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A poroelastic model valid in large strains with applications to perfusion in cardiac modeling

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Abstract

This paper is motivated by the modeling of blood flows through the beating myocardium, namely cardiac perfusion. As in other works, perfusion is modeled here as a flow through a poroelastic medium. The main contribution of this study is the derivation of a general poroelastic model valid for a nearly incompressible medium which experiences finite deformations. A numerical procedure is proposed to iteratively solve the porous flow and the nonlinear poroviscoelastic problems. Three-dimensional numerical experiments are presented to illustrate the model. The first test cases consist of typical poroelastic configurations: swelling and complete drainage. Finally, a simulation of cardiac perfusion is presented in an idealized left ventricle embedded with active fibers. Results show the complex temporal and spatial interactions of the muscle and blood, reproducing several key phenomena observed in cardiac perfusion.

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References

  1. Spaan J, Kolyva C, van den Wijngaard J, ter Wee R, van Horssen P, Piek J, Siebes M (2008) Coronary structure and perfusion in health and disease. Phil Trans R Soc A 366(1878): 3137–3153

    Article  Google Scholar 

  2. Horssen P, Wijngaard JPHM, Siebes M, Spaan JAE (2009) Improved regional myocardial perfusion measurement by means of an imaging cryomicrotome. In: 4th European conference of the international federation for medical and biological engineering. Springer, New York, pp 771–774

  3. Westerhof N, Boer C, Lamberts RR, Sipkema P (2006) Cross-talk between cardiac muscle and coronary vasculature. Physiol Rev 86(4): 1263–1308

    Article  Google Scholar 

  4. Smith N, Kassab G (2001) Analysis of coronary blood flow interaction with myocardial mechanics based on anatomical models. Phil Trans R Soc Lond A 359: 1251–1262

    Article  Google Scholar 

  5. Smith N (2004) A computational study of the interaction between coronary blood flow and myocardial mechanics. Physiol Meas 25(4): 863–877

    Article  Google Scholar 

  6. Terzaghi K (1943) Theoretical soil mechanics. Wiley, New York

    Book  Google Scholar 

  7. Biot MA (1956) Theory of propagation of elastic waves in a fluid-saturated porous solid. II Higher frequency range. J Acoust Soc Am 28: 179–191

    Article  MathSciNet  Google Scholar 

  8. Biot MA (1972) Theory of finite deformations of porous solids. Indiana Univ Math J 21: 597–620

    Article  MathSciNet  Google Scholar 

  9. May-Newman K, McCulloch AD (1998) Homogenization modeling for the mechanics of perfused myocardium. Prog Biophys Mol Biol 69: 463–481

    Article  Google Scholar 

  10. Almeida E, Spilker R (1998) Finite element formulations for hyperelastic transversely isotropic biphasic soft tissues. Comput Methods Appl Mech Eng 151(3–4): 513–538

    Article  MATH  Google Scholar 

  11. Yang Z, Smolinski P (2006) Dynamic finite element modeling of poroviscoelastic soft tissue. Comput Methods Biomech Biomed Eng 9(1): 7–16

    Article  Google Scholar 

  12. Borja R (2006) On the mechanical energy and effective stress in saturated and unsaturated porous continua. Int J Solids Struct 43(6): 1764–1786

    Article  MATH  Google Scholar 

  13. Badia S, Quaini A, Quarteroni A (2009) Coupling Biot and Navier–Stokes equations for modelling fluid–poroelastic media interaction. J Comput Phys (to appear)

  14. Koshiba N, Ando J, Chen X, Hisada T (2007) Multiphysics simulation of blood flow and LDL transport in a porohyperelastic arterial wall model. J Biomech Eng 129: 374

    Article  Google Scholar 

  15. Calo V, Brasher N, Bazilevs Y, Hughes T (2008) Multiphysics model for blood flow and drug transport with application to patient-specific coronary artery flow. Comput Mech 43(1): 161–177

    Article  MATH  Google Scholar 

  16. Feenstra P, Taylor C (2009) Drug transport in artery walls: a sequential porohyperelastic-transport approach. Comput Methods Biomech Biomed Eng 12(3): 263–276

    Article  Google Scholar 

  17. Huyghe JM, van Campen DH (1991) Finite deformation theory of hierarchically arranged porous solids: I. Balance of mass and momentum. Int J Eng Sci 33(13): 1861–1871

    Article  Google Scholar 

  18. Huyghe JM, van Campen DH (1991) Finite deformation theory of hierarchically arranged porous solids: II. Constitutive behaviour. Int J Eng Sci 33(13): 1861–1871

    Article  Google Scholar 

  19. Cimrman R, Rohan E (2003) Modelling heart tissue using a composite muscle model with blood perfusion. In: Bathe KJ (ed) Computational fluid and solid mechanics, 2nd MIT conference, pp 1642–1646

  20. Vankan W, Huyghe J, Janssen J, Huson A (1997) A finite element mixture model for hierarchical porous media. Int J Numer Methods Eng 40: 193–210

    Article  Google Scholar 

  21. Coussy O (1995) Mechanics of porous continua. Wiley, New York

    MATH  Google Scholar 

  22. de Buhan P, Chateau X, Dormieux L (1998) The constitutive equations of finite-strain poroelasticity in the light of a micro-macro approach. Eur J Mech A/Solids 17(6): 909–922

    Article  MATH  MathSciNet  Google Scholar 

  23. Ciarlet PG, Geymonat G (1982) Sur les lois de comportement en élasticité non linéaire. CRAS Série II 295: 423–426

    MATH  MathSciNet  Google Scholar 

  24. Sainte-Marie J, Chapelle D, Cimrman R, Sorine M (2006) Modeling and estimation of the cardiac electromechanical activity. Comput Struct 84: 1743–1759

    Article  MathSciNet  Google Scholar 

  25. Brezzi F, Fortin M (1991) Mixed and hybrid finite element method. Springer, New York

    Google Scholar 

  26. Irons B, Tuck R (1969) A version of the Aitken accelerator for computer implementation. Int J Numer Methods Eng 1: 275–277

    Article  MATH  Google Scholar 

  27. Bestel J, Clément F, Sorine M (2001) A biomechanical model of muscle contraction. In: Niessen WJ, Viergever MA (eds) Lectures Notes in Computer Science, vol 2208. Springer-Verlag, New York, pp 1159–1161

    Google Scholar 

  28. Krejčí P, Sainte-Marie J, Sorine M, Urquiza J (2005) Solutions to muscle fiber equations and their long time behaviour. Nonlinear Anal: Real World Anal 7(4): 535–558

    Article  Google Scholar 

  29. Chapelle D, Le Tallec P, Moireau P (2009) Mechanical modeling of the heart contraction. (in preparation)

  30. Chapelle D, Fernánde M, Gerbeau J-F, Moireau P, Sainte- Marie J, Zemzemi N (2009) Numerical simulation of the electromechanical activity of the heart. In: FIMH, vol 5528 of Lecture Notes in Computer Science, pp 357–365

  31. Boulakia M, Cazeau S, Fernández MA, Gerbeau J-F, Zemzemi N (2009) Mathematical modeling of electrocardiograms: a numerical study. Research Report RR-6977, INRIA. URL http://hal.inria.fr/inria-00400490/en/

  32. Zinemanas D, Beyar R, Sideman S (1995) An integrated model of LV muscle mechanics, coronary flow, and fluid and mass transport. Am J Physiol Heart Circ Physiol 268(2): H633–H645

    Google Scholar 

  33. Kassab GS, Le KN, Fung Y-CB (1999) A hemodynamic analysis of coronary capillary blood flow based on anatomic and distensibility data. Am J Physiol Heart Circ Physiol 277(6): H2158–H2166

    Google Scholar 

  34. Fronek K, Zweifach B (1975) Microvascular pressure distribution in skeletal muscle and the effect of vasodilation. Am J Physiol 228(3): 791–796

    Google Scholar 

  35. Berne R, Levy M (2001) Cardiovascular physiology. St Louis, Mosby

  36. Gonzalez F, Bassingthwaighte JB (1990) Heterogeneities in regional volumes of distribution and flows in rabbit heart. Am J Physiol Heart Circ Physiol 258(4): H1012–H1024

    Google Scholar 

  37. May-Newman K, Chen C, Oka R, Haslim R, DeMaria A (2001) Evaluation of myocardial perfusion using three-dimensional myocardial contrast echocardiography. In: Nuclear science symposium conference record, vol 3. IEEE, pp 1691–1694

  38. Ghista D, Ng E (2007) Cardiac perfusion and pumping engineering. World Scientific, Singapore

    Google Scholar 

  39. Huyghe JM, Arts T, van Campen DH, Reneman RS (1992) Porous medium finite element model of the beating left ventricle. Am J Physiol Heart Circ Physiol 262(4): H1256–H1267

    Google Scholar 

  40. Ashikaga H, Coppola BA, Yamazaki K, Villarreal FJ, Omens JH, Covell JW (2008) Changes in regional myocardial volume during the cardiac cycle: implications for transmural blood flow and cardiac structure. Am J Physiol Heart Circ Physiol 295(2): H610–H618

    Article  Google Scholar 

  41. Goto M, Flynn AE, Doucette JW, Jansen CM, Stork MM, Coggins DL, Muehrcke DD, Husseini WK, Hoffman JI (1991) Cardiac contraction affects deep myocardial vessels predominantly. Am J Physiol Heart Circ Physiol 261(5): H1417–H1429

    Google Scholar 

  42. Gregg D, Green H (1940) Registration and interpretation of normal phasic inflow into a left coronary artery by an improved differential manometric method. Am J Physiol 130: 114–125

    Google Scholar 

  43. Nichols W, O’Rourke M (2005) McDonald’s blood flow in arteries. Hodder Arnold

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Authors and Affiliations

  1. INRIA Paris-Rocquencourt, BP 105, 78153, Le Chesnay Cedex, France

    D. Chapelle, J.-F. Gerbeau, J. Sainte-Marie & I. E. Vignon-Clementel

  2. Saint-Venant Laboratory, 6 quai Watier, 78401, Chatou Cedex, France

    J. Sainte-Marie

Authors
  1. D. Chapelle
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  2. J.-F. Gerbeau
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  3. J. Sainte-Marie
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  4. I. E. Vignon-Clementel
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Correspondence to I. E. Vignon-Clementel.

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Chapelle, D., Gerbeau, JF., Sainte-Marie, J. et al. A poroelastic model valid in large strains with applications to perfusion in cardiac modeling. Comput Mech 46, 91–101 (2010). https://doi.org/10.1007/s00466-009-0452-x

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  • DOI: https://doi.org/10.1007/s00466-009-0452-x

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