Abstract
We have identified the simplest topology that will permit spontaneous oscillations in a model of microvascular blood flow that includes the plasma skimming effect and the Fahraeus–Lindqvist effect and assumes that the flow can be described by a first-order wave equation in blood hematocrit. Our analysis is based on transforming the governing partial differential equations into delay differential equations and analyzing the associated linear stability problem. In doing so we have discovered three dimensionless parameters, which can be used to predict the occurrence of nonlinear oscillations. Two of these parameters are related to the response of the hydraulic resistances in the branches to perturbations. The other parameter is related to the amount of time necessary for the blood to pass through each of the branches. The simple topology used in this study is much simpler than networks found in vivo. However, we believe our analysis will form the basis for understanding more complex networks.
Similar content being viewed by others
References
Azelvandre, F., and C. Oiknine. Fahraeus effect and Fahraeus Lindqvist effect—experimental results and theoretical models. Biorheology 13(6): 325–335, 1976.
Barbee, J. H., and G. R. Cokelet. Prediction of blood flow in tubes with diameters as small as 29 microns. Microvasc. Res. 5:17–21, 1971.
Barclay, K. D., G. A. Klassen, and C. Young. A method for detecting chaos in canine myocardial microcirculatory red cell flux. Microcirculation 7:335–346, 2000.
Biswall, B. B., and A. G. Hudetz. Synchronous oscillations in cerebrocortical capillary red blood cell velocity after nitric oxide synthase inhibition. Microvasc. Res. 52:1–12, 1996.
Carr, R. T., and M. LeCoin. Nonlinear dynamics in microvascular networks. Ann. Biomed. Eng. 28:641–652, 2000.
Cavalcanti, S., and M. Ursino. Chaotic oscillations in microvessel arterial networks. Ann. Biomed. Eng. 24:37–47, 1996.
Dellimore, J. W., M. J. Dunlop, and P. B. Canham. Ratio of cells and plasma in blood flowing past branches in small plastic channels. Am. J. Physiol. 244:H635–H643, 1983.
Fahraeus, R. Suspension stability of the blood. Physiol. Rev. 9:241–274, 1929.
Fahraeus, R., and T. Lindqvist. The viscosity of blood in narrow capillary tubes. Am. J. Physiol. 96:562–568, 1931.
Fenton, B. M., R. T. Carr, and G. R. Cokelet. Nonuniform red cell distribution in 20 to 100 μm bifurcations. Microvasc. Res. 29:103–126, 1985.
Fenton, B. M., D. W. Wilson, and G. R. Cokelet. Analysis of the effects of measured white cell entrance times on hemodynamics in a computer model of a microvascular bed. Pflueg. Arch. 403:396–401, 1985.
Glass, L., and M. C. Mackey. From Clocks to Chaos: The Rhythms of Life. Princeton University Press, NJ, 1988.
Glenny, R. W., N. L. Polissar, S. McKinney, and H. T. Robertson. Temporal heterogeneity of regional pulmonary perfusion is spatially clustered. J. Appl. Physiol. 79:986–1001, 1995.
Johnson, P. C., and H. Wayland. Regulation of blood flow in single capillaries. Am. J. Physiol. 212:1405–1415, 1967.
Kiani, M. F., and A. G. Hudetz. A semiempirical model of apparent blood viscosity as a function of vessel diameter and discharge hematocrit. Biorheology 28:65–73, 1991.
Kiani, M. F., A. R. Pries, L. L. Hsu, I. H. Sarelius, and G. R. Cokelet. Fluctuations in microvascular blood flow parameters caused by hemodynamic mechanisms. Am. J. Physiol. 266:H1822–H1828, 1994.
Krogh, A. The Anatomy and Physiology of Capillaries. Yale University Press, CT, 1922.
Mollica, F., R. K. Jain, and P. A. Netti. A model for temporal heterogeneities of tumor blood flow. Microvasc. Res. 65:56–60, 2003.
Parthimos, D., K. Osterloh, A. R. Pries, and T. M. Griffith. Deterministic nonlinear characteristics of in vivo blood flow velocity and arteriolar diameter fluctuations. Phys. Med. Biol. 49:1789–1802, 2004.
Pries, A. R., K. Ley, and P. Gaehtgens. Red cell distributions at microvascular bifurcations. Microvasc. Res. 38:81–101, 1989.
Pries, A. R., T. W. Secomb, T. Gessner, M. B. Sperandio, J. F. Gross, and P. Gaehtgens. Resistance to blood flow in microvessels in vivo. Circ. Res. 75:904–915, 1994.
Rodgers, G. P., A. N. Schechter, C. T. Noguchi, H. G. Klein, A. W. Niehuis, and R. F. Bonner. Periodic microcirculatory flow in patients with sickle-cell disease. New England J. Med. 311:1534–1538, 1984.
Slaaf, D. W., G. J. Tangelder, H. C. Teirlinck, and R. C. Reneman. Arteriolar vasomotion and arterial pressure reduction in rabbit tenissumus muscle. Microvasc. Res. 33:71–80, 1987.
Wiederhielm, C., J. W. Woodbury, S. Kirk, and R. F. Rushmer. Pulsatile pressures in the microcirculation of frog mesentery. Am. J. Physiol. 207:173–176, 1964.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Carr, R.T., Geddes, J.B. & Wu, F. Oscillations in a Simple Microvascular Network. Ann Biomed Eng 33, 764–771 (2005). https://doi.org/10.1007/s10439-005-2345-2
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10439-005-2345-2