Modeling of 2D diffusion processes based on microscopy data: parameter estimation and practical identifiability analysis

Abstract

BackgroundDiffusion is a key component of many biological processes such as chemotaxis, developmental differentiation and tissue morphogenesis. Since recently, the spatial gradients caused by diffusion can be assessed in-vitro and in-vivo using microscopy based imaging techniques. The resulting time-series of two dimensional, high-resolutions images in combination with mechanistic models enable the quantitative analysis of the underlying mechanisms. However, such a model-based analysis is still challenging due to measurement noise and sparse observations, which result in uncertainties of the model parameters.MethodsWe introduce a likelihood function for image-based measurements with log-normal distributed noise. Based upon this likelihood function we formulate the maximum likelihood estimation problem, which is solved using PDE-constrained optimization methods. To assess the uncertainty and practical identifiability of the parameters we introduce profile likelihoods for diffusion processes.Results and conclusionAs proof of concept, we model certain aspects of the guidance of dendritic cells towards lymphatic vessels, an example for haptotaxis. Using a realistic set of artificial measurement data, we estimate the five kinetic parameters of this model and compute profile likelihoods. Our novel approach for the estimation of model parameters from image data as well as the proposed identifiability analysis approach is widely applicable to diffusion processes. The profile likelihood based method provides more rigorous uncertainty bounds in contrast to local approximation methods.

Image based validation of dynamical models for cell reorientation

“…The estimation of model parameters from image data and identifiability analysis in the context of diffusion processes is an emerging area of research (29). We have reduced the Meinhardt model to a two-variable model, which is uniquely identifiable.…”

Radial Basis Function Approximations of Bayesian Parameter Posterior Densities for Uncertainty Analysis

“…In this section we consider a PDE model describing the formation of gradients of the cytokine CCL21 around lymphatic vessels [17]. Such gradients are, among other processes, responsible for the migration of dendritic cells towards the lymphatic vessels.…”

“…Of biological interest are the five unknown parameters: diffusion D, secretion α and degradation rate γ of CCL21 as well as the association k 1 and dissociation constant k −1 of the complex. For the detailed model, parameter inference and uncertainty analysis for those parameters we refer to the original publication [17]. In the original work the posterior probability was not calculate as it requires the simulation of the discretised PDE with several thousand state variables and is nearly infeasible with traditional methods.…”

Using a continuum model to decipher the mechanics of embryonic tissue spreading from time-lapse image sequences: An approximate Bayesian computation approach

“…ABC determines probability distributions for the model parameters for each individual tissue explant (see Estimating parameters for our model in a deterministic manner appeared infeasible as preliminary studies suggested that our model had parameter structural non-identifiability, or functionally related model parameters [50]. While parameter identifiability has been explored extensively for ordinary differential equations, for example determining whether different parameter sets give rise to different model predictions, it has not been studied as deeply in partial differential equation models [51]. Thus, from preliminary analysis of parameter space and the knowledge that models similar to our own have shown non-identifiability relations between parameter values [52], we adopted ABC rejection because it allows for quantification of the uncertainty of our parameter estimates.…”