Advanced data analysis for traction force microscopy and data-driven discovery of physical equations

The plummeting cost of collecting and storing data and the increasingly available computational power in the last decade have led to the emergence of new data analysis approaches in various scientific fields. Frequently, the new statistical methodology is employed for analyzing data involving incomplete or unknown information. In this thesis, new statistical approaches are developed for improving the accuracy of traction force microscopy (TFM) and data-driven discovery of physical equations. TFM is a versatile method for the reconstruction of a spatial image of the traction forces exerted by cells on elastic gel substrates. The traction force field is calculated from a linear mechanical model connecting the measured substrate displacements with the sought-for cell-generated stresses in real or Fourier space, which is an inverse and ill-posed problem. This inverse problem is commonly solved making use of regularization methods. Here, we systematically test the performance of new regularization methods and Bayesian inference for quantifying the parameter uncertainty in TFM. We compare two classical schemes, L1- and L2-regularization with three previously untested schemes, namely Elastic Net regularization, Proximal Gradient Lasso, and Proximal Gradient Elastic Net. We find that Elastic Net regularization, which combines L1 and L2 regularization, outperforms all other methods with regard to accuracy of traction reconstruction. Next, we develop two methods, Bayesian L2 regularization and Advanced Bayesian L2 regularization, for automatic, optimal L2 regularization. We further combine the Bayesian L2 regularization with the computational speed of Fast Fourier Transform algorithms to develop a fully automated method for noise reduction and robust, standardized traction-force reconstruction that we call Bayesian Fourier transform traction cytometry (BFTTC). This method is made freely available as a software package with graphical user-interface for intuitive usage. Using synthetic data and experimental data, we show that these Bayesian methods enable robust reconstruction of traction without requiring a difficult selection of regularization parameters specifically for each data set. Next, we employ our methodology developed for the solution of inverse problems for automated, data-driven discovery of ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). To find the equations governing a measured time-dependent process, we construct dictionaries of non-linear candidate equations. These candidate equations are evaluated using the measured data. With this approach, one can construct a likelihood function for the candidate equations. Optimization yields a linear, inverse problem which is to be solved under a sparsity constraint. We combine Bayesian compressive sensing using Laplace priors with automated thresholding to develop a new approach, namely automatic threshold sparse Bayesian learning (ATSBL). ATSBL is a robust method to identify ODEs, PDEs, and SDEs involving Gaussian noise, which is also referred to as type I noise. We extensively test the method with synthetic datasets describing physical processes. For SDEs, we combine data-driven inference using ATSBL with a novel entropy-based heuristic for discarding data points with high uncertainty. Finally, we develop an automatic iterative sampling optimization technique akin to Umbrella sampling. Therewith, we demonstrate that data-driven inference of SDEs can be substantially improved through feedback during the inference process if the stochastic process under investigation can be manipulated either experimentally or in simulations

Assisting digital volume correlation with mechanical image-based modeling: application to the measurement of kinematic fields at the architecture scale in cellular materials

La mesure de champs de déplacement et de déformation aux petites échelles dans des microstructures complexes représente encore un défi majeur dans le monde de la mécanique expérimentale. Ceci est en partie dû aux acquisitions d'images et à la pauvreté de la texture à ces échelles. C'est notamment le cas pour les matériaux cellulaires lorsqu'ils sont imagés avec des micro-tomographes conventionnels et qu'ils peuvent être sujets à des mécanismes de déformation complexes. Comme la validation de modèles numériques et l'identification des propriétés mécaniques de matériaux se base sur des mesures précises de déplacements et de déformations, la conception et l'implémentation d'algorithmes robustes et fiables de corrélation d'images semble nécessaire. Lorsque l'on s'intéresse à l'utilisation de la corrélation d'images volumiques (DVC) pour les matériaux cellulaires, on est confronté à un paradoxe: l'absence de texture à l'échelle du constituant conduit à considérer l'architecture comme marqueur pour la corrélation. Ceci conduit à l'échec des techniques ordinaires de DVC à mesurer des cinématiques aux échelles subcellulaires en lien avec des comportements mécaniques locaux complexes tels que la flexion ou le flambement de travées. L'objectif de cette thèse est la conception d'une technique de DVC pour la mesure de champs de déplacement dans des matériaux cellulaires à l'échelle de leurs architectures. Cette technique assiste la corrélation d'images par une régularisation élastique faible en utilisant un modèle mécanique généré automatiquement et basé sur les images. La méthode suggérée introduit une séparation d'échelles au dessus desquelles la DVC est dominante et en dessous desquelles elle est assistée par le modèle mécanique basé sur l'image. Une première étude numérique consistant à comparer différentes techniques de construction de modèles mécaniques basés sur les images est conduite. L'accent est mis sur deux méthodes de calcul particulières: la méthode des éléments finis (FEM) et la méthode des cellules finies (FCM) qui consiste à immerger la géométrie complexe dans une grille régulière de haut ordre sans utiliser de mailleurs. Si la FCM évite une première phase délicate de discrétisation, plusieurs paramètres restent néanmoins délicats à fixer. Dans ce travail, ces paramètres sont ajustés afin d'obtenir (a) la meilleure précision (bornée par les erreurs de pixellisation) tout en (b) assurant une complexité minimale. Pour l'aspect mesure par corrélation d'images régularisée, plusieurs expérimentations virtuelles à partir de différentes simulations numériques (en élasticité, en plasticité et en non-linéarité géométrique) sont d'abord réalisées afin d'analyser l'influence des paramètres de régularisation introduits. Les erreurs de mesures peuvent dans ce cas être quantifiées à l'aide des solutions de référence éléments finis. La capacité de la méthode à mesurer des cinématiques complexes en absence de texture est démontrée pour des régimes non-linéaires tels que le flambement. Finalement, le travail proposé est généralisé à la corrélation volumique des différents états de déformation du matériau et à la construction automatique de la micro-architecture cellulaire en utilisant soit une grille B-spline d'ordre arbitraire (FCM) soit un maillage éléments finis (FEM). Une mise en évidence expérimentale de l'efficacité et de la justesse de l'approche proposée est effectuée à travers de la mesure de cinématiques complexes dans une mousse polyuréthane sollicitée en compression lors d'un essai in situ.Measuring displacement and strain fields at low observable scales in complex microstructures still remains a challenge in experimental mechanics often because of the combination of low definition images with poor texture at this scale. The problem is particularly acute in the case of cellular materials, when imaged by conventional micro-tomographs, for which complex highly non-linear local phenomena can occur. As the validation of numerical models and the identification of mechanical properties of materials must rely on accurate measurements of displacement and strain fields, the design and implementation of robust and faithful image correlation algorithms must be conducted. With cellular materials, the use of digital volume correlation (DVC) faces a paradox: in the absence of markings of exploitable texture on/or in the struts or cell walls, the available speckle will be formed by the material architecture itself. This leads to the inability of classical DVC codes to measure kinematics at the cellular and a fortiori sub-cellular scales, precisely because the interpolation basis of the displacement field cannot account for the complexity of the underlying kinematics, especially when bending or buckling of beams or walls occurs. The objective of the thesis is to develop a DVC technique for the measurement of displacement fields in cellular materials at the scale of their architecture. The proposed solution consists in assisting DVC by a weak elastic regularization using an automatic image-based mechanical model. The proposed method introduces a separation of scales above which DVC is dominant and below which it is assisted by image-based modeling. First, a numerical investigation and comparison of different techniques for building automatically a geometric and mechanical model from tomographic images is conducted. Two particular methods are considered: the finite element method (FEM) and the finite-cell method (FCM). The FCM is a fictitious domain method that consists in immersing the complex geometry in a high order structured grid and does not require meshing. In this context, various discretization parameters appear delicate to choose. In this work, these parameters are adjusted to obtain (a) the best possible accuracy (bounded by pixelation errors) while (b) ensuring minimal complexity. Concerning the ability of the mechanical image-based models to regularize DIC, several virtual experimentations are performed in two-dimensions in order to finely analyze the influence of the introduced regularization lengths for different input mechanical behaviors (elastic, elasto-plastic and geometrically non-linear) and in comparison with ground truth. We show that the method can estimate complex local displacement and strain fields with speckle-free low definition images, even in non-linear regimes such as local buckling. Finally a three-dimensional generalization is performed through the development of a DVC framework. It takes as an input the reconstructed volumes at the different deformation states of the material and constructs automatically the cellular micro-architeture geometry. It considers either an immersed structured B-spline grid of arbitrary order or a finite-element mesh. An experimental evidence is performed by measuring the complex kinematics of a polyurethane foam under compression during an in situ test

Image Analysis for the Life Sciences - Computer-assisted Tumor Diagnostics and Digital Embryomics

Current research in the life sciences involves the analysis of such a huge amount of image data that automatization is required. This thesis presents several ways how pattern recognition techniques may contribute to improved tumor diagnostics and to the elucidation of vertebrate embryonic development. Chapter 1 studies an approach for exploiting spatial context for the improved estimation of metabolite concentrations from magnetic resonance spectroscopy imaging (MRSI) data with the aim of more robust tumor detection, and compares against a novel alternative. Chapter 2 describes a software library for training, testing and validating classification algorithms that estimate tumor probability based on MRSI. It allows flexible adaptation towards changed experimental conditions, classifier comparison and quality control without need for expertise in pattern recognition. Chapter 3 studies several models for learning tumor classifiers that allow for the common unreliability of human segmentations. For the first time, models are used for this task that additionally employ the objective image information. Chapter 4 encompasses two contributions to an image analysis pipeline for automatically reconstructing zebrafish embryonic development based on time-resolved microscopy: Two approaches for nucleus segmentation are experimentally compared, and a procedure for tracking nuclei over time is presented and evaluated