Solving 2D Fredholm Integral from Incomplete Measurements Using Compressive Sensing

Comparison of experimental times in T1-D and D-T2 correlation experiments in single-sided NMR

Diffusion-relaxation correlation experiments in nuclear magnetic resonance are a powerful technique for the characterization of fluid dynamics in confined geometries or soft matter, in which relaxation may be either spin-spin (T₂) or spin-lattice (T₁). The general approach is to acquire a set of bidimensional data in which diffusion is codified by the evolution of the magnetization with either Hahn or stimulated echoes (STE) in the presence of a constant magnetic field gradient. While T₂ is codified by a Carr-Purcell-Meiboom-Gil (CPMG) sequence, T₁ is either encoded by saturation or inversion-recovery methods. In this work, we analyse the measurement time of diffusion-relaxation times in single-sided NMR and show that T₁-D acquisition is always shorter than D-T₂. Depending on the hardware characteristics, this time reduction can be up to an order of magnitude. We present analytical calculations and examples in model porous media saturated with water and in a dairy product.

Combined diffusion-relaxometry microstructure imaging: Current status and future prospects

Microstructure imaging seeks to noninvasively measure and map microscopic tissue features by pairing mathematical modeling with tailored MRI protocols. This article reviews an emerging paradigm that has the potential to provide a more detailed assessment of tissue microstructure-combined diffusion-relaxometry imaging. Combined diffusion-relaxometry acquisitions vary multiple MR contrast encodings-such as b-value, gradient direction, inversion time, and echo time-in a multidimensional acquisition space. When paired with suitable analysis techniques, this enables quantification of correlations and coupling between multiple MR parameters-such as diffusivity, T 1 , T 2 , and T 2 ∗ . This opens the possibility of disentangling multiple tissue compartments (within voxels) that are indistinguishable with single-contrast scans, enabling a new generation of microstructural maps with improved biological sensitivity and specificity.

Ultrafast methods for relaxation and diffusion

Relaxation and diffusion NMR measurements offer an approach to studying rotational and translational motion of molecules non-invasively, and they also provide chemical resolution complementary to NMR spectra. Multidimensional experiments enable the correlation of relaxation and diffusion parameters as well as the observation of molecular exchange phenomena through relaxation or diffusion contrast. This review describes how to accelerate multidimensional relaxation and diffusion measurements significantly through spatial encoding. This so-called ultrafast Laplace NMR approach shortens the experiment time to a fraction and makes even single-scan experiments possible. Single-scan experiments, in turn, significantly facilitate the use of nuclear spin hyperpolarization methods to boost sensitivity. The ultrafast Laplace NMR method is also applicable with low-field, mobile NMR instruments, and it can be exploited in many disciplines. For example, it has been used in studies of the dynamics of fluids in porous materials, identification of intra- and extracellular metabolites in cancer cells, and elucidation of aggregation phenomena in atmospheric surfactant solutions.

A New Hybrid Inversion Method for 2D Nuclear Magnetic Resonance Combining TSVD and Tikhonov Regularization

This paper is concerned with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) relaxometry. This is a large-scale and ill-posed inverse problem with many potential applications in biology, medicine, chemistry, and other disciplines. However, the large amount of data and the consequently long inversion times, together with the high sensitivity of the solution to the value of the regularization parameter, still represent a major issue in the applicability of the NMR relaxometry. We present a method for two-dimensional data inversion (2DNMR) which combines Truncated Singular Value Decomposition and Tikhonov regularization in order to accelerate the inversion time and to reduce the sensitivity to the value of the regularization parameter. The Discrete Picard condition is used to jointly select the SVD truncation and Tikhonov regularization parameters. We evaluate the performance of the proposed method on both simulated and real NMR measurements.

Fast multicomponent 3D-T1ρ relaxometry

NMR relaxometry can provide information about the relaxation of the magnetization in different tissues, increasing our understanding of molecular dynamics and biochemical composition in biological systems. In general, tissues have complex and heterogeneous structures composed of multiple pools. As a result, bulk magnetization returns to its original state with different relaxation times, in a multicomponent relaxation. Recovering the distribution of relaxation times in each voxel is a difficult inverse problem; it is usually unstable and requires long acquisition time, especially on clinical scanners. MRI can also be viewed as an inverse problem, especially when compressed sensing (CS) is used. The solution of these two inverse problems, CS and relaxometry, can be obtained very efficiently in a synergistically combined manner, leading to a more stable multicomponent relaxometry obtained with short scan times. In this paper, we will discuss the details of this technique from the viewpoint of inverse problems.

Multidimensional correlation MRI

Multidimensional correlation spectroscopy is emerging as a novel MRI modality that is well suited for microstructure and microdynamic imaging studies, especially of biological specimens. Conventional MRI methods only provide voxel-averaged and mostly macroscopically averaged information; these methods cannot disentangle intra-voxel heterogeneity on the basis of both water mobility and local chemical interactions. By correlating multiple MR contrast mechanisms and processing the data in an integrated manner, correlation spectroscopy is able to resolve the distribution of water populations according to their chemical and physical interactions with the environment. The use of a non-parametric, phenomenological representation of the multidimensional MR signal makes no assumptions about tissue structure, thereby allowing the study of microscopic structure and composition of complex heterogeneous biological systems. However, until recently, vast data requirements have confined these types of measurement to non-localized NMR applications and prevented them from being widely and successfully used in conjunction with imaging. Recent groundbreaking advancements have allowed this powerful NMR methodology to be migrated to MRI, initiating its emergence as a promising imaging approach. This review is not intended to cover the entire field of multidimensional MR; instead, it focuses on pioneering imaging applications and the challenges involved. In addition, the background and motivation that have led to multidimensional correlation MR development are discussed, along with the basic underlying mathematical concepts. The goal of the present work is to provide the reader with a fundamental understanding of the techniques developed and their potential benefits, and to provide guidance to help refine future applications of this technology.

A Tutorial Introduction to Inverse Problems in Magnetic Resonance

There has been a tremendous increase in applications of the inverse problem framework to parameter estimation in magnetic resonance. Attempting to capture both the basics of this formalism and modern developments would require an article of inordinate length. Therefore, in the following, we provide basic material as a practical introduction to the topic and an entree to the literature. First, we describe the formulation of linear and nonlinear inverse problems, with an emphasis on signal equations arising in magnetic resonance. We then describe the Fredholm equation of the first kind as a paradigm for these problems. This is followed by much more detailed considerations for determining solutions in the linear case, including central concepts such as condition number, regularization, and stability. Solution methods for nonlinear inverse problems are described next, followed by a treatment of their stability and regularization. Finally, we provide an introduction to compressed sensing, with signal reconstruction formulated as the solution to an inverse problem, making use of much of the previous material. Throughout, the emphasis is on outlines of the theory and on numerical examples, rather than on mathematical rigor and completeness.

Microstructural imaging of the human brain with a 'super-scanner': 10 key advantages of ultra-strong gradients for diffusion MRI

The key component of a microstructural diffusion MRI 'super-scanner' is a dedicated high-strength gradient system that enables stronger diffusion weightings per unit time compared to conventional gradient designs. This can, in turn, drastically shorten the time needed for diffusion encoding, increase the signal-to-noise ratio, and facilitate measurements at shorter diffusion times. This review, written from the perspective of the UK National Facility for In Vivo MR Imaging of Human Tissue Microstructure, an initiative to establish a shared 300 mT/m-gradient facility amongst the microstructural imaging community, describes ten advantages of ultra-strong gradients for microstructural imaging. Specifically, we will discuss how the increase of the accessible measurement space compared to a lower-gradient systems (in terms of Δ, b-value, and TE) can accelerate developments in the areas of 1) axon diameter distribution mapping; 2) microstructural parameter estimation; 3) mapping micro-vs macroscopic anisotropy features with gradient waveforms beyond a single pair of pulsed-gradients; 4) multi-contrast experiments, e.g. diffusion-relaxometry; 5) tractography and high-resolution imaging in vivo and 6) post mortem; 7) diffusion-weighted spectroscopy of metabolites other than water; 8) tumour characterisation; 9) functional diffusion MRI; and 10) quality enhancement of images acquired on lower-gradient systems. We finally discuss practical barriers in the use of ultra-strong gradients, and provide an outlook on the next generation of 'super-scanners'.

Towards clinically feasible relaxation-diffusion correlation MRI using MADCO

Multidimensional relaxation-diffusion correlation (REDCO) NMR is an assumption-free method that measures how water is distributed within materials. Although highly informative, REDCO had never been used in clinical MRI applications because of the large amount of data it requires, leading to infeasible scan times. A recently suggested novel experimental design and processing framework, marginal distributions constrained optimization (MADCO), was used here to accelerate and improve the reconstruction of such MRI correlations. MADCO uses the 1D marginal distributions as a priori information, which provide powerful constraints when 2D spectra are reconstructed, while their estimation requires an order of magnitude less data than conventional 2D approaches. In this work we experimentally examined the impact the complexity of the correlation distribution has on the accuracy and robustness of the estimates. MADCO and a conventional method were compared using two T1-D phantoms that differ in the proximity of their peaks, leading to a relatively simple case as opposed to a more challenging one. The phantoms were used to vet the achievable data compression using MADCO under these conditions. MADCO required ~43 and ~30 less data than the conventional approach for the simple and complex spectra, respectively, making it potentially feasible for preclinical and even clinical applications.

Diffusion-relaxation correlation spectroscopic imaging: A multidimensional approach for probing microstructure

PURPOSE

To propose and evaluate a novel multidimensional approach for imaging subvoxel tissue compartments called Diffusion-Relaxation Correlation Spectroscopic Imaging.

THEORY AND METHODS

Multiexponential modeling of MR diffusion or relaxation data is commonly used to infer the many different microscopic tissue compartments that contribute signal to macroscopic MR imaging voxels. However, multiexponential estimation is known to be difficult and ill-posed. Observing that this ill-posedness is theoretically reduced in higher dimensions, diffusion-relaxation correlation spectroscopic imaging uses a novel multidimensional imaging experiment that jointly encodes diffusion and relaxation information, and then uses a novel constrained reconstruction technique to generate a multidimensional diffusion-relaxation correlation spectrum for every voxel. The peaks of the multidimensional spectrum are expected to correspond to the distinct tissue microenvironments that are present within each macroscopic imaging voxel.

RESULTS

Using numerical simulations, experiment data from a custom-built phantom, and experiment data from a mouse model of traumatic spinal cord injury, diffusion-relaxation correlation spectroscopic imaging is demonstrated to provide substantially better multicompartment resolving power compared to conventional diffusion- and relaxation-based methods.

CONCLUSION

The diffusion-relaxation correlation spectroscopic imaging approach provides powerful new capabilities for resolving the different components of multicompartment tissue models, and can be leveraged to significantly expand the insights provided by MRI in studies of tissue microstructure. Magn Reson Med 78:2236-2249, 2017. © 2017 International Society for Magnetic Resonance in Medicine.

Fast, accurate 2D-MR relaxation exchange spectroscopy (REXSY): Beyond compressed sensing

Previously, we showed that compressive or compressed sensing (CS) can be used to reduce significantly the data required to obtain 2D-NMR relaxation and diffusion spectra when they are sparse or well localized. In some cases, an order of magnitude fewer uniformly sampled data were required to reconstruct 2D-MR spectra of comparable quality. Nonetheless, this acceleration may still not be sufficient to make 2D-MR spectroscopy practicable for many important applications, such as studying time-varying exchange processes in swelling gels or drying paints, in living tissue in response to various biological or biochemical challenges, and particularly for in vivo MRI applications. A recently introduced framework, marginal distributions constrained optimization (MADCO), tremendously accelerates such 2D acquisitions by using a priori obtained 1D marginal distribution as powerful constraints when 2D spectra are reconstructed. Here we exploit one important intrinsic property of the 2D-MR relaxation exchange spectra: the fact that the 1D marginal distributions of each 2D-MR relaxation exchange spectrum in both dimensions are equal and can be rapidly estimated from a single Carr-Purcell-Meiboom-Gill (CPMG) or inversion recovery prepared CPMG measurement. We extend the MADCO framework by further proposing to use the 1D marginal distributions to inform the subsequent 2D data-sampling scheme, concentrating measurements where spectral peaks are present and reducing them where they are not. In this way we achieve compression or acceleration that is an order of magnitude greater than that in our previous CS method while providing data in reconstructed 2D-MR spectral maps of comparable quality, demonstrated using several simulated and real 2D T2 - T2 experimental data. This method, which can be called "informed compressed sensing," is extendable to other 2D- and even ND-MR exchange spectroscopy.

Compressed Sensing Electron Tomography for Determining Biological Structure

There has been growing interest in applying compressed sensing (CS) theory and practice to reconstruct 3D volumes at the nanoscale from electron tomography datasets of inorganic materials, based on known sparsity in the structure of interest. Here we explore the application of CS for visualizing the 3D structure of biological specimens from tomographic tilt series acquired in the scanning transmission electron microscope (STEM). CS-ET reconstructions match or outperform commonly used alternative methods in full and undersampled tomogram recovery, but with less significant performance gains than observed for the imaging of inorganic materials. We propose that this disparity stems from the increased structural complexity of biological systems, as supported by theoretical CS sampling considerations and numerical results in simulated phantom datasets. A detailed analysis of the efficacy of CS-ET for undersampled recovery is therefore complicated by the structure of the object being imaged. The numerical nonlinear decoding process of CS shares strong connections with popular regularized least-squares methods, and the use of such numerical recovery techniques for mitigating artifacts and denoising in reconstructions of fully sampled datasets remains advantageous. This article provides a link to the software that has been developed for CS-ET reconstruction of electron tomographic data sets.

Efficient 2D MRI relaxometry using compressed sensing

Potential applications of 2D relaxation spectrum NMR and MRI to characterize complex water dynamics (e.g., compartmental exchange) in biology and other disciplines have increased in recent years. However, the large amount of data and long MR acquisition times required for conventional 2D MR relaxometry limits its applicability for in vivo preclinical and clinical MRI. We present a new MR pipeline for 2D relaxometry that incorporates compressed sensing (CS) as a means to vastly reduce the amount of 2D relaxation data needed for material and tissue characterization without compromising data quality. Unlike the conventional CS reconstruction in the Fourier space (k-space), the proposed CS algorithm is directly applied onto the Laplace space (the joint 2D relaxation data) without compressing k-space to reduce the amount of data required for 2D relaxation spectra. This framework is validated using synthetic data, with NMR data acquired in a well-characterized urea/water phantom, and on fixed porcine spinal cord tissue. The quality of the CS-reconstructed spectra was comparable to that of the conventional 2D relaxation spectra, as assessed using global correlation, local contrast between peaks, peak amplitude and relaxation parameters, etc. This result brings this important type of contrast closer to being realized in preclinical, clinical, and other applications.