Abstract
The extracellular matrix of cartilage is a charged porous fibrous material. Transport phenomena in such a medium are very complex. In this study, solute diffusive flux and convective flux in porous fibrous media were investigated using a continuum mixture theory approach. The intrinsic diffusion coefficient of solute in the mixture was defined and its relation to drag coefficients was presented. The effect of mechanical loading on solute diffusion in cartilage under unconfined compression with a frictionless boundary condition was analyzed numerically using the model developed. Both strain-dependent hydraulic permeability and diffusivity were considered. Analyses and results show that (1) In porous media, the convective velocity for each solute phase is different. (2) The solute convection in tissue is governed by the relative convective velocity (i.e., relative to solid velocity). (3) Under the assumption that all the frictional interactions among solutes are negligible, the relative convective velocity for α-solute phase is equal to the relative solvent velocity multiplied by its convective coefficient (H α) which is also known as the hindrance factor in the literature. The relationship between the convective coefficient and the relative diffusivity of solute is presented. (4) Solute concentration profile within the cartilage sample depends on the phase of dynamic compression.
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References
Armstrong CG, Lai WM, Mow VC (1984) An analysis of the unconfined compression of articular cartilage. J Biomech Eng 106:165–173
Bird RB, Stewart WE, Lightfoot EN (2002) Transport phenomena. Wiley, New York
Bonassar LJ, Grodzinsky AJ, Srinivasan A, Davila SG, Trippel SB (2000) Mechanical and physicochemical regulation of the action of insulin-like growth factor-I on articular cartilage. Arch Biochem Biophys 379:57–63
Brenner H, Edwards DA (1993) Macrotransport processes. Butterworth-Heinemann, Stoneham
Burstein D, Gray ML, Hartman AL, Gipe R, Foy BD (1993) Diffusion of small solutes in cartilage as measured by nuclear magnetic resonance (NMR) spectroscopy and imaging. J Orthop Res 11:465–478
Deen WM (1998) Analysis of transport phenomena. Oxford University Press, New York
Deen WM (1987) Hindered transport of large molecules in liquid-filled pores. AIChE 33:1409–1425
Evans RC, Quinn TM (2006) Solute convection in dynamically compressed cartilage. J Biomech 39:1048–1055
Ferguson SJ, Ito K, Nolte LP (2004) Fluid flow and convective transport of solutes within the intervertebral disc. J Biomech 37:213–221
Garcia AM, Frank EH, Grimshaw PE, Grodzinsky AJ (1996) Contributions of fluid convection and electrical migration to transport in cartilage: relevance to loading. Arch Biochem Biophys 333:317–325
Grodzinsky AJ (1983) Electromechanical and physicochemical properties of connective tissue. Crit Rev Biomed Eng 9:133–199
Gu WY, Lai WM, Mow VC (1993) Transport of fluid and ions through a porous-permeable charged-hydrated tissue, and streaming potential data on normal bovine articular cartilage. J Biomech 26:709–723
Gu WY, Lai WM, Mow VC (1998) A mixture theory for charged-hydrated soft tissues containing multi-electrolytes: passive transport and swelling behaviors. J Biomech Eng 120:169–180
Gu WY, Yao H, Huang C-Y, Cheung HS (2003) New insight into deformation-dependent hydraulic permeability of gels and cartilage, and dynamic behavior of agarose gels in confined compression. J Biomech 36:593–598
Gu WY, Yao H, Vega AL, Flagler D (2004) Diffusivity of ions in agarose gels and intervertebral disc: effect of porosity. Ann Biomed Eng 32:1710–1717
Horner HA, Urban JP (2001) 2001 Volvo Award Winner in Basic Science Studies: effect of nutrient supply on the viability of cells from the nucleus pulposus of the intervertebral disc. Spine 26:2543–2549
Huyghe JM, Janssen JD (1997) Quadriphasic mechanics of swelling incompressible porous media. Int J Eng Sci 35:793–802
Johnston ST, Deen WM (1999) Hindered convection of proteins in agarose gels. J Membr Sci 153:271–279
Lai WM, Hou JS, Mow VC (1991) A triphasic theory for the swelling and deformation behaviors of articular cartilage. J Biomech Eng 113:245–258
Leddy HA, Guilak F (2003) Site-specific molecular diffusion in articular cartilage measured using fluorescence recovery after photobleaching. Ann Biomed Eng 31:753–760
Levenston ME, Eisenberg SR, Grodzinsky AJ (1998) A variational formulation for coupled physicochemical flows during finite deformations of charged porous media. Int Solids and Struct 35:4999–5019
Levenston ME, Frank EH, Grodzinsky AJ (1999) Electrokinetic and poroelastic coupling during finite deformations of charged porous media. J Appl Mech 66:323–333
Maroudas A (1979) Physicochemical properties of articular cartilage. In: Freeman MAR (ed) Adult Articular Cartilage. Pitman Medical, Tunbridge wells
Maroudas A (1975) Biophysical chemistry of cartilaginous tissues with special reference to solute and fluid transport. Biorheology 12: 233–248
Masaro L, Zhu XX (1999) Physical models of diffusion for polymer solutions, gels and solids. Prog Poly Sci 24:731–775
Mauck RL, Hung CT, Ateshian GA (2003) Modeling of neutral solute transport in a dynamically loaded porous permeable gel: implications for articular cartilage biosynthesis and tissue engineering. J Biomech Eng 125:602–614
Mow VC, Ratcliffe A, Poole AR (1992) Cartilage and diarthrodial joints as paradiams for hierarchical materials and structures. Biomaterials 13:67–97
Mow VC, Sun DN, Guo XE, Likhitpanichkul M, Lai WM (2002) Fixed negative charges modulate mechanical behavior and electrical signals in articular cartilage under unconfined compression a triphasic paradigm. In: Ehlers W, Bluhm J (eds) Porous media: theory, experiments and numerical application. Springer, Berlin Heidelberg New York
Nimer E, Schneiderman R, Maroudas A (2003) Diffusion and partition of solutes in cartilage under static load. Biophys Chem 106:125–146
O’Hara BP, Urban JP, Maroudas A (1990) Influence of cyclic loading on the nutrition of articular cartilage. Ann Rheum Dis 49:536–539
Quinn TM, Kocian P, Meister JJ (2000) Static compression is associated with decreased diffusivity of dextrans in cartilage explants. Arch Biochem Biophys 384:327–334
Quinn TM, Morel V, Meister JJ (2001) Static compression of articular cartilage can reduce solute diffusivity and partitioning: implications for the chondrocyte biological response. J Biomech 34:1463–1469
Quinn TM, Studer C, Grodzinsky AJ, Meister JJ (2002) Preservation and analysis of nonequilibrium solute concentration distributions within mechanically compressed cartilage explants. J Biochem Biophys Methods 31:83–95
Sun DN (2002) Theoretical and experimental investigations of the mechano-electrochemical properties of articular cartilage, a charged-hydrated-soft, biological tissue. PhD Dissertation, Columbia University, New York
Sun DN, Gu WY, Guo XE, Lai WM, Mow VC (1999) A mixed finite element formulation of triphasic mechano-electrochemical theory for charged, hydrated biological soft tissues. Int J Numer Methods Eng 45:1375–1402
Sun DN, Guo XE, Likhitpanichkul M, Lai WM, Mow VC (2004) The influence of the fixed negative charges on mechanical and electrical behaviors of articular cartilage under unconfined compression. J Biomech Eng 126:6–16
Torzilli PA, Adams TC, Mis RJ (1987) Transient solute diffusion in articular cartilage. J Biomech 20:203–214
Truskey GA, Fan Y, Katz DF (2004) Transport phenomena in biological systems. Pearson Education, Upper Saddle River
Urban JP, Holm S, Maroudas A (1978) Diffusion of small solutes into the intervertebral disc: as in vivo study. Biorheology 15:203–221
Yao H (2004) Physical signals and solute transport in cartilaginous tissues. PhD Dissertation, University of Miami, Coral Gables
Yao H, Gu WY (2004) Physical signals and solute transport in cartilage under dynamic unconfined compression: finite element analysis. Ann Biomed Eng 32:380–390
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Yao, H., Gu, W.Y. Convection and Diffusion in Charged Hydrated Soft Tissues: A Mixture Theory Approach. Biomech Model Mechanobiol 6, 63–72 (2007). https://doi.org/10.1007/s10237-006-0040-3
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DOI: https://doi.org/10.1007/s10237-006-0040-3