As mentioned above, when scanning a static phantom, there are two main sources of noise: background and instability. Scanning a human introduces a third type of noise due to physiological processes. Background, instability and physiological noise are temporally independent, so their variances will add linearly. The total temporal noise variance

at flip angle α is given by:

where

is the physiological variance at flip angle α,

is the scanner instability variance at flip angle α and

is the background noise variance (independent of flip angle). Signal-to-fluctuation-noise (SFNR) is defined as the mean intensity divided by the standard deviation of the total noise (

5). We will define several new SFNR measures below (see also for a summary).

| **Table 1**SFNR abbreviations and definitions. |

We model the physiological and instability fluctuations as

*signal-weighted* (SW), i.e., they are multiplicative fluctuations in signal and therefore variances are proportional to the square of the signal intensity (

2). These variances can therefore be grouped together into a single component:

, with the total variance (

Equation 1) re-expressed as

We can measure the total variance from the time series, but we also need the background variance in order to compute the signal-weighted variance. Some proposals ((

6),(

7),(

8)) suggest measuring the variance in the actual background, e.g., in the corners of the image or in other areas away from locations were EPI artifacts are suspected. There are several problems with this approach. First, it is difficult to find areas where no artifact signal is deposited due to ghosting and ramp sampling or non-Cartesian k-space trajectories such as spiral. Second, the variance of the noise as it appears in the background is not the same as that for when it appears in the foreground due to the complex-to-magnitude operation. For a single channel coil, the noise is Rayleigh distributed, and it is easy to convert the variance to foreground variance by dividing by 2−π/2 (

9). For multiple-channel coils, there is no simple conversion factor. Methods exist (

10–

12) to compute the background noise variance in multiple coils using a 0° flip angle scan (i.e., no RF power), but these methods require special pulse sequences, access to k-space data, and/or assume that there is no noise correlation across channels. Our method avoids these issues and may be implemented on any scanner.

The definition of signal weighted is that the variance of the SW component will simply scale when going from flip angle α_{1} to α_{2}, i.e.,

where

*μ*_{αi} is the mean intensity at flip angle α

_{i,.} Given two acquisitions at different flip angles, the mean intensities

*μ*_{αi} and total variances

can be measured empirically for each data set. This gives us three equations (

Equation 2 for each flip angle, plus

Equations 3) with three unknowns (

). Substituting and solving for the SW and background components, we get:

These variances can then be used to compute two new SFNR measures that can be used to assess scanner quality:

where

*swSFNR* is the signal weighted component, and

*bgSFNR* is the background component. When scanning a phantom, only instability and background noise are present, and so the

*swSFNR* will only represent instability noise. When scanning a human subject, all three components (physiological, instability, and background) will be present, and the physiological and instability components cannot be separated. However, we can estimate the amount of instability in human scans using the instability in the phantom scan because the instability is proportional to image intensity, so

where

*iSFNR* is the instability SFNR measured from the phantom scan,

*μ*_{α,H} is the voxel-wise mean intensity of the human scan, and

_{I α} is the voxel-wise estimate of the instability standard deviation in the human scan. As the total variance, the instability variance, and the background variance are now estimated,

Equation 1 can be used to solve for the estimate of the physiological variance

. This yields three new SFNR measures: bgSFNR and iSFNR (both defined above), and pSFNR=

*μ*_{α,H}/_{P,α}, the physiological SFNR.

In an fMRI study, these three sources of noise are independent of any task, so the increase of any source can be reversed by an increase in scanning time proportional to the increase in total variance that it causes. For example, if instability accounted for 10% of the total noise, one would need to scan 10% longer (with concomitant increase in scanner costs, the number stimuli needed, and subject fatigue). Alternatively, one could think of the scanner as running at 90% capacity. While this metric provides no absolute cutoff for scanner acceptance, it provides a clear link between instability and scanner performance, and researchers will be well prepared to set a threshold based on their own comfort level. For this study, we quantify the instability noise with respect to only the physiological noise (i.e., assume the background noise is, or can be made, negligible). This gives a more conservative measure and also simplifies the quantification because the increase in total noise variance can be expressed as the square of the ratio of iSFNR to pSFNR.