Kinematic models of interseismic deformation from inversion of GPS and InSAR measurements to estimate fault parameters and coupling degree

We have used kinematic models in two Italian regions to reproduce surface interseismic velocities obtained from InSAR and GPS measurements. We have considered a Block modeling, BM, approach to evaluate which fault system is actively accommodating the occurring deformation in both considered areas. We have performed a study for the Umbria-Marche Apennines, obtaining that the tectonic extension observed by GPS measurements is explained by the active contribution of at least two fault systems, one of which is the Alto Tiberina fault, ATF. We have estimated also the interseismic coupling distribution for the ATF using a 3D surface and the result shows an interesting correlation between the microseismicity and the uncoupled fault portions. The second area analyzed concerns the Gargano promontory for which we have used jointly the available InSAR and GPS velocities. Firstly we have attached the two datasets to the same terrestrial reference frame and then using a simple dislocation approach, we have estimated the best fault parameters reproducing the available data, providing a solution corresponding to the Mattinata fault. Subsequently we have considered within a BM analysis both GPS and InSAR datasets in order to evaluate if the Mattinata fault may accommodate the deformation occurring in the central Adriatic due to the relative motion between the North-Adriatic and South-Adriatic plates. We obtain that the deformation occurring in that region should be accommodated by more that one fault system, that is however difficult to detect since the poor coverage of geodetic measurement offshore of the Gargano promontory. Finally we have performed also the estimate of the interseismic coupling distribution for the Mattinata fault, obtaining a shallow coupling pattern. Both of coupling distributions found using the BM approach have been tested by means of resolution checkerboard tests and they demonstrate that the coupling patterns depend on the geodetic data positions.

Numerical simulations of the spine for patient-specific models of severe scoliosis

In silico methods, such as musculoskeletal modelling, may aid the selection of the optimal surgical treatment for highly complex pathologies such as scoliosis. Many musculoskeletal models use a generic, simplified representation of the intervertebral joints, which are fundamental to the flexibility of the spine. Therefore, to model and simulate the spine, a suitable representation of the intervertebral joint is crucial. The aim of this PhD was to characterise specimen-specific models of the intervertebral joint for multi-body models from experimental datasets. First, the project investigated the characterisation of a specimen-specific lumped parameter model of the intervertebral joint from an experimental dataset of a four-vertebra lumbar spine segment. Specimen-specific stiffnesses were determined with an optimisation method. The sensitivity of the parameters to the joint pose was investigate. Results showed the stiffnesses and predicted motions were highly depended on both the joint pose. Following the first study, the method was reapplied to another dataset that included six complete lumbar spine segments under three different loading conditions. Specimen-specific uniform stiffnesses across joint levels and level-dependent stiffnesses were calculated by optimisation. Specimen-specific stiffness show high inter-specimen variability and were also specific to the loading condition. Level-dependent stiffnesses are necessary for accurate kinematic predictions and should be determined independently of one another. Secondly, a framework to create subject-specific musculoskeletal models of individuals with severe scoliosis was developed. This resulted in a robust codified pipeline for creating subject-specific, severely scoliotic spine models from CT data. In conclusion, this thesis showed that specimen-specific intervertebral joint stiffnesses were highly sensitive to joint pose definition and the importance of level-dependent optimisation. Further, an open-source codified pipeline to create patient-specific scoliotic spine models from CT data was released. These studies and this pipeline can facilitate the specimen-specific characterisation of the scoliotic intervertebral joint and its incorporation into scoliotic musculoskeletal spine models.

Data driven regularization models of non-linear ill-posed inverse problems in imaging

Imaging technologies are widely used in application fields such as natural sciences, engineering, medicine, and life sciences. A broad class of imaging problems reduces to solve ill-posed inverse problems (IPs). Traditional strategies to solve these ill-posed IPs rely on variational regularization methods, which are based on minimization of suitable energies, and make use of knowledge about the image formation model (forward operator) and prior knowledge on the solution, but lack in incorporating knowledge directly from data. On the other hand, the more recent learned approaches can easily learn the intricate statistics of images depending on a large set of data, but do not have a systematic method for incorporating prior knowledge about the image formation model. The main purpose of this thesis is to discuss data-driven image reconstruction methods which combine the benefits of these two different reconstruction strategies for the solution of highly nonlinear ill-posed inverse problems. Mathematical formulation and numerical approaches for image IPs, including linear as well as strongly nonlinear problems are described. More specifically we address the Electrical impedance Tomography (EIT) reconstruction problem by unrolling the regularized Gauss-Newton method and integrating the regularization learned by a data-adaptive neural network. Furthermore we investigate the solution of non-linear ill-posed IPs introducing a deep-PnP framework that integrates the graph convolutional denoiser into the proximal Gauss-Newton method with a practical application to the EIT, a recently introduced promising imaging technique. Efficient algorithms are then applied to the solution of the limited electrods problem in EIT, combining compressive sensing techniques and deep learning strategies. Finally, a transformer-based neural network architecture is adapted to restore the noisy solution of the Computed Tomography problem recovered using the filtered back-projection method.

Incompressible limit and well-posedness of PDE models of tissue growth

Both compressible and incompressible porous medium models are used in the literature to describe the mechanical aspects of living tissues. Using a stiff pressure law, it is possible to build a link between these two different representations. In the incompressible limit, compressible models generate free boundary problems where saturation holds in the moving domain. Our work aims at investigating the stiff pressure limit of reaction-advection-porous medium equations motivated by tumor development. Our first study concerns the analysis and numerical simulation of a model including the effect of nutrients. A coupled system of equations describes the cell density and the nutrient concentration and the derivation of the pressure equation in the stiff limit was an open problem for which the strong compactness of the pressure gradient is needed. To establish it, we use two new ideas: an L3-version of the celebrated Aronson-Bénilan estimate, and a sharp uniform L4-bound on the pressure gradient. We further investigate the sharpness of this bound through a finite difference upwind scheme, which we prove to be stable and asymptotic preserving. Our second study is centered around porous medium equations including convective effects. We are able to extend the techniques developed for the nutrient case, hence finding the complementarity relation on the limit pressure. Moreover, we provide an estimate of the convergence rate at the incompressible limit. Finally, we study a multi-species system. In particular, we account for phenotypic heterogeneity, including a structured variable into the problem. In this case, a cross-(degenerate)-diffusion system describes the evolution of the phenotypic distributions. Adapting methods recently developed in the context of two-species systems, we prove existence of weak solutions and we pass to the incompressible limit. Furthermore, we prove new regularity results on the total pressure, which is related to the total density by a power law of state.