On the numerical treatment of viscous and convective effects in relative pressure reconstruction methods

The mechanism of many cardiovascular diseases can be understood by studying the pressure distribution in blood vessels. Direct pressure measurements, however, require invasive probing and provide only single‐point data. Alternatively, relative pressure fields can be reconstructed from imaging‐based velocity measurements by considering viscous and inertial forces. Both contributions can be potential troublemakers in pressure reconstruction: the former due to its higher‐order derivatives, and the latter because of the quadratic nonlinearity in the convective acceleration. Viscous and convective terms can be treated in various forms, which, although equivalent for ideal measurements, can perform differently in practice. In fact, multiple versions are often used in literature, with no apparent consensus on the more suitable variants. In this context, the present work investigates the performance of different versions of relative pressure estimators. For viscous effects, in particular, two new modified estimators are presented to circumvent second‐order differentiation without requiring surface integrals. In‐silico and in‐vitro data in the typical regime of cerebrovascular flows are considered, allowing a systematic noise sensitivity study. Convective terms are shown to be the main source of error, even for flows with pronounced viscous component. Moreover, the conservation (often integrated) form of convection exhibits higher noise sensitivity than the standard convective description, in all three families of estimators considered here. For the classical pressure Poisson estimator, the present modified version of the viscous term tends to yield better accuracy than the (recently introduced) integrated form, although this effect is in most cases negligible when compared to convection‐related errors

Topological Data Analysis in Sub-cellular Motion Reconstruction and Filament Networks Classification

Topological Data Analysis is a powerful tool in the image data analysis. In this dissertation, we focus on studying cell physiology by the sub-cellular motions of organelles and generation process of filament networks, relying on topology of the cellular image data. We first develop a novel, automated algorithm, which tracks organelle movements and reconstructs their trajectories on stacks of microscopy image data. Our tracking method proceeds with three steps: (i) identification, (ii) localization, and (iii) linking, and does not assume a specific motion model. This method combines topological data analysis principles with Ensemble Kalman Filtering in the computation of associated nerve during the linking step. Moreover, we show a great success of our method with several applications. We then study filament networks as a classification problem, and propose a distancebased classifier. This algorithm combines topological data analysis with a supervised machine learning framework, and is built based on the foundation of persistence diagrams on the data.We adopt a new metric, the dcp distance, on the space of persistence diagrams, and show it is useful in catching the geometric difference of filament networks. Furthermore, our classifier succeeds in classifying filament networks with high accuracy rate