Conference Agenda

We report on the main results of a recent work [1], in which we describe the reorientation of reinforcing fibers in a biological medium by blending together concepts of continuum mechanics and statistical mechanics.

Assuming saturation, fiber-reinforced tissues –such as articular cartilage– are often described macroscopically as triphasic media consisting of an interstitial fluid, a solid matrix and the fiber phase. Typically, the fibers are oriented non-uniformly in space and are assumed to be embedded in the matrix, which can be viewed as a double-porosity medium [2]. Moreover, while the matrix is often hypothesized to be isotropic, the complex matrix-fibers is generally anisotropic due to the orientation and material properties of the fibers. This space-dependent anisotropy also impacts the fluid flow by intervening, for example, on permeability.

In the literature (see, e.g., [3–6]), the orientational pattern of the fibers is often described by means of a probability density distribution, defined as a function of a field of unit vectors [7], and employed to determine the directional averages of the constitutive functions of interest written for a generic spatial direction.

Usually, the functional form of the probability density distribution is interpreted as a known property, assigned from the outset. This applies also to some models of fiber reorientation in which the probability density distribution maintains the same dependence on the field of unit vectors, although being parameterized by variables evolving in time [8,9]. Such variability, attributed for instance to the direction of the fibers’ most probable orientation, describes a type of tissue structural reorganization referred to as remodeling.

In our work, we deviate from the models in which the probability density distribution is assigned a priori and, by assuming that the fiber dynamics is governed by a Fokker–Planck equation, we compute it as the stationary solution of this equation. Specifically, we hypothesize that, at the fiber scale, reorientation is a dynamic process of Langevin type driven by a deterministic contribution and a stochastic one: the former is the deformation of the matrix; the latter resolves the short-range reciprocal actions among the fibers. By following this approach, we are able to retrieve relevant probability density distributions, such as the von Mises one, as particular stationary solutions of the Fokker–Planck equation. We also account for other structural transformations occurring in the matrix by assuming that they are represented by inelastic distortions.

References

[1] Giammarini, A., Pastore, A., Grillo, A.: Math. Mech. Solids, In production.

[2] Tomic, A., Grillo, A., Federico, S.: IMA J. Appl. Math., 79: 1027–1059 (2014).

[3] Federico, S., Gasser, T.: J. Roy. Soc. Interface, 7 955–966 (2010).

[4] Holzapfel, G.A., et al.: J. Roy. Soc. Interface, 12(106) 20150188 (2015).

[5] Gizzi, A., Pandolfi, A., Vasta, M.: J. Eng. Math., 109(1) 211–226 (2017).

[6] Holzapfel, G.A., Ogden, R.W.: J. Elasticity, 129(1–2) 49–68 (2017).

[7] Lanir, Y.: J. Biomech., 16 1–12 (1983).

[8] Baaijens, F., et al.: J. Biomech., 43 166–175 (2010).

[9] Grillo, A., et al.: J. Eng. Math., 109(1) 139–172 (2018).

Skeletal muscles are a biological tissue with unique mechanical properties, characterised by anisotropic effects determined by its fiber-like microstructure. It features a nonlinearly elastic response and, most importantly, its physiological motion is determined by activation phenomena that need to be included in the continuum models for its behaviour. Of particular relevance is also the multiscale nature of the muscle tissue. Indeed, sarcomeres, the smallest contractile units, are bundled to form miofibrils, which in turn constitute the muscle in its large-scale configuration, with non-trivial geometric arrangements. Experimental evidence indicates that the mechanical response of the smaller units is rather different from and simpler than what can be measured for the whole muscle. Hence, a multiscale approach is necessary to arrive at both an understanding and a proper continuum modelling of muscle dynamics.
We first present a theoretical framework for anisotropic nonlinear elasticity based on the decomposition of strain and stress tensors on a tensorial basis adapted to the local anisotropy of the material. The definition of a local orthogonal basis for the space of second-order symmetric tensors allows to express a generic constitutive prescription as a vector field on a six-dimensional space, in which each of the six components represents an independent and objective material function. The presence of local anisotropies is reflected on material symmetries, and we consider the corresponding restrictions on material functions for some important crystal classes and for transversely isotropic and isotropic materials. This formalism aims at a clear and mechanically motivated organisation of the degrees of freedom involved in describing nonlinear elasticity, to facilitate the experimental identification of material functions for their constitutive characterisation.
Within the said framework, we start from experimental data about the passive response of muscle specimens to identify nonlinear material functions for the stress-strain relation, thereby following a data-driven approach. The peculiar nonlinearity of the response can be ascribed to the microscopic heterogeneity that leads to a progressive recruitment of different fibers, ultimately leading to the strain-hardening evidenced by the data. A careful analysis of this phenomenon, driven in particular by its anisotropic nature, gives also important information about the asymmetry of the material response under traction or compression. We propose a basic model in the context of anisotropic Cauchy elasticity to capture these phenomena and then propose a strategy to include an activation mechanism. Finally, it is important to consider the significant sources of dissipation that affect the muscular dynamics. For this reason, we explore how to include in the constitutive law terms that produce a viscus damping during the passive and active motion.

The structural and refractive capabilities of the eye depend on the mechanical properties and imperfections that are present in its constituents, e.g. cornea, aqueous and vitreous humor. In particular, the cornea is a hydrated biological tissue with the structural role of balancing the mechanical loading exerted by the physiological intraocular pressure, and it is characterized by an underlying collagen microstructure [1]. Moreover, the cornea can be classified as a multi-constituent material, since it is possible to distinguish at least a solid and a fluid phase. In particular, the solid phase comprises various constituents as keratocytes, collagen fibers and extracellular matrix, while the fluid phase is constituted primarily by water and carries out electrostatic charges, which circulate due to passive transport and active pumping action of the corneal endothelium. For instance, these ionic charges contribute to the osmotic pressure experienced by the tissue [2].

In our work, by adhering to a thermodynamically consistent framework [3], we describe the cornea as a poroelastic medium undergoing finite deformation, and we show some numerical simulations within physiological parameters for a comparison with the results already present in the literature. Moreover, we are interested in investigating the predictive capabilities of our model when describing the pathological condition known as keratoconus, in which, due to a mechanical failure, the cornea assumes a conical shape [4]. While the cause of keratoconus insurgence is currently unclear, previous studies linked the change in shape of the cornea with an alteration of the fiber microstructure, and with a degradation of the elastic properties of the tissue [5].

In this respect, our aim is to improve the mechanical description of the progress of the disorder by introducing the effects related to the circulation of the fluid. The degradation of the mechanical properties of the cornea is accounted by introducing a damage variable, as is done in the literature for the monophasic models [5]. The introduction of the fluid phase requires the prescription of appropriate boundary conditions, and we perform finite element numerical simulations to assess qualitative and quantitative results. We remark that, to the best of our knowledge, this is the first attempt at formulating a poroelastic model of the cornea that accounts for the fluid’s mechanical influence, and it represents a first step towards a complete model that incorporates both the mechanical and hydraulic role of the endothelium.

References

[1] Meek K. M. et al. Current Eye Research, Vol. 6, No. 7, Informa UK Limited, p. 841-846 (1987).

[2] Lewis N. P. et al., Structure, Vol. 18, No. 2, Elsevier BV, p. 239-245 (2010).

[3] Hassanizadeh S. M., Advances in Water Resources, Vol. 9, p. 207-222 (1986).

[4] Rabinowitz Y. S., Survey of Ophthalmology, Vol. 42, No. 4, Elsevier BV, p. 297-319 (1998).

[5] Pandolfi, A., Mechanics of Materials, Vol. 199, Elsevier BV (2024).