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## On joint reconstruction of spin density, R2* decay and off-resonance frequency maps through a single-shot or multi-shot acquisition in magnetic resonance imaging

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##### Date

2015-07-28##### Author

Hu, Chenxi

##### Type of Degree

Dissertation##### Department

Electrical Engineering##### Metadata

Show full item record##### Abstract

Joint reconstruction of $R_2^*$ and off-resonance frequency maps is very important in many MR applications. For example, $R_2^*$ quantification can be applied to BOLD functional MRI, iron deposition measurement, and early detection of articular joint degeneration. Off-resonance quantification can be used to evaluate the severity of $B_0$ inhomogeneity, and it shows promise in MR thermometry. To reconstruct these parameter maps, a signal model must be specified, i.e., how the signal at each voxel evolves based on the spin density, $R_2^*$ decay, and off-resonance frequency at this voxel. A common signal model for this problem is the mono-exponential model, where the signal is a complex exponential function in time with the amplitude from the spin density and the decay rate from the other two parameters.
A common approach to reconstruct the three parameter maps given the model is based on a multi-echo sampling---sampling k-space at a series of echo times. After the sampling, each k-space frame is inverse Fourier transformed to reconstruct a series of images. A curve fitting is applied to the series of images on a voxel-by-voxel basis along the time domain to reconstruct the three parameter maps based on the mono-exponential model. This method is very straightforward in its reconstruction; however, it generally takes a relatively long time to acquire the multi-echo data, ranging from tens of seconds to tens of minutes. The long acquisition time reduces the practicality of the method and makes it difficult to use in some clinical applications such as cardiac MRI. Reduction of the acquisition time in MRI in general is very important and has attracted numerous researchers. Typical methods include parallel imaging, partial Fourier sampling, sparse multi-echo, and single- or multi-shot acquisition.
Sparse multi-echo is generally a direct application of compressed sensing techniques to this problem. The simplest idea is to sparsely sample each k-space frame in the multi-echo sampling, and then reconstruct each image by using compressed sensing. A difference from regular MR image reconstruction is the presence of a temporal dimension, and therefore a sparse representation of the signal in the temporal dimension is also very important. In fact, many authors have used sparse representations in both spatial and temporal domains so that a higher degree of undersampling is possible and a better reconstruction conditioning can be achieved.
Although the sparse multi-echo technique has been quite successful in its reconstruction quality, the time reduction achieved by the undersampling is typically only a factor of 4-6. Similar to the idea of undersampling, single- or multi-shot acquisition also undersamples the space that contains all k-space frames. However, a single- or multi-shot trajectory can achieve an even higher undersampling rate compared to sparse multi-echo. Typically, a single-shot trajectory only takes 40-80 milliseconds. Therefore, an important question is whether a quality reconstruction can be achieved with a single- or multi-shot k-space acquisition. In this dissertation, we propose two methods to do the underlying reconstruction and present analysis of the ill-conditioning of the problem under different single-shot trajectories. We present a new linear formulation of the mono-exponential model and demonstrate its power and potential applications in the reconstruction problem. In the following, we briefly introduce these contributions.
The first reconstruction method we propose in this dissertation is rooted in a classical nonlinear optimization strategy called trust-region methods. Specifically, we propose two trust region algorithms that use different local linearization techniques to address the same nonlinear optimization problem. A trust region is defined as a local area in the variable space where a local linear approximation is trustworthy. In each iteration, the method minimizes a local approximation within a trust region so that the step size can be kept in a suitable scale. A continuation scheme is applied to gradually reduce the regularization over the parameter maps and facilitate convergence from poor initializations. The two trust-region methods are compared to two other previously proposed methods---the nonlinear conjugate gradients algorithm and the gradual refinement algorithm. Experiments based on various synthetic data and real phantom data show that the two trust-region methods have a clear advantage in both speed and stability.
The second method we propose in this work employs variable splitting, which is very popular in dealing with convex optimization problems such as image reconstruction with nonsmooth regularizations. Most previous reconstruction algorithms are gradient-based iterative algorithms and the computational cost is high. We propose to reformulate the problem as a constrained optimization problem by employing auxiliary variables and use the well-known variable-splitting method to reduce the computational cost. We show that variable splitting for this problem is fundamentally different from variable splitting for many other applications, such as for regularized image reconstruction. As a result, the algorithm is very fast during the early stage of the iterations and much slower in the later stage. We propose a two-step method to address this issue. In the first step, we use the proposed variable-splitting method with regularization over both the auxiliary variable and the ordinary variables. We show the additional regularization is critical for the algorithm performance. In the second step, we employ our previously developed ordinary trust region algorithm to refine the estimate from the first step. We demonstrate that the hybrid method is faster than the ordinary trust region algorithm and the nonlinear conjugate gradient algorithm through simulation and \textit{in vivo} data.
The third contribution of this work deals with a new formulation of the mono-exponential model. The formulation is in linear form which can be utilized by some optimization methods such as alternating directions. The new formulation results in a new estimator for the spin density, $R_2^*$, and off-resonance. The new estimator works on multi-echo data. We compare the new estimator with other traditional curve fitting methods and demonstrate the speed and accuracy of the new estimator. We show at the end of the chapter how this new formulation as well as the new estimator can be applied to the second iterative reconstruction algorithm we developed in addressing the joint reconstruction problem.
The last contribution of this work deals with analysis of the ill-conditioning of the underlying problem under a variety of single-shot trajectories. The aim of the analysis is to find the optimal existing trajectory in terms of the conditioning of the reconstruction problem. We use tools such as the condition number and the singular value curve to study the ill-conditioning. We verify the analysis result by comparing the reconstruction accuracy of a reconstruction algorithm for every trajectory. Our results show that some trajectories are better than other trajectories due to certain strategies used to sample the data space. The findings can be useful to improve the reconstruction accuracy and convergence speed for any algorithm developed for the joint reconstruction problem.

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