A Mathematical Analysis of Red Blood Cell and Bone Marrow Stem Cell Control Mechanisms

Abstract

The oscillations, interactions and control mechanisms of the red blood cell and bone marrow stem cell system were studied. A mathematical formulation in the form of four difference‐differential simultaneous equations was established in the four variables of the system; (i) stem cell population, (ii) chalone mitotic inhibitor, (iii) red cell population, and (iv) erythropoietin; as postulated by Orr, Kirk, Gray and Anderson (1968). Numerical values of the constants in these equations were determined from a large range of data. Calculations on the behaviour of the formulation were carried out using an ICT 1902 digital computer. The predictions of the formulation were compared with three kinetic sets of experimental data, showing satisfactory agreement. Analysis of the system yields the value of the stem cell population. In conclusion, some properties of the oscillatory features of the system are discussed.

Regulation of Platelet Production: The Normal Response to Perturbation and Cyclical Platelet Disease

“…It is generally thought that these disorders involve a destabilization at the hematopoietic stem cell (HSC) level leading to an oscillatory e%ux of cells into all di!erentiation pathways (Mackey, 1978(Mackey, , 1979a(Mackey, , 1996. In other dynamic hematological diseases like periodic autoimmune hemolytic anemia (BeH lair et al, 1995;Kirk et al, 1968;Mackey, 1979b;Maha!y et al, 1998;Orr et al, 1968) there is only an oscillation in one of the major blood cell types. Cyclical thrombocytopenia (CT) is a rare blood disease apparently falling into this second category.…”

Blood Cell Dynamics: Half of a Century of Modelling

“…Let us finish the 1960's by the work of Kirk et al in [167] where oscillation as well as interactions and control mechanisms of the red blood cell and bone marrow stem cell were studied. The mathematical model consisted in four difference-differential equation and studied numerically (see Fig.…”

“…In this work, authors wanted to show that, this disorder was due to a loss of stability in the peripheral control process regulating neutrophil production through a negative feedback between mature neutrophils and their precursors. Based on the Kirk et al [167] approach, this mathematical model here was more computer oriented, but it was one of the first models dealing with a periodic chronic blood disease with oscillations due to the loss of stability. No mathematical analysis was done in this work but it inspired several authors, like Kazarinoff and van den Driessche in [164], nine years later, and many investigations of chronic blood diseases through Hopf bifurcation would follow in the next decades, becoming one of the modeling trends of that period.…”

A Cell Kinetic Study on the Cancer Problem Based on the Automatic Control Theory Using Digital Simulation

“…The papers of Blumenson (1973), De Vita (1971, Kiefer (1973), Kirk, Orr and Hope (1968), Orr et aI. (1968), Tarbutt and Blackett (1968), Wheldon, Kirk and Gray (1973), and Wheldon, Kirk and Orr (1974) deal with the clear method of representation by block diagrams and also show mathematical statements for the cell renewal process by using differential equations.…”