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PURPOSE: Computational simulation of phase-contrast MRI (PC-MRI) is an attractive way to physically interpret properties and errors in MRI-reconstructed flow velocity fields. Recent studies have developed PC-MRI simulators that solve the Bloch equation, with the magnetization transport being modeled using a Lagrangian approach. Because this method expresses the magnetization as spatial distribution of particles, influences of particle densities and their spatial uniformities on numerical accuracy are well known. This study developed an alternative method for PC-MRI modeling using an Eulerian approach in which the magnetization is expressed as a spatially smooth continuous function. METHODS: The magnetization motion was described using the Bloch equation with an advection term and computed on a fixed grid using a finite difference method, and k-space sampling was implemented using the spoiled gradient echo sequence. PC-MRI scans of a fully developed flow in straight and stenosed cylinders were acquired to provide numerical examples. RESULTS: Reconstructed flow in a straight cylinder showed excellent agreement with input velocity profiles and mean errors were less than 0.5% of the maximum velocity. Numerical cases of flow in a stenosed cylinder successfully demonstrated the velocity profiles, with displacement artifacts being dependent on scan parameters and intravoxel dephasing due to flow disturbances. These results were in good agreement with those obtained using the Lagrangian approach with a sufficient particle density. CONCLUSION: The feasibility of the Eulerian approach to PC-MRI modeling was successfully demonstrated.

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This paper presents an improved iterative algorithm (TOPS-2) for the design of broadband inversion, excitation and coherent transfer mixing sequence (TOCSY) pulses. The evolution of the Bloch vector is presented as a sequence of small constant flip angle pulses with varying phases and constant amplitude. This paper describes an improved algorithm for iterative optimization of piece-wise constant phases as we incorporate the quadratic terms in the propagators. In our iterative optimization we obtain a closed-form expression for each phase, and these phases are optimized sequentially using the new improved algorithm. This paper compares the simulation results of the TOPS vs TOPS-2 and shows that TOPS-2 perform better. Experimental validation of excitation and inversion TOPS-2 pulse sequence is performed with .5% H2O in 99.5% D2O, and experimental validation of TOPS-2 mixing (TOCSY) pulse sequence is done with 0.1% of Ethylbenzene (EB) in CDCl3 solvent.

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PURPOSE: This work aims to develop a fast and practical computation method for MR simulations. The computational cost of MR simulations is often high because magnetizations of many isochromats are updated using a small step size on the order of microseconds. There are two types of subsequences to be processed for the simulations: subsequences with and without RF pulses. While straightforward implementations spend most of their time calculating subsequences with RF pulses, there is a method which efficiently reuses the computation for repetitive RF pulses. THEORY AND METHODS: A new method for efficiently processing subsequences with RF pulses is proposed. Rather than using an iterative update approach, the proposed method computes the combined transition which combines all transitions applied iteratively for each subsequence with RF pulses. The combined transition is used again when the same subsequence is used later. The combined transitions are cached and managed using a least recently used algorithm. RESULTS: The proposed method was found to accelerate the simulation by ˜20 times when 3.9 million isochromats were simulated using spin-echo sequences. Even on a laptop computer, the proposed method was able to simulate these sequences in ˜3.5 min. CONCLUSION: An efficient method for simulating pulse sequences is proposed. The proposed method computes and manages combined transitions, making MR simulation practical on a wide range of computers, including laptops.

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To investigate the relationship between microscopic myocardial structures and macroscopic measurements of diffusion tensor imaging (DTI), we proposed a cardiac DTI simulation method using the Bloch equation and the Monte Carlo random walk in a realistic myocardium model reconstructed from polarized light imaging (PLI) data of the entire human heart. To obtain a realistic simulation, with the constraints of prior knowledge pertaining to the maturational change of the myocardium structure, appropriate microstructure modeling parameters were iteratively determined by matching DTI simulations and real acquisitions of the same hearts in terms of helix angle, fractional anisotropy (FA) and mean diffusivity (MD) maps. Once a realistic simulation was obtained, we varied the extra-cellular volume (ECV) ratio, myocyte orientation heterogeneity and myocyte size, and explored the effects of microscopic changes in tissue structure on macroscopic diffusion metrics. The experimental results demonstrated the feasibility of simulating the DTI of the whole heart using PLI measurements. When varying ECV from 15% to 55%, mean FA decreased from 0.55 to 0.26, axial diffusivity increased by 0.6 µm2/ms, and radial diffusivity increased by 0.7 µm2/ms. When orientation heterogeneity was varied from 0 to 20∘, mean FA decreased from 0.4 to 0.3, axial diffusivity decreased by 0.08 µm2/ms, and radial diffusivity increased by 0.03 µm2/ms. When mean diameter of myocytes was varied from 6 µm to 10 µm, FA decreased from 0.67 to 0.46, axial and radial diffusivities increased by 0.05 and 0.2 µm2/ms, respectively.

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Obtaining multiple series of magnetic resonance (MR) images with different contrasts is useful for accurate diagnosis of human spinal conditions. However, this can be time consuming and a burden on both the patient and the hospital. We propose a Bloch equation-based autoencoder regularization generative adversarial network (BlochGAN) to generate a fat saturation T2-weighted (T2 FS) image from T1-weighted (T1-w) and T2-weighted (T2-w) images of human spine. To achieve this, our approach was to utilize the relationship between the contrasts using Bloch equation since it is a fundamental principle of MR physics and serves as a physical basis of each contrasts. BlochGAN properly generated the target-contrast images using the autoencoder regularization based on the Bloch equation to identify the physical basis of the contrasts. BlochGAN consists of four sub-networks: an encoder, a decoder, a generator, and a discriminator. The encoder extracts features from the multi-contrast input images, and the generator creates target T2 FS images using the features extracted from the encoder. The discriminator assists network learning by providing adversarial loss, and the decoder reconstructs the input multi-contrast images and regularizes the learning process by providing reconstruction loss. The discriminator and the decoder are only used in the training process. Our results demonstrate that BlochGAN achieved quantitatively and qualitatively superior performance compared to conventional medical image synthesis methods in generating spine T2 FS images from T1-w, and T2-w images.

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Fast and accurate modeling of MR signal responses are typically required for various quantitative MRI applications, such as MR fingerprinting. This work uses a new extended phase graph (EPG)-Bloch model for accurate simulation of transient-state, gradient-spoiled MR sequences, and proposes a recurrent neural network (RNN) as a fast surrogate of the EPG-Bloch model for computing large-scale MR signals and derivatives. The computational efficiency of the RNN model is demonstrated by comparisons with other existing models, showing one to three orders of acceleration compared with the latest GPU-accelerated, open-source EPG package. By using numerical and in vivo brain data, two used cases, namely, MRF dictionary generation and optimal experimental design, are also provided. Results show that the RNN surrogate model can be efficiently used for computing large-scale dictionaries of transient-state signals and derivatives within tens of seconds, resulting in several orders of magnitude acceleration with respect to state-of-the-art implementations. The practical application of transient-state quantitative techniques can therefore be substantially facilitated.

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PURPOSE: A supervised learning framework is proposed to automatically generate MR sequences and corresponding reconstruction based on the target contrast of interest. Combined with a flexible, task-driven cost function this allows for an efficient exploration of novel MR sequence strategies. METHODS: The scanning and reconstruction process is simulated end-to-end in terms of RF events, gradient moment events in x and y, and delay times, acting on the input model spin system given in terms of proton density, T1 and T2 , and ΔB0 . As a proof of concept, we use both conventional MR images and T1 maps as targets and optimize from scratch using the loss defined by data fidelity, SAR penalty, and scan time. RESULTS: In a first attempt, MRzero learns gradient and RF events from zero, and is able to generate a target image produced by a conventional gradient echo sequence. Using a neural network within the reconstruction module allows arbitrary targets to be learned successfully. Experiments could be translated to image acquisition at the real system (3T Siemens, PRISMA) and could be verified in the measurements of phantoms and a human brain in vivo. CONCLUSIONS: Automated MR sequence generation is possible based on differentiable Bloch equation simulations and a supervised learning approach.

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Spin-lock preparation was studied to detect tiny oscillatory magnetic fields such as a neural magnetic field without the blood oxygen level-dependent effect. This approach is a direct measurement and independent of static magnetic field strength. Accordingly, it is anticipated as a feasible functional magnetic resonance imaging (fMRI) in low and ultra-low-field MRI. Several reports have been published on spin-lock preparation but reports on imaging simulation are rare. Research in this area can assist in investigating magnetic resonance signal changes and, accordingly, can help to develop new spin-lock methods. In this study, we propose an imaging simulation method with an analytical solution using the Bloch equation. To demonstrate the feasibility of our proposed method, we compared simulated images with experimental results in which the number of sub-voxels and the amplitude and phase of the target oscillatory magnetic fields varied. In addition, we also applied graphics processing unit parallel computing and investigated the feasibility of avoiding an impracticable calculation time by doing so.

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Several noninvasive techniques for the direct measurement of the neuronal activity using magnetic resonance imaging (MRI) have recently been reported. As a promising candidate, we focus on a spin-lock MRI sequence (i.e., stimulus-induced rotary saturation (SIRS)) directly measuring a tiny oscillating magnetic field. Previous phantom studies on SIRS have applied the target oscillating magnetic field parallel to the direction of the static magnetic field B0. However, in practice, the neuromagnetic fields are not always aligned in the same direction as in such a condition. This study investigates the MR signal changes during SIRS when the target magnetic field direction is not the same as that of the B0 field through both phantom experiments and Bloch simulations. The experimental results indicate that only the target magnetic field component along the B0 field affects the signal change, indicating that SIRS has partial sensitivity, even if the target magnetic fields are tilted from the B0 field. Furthermore, the simulation results show good agreements with the experimental results. These results clarify the sensitivity direction of SIRS-based fMRI and lead to the possibility that the direction of the generated neuromagnetic fields can be estimated, such that we can separate directional information from the other information contained in neuromagnetic fields (e.g., phase information).

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We demonstrated the feasibility of the spin-lock preparation sequence using low-field magnetic resonance (MR) imaging that prevents interference from blood-oxygenation-level-dependent effects. We focused on two spin-lock preparations: spin-lock Mz (SL-Mz) and stimulus-induced rotary saturation (SIRS) and analyzed the magnetization dynamics during the sequences using the Bloch equation. Next, we performed phantom experiments using a loop coil to investigate the MR signal change as a function of the target signal strength and phase. Furthermore, we performed curve fittings to consider the radio frequency, which agreed with the experimental results. Then, we investigated the detectable strength of the magnetic field, and the SL-Mz detected a signal strength of 2.34 nT. In conclusion, our experimental results showed good agreement with the results obtained using the Bloch equation.

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PURPOSE: To propose and evaluate a deep learning model for rapid and accurate calculation of myocardial T1 /T2 values based on a previously proposed Bloch equation simulation with slice profile correction (BLESSPC) method. METHODS: Deep learning Bloch equation simulations (DeepBLESS) models are proposed for rapid and accurate T1 estimation for the MOLLI T1 mapping sequence with balanced SSFP readouts and T1 /T2 estimation for a radial simultaneous T1 and T2 mapping (radial T1 -T2 ) sequence. The DeepBLESS models were trained separately based on simulated radial T1 -T2 and MOLLI data, respectively. The DeepBLESS T1 -T2 estimation accuracy was evaluated based on simulated data with different noise levels. The DeepBLESS model was compared with BLESSPC in simulation, phantom, and in vivo studies for the MOLLI sequence at 1.5 T and radial T1 -T2 sequence at 3 T. RESULTS: After DeepBLESS was trained, in phantom studies, DeepBLESS and BLESSPC achieved similar accuracy and precision in T1 -T2 estimations for both MOLLI and radial T1 -T2 (P > .05). For in vivo, DeepBLESS and BLESSPC generated similar myocardial T1 /T2 values for radial T1 -T2 at 3 T (T1 : 1366 ± 31 ms for both methods, P > .05; T2 : 37.4 ms ± 0.9 ms for both methods, P > .05), and similar myocardial T1 values for the MOLLI sequence at 1.5 T (1044 ± 20 ms for both methods, P > .05). DeepBLESS generated a T1 /T2 map in less than 1 second. CONCLUSION: The DeepBLESS model offers an almost instantaneous approach for estimating accurate T1 /T2 values, replacing BLESSPC for both MOLLI and radial T1 -T2 sequences, and is promising for multiparametric mapping in cardiac MRI.

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We propose a simultaneous myocardial T1 and T2 mapping technique using a radial sequence with inversion recovery and T2 preparation, which achieves high accuracy and precision, with T1 and T2 reproducibility similar to the Modified Look-Locker Inversion recovery (MOLLI) sequence and the conventional bright blood T2 mapping technique, respectively. The sequence was developed by incorporating gold angle radial fast low angle shot (FLASH) readout combined with an inversion pulse and T2prep pulses. The extended Bloch equation simulation with slice profile correction (BLESSPC) algorithm was proposed to reconstruct T1 and T2 maps at the same time in a few seconds, while maintaining good T1 and T2 estimation accuracy. Accuracy and precision were compared among the proposed technique, MOLLI and conventional T2 mapping techniques using phantom studies, 10 healthy volunteers and three patients. In phantom studies, the proposed technique was more accurate than MOLLI (P < 0.05) while achieving similar precision (P = 0.3) in T1 estimation, and was more accurate (P < 0.05) and precise (P < 0.001) than conventional T2 mapping (two-parameter fitting) in T2 estimation. In vivo, the proposed technique achieved significantly higher T1 values (P < 0.001) and similar reproducibility (P = 0.3) compared with MOLLI, with significantly lower T2 values (P < 0.001) and similar reproducibility (P = 0.6) compared with the conventional T2 mapping technique. Thus, the proposed radial T1-T2 mapping technique allows for accurate, precise, simultaneous myocardial T1 and T2 mapping in an 11-heartbeat single breath-hold acquisition.

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We studied stimulus-induced rotary-saturation preparation (which enables measurement of oscillating magnetic fields using MRI) and derived an analytical solution of the Bloch equation to understand magnetization dynamics mathematically and comprehensively and to conduct simulations without sequential-calculation techniques such as the Runge-Kutta method. We formulated the dynamics using the Bloch equation, introducing an additional rotating frame and some approximations to make it into a homogeneous differential equation. Moreover, we found that there are two modes depending on the target oscillating magnetic field. To confirm the validity of the solution, we experimentally investigated its characteristics and performed curve fitting using the analytical model. Considering the constraints on the frame, the analytical solution was found to agree with experimental data. The experimental data indicate that it is necessary to design robust sequences compensating B0 or B1lock spatial inhomogeneity to improve measurements. Therefore, experimenters should consider the dynamics of magnetization with RF pulses to rewind the spin phase for accurate measurements.

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The mathematics of imaging is a growing field of research and is evolving rapidly parallel to evolution in the field of imaging. Imaging, which is a sub-field of biomedical engineering, considers novel approaches to visualize biological tissues with the general goal of improving health. "Medical imaging research provides improved diagnostic tools in clinical settings and supports the development of drugs and other therapies. The data acquisition and diagnostic interpretation with minimum error are the important technical aspects of medical imaging. The image quality and resolution are really important in portraying the internal aspects of patient's body. Although there are several user friendly resources for processing image features, such as enhancement, colour manipulation and compression, the development of new processing methods is still worthy of efforts. In this article we aim to present the role of fractional calculus in imaging with the aid of practical examples.

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Exact solutions of the Bloch equations with T1- and T2-relaxation terms for piecewise constant magnetic fields are numerically challenging. We therefore investigate an approximation for the achieved magnetization in which rotations and relaxations are split into separate operations. We develop an estimate for its accuracy and explicit first and second order derivatives with respect to the complex excitation radio frequency voltages. In practice, the deviation between an exact solution of the Bloch equations and this rotation relaxation splitting approximation seems negligible. Its computation times are similar to exact solutions without relaxation terms. We apply the developed theory to numerically optimize radio frequency excitation waveforms with T1- and T2-relaxations in several examples.

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OBJECTIVE: Use of spectroscopically-acquired spin echoes typically involves Fourier transformation of the right side of the echo while largely neglecting the left side. For sufficiently long echo times, the left side may have enough spectral resolution to offer some utility. Since the acquisition of this side is "free", we deemed it worthy of attention and investigated the spectral properties and information content of this data. MATERIALS AND METHODS: Theoretical expressions for left- and right-side spectra were derived assuming Lorentzian frequency distributions. For left-side spectra, three regimes were identified based upon the relative magnitudes of reversible and irreversible transverse relaxation rates, R 2' and R 2, respectively. Point-resolved spectroscopy (PRESS) data from muscle, fat deposit and bone marrow were acquired at 1.5 T to test aspects of the theoretical expressions. RESULTS: For muscle water or methylene marrow resonances, left-side signals were substantially or moderately larger than right-side signals but were similar in magnitude for muscle choline and creatine resonances. Left- versus right-side spectral-peak amplitude ratios depend sensitively on the relative values of R 2 and R 2' , which can be estimated given this ratio and a right-side linewidth measurement. CONCLUSION: Left-side spectra can be used to augment signal-to-noise and to estimate spectral R 2 and R 2' values under some circumstances.

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Magnetic resonance fingerprinting (MR fingerprinting or MRF) is a newly introduced quantitative magnetic resonance imaging technique, which enables simultaneous multi-parameter mapping in a single acquisition with improved time efficiency. The current MRF reconstruction method is based on dictionary matching, which may be limited by the discrete and finite nature of the dictionary and the computational cost associated with dictionary construction, storage and matching. In this paper, we describe a reconstruction method based on Kalman filter for MRF, which avoids the use of dictionary to obtain continuous MR parameter measurements. With this Kalman filter framework, the Bloch equation of inversion-recovery balanced steady state free-precession (IR-bSSFP) MRF sequence was derived to predict signal evolution, and acquired signal was entered to update the prediction. The algorithm can gradually estimate the accurate MR parameters during the recursive calculation. Single pixel and numeric brain phantom simulation were implemented with Kalman filter and the results were compared with those from dictionary matching reconstruction algorithm to demonstrate the feasibility and assess the performance of Kalman filter algorithm. The results demonstrated that Kalman filter algorithm is applicable for MRF reconstruction, eliminating the need for a pre-define dictionary and obtaining continuous MR parameter in contrast to the dictionary matching algorithm.

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The three-dimension Fast Low Angle SHot Magnetic Resonance Imaging (3D FLASH) sequence has been widely adopted in medical diagnostic imaging because of its availability, simplicity, and high spatial resolution. To improve the quality of structural brain images acquired with the 3D FLASH sequence, we developed a parameter optimization scheme and image inhomogeneity correction methods. The optimal imaging parameters were determined by maximizing gray-matter and white-matter CNR efficiency. Compared to protocols based on published parameters, applying the proposed optimal imaging parameters increased CNR efficiency by >10%. Image inhomogeneity, including signal and CNR inhomogeneity, was corrected by the choice of an optimal flip angle, estimated transmit function, and estimated receive sensitivity. As a result, our optimization and image inhomogeneity correction greatly improved the quality of images acquired with the 3D FLASH sequence.

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PURPOSE: The frequency-swept pulse known as the hyperbolic-secant (HS) pulse is popular in NMR for achieving adiabatic spin inversion. The HS pulse has also shown utility for achieving excitation and refocusing in gradient-echo and spin-echo sequences, including new ultrashort echo-time imaging (e.g., Sweep Imaging with Fourier Transform, SWIFT) and B1 mapping techniques. To facilitate the analysis of these techniques, the complete theoretical solution of the Bloch equation, as driven by the HS pulse, was derived for an arbitrary state of initial magnetization. METHODS: The solution of the Bloch-Riccati equation for transverse and longitudinal magnetization for an arbitrary initial state was derived analytically in terms of HS pulse parameters. The analytical solution was compared with the solutions using both the Runge-Kutta method and the small-tip approximation. RESULTS: The analytical solution was demonstrated on different initial states at different frequency offsets with/without a combination of HS pulses. Evolution of the transverse magnetization was influenced significantly by the choice of HS pulse parameters. The deviation of the magnitude of the transverse magnetization, as obtained by comparing the small-tip approximation to the analytical solution, was < 5% for flip angles < 30 °, but > 10% for the flip angles > 40 °. CONCLUSION: The derived analytical solution provides insights into the influence of HS pulse parameters on the magnetization evolution. Magn Reson Med 77:1630-1638, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

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PURPOSE: To study the utility of fractional calculus in modeling gradient-recalled echo MRI signal decay in the normal human brain. METHODS: We solved analytically the extended time-fractional Bloch equations resulting in five model parameters, namely, the amplitude, relaxation rate, order of the time-fractional derivative, frequency shift, and constant offset. Voxel-level temporal fitting of the MRI signal was performed using the classical monoexponential model, a previously developed anomalous relaxation model, and using our extended time-fractional relaxation model. Nine brain regions segmented from multiple echo gradient-recalled echo 7 Tesla MRI data acquired from five participants were then used to investigate the characteristics of the extended time-fractional model parameters. RESULTS: We found that the extended time-fractional model is able to fit the experimental data with smaller mean squared error than the classical monoexponential relaxation model and the anomalous relaxation model, which do not account for frequency shift. CONCLUSIONS: We were able to fit multiple echo time MRI data with high accuracy using the developed model. Parameters of the model likely capture information on microstructural and susceptibility-induced changes in the human brain. Magn Reson Med 77:1485-1494, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

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