↗
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.
Anisotropy↗
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.
Adaptation, Physiological↗
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 tissue's 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.
Aged↗
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.
Biomechanical Phenomena↗
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.
Adaptation, Physiological↗
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.
Algorithms↗
A resolution restriction for Wolff's law of trabecular architecture.
A rigid constructionist view of Wolff's trajectorial theory of trabecular architecture, called the "orthodox theory," requires that the trabeculae of cancellous bone intersect at right angles exactly like the stress trajectories with which they are associated by the theory. It is well established that trabeculae do not always intersect at right angles. The author discusses the fallacy in the orthodox interpretation of Wolff's law, and suggests that a "resolution length restriction" be imposed on the trajectorial theory to avoid interpretations that lead to the fallacy.
Bone Development↗
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.
Biomechanical Phenomena↗
Errors induced by off-axis measurement of the elastic properties of bone.
Misalignment between the axes of measurement and the material symmetry axes of bone causes error in anisotropic elastic property measurements. Measurements of Poisson's ratio were strongly affected by misalignment errors. The mean errors in the measured Young's moduli were 9.5 and 1.3 percent for cancellous and cortical bone, respectively, at a misalignment angle of 10 degrees. Mean errors of 1.1 and 5.0 percent in the measured shear moduli for cancellous and cortical bone, respectively, were found at a misalignment angle of 10 degrees. Although, cancellous bone tissue was assumed to have orthotropic elastic symmetry, the possibility of the greater symmetry of transverse isotropy was investigated. When the nine orthotropic elastic constants were forced to approximate the five transverse isotropic elastic constants, errors of over 60 percent were introduced. Therefore, it was concluded that cancellous bone is truly orthotropic and not transversely isotropic. A similar but less strong result for cortical bone tissue was obtained.
Animals↗
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.
Animals↗
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.
Bone and Bones↗
Wolff's law of trabecular architecture at remodeling equilibrium.
An elastic constitutive relation for cancellous bone tissue is developed. This relationship involves the stress tensor T, the strain tensor E and the fabric tensor H for cancellous bone. The fabric tensor is a symmetric second rank tensor that is a quantitative stereological measure of the microstructural arrangement of trabeculae and pores in the cancellous bone tissue. The constitutive relation obtained is part of an algebraic formulation of Wolff's law of trabecular architecture in remodeling equilibrium. In particular, with the general constitutive relationship between T, H and E, the statement of Wolff's law at remodeling equilibrium is simply the requirement of the commutativity of the matrix multiplication of the stress tensor and the fabric tensor at remodeling equilibrium, T*H* = H*T*. The asterisk on the stress and fabric tensor indicates their values in remodeling equilibrium. It is shown that the constitutive relation also requires that E*H* = H*E*. Thus, the principal axes of the stress, strain and fabric tensors all coincide at remodeling equilibrium.
Bone and Bones↗
The mechanical properties of immature osteopetrotic bone.
The elastic properties of the longer bones of the fore and hind limbs of an osteopetrotic Angus calf were measured using an ultrasonic technique. The same bones of a second Angus calf with unaffected bone tissue were also studied. The osteopetrotic bones were found to exhibit statistically significant lower values of density and mineral content than did the unaffected bones (P = .0054 and P = .0492, respectively). This result is contrary to what has been reported in the literature. The variations of the elastic properties and the density around the circumference of both the immature osteopetrotic femur and the unaffected femur were found to be similar to those previously measured on normal adult bovine femora.
Animals↗
Functional adaptation in long bones: establishing in vivo values for surface remodeling rate coefficients.
In this paper we describe a computational means, based on beam theory, for application of the theory of adaptive elasticity to examples of real bone geometries. The results of the animal experiments were taken from the literature, and each documented the temporal evolution of a change in bone shape after a significant change in the mechanical loading environment of the bone. For each of these studies, we establish preliminary estimates of the in vivo values of the surface remodeling rate coefficients--the key parameters in the theory of surface remodeling. Our preliminary parameter estimates are established by comparison of published animal experimental results with surface remodeling theory predictions generated by the computational method.
Adaptation, Physiological↗