The Moore-Penrose Pseudoinverse. A Tutorial Review of the Theory

In the last decades the Moore-Penrose pseudoinverse has found a wide range of applications in many areas of Science and became a useful tool for physicists dealing, for instance, with optimization problems, with data analysis, with the solution of linear integral equations, etc. The existence of such applications alone should attract the interest of students and researchers in the Moore-Penrose pseudoinverse and in related sub jects, like the singular values decomposition theorem for matrices. In this note we present a tutorial review of the theory of the Moore-Penrose pseudoinverse. We present the first definitions and some motivations and, after obtaining some basic results, we center our discussion on the Spectral Theorem and present an algorithmically simple expression for the computation of the Moore-Penrose pseudoinverse of a given matrix. We do not claim originality of the results. We rather intend to present a complete and self-contained tutorial review, useful for those more devoted to applications, for those more theoretically oriented and for those who already have some working knowledge of the sub ject.Comment: 23 page

Parallel imaging: is GRAPPA a useful acquisition tool for MR imaging intended for volumetric brain analysis?

<p>Abstract</p> <p>Background</p> <p>The work presented here investigates parallel imaging applied to T1-weighted high resolution imaging for use in longitudinal volumetric clinical studies involving Alzheimer's disease (AD) and Mild Cognitive Impairment (MCI) patients. This was in an effort to shorten acquisition times to minimise the risk of motion artefacts caused by patient discomfort and disorientation. The principle question is, "Can parallel imaging be used to acquire images at 1.5 T of sufficient quality to allow volumetric analysis of patient brains?"</p> <p>Methods</p> <p>Optimisation studies were performed on a young healthy volunteer and the selected protocol (including the use of two different parallel imaging acceleration factors) was then tested on a cohort of 15 elderly volunteers including MCI and AD patients. In addition to automatic brain segmentation, hippocampus volumes were manually outlined and measured in all patients. The 15 patients were scanned on a second occasion approximately one week later using the same protocol and evaluated in the same manner to test repeatability of measurement using images acquired with the GRAPPA parallel imaging technique applied to the MPRAGE sequence.</p> <p>Results</p> <p>Intraclass correlation tests show that almost perfect agreement between repeated measurements of both segmented brain parenchyma fraction and regional measurement of hippocampi. The protocol is suitable for both global and regional volumetric measurement dementia patients.</p> <p>Conclusion</p> <p>In summary, these results indicate that parallel imaging can be used without detrimental effect to brain tissue segmentation and volumetric measurement and should be considered for both clinical and research studies where longitudinal measurements of brain tissue volumes are of interest.</p

Cardiovascular magnetic resonance physics for clinicians: part I

There are many excellent specialised texts and articles that describe the physical principles of cardiovascular magnetic resonance (CMR) techniques. There are also many texts written with the clinician in mind that provide an understandable, more general introduction to the basic physical principles of magnetic resonance (MR) techniques and applications. There are however very few texts or articles that attempt to provide a basic MR physics introduction that is tailored for clinicians using CMR in their daily practice. This is the first of two reviews that are intended to cover the essential aspects of CMR physics in a way that is understandable and relevant to this group. It begins by explaining the basic physical principles of MR, including a description of the main components of an MR imaging system and the three types of magnetic field that they generate. The origin and method of production of the MR signal in biological systems are explained, focusing in particular on the two tissue magnetisation relaxation properties (T1 and T2) that give rise to signal differences from tissues, showing how they can be exploited to generate image contrast for tissue characterisation. The method most commonly used to localise and encode MR signal echoes to form a cross sectional image is described, introducing the concept of k-space and showing how the MR signal data stored within it relates to properties within the reconstructed image. Before describing the CMR acquisition methods in detail, the basic spin echo and gradient pulse sequences are introduced, identifying the key parameters that influence image contrast, including appearances in the presence of flowing blood, resolution and image acquisition time. The main derivatives of these two pulse sequences used for cardiac imaging are then described in more detail. Two of the key requirements for CMR are the need for data acquisition first to be to be synchronised with the subject's ECG and to be fast enough for the subject to be able to hold their breath. Methods of ECG synchronisation using both triggering and retrospective gating approaches, and accelerated data acquisition using turbo or fast spin echo and gradient echo pulse sequences are therefore outlined in some detail. It is shown how double inversion black blood preparation combined with turbo or fast spin echo pulse sequences acquisition is used to achieve high quality anatomical imaging. For functional cardiac imaging using cine gradient echo pulse sequences two derivatives of the gradient echo pulse sequence; spoiled gradient echo and balanced steady state free precession (bSSFP) are compared. In each case key relevant imaging parameters and vendor-specific terms are defined and explained

Accelerated CMR using zonal, parallel and prior knowledge driven imaging methods

Accelerated imaging is highly relevant for many CMR applications as competing constraints with respect to spatiotemporal resolution and tolerable scan times are frequently posed. Three approaches, all involving data undersampling to increase scan efficiencies, are discussed in this review. Zonal imaging can be considered a niche but nevertheless has found application in coronary imaging and CMR flow measurements. Current work on parallel-transmit systems is expected to revive the interest in zonal imaging techniques. The second and main approach to speeding up CMR sequences has been parallel imaging. A wide range of CMR applications has benefited from parallel imaging with reduction factors of two to three routinely applied for functional assessment, perfusion, viability and coronary imaging. Large coil arrays, as are becoming increasingly available, are expected to support reduction factors greater than three to four in particular in combination with 3D imaging protocols. Despite these prospects, theoretical work has indicated fundamental limits of coil encoding at clinically available magnetic field strengths. In that respect, alternative approaches exploiting prior knowledge about the object being imaged as such or jointly with parallel imaging have attracted considerable attention. Five to eight-fold scan accelerations in cine and dynamic CMR applications have been reported and image quality has been found to be favorable relative to using parallel imaging alone

Application of kt-BLAST acceleration to reduce cardiac MR imaging time in healthy and infarcted mice

OBJECT: We evaluated the use of kt-broad-use linear acquisition speed-up technique (kt-BLAST) acceleration of mouse cardiac imaging in order to reduce scan times, thereby minimising physiological variation and improving animal welfare. MATERIALS AND METHODS: Conventional cine cardiac MRI data acquired from healthy mice (n = 9) were subsampled to simulate kt-BLAST acceleration. Cardiological indices (left ventricular volume, ejection fraction and mass) were determined as a function of acceleration factor. kt-BLAST threefold undersampling was implemented on the scanner and applied to a second group of mice (n = 6 healthy plus 6 with myocardial infarct), being compared with standard cine imaging (3 signal averages) and cine imaging with one signal average. RESULTS: In the simulations, sufficient accuracy was achieved for undersampling factors up to three. Cardiological indices determined from the implemented kt-BLAST scanning showed no significant differences compared with the values determined from the standard sequence, and neither did indices derived from the cine scan with only one signal average despite its lower signal-to-noise ratio. Both techniques were applied successfully in the infarcted hearts. CONCLUSION: For cardiac imaging of mice, threefold undersampling of kt-space, or a similar reduction in the number of signal averages, are both feasible with subsequent reduction in imaging time

Multishot versus Single-Shot Pulse Sequences in Very High Field fMRI: A Comparison Using Retinotopic Mapping

High-resolution functional MRI is a leading application for very high field (7 Tesla) human MR imaging. Though higher field strengths promise improvements in signal-to-noise ratios (SNR) and BOLD contrast relative to fMRI at 3 Tesla, these benefits may be partially offset by accompanying increases in geometric distortion and other off-resonance effects. Such effects may be especially pronounced with the single-shot EPI pulse sequences typically used for fMRI at standard field strengths. As an alternative, one might consider multishot pulse sequences, which may lead to somewhat lower temporal SNR than standard EPI, but which are also often substantially less susceptible to off-resonance effects. Here we consider retinotopic mapping of human visual cortex as a practical test case by which to compare examples of these sequence types for high-resolution fMRI at 7 Tesla. We performed polar angle retinotopic mapping at each of 3 isotropic resolutions (2.0, 1.7, and 1.1 mm) using both accelerated single-shot 2D EPI and accelerated multishot 3D gradient-echo pulse sequences. We found that single-shot EPI indeed led to greater temporal SNR and contrast-to-noise ratios (CNR) than the multishot sequences. However, additional distortion correction in postprocessing was required in order to fully realize these advantages, particularly at higher resolutions. The retinotopic maps produced by both sequence types were qualitatively comparable, and showed equivalent test/retest reliability. Thus, when surface-based analyses are planned, or in other circumstances where geometric distortion is of particular concern, multishot pulse sequences could provide a viable alternative to single-shot EPI

What scans we will read: imaging instrumentation trends in clinical oncology

Oncological diseases account for a significant portion of the burden on public healthcare systems with associated costs driven primarily by complex and long-lasting therapies. Through the visualization of patient-specific morphology and functional-molecular pathways, cancerous tissue can be detected and characterized non- invasively, so as to provide referring oncologists with essential information to support therapy management decisions. Following the onset of stand-alone anatomical and functional imaging, we witness a push towards integrating molecular image information through various methods, including anato-metabolic imaging (e.g., PET/ CT), advanced MRI, optical or ultrasound imaging. This perspective paper highlights a number of key technological and methodological advances in imaging instrumentation related to anatomical, functional, molecular medicine and hybrid imaging, that is understood as the hardware-based combination of complementary anatomical and molecular imaging. These include novel detector technologies for ionizing radiation used in CT and nuclear medicine imaging, and novel system developments in MRI and optical as well as opto-acoustic imaging. We will also highlight new data processing methods for improved non-invasive tissue characterization. Following a general introduction to the role of imaging in oncology patient management we introduce imaging methods with well-defined clinical applications and potential for clinical translation. For each modality, we report first on the status quo and point to perceived technological and methodological advances in a subsequent status go section. Considering the breadth and dynamics of these developments, this perspective ends with a critical reflection on where the authors, with the majority of them being imaging experts with a background in physics and engineering, believe imaging methods will be in a few years from now. Overall, methodological and technological medical imaging advances are geared towards increased image contrast, the derivation of reproducible quantitative parameters, an increase in volume sensitivity and a reduction in overall examination time. To ensure full translation to the clinic, this progress in technologies and instrumentation is complemented by progress in relevant acquisition and image-processing protocols and improved data analysis. To this end, we should accept diagnostic images as “data”, and – through the wider adoption of advanced analysis, including machine learning approaches and a “big data” concept – move to the next stage of non-invasive tumor phenotyping. The scans we will be reading in 10 years from now will likely be composed of highly diverse multi- dimensional data from multiple sources, which mandate the use of advanced and interactive visualization and analysis platforms powered by Artificial Intelligence (AI) for real-time data handling by cross-specialty clinical experts with a domain knowledge that will need to go beyond that of plain imaging

Simultaneous multi slice (SMS) balanced steady state free precession first-pass myocardial perfusion cardiovascular magnetic resonance with iterative reconstruction at 1.5T

Background: Simultaneous-Multi-Slice (SMS) perfusion imaging has the potential to acquire multiple slices, increasing myocardial coverage without sacrificing in-plane spatial resolution. To maximise signal-to-noise ratio (SNR), SMS can be combined with a balanced steady state free precession (bSSFP) readout. Furthermore, application of gradient-controlled local Larmor adjustment (GC-LOLA) can ensure robustness against off-resonance artifacts and SNR loss can be mitigated by applying iterative reconstruction with spatial and temporal regularisation. The objective of this study was to compare cardiovascular magnetic resonance (CMR) myocardial perfusion imaging using SMS bSSFP imaging with GC-LOLA and iterative reconstruction to 3 slice bSSFP. Methods: Two contrast-enhanced rest perfusion sequences were acquired in random order in 8 patients: 6-slice SMS bSSFP and 3 slice bSSFP. All images were reconstructed with TGRAPPA. SMS images were also reconstructed using a non-linear iterative reconstruction with L1 regularisation in wavelet space (SMS-iter) with 7 different combinations for spatial (λσ) and temporal (λτ) regularisation parameters. Qualitative ratings of overall image quality (0 = poor image quality, 1 = major artifact, 2 = minor artifact, 3 = excellent), perceived SNR (0 = poor SNR, 1 = major noise, 2 = minor noise, 3 = high SNR), frequency of sequence related artifacts and patient related artifacts were undertaken. Quantitative analysis of contrast ratio (CR) and percentage of dark rim artifact (DRA) was performed. Results: Among all SMS-iter reconstructions, SMS-iter 6 (λσ 0.001 λτ 0.005) was identified as the optimal reconstruction with the highest overall image quality, least sequence related artifact and higher perceived SNR. SMS-iter 6 had superior overall image quality (2.50 ± 0.53 vs 1.50 ± 0.53, p = 0.005) and perceived SNR (2.25 ± 0.46 vs 0.75 ± 0.46, p = 0.010) compared to 3 slice bSSFP. There were no significant differences in sequence related artifact, CR (3.62 ± 0.39 vs 3.66 ± 0.65, p = 0.88) or percentage of DRA (5.25 ± 6.56 vs 4.25 ± 4.30, p = 0.64) with SMS-iter 6 compared to 3 slice bSSFP. Conclusions: SMS bSSFP with GC-LOLA and iterative reconstruction improved image quality compared to a 3 slice bSSFP with doubled spatial coverage and preserved in-plane spatial resolution. Future evaluation in patients with coronary artery disease is warranted