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).

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).

Within the framework of "Hybrid Mixture Theory" [1], multicellular aggregates can be formalized as biphasic continuum media featuring a fluid phase, typically identified with an interstitial fluid carrying nutrients, and a solid phase representing, e.g., cells, protein filaments, extracellular matrix and other biological components.

Experimental evidence [2] shows that, under external loading, the solid phase of multicellular aggregates can undergo irreversible deformations, so that, after relaxation, these biological structures tend not to recover their shape prior to the application of the load [3]. Some authors [4,5] have proposed to attribute this behavior to internal structural reorganizations, which, in some degree, share similarities with the inelastic processes occurring in non-biological media. While these structural transformations belong to the class of phenomena named “remodeling” in the biomechanics community, their presumed resemblance with inelasticity has suggested, among other possibilities, to describe them by multiplicatively decomposing the deformation gradient tensor of a given multicellular aggregate into an elastic and an inelastic (remodeling) part (see, e.g., [3] and the references therein). On the other hand, also the fluid phase contributes to the overall dissipative behavior of the systems under study, for example through its exchange interactions with the solid phase. In the literature, the inelastic aspects of remodeling and the dissipation introduced by the fluid are usually addressed by resorting to models that are of grade zero in the distortions and that rely on flow laws of Darcian type for the fluid. In spite of their utility, however, these models are unable to resolve explicitly the interactions between the multicellular aggregates and the surfaces of the apparatuses with which they are in contact during experiments. An example of these interactions could be the change in ductility localized near the contact regions of a specimen with the testing device.

To cope with this lack of information, an interesting, yet relatively unexplored, research direction could be the introduction of higher-order descriptors both for the inelastic distortions and for the fluid kinematics. In particular, the flow of the interstitial fluid may be characterized by a non-negligible viscosity, thereby undermining the hypothesis of Darcian-like regimes. In this case, one possibility is to account for the so-called Brinkman correction in the expression of the overall fluid stress tensor [6].

In this presentation, we take a step in this direction by extending the model of Gurtin&Anand [7] to the theory of biphasic mixtures, and coupling it with a Darcy–Brinkman model for the fluid flow. We follow the paradigm of the Principle of Virtual Power [8] to obtain the dynamic equations of the system, and we discuss some related aspects of configurational and analytical mechanics. Finally, we compare numerically some preliminary results [9] with some well established models taken from the literature.

References:

[1] Bennethum, L.S., et al.: Transport in Porous Media 39(2): 187–225 (2000).

[2] Marmottant, P. et al.: Proc Natl Acad Sci USA. 106(41): 17271-5 (2009).

[3] Di Stefano, S., Giammarini, A., Giverso, C., et al.: Z. Angew. Math. Phys. 73: 79 (2022).

[4] Giverso, C., Preziosi, L.: Math Med Bio 29: 181-204 (2012).

[5] Ambrosi, D., Preziosi, L.: Biomech Model Mechanobiol 8(5): 397-413 (2009).

[6] Brinkman, H. C.: Applied Scientific Research. 1(1): 27–34 (1949).

[7] Gurtin, M. E., Anand, L.: Int. J. Plast., 21(12): 2297-2318 (2005).

[8] Germain, P.: SIAM J. Appl. Math. 25(3): 556–575 (1973).

[9] Giammarini, A., Pastore, A., Ramìrez-Torres, A., Grillo, A.: To be submitted.