Nitroxide biradicals have been prepared with electron-electron spin-spin exchange interaction, J, ranging from weak to very strong. EPR spectra of these biradicals in fluid solution depend on the ratio of J to the nitrogen hyperfine coupling, A, and the rates of interconversion between conformations with different values of J. For relatively rigid biradicals EPR spectra can be simulated as the superposition of AB splitting patterns arising from different combinations of nitrogen nuclear spin states. For more flexible biradicals spectra can be simulated with a Liouville representation of the dynamics that interconvert conformations with different values of J on the EPR timescale. Analysis of spectra, factors that impact J, and examples of applications to chemical and biophysical problems are discussed.

Protein NMR resonance assignment can be a tedious and error prone process, and it is often a limiting factor in biomolecular NMR studies. Challenges are exacerbated in larger proteins, disordered proteins, and often alpha-helical proteins, owing to an increase in spectral complexity and frequency degeneracies. Here, several multi-dimensional spectra must be inspected and compared in an iterative manner before resonances can be assigned with confidence. Over the last two decades, covariance NMR has evolved to become applicable to protein multi-dimensional spectra. The method, previously used to generate new correlations from spectra of small organic molecules, can now be used to recast assignment procedures as mathematical operations on NMR spectra. These operations result in multidimensional correlation maps combining all information from input spectra and providing direct correlations between moieties that would otherwise be compared indirectly through reporter nuclei. Thus, resonances of sequential residues can be identified and side-chain signals can be assigned by visual inspection of 4D arrays. This review highlights advances in covariance NMR that permitted to generate reliable 4D arrays and describes how these arrays can be obtained from conventional NMR spectra.

Multiple-pulse NMR experiments are a powerful tool for the investigation of molecules with coupled nuclear spins. The product operator formalism provides a way to understand the quantum evolution of an ensemble of weakly coupled spins in such experiments using some of the more intuitive concepts of classical physics and semi-classical vector representations. In this paper I present a new way in which to interpret the quantum evolution of an ensemble of spins. I recast the quantum problem in terms of mixtures of pure states of two spins whose expectation values evolve identically to those of classical moments. Pictorial representations of these classically evolving states provide a way to calculate the time evolution of ensembles of weakly coupled spins without the full machinery of quantum mechanics, offering insight to anyone who understands precession of magnetic moments in magnetic fields.

The extension of the Bloch equations for acid-base reactions in an aqueous solution is revisited. The acid-base reactions are second-order, and several reactions catalyzed by distinct catalysts may happen simultaneously. By constructing pseudo first-order reactions and assuming fast dissemination of protons from catalysts to solvent water, this extension converges to the well-known Bloch-McConnell equations for a two-site first-order exchange. Thus, explicit relationships between the parameters appearing in the reactions and the Bloch-McConnell equations are established. The dependencies of exchange rates and chemical exchange saturation transfer effects on pH were numerically and experimentally investigated for representative examples.

Abundant solvent nuclear spins, such as water protons in aqueous solution, cause radiation damping in NMR experiments. It is important to know how the effect of radiation damping appears in high-resolution protein NMR because macromolecular studies always require very high magnetic field strengths with a highly sensitive NMR probe that can easily cause radiation damping. Here, we show the behavior of water magnetization after a pulsed-field gradient (PFG) using nutation experiments at 900 MHz with a cryogenic probe: when water magnetization is located in the upper hemisphere (having +Z component, parallel to the external magnetic field), dephasing of the magnetization by a PFG effectively suppresses residual water magnetization in the transverse plane. In contrast, when magnetization is located in the lower hemisphere (having -Z component), the small residual transverse component remaining after a PFG is still sufficient to induce radiation damping. Based on this observation, we designed H-N HSQC experiments in which water magnetization is maintained in the upper hemisphere, but not necessarily along Z, and compared them with the conventional experiments, in which water magnetization is inverted during the t period. The result demonstrates moderate gain of signal-to-noise ratio, 0-28%. Designing the experiments such that water magnetization is maintained in the upper hemisphere allows shorter pulses to be used compared to the complete water flip-back and, thereby, is useful as a building block of protein NMR pulse programs in solution.

The influence of Gaussian diffusion on the magnetic resonance signal is determined by the apparent diffusion coefficient (ADC) and tensor (ADT) of the diffusing fluid as well as the gradient waveform applied to sensitize the signal to diffusion. Estimations of ADC and ADT from diffusion-weighted acquisitions necessitate computations of, respectively, the -value and -matrix associated with the employed pulse sequence. We establish the relationship between these quantities and the gradient waveform by expressing the problem as a path integral and explicitly evaluating it. Further, we show that these important quantities can be conveniently computed for any gradient waveform using a simple algorithm that requires a few lines of code. With this representation, our technique complements the multiple correlation function (MCF) method commonly used to compute the effects of restricted diffusion, and provides a consistent and convenient framework for studies that aim to infer the microstructural features of the specimen.

In biomedical imaging, edge sharpness is an important yet often overlooked image quality metric. In this work, a semi-automatic method to quantify edge sharpness in the presence of significant noise is presented with application to magnetic resonance imaging (MRI). The method is based on parametric modeling of image edges. First, an edge map is automatically generated and one or more edges-of-interest (EOI) are manually selected using graphical user interface. Multiple exclusion criteria are then enforced to eliminate edge pixels that are potentially not suitable for sharpness assessment. Second, at each pixel of the EOI, an image intensity profile is read along a small line segment that runs locally normal to the EOI. Third, the profiles corresponding to all EOI pixels are individually fitted with a sigmoid function characterized by four parameters, including one that represents edge sharpness. Last, the distribution of the sharpness parameter is used to quantify edge sharpness. For validation, the method is applied to simulated data as well as MRI data from both phantom imaging and cine imaging experiments. This method allows for fast, quantitative evaluation of edge sharpness even in images with poor signal-to-noise ratio. Although the utility of this method is demonstrated for MRI, it can be adapted for other medical imaging applications.

Fetal magnetic resonance imaging (MRI) examinations have become well-established procedures at many institutions and can serve as useful adjuncts to ultrasound (US) exams when diagnostic doubts remain after US. Due to fetal motion, however, fetal MRI exams are challenging and require the MR scanner to be used in a somewhat different mode than that employed for more routine clinical studies. Herein we review the techniques most commonly used, and those that are available, for fetal MRI with an emphasis on the physics of the techniques and how to deploy them to improve success rates for fetal MRI exams. By far the most common technique employed is single-shot T2-weighted imaging due to its excellent tissue contrast and relative immunity to fetal motion. Despite the significant challenges involved, however, many of the other techniques commonly employed in conventional neuro- and body MRI such as T1 and T2*-weighted imaging, diffusion and perfusion weighted imaging, as well as spectroscopic methods remain of interest for fetal MR applications. An effort to understand the strengths and limitations of these basic methods within the context of fetal MRI is made in order to optimize their use and facilitate implementation of technical improvements for the further development of fetal MR imaging, both in acquisition and post-processing strategies.

Lung parenchyma is challenging to image with proton MRI. The large air space results in ~l/5th as many signal-generating protons compared to other organs. Air/tissue magnetic susceptibility differences lead to strong magnetic field gradients throughout the lungs and to broad frequency distributions, much broader than within other organs. Such distributions have been the subject of experimental and theoretical analyses which may reveal aspects of lung microarchitecture useful for diagnosis. Their most immediate relevance to current imaging practice is to cause rapid signal decays, commonly discussed in terms of short T values of 1 ms or lower at typical imaging field strengths. Herein we provide a brief review of previous studies describing and interpreting proton lung spectra. We then link these broad frequency distributions to rapid signal decays, though not necessarily the exponential decays generally used to define T values. We examine how these decays influence observed signal intensities and spatial mapping features associated with the most prominent torso imaging sequences, including spoiled gradient and spin echo sequences. Effects of imperfect refocusing pulses on the multiple echo signal decays in single shot fast spin echo (SSFSE) sequences and effects of broad frequency distributions on balanced steady state free precession (bSSFP) sequence signal intensities are also provided. The theoretical analyses are based on the concept of explicitly separating the effects of reversible and irreversible transverse relaxation processes, thus providing a somewhat novel and more general framework from which to estimate lung signal intensity behavior in modern imaging practice.

In this article, we discuss the theoretical background for diffusion weighted imaging and diffusion tensor imaging. Molecular diffusion is a random process involving thermal Brownian motion. In biological tissues, the underlying microstructures restrict the diffusion of water molecules, making diffusion directionally dependent. Water diffusion in tissue is mathematically characterized by the diffusion tensor, the elements of which contain information about the magnitude and direction of diffusion and is a function of the coordinate system. Thus, it is possible to generate contrast in tissue based primarily on diffusion effects. Expressing diffusion in terms of the measured diffusion coefficient (eigenvalue) in any one direction can lead to errors. Nowhere is this more evident than in white matter, due to the preferential orientation of myelin fibers. The directional dependency is removed by diagonalization of the diffusion tensor, which then yields a set of three eigenvalues and eigenvectors, representing the magnitude and direction of the three orthogonal axes of the diffusion ellipsoid, respectively. For example, the eigenvalue corresponding to the eigenvector along the long axis of the fiber corresponds qualitatively to diffusion with least restriction. Determination of the principal values of the diffusion tensor and various anisotropic indices provides structural information. We review the use of diffusion measurements using the modified Stejskal-Tanner diffusion equation. The anisotropy is analyzed by decomposing the diffusion tensor based on symmetrical properties describing the geometry of diffusion tensor. We further describe diffusion tensor properties in visualizing fiber tract organization of the human brain.

A thorough review of the q-space technique is presented starting from a discussion of Fick's laws. The work presented here is primarily conceptual, theoretical and hopefully pedagogical. We offered the notion of molecular concentration to unify Fick's laws and diffusion MRI within a coherent conceptual framework. The fundamental relationship between diffusion MRI and the Fick's laws are carefully established. The conceptual and theoretical basis of the q-space technique is investigated from first principles.

Low-resolution nuclear magnetic resonance (LR-NMR) relaxometry is a powerful tool that can be harnessed for characterizing constituents in complex materials. Conversion of the relaxation signal into a continuous distribution of relaxation components is an ill-posed inverse Laplace transform problem. The most common numerical method implemented today for dealing with this kind of problem is based on -norm regularization. However, sparse representation methods via regularization and convex optimization are a relatively new approach for effective analysis and processing of digital images and signals. In this article, a numerical optimization method for analyzing LR-NMR data by including non-negativity constraints and regularization and by applying a convex optimization solver PDCO, a primal-dual interior method for convex objectives, that allows general linear constraints to be treated as linear operators is presented. The integrated approach includes validation of analyses by simulations, testing repeatability of experiments, and validation of the model and its statistical assumptions. The proposed method provides better resolved and more accurate solutions when compared with those suggested by existing tools. © 2013 Wiley Periodicals, Inc. Concepts Magn Reson Part A 42A: 72-88, 2013.

Segmented interleaved echo planar imaging (EPI) is a highly efficient data acquisition technique; however, EPI is sensitive to artifacts from off-resonance spins. The choice of k-space trajectories is important in determining how off-resonance spins contribute to image artifacts. Top-down and center-out trajectories are theoretically analyzed, simulated, implemented, and tested in phantom and volunteer experiments. Theoretical results show off-resonance artifact manifests as a simple positional shift for the top-down trajectory, while for the center-out trajectory off-resonance artifact manifests as a splitting of the object, which entails both shift and blurring. These results were validated using simulation and phantom scan data where a frequency-offset was introduced ranging from -300 Hz to +300 Hz. As predicted by the theoretical results, inferior image quality was observed for the center-out trajectory in a single volunteer. Off-resonance produces more severe and complex artifacts with the center-out trajectory than the top-down trajectory.

Electron paramagnetic resonance (EPR) experimental techniques produce absorption or first-derivative spectra. Uncertainty analysis provides the basis for comparison of spectra obtained by different methods. In this study it was used to derive analytical equations to relate uncertainties for integrated intensity and line widths obtained from absorption or first-derivative spectra to the signal-to-noise ratio (SNR), with the assumption of white noise. Predicted uncertainties for integrated intensities and line widths are in good agreement with Monte Carlo calculations for Lorentzian and Gaussian lineshapes. Conservative low-pass filtering changes the noise spectrum, which can be modeled in the Monte Carlo simulations. When noise is close to white, the analytical equations provide useful estimates of uncertainties. For example, for a Lorentzian line with white noise, the uncertainty in the number of spins obtained from the first-derivative spectrum is 2.6 times greater than from the absorption spectrum at the same SNR. Uncertainties in line widths obtained from absorption and first-derivative spectra are similar. The impact of integration or differentiation on SNR and on uncertainties in fitting parameters was analyzed. Although integration of the first-derivative spectrum improves the apparent smoothness of the spectrum, it also changes the frequency distribution of the noise. If the lineshape of the signal is known, the integrated intensity can be determined more accurately by fitting the first-derivative spectrum than by first integrating and then fitting the absorption spectrum. Uncertainties in integrated intensities and line widths are less when the parameters are determined from the original data than from spectra that have been either integrated or differentiated.

Sequences that encode the spatial information of an object using nonlinear gradient fields are a new frontier in MRI, with potential to provide lower peripheral nerve stimulation, windowed fields of view, tailored spatially-varying resolution, curved slices that mirror physiological geometry, and, most importantly, very fast parallel imaging with multichannel coils. The acceleration for multichannel images is generally explained by the fact that curvilinear gradient isocontours better complement the azimuthal spatial encoding provided by typical receiver arrays. However, the details of this complementarity have been more difficult to specify. We present a simple and intuitive framework for describing the mechanics of image formation with nonlinear gradients, and we use this framework to review some the main classes of nonlinear encoding schemes.

Multi-turn spiral surface coils are constructed in flat and cylindrical arrangements and used for high field (7.1 T) mouse cardiac MRI. Their electrical and imaging performances, based on experimental measurements, simulations, and MRI experiments in free space, and under phantom, and animal loading conditions, are compared with a commercially available birdcage coil. Results show that the four-turn cylindrical spiral coil exhibits improved relative SNR (rSNR) performance to the flat coil counterpart, and compares fairly well with a commercially available birdcage coil. Phantom experiments indicate a 50% improvement in the SNR for penetration depths ≤ 6.1 mm from the coil surface compared to the birdcage coil, and an increased penetration depth at the half-maximum field response of 8 mm in the 4-spiral cylindrical coil case, in contrast to 2.9 mm in the flat 4-turn spiral case. Quantitative comparison of the performance of the two spiral coil geometries in anterior, lateral, inferior, and septal regions of the murine heart yield maximum mean percentage rSNR increases of the order of 27-167% in vivo post-mortem (cylindrical compared to flat coil). The commercially available birdcage outperforms the cylindrical spiral coil in rSNR by a factor of 3-5 times. The comprehensive approach and methodology adopted to accurately design, simulate, implement, and test radiofrequency coils of any geometry and type, under any loading conditions, can be generalized for any application of high field mouse cardiac MRI.

A type of "matched filter" (MF), used extensively in the processing of one-dimensional spectra, is defined by multiplication of a free-induction decay (FID) by a decaying exponential with the same time constant as that of the FID. This maximizes, in a sense to be defined, the signal-to-noise ratio (SNR) in the spectrum obtained after Fourier transformation. However, a different entity known also as the matched filter was introduced by van Vleck in the context of pulse detection in the 1940's and has become widely integrated into signal processing practice. These two types of matched filters appear to be quite distinct. In the NMR case, the "filter", that is, the exponential multiplication, is defined by the characteristics of, and applied to, a time domain signal in order to achieve improved SNR in the spectral domain. In signal processing, the filter is defined by the characteristics of a signal in the spectral domain, and applied in order to improve the SNR in the temporal (pulse) domain. We reconcile these two distinct implementations of the matched filter, demonstrating that the NMR "matched filter" is a special case of the matched filter more rigorously defined in the signal processing literature. In addition, two limitations in the use of the MF are highlighted. First, application of the MF distorts resonance ratios as defined by amplitudes, although not as defined by areas. Second, the MF maximizes SNR with respect to resonance amplitude, while intensities are often more appropriately defined by areas. Maximizing the SNR with respect to area requires a somewhat different approach to matched filtering.

The product-operator formalism is the most commonly used tool for describing and designing multidimensional NMR experiments. In spite of its relative simplicity and sound theoretical underpinnings, however, students and practitioners often find it difficult to relate the mathematical manipulations to a physical picture. In an effort to address this pedagogical challenge, the present paper begins with a quantum-mechanical treatment of pure populations of scalar-coupled spin-pairs, rather than the equilibrium population of spin-pairs in different quantum states, which is the usual starting point for treatments based on the density matrix and product operators. In the context of pure populations, the product operators are shown to represent quantum correlations between the nuclei in individual molecules, and a new variation on the classical vector diagram is introduced to represent these correlations. The treatment is extended to mixed populations that begin at thermal equilibrium, and the density matrix is introduced as an efficient means of carrying out quantum calculations for a mixed population. Finally, it is shown that the operators for observable magnetization and correlations can be used as a basis set for the density matrix, providing the formal justification for the widely-used rules of the product-operator treatment. Throughout the discussion, the vector diagrams are used to help maintain a connection between the mathematics and the sometimes subtle physical principles. An electronic supplement created with the Mathematica computer program is used to provide additional mathematical details and the means to carry out further calculations.

The frequency dependence of the signal-to-noise ratio (S/N) that is theoretically possible for pulsed EPR experiments is the same as for continuous wave experiments. To select the optimum resonance frequency or frequencies for pulsed EPR experiments it is important to consider not only S/N, but also orientation selection, depth of spin echo modulation, and intensities of forbidden transitions. Evaluation of factors involved in selecting the optimum frequency for pulsed EPR measurements of distances between spins is discussed.

The time required to complete a multidimensional NMR experiment is directly proportional to the number of evolution times sampled in the indirect dimensions. A consequence when utilizing conventional methods of data acquisition and spectrum analysis is that resolution in the indirect dimensions is frequently sample-limited. The problem becomes more acute at very high magnetic fields, where increased chemical shift dispersion requires shorter time increments to avoid aliasing. It has long been recognized that a way to avoid this limitation is to utilize methods of spectrum analysis that do not require data to be sampled at uniform intervals, permitting the collection of data at long evolution times requisite for high resolution without requiring collection of data at all intervening multiples of the sampling interval. Several promising methods have evolved that are seemingly quite different, yet can be shown to yield similar results when applied to similar sampling strategies, emphasizing the importance of the choice of samples, regardless of the technique used to compute the spectrum. Maximum entropy (MaxEnt) reconstruction is a very general method for spectrum analysis of non-uniformly sampled data (NUS), and because it can be used with essentially arbitrary sampling strategies and makes no assumptions about the nature of the signal, it provides a convenient basis for exploring the influence of the choice of samples on spectral quality. In this article we use this versatility of MaxEnt reconstruction to compare different approaches to NUS in multidimensional NMR and suggest strategies for improving spectral quality by careful choice of sample times.