BackgroundPhase contrast magnetic resonance imaging (PC-MRI) is used clinically for quantitative assessment of cardiovascular flow and function, as it is capable of providing directly-measured 3D velocity maps. Alternatively, vascular flow can be estimated from model-based computation fluid dynamics (CFD) calculations. CFD provides arbitrarily high resolution, but its accuracy hinges on model assumptions, while velocity fields measured with PC-MRI generally do not satisfy the equations of fluid dynamics, provide limited resolution, and suffer from partial volume effects. The purpose of this study is to develop a proof-of-concept numerical procedure for constructing a simulated flow field that is influenced by both direct PC-MRI measurements and a fluid physics model, thereby taking advantage of both the accuracy of PC-MRI and the high spatial resolution of CFD. The use of the proposed approach in regularizing 3D flow fields is evaluated.MethodsThe proposed algorithm incorporates both a Newtonian fluid physics model and a linear PC-MRI signal model. The model equations are solved numerically using a modified CFD algorithm. The numerical solution corresponds to the optimal solution of a generalized Tikhonov regularization, which provides a flow field that satisfies the flow physics equations, while being close enough to the measured PC-MRI velocity profile. The feasibility of the proposed approach is demonstrated on data from the carotid bifurcation of one healthy volunteer, and also from a pulsatile carotid flow phantom.ResultsThe proposed solver produces flow fields that are in better agreement with direct PC-MRI measurements than CFD alone, and converges faster, while closely satisfying the fluid dynamics equations. For the implementation that provided the best results, the signal-to-error ratio (with respect to the PC-MRI measurements) in the phantom experiment was 6.56 dB higher than that of conventional CFD; in the in vivo experiment, it was 2.15 dB higher.ConclusionsThe proposed approach allows partial or complete measurements to be incorporated into a modified CFD solver, for improving the accuracy of the resulting flow fields estimates. This can be used for reducing scan time, increasing the spatial resolution, and/or denoising the PC-MRI measurements.

Fourier velocity encoding (FVE) is a promising magnetic resonance imaging (MRI) method for measurement of cardiovascular blood flow. FVE provides considerably higher SNR than phase contrast imaging, and is robust to partial-volume effects. FVE data is usually acquired with low spatial resolution, due to scan-time restrictions associated with its higher dimensionality. Thus, FVE is capable of providing the velocity distribution associated with a large voxel, but does not directly provides a velocity map. Velocity maps, however, are useful for calculating the actual blod flow through a vessel, or for guiding computational fluid dynamics simulations. This work proposes a method to derive velocity maps with high spatial resolution from low-resolution FVE data using a hyper-Laplacian prior deconvolution algorithm. Experiments using numerical phantoms, as well simulated spiral FVE data derived from real phase contrast data, acquired using a pulsatile carotid flow phantom, show that it is possible to obtain reasonably accurate velocities maps from low-resolution FVE distributions.

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Resumo Problemas de valores inicial e de contorno são muito comuns na Física, Matemática e Engenharia. Eles podem modelar diversos tipos de problemas relacionados a difusão de calor e a vibração de membranas, por exemplo. Quando se deseja encontrar a solução analítica desses problemas podemos encontrar dificuldades extras quando as equações e também as condições de contorno que descrevem os fenômenos são não-homogêneas. Desta forma, neste trabalho apresentamos uma técnica de solução de problemas de valores iniciais e de contorno por meio de transformações integrais. O diferencial da apresentação está na construção da transformada integral apropriada à solução do problema. Essas transformadas são conhecidas como transformadas finitas e neste caso elas estão relacionadas a um problema de Sturm–Liouville associado com o operador diferencial ligado à equação diferencial. Como exemplo do desenvolvimento e aplicação da ferramenta, resolvemos dois problemas de difusão de calor em coordenadas espaciais distintas. A apresentação do trabalho segue de forma pedagógica e autocontida. Sendo assim, esperamos que o leitor compreenda a técnica e possa utilizá-la na resolução de outros problemas envolvendo equações diferencias parciais.

Neste trabalho descrevemos um formalismo para descrever a Mecânica Clássica fracionária segundo o cálculo fracionário conformável. Assim, uma breve revisão sobre cálculo diferencial e integral conformável foi feita. Na sequência, as versões conformáveis dos formalismos Lagrangiano e Hamiltoniano são apresentadas. Como exemplo, o oscilador harmônico conformável e a equação de onda conformável são discutidos. Em todas as situações consideradas, os resultados usuais são obtidos se a ordem da derivada ( α) for igual a 1. Um resultado interessante é que se considerarmos 1 - α ≪ 1, o oscilador harmônico conformável torna-se análogo aos oscilador usual com massa variável e termo de dissipação.

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Knowledge about the input–output relations of a system can be very important in many practical situations in engineering. Linear systems theory comes from applied mathematics as an efficient and simple modeling technique for input–output systems relations. Many identification problems arise from a set of linear equations, using known outputs only. It is a type of inverse problems, whenever systems inputs are sought by its output only. This work presents a regularization method, called random matrix method, which is able to reduce errors on the solution of ill-conditioned inverse problems by introducing modifications into the matrix operator that rules the problem. The main advantage of this approach is the possibility of reducing the condition number of the matrix using the probability density function that models the noise in the measurements, leading to better regularization performance. The method described was applied in the context of a force identification problem and the results were compared quantitatively and qualitatively with the classical Tikhonov regularization method. Results show the presented technique provides better results than Tikhonov method when dealing with high-level ill-conditioned inverse problems.