### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 557-568 |

Number of pages | 12 |

Journal | Journal of Mathematical Biology |

Volume | 36 |

Issue number | 6 |

State | Published - Jun 1998 |

### Fingerprint

### Keywords

- Apparent density
- Bone remodeling
- Fabric tensor
- Theoretical model

### ASJC Scopus subject areas

- Agricultural and Biological Sciences (miscellaneous)
- Mathematics (miscellaneous)

### Cite this

*Journal of Mathematical Biology*,

*36*(6), 557-568.

**A theoretical model to predict distribution of the fabric tensor and apparent density in cancellous bone.** / Luo, Zong Ping; An, Kai Nan.

Research output: Contribution to journal › Article

*Journal of Mathematical Biology*, vol. 36, no. 6, pp. 557-568.

}

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

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.

KW - Apparent density

KW - Bone remodeling

KW - Fabric tensor

KW - Theoretical model

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

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

M3 - Article

VL - 36

SP - 557

EP - 568

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

IS - 6

ER -