Bayesian Methods in Functional Magnetic Resonance Imaging

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1 Bayesian Methods in Functional Magnetic Resonance Imaging Galin L. Jones Kuo-Jung Lee Brian S. Caffo Susan Spear Bassett Abstract: One of the major objectives of functional magnetic resonance imaging studies is to infer regional neuronal activity in response to a stimulus or task. We consider a Bayesian approach which incorporates spatial and temporal dependence and allows for the task-related response to change dynamically over the scanning session. AMS 2000 subject classifications: Primary 62-02; secondary 62P10. Keywords and phrases: BOLD response, functional MRI, neuroimaging, Variable selection. 1. Introduction Most functional neuroimaging experiments aim to either uncover localized regions where the brain activates during a task or describe the networks required for a particular brain function. For example, is there a common area of the brain which is active when presented with auditory stimuli? Or are there remote regions of the brain which interact to accomplish this task? Our focus is on functional magnetic resonance imaging (fmri) techniques to study brain activation. Neuronal activation occurs in milliseconds and is not observed directly in fmri experiments. However, the well-established principle of neurovascular coupling (that is, local neuronal activation is related to changes in cerebral blood flow) allows us to observe this phenomenon via the blood oxygenation level dependent (BOLD) signal contrast. The structure of a typical fmri experiment is straightforward. A subject in an MRI scanner performs a task in response to a stimulus. Three-dimensional images of the subject s brain are captured every 2-3 seconds. The experiment may be repeated on the same subject and is often performed on several subjects. The subjects may be divided into groups. Even if we limit attention to a single subject in a single session, the data collected from an fmri experiment has a complicated structure. Imagine that the subject s brain is divided into a regular grid of 100, ,000 volume elements, or voxels. The BOLD signal is observed at each voxel at each of say time points. Thus there is an enormous amount of data which exhibits both spatial and temporal dependence. Moreover, the data tend to be noisy and have a weak signal. There are few, if any, off-the-shelf methods for constructing 1

2 Jones et al./bayes for fmri 2 sensible, computationally feasible statistical models in this situation. Indeed, Lindquist (2008) considered only non-bayesian approaches and concluded The size and complexity of the data make it difficult to create a full statistical model for describing its behavior, and a number of shortcuts are required to balance computational feasibility with model efficiency. In reality, the situation is much more delicate than we have indicated. Raw fmri data require preprocessing to remove artifacts accrued during acquisition. Throughout we will use standard fmri preprocessing techniques so that we can assume the data have the basic characteristics described in the previous paragraph. The interested reader should consult Friston et al. (2007) and Lindquist (2008) for accessible introductions to this important topic. Now we consider a specific fmri experiment which motivated the development of the general Bayesian approach described in Section An example fmri experiment The following experiment is fully analyzed by Lee et al. (2011) and is part of a longitudinal study of Alzheimer s disease. (We will consider a partial analysis of the data arising from this experiment in Section 2.1.) However, none of the subjects we considered have clinically diagnosed neurologic disorders, including Alzheimer s disease. The goal is to develop useful models of single-subject taskrelated activation in order to develop effective imaging biomarkers. Each subject completed a Stroop exam in an MRI scanner. That is, the subject is shown a word and subsequently asked to identify the color of the ink. There are three possible characteristics of each word: Ink only; for example, when presented with, XXXX, the subject should answer red. Congruence; for example, when presented with BLUE, the subject should answer blue. Interference; for example, when presented with BLUE, the subject should answer red. The Stroop exam was administered in a block design (see e.g. Friston et al., 2007) with a two second scanning time repetition. The battery of Ink, Congruence and Interference tasks were repeated in sequence 3 times with observations taken at a total of 465 time points. The Stroop task is an attempt to assess directed attention and hence it may be unsurprising that there is a well-documented age effect. That is, younger adults will typically experience a smaller interference effect than older adults. Also, interference effects may decrease with practice (Davidson, Zacks and Williams, 2003). In this study all subjects are older, generally well-educated, and healthy so we will not model an age effect, although it could be easily incorporated into our model below, but we should try to account for the possibility of a temporal drift in the response over the scanning session.

3 Jones et al./bayes for fmri 3 2. A Bayesian approach to detecting activation Our goal is detecting activation of a voxel when a stimulus is presented. For voxel v = 1,..., N let y v,i be the BOLD image intensity at time i = 1,..., T v and set y v = (y v,1,..., y v,tv ) T. Let X v be a known T v p matrix of full column rank and β v = (β v,1,..., β v,p ) T be a p 1 vector. (Typically, X v is a stimulus function convolved with the assumed haemodynamic response function (Friston et al., 2007). ) We assume a linear model for the conditional mean so that E[y v X v ] = X v β v. Detecting activation is equivalent to identifying the nonzero β v, a variable selection problem. We build on the work of Smith and Fahrmeir (2007) who were the first to use a variable selection approach in fmri settings, but our model differs substantively from theirs in that we model both spatial and temporal correlation, while allowing for the possibility of temporal drift in the BOLD signal over the scanning session. Let γ v = (γ v,1,..., γ v,p ) T be binary random variables used to indicate whether the voxel is activated by a sequence of input stimuli. That is, the coefficient β v,j is equal to zero if γ v,j = 0 and β v,j is nonzero if γ v,j = 1. The zero of γ v,j implies no effect on voxel v is caused by the corresponding experimental task j. Therefore, the model in can be expressed as y v = X v (γ v )β v (γ v ) + ε v, ε v N Tv (0, σ 2 vλ v ), where β v (γ v ) is the vector of nonzero regression coefficients and X v (γ v ) is the corresponding matrix for a given indicator variable γ v. An appropriate choice for the structure of the matrix Λ v will allow us to account for the temporal dependence between observations on a given voxel. We now turn our attention to specification of prior distributions. First consider the prior on β v (γ v ) given γ v for a particular voxel, v. We will use Zellner s g-prior (Zellner, 1996) given by where β v (γ v ) y v, σ 2 v, Λ v, γ v N( ˆβ v (γ v ), T v σ 2 v[x T v (γ v )Λ 1 v X v (γ v )] 1 ), (1) ˆβ v (γ v ) = [Xv T (γ v )Λ 1 v X v (γ v )] 1 Xv T (γ v )Λ 1 v y v. (2) We assume the σ 2 v are independent with prior π(σ 2 v) 1 σ 2 v. (3) Lee et al. (2011) demonstrate that the choice of prior on Λ v is critically important to the resulting posterior inference. It is natural to think of using a Wishart distribution or to assume the elements of Λ v (i, j) = ρ i j j, that is an AR(1) structure, and further that ρ v Uniform( 1, 1). Other autoregressive structures (e.g. AR(2)) are possible and popular. However, all of these possibilities require substantial computational effort. Thus some have suggested that Λ v

4 Jones et al./bayes for fmri 4 be taken to be the identity matrix. Note that this assumes independence in the time series on each voxel. Lee et al. (2011) show that little inferential efficacy is lost, compared to using autoregressive structures, by taking the prior on ρ v to be a point mass at the maximum likelihood estimator while the computational effort is comparable to assuming Λ v to be the identity matrix. We will use this point mass prior in the rest of this paper. We assume a binary spatial Ising prior for the indicator variables. Let γ (j) = {γ 1,j,..., γ N,j } be the vector of indicator variables for regression j over locations {1, 2,..., N}. We specify a three-dimensional neighborhood structure where each interior voxel has 6 neighbors. Let v k denote that voxels v and k are neighbors. Next, let the ω v,k be prespecified constants that allow us to weigh the interaction between neighboring locations on lattices v and k and let θ j > 0 represent the strength of the interaction between any two voxels. The prior on γ is then π(γ θ) = p j=1 π(γ (j) θ j ) with { N } π(γ (j) θ j ) exp α v,j γ v,j + θ j ω v,k I(γ v,j = γ k,j ). (4) The term v=1 N α v,j γ v,j. v=1 is known as the external field and is where we incorporate anatomical prior information. Finally, we assume a uniform prior on θ = (θ 1,..., θ p ), i.e., π(θ) p j=1 I(0 < θ j < θmax), where θmax is a user-specified hyperparameter. Notice that we assume θ, ρ, and σ 2 are a priori independent, γ conditionally independent, and independence across voxels. Thus the posterior density is characterized by q(β(γ), γ, ρ, θ, σ 2 y) v k N [p(y v β v (γ v ), γ v, σv, 2 Λ v )π(β v (γ v ) y v, γ v, σv, 2 Λ v ) v=1 π(σ 2 v)]π(γ θ)π(ρ)π(θ). (5) The posterior quantities of interest in detecting activation are the activation probability and its magnitude for each voxel, specifically, q(γ v,j = 1 y) and E(β v y). These quantities are analytically intractable and hence sophisticated Markov chain Monte Carlo methods are required to perform posterior inference; see Lee et al. (2011) for the details Application to example fmri data In the fmri experiment described in Section 1.1 the Stroop tasks are repeated three times. Here we consider how to study possible changes to the activation pattern over time. Let x i,j be the transformed input function of task j in ith trial, z i is a vector used to remove low-frequency, stimulus independent effects,

5 Jones et al./bayes for fmri 5 β i,j is the parameter of interest corresponding to the jth task in ith trial, the α i are nonzero nuisance parameters to model the baseline of a brain signal. Set Y v =α 0 z 0 + α 1 x 1 + β 1,1 x 1,1 + β 1,2 x 1,2 + β 1,3 x 1,3 + β 2,1 x 2,1 + β 2,2 x 2,2 + β 2,3 x 2,3 + β 3,1 x 3,1 + β 3,2 x 3,2 + β 3,3 x 3,3 + ε v, (6) where ε v N(0, σ 2 vλ v ). A sequence of binary variables is used to indicate if the corresponding parameter is zero or not so that for voxel v we have γ v = {1, 1, γ 3,v, γ 4,v, γ 5,v, γ 6,v, γ 7,v, γ 8,v, γ 9,v, γ 10,v, γ 11,v }. The rest of the model specification is as described above. The results for the interference task for each of the 3 trials are presented in Figure 1. Notice that there does appear to be a difference in the activation patterns over the trials. 3. Bayesian modeling versus voxel-level analyses A standard non-bayesian approach to analyzing fmri data is statistical parametric mapping (SPM). A normal-theory general linear model is fit for the time series of observations for each voxel, often assuming an AR correlation structure for the time series and a working assumption that all voxels are independent. Detecting activation of a voxel is done with a voxel-specific hypothesis test, producing a p-value (equivalently a test statistic which is usually a t- or F -test). The collection of p-values forms the SPM. In order to obtain an activation map of the brain, the next step is to threshold the spatially dependent p-values at a given overall error rate, leading to a problem of multiplicity. The most common way of addressing this problem is to use random field theory in which one must assume that the SPM is a good approximation to an underlying continuous random field. However, even with this assumption, the thresholding is done indirectly as it is based on the so-called expected Euler characteristic. For a full description of the regularity conditions and the underlying theory the reader can consult Friston et al. (1994), Worsley et al. (1992), Worsley (1994), Worsley et al. (1996) and Worsley (2003). Attempts to avoid random field assumptions employ multivariate permutation tests (Nichols and Holmes, 2002). Even if we accept the SPM approach it does not allow us to employ a principled approach to including prior information. In some fmri experiments we have a lot of scientific prior information about which regions of the brain should activate. In the SPM approach we would just restrict attention to that region plus a little bit more to be safe. Of course, incorporating prior information are conceptually straightforward in the Bayesian framework. Permutation tests, error rate thresholding (Chen et al., 2009; Genovese, Lazar and Nichols, 2002) and SPM random field corrections all have a similar theme of not producing a full model for the neuroimaging data. As such, activities such as a jointly modeling activation and connectivity are not possible. In contrast,

6 Jones et al./bayes for fmri 6 Fig 1. Activation maps for the time-varying model for the Interference task. The top panel contains activation maps for the first trial, the middle panel for the second trial and the bottom panel for the third trial.

7 Jones et al./bayes for fmri 7 Bayesian efforts are capable of producing a far more complete, thorough and robust approach towards investigating fmri data. However, more research is required to ensure that Bayesian modeling approaches are as computationally feasible as voxel-level approaches. References Chen, S., Wang, C., Eberly, L. E., Caffo, B. S. and Schwartz, B. S. (2009). Adaptive control of the false discovery rate in voxel-based morphometry. Human Brain Mapping Davidson, D. J., Zacks, R. T. and Williams, C. C. (2003). Stroop interference, practice, and aging. Aging, Neuropsychology, and Cognition Friston, K. J., Worsley, K. J., Frackowiak, R. S. J., Mazziotta, J. C. and Evans, A. C. (1994). Assessing the significance of focal activations using their spatial extent. Human Brain Mapping Friston, K. J., Ashburner, J. T., Kiebel, S. J., Nichols, T. E. and Penny, W. D. (2007). Statistical Parametric Mapping: The analysis of functional brain images. Academic Press, Amsterdam. Genovese, C. R., Lazar, N. A. and Nichols, T. (2002). Thresholding of statistical maps in functional neuroimaging using the false discovery rate. Neuroimage Lee, K.-J., Jones, G. L., Caffo, B. S. and Bassett, S. S. (2011). Spatial Bayesian variable selection models for functional magnetic resonance imaging time-series data. Preprint. Lindquist, M. A. (2008). The statistical analysis of fmri data. Statistical Science Nichols, T. E. and Holmes, A. P. (2002). Nonparametric permutation tests for functional neuroimaging: a primer with examples. Human Brain Mapping Smith, M. and Fahrmeir, L. (2007). Spatial Bayesian variable selection with application to functional magnetic resonance imaging. Journal of the American Statistical Association Worsley, K. J. (1994). Local maxima and the expected Euler characteristic of excursion sets of χ 2, F, and t fields. Advances in Applied Probability Worsley, K. J. (2003). Detecting activation in fmri data. Statistical Methods in Medical Research Worsley, K. J., Marrett, S., Neelin, P. and Evans, A. C. (1992). A three-dimensional statistical analysis for CBF activation studies in human brain. Journal of Cerebral Blood Flow and Metabolism Worsley, K. J., Marrett, S., Neelin, P., Vandal, A., Friston, K. J. and Evans, A. C. (1996). A unified statistical approach for determining significant signals in images of cerebral activation. Human Brain Mapping Zellner, A. (1996). On assessing prior distributions and Bayesian regression analysis with g-prior distributions. In Bayesian Inference and Decision Tech-

8 Jones et al./bayes for fmri 8 niques: Essays in Honor of Bruno de Finetti North-Holland/Elsevier