Circadian rhythm parameters from actigraphy

In the cosinor analysis a cosine curve is fitted on the actigraph signal by adjust-ing the cosinor formula’s parameters. The fittadjust-ing is based on the least squares method utilizing trigonometry. From the cosinor analysis amplitude (AMPL), mesor (MESOR) and acrophase (PHASE) were extracted for the analysis in Study V and Study VI (Halberg et al. 1972) (Figure 6). Extended cosinor methods exist which infer more parameters (Martin et al. 2000) or cosine curve transformations, such as the Hill’s function or the anti-logistic function (Marler et al. 2006). We have left them out from the current analysis due to the vast amount of methods already included.

Figure 6: Cosinor analysis parameters. X-axis is minutes from the beginning of the recording. Y-axis presents absolute epoch values of the telemetric actigraph. Blue line is the original actigraphic time series data. Black line is the fitted cosinor data.

The cosinor analysis is performed in the following steps (Cornelissen 2014).

The cosine formula can be written

( ) = + (2πt/τ + φ) + e(t) (1) When τ is 24 hours we can write the formula as

( ) = + + + ( ) (2)

Where

= ( ); = − ( ); = cos ; = sin ( ) (3)

For solving the unknown parameters least squares equation can be written as (Y is now the original actigraphy)

= ∑[ −( + + ] (4)

RSS is minimal if the first derivative of each parameter (M, β and γ) is zero. The matrix form of the derivatives can written as

∑∑

∑ =

∑ ∑

∑ ∑ ∑

∑ ∑ ∑

(5)

After solving the equation amplitude (A) and acrophase (φ) can be determined as:

= ( + )^/ (6)

= − + (7)

where K is an integer and depends on the sign of a acrophase estimate (φ). M is the mean value of the actigraphy signal.

4.3.2.2 Autocorrelation

Because physical activity typically varies close to a 24-hour rhythm, the 24-hour autocorrelation (AUTOCORR) has been used to quantify the systematic of the 24-hour rhythm. The autocorrelation function has been referred to as “memory” of the time series (Taylor 1990). In practise the autocorrelation is calculated with a 1440-minute lag for the telemetric actigraph data. The algorithm is

= ^∑ _∑⁽ ₍ ⁾⁽ ₎ ⁾ (8) Where xt= activity value for minute t

k = lag i.e. to how much later the sample is moved (1440 in the analysis is fixed)

xt+k = sample at minute t with lag k N = number of observations in time series x = mean of the time-series signal

The divisor makes sure that the autocorrelation with zero lag is scaled to 1.0.

4.3.2.3 Parameters based on the hourly bins

Circadian rhythms parameters’ inter-daily stability (IS), intra-daily variability (IV) and relative amplitude (RA), require a minute-to-minute active/passive classifica-tion. According to the active/passive classification hour bins are formed by count-ing the number of active minutes durcount-ing the hour.

IS describes the stability between the days implying stable zeitgebers. It is the 24-hour value of chi-square periodogram. It is calculated as

= ^∑_∑ ⁽_{( )}⁾ (9)

where n = total number of hour bins

p = number of data per day (24 in this case)

xh= hourly mean over all hour bin (for example between 16:00 and 17:00 for all the days)

xi= individual hour bin x = mean of all hour bins

IS varies between zero for Gaussian noise and one for a perfect match between the days, that is, the averaged hour bins and daily hour bins have equal variance.

If daily hour bins would differ substantially from each other, the average hourly bin would approach the average of the data and IS would approach zero. The length of the measurement affects slightly the score.

IV quantifies the fragmentation of the rhythm and activity. It is the ratio between the variance of the consecutive hour bins and overall variance.

=₍ ^∑_)∑⁽ ₍ ⁾₎ (10)

IV gets values near zero for a perfect sine wave and about two for Gaussian noise meaning that more fragmented activity data yields larger values. For exam-ple, Meadows et al. (Meadows et al. 2010) reported that high IV (>1) was an indi-cation of daytime naps and/or night-time arousals.

From the averaged hour bins the least active five-hour period (L5) and most ac-tive ten-hour periods (M10) are extracted. The relaac-tive amplitude is calculated as

= (11)

In addition, the average M10 and the average L5 were included in the analysis.

Figure 7 presents an example of three different activity profiles in hourly bin data including IV, IS and RA values.

IS=0.58 IV=0.37 RA=0.73

IS=0.78 IV=0.27 RA=0.86

Figure 7: Three examples of hourly bin data with estimates for intra-daily variability (IV), inter-daily stability (IS) and relative amplitude (RA).

4.3.2.4 Circadian rhythm strength

Circadian rhythm strength (CRS) was calculated by dividing average night-time activity (11PM – 5AM) by the average activity of the previous day (8PM – 20AM) for each available circadian(Paavilainen et al. 2005). The average value of the daily CRS score is used in the analysis. CRS is utilized in the telemetric actigraphy system to describe a person’s health status.

4.3.3 Sleep patterns from actigraphy

According to the minute-to-minute activity epochs, sleep/wake classification is performed based on the algorithm reported in the related article (Lötjönen et al.

2003). The bedtime information (subjects or nursing personnel annotated) was present for Dataset 2 and partly for Dataset 1. In case the sleep diary was missing the long-term activity data were observed and constant bed times were selected (Study V, Study VI case demonstrations). The constant time points were selected based on the activity profile by observing double-plotted actograms.

Total sleep time (TST) is a sum of sleep classification during in-bedtime. NAP is the sum of sleep classifications during the daytime, between 9 AM and 9 PM. in the analysis. The same daytime period had been utilized in related research (Paavilainen et al. 2005). Awakenings are count of transitions from sleep to wake during the in-bed time. Sleep efficiency is a percentage of sleep during the in-bed time (only used with the sleep diary). The settings for sleep/wake classification procedure for older people with low functional capacity were done according to the manufacturer’s recommendations when it was applicable.

IS=0.29 IV=0.63 RA=0.28