Mathematical problems of the dynamics of the red blood cells

doi:10.14708/ma.v4i6.1173

Vol 4, No 6 (1976) , 23-40

Mathematical problems of the dynamics of the red blood cells

M. Ważewska-Czyżewska, A. Lasota

Digital Object Identifier (DOI): 10.14708/ma.v4i6.1173

Abstract

The purpose of this paper is to construct a model for the process of generation and degeneration of red blood cells, taking into account biological-medical experimental data. The paper contains four sections. The first section presents basic biological facts enabling the reader who is not a specialist to understand this process. Section 2 is devoted to constructing an analytic model which uses a first order linear partial differential equation and a nonlinear integral equation. Section 3 lists some of the simplest mathematical properties of this model and the biological consequences arising from it. Section 4 gives a simplified model which can be described in terms of a nonlinear ordinary differential equation with a delayed parameter. The properties of this simple equation are of interest from both a mathematical and a biological point of view. In particular, the proof of the existence of periodic equations requires the application of a nontrivial version of a theorem on fixed points. The problem of the stability of this periodic solution is open. The problem is important in that the existence of a stable periodic solution gives a theoretical explanation of certain types of blood diseases.

Keywords: 97M60

References

[1] S. N. Chow, Existence of Periodic Solutions of Autonomous Functional Differential Equations, J. Differential Equations 15 (1974), str. 350-378.

[2] H. F. Dorn, Mortality, rozdział w książce: The Study of Population; An Inventory and Appraisal, redaktorzy P. M. Hauser i O. D. Duncan, The University of Chicago Press, Chicago, Illinois, USA, 1959, str. 437-471.

[3] H. Von Foerster, Some Remarks on Changing Populations, The Kinetics of Cellular Proliferation, redaktor F. Stohlman, Jr., Grune Stratton, New York-London 1959, str. 382-407.

[4] J. K. Hale, Functional Differential Equations, Springer-Verlag, New York 1971.

[5] J. Kirk, J. S. Orr i C. S. Hope, A Mathematical Analysis of Red Blood Cell and Bone Marrow Stem Cell Control Mechanisms, Brit. J. Haemat. 15 (1968), str. 35-46.

[6] B. I. Lord i M. J. Murphy, Jr., Hematopoietic Stem Cell Regulation II. Chronic Effects of Hypoxic-Hypoxia on CFU Kinetics, Blood 42 (1973), str. 89-98.

[7] A. Morley, E. A. King-Smith i F. Stohlman, Jr., The Oscillatory Nature of Hemopoiesis, Symposium on Hemopoietic Cellular Proliferation, redaktor F. Stohlman, Jr., 1969, str. 3-14.

[8] J. S. Orr, J. Kirk, K. G. Gray i J. R. Anderson, A Study of the Interdependence of Red Cell and Bone Marrow Stem Cell Populations, Brit. J. Haemat. 15 (1968), str. 23-34.

[9] D. Shemin i D. Rittenberg, The Life Span of the Human Red Blood Cell, J. Biol. Chem. 166 (1946), str. 627-636.

[10] M. Ważewska-Czyżewska, A New Mathematical Formulation of the Process of Production and Destruction of Erythrocytes, Pol. Med. Sc. and Hist. Bulletin 9 (1966), str.148-151.

[11] G. H. Weiss i G. Zajicek, Kinetics of Red Blood Cells following Hemolysis, J. Theoret. Biol. 23 (1969), str. 475-491.

[12] W. J. Williams, E. Beutler, A.J. Erslew i R. W. Rundles, Hematology, Mc Graw-Hill Book Company, New York 1972.

Pages: 23-40

Full Text: PDF