The mineralization level is heterogeneous in cortical bone extracellular matrix as a consequence of remodeling. Models of the effective elastic properties at the millimeter scale have been developed based on idealizations of the vascular pore network and matrix properties. Some popular models do not take into account the heterogeneity of the matrix. However, the errors on the predicted elasticity when the difference in elastic properties between osteonal and interstitial tissues is not modeled have not been quantified. This work provides an estimation of the maximum error. We compare the effective elasticity of a representative volume element (RVE) assuming (1) different elastic properties in osteonal and interstitial tissues vs. (2) average matrix properties. In order to account for the variability of bone microstructure, we use a collection of high resolution images of the pore network to build RVEs. In each RVE we assumed a constant osteonal wall thickness and we artificially varied this thickness between 35 and 140 \(\upmu\)m to create RVEs with different amounts of osteonal tissue. The homogenization problem was solved with a fast Fourier transform (FFT)-based numerical scheme. We found that the error depends on pore volume fraction and varies on average from 1 to \(7\%\) depending on the assumed diameter of the osteons. The results suggest that matrix heterogeneity may be disregarded in cortical bone models in most practical cases.

由于重塑，皮质骨细胞外基质中的矿化水平是不均匀的。基于血管孔隙网络和基质特性的理想化，已经开发了毫米尺度的有效弹性特性模型。一些流行的模型没有考虑矩阵的异质性。然而，当没有对骨组织和间质组织之间的弹性特性差异进行建模时，预测弹性的误差尚未量化。这项工作提供了最大误差的估计。我们比较了代表性体积元素 (RVE) 的有效弹性，假设 (1) 骨和间质组织中的不同弹性特性与 (2) 平均基质特性。为了解释骨骼微观结构的可变性，我们使用孔隙网络的高分辨率图像集合来构建 RVE。在每个 RVE 中，我们假设一个恒定的骨壁厚度，我们在 35 到 140 之间人为地改变这个厚度 \(\upmu\) m 以创建具有不同数量骨组织的 RVE。使用基于快速傅里叶变换 (FFT) 的数值方案解决了均匀化问题。我们发现误差取决于孔体积分数，并且平均从 1 到\(7\%\) 变化，具体取决于假定的骨直径。结果表明，在大多数实际情况下，皮质骨模型中可能会忽略基质异质性。