Modeling both the magnitude and phase of complex-valued fMRI data

In MRI and fMRI, images or voxel measurement are complex valued or bivariate at each time point. Recently, (Rowe, D.B., Logan, B.R., 2004. A complex way to compute fMRI activation. NeuroImage 23 (3), 1078-1092) introduced an fMRI magnitude activation model that utilized both the real and imaginary data in each voxel. This model, following traditional beliefs, specified that the phase time course were fixed unknown quantities which may be estimated voxel-by-voxel. Subsequently, (Rowe, D.B., Logan, B.R., 2005. Complex fMRI analysis with unrestricted phase is equivalent to a magnitude-only model. NeuroImage 24 (2), 603-606) generalized the model to have no restrictions on the phase time course. They showed that this unrestricted phase model was mathematically equivalent to the usual magnitude-only data model including regression coefficients and voxel activation statistic but philosophically different due to it derivation from complex data. Recent findings by (Hoogenrad, F.G., Reichenbach, J.R., Haacke, E.M., Lai, S., Kuppusamy, K., Sprenger, M., 1998. In vivo measurement of changes in venous blood-oxygenation with high resolution functional MRI at .95 Tesla by measuring changes in susceptibility and velocity. Magn. Reson. Med. 39 (1), 97-107) and (Menon, R.S., 2002. Postacquisition suppression of large-vessel BOLD signals in high-resolution fMRI. Magn. Reson. Med. 47 (1), 1-9) indicate that the voxel phase time course may exhibit task related changes. In this paper, a general complex fMRI activation model is introduced that describes both the magnitude and phase in complex data which can be used to specifically characterize task related change in both. Hypotheses regarding task related magnitude and/or phase changes are evaluated using derived activation statistics. It was found that the Rowe-Logan complex constant phase model strongly biases against voxels with task related phase changes and that the current very general complex linear phase model can be cast to address several different hypotheses sensitive to different magnitude/phase changes.     

Parameter estimation in the magnitude-only and complex-valued fMRI data models

In functional magnetic resonance imaging, voxel time courses are complex-valued data but are traditionally converted to real magnitude-only data ones. At a large signal-to-noise ratio (SNR), the magnitude-only data Ricean distribution is approximated by a normal distribution that has been suggested as reasonable in magnitude-only data magnetic resonance images for an SNR of 5 and potentially as low as 3. A complex activation model has been recently introduced by Rowe and Logan [Rowe, D.B., and Logan, B.R. (2004). A complex way to compute fMRI activation. NeuroImage, 23 (3):1078-1092] that is valid for all SNRs. The properties of the parameter estimators and activation statistic for these two models and a more accurate Ricean approximation based on a Taylor series expansion are characterized in terms of bias, variance, and Cramer-Rao lower bound. It was found that the unbiased estimators in the complex model continued to be unbiased for lower SNRs while those of the normal magnitude-only data model became biased as the SNR decreased and at differing levels for the regression coefficients. The unbiased parameter estimators from the approximate magnitude-only Ricean Taylor model were unbiased to lower SNRs than the magnitude-only normal ones. Further, the variances of the parameter estimators achieved their minimum value in the complex data model regardless of SNR while the magnitude-only data normal model and Ricean approximation using a Taylor series did not as the SNR decreased. Finally, the mean activation statistic for the complex data model was higher and not SNR dependent while it decreased with SNR in the magnitude-only data models but less so for the approximate Ricean model. These results indicate that using the complex data model and not approximations to the true magnitude-only Ricean data model is more appropriate at lower SNRs. Therefore, since the computational cost is relatively low for the complex data model and since the SNR is not inherently known a priori for all voxels, the complex data model is recommended at all SNRs.     

Complex fMRI analysis with unrestricted phase is equivalent to a magnitude-only model

Due to phase imperfections, voxel time course measurements are complex valued. However, most fMRI studies measure activation using magnitude-only time courses. We show that magnitude-only analyses are equivalent to a complex fMRI activation model in which the phase is unrestricted, or allowed to dynamically change over time. This suggests that improvements to the magnitude-only model are possible by modeling the phase in each voxel over time.     

Cross-correlation: an fMRI signal-processing strategy

The discovery of functional MRI (fMRI), with the first papers appearing in 1992, gave rise to new categories of data that drove the development of new signal-processing strategies. Workers in the field were confronted with image time courses, which could be reshuffled to form pixel time courses. The waveform in an active pixel time-course was determined not only by the task sequence but also by the hemodynamic response function. Reference waveforms could be cross-correlated with pixel time courses to form an array of cross-correlation coefficients. From this array of numbers, colorized images could be created and overlaid on anatomical images. An early paper from the authors' laboratory is extensively reviewed here (Bandettini et al., 1993. Magn. Reson. Med. 30:161-173). That work was carried out using the vocabulary of vector algebra. Cross-correlation methodology was central to the discovery of functional connectivity MRI (fcMRI) by Biswal et al. (1995. Magn. Reson. Med. 34:537-541). In this method, a whole volume time course of images is collected while the brain is nominally at rest and connectivity is studied by cross-correlation of pixel time courses.     

Functional imaging of the human superior colliculus: An optimised approach

Effective functional imaging of the human Superior Colliculus (SC) has often been regarded as difficult because of the small size of the SC and its proximity to sources of pulsatile (cardiac) noise. An optimised approach to functional imaging of the SC with fMRI is presented, based upon the novel finding that visually-induced BOLD responses in the SC are qualitatively different from responses in both cortical (V1) and sub-cortical (LGN) comparison areas. An optimised model with a Haemodynamic Response Function (HRF) which peaks early (4–5 s) and then falls rapidly is shown to be best suited for revealing SC responses, while a model peaking at 6 s and falling more slowly was most sensitive in the two comparison areas. Additionally, a method of correcting for the noise characteristics of fMRI responses proposed recently by de Zwart et al. (de Zwart, J. A., van Gelderen, P., Fukunaga, M., & Duyn, J. H. (2008). Reducing correlated noise in fMRI data. Magn Reson Med, 59, 939–945) is modified for use in the SC, and shown to be highly effective at further improving the statistical detectability of responses by modelling out noise. Together these methods represent a significant advance over previous approaches to functional imaging of the human SC. They permit the routine detection of strong SC activity in single subjects at standard spatial resolutions.     

Low-frequency fluctuations in the cardiac rate as a source of variance in the resting-state fMRI BOLD signal

Heart rate fluctuations occur in the low-frequency range (<0.1 Hz) probed in functional magnetic resonance imaging (fMRI) studies of resting-state functional connectivity and most fMRI block paradigms and may be related to low-frequency blood-oxygenation-level-dependent (BOLD) signal fluctuations. To investigate this hypothesis, temporal correlations between cardiac rate and resting-state fMRI signal timecourses were assessed at 3 T. Resting-state BOLD fMRI and accompanying physiological data were acquired and analyzed using cross-correlation and regression. Time-shifted cardiac rate timecourses were included as regressors in addition to established physiological regressors (RETROICOR (Glover, G.H., Li, T.Q., Ress, D., 2000. Image-based method for retrospective correction of physiological motion effects in fMRI: RETROICOR. Magn Reson Med 44, 162-167) and respiration volume per unit time (Birn, R.M., Diamond, J.B., Smith, M.A., Bandettini, P.A., 2006b. Separating respiratory-variation-related fluctuations from neuronal-activity-related fluctuations in fMRI. NeuroImage 31, 1536-1548). Significant correlations between the cardiac rate and BOLD signal timecourses were revealed, particularly negative correlations in gray matter at time shifts of 6-12 s and positive correlations at time shifts of 30-42 s (TR=6 s). Regressors consisting of cardiac rate timecourses shifted by delays of between 0 and 24 s explained an additional 1% of the BOLD signal variance on average over the whole brain across 9 subjects, a similar additional variance to that explained by respiration volume per unit time and RETROICOR regressors, even when used in combination with these other physiological regressors. This suggests that including such time-shifted cardiac rate regressors will be beneficial for explaining physiological noise variance and will thereby improve the statistical power in future task-based and resting-state fMRI studies.     

Reducing the unwanted draining vein BOLD contribution in fMRI with statistical post-processing methods

Recent BOLD fMRI data analysis methods show promise in reducing contributions from draining veins. The phase regressor method developed by [Menon, R.S., 2002. Post-acquisition suppression of large-vessel BOLD signals in high-resolution fMRI. Magn. Reson. Med., 47, 1-9] creates phase and magnitude images, regresses magnitude as a function of phase, and subtracts phase-estimated magnitudes from the observed magnitudes. The corrected magnitude images are used to compute cortical activations. The complex constant phase method, developed by [Rowe, D.B., Logan, B.R., 2004. A complex way to compute fMRI activation. NeuroImage, 23, 1078-1092], uses complex-valued reconstructed images and a nonlinear regressor model to compute magnitude cortical activations assuming temporally constant phase. In both methods, the usage of the phase information is claimed to bias against voxels with task-related phase changes caused by some draining veins. The behavior of the statistical methods in data with several task-related magnitude and phase changes is compared. The power of the statistical methods for determining voxels with specific task-related magnitude and phase change combinations are determined in ideal simulated data. The phase regressor and complex constant phase activation determination techniques are examined to characterize the responses of the models to select task-related phase and magnitude change combinations in representative simulated time series. Possible draining veins in human preliminary data are discussed and analyzed with the models and the current challenges which prevent these methods from being reliably implemented are discussed.     

Using nonlinear models in fMRI data analysis: model selection and activation detection

There is an increasing interest in using physiologically plausible models in fMRI analysis. These models do raise new mathematical problems in terms of parameter estimation and interpretation of the measured data. In this paper, we show how to use physiological models to map and analyze brain activity from fMRI data. We describe a maximum likelihood parameter estimation algorithm and a statistical test that allow the following two actions: selecting the most statistically significant hemodynamic model for the measured data and deriving activation maps based on such model. Furthermore, as parameter estimation may leave much incertitude on the exact values of parameters, model identifiability characterization is a particular focus of our work. We applied these methods to different variations of the Balloon Model (Buxton, R.B., Wang, E.C., and Frank, L.R. 1998. Dynamics of blood flow and oxygenation changes during brain activation: the balloon model. Magn. Reson. Med. 39: 855-864; Buxton, R.B., Uluda?, K., Dubowitz, D.J., and Liu, T.T. 2004. Modelling the hemodynamic response to brain activation. NeuroImage 23: 220-233; Friston, K. J., Mechelli, A., Turner, R., and Price, C. J. 2000. Nonlinear responses in fMRI: the balloon model, volterra kernels, and other hemodynamics. NeuroImage 12: 466-477) in a visual perception checkerboard experiment. Our model selection proved that hemodynamic models better explain the BOLD response than linear convolution, in particular because they are able to capture some features like poststimulus undershoot or nonlinear effects. On the other hand, nonlinear and linear models are comparable when signals get noisier, which explains that activation maps obtained in both frameworks are comparable. The tools we have developed prove that statistical inference methods used in the framework of the General Linear Model might be generalized to nonlinear models.     

A circuit for synchronizing external stimuli and events to the pulse sequence of a clinical magnetic resonance scanner

Interfacing experiments to a clinical magnetic resonance imaging scanner which require synchronization with an imaging pulse sequence can be challenging unless access to a trigger‐out is available. We present a simple, alternative approach to synchronize an experiment to a scanner using an external trigger to drive the scanner through the peripheral optical pulse rate monitor and cardiac trigger available on most clinical imagers. A trigger circuit and pulse rate monitor interface are described for applying a stimulus in the form of a pulsed voltage to a sample at a specific time in a spin‐echo, echo planar imaging sequence. The apparatus and approach could be used for many other types and numbers of experimental stimuli or events. © 2014 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 44B: 50–52, 2014     

A unified framework for group independent component analysis for multi-subject fMRI data

Independent component analysis (ICA) is becoming increasingly popular for analyzing functional magnetic resonance imaging (fMRI) data. While ICA has been successfully applied to single-subject analysis, the extension of ICA to group inferences is not straightforward and remains an active topic of research. Current group ICA models, such as the GIFT [Calhoun, V.D., Adali, T., Pearlson, G.D., Pekar, J.J., 2001. A method for making group inferences from functional MRI data using independent component analysis. Hum. Brain Mapp. 14, 140–151.] and tensor PICA [Beckmann, C.F., Smith, S.M., 2005. Tensorial extensions of independent component analysis for multisubject FMRI analysis. Neuroimage 25, 294–311.], make different assumptions about the underlying structure of the group spatio-temporal processes and are thus estimated using algorithms tailored for the assumed structure, potentially leading to diverging results. To our knowledge, there are currently no methods for assessing the validity of different model structures in real fMRI data and selecting the most appropriate one among various choices. In this paper, we propose a unified framework for estimating and comparing group ICA models with varying spatio-temporal structures. We consider a class of group ICA models that can accommodate different group structures and include existing models, such as the GIFT and tensor PICA, as special cases. We propose a maximum likelihood (ML) approach with a modified Expectation–Maximization (EM) algorithm for the estimation of the proposed class of models. Likelihood ratio tests (LRT) are presented to compare between different group ICA models. The LRT can be used to perform model comparison and selection, to assess the goodness-of-fit of a model in a particular data set, and to test group differences in the fMRI signal time courses between subject subgroups. Simulation studies are conducted to evaluate the performance of the proposed method under varying structures of group spatio-temporal processes. We illustrate our group ICA method using data from an fMRI study that investigates changes in neural processing associated with the regular practice of Zen meditation.     

Selection of voxel size and slice orientation for fMRI in the presence of susceptibility field gradients: application to imaging of the amygdala

The impact of voxel geometry on the blood oxygenation level-dependent (BOLD) signal detectability in the presence of field inhomogeneity is assessed and a quantitative approach to selecting appropriate voxel geometry is developed in this report. Application of the developed technique to BOLD sensitivity improvement of the human amygdala is presented. Field inhomogeneity was measured experimentally at 1.5 T and 3 T and the dominant susceptibility field gradient in the human amygdala was observed approximately along the superior-inferior direction. Based on the field mapping studies, an optimal selection for the slice orientation would be an oblique pseudo-coronal plane with its frequency-encoding direction parallel to the field gradient measured from each subject. Experimentally this was confirmed by comparing the normalized standard deviation of time-series echo-planar imaging signals acquired with different slice orientations, in the absence of a functional stimulus. A further confirmation with a carefully designed functional magnetic resonance imaging study is needed. Although the BOLD sensitivity may generally be improved by a voxel size commensurable with the activation volume, our quantitative analysis shows that the optimal voxel size also depends on the susceptibility field gradient and is usually smaller than the activation volume. The predicted phenomenon is confirmed with a hybrid simulation, in which the functional activation was mathematically added to the experimentally acquired rest-period echo-planar imaging data.     

Mapping the MRI voxel volume in which thermal noise matches physiological noise--implications for fMRI

This work addresses the choice of the imaging voxel volume in blood oxygen level dependent (BOLD) functional magnetic resonance imaging (fMRI). Noise of physiological origin that is present in the voxel time course is a prohibitive factor in the detection of small activation-induced BOLD signal changes. If the physiological noise contribution dominates over the temporal fluctuation contribution in the imaging voxel, further increases in the voxel signal-to-noise ratio (SNR) will have diminished corresponding increases in temporal signal-to-noise (TSNR), resulting in reduced corresponding increases in the ability to detect activation induced signal changes. On the other hand, if the thermal and system noise dominate (suggesting a relatively low SNR) further decreases in SNR can prohibit detection of activation-induced signal changes. Here we have proposed and called the "suggested" voxel volume for fMRI the volume where thermal plus system-related and physiological noise variances are equal. Based on this condition we have created maps of fMRI suggested voxel volume from our experimental data at 3T, since this value will spatially vary depending on the contribution of physiologic noise in each voxel. Based on our fast EPI segmentation technique we have found that for gray matter (GM), white matter (WM), and cerebral spinal fluid (CSF) brain compartments the mean suggested cubical voxel volume is: (1.8 mm)3, (2.1 mm)3 and (1.4 mm)3, respectively. Serendipitously, (1.8 mm)3 cubical voxel volume for GM approximately matches the cortical thickness, thus optimizing BOLD contrast by minimizing partial volume averaging. The introduced suggested fMRI voxel volume can be a useful parameter for choice of imaging volume for functional studies.     

An advanced,integrated large‐volume high‐pressure autoclave and 1h/13c double‐tuned resonator for chemistry and materials nuclear magnetic resonance spectroscopy and microscopy investigations

Nuclear magnetic resonance spectroscopy and imaging are well‐established tools in chemistry, physics, and life sciences. Nevertheless, most applications are performed at room temperature and atmospheric pressure. To study the processes in supercritical fluids, sample containers and coils have to be redesigned to especially allow for higher pressures up to several hundred times the atmospheric pressure. In this study, we present a setup for performing spectroscopic and imaging experiments on wood immersed in supercritical CO₂ at up to 20 MPa for drying. A magnetic resonance‐compatible autoclave as well as a double‐tuned ¹H/¹³C‐birdcage coil was designed and a setup for regulating pressure and storing gases was assembled. We were able to successfully perform measurements on the wood and water during the drying process and gaininsights into the displacement of water and its chemical reactions with the highly pressurized CO₂. © 2013 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 43B: 49–58, 2013     

Voxel-based homogeneity probability maps of gray matter in groups: assessing the reliability of functional effects

A subject of increasing importance in magnetic resonance imaging (MRI) is the analysis of intersubject structural differences, particularly when comparing groups of subjects with different conditions or diagnoses. On the other hand, determining structural homogeneity across subjects using voxel-based morphological (VBM) methods has become even more important to investigators who test for group brain activation using functional magnetic resonance images (fMRI) or positron emission tomography (PET). In the absence of methods that evaluate structural differences, one does not know how much reliability to assign to the functional differences. Here, we describe a voxel-based method for quantitatively assessing the homogeneity of tissues from structural magnetic resonance images of groups. Specifically, this method determines the homogeneity of gray matter for a group of subjects. Homogeneity probability maps (HPMs) of a given tissue type (e.g., gray matter) are generated by using a confidence interval based on binomial distribution. These maps indicate for each voxel the probability that the tissue type is gray for the population being studied. Therefore, HPMs can accompany functional analyses to indicate the confidence one can assign to functional difference at any given voxel. In this paper, examples of HPMs generated for a group of control subjects are shown and discussed. The application of this method to functional analysis is demonstrated.     

Universal radio‐frequency receive‐coil connector: Concept and validation at 3 T

In most clinical magnetic resonance imaging systems, only commercial receive coils with the appropriate connector and encoding can be plugged in. When willing to use a dedicated receive coil for a specific study which cannot be achieved with commercial coils, the researcher faces the connecting issue related to the specificity of the proprietary connector. In this work, a universal device is proposed which allows for the connection of any single channel dedicated coil on any magnetic resonance (MR) system, as long as it is provided with at least one commercial receive coil. Technical feasibility of the universal connecting device was demonstrated on a 3 T MR clinical imager. The device included an independent active decoupling circuit while signal transmission to the data cabinet was achieved by electromagnetic coupling with a commercial receive coil plugged to the MR device. The universal connecting device was notably characterized in terms of signal‐to‐noise ratio (SNR) and compared to the standard connection. Image SNR was comparable using both means of connection. © 2015 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 45B: 125–133, 2015     

Estimation and classification of fMRI hemodynamic response patterns

In this paper, we propose an approach to modeling functional magnetic resonance imaging (fMRI) data that combines hierarchical polynomial models, Bayes estimation, and clustering. A cubic polynomial is used to fit the voxel time courses of event-related design experiments. The coefficients of the polynomials are estimated by Bayes estimation, in a two-level hierarchical model, which allows us to borrow strength from all voxels. The voxel-specific Bayes polynomial coefficients are then transformed to the times and magnitudes of the minimum and maximum points on the hemodynamic response curve, which are in turn used to classify the voxels as being activated or not. The procedure is demonstrated on real data from an event-related design experiment of visually guided saccades and shown to be an effective alternative to existing methods.     

Changes in fMRI magnitude data and phase data observed in block-design and event-related tasks

Functional magnetic resonance imaging (fMRI) data are acquired as a complex image pair including magnitude and phase information. The vast majority of fMRI experiments do not attempt to take advantage of the time varying phase information. The phase of the MRI signal is related to the local magnetic field changes, suggesting it may contain useful information about the source of hemodynamic activity. Analysis of phase data acquired from different fMRI experiments has shown the presence of activity in response to various stimuli. However, there have been no studies that have examined phase data in a larger group of subjects for multiple types of fMRI tasks nor have studies examined phase changes due to event-related stimuli. In this paper, we evaluate the correspondence between the magnitude and phase changes at a group level in a block-design motor tapping task and in an event-related auditory oddball task. The results for both block-design and event-related tasks indicate the presence of task-related information in the phase data with phase-only and magnitude-only approaches showing signal changes in the expected brain regions. Although there is more overall activity detected with magnitude data, the phase-only analysis also reveals activity in regions expected to be involved in the task, some of which were not significantly activated in the magnitude-only analysis, suggesting that the phase might provide some unique information. In addition, the phase can potentially increase sensitivity within regions also showing magnitude changes. Future work should focus on additional methods for combining the magnitude and phase data.     

Analytical expressions for magnetic fields and field gradients of current‐carrying triangles

We present an analytical method for calculating magnetic field gradients generated by arbitrary triangulated surfaces. Our work builds upon the results published by Pissanetzky and Xiang, who presented formulas for calculating the magnetic field of current‐carrying faceted surfaces. We show that the analytical gradient expressions can be computed considerably faster than finite field value differences. We also find that the aforementioned published expressions for the magnetic field can be simplified and optimized substantially. Closer inspection of the algorithms, for both field and gradient, reveals a number pathological parameter constellations, which require special treatment. We present a detailed discussion on this. Our results can be directly applied in the optimization of complex magnetic field coils, such as magnetic resonance gradient coils. © 2014 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 44B: 18–25, 2014     

Mixed-effects and fMRI studies

This note concerns mixed-effect (MFX) analyses in multisession functional magnetic resonance imaging (fMRI) studies. It clarifies the relationship between mixed-effect analyses and the two-stage "summary statistics" procedure (Holmes, A.P., Friston, K.J., 1998. Generalisability, random effects and population inference. NeuroImage 7, S754) that has been adopted widely for analyses of fMRI data at the group level. We describe a simple procedure, based on restricted maximum likelihood (ReML) estimates of covariance components, that enables full mixed-effects analyses in the context of statistical parametric mapping. Using this procedure, we compare the results of a full mixed-effects analysis with those obtained from the simpler two-stage procedure and comment on the situations when the two approaches may give different results.