O Transtorno de Humor Bipolar (THB) acomete até 3% dos adultos e os filhos desses pacientes constituem uma população em risco para o desenvolvimento de transtornos psiquiátricos. No entanto, faltam marcadores que permitam a identificação precoce dos indivíduos que apresentam maior vulnerabilidade para o desenvolvimento de psicopatologia. Estudos preliminares com Ressonância Magnética (RM) indicaram que alterações em substância branca estariam presentes não apenas em pacientes em episódio de alteração de humor, mas também em pacientes eutímicos e em seus familiares saudáveis, sugerindo que tais alterações poderiam constituir um endofenótipo potencial deste transtorno. A Imagem por Tensor de Difusão (Diffusion Tensor Magnetic Resonance Imaging - DT-MRI) é uma aquisição de RM que permite análise mais completa e detalhada das características da substância branca cerebral que a RM tradicional. A investigação de alterações na estrutura cerebral, particularmente de substância branca, de jovens portadores de THB não medicados e familiares saudáveis criteriosamente avaliados pode ajudar a elucidar a neurobiologia subjacente ao THB e, conseqüentemente, a identificar marcadores de vulnerabilidade ao transtorno. Objetivo: Avaliar se havia alterações em substância branca em crianças e adolescentes com THB e crianças e adolescentes filhos saudáveis de portadores de transtorno do humor bipolar quando comparados a controles saudáveis, utilizando a técnica de neuroimagem de DTMRI. Nossas hipóteses eram que jovens com THB e filhos de pacientes com THB, quando comparados a controles saudáveis, apresentariam (i) diminuição da fração de anisotropia (FA) e (ii) essas alterações seriam mais pronunciadas em crianças acometidas por THB do que em crianças saudáveis filhas de pacientes com THB. Métodos: Obtivemos imagens de DT-MRI de boa qualidade de 16 crianças e adolescentes com THB (média de idade ± D.P.= 12,7 ± 2,5 anos), 15 filhos saudáveis de pacientes com THB tipo I (média de idade ± D.P.= 13,5 ± 2,7 anos) e 15 controles saudáveis (média de idade ± D.P.= 13,5 ± 2,5 anos). Os diagnósticos foram formulados de acordo com os critérios do DSM-IV, usando as entrevistas Kiddie-SADS-PL (crianças) e Structured Clinical Interview for DSM-IV (adultos). A RM foi realizada em um scanner Philips 3,0 Tesla, com os seguintes parâmetros de aquisição: TR = 6106,0 ms, TE = 65,0ms, FOV = 224x224mm, espessura de corte = 2.0mm, sem gap, matriz de aquisição = 112x112 pixels e 3 médias, resultando em tamanho de voxel isotrópico = 2,0x2,0x2,0mm. As imagens de DTI foram pré-processadas com programas oriundos do FMRIB\'s software library (FSL), de acordo com o pipeline sugerido para o processamento de substância branca com Tract-Based Spatial Statistics (TBSS) e análise estatística com o programa Randomise (ambos integrantes do FSL). Resultados: Os grupos não diferiram em idade, gênero, grau de puberdade ou QI. Valores de FA de pacientes pediátricos com THB foram significativamente menores em relação aos de controles saudáveis (p < 0,05, corrigido para múltiplas comparações) em um cluster de 695 voxels no hemisfério direito que abrange a porção superiora da corona radiata e o corpo do corpo. Não houve diferença significativa entre pacientes com THB e filhos saudáveis de pacientes com THB, ou com filhos saudáveis de pacientes com THB e controles saudáveis. Discussão: Nossos dados corroboram a literatura de diminuição de FA em crianças e adolescentes com THB e avançam em mostrar esta alteração em pacientes não medicados. Mas nossos resultados não apoiam a hipótese de alterações em substância branca como endofenótipo de THB. Estudos de seguimento com amostras maiores e rigorosamente caracterizadas são necessários para se elucidar o papel das alterações em substância branca no THB. / Up to 3% of adults are affected with Bipolar Disorder (BD) and the offspring of these patients constitute a population at risk for the development of psychiatric disorders. Nevertheless, there are still no vulnerability markers to allow the early identification of those who are at greater risk of developing psychopathology among this population. Preliminary data indicate that white matter abnormalities may precede the disease onset and be present even in unaffected relatives - suggesting they could be further explored as an endophenotype for BD. Diffusion Tensor Magnetic Resonance Imaging (DTMRI) is an MRI acquisition that allows a thorough and detailed analysis of brain white matter characteristics. The investigation of white matter alterations in young, nonmedicated, BD patients and healthy relatives may help us understand the underlying neurobiology of BD. Objectives: evaluate white matter alterations in children at-risk and with BD using DT-MRI. Our hypothesis were that BD offspring, compared to healthy controls, would exhibit (i) reduced fractional anisotropy (FA) and (ii) these alterations would be more pronounced in children with BD than in those at-risk. Methods: We successfully scanned 16 BD patients (mean age ± S.D.= 12,7 ± 2,5 years) 15 healthy offspring with at least one parent with BD I diagnosis (mean age ± S.D.= 13,5 ± 2,7 years) and 15 healthy controls (mean age ± S.D.= 13,5 ± 2,5 years) with no history of psychiatric disorder in first-degree relatives. Psychiatric diagnosis were established according to the DSM-IV criteria, using the Kiddie-SADS-PL interview (children) and Structured Clinical Interview for DSM-IV (adults). The MRI was conducted at a 3.0 Tesla Philips scanner. Acquisition: the parameters were TR = 6106,0ms, TE = 65,0ms, FOV= 224x224mm, slice thickness = 2.0mm/no gap, matrix acquisition = 112x112 pixels, 3 averages, resulting in an isotropic voxel = 2,0x2,0x2,0 mm. DT-MRI images were preprocessed according to do FMRIB\'s software library\'s (FSL) pipeline for Tract-Based Spatial Statistics\' (TBSS) and Randomise analyses. Results: Groups did not differ in age, gender or pubertal status. Voxelwise analyses showed significant differences in FA values between BD patients and healthy controls (p < 0,05, FDR corrected for multiple comparisons) in a 695 voxels cluster comprising right corona radiata and corpus callosum . Discussion: This study was the first to evaluate a sample of non-medicated BD children and adolescents with DT-MRI and it corroborates extant literature data of lower FA in BD children compared to healthy controls. Nevertheless, our data do not support white matter alterations as an endophenotype for BD. More studies, with larger and well characterized samples are necessary to advance our understanding of the role of white matter alteration in BD.

Introdução: No campo do tratamento cirúrgico de Gliomas, tem se observado um interesse crescente no uso de novas tecnologias de suporte como métodos auxiliares na obtenção do equilíbrio entre radicalidade cirúrgica e preservação da funcionalidade cerebral. Na maior parte dos estudos, a localização tumoral tem sido pouco considerada e a real eficácia das tecnologias de suporte ainda está pouco investigada nos gliomas insulares. Objetivos: avaliar a eficácia da fluorescência intraoperatória com 5-ALA, monitorização neurofisiológica, neuronavegação e tractografia no grau de ressecção tumoral (GRT), escores de funcionalidade, sobrevida global e sobrevida livre de progressão em uma coorte retrospectiva de gliomas insulares. Métodos: revisamos todos os casos de tumores insulares operados no Departamento de Neurocirurgia da Universidade de Tübingen - Alemanha, entre maio de 2008 e novembro de 2013. O grau de ressecção foi avaliado por volumetria. Foram utilizados os testes de Mann Whitney, Qui-quadrado e funções de Kaplan Meier para análise do efeito de cada tecnologia nos desfechos primários e secundários. Resultados: 28 casos - 18 homens (64%) e 10 mulheres (36%); idade média 52,5 anos (12 - 59) - foram inclusos para análise. Gliomas de alto grau corresponderam a 20 casos (71%), com 8 lesões de baixo grau (29%). As tecnologias mais utilizadas foram monitorização neurofisiológica (64%) e neuronavegação (68%). 5-ALA foi a única modalidade associada a taxas de ressecção > 90% (p = 0,05). O uso de tractografia determinou melhora no KPS (50% vs 5%, p = 0,02). Houve associação positiva entre o uso de neuronavegação e sobrevida global (23 vs. 27,4 meses, p = 0,03), mas o uso de 5-ALA se associou a piora na sobrevida global (34,8 vs 21,1 meses, p = 0,01) e sobrevida livre de progressão (24,4 vs. 11,8 meses, p = 0,01). Conclusões: Considerando os gliomas insulares, o presente trabalho demonstra pioneiramente que o uso de 5-ALA tem papel na obtenção de maiores taxas de ressecção, ainda que este achado possa estar associado a piora nas sobrevidas global e livre de progressão. Tractografia e neuronavegação parecem desempenhar papel importante no tratamento dos gliomas insulares, na medida em que determinaram melhor sobrevida global e funcionalidade, respectivamente. Estudos prospectivos com uma amostra mais proeminente e análise multivariada permitirão a avaliação do benefício real destas tecnologias de suporte no tratamento dos gliomas insulares / Introduction: In the field of Glioma surgery, there has been an increasing interest in the use of assistive technologies to overcome the difficulty of preserving brain function while improving surgical radicality. In most reports, tumor localization has seldom been considered a variable and the role of intraoperative adjuncts is yet to be determined for gliomas of the insula. Objectives: to evaluate the efficacy of fluorescence-guided resection with 5-ALA, intraoperative neurophysiological monitoring (IOM), neuronavigation, and tractography in the Extent of Resection (EOR), functionality scores, overall survival (OS) and progression-free survival (PFS) in a retrospective cohort of insular gliomas. Methods: we reviewed all cases of insular tumors operated on at the Department of Neurosurgery, University Hospital of Tübingen - Germany, between May 2008 and November 2013. EOR was determined by volumetric analysis. Mann Whitney, Chi-square and Kaplan Meier functions were used for assessment of each technology\'s effect on primary and secondary outcomes. Results: 28 cases (18 men (64%) and 10 women (36%); median age at diagnosis: 52.5 years, range 12 - 59) were considered eligible for analysis. High grade and low grade gliomas accounted for 20 (71%) and 8 (29%) cases, respectively. The most used technologies were IOM (64%) and Neuronavigation (68%). 5-ALA was the only technique associated with EOR > 90% (p = 0.05). Tractography determined improvement in the Karnofsky Performance Scale (50% vs 5% cases improved, p = 0.02). There was a positive association between the use of neuronavigation and overall survival (23 vs. 27.4 months, p = 0.03), but the use of 5-ALA was associated with shorter OS (34.8 vs. 21.1 months, p = 0.01) and PFS (24.4 vs. 11.8, p = 0.01). Conclusions: we demonstrate for the first time that for insular gliomas 5-ALA plays a role in achieving higher EOR, although this technology was associated with poor OS and PFS; Also tractography and neuronavigation can be of great importance in the treatment of insular gliomas as they determined better functionality and overall survival in this study, respectively. Prospective studies with a more prominent sample and proper multivariate analysis will help determine the real benefit of these adjuncts in the setting of insular gliomas

Ossos trabeculares possuem uma microestrutura porosa e podem ser modelados como um sólido elástico linear, heterogêneo e anisotrópico. A microtomografia tridimensional por raios-x (uCT) tem sido mundialmente utilizada para a caracterização de osso trabecular em pesquisas relacionadas à qualidade óssea e à doenças do metabolismo ósseo como a osteoporose. Na literatura há poucas investigações comparando as análises morfométricas 2D e 3D de osso trabecular. Nesta investigação amostras de osso trabeculares cilíndricas extraídas da cabeça femoral bovina foram utilizadas para a comparação dessas análises por \'mü\'CT. O software CT-Analyser foi utilizado para medidas em três direções (crânio-caudal, lateral-medial e anterior-posterior) de diversos parâmetros da microestrutura trabecular tais como espessura trabecular, separação trabecular, número trabecular e os autovalores do tensor de anisotropia (M). A comparação entre os valores dos parâmetros medidos por análises morfométricas 2D e 3D foi realizada pelo teste-t pareado com nível de significância p < 0,05 e por correlação linear de Pearson. Os autovalores da matriz M mostram que a microestrutura trabecular bovina tem uma tendência para a simetria transversalmente isotrópica. Os resultados obtidos mostram que a microtomografia tridimensional por raios-X é uma técnica de grande potencial para caracterização da qualidade óssea gerando bons parâmetros para o diagnóstico de doenças do metabolismo ósseo. / Trebecular bones have a porous microstructure and can be modeled as a linear elastic solid, heterogenous and anisotropic. The x-ray tridimensional microtomography (\'mü\'CT) has been worldwide used for the assessment of trabecular bone in investigations related to bone quality and aimed to the diagnostic of bone metabolic diseases such as osteoporosis. In the literature few investigations have compared the 2D and 3D morphometric analysis of trabecular bone. In this investigation cylindrical trabecular bone samples were harvested from bovine head femur to carry out that comparison by \'mü\'CT. The CT-Analyser software was used to measure in three directions (superior-inferior, lateral-medial and anterior-posterior) several microstructural trabecular parameters such as trabecular thickness, trabecular separation, trabecular number and the eigenvalues of the fabric tensor (M). The comparison between the parameters values measured by 2D and 3D morphometric analysis was performed by the paired-sample t test with a level of significance p < 0,05 and the Pearson\'s linear correlation. The eigenvalues show that the bovine trabecular microstructure has a tendence to transverse isotropy symmetry. The results show that x-ray tridimensional microtomography (\'mü\'CT) is a technique of great potential for characterization of generating good bone quality parameters for the diagnosis of diseases of bone metabolism.

Introdução: Os processos degenerativos vêm sendo considerados determinantes da progressão do déficit neurológico na esclerose múltipla (EM) e são associados sobretudo à perda neuronal e axonal. A patologia na substância branca (SB) manifesta-se pela quebra de membranas e perda da complexidade microestrutural dos tratos cerebrais, o que pode ser estudado indiretamente pelas alterações nos índices de fração de anisotropia (FA) e difusividade média (DM), obtidos por meio das análises das imagens por tensores de difusão (diffusion tensor imaging - DTI). Essa técnica oferece outros dois índices mais específicos, a difusividade axial () e difusividade radial (), que são associados aos processos de perda axonal e desmielinização, respectivamente. A perda neuronal na substância cinzenta (SC) pode ser avaliada pelo grau de atrofia do córtex cerebral. Este estudo tem como objetivos mensurar os índices de DTI na maior comissura cerebral, o corpo caloso (CC), e o grau e distribuição da atrofia cortical em indivíduos com EM remitente-recorrente (EMRR) e baixos escores de incapacidade funcional, correlacionando essas alterações com o volume de lesões macroscópicas e os principais parâmetros clínicos. Método: 31 indivíduos (22 mulheres, idade média 30,5 anos ± 8,7) com EMRR e um grupo controle (GC) composto por 34 indivíduos saudáveis (27 mulheres, idade média 32,3 anos ± 7,8) realizaram exames de crânio em aparelho de ressonância magnética de 3 Tesla (3T), sendo adquiridas imagens de DTI com 32 direções de gradiente, obtendo-se os índices de FA, DM, e de cinco segmentos na secção sagital do corpo caloso (CC). Através da segmentação de imagens volumétricas ponderadas em T1 foram obtidas as espessuras corticais regionais nos grupos. Esses resultados foram correlacionados com os volumes lesionais de imagens ponderadas em T1 e T2/FLAIR e os escores da escala expandida do estado de incapacidade (Expanded Disability Status Scale - EDSS), considerando-se significativos resultados com p< 0,05. Resultados: Os índices de FA, DM e do CC estavam difusamente alterados no grupo EMRR e a , alterada significativamente no esplênio, tronco médio anterior e tronco médio posterior do CC. Observou-se atrofia cortical significativa no terço anterior dos lobos temporais, bilateralmente, e nas regiões parietal inferior, insular e fronto-orbitária direitas, com uma tendência à atrofia no giro frontal superior esquerdo. As FA, DM e correlacionaram-se com os volumes lesionais T1 e, mais significativamente, com os volumes lesionais T2/FLAIR, porém não houve correlação entre os volumes lesionais e a . A espessura cortical no grupo EMRR apresentou correlações com ambos os volumes lesionais, mais significativamente com as lesões em T1. O escore médio da EDSS era 1,1 ± 0,9 (variando de 0-3), apresentando correlações com a DM e a no esplênio, tronco médio anterior e posterior do CC, com uma correlação com a no tronco médio posterior. O EDSS correlacionou-se com a espessura cortical na topografia do giro frontal superior esquerdo. Discussão e conclusão: Houve alteração difusa nos índices de FA, DM e nos segmentos do CC, com acometimento mais localizado, predominantemente médio posterior, da , o que pode sugerir desmielinização difusa do CC, porém axonopatia ou degeneração mais acentuada em algumas regiões da SB. A atrofia cortical também apresentou uma distribuição regional característica, afetando sobretudo as regiões temporais, bilateralmente, parietal inferior, insular e fronto-orbitária direitas. As correlações encontradas entre os índices de DTI e a espessura cortical e os volumes lesionais demonstraram que, ao menos em parte, as degenerações das SB e SC podem ser relacionadas à degeneração Walleriana, secundária ao acúmulo de placas lesionais. As correlações entre a DM, de alguns segmentos do CC e a espessura cortical do giro frontal superior com os escores da EDSS favoreceram à hipótese de que a degeneração tecidual na EM foi um fator preponderante na progressão do déficit neurológico na EMRR / Introduction: The degenerative processes are gaining attention as predictors of the neurological deficit in multiple sclerosis (MS), being reflected by the degree of axon loss and central nervous system atrophy. The white matter pathology (WM) is characterized by cellular membranes disruption and loss of the microstructural complexity, which can be accessed by the diffusion tensor imaging (DTI) indices of fractional anisotropy (FA) and mean diffusivity (MD). This imaging technique also offers two more specific indices: the axial diffusivity () and radial diffusivity (), which are useful to differentiate between axon loss and demyelination, respectively. The gray matter (GM) neuronal loss can be accessed by the degree of cortical atrophy. The aim of this study is to measure the DTI indices in the greatest WM commisure, the corpus callosum (CC), and the degree and distribution of cortical atrophy in patients with relapsing remitting MS (RRMS) and low disability scores, correlating them to the macroscopic lesion load and the main clinical scores. Method: 31 RRMS patients (22 women, mean age 30.5 years ± 8.7) and 34 healthy control (HC) subjects (27 women, mean age 32.3 years ± 7.8) were submitted to brain examinations in a 3T magnetic resonance image scanner. From DTI with 32 gradient encoding directions were extracted the indices of FA, MD, and , which were measured in 5 segments of the mid-sagital section of the corpus callosum (CC). The cortical thickness was obtained from the segmentation of volumetric T1 images. These results were correlated with the macroscopic lesion loads in the T1 and T2/FLAIR images and the scores in the Expanded Disability Status Scale EDSS, considering significant the results with p< 0.05. Results: The FA, MD and were diffusively abnormal in all 5 segments of the CC in the RRMS group and the was abnormal only in the splenium, anterior midbody and posterior mid-body. The anterior area of the both temporal lobes and right inferior parietal, some orbital-frontal and insular regions showed significant atrophy, with a tendency of atrophy in the superior frontal gyrus. The FA, MD and correlated with the T1 lesion load and, more significantly, with the T2/FLAIR lesion load. The cortical thickness correlated with T1 and T2/FLAIR lesion loads, more significantly with the T1 lesion load. The mean EDSS in the RRMS group was 1.1 ± 0.9 (range 0-3), correlating with the MD and of the splenium, anterior and posterior mid-body of the CC. The EDSS correlated to cortical thickness in the topography of the superior frontal gyrus. Discussion and conclusion: The FA, MD and are diffusively abnormal in the CC, with abnormalities in the , restricted to the medial and posterior segments. These results can be interpreted as signs of diffuse demyelination in the CC and a predominance of axonopathy or more advanced degeneration in some segments. The cortical atrophy also followed a characteristic regional distribution, affecting predominantly the bilateral temporal lobes, and inferior parietal, orbital-frontal and insular regions, in the right hemisphere. The correlations found between the DTI indices and the cortical thickness and the macroscopic lesion loads show that, at least partially, the WM and GM degeneration can be related to Wallerian degeneration secondary to macroscopic lesion accumulation. The correlations between the DM, , in some of the CC segments, and cortical thickness, in the superior frontal gyrus, and the EDSS scores reinforces the hypothesis that the degenerative processes in MS can play a role in the disability status of the patients

Diffusion of water molecules in biological tissues depends on several microstructural properties. Therefore, diffusion Magnetic Resonance Imaging (dMRI) is a useful tool to infer and study microstructural brain changes in the context of human development, ageing and neuropathology. In this thesis, the state-of-the-art of advanced dMRI techniques is explored and strategies to overcome or reduce its pitfalls are developed and validated. Firstly, it is shown that PCA denoising and Gibbs artefact suppression algorithms provide an optimal compromise between increased precision of diffusion measures and the loss of tissue's diffusion non-Gaussian information. Secondly, the spatial information provided by the diffusion kurtosis imaging (DKI) technique is explored and used to resolve crossing fibres and generalize diffusion measures to cases not limited to well-aligned white matter fibres. Thirdly, as an alternative to diffusion microstructural modelling techniques such as the neurite orientation dispersion and density imaging (NODDI), it is shown that spherical deconvolution techniques can be used to characterize fibre crossing and dispersion simultaneously. Fourthly, free water volume fraction estimates provided by the free water diffusion tensor imaging (fwDTI) are shown to be useful to detect and remove voxels corrupted by cerebrospinal fluid (CSF) partial volume effects. Finally, dMRI techniques are applied to the diffusion data from the large collaborative Cambridge Centre for Ageing and Neuroscience (CamCAN) study. From these data, the inference provided by diffusion anisotropy measures on maturation and degeneration processes is shown to be biased by age-related changes of fibre organization. Inconsistencies of previous NODDI ageing studies are also revealed to be associated with the different age ranges covered. The CamCAN data is also processed using a novel non-Gaussian diffusion characterization technique which is invariant to different fibre configurations. Results show that this technique can provide indices specific to axonal water fraction which can be linked to age-related fibre density changes.

Corvino, Corvino and Schoen, Chruściel and Delay have shown the existence of a large class of asymptotically flat vacuum initial data for Einstein's field equations which are static or stationary in a neighborhood of space-like infinity, yet quite general in the interior. The proof relies on some abstract, non-constructive arguments which makes it difficult to calculate such data numerically by using similar arguments. A quasilinear elliptic system of equations is presented of which we expect that it can be used to construct vacuum initial data which are asymptotically flat, time-reflection symmetric, and asymptotic to static data up to a prescribed order at space-like infinity. A perturbation argument is used to show the existence of solutions. It is valid when the order at which the solutions approach staticity is restricted to a certain range. Difficulties appear when trying to improve this result to show the existence of solutions that are asymptotically static at higher order. The problems arise from the lack of surjectivity of a certain operator. Some tensor decompositions in asymptotically flat manifolds exhibit some of the difficulties encountered above. The Helmholtz decomposition, which plays a role in the preparation of initial data for the Maxwell equations, is discussed as a model problem. A method to circumvent the difficulties that arise when fast decay rates are required is discussed. This is done in a way that opens the possibility to perform numerical computations. The insights from the analysis of the Helmholtz decomposition are applied to the York decomposition, which is related to that part of the quasilinear system which gives rise to the difficulties. For this decomposition analogous results are obtained. It turns out, however, that in this case the presence of symmetries of the underlying metric leads to certain complications. The question, whether the results obtained so far can be used again to show by a perturbation argument the existence of vacuum initial data which approach static solutions at infinity at any given order, thus remains open. The answer requires further analysis and perhaps new methods. / Corvino, Corvino und Schoen als auch Chruściel und Delay haben die Existenz einer grossen Klasse asymptotisch flacher Anfangsdaten für Einsteins Vakuumfeldgleichungen gezeigt, die in einer Umgebung des raumartig Unendlichen statisch oder stationär aber im Inneren der Anfangshyperfläche sehr allgemein sind. Der Beweis beruht zum Teil auf abstrakten, nicht konstruktiven Argumenten, die Schwierigkeiten bereiten, wenn derartige Daten numerisch berechnet werden sollen. In der Arbeit wird ein quasilineares elliptisches Gleichungssystem vorgestellt, von dem wir annehmen, dass es geeignet ist, asymptotisch flache Vakuumanfangsdaten zu berechnen, die zeitreflektionssymmetrisch sind und im raumartig Unendlichen in einer vorgeschriebenen Ordnung asymptotisch zu statischen Daten sind. Mit einem Störungsargument wird ein Existenzsatz bewiesen, der gilt, solange die Ordnung, in welcher die Lösungen asymptotisch statische Lösungen approximieren, in einem gewissen eingeschränkten Bereich liegt. Versucht man, den Gültigkeitsbereich des Satzes zu erweitern, treten Schwierigkeiten auf. Diese hängen damit zusammen, dass ein gewisser Operator nicht mehr surjektiv ist. In einigen Tensorzerlegungen auf asymptotisch flachen Räumen treten ähnliche Probleme auf, wie die oben erwähnten. Die Helmholtzzerlegung, die bei der Bereitstellung von Anfangsdaten für die Maxwellgleichungen eine Rolle spielt, wird als ein Modellfall diskutiert. Es wird eine Methode angegeben, die es erlaubt, die Schwierigkeiten zu umgehen, die auftreten, wenn ein schnelles Abfallverhalten des gesuchten Vektorfeldes im raumartig Unendlichen gefordert wird. Diese Methode gestattet es, solche Felder auch numerisch zu berechnen. Die Einsichten aus der Analyse der Helmholtzzerlegung werden dann auf die Yorkzerlegung angewandt, die in den Teil des quasilinearen Systems eingeht, der Anlass zu den genannten Schwierigkeiten gibt. Für diese Zerlegung ergeben sich analoge Resultate. Es treten allerdings Schwierigkeiten auf, wenn die zu Grunde liegende Metrik Symmetrien aufweist. Die Frage, ob die Ergebnisse, die soweit erhalten wurden, in einem Störungsargument verwendet werden können um die Existenz von Vakuumdaten zu zeigen, die im räumlich Unendlichen in jeder Ordnung statische Daten approximieren, bleibt daher offen. Die Antwort erfordert eine weitergehende Untersuchung und möglicherweise auch neue Methoden.

All procedures in the literature for decomposing symmetric second rank (stress) tensor and symmetric fourth rank (elastic constant) tensor are elaborated and compared which have many engineering and scientific applications for anisotropic materials. The decomposition methods for symmetric second rank tensors are orthonormal tensor basis method, complex variable representation and spectral method. For symmetric fourth rank (elastic constant) tensor, there are four mainly decomposition methods namely as, orthonormal tensor basis, irreducible, harmonic decomposition and spectral. Those are applied to anisotropic materials possessing various symmetry classes which are isotropic, cubic, transversely isotropic, tetragonal, trigonal and orthorhombic. For isotropic materials, an expression for the elastic constant tensor different than the traditionally known form is given. Some misprints found in the literature are corrected. For comparison purposes, numerical examples of each decomposition process are presented for the materials possessing different symmetry classes. Some applications of these decomposition methods are given. Besides, norm and norm ratio concepts are introduced to measure and compare the anisotropy degree for various materials with the same or di&curren / erent symmetries. For these materials,norm and norm ratios are calculated. It is suggested that the norm of a tensor may be used as a criterion for comparing the overall e&curren / ect of the properties of anisotropic materials and the norm ratios may be used as a criterion to represent the anisotropy degree of the properties of materials. Finally, comparison of all methods are done in order to determine similarities and differences between them. As a result of this comparison process, it is proposed that the spectral method is a non-linear decomposition method which yields non-linear orthogonal decomposed parts. For symmetric second rank and fourth rank tensors, this case is a significant innovation in decomposition procedures in the literature.

An important research area in computer vision is parameter estimation. Given a mathematical model and a sample of image measurement data, key parameters are sought to encapsulate geometric properties of a relevant entity. An optimisation problem is often formulated in order to find these parameters. This thesis presents an elaboration of fundamental numerical algorithms for estimating parameters of multi-objective models of importance in computer vision applications. The work examines ways to solve unconstrained and constrained minimisation problems from the view points of theory, computational methods, and numerical performance. The research starts by considering a particular form of multi-equation constraint function that characterises a wide class of unconstrained optimisation tasks. Increasingly sophisticated cost functions are developed within a consistent framework, ultimately resulting in the creation of a new iterative estimation method. The scheme operates in a maximum likelihood setting and yields near-optimal estimate of the parameters. Salient features of themethod are that it has simple update rules and exhibits fast convergence. Then, to accommodate models with functional dependencies, two variant of this initial algorithm are proposed. These methods are improved again by reshaping the objective function in a way that presents the original estimation problem in a reduced form. This procedure leads to a novel algorithm with enhanced stability and convergence properties. To extend the capacity of these schemes to deal with constrained optimisation problems, several a posteriori correction techniques are proposed to impose the so-called ancillary constraints. This work culminates by giving two methods which can tackle ill-conditioned constrained functions. The combination of the previous unconstrained methods with these post-hoc correction schemes provides an array of powerful constrained algorithms. The practicality and performance of themethods are evaluated on two specific applications. One is planar homography matrix computation and the other trifocal tensor estimation. In the case of fitting a homography to image data, only the unconstrained algorithms are necessary. For the problem of estimating a trifocal tensor, significant work is done first on expressing sets of usable constraints, especially the ancillary constraints which are critical to ensure that the computed object conforms to the underlying geometry. Evidently here, the post-correction schemes must be incorporated in the computational mechanism. For both of these example problems, the performance of the unconstrained and constrained algorithms is compared to existing methods. Experiments reveal that the new methods perform with high accuracy to match a state-of-the-art technique but surpass it in execution speed. / Thesis (Ph.D.) - University of Adelaide, School of Mathemtical Sciences, Discipline of Pure Mathematics, 2008

Introdução: Os processos degenerativos vêm sendo considerados determinantes da progressão do déficit neurológico na esclerose múltipla (EM) e são associados sobretudo à perda neuronal e axonal. A patologia na substância branca (SB) manifesta-se pela quebra de membranas e perda da complexidade microestrutural dos tratos cerebrais, o que pode ser estudado indiretamente pelas alterações nos índices de fração de anisotropia (FA) e difusividade média (DM), obtidos por meio das análises das imagens por tensores de difusão (diffusion tensor imaging - DTI). Essa técnica oferece outros dois índices mais específicos, a difusividade axial () e difusividade radial (), que são associados aos processos de perda axonal e desmielinização, respectivamente. A perda neuronal na substância cinzenta (SC) pode ser avaliada pelo grau de atrofia do córtex cerebral. Este estudo tem como objetivos mensurar os índices de DTI na maior comissura cerebral, o corpo caloso (CC), e o grau e distribuição da atrofia cortical em indivíduos com EM remitente-recorrente (EMRR) e baixos escores de incapacidade funcional, correlacionando essas alterações com o volume de lesões macroscópicas e os principais parâmetros clínicos. Método: 31 indivíduos (22 mulheres, idade média 30,5 anos ± 8,7) com EMRR e um grupo controle (GC) composto por 34 indivíduos saudáveis (27 mulheres, idade média 32,3 anos ± 7,8) realizaram exames de crânio em aparelho de ressonância magnética de 3 Tesla (3T), sendo adquiridas imagens de DTI com 32 direções de gradiente, obtendo-se os índices de FA, DM, e de cinco segmentos na secção sagital do corpo caloso (CC). Através da segmentação de imagens volumétricas ponderadas em T1 foram obtidas as espessuras corticais regionais nos grupos. Esses resultados foram correlacionados com os volumes lesionais de imagens ponderadas em T1 e T2/FLAIR e os escores da escala expandida do estado de incapacidade (Expanded Disability Status Scale - EDSS), considerando-se significativos resultados com p< 0,05. Resultados: Os índices de FA, DM e do CC estavam difusamente alterados no grupo EMRR e a , alterada significativamente no esplênio, tronco médio anterior e tronco médio posterior do CC. Observou-se atrofia cortical significativa no terço anterior dos lobos temporais, bilateralmente, e nas regiões parietal inferior, insular e fronto-orbitária direitas, com uma tendência à atrofia no giro frontal superior esquerdo. As FA, DM e correlacionaram-se com os volumes lesionais T1 e, mais significativamente, com os volumes lesionais T2/FLAIR, porém não houve correlação entre os volumes lesionais e a . A espessura cortical no grupo EMRR apresentou correlações com ambos os volumes lesionais, mais significativamente com as lesões em T1. O escore médio da EDSS era 1,1 ± 0,9 (variando de 0-3), apresentando correlações com a DM e a no esplênio, tronco médio anterior e posterior do CC, com uma correlação com a no tronco médio posterior. O EDSS correlacionou-se com a espessura cortical na topografia do giro frontal superior esquerdo. Discussão e conclusão: Houve alteração difusa nos índices de FA, DM e nos segmentos do CC, com acometimento mais localizado, predominantemente médio posterior, da , o que pode sugerir desmielinização difusa do CC, porém axonopatia ou degeneração mais acentuada em algumas regiões da SB. A atrofia cortical também apresentou uma distribuição regional característica, afetando sobretudo as regiões temporais, bilateralmente, parietal inferior, insular e fronto-orbitária direitas. As correlações encontradas entre os índices de DTI e a espessura cortical e os volumes lesionais demonstraram que, ao menos em parte, as degenerações das SB e SC podem ser relacionadas à degeneração Walleriana, secundária ao acúmulo de placas lesionais. As correlações entre a DM, de alguns segmentos do CC e a espessura cortical do giro frontal superior com os escores da EDSS favoreceram à hipótese de que a degeneração tecidual na EM foi um fator preponderante na progressão do déficit neurológico na EMRR / Introduction: The degenerative processes are gaining attention as predictors of the neurological deficit in multiple sclerosis (MS), being reflected by the degree of axon loss and central nervous system atrophy. The white matter pathology (WM) is characterized by cellular membranes disruption and loss of the microstructural complexity, which can be accessed by the diffusion tensor imaging (DTI) indices of fractional anisotropy (FA) and mean diffusivity (MD). This imaging technique also offers two more specific indices: the axial diffusivity () and radial diffusivity (), which are useful to differentiate between axon loss and demyelination, respectively. The gray matter (GM) neuronal loss can be accessed by the degree of cortical atrophy. The aim of this study is to measure the DTI indices in the greatest WM commisure, the corpus callosum (CC), and the degree and distribution of cortical atrophy in patients with relapsing remitting MS (RRMS) and low disability scores, correlating them to the macroscopic lesion load and the main clinical scores. Method: 31 RRMS patients (22 women, mean age 30.5 years ± 8.7) and 34 healthy control (HC) subjects (27 women, mean age 32.3 years ± 7.8) were submitted to brain examinations in a 3T magnetic resonance image scanner. From DTI with 32 gradient encoding directions were extracted the indices of FA, MD, and , which were measured in 5 segments of the mid-sagital section of the corpus callosum (CC). The cortical thickness was obtained from the segmentation of volumetric T1 images. These results were correlated with the macroscopic lesion loads in the T1 and T2/FLAIR images and the scores in the Expanded Disability Status Scale EDSS, considering significant the results with p< 0.05. Results: The FA, MD and were diffusively abnormal in all 5 segments of the CC in the RRMS group and the was abnormal only in the splenium, anterior midbody and posterior mid-body. The anterior area of the both temporal lobes and right inferior parietal, some orbital-frontal and insular regions showed significant atrophy, with a tendency of atrophy in the superior frontal gyrus. The FA, MD and correlated with the T1 lesion load and, more significantly, with the T2/FLAIR lesion load. The cortical thickness correlated with T1 and T2/FLAIR lesion loads, more significantly with the T1 lesion load. The mean EDSS in the RRMS group was 1.1 ± 0.9 (range 0-3), correlating with the MD and of the splenium, anterior and posterior mid-body of the CC. The EDSS correlated to cortical thickness in the topography of the superior frontal gyrus. Discussion and conclusion: The FA, MD and are diffusively abnormal in the CC, with abnormalities in the , restricted to the medial and posterior segments. These results can be interpreted as signs of diffuse demyelination in the CC and a predominance of axonopathy or more advanced degeneration in some segments. The cortical atrophy also followed a characteristic regional distribution, affecting predominantly the bilateral temporal lobes, and inferior parietal, orbital-frontal and insular regions, in the right hemisphere. The correlations found between the DTI indices and the cortical thickness and the macroscopic lesion loads show that, at least partially, the WM and GM degeneration can be related to Wallerian degeneration secondary to macroscopic lesion accumulation. The correlations between the DM, , in some of the CC segments, and cortical thickness, in the superior frontal gyrus, and the EDSS scores reinforces the hypothesis that the degenerative processes in MS can play a role in the disability status of the patients

Made available in DSpace on 2015-05-15T11:46:16Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 713593 bytes, checksum: 9a4eca1dfe2fca22fc63a3364773af08 (MD5) Previous issue date: 2013-09-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Cook's local infuence approach based on normal curvature is an important diagnostic tool for assessing local infuence of minor perturbations to a statistical model. However, no rigorous approach has been developed to address two fundamental issues: the selection of an appropriate perturbation and the development of infuence measures for objective functions at a point with a nonzero rst derivative. The aim of this paper is to develop a diferential-geometrical framework of a perturbation model (called the perturbation manifold) and utilize associated metric tensor and ane curvatures to resolve these issues. We will show that the metric tensor of the perturbation manifold provides important information about selecting an appropriate perturbation of a model. / A Abordagem de influência local de Cook [2] com base em curvatura normal é uma importante ferramenta de diagnóstico para avaliar a influência local de pequenas perturbações de um modelo estatístico. No entanto, tem sido desenvolvida nenhuma abordagem rigorosa para abordar duas questões fundamentais: a escolha de uma perturbação apropriada e o desenvolvimento de medidas de influência para funções objetos em um ponto com a primeira derivada diferente de zero. O objetivo deste trabalho é desenvolver uma estrutura diferencial-geométrica de um modelo de perturbação (chamado de variedade de perturbação) e utilizar o tensor métrico associado e as curvaturas afins para resolver esses problemas. Vamos mostrar que o tensor métrico da variedade de perturbação fornece informações importantes sobre a seleção de uma perturbação apropriada de um modelo.