Probabilistic methods for addressing uncertainty and variability in biological models: application to a toxicokinetic model

Abstract

Population variability and uncertainty are important features of biological systems that must be considered when developing mathematical models for these systems. In this paper we present probability-based parameter estimation methods that account for such variability and uncertainty. Theoretical results that establish well-posedness and stability for these methods are discussed. A probabilistic parameter estimation technique is then applied to a toxicokinetic model for trichloroethylene using several types of simulated data. Comparison with results obtained using a standard, deterministic parameter estimation method suggests that the probabilistic methods are better able to capture population variability and uncertainty in model parameters.

A Division-Dependent Compartmental Model for Computing Cell Numbers in CFSE-based Lymphocyte Proliferation Assays

“…To better understand rates at the generation number cohort or division number cohort level, one should attempt to develop individual (cohort) dynamics to investigate the CFSE data in a Type I framework of Aggregate Data/Individual (Cohort) Dynamics inverse problems such as those discussed in [1,Chapter 14] and [5]. Similar approaches have been successfully pursued in marine and insect population models [3,6,10,12,22] as well as in physiologically based pharmacokinetic (PBPK) models in toxicology [5,17]. Fortunately, a simple reformulation of (1) allows such an approach and permits both the accurate quantification of total cells per division number and the accurate estimation of proliferation and death rates in terms of division number in such a framework.…”

Mathematical Models of Dividing Cell Populations: Application to CFSE Data

“…That is, one must use aggregate (or population level) data in an attempt to describe what are ultimately the dynamics of individual cells. This type of inverse problem is well known in mathematics, and successful mathematical models have been developed and fit to data in a variety of applications such as size-structured marine and insect population models [4,7,11,12], wave propagation models for viscoelastic solids [19], electromagnetic wave propagation [13,14], physiologically-based pharmacokinetics models [6,20], and HIV models [5]. In addition to these applications, theory for such inverse problems is well-developed [3,6,18].…”

A Simulation-based comparison between parametric and nonparametric estimation methods in PBPK Models

“…In the best case scenario, the final estimate produces a model that fits the data very poorly. In a more serious outcome, the assumed distribution may produce a reasonable model fit to data even when it is an incorrect distribution (see [9,25] for examples).…”

“…However, increased levels of noise in the data do degrade the ability of the nonparametric method to yield results that clearly suggest the presence of a bimodal distribution as the "true" distribution (compare Figures 7 and 8). This can be partially compensated for by taking data over a longer period (compare Figures 8 and 9 and see also the examples in [9]). However, it is rather clear that a smoother family of approximations (for example, piecewise linear or cubic splines as mentioned in Section 4 and discussed in [8]) would be more appropriate than the sum of Dirac approximations of (19) in examples such as these wherein one is attempting to estimate a distribution possessing a smooth density function.…”