Generalized likelihood ratio detection for fMRI using complex data

Enhanced clinical task-based fMRI metrics through locally low-rank denoising of complex-valued data

OBJECTIVE

This study investigates a locally low-rank (LLR) denoising algorithm applied to source images from a clinical task-based functional MRI (fMRI) exam before post-processing for improving statistical confidence of task-based activation maps.

METHODS

Task-based motor and language fMRI was obtained in eleven healthy volunteers under an IRB approved protocol. LLR denoising was then applied to raw complex-valued image data before fMRI processing. Activation maps generated from conventional non-denoised (control) data were compared with maps derived from LLR-denoised image data. Four board-certified neuroradiologists completed consensus assessment of activation maps; region-specific and aggregate motor and language consensus thresholds were then compared with nonparametric statistical tests. Additional evaluation included retrospective truncation of exam data without and with LLR denoising; a ROI-based analysis tracked t-statistics and temporal SNR (tSNR) as scan durations decreased. A test-retest assessment was performed; retest data were matched with initial test data and compared for one subject.

RESULTS

fMRI activation maps generated from LLR-denoised data predominantly exhibited statistically significant (p = 4.88×10-4 to p = 0.042; one p = 0.062) increases in consensus t-statistic thresholds for motor and language activation maps. Following data truncation, LLR data showed task-specific increases in t-statistics and tSNR respectively exceeding 20 and 50% compared to control. LLR denoising enabled truncation of exam durations while preserving cluster volumes at fixed thresholds. Test-retest showed variable activation with LLR data thresholded higher in matching initial test data.

CONCLUSION

LLR denoising affords robust increases in t-statistics on fMRI activation maps compared to routine processing, and offers potential for reduced scan duration while preserving map quality.

COMPLEX-VALUED TIME SERIES MODELING FOR IMPROVED ACTIVATION DETECTION IN FMRI STUDIES

A complex-valued data-based model with pth order autoregressive errors and general real/imaginary error covariance structure is proposed as an alternative to the commonly-used magnitude-only data-based autoregressive model for fMRI time series. Likelihood-ratio-test-based activation statistics are derived for both models and compared for experimental and simulated data. For a dataset from a right-hand finger-tapping experiment, the activation map obtained using complex-valued modeling more clearly identifies the primary activation region (left functional central sulcus) than the magnitude-only model. Such improved accuracy in mapping the left functional central sulcus has important implications in neurosurgical planning for tumor and epilepsy patients. Additionally, we develop magnitude and phase detrending procedures for complex-valued time series and examine the effect of spatial smoothing. These methods improve the power of complex-valued data-based activation statistics. Our results advocate for the use of the complex-valued data and the modeling of its dependence structures as a more efficient and reliable tool in fMRI experiments over the current practice of using only magnitude-valued datasets.

Structural architecture supports functional organization in the human aging brain at a regionwise and network level

Functional interactions in the brain are constrained by the underlying anatomical architecture, and structural and functional networks share network features such as modularity. Accordingly, age-related changes of structural connectivity (SC) may be paralleled by changes in functional connectivity (FC). We provide a detailed qualitative and quantitative characterization of the SC-FC coupling in human aging as inferred from resting-state blood oxygen-level dependent functional magnetic resonance imaging and diffusion-weighted imaging in a sample of 47 adults with an age range of 18-82. We revealed that SC and FC decrease with age across most parts of the brain and there is a distinct age-dependency of regionwise SC-FC coupling and network-level SC-FC relations. A specific pattern of SC-FC coupling predicts age more reliably than does regionwise SC or FC alone (r = 0.73, 95% CI = [0.7093, 0.8522]). Hence, our data propose that regionwise SC-FC coupling can be used to characterize brain changes in aging. Hum Brain Mapp 37:2645-2661, 2016. © 2016 Wiley Periodicals, Inc.

Enhanced phase regression with Savitzky-Golay filtering for high-resolution BOLD fMRI

Phase regression exploits the temporal evolution of phase in individual voxels to suppress blood oxygenation level dependent (BOLD) signal fluctuations caused by larger vessels and draining veins while preserving signal changes from microvascular effects. However, this process does not perform well when phase time series have low signal-to-noise ratios because of high levels of physiological noise. We demonstrate that Savitzky-Golay filters may be used to recover the underlying change in phase and completely restore the efficacy of phase regression. We do not make a priori assumptions regarding phase evolution and perform a data-driven exploration of parameter space to select the Savitzky-Golay filter parameters that minimize temporal variance in each voxel after phase regression. This approach is shown to work well on data acquired with single-shot and multi-shot pulse sequences, and should therefore be useful for both human and animal gradient-echo fMRI at high spatial resolutions at high fields. The ability to improve the spatial specificity of BOLD activation may be especially advantageous for clinical applications of fMRI that rely upon the accuracy of individual subject's activation maps to assist with presurgical planning and clinical decision-making. Enhanced phase regression with Savitzky-Golay filtering may also find other uses in analyses of resting state functional connectivity.

Ricean over Gaussian modelling in magnitude fMRI Analysis-Added Complexity with Negligible Practical Benefits

It is well-known that Gaussian modeling of functional Magnetic Resonance Imaging (fMRI) magnitude time-course data, which are truly Rice-distributed, constitutes an approximation, especially at low signal-to-noise ratios (SNRs). Based on this fact, previous work has argued that Rice-based activation tests show superior performance over their Gaussian-based counterparts at low SNRs and should be preferred in spite of the attendant additional computational and estimation burden. Here, we revisit these past studies and after identifying and removing their underlying limiting assumptions and approximations, provide a more comprehensive comparison. Our experimental evaluations using ROC curve methodology show that tests derived using Ricean modeling are substantially superior over the Gaussian-based activation tests only for SNRs below 0.6, i.e SNR values far lower than those encountered in fMRI as currently practiced.

On the definition of signal-to-noise ratio and contrast-to-noise ratio for FMRI data

Signal-to-noise ratio, the ratio between signal and noise, is a quantity that has been well established for MRI data but is still subject of ongoing debate and confusion when it comes to fMRI data. fMRI data are characterised by small activation fluctuations in a background of noise. Depending on how the signal of interest and the noise are identified, signal-to-noise ratio for fMRI data is reported by using many different definitions. Since each definition comes with a different scale, interpreting and comparing signal-to-noise ratio values for fMRI data can be a very challenging job. In this paper, we provide an overview of existing definitions. Further, the relationship with activation detection power is investigated. Reference tables and conversion formulae are provided to facilitate comparability between fMRI studies.

A statistical fMRI model for differential T2* contrast incorporating T1 and T2* of gray matter

Relaxation parameter estimation and brain activation detection are two main areas of study in magnetic resonance imaging (MRI) and functional magnetic resonance imaging (fMRI). Relaxation parameters can be used to distinguish voxels containing different types of tissue whereas activation determines voxels that are associated with neuronal activity. In fMRI, the standard practice has been to discard the first scans to avoid magnetic saturation effects. However, these first images have important information on the MR relaxivities for the type of tissue contained in voxels, which could provide pathological tissue discrimination. It is also well-known that the voxels located in gray matter (GM) contain neurons that are to be active while the subject is performing a task. As such, GM MR relaxivities can be incorporated into a statistical model in order to better detect brain activation. Moreover, although the MR magnetization physically depends on tissue and imaging parameters in a nonlinear fashion, a linear model is what is conventionally used in fMRI activation studies. In this study, we develop a statistical fMRI model for Differential T2(*) ConTrast Incorporating T1 and T2(*) of GM, so-called DeTeCT-ING Model, that considers the physical magnetization equation to model MR magnetization; uses complex-valued time courses to estimate T1 and T2(*) for each voxel; then incorporates gray matter MR relaxivities into the statistical model in order to better detect brain activation, all from a single pulse sequence by utilizing the first scans.

Optimal compressed sensing reconstructions of fMRI using 2D deterministic and stochastic sampling geometries

BACKGROUND

Compressive sensing can provide a promising framework for accelerating fMRI image acquisition by allowing reconstructions from a limited number of frequency-domain samples. Unfortunately, the majority of compressive sensing studies are based on stochastic sampling geometries that cannot guarantee fast acquisitions that are needed for fMRI. The purpose of this study is to provide a comprehensive optimization framework that can be used to determine the optimal 2D stochastic or deterministic sampling geometry, as well as to provide optimal reconstruction parameter values for guaranteeing image quality in the reconstructed images.

METHODS

We investigate the use of frequency-space (k-space) sampling based on: (i) 2D deterministic geometries of dyadic phase encoding (DPE) and spiral low pass (SLP) geometries, and (ii) 2D stochastic geometries based on random phase encoding (RPE) and random samples on a PDF (RSP). Overall, we consider over 36 frequency-sampling geometries at different sampling rates. For each geometry, we compute optimal reconstructions of single BOLD fMRI ON & OFF images, as well as BOLD fMRI activity maps based on the difference between the ON and OFF images. We also provide an optimization framework for determining the optimal parameters and sampling geometry prior to scanning.

RESULTS

For each geometry, we show that reconstruction parameter optimization converged after just a few iterations. Parameter optimization led to significant image quality improvements. For activity detection, retaining only 20.3% of the samples using SLP gave a mean PSNR value of 57.58 dB. We also validated this result with the use of the Structural Similarity Index Matrix (SSIM) image quality metric. SSIM gave an excellent mean value of 0.9747 (max = 1). This indicates that excellent reconstruction results can be achieved. Median parameter values also gave excellent reconstruction results for the ON/OFF images using the SLP sampling geometry (mean SSIM > =0.93). Here, median parameter values were obtained using mean-SSIM optimization. This approach was also validated using leave-one-out.

CONCLUSIONS

We have found that compressive sensing parameter optimization can dramatically improve fMRI image reconstruction quality. Furthermore, 2D MRI scanning based on the SLP geometries consistently gave the best image reconstruction results. The implication of this result is that less complex sampling geometries will suffice over random sampling. We have also found that we can obtain stable parameter regions that can be used to achieve specific levels of image reconstruction quality when combined with specific k-space sampling geometries. Furthermore, median parameter values can be used to obtain excellent reconstruction results.

Order Selection of the Linear Mixing Model for Complex-valued FMRI Data

Functional magnetic resonance imaging (fMRI) data are originally acquired as complex-valued images, which motivates the use of complex-valued data analysis methods. Due to the high dimension and high noise level of fMRI data, order selection and dimension reduction are important procedures for multivariate analysis methods such as independent component analysis (ICA). In this work, we develop a complex-valued order selection method to estimate the dimension of signal subspace using information-theoretic criteria. To correct the effect of sample dependence to information-theoretic criteria, we develop a general entropy rate measure for complex Gaussian random process to calibrate the independent and identically distributed (i.i.d.) sampling scheme in the complex domain. We show the effectiveness of the approach for order selection on both simulated and actual fMRI data. A comparison between the results of order selection and ICA on real-valued and complex-valued fMRI data demonstrates that a fully complex analysis extracts more information about brain activation.

Complex and magnitude-only preprocessing of 2D and 3D BOLD fMRI data at 7 T

A challenge to ultra high field functional magnetic resonance imaging is the predominance of noise associated with physiological processes unrelated to tasks of interest. This degradation in data quality may be partially reversed using a series of preprocessing algorithms designed to retrospectively estimate and remove the effects of these noise sources. However, such algorithms are routinely validated only in isolation, and thus consideration of their efficacies within realistic preprocessing pipelines and on different data sets is often overlooked. We investigate the application of eight possible combinations of three pseudo-complementary preprocessing algorithms - phase regression, Stockwell transform filtering, and retrospective image correction - to suppress physiological noise in 2D and 3D functional data at 7 T. The performance of each preprocessing pipeline was evaluated using data-driven metrics of reproducibility and prediction. The optimal preprocessing pipeline for both 2D and 3D functional data included phase regression, Stockwell transform filtering, and retrospective image correction. This result supports the hypothesis that a complex preprocessing pipeline is preferable to a magnitude-only pipeline, and suggests that functional magnetic resonance imaging studies should retain complex images and externally monitor subjects' respiratory and cardiac cycles so that these supplementary data may be used to retrospectively reduce noise and enhance overall data quality.

Enhancing the utility of complex-valued functional magnetic resonance imaging detection of neurobiological processes through postacquisition estimation and correction of dynamic B(0) errors and motion

Functional magnetic resonance imaging (fMRI) time series analysis is typically performed using only the magnitude portion of the data. The phase information remains unused largely due to its sensitivity to temporal variations in the magnetic field unrelated to the functional response of interest. These phase changes are commonly the result of physiologic processes such as breathing or motion either inside or outside the imaging field of view. As a result, although the functional phase response carries pertinent physiological information concerning the vasculature, one aspect of which is the location of large draining veins, the full hemodynamic phase response is understudied and is poorly understood, especially in comparison with the magnitude response. It is likely that the magnitude and phase contain disjoint information, which could be used in tandem to better characterize functional hemodynamics. In this work, simulated and human fMRI experimental data are used to demonstrate how statistical analysis of complex-valued fMRI time series can be problematic, and how robust analysis using these powerful and flexible complex-valued statistics is possible through postprocessing with correction for dynamic magnetic field fluctuations in conjunction with estimated motion parameters. These techniques require no special pulse sequence modifications and can be applied to any complex-valued echo planar imaging data set. This analysis shows that the phase component appears to contain information complementary to that in the magnitude and that processing and analysis techniques are available to investigate it in a robust and flexible manner.

Magnitude and phase behavior of multiresolution BOLD signal

High spatial resolution fMRI provides a more precise estimate of brain activity than low resolution fMRI. The magnitude and phase parts of the BOLD signals are impacted differently by changes in the scan resolution. In this paper, we report on a numerical simulation to show the impact of spatial resolution upon the complex-valued BOLD signal in terms of magnitude and phase variation. We generate realistic capillary networks in cortex voxels, calculate the BOLD-induced magnetic field disturbance and the complex BOLD signals for the voxel and its subvoxels, and thereby characterize the magnitude and phase behaviors across multiple grid resolutions. Our results show that: 1) at higher spatial resolution there is greater spatial variation in the phase of the BOLD signal as compared to its magnitude; 2) the spatial variation of the phase signal monotonically increases with respect to spatial resolution while for the magnitude the spatial variation may reach a maximum at some resolution level; 3) voxels containing large capillaries have higher phase spatial variation than those with smaller capillaries; 4) the amplitude spatial variation at a resolution level increases with respect to relaxation time whereas the phase variation is generally unaffected.

Functional magnetic resonance imaging brain activation directly from k-space

In functional magnetic resonance imaging (fMRI), the process of determining statistically significant brain activation is commonly performed in terms of voxel time series measurements after image reconstruction and magnitude-only time series formation. The image reconstruction and statistical activation processes are treated separately. In this manuscript, a framework is developed so that statistical analysis is performed in terms of the original, prereconstruction, complex-valued k-space measurements. First, the relationship between complex-valued (Fourier) encoded k-space measurements and complex-valued image measurements from (Fourier) reconstructed images is reviewed. Second, the voxel time series measurements are written in terms of the original spatiotemporal k-space measurements utilizing this k-space and image relationship. Finally, voxelwise fMRI activation can be determined in image space in terms of the original k-space measurements. Additionally, the spatiotemporal covariance between reconstructed complex-valued voxel time series can be written in terms of the spatiotemporal covariance between complex-valued k-space measurements. This allows one to utilize the originally measured data in its more natural, acquired state rather than in a transformed state. The effects of modeling preprocessing in k-space on voxel activation and correlation can then be examined.

Biophysical modeling of phase changes in BOLD fMRI

In BOLD fMRI, stimulus related phase changes have been repeatedly observed in humans. However, virtually all fMRI processing utilizes the magnitude information only, while ignoring the phase. This results in an unnecessary loss of physiological information and signal-to-noise efficiency. A widely held view is that the BOLD phase change is zero for a voxel containing randomly orientated blood vessels and that phase changes are only due to the presence of large vessels. Based on a previously developed theoretical model, we show through simulations and experimental human BOLD fMRI data that a non-zero phase change can be present in a region with randomly oriented vessels. Using simulations of the model, we first demonstrate that a spatially distributed susceptibility results in a non-zero phase distribution. Next, experimental data in a finger-tapping experiment show consistent bipolar phase distribution across multiple subjects. This model is then used to show that in theory a bipolar phase distribution can also be produced by the model. Finally, we show that the model can produce a bipolar phase pattern consistent with that observed in the experimental data. Understanding of the mechanisms behind the experimentally observed phase changes in BOLD fMRI would be an important step forward and will enable biophysical model based methods for integrating the phase and magnitude information in BOLD fMRI experiments.

Likelihood-based hypothesis tests for brain activation detection from MRI data disturbed by colored noise: a simulation study

Functional magnetic resonance imaging (fMRI) data that are corrupted by temporally colored noise are generally preprocessed (i.e., prewhitened or precolored) prior to functional activation detection. In this paper, we propose likelihood-based hypothesis tests that account for colored noise directly within the framework of functional activation detection. Three likelihood-based tests are proposed: the generalized likelihood ratio (GLR) test, the Wald test, and the Rao test. The fMRI time series is modeled as a linear regression model, where one regressor describes the task-related hemodynamic response, one regressor accounts for a constant baseline and one regressor describes potential drift. The temporal correlation structure of the noise is modeled as an autoregressive (AR) model. The order of the AR model is determined from practical null data sets using Akaike's information criterion (with penalty factor 3) as order selection criterion. The tests proposed are based on exact expressions for the likelihood function of the data. Using Monte Carlo simulation experiments, the performance of the proposed tests is evaluated in terms of detection rate and false alarm rate properties and compared to the current general linear model (GLM) test, which estimates the coloring of the noise in a separate step. Results show that theoretical asymptotic distributions of the GLM, GLR, and Wald test statistics cannot be reliably used for computing thresholds for activation detection from finite length time series. Furthermore, it is shown that, for a fixed false alarm rate, the detection rate of the proposed GLR test statistic is slightly, but (statistically) significantly improved compared to that of the common GLM-based tests. Finally, simulations results reveal that all tests considered show seriously inferior performance if the order of the AR model is not chosen sufficiently high to give an adequate description of the correlation structure of the noise, whereas the effects of (slightly) overmodeling are observed to be less harmful.

Bayesian kernel methods for analysis of functional neuroimages

We propose an approach to analyzing functional neuroimages in which 1) regions of neuronal activation are described by a superposition of spatial kernel functions, the parameters of which are estimated from the data and 2) the presence of activation is detected by means of a generalized likelihood ratio test (GLRT). Kernel methods have become a staple of modern machine learning. Herein, we show that these techniques show promise for neuroimage analysis. In an on-off design, we model the spatial activation pattern as a sum of an unknown number of kernel functions of unknown location, amplitude, and/or size. We employ two Bayesian methods of estimating the kernel functions. The first is a maximum a posteriori (MAP) estimation method based on a Reversible-Jump Markov-chain Monte-Carlo (RJMCMC) algorithm that searches for both the appropriate model complexity and parameter values. The second is a relevance vector machine (RVM), a kernel machine that is known to be effective in controlling model complexity (and thus discouraging overfitting). In each method, after estimating the activation pattern, we test for local activation using a GLRT. We evaluate the results using receiver operating characteristic (ROC) curves for simulated neuroimaging data and example results for real fMRI data. We find that, while RVM and RJMCMC both produce good results, RVM requires far less computation time, and thus appears to be the more promising of the two approaches.

Complex data analysis in high-resolution SSFP fMRI

In transition-band steady-state free precession (SSFP) functional MRI (fMRI), functional contrast originates from a bulk frequency shift induced by a deoxygenated hemoglobin concentration change in the activated brain regions. This frequency shift causes a magnitude and/or phase-signal change depending on the off-resonance distribution of a voxel in the balanced-SSFP (bSSFP) profile. However, in early low-resolution studies, only the magnitude signal activations were shown. In this paper the task-correlated phase-signal change is presented in a high-resolution (1 x 1 x 1 mm3) study. To include this phase activation in a functional analysis, a new complex domain data analysis method is proposed. The results show statistically significant phase-signal changes in a large number of voxels comparable to that of the magnitude-activated voxels. The complex-data analysis method successfully includes these phase activations in the activation map and thus provides wider coverage compared to magnitude-data analysis results.

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.

Characterizing phase-only fMRI data with an angular regression model

FMRI voxel time series are complex-valued with real and imaginary parts that are usually converted to magnitude-phase polar coordinates. Magnitude-only data models that discard the phase portion of the data have dominated fMRI analysis. However, when such analyses are performed, the data that is discarded may contain valuable biologic information that is not in the magnitude data. This biologic information from BOLD fMRI data may be vascular [Menon RS. Postacquisition suppression of large-vessel BOLD signals in high-resolution fMRI. Magn Reson Med 2002;47(1):1-9] or neuronal [Bodurka J, Jesmanowicz A, Hyde JS, Xu H, Estowski L, Li S-J. Current-induced magnetic resonance phase imaging. J Magn Reson 1999;137(1):265-71] in origin. When phase-only time series that discard the magnitude portion of the data have been analyzed, ordinary least squares (OLS) regression has been the technique of choice. However, OLS models may fit poorly when phase-wrap or low signal-to-noise ratio (SNR) is present. We have explored alternatives to the OLS model which will account for the angular response of the phase while also allowing us the flexibility to develop similar hypothesis tests. We adopt an angular regression model by Fisher and Lee [Fisher NI, Lee AJ. Regression models for an angular response. Biometrics 1992;48:665-77] for our analysis and show its improvement over the OLS model at low SNR in terms of both parameter estimation and inferences. We found an improvement in parameter estimation along with modeling for the Fisher and Lee method in simulated data while detailing potential benefits when used with experimentally acquired data. Finally, we look at a map of the statistics testing the association of the observed voxel phase time course and the reference function in our acquired data. This shows the possible detection of biological information in the generally discarded phase.

Signal and noise of Fourier reconstructed fMRI data

In magnetic resonance imaging, complex-valued measurements are acquired in time corresponding to spatial frequency measurements in space generally placed on a Cartesian rectangular grid. These complex-valued measurements are transformed into a measured complex-valued image by an image reconstruction method. The most common image reconstruction method is the inverse Fourier transform. It is known that image voxels are spatially correlated. A property of the inverse Fourier transformation is that uncorrelated spatial frequency measurements yield spatially uncorrelated voxel measurements and vice versa. Spatially correlated voxel measurements result from correlated spatial frequency measurements. This paper describes the resulting correlation structure between voxel measurements when inverse Fourier reconstructing correlated spatial frequency measurements. A real-valued representation for the complex-valued measurements is introduced along with an associated multivariate normal distribution. One potential application of this methodology is that there may be a correlation structure introduced by the measurement process or adjustments made to the spatial frequencies. This would produce spatially correlated voxel measurements after inverse Fourier transform reconstruction that have artificially inflated spatial correlation. One implication of these results is that one source of spatial correlation between voxels termed connectivity may be attributed to correlated spatial frequencies. The true voxel connectivity may be less than previously thought. This methodology could be utilized to characterize noise correlation in its original form and adjust for it. The exact statistical relationship between spatial frequency measurements and voxel measurements has now been established.

Multisubject activation detection in fMRI by testing correlation of data with a signal subspace

In this article, a generalized likelihood ratio test is proposed to assess the correlation between multisubject functional MRI (fMRI) time series and bases of a signal subspace for detecting the existence of group activation in each voxel of the brain. The signal subspace is generated by a design matrix using the time series of the desired effects. The proposed method leads to testing the product of eigenvalues of a specific matrix. The eigenvector corresponding to the largest eigenvalue is the weighting vector for the linear combination of time series of various subjects that has the maximum correlation with the signal subspace. In another method, namely, canonical correlation analysis, the largest eigenvalue of the above matrix is tested for activation detection. Surrogate data on resting state (no activation) are generated by randomization and used to estimate the statistical distribution of these parameters under the null hypothesis condition. A postprocessing step is applied to prevent false detection of voxels that are not sufficiently active (among subjects) by defining a minimum ratio for the active population. The proposed methods are applied on simulated and experimental fMRI data, and the results are compared with those of the general linear model (GLM; using the SPM and FMRISTAT toolboxes). The proposed methods showed higher detection sensitivity as compared with the GLM for activation detection in simulated data. Similarly, they detected more activated regions than did the GLM from multisubject experimental fMRI data on a visual (sensorimotor) event-related task.

Implications of the Rician distribution for fMRI generalized likelihood ratio tests

In functional magnetic resonance imaging (fMRI), the general linear model test (GLMT) is widely used for brain activation detection. However, the GLMT relies on the assumption that the noise corrupting the data is Gaussian distributed. Because the majority of fMRI studies employ magnitude image reconstructions, which are Rician distributed, this assumption is invalid and has significant consequences in case the signal-to-noise ratio (SNR) is low. In this study, we show that the GLMT should not be used at low SNR. Furthermore, we propose a generalized likelihood ratio test for magnitude MR data that has the same performance compared to the GLMT for high SNR, but performs significantly better than the GLMT for low SNR.

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.

Ultrawideband Microwave Breast Cancer Detection: A Detection-Theoretic Approach Using the Generalized Likelihood Ratio Test

Microwave imaging has been suggested as a promising modality for early-stage breast cancer detection. In this paper, we propose a statistical microwave imaging technique wherein a set of generalized likelihood ratio tests (GLRT) is applied to microwave backscatter data to determine the presence and location of strong scatterers such as malignant tumors in the breast. The GLRT is formulated assuming that the backscatter data is Gaussian distributed with known covariance matrix. We describe the method for estimating this covariance matrix offline and formulating a GLRT for several heterogeneous two-dimensional (2-D) numerical breast phantoms, several three-dimensional (3-D) experimental breast phantoms, and a 3-D numerical breast phantom with a realistic half-ellipsoid shape. Using the GLRT with the estimated covariance matrix and a threshold chosen to constrain the false discovery rate (FDR) of the image, we show the capability to detect and localize small (<0.6 cm) tumors in our numerical and experimental breast phantoms even when the dielectric contrast of the malignant-to-normal tissue is below 2:1.

Generalized likelihood ratio tests for complex fMRI data: a simulation study

Statistical tests developed for the analysis of (intrinsically complex valued) functional magnetic resonance time series, are generally applied to the data's magnitude components. However, during the past five years, new tests were developed that incorporate the complex nature of fMRI data. In particular, a generalized likelihood ratio test (GLRT) was proposed based on a constant phase model. In this work, we evaluate the sensitivity of GLRTs for complex data to small misspecifications of the phase model by means of simulation experiments. It is argued that, in practical situations, GLRTs based on magnitude data are likely to perform better compared to GLRTs based on complex data in terms of detection rate and constant false alarm rate properties.

A complex way to compute fMRI activation

In functional magnetic resonance imaging, voxel time courses after Fourier or non-Fourier "image reconstruction" are complex valued as a result of phase imperfections due to magnetic field inhomogeneities. Nearly all fMRI studies derive functional "activation" based on magnitude voxel time courses [Bandettini, P., Jesmanowicz, A., Wong, E., Hyde, J.S., 1993. Processing strategies for time-course data sets in functional MRI of the human brain. Magn. Reson. Med. 30 (2): 161-173 and Cox, R.W., Jesmanowicz, A., Hyde, J.S., 1995. Real-time functional magnetic resonance imaging. Magn. Reson. Med. 33 (2): 230-236]. Here, we propose to directly model the entire complex or bivariate data rather than just the magnitude-only data. A nonlinear multiple regression model is used to model activation of the complex signal, and a likelihood ratio test is derived to determine activation in each voxel. We investigate the performance of the model on a real dataset, then compare the magnitude-only and complex models under varying signal-to-noise ratios in a simulation study with varying activation contrast effects.

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.

Model-independent method for fMRI analysis

This paper presents a fast method for delineation of activated areas of the brain from functional magnetic resonance imaging (fMRI) time series data. The steps of the work accomplished are as follows. 1) It is shown that the detection performance evaluated by the area under the receiver operating characteristic curve is directly related to the signal-to-noise ratio (SNR) of the composite image generated in the detection process. 2) Detection and segmentation of activated areas are formulated in a vector space framework. In this formulation, a linear transformation (image combination method) is shown to be desirable to maximize the SNR of the activated areas subject to the constraint of removing inactive areas. 3) An analytical solution for the problem is found. 4) Image pixel vectors and expected time series pattern (signature) for inactive pixels are used to calculate weighting vector and identify activated regions. 5) Signatures of the activated regions are used to segment different activities. 6) Segmented images by the proposed method are compared with those generated by the conventional methods (correlation, t-statistic, and z statistic). Detection performance and SNRs of the images are compared. The proposed approach outperforms the conventional methods of fMRI analysis. In addition, it is model-independent and does not require a priori knowledge of the fMRI response to the paradigm. Since the method is linear and most of the work is done analytically, numerical implementation and execution of the method are much faster than the conventional methods.

Activation detection in fMRI using a maximum energy ratio statistic obtained by adaptive spatial filtering

An adaptive spatial filtering method is proposed that takes into account contextual information in fMRI activation detection. This filter replaces the time series of each voxel with a weighted average of time series of a small neighborhood around it. The filter coefficients at each voxel are derived so as to maximize a test statistic designed to indicate the presence of activation. This statistic is the ratio of the energy of the filtered time series in a signal subspace to the energy of the residuals. It is shown that the filter coefficients and the maximum energy ratio can be found through a generalized eigenproblem. This approach equates the filter coefficients to the elements of an eigenvector corresponding to the largest eigenvalue of a specific matrix, while the largest eigenvalue itself becomes the maximum energy ratio that can be used as a statistic for detecting activation. The distribution of this statistic under the null hypothesis is derived by a nonparametric permutation technique in the wavelet domain. Also, in this paper we introduce a new set of basis vectors that define the signal subspace. The space spanned by these basis vectors covers a wide range of possible hemodynamic response functions (HRF) and is applicable to both event related and block design fMRI signal analysis. This approach circumvents the need for a priori assumptions about the exact shape of the HRF. Resting-state experimental fMRI data were used to assess the specificity of the method, showing that the actual false-alarm rate of the proposed method is equal or less than its expected value. Analysis of simulated data and motor task fMRI datasets from six volunteers using the method proposed here showed an improved sensitivity as compared to a conventional test with a similar statistic applied to spatially smoothed data.

Functional MRI activity characterization using response time shift estimates from curve evolution

Characterizing the response of the brain to a stimulus based on functional magnetic resonance imaging data is a major challenge due to the fact that the response time delay of the brain may be different from one stimulus phase to the next and from pixel to pixel. To enhance detectability, this work introduces the use of a curve evolution approach that provides separate estimates of the response time shifts at each phase of the stimulus on a pixel-by-pixel basis. The approach relies on a parsimonious but simple model that is nonlinear in the time shifts of the response relative to the stimulus and linear in the gains. To effectively use the response time shift estimates in a subspace detection framework, we implement a robust hypothesis test based on a Laplacian noise model. The algorithm provides a pixel-by-pixel functional characterization of the brain's response. The results based on experimental data show that response time shift estimates, when properly implemented, enhance detectability without sacrificing robustness.

Independent component analysis of fMRI data in the complex domain

In BOLD fMRI a series of MR images is acquired and examined for task-related amplitude changes. These functional changes are small, so it is important to maximize detection efficiency. Virtually all fMRI processing strategies utilize magnitude information and ignore the phase, resulting in an unnecessary loss of efficiency. As the optimum way to model the phase information is not clear, a flexible modeling technique is useful. To analyze complex data sets, independent component analysis (ICA), a data-driven approach, is proposed. In ICA, the data are modeled as spatially independent components multiplied by their respective time-courses. There are thus three possible approaches: 1) the time-courses can be complex-valued, 2) the images can be complex-valued, or 3) both the time-courses and the images can be complex-valued. These analytic approaches are applied to data from a visual stimulation paradigm, and results from three complex analysis models are presented and compared with magnitude-only results. Using the criterion of the number of contiguous activated voxels at a given threshold, an average of 12-23% more voxels are detected by complex-valued ICA estimation at a threshold of /Z/ > 2.5. Additionally, preliminary results from the complex models reveal a phase modulation similar to the magnitude time-course in some voxels, and oppositely modulated in other voxels.