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Unrolled Neural Networks (NNs) have been extensively applied to different image reconstruction problems across all imaging modalities. A key component of the latter is that they allow for physics-informed learning of the regularization method which is parametrized by the NN. However, due to the lack of understanding of deep NNs from a theoretical point of view, unrolled NNs are still black-boxes when the regularizers are given by deep NNs, e.g. U-Nets.Dictionary Learning (DL) is a well-established regularization method which is based on learning a transform to sparsely approximate the signals of interest. Typically, DL-based image reconstruction either employs a dictionary which was pre-trained on a set of patches which were extracted from ground-truth images or a dictionary which is jointly trained during the reconstruction. However, in both cases, the used DL-algorithms are not designed to take into account the reconstruction problem or the underlying physical model which describes the imaging process. In this work, we propose a DL-algorithm based on unrolled NNs to overcome these limitations.We construct an unrolled NN which corresponds to an unrolled DL-based reconstruction algorithm and train the unrolled NN to optimize its weights, i.e. the atoms of the dictionary, by back-propagation in a supervised manner. Further, we propose a new way to employ a 2D dictionary in the spatio-temporal domain. We tested and evaluated the method on an accelerated cardiac cine MR image reconstruction problem using 216/36/36 dynamic images for training, validation and testing and compared it to two well-known state-of-the-art approaches for cardiac cine MRI based on deep iterative CNNs. Further, we analyze the obtained dictionaries in terms of dictionary-coherence and structure of the atoms. Last, we compare the reported methods in terms of stability by applying them to an entirely different dataset consisting of 49 different test images.The investigated physics-informed DL-approach yields significantly more accurate reconstructions compared to the DL-method which uses dictionaries obtained by decoupled pre-training, thereby providing an improvement of up to 4.90 dB in terms of PSNR and 5% in terms of SSIM. Further, the proposed spatio-temporal 2D dictionary outperforms the 1D and 3D dictionaries by preventing smoothing of image details while still accurately removing undersampling artefacts and noise resulting in an increase of up to 1.10 dB in terms of PSNR and 4% in terms of SSIM. Although being surpassed by the CNNs on the first dataset, the proposed NNs-based DL-method is more stable compared to the latter approach and yields comparable results on the second dataset. Last, it has the advantage of being entirely interpretable in each component.The presented physics-informed NN can be used as training algorithm for a classical and interpretable data-driven regularization method based a learned dictionary which can then not only be linked to the considered data but also to the reconstruction method that the NN defines. This article is protected by copyright. All rights reserved.

Objective. To provide 3D high-resolution cardiac T1 maps using model-based super-resolution reconstruction (SRR). Approach. Due to signal-to-noise ratio limitations and the motion of the heart during imaging, often 2D T1 maps with only low through-plane resolution (i.e. slice thickness of 6–8 mm) can be obtained. Here, a model-based SRR approach is presented, which combines multiple stacks of 2D acquisitions with 6–8 mm slice thickness and generates 3D high-resolution T1 maps with a slice thickness of 1.5–2 mm. Every stack was acquired in a different breath hold (BH) and any misalignment between BH was corrected retrospectively. The novelty of the proposed approach is the BH correction and the application of model-based SRR on cardiac T1 Mapping. The proposed approach was evaluated in numerical simulations and phantom experiments and demonstrated in four healthy subjects. Main results. Alignment of BH states was essential for SRR even in healthy volunteers. In simulations, respiratory motion could be estimated with an RMS error of 0.18 ± 0.28 mm. SRR improved the visualization of small structures. High accuracy and precision (average standard deviation of 69.62 ms) of the T1 values was ensured by SRR while the detectability of small structures increased by 40%. Significance. The proposed SRR approach provided T1 maps with high in-plane and high through-plane resolution (1.3 × 1.3 × 1.5–2 mm3). The approach led to improvements in the visualization of small structures and precise T1 values.

Assess and optimise acquisition parameters for continuous cardiac Magnetic Resonance Fingerprinting (MRF).Different acquisition schemes (flip angle amplitude, lobe size, T2-preparation pulses) for cardiac MRF were assessed in simulations and phantom and demonstrated in one healthy volunteer. Three different experimental designs were evaluated using central composite and fractional factorial designs. Relative errors for T1 and T2 were calculated for a wide range of realistic T1 and T2 value combinations. The effect of different designs on the accuracy of T1 and T2 was assessed using response surface modelling and Cohen's f calculations.Larger flip angle amplitudes lead to an improvement of T2 accuracy and precision for simulations and phantom experiments. Similar effects could also be shown qualitatively in in-vivo scans. Accuracy and precision of T1 were robust to different design parameters with improved values for faster flip angle variation. Cohen's f showed that T2-preparation pulses influence the accuracy of T2. The number of pulses used is the most important parameter. Without T2-preparation pulses, RMSE were 3.0 ± 8.09 % for T1 and 16.24 ± 14.47 % for T2. Using those pulses reduced the RMSE to 2.3 ± 8.4 % for T1 and 14.11 ± 13.46 % for T2. Nonetheless, even if the improvement is significant, RMSE are still too high for reliable quantification.In contrast to previous study using triggered MRF sequences using 30° flip angles, large flip angle amplitudes led to better results for continuous cardiac MRF sequences. T2-preparation pulse can improve the accuracy of T2 estimation but lead to longer scan times.

Background Cardiac PET has recently found novel applications in coronary atherosclerosis imaging using [18F]NaF as a radiotracer, highlighting vulnerable plaques. However, the resulting uptakes are relatively small, and cardiac motion and respiration-induced movement of the heart can impair the reconstructed images due to motion blurring and attenuation correction mismatches. This study aimed to apply an MR-based motion compensation framework to [18F]NaF data yielding high-resolution motion-compensated PET and MR images. Methods Free-breathing 3-dimensional Dixon MR data were acquired, retrospectively binned into multiple respiratory and cardiac motion states, and split into fat and water fraction using a model-based reconstruction framework. From the dynamic MR reconstructions, both a non-rigid cardiorespiratory motion model and a motion-resolved attenuation map were generated and applied to the PET data to improve image quality. The approach was tested in 10 patients and focal tracer hotspots were evaluated concerning their target-to-background ratio, contrast-to-background ratio, and their diameter. Results MR-based motion models were successfully applied to compensate for physiological motion in both PET and MR. Target-to-background ratios of identified plaques improved by 7 ± 7%, contrast-to-background ratios by 26 ± 38%, and the plaque diameter decreased by −22 ± 18%. MR-based dynamic attenuation correction strongly reduced attenuation correction artefacts and was not affected by stent-related signal voids in the underlying MR reconstructions. Conclusions The MR-based motion correction framework presented here can improve the target-to-background, contrast-to-background, and width of focal tracer hotspots in the coronary system. The dynamic attenuation correction could effectively mitigate the risk of attenuation correction artefacts in the coronaries at the lung-soft tissue boundary. In combination, this could enable a more reproducible and reliable plaque localisation.

PURPOSE To introduce and investigate a method for free-breathing three-dimensional (3D) B1+ mapping of the human body at ultrahigh field (UHF), which can be used to generate homogenous flip angle (FA) distributions in the human body at UHF. METHODS A 3D relative B1+ mapping sequence with a radial phase-encoding (RPE) k-space trajectory was developed and applied in 11 healthy subjects at 7T. An RPE-based actual flip angle mapping method was applied with a dedicated B1+ shim setting to calibrate the relative B1+ maps yielding absolute B1+ maps of the individual transmit channels. The method was evaluated in a motion phantom and by multidimensional in vivo measurements. Additionally, 3D gradient echo scans with and without static phase-only B1+ shims were used to qualitatively validate B1+ shim predictions. RESULTS The phantom validation revealed good agreement for B1+ maps between dynamic measurement and static reference acquisition. The proposed 3D method was successfully validated in vivo by comparing magnitude and phase distributions with a 2D Cartesian reference. 3D B1+ maps free from visible motion artifacts were successfully acquired for 11 subjects with body mass indexes ranging from 19 kg/m2 to 34 kg/m2 . 3D respiration-resolved absolute B1+ maps indicated FA differences between inhalation and exhalation up to 15% for one channel and up to 24% for combined channels for shallow breathing. CONCLUSION The proposed method provides respiration-resolved absolute 3D B1+ maps of the human body at UHF, which enables the investigation and development of 3D B1+ shimming and parallel transmission methods to further enhance body imaging at UHF.

Objective To provide respiratory motion correction for free-breathing myocardial T1 mapping using a pilot tone (PT) and a continuous golden-angle radial acquisition. Materials and methods During a 45 s prescan the PT is acquired together with a dynamic sagittal image covering multiple respiratory cycles. From these images, the respiratory heart motion in head-feet and anterior–posterior direction is estimated and two linear models are derived between the PT and heart motion. In the following scan through-plane motion is corrected prospectively with slice tracking based on the PT. In-plane motion is corrected for retrospectively. Our method was evaluated on a motion phantom and 11 healthy subjects. Results Non-motion corrected measurements using a moving phantom showed T1 errors of 14 ± 4% (p p p Discussion Our proposed PT-based motion correction approach allows for T1 mapping during free-breathing with the same accuracy as a corresponding breathhold T1 mapping scan.

Objective. T1 mapping of the liver is time consuming and can be challenging due to respiratory motion. Here we present a prospective slice tracking approach, which utilizes an external ultra-wide band radar signal and allows for efficient T1 mapping during free-breathing. Approach. The fast radar signal is calibrated to an MR-based motion signal to create a motion model. This motion model provides motion estimates, which are used to carry out slice tracking for any subsequent clinical scan. This approach was evaluated in simulations, phantom experiments and in vivo scans. Main results. Radar-based slice tracking was implemented on an MR system with a total latency of 77 ms. Moving phantom experiments showed accurate motion prediction with an error of 0.12 mm in anterior-posterior and 0.81 mm in head-feet direction. The model error remained stable for up to two hours. In vivo experiments showed visible image improvement with a motion model error three times smaller than with a respiratory bellow. For T1 mapping during free-breathing the proposed approach provided similar results compared to reference T1 mapping during a breathhold. Significance. The proposed radar-based approach achieves accurate slice tracking and enables efficient T1 mapping of the liver during free-breathing. This motion correction approach is independent from scanning parameters and could also be used for applications like MR guided radiotherapy or MR Elastography.

Purpose: Respiratory motion‐compensated (MC) 3D cardiac fat‐water imaging at 7T. Methods: Free‐breathing bipolar 3D triple‐echo gradient‐recalled‐echo (GRE) data with radial phase‐encoding (RPE) trajectory were acquired in 11 healthy volunteers (7M\4F, 21–35 years, mean: 30 years) with a wide range of body mass index (BMI; 19.9–34.0 kg/m2) and volunteer tailored B1+ shimming. The bipolar‐corrected triple‐echo GRE‐RPE data were binned into different respiratory phases (self‐navigation) and were used for the estimation of non‐rigid motion vector fields (MF) and respiratory resolved (RR) maps of the main magnetic field deviations (ΔB0). RR ΔB0 maps and MC ΔB0 maps were compared to a reference respiratory phase to assess respiration‐induced changes. Subsequently, cardiac binned fat‐water images were obtained using a model‐based, respiratory motion‐corrected image reconstruction. Results: The 3D cardiac fat‐water imaging at 7T was successfully demonstrated. Local respiration‐induced frequency shifts in MC ΔB0 maps are small compared to the chemical shifts used in the multi‐peak model. Compared to the reference exhale ΔB0 map these changes are in the order of 10 Hz on average. Cardiac binned MC fat‐water reconstruction reduced respiration induced blurring in the fat‐water images, and flow artifacts are reduced in the end‐diastolic fat‐water separated images. Conclusion: This work demonstrates the feasibility of 3D fat‐water imaging at UHF for the entire human heart despite spatial and temporal B1+ and B0 variations, as well as respiratory and cardiac motion.

PURPOSE: Myocardial fat infiltrations are associated with a range of cardiomyopathies. The purpose of this study was to perform cardio-respiratory motion-correction for model-based water-fat separation to image fatty infiltrations of the heart in a free-breathing, non-cardiac-triggered high-resolution 3D MRI acquisition. METHODS: Data were acquired in nine patients using a free-breathing, non-cardiac-triggered high-resolution 3D Dixon gradient-echo sequence and radial phase encoding trajectory. Motion correction was combined with a model-based water-fat reconstruction approach. Respiratory and cardiac motion models were estimated using a dual-mode registration algorithm incorporating both motion-resolved water and fat information. Qualitative comparisons of fat structures were made between 2D clinical routine reference scans and reformatted 3D motion-corrected images. To evaluate the effect of motion correction the local sharpness of epicardial fat structures was analyzed for motion-averaged and motion-corrected fat images. RESULTS: The reformatted 3D motion-corrected reconstructions yielded qualitatively comparable fat structures and fat structure sharpness in the heart as the standard 2D breath-hold. Respiratory motion correction improved the local sharpness on average by 32% ± 24% with maximum improvements of 81% and cardiac motion correction increased the sharpness further by another 15% ± 11% with maximum increases of 31%. One patient showed a fat infiltration in the myocardium and cardio-respiratory motion correction was able to improve its visualization in 3D. CONCLUSION: The 3D water-fat separated cardiac images were acquired during free-breathing and in a clinically feasible and predictable scan time. Compared to a motion-averaged reconstruction an increase in sharpness of fat structures by 51% ± 27% using the presented motion correction approach was observed for nine patients.

PURPOSE To improve the efficiency of native and postcontrast high-resolution cardiac T1 mapping by utilizing cardiac motion correction. METHODS Common cardiac T1 mapping techniques only acquire data in a small part of the cardiac cycle, leading to inefficient data sampling. Here, we present an approach in which 80% of each cardiac cycle is used for T1 mapping by integration of cardiac motion correction. Golden angle radial data was acquired continuously for 8 s with in-plane resolution of 1.3 × 1.3 mm2 . Cine images were reconstructed for nonrigid cardiac motion estimation. Images at different TIs were reconstructed from the same data, and motion correction was performed prior to T1 mapping. Native T1 mapping was evaluated in healthy subjects. Furthermore, the technique was applied for postcontrast T1 mapping in 5 patients with suspected fibrosis. RESULTS Cine images with high contrast were obtained, leading to robust cardiac motion estimation. Motion-corrected T1 maps showed myocardial T1 times similar to cardiac-triggered T1 maps obtained from the same data (1288 ± 49 ms and 1259 ± 55 ms, respectively) but with a 34% improved precision (spatial variation: 57.0 ± 12.5 ms and 94.8 ± 15.4 ms, respectively, P < 0.0001) due to the increased amount of data. In postcontrast T1 maps, focal fibrosis could be confirmed with late contrast-enhancement images. CONCLUSION The proposed approach provides high-resolution T1 maps within 8 s. Data acquisition efficiency for T1 mapping was improved by a factor of 5 by integration of cardiac motion correction, resulting in precise T1 maps.

This special issue is the second part of a themed issue that focuses on synergistic tomographic image reconstruction and includes a range of contributions in multiple disciplines and application areas. The primary subject of study lies within inverse problems which are tackled with various methods including statistical and computational approaches. This volume covers algorithms and methods for a wide range of imaging techniques such as spectral X-ray computed tomography (CT), positron emission tomography combined with CT or magnetic resonance imaging, bioluminescence imaging and fluorescence-mediated imaging as well as diffuse optical tomography combined with ultrasound. Some of the articles demonstrate their utility on real-world challenges, either medical applications (e.g. motion compensation for imaging patients) or applications in material sciences (e.g. material decomposition and characterization). One of the desired outcomes of the special issues is to bring together different scientific communities which do not usually interact as they do not share the same platforms such as journals and conferences. This article is part of the theme issue ‘Synergistic tomographic image reconstruction: part 2’.

Coronary artery disease (CAD) is caused by the formation of plaques in the coronary arteries and is one of the most common cardiovascular diseases. NaF-PET can be used to assess plaque composition, which could be important for therapy planning. One of the main challenges of NaF-PET is cardiac and respiratory motion which can strongly impair diagnostic accuracy. In this study, we investigated the use of a synergistic image registration approach which combined motion-resolved MR and PET data to estimate cardiac and respiratory motion. This motion estimation could then be used to improve the NaF-PET image quality. The approach was evaluated with numerical simulations and in vivo scans of patients suffering from CAD. In numerical simulations, it was shown, that combining MR and PET information can improve the accuracy of motion estimation by more than 15%. For the in vivo scans, the synergistic image registration led to an improvement in uptake visualization. This is the first study to assess the benefit of combining MR and NaF-PET for cardiac and respiratory motion estimation. Further patient evaluation is required to fully evaluate the potential of this approach. This article is part of the theme issue ‘Synergistic tomographic image reconstruction: part 1’.

PURPOSE Iterative convolutional neural networks (CNNs) which resemble unrolled learned iterative schemes have shown to consistently deliver state-of-the-art results for image reconstruction problems across different imaging modalities. However, because these methods include the forward model in the architecture, their applicability is often restricted to either relatively small reconstruction problems or to problems with operators which are computationally cheap to compute. As a consequence, they have not been applied to dynamic non-Cartesian multi-coil reconstruction problems so far. METHODS In this work, we propose a CNN architecture for image reconstruction of accelerated 2D radial cine MRI with multiple receiver coils. The network is based on a computationally light CNN component and a subsequent conjugate gradient (CG) method which can be jointly trained end-to-end using an efficient training strategy. We investigate the proposed training strategy and compare our method with other well-known reconstruction techniques with learned and non-learned regularization methods. RESULTS Our proposed method outperforms all other methods based on non-learned regularization. Further, it performs similar or better than a CNN-based method employing a 3D U-Net and a method using adaptive dictionary learning. In addition, we empirically demonstrate that even by training the network with only iteration, it is possible to increase the length of the network at test time and further improve the results. CONCLUSIONS End-to-end training allows to highly reduce the number of trainable parameters of and stabilize the reconstruction network. Further, because it is possible to change the length of the network at the test time, the need to find a compromise between the complexity of the CNN-block and the number of iterations in each CG-block becomes irrelevant.

Simultaneous positron-emission tomography (PET)-magnetic resonance (MR) imaging is a hybrid technique in oncological hepatic imaging combining soft-tissue and functional contrast of dynamic contrast enhanced MR (DCE-MR) with metabolic information from PET. In this context, respiratory motion represents a major challenge by introducing blurring, artifacts and misregistration in the liver. In this work, we propose a free-breathing 3D non-rigid respiratory motion correction framework for simultaneously acquired DCE-MR and PET data, which makes use of higher spatial resolution MR data to derive motion information used directly during image reconstruction to minimize image blurring and motion artifacts. The main aim was to increase contrast of hepatic metastases to improve their detection and characterization. DCE-MR data were acquired at 3T through a golden radial phase encoding scheme, enabling derivation of motion fields. These were used in the motion compensated image reconstruction of DCE-MR time-series (48 time-points, 6 s temporal resolution, 1.5 mm isotropic spatial resolution) and 3D PET activity map, which was subsequently interpolated to the DCE-MR resolution. The extended Tofts model was fitted to DCE-MR data, obtaining functional parametric maps related to perfusion such as the endothelial permeability ( K t ). Fifty-seven hepatic metastases were identified and analyzed. Quantitative evaluations of motion correction in PET images demonstrated average percentage increases of 16% ± 5% (mean ± SD) in Contrast (C), 18% ± 6% in SUVmean and 14% ± 2% in SUVmax, while DCE-MR and K t scored contrast-to-noise-ratio increases of 64% ± 3% and 90% ± 6%, respectively. Motion-corrected data visually showed improved image contrast of hepatic metastases and effectively reduced blurring and motion artefacts. Scatter plots of SUVmean versus K t suggested that the proposed framework improved differentiation of K t measurements. The presented motion correction framework for simultaneously acquired PET-DCE-MR data provides accurately aligned images with increased contrast of hepatic lesions allowing for improved detection and characterization.

PURPOSE In the past, dictionary learning (DL) and sparse coding (SC) have been proposed for the regularization of image reconstruction problems. The regularization is given by a sparse approximation of all image patches using a learned dictionary, that is, an overcomplete set of basis functions learned from data. Despite its competitiveness, DL and SC require the tuning of two essential hyperparameters: the sparsity level S - the number of basis functions of the dictionary, called atoms, which are used to approximate each patch, and K - the overall number of such atoms in the dictionary. These two hyperparameters usually have to be chosen a priori and are determined by repetitive and computationally expensive experiments. Furthermore, the final reported values vary depending on the specific situation. As a result, the clinical application of the method is limited, as standardized reconstruction protocols have to be used. METHODS In this work, we use adaptive DL and propose a novel adaptive sparse coding algorithm for two-dimensional (2D) radial cine MR image reconstruction. Using adaptive DL and adaptive SC, the optimal dictionary size K as well as the optimal sparsity level S are chosen dependent on the considered data. RESULTS Our three main results are the following: First, adaptive DL and adaptive SC deliver results which are comparable or better than the most widely used nonadaptive version of DL and SC. Second, the time needed for the regularization is accelerated due to the fact that the sparsity level S is never overestimated. Finally, the a priori choice of S and K is no longer needed but is optimally chosen dependent on the data under consideration. CONCLUSIONS Adaptive DL and adaptive SC can highly facilitate the application of DL- and SC-based regularization methods. While in this work we focused on 2D radial cine MR image reconstruction, we expect the method to be applicable to different imaging modalities as well.

Purpose To evaluate prospective motion correction using the pilot tone (PT) as a quantitative respiratory motion signal with high temporal resolution for cardiac cine images during free breathing. Methods Before cine data acquisition, a short prescan was performed, calibrating the PT to the respiratory-induced heart motion using respiratory-resolved real-time images. The calibrated PT was then applied for nearly real-time prospective motion correction of cine MRI through slice tracking (ie, updating the slice position before every readout). Additionally, in-plane motion correction was performed retrospectively also based on the calibrated PT data. The proposed method was evaluated in a moving phantom and 10 healthy volunteers. Results The PT showed very good correlation to the phantom motion. In volunteer studies using a long-term scan over 7.96 ± 1.40 min, the mean absolute error between registered and predicted motion from the PT was 1.44 ± 0.46 mm in head-feet and 0.46 ± 0.07 mm in anterior-posterior direction. Irregular breathing could also be corrected well with the PT. The PT motion correction leads to a significant improvement of contrast-to-noise ratio by 68% (P ≤ .01) between blood pool and myocardium and sharpness of endocardium by 24% (P = .04) in comparison to uncorrected data. The image score, which refers to the cine image quality, has improved with the utilization of the proposed PT motion correction. Conclusion The proposed approach provides respiratory motion-corrected cine images of the heart with improved image quality and a high scan efficiency using the PT. The PT is independent of the MR acquisition, making this a very flexible motion-correction approach.

We develop a hierarchical Bayesian approach for the inference of large-scale spatial-temporal regression as often encountered in the analysis of imaging data. For each spatial location a linear tem...

Magnetic resonance imaging (MRI) is a medical imaging technique which is widely used in clinical routine. Standard imaging methods lead to so-called contrast-weighted images. The contrast arises from several tissue-related parameters such as the relaxation times T 1 and T 2. The aim in quantitative MRI (qMRI) is an estimation of these quantitative parameters. Magnetic resonance fingerprinting (MRF) is a promising technique in qMRI that allows for the simultaneous determination of multiple tissue-related parameters within a short acquisition time. The conventional MRF method utilizes an approximate inverse Fourier transformation for the highly undersampled data in the Fourier domain, leading to aliasing errors in the reconstructed magnetization courses. Computationally expensive statistical MRF modeling approaches overcome this issue by modeling the data directly in the Fourier domain. However, this leads to a non-convex and large-scale optimization task that is challenging to solve and requires expensive calculations. We compare two recent approaches, namely the conventional MRF method and a statistical MRF modeling approach based on Bayesian statistics in terms of their accuracy, reliability and computational costs. The comparison is carried out for simulated data with known ground truth for different signal-to-noise ratios, in the presence of errors in the physical model, and for several Fourier domain sampling schemes. It is demonstrated that a residual analysis can help to decide if the conventional MRF method is sufficient or if the complex Bayesian Fourier domain modeling approach can lead to a significant improvement.