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Homogenized modeling for vascularized poroelastic materials

  • Active behavior in soft matter and Mechanobiology
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Abstract

A new mathematical model for the macroscopic behavior of a material composed of a poroelastic solid embedding a Newtonian fluid network phase (also referred to as vascularized poroelastic material), with fluid transport between them, is derived via asymptotic homogenization. The typical distance between the vessels/channels (microscale) is much smaller than the average size of a whole domain (macroscale). The homogeneous and isotropic Biot’s equation (in the quasi-static case and in absence of volume forces) for the poroelastic phase and the Stokes’ problem for the fluid network are coupled through a fluid-structure interaction problem which accounts for fluid transport between the two phases; the latter is driven by the pressure difference between the two compartments. The averaging process results in a new system of partial differential equations that formally reads as a double poroelastic, globally mass conserving, model, together with a new constitutive relationship for the whole material which encodes the role of both pore and fluid network pressures. The mathematical model describes the mutual interplay among fluid filling the pores, flow in the network, transport between compartments, and linear elastic deformation of the (potentially compressible) elastic matrix comprising the poroelastic phase. Assuming periodicity at the microscale level, the model is computationally feasible, as it holds on the macroscale only (where the microstructure is smoothed out), and encodes geometrical information on the microvessels in its coefficients, which are to be computed solving classical periodic cell problems. Recently developed double porosity models are recovered when deformations of the elastic matrix are neglected. The new model is relevant to a wide range of applications, such as fluid in porous, fractured rocks, blood transport in vascularized, deformable tumors, and interactions across different hierarchical levels of porosity in the bone.

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Notes

  1. These two paper differ in some scaling assumptions concerning the drug transport analysis, but the double porosity fluid transport models, for a purely Newtonian blood vessels’ rheology and for macroscopically uniform media, coincide, as actually demonstrated in [35].

  2. The vector field \(\varvec{w}_p\) that satisfies Darcy’s law (8) actually represents the average relative fluid velocity in the porous medium and is also referred to as the discharge, flux, or filtration velocity. It is indeed related to the specific relative velocity \(\varvec{v}_p-\dot{\varvec{u}}\) via the interstitial porosity \(\phi\). However, since we conveniently carry out our analysis in terms of \(\varvec{w}_p\) hereinafter, we simply refer to \(\varvec{w}_p\) as the relative interstitial velocity.

  3. Our admissibility constraint (20) is equivalent to equation (2.a), page 7, [43], setting their \(\beta =0\) and considering that \(c_0\), \(\varvec{v}^1\), \(\varvec{v}^2\), and \(\varvec{q}\) are denoted by \(\tilde{\alpha }\), \(\dot{\varvec{u}}\), \(\varvec{u}_n\), \(\varvec{w}_p\) in our manuscript.

  4. Note that in the appendix reported in [13] the third rank tensor \(v^{ij}_p\) is equivalent to our \(\mathbb {{\mathcal {A}}}\) and its gradient to our \({\mathbb {M}}\). However, the latter is always identified to \({\mathbb {LC}}\), although relationship (138) actually rigorously holds only when \({\mathbb {C}}\) is locally constant.

  5. See Section 4.3.1, pages 24–25, [35], and the model (4.100-4.103) therein, which is in turn proved to be equivalent to Eqs. (54), (56), (70), (75) in [42]. In [35], the quantities \({\tilde{{\mathsf {K}}}}\), \({\tilde{{\mathsf {G}}}}\), \(p_p\), \(\varvec{v}_n\), \(\phi \varvec{v}_p\), \(|\varOmega _p|\), \(|\varGamma |\) are denoted by \({\mathsf {K}}\), \({\mathsf {E}}\), \(p_t\), \(\varvec{u}_n\), \(\varvec{u}_t\), \(|\varOmega _t|\), S, respectively. Both in [42] and [35], only specific averages are used, and they are denoted as we denote non-specific averages, i.e. \(\displaystyle \left\langle \,\bullet \,\right\rangle _k\).

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Acknowledgements

The authors are indebted to Davide Ambrosi, Alf Gerisch, and Rebecca Shipley for useful hints and discussions concerning the content of this manuscript.

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Authors and Affiliations

  1. Departamento de Mecanica de los Medios Continuos y T. Estructuras, E.T.S. de caminos, canales y puertos, Universidad Politecnica de Madrid, Calle Profesor Aranguren S/N, 28040, Madrid, Spain

    Raimondo Penta & José Merodio

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  1. Raimondo Penta
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  2. José Merodio
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Correspondence to Raimondo Penta.

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The authors declare that they have no conflict of interest.

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This study was funded by the Ministry of Economy in Spain, under the project reference DPI2014-58885-R.

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Penta, R., Merodio, J. Homogenized modeling for vascularized poroelastic materials. Meccanica 52, 3321–3343 (2017). https://doi.org/10.1007/s11012-017-0625-1

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