TY - JOUR

T1 - A theoretical model to predict distribution of the fabric tensor and apparent density in cancellous bone

AU - Luo, Zong Ping

AU - An, Kai Nan

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 1998/6

Y1 - 1998/6

N2 - The adaptation of cancellous bone to mechanical forces is well recognized. Theoretical models for predicting cancellous bone architecture have been developed and have mainly focused on the distribution of trabecular mass or the apparent density. The purpose of this study was to develop a theoretical model which can simultaneously predict the distribution of trabecular orthotropy/orientation, as represented by the fabric tensor, along with apparent density. Two sets of equations were derived under the assumption that cancellous bone is a biological self-optimizing material which tends to minimize strain energy. The first set of equations provide the relationship between the fabric tensor and stress tensor, and have been verified to be consistent with Wolff's law of trabecular architecture, that is, the principal directions of the fabric tensor coincide with the principal stress trajectories. The second set of equations yield the apparent density from the stress tensor, which was shown to be identical to those obtained based on local optimization with strain energy density of true bone tissue as the objective function. These two sets of equations, together with elasticity field equations, provide a complete mathematical formulation for the adaptation of cancellous bone.

AB - The adaptation of cancellous bone to mechanical forces is well recognized. Theoretical models for predicting cancellous bone architecture have been developed and have mainly focused on the distribution of trabecular mass or the apparent density. The purpose of this study was to develop a theoretical model which can simultaneously predict the distribution of trabecular orthotropy/orientation, as represented by the fabric tensor, along with apparent density. Two sets of equations were derived under the assumption that cancellous bone is a biological self-optimizing material which tends to minimize strain energy. The first set of equations provide the relationship between the fabric tensor and stress tensor, and have been verified to be consistent with Wolff's law of trabecular architecture, that is, the principal directions of the fabric tensor coincide with the principal stress trajectories. The second set of equations yield the apparent density from the stress tensor, which was shown to be identical to those obtained based on local optimization with strain energy density of true bone tissue as the objective function. These two sets of equations, together with elasticity field equations, provide a complete mathematical formulation for the adaptation of cancellous bone.

UR - http://www.scopus.com/inward/record.url?scp=0032093047&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032093047&partnerID=8YFLogxK

U2 - 10.1007/s002850050114

DO - 10.1007/s002850050114

M3 - Article

C2 - 9710972

AN - SCOPUS:0032093047

SN - 0303-6812

VL - 36

SP - 557

EP - 568

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

IS - 6

ER -