HST 583Functional Magnetic Resonance Imaging Data Analysis and AcquisitionA Review of Statistics for fMRI Data AnalysisRevisedHemodynamic Response/MR PhysicsD. Complete Model isAlternative fMRI Signal ModelHarmonic Regression ModelE. Maximum Likelihood Estimation for the fMRI Signal + Noise ModelRemarksG.1 Harmonic Regression Hemodynamic Response ModelReferencesHST 583FUNCTIONAL MAGNETIC RESONANCE IMAGING DATA ANALYSIS AND ACQUISITIONA REVIEW OF STATISTICS FOR FMRI DATA ANALYSISEMERY N. BROWN AND CHRIS LONGNEUROSCIENCE STATISTICS RESEARCH LABORATORYDEPARTMENT OF ANESTHESIA AND CRITICAL CAREMASSACHUSETTS GENERAL HOSPITALDIVISION OF HEALTH SCIENCES AND TECHNOLOGYHARVARD MEDICAL SCHOOL / MITREVISEDTHURSDAY, NOVEMBER 29, 2001page 2: HST 583; A Review of Statistics for fMRI Data Analysis; EN Brown, C. LongSTATISTICSThe science of making decisions under uncertainty using mathematical models fromprobability theory.1. Statistical Analysis Paradigm (Box, Tukey)QuestionPreliminary Data (Exploration Data Analysis)ModelsExperiment (Confirmatory Analysis)Model FitGoodness-of-fit not satisfactoryAssessment SatisfactoryMake an InferenceMake a Decisionpage 3: HST 583; A Review of Statistics for fMRI Data Analysis; EN Brown, C. Long c) d)Figure 1: a) This panel shows a slice taken from a combined visual and motor fMRI experiment.The subject was presented with a full-field flickering checkerboard, in a 12.8-s OFF, 12.8-s ONpattern, repeated 8 times. The slice shown was chosen to transect both the visual and motorcortices, and was imaged once every 800ms for the duration of the experiment. Three regions ofinterest have been selected, corresponding to the motor cortex 1), the visual cortex 2), and the whitematter 3). Figures b) – d) illustrate the raw timeseries taken from each of these regions, along withtiming diagrams of the input stimulus.Question: Is there significant activation in the visual and motor cortices duringcombined motor and visual tasks? Is the level of activation greater in the visual area thanin the motor area?a)b)page 4: HST 583; A Review of Statistics for fMRI Data Analysis; EN Brown, C. Long2. What Makes Up An fMRI SignalHemodynamic Response/MR Physics i) stimulus paradigma) event-relatedb) block ii) blood flow iii) blood volume iv) hemoglobin and deoxy hemoglobin contentNoise Stochastic i) physiologic ii) scanner noise Systematic i) motion artifact ii) drift iii) [distortion] iv) [registration]v) [susceptibility]3. Statistical ModelA fMRI Bold Signal (Neurovascular Coupling)In a small time interval t∆ we have on a given pixel00() () ( ()) ()Vt Hbt V Vt Hb Hbt×=+∆×+∆.(1)0V initial blood volume()Vt∆ change in blood volume0Hb is the baseline deoxygenated hemoglobin()Hb t∆change in deoxygenated hemoglobinAssume that there is linear coupling of the stimulus to12() ( )aHb t k k g c=+ ∗(2)//2() (1 )( 1)aatd tdagt e t e−−=− + ,(3)where 1k and 2k are constants, ()ct is the stimulus input, ()agt is a hemodynamic impulseresponse, chosen to be a discrete gamma function, ad is a time constant0()( )aagc guctu∞∗= −∑.(4)page 5: HST 583; A Review of Statistics for fMRI Data Analysis; EN Brown, C. LongFigure 2: Examples of the gamma function when calculated with some different choicesof α.page 6: HST 583; A Review of Statistics for fMRI Data Analysis; EN Brown, C. LongNow, the response to the blood volume is34() ( )bVt k k g c=+ ∗(5)//() (1 )abtd tdbgt e e−−=− ,(6)where 3k and 4k are constants, ()ct is the stimulus, ()bgt is an exponential impulseresponse function with time constant bd.The responses most likely have a delay D. Multiplying Eq. 5 by Eq. 6 andcollecting terms, we obtain() () ()()t aa tDbb tDca tDb tDs fgc fgc fgc gc−−−−=∗+∗+∗ ∗,where 23afkk=, 14bfkk= and 24cfkk=. Physiologically af corresponds to the flowresponse, bf the volume response, and cfrepresents their interaction.Figure 3 Sequence of steps followed when estimating a model for an fMRI experiment.We take advantage of the input function (top) and the physiology of the bold response(second down), to formulate a likely response for the brain (third and fourth panels). Thisis shown for the simplest case of convolution with one basis function.page 7: HST 583; A Review of Statistics for fMRI Data Analysis; EN Brown, C. LongFigure 4 Same sequence of steps as for Figure 3 except the input function is now eventrelated rather than blocked-periodic. Furthermore the time intervals between consecutiveresponses is random.page 8: HST 583; A Review of Statistics for fMRI Data Analysis; EN Brown, C. Long0 20 40 60 80 100 12000.51Flow Term0 20 40 60 80 100 12000.51Volume Term0 20 40 60 80 100 12000.51Interaction Term0 20 40 60 80 100 120-0.200.20.40.6Modeled BOLD Signalfa=1 fb=-0.5fc=0.2Figure 5 Illustration of BOLD signal model for a block-paradigm stimulus. Thecontribution of flow (fa), volume (fb), and interaction (fc) terms combine to form anoverall signal.page 9: HST 583; A Review of Statistics for fMRI Data Analysis; EN Brown, C. LongB. fMRI noise model (stochastic)Physiologic low frequency noise and scanner noise is generally modeled as1,ttttttvwnwwρε−=+=+tw first order Gaussian AR(1) process, ρis the correlation coefficient, tε is zero meanGaussian white noise with variance 2tσ, tn is zero mean Gaussian white noise withvariance 2ησ.Figure 6 Illustration of two major noise classes encountered in fMRI. The left handcolumn summarizes the associated spectra.page 10: HST 583; A Review of Statistics for fMRI Data Analysis; EN Brown, C. LongC. Drift TermSlow drifts of the static magnetic field and residual motion not accounted for by priormotion correction.drift = mbt+D. Complete Model isfMRI signal = drift + hemodynamic response + noisetttYmbtsv=+++for 1, ,tN= …. In matrix notation11111 111 * * (*)(*)1* *(*)(*)aDbD aDbDabcN a ND b ND a ND b ND NYgcgcgcgc vmbfffY Ngc gc gc gc v−− −−−− −−=          +=              """ " " " "Or we have22(,,) (,, )abYXDdd vεηβρσσ=+,(7)where [,, , , ]abcmb f f fβ= is the set of linear parameters and ,,abDd d