What is the response vector in the fMRI GLM model?

Question

In fMRI, univariate analysis typically makes use of the general linear model (GLM), where the relation between experimental condition and BOLD activity is estimated with a linear regression model for each voxel separately ("mass univariate model"). However, I found it surprisingly hard to find out what the response vector $Y$ contains. I am aware that it is some measure of the blood oxygenation level dependent signal (BOLD signal) - but what is measured exactly, i.e. what are the units? Is it the oxy/deoxy ratio, or is it another property?

Answer 1

Short answer

The units aren't important, and are typically normalized.

Longer answer

The name "BOLD" is actually very instructive: Blood Oxygenation Level Dependent contrast (or signal). BOLD is not directly measuring anything like oxygenated/deoxygenated ratio, or we would probably call it something more similar to that: oxy/deoxy ratio.

Instead, "BOLD" is just the MRI image you get when you set up the scanner properties in a way that happens to be influenced by the oxygenation level of blood. Oxygenation level is far from the only contributor to the BOLD signal, but experimentally you try to factor out any influences that are time-varying but not interesting, and you ignore baseline levels that don't change.

fMRI works because the transverse relaxation time T2 of water in blood depends on oxygenation of the blood. From the original paper showing BOLD contrast in a rat brain Ogawa et al. 1990:

At the 7-T field strength used in this study, T2 varies from 50 msec at 100% oxygenation to 4 msec at 0% oxygenation. The venous blood signal becomes very weak when the T2 value becomes comparable to or shorter than the echo time of the signal acquisition. At 60% oxygenation level, the estimated T2 value is 18 msec, similar to the echo time used in this report.

In other words, if you produce gradient-echo images with an echo time of 18 ms, the images you get show very different signal levels at 100% vs 60% oxygenation. Depending on the strength of your magnet, you can tune this time to go after the differences you want to see.

There is no straightforward way to compare images recorded in different conditions (i.e., different scanners, different parameters) on any useful scale of measurement. Instead, people typically report relative values, either in time or space. For this reason, you often see BOLD signals expressed as a % change or in terms of standard deviation (i.e., z-scores).

Ultimately, for fitting a general linear model it doesn't matter: you can freely scale a dependent variable without changing the structure of the model fit, all you influence is the interpretation of the magnitudes of the intercept and coefficients.

References

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Ogawa, S., Lee, T. M., Kay, A. R., & Tank, D. W. (1990). Brain magnetic resonance imaging with contrast dependent on blood oxygenation. Proceedings of the National Academy of Sciences, 87(24), 9868-9872.