Mathematical Biosciences 274 (2016) 83–93 Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs Mathematical modeling of bone marrow – peripheral blood dynamics in the disease state based on current emerging paradigms, part I Evans K. Afenya a,∗ , Rachid Ouifki b , Baba I. Camara c , Suneel D. Mundle d a Department of Mathematics, Elmhurst College, 190 Prospect Avenue, Elmhurst, IL 60126, USA b DST/NRF Centre of Excellence in Epidemiological Modelling and Analysis (SACEMA), Stellenbosch University, 19 Jonkershoek Rd, Stellenbosch, 7600, South Africa c Laboratoire Interdisciplinaire des Environnements Continentaux, Universit de Lorraine, CNRS UMR 7360, 8 rue du General Delestraint, Metz 57070, France d Department of Biochemistry, Rush University Medical Center, 1735 W. Harrison St, Chicago, IL 60612, USA a r t i c l e i n f o Article history: Received 1 September 2015 Revised 8 January 2016 Accepted 28 January 2016 Available online 11 February 2016 Keywords: Normal cells Cancer stem cells Mathematical models a b s t r a c t Stemming from current emerging paradigms related to the cancer stem cell hypothesis, an existing math- ematical model is expanded and used to study cell interaction dynamics in the bone marrow and periph- eral blood. The proposed mathematical model is described by a system of nonlinear differential equations with delay, to quantify the dynamics in abnormal hematopoiesis. The steady states of the model are an- alytically and numerically obtained. Some conditions for the local asymptotic stability of such states are investigated. Model analyses suggest that malignancy may be irreversible once it evolves from a nonma- lignant state into a malignant one and no intervention takes place. This leads to the proposition that a great deal of emphasis be placed on cancer prevention. Nevertheless, should malignancy arise, treatment programs for its containment or curtailment may have to include a maximum and extensive level of ef- fort to protect normal cells from eventual destruction. Further model analyses and simulations predict that in the untreated disease state, there is an evolution towards a situation in which malignant cells dominate the entire bone marrow - peripheral blood system. Arguments are then advanced regarding requirements for quantitatively understanding cancer stem cell behavior. Among the suggested require- ments are, mathematical frameworks for describing the dynamics of cancer initiation and progression, the response to treatment, the evolution of resistance, and malignancy prevention dynamics within the bone marrow – peripheral blood architecture. © 2016 Elsevier Inc. All rights reserved. 1. Introduction In a recent article [1], we discussed the hematologic disorders such as the anemias, the leukemias, and the myelodysplastic syn- dromes (MDS) and proposed a simple delay differential equation model for quantifying the normal hematopoietic state. This was a prelude to effectively modeling and understanding the cancerous disorders that arise due to breaches of the massive physiological activity of hematopoiesis. Our modeling activities in [1] were based on considerations of cell systems in the bone marrow (BM) and peripheral blood (PB) since a number of studies, as in [2–6], deal with or generate data that are based on these cell systems. These studies provide us with the motivational impetus to proceed with modeling the disease state within the context of current emerging ∗ Corresponding author. E-mail addresses: evansa@elmhurst.edu (E.K. Afenya), ouifkir@sun.ac.za (R. Ouifki), baba-issa.camara@univ-lorraine.fr (B.I. Camara), smundle@its.jnj.com (S.D. Mundle). URL: http://evansafenya.elmhurst.edu (E.K. Afenya) paradigms about cancer development by dwelling on cell behaviors in the BM and PB. In earlier studies [7–11], we only considered single cell popula- tions lumped together as normal and abnormal cells, respectively. Lumping together of the cell populations served as a simple first approximation to modeling and seeking insight into cell behav- ior but this had its limitations since it tended to hide other in- tricate properties such as cell cycle mechanisms but that approach served the purpose of uncovering and shedding light on the mech- anisms of phenomena such as the evolution of contact inhibition. We note that new understanding of the fundamental mechanisms of tumor initiation and propagation have led to ideas such as the cancer stem cell hypothesis of which, some elucidations can be found in works such as [12–20], that may hold keys to shifts in current thinking regarding cancer treatment paradigms [21,22]. The growing evidence supporting this hypothesis is becoming exten- sive in scope as highlighted in a number of studies for which we herein cite a few, [18–37], a list that is in no way exhaustive but serves as a representation of active work going on in this field of endeavor. This makes it important and imperative for us to seek http://dx.doi.org/10.1016/j.mbs.2016.01.010 0025-5564/© 2016 Elsevier Inc. All rights reserved.