Kwan-Jin Jung^{1} and Hae-Min Jung^{2}

The temporal derivative has been considered as a mathematical solution for the latency variation of the hemodynamic response function (HRF) in the general linear model (GLM) analysis of the task-based functional MRI (fMRI). A method of combining the primary and derivate estimates was developed by Calhoun and its implementation was introduced. However, serious defects were revealed in the existing methods from a GLM analysis of an event-related fMRI. Here, the method is revised to provide a correct combined estimate using a weighted square average method. The proposed method was confirmed with event-related fMRI studies at various phases of the double Gamma HRF.

1. Calhoun VD, et al. fMRI analysis with the general linear model: removal of latency-induced amplitude bias by incorporation of hemodynamic derivative terms. Neuroimage 2004;22(1):252-257.

2. Steffener J, et al. Investigating hemodynamic response variability at the group level using basis functions. Neuroimage 2010;49(3):2113-2122.

3. Pernet CR. Misconceptions in the use of the General Linear Model applied to functional MRI: a tutorial for junior neuro-imagers. Front Neurosci 2014;8:1.

Equations.
A mathematical description of the proposed weighted square averaging (WSA)
method. The data, *y*_{t}, is expressed as a GLM fit with a new combined
estimate α and a new combined design model given in the parenthesis in Eq. (1).
The new estimate *α* is a square root of the weighted average of squared *β*_{1}
and *β*_{2} in Eq. (2) with the sign assigned by Eq. (3). Note that the original
equation of the weighted square summation (WSS) in the Calhoun’s paper lacked
the denominator in Eq. (2) and the sign was determined from *β*_{1}.

Fig. 1. Z-score
maps of the GLM analysis of the blocked and event-related fMRI before combining
the primary and derivative estimates. The green-colored cross is at the cortex
region stimulated by the right button press. The derivative estimates (RD and
LD) were negligible compared to the primary estimates (R and L) in the blocked
fMRI, but they were more significant than the primary estimates in the
event-related fMRI particularly for the right button presses. Note the lack of
BOLD contrast of ‘R>L’ at the crossed region in the event-related BOLD maps.

Fig. 2. A
comparison of the proposed WSA and existing combining methods from the z-score
maps of the R estimate of the even-related fMRI. WSA-β1 was obtained using the WSA method but with the
sign given by the primary estimate *β*_{1}. WSS-β1 was obtained using the existing summation method
and with the sign given by the primary estimate *β*_{1}. Note that the inverted polarity at the region pointed
by the green crosses in the bottom two rows, compared to the blocked fMRI
results shown in Fig. 1. The
inverted polarity was caused from the negative primary estimate *β*_{1} at those voxels.

Table 1. Z scores at the voxels pointed in
the event-related BOLD map figures for various phases of HRF. The primary (L
and R) and their derivative estimates (LD and RD) as well as the contrast
‘R>L’ are listed before being combined. In addition, the combined estimates
of R, L, and their ‘R>L’ are compared between the proposed WSA and the WSA-β1 that is WSA with the sign of the primary
estimate β_{1}. The z scores
with a negative polarity are marked in red. Note the incorrect estimates of R
and ‘R>L’ in the WSA-β1 results at HRF’s
phases = -1s and 0s.

Fig. 3. Z-scores of BOLD maps for the R
estimate and the contrast estimate of ‘R>L’ over a range of HRF’s phases
after being combined of the primary and derivative estimates using the proposed
WSA method. The z scores at the green crosses are listed in Table 1.
Note that the sign and the magnitude of the z scores at the green crossed
voxels were maintained regardless of the HRF’s phases.