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Journal of Biomechanical Engineering | Volume 131 | Issue 4 | Technical Brief
Technical Briefs

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

[+] Author and Article Information
J. M. Huyghe1

Department of Biomedical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlandsj.m.r.huyghe@tue.nl

W. Wilson

Department of Biomedical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands

K. Malakpoor

Department of Mathematics, University of Amsterdam,1081TV Amsterdam, The Netherlands

1

Corresponding author.

J Biomech Eng 131(4), 044504 (Feb 03, 2009) (5 pages) doi:10.1115/1.3049531 History: Received October 17, 2007; Revised September 08, 2008; Published February 03, 2009
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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.
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References

1
Maroudas, A., and Bannon, C., 1981, “Measurement of Swelling Pressure in Cartilage and Comparison With the Osmotic Pressure of Constituent Proteoglycans,” Biorheology, 18 , pp. 619–632.
2
Wilson, W., Huyghe, J. M., and van Donkelaar, C. C., 2006, “A Composition-Based Cartilage Model for the Assessment of Compositional Changes During Cartilage Damage and Adaptation,” Osteoarthritis Cartilage, 14 , pp. 554–560.
3
Eisenberg, S. R., and Grodzinsky, A. J., 1985, “Swelling of Articular Cartilage and Other Connective Tissues: Electromechanical Forces,” J. Orthop. Res.  [CrossRef], 3 , pp. 148–159.
4
Eisenberg, S. R., and Grodzinsky, A. J., 1987, “The Kinetics of Chemically Induced Nonequilibrium Swelling of Articular Cartilage and Corneal Stroma,” ASME J. Biomech. Eng., 109 , pp. 79–89.
5
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.  [CrossRef], 113 , pp. 245–258.
6
Hascall, V. C., and Hascall, G. K., 1983, “Proteoglycans,” "Cell Biology of Extracellular Matrix", 1st ed., E.D.Hay, ed., Plenum, New York, pp. 39–60.
7
Muir, H., 1983, “Proteoglycans as Organizers of the Extracellular Matrix,” Biochem. Soc. Trans., 11 , pp. 613–622.
8
Jin, M. S., and Grodzinsky, A. J., 2001, “Effect of Electrostatic Interactions Between Glycosaminoglycans on the Shear Stiffness of Cartilage,” Macromolecules  [CrossRef], 34 , pp. 8330–8339.
9
Lanir, Y., 1987, “Biorheology and Fluid Flux in Swelling Tissues. I. Bicomponent Theory for Small Deformations, Including Concentration Effects,” Biorheology, 24 , pp. 173–187.
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Lai, W. M., Gu, W., Setton, L., and Mow, V. C., 1991, “Conditional Equivalence of Chemical Loading and Mechanical Loading on Articular Cartilage,” Adv. Bioeng., 20 , pp. 481–484.
11
Setton, L. A., Lai, W. M., and Mow, V. C., 1993, “Swelling Induced Residual Stress and the Mechanism of Curling in Articular Cartilage,” Adv. Bioeng., 26 , pp. 59–62.
12
Gu, W. Y., Lai, W. M., and Mow, V. C., 1993, “Transport of Fluid and Ions Through a Porous-Permeable Charged-Hydrated Tissue, and Streaming Potential Data on Normal Bovine Articular Cartilage,” J. Biomech.  [CrossRef], 26 (6), pp. 709–723.
13
Gu, W. Y., Lai, W. M., and Mow, V. C., 1997, “A Triphasic Analysis of Negative Osmotic Flows Through Charged Hydrated Soft Tissues,” J. Biomech.  [CrossRef], 30 (1), pp. 71–78.
14
Simon, B. R., Liable, J. P., Pflaster, D., Yuan, Y., and Krag, M. H., 1996, “A Poroelastic Finite Element Formulation Including Transport and Swelling in Soft Tissue Structures,” ASME J. Biomech. Eng.  [CrossRef], 118 , pp. 1–9.
15
Lai, W. M., Gu, W. Y., and Mow, V. C., 1998, “On the Conditional Equivalence of Chemical Loading and Mechanical Loading in Articular Cartilage,” J. Biomech., 31 (12), pp. 1181–1185.
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Sun, D. N., Gu, W. Y., Lai, W. M., and Mow, V. C., 1999, “A Mixed Finite Element Formulation of Triphasic Mechano-Electrochemical Theory for Charged, Hydrated Biological Soft Tissues,” Int. J. Numer. Methods Eng.  [CrossRef], 45 (10), pp. 1375–1402.
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Schugart, R. C., 2005, “Mathematical Models and Numerical Methods for Analysis of Mechanical and Chemical Loading in Articular Cartilage,” Ph.D. thesis, North Carolina State University, Raleigh.
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Huyghe, J. M., and Janssen, J. D., 1997, “Quadriphasic Mechanics of Swelling Incompressible Porous Media,” Int. J. Eng. Sci.  [CrossRef], 35 , pp. 793–802.
19
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20
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21
van Loon, R., Huyghe, J. M., Wijlaars, M. W., and Baaijens, F. P. T., 2003, “3d fe Implementation of an Incompressible Quadriphasic Mixture Model,” Int. J. Numer. Methods Eng.  [CrossRef], 57 , pp. 1243–1258.
22
van Meerveld, J., Molenaar, M. M., Huyghe, J. M., and Baaijens, F. T. P., 2003, “Analytical Solution of Compression, Free Swelling and Electrical Loading of Saturated Charged Porous Media,” Transp. Porous Media  [CrossRef], 50 , pp. 111–126.
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26
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27
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Figures

Grahic Jump Location
+Prediction of the triphasic theory of charged hydrated tissues. The ratio ΔW/ΔWmech of the net free energy ΔW produced in the closed loop shown in Fig. 1 over the mechanical work associated with phase 1 of the loop ΔWmech is plotted as a function of the external salt concentration step Δce. All positive values of ΔW/ΔWmech are unphysical.
View Large  |  Save Figure  |  Download Slide (.ppt)
Prediction of the triphasic theory of charged hydrated tissues. The ratio ΔW/ΔWmech of the net free energy ΔW produced in the closed loop shown in Fig. 1 over the mechanical work associated with phase 1 of the loop ΔWmech is plotted as a function of the external salt concentration step Δce. All positive values of ΔW/ΔWmech are unphysical.
Figure 2

Prediction of the triphasic theory of charged hydrated tissues. The ratio ΔW/ΔWmech of the net free energy ΔW produced in the closed loop shown in Fig. 1 over the mechanical work associated with phase 1 of the loop ΔWmech is plotted as a function of the external salt concentration step Δce. All positive values of ΔW/ΔWmech are unphysical.

Grahic Jump Location
+The cartilage sample is subjected to four subsequent changes in boundary conditions: (1) increase in salt concentration, (2) increase in strain, (3) decrease in salt concentration, and (4) decrease in strain. The tissue returns after the four changes to its initial state in the lower left corner of the square, E=0, ϕw=ϕ0w, and cF=c0F.
View Large  |  Save Figure  |  Download Slide (.ppt)
The cartilage sample is subjected to four subsequent changes in boundary conditions: (1) increase in salt concentration, (2) increase in strain, (3) decrease in salt concentration, and (4) decrease in strain. The tissue returns after the four changes to its initial state in the lower left corner of the square, E=0, ϕw=ϕ0w, and cF=c0F.
Figure 1

The cartilage sample is subjected to four subsequent changes in boundary conditions: (1) increase in salt concentration, (2) increase in strain, (3) decrease in salt concentration, and (4) decrease in strain. The tissue returns after the four changes to its initial state in the lower left corner of the square, E=0, ϕw=ϕ0w, and cF=c0F.

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