Recovery of cellular traction in three-dimensional nonlinear hyperelastic matrices

The traction exerted by a cell on the extra-cellular matrix (ECM) is critical to understanding and manipulating important biological processes such as stem cell differentiation, cancer cell metastasis, and embryonic morphogenesis. This traction is typically quantified through traction force microscopy (TFM). In TFM, the displacement of select markers inside the ECM is tracked, and is used in conjunction with an elasticity problem to reconstruct the traction field. Most applications of this technique thus far have assumed that the matrix behaves as a linear elastic solid that undergoes small deformation and infinitesimal strains. In this manuscript, we develop and implement a robust and efficient TFM methodology that overcomes these limitations by accounting for geometric and material nonlinearities in the ECM. We pose the TFM problem as an inverse problem and develop efficient adjoint-based minimization techniques to solve it. We test the effect of measurement noise on the proposed method, and examine the error incurred by not including nonlinear effects when solving the TFM problem. We present these results for in-silico traction fields that are applied to realistic geometric models of microglial and neuronal cells

A few-shot graph Laplacian-based approach for improving the accuracy of low-fidelity data

Low-fidelity data is typically inexpensive to generate but inaccurate. On the other hand, high-fidelity data is accurate but expensive to obtain. Multi-fidelity methods use a small set of high-fidelity data to enhance the accuracy of a large set of low-fidelity data. In the approach described in this paper, this is accomplished by constructing a graph Laplacian using the low-fidelity data and computing its low-lying spectrum. This spectrum is then used to cluster the data and identify points that are closest to the centroids of the clusters. High-fidelity data is then acquired for these key points. Thereafter, a transformation that maps every low-fidelity data point to its bi-fidelity counterpart is determined by minimizing the discrepancy between the bi- and high-fidelity data at the key points, and to preserve the underlying structure of the low-fidelity data distribution. The latter objective is achieved by relying, once again, on the spectral properties of the graph Laplacian. This method is applied to a problem in solid mechanics and another in aerodynamics. In both cases, this methods uses a small fraction of high-fidelity data to significantly improve the accuracy of a large set of low-fidelity data

Final Report on the “New Directions in the Variational Multiscale Formulation of Large Eddy Simulation of Turbulence”

In the report we present a summary of the new models and algorithms developed by the PI and the students supported by this grant. These developments are described in detail in ten peer-reviewed journal articles that acknowledge support from this grant

Solution of Physics-based Bayesian Inverse Problems with Deep Generative Priors

Inverse problems are notoriously difficult to solve because they can have no solutions, multiple solutions, or have solutions that vary significantly in response to small perturbations in measurements. Bayesian inference, which poses an inverse problem as a stochastic inference problem, addresses these difficulties and provides quantitative estimates of the inferred field and the associated uncertainty. However, it is difficult to employ when inferring vectors of large dimensions, and/or when prior information is available through previously acquired samples. In this paper, we describe how deep generative adversarial networks can be used to represent the prior distribution in Bayesian inference and overcome these challenges. We apply these ideas to inverse problems that are diverse in terms of the governing physical principles, sources of prior knowledge, type of measurement, and the extent of available information about measurement noise. In each case we apply the proposed approach to infer the most likely solution and quantitative estimates of uncertainty.Comment: Paper: 18 pages, 5 figures. Supplementary: 9 pages, 6 Figures, 2 Table

Final Report on the “New Directions in the Variational Multiscale Formulation of Large Eddy Simulation of Turbulence”

In the report we present a summary of the new models and algorithms developed by the PI and the students supported by this grant. These developments are described in detail in ten peer-reviewed journal articles that acknowledge support from this grant

Final Report on the “ New Directions in the Variational Multiscale Formulation of Large Eddy Simulation of Turbulence”

The work performed as part of this grant may be described in the context of developing numerical methods for solving an abstract, nonlinear variational problem for which the straightforward Galerkin approximation on a space of reasonable dimension is inaccurate. Target applications include flows with turbulence and systems with shocks. With this in mind, an alternate finite dimensional problem is proposed in which a model term (with unknown parameters) is added to the Galerkin approximation to improve its performance. We have developed new formulations for this model term especially for turbulent flows and for systems with shocks. We have also developed a dynamic approach for evaluating the parameters that appear in this model term. We have conducted several numerical studies to confirm the usefulness of the proposed approach

Towards the mechanical characterization of abdominal wall by inverse analysis

The aim of this study is to characterize the passive mechanical behaviour of abdominal wall in vivo in an animal model using only external cameras and numerical analysis. The main objective lies in defining a methodology that provides in vivo information of a specific patient without altering mechanical properties. It is demonstrated in the mechanical study of abdomen for hernia purposes. Mechanical tests consisted on pneumoperitoneum tests performed on New Zealand rabbits, where inner pressure was varied from 0 mmHg to 12 mmHg. Changes in the external abdominal surface were recorded and several points were tracked. Based on their coordinates we reconstructed a 3D finite element model of the abdominal wall, considering an incompressible hyperelastic material model defined by two parameters. The spatial distributions of these parameters (shear modulus and non linear parameter) were calculated by inverse analysis, using two different types of regularization: Total Variation Diminishing (TVD) and Tikhonov (H1). After solving the inverse problem, the distribution of the material parameters were obtained along the abdominal surface. Accuracy of the results was evaluated for the last level of pressure. Results revealed a higher value of the shear modulus in a wide stripe along the craneo-caudal direction, associated with the presence of linea alba in conjunction with fascias and rectus abdominis. Non linear parameter distribution was smoother and the location of higher values varied with the regularization type. Both regularizations proved to yield in an accurate predicted displacement field, but H1 obtained a smoother material parameter distribution while TVD included some discontinuities. The methodology here presented was able to characterize in vivo the passive non linear mechanical response of the abdominal wall