Time-domain NMR, in one and higher dimensionalities, makes routine use of inversion algorithms to generate results called \T2-distributions' or joint distributions in two (or higher) dimensions of other NMR parameters, T1, diffusivity D, pore size a, etc. These are frequently referred to as \Inverse Laplace Transforms' although the standard inversion of the Laplace Transform long-established in many textbooks of mathematical physics does not perform (and cannot perform) the calculation of such distributions. The operations performed in the estimation of a \T2-distribution' are the estimation of solutions to a Fredholm Integral Equation (of the First Kind), a different and more general object whose discretization results in a standard problem in linear algebra, albeit suffering from well-known problems of ill-conditioning and computational limits for large problem sizes. The Fredholm Integral Equation is not restricted to exponential kernels; the same solution algorithms can be used with kernels of completely different form. On the other hand, (true) Inverse Laplace Transforms, treated analytically, can be of real utility in solving the diffusion problems highly relevant in the subject of NMR in porous media.

O presente trabalho teve como objetivo a implementação de técnicas de Ressonância Magnética Nuclear (RMN) para o estudo de meios porosos em geral, com potenciais aplicações no estudo de rochas, ossos, polímeros e materiais cerâmicos porosos. Devido à heterogeneidade e à complexidade desses meios, torna-se importante correlacionar variados parâmetros físicos que contenham informações tanto sobre a dinâmica molecular quanto sobre a estrutura físico-química dos poros e dos fluidos que os permeiam. Utilizando técnicas de RMN, é possível investigar esses parâmetros através da medida dos coeficientes de difusão e dos tempos de relaxação longitudinal e transversal dos fluidos presentes nos poros, que apresentam diferentes estados dinâmicos, variando desde um líquido isotrópico até um líquido com alta viscosidade ou organizado devido à sua interação com a superfície dos poros ou com partículas presentes nos fluidos. Para isso, foram empregadas técnicas de RMN baseadas especialmente em medidas de tempos de relaxação transversal que, quando adequadamente combinadas, permitem a construção de mapas bidimensionais que correlacionam propriedades da dinâmica molecular de fluidos saturantes nessas amostras em instantes diferentes do experimento, provendo informações sobre as dimensões e escalas temporais envolvidas na dinâmica dos fluidos saturantes dos meios porosos. Esses mapas foram obtidos a partir de dados experimentais obtidos com rochas petrolíferas saturadas com água via algoritmos de inversão da transformada de Laplace bidimensional. Este trabalho foi realizado utilizando a infraestrutura existente nos laboratórios de RMN do IFSC, sendo grande parte do hardware e software desenvolvidos durante o projeto de doutorado. Foi desenvolvido um programa para processamento da transformada inversa de Laplace de dados unidimensionais, sendo este utilizado para o estudo de propriedades de adesão de pastilhas de gesso quando submetidas a diferentes pressões. Além disso, foram desenvolvidos modelos computacionais para simular essas propriedades relacionadas aos experimentos utilizados para o estudo dos meios porosos. Parte deste trabalho contou com a realização de dos estudos na Victoria University of Wellington em colaboração com o professor Paul T. Callaghan através do Programa de Doutorado e Estágio no Exterior (PDEE) da CAPES. / This study aimed to implement Nuclear Magnetic Resonance (NMR) for the study of porous media in general, with potential applications in the study of rocks, bones, porous polymers and ceramics. Due to the heterogeneity and complexity of these media, it becomes important to correlate various physical parameters that contain information about both the molecular dynamics on the structure and physical chemistry of pore fluid and the permeate. Using NMR techniques, it is possible to investigate these parameters by measuring diffusion coefficients, transverse and longitudinal relaxation times of fluids within the pores, which have different dynamical states, ranging from an isotropic liquid to a liquid with high viscosity or organized due to its interaction with the surface of the pores or particles suspended in fluid. For this, we employed NMR techniques based especially on measurements of transverse relaxation times, which when properly combined, allow the construction of two-dimensional maps that correlate properties of the molecular dynamics of fluids saturating these samples at different times of the experiment, providing information on dimensions and time scales involved in the dynamics of fluids saturating the porous media. These maps were obtained from experimental data on water-saturated oil rocks via inversion of two-dimensional Laplace transform. This work was performed using the existing infrastructure in the IFSC NMR laboratories and much of the hardware and software was developed during the doctoral project. We developed a program for processing the one-dimensional inverse Laplace transform data, which was used for studying the adhesion properties of gypsum pellets when subjected to different pressures. Additionally, we developed computational models to simulate physical-chemical properties related to the experiments used for the studies of fluids within porous media. Part of this work included studies performed in Victoria University of Wellington in collaboration with Professor Paul T. Callaghan through the Programa de Doutorado e Estágio no Exterior (PDEE) from CAPES.

O presente trabalho teve como objetivo a implementação de técnicas de Ressonância Magnética Nuclear (RMN) para o estudo de meios porosos em geral, com potenciais aplicações no estudo de rochas, ossos, polímeros e materiais cerâmicos porosos. Devido à heterogeneidade e à complexidade desses meios, torna-se importante correlacionar variados parâmetros físicos que contenham informações tanto sobre a dinâmica molecular quanto sobre a estrutura físico-química dos poros e dos fluidos que os permeiam. Utilizando técnicas de RMN, é possível investigar esses parâmetros através da medida dos coeficientes de difusão e dos tempos de relaxação longitudinal e transversal dos fluidos presentes nos poros, que apresentam diferentes estados dinâmicos, variando desde um líquido isotrópico até um líquido com alta viscosidade ou organizado devido à sua interação com a superfície dos poros ou com partículas presentes nos fluidos. Para isso, foram empregadas técnicas de RMN baseadas especialmente em medidas de tempos de relaxação transversal que, quando adequadamente combinadas, permitem a construção de mapas bidimensionais que correlacionam propriedades da dinâmica molecular de fluidos saturantes nessas amostras em instantes diferentes do experimento, provendo informações sobre as dimensões e escalas temporais envolvidas na dinâmica dos fluidos saturantes dos meios porosos. Esses mapas foram obtidos a partir de dados experimentais obtidos com rochas petrolíferas saturadas com água via algoritmos de inversão da transformada de Laplace bidimensional. Este trabalho foi realizado utilizando a infraestrutura existente nos laboratórios de RMN do IFSC, sendo grande parte do hardware e software desenvolvidos durante o projeto de doutorado. Foi desenvolvido um programa para processamento da transformada inversa de Laplace de dados unidimensionais, sendo este utilizado para o estudo de propriedades de adesão de pastilhas de gesso quando submetidas a diferentes pressões. Além disso, foram desenvolvidos modelos computacionais para simular essas propriedades relacionadas aos experimentos utilizados para o estudo dos meios porosos. Parte deste trabalho contou com a realização de dos estudos na Victoria University of Wellington em colaboração com o professor Paul T. Callaghan através do Programa de Doutorado e Estágio no Exterior (PDEE) da CAPES. / This study aimed to implement Nuclear Magnetic Resonance (NMR) for the study of porous media in general, with potential applications in the study of rocks, bones, porous polymers and ceramics. Due to the heterogeneity and complexity of these media, it becomes important to correlate various physical parameters that contain information about both the molecular dynamics on the structure and physical chemistry of pore fluid and the permeate. Using NMR techniques, it is possible to investigate these parameters by measuring diffusion coefficients, transverse and longitudinal relaxation times of fluids within the pores, which have different dynamical states, ranging from an isotropic liquid to a liquid with high viscosity or organized due to its interaction with the surface of the pores or particles suspended in fluid. For this, we employed NMR techniques based especially on measurements of transverse relaxation times, which when properly combined, allow the construction of two-dimensional maps that correlate properties of the molecular dynamics of fluids saturating these samples at different times of the experiment, providing information on dimensions and time scales involved in the dynamics of fluids saturating the porous media. These maps were obtained from experimental data on water-saturated oil rocks via inversion of two-dimensional Laplace transform. This work was performed using the existing infrastructure in the IFSC NMR laboratories and much of the hardware and software was developed during the doctoral project. We developed a program for processing the one-dimensional inverse Laplace transform data, which was used for studying the adhesion properties of gypsum pellets when subjected to different pressures. Additionally, we developed computational models to simulate physical-chemical properties related to the experiments used for the studies of fluids within porous media. Part of this work included studies performed in Victoria University of Wellington in collaboration with Professor Paul T. Callaghan through the Programa de Doutorado e Estágio no Exterior (PDEE) from CAPES.

Knowledge of the indoor propagation of sound has many important applications including acoustic prediction in homes, office buildings, stores, and schools, and the design of concert halls, auditoriums, classrooms, and factories. At low frequencies, interior acoustics are analyzed with the wave equation, but significant computational expense imposes an upper frequency limit. Thus, energy methods are often sought for high frequency analysis. However, conventional energy methods are significantly limited by vast simplifications or computational costs. Therefore, new improvements are still being sought. The basis of this dissertation is a recently developed mathematical model for interior acoustics known as the acoustic diffusion model. The model extends statistical methods in high frequency acoustics to predict the spatial distribution of acoustic energy in the volume over time as a diffusion process. Previously, solutions to the acoustic diffusion model have been limited to one dimensional (1-D) analytical solutions and to the use of the finite element method (FEM). This dissertation focuses on a new, efficient method for solving the acoustic diffusion model based on a boundary element method (BEM) solution using the Laplace transform. First, a Laplace domain solution to the diffusion model is obtained using the BEM. Then, a numerical inverse Laplace transform is used to efficiently compute the time domain response. The diffusion boundary element-Laplace transform solution (BE-LTS) is validated through comparisons with Sabine theory, ray tracing, and a diffusion FEM solution. All methods demonstrate excellent agreement for three increasingly complex acoustic volumes and the computational efficiency of the BE-LTS is exposed. Structural coupling is then incorporated in the diffusion BE-LTS using two methods. First, a simple transmission coefficient separating two acoustic volumes is implemented. Second, a structural power flow model represents the coupling partition separating acoustic volumes. The validation of these methods is successfully performed in an example through comparisons with statistical theory, a diffusion FEM solution, ray tracing, and experimental data. Finally, the diffusion model and the BE-LTS are shown to possess capabilities beyond that of room acoustics. The acoustic transmission through a heat exchanger, acoustic foam, and mufflers is successfully modeled using the diffusion BE-LTS and compared to experimental data. / Ph. D.

Na condução por saltos dispersiva, os portadores de carga são caracterizados por tempos de residência não exponenciais. Como uma conseqüência, efeitos hereditários aparecem e o problema deve ser conduzido no espaçõ de Laplace. Distribuições dielétricas conhecidas e algumas outras foram usadas como possíveis funções capazes de descrever o meio desordenado. Um estudo cuidadoso de seus espectros de freqüência foi realizado. O método de inversão de Widder truncado foi desenvolvido para possibilitar a volta do espaço de Laplace para o do tempo real. Dois tipos de problemas práticos foram abordados: condução dispersiva na qual o campo elétrico das cargas móveis pode ser desprezado e o caso de carga espacial para o qual essa aproximação não pode ser feita. Duas configurações de interesse foram escolhidas: decaimento do potencial de superfície de amostras carregadas por descarga corona e a subida do potencial no carregamento por corrente constante. / Dispersive conductive process are characterized by non-exponential residence times of the hopping carrier. As a consequence, hereditary effects appear and the problem must be conducted in the Laplace space. A bunch of known dielectric distributions functions and others were used as possible functions describing the disordered medium. A careful study of their frequency spectrum was carried out. The truncate Widder inversion was developed in order to allow returning from the Lapace to the real time space. Two kind of practical problems were studied: dispersive conduction in which the electric field of the mobile charge may be neglected and the space charge ones when that approximation cannot be made. Two configurations of interest were selected: the corona discharge potential decay and the constant current potentinal build-up.

Na condução por saltos dispersiva, os portadores de carga são caracterizados por tempos de residência não exponenciais. Como uma conseqüência, efeitos hereditários aparecem e o problema deve ser conduzido no espaçõ de Laplace. Distribuições dielétricas conhecidas e algumas outras foram usadas como possíveis funções capazes de descrever o meio desordenado. Um estudo cuidadoso de seus espectros de freqüência foi realizado. O método de inversão de Widder truncado foi desenvolvido para possibilitar a volta do espaço de Laplace para o do tempo real. Dois tipos de problemas práticos foram abordados: condução dispersiva na qual o campo elétrico das cargas móveis pode ser desprezado e o caso de carga espacial para o qual essa aproximação não pode ser feita. Duas configurações de interesse foram escolhidas: decaimento do potencial de superfície de amostras carregadas por descarga corona e a subida do potencial no carregamento por corrente constante. / Dispersive conductive process are characterized by non-exponential residence times of the hopping carrier. As a consequence, hereditary effects appear and the problem must be conducted in the Laplace space. A bunch of known dielectric distributions functions and others were used as possible functions describing the disordered medium. A careful study of their frequency spectrum was carried out. The truncate Widder inversion was developed in order to allow returning from the Lapace to the real time space. Two kind of practical problems were studied: dispersive conduction in which the electric field of the mobile charge may be neglected and the space charge ones when that approximation cannot be made. Two configurations of interest were selected: the corona discharge potential decay and the constant current potentinal build-up.

While great strides have been made in treating many classes of human disease, the late-onset neurodegenerative diseases continue to elude modern medicine. These diseases, which include Alzheimer's disease (AD), Parkinson's disease (PD), Huntington's disease (HD), the transmissible spongiform encephalopathies (TSEs), and amyotrophic lateral sclerosis (ALS), involve accumulation of insoluble aggregates of one or more causative proteins, leading to progressive loss of central nervous system neurons, progressively worsening neurological symptoms, and eventual patient death. All of these diseases are currently incurable and fatal. In the case of ALS, progressive death of upper and lower motor neurons leads to full-body paralysis, respiratory difficulty, and patient death. Of the subset of ALS cases showing familial inheritance, approximately 20% are caused by mutations in the SOD1 gene, encoding the Cu, Zn superoxide dismutase (SOD1). These mutations do not have the common property of impairing SOD1's normal function as a free radical scavenger. Instead, they are thought to increase the protein's likelihood of misfolding and aggregating via a poorly-understood aggregation cascade. It is believed that species populated along the misfolding and aggregation pathway may prove to be good targets for therapies designed to block accumulation of downstream toxic species, or to prevent aberrant protein-protein interactions responsible for neurotoxicity. In this thesis, several new techniques are developed to enable detailed elucidation of the SOD1 unfolding and misfolding pathways. Time-resolved measurements collected during SOD1 unfolding or misfolding of release of bound Cu and Zn, of changes in intrinsic fluorescence, of exposure of hydrophobic surface area, and of alterations in the chemical environment of histidine residues, are presented. A new mathematical analysis technique named the Analytical Laplace Inversion Algorithm is developed for rapid extraction of mechanistic information from these time-resolved signals. These tools are applied to the construction of the most detailed models to date of the unfolding and misfolding mechanisms of WT and ALS-causing mutant SOD1. The models presented identify several well-populated unfolding and misfolding intermediates that could serve as good targets for therapies designed to address the fundamental molecular mechanisms underlying SOD1-associated ALS, and to treat what is currently a devastating and incurable disease.

While great strides have been made in treating many classes of human disease, the late-onset neurodegenerative diseases continue to elude modern medicine. These diseases, which include Alzheimer's disease (AD), Parkinson's disease (PD), Huntington's disease (HD), the transmissible spongiform encephalopathies (TSEs), and amyotrophic lateral sclerosis (ALS), involve accumulation of insoluble aggregates of one or more causative proteins, leading to progressive loss of central nervous system neurons, progressively worsening neurological symptoms, and eventual patient death. All of these diseases are currently incurable and fatal. In the case of ALS, progressive death of upper and lower motor neurons leads to full-body paralysis, respiratory difficulty, and patient death. Of the subset of ALS cases showing familial inheritance, approximately 20% are caused by mutations in the SOD1 gene, encoding the Cu, Zn superoxide dismutase (SOD1). These mutations do not have the common property of impairing SOD1's normal function as a free radical scavenger. Instead, they are thought to increase the protein's likelihood of misfolding and aggregating via a poorly-understood aggregation cascade. It is believed that species populated along the misfolding and aggregation pathway may prove to be good targets for therapies designed to block accumulation of downstream toxic species, or to prevent aberrant protein-protein interactions responsible for neurotoxicity. In this thesis, several new techniques are developed to enable detailed elucidation of the SOD1 unfolding and misfolding pathways. Time-resolved measurements collected during SOD1 unfolding or misfolding of release of bound Cu and Zn, of changes in intrinsic fluorescence, of exposure of hydrophobic surface area, and of alterations in the chemical environment of histidine residues, are presented. A new mathematical analysis technique named the Analytical Laplace Inversion Algorithm is developed for rapid extraction of mechanistic information from these time-resolved signals. These tools are applied to the construction of the most detailed models to date of the unfolding and misfolding mechanisms of WT and ALS-causing mutant SOD1. The models presented identify several well-populated unfolding and misfolding intermediates that could serve as good targets for therapies designed to address the fundamental molecular mechanisms underlying SOD1-associated ALS, and to treat what is currently a devastating and incurable disease.

A aplicação de uma sequência de pulsos com tempo de repetição muito menor que os tempos de relaxação Tp << T2; T1, faz com que a magnetização atinja um estado estacionário descrito por H.Y. Carr como Estado Estacionário em Precessão Livre, Steady State Free Precession (SSFP). Nessa condição, o sinal é composto pela complexa sobreposição das componentes FID e eco. Sequências tipo SSFP são utilizadas na aquisição rápida de sinais, resultando em uma boa razão sinal ruído (s/r) em curto intervalo de tempo, porém introduzem fortes anomalias de fase e amplitude devido a complexa interação das componentes que formam o estado estacionário. Neste trabalho, desenvolvemos sequências de pulsos tipo SSFP para RMN em alta e baixa resolução com alternância e incremento de fase. Em alta resolução desenvolvemos as sequências SSFPdx e SSFPdxdt com incremento de fase linear e quadrático respectivamente. Os resultados mostram que espectros de núcleos com baixa sensibilidade podem ser obtidos com mesma razão s/r em menor tempo experimental e as sequências desenvolvidas removem as anomalias espectrais. Em baixa resolução, os resultados mostram que a introdução de alternâncias de fase na Continuous Wave Free Precession (CWFP) possibilita a remoção da dependência da sequência com o offset de frequência e com o tempo entre pulsos. Além disso, mostramos que a sequência CP-CWFPx-x com ângulo de refocalização pequeno (5&deg; a 10&deg;) possibilita a estimativa rápida do tempos de relaxação longitudinal. Apresentamos também resultados dos estudos e desenvolvidos no estágio de pesquisa no exterior, onde as sequências de pulsos no estado estacionário &ndash; DECPMG e Split 180&deg; &ndash; foram estudas numericamente e implementadas nos sistemas magnéticos compactos: mini-Halbach e MOUSE-NMR. Por fim, são apresentados resultados com os métodos de processamento de dados Krylov Basis Diagonalization Method (KBDM) e a Transformada Inversa de Laplace aplicados na análise de sinais SSFP. Resultados mostram que KBDM é uma ferramenta útil no processamento de dados em alta e baixa resolução, tanto na obtenção de espectros como na determinação da distribuição dos tempos de relaxação. / The application of a pulse sequence with repetition time much smaller than the relaxation times, Tp << T2; T1, causes the magnetization to reach a steady state, described by H. Y. Carr as a Steady State Free Precession (SSFP). In this condition, the signal is composed of the complex overlapping of the FID and eco components. SSFP type sequences are used in fast acquisition of NMR signals, resulting in a good signal to noise ratio (s/r) in a short time interval, however, they introduce phase and amplitude anomalies due to the complex interaction between the components of the steady state. In this work, we develop SSFP type pulse sequences for NMR in high and low resolution, with alternation and increment of phase. In high resolution, we develop SSFPdx and SSFPdxdt sequences, with linear and quadratic phase increment respectively. Results show that the low sensitivity nuclei spectra can be obtained with the same s/r ratio in smaller experimental time, about an order of magnitude, and the developed sequences can remove the spectral anomalies. In low resolution, the results show that the introduction of a phase alternation in the Continuous Wave Free Precession (CWFP) allows the elimination of the dependence of the sequence with the offset frequency and the time between pulses. Besides, we show that the CP-CWFPx-x sequence with a small refocalization angle (5° to 10°) allows the fast estimative of the longitudinal relaxation time in a single experiment. The results of the studies conducted during an international research internship are also presented. Steady state pulse sequences &ndash; DECPMG and Split 180° &ndash; were studied and implemented in compact magnetic systems: mini-Halbach and MOUSE-NMR. Finally, the results of the application of the Krylov Basis Diagonalization Method (KBDM) and the Inverse Laplace Transform for the analysis of SSFP signals are presented. The results show that KBDM is a useful tool in data processing for low and high resolution, both for obtaining spectra and determining the relaxation times distribution.

A aplicação de uma sequência de pulsos com tempo de repetição muito menor que os tempos de relaxação Tp << T2; T1, faz com que a magnetização atinja um estado estacionário descrito por H.Y. Carr como Estado Estacionário em Precessão Livre, Steady State Free Precession (SSFP). Nessa condição, o sinal é composto pela complexa sobreposição das componentes FID e eco. Sequências tipo SSFP são utilizadas na aquisição rápida de sinais, resultando em uma boa razão sinal ruído (s/r) em curto intervalo de tempo, porém introduzem fortes anomalias de fase e amplitude devido a complexa interação das componentes que formam o estado estacionário. Neste trabalho, desenvolvemos sequências de pulsos tipo SSFP para RMN em alta e baixa resolução com alternância e incremento de fase. Em alta resolução desenvolvemos as sequências SSFPdx e SSFPdxdt com incremento de fase linear e quadrático respectivamente. Os resultados mostram que espectros de núcleos com baixa sensibilidade podem ser obtidos com mesma razão s/r em menor tempo experimental e as sequências desenvolvidas removem as anomalias espectrais. Em baixa resolução, os resultados mostram que a introdução de alternâncias de fase na Continuous Wave Free Precession (CWFP) possibilita a remoção da dependência da sequência com o offset de frequência e com o tempo entre pulsos. Além disso, mostramos que a sequência CP-CWFPx-x com ângulo de refocalização pequeno (5&deg; a 10&deg;) possibilita a estimativa rápida do tempos de relaxação longitudinal. Apresentamos também resultados dos estudos e desenvolvidos no estágio de pesquisa no exterior, onde as sequências de pulsos no estado estacionário &ndash; DECPMG e Split 180&deg; &ndash; foram estudas numericamente e implementadas nos sistemas magnéticos compactos: mini-Halbach e MOUSE-NMR. Por fim, são apresentados resultados com os métodos de processamento de dados Krylov Basis Diagonalization Method (KBDM) e a Transformada Inversa de Laplace aplicados na análise de sinais SSFP. Resultados mostram que KBDM é uma ferramenta útil no processamento de dados em alta e baixa resolução, tanto na obtenção de espectros como na determinação da distribuição dos tempos de relaxação. / The application of a pulse sequence with repetition time much smaller than the relaxation times, Tp << T2; T1, causes the magnetization to reach a steady state, described by H. Y. Carr as a Steady State Free Precession (SSFP). In this condition, the signal is composed of the complex overlapping of the FID and eco components. SSFP type sequences are used in fast acquisition of NMR signals, resulting in a good signal to noise ratio (s/r) in a short time interval, however, they introduce phase and amplitude anomalies due to the complex interaction between the components of the steady state. In this work, we develop SSFP type pulse sequences for NMR in high and low resolution, with alternation and increment of phase. In high resolution, we develop SSFPdx and SSFPdxdt sequences, with linear and quadratic phase increment respectively. Results show that the low sensitivity nuclei spectra can be obtained with the same s/r ratio in smaller experimental time, about an order of magnitude, and the developed sequences can remove the spectral anomalies. In low resolution, the results show that the introduction of a phase alternation in the Continuous Wave Free Precession (CWFP) allows the elimination of the dependence of the sequence with the offset frequency and the time between pulses. Besides, we show that the CP-CWFPx-x sequence with a small refocalization angle (5° to 10°) allows the fast estimative of the longitudinal relaxation time in a single experiment. The results of the studies conducted during an international research internship are also presented. Steady state pulse sequences &ndash; DECPMG and Split 180° &ndash; were studied and implemented in compact magnetic systems: mini-Halbach and MOUSE-NMR. Finally, the results of the application of the Krylov Basis Diagonalization Method (KBDM) and the Inverse Laplace Transform for the analysis of SSFP signals are presented. The results show that KBDM is a useful tool in data processing for low and high resolution, both for obtaining spectra and determining the relaxation times distribution.