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


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.


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.



deformation gradient


Faraday constant


universal gas constant


absolute temperature

ρ wT

true density of water




total intracellular mobile ion concentration


intracellular fluid pressure


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


osmotically active volume fraction in cell


cell radius in hypertonic reference state


cell volume


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):



total PCM mobile ion concentration


PCM mobile cation concentration


PCM mobile anion concentration


PCM fixed charge density (equivalent per unit extrafibrillar water)


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


diffusive drag coefficient between phase j and phase k


PCM volume change relative to hypertonic reference state


volumetric strain measure of PCM deformation, relative to cell


PCM fluid pressure


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):



total mobile ion concentration in external solution


Na+ or Cl concentration in external solution


anion (or cation) concentration of isotonic NaCl solution


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 radius in hypertonic reference state


chondron volume


chondron volume in isotonic solution


ratio of chondron to cell radius in hypertonic reference state


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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

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