A Time-Varying Branching Process Approach to Model Self-Renewing Cells

Stem cells, through their ability to produce daughter stem cells and differentiate into specialized cells, are essential in the growth, maintenance, and repair of biological tissues. Understanding the dynamics of cell populations in the proliferation process not only uncovers proliferative properties of stem cells, but also offers insight into tissue development under both normal conditions and pathological disruption. In this paper, we develop a continuous time branching process model with time-dependent offspring distribution to characterize stem cell proliferation process. We derive analytical expressions for mean, variance, and autocovariance of the stem cell counts, and develop likelihood-based inference procedures to estimate model parameters. Particularly, we construct a forward algorithm likelihood to handle situations when some cell types cannot be directly observed. Simulation results demonstrate that our estimation method recovers the time-dependent division probabilities with good accuracy.

💡 Research Summary

This paper introduces a continuous‑time branching‑process framework for modeling stem‑cell proliferation in which the probabilities of the four possible division outcomes—symmetric self‑renewal, asymmetric division, symmetric differentiation, and self‑renewal with a non‑viable daughter—are allowed to vary deterministically with time. The authors parameterize the time‑dependent probabilities p₁(t), p₂(t), and p₄(t) using Lorentzian‑shaped functions characterized by a peak height, a decay rate, and a peak time; p₃(t) is defined implicitly so that the probabilities sum to one. This functional form captures the biologically observed shift from early self‑renewal dominance to later differentiation dominance.

The stochastic dynamics are built on the assumption that each viable stem cell divides independently with a constant per‑cell rate r. Consequently, when X(t) viable stem cells are present, the population‑wide division events follow a Poisson process with intensity r·X(t). At each division time the change in the three cell compartments (viable stem cells X, differentiated cells Y, and non‑viable stem cells Z) is drawn from a discrete distribution determined by the current values of p₁(t)–p₄(t).

Analytically, the authors first prove that the expected number of viable stem cells, S(t)=E