Daniel B. Rowe

Characterization of continuously distributed cortical water diffusion rates with a stretched-exponential model

Experience with diffusion‐weighted imaging (DWI) shows that signal attenuation is consistent with a multicompartmental theory of water diffusion in the brain. The source of this so‐called nonexponential behavior is a topic of debate, because the cerebral cortex contains considerable microscopic heterogeneity and is therefore difficult to model. To account for this heterogeneity and understand its implications for current models of diffusion, a stretched‐exponential function was developed to describe diffusion‐related signal decay as a continuous distribution of sources decaying at different rates, with no assumptions made about the number of participating sources. DWI experiments were performed using a spin‐echo diffusion‐weighted pulse sequence with b‐values of 500–6500 s/mm2 in six rats. Signal attenuation curves were fit to a stretched‐exponential function, and 20% of the voxels were better fit to the stretched‐exponential model than to a biexponential model, even though the latter model had one more adjustable parameter. Based on the calculated intravoxel heterogeneity measure, the cerebral cortex contains considerable heterogeneity in diffusion. The use of a distributed diffusion coefficient (DDC) is suggested to measure mean intravoxel diffusion rates in the presence of such heterogeneity. Magn Reson Med 50:727–734, 2003.

An evaluation of thresholding techniques in fMRI analysis

This paper reviews and compares individual voxel-wise thresholding methods for identifying active voxels in single-subject fMRI datasets. Different error rates are described which may be used to calibrate activation thresholds. We discuss methods which control each of the error rates at a prespecified level alpha, including simple procedures which ignore spatial correlation among the test statistics as well as more elaborate ones which incorporate this correlation information. The operating characteristics of the methods are shown through a simulation study, indicating that the error rate used has an important impact on the sensitivity of the thresholding method, but that accounting for correlation has little impact. Therefore, the simple procedures described work well for thresholding most single-subject fMRI experiments and are recommended. The methods are illustrated with a real bilateral finger tapping experiment.

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.

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.

A Bayesian approach to blind source separation

Abstract This paper presents a Bayesian statistical approach to the blind source separation problem. The blind source separation model is described; the source distribution is discussed; other approaches such as Principal Components, Independent Components, and Factor Analysis are detailed; prior distributions are introduced to incorporate available prior knowledge; the posterior distribution for the model parameters (including the number of sources) is derived; and the parameter estimation procedure is outlined. Finally Bayesian blind source separation is applied in a simulated example and its advantages over the other methods are stated.

Regression models for identifying noise sources in magnetic resonance images

Stochastic noise, susceptibility artifacts, magnetic field and radiofrequency inhomogeneities, and other noise components in magnetic resonance images (MRIs) can introduce serious bias into any measurements made with those images. We formally introduce three regression models including a Rician regression model and two associated normal models to characterize stochastic noise in various magnetic resonance imaging modalities, including diffusion-weighted imaging (DWI) and functional MRI (fMRI). Estimation algorithms are introduced to maximize the likelihood function of the three regression models. We also develop a diagnostic procedure for systematically exploring MR images to identify noise components other than simple stochastic noise, and to detect discrepancies between the fitted regression models and MRI data. The diagnostic procedure includes goodness-of-fit statistics, measures of influence, and tools for graphical display. The goodness-of-fit statistics can assess the key assumptions of the three regression models, whereas measures of influence can isolate outliers caused by certain noise components, including motion artifacts. The tools for graphical display permit graphical visualization of the values for the goodness-of-fit statistic and influence measures. Finally, we conduct simulation studies to evaluate performance of these methods, and we analyze a real dataset to illustrate how our diagnostic procedure localizes subtle image artifacts by detecting intravoxel variability that is not captured by the regression models.

An Evaluation of Spatial Thresholding Techniques in fMRI Analysis

Many fMRI experiments have a common objective of identifying active voxels in a neuroimaging dataset. This is done in single subject experiments, for example, by performing individual voxel‐wise tests of the null hypothesis that the observed time course is not significantly related to an assigned reference function. A voxel activation map is then constructed by applying a thresholding rule to the resulting statistics (e.g., t‐statistics). Typically the task‐related activation is expected to occur in clusters of voxels rather than in isolated single voxels. A variety of spatial thresholding techniques have been proposed to reflect this belief, including smoothing the raw t‐statistics, cluster size inference, and spatial mixture modeling. We study two aspects of these spatial thresholding procedures applied to single subject fMRI analysis through simulation. First, we examine the performance of these procedures in terms of sensitivity to detect voxel activation, using receiver operating characteristic curves. Second, we consider the accuracy of these spatial thresholding procedures in estimation of the size of the activation region, both in terms of bias and variance. The findings indicate that smoothing has the highest sensitivity to modest magnitude signals, but tend to overestimate the size of the activation region. Spatial mixture models estimate the size of a spatially distributed activation region well, but may be less sensitive to modest magnitude signals, indicating that additional research into more sensitive spatial mixture models is needed. Finally, the methods are illustrated with a real bilateral finger‐tapping fMRI experiment. Hum Brain Mapp, 2008.

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.