Elsevier

NeuroImage

Volume 29, Issue 1, 1 January 2006, Pages 54-66
NeuroImage

Non-white noise in fMRI: Does modelling have an impact?

https://doi.org/10.1016/j.neuroimage.2005.07.005Get rights and content

Abstract

The sources of non-white noise in Blood Oxygenation Level Dependent (BOLD) functional magnetic resonance imaging (fMRI) are many. Familiar sources include low-frequency drift due to hardware imperfections, oscillatory noise due to respiration and cardiac pulsation and residual movement artefacts not accounted for by rigid body registration. These contributions give rise to temporal autocorrelation in the residuals of the fMRI signal and invalidate the statistical analysis as the errors are no longer independent. The low-frequency drift is often removed by high-pass filtering, and other effects are typically modelled as an autoregressive (AR) process. In this paper, we propose an alternative approach: Nuisance Variable Regression (NVR). By inclusion of confounding effects in a general linear model (GLM), we first confirm that the spatial distribution of the various fMRI noise sources is similar to what has already been described in the literature. Subsequently, we demonstrate, using diagnostic statistics, that removal of these contributions reduces first and higher order autocorrelation as well as non-normality in the residuals, thereby improving the validity of the drawn inferences. In addition, we also compare the performance of the NVR method to the whitening approach implemented in SPM2.

Introduction

Non-white noise in fMRI has been a known issue ever since the earliest days of fMRI. Weisskoff et al. (1993) presented power-spectra from different regions of interest acquired using fast (7 Hz) imaging of a single slice in visual cortex. The spectra showed white noise in white matter (WM), cardiac, respiratory and low-frequency noise in grey matter (GM) and cardiac noise in cerebrospinal fluid (CSF). Since a convenient assumption when making statistical inference is to assume that errors in the measurement are independent and identically normally distributed (i.i.d.), this observation is important and has had large impact on paradigm design and data analyses.

With non-white noise, the i.i.d. assumption is no longer fulfilled, and if this is ignored, the estimated standard deviations will typically be negatively biased, resulting in invalid (liberal) statistical inferences. Another consequence is the difficulty in detecting signals when covered in noise. As we are normally interested in the GM signal, it is problematic that this is the region where structured noise is most pronounced. With physiological noise increasing with field strength (Krüger and Glover, 2001, Krüger et al., 2001), the full benefit in signal to noise ratio (SNR) by using high field scanners cannot be achieved without proper handling of physiological noise.

Whilst it has been argued that the low-frequency oscillations or drifts are physiological in nature, they have also been observed in cadavers (Smith et al., 1999) and phantoms (Lund and Larsson, 1999). Noise contributions with these frequencies can be avoided by a combination of a high-pass filter and a stimulation paradigm fast enough to “escape” the region of the spectra where low-frequency noise is dominant. The system noise makes it difficult to study states of neural activity with BOLD fMRI, if the state remains unchanged for more than a minute. Being a differential, measurement arterial spin-labelling (ASL) experiments are less prone to these drift problems, and thus low paradigm frequency can be used in ASL (Wang et al., 2003).

In typical whole brain acquisitions, the physiological noise components are heavily aliased and possibly non-stationary making it difficult to identify directly. This is perhaps the reason why no commonly accepted model of noise in fMRI exists. Recent versions of most fMRI analysis software try to estimate the covariance due to these effects (e.g. using an autoregressive model), and use the estimated process to pre-whiten the errors. If the estimated covariance structure is correct, this procedure is unbiased and efficient. However, the modelling is time consuming and often global and/or low-order models need to be used in order to get a robust estimate. From the power spectra by Weisskoff et al. (1993), it is known that the noise in part consists of oscillations at respiratory and cardiac frequencies and their higher harmonics. It is also known (Dagli et al., 1999) that the non-white noise is structured in space. Nevertheless, a first order process stationary in time and often in space as well, is often assumed when the coefficients of the model are to be estimated (e.g. SPM2; Friston et al., 2002). As it takes two AR coefficients to model a single oscillator, the approach will, at best, only be valid when the noise is heavily aliased and smeared due to long repetition times (TR) and non-stationary heart and respiratory rates. Using a variational Bayesian approach for AR model-order estimation, Penny et al. (2003) have demonstrated that higher order processes are necessary to describe the auto-correlation in regions close to the vasculature e.g. medial cerebral artery (MCA), and circle of Willis (CW). However, with current high-end computing power, this estimation takes about 30 min per slice.

As an alternative to estimating the correlation in the errors, a number of techniques have been suggested which try to predict the effect of various noise sources on the measured signal. Whilst these methods were primarily introduced for noise reduction purposes, they could potentially serve as whitening filters too and substitute autoregressive modelling which is both difficult and time consuming.

The first methods (Hu et al., 1995, Biswal et al., 1996) for removing cardiac and respiratory noise both used pulse-oximeter time-courses. The method by Biswal et al. (1996) uses the pulse-oximeter time-course to construct a band-stop filter, which is only appropriate when the noise is critically sampled or very stationary. The method by Hu et al. (1995) conversely uses retrospective gating of raw k-space data, and also works for long TR, and multi-shot sequences, as long as the noise is quasi-periodic. A drawback of this method is that it performs best for data points with high SNR which would typically mean mainly points in central k-space. Central k-space is where global effects like movement (e.g. due to respiration) manifest themselves. Effects of pulsating blood, on the other hand, are localised in real space to areas near vessels, and hence spread out over the entire k-space. This is perhaps the reason why Hu et al. (1995) find effects of respiration to be more dominant than effects of cardiac pulsation. On the other hand, the method by Hu et al. (1995) was used by Dagli et al. (1999) to show that cardiac noise is dominant near major arteries e.g. MCA and CW. Josephs et al. (1997) and Glover et al. (2000) suggested performing retrospective correction in the reconstructed images in a way similar to that proposed by Hu et al. (1995). The two implementations differ slightly. Josephs et al. (1997) implemented the correction as a part of the design matrix in a GLM whereas Glover et al. (2000) performed the correction during the preprocessing. Both methods assume that physiological noise is fixed during acquisition of a single slice. Whilst this is a good approximation for single-shot 2D echo planar imaging (EPI), it is not the case for multi-shot acquisitions in general.1

Using the property that respiration-induced noise is a global effect, Frank et al. (2001) proposed estimating and correcting this effect from the k-space data itself. As global effects (centre of k-space) are actually sampled for each slice in an EPI volume, the sampling rate for global effects equals the number of slices per volume divided by TR instead of 1/TR, thereby making global noise-effects critically sampled. A similar approach is to use time-courses from regions of no interest to regress out behaviour similar to that of major vessels or CSF time-courses (Petersen et al., 1998, Lund and Hanson, 2001). This can be achieved by identifying voxels located in CSF and vessels by inspecting a variance image of the fMRI time-course (Lund and Hanson, 2001). Buonocore and Maddock (1997) used time-courses from “truly inactive” and “truly active” voxels to construct a Wiener-filter and demonstrated how this approach is superior to notch filtering. This filter is, however, still stationary and will not be optimal in the presence of variations in the heart and respiratory rates. Independent component analysis (ICA) has great potential for identifying patterns of structured noise in fMRI data. However, so far, it has only been evaluated for TRs short enough to critically sample the cardiac noise (Thomas et al., 2002) in which case, a simple low-pass filter is adequate for removing physiological noise.

The purpose of this paper is to suggest a unified theory for physiological noise in fMRI. We will show how relevant nuisance regressors in a general linear model can be used to describe spatiotemporal behaviour of low-frequency, residual movement, respiratory and cardiac related effects. Using Statistical Parametric Mapping diagnosis (SPMd) (Luo and Nichols, 2003), we hereafter demonstrate the impact of physiological noise correction on the i.i.d. assumption and compare the performance of the suggested method to the whitening approach implemented in SPM2. Finally, the findings of this paper are discussed in relation to assumptions made during common fMRI preprocessing and analysis including slice-timing and functional connectivity analysis.

Section snippets

General linear models or non-white noise in fMRI images

In this section, we describe how the effect of different of non-white noise sources in fMRI can be described in the GLM framework where the significance of each noise source can be tested using an F test.

Low-frequency drift due to hardware instabilities

A commonly used model for low-frequency drift is to include a basis-set of slowly varying functions in the design matrix. This will serve as a high-pass filter, removing oscillations which can be modelled as a linear combination of the basis-set. The set can be constructed using polynomials, as

Simulated data

In Fig. 2, we show as a function of SNR, for different values of jitter in the phase of the reference time-course, the result of the SPMd analysis as applied to the residuals after model fitting. With regard to the whiteness of the errors (Dep and Corr), it is seen that for a range of white noise levels the nominal number of rejections is achieved with a phase precision around 30 ms whereas 70 ms will lead to 10 times more voxels being rejected. With regard to the normality of errors (Norm), it

Simulated data

Our simulations show that aliased physiological noise can give rise to non-white non-normal noise, suggesting that these can be modelled satisfactorily using RETROICOR or similar methods. The results show, as expected, that the necessary precision of the phase measurement provided by external recording (ECG, pulse-oximeter, respiratory belt) increases with the frequency of the oscillation in question. Correspondingly, a longer TR will also require higher precision. The results indicate that 1

Conclusion

In the current paper, we have shown that the NVR model composed of a comprehensive set of nuisance regressors substantially reduces the structured noise in fMRI residuals. The NVR model is based on a number of effects which are known to contribute to the non-white noise in fMRI (hardware drift, residual movement artefacts, respiration and cardiac pulsation). In fact, the proposed NVR model is only new in the sense that we for the first time have used a combination of several already published

Acknowledgments

Mark Griffin, Montreal Neurological Institute, Canada and Gunnar Krüger, Siemens Medical, Germany, are acknowledged for their help with accessing the physiological recordings from the scanner. The Simon Spies Foundation is acknowledged for donation of the Siemens Trio scanner.

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