Academic literature on the topic 'Bloch-Torrey equation'
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Journal articles on the topic "Bloch-Torrey equation"
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Yu, Qiang, Fawang Liu, Ian Turner, and Kevin Burrage. "Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1990 (May 13, 2013): 20120150. http://dx.doi.org/10.1098/rsta.2012.0150.
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Fractional-order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brownian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As magnetic resonance imaging is applied with increasing temporal and spatial resolution, the spin dynamics is being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here, the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments, where processes are often anisotropic. Anomalous diffusion in the human brain using fractional-order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch–Torrey equation using fractional-order calculus with respect to time and space. However, effective numerical methods and supporting error analyses for the fractional Bloch–Torrey equation are still limited. In this paper, the space and time fractional Bloch–Torrey equation (ST-FBTE) in both fractional Laplacian and Riesz derivative form is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE in fractional Laplacian form with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE based on the Riesz form, and the stability and convergence of the INM are investigated. We prove that the INM for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.
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Caubet, Fabien, Houssem Haddar, Jing-Rebecca li, and Dang Van Nguyen. "New transmission condition accounting for diffusion anisotropy in thin layers applied to diffusion MRI." ESAIM: Mathematical Modelling and Numerical Analysis 51, no. 4 (June 30, 2017): 1279–301. http://dx.doi.org/10.1051/m2an/2016060.
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The Bloch-Torrey Partial Differential Equation (PDE) can be used to model the diffusion Magnetic Resonance Imaging (dMRI) signal in biological tissue. In this paper, we derive an Anisotropic Diffusion Transmission Condition (ADTC) for the Bloch-Torrey PDE that accounts for anisotropic diffusion inside thin layers. Such diffusion occurs, for example, in the myelin sheath surrounding the axons of neurons. This ADTC can be interpreted as an asymptotic model of order two with respect to the layer thickness and accounts for water diffusion in the normal direction that is low compared to the tangential direction. We prove uniform stability of the asymptotic model with respect to the layer thickness and a mass conservation property. We also prove the theoretical quadratic accuracy of the ADTC. Finally, numerical tests validate these results and show that our model gives a better approximation of the dMRI signal than a simple transmission condition that assumes isotropic diffusion in the layers.
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Rotkopf, L. T., E. Wehrse, F. T. Kurz, H. P. Schlemmer, and C. H. Ziener. "Efficient discretization scheme for semi-analytical solutions of the Bloch-Torrey equation." Journal of Magnetic Resonance Open 6-7 (June 2021): 100010. http://dx.doi.org/10.1016/j.jmro.2021.100010.
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Seroussi, Inbar, Denis S. Grebenkov, Ofer Pasternak, and Nir Sochen. "Microscopic interpretation and generalization of the Bloch-Torrey equation for diffusion magnetic resonance." Journal of Magnetic Resonance 277 (April 2017): 95–103. http://dx.doi.org/10.1016/j.jmr.2017.01.018.
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Magin, Richard L., Osama Abdullah, Dumitru Baleanu, and Xiaohong Joe Zhou. "Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation." Journal of Magnetic Resonance 190, no. 2 (February 2008): 255–70. http://dx.doi.org/10.1016/j.jmr.2007.11.007.
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Zhu, Yun, and Zhi-Zhong Sun. "A High-Order Difference Scheme for the Space and Time Fractional Bloch–Torrey Equation." Computational Methods in Applied Mathematics 18, no. 1 (January 1, 2018): 147–64. http://dx.doi.org/10.1515/cmam-2017-0034.
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AbstractIn this paper, a high-order difference scheme is proposed for an one-dimensional space and time fractional Bloch–Torrey equation. A third-order accurate formula, based on the weighted and shifted Grünwald–Letnikov difference operators, is used to approximate the Caputo fractional derivative in temporal direction. For the discretization of the spatial Riesz fractional derivative, we approximate the weighed values of the Riesz fractional derivative at three points by the fractional central difference operator. The unique solvability, unconditional stability and convergence of the scheme are rigorously proved by the discrete energy method. The convergence order is 3 in time and 4 in space in {L_{1}(L_{2})}-norm. Two numerical examples are implemented to testify the accuracy of the numerical solution and the efficiency of the difference scheme.
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Xu, Tao, Shujuan Lü, and Haonan Li. "An implicit numerical method for the space-time variable-order fractional Bloch-Torrey equation." Journal of Physics: Conference Series 1039 (June 2018): 012008. http://dx.doi.org/10.1088/1742-6596/1039/1/012008.
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Barzykin, A. V. "Exact solution of the Torrey-Bloch equation for a spin echo in restricted geometries." Physical Review B 58, no. 21 (December 1, 1998): 14171–74. http://dx.doi.org/10.1103/physrevb.58.14171.
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Beltrachini, Leandro, Zeike A. Taylor, and Alejandro F. Frangi. "A parametric finite element solution of the generalised Bloch–Torrey equation for arbitrary domains." Journal of Magnetic Resonance 259 (October 2015): 126–34. http://dx.doi.org/10.1016/j.jmr.2015.08.008.
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Zhao, Yue, Weiping Bu, Xuan Zhao, and Yifa Tang. "Galerkin finite element method for two-dimensional space and time fractional Bloch–Torrey equation." Journal of Computational Physics 350 (December 2017): 117–35. http://dx.doi.org/10.1016/j.jcp.2017.08.051.
Full textDissertations / Theses on the topic "Bloch-Torrey equation"
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Mekkaoui, Imen. "Analyse numérique des équations de Bloch-Torrey." Thesis, Lyon, 2016. http://www.theses.fr/2016LYSEI120/document.
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L’imagerie par résonance magnétique de diffusion (IRMd) est une technique non-invasive permettant d’accéder à l'information structurelle des tissus biologiques à travers l’étude du mouvement de diffusion des molécules d’eau dans les tissus. Ses applications sont nombreuses en neurologie pour le diagnostic de certaines anomalies cérébrales. Cependant, en raison du mouvement cardiaque, l’utilisation de cette technique pour accéder à l’architecture du cœur in vivo représente un grand défi. Le mouvement cardiaque a été identifié comme une des sources majeures de perte du signal mesuré en IRM de diffusion. A cause de la sensibilité au mouvement, il est difficile d’évaluer dans quelle mesure les caractéristiques de diffusion obtenues à partir de l’IRM de diffusion reflètent les propriétés réelles des tissus cardiaques. Dans ce cadre, la modélisation et la simulation numérique du signal d’IRM de diffusion offrent une approche alternative pour aborder le problème. L’objectif de cette thèse est d’étudier numériquement l’influence du mouvement cardiaque sur les images de diffusion et de s’intéresser à la question d’atténuation de l’effet du mouvement cardiaque sur le signal d’IRM de diffusion. Le premier chapitre est consacré à l’introduction du principe physique de l'imagerie par résonance magnétique(IRM). Le deuxième chapitre présente le principe de l’IRM de diffusion et résume l’état de l’art des différents modèles proposés dans la littérature pour modéliser le signal d’IRM de diffusion. Dans le troisième chapitre un modèle modifié de l’équation de Bloch-Torrey dans un domaine qui se déforme au cours du temps est introduit et étudié. Ce modèle représente une généralisation de l’équation de Bloch-Torrey utilisée dans la modélisation du signal d’IRM de diffusion dans le cas sans mouvement. Dans le quatrième chapitre, l’influence du mouvement cardiaque sur le signal d’IRM de diffusion est étudiée numériquement en utilisant le modèle de Bloch-Torrey modifié et un champ de mouvement analytique imitant une déformation réaliste du cœur. L’étude numérique présentée, permet de quantifier l’effet du mouvement sur la mesure de diffusion en fonction du type de la séquence de codage de diffusion utilisée, de classer ces séquences en terme de sensibilité au mouvement cardiaque et d’identifier une fenêtre temporelle par rapport au cycle cardiaque où l’influence du mouvement est réduite. Enfin, dans le cinquième chapitre, une méthode de correction de mouvement est présentée afin de minimiser l’effet du mouvement cardiaque sur les images de diffusion. Cette méthode s’appuie sur un développement singulier du modèle de Bloch-Torrey modifié pour obtenir un modèle asymptotique qui permet de résoudre le problème inverse de récupération puis correction de la diffusion influencée par le mouvement cardiaqueDiffusion magnetic resonance imaging (dMRI) is a non-invasive technique allowing access to the structural information of the biological tissues through the study of the diffusion motion of water molecules in tissues. Its applications are numerous in neurology, especially for the diagnosis of certain brain abnormalities, and for the study of the human cerebral white matter. However, due to the cardiac motion, the use of this technique to study the architecture of the in vivo human heart represents a great challenge. Cardiac motion has been identified as a major source of signal loss. Because of the sensitivity to motion, it is difficult to assess to what extent the diffusion characteristics obtained from diffusion MRI reflect the real properties of the cardiac tissue. In this context, modelling and numerical simulation of the diffusion MRI signal offer an alternative approach to address the problem. The objective of this thesis is to study numerically the influence of cardiac motion on the diffusion images and to focus on the issue of attenuation of the cardiac motion effect on the diffusion MRI signal. The first chapter of this thesis is devoted to the introduction of the physical principle of nuclear magnetic resonance (NMR) and image reconstruction techniques in MRI. The second chapter presents the principle of diffusion MRI and summarizes the state of the art of the various models proposed in the litera- ture to model the diffusion MRI signal. In the third chapter a modified model of the Bloch-Torrey equation in a domain that deforms over time is introduced and studied. This model represents a generalization of the Bloch-Torrey equation used to model the diffusion MRI signal in the case of static organs. In the fourth chapter, the influence of cardiac motion on the diffusion MRI signal is investigated numerically by using the modified Bloch-Torrey equation and an analytical motion model mimicking a realistic deformation of the heart. The numerical study reported here, can quantify the effect of motion on the diffusion measurement depending on the type of the diffusion coding sequence. The results obtained allow us to classify the diffusion encoding sequences in terms of sensitivity to the cardiac motion and identify for each sequence a temporal window in the cardiac cycle in which the influence of motion is reduced. Finally, in the fifth chapter, a motion correction method is presented to minimize the effect of cardiac motion on the diffusion images. This method is based on a singular development of the modified Bloch-Torrey model in order to obtain an asymptotic model of ordinary differential equation that gives a relationship between the true diffusion and the diffusion reconstructed in the presence of motion. This relationship is then used to solve the inverse problem of recovery and correction of the diffusion influenced by the cardiac motion
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Moutal, Nicolas. "Study of the Bloch-Torrey equation associated to diffusion magnetic resonance imaging." Thesis, Institut polytechnique de Paris, 2020. http://www.theses.fr/2020IPPAX031.
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L'imagerie de résonance magnétique nucléaire pondérée par diffusion (dMRI) est une technique expérimentale qui a pour but d'indentifier les propriétés microstructurales d'un échantillon bien en-dessous de la résolution conventionnelle de l'IRM "classique''. Bien que cette technique ait été introduite et appliquée dans divers contextes depuis plusieurs décennies, de nombreux éléments théoriques restent à élucider, et ce d'autant plus avec l'amélioration constante des appareils d'imagerie et des techniques expérimentales. Notablement, les mécanismes de formation du signal d'IRM aux forts gradients sont encore largement incompris, malgré une tendance ``naturelle'' à l'augmentation des gradients pour sonder des échelles structurales de plus en plus fines.Nous revisitons dans un premier temps les effets d'anisotropie géométrique. Tandis que l'anisotropie aux échelles micro- et macroscopiques a été l'objet de beaucoup d'attention ces dernières années, l'échelle intermédiaire, ``mésoscopique'', n'avait pas encore été étudiée systématiquement. Nous avons obtenu une généralisation de la formule de Mitra qui permet d'améliorer significativement l'estimation du rapport surface-volume de domaines arbitraires quelle que soit la séquence de gradient utilisée.Dans un second temps, nous étudions les effets de perméabilité, qui sont cruciaux pour les applications biomédicales. Nous proposons une analyse critique de trois modèles classiques de l'effet de l'échange sur le signal d'IRM de diffusion. De plus, nous formulons une méthode numérique et théorique générale et flexible pour étudier la diffusion à travers plusieurs membranes perméables parallèles.Le dernier chapitre constitue le cœur de la thèse et aborde l'étude non-perturbative de l'équation de Bloch-Torrey qui régit l'évolution du signal d'IRM de diffusion. Aux forts gradients, nous montrons théoriquement, numériquement, et expérimentalement l'universalité du phénomène de localisation, qui ouvre des perspectives prometteuses pour augmenter la sensibilité du signal d'IRM à la microstructureDiffusion magnetic resonance imaging (dMRI) is an experimental technique which aims at unraveling the microstructural properties of a sample well below the conventional spatial resolution of ``classic'' MRI. Although this technique has been proposed and applied in various contexts for several decades, many theoretical points remain to be clarified, even more with the permanent improvement of MRI scanners and experimental protocols. Notably, the understanding of the signal formation at high gradients is largely incomplete, in spite of the ``natural'' tendency to increase the gradient in order to probe finer and finer structural scales.We first revisit anisotropy effects. While micro- and macroscopic anisotropy have been largely studied over past years, the intermediate, ``mesocopic'' scale had not been investigated in a systematic way. We have obtained a generalized Mitra formula which improves significantly surface-to-volume ratio estimations for arbitrary domains and gradient waveforms.In a second chapter, we investigate permeability effects, that are crucial for biomedical applications. We critically revise three classical models of exchange for dMRI. Moreover, we obtain a general and flexible numerical and theoretical method to study diffusion trough several parallel permeable membranes.The last chapter is the heart of the thesis and contains a non-perturbative study of Bloch-Torrey equation, which governs the evolution of dMRI signal. At high gradient strength, we reveal theoretically, numerically, and experimentally the universality of the localization phenomenon, which opens promising perspectives to improve the sensitivity of the signal to the microstructure
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Fang, Chengran. "Neuron modeling, Bloch-Torrey equation, and their application to brain microstructure estimation using diffusion MRI." Electronic Thesis or Diss., université Paris-Saclay, 2023. http://www.theses.fr/2023UPASG010.
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L'estimation non invasive de la microstructure du cerveau, qui se compose de nombreux neurites, de somas et de cellules gliales, est essentielle pour l'imagerie cérébrale. L'IRM de diffusion (IRMd) est une technique prometteuse pour sonder les propriétés microstructurelles du cerveau en dessous de la résolution spatiale des scanners IRM. En raison de la complexité structurelle du tissu cérébral et du mécanisme complexe de l'IRM de diffusion, l'estimation de la microstructure in vivo est un défi. Les méthodes existantes utilisent généralement des géométries simplifiées, notamment des sphères et des bâtons, pour modéliser les structures neuronales et obtenir des expressions analytiques des signaux intracellulaires. La validité des hypothèses faites par ces méthodes reste indéterminée. Cette thèse vise à faciliter l'estimation de la microstructure du cerveau par simulation en remplaçant les géométries simplifiées par des modèles réalistes de la géométrie des neurones et les expressions analytiques des signaux intracellulaires par des simulations d'IRM de diffusion. Combinées à des modèles précis de la géométrie des neurones, les simulations numériques d'IRMd peuvent donner des signaux intracellulaires précis et incorporer les effets dus, par exemple, à l'ondulation des neurites ou à l'échange d'eau entre le soma et les neurites.Malgré ces avantages, les simulations d'IRMd n'ont pas été largement adoptées en raison de l'inaccessibilité des fantômes numériques, de la faible efficacité de calcul des simulateurs d'IRMd et de la difficulté d'approximer les mappings implicites entre les signaux d'IRMd et les propriétés de la microstructure. Cette thèse contribue à la résolution des problèmes susmentionnés de la manière suivante : (1) en développant un générateur de maillage de neurones open-source et en rendant accessibles au public plus d'un millier de maillages cellulaires réalistes ; (2) en augmentant d'un facteur dix l'efficacité de calcul de la méthode du formalisme matriciel numérique ; (3) en mettant en œuvre une nouvelle méthode de simulation qui fournit une représentation de type Fourier des signaux IRMd ; (4) en proposant un cadre d'apprentissage supervisé basé sur la simulation pour estimer la microstructure du cerveau par IRM de diffusionNon-invasively estimating brain microstructure that consists of a very large number of neurites, somas, and glial cells is essential for future neuroimaging. Diffusion MRI (dMRI) is a promising technique to probe brain microstructural properties below the spatial resolution of MRI scanners. Due to the structural complexity of brain tissue and the intricate diffusion MRI mechanism, in vivo microstructure estimation is challenging.Existing methods typically use simplified geometries, particularly spheres, and sticks, to model neuronal structures and to obtain analytical expressions of intracellular signals. The validity of the assumptions made by these methods remains undetermined. This thesis aims to facilitate simulationdriven brain microstructure estimation by replacing simplified geometries with realistic neuron geometry models and the analytical intracellular signal expressions with diffusion MRI simulations. Combined with accurate neuron geometry models, numerical dMRI simulations can give accurate intracellular signals and seamlessly incorporate effects arising from, for instance, neurite undulation or water exchange between soma and neurites.Despite these advantages, dMRI simulations have not been widely adopted due to the difficulties in constructing realistic numerical phantoms, the high computational cost of dMRI simulations, and the difficulty in approximating the implicit mappings between dMRI signals and microstructure properties. This thesis addresses the above problems by making four contributions. First, we develop a high-performance opensource neuron mesh generator and make publicly available over a thousand realistic cellular meshes.The neuron mesh generator, swc2mesh, can automatically and robustly convert valuable neuron tracing data into realistic neuron meshes. We have carefully designed the generator to maintain a good balance between mesh quality and size. A neuron mesh database, NeuronSet, which contains 1213 simulation-ready cell meshes and their neuroanatomical measurements, was built using the mesh generator. These meshes served as the basis for further research. Second, we increased the computational efficiency of the numerical matrix formalism method by accelerating the eigendecomposition algorithm and exploiting GPU computing. The speed was increased tenfold. With similar accuracy, the optimized numerical matrix formalism is 20 times faster than the FEM method and 65 times faster than a GPU-based Monte Carlo method. By performing simulations on realistic neuron meshes, we investigated the effect of water exchange between somas and neurites, and the relationship between soma size and signals. We then implemented a new simulation method that provides a Fourier-like representation of the dMRI signals. This method was derived theoretically and implemented numerically. We validated the convergence of the method and showed that the error behavior is consistent with our error analysis. Finally, we propose a simulation-driven supervised learning framework to estimate brain microstructure using diffusion MRI. By exploiting the powerful modeling and computational capabilities that are mentioned above, we have built a synthetic database containing the dMRI signals and microstructure parameters of 1.4 million artificial brain voxels. We have shown that this database can help approximate the underlying mappings of the dMRI signals to volume and surface fractions using artificial neural networks
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Nguyen, Van Dang. "A finite elements method to solve the Bloch-Torrey equation applied to diffusion magnetic resonance imaging of biological tissues." Palaiseau, Ecole polytechnique, 2014. http://pastel.archives-ouvertes.fr/docs/00/95/77/50/PDF/thesis_Dang.pdf.
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L'imagerie de résonance magnétique de diffusion (IRMD) est une technique d'imagerie non-invasive qui donne l'accès aux caractéristiques de diffusion de l'eau dans des tissus biologiques, notamment, dans le cerveau. Les restrictions que la structure cellulaire microscopique impose à la diffusion des molécules d'eau, sont agrégées statistiquement dans un mesurable signal d'IRMD macroscopique. L'inférence de la structure microscopique du tissu à partir du signal d'IRMD permet de détecter des régions pathologiques et d'observer les propriétés fonctionnelles du cerveau. A cet effet, il est important de mieux comprendre la relation entre la microstructure du tissu et le signal d'IRMD ce qui nécessite des nouvelles outils numériques capable de faire les calculs dans des géométries complexes modèles des tissus. Nous proposons une telle méthode numérique basée sur les éléments finis linéaires ce qui permet de décrire précisément des géométries complexes. La discrétisation par des éléments finis est couplée à la méthode adaptative des pas de temps de Runge-Kutta Chebyshev. Cette méthode qui assure la convergence du second ordre à la fois en temps et en espace, est implémentée sous la plateforme FeniCS C++. Nous utilisons aussi le générateur de maillage Salome pour travailler de manière efficace avec des géométries périodiques à plusieurs compartiments. Nous considérons quatre applications de la méthode pour étudier la diffusion dans des modèles à plusieurs compartiments. Dans la première application, nous étudions le comportement au temps long et démontrons la convergence d'un coefficient de diffusion apparent vers un tenseur de diffusion effectif obtenu par l'homogénéisation. La deuxième application vise à vérifier numériquement qu'un modèle à deux compartiments permet d'approximer le modèle à trois compartiments dans lequel le compartiment cellulaire et le compartiment extra-cellulaire sont complétés par un compartiment membranaire. La troisième application consiste à valider le modèle de Karger du signal d'IMRD macroscopique qui prend en compte l'échange entre compartiments. La dernière application se focalise sur le signal d'IMRD issu des neurones isoles. Nous proposons un modèle efficace unidimensionnel pour calculer le signal d'IRMD de manière précise dans un réseau des neurites de rayons variés. Nous testons la validité d'une expression semi-analytique du signal d'IRMD issu des réseaux de neuritesDiffusion magnetic resonance imaging (dMRI) is a non-invasive imaging technique that gives a measure of the diffusion characteristics of water in biological tissues, notably, in the brain. The hindrances that the microscopic cellular structure poses to water diffusion are statistically aggregated into the measurable macroscopic dMRI signal. Inferring the microscopic structure of the tissue from the dMRI signal allows one to detect pathological regions and to monitor functional properties of the brain. For this purpose, one needs a clearer understanding of the relation between the tissue microstructure and the dMRI signal. This requires novel numerical tools capable of simulating the dMRI signal arising from complex microscopic geometrical models of tissues. We propose such a numerical method based on linear finite elements that allows for a more accurate description of complex geometries. The finite elements discretization is coupled to the adaptive Runge-Kutta Chebyshev time stepping method. This method, which leads to the second order convergence in both time and space, is implemented on FeniCS C++ platform. We also use the mesh generator Salome to work efficiently with multiple-compartment and periodic geometries. Four applications of the method for studying the dMRI signal inside multi-compartment models are considered. In the first application, we investigate the long-time asymptotic behavior of the dMRI signal and show the convergence of the apparent diffusion coefficient to the effective diffusion tensor computed by homogenization. The second application aims to numerically verify that a two-compartment model of cells accurately approximates the three-compartment model, in which the interior cellular compartment and the extracellular space are separated by a finite thickness membrane compartment. The third application consists in validating the macroscopic Karger model of dMRI signals that takes into account compartmental exchange. The last application focuses on the dMRI signal arising from isolated neurons. We propose an efficient one-dimensional model for accurately computing the dMRI signal inside neurite networks in which the neurites may have different radii. We also test the validity of a semi-analytical expression for the dMRI signal arising from neurite networks
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Yu, Qiang. "Numerical simulation of anomalous diffusion with application to medical imaging." Thesis, Queensland University of Technology, 2013. https://eprints.qut.edu.au/62068/1/Qiang_Yu_Thesis.pdf.
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The first objective of this project is to develop new efficient numerical methods and supporting error and convergence analysis for solving fractional partial differential equations to study anomalous diffusion in biological tissue such as the human brain. The second objective is to develop a new efficient fractional differential-based approach for texture enhancement in image processing. The results of the thesis highlight that the fractional order analysis captured important features of nuclear magnetic resonance (NMR) relaxation and can be used to improve the quality of medical imaging.
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Qin, Shanlin. "Fractional order models: Numerical simulation and application to medical imaging." Thesis, Queensland University of Technology, 2017. https://eprints.qut.edu.au/115108/1/115108_9066888_shanlin_qin_thesis.pdf.
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This thesis is primarily concerned with developing new models and numerical methods based on the fractional generalisation of the Bloch and Bloch-Torrey equations to account for anomalous MRI signal attenuation. The two main contributions of the research are to investigate the anomalous evolution of MRI signals via the fractionalised Bloch equations, and to develop new effective numerical methods with supporting analysis to solve the time-space fractional Bloch-Torrey equations.
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Wales, David H. "Symmetry solutions for variations of the Torrey-Bloch equation." Thesis, 2018. http://hdl.handle.net/1959.7/uws:52075.
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A nuclear magnetic resonance signal is generated when a net magnetic moment precesses in the presence of a detector coil. The Torrey-Bloch equation models the change in net magnetic moment over time, while accounting for self-diffusion of nuclear spins. However, more general forms make the Torrey-Bloch equation difficult to solve analytically. Symmetry methods provide a unified approach to solving differential equations, which can remove some of the guesswork, and sometimes yield new analytical solutions. In this thesis, classical symmetry solutions are investigated for variations of the Torrey-Bloch equation involving relaxation, linear gradient, and Fokker-Planck terms.
Book chapters on the topic "Bloch-Torrey equation"
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Dela Haije, T. C. J., A. Fuster, and L. M. J. Florack. "Finslerian Diffusion and the Bloch–Torrey Equation." In Visualization and Processing of Higher Order Descriptors for Multi-Valued Data, 21–35. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-15090-1_2.
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Grebenkov, Denis, Bernard Helffer, and Nicolas Moutal. "On the spectral properties of the Bloch–Torrey equation in infinite periodically perforated domains." In Partial Differential Equations, Spectral Theory, and Mathematical Physics, 177–96. Zuerich, Switzerland: European Mathematical Society Publishing House, 2021. http://dx.doi.org/10.4171/ecr/18-1/10.
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Beltrachini, Leandro, Zeike A. Taylor, and Alejandro F. Frangi. "An Efficient Finite Element Solution of the Generalised Bloch-Torrey Equation for Arbitrary Domains." In Computational Diffusion MRI, 3–14. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-28588-7_1.
Full textConference papers on the topic "Bloch-Torrey equation"
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Yu, Q., F. Liu, I. Turner, and K. Burrage. "Analytical and Numerical Solutions of the Space and Time Fractional Bloch-Torrey Equation." In ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/detc2011-47613.
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Fractional order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brownian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As MRI is applied with increasing temporal and spatial resolution, the spin dynamics are being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments where processes are often anisotropic. Anomalous diffusion in the human brain using fractional order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (see R.L. Magin et at, J. Magnetic Resonance, 190 (2008) 255–270). However effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE, and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.
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Beltrachini, L., Z. A. Taylor, and A. F. Frangi. "A Parametrical Finite Element Formulation of the Bloch-Torrey Equation for NMR Applications." In University of Sheffield Engineering Symposium. USES, 2015. http://dx.doi.org/10.15445/01012014.120.
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Magin, Richard L., Viktor Kovacs, and Andrzej Hanyga. "Comparison of analytical and numerical models for anomalous diffusion in the Bloch-Torrey equation." In 2014 International Conference on Fractional Differentiation and its Applications (ICFDA). IEEE, 2014. http://dx.doi.org/10.1109/icfda.2014.6967358.
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Magin, Richard L., and Dumitru Baleanu. "NMR Measurements of Anomalous Diffusion Reflect Fractional Order Dynamics." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34224.
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Diffusion weighted MRI is often used to detect and stage neurodegenerative, malignant and ischemic diseases. The correlation between developing pathology and localized diffusion measurements relies on the design of selective phase encoding pulses that alter the intensity of the acquired signal according to biophysical models of spin diffusion in tissue. The most common approach utilizes a bipolar or Stejskal-Tanner gradient pulse sequence to encode the apparent diffusion coefficient as an exponential, multi-exponential or stretched exponential function of experimentally-controlled parameters. Several studies have investigated the ability of the stretched exponential to provide an improved fit to diffusion-weighted imaging data. These results were recently analyzed by establishing a direct link between water diffusion, as measured using NMR, and fractal structural models of tissues. In this paper we suggest an alternative description for stretched exponential behavior that reflects fractional order dynamics of a generalized Bloch-Torrey equation in either space or time. Such generalizations are the basis for similar anomalous diffusion phenomena observed in optical spectroscopy, polymer dynamics and electrochemistry. Here we demonstrate a correspondence between the detected NMR signal and anomalous diffusional dynamics of water through the Riesz fractional order space derivative and the Caputo form of the fractional order Riemann-Liouville time derivative.
Reports on the topic "Bloch-Torrey equation"
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Rohmer, Damien, and Grant T. Gullberg. A Bloch-Torrey Equation for Diffusion in a Deforming Media. Office of Scientific and Technical Information (OSTI), December 2006. http://dx.doi.org/10.2172/919380.
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