The Heart and Circulation

  • Frank C. Hoppensteadt
  • Charles S. Peskin
Part of the Texts in Applied Mathematics book series (TAM, volume 10)


This chapter begins a discussion of mathematics in physiology to which the remainder of the book is devoted. The discussion begins with the heart and blood circulation in the body. We first outline the structure of the circulation and then we derive models of blood flow and pressure and mechanisms for controlling them.


Cardiac Output Arterial Pressure Stroke Volume Pulse Pressure Diastolic Pressure 
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Annotated References

Many of the concepts used in this Chapter were pioneered by A.C. Guyton. See, for example

  1. Guyton A.C.: Circulatory Physiology: Cardiac Output and its Regulation. Saunders, Philadelphia, PA, 1963.Google Scholar

The idea that each cardiac ventricle can be modeled as a time-varying compliance and hence that the stroke volume can be determined from a pressure-volume loop (Fig. 5.5) comes from the work of K. Sagawa and his colleagues

  1. Sagawa, K., Suga, H. and Nakayama, K.: Instantaneous pressure-volume ratio of the left ventricle versus instantaneous force-length relation of papillary muscle. In: Cardiovascular System Dynamics ( Baan, J., Noordergraaf, A., and Raines, J., eds.), M.I.T. Press, Cambridge, MA, 1978, 99–105.Google Scholar

Neural control of the circulation is discussed in the following references

  1. Karloff, et al.: Adaptation of the left ventricle to sudden changes in heart rate in patients with artificial pacemakers. Cardiovascular Research, 7: 322, 1973.CrossRefGoogle Scholar
  2. Korner, P.I.: Integrative neural control of the circulation, Physiological Reviews 51, 312–367, 1971.Google Scholar
  3. Rowell, L.B.: Human cardiovascular adjustments to exercise and thermal stress, Physiological Reviews, 54, 75–159, 1974.Google Scholar
  4. Topham, W.S. and Warner, H.R.: The control of cardiac output during exercise. In: Physical Bases of Circulatory Transport: Regulation and Exchange ( Reeve and Guyton, eds.), Saunders, Philadelphia, PA, 1967.Google Scholar

Our emphasis on oxygen as the key factor in autoregulation (Section 5.9) can be traced back to the work of Guyton

  1. Guyton, A.C., Ross, J.M., Carrier, O., Jr. and Walker, J.R.: Evidence for tissue oxygen demand as the major factor causing autoregulation, Circulation Research, 14, 60, 1964.Google Scholar

The particular model of autoregulation that we use is simplified from

  1. Huntsman, L.L., Attinger, E.O., and Noordergraaf, A.: Metabolic autoregulation of blood flow in skeletal muscle: A model. In: Cardiovascular System Dynamics ( Baan, J., Noordergraaf, A., and Raines, J., eds.), M.I.T. Press, Cambridge, MA, 1978, 400–414.Google Scholar
  2. Rudolph, A.M.: Congenital diseases of the heart, Year Book Medical Publishers, Chicago, IL, 1974.Google Scholar
  3. Rudolph’s book is written in such a way that the mathematically inclined reader will find many opportunities for the construction of medically relevant mathematical models.Google Scholar

Finally, for a slightly more advanced look at some of the material presented in this Chapter, see

  1. Peskin, C.S.: Control of the heart and circulation, In: Mathematical Aspects of Physiology (Hoppensteadt, F.C., ed.), Lectures in Applied Mathematics, 19, American Mathematical Society, Providence, RI, 1981, 138.Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Frank C. Hoppensteadt
    • 1
  • Charles S. Peskin
    • 2
  1. 1.College of Natural ScienceMichigan State UniversityEast LansingUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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