Abstract
A numerical method is implemented for computing blood flow through a branching microvascular capillary network. The simulations follow the motion of individual red blood cells as they enter the network from an arterial entrance point with a specified tube hematocrit, while simultaneously updating the nodal capillary pressures. Poiseuille’s law is used to describe flow in the capillary segments with an effective viscosity that depends on the number of cells residing inside each segment. The relative apparent viscosity is available from previous computational studies of individual red blood cell motion. Simulations are performed for a tree-like capillary network consisting of bifurcating segments. The results reveal that the probability of directional cell motion at a bifurcation (phase separation) may have an important effect on the statistical measures of the cell residence time and scattering of the tube hematocrit across the network. Blood cells act as regulators of the flow rate through the network branches by increasing the effective viscosity when the flow rate is high and decreasing the effective viscosity when the flow rate is low. Comparison with simulations based on conventional models of blood flow regarded as a continuum indicates that the latter underestimates the variance of the hematocrit across the vascular tree.
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Apelblat, A., Katzir-Katchalsky, A., Silberberg, A., 1974. A mathematical analysis of capillary-tissue fluid exchange. Biorheology 11, 1–49.
Bassingthwaighte, J.B., Leibovitch, L.S., West, B.J., 1994. Fractal Physiology. Oxford University Press, London.
Beard, D.A., Bassingthwaighte, J.B., 2000. The fractal nature of myocardial blood flow emerges from a whole-organ model of arterial network. J. Vasc. Res. 37, 282–296.
Beard, D.A., Bassingthwaighte, J.B., 2001. Modeling advection and diffusion of oxygen in complex vascular networks. Ann. Biomed. Eng. 29, 298–310.
Blake, T.R., Gross, J.F., 1982. Analysis of coupled intra- and extraluminal flows for single and multiple capillaries. Math. Biosci. 59, 173–206.
Carr, R.T., Lacoin, M., 2000. Nonlinear dynamics of microvascular blood flow. Ann. Biomed. Eng. 28, 641–652.
Carr, R.T., Geddes, J.B., Wu, F., 2005. Oscillations in a simple microvascular network. Ann. Biomed. Eng. 33, 764–771.
Enden, G., Popel, A.S., 1994. A numerical study of plasma skimming in small vascular bifurcations. J. Biomech. Eng. 116, 79–88.
Fenton, B.M., Wilson, D.W., Cokelet, G.R., 1985. Analysis of the effects of measured white blood cell entrance times on hemodynamics in a computer model of a microvascular bed. Pflügers Arch. 403, 396–400.
Fung, Y.C., 1973. Stochastic flow in capillary blood vessels. Microvasc. Res. 5, 34–48.
Furman, M.B., Olbricht, W.L., 1985. Unsteady cell distributions in capillary networks. Biotechnol. Prog. 1, 26–32.
Geddes, J.B., Carr, R.T., Karst, N.J., Wu, F., 2007. The onset of oscillations in microvascular blood flow. SIAM J. Appl. Dyn. Syst. 6, 694–727.
Gazit, Y., Baish, J.W., Safabakhsh, N., Leunig, M., Baxter, L.T., Jain, R.K., 1997. Fractal characteristics of tumor vascular architecture during tumor growth and regression. Microcirculation 4, 395–402.
Goldman, D., Popel, A.S., 2000. A computational study of the effect of capillary network anastomoses and tortuosity on oxygen transport. J. Theor. Biol. 206, 181–194.
Hsu, R., Secomb, T.W., 1989. A Green’s function method for analysis of oxygen delivery to tissue by microvascular networks. Math. Biosci. 96, 61–78.
Karshafian, R., Burns, P.N., Henkelman, M.R., 2003. Transit time kinetics in ordered and disordered vascular trees. Phys. Med. Biol. 48, 3225–3237.
Kassab, G.S., Fung, Y.C., 1994. Topology and dimensions of pig coronary capillary network. Am. J. Physiol. 267, H319–H325.
Kassab, G.S., Rider, C.A., Tang, N.J., Fung, Y.C., 1993. Morphometry of pig coronary arterial trees. Am. J. Physiol. 265, H350–H365.
Kiani, M.F., Pries, A.R., Hsu, L.L., Sarelius, I.H., Cokelet, G.R., 1994. Fluctuations in microvascular blood flow parameters caused by hemodynamic mechanisms. Am. J. Physiol. 266, H1822–H1828.
Klitzman, B., Johnson, P.C., 1982. Capillary network geometry and red cell distribution in the hamster cremaster muscle. Am. J. Physiol., Heart Circ. Physiol. 242(11), H211–H219.
Krogh, A., 1919. The number and the distribution of capillaries in muscle with the calculation of the oxygen pressure necessary for supplying the tissue. J. Physiol. (Lond.) 52, 409–515.
Less, J.R., Skalak, T.C., Sevick, E.M., Jain, R.K., 1991. Microvascular architecture in a mammary carcinoma: branching patterns and vessel dimensions. Cancer Res. 51, 265–273.
Lipowsky, H.H., Zweifach, B.W., 1974. Network analysis of microcirculation of cat mesentery. Microvasc. Res. 7, 73–83.
Popel, A.S., 1989. Theory of oxygen transport to tissue. Crit. Rev. Biomed. Eng. 17, 257–321.
Popel, A.S., Johnson, P.C., 2005. Microcirculation and hemorheology. Annu. Rev. Fluid Mech. 37, 43–49.
Pozrikidis, C., 2005. Axisymmetric motion of a file of red blood cells through capillaries. Phys. Fluids 17, 031503.
Pozrikidis, C., 2008. Numerical Computation in Science and Engineering, 2nd edn. Oxford University Press, London.
Price, R.J., Skalak, T.C., 1995. A circumferential stress-growth rule predicts arcade arteriole formation in a network model. Microcirculation 2, 41–51.
Pries, A.R., Secomb, T.W., Gaehtgens, P., Gross, J.F., 1990. Blood flow in microvascular networks. Experiments and simulation. Circ. Res. 67, 826–834.
Pries, A.R., Neuhaus, D., Gaetgens, P., 1992. Blood viscosity in tube flow: dependence on diameter and hematocrit. Am. J. Physiol. 263, H1770–H1778.
Pries, A.R., Secomb, T.W., Gaehtgens, P., 1996. Biophysical aspects of blood flow in the microvasculature. Cardiovasc. Res. 32, 654–667.
Schmid-Schönbein, G.W., Skalak, R., Usami, S., Chien, S., 1980. Cell distribution in capillary networks. Microvascular Res. 19, 18–44.
Secomb, T.W., 2005. Microvascular networks: 3D structural information. Available from the Internet site: http://www.physiology.arizona.edu/people/secomb/network.html.
Secomb, T.W., Hsu, R., 1988. Analysis of oxygen delivery to tissue by microvascular networks. Adv. Exp. Med. Biol. 222, 95–103.
Secomb, T.W., Hsu, R., 1994. Simulation of O2 transport in skeletal muscle: diffusive exchange between arterioles and capillaries. Am. J. Physiol. 267, H1214–H1221.
Secomb, T.W., Skalak, R., Özkaya, N., Gross, J.F., 1986. Flow of axisymmetric red blood cells in narrow capillaries. J. Fluid Mech. 163, 405–423.
Secomb, T.W., Hsu, R., Dewhirst, M.W., Klitzman, B., Gross, J.F., 1993. Analysis of oxygen transport to tumor tissue by microvascular networks. Int. J. Radiat. Oncol. Biol. Phys. 25, 481–489.
Secomb, T.W., Hsu, R., Ong, E.T., Gross, J.F., Dewhirst, M.W., 1995. Analysis of the effects of oxygen supply and demand on hypoxic fraction in tumors. Acta Oncol. 34, 313–316.
Secomb, T.W., Hsu, R., Beamer, N.B., Coull, B.M., 2000. Theoretical simulation of oxygen transport to brains by networks of microvessels: Effects of oxygen supply and demand on tissue hypoxia. Microcirculation 7, 237–247.
Secomb, T.W., Hsu, R., Pries, A.R., 2001. Motion of red blood cells in a capillary with an endothelial surface layer: effect of flow velocity. Am. J. Physiol. Heart. Circ. Physiol. 281, H629–H636.
Secomb, T.W., Hsu, R., Pries, A.R., 2002. Blood flow and red blood cell deformation in nonuniform capillaries: effects of the endothelial surface layer. Microcirculation 9, 189–196.
Secomb, T.W., Hsu, R., Park, E.Y.H., Dewhirst, M.W., 2004. Greenâs function methods for analysis of oxygen delivery to tissue by microvascular networks. Ann. Biomed. Eng. 32, 1519–1529.
Sutera, S.P., Skalak, R., 1993. The history of Poiseuilleâs law. Annu. Rev. Fluid Mech. 25, 1–19.
Sutera, S.P., Seshadri, V., Croce, P.A., Hochmuth, R.M., 1970. Capillary blood flow. II. Deformable model cells in tube flow. Microvasc. Res. 2, 420–433.
Tsafnat, N., Tsafnat, G., Lambert, T.D., 2004. A three-dimensional fractal model of tumour vasculature. In: Proc. 26th Ann. Int. Conf. IEEE EMBS, pp. 683–686.
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Pozrikidis, C. Numerical Simulation of Blood Flow Through Microvascular Capillary Networks. Bull. Math. Biol. 71, 1520–1541 (2009). https://doi.org/10.1007/s11538-009-9412-z
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DOI: https://doi.org/10.1007/s11538-009-9412-z