Regular dendritic patterns induced by non-local time-periodic forcing

The dynamic response of dendritic solidification to spatially homogeneous time-periodic forcing has been studied. Phase-field calculations performed in two dimensions (2D) and experiments on thin (quasi 2D) liquid crystal layers show that the frequency of dendritic side-branching can be tuned by oscillatory pressure or heating. The sensitivity of this phenomenon to the relevant parameters, the frequency and amplitude of the modulation, the initial undercooling and the anisotropies of the interfacial free energy and molecule attachment kinetics, has been explored. It has been demonstrated that besides the side-branching mode synchronous with external forcing as emerging from the linear Wentzel-Kramers-Brillouin analysis, modes that oscillate with higher harmonic frequencies are also present with perceptible amplitudes.Comment: 15 pages, 23 figures, Submitted to Phys. Rev.

Doctor of Philosophy

dissertationDynamic contrast enhanced magnetic resonance imaging (DCE-MRI) is a powerful tool to detect cardiac diseases and tumors, and both spatial resolution and temporal resolution are important for disease detection. Sampling less in each time frame and applying sophisticated reconstruction methods to overcome image degradations is a common strategy in the literature. In this thesis, temporal TV constrained reconstruction that was successfully applied to DCE myocardial perfusion imaging by our group was extended to three-dimensional (3D) DCE breast and 3D myocardial perfusion imaging, and the extension includes different forms of constraint terms and various sampling patterns. We also explored some other popular reconstruction algorithms from a theoretical level and showed that they can be included in a unified framework. Current 3D Cartesian DCE breast tumor imaging is limited in spatiotemporal resolution as high temporal resolution is desired to track the contrast enhancement curves, and high spatial resolution is desired to discern tumor morphology. Here temporal TV constrained reconstruction was extended and different forms of temporal TV constraints were compared on 3D Cartesian DCE breast tumor data with simulated undersampling. Kinetic parameters analysis was used to validate the methods

Level Set Methods for MRE Image Processing and Analysis

Ph.DDOCTOR OF PHILOSOPH

Sparse variational regularization for visual motion estimation

The computation of visual motion is a key component in numerous computer vision tasks such as object detection, visual object tracking and activity recognition. Despite exten- sive research effort, efficient handling of motion discontinuities, occlusions and illumina- tion changes still remains elusive in visual motion estimation. The work presented in this thesis utilizes variational methods to handle the aforementioned problems because these methods allow the integration of various mathematical concepts into a single en- ergy minimization framework. This thesis applies the concepts from signal sparsity to the variational regularization for visual motion estimation. The regularization is designed in such a way that it handles motion discontinuities and can detect object occlusions

Modelling cytosolic flow and vesicle transport in the growing pollen tube

Scientific interest in the mathematical modelling of pollen tube growth has increased steadily over the last few decades. The highly localized and rapid nature of this growth necessitates large--scale actomyosin transport of cellular material throughout the cell cytoplasm. This directed movement of cellular material induces a flow in the cytosol, also known as 'cyclosis'. The extent to which inclusion of this flow is important to modelling the distribution of elements in the cytoplasm is currently unclear, with its effect often conflated with that of actomyosin transport. In this thesis, a finite volume method (FVM) is developed for the numerical evaluation of transport equations describing vesicle distribution in the pollen tube cytoplasm. This is coupled with a novel method of regularized ringlets, derived via analytical azimuthal integration of the regularized Stokeslet, for obtaining numerical solutions to axisymmetric Stokes flows. Using this method of regularized ringlets, we present an axisymmetric velocity profile for cytosolic flow in the pollen tube based on the drag induced by actomyosin vesicle transport. When used in the transport equation for vesicle distribution, we find that recreation of the apical `inverted vesicle cone' requires the use of an enlarged effective fluid viscosity amongst other results

Reaction-diffusion systems in and out of equilibrium - methods for simulation and inference

Reaction-diffusion methods allow treatment of mesoscopic dynamic phenomena of soft condensed matter especially in the context of cellular biology. Macromolecules such as proteins consist of thousands of atoms, in reaction-diffusion models their interaction is described by effective dynamics with much fewer degrees of freedom. Reaction-diffusion methods can be categorized by the spatial and temporal length-scales involved and the amount of molecules, e.g. classical reaction kinetics are macroscopic equations for fast diffusion and many molecules described by average concentrations. The focus of this work however is interacting-particle reaction-dynamics (iPRD), which operates on length scales of few nanometers and time scales of nanoseconds, where proteins can be represented by coarse-grained beads, that interact via effective potentials and undergo reactions upon encounter. In practice these systems are often studied using time-stepping computer simulations. Reactions in such iPRD simulations are discrete events which rapidly interchange beads, e.g. in the scheme A + B C the two interacting particles A and B will be replaced by a C complex and vice-versa. Such reactions in combination with the interaction potentials pose two practical problems: 1. To achieve a well defined state of equilibrium, it is of vital importance that the reaction transitions obey microscopic reversiblity (detailed balance). 2. The mean rate of a bimolecular association reaction changes when the particles interact via a pair-potential. In this work the first question is answered both theoretically and algorithmically. Theoretically by formulating the state of equilibrium for a closed iPRD system and the requirements for detailed balance. Algorithmically by implementing the detailed balance reaction scheme in a publicly available simulator ReaDDy~2 for iPRD systems. The second question is answered by deriving concrete formulae for the macroscopic reaction rate as a function of the intrinsic parameters for the Doi reaction model subject to pair interactions. Especially this work addresses two important scenarios: Reversible reactions in a closed container and irreversible bimolecular reactions in the diffusion-influenced regime. A characteristic of reactions occurring in cellular environments is that the number of species involved in a physiological response is very large. Unveiling the network of necessary reactions is a task that can be addressed by a data-driven approach. In particular, analyzing observation data of such processes can be used to learn the important governing dynamics. This work gives an overview of the inference of dynamical reactive systems for the different reaction-diffusion models. For the case of reaction kinetics a method called Reactive Sparse Identification of Nonlinear Dynamics (Reactive SINDy) is developed that allows to obtain a sparse reaction network out of candidate reactions from time-series observations of molecule concentrations

Graph Deep Learning for Time Series Forecasting

Graph-based deep learning methods have become popular tools to process collections of correlated time series. Differently from traditional multivariate forecasting methods, neural graph-based predictors take advantage of pairwise relationships by conditioning forecasts on a (possibly dynamic) graph spanning the time series collection. The conditioning can take the form of an architectural inductive bias on the neural forecasting architecture, resulting in a family of deep learning models called spatiotemporal graph neural networks. Such relational inductive biases enable the training of global forecasting models on large time-series collections, while at the same time localizing predictions w.r.t. each element in the set (i.e., graph nodes) by accounting for local correlations among them (i.e., graph edges). Indeed, recent theoretical and practical advances in graph neural networks and deep learning for time series forecasting make the adoption of such processing frameworks appealing and timely. However, most of the studies in the literature focus on proposing variations of existing neural architectures by taking advantage of modern deep learning practices, while foundational and methodological aspects have not been subject to systematic investigation. To fill the gap, this paper aims to introduce a comprehensive methodological framework that formalizes the forecasting problem and provides design principles for graph-based predictive models and methods to assess their performance. At the same time, together with an overview of the field, we provide design guidelines, recommendations, and best practices, as well as an in-depth discussion of open challenges and future research directions