[Paper Review] Image-based Parameter Inference for Spatio-temporal models of Organogenesis

Britta Velten, Erkan Uenal+1 more|arXiv (Cornell University)|Jun 10, 2014

Pluripotent Stem Cells Research4 references3 citations

TL;DR

This paper presents a hybrid computational framework for image-based parameter inference in spatio-temporal models of organogenesis, using a decision tree to guide initial parameter estimation and SNOPT optimization to refine model outputs. The method successfully recovers target gene expression patterns in a mouse limb bud model, reducing the fitness function from 13,989 to 4,526, demonstrating feasibility for automated parameter inference from imaging data.

ABSTRACT

Advances in imaging technology now provide us with detailed 3D data on gene expression patterns in developing embryos. This information can be used to build predictive mathematical models of embryogenesis. Current modelling approaches are, however, limited by lack of methods to automatically infer the regulatory networks and the parameter values from the image-based information. Here we make a first step to the development of such methods. We use limb bud development as a model system. For a given regulatory network we developed a decision tree based algorithm to automatically determine parameter values for which the model reproduces the expression patterns. Starting from this parameter set, local optimization was performed to further reduce the chosen goodness-of-fit measure. This approach allowed us to recover the target expression patterns, as judged by eye, and thus provides a first step towards the automated inference of parameter values for a given regulatory network.

Motivation & Objective

• To address the lack of automated methods for inferring regulatory network parameters from 3D image data of embryonic development.
• To develop a computationally efficient approach that integrates image-based expression patterns into predictive mathematical models of organogenesis.
• To enable systematic, step-by-step parameter inference by identifying key regulatory components and their influence on spatial patterning.
• To reduce computational cost while maintaining accuracy through hierarchical parameter tuning and gradient-based optimization.
• To validate the method using in silico data, demonstrating its ability to reproduce experimentally observed expression patterns.

Proposed method

• A 2D reaction-diffusion PDE model simulates spatio-temporal dynamics of FGF10, FGF8, BMP, WNT, and their receptor complexes in the limb bud.
• The model incorporates non-linear feedback via Hill functions and includes diffusion, degradation, and receptor-ligand binding kinetics.
• A decision tree algorithm sequentially infers parameters by testing whether key components (e.g., WNT, FGF8) are correctly localized in specific tissues (e.g., ectoderm, AER).
• Parameters are adjusted by doubling or halving based on directional influence (e.g., increase production rate if target is absent) until spatial patterns match the reference.
• SNOPT, a gradient-based optimization algorithm using the adjoint method, refines parameters using a fitness function that minimizes differences between simulated and target expression patterns.
• Scaling factors (λi) are introduced as additional parameters to account for relative expression levels, avoiding absolute value constraints.

Experimental results

Research questions

• RQ1 Can a decision tree-based algorithm effectively guide the initial estimation of parameters in a spatio-temporal model of organogenesis?
• RQ2 To what extent can gradient-based optimization with SNOPT improve model accuracy when initialized with decision tree outputs?
• RQ3 Can the combined approach reproduce complex, multi-component gene expression patterns observed in limb bud development?
• RQ4 How can computational cost be reduced without sacrificing model fidelity in parameter inference for PDE-based developmental models?
• RQ5 What role do sequential parameter updates, based on regulatory hierarchy, play in achieving convergence to biologically plausible parameter sets?

Key findings

• The decision tree approach reduced the fitness function from 13,989 (initial parameters) to 5,598, significantly improving pattern fidelity.
• Subsequent SNOPT optimization further reduced the fitness function to 4,526, approaching the target pattern value of 3,852.
• The optimized model successfully reproduced the spatial expression patterns of FGF10, FGF8, BMP, and WNT in the correct tissue domains (mesenchyme, AER, ectoderm).
• The method achieved a 10x reduction in computation time (from ~24 hours to ~2 hours) by excluding Hill constants and reducing simulation accuracy during optimization.
• The inclusion of scaling factors (λi) as parameters enabled robust comparison of relative expression levels without requiring absolute concentration values.
• The hierarchical, component-wise parameter inference strategy proved effective in navigating the high-dimensional parameter space of complex regulatory networks.