Wolff’s Law of Trabecular Architecture at Remodeling Equilibrium

Abstract

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.