Abstract
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.
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Annotated References
Many of the concepts used in this Chapter were pioneered by A.C. Guyton. See, for example
Guyton A.C.: Circulatory Physiology: Cardiac Output and its Regulation. Saunders, Philadelphia, PA, 1963.
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
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.
Neural control of the circulation is discussed in the following references
Karloff, et al.: Adaptation of the left ventricle to sudden changes in heart rate in patients with artificial pacemakers. Cardiovascular Research, 7: 322, 1973.
Korner, P.I.: Integrative neural control of the circulation, Physiological Reviews 51, 312–367, 1971.
Rowell, L.B.: Human cardiovascular adjustments to exercise and thermal stress, Physiological Reviews, 54, 75–159, 1974.
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.
Our emphasis on oxygen as the key factor in autoregulation (Section 5.9) can be traced back to the work of Guyton
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.
The particular model of autoregulation that we use is simplified from
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.
Rudolph, A.M.: Congenital diseases of the heart, Year Book Medical Publishers, Chicago, IL, 1974.
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.
Finally, for a slightly more advanced look at some of the material presented in this Chapter, see
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.
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© 1992 Springer Science+Business Media New York
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Hoppensteadt, F.C., Peskin, C.S. (1992). The Heart and Circulation. In: Mathematics in Medicine and the Life Sciences. Texts in Applied Mathematics, vol 10. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4131-5_6
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DOI: https://doi.org/10.1007/978-1-4757-4131-5_6
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