# Mathematical models of granulopoiesis

Wheldon, Thomas E (1973) Mathematical models of granulopoiesis. PhD thesis, University of Glasgow.

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## Abstract

Haemopoiesis (blood cell production) is a process subject to active physiological regulation. It constitutes one example of a biological process controlled through a hierarchy of feedback loops acting at a range of levels from the molecular to the macroscopic. The thesis describes mathematical studies of the more macroscopic (physiological) levels of the control of haemopoiesis, with special emphasis on granulopoiesis. Following review of pertinent background material in cybernetics, physiology and pathology, attention is focussed on the mathematical representation of cellular proliferation and maturation, and a representation formulated in tennis of experimental observables is proposed. This leads to the study of a non-linear transcendental equation of the form with the unknown quantity. An iterative method of solution is proposed for this equation, which permits kinetic analysis of a certain class of non-steady-state processes. The method is used for the analysis of maturation kinetics of embryonic erythroid cells. Attention is then turned to the causal basis of the control of haemopoiesis. It is pointed out that certain features of haemopoietic regulation lead to the expectation of oscillatory phenomena and th.at observation of such phenomena can be revealing. With this in mind, a simple model is advanced for the control of granulocyte production. The model comprises two feedback loops, one regulating 'de novo' granulopoiesis in accordance with marrow granulocyte numbers and one regulating release from the marrow in accordance with blood granu1ocyte numbers. The model is described by the system of delay-differential equations where Gm , GB are (respectively) the marrow granulocyte number and blood granulocyte number, both at time, and to are parameters, chosen to give maxim.um physiological realism. Since 'de novo' granulopoiesis is believed to constitute a drain on primitive 'stem, cells', regulation of stem cell number appears necessary. A model for stem cell mitotic autoregulation based on a diffusible inhibitor concept is proposed. This theory leads to the study of a pair of differential equations which, utilizing probable order-of-magnitude differences in relaxation times, may be reduced to the single equation which admits of a closed analytic solution. This equation exhibits both self-limiting and non-self-limiting modes of behaviour, the biological implications of which are discussed. With parameters selected for stability, the stem cell mitotic autoreguation loop may be adjoined to the granulopoiesis control system, previously considered, to obtain a composite model. However, the unstable and capricious behaviour of this multi-loop combination renders it physiologically unacceptable, Modifications which restore stability seem to imply heterogeneity of the stem cell population, representation of which lies outside the scope of the study described. In the following chapter, consideration is given to the physical, basis of regulation of 'de novo' granulopoiesis with emphasis on 'in vitro' evidence relating to 'colony stimulating factor' and the possible role of positive feedback in regulating granulocyte numbers in infection. By mathematical formulation of a. recently proposed pcsitive- feedback model it is shown that positive feedback systems can be stable in the absence of overt negative feedback loops provided passive damping elements (e.g, cell death) exist and satisfy'' certain criteria. It is shown, however, that the criteria concerned are incompa.tib3.e with known featu.res of the regulation of granulopoiesis in infection and the existence of additional, negatively-acting, loops is deduced. Some possibilities in this direction are proposed. The model studies may illuminate the pathogenesis of some disorders of the control of granulopoiesis, notably cyclical neutropenia and myeloid leukaemia.

Item Type: Thesis (PhD) Doctoral Adviser: J Anderson Applied mathematics, Bioengineering 1973 Enlighten Team glathesis:1973-74179 Copyright of this thesis is held by the author. 23 Sep 2019 15:33 23 Sep 2019 15:33 https://theses.gla.ac.uk/id/eprint/74179