The Human Meniscus Behaves as a Functionally Graded Fractional Porous Medium under Confined Compression Conditions

Abstract

In this study, we observe that the poromechanical parameters in human meniscus vary spatially throughout the tissue. The response is anisotropic and the porosity is functionally graded. To draw these conclusions, we measured the anisotropic permeability and the “aggregate modulus” of the tissue, i.e., the stiffness of the material at equilibrium, after the interstitial fluid has ceased flowing. We estimated those parameters within the central portion of the meniscus in three directions (i.e., vertical, radial and circumferential) by fitting an enhanced model on stress relation confined compression tests. We noticed that a classical biphasic model was not sufficient to reproduce the observed experimental behaviour. We propose a poroelastic model based on the assumption that the fluid flow inside the human meniscus is described by a fractional porous medium equation analogous to Darcy’s law, which involves fractional operators. The fluid flux is then time-dependent for a constant applied pressure gradient (in contrast with the classical Darcy’s law, which describes a time independent fluid flux relation). We show that a fractional poroelastic model is well-suited to describe the flow within the meniscus and to identify the associated parameters (i.e., the order of the time derivative and the permeability). The results indicate that mean values of λβ,β in the central body are λβ=5.5443×10−10m4Ns1−β, β=0.0434, while, in the posterior and anterior regions, are λβ=2.851×10−10m4Ns1−β, β=0.0326 and λβ=1.2636×10−10m4Ns1−β, β=0.0232, respectively. Furthermore, numerical simulations show that the fluid flux diffusion is facilitated in the central part of the meniscus and hindered in the posterior and anterior regions.

On the Fractional Transversely Isotropic - Functionally Graded - Nature of Soft Biological Tissues Under Confined Consolidation

“…Whereas the internal layer of the meniscus relies on a hierarchical anisotropic network of collagen channels mainly aligned along the circumferential direction in which fluid flows during the physiological deformation process (Figure 1). In [5], we have shown that the permeability changes from the anterior/posterior horns to the body region, with the body region being more permeable. Here, we show that the permeability tensor in this region is also transversely isotropic and that the permeability in the circumferential direction (the preferential direction of the collagen channel) is higher than the other two directions (Figure 1).…”

“…Biological soft tissues, such as the meniscal tissue (Figure 1), exhibit a hierarchical porous solid matrix and an interstitial fluid flowing into the pores [1][2][3][4][5]. The overall mechanical behaviour depends not only on the solid matrix deformation but also on the movement of the fluid in and out of the pores during the deformation.…”

“…Our recent experimental findings on human meniscal samples show that the biphasic model was not sufficient to reproduce the observed experimental behaviour [5]. To address this, [17] proposed a poroelastic model wherein the pore pressure diffusion equation is derived by adopting a modified version of Darcy's law involving fractional derivatives.…”

On the microstructurally driven heterogeneous response of brain white matter to drug infusion pressure

“…To describe this type of porous-media behaviour, Barrera's group introduced Caputo fractional derivatives into Darcy's law, and implemented this fractional pore pressure diffusion equation in ABAQUS (Barrera 2021). This modelling scheme was then adopted to successfully predict the human meniscus behaviour under confined compression conditions against experiments (Bulle et al 2021). However, these methodologies most often adopted poroelastic theory, and thus integrating the solid and fluid phases in a continuum model (Jamal et al 2022b).…”

Fractional-Order Shell Theory: Formulation and Application to the Analysis of Nonlocal Cylindrical Panels

“…For example, size-effects within lowdimensional (micro-and nano-scale) structures, such as carbon nanotubes, emerge due to the long-range forces that become more prevalent at these scales [5,6]. Several studies have also demonstrated the presence of nonlocal interactions at the macro-scale in complex structures like porous solids [7][8][9], periodic structures [10,11], intentionally engineered nonlocal structures [12], and sandwiched structures [13]. The inability of the classical (local) continuum theory to capture the nonlocal effects has been one of the main drivers fostering the development of nonlocal continuum theories.…”