A prototype compartmental model of blood pressure distribution. (English) Zbl 1204.34060
In this interesting paper, the authors construct a nonlinear compartmental model which describes the blood flow through the cardiovascular system in terms of pressure variables and assume that the blood pressure in the aorta and arteries is subject to impulsive perturbations, possibly modelling the regulation of the blood flow through the use of a pacemaker.
The existence of a bounded solution with various periodicity properties (periodic, eventually periodic, discontinuous almost periodic) is then established through the use of a fixed point argument. The uniform exponential stability of this solution is then proved, again via fixed point theory, it is also shown that the first coordinate of the solution is ultimately larger than all others. These properties are then interpreted as regulatory properties for arrhythmia processes.
The existence of a bounded solution with various periodicity properties (periodic, eventually periodic, discontinuous almost periodic) is then established through the use of a fixed point argument. The uniform exponential stability of this solution is then proved, again via fixed point theory, it is also shown that the first coordinate of the solution is ultimately larger than all others. These properties are then interpreted as regulatory properties for arrhythmia processes.
Reviewer: Paul Georgescu (Iaşi)
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 34A37 | Ordinary differential equations with impulses |
| 92C35 | Physiological flow |
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |
Keywords:
blood pressure distribution model; almost periodicity; eventually periodic solutions; stabilityReferences:
| [1] | Abbiw-Jackson, R. M.; Langford, W. F., Gain-induced oscillations in blood pressure, J. Math. Biol., 37, 203-234 (1998) · Zbl 0903.92017 |
| [2] | M.U. Akhmet, G.A. Bekmukhambetova, and Y. Serinagaoglu, Regular and irregular discontinuous dynamics of the systemic arterial pressure, in: Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2005, 16-20 August, Rodos, 2005, pp. 44-46; M.U. Akhmet, G.A. Bekmukhambetova, and Y. Serinagaoglu, Regular and irregular discontinuous dynamics of the systemic arterial pressure, in: Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2005, 16-20 August, Rodos, 2005, pp. 44-46 · Zbl 1081.92011 |
| [3] | V.I. Arnold, Cardiac arrhythmias and circle mappings, Chaos 1, pp. 20-24; V.I. Arnold, Cardiac arrhythmias and circle mappings, Chaos 1, pp. 20-24 · Zbl 0900.92094 |
| [4] | Bellman, R., (Mathematical Mmethods in Medicine. Mathematical Mmethods in Medicine, Series in Modern Applied Mathematics, vol. 1 (1983), World Scientific Publishing Co.: World Scientific Publishing Co. Singapore) · Zbl 0527.92004 |
| [5] | Cavalcanti, A.; Belardinelli, E., Modelling cardiovascular variability using differential delay equation, IEEE Trans. Biomed. Eng., 43, 982 (1996) |
| [6] | Fung, Y. C., Biomechanics Circulation, ((1997), Springer-Verlag: Springer-Verlag New York) · Zbl 0366.73091 |
| [7] | L. Glass, Cardiac arrhythmias and circle maps —A classical problem, Chaos 1, pp. 13-19; L. Glass, Cardiac arrhythmias and circle maps —A classical problem, Chaos 1, pp. 13-19 · Zbl 0900.92093 |
| [8] | Hoppensteadt, F. C.; Peskin, C. S., Mathematics in Medicine and in the Life Sciences (1992), Springer-Verlag: Springer-Verlag New York, Berlin, Heidelberg · Zbl 0744.92001 |
| [9] | Lerma, C.; Minzoni, A.; Infante, O.; Jose, M. V., A mathematical analysis for the cardiovascular control adaptations in chronic renal failure, Artificial Organs, 28, 4, 398-409 (2004) |
| [10] | McDonald, D. A., The relation of pulsative pressure to flow in arteries, J. Physiol., 127, 3, 533-552 (1955) |
| [11] | Nicols, W. W.; O’Rourke, M. F., McDonald’s Blood Flow in Arteries: Theoretical, Experimental and Clinical Principles (1990), Lea & Febier: Lea & Febier London |
| [12] | Ottesen, J. T., Modelling of the baroreflex-feedback mechanism with time-delay, J. Math. Biol., 36, 41-63 (1997) · Zbl 0887.92015 |
| [13] | Pedley, T., The Fluid Mechanics of Large Blood Vessels (1980), Cambridge University Press: Cambridge University Press London, New York · Zbl 0449.76100 |
| [14] | Stefanovska, A.; Luchinsky, D. G.; McClintock, P., Modeling couplings among the oscillators of the cardiovascular system, Physiol. Meas., 22, 551-564 (2001) |
| [15] | Gyori, I.; Eller, J., Compartmental systems with pipes, Math. Biosci., 53, 223-247 (1981) · Zbl 0488.92001 |
| [16] | Nicolis, G.; Prigogine, I., Self-organization in Non-equilibrium Systems (1977), Wiley: Wiley New York · Zbl 0363.93005 |
| [17] | Anderson, D. H., Compartmental Modeling and Tracer Kinetics (1983), Springer-Verlag · Zbl 0509.92001 |
| [18] | Jacquez, J. A., Compartmental Analysis in Biology and Medicine (1972), Elsevier: Elsevier New York · Zbl 0703.92001 |
| [19] | Ladde, G. S., Cellular systems, II. Stability of compartmental systems, Math. Biosci., 30, 1-21 (1976) · Zbl 0327.92004 |
| [20] | Murray, J. D., Mathematical Biology (1989), Springer-Verlag · Zbl 0682.92001 |
| [21] | (Matis, J. M.; Patten, B. C.; White, G. C., Compartmental Analysis of Ecosystem Models (1979), International Cooperative Pub. House) |
| [22] | Rescigno, A.; Segre, G., Drug and Tracer Kinetics (1966), Blaisdel: Blaisdel Waltham |
| [23] | Solimano, F.; Bischi, G. I.; Bianchi, M.; Rossi, L.; Magnani, M., A nonlinear three-compartment model for the administration of \(2^\prime, 3^\prime \)-Dideoxycytidine by using red blood cells as bioreactors, Bull. Math. Biol., 52, 785-796 (1990) · Zbl 0707.92013 |
| [24] | Akhmet, M. U., Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Anal., 60, 163-178 (2005) · Zbl 1066.34008 |
| [25] | M.U. Akhmetov, N.A. Perestyuk, A.M. Samoilenko, Almost-periodic solutions of differential equations with impulse action, Akad. Nauk Ukrain. SSR Inst., Mat. Preprint (26) (1983) 49 pp (in Russian); M.U. Akhmetov, N.A. Perestyuk, A.M. Samoilenko, Almost-periodic solutions of differential equations with impulse action, Akad. Nauk Ukrain. SSR Inst., Mat. Preprint (26) (1983) 49 pp (in Russian) |
| [26] | Akhmetov, M.; Perestyuk, N., Periodic and almost periodic solutions of strongly nonlinear impulse systems, J. Appl. Math. Mech., 56, 829-837 (1992) · Zbl 0791.34034 |
| [27] | Halanay, A.; Wexler, D., Qualitative Theory of Impulsive Systems (1968), Republici Socialiste Romania: Republici Socialiste Romania Bucuresti, (in Romanian) · Zbl 0176.05202 |
| [28] | Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002 |
| [29] | Samoilenko, A. M.; Perestyuk, N. A., Impulsive Differential Equations (1995), World Scientific: World Scientific Singapore · Zbl 0837.34003 |
| [30] | Akhmet, M. U.; Bekmukhambetova, G., On modeling of blood pressure distribution, Proceedings of the Fifth International Conference on Dynamic Systems and Applications held at Morehouse College, Atlanta, USA, May 30-June 2, 2007. Proceedings of the Fifth International Conference on Dynamic Systems and Applications held at Morehouse College, Atlanta, USA, May 30-June 2, 2007, Proc. Dyn. Syst. Appl., 5, 17-20 (2008) · Zbl 1203.76181 |
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