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A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia
Mostafa Adimy, Fabien Crauste, Shigui Ruan
TL;DR
Periodic chronic myelogenous leukemia exhibits blood-cell oscillations lasting much longer than the average pluripotential cell cycle, a difference that remains poorly understood. The paper models stem-cell phases and variable division ages with distributed delay, then analyzes stability and bifurcation; simulations produce oscillations resembling those observed in PCML.
Problem
In periodic chronic myelogenous leukemia, oscillations last 30 to 100 days, compared with an average pluripotential cell cycle of 1 to 4 days, and this difference is not well understood.
Source: §5 · p. 22Method
The paper models resting and proliferating stem-cell phases with division ages distributed over time, then analyzes equilibria and Hopf bifurcation.
Source: §1 · p. 2, §5 · p. 22Results
As sensitivity n increases, simulated oscillation periods increase, reaching about 70 days and amplitudes from low to high values.
Source: §5 · p. 22Takeaways & Limitations
The simulated periods, amplitudes, and relaxation-like shape correspond to features reported for periodic chronic myelogenous leukemia.
Source: §5 · pp. 22, 24At the stated parameter condition, the trivial equilibrium has a zero characteristic root and the analysis cannot determine its stability or instability.
Source: §3.1 · p. 12
Abstract
from arXiv · showhide
This paper is devoted to the analysis of a mathematical model of blood cells production in the bone marrow (hematopoiesis). The model is a system of two age-structured partial differential equations. Integrating these equations over the age, we obtain a system of two nonlinear differential equations with distributed time delay corresponding to the cell cycle duration. This system describes the evolution of the total cell populations. By constructing a Lyapunov functional, it is shown that the trivial equilibrium is globally asymptotically stable if it is the only equilibrium. It is also shown that the nontrivial equilibrium, the most biologically meaningful one, can become unstable via a Hopf bifurcation. Numerical simulations are carried out to illustrate the analytical results. The study maybe helpful in understanding the connection between the relatively short cell cycle durations and the relatively long periods of peripheral cell oscillations in some periodic hematological diseases.
1 Introduction
Hematopoiesis is regulated through feedback between blood and bone marrow, and abnormalities in that feedback are linked to periodic blood diseases. Building on earlier stem-cell models, the paper allows division ages to vary and represents cell-cycle timing with a distributed delay.
1 Introduction
- Abnormalities in blood-to-marrow feedback are considered major suspects in periodic hematological diseases, including chronic myelogenous leukemia.The introduction describes blood-cell production as responding to oxygen levels through erythropoietin and feedback to the marrow.Source: §1 · p. 2
- Stem cells are modeled as proliferating cells or quiescent G0-cells, with resting cells able to re-enter proliferation or be lost.Proliferating cells can also be lost through apoptosis.Source: §1 · p. 2
- The paper extends Mackey’s model by representing resting and proliferating phases while allowing cytokinesis ages to vary across cells.The authors motivate age variation by differences between pluripotent and more mature committed stem cells.Source: §1 · p. 2
- The resulting population model uses a distributed delay for cell-cycle duration, addressing prior stability results that lack necessary conditions and do not directly apply to this model.Earlier work also numerically associated stem-cell population models with oscillations and bifurcations.Source: §1 · pp. 2–3
2 The hematopoiesis process: presentation of the model
The model tracks resting and proliferating stem-cell populations, with transitions, losses, and division represented through age-dependent dynamics. Integrating over age yields a distributed-delay system whose analysis establishes solution properties and identifies conditions governing equilibria.
2 The hematopoiesis process: presentation of the model
- Resting cells can be lost or enter proliferation, while proliferating cells can undergo apoptosis or divide into two resting daughter cells.The model assigns resting-cell loss and entry rates δ and β, and proliferating-cell apoptosis rate γ.Source: §2 · p. 3
- The re-entry rate β decreases with the resting-cell population and tends to zero as that population grows.The model also commonly represents β as a decreasing Hill function.Source: §2 · p. 3
- Integrating the age-structured equations and using characteristics produces equations for total populations with division-time distributed delay.The delay density represents the distribution of proliferating-cell division ages.Source: §2 · p. 4
- If xβ(x) is Lipschitz continuous, the system has a unique nonnegative continuous solution; when δ > 0, solutions for resting cells are bounded.When δ = 0, resting-cell solutions may instead explode under some conditions, although proliferating-cell solutions may still be stable.Source: §2 · pp. 6–8
- After one generation, proliferating-cell population y no longer depends on the initial proliferating population, and periodic x yields periodic y with the same period.The initial proliferating cells have divided or died by then, leaving new proliferating cells to come from resting cells.Source: §2 · p. 7
- The system has an extinction equilibrium and at most one nontrivial equilibrium representing the expected population, which exists only under a stated condition.The nontrivial equilibrium is denoted E* = (x*, y*).Source: §2 · p. 10
3 Asymptotic stability
The section establishes stability behavior for both equilibria: the trivial equilibrium is globally asymptotically stable when it is the only equilibrium, while the nontrivial equilibrium is locally stable under stated conditions but can be destabilized via Hopf bifurcation.
3 Asymptotic stability
- When the nontrivial equilibrium appears, the trivial equilibrium becomes unstable; when it is the only equilibrium, it is globally asymptotically stable.Its loss of stability occurs through a transcritical bifurcation.Source: §3 · p. 10
3.1 Stability of the trivial equilibrium
- A Lyapunov-functional theorem gives a necessary and sufficient condition for global asymptotic stability of the trivial equilibrium.Source: §3.1 · p. 10
- At the equality boundary, λ = 0 is a characteristic root and the paper cannot conclude whether the trivial equilibrium is stable.All other characteristic roots have negative real parts; the boundary case is left for further analysis.Source: §3.1 · p. 12
3.2 Stability of the nontrivial equilibrium
- When condition (34) holds, the nontrivial equilibrium E∗ is locally asymptotically stable.The analysis concludes that all characteristic roots have negative real parts under this condition.Source: §3.2 · pp. 14–15
- When condition (34) fails, β∗ < 0, and the nontrivial equilibrium can be destabilized via Hopf bifurcation.Source: §3.2 · p. 15
4 Hopf bifurcation and periodic solutions
The section establishes conditions for Hopf bifurcation of the nontrivial equilibrium and examines the resulting periodic solutions. It also relates their appearance to the sensitivity parameter n and numerical examples.
4 Hopf bifurcation and periodic solutions
- The trivial equilibrium is globally asymptotically stable when it is the only equilibrium, and becomes unstable when the nontrivial equilibrium appears via a transcritical bifurcation.Because solutions are bounded, instability in (12a) occurs only via oscillatory solutions.Source: §4 · p. 16
- When τ ↦ e^-γτ f(τ) is decreasing and δ satisfies (25), the nontrivial equilibrium undergoes a Hopf bifurcation at β*=β*c and is locally stable for β*c<β*≤eδ.The bifurcation occurs at the boundary of this stability interval.Source: §4 · p. 20
- The sensitivity n strongly affects β* and plays a crucial role in the appearance of periodic solutions, while mortality rates and delay bounds remain possible parameters for further study.Here n describes how the proliferating-phase introduction rate responds to changes in the resting-phase population.Source: §4 · p. 20
- About 33 days is the oscillation period at the illustrated Hopf bifurcation, for n=2.53 and τ=7 days.The corresponding trajectories reach a limit cycle surrounding the equilibrium in the (x,y)-plane.Source: §4 · p. 20, Figure 2 · p. 21, Figure 3 · p. 21
5 Discussion
Periodic chronic myelogenous leukemia (PCML) features oscillations much longer than the observed pluripotential cell-cycle duration. The simulations show that the model can produce long-period oscillations with short cell cycles, with periods and amplitudes resembling those described for PCML.
5 Discussion
- PCML oscillations span 30–100 days, whereas the average pluripotential cell-cycle duration is 1–4 days, a difference that remains poorly understood.PCML oscillations are observed with the same period in white cells, red blood cells, and platelets.Source: §5 · p. 22
- With n=3, simulated oscillations have periods around 45 days and amplitudes from low to normal values.These parameters correspond to values for which abnormal periodic behavior is usually observed in circulating blood-cell types.Source: §5 · p. 22, Figure 4 · p. 23
- The model produces oscillations around 70 days as n increases, with amplitudes ranging from low to high values, a pattern described as characterizing PCML.Figure 5 uses n=4.Source: §5 · p. 22, Figure 5 · p. 23
- The simulated oscillations in Figures 4 and 5 resemble relaxation oscillations, a shape also suggested by patient data for PCML.A discrete-delay hematopoiesis model may also exhibit relaxation oscillations.Source: §5 · p. 24
- The simulations demonstrate that long-period oscillations in circulating cells are possible in the model despite short cell-cycle durations.The model is proposed to characterize hematological diseases with periodic behavior across circulating blood cells.Source: §5 · pp. 22, 24