Estimation of respiration-induced noise fluctuations from undersampled multislice fMRI data

Authors

  • Lawrence R. Frank,

    Corresponding author
    1. Department of Radiology, University of California at San Diego, San Diego, California
    2. San Diego VA Healthcare System, San Diego, California
    • VA Medical Center, 9114/MRI, 3350 La Jolla Village Drive, San Diego, CA 92161
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  • Richard B. Buxton,

    1. Department of Radiology, University of California at San Diego, San Diego, California
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  • Eric C. Wong

    1. Department of Radiology, University of California at San Diego, San Diego, California
    2. Department of Psychiatry, University of California at San Diego, San Diego, California
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  • This article is a US Government work and, as such, is in the public domain in the United States of America.

Abstract

Functional MRI time series data are known to be contaminated by highly structured noise due to physiological fluctuations. Significant components of this noise are at frequencies greater than those critically sampled in standard multislice imaging protocols and are therefore aliased into the activation spectrum, compromising the estimation of functional activations and the determination of their significance. However, in this work it is demonstrated that unaliased noise information is available in multislice data, and can be used to estimate and reduce noise due to high-frequency respiratory-related fluctuations. Magn Reson Med 45:635–644, 2001. Published 2001 Wiley-Liss, Inc.

The analysis of functional magnetic resonance imaging (fMRI) time series data is complicated by the fact that the noise is not Gaussian (1–5). This is a consequence of the fact that the dominant contributions to the noise in fMRI are signal variations produced by physiological processes, rather than by the thermal noise. These physiologically related signal fluctuations are generally quite complicated and can have significant power over a wide range of frequencies. Moreover, these fluctuations can be correlated with one another, producing sidebands with significant power. The dominant contributions appear to be high-frequency fluctuations related to the quasiperiodic processes of respiration and cardiac pulsations, and low-frequency fluctuations, which can result from slow drifts in the time series, but have also been hypothesized to be related to noise correlations produced by the hemodynamic response of the brain (1, 6).

In spite of the complexities of the spectrum of noise fluctuations, estimation of functional activation can still be relatively straightforward even in the presence of such fluctuations, provided that its spectrum is not overlapped by that of the noise fluctuations, and that the data sampling rate is sufficient to critically sample the spectrum of the noise. If these conditions hold, standard methods of filtering can be applied (7) to reduce unwanted noise components. Unfortunately, while it is possible to collect data in a single slice at a rate sufficient to critically sample high-frequency physiological fluctuations, virtually no fMRI experiments are actually done in this way. Rather, multislice acquisitions are performed in order to achieve adequate spatial coverage. With the typical imaging parameters used in multislice studies, the sampling rate for each slice is not sufficient to critically sample the physiological fluctuations, and the resulting time series noise is contaminated by aliased spectral components from the high-frequency physiological fluctuations. This not only reduces both functional signal-to-noise ratio (SNR) and significance of the estimates, but makes improper the use of many well-developed standard estimation techniques based on Gaussian noise models.

However, in this work we show that it is possible to obtain unaliased information about the noise structure directly from multislice echo-planar imaging (EPI) fMRI time series data. This is achieved by noting two important features of such fMRI data: 1) Some physiological fluctuations are relatively global in nature and are therefore present in the central k-space components. 2) Surprisingly, physiological fluctuations are critically sampled and obtainable in most multislice EPI data acquisitions if the data is reordered into temporal, rather than spatial, order (8).

EXPERIMENTAL METHODS

A natural way to assess the problem of noise in multislice fMRI is to compare two data sets acquired during the same stimulation paradigm: one acquired in a multislice fashion, and one in a rapidly sampled acquisition in a single-slice location as one of the multislice locations. To make the data sets more easily comparable, the same imaging parameters (except those related to slice acquisition) are used and, for reasons that will become clear, the repetition time in the multislice acquisition is set to the repetition time of the single-slice acquisition times the number of slices, so that the time between each data acquisition step, which we term the “effective sampling rate,” is the same in both experiments.

All images were acquired on a 1.5T GE Signa LX system using a single-shot EPI acquisition. Images in both data sets acquired in the sagittal plane with imaging parameters FOV = 24 cm, slice thickness = 7 mm, TE = 40 msec, 64 matrix. The stimulation in both data sets was a simple visual field stimulation study (8 Hz flashing checkerboard) presented in a block design paradigm in order to simplify the spectrum of the activation: the stimulus was presented at a rate of 1 per sec, with the stimulus on for 16 sec and off for 16 sec, and thus a stimulation period = .03125 Hz.

In the first data set, data were collected on a single slice with a very short repetition time (TR = .25 sec) in order to critically sample what we hypothesize to be the largest frequency components—those related to cardiac pulsations (≈ 1 Hz). Thus, 64 images were collected during the “on” period and 64 were collected during the “off” period. In the second experiment, the same activation protocol was performed, but data was collected in eight slices at a repetition rate of TR = .25 sec × 8 slices = 2.0 sec so that the effective sampling rate for the temporally reordered data was then equivalent to the single-slice experiment: 2 sec/(8 slices) = .25 sec. Thus eight images were collected during the “on” period and eight were collected during the “off” period.

STRUCTURE OF fMRI NOISE

The structure of the noise in fMRI is complex because it is dominated by signal variations produced by physiological processes. In addition, since there are multiple mechanisms by which the MR signal can be altered, the physiologically related noise fluctuations are not likely to be simply related to any one physiological process, but rather a combination of effects. Therefore, while it is important to understand the mechanisms that generate these noise variations, it may be difficult to accurately model them. Moreover, since most physiological processes are only quasiperiodic, simple periodic representations are inadequate. Perhaps the most difficult aspect of the noise characterization is that its domination by physiological processes may obviate the most obvious method of characterization: collecting data in the absence of functional activation. For if the noise is generated by physiological activity, it is likely altered by the functional activity.

An example of these complications is seen in the power spectrum from an activated voxel within the visual cortex from the single-slice data (Fig. 1). Also shown in Fig. 1 are images of the spatial distribution of the power spectra at the peak frequencies in the power spectrum plot.

Figure 1.

Power spectrum from an activated voxel within the visual cortex. Data was collected from a single slice with a TR of .25 sec. The stimulus was a flashing checkerboard. Evident are strong spectral components from the activation, respiratory, and cardiac fluctuations.

There are a few important characteristics of the noise to note in Fig. 1. It is evident that not only is there significant power at frequencies other than the activation, but that the spatial distribution of these components can overlap regions of functional activation. Also, there is a distinct low-frequency spectrum whose amplitude appears to fall off with increasing frequency. This is sometimes termed a “1/f” spectrum, although this is perhaps an unfortunate nomenclature because there are processes in fMRI that can generate such low-frequency spectra but are not 1/f processes. The most obvious are slowly varying temporal variations of the signal, or “trends,” that can be caused by a variety of reasons, such as slow patient motion (e.g., sinking into the padding) or scanner drift. Most relevant for the present discussion is the existence of distinct peaks at frequencies much higher than the activation due to signal variations related to both respiratory and cardiac pulsations. Note, however, that the power spectrum near these frequencies does not reflect a purely sinusoidal variation, but a distribution of frequencies about the peak. This is to be expected, since physiological processes are rarely purely periodic. The heart rate, for instance, has significant temporal variability in healthy humans. Moreover, the respiratory and cardiac rates are coupled, so that situations in which, for example, the respiratory rate increases with the cardiac rate, are not uncommon.

Physiological processes produce signal variations in both the image magnitude and phase time series. Both of these are, of course, manifestations of magnitude and phase variations in the k-space data actually acquired. A remarkable fact that has been noted before is that these physiologically induced k-space phase variations are relatively uncontaminated in the phase channel, and can be utilized for retrospective correction of either k-space phase or image space magnitude time series data (9, 10). The fluctuations in the k-space phase produced by respiratory processes are relatively global, and therefore are expected to be present to some degree in most k-space components. This is to a lesser degree true of cardiac fluctuations, which are more punctate spatially. For the present we will assume, however, that both components are relatively global. Difficulties with this assumption are addressed in the Discussion section. The waveform of a particular spectral component will be a complicated mixture of effects due to both the natural variations in the physiological process (e.g., variation in the cardiac frequency during the experiment) and variations in the MR signal produced by the physiological processes (e.g., variations in blood oxygenation with respiration). In addition, the variability of physiological processes is likely to be different in every subject, and possibly different even for the same subject during different studies. This is a topic in its own right, and will be addressed in another work.

NOISE ALIASING

The difficulty imposed by multislice imaging is related to the issue of critical sampling. Frequencies higher than 1/(2τN), where τN is the Nyquist sampling interval (i.e., the maximum time allowed between samples to still correctly reconstruct a frequency) are not accurately reconstructed and appear at another frequency (their “alias”) (see, for example, Ref. 11). Aliased power from frequency components higher than the sampling bandwidth can obscure activations (if their spectra overlap), reducing the significance of the detected signal. Even if there is no overlap, aliased noise makes estimation of the noise, which is important for determining significance, difficult.

This effect is illustrated in Fig. 2, in which the effect of a lower sampling rate on the noise spectrum is illustrated by resampling the data collected from a single slice with TR of .25 sec (top) at every eighth time point to simulate the data for an eight-slice acquisition with a TR of 2 sec. These multislice parameters were chosen so that the time between successive data collections (i.e., between slices) is the same as in the single slice data (2 sec/8 = .25 sec). Frequency components higher than .25 Hz are now aliased across the power spectrum, and are less clearly delineated because the aliased peaks are broad. Note that because each component of the physiological fluctuations covers a range of frequencies, the aliasing of frequencies can be somewhat subtle since a multislice experiment that critically samples the nominal peak frequency may not critically sample all of the frequency components in the waveform of fluctuations. For our previous examples of an eight-slice acquisition with a TR of 2.0 sec, a respiratory frequency of .25 Hz ⇒ τN = 2 sec would just be critically sampled. But the spread of frequencies about .25 Hz would mean that some frequencies are not critically sampled. These then can be aliased into the spectrum even if the main peak is not. In the data shown in Fig. 1, the respiratory peak is at approximately .3 Hz, which corresponds to a period of τ = 1.7 sec, and so is not critically sampled in our eight-slice experiments; nor is the cardiac peak, which is at about 1 Hz ⇒ τ = 0.5 sec.

Figure 2.

Top: Estimated noise spectrum from data collected from a single slice with a TR of .25 sec. Bottom: Same spectrum resampled at every 8th time point to simulate the power spectrum for an eight-slice acquisition with the same imaging parameters. High-frequency components are now aliased across the power spectrum.

RECOVERY OF ALIASED INFORMATION

The representation of the physiological fluctuations in the data depend on the sampling rate: if the sampling rate is not sufficient to critically sample the fluctuations, they will be aliased and therefore no longer take on their true physiological waveform or spectral properties.

Given that the frequency content of the dominant physiological fluctuations are much higher than that of a simple block design paradigm, and can therefore be removed from the data if it is critically sampled, the question remains: Can anything be done if the fMRI data is collected in a multislice fashion where the sampling rate for a voxel time series is the TR? This TR is determined primarily by the requirements of achieving sufficient spatial coverage, and so is usually relatively long compared to the time scale of physiological fluctuations. As a consequence, the higher-frequency components of the time series contributed by the physiological fluctuations are not sampled fast enough to be accurately reconstructed.

However, although the sampling rate in any one slice is relatively slow, the rate at which data is actually being collected is much faster than the sampling rate at any particular slice. In the eight-slice example above, for instance, data is being collected every (2 sec)/(8 slice) = .25 sec. This rate is fast enough to sample physiological fluctuations since the Nyquist frequency (i.e., the highest frequency that can be accurately reconstructed for a give sampling rate) is 1/(2 × .25) = 2 Hz. Therefore, if the data is reordered into the order in time in which it was collected, the sampling rate is sufficient to reconstruct the spectrum of physiological fluctuations in the data. This is illustrated in Fig. 3. Of course, there is a slight complication: The temporal order is not the spatial order! That is, we are looking at a time series composed of data points from different locations.

Figure 3.

Reordering the data from slice ordering (a) to time ordering (b). The sampling interval in the slice-ordering method is just TR, while in the temporal ordering scheme it is TR/(# slices).

The situation is not so dire as it might at first appear. If the physiological fluctuations are, as we hypothesize, relatively global in nature, and therefore well represented in the central k-space components, then this should also be true of the reordered data. To test this hypothesis, we can compare the spectrum of the phase variations in a single-slice, critically sampled experiment with a multislice experiment with an equivalent effective sampling rate. This is shown in Fig. 4, with data from the visual cortex taken during the two visual stimulation experiments.

Figure 4.

Comparison of k-space phase from single-slice and multislice data. Left: Phase time course and corresponding power spectrum from a single-slice experiment through the visual cortex with TR = .25 sec taken during visual stimulation. Right: Phase time course and corresponding power spectrum from reordered data from an eight-slice experiment through the visual cortex with TR = 2 sec taken during visual stimulation. The similarity between the spectra on the left and the right suggests that the temporal reordering strategy can produce a good representation of the physiological fluctuations.

The multislice data were processed by transforming the complex image data back to k-space, reordering the slices in the temporal order of acquisition, being careful to remove the mean phase and magnitude values from each slice so that the spectrum was not dominated by slice-to-slice variations. A single time series of the phase variations was estimated from the time series of the phase at the central k-space point. Because the effective sampling rate for the temporally reordered data is equivalent to the single slice experiment: 2 sec/(8 slices) = .25 sec, the time steps in Fig. 4 are equal. The resulting phase time courses and corresponding power spectra from both these experiments are seen in Fig. 4 to be remarkably similar, suggesting that the temporal reordering strategy can produce a good representation of the physiological fluctuations.

ESTIMATION OF PHYSIOLOGICAL FLUCTUATIONS

Since it appears from Fig. 4 that it is possible to recover “unaliased” information about the noise fluctuations from temporally reordered data, the next step is to use this information to estimate these noise waveforms and remove them from the data. There are two things that we specifically do not want to assume about the data: 1) Physiological fluctuations are simple sinusoids, and 2) physiological fluctuations are the same in each voxel time series. Both of these assumptions are clearly not true in fMRI data and, in fact, pose a significant source of complication in the data analysis problem.

The k-space phase shown in Fig. 4B contains contributions from several physiological noise sources, each of which may vary independently from voxel to voxel. We must therefore estimate the contributions of the individual physiological components. If the individual components were purely sinusoidal variations, they would be relatively straightforward to account for, as long as they are well separated in frequency from each other and from the spectrum of the activation. Physiological waveforms are not generally sinusoidal. Usually they are merely quasiperiodic, so that estimating these noise waveforms is a difficult problem in its own right, even if they were critically sampled. A more complete attempt at solving this problem will be presented elsewhere, but for the moment the key issue is that these waveforms have a significant spread in frequency, which produces a much broader range of spectral contamination than do purely sinusoidal variations.

The scheme used in this study to estimate the waveforms is admittedly simplistic, and its improvement is currently the subject of our investigation. For the present, we employ the following method: The peaks in the reordered k-space phase power spectrum near the nominal respiratory and cardiac peak frequencies are automatically determined and a user-specified width of frequencies about these peaks is extracted. In the current implementation, this width was simply chosen by eye to be larger than the observed spread of frequencies about the noise peaks. The Fourier transforms of these individual frequency bands are used as the estimates of the individual time courses of the physiological fluctuations.

Once these waveforms have been estimated, they must then be removed from the data. Now, the waveforms are estimated from the time-ordered data, so the correction to the data is also performed on the time-ordered data. However, while the waveform estimation was performed in k-space to get a globally averaged waveform estimate, the correction is done in the time-ordered image space, since that is where spatial localization is represented, and the contribution (i.e., the amplitude) of the physiological fluctuations (but, by hypothesis, not the waveforms) can be quite different amongst voxels.

For this reason we do not want to simply cut out ranges of frequencies from the spectrum of the individual voxel time courses, for this act implicitly assumes that all frequencies within the selected bands are artifactual. Rather, our model assumes only that there may be contributions from physiological sources. We therefore want to eliminate the existing components of the fluctuations. Mathematically, this translates into projecting out the unwanted components. Specifically, the projection of the data onto the subspace spanned by the estimated waveforms is removed from the data: the standard least-squares procedure. Projecting out specified waveforms is distinctly different from simply cutting out blocks of the Fourier spectrum. If there is no contribution from such a fluctuation, the component projected out is essentially zero, and no correction is made.

The basic computational approach is fairly straightforward. We use the following notation: FTl denotes Fourier transform with respect to coordinate l, s is the time step in each slice (i.e., repetitions of slice j are collected at times s1, s2, … ,sn), and t represents the true time coordinates (i.e., slice 1, repetition 1 is collected at t1; slice 2, repetition 1 is collected at t2, etc), and primed coordinates (x′,y′) and (kmath image,kmath image) denote pixel locations in the time-reordered image and k-space, respectively, which actually correspond to several true spatial or k-space locations, and Ic denotes the corrected version of the data. Here we are using the term “pixel” simply to denote locations in the image or k-space data array. Thus:

  • 1Transform multislice image data I(x,y,s) to k-space: I(kx,ky,s) = FTxy[I(x,y,s)], reorder from slice order to temporal order: I(kx,ky,s) → I(kmath image,kmath image,t), being careful to remove slice-dependent means and trends. Means and trends were removed from each slice of the image data by fitting (in a least-squares sense) a low-order (3) polynomial to the time series in that slice.
  • 2Compute time-ordered magnitude and phase-time series, mk(t) and ϕk(t) from center k-space point.
  • 3Use resulting time series to find locations in frequency spectrum ϕk(ω) = FTtk(t)] of peak power, being careful to avoid harmonics of the activation. Our current method finds the peak from a user-supplied initial guess of the location of a specified number of fluctuation components, locates the sidebands, and chooses a fixed range about them.
  • 4Determine the time courses of the physiological fluctuations g(t) in the image magnitude data using the frequencies and ranges found in the k-space data: g(t) = Fk(ω)] where ϕ denotes the estimated spectrum of the waveform, found from the location of the peak and a given width in frequency about the peak.
  • 5Transform back to image space I(x′,y′,t), keeping data in temporal order, and in each location (x′,y′) project out estimated noise components from time-ordered original magnitude data. So at location (i,j), Ic(i,j,t) = I(i,j,t) − Pf(t)[I(i,j,t)] where Pf(t)[I] symbolizes the projection of the data I onto the model function f. This is equivalent to reordering back into slice order, resampling the physiological fluctuations at the sampling rate of the slices, and projecting out these components on a slice-by-slice basis.
  • 6Reorder data back into slice order: Ic(kmath image,kmath image,t) → Ic(x,y,s), putting back slice-dependent means.

RESULTS

The method was tested on four normal human subjects, with approval from the Humans Subject Committee at UC–San Diego. The multislice protocol used was that described above, in which eight slices were collected at a TR of 2.0 sec for an effective sampling rate of .25 sec. The data were first registered (IMREG, part of the AFNI suite [12]) to reduce effects due to subject motion. In Fig. 5 is shown the power spectrum of the image space magnitude of a single pixel reordered in time, before (top) and after (bottom) processing. The prominent respiratory peak and the two respiratory sidebands of the cardiac fluctuation have been virtually eliminated. The respiratory waveform was estimated automatically from the multiplexed k-space phase data. The sideband waveforms were assumed to be the same form as the primary respiratory spectrum, and their locations were automatically determined since they are the same distance (in frequency) from the center of the cardiac peak (determined from the multiplexed k-space magnitude data) as the primary respiratory peak is from the origin. The validity of these assumptions is borne out by the efficiency with which these peaks are removed. The current algorithm requires only an initial guess (within specified limits) at the locations of the primary respiratory and cardiac fluctuation frequencies. The user can specify the number of respiratory sidebands of cardiac or cardiac sidebands of respiratory to estimate. Their location and estimation is performed automatically.

Figure 5.

Removal of the respiratory peak and respiratory sidebands of the cardiac peak from the time-reordered data from the eight-slice experiment.

A useful way to visualize the spatial distribution of the improvements afforded by the method is to display the difference in voxel time series variance before and after processing. As shown in Fig. 6, there are regions of significant spatial extent that have been improved. These regions bear a close resemblance to those specified in Fig. 1, showing that the procedure has correctly extracted the known noise components. Plotting the individual voxel variance values before and after processing against one another (Fig. 7) shows that there is nearly a global decrease in image noise variance. The overall improvement reflected in this plot is approximately 15%.

Figure 6.

Spatial maps in eight slices from a single subject of the difference in temporal variance between the pre- and postprocessed data.

Figure 7.

Plot of image variance before vs. after processing, showing the trend towards improved image variance. Data shown are combined from all four subjects. The dashed line is for reference and has a slope of 1. The solid line is the regression line through the data, showing an overall improvement of approximately 15%.

DISCUSSION

The existence of highly structured signal components due to physiological fluctuations in fMRI time series data is a significant complication in the detection and estimation of functional activations. The main difficulty in multislice data is that the sampling rate in any one slice is not sufficient to critically sample the majority of the physiological fluctuations, resulting in significant power being aliased on top of the spectral components of the activation itself. As we have demonstrated, it is possible to estimate these fluctuations directly from the undersampled data and reduce these contributions. If these physiological fluctuations are critically sampled and their spectral components do not overlap the spectrum of the activation, then the detection of the fMRI signal and the estimation of its amplitude are much simpler. However, the determination of the functional SNR is still problematic, for this requires an estimate of the structured noise components. The oft-reported non-Gaussian nature of the noise is in large part due to the presence of physiological fluctuations. In taking these fluctuations into account in our current model, the remaining noise after removal of the physiological fluctuations (as well as trends) results in noise that is expected to be much closer to true Gaussian noise, thereby providing stronger justification for the use of Gaussian noise models in data analysis performed subsequent to our correction method. This is illustrated in the normal probability plots shown in Fig. 8 for the uncorrected and corrected data in Fig. 5. (In a normal probability, the data are plotted vs. a theoretical normal distribution in such a way that normally distributed data forms a straight line. See, for example, Ref. 17).

Figure 8.

Normal probability plots (see, for example, Ref. 17) for the uncorrected and corrected data in Fig. 5. Normally distributed data falls along the dashed line. The corrected noise on the right plot is seen to be much closer to Gaussian.

The detection and estimation of physiological fluctuations directly from the data is important for several reasons. First, this allows for the retrospective correction of previously acquired data. Second, it greatly simplifies the experimental setup, since it does not require monitoring systems necessary for direct measurement of cardiac and respiratory fluctuations. That the fluctuations in the data are consistent with externally monitored variations has been shown (9, 10). In addition, time courses from external monitoring systems are not necessarily the most helpful data, since the processes by which the MR signal varies as a result of these processes is complicated and their relationship to the MR signal variations is not straightforward. What is desired are estimates of the MR signal fluctuations themselves (9). Several investigators have utilized the phase information to estimate the physiologically induced signal fluctuations. Wowk et al. (10) estimated the time courses of both respiratory and cardiac fluctuations from the k-space phase data in conventional imaging and used the estimates to correct phase errors between phase-encoding lines in a standard acquisition. Le and Hu (9) also used the phase information to retrospectively correct for physiologically induced fluctuations in fMRI time-series data. Recently, Glover and coworkers (13) have developed a method by which physiological monitoring data can be used for reducing physiological fluctuations.

In the present work, we have shown that reordering the data into the temporal order in which it was acquired regains the temporal resolution necessary to critically sample the physiological fluctuations. However, since the temporal order is not the spatial order, estimation of the physiologically induced signal fluctuations from the data is not straightforward. The data contains gaps in the Fourier spectrum of the time series that are contaminated by the reordering process. This occurs at the harmonics of the reordering frequency. For instance, for 128 time points collected in each of eight slices, the reordered 1024-point time series will be contaminated at the frequency related to the reordering of the eight slices—that is, every 128th point. In addition, baseline inconsistencies between the slices, such as are produced by the slice-dependent trends, will widen the contaminated region about the reordering frequencies. It is therefore important that such slice-dependent trends be removed prior to reordering. Since the magnitude fluctuations produced by the activations vary amongst the slices, the resampling frequency artifacts can also contain harmonics from the activation, which must be avoided if the activation changes are not to be reduced.

The estimate of the respiratory waveform was shown to be robustly found from the k-space phase. This was evident by the completeness with which this component was removed from the time series spectrum. This is consistent with the notion that respiratory effects really are fairly global in nature, and therefore will be well represented by the low-frequency k-space components, as well as consistently present in the time-series spectrum. However, the estimation and removal of the cardiac waveform was more problematic. Cardiac-related fluctuations are spatially focal, so that their representation in the low spatial frequencies, as an average over all the voxels, can only vaguely represent the signal in any one voxel and are therefore more problematic to estimate from global data. Importantly, however, the respiratory sidebands of the cardiac fluctuations are removed. Since these are of high frequency and localized in regions of cardiac fluctuations, they produce significant aliased noise in the spectrum.

Generally speaking, physiological waveforms are complicated since they vary in response to several physiological factors (e.g., cardiac varies with respiration, exertion level, etc). Since these factors may themselves produce fluctuations (e.g., as does respiration), the waveforms may be highly correlated. The most pertinent example for the present study is the coupling of the cardiac and respiratory waveforms. Cardiac and respiratory waveforms vary in time, in part due to each other. This is reflected in the strong respiratory sidebands on the cardiac peaks. Because the frequencies in these components vary during the experiment, the spectrum over the course of an experiment represents a combination of spectral components that existed during the experiment. The relationship of these variations to the slice acquisition schedule will clearly affect the waveform in any particular slice time series. The improvement of the estimation of the physiological waveforms will be an important aspect of our continuing refinement of this technique. However, the global nature of the respiratory waveform still allows a robust estimation of its waveform.

Estimation of the waveforms is clearly an area that could be improved. In the current method, the width of the frequency range is chosen by the user to cover the apparent width of the distribution of frequencies in the power spectrum. This entire frequency band is used without modification to estimate the waveform because the shape of these distributions is quite complicated and not amenable to modeling with some simple distribution (such as a Gaussian). A more proper approach (which we are pursuing) is to incorporate a physical model for the frequency spectrum (say, a model for the cardiac pulsations based on cardiac characteristics) and estimate the parameters of such a model and use this in constructing the estimated waveform from the spectrum centered about the primary frequency components.

It should also be pointed out that in the present implementation, the noise components were estimated, then projected out of the data. More properly, one should estimate the noise fluctuations and use them as components of the signal model in the estimation of the functional activation components of this signal in order to properly take them into account in the estimation of the activation. This is the proper way to “remove” the noise from the data. However, it is computationally more efficient to project the noise out first, and is of no consequence if the spectral components of the noise are well separated from those of the activation.

One caveat for this technique is that regular slice sampling intervals have been assumed. In certain applications (e.g., fMRI of the auditory system) it is beneficial to collect clustered volume multislice acquisitions (e.g., Ref. 14) that employ a nonuniform temporal sampling scheme. In principle, the current technique is still applicable, and in fact may be improved, as irregular sampling tends to disambiguate aliased frequencies. However, this would require a significantly more complex estimation scheme for frequency identification.

CONCLUSIONS

The primary focus of this study was to address the issue of physiological fluctuations that contaminate multislice fMRI time-series data. Since the sampling rate for any particular slice is usually long so as to accommodate several slices within the study, components of the physiological fluctuations are aliased into the spectrum of the time series, making estimation and removal difficult. We have shown that it is possible to recover unaliased information about the respiratory-induced fluctuations, including high-frequency respiratory sidebands of cardiac fluctuation, by viewing the data in the precise temporal order in which it was collected, rather than the usual spatial order. Estimates of the time courses of components of these fluctuations are made from the k-space phase viewed in temporal order, since the k-space time courses contain spectral components of the physiological fluctuations. These estimated time courses are then used as model functions to be projected out of the temporally-ordered image magnitude data. Although respiratory fluctuations that were superimposed on cardiac fluctuations (i.e., the respiratory sidebands of cardiac) were significantly reduced, cardiac fluctuations themselves were not because they are highly localized in space and therefore do not satisfy the “global” nature of the artifacts upon which this technique is based. However, the presence of cardiac-related peak in the reordered data does suggest that our reordering technique might yet provide some useful information on cardiac fluctuations.

The results suggest that this method can provide a significant improvement in the functional signal-to-noise, thereby improving both the detection and significance of functional activations. In addition, the growing sentiment that low-frequency components are of physiological interest (15, 16) will require a similar reduction of the aliasing of high-frequency components in order that the broad low-frequency region of the spectrum be interpretable.

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