Reply to Discussion: “On the Thermodynamical Admissibility of the Triphasic Theory of Charged Hydrated Tissues” (Mow, V. C., Lai, W. M., Setton, L. A., Gu, W., Yao, H., and Lu, X. L., 2009, ASME J. Biomech. Eng., 131, p. 095501)

2010 ◽  
Vol 132 (6) ◽  
Author(s):  
Jacques M. Huyghe ◽  
Wouter Wilson ◽  
Kamyar Malakpoor
Keyword(s):  
Triphasic Theory

Direct Osmotic Pressure Measurements in Articular Cartilage Demonstrate Nonideal and Concentration-Dependent Phenomena

2020 ◽  
Vol 143 (4) ◽  
Author(s):  
Brandon K. Zimmerman ◽  
Robert J. Nims ◽  
Alex Chen ◽  
Clark T. Hung ◽  
Gerard A. Ateshian
Keyword(s):  
Articular Cartilage ◽  
Stress Relaxation ◽  
Osmotic Pressure ◽  
Electrostatic Interactions ◽  
Swelling Pressure ◽  
Confined Compression ◽  
Human Cartilage ◽  
Poisson Boltzmann ◽  
Triphasic Theory

Abstract The osmotic pressure in articular cartilage serves an important mechanical function in healthy tissue. Its magnitude is thought to play a role in advancing osteoarthritis. The aims of this study were to: (1) isolate and quantify the magnitude of cartilage swelling pressure in situ; and (2) identify the effect of salt concentration on material parameters. Confined compression stress-relaxation testing was performed on 18 immature bovine and six mature human cartilage samples in solutions of varying osmolarities. Direct measurements of osmotic pressure revealed nonideal and concentration-dependent osmotic behavior, with magnitudes approximately 1/3 those predicted by ideal Donnan law. A modified Donnan constitutive behavior was able to capture the aggregate behavior of all samples with a single adjustable parameter. Results of curve-fitting transient stress-relaxation data with triphasic theory in febio demonstrated concentration-dependent material properties. The aggregate modulus HA increased threefold as the external concentration decreased from hypertonic 2 M to hypotonic 0.001 M NaCl (bovine: HA=0.420±0.109 MPa to 1.266±0.438 MPa; human: HA=0.499±0.208 MPa to 1.597±0.455 MPa), within a triphasic theory inclusive of osmotic effects. This study provides a novel and simple analytical model for cartilage osmotic pressure which may be used in computational simulations, validated with direct in situ measurements. A key finding is the simultaneous existence of Donnan osmotic and Poisson–Boltzmann electrostatic interactions within cartilage.

Implementation of Finite Deformation Triphasic Modeling in the Finite Element Code FEBio

2012 ◽  
Author(s):  
Gerard A. Ateshian ◽  
Steve Maas ◽  
Jeffrey A. Weiss
Keyword(s):  
Soft Tissues ◽  
Mixture Theory ◽  
Solid Matrix ◽  
Osmotic Swelling ◽  
Solute Content ◽  
Counter Ions ◽  
The Matrix ◽  
Triphasic Theory ◽  
Ion Species ◽  
And Diffusion

Many biological soft tissues exhibit a charged solid matrix, most often due to the presence of proteoglycans enmeshed within the matrix. The predominant solute content of the interstitial fluid of these tissues consists of the monovalent counter-ions Na+ and Cl−. The electrical interactions between the mobile ion species and fixed charge density of the solid matrix produces an array of mechano-electrochemical effects, including Donnan osmotic swelling, and streaming and diffusion potentials and currents. These phenomena have been successfully modeled by the triphasic theory of Lai et al. [1], which is based on the framework of mixture theory [2]. Other similar frameworks have also been proposed [3, 4]. The equations of triphasic theory are nonlinear, even in the range of infinitesimal strains. Therefore, numerical schemes are generally needed to solve all but the simplest problems using this framework.

A Triphasic Theory for the Swelling and Deformation Behaviors of Articular Cartilage

1991 ◽  
Vol 113 (3) ◽  
pp. 245-258 ◽  
Author(s):  
W. M. Lai ◽  
J. S. Hou ◽  
V. C. Mow
Keyword(s):  
Articular Cartilage ◽  
Charge Density ◽  
Fixed Charge ◽  
Solid Matrix ◽  
Confined Compression ◽  
Fixed Charge Density ◽  
Chemical Expansion ◽  
Chemical Potentials ◽  
Free Swelling ◽  
Triphasic Theory

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 μi = μoi + (RT/Mi)ln[γ±2c (c + c F)] and μW = μow + [p − RTφ(2c + cF) + Bwe]/ρTw, where R, T, Mi, γ±, φ, ρTw 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 σ = − pI − TcI + λs(trE)I + 2μsE, where E is the strain tensor and (λs,μs) are the Lame´ 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.

The Strain-Softening of Bovine Articular Cartilage Under Infinitesimal Deformation in Unconfined Compression

2001 ◽  
Author(s):  
Christopher C.-B. Wang ◽  
Nadeen O. Chahine ◽  
Terri-Ann N. Kelly ◽  
W. Michael Lai ◽  
Clark T. Hung ◽  
...  
Keyword(s):  
Articular Cartilage ◽  
Strain Softening ◽  
Articular Surface ◽  
Osmotic Swelling ◽  
Bovine Articular Cartilage ◽  
Softening Mechanism ◽  
Softening Behavior ◽  
Depth Direction ◽  
Swelling Equilibrium ◽  
Triphasic Theory

Abstract Significant strain-softening of articular cartilage along its depth direction has been observed in several studies [1,2], where the free-swelling equilibrium state of the tissue was taken to be the reference configuration for subsequent deformation. Microstructural models have been proposed to interpret this softening as buckling of pre-stressed collagen fibers [3,4]. In this study, we propose that a constitutive model which accounts for the disparity in tensile and compressive moduli of cartilage as well as the osmotic swelling response, can explain this experimentally observed strain-softening mechanism. The strain-softening behavior of the tissue is investigated experimentally and theoretically along three mutually perpendicular directions: directions parallel and perpendicular to the split line direction (1- and 2-direction), and normal to the articular surface (3-direction). A conewise nonlinear elasticity model is incorporated into the triphasic theory [5] to interpret this strain-softening in the context of Donnan osmotic swelling [6].

On the Thermodynamical Admissibility of the Triphasic Theory of Charged Hydrated Tissues

2009 ◽  
Vol 131 (4) ◽  
Author(s):  
J. M. Huyghe ◽  
W. Wilson ◽  
K. Malakpoor
Keyword(s):  
Free Energy ◽  
Articular Cartilage ◽  
Second Law Of Thermodynamics ◽  
Volume Effect ◽  
Closed Cycle ◽  
Chemical Expansion ◽  
Deformation Behaviors ◽  
Triphasic Theory ◽  
Loading And Unloading ◽  
Stress Term

The triphasic theory on soft charged hydrated tissues (Lai, W. M., Hou, J. S., and Mow, V. C., 1991, “A Triphasic Theory for the Swelling and Deformation Behaviors of Articular Cartilage,” ASME J. Biomech. Eng., 113, pp. 245–258) attributes the swelling propensity of articular cartilage to three different mechanisms: Donnan osmosis, excluded volume effect, and chemical expansion stress. The aim of this study is to evaluate the thermodynamic plausibility of the triphasic theory. The free energy of a sample of articular cartilage subjected to a closed cycle of mechanical and chemical loading is calculated using the triphasic theory. It is shown that the chemical expansion stress term induces an unphysiological generation of free energy during each closed cycle of loading and unloading. As the cycle of loading and unloading can be repeated an indefinite number of times, any amount of free energy can be drawn from a sample of articular cartilage, if the triphasic theory were true. The formulation for the chemical expansion stress as used in the triphasic theory conflicts with the second law of thermodynamics.

Triphasic Theory for Swelling Properties of Hydrated Charged Soft Biological Tissues

1990 ◽  
Vol 43 (5S) ◽  
pp. S134-S141 ◽  
Author(s):  
V. C. Mow ◽  
W. M. Lai ◽  
J. S. Hou
Keyword(s):  
Osmotic Pressure ◽  
Soft Tissues ◽  
Interstitial Water ◽  
Biological Tissues ◽  
Ion Concentration ◽  
Solid Matrix ◽  
Swelling Properties ◽  
Chemical Potentials ◽  
Triphasic Theory ◽  
Repulsive Forces

Swelling phenomenon of biological soft tissues, such as articular cartilage, depends on their fixed charge densities, the stiffness of their collagen-proteoglycan solid matrix and the ion concentration in the interstitium. Based on the thermodynamic continuum mixture theory, a multiphasic mixture model is developed to describe the equilibrium and transient swelling properties. For articular cartilages in a single salt environment (e.g. NaCl), a three phase model (triphasic theory) suffices to describe its swelling behavior. The three phases are: solid matrix, interstitial water and the mobile salt. The equations of motion in this theory shows that the driving forces for interstitial water and salt are the gradients of their chemical potentials. Constitutive equations for the chemical potentials of the phases and for the total stress under infinitesimal strain but large variation of salt concentration are presented based on the physico-chemical theory for polyelectrolytic solutions and continuum theory. Application of this theory to equilibrium problems yields the well known Donnan equilibrium ion distribution and osmotic pressure equations. The theory indicates that at equilibrium the applied load on the tissue is shared by 1) the solid matrix elastic stress due to deformation; 2) the Donnan osmotic pressure; and 3) the chemical expansion stress due to the charge-to-charge repulsive forces between the charged groups in the solid matrix. For the transient isometric swelling problem, the theory is shown to describe the experimentally observed responses very well.

Sign in / Sign up

Export Citation Format

Share Document