Review on lumped parameter method for modeling the blood flow in systemic arteries


The cardiovascular system is characterized by complex interactions between various control mechanisms and physiological processes. Different approaches are used to provide better diagnostics and physiological understanding, cardiac prosthesis and medical planning. The mathematical description and modelling of the human cardiovascular system plays nowadays an important role in the comprehension of the genesis and development of cardiovascular disorders by providing computer based simulation of dynamic processes in this system. This paper aims to give an overview on lumped parameter models that have been developed by many researchers all over the world, to simulate the blood flow in systemic arteries. Surveying various references we make a review of different approaches to arterial tree modelling and discuss on the applications of such models.

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Kokalari, I. , Karaja, T. and Guerrisi, M. (2013) Review on lumped parameter method for modeling the blood flow in systemic arteries. Journal of Biomedical Science and Engineering, 6, 92-99. doi: 10.4236/jbise.2013.61012.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] WHO Fact Sheet No. 317, (2011).
[2] Guyton, A.C. and Hall, J.E. (2006) Textbook of medical physiology. 11th Edition., Elsevier Saunders, Philadelphia.
[3] Guyton, A.C., Coleman, T.G. and Granger, H.J. (1972) Circulation: Overall regulation. Annual Review of Physiology, 34, 13-44. doi:10.1146/
[4] Wu, X.M. (1994) Modeling and simulation of cardiovascular circulation system: Status and prospect. Science- paper Online, 2, 112-116.
[5] Cellier, F.E. and Kofman, E. (2006) Continuous system simulation. Springer-Science, New York.
[6] Werner, J., B?hringer, D. and Hexamer, M. (2002) Simulation and Prediction of cardiotherapeutical phenomena from a pulsatile model coupled to the guyton circulation model. IEEE Transaction on Biomedical Engineering, 49, 430-439. doi:10.1109/10.995681
[7] Simanski, O., K?hler, R., Schubert, A., Janda, M., Bajorat, J., Hofmockel, R. and Lampe, B.P. (2008) Proceedings of the 17th IFAC World Congress. In: Chung, M.J., Misra, P. and Shim, H., Eds, The International Federation of Automatic Control, Seoul, 9601-9606.
[8] Coleman, T.G. and Randall, J.E. (1983) HUMAN: A comprehensive physiological model. Physiologist, 26, 15- 21.
[9] Lu, K., Clark, J.W., Ghorbel, F.H., Ware, D.L. and Bidani, A. (2001) A human cardiopulmonary system model applied to the analysis of the Valsalva maneuver. American Journal of Physiology—Heart and Circulatory Physiology, 281, 2661-2679.
[10] Magosso, E., Biavati, V. and Ursino, M. (2001) Role of the baroreflex in cardiovascular instability: A modeling study. Cardiovascular Engineering, 1, 101-115. doi:10.1023/A:1012574513589
[11] Abram, S.R., Hodnett, B.L., Summers, R.L., Coleman, T.G. and Hester, R.L. (2007) Quantitative circulatory physiology: An integrative mathematical model of human physiology for medical education. Advances in Physiology Education, 31, 202-210. doi:10.1152/advan.00114.2006
[12] Mili?i?, V. and Quarteroni, A. (2004) Analysis of lumped parameter models for blood flow simulations and their relation with 1D models. Mathematical Modeling and Numerical Analysis, 38, 613-632. doi:10.1051/m2an:2004036
[13] Grodins, F.S. (1959) Integra-tive cardiovascular physiol- ogy: A mathematical synthesis of cardiac and blood ves- sel hemodynamics. Quarterly Review of Biology, 34, 93- 116. doi:10.1086/402631
[14] Hales, S. (1733) Statistical essays: Containing haemostaticks, or an account of some hydraulick and hydrostatical experiments on the blood and blood-vessels of animals, etc. 2, Innys & Manby, Lon-don.
[15] Frank, O. (1899) Die grundform des arterielen pulses erste abhandlung: Mathematische analyse. Zeitschrift für Bi-ologie, 37, 483-526.
[16] Wetterer, E. (1954) Flow and pres-sure in the arterial system, their hemodynamic relationship, and the principles of their measurement. Minnesota Medicine, 37, 77-86.
[17] Westerhof, N., Stergiopulos, N. and Noble, I.M. (2010) Snapshots of hemodynamics an aid for clinical research and graduate education. 2nd Edition, Springer, New York. doi:10.1007/978-1-4419-6363-5
[18] Westerhof, N., Bosman, F., DeVries, C.J. and Noordergraaf, A. (1969) Analogue studies of the human systemic arterial tree. Journal of Biomechanics, 2, 121-208. doi:10.1016/0021-9290(69)90024-4
[19] Westerhof, N., El-zinga, G. and Sipkema, P. (1971) An artificial arterial system for pumping hearts. Journal of Applied Physiology, 31, 776-778.
[20] Wang, J.J., O’Brien, A.B., Shrive, N.G., Parker, K.H. and Tyberg, J.V. (2003) Time-domain representation of ventricular-arterial coupling as a Windkessel and wave system. American Journal of Physiology—Heart and Circulatory Phy-siology, 284, 1358-1368.
[21] Burkhoff, D., Alexander, J. and Schipke, J. (1988) Assessment of windkessel as a model of aortic input impedance. American Journal of Physiology—Heart and Circulatory Physiology, 255, 742-753.
[22] Burattini, R. and Natalucci, S. (1998) Complex and frequency-dependent compliance of viscoelastic Windkessel resolves contradictions in elastic Windkessels. Medical Engineering & Physics, 20, 502-514. doi:10.1016/S1350-4533(98)00055-1
[23] Stergiopulos, N., Westerhof, B.E. and Westerhof, N. (1992) Total arterial iner-tance as the fourth element of the Windkessel model. American Journal of Physiology— Heart and Circulatory Physiology, 276, 81-88.
[24] Grant, B.J.B. and Paradowski, L.J. (1987) Characterization of pulmonary arterial input impedance with lumped parameter models. American Journal of Physiology-Heart and Circulatory Physiology, 252, 585-593.
[25] Burattini, R. and Gnudi, R. (1982) Computer identification of models for the arterial tree input impedance: Comparison between two new simple models and first experimental results. Medical and Biological Engineering and Computing, 20, 134-144. doi:10.1007/BF02441348
[26] Burattini, R. and Di Salvia, P.O. (2007) Development of systemic arterial mechanical properties from infancy to adulthood interpreted by four-element Wind-kessel models. Journal of Applied Physiology, 103, 66-79. doi:10.1152/japplphysiol.00664.2006
[27] Kolh, P., D’Orio, V., Bernard, L., Gerard, P., Gommes, C. and Limet, R. (2000) Increased aortic compliance maintains left ventricular performance at lower energetic cost. European Journal Cardio-Thoracic Surgery, 17, 272-278. doi:10.1016/S1010-7940(00)00341-9
[28] Sharp, K.M., Pantalos, G.M., Minich, L., Tani, L.Y., McGough, E.C. and Haw-kins, J.A. (2000) Aortic input impedance in infants and children. Journal of Applied Physiology, 88, 2227-2239.
[29] Rager, G.N., Westerhof, N. and Noordergraaf, A. (1965) Oscillatory flow impedance in electrical analog of arterial system: Representation of sleeve effect and non-newtonian properties of blood. Circulation Research, 16, 121- 133. doi:10.1161/01.RES.16.2.121
[30] Frasch, H.F., Kresh, Y.J. and Noordergraaf, A. (1996) Two-port analysis of microcircu-lation: An extension of windkessel. American Journal of Physiology—Heart and Circulatory Physiology, 270, 376-385.
[31] Formaggia, L. and Veneziani, A. (2003) Reduced and multiscale models for the human CVS. Technical Report, PoliMI, Milan.
[32] Avolio, A.P. (1980) Multi-branched model of the human arterial system. Medical and Biological Engineering and Computing, 18, 709-718. doi:10.1007/BF02441895
[33] Beyar, R., Hausknecht, M.J., Halperin, H.R., Yin, F.C. and Weisfeldt, M.L. (1987) Interaction between cardiac chambers and thoracic pressure in intact circulation. American Journal of Physiology—Heart and Circulatory Physiology, 253, 1240-1252.
[34] Santamore, W.P. and Burkhoff, D. (1991) Hemodynamic consequences of ven-tricular interaction as assessed by model analysis. American Journal of Physiology—Heart and Circulatory Physiology, 260, 146-157.
[35] Ursino, M. (1998) Interaction between carotid baroregulation and the pulsating heart: A mathematical model. American Journal of Physiology—Heart and Circulatory Phy-siology, 275, 1733-1747.
[36] ?á?ek, M. and Krause, E. (1996) Numerical simulation of the blood flow in the human CVS. Journal of Biomechanics, 29, 13-20. doi:10.1016/0021-9290(95)00027-5
[37] Chen, S., Zhang, S., Gong, Y., Dai, K., Sui, M., Yu, Y. and Ning, G. (2008) The role of the autonomic nervous system in hypertension: A bond graph model study. Physiological Measurement, 29, 473-495. doi:10.1088/0967-3334/29/4/005
[38] Heldt, T., Shim, E.B., Kamm, R.D. and Mark, R.G. (2002) Computational modeling of cardiovascular response to orthostatic stress. Journal of Applied Physiology, 92, 1239- 1254.
[39] Noordergraaf, A., Verdouw, P.D. and Boom, H.B.K. (1963) The use of an analog computer in a circulation model. Progress in Cardiovascular Diseases, 5, 419-439. doi:10.1016/S0033-0620(63)80009-2
[40] O’Rourke, M.F. and Avolio, A.P. (1980) Pulsatile flow and pressure in human systemic arteries. Studies in man and in a multibranched model of the human systemic arterial tree. Circulation Research, 46, 363-372. doi:10.1161/01.RES.46.3.363
[41] Olansen, J.B., Clark, J.W., Khoury, D., Ghorbel, F.H. and Bidani, A. (2000) A closed-loop model of the canina cvs that includes ventricular interaction. Computer and Biomedical Research, 33, 260-295. doi:10.1006/cbmr.2000.1543
[42] Ursino, M. and Magosso, E. (2003) Role of short-term cardiovascular regulation in heart period variability: A model study. American Journal of Physiology—Heart and Circulatory Physiology, 284, 1479-1493.
[43] Ursino, M. (1999) A mathematical model of the carotid baroregulation in pulsating conditions. IEEE Transactions on Biomedical Engineering, 46, 382-392. doi:10.1109/10.752935
[44] Westerhof, N., Lankhaar, J.W. and Westerhof, B.E. (2009) The arterial Windkessel. Medical and Biological Engineering and Computing, 47, 131-141. doi:10.1007/s11517-008-0359-2

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