Theory of small on large: Potential utility in computations of fluid-solid interactions in arteries. (English) Zbl 1127.74026

Summary: Recent advances in medical imaging, computational methods and biomechanics promise to enable significant improvements in engineering-based decision making in vascular medicine, surgery, and training. To realize the potential of this approach, however, we must better synthesize the separate advances, particularly those in biofluid mechanics and arterial wall mechanics. In this paper, we describe a method for exploiting the typically small deformations experienced by arteries during the cardiac cycle while retaining essential features of complex nonlinear anisotropic behavior of the wall relative to unloaded configurations. In particular, we show that the well-known theory of small deformations superimposed on large can facilitate computations of fluid-solid interactions by exploiting methods familiar in linearized elasticity without compromising the description of nonlinear wall mechanics. Indeed, the theory reveals potential errors when one simply tries to employ standard linearized results straight away. For purposes of illustration, small on large results are provided for the rabbit basilar artery and for a constitutive relation for arteries recently proposed by G. A. Holzapfel et al. [Ann. Biomed. Eng. 30, 753–767 (2002)]. It now remains for future studies to implement this approach in coupled fluid-solid problems.


74L15 Biomechanical solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
92C10 Biomechanics
Full Text: DOI


[1] Dzau, V.J.; Gibbons, G.H., Vascular remodeling – mechanisms and implications, J. cardiovasc. pharmacol., 21, S1-S5, (1993)
[2] Langille, B.L., Arterial remodeling: relation to hemodynamics, Canad. J. physiol. pharmacol., 74, 834-841, (1996)
[3] Levy, B.; Tedgui, A., Biology of the arterial wall, (1999), Kluwer Academic Publishers Dordrecht
[4] Davies, P.F., Flow-mediated endothelial mechanotransduction, Physiol. rev., 75, 519-560, (1995)
[5] Humphrey, J.D., Cardiovascular solid mechanics: cells, tissues, and organs, (2002), Springer-Verlag New York
[6] Taylor, C.A.; Hughes, T.J.R.; Zarins, C.K., Finite element modeling of three-dimensional pulsatile flow in the abdominal aorta: relevance to atherosclerosis, Ann. biomed. engrg., 26, 975-987, (1998)
[7] Holzapfel, G.A.; Stadler, M.; Schulze-Bauer, C.A., A layer specific three-dimensional model for the simulation of balloon angioplasty using magnetic resonance imaging and mechanical testing, Ann. biomed. engrg., 30, 753-767, (2002)
[8] Vito, R.P.; Dixon, S.A., Blood vessel constitutive model - 1995-2002, Ann. rev. biomed. engrg., 5, 413-439, (2003)
[9] Berger, S.A.; Jou, L.D., Flows in stenotic vessels, Ann. rev. fluid mech., 32, 347-382, (2000) · Zbl 0989.76096
[10] Taylor, C.A.; Draney, M.T., Experimental and computational methods in cardiovascular fluid mechanics, Ann. rev. fluid mech., 36, 197-231, (2004) · Zbl 1125.76414
[11] Heil, K., An efficient solver for the fully coupled solution of large-displacement fluid – structure interaction problems, Comput. methods appl. mech. engrg., 193, 1-23, (2004) · Zbl 1137.74439
[12] Figueroa, C.A.; Vignon-Clemental, I.E.; Jansen, K.E.; Hughes, T.J.R.; Taylor, C.A., A coupled momentum method for modeling blood flow in three-dimensional deformable arteries, Comput. methods appl. mech. engrg., 195, 5685-5706, (2006) · Zbl 1126.76029
[13] Holzapfel, G.A.; Gasser, T.G.; 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
[14] Rachev, A.; Hayashi, K., Theoretical study of the effects of vascular smooth muscle contraction on strain and stress distributions in arteries, Ann. biomed. engrg., 27, 459-468, (1999)
[15] Fung, Y.C., Biomechanics: motion, flow, stress, and growth, (1990), Springer-Verlag New York · Zbl 0743.92007
[16] Renton, J.D., Applied elasticity, (1987), John Wiley & Sons New York · Zbl 0703.73015
[17] Humphrey, J.D.; Na, S., Elastodynamics and arterial wall stress, Ann. biomed. engrg., 30, 509-523, (2002)
[18] Finlay, H.M.; McCullough, L.; Canham, P.B., Three-dimensional collagen organization of human brain arteries at different transmural pressures, J. vascular res., 32, 301-312, (1995)
[19] Gleason, R.L.; Gray, S.P.; Wilson, E.; Humphrey, J.D., A multiaxial computer-controlled organ culture and biomechanical device for mouse carotid arteries, J. biomech. engrg., 126, 787-795, (2004)
[20] Press, W.; Flannery, B.; Teukolsky, S.; Vettering, W., Numerical recipes in C, (1988), Cambridge University Press Cambridge, UK
[21] Chuong, C.J.; Fung, Y.C., Three-dimensional stress distribution in arteries, ASME J. biomech. engrg., 105, 268-274, (1983)
[22] Milnor, W.R., Hemodynamics, (1989), Williams & Wilkins Baltimore · Zbl 0354.76079
[23] Baek, S.; Rajagopal, K.R.; Humphrey, J.D., A theoretical model of enlarging intracranial fusiform aneurysms, ASME J. biomed. engrg., 128, 142-149, (2006)
[24] Pontrelli, G., Blood flow through an axisymmetric stenosis, Proc. inst. mech. engrs. part H - J. engrg. med., 215, 1-10, (2001)
[25] Oshima, M.; Torii, R.; Kobayashi, T.; Taniguchi, N.; Takagi, K., Finite element simulation of blood flow in the cerebral artery, Comput. methods appl. mech. engrg., 191, 661-671, (2001) · Zbl 0999.76081
[26] Salmon, S.; Thiriet, M.; Gerbeau, J.F., Medical image-based computational model of pulsatile flow in saccular aneurysms, Math. model. numer. anal., 37, 663-679, (2003) · Zbl 1065.92029
[27] Olufsen, M.S.; Peskin, C.S.; Kim, W.Y.; Pedersen, E.M.; Nadim, A.; Larsen, J., Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions, Ann. biomed. engrg., 28, 1281-1299, (2000)
[28] Zhao, S.Z.; Xu, X.Y.; Hughes, A.D.; Thom, S.A.; Stanton, A.V.; Ariff, B.; Long, Q., Blood flow and vessel mechanics in a physiologically realistic model of a human carotid arterial bifurcation, J. biomech., 33, 975-984, (2000)
[29] Cebral, J.R.; Yim, P.J.; Löhner, R.; Soto, O.; Choyke, P.L., Blood flow modeling in carotid arteries with computational fluid dynamics and MR imaging, Acad. radiol., 9, 1286-1299, (2002)
[30] Canfield, T.R.; Dobrin, P.B., Static elastic properties of blood vessels, ()
[31] Bathe, M.; Kamm, R.D., A fluid – structure interaction finite element analysis of pulsatile blood flow through a compliant stenotic artery, ASME J. biomech. engrg., 121, 361-369, (1999)
[32] Younis, H.F.; Kaazempur-Mofrad, M.R.; Chung, C.; Chan, R.C.; Kamm, R.D., Computational analysis of the effects of exercise on hemodynamics in the carotid bifurcation, Ann. biomed. engrg., 31, 995-1006, (2003)
[33] Truesdell, C.; Noll, W., The non-linear field theories of mechanics, encyclopedia of physics III/3, (1965), Springer-Verlag Berlin
[34] Green, A.E.; Rivlin, R.S.; Shield, R.T., General theory of small elastic deformations superposed on finite elastic deformations, Proc. royal soc. A, 211, 128-154, (1951) · Zbl 0046.41208
[35] Green, A.E.; Zerna, W., Theoretical elasticity, (1968), Clarendon Press Oxford, London · Zbl 0155.51801
[36] Demiray, H., Wave propagation through a viscous fluid contained in a prestressed thin elastic tube, Int. J. engrg. sci., 30, 1607-1620, (1992) · Zbl 0764.73065
[37] Demiray, H., Incremental elastic-modulus for isotropic elastic body with application to arteries, J. biomech. engrg., 105, 308-309, (1983)
[38] Demiray, H.; Erbay, H.A.; Erbay, S., Effect of prestress on pulse waves in arteries, Zamm, 67, 473-485, (1987) · Zbl 0636.76138
[39] Kelly, B.A.A.; Chowienczyk, P., Vascular compliance, (), 33-48
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.