Magnetohydrodynamic Waves and Instabilities in Rotating Tokamak Plasmas

One of the most promising ways to achieve controlled nuclear fusion for the commercial production of energy is the tokamak design. In such a device, a hot plasma is confined in a toroidal geometry using magnetic fields. The present generation of tokamaks shows significant plasma rotation, primarily in the toroidal direction. This plasma flow has an important impact on stability and confinement, aspects of which can be described quite well by the theory of magnetohydrodynamics (MHD). This work contains a comprehensive theoretical analysis, supported by numerical simulations, of the MHD equilibrium, waves, and instabilities of rotating tokamak plasmas. A new general description of the thermodynamic state of the equilibrium is presented. Next, a stability criterion is derived that generalizes various previous results by including toroidal rotation. This criterion shows that a radially decreasing rotation profile can be stabilizing. The previously unknown origin of this stabilization is shown to be the Coriolis effect, with a mediating role for the pressure. Various factors that affect stability also influence stable waves and eigenmodes of the plasma. New modes that are created by rotation are found, and the effect of rotation on a type of experimentally well-known modes is described. Finally, the step to nonlinear magnetohydrodynamics is made by extending an existing reduced MHD code to the full viscoresistive MHD equations. This allows a study of the nonlinear evolution of the equilibria, waves, and instabilities described in this thesis

Progress in DNP theory and hardware

Dynamic nuclear polarisation is a technique that allows one to increase the signal-to-noise ratio in an NMR experiment substantially, by transferring the inherently larger electron polarisation to the nuclei. Quantum mechanical models of this effect have thus far been limited to the description of only a few nuclei. This is due to the exponential scaling of the matrices involved in the description of the system. In this thesis methods of reducing the state space needed to accurately describe the simulation of solid effect DNP were explored and tested. Krylov Bogoliubov averaging has been used to remove high frequency oscillations from the system Hamiltonian and confine the trajectory of the dynamics to the zero quantum coherence subspace. Truncation of the basis spanning the Liouville space to low spin correlation orders has been tested and a condition for a minimum truncation level was found. A strategy based on a projection method, which allows one to describe the spin polarisation transient with multi-exponential functions, is introduced. This results in a linear scaling of the propagator with the number of spins. The influence of the parameters involved in the solid effect on the dynamics of the polarisation build up is discussed. The second part of this thesis is concerned with a novel approach to detecting fast molecular dynamics with the use of multiple RF receive and transmit coils. A proof of principle probe with two decoupled RF coils is presented, as well as a field map based shimming strategy and fast 2D data acquired with the probe. Lastly a probe with six RF coils, based on the design of the dual coil probe, will be presented, and initial data shown. The potential for using this probe in hyper-polarisation experiments for protein binding and folding studies will be discussed

Progress in DNP theory and hardware

Dynamic nuclear polarisation is a technique that allows one to increase the signal-to-noise ratio in an NMR experiment substantially, by transferring the inherently larger electron polarisation to the nuclei. Quantum mechanical models of this effect have thus far been limited to the description of only a few nuclei. This is due to the exponential scaling of the matrices involved in the description of the system. In this thesis methods of reducing the state space needed to accurately describe the simulation of solid effect DNP were explored and tested. Krylov Bogoliubov averaging has been used to remove high frequency oscillations from the system Hamiltonian and confine the trajectory of the dynamics to the zero quantum coherence subspace. Truncation of the basis spanning the Liouville space to low spin correlation orders has been tested and a condition for a minimum truncation level was found. A strategy based on a projection method, which allows one to describe the spin polarisation transient with multi-exponential functions, is introduced. This results in a linear scaling of the propagator with the number of spins. The influence of the parameters involved in the solid effect on the dynamics of the polarisation build up is discussed. The second part of this thesis is concerned with a novel approach to detecting fast molecular dynamics with the use of multiple RF receive and transmit coils. A proof of principle probe with two decoupled RF coils is presented, as well as a field map based shimming strategy and fast 2D data acquired with the probe. Lastly a probe with six RF coils, based on the design of the dual coil probe, will be presented, and initial data shown. The potential for using this probe in hyper-polarisation experiments for protein binding and folding studies will be discussed

Local Geometry Processing for Deformations of Non-Rigid 3D Shapes

Geometry processing and in particular spectral geometry processing deal with many different deformations that complicate shape analysis problems for non-rigid 3D objects. Furthermore, pointwise description of surfaces has increased relevance for several applications such as shape correspondences and matching, shape representation, shape modelling and many others. In this thesis we propose four local approaches to face the problems generated by the deformations of real objects and improving the pointwise characterization of surfaces. Differently from global approaches that work simultaneously on the entire shape we focus on the properties of each point and its local neighborhood. Global analysis of shapes is not negative in itself. However, having to deal with local variations, distortions and deformations, it is often challenging to relate two real objects globally. For this reason, in the last decades, several instruments have been introduced for the local analysis of images, graphs, shapes and surfaces. Starting from this idea of localized analysis, we propose both theoretical insights and application tools within the local geometry processing domain. In more detail, we extend the windowed Fourier transform from the standard Euclidean signal processing to different versions specifically designed for spectral geometry processing. Moreover, from the spectral geometry processing perspective, we define a new family of localized basis for the functional space defined on surfaces that improve the spatial localization for standard applications in this field. Finally, we introduce the discrete time evolution process as a framework that characterizes a point through its pairwise relationship with the other points on the surface in an increasing scale of locality. The main contribute of this thesis is a set of tools for local geometry processing and local spectral geometry processing that could be used in standard useful applications. The overall observation of our analysis is that localization around points could factually improve the geometry processing in many different applications

Mathematical Challenges in Electron Microscopy

Development of electron microscopes first started nearly 100 years ago and they are now a mature imaging modality with many applications and vast potential for the future. The principal feature of electron microscopes is their resolution; they can be up to 1000 times more powerful than a visible light microscope and resolve even the smallest atoms. Furthermore, electron microscopes are also sensitive to many material properties due to the very rich interactions between electrons and other matter. Because of these capabilities, electron microscopy is used in applications as diverse as drug discovery, computer chip manufacture, and the development of solar cells. In parallel to this, the mathematical field of inverse problems has also evolved dramatically. Many new methods have been introduced to improve the recovery of unknown structures from indirect data, typically an ill-posed problem. In particular, sparsity promoting functionals such as the total variation and its extensions have been shown to be very powerful for recovering accurate physical quantities from very little and/or poor quality data. While sparsity-promoting reconstruction methods are powerful, they can also be slow, especially in a big-data setting. This trade-off forms an eternal cycle as new numerical tools are found and more powerful models are developed. The work presented in this thesis aims to marry the tools of inverse problems with the problems of electron microscopy: bringing state-of-the-art image processing techniques to bear on challenges specific to electron microscopy, developing new optimisation methods for these problems, and modelling new inverse problems to extend the capabilities of existing microscopes. One focus is the application of a directional total variation to overcome the limited angle problem in electron tomography, another is the proposal of a new inverse problem for the reconstruction of 3D strain tensor fields from electron microscopy diffraction data. The remaining contributions target numerical aspects of inverse problems, from new algorithms for non-convex problems to convex optimisation with adaptive meshes.Cantab Capital Institute for Mathematics of Informatio

On the electromagnetic field of a proton beam as a basis for range verification in particle therapy

Targeting tumor cells with ionizing radiation in an effort to eliminate them is a mainstay of cancer treatment. External beams of heavy charged particles, such as protons, are applied for such purposes and have the potential to enable highly conformal dose delivery due to their favorable depth dose profile, i.e. the so-called Bragg peak. This allows an effective sparing of healthy tissues and organs at risk, especially when compared to the conventional approach with x-rays. The finite penetration depth of protons within the patient, known as the range, is however subject to uncertainties, which can lead to significant underdosage in the tumor and excessive dose to critical structures. Inaccuracies can originate from imaging, anatomical changes, patient positioning, just to name a few. Such risk factors limit the full potential of proton therapy and necessitate the utilization of safety margins around the tumor volume, which increases the overall dose to healthy tissue. Hence, the development of methodologies to verify the proton range in vivo is an active field of research. The most prominent candidates rely on positron emission tomography (PET), prompt gamma (PG) imaging or the detection of thermoacoustic waves. These methods, however, are limited in several aspects, such as low signal-to-noise ratios or challenging detection, leaving room for new ideas and methods to be developed. Recently, it has been suggested to use the electric field of the primary protons as a basis for an alternative range verification method. The present work aims to investigate the possibilities and limitations of such an approach. The first part is concerned with an exhaustive analytical characterization of the electromagnetic field that originates from a proton pencil beam and how it is affected by biological tissues. The impact of the beam pulse shape, permittivity, conductivity and tissue boundaries are considered. Contradictory to previous results, it has been found that the charge relaxation, which originates from the ionic conductivity of biological tissues, has a huge impact on the electric field, causing it to diminish in a nanosecond time scale. The electric field is thus not suitable as a basis for range verification, considering also the washout effect, that the rapid redistribution of charges creates. The magnetic field, on the other hand, is not affected by the latter and benefits from the approximate constancy of the current density. It does not drop together with the decreasing particle velocity, but is upheld due to the equally increasing charge density towards the range. The associated magnetic field does not show a distinctive peak at the range but follows a smooth yet characteristic profile along the beam axis, from which the range could be determined. Finally, an in-depth analysis of the frequency spectrum has been carried out, separating it into well-known constituents. The second part aimed to lift some of the simplifying assumptions, investigating the impact of nuclear reactions, energy and range straggling, lateral scattering, beam spot size and secondary particles. With an emphasis on the secondary electrons, dedicated Monte Carlo (MC) simulations were conducted, tracking them down to 10 eV. Despite being significantly more numerous than the primary protons, they reduce the overall current density by only 10%. The main reasons are their mostly isotropic flow and short lifetimes, which followed from a thorough phase space analysis. The current density extracted from the MC simulations served as an input for a numerical magnetic field estimation via finite element analysis. Thereby, it has been found that the loss of intensity from nuclear reactions, the electron current and the radial proton current introduce a small but non-negligible longitudinal shift with respect to the analytical result from the first part. In addition, the random current density fluctuations were quantified and deemed negligible in the context of a measurement. Finally, it has been shown that the beam spot size has no impact on the detectable magnetic field. In summary, barring minor deviations, the findings from the first part have been confirmed under more realistic assumptions. The last part expand the applicability of the analytical approach to simple inhomogeneous targets. Through a Green's function approach, the impact of boundaries for a more realistic beam, which includes the RF structure from the accelerator, has been examined. Also, the possibility to modulate the beam intensity artificially in an effort to separate the sought signal from ambient noise (bioelectricity) can be investigated with the same method. Preliminary results indicated that the boundaries cannot be neglected causing an overall reduction of the transmitted signal due to the comparatively large reflection coefficients. Also, the longitudinal magnetic field profile depends on the modulation frequency. Finally, the potential of the range verification method under consideration has been evaluated with respect to current technological capabilities.Die Bestrahlung von Tumorzellen mit ionisierenden Strahlen mit dem Ziel sie unschädlich zu machen gehört zu den Hauptpfeilern der Krebsbehandlung. Zu diesem Zweck werden externe Strahlen bestehend aus schweren geladenen Teilchen wie Protonen eingesetzt, die aufgrund ihres vorteilhaften Tiefendosisprofils, des so genannten Bragg-Peaks, eine sehr konforme Dosisabgabe ermöglichen. Dies gestattet eine effektive Schonung von gesundem Gewebe und gefährdeten Organen, insbesondere im Vergleich zum konventionellen Ansatz mit Röntgenstrahlen. Die endliche Eindringtiefe der Protonen in den Patienten, die so genannte Reichweite, ist jedoch mit Unsicherheiten behaftet, die zu einer erheblichen Unterdosierung im Tumor und einer übermäßigen Dosis in lebensnotwendigen Organen führen können. Ungenauigkeiten können von der Bildgebung, anatomischen Veränderungen, der Positionierung des Patienten, um nur einige zu nennen, herrühren. Solche Risikofaktoren schränken das volle Potenzial der Protonentherapie ein und machen die Verwendung von Sicherheitsmargen um das Tumorvolumen herum erforderlich, was die Gesamtdosis für das gesunde Gewebe erhöht. Daher ist die Entwicklung von Methoden zur Überprüfung der Protonenreichweite in vivo ein aktives Forschungsgebiet. Die bekann-testen Herangehensweisen stützen sich auf die Positronen-Emissions-Tomographie (PET), die prompt gamma (PG) Bildgebung oder die Messung von thermoakustischen Wellen. Diese Methoden sind jedoch in vielerlei Hinsicht eingeschränkt, z. B. durch ein geringes Signal-Rausch-Verhältnis oder eine schwierige Detektion, was Raum für die Entwicklung neuer Ideen und Methoden lässt. Kürzlich wurde vorgeschlagen, das elektrische Feld der Primärprotonen als Grundlage für eine alternative Methode zur Überprüfung der Reichweite zu verwenden. Die vorliegende Arbeit zielt darauf ab, die Möglichkeiten und Grenzen eines solchen Ansatzes zu untersuchen. Der erste Teil befasst sich mit einer umfassenden analytischen Charakterisierung des elektromagnetischen Feldes, das von einem Protonenstrahl ausgeht und wie es von biologischem Gewebe beeinflusst wird. Dabei werden die Auswirkungen der Form des Strahlpulses, der Permittivität, der Leitfähigkeit und der Gewebegrenzen berücksichtigt. Im Gegensatz zu bisherigen Ergebnissen wurde festgestellt, dass die Ladungsrelaxation, die auf die Ionenleitfähigkeit von biologischem Gewebe zurückzuführen ist, einen enormen Einfluss auf das elektrische Feld hat, so dass es sich innerhalb von Nanosekunden abschwächt. Das elektrische Feld eignet sich daher nicht als Grundlage für die Reichwei-tenüberprüfung, auch unter Berücksichtigung des Auswascheffekts, der durch die schnelle Umverteilung der Ladungen entsteht. Das magnetische Feld hingegen wird davon nicht beeinflusst und profitiert von der annähernd konstanten Stromdichte. Sie nimmt nicht mit der abnehmenden Teilchengeschwindigkeit ab, sondern wird aufgrund der ebenfalls zunehmenden Ladungsdichte zur Reichweite hin aufrechterhalten. Das zugehörige Magnetfeld weist keinen ausgeprägten Peak im Bereich der Reichweite auf, sondern folgt einem flachem, aber charakteristischen Profil entlang der Strahlachse, aus dem die Reichweite be- stimmt werden könnte. Schließlich wurde eine eingehende Analyse des Frequenzspektrums durchgeführt, wobei es in die bekannten Bestandteile zerlegt wurde. Der zweite Teil zielte darauf ab, einige der vereinfachenden Annahmen aufzuheben und die Auswirkungen von Kernreaktionen, Energie- und Reichweitenstreuung, lateraler Streuung, Strahldurchmesser und Sekundärteilchen zu untersuchen. Mit Schwerpunkt auf den Sekundärelektronen wurden speziell dafür vorgesehene Monte-Carlo-Simulationen (MC) durchgeführt, bei denen die Elektronen bis hinunter zu 10 eV nachverfolgt wurden. Obwohl sie wesentlich zahlreicher sind als die primären Protonen, reduzieren sie die Gesamtstromdichte nur um 10%. Die Hauptgründe dafür sind ihr überwiegend isotroper Fluss und ihre kurze Lebensdauer, was sich aus einer eingehenden Phasenraumanalyse ergeben hat. Die aus den MC-Simulationen extrahierte Stromdichte diente als Ausgangspunkt für eine numerische Magnetfeldbestimmung mittels Finite-Elemente-Analyse. Dabei wurde festgestellt, dass der Intensitätsverlust aus Kernreaktionen, der Elektronenstrom und der radiale Protonenstrom eine kleine, aber nicht vernachlässigbare Längsverschiebung gegenüber dem analytischen Ergebnis aus dem ersten Teil verursachen. Darüber hinaus wurden die Zufallsschwankungen der Stromdichte quantifiziert und im Rahmen einer Messung als vernachlässigbar eingestuft. Schließlich wurde gezeigt, dass der Strahldurchmesser keinen Einfluss auf das messbare Magnetfeld hat. Zusammenfassend kann gesagt werden, dass die Ergebnisse des ersten Teils, abgesehen von geringfügigen Abweichungen, unter realistischeren Annahmen bestätigt werden konnten. Im letzten Teil wird die Anwendbarkeit des analytischen Ansatzes auf einfache inhomogene Targets erweitert. Mit Hilfe eines Green'schen Funktionsansatzes wurden die Auswirkungen von Gewebegrenzen für einen realistischeren Strahl, der die HF-Struktur des Beschleunigers einschließt, untersucht. Auch die Möglichkeit, die Strahlintensität künst-lich zu modulieren, um das gesuchte Signal vom Umgebungsrauschen (Bioelektrizität) zu trennen, kann mit der gleichen Methode untersucht werden. Vorläufige Ergebnisse zeigten, dass die Grenzflächen nicht vernachlässigt werden können, was aufgrund der vergleichsweise großen Reflexionskoeffizienten zu einer Gesamtverringerung des transmittierten Signals führt. Außerdem hängt das longitudinale Profil des magnetischen Feldes von der Modulationsfrequenz ab. Schließlich wurde das Potenzial der untersuchten Methode zur Reichweitenverifizierung im Hinblick auf die derzeitigen technischen Möglichkeiten evaluiert

Development of a non-contrast-enhanced method for spatially resolved lung ventilation and perfusion measurement using Magnetic Resonance Imaging

Assessment of the pulmonary function remains a challenge for the development of suitable MRI techniques due to the unique lung tissue structure and its short effective transverse relaxation time (T2* = 1 ms). In this work, a new method of non-contrast-enhanced lung ventilation and perfusion MRI is presented. A 2D bSSFP pulse sequence (TR/TE/TA = 1.9/0.8/116 ms, 3-7 images/s, FA = 75°, ST = 10 mm, matrix = 128 x 128, GRAPPA 3) was implemented on a 1.5 T MR-scanner. The method uses fast image acquisition and submillisecond echo sampling to enhance the signal intensity in the pulmonary tissue. The proposed technique does not rely on respiratory and ECG-triggering. Application of non-rigid image registration was mandatory to compensate for the breathing motion. The rapid acquisition of time-resolved MR-data allowed observing intensity changes in corresponding lung areas modulated with respiratory and cardiac frequencies. Two different spectral analysis methods, Fourier decomposition (FD) and wavelet analysis (WA) were used to produce ventilation- and perfusion-weighted images by retrieving information associated with both physiological frequencies (FD/WA-MRI). The imaging technique was used in volunteers to test the technical and medical reproducibility. For validation purposes a group of cystic fibrosis patients was examined using FD-MRI and dynamic Contrast-Enhanced MRI. A good correlation between both methods (r = 0.82, P < 0.05) was determined. Animal experiments were conducted for validation of FD-MRI against other imaging modalities (CT and SPECT/CT)