Mathematical Biosciences 282 (2016) 174–180 Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs About the discrete-continuous nature of a hematopoiesis model for Chronic Myeloid Leukemia Marcos E. Gaudiano a,b,∗ , Tom Lenaerts c,d , Jorge M. Pacheco b,e a CIEM-CONICET & Universidad Nacional de Córdoba, Ciudad Universitaria CP 5000, Córdoba, Argentina b ATP-group, P-2744-016 Porto Salvo, Portugal c MLG, Université Libre de Bruxelles, Boulevard du Triomphe CP212, Building O, 8th floor, 1050 Brussels, Belgium d AI lab, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium e CBMA & DMA, Universidade do Minho, 4710-057 Braga, Portugal a r t i c l e i n f o Article history: Received 3 August 2016 Revised 26 October 2016 Accepted 1 November 2016 Available online 2 November 2016 Keywords: Hematopoiesis Chronic Myeloid Leukemia Discrete-continuous model Moran’s process Peto’s paradox a b s t r a c t Blood of mammals is composed of a variety of cells suspended in a fluid medium known as plasma. Hematopoiesis is the biological process of birth, replication and differentiation of blood cells. Despite of being essentially a stochastic phenomenon followed by a huge number of discrete entities, blood forma- tion has naturally an associated continuous dynamics, because the cellular populations can – on average – easily be described by (e.g.) differential equations. This deterministic dynamics by no means contemplates some important stochastic aspects related to abnormal hematopoiesis, that are especially significant for studying certain blood cancer deceases. For instance, by mere stochastic competition against the normal cells, leukemic cells sometimes do not reach the population thereshold needed to kill the organism. Of course, a pure discrete model able to follow the stochastic paths of billons of cells is computationally impossible. In order to avoid this difficulty, we seek a trade-off between the computationally feasible and the biologically realistic, deriving an equation able to size conveniently both the discrete and continuous parts of a model for hematopoiesis in terrestrial mammals, in the context of Chronic Myeloid Leukemia. Assuming the cancer is originated from a single stem cell inside of the bone marrow, we also deduce a theoretical formula for the probability of non-diagnosis as a function of the mammal average adult mass. In addition, this work cellular dynamics analysis may shed light on understanding Peto’s paradox, which is shown here as an emergent property of the discrete-continuous nature of the system. © 2016 Elsevier Inc. All rights reserved. 1. Introduction Hematopoiesis is the process for the generation of all cellular blood elements. A continuous supply of cells is necessary to com- pensate for the loss of cells due to apoptotic senescence or migra- tion out of the circulating compartment. Blood cell formation has at its root hematopoietic stem cells (HSC) that have the dual prop- erty of self renewal and the ability to differentiate into all types of blood cells [1–3]. Allometric scaling laws of observables in biological organisms are widely known and a general model for the origin of many of them can be found (e.g.) in [4]. The number N of HSC that a mam- mal possesses is an example of this, because it can be written as a function of the adult average mass M in the form: N = N SC M 3/4 , (1) ∗ Corresponding author. E-mail address: marcosgaudiano@gmail.com (M.E. Gaudiano). with N SC = 15.9 kg −3/4 , as it was stated in [5]. In [6] the authors provided a simple model for human hematopoiesis in which the observed exponential expansion of cells from the active stem cell pool to the mature cells is naturally incorporated, as 32 cellular differentiation stages (compartments) composed of approximately N i cells satisfying N i+1 /N i ≈ 1.93 (i = 1, . . . , 31). The same idea was later generalized for mammals [7]. A discrete-continuous model for Chronic Myeloid Leukemia (CML) in humans was developed in [8], able to reproduce the rarely (but statistically significant) cases in which the pacient does not die of cancer, simply because leukemic cancerous cells sometimes by chance do not proliferate. The authors concluded that it was enough to assume just the first k = 7 differentia- tion compartments as discrete/stochastic quantities and continu- ous/deterministic the rest. Using a similar mathematical model, CML and other hematolog- ical deceases were studied across mammals of arbitrary mass M in http://dx.doi.org/10.1016/j.mbs.2016.11.001 0025-5564/© 2016 Elsevier Inc. All rights reserved.