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A constrained mixture model for arterial adaptations to a sustained step change in blood flow

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Abstract

A sustained change in blood flow results in an arterial adaptation that can be thought to consist of two general steps: an immediate vasoactive response that seeks to return the wall shear stress to its homeostatic value, and a long-term growth and remodeling process that seeks to restore the intramural stresses and, if needed, the wall shear stress toward their homeostatic values. Few papers present mathematical models of arterial growth and remodeling in general, and fewer yet address flow-induced changes. Of these, most prior models build upon the concept of “kinematic growth” proposed by Skalak in the early 1980s (Skalak R (1981) In: Proceedings of the IUTAM Symposium on finite elasticity. Martinus Nijhoff, The Hague, pp 347–355). Such approaches address important consequences of growth and remodeling, but not the fundamental means by which such changes occur. In this paper, therefore, we present a new approach for mathematically modeling arterial adaptations and, in particular, flow-induced alterations. The model is motivated by observations reported in the literature and is based on a locally homogenized, constrained mixture theory. Specifically, we develop a 3-D constitutive relation for stress in terms of the responses of the three primary load-bearing constituents and their time-varying mass fractions, with the latter accounting for the kinetics of the turnover of cells and extracellular matrix in changing, stressed configurations. Of particular importance is the concept that the natural configurations of the individual constituents can evolve separately and that this leads to changes in the overall material properties and empirically inferred residual stress field of the vessel. Potential applications are discussed, but there is a pressing need for new, theoretically motivated data to allow the prescription of specific functional forms of the requisite constitutive relations and the values of the associated material parameters.

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Notes

  1. Whereas homeostasis is a process that acts to maintain or restore a stable condition or state, we use the modifier homeostatic to refer to quantities in that stable condition or state.

  2. See Humphrey and Rajagopal (2002) for detailed definitions of terms such as adaptation, growth, and remodeling. Briefly, however, growth implies a change in mass and remodeling a change in microstructure.

  3. That is, the presence of self-equilibrating internal stresses in the absence of boundary tractions.

  4. Clearly, s < 0 represents development, which is not considered explicitly herein.

  5. Many consider mass production and removal via reaction (proliferation/apoptosis) and diffusion (migration) equations, but we focus on the net effect independent of the specific means.

References

  • Barocas VH, Tranquillo RT (1997) An anisotropic biphasic theory of tissue-equivalent mechanics: the interplay among cell traction, fibrillar network deformation, fibril alignment, and cell contact guidance. ASME J Biomech Eng 119:137–145

    CAS  Google Scholar 

  • Busse R, Fleming I (1998) Pulsatile stretch and shear stress: physical stimuli determining the production of endothelium-derived relaxing factors. J Vasc Res 35:73–84

    Article  CAS  PubMed  Google Scholar 

  • Cho A, Courtman DW, Langille BL (1995) Apoptosis (programmed cell death) in arteries of the neonatal lamb. Circ Res 76:168–175

    CAS  PubMed  Google Scholar 

  • Cho A, Mitchell L, Koopmans D, Langille BL (1997) Effects of changes in blood flow rate on cell death and cell proliferation in carotid arteries of immature rabbits. Circ Res 81:328–337

    CAS  PubMed  Google Scholar 

  • Chuong CJ, Fung YC (1986) On residual stress in arteries. ASME J Biomech Eng 108:189–192

    CAS  Google Scholar 

  • Davies PF (1995) Flow-mediated endothelial mechanotransduction. Physiol Rev 75:519–560

    CAS  PubMed  Google Scholar 

  • Davis MJ, Hill MA (1999) Signaling mechanisms underlying the vascular myogenic response. Physiol Rev 79:387–423

    CAS  PubMed  Google Scholar 

  • Driss AB, Benessiano J, Poitevin P, Levy BI, Michel J-B (1997) Arterial expansive remodeling induced by high flow rates. Am J Physiol 272:H851–H858

    PubMed  Google Scholar 

  • Epstein M, Maugin GA (2000) Thermomechanics of volumetric growth in uniform bodies. Int J Plast 16:951–978

    Article  Google Scholar 

  • Fath SW, Burkhart HM, Miller SC, Dalsong MC, Unthank JL (1998) Wall remodeling after wall shear rate normalization in rat mesenteric arterial collaterals. J Vasc Res 35:257–264

    Article  CAS  PubMed  Google Scholar 

  • Frangos JA, McIntire LV, Ives CL (1985) Flow effects on prostacyclin by cultured human endothelial cells. Science 227:1477–1479

    CAS  PubMed  Google Scholar 

  • Fridez P, Rachev A, Meister J-J, Hayashi K, Stergiopulos N (2001) Model of geometric and smooth muscle tone adaptation of carotid artery subject to step change in pressure. Am J Physiol 280:H2752–H2760

    CAS  Google Scholar 

  • Fung YC (1990) Biomechanics. Motion, flow, stress, and growth. Springer Verlag, New York Berlin Heidelberg

  • Fung YC, Liu SQ (1989) Change of residual strains in arteries due to hypertrophy caused by aortic constriction. Circ Res 65:1340–1349

    CAS  PubMed  Google Scholar 

  • Furchgott RF, Zawadzki JV (1980) The obligatory role of endothelial cells in the relaxation of arterial smooth muscle by acetylcholine. Nature 288:373–376

    CAS  PubMed  Google Scholar 

  • Gibbons GH, Dzau VJ (1994) The emerging concept of vascular remodeling. N Eng J Med 330:1431–1438

    Article  CAS  Google Scholar 

  • Girerd X, London G, Boutouyrie P, Mourad J-J, Safar M, Laurent S (1996) Remodeling of the radial artery in response to a chronic increase in shear stress. Hypertension 27:799–803

    PubMed  Google Scholar 

  • Greenwald SE, Moore JE, Jr, Rachev A, Kane TPC, Meister J-J (1997) Experimental investigation of the distribution of residual strains in the artery wall. J Biomech Eng 119:438–444

    CAS  PubMed  Google Scholar 

  • Grinnell F (1994) Fibroblasts, myofibroblasts, and wound contraction. J Cell Biol 124:401–404

    CAS  PubMed  Google Scholar 

  • Guyton JR, Hartley CJ (1985) Flow restriction of one carotid artery in juvenile rats inhibits growth of arterial diameter. Am J Physiol 248:H540–H546

    CAS  PubMed  Google Scholar 

  • Guzman RJ, Abe K, Zarins C (1997) Flow-induced arterial enlargement is inhibited by suppression of nitric oxide synthase activity in vivo. Surgery 122:273–280

    CAS  PubMed  Google Scholar 

  • Harris AK, D Stopak, P Wild (1981) Fibroblast traction as a mechanism for collagen morphogenesis. Nature 290:249–251

    Google Scholar 

  • Holtz J, Forstermann U, Pohl U, Giesler M, Bassenge E (1984) Flow-dependent, endothelium-mediated dilation of epicardial coronary arteries in conscious dogs: effects of cyclooxygenase inhibition. J Cardiovasc Pharm 6:1161–1169

    CAS  Google Scholar 

  • Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61:1–48

    Article  Google Scholar 

  • Humphrey JD (1999) Remodeling of a collagenous tissue at fixed lengths. ASME J Biomech Eng 121:591–597

    CAS  Google Scholar 

  • Humphrey JD (2001) Stress, strain, and mechanotransduction in cells. ASME J Biomech Eng 123:638–641

    Article  CAS  Google Scholar 

  • Humphrey JD (2002) Cardiovascular solid mechanics: cells, tissues, and organs. Springer-Verlag, New York Berlin Heidelber

    Google Scholar 

  • Humphrey JD, Na S (2002) Elastodynamics and arterial wall stress. Ann Biomed Eng 30:509–523

    Article  CAS  PubMed  Google Scholar 

  • Humphrey JD, Rajagopal KR (2002) A constrained mixture model for growth and remodeling of soft tissues. Math Model Meth Appl Sci 12:407–430

    Article  Google Scholar 

  • Kamiya A, Togawa T (1980) Adaptive regulation of wall shear stress to flow change in the canine carotid artery. Am J Physiol 239:H14–H21

    CAS  PubMed  Google Scholar 

  • Klisch SM, Sah RL, Hoger A (2000) A growth mixture theory for cartilage. In: Casey J and Bao G (eds) Mechanics in biology. ASME AMD 242: 229–242

    Google Scholar 

  • Langille BL (1993) Remodeling of developing and mature arteries: endothelium, smooth muscle, and matrix. J Cardio Pharm 21:S11–S17

    Google Scholar 

  • Langille BL, Bendeck MP, Keeley FW (1989) Adaptations of carotid arteries of young and mature rabbits to reduced carotid blood flow. Am J Physiol 256:H931–H939

    CAS  PubMed  Google Scholar 

  • Langille BL, O’Donnell F (1986) Reductions in arterial diameter produced by chronic decreases in blood flow are endothelium dependent. Science 231:405–407

    CAS  PubMed  Google Scholar 

  • Lanir Y (1979) Constitutive equations for fibrous connective tissues. J Biomech 16:1–12

    Google Scholar 

  • Lehman RM, Owens GK, Kassel NF, Hongo K (1991) Mechanism of enlargement of major cerebral collateral arteries in rabbits. Stroke 22:499–504

    CAS  PubMed  Google Scholar 

  • Libby P (2000) Changing concepts of atherogenesis. J Int Med 247:349–358

    Article  CAS  Google Scholar 

  • Lu X, Zhao JB, Wang GR, Gregersen H, Kassab GS (2001) Remodeling of the zero-stress state of femoral arteries in response to flow overload. Am J Physiol 280:H1547–1559

    CAS  Google Scholar 

  • Masuda H, Bassiouny H, Glagov S, Zarins CK (1989) Artery wall restructuring in response to increased flow. Surg Forum 40:285–286

    Google Scholar 

  • McCarthy NJ, Bennett MR (2000) The regulation of vascular smooth muscle cell apoptosis. Cardiovasc Res 45:747–755

    Article  CAS  PubMed  Google Scholar 

  • Niedermuller H, Skalicky M, Hofecker G, Kment A (1977) Investigations on the kinetics of collagen metabolism in young and old rats. Exp Gerontol 21:159–168

    Article  Google Scholar 

  • Nissen R, Cardinale GJ, Undenfriend S (1978) Increased turnover of arterial collagen in hypertensive rats. Proc Natl Acad Sci USA 75:451–453

    CAS  PubMed  Google Scholar 

  • Papadaki M, Eskin SG (1997) Effects of fluid shear stress on gene regulation of vascular cells. Biotechnol Prog 13:209–221

    Article  CAS  PubMed  Google Scholar 

  • Rachev A (2000) A model of arterial adaptation to alterations in blood flow. J Elast 61:83–111

    Article  Google Scholar 

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

    Article  CAS  PubMed  Google Scholar 

  • Rachev A, Stergiopulos N, Meister J-J (1998) A model for geometric and mechanical adaptation of arteries to sustained hypertension. ASME J Biomech Eng 120:9–17

    CAS  Google Scholar 

  • Rajagopal KR, Tao L (1995) Mechanics of mixtures. World Scientific, Singapore

  • Rao IJ, Humphrey JD, Rajagopal KR (2003) Growth and remodeling in a dynamically loaded uniaxial tissue. Comput Model Eng Sci 4:439–456

    Google Scholar 

  • Resnick N, Gimbrone MA (1995) Hemodynamic forces are complex regulators of endothelial gene expression. FASEB J 9:874–882

    CAS  PubMed  Google Scholar 

  • Rodbard S (1975) Vascular caliber. Cardiol 60:4–49

    CAS  Google Scholar 

  • Rodriguez EK, Hoger A, McCulloch AD (1994) Stress-dependent finite growth in soft elastic tissues. J Biomech 27:455–467

    CAS  PubMed  Google Scholar 

  • Rosen LA, Hollis TH, Sharma MG (1974) Alterations in bovine endothelial histidine decarboxylase activity following exposure to shear stress. Exp Mol Pathol 20:329–343

    CAS  PubMed  Google Scholar 

  • Roy P, Petroll WM, Chuong CJ, Cavanagh HD, Jester JV (1999) Effect of cell migration on the maintenance of tension on a collagen matrix. Ann Biomed Eng 27:721–730

    Article  CAS  PubMed  Google Scholar 

  • Schretzenmayr A (1933) Uber kreislaufregulatorische Vorgange an den grossen Arterien bei der Muskelarbeit. Pflugers Arch Ges Physiol 232:743–748

    Google Scholar 

  • Skalak R (1981) Growth as a finite displacement shield. In: Carlson DE, Shield RT (eds) Proceedings of the IUTAM Symposium on finite elasticity. Martinus Nijhoff, The Hague, pp 347–355

  • Stenmark KR, Mecham RP (1997) Cellular and molecular mechanisms of pulmonary vascular remodeling. Ann Rev Physiol 59:89–144

    Article  CAS  Google Scholar 

  • Stergiopulos N, Vulliemoz S, Rachev A, Meister J-J, Greenwald SE (2001) Assessing the homogeneity of the elastic properties and composition of the pig aortic media. J Vasc Res 38:237–246

    Article  CAS  PubMed  Google Scholar 

  • Taber LA (1995) Biomechanics of growth, remodeling, and morphogenesis. Appl Mech Rev 48:487–545

    Google Scholar 

  • Taber LA (1998) A model of aortic growth based on fluid shear and fiber stresses. ASME J Biomech Eng 120:348–354

    CAS  Google Scholar 

  • Taber LA, Humphrey JD (2001) Stress modulated growth, residual stress, and vascular heterogeneity. ASME J Biomech Eng 123:528–535

    Article  CAS  Google Scholar 

  • Thoma R (1893) Untersuchagen über die histogenese und histomechanik des Gefassystems. Enke, Stuttgart

  • Tozeren A, Skalak R (1988) Interaction of stress and growth in fibrous tissue. J Theor Biol 130:337–350

    PubMed  Google Scholar 

  • Tronc F, Wassef M, Esposito B, Henrion D, Glagov S, Tedgui A (1996) Role of NO in flow-induced remodeling of the rabbit common carotid artery. Arterioscler Thromb Vasc Biol 16:1256–1262

    CAS  PubMed  Google Scholar 

  • Truesdell C, Noll W (1965) The nonlinear field theories of mechanics. In: Flugge S (ed) Handbuch der Physik, vol III:3. Springer-Verlag, Berlin Heidelber New York

  • von Maltzahn WW, Warriyar RG, Keitzer WF (1984) Experimental measurements of elastic properties of media and adventitia of bovine carotid arteries. J Biomech 17:839–847

    PubMed  Google Scholar 

  • Vossoughi J, Hedjazi Z, Borris FS (1993) Intimal residual stress and strain in large arteries. In: 1993 ASME Advances in bioengineering. Amercan Society of Mechanical Engineers, New York

  • Wilson E, Meininger GA (2002) Extracellular matrix changes and vascular smooth muscle signaling. In: Hagen T (ed) Mechanisms of cardiovascular aging 11:183–199

  • Zhao S, Suciu A, Ziegler T, Moore JE, Burki E, Meister J-J, Brunner HR (1995) Synergistic effects of fluid shear stress and cyclic circumferential stretch on vascular endothelial cell morphology and cytoskeleton. Arterioscler Thromb Vasc Biol 15:1781–1786

    CAS  PubMed  Google Scholar 

  • Zeller PJ, Skalak TC (1998) Contribution of individual structural components in determining the zero-stress state in small arteries. J Vasc Res 35:8–17

    Article  CAS  PubMed  Google Scholar 

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Acknowledgements

This work was supported, in part, by NSF grant BES-0084644 and NIH grants HL-64372 and HL-58856 (sub-contract from Duke University, M. Friedman, PI).

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  1. Department of Biomedical Engineering and M.E. DeBakey Institute, Texas A&M University, College Station, TX 77843-3120, USA

    J. D. Humphrey

  2. Departments of Mechanical Engineering and Biomedical Engineering, Texas A&M University, College Station, TX 77843-3123, USA

    K. R. Rajagopal

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Correspondence to J. D. Humphrey.

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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, 109–126 (2003). https://doi.org/10.1007/s10237-003-0033-4

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