The impact of temporal regularization on estimates of the BOLD hemodynamic response function: A comparative analysis

Abstract

In fMRI data analysis it has been shown that for a wide range of situations the hemodynamic response function (HRF) can be reasonably characterized as the impulse response function of a linear and time invariant system. An accurate and robust extraction of the HRF is essential to infer quantitative information about the relative timing of the neuronal events in different brain regions. When no assumptions are made about the HRF shape, it is most commonly estimated using time windowed averaging or a least squares estimated general linear model based on either Fourier or delta basis functions. Recently, regularization methods have been employed to increase the estimation efficiency of the HRF; typically these methods produce more accurate HRF estimates than the least squares approach [Goutte, C., Nielsen, F.A., Hansen, L.K., 2000. Modeling the Haemodynamic Response in fMRI Using Smooth FIR Filters. IEEE Trans. Med. Imag. 19(12), 1188-1201.]. Here, we use simulations to clarify the relative merit of temporal regularization based methods compared to the least squares methods with respect to the accuracy of estimating certain characteristics of the HRF such as time to peak (TTP), height (HR) and width (W) of the response. We implemented a Bayesian approach proposed by Marrelec et al. [Marrelec, G., Benali, H., Ciuciu, P., Pelegrini-Issac, M., Poline, J.-B., 2003. Robust Estimation of the Hemodynamic Response Function in Event-Related BOLD fMRI Using Basic Physiological Information. Hum. Brain Mapp. 19, 1-17., Marrelec, G., Benali, H., Ciuciu, P., Poline, J.B. Bayesian estimation of the hemodynamic of the hemodynamic response function in functional MRI. In: R. F, editor; 2001; Melville. p 229-247.] and its deterministic counterpart based on a combination of Tikhonov regularization [Tikhonov, A.N., Arsenin, V.Y., 1977. Solution of ill-posed problems. Washington DC: W.H. Winston.] and generalized cross-validation (GCV) [Wahba, G., 1990. Spline Models for Observational Data. Philadelphia: SIAM.] for selecting the regularization parameter. The performance of both methods is compared with least square estimates as a function of temporal resolution, color and strength of the noise, and the type of stimulus sequences used. In almost all situations, under the considered assumptions (e.g. linearity, time invariance and smooth HRF), the regularization-based techniques more accurately characterize the HRF compared to the least-squares method. Our results clarify the effects of temporal resolution, noise color, and experimental design on the accuracy of HRF estimation.

Variational Solution to the Joint Detection Estimation of Brain Activity in fMRI

“…In the latter case, the two AR parameters are varied while maintaining a stable AR(2) process. As already observed in [3], at fixed input SNR, the impact of large autocorrelation is stronger than that of large noise variance irrespective of the inference scheme. Moreover, the two inference methods perform very similarly on a large scale of input SNR (SNR > 5 dB).…”

Nonparametric inference of the hemodynamic response using multi-subject fMRI data

“…Let Δ be the time unit representing the discretization of the HRF temporal resolution. Since it is possible to have the temporal resolution of the HRF shorter than that of the fMRI data (Casanova et al, 2008;Ciuciu et al, 2003), Δ can be smaller than the repetition time unit (TR) of the experimental design. For each subject i, let Y i = (y i (1), …, y i (T)) ' be the observed fMRI time series.…”

“…Different choices of Γ defines different Tikhonov-regularized estimators. One choice of Г in the fMRI literature is the discrete second derivative matrix, as adopted in Marrelec et al (2003), and Casanova et al (2008Casanova et al ( , 2009. Another choice of Γ is the scalar matrix αI dim(η i ) , where I dim(η i ) is an identity matrix with the dimension of η i .…”

“…We note that even thoughβ cor i;k is the estimator we use in analysis, parameter selection is easier to be conducted on the intermediate Tik-Kern estimatorβ r i;k (hereβ r i;k denotes the sub-vector ofβ r i corresponding to β i,k ). Generalized crossvalidation (GCV) (Wahba, 1990) is a standard method for choosing the regularization parameter, and was employed by Casanova et al (2008Casanova et al ( , 2009. GCV improves upon the time-consuming leave-one-out ordinary cross-validation (OCV).…”

“…Similarly, representing the HRF by spline bases, Vakorin et al (2007) and Zhang et al (2007) used Tikhonov regularization (Tikhonov and Arsenin, 1977). The estimator proposed by Casanova et al (2008Casanova et al ( , 2009) combines Tikhonov regularization and generalized cross validation (Wahba, 1990) (referred to Tik-GCV hereafter), greatly reducing the computational burden involved in parameter selection. Motivated by these developments, a second goal of this paper is to propose a new nonparametric estimator that combines kernel smoothing with Tikhonov regularization.…”