Quantitative MRI by nonlinear inversion of the Bloch equations.

PURPOSE: Development of a generic model-based reconstruction framework for multiparametric quantitative MRI that can be used with data from different pulse sequences. METHODS: Generic nonlinear model-based reconstruction for quantitative MRI estimates parametric maps directly from the acquired k-space by numerical optimization. This requires numerically accurate and efficient methods to solve the Bloch equations and their partial derivatives. In this work, we combine direct sensitivity analysis and pre-computed state-transition matrices into a generic framework for calibrationless model-based reconstruction that can be applied to different pulse sequences. As a proof-of-concept, the method is implemented and validated for quantitative T 1 $$ {\mathrm{T}}_1 $$ and T 2 $$ {\mathrm{T}}_2 $$ mapping with single-shot inversion-recovery (IR) FLASH and IR bSSFP sequences in simulations, phantoms, and the human brain. RESULTS: The direct sensitivity analysis enables a highly accurate and numerically stable calculation of the derivatives. The state-transition matrices efficiently exploit repeating patterns in pulse sequences, speeding up the calculation by a factor of 10 for the examples considered in this work, while preserving the accuracy of native ordinary differential equations solvers. The generic model-based method reproduces quantitative results of previous model-based reconstructions based on the known analytical solutions for radial IR FLASH. For IR bSFFP it produces accurate T 1 $$ {\mathrm{T}}_1 $$ and T 2 $$ {\mathrm{T}}_2 $$ maps for the National Insitute of Standards and Technology (NIST) phantom in numerical simulations and experiments. Feasibility is also shown for human brain, although results are affected by magnetization transfer effects. CONCLUSION: By developing efficient tools for numerical optimizations using the Bloch equations as forward model, this work enables generic model-based reconstruction for quantitative MRI.

Nonlinear dipole inversion (NDI) enables robust quantitative susceptibility mapping (QSM).

High-quality Quantitative Susceptibility Mapping (QSM) with Nonlinear Dipole Inversion (NDI) is developed with pre-determined regularization while matching the image quality of state-of-the-art reconstruction techniques and avoiding over-smoothing that these techniques often suffer from. NDI is flexible enough to allow for reconstruction from an arbitrary number of head orientations and outperforms COSMOS even when using as few as 1-direction data. This is made possible by a nonlinear forward-model that uses the magnitude as an effective prior, for which we derived a simple gradient descent update rule. We synergistically combine this physics-model with a Variational Network (VN) to leverage the power of deep learning in the VaNDI algorithm. This technique adopts the simple gradient descent rule from NDI and learns the network parameters during training, hence requires no additional parameter tuning. Further, we evaluate NDI at 7 T using highly accelerated Wave-CAIPI acquisitions at 0.5 mm isotropic resolution and demonstrate high-quality QSM from as few as 2-direction data.

Dynamic water/fat separation and B0 inhomogeneity mapping-joint estimation using undersampled triple-echo multi-spoke radial FLASH.

PURPOSE: To achieve dynamic water/fat separation and B0 field inhomogeneity mapping via model-based reconstructions of undersampled triple-echo multi-spoke radial FLASH acquisitions. METHODS: This work introduces an undersampled triple-echo multi-spoke radial FLASH sequence, which uses (i) complementary radial spokes per echo train for faster spatial encoding, (ii) asymmetric echoes for flexible and nonuniform echo spacing, and (iii) a golden angle increment across frames for optimal k-space coverage. Joint estimation of water, fat, B0 inhomogeneity, and coil sensitivity maps from undersampled triple-echo data poses a nonlinear and non-convex inverse problem which is solved by a model-based reconstruction with suitable regularization. The developed methods are validated using phantom experiments with different degrees of undersampling. Real-time MRI studies of the knee, liver, and heart are conducted without prospective gating or retrospective data sorting at temporal resolutions of 70, 158, and 40 ms, respectively. RESULTS: Up to 18-fold undersampling is achieved in this work. Even in the presence of rapid physiological motion, large B0 field inhomogeneities, and phase wrapping, the model-based reconstruction yields reliably separated water/fat maps in conjunction with spatially smooth inhomogeneity maps. CONCLUSIONS: The combination of a triple-echo acquisition and joint reconstruction technique provides a practical solution to time-resolved and motion robust water/fat separation at high spatial and temporal resolution.

Weak-harmonic regularization for quantitative susceptibility mapping.

PURPOSE: Background-field removal is a crucial preprocessing step for quantitative susceptibility mapping (QSM). Remnants from this step often contaminate the estimated local field, which in turn leads to erroneous tissue-susceptibility reconstructions. The present work aimed to mitigate this undesirable behavior with the development of a new approach that simultaneously decouples background contributions and local susceptibility sources on QSM inversion. METHODS: Input phase data for QSM can be seen as a composite scalar field of local effects and residual background components. We developed a new weak-harmonic regularizer to constrain the latter and to separate the 2 components. The resulting optimization problem was solved with the alternating directions of multipliers method framework to achieve fast convergence. In addition, for convenience, a new alternating directions of multipliers method-based preconditioned nonlinear projection onto dipole fields solver was developed to enable initializations with wrapped-phase distributions. Weak-harmonic QSM, with and without nonlinear projection onto dipole fields preconditioning, was compared with the original (alternating directions of multipliers method-based) total variation QSM algorithm in phantom and in vivo experiments. RESULTS: Weak-harmonic QSM returned improved reconstructions regardless of the method used for background-field removal, although the proposed nonlinear projection onto dipole fields method often obtained better results. Streaking and shadowing artifacts were substantially suppressed, and residual background components were effectively removed. CONCLUSION: Weak-harmonic QSM with field preconditioning is a robust dipole inversion technique and has the potential to be extended as a single-step formulation for initialization with uncombined multi-echo data.

Simultaneous multi-slice MRI using cartesian and radial FLASH and regularized nonlinear inversion: SMS-NLINV.

PURPOSE: The development of a calibrationless parallel imaging method for accelerated simultaneous multi-slice (SMS) MRI based on Regularized Nonlinear Inversion (NLINV), evaluated using Cartesian and radial fast low-angle shot (FLASH). THEORY AND METHODS: NLINV is a parallel imaging method that jointly estimates image content and coil sensitivities using a Newton-type method with regularization. Here, NLINV is extended to SMS-NLINV for reconstruction and separation of all simultaneously acquired slices. The performance of the extended method is evaluated for different sampling schemes using phantom and in vivo experiments based on Cartesian and radial SMS-FLASH sequences. RESULTS: The basic algorithm was validated in Cartesian experiments by comparison with ESPIRiT. For Cartesian and radial sampling, improved results are demonstrated compared to single-slice experiments, and it is further shown that sampling schemes using complementary samples outperform schemes with the same samples in each partition. CONCLUSION: The extension of the NLINV algorithm for SMS data was implemented and successfully demonstrated in combination with a Cartesian and radial SMS-FLASH sequence. Magn Reson Med 79:2057-2066, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

Fast nonlinear susceptibility inversion with variational regularization.

PURPOSE: Quantitative susceptibility mapping can be performed through the minimization of a function consisting of data fidelity and regularization terms. For data consistency, a Gaussian-phase noise distribution is often assumed, which breaks down when the signal-to-noise ratio is low. A previously proposed alternative is to use a nonlinear data fidelity term, which reduces streaking artifacts, mitigates noise amplification, and results in more accurate susceptibility estimates. We hereby present a novel algorithm that solves the nonlinear functional while achieving computation speeds comparable to those for a linear formulation. METHODS: We developed a nonlinear quantitative susceptibility mapping algorithm (fast nonlinear susceptibility inversion) based on the variable splitting and alternating direction method of multipliers, in which the problem is split into simpler subproblems with closed-form solutions and a decoupled nonlinear inversion hereby solved with a Newton-Raphson iterative procedure. Fast nonlinear susceptibility inversion performance was assessed using numerical phantom and in vivo experiments, and was compared against the nonlinear morphology-enabled dipole inversion method. RESULTS: Fast nonlinear susceptibility inversion achieves similar accuracy to nonlinear morphology-enabled dipole inversion but with significantly improved computational efficiency. CONCLUSION: The proposed method enables accurate reconstructions in a fraction of the time required by state-of-the-art quantitative susceptibility mapping methods. Magn Reson Med 80:814-821, 2018. © 2018 International Society for Magnetic Resonance in Medicine.

Particle Swarm Optimization and Uncertainty Assessment in Inverse Problems.

Most inverse problems in the industry (and particularly in geophysical exploration) are highly underdetermined because the number of model parameters too high to achieve accurate data predictions and because the sampling of the data space is scarce and incomplete; it is always affected by different kinds of noise. Additionally, the physics of the forward problem is a simplification of the reality. All these facts result in that the inverse problem solution is not unique; that is, there are different inverse solutions (called equivalent), compatible with the prior information that fits the observed data within similar error bounds. In the case of nonlinear inverse problems, these equivalent models are located in disconnected flat curvilinear valleys of the cost-function topography. The uncertainty analysis consists of obtaining a representation of this complex topography via different sampling methodologies. In this paper, we focus on the use of a particle swarm optimization (PSO) algorithm to sample the region of equivalence in nonlinear inverse problems. Although this methodology has a general purpose, we show its application for the uncertainty assessment of the solution of a geophysical problem concerning gravity inversion in sedimentary basins, showing that it is possible to efficiently perform this task in a sampling-while-optimizing mode. Particularly, we explain how to use and analyze the geophysical models sampled by exploratory PSO family members to infer different descriptors of nonlinear uncertainty.

Image denoising for real-time MRI.

PURPOSE: To develop an image noise filter suitable for MRI in real time (acquisition and display), which preserves small isolated details and efficiently removes background noise without introducing blur, smearing, or patch artifacts. THEORY AND METHODS: The proposed method extends the nonlocal means algorithm to adapt the influence of the original pixel value according to a simple measure for patch regularity. Detail preservation is improved by a compactly supported weighting kernel that closely approximates the commonly used exponential weight, while an oracle step ensures efficient background noise removal. Denoising experiments were conducted on real-time images of healthy subjects reconstructed by regularized nonlinear inversion from radial acquisitions with pronounced undersampling. RESULTS: The filter leads to a signal-to-noise ratio (SNR) improvement of at least 60% without noticeable artifacts or loss of detail. The method visually compares to more complex state-of-the-art filters as the block-matching three-dimensional filter and in certain cases better matches the underlying noise model. Acceleration of the computation to more than 100 complex frames per second using graphics processing units is straightforward. CONCLUSION: The sensitivity of nonlocal means to small details can be significantly increased by the simple strategies presented here, which allows partial restoration of SNR in iteratively reconstructed images without introducing a noticeable time delay or image artifacts. Magn Reson Med 77:1340-1352, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

Reducing slab boundary artifacts in three-dimensional multislab diffusion MRI using nonlinear inversion for slab profile encoding (NPEN).

PURPOSE: To propose a method to reduce the slab boundary artifacts in three-dimensional multislab diffusion MRI. METHODS: Bloch simulation is used to investigate the effects of multiple factors on slab boundary artifacts, including characterization of residual errors on diffusion quantification. A nonlinear inversion method is proposed to simultaneously estimate the slab profile and the underlying (corrected) image. RESULTS: Correction results of numerical phantom and in vivo data demonstrate that the method can effectively remove slab boundary artifacts for diffusion data. Notably, the nonlinear inversion is also successful at short TR, a regimen where previously proposed methods (slab profile encoding and weighted average) retain residual artifacts in both diffusion-weighted images and diffusion metrics (mean diffusion coefficient and fractional anisotropy). CONCLUSION: The nonlinear inversion for removing slab boundary artifacts provides improvements over existing methods, particularly at the short TRs required to maximize SNR efficiency. Magn Reson Med 76:1183-1195, 2016. © 2015 The Authors. Magnetic Resonance in Medicine published by Wiley Periodicals, Inc. on behalf of International Society for Magnetic Resonance in Medicine. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Chemical shift separation with controlled aliasing for hyperpolarized (13) C metabolic imaging.

PURPOSE: A chemical shift separation technique for hyperpolarized (13) C metabolic imaging with high spatial and temporal resolution was developed. Specifically, a fast three-dimensional pulse sequence and a reconstruction method were implemented to acquire signals from multiple (13) C species simultaneously with subsequent separation into individual images. THEORY AND METHODS: A stack of flyback echo-planar imaging readouts and a set of multiband excitation radiofrequency pulses were designed to spatially modulate aliasing patterns of the acquired metabolite images, which translated the chemical shift separation problem into parallel imaging reconstruction problem. An eight-channel coil array was used for data acquisition and a parallel imaging method based on nonlinear inversion was developed to separate the aliased images. RESULTS: Simultaneous acquisitions of pyruvate and lactate in a phantom study and in vivo rat experiments were performed. The results demonstrated successful separation of the metabolite distributions into individual images having high spatial resolution. CONCLUSION: This method demonstrated the ability to provide accelerated metabolite imaging in hyperpolarized (13) C MR using multichannel coils, tailored readout, and specialized RF pulses.

Aggregated motion estimation for real-time MRI reconstruction.

PURPOSE: In real-time MRI serial images are generally reconstructed from highly undersampled datasets as the iterative solutions of an inverse problem. While practical realizations based on regularized nonlinear inversion (NLINV) have hitherto been surprisingly successful, strong assumptions about the continuity of image features may affect the temporal fidelity of the estimated reconstructions. THEORY AND METHODS: The proposed method for real-time image reconstruction integrates the deformations between nearby frames into the data consistency term of the inverse problem. The aggregated motion estimation (AME) is not required to be affine or rigid and does not need additional measurements. Moreover, it handles multi-channel MRI data by simultaneously determining the image and its coil sensitivity profiles in a nonlinear formulation which also adapts to non-Cartesian (e.g., radial) sampling schemes. The new method was evaluated for real-time MRI studies using highly undersampled radial gradient-echo sequences. RESULTS: AME reconstructions for a motion phantom with controlled speed as well as for measurements of human heart and tongue movements demonstrate improved temporal fidelity and reduced residual undersampling artifacts when compared with NLINV reconstructions without motion estimation. CONCLUSION: Nonlinear inverse reconstructions with aggregated motion estimation offer improved image quality and temporal acuity for visualizing rapid dynamic processes by real-time MRI.

Geobody estimation by Bhattacharyya method utilizing nonlinear inverse modeling of magnetic data in Baba-Ali iron deposit, NW Iran.

One of the essential geophysical concerns is the estimation of the physical and geometrical parameters of the reserve (geobody), which is done by exploiting the nonlinear inverse modeling of magnetic data. The present study includes preparing and modeling magnetic data to suggest the Baba Ali Iron ore deposit's drilling locations in NW Iran. The area is covered with 1000 points of geomagnetic reading with an almost 5 × 10 m2 regularly spaces grid trending WE. The areal and depth extent of the iron ore geobody was unknown. The Bhattacharyya method by MATLAB software coding was employed to underestimate the misfit function and re-construct potential field data, providing the most suitable fitting with measured magnetic data. In this order, the residual calculated anomaly exhibited an excellent two-dimensional conformation with forward modeling. Furthermore, 3D modeling correctly reconstructs the anomalies' productive resources' properties. After preparing full magnetic maps, the magnetic lenses distinguished in four anomalies of surface depths, 20, 50, and deeper than 50 m for this zone. This magnetite lens for the first zone was estimated based on analytical signal filters applied on the entire magnetic map so that the lens's depth is trivial and almost zero. Due to specific gravity calculated as 4.77 t/m3, initial storage capacity is suggested to be about 95,400 tons of magnetite, pyrite, and hematite minerals at most in an area about 6 Km2. Finally, to complete the preliminary explorations of the specified area, exploratory drilling is suggested for three points by inverse modeling. Regarding this study as the first try in magnetic reconnaissance step of Iron mineral exploration in the study area, there is no geological constraints available based on drilling evidences. However, the model is well satisfies the surface anomalies considering residual magnetic property.

Nonlinear Inversion of Ultrasonic Dispersion Curves for Cortical Bone Thickness and Elastic Velocities.

In this study, a nonlinear grid-search inversion has been developed to estimate the thickness and elastic velocities of long cortical bones, which are important determinants of bone strength, from axially-transmitted ultrasonic data. The inversion scheme is formulated in the dispersive frequency-phase velocity domain to recover bone properties. The method uses ultrasonic guided waves to retrieve overlying soft tissue thickness, cortical thickness, compressional, and shear-wave velocities of the cortex. The inversion strategy requires systematic examination of a large set of trial dispersion-curve solutions within a pre-defined model space to match the data with minimum cost in a least-squares sense. The theoretical dispersion curves required to solve the inverse problem are computed for bilayered bone models using a semi-analytical finite-element method. The feasibility of the proposed approach was demonstrated by the numerically simulated data for a 1 mm soft tissue-5 mm bone bilayer and ex-vivo data from a bovine femur plate with an overlying 2 mm-thick soft-tissue mimic. The bootstrap method was employed to evaluate the inversion uncertainty and stability. Our results have shown that the cortical thickness and wave speeds could be recovered with fair accuracy.

Zoeppritz-based AVO inversion using an improved Markov chain Monte Carlo method.

The conventional Markov chain Monte Carlo (MCMC) method is limited to the selected shape and size of proposal distribution and is not easy to start when the initial proposal distribution is far away from the target distribution. To overcome these drawbacks of the conventional MCMC method, two useful improvements in MCMC method, adaptive Metropolis (AM) algorithm and delayed rejection (DR) algorithm, are attempted to be combined. The AM algorithm aims at adapting the proposal distribution by using the generated estimators, and the DR algorithm aims at enhancing the efficiency of the improved MCMC method. Based on the improved MCMC method, a Bayesian amplitude versus offset (AVO) inversion method on the basis of the exact Zoeppritz equation has been developed, with which the P- and S-wave velocities and the density can be obtained directly, and the uncertainty of AVO inversion results has been estimated as well. The study based on the logging data and the seismic data demonstrates the feasibility and robustness of the method and shows that all three parameters are well retrieved. So the exact Zoeppritz-based nonlinear inversion method by using the improved MCMC is not only suitable for reservoirs with strong-contrast interfaces and long-offset ranges but also it is more stable, accurate and anti-noise.

Observation of direction-dependent mechanical properties in the human brain with multi-excitation MR elastography.

Magnetic resonance elastography (MRE) has shown promise in noninvasively capturing changes in mechanical properties of the human brain caused by neurodegenerative conditions. MRE involves vibrating the brain to generate shear waves, imaging those waves with MRI, and solving an inverse problem to determine mechanical properties. Despite the known anisotropic nature of brain tissue, the inverse problem in brain MRE is based on an isotropic mechanical model. In this study, distinct wave patterns are generated in the brain through the use of multiple excitation directions in order to characterize the potential impact of anisotropic tissue mechanics on isotropic inversion methods. Isotropic inversions of two unique displacement fields result in mechanical property maps that vary locally in areas of highly aligned white matter. Investigation of the corpus callosum, corona radiata, and superior longitudinal fasciculus, three highly ordered white matter tracts, revealed differences in estimated properties between excitations of up to 33%. Using diffusion tensor imaging to identify dominant fiber orientation of bundles, relationships between estimated isotropic properties and shear asymmetry are revealed. This study has implications for future isotropic and anisotropic MRE studies of white matter tracts in the human brain.

Non-identifiability of the Rayleigh damping material model in magnetic resonance elastography.

Magnetic Resonance Elastography (MRE) is an emerging imaging modality for quantifying soft tissue elasticity deduced from displacement measurements within the tissue obtained by phase sensitive Magnetic Resonance Imaging (MRI) techniques. MRE has potential to detect a range of pathologies, diseases and cancer formations, especially tumors. The mechanical model commonly used in MRE is linear viscoelasticity (VE). An alternative Rayleigh damping (RD) model for soft tissue attenuation is used with a subspace-based nonlinear inversion (SNLI) algorithm to reconstruct viscoelastic properties, energy attenuation mechanisms and concomitant damping behavior of the tissue-simulating phantoms. This research performs a thorough evaluation of the RD model in MRE focusing on unique identification of RD parameters, µI and ρI. Results show the non-identifiability of the RD model at a single input frequency based on a structural analysis with a series of supporting experimental phantom results. The estimated real shear modulus values (µR) were substantially correct in characterising various material types and correlated well with the expected stiffness contrast of the physical phantoms. However, estimated RD parameters displayed consistent poor reconstruction accuracy leading to unpredictable trends in parameter behaviour. To overcome this issue, two alternative approaches were developed: (1) simultaneous multi-frequency inversion; and (2) parametric-based reconstruction. Overall, the RD model estimates the real shear shear modulus (µR) well, but identifying damping parameters (µI and ρI) is not possible without an alternative approach.