Propagation of Growth Uncertainty in a Physiologically Structured Population

Abstract

Abstract. In this review paper we consider physiologically structured population models that have been widely studied and employed in the literature to model the dynamics of a wide variety of populations. However in a number of cases these have been found inadequate to describe some phenomena arising in certain real-world applications such as dispersion in the structure variables due to growth uncertainty/variability. Prompted by this, we described two recent approaches that have been investigated in the literature to describe this growth uncertainty/variability in a physiologically structured population. One involves formulating growth as a Markov diffusion process while the other entails imposing a probabilistic structure on the set of possible growth rates across the entire population. Both approaches lead to physiologically structured population models with nontrivial dispersion. Even though these two approaches are conceptually quite different, they were found in [17] to have a close relationship: in some cases with properly chosen parameters and coefficient functions, the resulting stochastic processes have the same probability density function at each time.

On analytical and numerical approaches to division and label structured population models

“…Since there is growing evidence that cellular populations exhibit heterogeneity, these methods will be of increasing importance when applied to estimating inter-cellular variability. One particular technique that will benefit from decomposition methods is based on using a Prohorov-metric based framework to estimate interindividual variability in population models [2][3][4][5][6][7]. This framework captures variability by non-parametrically estimating parameter distributions and relies on random differential equations to temporally propagate that variability.…”

“…The use of random differential equations requires solving many instances of the model evaluated at a mesh of parameters values used to describe a distribution. In previous work, Prohorov-metric based estimation methods have been applied to structured (Sinko-Streifer) population models [3][4][5], but without the added complexity of cell division. We postulate that the methodology described here will enable the exploration of Prohorov-metric based estimation methods within the realm of division and label structured cell population models.…”

Mathematical Models of Dividing Cell Populations: Application to CFSE Data

“…The fitted curves can be used to approximate the numbers of cells having divided a given number of times. variable environment vs. individual stochastic mechanisms has been treated specifically in [4,11,12,25] as well as more generically in [9,10,16,17]. While probability and stochasticity is fundamental to all of these models, the mathematical constructs used to incorporate asynchrony/variability into the modeling is strikingly different conceptually and computationally.…”

Preface. Distributed Parameter Systems in Immunology

“…The practical implementation of such an approach raises genuine challenges in a range of applicable mathematics (e.g., formulation of models for hierarchical and distributed systems, identification of the best mathematical description, representation of biological variation in the models). The study by Banks and Hu [3] compares two approaches to model growth variability in physiologically structured population models, one resulting in Fokker-Planck formulations and the other one entailing a probabilistic structure on the set of the physiological parameters across the entire population. The relationship between these two conceptually distinct approaches is examined.…”