Abstract
Materials are distributed throughout the body of mammals by fractal networks of branching tubes. Based on the scaling laws of the fractal structure, the vascular tree is reduced to an equivalent one-dimensional, tube model. A dispersion–convection partial differential equation with constant coefficients describes the heterogeneous concentration profile of an intravascular tracer in the vascular tree. A simple model for the mammalian circulatory system is built in entirely physiological terms consisting of a ring shaped, one-dimensional tube which corresponds to the arterial, venular, and pulmonary trees, successively. The model incorporates the blood flow heterogeneity of the mammalian circulatory system. Model predictions are fitted to published concentration-time data of indocyanine green injected in humans and dogs. Close agreement was found with parameter values within the expected physiological range. © 2003 Biomedical Engineering Society.
PAC2003: 8710+e, 8719Hh, 8719Uv
Similar content being viewed by others
REFERENCES
Audi, S. H., J. H. Linehan, G. S. Krenz, and C. A. Dawson. Accounting for the heterogeneity of capillary transit times in modeling multiple indicator dilution data. Ann. Biomed. Eng.26:914–930, 1998.
Bassingthwaighte, J. B., L. Liebovitch, and B. J. West. Fractal Physiol. Oxford: Oxford University Press, 1994.
Beard, D., and J. B. Bassingthwaighte. Advection and diffusion of substances in biological tissues with complex vascular networks. Ann. Biomed. Eng.28:253–268, 2000.
Boxenbaum, H.Interspecies scaling, allometry, physiological time, and the ground plan of pharmacokinetics. J. Pharmacokinet Biopharm.10:201–227, 1982.
Brown, J. H., and G. B. West, eds. Scaling in Biology. Oxford: Oxford University Press, 2000.
Edwards, D. A.A general theory of the macrotrasport of nondepositing particles in the lung by convective dispersion. J. Aerosol Sci.25:543–565, 1994.
Ellsworth, M. L., A. Liu, B. Dawamt, A. S. Popel, and R. N. Pittman. Analysis of vascular pattern and dimensions in arteriolar networks of the retractor muscle in young hamsters. Microvasc. Res.34:168–183, 1987.
Fung, Y. C. Biomechanics: Circulation. New York: Springer, 1997.
Gerlowski, L. E., and R. K. Jain. Physiologically based pharmacokinetic modeling: principles and applications. J. Pharm. Sci.72:1103–1127, 1983.
Guyton, A. C., and J. E. Hall. Textbook of Medical Physiology. Philadelphia: Saunders, 2000.
Hashimoto, M., and G. Watanabe. Simultaneous measurement of effective hepatic blood flow and systemic circulation. Hepatogastroenterology47:1669–1674, 2000.
Horsfield, K.Diameters, generations, and orders of branches in the bronchial tree. J. Appl. Physiol.68:457–461, 1990.
Huang, W., R. T. Yen, M. McLaurine, and G. Bledsoe. Morphometry of the human pulmonary vasculature. J. Appl. Physiol.81:2123–2133, 1996.
King, R. B., G. M. Raymond, and J. B. Bassingthwaighte. Modeling blood flow heterogeneity. Ann. Biomed. Eng.24:352–372, 1996.
Krejcie, T. C., M. J. Avram, W. B. Gentry, C. U. Niemann, M. P. Janowski, and T. K. Henthorn. A recirculatory model of the pulmonary uptake and pharmacokinetics of lidocaine based on analysis of arterial and mixed venous data from dogs. J. Pharmacokinet Biopharm.25:169–190, 1997.
LaBarbera, M.Principles of design of fluid transport systems in zoology. Science249:992–1000, 1990.
Lefevre, J.Teleonomical optimization of a fractal model of the pulmonary arterial bed. J. Theor. Biol.102:225–248, 1983.
Niemann, C. U., T. K. Henthorn, T. C. Krejcie, C. A. Shanks, C. Enders-Klein, and M. Avram. Indocyanine green kinetics characterize blood volume and flow distribution and their alteration by propranolol. J. Clin. Pharm. Ther.67:342–350, 2000.
Oliver, R. E., A. F. Jones, and M. Rowland. A whole-body physiologically based pharmacokinetic model incorporating dispersion concepts: Short and long time characteristics. J. Pharmacokinet Biopharm.28:27–55, 2001.
Olufsen, M. S., C. S. Peskin, W. Y. Kim, E. M. Pedersen, A. Nadim, and J. Larsen. Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann. Biomed. Eng.28:1281–1299, 2000.
Perl, W., and F. P. Chinard. A convection–diffusion model of indicator transport through an organ. Circ. Res.12:273–298, 1968.
Picker, O., G. Wietasch, T. W. Scheeren, and J. O. Arndt. Determination of total blood volume by indicator dilution: A comparison of mean transit time and mass conservation principle. Intensive Care Med.27:767–774, 2001.
Roberts, M. S., and M. Rowland. A dispersion model of hepatic elimination: 1. Formulation of the model and bolus considerations. J. Pharmacokinet Biopharm.14:227–260, 1986.
Scherer, P. W., L. H. Shendalman, and N. M. Greene. Simultaneous diffusion and convection in single breath lung washout. Bull. Math. Biophys.34:393–412, 1972.
Thomée, V. In: Handbook of Numerical Analysis, edited by Ciarlet, P. G., and J. L. Lions. Amsterdam: Elsevier, 1990, Vol. 1, pp. 5–195.
Unice, K. M., and B. E. Logan. The insignificant role of hydrodynamic dispersion on bacterial transport. J. Environ. Eng.126:491–500, 2000.
van Beek, J. H., S. A. Roger, and J. B. Bassingthwaighte. Regional myocardial flow heterogeneity explained with fractal networks. Am. J. Physiol.257:H1670-H1680, 1989.
Vicini, P., R. C. Bonadonna, M. Lehtovirta, L. C. Groop, and C. Cobelli. Estimation of blood flow heterogeneity in human skeletal muscle using intravascular tracer data: Importance for modeling transcapillary exchange. Ann. Biomed. Eng.26:764–774, 1998.
Wagner, J. G. Pharmacokinetics for the Pharmaceutical Scientist. Lancaster, PA: Technomic, 1993.
Weibel, E. R. Morphometry of the Human Lung. Berlin: Springer, 1963.
Weiss, M., and W. Foster. Pharmacokinetic model based on circulatory transport. Eur. J. Clin. Pharmacol.16:287–293, 1979.
West, G. B., J. H. Brown, and B. J. Enquist. A general model for the origin of allometric scaling laws in biology. Science276:122–126, 1997.
West, G. B., J. H. Brown, and B. J. Enquist. The fourth dimension of life: Fractal geometry and allometric scaling of organisms. Science284:1677–1679, 1999.
Zamir, M., P. Sinclair, and T. H. Wonnacott. Relation between diameter and flow in major branches of the arch of the aorta. J. Biomech.25:1303–1310, 1992.
Zamir, M.On fractal properties of arterial trees. J. Theor. Biol.197:517–526, 1999.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dokoumetzidis, A., Macheras, P. A Model for Transport and Dispersion in the Circulatory System Based on the Vascular Fractal Tree. Annals of Biomedical Engineering 31, 284–293 (2003). https://doi.org/10.1114/1.1555627
Issue Date:
DOI: https://doi.org/10.1114/1.1555627