Advertisement

A Mechano-chemical Model for the Passive Swelling Response of an Isolated Chondron under Osmotic Loading

  • Mansoor A. HaiderEmail author
  • Richard C. Schugart
  • Lori A. Setton
  • Farshid Guilak
Original Paper

Abstract

The chondron is a distinct structure in articular cartilage that consists of the chondrocyte and its pericellular matrix (PCM), a narrow tissue region surrounding the cell that is distinguished by type VI collagen and a high glycosaminoglycan concentration relative to the extracellular matrix. We present a theoretical mechano-chemical model for the passive volumetric response of an isolated chondron under osmotic loading in a simple salt solution at equilibrium. The chondrocyte is modeled as an ideal osmometer and the PCM model is formulated using triphasic mixture theory. A mechano-chemical chondron model is obtained assuming that the chondron boundary is permeable to both water and ions, while the chondrocyte membrane is selectively permeable to only water. For the case of a neo-Hookean PCM constitutive law, the model is used to conduct a parametric analysis of cell and chondron deformation under hyper- and hypo-osmotic loading. In combination with osmotic loading experiments on isolated chondrons, model predictions will aid in determination of pericellular fixed charge density and its relative contribution to PCM mechanical properties.

Keywords

Articular Cartilage Pericellular Matrix Normalize Cell Volume Osmotic Loading Simple Salt Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

F

deformation gradient

Fc

Faraday constant

R

universal gas constant

T

absolute temperature

ρ wT

true density of water

Cell:

 

CC

total intracellular mobile ion concentration

pC

intracellular fluid pressure

J

cell volumetric strain (change in cell volume relative to hypertonic reference state)

\(\tilde{R}\)

osmotically active volume fraction in cell

RC0

cell radius in hypertonic reference state

VC

cell volume

VCiso

cell volume in isotonic solution

\({\left(\tilde{\mu}^{w} \right)}^{C}, {\left({ {\tilde{\mu}}^{w}_{0} } \right)}^{C} \)

intracellular water chemical potential at general and hypertonic reference states, respectively

σ C

intracellular mixture stress

ϕ C

intracellular osmotic coefficient

Pericellular matrix (PCM):

 

CP

total PCM mobile ion concentration

c+

PCM mobile cation concentration

cc

PCM mobile anion concentration

cF

PCM fixed charge density (equivalent per unit extrafibrillar water)

cF0

PCM fixed charge density in hypertonic reference state (equivalent per unit extrafibrillar water)

fjk

diffusive drag coefficient between phase j and phase k

\(\bar{J}\)

PCM volume change relative to hypertonic reference state

\({\hat{J}}\)

volumetric strain measure of PCM deformation, relative to cell

pP

PCM fluid pressure

vj

velocity of phase j

γ P+, γ P

PCM mean cation and anion activity coefficients, respectively

\({\left(\tilde{\mu}^{w}\right)}^{P}, {\left(\tilde{\mu}^{w}_{0} \right)}^{P} \)

PCM water chemical potentials at general and hypertonic reference states, respectively

\({\left(\tilde{\mu}^{ + }\right)}^{P}, {\left(\tilde{\mu}^{ + }_{0}\right)}^{P} \)

PCM cation electrochemical potentials at general and hypertonic reference states, respectively

\({\left(\tilde{\mu}^{ - }\right)}^{P}, {\left(\tilde{\mu}^{ - }_{0} \right)}^{P} \)

PCM anion electrochemical potentials at general and hypertonic reference states, respectively

ρ j

apparent density for phase j

σ P

PCM mixture stress

ϕ P

PCM osmotic coefficient

φ w0

PCM porosity in hypertonic reference state

ψ P

PCM electric potential

External solution (NaCl):

 

C*

total mobile ion concentration in external solution

c*

Na+ or Cl concentration in external solution

c*iso

anion (or cation) concentration of isotonic NaCl solution

p*

fluid pressure in external solution

γ *+, γ *

mean cation and anion activity coefficients in external solution, respectively

\( \tilde{\mu}^{w*}, \tilde{\mu}^{w*}_{0} \)

water chemical potentials in external solution at general and hypertonic reference states, respectively

\( \tilde{\mu}^{ + *}, \tilde{\mu}^{ + *}_{0} \)

cation electrochemical potentials in external solution at general and hypertonic reference states, respectively

\(\tilde{\mu}^{ - *}, \tilde{\mu}^{ - *}_{0} \)

anion electrochemical potentials in external solution at general and hypertonic reference states, respectively

ϕ *

osmotic coefficient in external solution

ψ *

electric potential in external solution

Chondron

 

RCh0

chondron radius in hypertonic reference state

VCh

chondron volume

VChiso

chondron volume in isotonic solution

δ0

ratio of chondron to cell radius in hypertonic reference state

References

  1. Alexopoulos LG, Haider MA, Vail TP, Guilak F (2003) Alterations in the mechanical properties of the human chondrocyte pericellular matrix with osteoarthritis. J Biomech Eng 125:323–333CrossRefPubMedGoogle Scholar
  2. Alexopoulos LG, Williams GM, Upton ML, Setton LA, Guilak F (2005) Osteoarthritic changes in the biphasic mechanical properties of the chondrocyte pericellular matrix in articular cartilage. J Biomech 38:509–517CrossRefPubMedGoogle Scholar
  3. Armstrong CG, Bahrani AS, Gardner DL (1979) In vitro measurement of articular cartilage deformations in the intact human hip joint under load. J Bone Joint Surg Am 61:744–755PubMedGoogle Scholar
  4. Ateshian GA, Chahine NO, Basalo IM, Hung CT (2004) The correspondence between equilibrium biphasic and triphasic material properties in mixture models of articular cartilage. J Biomech 37:391–400CrossRefPubMedGoogle Scholar
  5. Ateshian GA, Likhitpanichkul M, Hung CT (2006) A mixture theory analysis for passive transport in osmotic loading of cells. J Biomech 39:464–475PubMedGoogle Scholar
  6. Benninghoff A (1925) Form und Bau der Gelenkknorpel in ihren Beziehungen Zur Funktion. Zweiter Teil: der Aufbau des Gelenkknorpels in seinen Beziehungen zur Funktion 2:783Google Scholar
  7. Buschmann MD, Hunziker EB, Kim YJ, Grodzinsky AJ (1996) Altered aggrecan synthesis correlates with cell and nucleus structure in statically compressed cartilage. J Cell Sci 109:499–508PubMedGoogle Scholar
  8. Bush PG, Hall AC (2001) The osmotic sensitivity of isolated and in situ bovine articular chondrocytes. J Orthop Res 29:768–778CrossRefGoogle Scholar
  9. Erickson GR, Northrup DL, Guilak F (2003) Hypo-osmotic stress induces calcium-dependent actin reorganization in articular chondrocytes. Osteoarth Cartil 11:187–197CrossRefGoogle Scholar
  10. Flahiff CM, Narmoneva DA, Huebner JL, Kraus VB, Guilak F, (2002) Osmotic loading to determine the intrinsic material properties of guinea pig knee cartilage. J Biomech 35:1285–1290CrossRefPubMedGoogle Scholar
  11. Greco F, Specchia N, Falciglia F, Toesca A, Nori S (1992) Ultrastructural analysis of the adaptation of articular cartilage to mechanical stimulation. Ital J Orthop Traumatol 18:311–321PubMedGoogle Scholar
  12. 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 26:709–723CrossRefPubMedGoogle Scholar
  13. 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 120:169–180PubMedCrossRefGoogle Scholar
  14. Guilak F (1995) Compression-induced changes in the shape and volume of the chondrocyte nucleus. J Biomech 28:1529–1542CrossRefPubMedGoogle Scholar
  15. Guilak F, Erickson GR, Ting-Beall HP (2002) The effects of osmotic stress on the viscoelastic and physical properties of articular chondrocytes. Biophys J 82:720–727PubMedGoogle Scholar
  16. Guilak F, Jones WR, Ting-Beall HP, Lee GM (1999) The deformation behavior and mechanical properties of chondrocytes in articular cartilage. Osteoarthr Cartil 7(1):59–70CrossRefPubMedGoogle Scholar
  17. Guilak F, Mow VC (2000) The mechanical environment of the chondrocyte: a biphasic finite element model of cell-matrix interactions in articular cartilage. J Biomech 33:1663–1673CrossRefPubMedGoogle Scholar
  18. Guilak F, Ratcliffe A, Lane N, Rosenwasser MP, Mow VC (1994) Mechanical and biochemical changes in the superficial zone of articular cartilage in canine experimental osteoarthritis. J Orthop Res 12:474–484CrossRefGoogle Scholar
  19. Guilak F, Ratcliffe A, Mow VC (1995) Chondrocyte deformation and local tissue strain in articular cartilage: a confocal microscopy study. J Orthop Res 13:410–421CrossRefPubMedGoogle Scholar
  20. Guilak F, Sah RL, Setton LA (1997) Physical regulation of cartilage metabolism. In: Mow VC, Hayes WC (ed) Basic orthopaedic biomechanics. Lippincott-Raven, Philadelphia, pp 179–207Google Scholar
  21. Haider MA (2004) A radial biphasic model for local cell-matrix mechanics in articular cartilage. SIAM J Appl Math 64:1588–1608CrossRefzbMATHMathSciNetGoogle Scholar
  22. Helminen HJ, Jurvelin J, Kiviranta I, Paukkonen K, Saamanen AM, Tammi M (1987) Joint loading effects on articular cartilage: a historical review. In: Helminen HJ et al (eds) Joint loading. Wright, Bristol, pp 1–25Google Scholar
  23. Hing WA, Sherwin AF, Ross JM, Poole CA (2002) The influence of the pericellular microenvironment on the chondrocyte response to osmotic challenge. Osteoarthr Cartil 10:297–307CrossRefPubMedGoogle Scholar
  24. Hoffmann EK, Dunham PB (1995) Membrane mechanisms and intracellular signalling in cell volume regulation. Int Rev Cytol 161: 173–262PubMedGoogle Scholar
  25. Hung CT, Jamieson KV, Roy R, Wong DD, Chao PG, Sun DN, Guo XE (2001) Comparison of transient chondrocyte swelling and shrinking behavior. Trans Orthop Res Soc 26:559Google Scholar
  26. Jones WR, Ting-Beall HP, Lee GM, Kelley SS, Hochmuth RM, Guilak F (1999) Alterations in the Young’s modulus and volumetric properties of chondrocytes isolated from normal and osteoarthritic human cartilage. J Biomech 32:119–127CrossRefPubMedGoogle Scholar
  27. Lai WM, Gu WY, Mow VC (1998) On the conditional equivalence of chemical loading and mechanical loading on articular cartilage. J Biomech 31:1181–1185CrossRefPubMedGoogle Scholar
  28. Lai WM, Hou JS, Mow VC (1991) A triphasic theory for the swelling and deformation behaviors of articular cartilage. J Biomech Eng 113:245–258PubMedCrossRefGoogle Scholar
  29. Lee GM, Paul TA, Slabaugh M, Kelley SS (2000) The incidence of enlarged chondrons in normal and osteoarthritic human cartilage and their relative matrix density. Osteoarthr Cartil 8:44–52CrossRefPubMedGoogle Scholar
  30. Lucke B, McCutcheon M (1932) The living cell as an osmotic system and its permeability to water. Physiol Rev 12:68–138Google Scholar
  31. Maroudas A (1979) Physicochemical properties of articular cartilage. In: Freeman M (eds) Adult articular cartilage. Pitman Medical, Tunbridge Wells, pp 215–290Google Scholar
  32. Maroudas A, Ziv I, Weisman N, Venn M. (1985) Studies of hydration and swelling pressure in normal and osteoarthritic cartilage. Biorheology 22:159–169PubMedGoogle Scholar
  33. McGann LE, Stevenson M, Muldrew K, Schachar N (1988) Kinetics of osmotic water movement in chondrocytes isolated from articular cartilage and applications to cryopreservation. J Orthop Res 6: 109–115CrossRefPubMedGoogle Scholar
  34. Mobasheri A, Mobasheri R, Francis MJO, Trujillo E, Delarosa DA, Martinvasallo P (1998) Ion transport in chondrocytes - membrane transporters involved in intracellular ion homeostasis and the regulation of cell volume, free [Ca2+] and pH. Histol Histopathol 13: 893–910PubMedGoogle Scholar
  35. Mow VC, Holmes MH, Lai WM (1984) Fluid transport and mechanical properties of articular cartilage: a review. J Biomech 17:377–394CrossRefPubMedGoogle Scholar
  36. Mow VC, Ratcliffe A, Poole AR (1992) Cartilage and diarthrodial joints as paradigms for hierarchical materials and structures. Biomaterials 13:67–97CrossRefPubMedGoogle Scholar
  37. Mow VC, Sun DN, Guo XE, Hung CT, Lai WM (1999) Chondrocyte-extracellular matrix interactions during osmotic swelling. ASME Bioeng Conf BED42:133–134Google Scholar
  38. Ponder E (1948) Hemolysis and related phenomena. Grune and Stratton, New YorkGoogle Scholar
  39. Poole CA, Flint MH, Beaumont BW (1987) Chondrons in cartilage: ultrastructural analysis of the pericellular microenvironment in adult human articular cartilages. J Orthop Res 5:509–522CrossRefPubMedGoogle Scholar
  40. Poole CA, Flint MH, Beaumont BW (1988) Chondrons extracted from canine tibial cartilage: preliminary report on their isolation and structure. J Orthop Res 6:408–419CrossRefPubMedGoogle Scholar
  41. Poole CA, Honda T, Skinner SJ, Schofield JR, Hyde KF, Shinkai H (1990) Chondrons from articular cartilage (II): analysis of the glycosaminoglycans in the cellular microenvironment of isolated canine chondrons. Connect Tissue Res 24:319–330PubMedCrossRefGoogle Scholar
  42. Setton LA, Mow VC, Muller FJ, Pita JC, Howell DS (1994) Mechanical properties of canine articular cartilage are significantly altered following transection of the anterior cruciate ligament. J Orthop Res 12:451–463CrossRefGoogle Scholar
  43. Szirmai JA (1974) The concept of the chondron as a biomechanical unit. In: Hartmann F(eds) Biopolymer und Biomechanik von Bindegewebssystemen. Academic, Berlin, pp 87Google Scholar
  44. Trickey WR, Vail TP, Guilak F (2004) The role of the cytoskeleton in the viscoelastic properties of human articular chondrocytes. J Orthop Res 22(1):131–139CrossRefPubMedGoogle Scholar
  45. Urban JP, Hall AC, Gehl KA (1993) Regulation of matrix synthesis rates by the ionic and osmotic environment of articular chondrocytes. J Cell Physiol 154:262–270CrossRefPubMedGoogle Scholar
  46. Wang CCB, Guo XE, Sun DN, Mow VC, Ateshian GA, Hung CT (2002) The functional environment of chondrocytes within cartilage subjected to compressive loading: a theoretical and experimental approach. Biorheology 39:11–25PubMedGoogle Scholar
  47. Wong M, Wuethrich P, Buschmann MD, Eggli P, Hunziker E (1997) Chondrocyte biosynthesis correlates with local tissue strain in statically compressed adult articular cartilage. J Orthop Res 15: 189–196CrossRefPubMedGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Mansoor A. Haider
    • 1
    Email author
  • Richard C. Schugart
    • 4
  • Lori A. Setton
    • 2
    • 3
  • Farshid Guilak
    • 2
    • 3
  1. 1.Department of MathematicsNorth Carolina State UniversityRaleighUSA
  2. 2.Department of Surgery, Division of Orthopaedic SurgeryDuke University Medical CenterDurhamUSA
  3. 3.Department of Biomedical EngineeringDuke UniversityDurhamUSA
  4. 4.Mathematical Biosciences InstituteThe Ohio State UniversityColumbusUSA

Personalised recommendations