Nonlinear Optimization Methods for Accelerating Magnetic Resonance Imaging

Abstract

Magnetic Resonance Imaging (MRI) is a non-invasive clinical imaging technique that does not use ionizing radiation and provides essential diagnosis information from tissues and structures of the body, which otherwise cannot be obtained with other popular medical imaging modalities like x-ray or computed tomography. These properties, in addition to its flexibility, explain why MRI is nowadays one of most widely used medical imaging techniques. Accordingly, MRI improvement and development is an important and multi-disciplinary research topic. This imaging technique suffers from the main limitation of being inherently slow, which provokes patient discomfort so increasing the possibility of motion during the scan time. This dissertation focuses on solving different challenges that arise from accelerating MRI through mathematical modeling and related optimization problems. Noise strength and image resolution in MRI directly depend on acquisition time, so accurate denoising and resolution enhancement are important tools to mitigate the effect of scan time reduction. The consideration of noise removal in a bayesian framework leads to the study of non-convex likelihood functions derived from the rician and non-central- distributions that govern MRI noise. In this work, variational models for image denoising are formulated using these likelihood terms with Total Variation (TV) or Total Generalized Variation (TGV) regularization. These are non-smooth and non-convex problems that are solved using efficient numerical techniques that include primal-dual and proximal point algorithms. The theoretical study for the case of TV-based rician denoising presented here proves the existence of at least one positive global minimizer of the functional. Successful comparison results with other TV-based rician denoising methods and in-vivo MRI tests show the effectiveness of the denoising algorithms here presented. Moreover, resolution enhancement is also considered in a variational framework, which results in the development of TGV-based super-resolution models that take into account the special properties of MRI noise. When tested on phantom and in-vivo MRI, the proposed methods outperform standard interpolation techniques, specially in very noisy conditions. These image processing techniques can alleviate undesired effects of shortened MRI scans, however, there are other approaches that attempt to find ways to accelerate the process of MRI data collection while preserving image quality. Among them, two techniques clearly stand out: parallel imaging and compressed sensing. Parallel imaging consist in using multiple receiver coils to acquire the data, which is subsequently combined to form the image. In compressed sensing, nonlinear methods are used to reconstruct randomly undersampled acquired data. In this work, a vectorial TGV based variational reconstruction model is proposed combining parallel imaging and compressed sensing with the use of shared information from different image contrasts of the same object. Its implementation using Alternating Direction Method of Multipliers (ADMM) and a primal-dual algorithm is explored, finding a more efficient performance of the ADMM method for this problem. Multi-contrast in-vivo results demonstrate an improvement of the proposed approach up to 17% in reconstruction accuracy (measured as root-mean-square error) over other widely used reconstruction methods. The last problem considered here is originated in the context of high-field MRI. By increasing the main magnetic field, among other benefits, the total amount of signal increases as well and consequently, less acquisition time is required. Nevertheless, high-field MRI suffers from some drawbacks such as augmented spatial inhomogeneity and the possibility of tissue heating, if the electrical fields generated are not properly controlled. Parallel transmission (pTx) pulse design has been demonstrated to provide a solution to both problems. Here, a robust approach for pTx pulse design is developed that allows to obtain a good field homogeneity while constraining electrical fields to be under safety limits even in the case of large radiofrequency errors (up to 20%). The nonlinear constrained optimization problem considered is solved using interior-point methods. Simulations of real pulse design scenarios using measured data errors confirmed the effectiveness of the proposed model and numerical resolution. The models and applications presented above show that mathematical optimization is a powerful tool for solving real problems arising in the continuous technological development of MRI. In particular, variational methods can provide solutions to different imaging challenges that result from MRI acceleration such as noise increasing, loss of image resolution or reconstruction from very undersampled data. The application-driven research presented in this manuscript opens new theoretical mathematical problems, which shall be considered in the future, following the path traced here for the case of TV operator. The results presented in this dissertation for multi-contrast MRI reconstruction and robust pulse design constitute the state-of-the art of these particular fields.