Introduction

Cerebrovascular reactivity (CVR), measuring the ability of vessel dilation or constriction in response to a stimulus, is an important indicator of cerebrovascular health¹. CVR typically uses a metered physiological challenge such as acetazolamide injection or carbon dioxide (CO₂) inhalation, although measures can also be derived from endogenous fluctuations in end-tidal CO₂ levels². CO₂ inhalation has been increasingly used as the stimulus in CVR study because it is well-tolerated, efficiently produces vasodilation and is compatible with magnetic resonance imaging (MRI). Blood oxygen level dependent (BOLD) MRI, is commonly used to record the cerebrovascular response¹ because of its speed and brain coverage. CVR is often calculated from the linear fit between CO₂ stimulus and MRI signal response in the time-domain, assuming the brain is a linear time-invariant system¹. However, this requires temporal alignment of the input and output signals, which can be difficult, when the signal is noisy or distorted. To avoid signal misalignment, modeling-based methods^3,4 are proposed, but they need a designed stimulus paradigm, typically, a sinusoidal or block input signal. In this article, we propose a frequency-based approach, which applies general linear model (GLM) in the magnitude of the signal spectrum, to avoid challenges with signal alignment and improve robustness to noise in CVR estimation.

Methods

CVR analysis: In a CVR study with CO₂ challenge, CVR can be defined by the following equation: $$CVR = \frac{\Delta(MRI\:measurement)}{\Delta{EtCO_2}}\qquad(1)$$ where $$$\Delta(MRI\:measurement)$$$ and $$$\Delta{EtCO_2}$$$ represent the changes of MRI signal and end-tidal CO₂ between baseline and hypercapnic measurements. Let $$$y(t)$$$ be the MRI response, $$$x(t)$$$ be the CO₂ stimulus, and $$$\tau$$$ be the voxelwise delay between the MRI and CO₂ signals. According to equation (1), we can have: $$y(t-\tau)=CVR\cdot{x(t)}\qquad(2)$$ Apply Fourier transform on both side: $$Y(\omega)e^{-j\omega\tau}=CVR\cdot{X(\omega)}\qquad(3)$$ If we only consider the magnitude and apply a general linear model: $$\mid{Y(\omega)}\!\mid=CVR\cdot\mid\!{X(\omega)}\mid+c+\varepsilon\qquad(4)$$ where $$$\mid{Y(\omega)}\mid$$$ and $$$\mid{X(\omega)}\mid$$$ are the magnitudes, $$$c$$$ is the estimated intercept, and $$$\varepsilon$$$ is the estimated error. Therefore, the CVR is estimated only by the magnitude, and the effect of signal time shifting is eliminated. Note that equation (4) estimates the mean of the gain in each frequency, which differs from the gain of transfer function².

Data description and preprocessing: 24 BOLD MRI (TR/TE = 1500/30 ms) scans were acquired using a 3T Philips Achieva D-Stream with a 32 element head coil. The BOLD images were registered to MNI space and slice-time corrected using FSL⁵ and AFNI⁶. The corresponding CO₂ stimulus were recorded by RespirAct. We use a 4-cycle sinusoidal paradigm of CO₂ with a peak-to-peak amplitude of 5 torr and a period of 1 minute. The CO₂ sinusoid was biased +5 torr from the resting end-tidal CO₂, such that CO₂ fluctuations range from 0 to 10 torr greater than resting end-tidal CO₂. The CO₂ data was resampled and interpolated at the BOLD sampling.

Evaluation: We compared our method with a modeling-based approach⁴ and time domain general linear model estimation¹. All the algorithms are running at voxel by voxel, and Figure 1 shows an example of the color-coding CVR map comparison. For each subject, we calculate the mean CVR of white matter and gray matter, respectively. One sample T-test and Bland-Altman plot⁷ are used to pairwise compare our method with other two.

Results and Discussion

Figure 2 shows the relative Bland-Altman plot between our frequency GLM and the other two methods. The method differences are normalized to their average value. Table 1 summarizes the mean and standard deviations of these comparisons. The time and frequency based GLMs (Figure 2b, 2d) are unbiased with one another in both the gray and white matter and have limits of agreement of 18% and 42%, respectively. In contrast, the limits of agreement between the model-based and frequency GLM in the gray and white matter are much broader, measuring 70% and 81%, respectively. Model-based CVR (Figure 2a, 2c) is greater than frequency-based GLM (8.8% and 36.2%), but this difference was only significant in the white matter; similar disagreements were observed between the model based and time-domain GLM (not shown).

The close agreement between time and frequency-based GLM’s was expected because they are duals of one-another and it is relatively easy to align the powerful, sinusoidal BOLD fluctuations using correlation analysis. However, frequency domain GLM does not require signal alignment, making it better suited to noisier signals where temporal alignment is more challenging.

The disagreement between both GLM methods and the model-based method likely arises from measurement and system non-idealities. The modeling-based method is designed to fit a specific signal shape, the performance highly depends on the fidelity of signals as well as system linearity. Large errors caused by signal distortion will enlarge the difference with other non-modeling-based methods. And it is worse when estimating the CVR of white matter, because the white matter has a longer group delay⁸, which could make signals more noisy.

Conclusion

We have validated frequency-domain GLM against time-domain GLM using a powerful CO₂ stimulus paradigm where time-domain GLM performs exceedingly well. We postulate that frequency-domain GLM will more robust than time-domain GLM for less powerful CO₂ stimuli where temporal alignment is challenging. The disagreement between GLM and model-based methods requires further exploration before concluding superiority or inferiority of either technique.

Acknowledgements

This work was supported by the Additional Ventures Single Ventricle Fund, National Heart, Lung, and Blood Institute (grant 1U01-HL-117718-01, 1R01-HL136484-01A1), and the National Center for Research (5UL1-TR000130-05) through the Clinical Translational Science Institute at Children’s Hospital Los Angeles. Philips Healthcare provided support for protocol development and applications engineering on a support-in-kind basis. Chau Vu was supported by a Research Career Development Fellowship from the Saban Research Institute at Children’s Hospital Los Angeles.