Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK.
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Rd, Oxford OX2 6GG, UK.
Math Med Biol. 2020 Sep 10;37(3):389-428. doi: 10.1093/imammb/dqaa001.
Swelling pressure in the interstitial fluid within the pores of cartilage tissue is known to have a significant effect on the rheology of cartilage tissue. The swelling pressure varies rapidly within thin regions inside pores known as Debye layers, caused by the presence of fixed charge, as observed in cartilage. Tissue level calculation of cartilage deformation therefore requires resolution of three distinct spatial scales: the Debye lengthscale within individual pores; the lengthscale of an individual pore; and the tissue lengthscale. We use asymptotics to construct a leading order approximation to the swelling pressure within pores, allowing the swelling pressure to be systematically included within a fluid-solid interaction model at the level of pores in cartilage. We then use homogenization to derive tissue level equations for cartilage deformation that are very similar to those governing the finite deformation of a poroviscoelastic body. The equations derived permit the spatial variations in porosity and electric charge that occur in cartilage tissue. Example solutions are then used to confirm the plausibility of the model derived and to consider the impact of fixed charge heterogeneity, illustrating that local fixed charge loss is predicted to increase deformation gradients under confined compression away from, rather than at, the site of loss.
众所周知，软骨组织孔隙间的间质液中的膨胀压力对软骨组织的流变学有显著影响。由于固定电荷的存在，在称为德拜层的薄区域内，孔隙内的膨胀压力会迅速变化，如软骨中观察到的那样。因此，软骨组织变形的组织水平计算需要解决三个不同的空间尺度：单个孔隙内的德拜长度尺度；单个孔隙的长度尺度；以及组织长度尺度。我们使用渐近法构建了一个在孔内膨胀压力的主导阶近似，从而可以在软骨中孔级别的流体-固体相互作用模型中系统地包含膨胀压力。然后，我们使用均匀化来推导出软骨变形的组织水平方程，这些方程与控制多孔黏弹性体的有限变形的方程非常相似。所推导的方程允许出现软骨组织中发生的孔隙率和电荷的空间变化。然后使用示例解来确认所推导模型的合理性，并考虑固定电荷异质性的影响，说明局部固定电荷损失预计会增加受限压缩下的变形梯度，远离而不是在损失部位。
