Parallel algorithms for maximum a posteriori estimation of spin density and spin-spin decay in magnetic resonance imaging

Abstract

A maximum a posteriori (MAP) algorithm is presented for the estimation of spin-density and spin-spin decay distributions from frequency and phase-encoded magnetic resonance imaging data. Linear spatial localization gradients are assumed: the y-encode gradient applied during the phase preparation time of duration tau before measurement collection, and the x-encode gradient applied during the full data collection time t>/=0. The MRI signal model developed in M.I. Miller et al., J. Magn. Reson., ser. B (Apr. 1995) is used in which a signal resulting from M phase encodes (rows) and N frequency encode dimensions (columns) is modeled as a superposition of MN sinc-modulated exponentially decaying sinusoids with unknown spin-density and spin-spin decay parameters. The nonlinear least-squares MAP estimate of the spin density and spin-spin decay distributions solves for the 2MN spin-density and decay parameters minimizing the squared-error between the measured data and the sine-modulated exponentially decay signal model using an iterative expectation-maximization algorithm. A covariance diagonalizing transformation is derived which decouples the joint estimation of MN sinusoids into M separate N sinusoid optimizations, yielding an order of magnitude speed up in convergence. The MAP solutions are demonstrated to deliver a decrease in standard deviation of image parameter estimates on brain phantom data of greater than a factor of two over Fourier-based estimators of the spin density and spin-spin decay distributions. A parallel processor implementation is demonstrated which maps the N sinusoid coupled minimization to separate individual simple minimizations, one for each processor.

A stochastic model for studying the laminar structure of cortex from MRI

“…Because cortex is not much thicker than the size of the voxels in an MR image, there is considerable blurring of the edges of the cortex, and we must take this "partial volume" artifact into account. In MRI the voxel resolution is determined by the distribution of phases across a voxel; the resulting voxel intensity resolution corresponds to twice the full-width at half-maximum of the function [46], [47] if we assume that the voxel size corresponds to sampling at the Nyquist rate. We model this partial volume effect function of MRI data by the convolution of the ideal intensity profile with a Gaussian density function having zero mean and constant standard deviation .…”

Bayesian $k$-Space–Time Reconstruction of MR Spectroscopic Imaging for Enhanced Resolution

“…Although it is impractical to achieve full convergence to the level of machine tolerance, the statistical metrics of the reconstructions did not change significantly after this time. Parallelization similar to that used in [10] will help to significantly reduce computational time. Because the algorithm is highly parallelizable, a large reduction in computational time can be achieved by distributing the processing and spreading the memory load amongst multiple processors.…”

“…Rearranging [10] in terms of A m ( p , q , r ), leads to $$\begin{array}{cc}right{A}_m(p,q,r)=& left\\ right& left\frac{1}{{\sigma }^2}\sum _{k_x,k_y}\mathrm{sinc}(\pi k_x∕P)\mathrm{sinc}(\pi k_y∕Q)\sum _{(t∕\delta )=0}^{T-1}\exp \left\{-\left[t∕T_a(p,q,r)\right]+{true[t∕T_b(p,q,r)true]}^2\right\}\\ right& left\sum _{n=0}^{N\left(m\right)-1}{L}_{mn}(\begin{array}{c}\mathrm{Re}true[z(k_x,k_y,p,q,r,w,t,m)true]\cos true(\begin{array}{c}left({\omega }_m+{\omega }_{mn})B_0(p,q,r)t+{\varphi }_{mn}\\ left+{\varphi }_A(p,q,r)+2\pi \left(k_{x}p∕P+k_{y}q∕Q\right)\end{array}true)-\hfill \\ \mathrm{Im}true[z(k_x,k_y,p,q,r,w,t,m)true]\sin true(\begin{array}{c}left({\omega }_m+{\omega }_{mn})B_0(p,q,r)t+{\varphi }_{mn}\\ left+{\varphi }_A(p,q,r)+2\pi (k_{x}p∕\mathrm{P...}\end{array}\hfill \end{array}\end{array}$$…”

“…Reconstruction to higher resolution using Bayesian image analysis methods for multi-contrast structural MRI is presented in [8]; however, they do not incorporate a full signal model and hence are subject to all of the problems with artifacts that are incurred from use of the DFT for reconstruction of low-resolution modalities. A model that incorporates a raw signal model for structural MRI was presented for maximum likelihood estimation in [9] and Bayesian maximum a posteriori reconstruction in [10]; however, the first model's focus was on high-resolution structural MRI, and the only prior information considered in [10] was a notion of overall a priori smoothness. Denney and Reeves [11] have developed a Bayesian approach for MRSI reconstruction that models the data in k -space-time and utilizes an edge preserving prior.…”

K-Bayes Reconstruction for Perfusion MRI I: Concepts and Application

“…However, the procedure was not designed to improve the reconstruction of low-resolution data and does not incorporate prior information. The method is extended in Schaewe and Miller 17 to incorporate a MRF prior model that encourages smooth image reconstructions. However, this prior is limited by not varying smoothness levels according to whether neighboring voxels are of the same tissue type or not.…”