A structural model for the viscoelastic behavior of arterial walls: continuum formulation and finite element analysis.

*(English)*Zbl 1100.74597Summary: In this paper we present a two-layer structural model suitable for predicting reliably the passive (unstimulated) time-dependent three-dimensional stress and deformation states of healthy young arterial walls under various loading conditions. It extends to the viscoelastic regime a recently developed constitutive framework for the elastic strain response of arterial walls (see Holzapfel et al. (2001)). The structural model is formulated within the framework of nonlinear continuum mechanics and is well-suited for a finite element implementation. It has the special merit that it is based partly on histological information, thus allowing the material parameters to be associated with the constituents of each mechanically-relevant arterial layer. As one essential ingredient from the histological information the constitutive model requires details of the directional organization of collagen fibers as commonly observed under a microscope. We postulate a fully automatic technique for identifying the orientations of cellular nuclei, these coinciding with the preferred orientations in the tissue. The biological material is assumed to behave incompressibly so that the constitutive function is decomposed locally into volumetric and isochoric parts. This separation turns out to be advantageous in avoiding numerical complications within the finite element analysis of incompressible materials. For the description of the viscoelastic behavior of arterial walls we employ the concept of internal variables. The proposed viscoelastic model admits hysteresis loops that are known to be relatively insensitive to strain rate, an essential mechanical feature of arteries of the muscular type. To enforce incompressibility without numerical difficulties, the finite element treatment adopted is based on a three-field Hu–Washizu variational approach in conjunction with an augmented Lagrangian optimization technique. Two numerical examples are used to demonstrate the reliability and efficiency of the proposed structural model for arterial wall mechanics as a basis for large scale numerical simulations.

##### MSC:

74L15 | Biomechanical solid mechanics |

74S05 | Finite element methods applied to problems in solid mechanics |

92C10 | Biomechanics |

##### Software:

FEAP
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\textit{G. A. Holzapfel} et al., Eur. J. Mech., A, Solids 21, No. 3, 441--463 (2002; Zbl 1100.74597)

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##### References:

[1] | () |

[2] | Arrow, K.J.; Hurwicz, L.; Uzawa, H., Studies in non-linear programming, (1958), Stanford University Press Stanford, CA · Zbl 0091.16002 |

[3] | Bergel, D.H., The dynamic elastic properties of the arterial wall, J. physiol., 156, 458-469, (1961) |

[4] | Bergel, D.H., The static elastic properties of the arterial wall, J. physiol., 156, 445-457, (1961) |

[5] | Billiar, K.L.; Sacks, M.S., A method to quantify the fiber kinematics of planar tissues under biaxial stretch, J. biomech., 30, 753-756, (1997) |

[6] | Canham, P.B., Orientation of cerebral vascular smooth muscle, mathematically modelled, J. biomech., 10, 241-251, (1977) |

[7] | Carew, T.E.; Vaishnav, R.N.; Patel, D.J., Compressibility of the arterial wall, Circ. res., 23, 61-68, (1968) |

[8] | Carmines, D.V.; McElhaney, J.H.; Stack, R., A piece-wise non-linear elastic stress expression of human and pig coronary arteries tested in vitro, J. biomech., 24, 899-906, (1991) |

[9] | Chen, Y.L.; Fung, Y.C., Stress-history relations of rabbit mesentery in simple elongation, (), 9-10 |

[10] | Chuong, C.J.; Fung, Y.C., Residual stress in arteries, (), 117-129 |

[11] | Clark, J.M.; Glagov, S., Transmural organization of the arterial media: the lamellar unit revisited, Arteriosclerosis, 5, 19-34, (1985) |

[12] | Coleman, B.D.; Gurtin, M.E., Thermodynamics with internal state variables, J. chem. phys., 47, 597-613, (1967) |

[13] | Coleman, B.D.; Noll, W., The thermodynamics of elastic materials with heat conduction and viscosity, Arch. ration. mech. anal., 13, 167-178, (1963) · Zbl 0113.17802 |

[14] | Cope, D.A.; Roach, M.R., A scanning electron microscopy study of human cerebral arteries, Canad. J. physiol. pharm., 53, 651-659, (1975) |

[15] | Dobrin, P.B., Distribution of lamellar deformations. implications for properties of the arterial media, Hypertension, 33, 806-810, (1999) |

[16] | Ferdman, A.G.; Yannas, I.V., Scattering of light from histologic sections: a new method for the analysis of connective tissue, J. invest. dermatol., 100, 710-716, (1993) |

[17] | Finlay, H.M.; Whittaker, P.; Canham, P.B., Collagen organization in the branching region of human brain arteries, Stroke, 29, 1595-1601, (1998) |

[18] | Flory, P.J., Thermodynamic relations for highly elastic materials, Trans. Faraday soc., 57, 829-838, (1961) |

[19] | Fung, Y.C., Stress-strain-history relations of soft tissues in simple elongation, (), 181-208, Chapter 7 |

[20] | Fung, Y.C., On pseudo-elasticity of living tissues, () · Zbl 0446.73086 |

[21] | Fung, Y.C., Biomechanics. mechanical properties of living tissues, (1993), Springer-Verlag New York |

[22] | Fung, Y.C.; Fronek, K.; Patitucci, P., Pseudoelasticity of arteries and the choice of its mathematical expression, Am. J. physiol., 237, H620-H631, (1979) |

[23] | Gasser, T.C., Holzapfel, G.A., 2001. A rate-independent elastoplastic constitutive model for fiber-reinforced composites at finite strains: continuum basis, algorithmic formulation and finite element implementation, submitted · Zbl 1146.74342 |

[24] | Gow, B.S.; Hadfield, C.D., The elasticity of canine and human coronary arteries with reference to postmortem changes, Circ. res., 45, 588-594, (1979) |

[25] | Hayashi, K., Experimental approaches on measuring the mechanical properties and constitutive laws of arterial walls, J. biomech. engrg., 115, 481-488, (1993) |

[26] | Holzapfel, G.A., Nonlinear solid mechanics. A continuum approach for engineering, (2000), Wiley Chichester · Zbl 0980.74001 |

[27] | Holzapfel, G.A.; Simo, J.C., Entropy elasticity of isotropic rubber-like solids at finite strains, Comput. methods appl. mech. engrg., 132, 17-44, (1996) · Zbl 0890.73022 |

[28] | Holzapfel, G.A.; Gasser, T.C., A viscoelastic model for fiber-reinforced composites at finite strains: continuum basis, computational aspects and applications, Comput. methods appl. mech. engrg., 190, 4379-4403, (2001) |

[29] | Holzapfel, G.A.; Gasser, T.C.; Ogden, R.W., A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. elasticity, 61, 1-48, (2000) · Zbl 1023.74033 |

[30] | Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (2000), Dover New York |

[31] | Humphrey, J.D., Mechanics of the arterial wall: review and directions, Critical rev. biomed. engr., 23, 1-162, (1995) |

[32] | Langewouters, G.J.; Wesseling, K.H.; Goedhard, W.J.A., The static elastic properties of 45 human thoracic and 20 abdominal aortas in vitro and the parameters of a new model, J. biomech., 17, 425-435, (1984) |

[33] | Learoyd, B.M.; Taylor, M.G., Alterations with age in the viscoelastic properties of human arterial walls, Circ. res., 18, 278-292, (1966) |

[34] | Lubliner, J., Plasticity theory, (1990), Macmillan Publishing Company New York · Zbl 0745.73006 |

[35] | Marsden, J.E.; Hughes, T.J.R., Mathematical foundations of elasticity, (1994), Dover New York |

[36] | Miehe, C., Numerical computation of algorithmic (consistent) tangent moduli in large-strain computational inelasticity, Comput. methods appl. mech. engrg., 134, 223-240, (1996) · Zbl 0892.73012 |

[37] | Nagtegaal, J.C.; Parks, D.M.; Rice, J.R., On numerically accurate finite element solutions in the fully plastic range, Comput. methods appl. mech. engrg., 4, 153-177, (1974) · Zbl 0284.73048 |

[38] | Nichols, W.W.; O’Rourke, M.F., Mcdonald’s blood flow in arteries, (1998), Arnold London, Chapter 4, pp. 73-97 |

[39] | Ogden, R.W., Nearly isochoric elastic deformations: application to rubberlike solids, J. mech. phys. solids, 26, 37-57, (1978) · Zbl 0377.73044 |

[40] | Ogden, R.W.; Schulze-Bauer, C.A.J., Phenomenological and structural aspects of the mechanical response of arteries, (), 125-140, AMD-Vol. 242/BED-Vol. 46 |

[41] | Patel, D.J.; Fry, D.L., The elastic symmetry of arterial segments in dogs, Circ. res., 24, 1-8, (1969) |

[42] | Peters, M.W.; Canham, P.B.; Finlay, H.M., Circumferential alignment of muscle cells in the tunica media of the human brain artery, Blood vessels, 20, 221-233, (1983) |

[43] | Ranvier, L., Leçons d’anatomie générale sur le système musculaire, Progrès médical, (1880), Delahaye |

[44] | Rhodin, J.A.G., Fine structure of vascular walls in mammals, with special reference to smooth muscle component, Physiol. rev., 42, 48-81, (1962) |

[45] | Rhodin, J.A.G., The ultrastructure of Mammalian arterioles and precapillary sphincters, J. ultrastruct. res., 18, 181-223, (1967) |

[46] | Rhodin, J.A.G., Architecture of the vessel wall, (), 1-31 |

[47] | Roach, M.R.; Burton, A.C., The reason for the shape of the distensibility curve of arteries, Canad. J. biochem. physiol., 35, 681-690, (1957) |

[48] | Roy, C.S., The elastic properties of the arterial wall, J. physiol., 3, 125-159, (1880-1882) |

[49] | Schulze-Bauer, C.A.J.; Regitnig, P.; Holzapfel, G.A., Mechanics of the human femoral adventitia including high-pressure response, Am. J. physiol. heart circ. physiol., 003973.2001, (2002) |

[50] | Silver, F.H.; Christiansen, D.L.; Buntin, C.M., Mechanical properties of the aorta: A review, Critical rev. biomed. engrg., 17, 323-358, (1989) |

[51] | Simo, J.C., On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects, Comput. methods appl. mech. engrg., 60, 153-173, (1987) · Zbl 0588.73082 |

[52] | Simo, J.C.; Miehe, C., Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation, Comput. methods appl. mech. engrg., 98, 41-104, (1992) · Zbl 0764.73088 |

[53] | Simo, J.C.; Taylor, R.L., Quasi-incompressible finite elasticity in principal stretches. continuum basis and numerical algorithms, Comput. methods appl. mech. engrg., 85, 273-310, (1991) · Zbl 0764.73104 |

[54] | Simo, J.C.; Taylor, R.L.; Pister, K.S., Variational and projection methods for the volume constraint in finite deformation elasto-plasticity, Comput. methods appl. mech. engrg., 51, 177-208, (1985) · Zbl 0554.73036 |

[55] | Simo, J.C.; Hughes, T.J.R., Computational inelasticity, (1998), Springer-Verlag New York · Zbl 0934.74003 |

[56] | Somlyo, A.P.; Somlyo, A.V., Vascular smooth muscle. I: normal structure, pathology, biochemistry and biophysics, Physiol. rev., 20, 197-272, (1968) |

[57] | Spencer, A.J.M., Constitutive theory for strongly anisotropic solids, (), 1-32, CISM Courses and Lectures No. 282, International Centre for Mechanical Sciences |

[58] | Strong, K.C., A study of the structure of the media of the distributing arteries by the method of microdissection, Anat. rec., 72, 151-168, (1938) |

[59] | Takamizawa, K.; Hayashi, K., Strain energy density function and uniform strain hypothesis for arterial mechanics, J. biomech., 20, 7-17, (1987) |

[60] | Tanaka, T.T.; Fung, Y.C., Elastic and inelastic properties of the canine aorta and their variation along the aortic tree, J. biomech., 7, 357-370, (1974) |

[61] | Taylor, R.L., FEAP - A finite element analysis program – version 7.3, (2000), University of California Press Berkeley |

[62] | Todd, M.E.; Laye, C.G.; Osborne, D.N., The dimensional characteristics of smooth muscle in rat blood vessels: A computer-assisted analysis, Circ. res., 53, 319-331, (1983) |

[63] | Vaishnav, R.N.; Vossoughi, J., Estimation of residual strains in aortic segments, (), 330-333 |

[64] | Valanis, K.C., Irreversible thermodynamics of continuous media, internal variable theory, (1972), Springer-Verlag Wien · Zbl 0277.73013 |

[65] | Vossoughi, J., Longitudinal residual strains in arteries, (), 17-19 |

[66] | Walmsley, J.G.; Canham, P.B., Orientation of nuclei as indicators of smooth muscle cell alignment in the cerebral artery, Blood vessels, 16, 43-51, (1979) |

[67] | Weiss, J.A.; Maker, B.N.; Govindjee, S., Finite element implementation of incompressible, transversely isotropic hyperelasticity, Comput. methods appl. mech. engrg., 135, 107-128, (1996) · Zbl 0893.73071 |

[68] | Wertheim, M.G., Mémoire sur l’élasticité et la cohésion des principaux tissus du corps humain, Ann. chim. phys., 21, 385-414, (1847) |

[69] | Wolinsky, H.; Glagov, S., Structural basis for the static mechanical properties of the aortic media, Circ. res., 14, 400-413, (1964) |

[70] | Wolinsky, H.; Glagov, S., A lamellar unit of aortic medial structure and function in mammals, Circ. res., 20, 90-111, (1967) |

[71] | Woo, S.L.Y.; Simon, B.R.; Kuei, S.C.; Akeson, W.H., Quasi-linear viscoelastic properties of normal articular cartilage, J. biomech. engrg., 102, 85-90, (1979) |

[72] | Xie, J.; Zhou, J.; Fung, Y.C., Bending of blood vessel wall: stress-strain laws of the intima-media and adventitia layers, J. biomech. engrg., 117, 136-145, (1995) |

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