Skip to main content
SpringerLink
Log in

A hybrid biphasic mixture formulation for modeling dynamics in porous deformable biological tissues

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

The primary aim of this study is to establish the theoretical foundations for a solid–fluid biphasic mixture domain that can accommodate inertial effects and a viscous interstitial fluid, which can interface with a dynamic viscous fluid domain. Most mixture formulations consist of constituents that are either all intrinsically incompressible or compressible, thereby introducing inherent limitations. In particular, mixtures with intrinsically incompressible constituents can only model wave propagation in the porous solid matrix, whereas those with compressible constituents require internal variables, and related evolution equations, to distinguish the compressibility of the solid and fluid under hydrostatic pressure. In this study, we propose a hybrid framework for a biphasic mixture where the skeleton of the porous solid is intrinsically incompressible but the interstitial fluid is compressible. We define a state variable as a measure of the fluid volumetric strain. Within an isothermal framework, the Clausius–Duhem inequality shows that a function of state arises for the fluid pressure as a function of this strain measure. We derive jump conditions across hybrid biphasic interfaces, which are suitable for modeling hydrated biological tissues. We then illustrate this framework using confined compression and dilatational wave propagation analyses. The governing equations for this hybrid biphasic framework reduce to those of the classical biphasic theory whenever the bulk modulus of the fluid is set to infinity and inertia terms and viscous fluid effects are neglected. The availability of this novel framework facilitates the implementation of finite element solvers for fluid-structure interactions at interfaces between viscous fluids and porous-deformable biphasic domains, which can include fluid exchanges across those interfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Truesdell, C., Toupin, R.: Encyclopedia of Physics. Springer, Berlin (1960). vol. III/1, chap. The classical field theories

    MATH  Google Scholar 

  2. Green, A.E., Naghdi, P.M.: On basic equations for mixtures. Q. J. Mech. Appl. Math. 22(4), 427 (1969)

    Article  Google Scholar 

  3. Bowen, R.M.: Theory of Mixtures. Continuum Physics. Academic Press, New York (1976)

    Google Scholar 

  4. Bedford, A., Drumheller, D.S.: Theories of immiscible and structured mixtures. Int. J. Eng. Sci. 21(8), 863 (1983)

    Article  MathSciNet  Google Scholar 

  5. Mow, V.C., Kuei, S.C., Lai, W.M., Armstrong, C.G.: Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments. J. Biomech. Eng. 102(1), 73 (1980). https://doi.org/10.1115/1.3138202

    Article  Google Scholar 

  6. Oomens, C.W., van Campen, D.H., Grootenboer, H.J.: A mixture approach to the mechanics of skin. J. Biomech. 20(9), 877 (1987)

    Article  Google Scholar 

  7. Lai, W.M., Hou, J.S., Mow, V.C.: A triphasic theory for the swelling and deformation behaviors of articular cartilage. J. Biomech. Eng. 113(3), 245 (1991)

    Article  Google Scholar 

  8. Huyghe, J.M., Janssen, J.: Quadriphasic mechanics of swelling incompressible porous media. Int. J. Eng. Sci. 35(8), 793 (1997)

    Article  Google Scholar 

  9. Humphrey, J.D., Rajagopal, K.R.: A constrained mixture model for arterial adaptations to a sustained step change in blood flow. Biomech. Model. Mechanobiol. 2(2), 109 (2003). https://doi.org/10.1007/s10237-003-0033-4

    Article  Google Scholar 

  10. Ateshian, G.A., Maas, S., Weiss, J.A.: Multiphasic finite element framework for modeling hydrated mixtures with multiple neutral and charged solutes. J. Biomech. Eng. 135, 11 (2013). https://doi.org/10.1115/1.4024823

    Article  Google Scholar 

  11. Kenyon, D.E.: The theory of an incompressible solid-fluid mixture. Arch. Ration. Mech. Anal. 62(2), 131 (1976). https://doi.org/10.1007/bf00248468

    Article  MathSciNet  MATH  Google Scholar 

  12. Bowen, R.M.: Incompressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 18(9), 1129 (1980)

    Article  Google Scholar 

  13. Ateshian, G.A.: The role of interstitial fluid pressurization in articular cartilage lubrication. J. Biomech. 42, 1163 (2009). https://doi.org/10.1016/j.jbiomech.2009.04.040

    Article  Google Scholar 

  14. Ateshian, G.A., Warden, W.H., Kim, J.J., Grelsamer, R.P., Mow, V.C.: Finite deformation biphasic material properties of bovine articular cartilage from confined compression experiments. J. Biomech. 30, 1157 (1997). https://doi.org/10.1016/s0021-9290(97)85606-0

    Article  Google Scholar 

  15. Huang, C.Y., Soltz, M.A., Kopacz, M., Mow, V.C., Ateshian, G.A.: Experimental verification of the roles of intrinsic matrix viscoelasticity and tension-compression nonlinearity in the biphasic response of cartilage. J. Biomech. Eng. 125, 84 (2003). https://doi.org/10.1115/1.1531656

    Article  Google Scholar 

  16. Park, S., Krishnan, R., Nicoll, S.B., Ateshian, G.A.: Cartilage interstitial fluid load support in unconfined compression. J. Biomech. 36, 1785 (2003). https://doi.org/10.1016/s0021-9290(03)00231-8

    Article  Google Scholar 

  17. Smith, J.H., García, J.J.: A nonlinear biphasic model of flow-controlled infusion in brain: fluid transport and tissue deformation analyses. J. Biomech. 42(13), 2017 (2009). https://doi.org/10.1016/j.jbiomech.2009.06.014

    Article  Google Scholar 

  18. Lande, B., Mitzner, W.: Analysis of lung parenchyma as a parametric porous medium. J. Appl. Physiol. 101(3), 926 (2006). https://doi.org/10.1152/japplphysiol.01548.2005

    Article  Google Scholar 

  19. Ricken, T., Dahmen, U., Dirsch, O.: A biphasic model for sinusoidal liver perfusion remodeling after outflow obstruction. Biomech. Model. Mechanobiol. 9(4), 435 (2010). https://doi.org/10.1007/s10237-009-0186-x

    Article  Google Scholar 

  20. Chapelle, D., Gerbeau, J.F., Sainte-Marie, J., Vignon-Clementel, I.E.: A poroelastic model valid in large strains with applications to perfusion in cardiac modeling. Comput. Mech. 46(1), 91 (2009). https://doi.org/10.1007/s00466-009-0452-x

    Article  MathSciNet  MATH  Google Scholar 

  21. Cimrman, R., Rohan, E. Modelling heart tissue using a composite muscle model with blood perfusion. In Computational Fluid and Solid Mechanics 2003, Elsevier, pp. 1642–1646. (2003) https://doi.org/10.1016/b978-008044046-0.50400-0

  22. Ateshian, G.A., Costa, K.D., Hung, C.T.: A theoretical analysis of water transport through chondrocytes. Biomech. Model. Mechanobiol. 6(1–2), 91 (2006). https://doi.org/10.1007/s10237-006-0039-9

    Article  Google Scholar 

  23. Barocas, V.H., Tranquillo, R.T.: An anisotropic biphasic theory of tissue-equivalent mechanics: the interplay among cell traction, fibrillar network deformation, fibril alignment, and cell contact guidance. J. Biomech. Eng. 119(2), 137 (1997). https://doi.org/10.1115/1.2796072

    Article  Google Scholar 

  24. Guilak, F., Mow, V.C.: The mechanical environment of the chondrocyte: a biphasic finite element model of cell–matrix interactions in articular cartilage. J. Biomech. 33(12), 1663 (2000). https://doi.org/10.1016/s0021-9290(00)00105-6

    Article  Google Scholar 

  25. Causin, P., Guidoboni, G., Harris, A., Prada, D., Sacco, R., Terragni, S.: A poroelastic model for the perfusion of the lamina cribrosa in the optic nerve head. Math. Biosci. 257, 33 (2014). https://doi.org/10.1016/j.mbs.2014.08.002

    Article  MathSciNet  MATH  Google Scholar 

  26. Chen, X., Dunn, A.C., Sawyer, W.G., Sarntinoranont, M.: A biphasic model for micro-indentation of a hydrogel-based contact lens. J. Biomech. Eng. 129(2), 156 (2006). https://doi.org/10.1115/1.2472373

    Article  Google Scholar 

  27. Tandon, P.N., Autar, R.: Biphasic model of the trabecular meshwork in the eye. Med. Biol. Eng. Comput. 29(3), 281 (1991). https://doi.org/10.1007/bf02446710

    Article  Google Scholar 

  28. Biot, M.A., Temple, G.: Theory of finite deformations of porous solids. Indiana Univ. Math. J. 21(7), 597–620 (1972)

    Article  Google Scholar 

  29. Hou, J.S., Holmes, M.H., Lai, W.M., Mow, V.C.: Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verifications. J. Biomech. Eng. 111(1), 78–87 (1989). https://doi.org/10.1115/1.3168343

    Article  Google Scholar 

  30. Hou, J., Mow, V., Lai, W., Holmes, M.: An analysis of the squeeze-film lubrication mechanism for articular cartilage. J. Biomech. 25(3), 247–259 (1992). https://doi.org/10.1016/0021-9290(92)90024-u

    Article  Google Scholar 

  31. Chan, B., Donzelli, P.S., Spilker, R.L.: A mixed-penalty biphasic finite element formulation incorporating viscous fluids and material interfaces. Ann. Biomed. Eng. 28(6), 589–597 (2000). https://doi.org/10.1114/1.1305529

    Article  Google Scholar 

  32. Badia, S., Quaini, A., Quarteroni, A.: Coupling Biot and Navier-Stokes equations for modelling fluid-poroelastic media interaction. J. Comput. Phys. 228(21), 7986–8014 (2009). https://doi.org/10.1016/j.jcp.2009.07.019

    Article  MathSciNet  MATH  Google Scholar 

  33. Unnikrishnan, G., Unnikrishnan, V., Reddy, J.: Tissue–fluid interface analysis using biphasic finite element method. Comput. Methods Biomech. Biomed. Eng. 12(2), 165–172 (2009). https://doi.org/10.1080/10255840802372045

    Article  Google Scholar 

  34. Brinkman, H.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Flow Turbul. Combust. 1(1), 27 (1949)

    Article  Google Scholar 

  35. Bukac, M., Yotov, I., Zakerzadeh, R., Zunino, P.: Effects of poroelasticity on fluid-structure interaction in arteries: a computational sensitivity study. In: MS&A, pp. 197–220. Springer, (2015). https://doi.org/10.1007/978-3-319-05230-4_8

  36. Yang, M., Taber, L.A., Clark, E.B.: A nonlinear poroelastic model for the trabecular embryonic heart. J. Biomech. Eng. (1994)

  37. Berger, L., Bordas, R., Burrowes, K., Grau, V., Tavener, S., Kay, D.: A poroelastic model coupled to a fluid network with applications in lung modelling. Int. J. Numer. Methods Biomed. Eng. (2015). https://doi.org/10.1002/cnm.2731

    Article  Google Scholar 

  38. Tully, B., Ventikos, Y.: Coupling poroelasticity and CFD for cerebrospinal fluid hydrodynamics. IEEE Trans. Biomed. Eng. 56(6), 1644–1651 (2009). https://doi.org/10.1109/tbme.2009.2016427

    Article  Google Scholar 

  39. Bowen, R.M.: Compressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 20(6), 697–735 (1982). https://doi.org/10.1016/0020-7225(82)90082-9

    Article  MATH  Google Scholar 

  40. de Boer, R.: Theory of Porous Media. Springer, Berlin (2000). https://doi.org/10.1007/978-3-642-59637-7

    Book  MATH  Google Scholar 

  41. Mow, V.C., Lai, W.M.: Recent developments in synovial joint biomechanics. SIAM Rev. 22(3), 275–317 (1980)

    Article  MathSciNet  Google Scholar 

  42. Ateshian, G.A., Ricken, T.: Multigenerational interstitial growth of biological tissues. Biomech. Model. Mechanobiol. 9(6), 689–702 (2010). https://doi.org/10.1007/s10237-010-0205-y

    Article  Google Scholar 

  43. Ateshian, G.A.: On the theory of reactive mixtures for modeling biological growth. Biomech. Model. Mechanobiol. 6(6), 423–445 (2007). https://doi.org/10.1007/s10237-006-0070-x

    Article  Google Scholar 

  44. Holmes, M.H.: Finite deformation of soft tissue: analysis of a mixture model in uni-axial compression. J. Biomech. Eng. 108(4), 372–81 (1986). https://doi.org/10.1115/1.3138633

    Article  Google Scholar 

  45. Lai, W.M., Hou, J.S., Mow, V.C.: A triphasic theory for the swelling and deformation behaviors of articular cartilage. J. Biomech. Eng. 113(3), 245–258 (1991). https://doi.org/10.1115/1.2894880

    Article  Google Scholar 

  46. Ateshian, G.A., Albro, M.B., Maas, S., Weiss, J.A.: Finite element implementation of mechanochemical phenomena in neutral deformable porous media under finite deformation. J. Biomech. Eng. 133, 8 (2011). https://doi.org/10.1115/1.4004810

    Article  Google Scholar 

  47. Shim, J.J., Maas, S.A., Weiss, J.A., Ateshian, G.A.: A formulation for fluid-structure interactions in febio using mixture theory. J. Biomech. Eng. (2019). https://doi.org/10.1115/1.4043031

    Article  Google Scholar 

  48. Holmes, M.H., Mow, V.C.: The nonlinear characteristics of soft gels and hydrated connective tissues in ultrafiltration. J. Biomech. 23(11), 1145–56 (1990). https://doi.org/10.1016/0021-9290(90)90007-p

    Article  Google Scholar 

  49. Ateshian, G.A.: Mixture Theory for Modeling Biological Tissues: Illustrations from Articular Cartilage, pp. 1–51. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-41475-1_1

    Book  Google Scholar 

  50. Mansour, J.M., Mow, V.C.: The permeability of articular cartilage under compressive strain and at high pressures. J. Bone Joint Surg. Am. 58(4), 509–16 (1976)

    Article  Google Scholar 

  51. Mow, V.C., Mansour, J.M.: The nonlinear interaction between cartilage deformation and interstitial fluid flow. J. Biomech. 10(1), 31–9 (1977). https://doi.org/10.1016/0021-9290(77)90027-6

    Article  Google Scholar 

  52. Holmes, M.H., Lai, W.M., Mow, V.C.: Singular perturbation analysis of the nonlinear, flow-dependent compressive stress relaxation behavior of articular cartilage. J. Biomech. Eng. 107(3), 206–18 (1985). https://doi.org/10.1115/1.3138545

    Article  Google Scholar 

  53. Ateshian, G.A., Shim, J.J., Maas, S.A., Weiss, J.A.: Finite element framework for computational fluid dynamics in FEBio. J. Bio. Eng. (2018). https://doi.org/10.1115/1.4038716

    Article  Google Scholar 

  54. Eringen, A.C., Ingram, J.D.: A continuum theory of chemically reacting media–i. Int. J. Eng. Sci. 3(2), 197–212 (1965)

    Article  Google Scholar 

  55. Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30(1), 197–207 (1967)

    Article  Google Scholar 

  56. Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid I Low-frequency range. J. Acoust. Soc. Am. 28(2), 168–178 (1956). https://doi.org/10.1121/1.1908239

    Article  MathSciNet  Google Scholar 

  57. Maas, S.A., Ateshian, G.A., Weiss, J.A.: FEBio: History and advances. Annu. Rev. Biomed. Eng. 19(1), 279–299 (2017). https://doi.org/10.1146/annurev-bioeng-071516-044738

    Article  Google Scholar 

  58. Brooks, A.N., Hughes, T.J.: Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier-stokes equations. Comput. Methods Appl. Mech. Eng. 32(1–3), 199–259 (1982). https://doi.org/10.1016/0045-7825(82)90071-8

    Article  MathSciNet  MATH  Google Scholar 

  59. Bazilevs, Y., Calo, V.M., Hughes, T.J.R., Zhang, Y.: Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput. Mech. 43(1), 3–37 (2008). https://doi.org/10.1007/s00466-008-0315-x

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

Division of Graduate Education, U.S. National Science Foundation (Grant No. NSF GRFP DGE-16-44869). National Institute of General Medical Sciences, U.S. National Institutes of Health (Award No. R01GM083925).

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Columbia University, New York, NY, 10027, USA

    Jay J. Shim & Gerard A. Ateshian

Authors
  1. Jay J. Shim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gerard A. Ateshian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gerard A. Ateshian.

Ethics declarations

Disclaimer

The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health or the National Science Foundation.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shim, J.J., Ateshian, G.A. A hybrid biphasic mixture formulation for modeling dynamics in porous deformable biological tissues. Arch Appl Mech 92, 491–511 (2022). https://doi.org/10.1007/s00419-020-01851-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-020-01851-8

Keywords

Search

Navigation