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

Hock et al. BMC Bioinformatics 2013, 14(Suppl 10):S7 http://www.biomedcentral.com/1471-2105/14/S10/S7 RESEARCH Open Access Modeling of 2D diffusion processes based on microscopy data: parameter estimation and practical identifiability analysis Sabrina Hock1,2, Jan Hasenauer1,2, Fabian J Theis1,2* From 10th International Workshop on Computational Systems Biology Tampere, Finland. 10-12 June 2013 Abstract Background: Diffusion 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. Methods: We 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 conclusion: As 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. Introduction Diffusion is assumed to be the basis of many spatial organization processes for multi-cellular organisms. Crucial processes such as developmental pattern formation or chemotaxis rely on gradient information arising from diffusion and transport processes [1,2]. In the last decades, diffusion processes have been of great interest not only for experimentalist but also for theoreticians. Turing [3] was the first to break ground, followed by Gierer and Meinhardt [4], who introduced models for such processes based on partial differential equations (PDEs). A prominent aspect is the diffusion of extracellular * Correspondence: 1 Institute of Computational Biology, Helmholtz Center Munich, Ingolstädter Landstr. 1, 85764 Neuherberg, Germany Full list of author information is available at the end of the article signaling molecules. Such molecules are synthesized and secreted by cells and spread through the surrounding tissue, forming a gradient. A biological prominent example is guided cell movement along such gradients. In this case, the cell senses the concentration difference between front and back, and moves along the gradient. Gradients of signaling molecules can be made visible in-vivo via antibody stainings (see Figure 1 and [5-7]). Combined with microscopy, this yields two-dimensional (2D) images. The color intensity of each pixel provides informations about the concentration (or the number) of signaling molecules. Modern microscopy devices can also generate stacks of images, providing information about the distribution of signaling molecules in three-dimensions (3D) [5,8]. © 2013 Hock et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Hock et al. BMC Bioinformatics 2013, 14(Suppl 10):S7 http://www.biomedcentral.com/1471-2105/14/S10/S7 Page 2 of 6 Figure 1 Haptotaxis: Data and schematic description of the process. Haptotaxis: Data and schematic description of the process. (A) Fluorescence staining image taken from [7], which shows the Z-stack projection of non-permeabilized ear dermis stained for CCL21. Left image is the maximum intensity projection and the right image shows same staining as color-coded average projection. Lymphoid vessel boundaries are indicated by the blue dotted line (scale bars: 100µm). (B) Schematic of the dendritic haptotaxis process adapted from [6]. Dendritic cells move along a gradient of immobilized CCL21 towards the lymphatic vessels. Despite these high-resolution imaging data, the number of quantitative models of biological diffusion processes is limited. While quantitative modeling with ordinary differential equations (ODEs) is a common method and the theory of parameter estimation and identifiability is sound, those results have yet to be transferred to the quantitative modeling with PDEs [9,10]. In recent years the field of PDE-constrained optimization emerged, providing the theory and methods to estimate parameters of PDEs [11]. Nevertheless, specific problems occurring in biological problems, like partial observations, sparse measurements and high noise levels, have yet to be addressed. This has already been done for ODE parameter estimation techniques [12] but is an open problem in the PDE context. In particular, appropriate likelihood functions and methods for the efficient and reliable analysis of practical identifiability [9] are not available. In this paper, we propose a likelihood function for the estimation of parameters of 2D diffusion process from image data. Furthermore, we transfer the concept of profile likelihood based identifiability analysis introduced by Raue et al. [9] from ODEs to PDEs. This allows us to go beyond the classical uncertainty analysis methods based on local approximation towards global uncertainty bounds. Finally, we evaluate the methods by studying a model for diffusion processes involved in the migration of dendritic cells towards lymphatic vessels (see schematic picture Figure 1B). Methods In the following section we shortly introduce the considered class of PDEs and the available types of measurement data. Afterwards, the parameter estimation and identifiability analysis methods are presented. Api for i = 1, . . . , M and k = 1, . . . , N . Here b ∈ R+ denotes a constant off-set due to background luminescence and h defines the observables and could for instance be a mapping onto the first component of u. With our assumptions made about existence, uniqueness and integrability this is a well-defined function. Biological measurement data are in general noise corrupted. The noise distribution depends on the measurement techniques. As measured fluorescence intensities are always positive and as image acquisition is basically a counting process, we (...truncated)