Lai W M, Hou J S, Mow V C
Department of Mechanical Engineering, Columbia University, New York, NY 10032.
J Biomech Eng. 1991 Aug;113(3):245-58. doi: 10.1115/1.2894880.
Swelling of articular cartilage depends on its fixed charge density and distribution, the stiffness of its collagen-proteoglycan matrix, and the ion concentrations in the interstitium. A theory for a tertiary mixture has been developed, including the two fluid-solid phases (biphasic), and an ion phase, representing cation and anion of a single salt, to describe the deformation and stress fields for cartilage under chemical and/or mechanical loads. This triphasic theory combines the physico-chemical theory for ionic and polyionic (proteoglycan) solutions with the biphasic theory for cartilage. The present model assumes the fixed charge groups to remain unchanged, and that the counter-ions are the cations of a single-salt of the bathing solution. The momentum equation for the neutral salt and for the intersitial water are expressed in terms of their chemical potentials whose gradients are the driving forces for their movements. These chemical potentials depend on fluid pressure p, salt concentration c, solid matrix dilatation e and fixed charge density cF. For a uni-uni valent salt such as NaCl, they are given by mu i = mu io + (RT/Mi)ln[gamma 2 +/- c(c + cF)] and mu w = mu wo + [p-RT phi (2c + cF) + Bwe]/pwT, where R, T, Mi, gamma +/-, phi, pwT and Bw are universal gas constant, absolute temperature, molecular weight, mean activity coefficient of salt, osmotic coefficient, true density of water, and a coupling material coefficient, respectively. For infinitesimal strains and material isotropy, the stress-strain relationship for the total mixture stress is sigma = - pI-TcI + lambda s(trE)I + 2 musE, where E is the strain tensor and (lambda s, mu s) are the Lamé constants of the elastic solid matrix. The chemical-expansion stress (-Tc) derives from the charge-to-charge repulsive forces within the solid matrix. This theory can be applied to both equilibrium and non-equilibrium problems. For equilibrium free swelling problems, the theory yields the well known Donnan equilibrium ion distribution and osmotic pressure equations, along with an analytical expression for the "pre-stress" in the solid matrix. For the confined-compression swelling problem, it predicts that the applied compressive stress is shared by three load support mechanisms: 1) the Donnan osmotic pressure; 2) the chemical-expansion stress; and 3) the solid matrix elastic stress. Numerical calculations have been made, based on a set of equilibrium free-swelling and confined-compression data, to assess the relative contribution of each mechanism to load support. Our results show that all three mechanisms are important in determining the overall compressive stiffness of cartilage.
关节软骨的肿胀取决于其固定电荷密度和分布、胶原 - 蛋白聚糖基质的刚度以及间质中的离子浓度。已开发出一种三元混合物理论，包括两个流 - 固相（双相）和一个离子相，该离子相代表单一盐的阳离子和阴离子，以描述化学和/或机械载荷下软骨的变形和应力场。这种三相理论将离子和聚离子（蛋白聚糖）溶液的物理化学理论与软骨的双相理论结合起来。本模型假设固定电荷基团保持不变，且反离子是浴液中单一盐的阳离子。中性盐和间质水的动量方程用它们的化学势表示，其梯度是它们运动的驱动力。这些化学势取决于流体压力p、盐浓度c、固体基质膨胀e和固定电荷密度cF。对于像NaCl这样的单价盐，它们由μi = μio + (RT/Mi)ln[γ±c(c + cF)]和μw = μwo + [p - RTφ(2c + cF) + Bwe]/pwT给出，其中R、T、Mi、γ±、φ、pwT和Bw分别是通用气体常数、绝对温度、分子量、盐的平均活度系数、渗透系数、水的真实密度和耦合材料系数。对于无穷小应变和材料各向同性，总混合物应力的应力 - 应变关系为σ = - pI - TcI + λs(trE)I + 2μsE，其中E是应变张量，(λs, μs)是弹性固体基质的拉梅常数。化学膨胀应力（-Tc）源自固体基质内的电荷 - 电荷排斥力。该理论可应用于平衡和非平衡问题。对于平衡自由膨胀问题，该理论得出了著名的唐南平衡离子分布和渗透压方程，以及固体基质中“预应力”的解析表达式。对于受限压缩膨胀问题，它预测施加的压缩应力由三种载荷支撑机制分担：1）唐南渗透压；2）化学膨胀应力；3）固体基质弹性应力。基于一组平衡自由膨胀和受限压缩数据进行了数值计算，以评估每种机制对载荷支撑的相对贡献。我们的结果表明，所有三种机制在确定软骨的整体压缩刚度方面都很重要。
