Reconstruction of Parametric Image Maps in Single- and Multiple-Coil Functional Magnetic Resonance Imaging

Abstract

Functional Magnetic Resonance Imaging (fMRI) is a standard tool to measure the hemodynamic response which is related to activation patterns in the human and animal brain. In conventional anatomical MRI, the decay and precession rates are regarded as sources of artifacts, but in applications such as functional MRI (fMRI), they are physiological quantities of interest. Single-shot parameter assessment by retrieval from signal encoding (SS-PARSE) acknowledges local decay and phase evolution in MRI, so it models each datum as a sample from (k,t)-space rather than $k$-space. Because local decay and frequency vary continuously in space, discrete models in space can cause artifacts in the reconstructed parameters. Increasing the resolution of the reconstructed parameters can more accurately capture the spatial variations, but the resolution is limited not only by computational complexity but also by the size of the acquired data. For a limited data set used for reconstruction, simply increasing the model resolution may cause the reconstruction to become an underdetermined problem. This dissertation presents a solution to this problem based on cubic convolution interpolation. Because the local decay and frequency are exponential time functions, FFTs can not be directly applied to the reconstruction algorithm. A polynomial expansion is proposed so that FFTs can be used to accelerate reconstruction.

The second contribution of this dissertation is a new method to optimize the nonuniform FFT (NUFFT). This work was motivated by the nonuniform k-space trajectory in SS-PARSE. With the polynomial expansion, the cost function of the reconstruction of SS-PARSE is represented by a linear combination of 2-D Fourier transforms whose inputs are uniformly distributed data and outputs are nonuniformly distributed frequency responses. The gradient of the cost function in the reconstruction is also a linear combination of 2-D Fourier transforms whose inputs are nonuniformly distributed data on the frequency domain and outputs are functions on a 2-D nonuniform grid. FFTs can be applied to neither the cost function nor the gradients function because of the nonequally spaced inputs or outputs. In this dissertation, we focused on the 1-D Fourier transforms with uniform inputs and nonuniform outputs. The basic form of the optimization of the NUFFT is a nonlinear problem. In this dissertation, this nonlinear problem was reformulated to find the least-square solution of a linear problem. The computational accuracy of the NUFFT is also improved by the new method. The results can be easily extended to 2-D or the case with nonuniform inputs and uniform outputs.

After validating and testing these ideas with a single-coil MRI system, we extended the framework to parallel MRI systems, which have multiple receiving coils. Existing reconstruction methods estimate the maps of coil sensitivities by imaging ``standard" objects. These convenient methods do not account for the change of coil sensitivities caused by the imaged objects. We propose a new algorithm that concurrently reconstructs the coil sensitivities along with magnitude, decay and field map. The core of this algorithm is the fast approach and the interpolation method we developed for SS-PARSE. From the simulation results, we observed significant improvement in the reconstruction accuracy of the decay function that is of the interest in fMRI.