A TRIPHASIC THEORY FOR THE SWELLING AND DEFORMATION BEHAVIORS OF ARTICULAR-CARTILAGE
A TRIPHASIC THEORY FOR THE SWELLING AND DEFORMATION BEHAVIORS OF ARTICULAR-CARTILAGE
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DOI：
10.1115/1.2894880
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发表时间：
1991-08-01
影响因子：
1.7
通讯作者：
MOW, VC
中科院分区：
文献类型：
--
作者：
LAI, WM;HOU, JS;MOW, VC
关节软骨的肿胀取决于其固定的电荷密度和分布，其胶原-蛋白多糖基质的硬度，以及间质中的离子浓度。为了描述软骨在化学和/或机械载荷作用下的变形和应力场，已经发展了一种三级混合物理论，它包括两个流固相(两相)和一个离子相(代表单一盐的阳离子和阴离子)。这一三相理论结合了离子和多离子(蛋白多糖)溶液的物理化学理论和软骨的两相理论。本模型假定固定电荷基团保持不变，反离子是浴液中单一盐的阳离子。中性盐和中间水的动量方程用化学势表示，化学势的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、固体基质膨胀系数e和固定电荷密度c(F)。对于一价盐如氯化钠，它们由My(I)=Mu(O)i at(Rt/M(I))ln[Gamma(2)+/-c(c+c(F))]和Mu(W)=Mu(O)w+[p-Rt-Phi(2c+c(F))+B(W)e]/Rho(T)w给出，其中R，T，M(I)，Gamma+/-，Phi，Rho(T)w和B(W)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、水的真密度和偶联物质系数。对于无限小的应变和材料的各向同性，总混合应力的应力-应变关系为Sigma=-Pi-Tci+lambda(S)(Tre)i+2-u(S)E，其中E是应变张量，(Lamba(S)，u(S))是弹性固体基质的Lame常数。化学膨胀应力(-T(C))来自固体基质中电荷间的排斥力。这一理论既适用于均衡问题，也适用于非均衡问题。对于平衡自由膨胀问题，该理论给出了众所周知的Donnan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。对于侧限压缩膨胀问题，预测了外加压应力由三种载荷支撑机制分担：1)Donnan渗透压力；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约束压缩数据，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，这三种机制在决定软骨的整体压缩刚度方面都是重要的。
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 c(F). For a uni-uni valent salt such as NaCl, they are given by my(i) = mu(o)i at (RT/M(i))ln[gamma(2) +/- c(c + c(F))] and mu(w) = mu(o)w + [p - RT-phi(2c + c(F)) + B(w)e]/rho(T)w, where R, T, M(i), gamma +/-, phi, rho(T)w and B(w) 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-mu(s)E, where E is the strain tensor and (lamba(s),mu(s)) are the Lame constants of the elastic solid matrix. The chemical-expansion stress (- T(c)) 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.