A multilevel hierarchical framework for quantification of experimental heterogeneity in population snapshot data

Biological systems exhibit substantial heterogeneity: that is, variation in specific characteristics of individuals within a population. As a result, it is of critical importance to appropriately account for biological heterogeneity when calibrating mathematical models to infer cellular processes and predict behaviour. Recent approaches consider ordinary differential equations with random parameters to quantify heterogeneity in dynamical processes of cells. In this setting, statistical inference is performed to characterise the distribution of these random parameters within a cell population. One significant limitation of this approach is the tacit assumption that there are no substantial deviations in these distributions across experimental replicates. In this work, we propose a flexible Bayesian hierarchical differential equation modelling framework that quantifies and distinguishes both inter-experimental heterogeneity (heterogeneity between experimental replicates) and intra-experimental heterogeneity (biological heterogeneity within replicate populations). We consider two recent studies that employ mathematical models to interpret flow cytometry snap-shot data and quantify heterogeneity in nano-particle cell interactions and cell internalisation processes. Using simulation data, we demonstrate that substantial inaccuracy in the inferred dynamics can arise when experimental heterogeneity is not accounted for. By contrast, our hierarchical approach is robust to variability in inter-experimental and intra-experimental heterogeneity and our method simplifies to previous methods when inter-experimental heterogeneity is negligible. Our approach is flexible and widely applicable to applications involving replicate populations and snapshot data. We provide open-source implementations of our methods on GitHub.Author summary: In the design of personalised and targeted drugs and therapeutics it is essential to quatify the variability in the response of cellular processes involved in drug uptake. In practice, the experimental instruments and protocols used to collect cell population data for the study of such cellular processes are complex. As a result, it is important for statistical analyses that use these data to correctly account for variability between experiments in addition to the variability of cellular behaviour within the same experimental replicate. We present a flexible model of cellular dynamic that accounts for variability in biological, technical, and experimental replicate. We show that this leads to more accurate and robust results over standard approaches that pool replicates together.