Ronan Duchesne (1,2), Anissa Guillemin (1), Fabien Crauste (3), and Olivier Gandrillon (1,2)
(1) Laboratory of Biology and Modelling of the Cell, Lyon, France, (2) Inria team Dracula, Inria, Villeurbanne, France, (3) Department of Mathematics, University of Bordeaux, Talence, France.
Introduction:
Mounting evidence demonstrates the importance of heterogeneity in biological processes. This increasing awareness has accompanied the development of Nonlinear Mixed Effect Models (NLME) in the last decades, to describe data involving an important amount of variability.
NLME consist in models in which parameters are defined by distributions of random variables instead of constant values. Different samples from these random variables model the repeated measurement of the same process on different individuals belonging to the same population.
Yet, the choice of the parameter distributions to be used in NLME might not be straightforward from raw data. More generally, it might be difficult to recover precise parameter estimates from small datasets. This difficulty at estimating precise parameter values is known as identifiability issues.
When confronted with a unidentifiable model (i.e. data are not sufficient to estimate all parameters), one essentially has two options: trying to generate novel, more informative, data in order to characterize all parameters; or trying to reduce the model in order to have fewer parameters to estimate, and ultimately facilitate their estimation.
The first one can usually be completed by using a step of experimental design, for which several algorithms already exist (1). However, it remains unclear how to proceed when reducing a NLME to make it identifiable. More importantly, the question of how to assess, or even define, the identifiability of NLME is mostly an open problem (2). Most of the current methods quantify the uncertainty on the parameter estimates using the Fisher Information Matrix (FIM), which is proven to render biased results (3).
Objectives:
This work aims at two objectives. Most importantly, we want to devise a procedure to reduce a NLME in order to make it identifiable. Thus, we also look for an empirical way of assessing the identifiability of a NLME without using the FIM.
Methods:
We address these issues through the example of a NLME for the dynamics of the in vitro erythropoiesis. Erythropoiesis is the process by which mature red blood cells are produced by the differentiation of immature progenitors in the bone marrow. These progenitors can either keep self-renewing, or engage into differentiation. A variety of mathematical models have focused on describing the dynamics of erythropoiesis in vivo, and we recently described a model focusing on the kinetics of cell populations differentiating in vitro (4), which we proved to be identifiable. Our NLME is based upon this previously described dynamical model.
We use experimental cell counts of different cell populations, at regularly spaced time points during the course of erythroid differentiation, to estimate model parameters. These parameters are the proliferation and differentiation rates of the cells in the culture. The population of individuals to be fitted by the model is made of repeated samples of this experiment, each repetition giving qualitatively similar though quantitatively different results due to inter-individual (i.e. inter-experiments) heterogeneity. We implemented the model in Monolix (5).
Results:
We illustrate the difficulty of fitting whole parameter distributions from such experimental datasets (meaning that our model is unidentifiable). First, we use an approximation of identifiability based on the random sampling of the initial guesses of the parameters estimates, and the comparison of the resulting estimated parameter values (2). We say that a parameter is unidentifiable when different initial guesses lead to different parameter values which render the same likelihood value (3). We then elaborate a simplified version of the model, with less parameters to be estimated, that is identifiable. Using the correlations between population parameters estimates, we reduce the number of fixed effects parameters to estimate. Then, we compare the population and individual parameters distributions, by the empirical shrinkage of the individual random effects, to simplify the random effects structure. The improvement of the parameter identifiability can be measured by the decrease in the standard deviation of the parameter estimates. In our final model, depending on the parameters, this ranges from 10% to 50% of the standard deviation in the initial model.
Conclusions:
We developed an empirical stand-in of identifiability for NLME, and we demonstrate how to use it for model reduction. In the end, we improve the estimation of parameters in our NLME, which we will use for predictive purposes in the future.
References:
[1] Bazzoli C, Retout S, Mentré F. 2010. Design evaluation and optimisation in multiple response nonlinear mixed effect models: PFIM 3.0. Comput Methods Programs Biomed 98:55–65.
[2] Lavielle M, Aarons L. 2016. What do we mean by identifiability in mixed effects models? J Pharmacokinet Pharmacodyn 43:111–122.
[3] Raue A, Kreutz C, Maiwald T, Bachmann J, Schilling M, Klingmüller U, Timmer J. 2009. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics 25:1923–1929.
[4] Duchesne R, Guillemin A, Gandrillon O, Crauste F. 2018. Calibration, Selection and Identifiability Analysis of a Mathematical Model of the in vitro Erythropoiesis in Normal and Perturbed Contexts. BioRxiv.
[5] Lixoft SAS. Monolix version 2018r1. http://lixoft.com/products/monolix/, 2018.
Reference: PAGE 28 (2019) Abstr 9145 [www.page-meeting.org/?abstract=9145]
Poster: Methodology - New Modelling Approaches