Theoretical and numerical study of a bone remodeling model: The effect of osteocyte cells distribution

S. Baiotto; M. Zidi

doi:10.1007/s10237-004-0042-y

Biomechanics and Modeling in Mechanobiology

September 2004, Volume 3, Issue 1, pp 6–16 | Cite as

Theoretical and numerical study of a bone remodeling model: The effect of osteocyte cells distribution

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S. Baiotto
M. Zidi

Original Paper

First Online: 09 July 2004

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15 Citations

Abstract

It is well argued that osteocytes are mechanosensory cells and are involved in the regulation of bone remodeling. In previous works, the predictions from a simulation model have suggested that both the influencing distance of osteocytes and the magnitude of the mechanical loads determine the thickness of trabeculae whereas the number of osteocytes primarily affects the rate of bone remodeling. The question that remains not completely answered is: for the same number of osteocytes, what is the effect of different distributions on the remodeling process? Based on a particular regulatory bone remodeling model, the question is addressed, in part, by performing a stability analysis in connection with numerical simulations. The results allow us to demonstrate that, on one hand, we cannot reach a conclusion about the stability of the model for a nonuniform osteocyte distribution. This implies that there is no relationship between the different parameters conveying the stability of the model. On the other hand, we show that the osteocyte cell distribution has a significant influence on the bone morphology, which seems to be confirmed by simulations with real data obtained from rat tibia.

Keywords

Bone Remodel Apparent Density Trabecular Architecture Bone Remodel Process Bone Mass Loss

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

Acknowledgement

We thank Dr. Laurence VICO from the Laboratoire de Biologie du Tissu Osseux (INSERM EMI 366) of Saint Etienne, FRANCE, for using the experimental data.

Appendix

Determination of the stationary states

By noting $ \overline{{W_{i} }} = \frac{{W_{i} }} {{\rho ^{q}_{i} }} $, Eq. 2 becomes:

$$ \frac{{\partial \rho _{i} }} {{\partial t}} = \tau {\sum\limits_{k = 1}^m {{\text{e}}^{{ - [d{\left( {i,I_{k} } \right)}/do]}} } }{\left( {\overline{{W_{{I_{k} }} }} - \overline{{W_{0} }} } \right)} $$

(12)

The strain energy is written:

$$ W_{i} = \frac{{\sigma _{i} \varepsilon _{i} }} {{2\rho ^{{}}_{i} }} $$

(13)

We assume that the strains are the same in all the elements:

$$ \varepsilon _{1} = \varepsilon _{2} = ... = \varepsilon = \frac{F} {{A{\sum\limits_i {E_{i} } }}} $$

(14)

where A is the area of one element.

Consequently, Eqs. 4, 13, and 14 give, for the ith element:

$$ W_{i} = \frac{{\rho ^{p}_{{_{i} }} F^{2} }} {{2A^{2} K{\left( {{\sum\limits_{j = 1}^n {\rho ^{p}_{j} } }} \right)}^{2} }} $$

(15)

We have also:

$$ \overline{{W_{i} }} = \frac{{C\rho ^{{p - q}}_{{_{i} }} }} {{{\left( {{\sum\limits_{j = 1}^n {\rho ^{p}_{j} } }} \right)}^{2} }},\quad {\text{where}}\;C = \frac{{F^{2} }} {{2A^{2} K}} $$

(16)

The stationary states of the Eq. 12 are reached for ∂ρ_i/∂t=0. That means that $ \overline{{W_{{I_{k} }} }} = \overline{{W_{0} }} $. We obtain:

$$ {\text{ }}\left\{ \begin{aligned} & \rho _{{I_{1} }} = \rho _{{I_{2} }} = ... = \rho _{{I_{p} }} = \rho ^{{{\left( 1 \right)}}}_{{\text{S}}} {\text{ }};{\text{ }}\rho _{{\bar{I}_{l} }} = \rho ^{{{\left( 2 \right)}}}_{{\text{S}}} {\text{ }} \\ & \rho ^{{p - q}}_{{I_{k} }} = \frac{{\bar{W}_{0} }} {C}{\left( {m\rho ^{p}_{{I_{k} }} + {\sum\limits_{l = 1}^{n - m} {\rho ^{p}_{{\bar{I}_{l} }} } }} \right)}^{2} \\ \end{aligned} \right. $$

(17)

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Authors and Affiliations

S. Baiotto
- 1
M. Zidi
- 2
Email author

1.Institut Supérieur des Biosciences de ParisUniversité Paris 12 Val de MarneCréteil cedexFrance
2.Institut Supérieur des Biosciences de Paris, Laboratoire Biosciences et Médecine (BIOSEM)Université Paris 12 Val de MarneNoisy le Grand CedexFrance

Theoretical and numerical study of a bone remodeling model: The effect of osteocyte cells distribution

Abstract

Keywords

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Acknowledgement

References

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