Optimizing the measurement of sample entropy in resting-state fMRI data

Abstract

Introduction: The complexity of brain signals may hold clues to understand brain-based disorders. Sample entropy, an index that captures the predictability of a signal, is a promising tool to measure signal complexity. However, measurement of sample entropy from fMRI signals has its challenges, and numerous questions regarding preprocessing and parameter selection require research to advance the potential impact of this method. For one example, entropy may be highly sensitive to the effects of motion, yet standard approaches to addressing motion (e.g., scrubbing) may be unsuitable for entropy measurement. For another, the parameters used to calculate entropy need to be defined by the properties of data being analyzed, an issue that has frequently been ignored in fMRI research. The current work sought to rigorously address these issues and to create methods that could be used to advance this field.

Methods: We developed and tested a novel windowing approach to select and concatenate (ignoring connecting volumes) low-motion windows in fMRI data to reduce the impact of motion on sample entropy estimates. We created utilities (implementing autoregressive models and a grid search function) to facilitate selection of the matching length m parameter and the error tolerance r parameter. We developed an approach to apply these methods at every grayordinate of the brain, creating a whole-brain dense entropy map. These methods and tools have been integrated into a publicly available R package ("powseR"). We demonstrate these methods using data from the ABCD study. After applying the windowing procedure to allow sample entropy calculation on the lowest-motion windows from runs 1 and 2 (combined) and those from runs 3 and 4 (combined), we identified the optimal m and r parameters for these data. To confirm the impact of the windowing procedure, we compared entropy values and their relationship with motion when entropy was calculated using the full set of data vs. those calculated using the windowing procedure. We then assessed reproducibility of sample entropy calculations using the windowed procedure by calculating the intraclass correlation between the earlier and later entropy measurements at every grayordinate.

Results: When applying these optimized methods to the ABCD data (from the subset of individuals who had enough windows of continuous "usable" volumes), we found that the novel windowing procedure successfully mitigated the large inverse correlation between entropy values and head motion seen when using a standard approach. Furthermore, using the windowed approach, entropy values calculated early in the scan (runs 1 and 2) are largely reproducible when measured later in the scan (runs 3 and 4), although there is some regional variability in reproducibility.

Discussion: We developed an optimized approach to measuring sample entropy that addresses concerns about motion and that can be applied across datasets through user-identified adaptations that allow the method to be tailored to the dataset at hand. We offer preliminary results regarding reproducibility. We also include recommendations for fMRI data acquisition to optimize sample entropy measurement and considerations for the field.

Keywords: R software; brain dynamics; complexity; fMRI; sample entropy (SampEn).

Figure 1

Overview of sample entropy. This demonstration of entropy estimation using SampEn was adapted with permission from Richman et al. (; page 10). In this example, m is 2 and the threshold for accepting a match is r multiplied by the standard deviation (see error bars). Note that the template (first two points) is matched by the 11th and 12th points (solid box), and that the m + 1st points also match (dashed box). In this case, quantities of A and B both increase by 1.

Figure 2

Schematic of the windowing procedure. Low-motion windows are selected and concatenated, with an ignored volume in between windows.

Figure 3

Time series processing for a single subject. Panel (A) illustrates the variation in Framewise Displacement for a single participant over the course of the scan. Red circles indicate the volumes where FD values exceeded the acceptable threshold of 0.2 mm. These windows were masked out prior to application of the bandpass, and were never included in the “best” windows that were selected for analysis (indicated with purple lines). Panels (B) and (C) illustrate BOLD signal patterns from a single subject in the left nucleus accumbens (B) and left anterior cingulate cortex (C). The gray line shows the raw BOLD signal time series; the blue line shows the time series after applying the bandpass filter. Red x marks indicate volumes that were masked out prior to bandpass application because of excessive motion (FD > 0.2 mm).

Figure 4

Results from search_mr_grid. Violin plots showing the distribution of median error criterion for m = 2 and various r values across subjects. The top panel shows median error criterion across all grayordinates. The lower three panels show each of the cortical hemispheres and the subcortex separately.

Figure 5

Whole-brain sample entropy in the ABCD Study data. Top: Average sample entropy across 3,058 participants for runs 1 + 2. Bottom: Average for runs 3 + 4.

Figure 6

Relationship between SampEn and FD in “full” and “windowed” methods. Panel (A) shows the relationship between SampEn values and head motion after using powseR’s windowing approach. Panel (B) shows the same relationship, using a conventional approach (where the full, uninterrupted time series is used to calculate SampEn). Panel (C) shows the relationship between SampEn values derived from these two approaches, with a color scale applied to indicate the mean FD of each observation.

Figure 7

Reliability of sample entropy measurement across the scan. Intraclass correlation values representing the similarity in measurements of sample entropy as calculated from runs 1 + 2 vs. runs 3 + 4 are shown for each vertex in the cortex and each voxel in the subcortex. Warm colors represent high reliability, darker colors reflect lower reliability.