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Biomechanical Modelling at the Molecular, Cellular and Tissue Levels pp 179–258Cite as

Anisotropy and Nonlinear Elasticity in Arterial Wall Mechanics

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Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 508))

Abstract

In this chapter we provide a theoretical framework based on the nonlinear theory of elasticity that can be used as the background against which the mechanical properties of soft biological tissue can be analyzed by comparing theory with experimental data. Of particular concern will be the elastic properties of arterial wall tissue. The results of mechanical testing are important for the characterization of the material properties through appropriate consitutive laws that are essential for the simulation of, for example, clinical procedures that involve deformation of the tissue. An important constituent of soft tissue is its fibrous structure, particularly the arrangement of collagen fibres. These fibres have a strong influence on the mechanical properties of the tissue and, in particular, they endow the material with anisotropic properties. This anisotropy features in our analysis and is accounted for by one or more families of preferred directions within the constitutive description. Particular attention is focused on biaxial deformations for both transversely isotropic materials and a general class of anisotropic materials. Specific constitutive laws are discussed for models based on invariants, incorporating the influence of fibre orientation and dispersion, and on constitutive formulations based directly on the Green strain tensor. The theory is illustrated by application to a prototype boundary-value problem, namely the extension and inflation of a circular cylindrical tube, representing an artery. Some attention is focused on questions related to the convexity and strong ellipticity of the constitutive laws since these issues are important for the appropriateness of the models from the mathematical point of view, for questions of material and structural stability and for stability of numerical schemes, including finite element computations.

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Bibliography

  • H. Abè, K. Hayashi, and M. Sato, editors, Data Book on Mechanical Properties of Living Cells, Tissues and Organs. Springer, 1996.

    Google Scholar 

  • P. B. Canham, H. M. Finlay, J. G. Dixon, D. R. Boughner, and A. Chen Measurements from light and polarised light microscopy of human coronary arteries fixed at distending pressure. Cardiovasc. Res. 23:973–982, 1989.

    Article  Google Scholar 

  • C. J. Chuong and Y. C. Fung. Three-dimensional stress distribution in arteries. J. Biomech. Eng. 105:268–274, 1983.

    Google Scholar 

  • R. H. Cox. Regional variation of series elasticity in canine arterial smooth muscles. Am. J. Physiol. 234:H542–H551, 1978.

    Google Scholar 

  • A. Delfino, N. Stergiopulos, J. E. Moore, and J.-J. Meister., Residual strain effects on the stress field in a thick wall finite element model of the human carotid bifurcation. J. Biomech. 30:777–786, 1997.

    Article  Google Scholar 

  • H. Demiray. A note on the elasticity of soft biological tissues. J. Biomech. 5:309–311, 1972.

    Article  Google Scholar 

  • H. Demiray and R. P. Vito. A layered cylindrical shell model for an aorta. Int. J. Eng. Sci. 29:47–54, 1991.

    Article  MATH  Google Scholar 

  • H. Demiray, H. W. Weizsäcker, K. Pascale, and H. A. Erbay. A stress-strain relation for a rat abdominal aorta. J. Biomech. 21:369–374, 1988.

    Article  Google Scholar 

  • S. X. Deng, J. Tomioka, J. C. Debes, and Y. C. Fung. New experiments on shear modulus of elastic arteries. Am. J. Physiol. Heart Circ. Physiol. 266:H1–H10, 1994.

    Google Scholar 

  • S. Dokos, I. J. LeGrice, B. H. Smaill, J. Kar, and A. A. Young. A triaxialmeasurement shear-test device for soft biological tissues. J. Biomech. Eng. 122:471–478, 2000.

    Article  Google Scholar 

  • S. Dokos, B. H. Smaill, A. A. Young, and I. J. LeGrice. Shear properties of passive ventricular myocardium. Am. J. Physiol. 283:H2650–H2659, 2002.

    Google Scholar 

  • H. M. Finlay, L. McCullough, and P. B. Canham., Three-dimensional collagen organization of human brain arteries at different transmural pressures. J. Vasc. Res. 32:301–312, 1995.

    Google Scholar 

  • H. M. Finlay, P. Whittaker, and P. B. Canham. Collagen organization in the branching region of human brain arteries. Stroke 29:1595–1601, 1998.

    Google Scholar 

  • A. D. Freed, D. R. Einstein, and I. Vesely. Invariant formulation for dispersed transverse isotropy in aortic heart valves. An efficient means for modeling fiber splay. Biomech. Model. Mechanobio. 4:100–117, 2005.

    Article  Google Scholar 

  • Y. B. Fu and R. W. Ogden, editors. Nonlinear Elasticity: Theory and Applications. Cambridge University Press, 2001.

    Google Scholar 

  • Y. C. Fung. Elasticity of soft tissues in simple elongation. Am. J. Physiol. 213:1532–1544, 1967.

    Google Scholar 

  • Y. C. Fung, K. Fronek, and P. Patitucci. Pseudoelasticity of arteries and the choice of its mathematical expression. Am. J. Physiol. 237:H620–H631, 1979.

    Google Scholar 

  • T. C. Gasser, R. W. Ogden, and G. A. Holzapfel. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J. R. Soc. Interface 3:15–35, 2006.

    Article  Google Scholar 

  • A. N. Gent. A new constitutive relation for rubber. Rubber Chem. Technol. 69:59–61, 1996.

    MathSciNet  Google Scholar 

  • P. F. Gou. Strain energy function for biological tissues. J. Biomech. 3:547–550, 1970.

    Article  Google Scholar 

  • D. M. Haughton and R. W. Ogden. On the incremental equations in nonlinear elasticity—II. Bifurcation of pressurized spherical shells. J. Mech. Phys. Solids 26:111–138, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  • D. M. Haughton and R. W. Ogden. Bifurcation of inflated circular cylinders of elastic material under axial loading —I. Membrane theory for thinwalled tubes. J. Mech. Phys. Solids 27:179–212, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Hayashi. Experimental Approaches on Measuring the Mechanical Properties and Constitutive Laws of Arterial Walls. J. Biomech. Eng. 115:481–488, 1993.

    Article  Google Scholar 

  • G. A. Holzapfel. Nonlinear Solid Mechanics. Wiley, 2000.

    Google Scholar 

  • G. A. Holzapfel. Biomechanics of soft tissue. In J. Lemaitre, editor, Handbook of Materials Behavior Models, pages 1049–1063. Academic Press, 2001.

    Google Scholar 

  • G. A. Holzapfel. Structural and numerical models for the (visco) elastic response of arterial walls with residual stresses. In G. A. Holzapfel and R. W. Ogden, editors, Biomechanics of Soft Tissue in Cardiovascular Systems, CISM Courses and Lectures no. 441, pages 109–184. Springer Verlag, Wien, 2003.

    Google Scholar 

  • G. A. Holzapfel. Similarities between soft biological tissues and rubberlike materials. In P.-E. Austreall and L. Kari editors, Constitutive Models for Rubber IV, pages 607–617. Balkema, Taylor and Francis, London, 2005.

    Google Scholar 

  • G. A. Holzapfel. Determination of material models for arterial walls from uniaxial extension tests and histological structure. J. Theor. Biol. 238:290–302, 2006.

    Article  MathSciNet  Google Scholar 

  • G. A. Holzapfel and T. C. Gasser. A viscoelastic model for fiber-reinforced materials at finite strains: continuum basis, computational aspects and applications. Comput. Meth. Appl. Mech. Engr. 190:4379–4403, 2001.

    Article  Google Scholar 

  • G. A. Holzapfel, T. C. Gasser, and R. W. Ogden. A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elasticity. 61:1–48, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  • G. A. Holzapfel, T. C. Gasser, and R. W. Ogden. Comparison of a multilayer structural model for arterial walls with a Fung-type model, and issues of material stability. J. Biomech. Eng. 126:264–275, 2004.

    Article  Google Scholar 

  • G. A. Holzapfel and R. W. Ogden, editors, Biomechanics of Soft Tissue in Cardiovascular Systems. CISM Courses and Lectures no. 441. Springer, Wien, 2003.

    MATH  Google Scholar 

  • G. A. Holzapfel and R. W. Ogden, editors, Proceedings of the IUTAM Symposium on Mechanics of Biological Tissue, Graz 2004. Springer, Heidelberg, 2006.

    Google Scholar 

  • G. A. Holzapfel and R. W. Ogden. On planar biaxial tests for anisotropic nonlinearly elastic solids. A continuum mechanical framework. Mathematics and Mechanics of Solids, in press.

    Google Scholar 

  • G. A. Holzapfel, G. Sommer, T. C. Gasser, and P. Regitnig. Determination of the layer-specific mechanical properties of human coronary arteries with non-atherosclerotic intimal thickening, and related constitutive modelling. Am. J. Physiol. Heart Circ. Physiol. 289:H2048–2058, 2005.

    Article  Google Scholar 

  • G. A. Holzapfel, G. Sommer, and P. Regitnig. Anisotropic mechanical properties of tissue components in human atherosclerotic plaques. J. Biomech. Eng. 126:657–665, 2004.

    Article  Google Scholar 

  • C. O. Horgan and G. Saccomandi. A description of arterial wall mechanics using limiting chain extensibility constitutive models. Biomechan. Model. Mechanobiol. 1:251–266, 2003.

    Article  Google Scholar 

  • J. D. Humphrey. Mechanics of the arterial wall: review and directions. Critical Reviews in Biomed. Engr. 23:1–162, 1995.

    Google Scholar 

  • J. D. Humphrey. Cardiovascular Solid Mechanics. Cells, Tissues, and Organs. Springer, New York, 2002.

    Google Scholar 

  • J. D. Humphrey. Intercranial saccular aneurysms. In G. A. Holzapfel and R. W. Ogden, editors, Biomechanics of Soft Tissue in Cardiovasular Systems, CISM Courses and Lectures no. 441, pages 185–220, Springer, Wien, 2003.

    Google Scholar 

  • J. D. Humphrey, T. Kang, P. Sakarda, and M. Anjanappa. Computer-aided vascular experimentation: A new electromechanical test system. Ann. Biomed. Eng. 21:33–43, 1993.

    Article  Google Scholar 

  • J. D. Humphrey and F. C. P. Yin. A new constitutive formulation for characterizing the mechanical behavior of soft tissues. Biophys. J. 52:563–570, 1987.

    Article  Google Scholar 

  • D. F. Jones and L. R. G. Treloar. The properties of rubber in pure homogeneous strain. J. Phys. D: Appl. Phys. 8:1285–1304, 1975.

    Article  Google Scholar 

  • J. K. Knowles. The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids. Int. J. Fracture 13:611–639, 1977.

    Article  MathSciNet  Google Scholar 

  • Y. Lanir. Constitutive equations for fibrous connective tissues. J. Biomech. 16:1–12, 1983.

    Article  Google Scholar 

  • I.-S. Liu. On representations of anisotropic invariants. Int. J. Eng. Sci. 20:1099–1109, 1982.

    Article  MATH  Google Scholar 

  • J. Merodio and R. W. Ogden. Material instabilities in fiber-reinforced non-linearly elastic solids under plane deformation. Arch. Mech. 54:525–552, 2002.

    MathSciNet  MATH  Google Scholar 

  • J. Merodio and R. W. Ogden. Instabilities and loss of ellipticity in fibre-reinforced compressible non-linearly elastic solids under plane deformation. Int. J. Solids Structures 40:4707–4727, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Merodio and R. W. Ogden. Mechanical response of fibre-reinforced incom-pressible non-linearly elastic solids. Int. J. Non-Linear Mech. 40:213–227, 2005.

    Article  MATH  Google Scholar 

  • W. W. Nichols and M. F. O’Rourke. McDonald’s Blood Flow in Arteries, 4th edition, chapter 4, pages 73–97. Arnold, London, 1998.

    Google Scholar 

  • R. W. Ogden. Non-linear Elastic Deformations. Dover, New York, 1997.

    Google Scholar 

  • R. W. Ogden. Elements of the theory of finite elasticity. In Y. B. Fu and R. W. Ogden, editors, Nonlinear Elasticity: Theory and Applications, pages 1–57. Cambridge University Press, 2001.

    Google Scholar 

  • R. W. Ogden. Nonlinear elasticity, anisotropy and residual stresses in soft tissue. In G. A. Holzapfel and R. W. Ogden, editors, Biomechanics of Soft Tissue in Cardiovascular Systems, CISM Courses and Lectures no. 441, pages 65–108. Springer, Wien, 2003.

    Google Scholar 

  • R. W. Ogden and G. Saccomandi. Introducing mesoscopic information into constitutive equations for arterial walls. Biomechan. Model. Mechanobiol. 6:333–344, 2007.

    Article  Google Scholar 

  • R. W. Ogden, G. Saccomandi, and I. Sgura. Fitting hyperelastic models to experimental data. Computational Mechanics 34:484–502, 2004.

    Article  MATH  Google Scholar 

  • R. W. Ogden and C. A. J. Schulze-Bauer. Phenomenological and structural aspects of the mechanical response of arteries. In J. Casey and G. Bao, editors, Mechanics in Biology. AMD-Vol. 242/BED-Vol. 46, pages 125–140. ASME, New York, 2000.

    Google Scholar 

  • A. Rachev and K. Hayashi. Theoretical study of the effects of vascular smooth muscle contraction on strain and stress distributions in arteries. Ann. Biomed. Engr. 27:459–468, 1999.

    Article  Google Scholar 

  • R. S. Rivlin and D. W. Saunders. Large elastic deformations of isotropic materials. VII. Experiments on the deformation of rubber. Phil. Trans. R. Soc. Lond. A 243:251–288, 1951.

    Article  Google Scholar 

  • M. R. Roach and A. C. Burton. The reason for the shape of the distensibility curve of arteries. Canad. J. Biochem. Physiol. 35:681–690, 1957.

    Google Scholar 

  • M. S. Sacks. A method for planar biaxial mechanical testing that includes in-plane shear. J. Biomech. Eng. 121:551–555, 1999.

    Article  Google Scholar 

  • C. A. J. Schulze-Bauer, C. Mörth, and G. A. Holzapfel. Passive biaxial mechanical response of aged human iliac arteries. J. Biomech. Eng. 125:395–406, 2003.

    Article  Google Scholar 

  • C. A. J. Schulze-Bauer, P. Regitnig, and G. A. Holzapfel. Mechanics of the human femoral adventitia including high-pressure response. Am. J. Physiol. Heart Circ. Physiol. 282:H2427–H2440, 2002.

    Google Scholar 

  • A. J. M. Spencer. Deformations of Fibre-reinforced Materials. Oxford University Press, 1972.

    Google Scholar 

  • A. J. M. Spencer. Constitutive theory for strongly anisotropic solids. In A. J. M. Spencer, editors, Continuum Theory of the Mechanics of Fibrereinforced Composites, CISM Courses and Lectures no. 282, pages 1–32. Springer, Wien, 1984.

    Google Scholar 

  • K. Takamizawa and K. Hayashi. Strain energy density function and uniform strain hypothesis for arterial mechanics. J. Biomech. 20:7–17, 1987.

    Article  Google Scholar 

  • A. Tözeren. Elastic properties and their influence on the cardiovascular system. J. Biomech. Eng. 106:182–185, 1984.

    Article  Google Scholar 

  • H. Vangerko and L. R. G. Treloar. The inflation and extension of rubber tube for biaxial strain studies. J. Phys. D: Appl. Phys. 11:1969–1978, 1978.

    Article  Google Scholar 

  • W. W. von Maltzahn, D. Besdo, and W. Wiemer. Elastic properties of arteries: a nonlinear two-layer cylindrical model. J. Biomech. 14:389–397, 1981.

    Article  Google Scholar 

  • P. B. Wells, J. L. Harris, and J. D. Humphrey. Altered mechanical behavior of epicardium under isothermal biaxial loading. J. Biomech. Eng. 126:492–497, 2004.

    Article  Google Scholar 

  • F. L. Wuyts, V. J. Vanhuyse, G. J. Langewouters, W. F. Decraemer, E. R. Raman, and S. Buyle. Elastic properties of human aortas in relation to age and atherosclerosis: a structural model. Physics Medicine Biol. 40:1577–1597, 1995.

    Article  Google Scholar 

  • M. A. Zulliger, P. Fridez, F. Hayashi, and N. Stergiopulos. A strain energy function for arteries accounting for wall composition and structure. J. Biomech. 37:989–1000, 2004.

    Article  Google Scholar 

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  1. Department of Mathematics, University of Glasgow, UK

    Ray W. Ogden

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  1. Ray W. Ogden
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  1. Graz University of Technology, Austria

    Gerhard A. Holzapfel

  2. Royal Institute of Technology-KTH, Stockholm, Sweden

    Gerhard A. Holzapfel

  3. University of Glasgow, UK

    Ray W. Ogden

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Ogden, R.W. (2009). Anisotropy and Nonlinear Elasticity in Arterial Wall Mechanics. In: Holzapfel, G.A., Ogden, R.W. (eds) Biomechanical Modelling at the Molecular, Cellular and Tissue Levels. CISM International Centre for Mechanical Sciences, vol 508. Springer, Vienna. https://doi.org/10.1007/978-3-211-95875-9_3

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  • DOI: https://doi.org/10.1007/978-3-211-95875-9_3

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  • Online ISBN: 978-3-211-95875-9

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