Skip to main content
Log in

A mathematical model for fundamental regulation processes in the cardiovascular system

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

Based on the four compartment model by Grodins we develop a model for the response of the cardiovascular system to a short term submaximal workload. Basic mechanisms included in the model are Starling's law of the heart, the Bowditch effect and autoregulation in the peripheral regions. A fundamental assumption is that the action of the feedback control is represented by the baroceptor loop and minimizes a quadratic cost functional. Simulation results show that the model provides a satisfactory description of data obtained in bicycle ergometer tests.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Altman, P. L. E., Dittmer, D. S. (eds.): Biological Handbook, Respiration and Circulation. Bethesda, MD: Federation of American Societies for Experimental Biology 1971

    Google Scholar 

  2. Brown, K. M., Dennis, J. E., Jr.: Derivative free analogues of the Levenberg-Marquardt and Gauss algorithms for nonlinear least squares approximation. Numer. Math. 18, 289–297 (1972)

    Google Scholar 

  3. Doubek, E. D., Jr.: Least energy regulation of the arteial system. Bull. Math. Biol. 40, 79–93 (1978)

    Google Scholar 

  4. Grimby, G., Nilsson, N. J., Saltin, B.: Cardiac output during submaximal and maximal exercise in active middle-aged athletes. J. Appl. Physiol. 21 (4), 1150–1156 (1960)

    Google Scholar 

  5. Grodins, F. S.: Integrative cardiovascular physiology: a mathematical synthesis of cardiac and blood vessel hemodynamics Q. Rev. Biol. 34 (2), 93–116 (1959)

    Google Scholar 

  6. Grodins, F.S.: Control Theory and Biological Systems. New York London: Columbia University Press 1963

    Google Scholar 

  7. Keidel, W. D. (ed.) Kurzgefaβtes Lehrbuch der Physiologie. Stuttgart New York: Thieme 1985

    Google Scholar 

  8. Kenner, T.: Physical and mathematical modeling in cardiovascular systems. In: Hwang, N. H. C., Gross, D. R., Patel, D. J. (eds.) Clinical and Research Applications of Engineering Principles, pp. 41–109. Baltimore: University Park Press 1979

    Google Scholar 

  9. Kenner, T., Pfeiffer, K. P.: Studies on the optimal matching between heart and arterial system. In: Baan, J., Artnezenius, A. C., Yellin, T., Nijhoff, M. (eds.) Cardiovascular Dynamics. The Hague 1980

  10. Kwakernaak, H., Sivan, R.: Linear Optimal Control Systems. New York London Sydney Toronto: Wiley-Interscience 1972

    Google Scholar 

  11. Magrini, A., Guerrisi, M., Arabia, M., Staderini, E. M., Franconi, C.: Control of the heart. In: Alberi, G., Bajzer, Z., Baxa, P. (eds.) Proceedings of the International Conference on Applications of Physics to Medicine and Biology. Trieste, Italy, 30.3–3.4.1982, pp. 247–276, Singapore: World Scientific 1983

    Google Scholar 

  12. Mö1ler, D.: Ein geschlossenes nichtlineares Modell zur Simulation des Kurzzeitverhaltens des Kreislaufsystems und seine Anwendung zur Identifikation. (Med. Inf. Stat., vol. 30) Berlin Heidelberg New York: Springer 1981

    Google Scholar 

  13. More, J. J.: The Levenberg-Marquardt algorithm: implementation and theory. In: Watson, G. A., (ed.) Numerical Analysis (Lee. Notes Math., vol. 630) pp. 105–116. Berlin Heidelberg New York: Springer 1977

    Google Scholar 

  14. Noldus, E. J.: Optimal control aspects of left ventricular ejection dynamics. J. Theor. Biol. 63, 275–309 (1976)

    Google Scholar 

  15. Ono, K., Uozumi, T., Yoshimoto, C., Kenner, T.: The optimal cardiovascular regulation of the arterial blood pressure. In: Kenner, T., Busse, R., Hinghofer-Szalkay, H. (eds.) pp. 119–131. London New York: Plenum Press 1982

    Google Scholar 

  16. Pavel, N. H.: Differential Equations, Flow Invariance and Applications. (Pitman Adv. Publ. Program) Boston London Melbourne: Longman 1984

    Google Scholar 

  17. Peer, R. O.: Matematische Modellierung von grundlegenden Regelungsvorgängen im Herzkreislauf-System. Technical Report, Technische Universität Graz (1989)

  18. Peskin, C. S.: Lectures on mathematical aspects of physiology. (Lect. Appl. Math., vol. 19, pp. 1–107). Providence, RI: Am. Math. Soc. 1981

    Google Scholar 

  19. Pessenhofer, H.: Personal communication, Institut für Physiologie, KFU-Graz

  20. Pfeiffer, K. P., Kenner, T.: On the optimal strategy of cardiac ejection. In: Kenner, T., Busse, R., Hinghofer-Szalkay, H. (eds.) Cardiovascular System Dynamics: Models and Measurements, pp. 133–136. London New York: Plenum Press 1981

    Google Scholar 

  21. Powell, M. J. D.: A hybrid method for nonlinear equations. In: Rabinowitz, P. (ed.) Numerical Methods for Nonlinear Algebraic Equations, pp. 115–161. Boston: Gordon and Breach 1970

    Google Scholar 

  22. Ranft, U.: Zur Mechanik und Regelung des Herzkreislaufsystems. (Med. Inf. Stat. vol. 6) Berlin, Heidelberg New York: Springer 1978

    Google Scholar 

  23. Russell, D. L.: Mathematics of Finite-Dimensional Control Systems — Theory and Design. New York Basel: Marcel Dekker 1979

    Google Scholar 

  24. Swan, G. W.: Applications of Optimal Control Theory in Biomedicin. New York Basel: Marcel Dekker 1984

    Google Scholar 

  25. Wesseling, K. H., Settels, J. J., Walstra, H. G., van Esch, H. J., Donders, J. J. H.: Baromodulation as the cause of short term blood pressure variability? In: Alberti, G., Bajzer, Z., Baxa, P., (eds.) Proc. I. Int. Conf. Applications of Physics to Medicine and Biology, pp. 247–276. Singapur: World Scientific 1983

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kappel, F., Peer, R.O. A mathematical model for fundamental regulation processes in the cardiovascular system. J. Math. Biol. 31, 611–631 (1993). https://doi.org/10.1007/BF00161201

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00161201

Key words

Navigation