The combined impact of tissue heterogeneity and fixed charge for models of cartilage: the one-dimensional biphasic swelling model revisited

Sensitivity of cartilage mechanical behaviour to spatial variations in material properties

Articular cartilage tissue exhibits a spatial dependence in material properties that govern mechanical behaviour. A mathematical model of cartilage tissue under one dimensional confined compression testing is developed for normal tissue that takes account of these variations in material properties. Modifications to the model representative of a selection of mechanisms driving osteoarthritic cartilage are proposed, allowing application of the model to both physiological and pathophysiological, osteoarthritic tissue. Incorporating spatial variations into the model requires the specification of more parameters than are required in the absence of these variations. A global sensitivity analysis of these parameters is implemented to identify the dominant mechanisms of mechanical response, in normal and osteoarthritic cartilage tissue, to both static and dynamic loading. The most sensitive parameters differ between dynamic and static mechanics of the cartilage, and also differ between physiological and osteoarthritic pathophysiological cartilage. As a consequence changes in cartilage mechanics in response to alterations in cartilage structure are predicted to be contingent on the nature of loading and the health, or otherwise, of the cartilage. In particular the mechanical response of cartilage, especially deformation, is predicted to be much more sensitive to cartilage stiffness in the superficial zone given the onset of osteoarthritic changes to material properties, such as superficial zone increases in permeability and reductions in fixed charge. In turn this indicates that any degenerative changes in the stiffness associated with the superficial cartilage collagen mesh are amplified if other elements of osteoarthritic disease are present, which provides a suggested mechanism-based explanation for observations that the range of mechanical parameters representative of normal and osteoarthritic tissue can overlap substantially.

Modelling articular cartilage: the relative motion of two adjacent poroviscoelastic layers

In skeletal joints two layers of adjacent cartilage are often in relative motion. The individual cartilage layers are often modelled as a poroviscoelastic material. To model the relative motion, noting the separation of scales between the pore level and the macroscale, a homogenization based on multiple scale asymptotic analysis has been used in this study to derive a macroscale model for the relative translation of two poroviscoelastic layers separated by a very thin layer of fluid. In particular the fluid layer thickness is essentially zero at the macroscale so that the two poroviscoelastic layers are effectively in contact and their interaction is captured in the derived model via a set of interfacial conditions, including a generalization of the Beavers-Joseph condition at the interface between a viscous fluid and a porous medium. In the simplifying context of a uniform geometry, constant fixed charge density, a Newtonian interstitial fluid and a viscoelastic scaffold, modelled via finite deformation theory, we present preliminary simulations that may be used to highlight predictions for how oscillatory relative movement of cartilage under load influences the peak force the cartilage experiences and the extent of the associated deformations. In addition to highlighting such cartilage mechanics, the systematic derivation of the macroscale models will enable the study of how nanoscale cartilage physics, such as the swelling pressure induced by fixed charges, manifests in cartilage mechanics at much higher lengthscales.

Multiscale Strain Transfer in Cartilage

The transfer of stress and strain signals between the extracellular matrix (ECM) and cells is crucial for biochemical and biomechanical cues that are required for tissue morphogenesis, differentiation, growth, and homeostasis. In cartilage tissue, the heterogeneity in spatial variation of ECM molecules leads to a depth-dependent non-uniform strain transfer and alters the magnitude of forces sensed by cells in articular and fibrocartilage, influencing chondrocyte metabolism and biochemical response. It is not fully established how these nonuniform forces ultimately influence cartilage health, maintenance, and integrity. To comprehend tissue remodelling in health and disease, it is fundamental to investigate how these forces, the ECM, and cells interrelate. However, not much is known about the relationship between applied mechanical stimulus and resulting spatial variations in magnitude and sense of mechanical stimuli within the chondrocyte’s microenvironment. Investigating multiscale strain transfer and hierarchical structure-function relationships in cartilage is key to unravelling how cells receive signals and how they are transformed into biosynthetic responses. Therefore, this article first reviews different cartilage types and chondrocyte mechanosensing. Following this, multiscale strain transfer through cartilage tissue and the involvement of individual ECM components are discussed. Finally, insights to further understand multiscale strain transfer in cartilage are outlined.

Modelling the inclusion of swelling pressure in a tissue level poroviscoelastic model of cartilage deformation

Swelling pressure in the interstitial fluid within the pores of cartilage tissue is known to have a significant effect on the rheology of cartilage tissue. The swelling pressure varies rapidly within thin regions inside pores known as Debye layers, caused by the presence of fixed charge, as observed in cartilage. Tissue level calculation of cartilage deformation therefore requires resolution of three distinct spatial scales: the Debye lengthscale within individual pores; the lengthscale of an individual pore; and the tissue lengthscale. We use asymptotics to construct a leading order approximation to the swelling pressure within pores, allowing the swelling pressure to be systematically included within a fluid-solid interaction model at the level of pores in cartilage. We then use homogenization to derive tissue level equations for cartilage deformation that are very similar to those governing the finite deformation of a poroviscoelastic body. The equations derived permit the spatial variations in porosity and electric charge that occur in cartilage tissue. Example solutions are then used to confirm the plausibility of the model derived and to consider the impact of fixed charge heterogeneity, illustrating that local fixed charge loss is predicted to increase deformation gradients under confined compression away from, rather than at, the site of loss.