The number of cells of a given type in a tissue section can change by division or death, moving in or out of the section (flux) or transdifferentiation40. If we restrict ourselves to modelling cell types that do not transdifferentiate at appreciable rates, such as T cells and macrophages, and analyse tissues in which flux is negligible relative to rates of death or division (or where we can add flux terms post hoc), we remain with the following equation for the rate of change of a population of cells of type \(i\):
$$\frac{d{X}_{i}}{{dt}}=\frac{{\rm{\#}}\mathrm{Divisions}}{{dt}}-\frac{{\rm{\#}}\mathrm{Deaths}}{{dt}}$$
OSDR aims to transition from static observations of cell division or death in a tissue into rates. The key insight is: if we obtain a marker for cell division (or death), and in each cell division the marker remains above a defined threshold for a time period \({dt}\), then all observed divisions occurred within the last \({dt}\) hours. Thus:
$$\frac{d{X}_{i}}{{dt}}=\frac{{\rm{\#}}\mathrm{Divisions}}{\mathrm{Time}\,{\rm{a}}\,\mathrm{division}\,\mathrm{remains}\,\mathrm{observable}}-\frac{{\rm{\#}}\mathrm{Deaths}}{\mathrm{Time}\,{\rm{a}}\,\mathrm{death}\,\mathrm{remains}\,\mathrm{observable}}$$
The rate of division or death of a cell is influenced by the signals that it receives from its environment, its access to nutrients, its genetics and factors such as physical contact with other cells. We call this complete set of factors the ‘neighbourhood’ of the cell. This definition sets an ideal, and we denote the particular set of features used to approximate this ideal as \(N(x)\), where \(x\) is some cell in the tissue.
If we consider cells with identical neighbourhoods, a fraction of them will be dividing. This fraction is higher if the neighbourhood induces a high rate of division. We thus viewed the observations of division or death as random events whose probabilities are determined by the neighbourhood of the cell. Thus, for cell \(x\) of type \(i\), the distribution of the observation \({O}_{i,t}(x)\) of a division or death event is modelled as:
$${O}_{i,t}(x)=\left\{\begin{array}{cc}+1/d{t}^{+} & \text{with probability}\,{p}_{i}^{+1}(N(x))\\ -1/d{t}^{-} & \text{with probability}\,{p}_{i}^{-1}(N(x))\\ 0 & \text{remaining}\end{array}\right..$$
Where p i +1 and p i −1 are the statistical inference models for division or death, and dt+ and dt− are the durations of observed division or death markers, respectively. In this study, dt+ is defined as 1 time unit (roughly a few hours; ref. 27), and the approximation of the death rate as the mean division rate is defined using the same time units. Thus, in this implementation dt+ = dt− = 1. We divided by the durations so that the (stochastic) change in the number of cells in each timestep is:
$$\frac{d{X}_{i}}{{dt}}=\sum _{x\in {X}_{i}}{O}_{i,t}(x)$$
Tissues are heterogenous, and the diverse cellular compositions in different regions result in various directions of change. To analyse the change at a certain state, rather than the change in complete tissues, we computed the expected change with respect to an initial condition where cells share the same neighbourhood:
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