A Mathematical Model of Granulopoiesis Incorporating the Negative Feedback Dynamics and Kinetics of G-CSF/Neutrophil Binding and Internalization

Abstract

We develop a physiological model of granulopoiesis which includes explicit modelling of the kinetics of the cytokine granulocyte colony-stimulating factor (G-CSF) incorporating both the freely circulating concentration and the concentration of the cytokine bound to mature neutrophils. G-CSF concentrations are used to directly regulate neutrophil production, with the rate of differentiation of stem cells to neutrophil precursors, the effective proliferation rate in mitosis, the maturation time, and the release rate from the mature marrow reservoir into circulation all dependent on the level of G-CSF in the system. The dependence of the maturation time on the cytokine concentration introduces a state-dependent delay into our differential equation model, and we show how this is derived from an age-structured partial differential equation model of the mitosis and maturation, and also detail the derivation of the rest of our model. The model and its estimated parameters are shown to successfully predict the neutrophil and G-CSF responses to a variety of treatment scenarios, including the combined administration of chemotherapy and exogenous G-CSF. This concomitant treatment was reproduced without any additional fitting to characterise drug-drug interactions.

Dynamics of a Mathematical Hematopoietic Stem-Cell Population Model

“…In Table I we state homeostatic values of the parameters for the model (1)- (3). The values in Table I are all taken from Craig et al 18 . The first two parameters in the table, Q h the homeostatic concentration of HSCs, and β(Q h ) the homeostatic rate that cells enter the cell cycle, do not appear explicitly in the model (1)-(3), but are used to calculate the last two parameters.…”

“…As noted after (4), the dynamics of the HSC DDE (1) only depend on the four parameters τ , γ, κ and s. The dependence of the dynamics on the parameter s has previously been studied for small integer values of s (through numerical simulation) and analytically in the limit as s → ∞ (in which case the Hill function (2) simplifies to a Heaviside function) 40,49,50 . In the current work we will keep s = 2 fixed and equal to its value in Craig et al 18 , and we will only vary the parameters τ , γ, and κ. In the following, unless mentioned otherwise, all parameters take the values stated in Table I.…”

“…For the parameters used in Figure 12 we have ε = 6.132 × 10 −3 and C = 2.23 with Q * = 0.0896868 when γ = 0.2453692. The parameter definitions in (18) could be applied to the non-dimensionalised equation (4) with = ε andκ = ε/C, but we prefer to continue to study (1) directly. Letting h(Q) = Qβ(Q), as in (11), and using (18) we re-write (1) as…”

Response of an oscillatory differential delay equation to a periodic stimulus

“…For cyclic neutropenia the circulating neutrophil numbers typically oscillate from normal levels to very low levels with a period of about 19 to 21 days [3]. The period for cycling induced by periodic chemotherapy is approximately equal to the period of the chemotherapy [5].…”

“…wherex (p) andx (p) are respectively given by (28) and (29). For a healthy adult human the normal circulating neutrophil level fluctuates around 0.22 − 0.85 × 10 9 cells/kg of body mass [5]. Assuming that we want to administer G-CSF to maintain the oscillation within these normal bounds and taking x norm ≈ 0.4, then we have f min ≈ 0.5 and f max ≈ 1.5.…”