Hematopoiesis as a continuum: from stochastic compartmental model to hydrodynamic limit
Vincent Bansaye, Ana Fernández Baranda, Stéphane Giraudier, Sylvie Méléard
TL;DR
The paper develops a multiscale stochastic compartmental model of hematopoiesis with regulatory feedback from mature cells and derives a hydrodynamic limit as the number of immature compartments $N\to\infty$. The limit consists of a stem-cell ODE $a'(t)=[r(0,z(t))-m(0,z(t))]a(t)$, a measure-valued maturation component $\mu_t$ governed by a transport-type equation with renewal terms, and a mature-cell ODE $z'(t)=\langle\mu_t,r(\cdot,z(t))\rangle+m(0,z(t))a(t)-d z(t)$, with $\mu_t(\{0\})=\mu_t(\{1\})=0$, and under density existence, a PDE for the immersion density $u(t,x)$: $\partial_t u+\partial_x(m(x,z(t))u)=r(x,z(t))u$. The authors prove convergence in law to this deterministic limit, establish uniqueness via a maturation-flow mild formulation, and show that the limit can be described as a density PDE with boundary conditions, extending to a density-formulation of prior deterministic models. Numerical simulations validate the continuum model against stochastic simulations and reveal maturation-front propagation and amplification of mature cells, supporting the continuum description’s biological relevance.
Abstract
We consider a multiscale stochastic compartmental model with three types of cells (stem cells, immature cells and mature cells) which combines cell proliferation and cell differentiation. We derive a hydrodynamic limit when the number of immature compartments goes to infinity obtaining a partial differential equations system with boundary conditions, modelling hematopoiesis as a continuum. We assume that proliferation and differentiation are regulated and let the corresponding rates depend on the number of mature cells. This leads us to model the dynamics of the population by a Markov process in continuous time and discrete space, which does not satisfy the branching property. We prove the convergence in law of the stem and mature cells population size processes and of the empirical measures of the immature cells dynamics, conveniently rescaled, to the unique triplet involving coupled functions and a measure, which are solutions of a deterministic measure valued equation with boundary dynamics. The cell differentiation induces a transport term in space and the main difficulty comes from the boundary effects coming from stem and mature cells. We also prove that the limiting measure admits at each time a density with respect to Lebesgue measure and can be characterized as solution of a partial differential equation.