What is NIRS? First-Level Statistical Models 5/18/18
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1 First-Level Statistical Models Theodore Huppert, PhD University of Pittsburgh Departments of Radiology and Bioengineering What is NIRS? Light Intensity SO 2 and Heart Rate 2 1
2 5/18/18 FunctionalWhat near-ir isspectroscopy NIRS? From Light Sources Light Intensity To Light Detectors SO2 and Heart Rate 3 FunctionalWhat near-ir isspectroscopy NIRS? Thicknesses Scalp 5-8 mm Skull 3-7 mm CSF 0-2 mm Cortex limited to outer ~8mm of brain 4 2
3 5/18/18 What is NIRS? NIRS measures changes in oxygenated (HbO2) and deoxygenated (HbR) hemoglobin HbO2 HbR HbT (volume) 2-second duration finger-tapping 5 What is NIRS? NIRS measures changes in oxygenated (HbO2) and deoxygenated (HbR) hemoglobin Due to CMRO2 Due to Flow 2-second duration finger-tapping 6 3
4 Assumptions in model Responses sum linearly. Noise (ε) is: normally distributed zero mean. Homoscedasticity i.i.d. No autocorrelation Stationary and ergodic Assumptions in model Responses sum linearly. Noise (ε) is: normally distributed zero mean. Homoscedasticity i.i.d. No autocorrelation Stationary and ergodic Translation - Every time the task is done, the signal changes by the same amount. - Independent of the baseline level - Not true for really short stimulus intervals <4s ISI (neural-refraction) or really long stimuli (habituation/windkessel effect) 4
5 Assumptions in model Responses sum linearly. Noise (ε) is: normally distributed zero mean. Homoscedasticity i.i.d. No autocorrelation Stationary and ergodic Translation - The error in the measurements is random and centered around zero. Normal (Good) noise Heteroscedastic (Bad) noise Assumptions in model Responses sum linearly. Noise (ε) is: normally distributed zero mean. Homoscedasticity i.i.d. No autocorrelation Stationary and ergodic True in NIRS? NIRS noise is heteroscedastic (Temporal) Motion artifacts are heavy tailed (Spatial) Channel noise determined by probe placement and contact 5
6 Assumptions in model Responses sum linearly. Temporal non-sphericity Noise (ε) is: NIRS noise is colored. normally distributed Specific frequencies (cardiac, respiratory, Meyer waves) are over zero mean. represented Homoscedasticity Noise has autocorrelation i.i.d. No autocorrelation Stationary and ergodic True in NIRS? Assumptions in model Responses sum linearly. Noise (ε) is: normally distributed zero mean. Homoscedasticity i.i.d. No autocorrelation Stationary and ergodic True in NIRS? Spatial non-sphericity NIRS noise has low spatial frequency (superficial) Noise between channels is correlated HbO 2 -Hb noise is correlated 6
7 Assumptions in model True in NIRS? Responses sum linearly. Physiological noise can be Noise (ε) is: stimulus-locked. Thus, noise can be non-stationary. normally distributed zero mean. Homoscedasticity i.i.d. No autocorrelation Stationary and ergodic Dealing with non-sphericity Pre-filtering Example: Band-pass filtering Only applied to data (Y) Can distort estimate of β Based on prior assumptions (may not match noise) Pre-coloring (low-pass filtering) Example: Spatial or Temporal smoothing Convolve noise with a known structure Correct the statistics to match structure Applied to both sides of equation Used in fmri and NIRS-SPM Should not be applied to heteroscedastic data Pre-whitening (high-pass filtering) 7
8 Dealing with non-sphericity Pre-whitening Apply a filter to remove colored noise Similar to pre-filtering but applied to BOTH sides of equation. Filter either pre-defined (e.g. NIRS- SPM) or iterative from noise Examples Autoregressive (AR), ARMA, or ARIMA models Auto-regressive model Whitening filter Dealing with non-sphericity Pre-whitening Apply a filter to remove colored noise Similar to pre-filtering but applied to BOTH sides of equation. Filter either pre-defined (e.g. NIRS- SPM) or iterative from noise Examples Autoregressive (AR), ARMA, or ARIMA models 8
9 Dealing with heteroscedastic noise Robust-regression Iteratively weighted solution Detect and discount outliers Apply weights to both sides of the equation Noise is still heavy-tailed after whitening Pre-whitening sharpens outliers Pre-coloring smears outliers 1. Set S =1 (no weighting) 2. Solve equation 3. Compute studentized residual (e.g. prob(outliers) ) 4. Update S 5. Repeat 2-4 Barker JW1, Aarabi A, Huppert TJ. (2013) Autoregressive model based algorithm for correcting motion and serially correlated errors in fnirs. Biomed Opt Express Jul 17;4(8): No Motion Spike Artifacts Shift Artifacts 9
10 The fnirs linear model is given by (or in shortened-notation) The unbiased solution is given by β With parameter uncertainty Canonical vs Deconvolution models Canonical model Assumes the shape of the response Estimates only the magnitude Widely used in fmri/nirs-spm Deconvolution (FIR) Estimates the full time-course of response One parameter per time point Implemented in HOMER1/2 Requires a contrast window to calculate effects β 10
11 11
12 Canonical vs Deconvolution models For either model: And they are equal* to each other when Which for ideal stimulus designs reduces to this: Remarks: 1. The canonical model is equivalent to a FIR model followed by using the prior response as the contrast window 2. The contrast is maximized when c has a tapered shape matching the response 3. Errors in the canonical shape are equivalent to using the wrong window (increase type-ii error) 4. A boxcar canonical model is equivalent to using a uniform contrast window. Data BH (locked) Preprocessing Nothing GLM OLS BH (locked) BH (random) BH (random) PCA-same AR-IRLS Resting PCAseparate SD-filter OLS + SD-regression AR-IRLS + SDregression ME + AR-IRLS + SDregression 12
13 13
14 A look at general linear models Whitening, coloring, and noise distributions Canonical and deconvolution models Group-level models Mixed Effects and ANOVA Extension to image reconstruction Huppert Lab: NIRS-toolbox Overview of toolbox structure Example of GLM analysis Example of group-level methods Example of forward model and image reconstructions GLM Coefficients Average Response Subject Response Condition Subj1 β 1A A Subj1 β 1B B Subj2 β 2A A Subj2 β 2B B Subj3 β 3A A Subj3 β 3B B 14
15 GLM Coefficients Average Response Similar assumptions Noise (ε) is: normally distributed Homoscedasticity i.i.d. So we should use the same pre-whitening and robust regression concepts as before Is this true for NIRS? Noise across subjects is not normal Heteroscedasticity e.g. Bad contact in sensors Channels are not independent and we already know the whitening from the GLM model * W = 1/cholesky(Cov β) Average of Conditions Does response vary with age? Subject Response Condition Age Subj1 β 1A A 4 Subj1 β 1B B 4 Subj2 β 2A A 7 Subj2 β 2B B 7 Subj3 β 3A A 6 Subj3 β 3B B 6 With Subject as a random variable 15
16 Model for a single source-detector pair Model for whole NIRS probe at once Accounts for covariance between src-dets Allows region-of-interest analysis Model for image reconstruction 16
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