Department of Mathematics, Elmhurst College. 190 Prospect Avenue, Elmhurst, IL 60126 USA,. Keywords: mathematical modeling, myelodysplastic syndromes, ...
HEMOPOIETIC DYNAMICS IN THE BONE MARROW, THE MYELODYSPLASTIC SYNDROMES, AND ISSUES RELATED TO CONTROL OF THIS DISEASE Evans Afenya Department of Mathematics, Elmhurst College 190 Prospect Avenue, Elmhurst, IL 60126 USA Keywords: mathematical modeling, myelodysplastic syndromes, control
1. INTRODUCTION Hematopoiesis has remained a subject of intense study over the years because it is basically a massive process in the human body that involves various distinct cell lineages and results in the production of billions of different cell types each day. The genesis of this phenomenon is rooted in pluripotent stem cells in the bone marrow (BM) and involves a pool of totipotent stem cells that provide unipotent stem cells to the granulocytic, erythrocytic, thrombocytic and other lines. In each line, unipotent stem cells supply cells to a number of nonproliferating differentiation compartments in the BM before the release of mature neutrophils, erythrocytes, platelets, and other cell types into the blood. There are a number of disorders such as the myelodysplastic syndromes (MDS) that negatively affect the normal functioning of the hemopoietic system and that are not yet completely understood as is evidenced by the absence of a known cure in certain instances (Anderson et. al., 1993; Shimazaki et. al., 2000). The myelodysplastic syndromes can be described as a group of acquired hematopoietic disorders with evidence of trilineage dysplasia and an approximately 30% incidence of eventual transformation into acute myeloid leukemia (Raza et al., 1995a). These disorders are clonal in nature and involve one or more clones. They evolve from a transformation of the normal hematopoietic state into the precancerous disease state. While some investigations of MDS have led to observations of ineffective hematopoiesis (Raza
et. al., 1995a, b; Parcharidou et. al., 1999) in the BM arising from massive apoptosis of cells in this compartment others have suggested that the precise relationship between increased apoptosis of myeloid precursors and cytopenias should be more precisely explored (Lepelley et. al., 1996). Our aim is to contribute to deepening and enriching the understanding of MDS and its treatment through the use of biomathematical models that give insight into its etiology and evolutionary dynamics. In the process of doing this we hope to address some of the discrepancies arising from the MDS investigations with the view to proposing possible resolutions of such discrepancies. Consequently, we propose to start our investigations by considering hematopoietic dynamics in the peripheral blood and marrow since these compartments understandably form the overwhelming focus of MDS research. We then end up discussing ways in which this disease can be controlled. 2. MODEL DESIGN AND ANALYSES By relying on information from the literature regarding hematopoiesis (Mackey & Glass, 1977; Kazarinoff & denDriessche, 1979; Lord et. al., 1992; Schmitz et. al., 1993; Dale et. al., 1998; Price et. al., 1996; Schrier, 1988; Afenya, 1996; Marer & Skacel, 1999) and drawing upon investigations related to MDS (Anderson et. al., 1993; Hellstrom-Lindberg et. al., 1997; Shimazaki et. al., 2000; Raza et. al., 1995a, b; Khan et. al., 1991; Parcharidou et. al., 1999; Lepelley et. al., 1996; Mundle et. al., 1996, 2000), it is
appropriate to consider a model that comprises two broad compartments –a BM compartment and a peripheral blood (PB) compartment. Since the BM is said to be surprisingly uniform (Schrier, 1988), it will be assumed to a reasonable first approximation that the cells in this tissue are homogeneously distributed. This assumption is stretched to the PB compartment. It is well known that cells in the BM spend some time maturing (Lord et. al., 1992; Schmitz et. al., 1993; Dale et. al., 1998) in this tissue before entering the blood to perform various functions during hemopoiesis. This means that a time lag due to cell maturation exists during the movement of cells from the BM compartment to the PB. Also in existence is a feedback mechanism through which cells in the BM are instructed to reproduce to account for shortfalls in the cell population of the PB compartment when situations that entail such developments arise. A schematic description of hemopoietic function is shown in Figure 1. An interpretation of this description that yields the model can be stated in words as follows: [Rate of Change of the BM Cell Population] = [Growth Rate of Marrow Cells] + [Feedback from the Blood to the BM] –[Rate of BM Cell Apoptosis] – [Release Rate of BM Cells to the Blood]. [Rate of Change of the Blood Cell Population] = [Rate of Influx into and Turnover of Cells in the PB] –[Rate of Efflux of Cells out of the PB].
We note that the rate of influx into and turnover of cells in the PB encompass the rate at which BM cells are released into this compartment and the rate at which PB cells are produced (Lord et. al., 1992) in this compartment. The efflux rate of cells out of the PB include the rate of cell loss or cell disappearance (Lord et. al., 1992) out of this compartment in addition to the feedback sent from the blood to the BM. In mathematical terms we obtain the following system of equations: N m = α m (N m ) + F (N b ) (1) − α ( N m (t − Tm )) N m (t − Tm ) − α md N m N b = α ( N m (t − Tm )) N m (t − Tm ) − α bd N b (2) with N m (0) = N m 0 , N b (0) = N b 0 , and N m (t ) = N mc when − Tm ≤ t < 0 (3) where the parameters and variables in the equations above are described as follows: m (Nm ) = state-dependent growth rate of the BM cells per unit time, F(Nb) = state-feedback from the PB to the BM, md = fractional apoptotic rate of BM cells per unit time, bd = fractional rate of PB cell loss per unit time, Tm = transit time of cells in the BM due to maturation, (Nm(t –Tm)) = release rate of cells from the BM into the PB, Nm(t) = population of BM cells/liter at time t,
and Nb(t) = population of PB cells/liter at time t. The quantity Nmc = critical homeostatic level of cells per liter in the BM. In analyzing the model, functional representations are obtained for m (Nm ) and F(Nb ) and system behavior is considered with regards to the parameters. The issues of massive apoptosis and ineffective hematopoiesis are placed within our analytical considerations. Control of MDS is analyzed and simulated by focusing on the mechanisms that could influence maturation delays. Feedback
Bone Marrow (BM)
Maturation Delay
BM Cell Apoptosis
Peripheral Blood (PB) PB Cell Loss
Fig.1. Schematic description of hemopoietic function
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