Abstract
Numerical simulation of arterial hemodynamics and mass transport have become an important tool in recent years due to the significant advances in numerical mathematics, scientific computation and due to the increased power of computers. Over the past years increasingly more elaborate models have been developed in order to gain a better insight into the physiological processes in the vascular system and the initiation and development of arterial diseases. Hemodynamic factors apparently play an important role in the development of these diseases, and therefore, local arterial flow dynamics, such as flow separation, flow recirculation, low and oscillatory wall shear stress, and the influence on mass transport in the arterial lumen and in the artery wall are subjects of intensive research. Corresponding studies include rheological effects in blood flow resulting from the interactions between blood phases up to the transport processes of macromolecules in the arteries and in the artery wall layers. The problems discussed are mathematically described by systems of coupled nonlinear partial differential equations mostly in large parameter range. The numerical approach uses the finite element method, which is often the most suitable approximation technique due to its high flexibility.
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References
Anderson, J.L., and Malone, D.M. (1974). Mechanism of osmotic flow in porous membranes. Biophys. J. 14: 957–982.
Back, L.H., Radbill, J.R., and Crawford, D.W. (1977). Analysis of oxygen transport from pulsatile, viscous blood flow to diseased coronary arteries of man. J. Biomechanics 10:763–774.
Bassiouny, H.S., White, S., Glagov, S., Choi, E., Giddens, D.P., and Zarins, C.K. (1992). Anastomotic intimal hyperplasia: mechanical injury or flow induced. J. Vasc. Surg. 15: 708–717.
Batson, R.C., Sottiurai, V.S., and Craighead, C.C. (1984). Linton patch angioplasty: An adjunct to distal bypass with polytetrafluoroethylene grafts. Ann. Surg. 199: 684–693.
Böhme, G., and Rubart, L. (1993). Einblick in die theoretische Analyse der Stroemungen viskoelastischer Fluessigkeiten. In Mennicken, R., ed., GA MM Mitteilungen 16: 59–97.
Brezzi, F., and Fortin, M. (1991). Mixed and Hybrid Finite Elements. SSCM n. 5, Springer-Verlag, Berlin.
Brooks, A.N., and Hughes, T.J.R. (1982). Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32: 199–259.
Caro, C.G., Fitz-Gerald, J.M., and Schroter, R.C. (1971). Atheroma and arterial wall shear: Observation, correlation and proposal of a shear dependent mass transfer mechanism of atherogenesis. In Proc. Roy. Soc. Lond. 177:109–159.
Chien, S., Usami, S., Dellenback, R.J., and Gregersen, M.I. (1970). Shear-dependent deformation of erythrocytes in rheology of human blood. American Journal of Physiology 219: 136–142.
Chorin A. (1968). Numerical solution of the Navier-Stokes equations. Math. Comp. 22: 745–762.
Cokelet, G.R. (1980). Rheology and hemodynamics. Ann. Rev. Physiol. 42: 311–324.
Crone, C., and Levitt, D.G. (1984). Capillary permeability to small solutes. In Handbook of Physiology. Microcirculation. The Cardiovascular System. Bethesda, MD: Am. Physiol. Soc., sect. 2, vol. IV, pt. 1, Chapter 8, 411–466.
Curry, F. E. (1984). Mechanics and thermodynamics of transcapillary exchange. In Handbook of Physiology. Microcirculation. The Cardiovascular System. Bethesda, MD: Am. Physiol. Soc., sect. 2, vol. IV, pt. 1, Chapter 8, 309–374.
Cuvelier, C., Segal, A., and van Steenhoven, A.A. (1986). Finite Element Methods and Navier-Stokes Equations. D. Reidel Pulishing Company, Dordrecht/Boston/Lancaster/Tokyo.
Delfino, A., Stergiopulos, N., Moore, Jr., J.E., and Meister, J.-J. (1997). Residual strain effects on the stress field in a thick wall finite element model of the human carotid bifurcation. J. Biomechanics 30: 777–786.
Donea, J., Giuliani, S., Laval, H., and Quartapelle, L. (1981). Solution of the unsteady Navier-Stokes equations by a finite element projections method. In Taylor, C., and Morgan, K., eds., Computational Techniques in Transient and Turbulent Flow. Pineridge Press, Swansea. 97–132.
Ethier, C.R., Steinman, D.A., Zhang, X., Karpik, S.R., and Ojha, M. (1998). Flow waveform effects on end-to-side anastomotic flow patterns. J. Biomechanics 31: 609–617.
Fahraeus, R., and Lindquist, T. (1931). The viscosity of the blood in narrow capillary tubes. Am. J. Physiol. 96: 562–568.
Formaggia, L., Gerbeau, J.-F., Nobile, F., and Quarteroni, A. (2001). On a coupling of 3D and 1D NavierStokes equations for flow problems in compliant vessels. Comp. Methods in Appl. Mech. Engng. 191: 561–582.
Formaggia, L., and Nobile, F. (1999). A stability analysis for the Arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 7: 105–131.
Friedman, M.H., Bargeron, C.B., Deters, O.J., Hutchins, G.M., and Mark, F.F. (1987). Correlation between wall shear and intimal thickness at a coronary artery branch. Atherosclerosis 68: 27–33.
Friedman, M.H., Deters, O.J., Mark, F.F., Bargeron, C.B., and Hutchins, G.M. (1983). Arterial geometry affects hemodynamics–a potential risk factor for atherosclerosis. Atherosclerosis 46: 225–231.
Friedman, M.H., and Fry, D.L. (1993). Arterial permeability dynamics and vascular disease. Atherosclerosis 104: 189–194.
Fry, D.L. (1985). Mathematical models of arterial transmural transport. Am. J Physiol. 248:H240–H263. Fry, D.L. (1987). Mass transport, atherogenesis, and risk. Arteriosclerosis 7: 88–100.
Fung, Y.C. (1965). Foundations of Solid Mechanics. Prentice-Hall International Series in Dynamics: Prentice-Hall, Inc., Englewood Cliffs, N.J.
Galdi, G.P., Heywood, J.G., and Rannacher, R. (2000). Fundamental Directions in Mathematical Fluid Mechanics. Birkhaeuser Verlag, Basel.
Gijsen, F.J.H. (1998). Modeling of wall shear stress in large arteries. Thesis, TU-Eindhoven.
Gijsen, F.J.H., Allanic, E., van de Vosse, F.N., and Janssen, J.D. (1999). The influence of the non Newtoniean properties of blood on the flow in large arteries: unsteady flow in a 900 curved tube. J. Biomechanics 32: 705–713.
Girault, V., and Raviart, P.-A. (1986). Finite element methods for Navier-Stokes equations. Springer-Verlag, Berlin Heidelberg New York Tokyo.
Goldsmith, H.L, and Marlow, J. (1979). Flow behaviour of erythrocytes. II. Particle motions in concentrated suspensions of ghost cells. Journal of Colloid Interface Science 71: 383–407.
Gresho, P.M., Chan, S.T., Lee, RL., and Upson, C.D. (1984). A modified finite element method for solving the time-dependent incompressible Navier-Stokes equations. Int. J. Num. Meth. Fluids 4: 557–598.
Gresho, P.M., and Sani, R.L. (2000). Incompressible Flow and the Finite Element Method, Vol. 2, John Wiley and Sons, Chichester.
Gunzberger, M.D. (1989). Finite Element Methods for Viscous Incompressible Flows, A Giude to Theory, Practice and Algorithms. Academic Press Inc., San Diego.
Hofer, M. (1998). Numerische Simulation von Multiphasenstroemungen und Anwendungen auf die Blutstroemung. Dissertation, TU-Graz.
Hofer, M., and Perktold, K. (1995). Vorkonditionierter konjugierter Gradienten Algorithmus für große schlecht konditionierte unsymmetrische Gleichungssysteme. Suppl. Vol. ZAMM Z. angew. Math. Mech. 75 SII: 641–642.
Hofer, M., and Perktold, K. (1997). Computer simulation of concentrated fluid-particle suspension flows in axisymmetric geometries. Biorheology 34: 261–279.
Huang, Y., Rumschitzki, D., Chien, S., and Weinbaum, S. (1997). A fiber matrix model for the filtration through fenestral pores in a compressible arterial intima. Am. J. Physiol. 272: H2023 - H2039.
Huang, Z.J., and Tarbell, J.M. (1997). Numerical simulation of mass transfer in porous media of blood vessel walls. Am. J. Physiol. 273: H464 - H477.
Huang, Y., Weinbaum, S., Rumschitzki, D., and Chien, S. (1992). A fiber matrix model for the growth of macromolecular leakage spots in the arterial intima. Advances in Biological Heat and Mass Transfer, HTD-Vol. 231, ASME 1992: 81–92.
Hughes, T.J.R., Liu, W.K., and Zimmermann, T.K. (1991). Lagrangian-Eulerian finite element formulation in incompressible viscous flows. Comput. Methods Appl. Mech. Engrg, 29: 329–349.
Hughes, T.J.R., Mallet, M., and Mizukami, A. (1986). A new finite element formulation for computational fluid dynamics: II. Beyond SUPG. Comput. Methods Appl. Mech. Engrg. 54: 341–355.
Joseph, D.D. (1990). Fluid Dynamics of Viscoelastic Liquids. Vol. 84 of Applied Mathematical Science. Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, Hong Kong.
Kamer, G., and Perktold, K. (1998). The influence of flow on the concentration of platelet active substances in the vicinity of mural microthrombi. Computer Methods in Biomechanics and Biomedical Engineering 1: 285–301.
Karner, G., and Perktold, K. (2000). Effect of endothelial injury and increased blood pressure on albumin accumulation in the arterial wall: a numerical study. J. Biomechanics 33: 709–715.
Karner, G., Perktold, K., Hofer, M., and Liepsch, D. (1999). Flow characteristics in an anatomically realistic compliant carotid artery bifurcation model. Computer Methods in Biomechanics and Biomedical Engineering 2: 171–185.
Karner, G., Perktold, K., and Zehentner, H.P. (2001). Computational modeling of macromolecule transport in the arterial wall. Computer Methods in Biomechanics and Biomedical Engineering 3: 491–504.
Keunings, R. (1989). Simulation of viscoelastic fluid flow. In Tucker III, C.L., ed., Computer Modeling for Polymer Processing. Hanser Publishers, Munich, Vienna, New York. 403–469.
Kissin, M., Kansal, N., Pappas, P.J., DeFouw, D.O., Duran, W.N., and Hobsen, R.W. (2000). Vein interposition cuffs decrease the intimal hyperplastic response of polytetrafluorethylene bypass grafts. J. Vasc. Surg. 31: 69–83.
Koiter, W.T., and Simmonds, J.C. (1973). Foundations of shell theory. In Proc. 13th Int. Congress Theor. Appl. Mech. Berlin: Springer-Verlag, 150–176.
Ku, D., Giddens, D.P., Zarins, C.K., and Glagov, S. (1985). Pulsatile flow and atherosclerosis in the human carotid bifurcation. Atherosclerosis 5: 293–302.
Lei, M., Kleinstreuer, C., and Archie, Jr., J.P. (1996). Geometric design improvements for femoral graft-artery junctions mitigating restenosis. J. Biomechanics 29: 1605–1614.
Lemson, M.S., Tordoir, J.H.M., Daemen, M.J.A.P., and Kitslaar, P.J.E.H.M. (2000). Intimal hyperplasia invascular grafts. Eur. J. Vasc. Endovasc. Surg. 19: 336–350.
Leuprecht, A., and Perktold, K. (2001). Computer simulation of non-Newtonian effects on blood flow in large arteries. Computer Methods in Biomechanics and Biomedical Engineering 4: 149–163.
Leuprecht, A., Perktold, K., Prosi, M., Berk, T., Trubel, W., and Schima, H. (2001). Numerical study of hemodynamics and wall mechanics in distal end-to-side anastomoses of bypass grafts. J. Biomechanics 35: 225–236.
Lever, M.J. (1995). Mass transport through the walls of arteries and veins. In Jaffrin, M.Y., and Caro, C.G., eds., Biological Flow, Plenum Press: New York, 177–197.
Lever, M.J., and Coleman, P.J. (1995). Fractionation of plasma proteins during their passage through blood vessel walls. In Hochmuth, R.M., Langrana, N.A., and Hefzy, M.S., eds., Proc. 1995 Bioengineering Conference, BED-Vol. 29, ASME, New York, 133–134.
Liepsch, D.W., Thurston, G., and Lee, M. (1991). Studies of fluids simulating blood-like rheological properties and applications in models of arterial branches. Biorheology 28: 39–52.
Ma, P., Li, X., and Ku, D.N. (1997). Convective mass transfer at the carotid bifurcation. J. Biomechanics 30: 565–571.
McIntire, L.V., and Tran-Son Tay, R. (1989). Concentration of materials released from mural platelet aggregates: flow effects. In Yang, W.J., and Chun, J.L, eds., Biomedical Engineering, Hemisphere Publishing Corporation, New York–Washington, 229–245.
Meyer, G., Merval, R., and Tedgui, A. (1996). Effects of pressure-induced stretch and convection on low density lipoprotein and albumin uptake in the rabbit aortic wall. Circ. Res. 79:532–540.
Miller, J.H., Foreman, R.K., Ferguson, L., and Faris, I. (1984). Interposition vein cuff for anastomosis of prosthesis to small artery. Aust. N.Z. J. Surg. 54:283–285.
Nerem, R.M., and Cornhill, J.F. (1980). The role of fluid mechanics in atherogenesis. ASME J. Biomech. Eng. 102: 181–189.
Nobile, F. (2001). Numerical Approximation of Fluid-Structure Interaction Problems with Application to Hemodynamics. PhD thesis, École Polytechnique Fédérale de Lausanne (EPFL), Thesis N. 2458.
Nunziato, J.W. (1983). A multiphase mixture theory for fluid-particle flows. In Meyer R.E., ed., The Theory of Dispersed Multiphase Flow. Proc. of an Advanced Seminar Conducted by the Mathematics Research Center, University of Wisconsin-Madison. Academic Press, New York. 191–226.
Ogston, A.G., Preston, B.N., and Wells, J.D. (1973). On the transport of compact particles through solutions of chainpolymers. Proc. R. Soc. London Ser. A 333: 297–316.
Ojha, M. (1994). Wall shear stress temporal gradient and anastomotic intimai hyperplasia. Circ. Res. 74: 1227–1231.
Osenberg, H.P. (1991). Simulation des arteriellen Blutflusses - Ein allgemeines Modell mit Anwendung auf das menschliche Hirngefäßsystem. Dissertation, ETH Zürich 9342, IBT Zürich.
Penn, M.S., Saidel, G.M., and Chisolm, G.M. (1994). Relative significance of endothelium and internal elastic lamina in regulating the entry of macromolecules into arteries in vivo. Circ. Res. 74: 74–82.
Perktold, K. (1987). On numerical simulation of three-dimensional physiological flow problems. Ber. Math.-Stat. Sektion, Forschungsges. Johanneum Graz, Nr. 280, Graz.
Perktold, K., and Hofer, M. (1999). Mathematical modelling of flow effects and transport processes in arterial bifurcation models. In Xu, X.Y., and Collins, M.W., eds., Haemodynamics of Arterial Organs. WIT Press. Southampton, Boston. 43–84.
Perktold, K., Hofer, M., Rappitsch, G., Low, M., Kuban, B.D., and Friedman, M.H. (1998). Validated computation of physiologic flow in a realistic coronary artery branch. J. Biomechanics 31: 217–228.
Perktold, K., and Karner, G. (2001). Computational principles and models of hemodynamics. In Hennerici, M., and Meairs, S., eds., Cerebrovascular Ultrasound-Theory, Practice and Future Developments. Cambridge University Press, 63–76.
Perktold, K., Leuprecht, A., Prosi, M., Berk, T., Czerny, M., Trubel, W., and Schima, H. (2002). Fluid dynamics, wall mechanics and oxygen transfer in peripheral bypass anastomoses. Annals of Biomedical Engineering 30: 447–460.
Perktold, K., Prosi, M., Leuprecht, A., Ding, Z., and Friedman, M.H. (2001). Curvature effects on bifurcating coronary artery flow. BED-Vol. 50, 2001 Bioengineering Conference, ASME 2001: 69–70.
Perktold, K., and Rappitsch, G. (1994). Mathematical modeling of local arterial flow and vessel mechanics. In Crolet, J.M., and Ohayon, R., eds., Computational methods for fluid-structure interaction. Pitman Research Notes in Mathematics Series 306, Longman Scientific and Technical, J. Wiley and Sons, New York. 230–245.
Perktold, K., Resch, M., and Florian, H. (1991). Pulsatile non-Newtonian flow characteristics in a threedimensional human carotid bifurcation model. Jounal ofBiomechanical Engineering 113: 464–475.
Phillips, W., and Deutsch, S. (1975). Towards a constitutive equation for blood. Biorheology 12: 383–389.
Quarteroni, A., and Formaggia, L. (2002). Mathematical Modelling and Numerical Simulation of the Cardiovascular System. Modeling and Scientific Computing, MOX-Report No. 01.
Quarteroni, A., Ragni, S., and Veneziani, A. (2001). Coupling between lumped and distributed models for blood problems. Computing and Visualisation in Science 4: 111–124.
Quarteroni, A., and Valli, A. (1994). Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin, Heidelberg, New York.
Quarteroni, A., Veneziani, A., and Zunino, P. (2002). Mathematical and numerical modelling of solute dynamics in blood flow and arterial walls. SIAM J. Numer. Anal. 39: 1488–1511.
Rappitsch, G., and Perktold, K. (1996). Computer simulation of convective diffusion processes in large arteries. J. Biomechanics 29: 207–215.
Rappitsch, G., Perktold, K., and Pernkopf, E. (1997). Numerical modelling of shear-dependent mass transfer in large arteries. Int. J. Numer. Meth. Fluids 25: 847–857.
Reddy, J.N., and Gartling, D.K. (1994). The Finite Element Method in Heat Transfer and Fluid Dynamics. CRC Press, Boca Raton.
Reneman, R.S., van Merode, T., Hick, P.J.J., and Hoeks, A.P.G. (1985). Flow velocity pattern in and distensibility of the carotid artery bulb in subjects of various ages. Circulation 71: 500–509.
Santamaria, A., Siegel, J.M., and Moore Jr., J.E. (1998). Computational analysis of flow in a curved tube model of the coronary arteries: Effects of time-varying curvature. Annals of Biomedical Engineering 26: 944–954.
Schmid-Schoenbein, H., Grunau, G., and Braeuer, H. (1980). Exempla haemorheologica. Albert-Roussel Pharma GmbH.
Segre, G., and Silberberg, A. (1962). Behaviour of macroscopic rigid sheres in Poiseulle flow, parts 1 and 2. Journal of Fluid Mechanics 12: 115–157.
Sharp, M.K., Thurston, G.B., and Moore, Jr., J.E. (1996). The effect of blood viscoelasticity on pulsatile flow in stationary and axially moving tubes. Biorheology 33: 185–208.
Sottiurai, V.S., Yao, J.S.T., Baston, R.C., Sue, S.L., Jones, R., and Nakamura, Y.A. (1989). Distal anastomotic intimal hyperplasia: histological character and biogenesis. Ann. Vasc. Surg. 3: 26–33.
Shyy, W., Thakur, S., Ouyang, H., Liu, J., and Blosch, E. (1997). Computational techniques for complex transport phenomena. Cambridge University Press, Cambridge, UK.
Tanner, R.I. (1985). Engineering Rheology. Clarendon Press, Oxford.
Tarbell, J.M. (1993). Bioengineering studies of the endothelial transport barrier. Bioengineering Science News, BMES Bulletin 17: 35–39.
Tarbell, J.M., Lever, M.J., and Caro, C.G. (1988). The effect of varying albumin concentration on the hydraulic conductivity of the rabbit common carotid artery. Microvascular Research 35: 204–220.
Taylor, R.S., Loh, A., McFarland, R.J., Cox, M., and Chester, J.F. (1992). Improved technique for polytetrafluoroethylene bypass grafting: long-term results using anastomotic vein patches. Br. J. Surg. 79: 348–354.
Temam, R. (1984). Navier-Stokes Equations, Theory and Numerical Analysis. North Holland, Amsterdam. Thurston, G.B. (1979). Rheological parameters for the viscosity, viscoelasticity and thixotropy of blood. Biorheology 16: 149–162.
Trubel, W., Schima, H., Moritz, A., Raderer, F., Windisch, A., Ullrich, R., Windberger, U., Losert, U., and Polterauer, P. (1995). Compliance mismatch and formation of distal anastomotic intimal hyperplasia in externally stiffened and lumen-adapted venous grafts. Eur. J. Vasc. Endovasc. Surg. 10: 1–9.
Van Merode, T., Hick, P.J.J., Hoeks, A.P.G., Rahn, K.H., and Reneman, R.S. (1988). Carotid artery wall properties in normotensive and borderline hypertensive subjects of various ages. Ultrasound in Med. and Biol. 14: 563–569.
Wada, S., and Karino, T. (2000). Computational study on LDL transfer from flowing blood to arterial walls. In Yamaguchi, T., ed., Clinical Application of Computational Mechanics to the Cardiovascular System. Springer-Verlag, Tokyo. 157–173.
Walitza, E. (1990). Zum nicht-Newtonschen Flieszverhalten von Blut und einigen damit verbundenen Konsequenzen fuer laminare Stroemungen. Dissertation, Universitaet Stuttgart.
Weydahl, E.S., and Moore, Jr., J.E. (2001). Dynamic curvature strongly affects wall shear rates in a coronary artery bifurcation model. J.Biomechanics 34: 1189–1196.
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Perktold, K., Prosi, M. (2003). Computational Models of Arterial Flow and Mass Transport. In: Pedrizzetti, G., Perktold, K. (eds) Cardiovascular Fluid Mechanics. International Centre for Mechanical Sciences, vol 446. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2542-7_2
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