Stochastic dynamics of two-compartment cell proliferation models with regulatory mechanisms for hematopoiesis

We present an asymptotic analysis of a stochastic two-compartmental cell division system with regulatory mechanisms inspired by Getto et al. (2013). The hematopoietic system is modeled as a two-compartment system, where the first compartment consists of dividing cells in the bone marrow, referred to as type $0$ cells, and the second compartment consists of post-mitotic cells in the blood, referred to as type $1$ cells. Division and self-renewal of type $0$ cells are regulated by the population density of type $1$ cells. By scaling up the initial population, we demonstrate that the scaled dynamics converges in distribution to the solution of a system of ordinary differential equations (ODEs). This system of ODEs exhibits a unique non-trivial equilibrium that is globally stable. Furthermore, we establish that the scaled fluctuations of the density dynamics converge in law to a linear diffusion process with time-dependent coefficients. When the initial data is Gaussian, the limit process is a Gauss-Markov process. We analyze its asymptotic properties to elucidate the joint structure of both compartments over large times. This is achieved by proving exponential convergence in the 2-Wasserstein metric for the associated Gaussian measures on an $L^{2}$ Hilbert space. Finally, we apply our results to compare the effects of regulating division and self-renewal of type $0$ cells, providing insights into their respective roles in maintaining hematopoietic system stability.