A mathematical model of clonal hematopoiesis explaining phase transitions in chronic myeloid leukemia

TL;DR

The paper develops a ten-dimensional nonlinear compartmental model of clonal hematopoiesis in chronic myeloid leukemia, distinguishing five cell-level stages for healthy and malignant lineages and incorporating self-regulation within cycling stem cells. It identifies four steady states, including three nonzero states that capture transitions among healthy, chronic, and accelerated-acute phases, and provides local and global stability analyses that reveal bifurcations driven by the parameter $R$ and the ratio $b_{1}/b_{2}$. Numerical simulations with literature-inspired parameters illustrate transitions between states, showing healthy maintenance with malignant extinction, coexistence in the chronic phase, and leukemic dominance in the accelerated-acute phase. The model offers mechanistic insight into phase transitions in CML and suggests factors (symmetric self-renewal, death and differentiation rates) that could inform therapeutic strategies and cross-disease applicability, with code available in the accompanying Maple Supplementary file.

Abstract

This study presents a mathematical model describing cloned hematopoiesis in chronic myeloid leukemia (CML) through a nonlinear system of differential equations. The primary objective is to understand the progression from healthy hematopoiesis to the chronic and accelerated-acute phases in myeloid leukemia. The model incorporates intrinsic cellular division events in hematopoiesis and delineates the evolution of chronic myeloid leukemia into five compartments: cycling stem cells, quiescent stem cells, progenitor cells, differentiated cells and terminally differentiated cells. Our analysis reveals the existence of three distinct non-zero steady states within the dynamical system, representing healthy hematopoiesis, the chronic phase and the accelerated-acute stage of the disease. We investigate the local and global stability of these steady states and provide a characterization of the hematopoietic states based on this analysis. Additionally, numerical simulations are included to illustrate the theoretical results.

Figures (3)

• Figure 1: The proposed model is represented using compartments, where a specific rate constant influences each cellular event denoted as $k_{j}$. Both healthy and malignant cycling stem cells (HCSCs/MCSCs) possess the ability to self-renew, which is indicated by the rate constants multiplied by the corresponding self-regulatory functions ($k_{1}\varphi_{H})/(k_{17}\varphi_{M}$), respectively. Additionally, they can enter a resting phase with rate constants ($k_{2})/(k_{18}$), during which they become healthy/malignant quiescent stem cells (HQSCs/MQSCs). After a period of time, the quiescent stem cells can reactivate, denoted by rate constants ($k_{3})/(k_{19}$), and return to an active state as healthy/malignant cycling stem cells. The healthy/malignant cycling stem cells can give rise to intermediate healthy/malignant progenitor cells (HPCs/MPCs) through asymmetric ($k_{4})/(k_{20}$) division and symmetric ($k_{6})/(k_{22}$) differentiation. Additionally, they can undergo direct differentiation ($k_{5})/(k_{21}$) into intermediate healthy/malignant progenitor cells. These intermediate healthy/malignant progenitor cells, in turn, can proliferate and give rise to later healthy/malignant progenitor cells (HPCs/MPCs) through symmetric ($k_{7})/(k_{23}$) and asymmetric ($k_{8})/(k_{24}$) divisions. Similarly, they can give rise to healthy/malignant differentiated cells (HDCs/MDCs) through asymmetric ($k_{8}$)/($k_{24}$) division. Healthy/malignant later progenitor cells can directly differentiate ($\widetilde{k}_{7})/(\widetilde{k}_{23}$) into other types of later healthy/malignant progenitor cells. Additionally, they can undergo direct differentiation ($k_{9}$)/($k_{25}$) into healthy/malignant differentiated cells, and they can give rise to healthy/malignant differentiated cells through symmetric ($k_{10})/(k_{26}$) differentiation. The healthy/malignant differentiated cells can undergo direct differentiation ($\widetilde{k}_{10})/(\widetilde{k}_{26}$) into other types of healthy/malignant differentiated cells. Furthermore, they can give rise to healthy/malignant terminally differentiated cells (HTDCs/MTDCs) through direct ($k_{11})/(k_{27}$) and symmetric ($k_{12})/(k_{28}$) differentiation. Each cell type, except for quiescent stem cells, whether healthy or malignant, has a death rate represented by the rate constants ($k_{13})/(k_{29}$) for cycling stem cells, ($k_{14})/(k_{30}$) for progenitor cells, ($k_{15})/(k_{31}$) for differentiated cells and ($k_{16})/(k_{32}$) for terminally differentiated cells.
• Figure 2: This diagram illustrates the transition from healthy hematopoiesis to the chronic and accelerated-acute stages in myeloid leukemia. In this context, $r$ and $R$ represent the homeostatic quantities of healthy and malignant cycling hematopoietic stem cells, respectively, as defined by equation (\ref{['eq:star']}). Values of $R$ less than $r$ correspond to the healthy hematopoietic state; values of $R$ between $r$ and $(b_{1}/b_{2})r$ correspond to the chronic phase of leukemia; values of $R$ greater than $(b_{1}/b_{2})r$ characterize the accelerated-acute phase of the disease.
• Figure 3: Behavior of healthy and malignant (leukemic) cell populations. Initial conditions: (a) healthy cell populations: cycling stem cells $x_{0}(0)=9\times10^5$, quiescent stem cells $x_{1}(0)=10^5$, progenitor cells $x_{2}(0)=10^8$, differentiated cells $x_{3}(0)=10^{10}$, terminally differentiated cells $x_{4}(0)=10^{12}$; (b) - (c) healthy and malignant cells: cycling stem cells $x_{0}(0)=9\times10^5$ and $y_{0}(0)=1$, quiescent stem cells $x_{1}(0)=10^5$ and $y_{1}(0)=1$, progenitor cells $x_{2}(0)=10^8$ and $y_{2}(0)=1$, differentiated cells $x_{3}(0)=10^{10}$ and $y_{3}(0)=1$, terminally differentiated cells $x_{4}(0)=10^{12}$ and $y_{4}(0)=1$.

Theorems & Definitions (3)

• Theorem 2.1
• Remark 2.2
• Remark 2.3