Bayesian fMRI time series analysis with spatial priors

We describe a Bayesian estimation and inference procedure for fMRI time series based on the use of General Linear Models (GLMs). Importantly, we use a spatial prior on regression coefficients which embodies our prior knowledge that evoked responses are spatially contiguous and locally homogeneous. Further, using a computationally efficient Variational Bayes framework, we are able to let the data determine the optimal amount of smoothing. We assume an arbitrary order Auto-Regressive (AR) model for the errors. Our model generalizes earlier work on voxel-wise estimation of GLM-AR models and inference in GLMs using Posterior Probability Maps (PPMs). Results are shown on simulated data and on data from an event-related fMRI experiment.     

Multiple sparse priors for the M/EEG inverse problem

This paper describes an application of hierarchical or empirical Bayes to the distributed source reconstruction problem in electro- and magnetoencephalography (EEG and MEG). The key contribution is the automatic selection of multiple cortical sources with compact spatial support that are specified in terms of empirical priors. This obviates the need to use priors with a specific form (e.g., smoothness or minimum norm) or with spatial structure (e.g., priors based on depth constraints or functional magnetic resonance imaging results). Furthermore, the inversion scheme allows for a sparse solution for distributed sources, of the sort enforced by equivalent current dipole (ECD) models. This means the approach automatically selects either a sparse or a distributed model, depending on the data. The scheme is compared with conventional applications of Bayesian solutions to quantify the improvement in performance.     

Bayesian fMRI data analysis with sparse spatial basis function priors

In previous work we have described a spatially regularised General Linear Model (GLM) for the analysis of brain functional Magnetic Resonance Imaging (fMRI) data where Posterior Probability Maps (PPMs) are used to characterise regionally specific effects. The spatial regularisation is defined over regression coefficients via a Laplacian kernel matrix and embodies prior knowledge that evoked responses are spatially contiguous and locally homogeneous. In this paper we propose to finesse this Bayesian framework by specifying spatial priors using Sparse Spatial Basis Functions (SSBFs). These are defined via a hierarchical probabilistic model which, when inverted, automatically selects an appropriate subset of basis functions. The method includes non-linear wavelet shrinkage as a special case. As compared to Laplacian spatial priors, SSBFs allow for spatial variations in signal smoothness, are more computationally efficient and are robust to heteroscedastic noise. Results are shown on synthetic data and on data from an event-related fMRI experiment.     

Parametric imaging with Bayesian priors: a validation study with (11)C-Altropane PET

It has been suggested that Bayesian estimation methods may be used to improve the signal-to-noise ratio of parametric images. However, there is little experience with the method and some of the underlying assumptions and performance properties of Bayesian estimation remain to be investigated. We used a sample population of 54 subjects, studied previously with (11)C-Altropane, to empirically evaluate the assumptions, performance and some practical issues in forming parametric images. By using normality tests, we showed that the underpinning normality assumptions of data and parametric distribution apply to more than 80% of voxels. The standard deviation of the binding potential can be reduced 30-50% using Bayesian estimation, without introducing substantial bias. The sample size required to form the a priori information was found to be modest; as little as ten subjects may be sufficient and the choice of specific subjects has little effect on Bayesian estimation. A realistic simulation study showed that detection of localized differences in parametric images, e.g. by statistical parametric mapping (SPM), could be made more reliable and/or conducted with smaller sample size using Bayesian estimation. In conclusion, Bayesian estimation can improve the SNR of parametric images and better detect localized changes in cohorts of subjects.     

Solution of the inverse problem of magnetic induction tomography (MIT)

Magnetic induction tomography (MIT) of biological tissue is used to reconstruct the changes in the complex conductivity distribution inside an object under investigation. The measurement principle is based on determining the perturbation DeltaB of a primary alternating magnetic field B0, which is coupled from an array of excitation coils to the object under investigation. The corresponding voltages DeltaV and V0 induced in a receiver coil carry the information about the passive electrical properties (i.e. conductivity, permittivity and permeability). The reconstruction of the conductivity distribution requires the solution of a 3D inverse eddy current problem. As in EIT the inverse problem is ill-posed and on this account some regularization scheme has to be applied. We developed an inverse solver based on the Gauss-Newton-one-step method for differential imaging, and we implemented and tested four different regularization schemes: the first and second approaches employ a classical smoothness criterion using the unit matrix and a differential matrix of first order as the regularization matrix. The third method is based on variance uniformization, and the fourth method is based on the truncated singular value decomposition. Reconstructions were carried out with synthetic measurement data generated with a spherical perturbation at different locations within a conducting cylinder. Data were generated on a different mesh and 1% random noise was added. The model contained 16 excitation coils and 32 receiver coils which could be combined pairwise to give 16 planar gradiometers. With 32 receiver coils all regularization methods yield fairly good 3D-images of the modelled changes of the conductivity distribution, and prove the feasibility of difference imaging with MIT. The reconstructed perturbations appear at the right location, and their size is in the expected range. With 16 planar gradiometers an additional spurious feature appears mirrored with respect to the median plane with negative sign. This demonstrates that a symmetrical arrangement with one ring of planar gradiometers cannot distinguish between a positive conductivity change at the true location and a negative conductivity change at the mirrored location.     

Review on solving the inverse problem in EEG source analysis

In this primer, we give a review of the inverse problem for EEG source localization. This is intended for the researchers new in the field to get insight in the state-of-the-art techniques used to find approximate solutions of the brain sources giving rise to a scalp potential recording. Furthermore, a review of the performance results of the different techniques is provided to compare these different inverse solutions. The authors also include the results of a Monte-Carlo analysis which they performed to compare four non parametric algorithms and hence contribute to what is presently recorded in the literature. An extensive list of references to the work of other researchers is also provided.     

Solution of the inverse problem of magnetic induction tomography (MIT) with multiple objects: analysis of detectability and statistical properties with respect to the reconstructed conducting region

Magnetic induction tomography (MIT) is a technique to image the passive electrical properties (i.e. conductivity, permittivity, permeability) of biological tissues. The inverse eddy current problem is nonlinear and ill-posed, thus a Gauss-Newton one-step method in combination with four different regularization schemes is used to obtain stable solutions. Simulations with 16 excitation coils, 32 receiving coils and different spherical perturbations inside a homogeneous cylinder were computed. In order to compare the statistical properties of the reconstructed results a Monte Carlo study with a SNR of 40 dB and 20 dB was carried out. Simulated conductivity perturbations inside a homogeneous cylinder can be localized and resolved and the results prove the feasibility of difference imaging with MIT.     

Quantification of the benefit from integrating MEG and EEG data in minimum l2-norm estimation

Source current estimation from electromagnetic (MEG and EEG) signals is an ill-posed problem that often produces blurry or inaccurately positioned estimates. The two modalities have distinct factors limiting the resolution, e.g., MEG cannot detect radially oriented sources, while EEG is sensitive to accuracy of the head model. This makes combined EEG + MEG estimation techniques desirable, but different acquisition noise statistics, complexity of the head models, and lack of pertinent metrics all complicate the assessment of the resulting improvements. We investigated analytically the effect of including EEG recordings in MEG studies versus the addition of new MEG channels when computing noise-normalized minimum ℓ₂-norm estimates. Three-compartment boundary-element forward models were constructed using structural MRI scans for four subjects. Singular value analysis of the resulting forward models predicted better performance of the EEG + MEG case in the form of higher matrix rank. MNE inverse operators for EEG, MEG and EEG + MEG were constructed using the sensor noise covariance estimated from data. Metrics derived from the resolution matrices predicted higher spatial resolution in EEG + MEG as compared to MEG due to decreased spread (lower spatial dispersion, higher resolution index) with no reduction in dipole localization error. The effect was apparent in all source locations, with increased magnitude for deep areas such as the cingulate cortex. We were also able to corroborate the results for the somatosensory cortex using median nerve responses.     

Spatiotemporal analysis of neuromagnetic events underlying the emergence of coordinative instabilities

A full-head 143-channel superconducting quantum interference device was used to study changes occurring in the magnetic activity of the human brain during performance of an auditory-motor coordination task in which the rate of coordination was systematically increased. Previous research using the same task paradigm demonstrated that spontaneous switches in timing behavior that arise with higher coordination rates are accompanied by qualitative changes in spatiotemporal brain activity measured by electro- and magnetoencephalography. Here we show how these patterns can be decomposed into basic physiological events, i.e., evoked brain responses to acoustic tones and self-initiated finger movements. The frequency dependence of the amplitudes of these component responses may shed new light onto why spontaneous timing transitions occur in the first place.     

Anatomically accurate hard priors for transrectal electrical impedance tomography (TREIT) of the prostate

Current prostate biopsy procedures entail sampling tissues at template-based locations that are not patient specific. Ultrasound (US)-coupled transrectal electrical impedance tomography (TREIT), featuring an endorectal US probe retrofitted with electrodes, has been developed for prostate imaging. This multi-modal imaging system aims to identify suspicious tumor regions based on their electrical properties and ultimately provide additional patient-specific locations where to take biopsy samples. Unfortunately, the open-domain geometry associated with TREIT results in a severely ill-posed problem due to the small number of measurements and unbounded imaging domain. Furthermore, reconstructing contrasts within the prostate volume is challenging because the conductivity differences between the prostate and surrounding tissues are much larger than the conductivity differences between benign and malignant tissues within the prostate. To help overcome these problems, anatomically accurate hard priors can be employed to limit estimation of the electrical property distribution to within the prostate volume; however, this requires the availability of structural information. Here, a method that extracts the prostate surface from US images and incorporates this surface into the image reconstruction algorithm has been developed to enable estimation of electrical parameters within the prostate volume. In this paper, the performance of this algorithm is evaluated against a more traditional EIT algorithm that does not use anatomically accurate structural information, in the context of numerical simulations and phantom experiments. The developed anatomically accurate hard-prior algorithm demonstrably identifies contrasts within the prostate volume while an algorithm that does not rely on anatomically accurate structural information is unable to localize these contrasts. While inclusions are identified in the correct locations, they are found to be smaller in size than the actual object due to the rapid decay in sensitivity at increasing distances from the probe surface. Despite this, identifying the size of the inclusion accurately may not be essential for biopsy guidance in a clinical setting; instead, knowledge of the general vicinity of a cancerous lesion may be sufficient for suggesting and guiding clinicians to extract additional biopsy cores.     

Hierarchical Bayesian inference for the EEG inverse problem using realistic FE head models: depth localization and source separation for focal primary currents

The estimation of the activity-related ion currents by measuring the induced electromagnetic fields at the head surface is a challenging and severely ill-posed inverse problem. This is especially true in the recovery of brain networks involving deep-lying sources by means of EEG/MEG recordings which is still a challenging task for any inverse method. Recently, hierarchical Bayesian modeling (HBM) emerged as a unifying framework for current density reconstruction (CDR) approaches comprising most established methods as well as offering promising new methods. Our work examines the performance of fully-Bayesian inference methods for HBM for source configurations consisting of few, focal sources when used with realistic, high-resolution finite element (FE) head models. The main foci of interest are the correct depth localization, a well-known source of systematic error of many CDR methods, and the separation of single sources in multiple-source scenarios. Both aspects are very important in the analysis of neurophysiological data and in clinical applications. For these tasks, HBM provides a promising framework and is able to improve upon established CDR methods such as minimum norm estimation (MNE) or sLORETA in many aspects. For challenging multiple-source scenarios where the established methods show crucial errors, promising results are attained. Additionally, we introduce Wasserstein distances as performance measures for the validation of inverse methods in complex source scenarios.     

An inverse problem for the motion of blood in small vessels

In this paper, the blood motion in vessels with small radius is analyzed. The blood is modeled as a micromorphic fluid containing deformable material particles with 12 degrees of freedom: three translations, three rotations and six stretch and shears. Seven micromorphic viscosity coefficients are introduced as a function of the initial particle concentration and are reconstructed by a genetic algorithm based on experimental data.     

Systematic regularization of linear inverse solutions of the EEG source localization problem

Distributed linear solutions of the EEG source localization problem are used routinely. Here we describe an approach based on the weighted minimum norm method that imposes constraints using anatomical and physiological information derived from other imaging modalities to regularize the solution. In this approach the hyperparameters controlling the degree of regularization are estimated using restricted maximum likelihood (ReML). EEG data are always contaminated by noise, e.g., exogenous noise and background brain activity. The conditional expectation of the source distribution, given the data, is attained by carefully balancing the minimization of the residuals induced by noise and the improbability of the estimates as determined by their priors. This balance is specified by hyperparameters that control the relative importance of fitting and conforming to prior constraints. Here we introduce a systematic approach to this regularization problem, in the context of a linear observation model we have described previously. In this model, basis functions are extracted to reduce the solution space a priori in the spatial and temporal domains. The basis sets are motivated by knowledge of the evoked EEG response and information theory. In this paper we focus on an iterative "expectation-maximization" procedure to jointly estimate the conditional expectation of the source distribution and the ReML hyperparameters on which this solution rests. We used simulated data mixed with real EEG noise to explore the behavior of the approach with various source locations, priors, and noise levels. The results enabled us to conclude: (i) Solutions in the space of informed basis functions have a high face and construct validity, in relation to conventional analyses. (ii) The hyperparameters controlling the degree of regularization vary largely with source geometry and noise. The second conclusion speaks to the usefulness of using adaptative ReML hyperparameter estimates.     

Bayesian analysis of a dose-response experiment with serial sacrifices

This paper presents analysis and comments which are believed to be appropriate for certain carcinogenesis studies where sacrifices are performed throughout the experiment. Estimates of the risk probability for each dose level and sacrifice time are found utilizing the sample likelihood as the posterior density. The dose-response relationship is investigated with these estimates as the response. In order to test if the dose is effective and to check the appropriateness of the time-to-incidence model a Bayesian multiple comparisons technique is introduced.     

A solution to the dynamical inverse problem of EEG generation using spatiotemporal Kalman filtering

We present a new approach for estimating solutions of the dynamical inverse problem of EEG generation. In contrast to previous approaches, we reinterpret this problem as a filtering problem in a state space framework; for the purpose of its solution, we propose a new extension of Kalman filtering to the case of spatiotemporal dynamics. The temporal evolution of the distributed generators of the EEG can be reconstructed at each voxel of a discretisation of the gray matter of brain. By fitting linear autoregressive models with neighbourhood interactions to EEG time series, new classes of inverse solutions with improved resolution and localisation ability can be explored. For the purposes of model comparison and parameter estimation from given data, we employ a likelihood maximisation approach. Both for instantaneous and dynamical inverse solutions, we derive estimators of the time-dependent estimation error at each voxel. The performance of the algorithm is demonstrated by application to simulated and clinical EEG recordings. It is shown that by choosing appropriate dynamical models, it becomes possible to obtain inverse solutions of considerably improved quality, as compared to the usual instantaneous inverse solutions.