A Bayesian GLM for Cortical Surface fMRI Task Analysis Amanda F. 1 Mejia , Finn 4 Lindgren and Martin A. methods corticalsurfacefMRI L results 1 β’ Reduceddimensionality β’ Improvedalignmentofcorticalareas β’ Moremeaningfulsmoothing& spatialmodeling 2 AssumeSPDE prioroneachπ· 3 UseINLA toestimate posteriors Identifyareasof activationusing jointPPM Jointgroup-level model R 1. Stochastic partial differential equation prior (Lindgren et al. 2011) fMRItimeseries atlocationπ£ Taskdesign Activationat Nuisance βActivationβdue signals tonuisanceat locationπ£ matrix locationπ£ π¦: = ππ½: + ππ: + π: (π×1) (π×πΎ) πΎ×1 (π×π½) π½×1 (π×1) π β lengthoffMRItimeseries π β numberoflocations πΎ β numberoftasks+baseline π½ β numberofnuisancesignals fMRItime seriesatall locations - Finite approximation to a continuous Matérn Gaussian random field - Markov property due to sparse inverse covariance structure π π π π β P πR π π€R - Constructed on a triangular mesh R (format of cs-fMRI data) BayesianGLM Taskdesign Activationat Nuisance βActivationβdue signals tonuisanceatall alllocations matrix locations π = ππ½ + ππ΅ + πΈ (π×π) (π×πΎ) πΎ×π (π×π½) π½×π (π×π) 2. Integrated nested Laplace approximation (Rue et al. 2009) π½ βΌ π(π½|π) Spatialprocess prior(e.g.GMRF) - Accurately estimates posterior densities using Laplace approximation - Much faster than MCMC for Bayesian computation numerical - Implemented in R (www.r-inla.org) Hyperparameters Pitfallsofβmassiveunivariateβ AdvantagesofBayesianGLM: β’ Smoothercoefficientestimates classicalGLMapproach: π π½π π¦ = β« π π½π π, π¦ π π π¦ ππ β’ Degreeofsmoothnessdetermined β’ Failstoaccountforspatial separatelyforeachactivationfield dependenceinactivation β’ Possibletoavoidmultiple β’ SmoothingincreasesSNR,but comparisonscorrection,sinceall shapeanddegreead-hocand locationsmodeledjointly commonacrosstasks β’ Multiplecomparisoncorrection β’ Morepowertodetectactivation essentialforfalsepositive ChallengesofBayesianGLM: control 1. Assumeanappropriateprioronπ½s - Complexspatialandtemporal dependenciesmakepropercorrection challenging - Smoothingincreasesdependence betweentests - Populartechniqueshavebeenfound tolackpowerorhavehighfalse positiverates - Excursion set is the largest set of locations πΈπΎ,πΌ such that every location exceeds activation level πΎ with joint probability 1 β πΌ - Excursion function is the largest value 1 β πΌ for which each location would be included in πΈπΎ,πΌ - Estimate excursion function using INLA techniques, then threshold to identify regions of activation at a given significance level πΌ 2. PerformBayesiancomputation Group-level model - MCMCslow,maymixpoorly - VBfastbutunderestimatesvariance 3. Thresholdposteriorprobabilitymap (PPM)todetectareasofactivation - Define group-level activations π½πΊ as linear combination (e.g., mean) of subject-level π½π βs: π½πΊ = π΄π½, π½ = π½1β² , β¦ , π½πβ² β² - Estimate posterior of π½πΊ using importance sampling useINLAGaussianapprox. β’ β’ fromsubject-levelmodels π(π½πΊ π¦ = β« π(π½πΊ π¦, π π π π¦ ππ 4. Modeldatafrommultiplesubjects? Use original subject-level surfaces for more accurate distance-based spatial dependence modeling Relax assumption of stationarity in SPDE prior Use empirical spatial dependence structure for each task to improve computational efficiency and incorporate long-range dependencies \ 3. Identify areas of activation with joint PPM (Bolin & Lindgren 2015) - Shouldcapturespatialdependence structurewhilehavingsparseprecision - cs-fMRIdatahasmeshformat,butmany priorsassumelatticestructure future work approximation π π½, π, π¦ πU π π¦ β X πUW π½ π, π¦ YZYβ - MarginalPPMconvenientbutreintroduces multiplecomparisonsissue β’ simulation study β’ Generated a time series of length 200 for a single slice consisting of baseline (GM probability map) + two activation profiles + AR(1) noise β’ Fit classical GLM (after smoothing & prewhitening) and Bayesian GLM with AR(1) errors β’ Classical GLM: corrected for multiple comparisons by - FDR correction using Benjamini-Yekutieli procedure (π = 0.01) - FWER correction using a permutation test (πΌ = 0.01) β’ Bayesian GLM: marginal and joint PPMs used to ID activations (πΌ = 0.01) Estimateasamixtureof Gaussiansforeach samplefromπ π π¦ βπ π ]^_ ` π π π¦a a βGaussiandensityπ π π¦ HCP motor task study β’ 20 randomly sampled subjects from HCP 500 performed 5 motor tasks (12 sec) following visual cue (3 sec) over 3.5 min (284 volumes) β’ Data minimally preprocessed according to HCP fMRISurface pipeline, plus: - Classical GLM: surface smoothing (6mm FWHM) using Connectome Workbench - Bayesian GLM: resampling (32K to 6K) using Connectome Workbench β’ Prewhitened data using an AR(6) model with spatially-varying coefficients β’ Included 14 nuisance covariates in models to account for motion and drift β’ To identify areas of activation, used FDR and FWER correction as in simulation for classical GLM, joint PPM approach for Bayesian GLM at 0.01 Subjectestimates ClassicalGLM ClassicalGLM 5 Lindquist Groupestimates Areasofactivation(πΌ, π = 0.01) FWER FDR JointPPM(πΌ = 0.01) BayesianGLM β’ Veryhighdimensional β’ Difficulttoaligncorticalareas β’ Complexspatialdependencies 1 Indiana David 3 Bolin , University, Bloomington, Indiana, USA 2 Baruch College, The City University of New York, New York, USA 3 Gothenburg University, Gothenburg, Sweden 4 The University of Edinburgh, Edinburgh, UK 5 Johns Hopkins University, Baltimore, Maryland, USA background volumetricfMRI Yu Ryan 2 Yue , πΎ=0 πΎ = 1%baseline references β’ β’ β’ β’ β’ Mejia AF, Yue Y, Bolin D, Lindgren F & Lindquist MA (2017+). A Bayesian GLM approach to cortical surface fMRI data analysis. Lindgren F, Rue H and Lindström J (2011). An explicit link between Gaussian fields and GMRFs: the SPDE approach (with discussion). JRSSB. Whittle, P. (1954). On stationary processes in the plane. Biometrika. Rue H, Martino S and Chopin N (2009). Approximate Bayesian inference for latent Gaussian models using integrated nested Laplace approximations (INLA) (with discussion). JRSSB. Bolin D and Lindgren F (2015). Excursion and contour uncertainty regions for latent Gaussian models. JRSSB.
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