\[ S_{TE_n} = S_0 e^{-TE_n R_2^{\star}} + n \]

\[ \ln(S_{TE_n}) = \ln(S_0 e^{-TE_n R_2^{\star}}) \]

\[ \ln(S_{TE_n}) = \ln(S_0) -R_2^{\star} TE_n \]

\[ \begin{cases} \ln(S_{TE_1}) = \ln(S_0) -R_2^{\star} TE_1 \\ \ln(S_{TE_2}) = \ln(S_0) -R_2^{\star} TE_2 \\\ln(S_{TE_3}) = \ln(S_0) -R_2^{\star} TE_3 \end{cases} \]
GLM: \( Y = a_0 - a_1 TE \)

$\overline{S_{0_{(fit)}}}$ vs $\overline{T_{2_{(fit)}}^{\star}}$

Improved coregistration with anatomical!

\[\small S_{OC}(t)=\sum{S_{TE_n}(t) w(T^{\star}_2)}_n \]

\[ w(T^{\star}_2)_n = \frac{TE_n \cdot e^{-TE_n/T^{\star}_{2(fit)}}}{\sum {TE_n \cdot e^{-TE_n/T^{\star}_{2(fit)}}}} \]

In this way, spatial CNR is maximised and the signal can be recovered in areas of drop-out (Posse, 1999)

One subject from OpenfMRI ds000258
TR: 2.47s, TEs: 12,28,44,60 ms, Res: 3.75x3.75x4.4 mm

Preprocessing:

• Discard first 10 tps
• Motion computation
• Realignment
• Smoothing (6mm)
• Nuisance regression (avg. WM, avg. CSF, 6 DoF Motion) & Detrend (Fifth Order)
• Optimal Combination of the signal
• Coregistration, Normalisation to MNI 152

Seed-based CAPs, 9 mm ROI around MNI coord: 3, -52, 21 mm (PCC)

\(\small S_{TE_n} = S_0 e^{-TE_n R_2^{\star}} + n\)\(\small \qquad\qquad \overline{S_{TE_n}} = \overline{S_0} e^{-TE_n \overline{R_2^{\star}}} \)

\[\scriptsize S_0 = \overline{S_0}+\Delta S_0 \qquad\qquad \overline{S_0}>>\Delta S_0 \] \[\scriptsize R_2^{\star} = \overline{R_2^{\star}}+\Delta R_2^{\star} \qquad\qquad \overline{R_2^{\star}}>>\Delta R_2^{\star} \]

\[\small S_{TE_n} = (\overline{S_0} + \Delta S_0)e^{-TE_n(\overline{R_2^{\star}}+ \Delta R_2^{\star})} \]

\[\small S_{TE_n} = \overline{S_0} e^{-TE_n\overline{R_2^{\star}}}e^{-TE_n \Delta R_2^{\star}} + \Delta S_0 e^{-TE_n\overline{R_2^{\star}}}e^{-TE_n \Delta R_2^{\star}} \]

\[\small S_{TE_n} = (\overline{S_{TE_n}} / \overline{S_{TE_n}}) ( \overline{S_0} e^{-TE_n\overline{R_2^{\star}}} + \Delta S_0 e^{-TE_n\overline{R_2^{\star}}}) e^{-TE_n \Delta R_2^{\star}} \]

\[\small S_{TE_n} = \overline{S_{TE_n}}( 1 + \Delta S_0 / \overline{S_0}) e^{-TE_n \Delta R_2^{\star}} \]

\[\small S_{TE_n} = \overline{S_{TE_n}}( 1 + \Delta S_0 / \overline{S_0}) (1 -TE_n \Delta R_2^{\star}) \]

\[\small S_{TE_n} = \overline{S_{TE_n}} \big( 1 + \frac{\Delta S_0}{\overline{S_0}}\big)(1 -TE_n \Delta R_2^{\star}) \]

\[\small S_{TE_n} = \overline{S_{TE_n}} \big( 1 -TE_n \Delta R_2^{\star} + \frac{\Delta S_0}{\overline{S_0}} - \frac{\Delta S_0 TE_n \Delta R_2^{\star}}{\overline{S_0}} \big) \]

\[\small \frac{S_{TE_n}-\overline{S_{TE_n}}}{\overline{S_{TE_n}}} = -TE_n \Delta R_2^{\star} + \frac{\Delta S_0}{\overline{S_0}} \]

\[\small \Delta\rho=\Delta S_0 / \overline{S_0} \]

\[\small \Delta\kappa=\Delta R_2^{\star}\overline{TE} \]

 

\[\small S_{SPC} \approx \Delta\rho - \frac{TE_n}{\overline{TE}}\Delta\kappa\]

\[\small S_{SPC} \approx \Delta\rho - \frac{TE_n}{\overline{TE}}\Delta\kappa\]
\[\small S_{SPC} \approx\Delta\rho \]
\[\small S_{SPC} \approx \frac{TE_n}{\overline{TE}}\Delta\kappa=TE_n\Delta R_2^{\star} \]
$\Delta\kappa$ is dependent from the TE, while $\Delta\rho$ is independent (Kundu et al., 2012)

Independent Component Analysis can be use to decompose a matrix in ortogonal spatial components (with relative timecourse) FSL MELODIC webpage

A widely accepted denoising approach is ICA denoise (Griffanti et al., 2014, 2017, Pruim et al., 2015a, 2015b, Salimi-Khorshidi et al., 2014)

It is possible to use the content of $\Delta\kappa$ (and $\Delta\rho$) to estimate the number of and classify ICA components

(Kundu et al., 2012, 2013)

\[\small S_{SPC} \approx \frac{TE_n}{\overline{TE}}\Delta\kappa \qquad\qquad S_{SPC} \approx\Delta\rho \]
\[\small \alpha_v = \sum \Delta S^2_{TE_n} \]
\[\small \kappa = \frac{\sum{\alpha_v F_v,\Delta \kappa}}{\sum \alpha_v} \qquad \qquad \rho = \frac{\sum{\alpha_v F_v,\Delta \rho}}{\sum \alpha_v} \]

$F_v$ is the result of a voxelwise F-test between the residual of the fit of a model to the residual of the null model ($\alpha_v$)

• Preproc:
  • (Slice timing, despiking, motion realignment)
  • Brain Extraction
• Optimal Combination
• P-PCA or ted-PCA:
  • PCA decomposition
  • Component selection: variance or $\kappa / \rho$ threshold
• ted-ICA:
  • fast-ICA (MELODIC) decomposition
  • $\kappa$ and $\rho$ rank
  • Decision tree
  • GLM to remove bad components

(DuPre et al., 2019, Kundu et al., 2012, 2013)

The components are sorted by their $\kappa$ value (and by $\rho$), $\kappa$ and $\rho$ thresholds selected by the "elbow" method

(Kundu et al., 2012, 2013)

Kundu et al., 2013: use these for ME-ICR

Current approach: use these for denoise

• The original algorithm "misfire":
  • It's probably due to the original decision tree
  • There's a group of volunteers openly developing and mantaining a new tedana version
• The linear model is an approximation:
  • We introduced approximations twice, but most importantly, we "forgot" about the noise term, thus not separating it
  • The model works only with "whole brain" analysis

• The monoexponential model is an assumption (actually two):
  • The brain has different compartments
  • Biexponential models reveal non-linear IV/ TE dependence(Havlicek et al., 2017)
• Data seem to be more correlated to motion after ME-ICA:
  • ICA detects and removes "local" sources, thus global sources "arise"
  • Solution 1: GODEC or Robust PCA(Power et al., 2018)
  • Solution 2: ME-SPFM(Caballero-Gaudes et al., 2019)
• Spatial ICA might not be the best candidate to model noise:
  • It assumes noise being spatially independent from BOLD
  • Bayesian estimation based on "shape" of fluctuation(Cai et al., 2019)

Biexponential models reveal non-linear IV/ TE dependence(Havlicek et al., 2017)

GODEC can further clean the timeseries(Power et al., 2018)

Bayesian estimation of signals(Cai et al., 2019)