An Evolutionary Wolff's Law for Trabecular Architecture S. C. Cowin A. M. Sadegh G. M. Luo Department of Mechanical Engineering, The City College of The City University of New York, New York, NY 10031
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 this 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.
Introduction Wolff's law suggests that the architecture of trabecular bone tissue in a local region is structurally adapted to the local stresses experienced by the bone tissue. Specifically, Wolff's law is the observation that the axes of the individual bony trabeculae coincide with the local principal directions of stress. The law is described in many places, (Wolff, 1892; Murray, 1936; Oxnard and Yang, 1981; Cowin, 1986; Cowin, 1989). A continuum formulation of Wolff's law at remodeling equilibrium (RE) was suggested by (Cowin, 1986). This formulation is based on an elastic constitutive relation for cancellous bone that includes a measure of microstructure, called the fabric tensor, as well as the stress and the strain tensors and the solid volume fraction of bone. The constitutive equation for this continuum formulation is T = C(H,*/)[E], or 7') which depends upon the architecture parameters H and v. One can view increasing values of v as generally stiffening the elastic coefficients C(H,»
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 this 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.
Introduction Wolff's law suggests that the architecture of trabecular bone tissue in a local region is structurally adapted to the local stresses experienced by the bone tissue. Specifically, Wolff's law is the observation that the axes of the individual bony trabeculae coincide with the local principal directions of stress. The law is described in many places, (Wolff, 1892; Murray, 1936; Oxnard and Yang, 1981; Cowin, 1986; Cowin, 1989). A continuum formulation of Wolff's law at remodeling equilibrium (RE) was suggested by (Cowin, 1986). This formulation is based on an elastic constitutive relation for cancellous bone that includes a measure of microstructure, called the fabric tensor, as well as the stress and the strain tensors and the solid volume fraction of bone. The constitutive equation for this continuum formulation is T = C(H,*/)[E], or 7') which depends upon the architecture parameters H and v. One can view increasing values of v as generally stiffening the elastic coefficients C(H,»
