Shannon Entropy Applied to the Measurement of the Electroencephalographic Effects of Desflurane

We used electroencephalographic data recorded in a previous study, which was performed by our research group after approval by the local Ethics Committee (Bonn, Germany). 1The anesthetic management of the patients is described in detail in a previous article by our group. 1Briefly, after induction with 2 mg/kg propofol, anesthesia was maintained with desflurane as the sole anesthetic agent in 12 female patients undergoing gynecologic laparotomies. Only the electroencephalographic data recorded between opening and closure of the peritoneum was used for further analysis. In all cases, at least 60 min was allowed between induction of anesthesia and start of measurements to minimize the effects of propofol administered for induction. After opening of the peritoneum, the end-tidal desflurane concentration was decreased until 0.5 minimum alveolar concentration (MAC) (= 1.3 · MAC awake) was achieved or until the attending anesthesiologist no longer considered there to be clinically adequate depth of anesthesia. Subsequently, the desflurane vapor setting was increased until an end-tidal desflurane concentration of 1.6 MAC was achieved or until the attending anesthesiologist considered the level of anesthesia as too deep. Subsequently, the desflurane vapor setting was decreased again. Several of these cycles were performed in each patient.

The electroencephalograph was recorded continuously with a frontal montage (Fp1-Fpz, Fp2-Fpz; international 10-20 system; Sirecust 404; Siemens, Erlangen, Germany). The amplitude resolution of the A/D converter was 12 bit. The raw signal was filtered between 0.5 and 32 Hz and divided into epochs of 8.192 s. The raw signal was digitized at a rate of 125 Hz and stored on a hard drive for further off-line analysis. The Shannon entropy (see below), approximate entropy (quantifies the predictability of subsequent amplitude values of the electroencephalograph based on the knowledge of the previous amplitude values as described previously 1), median electroencephalographic frequency (50% quantile of the power spectrum), spectral edge frequency 95 (SEF 95; 95% quantile of the power spectrum), total power, and log total power were calculated from 2¹⁰data points (= 8.192-s epochs). A moving average over seven epochs (three forward and three backward epochs) was used for data smoothing for Shannon entropy, approximate entropy, and all power spectrum–derived parameters. Additionally, we used the Aspect A-1000 (version 3.12; Aspect, Natick, MA) to determine the Bispectral Index (BIS). The BIS was internally averaged over 60 s. For each electroencephalographic epoch, the corresponding end-tidal desflurane concentration was recorded.

The Shannon entropy was calculated off-line on a personal computer according to the following algorithm 2:

where i extends over all observed amplitude values of the data time series and pⁱis the probability that the amplitude value vⁱoccurs anywhere in the data time series. Therefore, pⁱis the ratio of the number of data points with the amplitude value vⁱto the total number of data points in the data time series.

For example, if four amplitude values occur in a time series of 12 data points two times, three times, five times, and two times, respectively: Shannon entropy H =−1 ×[2/12 × log (2/12) + 3/12 × log (3/12) + 5/12 × log (5/12) + 2/12 × log (2/12)]. The length of the epoch and the sampling rate determine the total number of samples, whereas the resolution determines the possible number of observable amplitude values. We evaluated the influence of different epoch lengths (N = 1024, 2048, 512, and 256) and different amplitude resolutions (12 bit to 1 bit) for the correlation coefficient R²of electroencephalographic Shannon entropy versus  desflurane effect compartment concentration. The software for electroencephalographic analysis has been programmed by one of the authors (J. B.) in Visual Basic and is available on request.

Desflurane effect compartment concentrations were obtained by simultaneous pharmacokinetic–pharmacodynamic modeling. 5To eliminate the hysteresis between the end-tidal concentrations of desflurane and the electroencephalographic effect, an effect compartment was introduced into the model:

where C^etis the end-tidal concentration of the respective volatile anesthetic, C^effis the effect compartment concentration of the respective volatile anesthetic, and k^e0is the first order rate constant determining the efflux from the effect compartment.

The relation between effect compartment concentration and electroencephalograph was modeled with a fractional sigmoid E^maxmodel (Hill equation) 6:

where E⁰is the measured baseline effect of each individual, c^effis the apparent effect side concentration, C⁵⁰is the concentration that causes 50% of the maximum effect, and λ describes the slope of the concentration response relation.

Electroencephalographic parameter values from each individual and for each electroencephalographic parameter were fitted separately. The parameters of the described models were estimated using nonlinear regression with ordinary least-squares. The computations were performed on a spreadsheet using the Excel software program (Microsoft, Redmond, WA), and the parameters were optimized with the Solver tool within Excel.

No data were excluded from the pharmacodynamic analysis, but some parameter estimates were excluded from the summary values and from the calculation of the median, according to the following criteria 7:

• 1. The estimate of T^1/2k^e0had to be shorter than 10 min. Values longer than this could not reasonably be supported.

• 2. The C⁵⁰had to be less than 10 vol%. Values larger than this usually indicated failure of the model to identify E^max.

The pharmacodynamic model used in this analysis required determination of E^maxto obtain accurate estimates of C⁵⁰and k^e0. The study design precluded estimation of E^maxin several patients, resulting in poor estimates in some individuals.

Because our aim was to maximize the correlation between the measure of drug effect and the drug concentration at the effect site, we chose the coefficient of determination (R²) as the objective function. 8 

SSE, the sum of squared errors, represents the sum of the squares of the differences between the observed measurements yⁱfor a given time and the corresponding model prediction, ŷⁱ. SST, the total sum of squares, stands for the sum of the squares of the differences between each actual measurement and the average of all the measurements, ¯yⁱ.

Because SST is independent of the model parameters, maximizing R²is equivalent to minimizing SSE, i.e. , it is equivalent to nonlinear regression with ordinary least-squares. A value of R²close to 1 means that changes in effect can be entirely explained by changes in the apparent effect compartment concentrations. A value of R²close to 0 means that there is no relation between effect compartment concentration and effect. 8 

We compared the values of R²between Shannon entropy and approximate entropy, SEF 95, BIS, median electroencephalographic frequency, total power, and log total power using the Wilcoxon matched pairs signed rank test. Statistical significance was assumed at probability levels of P ≤ 0.05.

With increasing desflurane concentrations, the electroencephalographic Shannon entropy decreased within a time delay (fig. 1). Plotting Shannon entropy versus  end-tidal desflurane concentration revealed hysteresis (fig. 2, left side), which was collapsed by introduction of an effect compartment (fig. 2, right side).

The Shannon entropy (R²= 0.84 ± 0.08, mean ± SD) performed as well as approximate entropy (R²= 0.85 ± 0.12), SEF 95 (R²= 0.85 ± 0.10), and BIS (R²= 0.82 ± 0.13) as a measure of anesthetic drug effect. Electroencephalographic median frequency (R²= 0.72 ± 0.17), total power (R²= 0.67 ± 0.18), and log total power (R²= 0.80 ± 0.09) performed significantly worse. Figure 3shows the R²values for each patient.

Calculating electroencephalographic Shannon entropy for 2,048 data points did not improve the correlation with desflurane effect compartment concentrations (R²= 0.84 ± 0.08) compared with 1,024 data points. Decreasing the number of data points by 50% (75%) substantially worsened the correlation R²(N = 512: R²= 0.80 ± 0.06; N = 256: R²= 0.69 ± 0.12).

The influence of different amplitude resolutions (12 bit to 1 bit) on the correlation coefficient R²for Shannon entropy versus  desflurane effect compartment concentrations is displayed in table 1. Choosing amplitude resolutions from 12 bit down to 5 bit yielded different Shannon entropy values and a decreasing number of different observed amplitude values but changed R²only slightly. Further decreasing the amplitude resolution substantially worsened the correlation R².

The median T^1/2k^e0and C⁵⁰for desflurane, calculated using the Shannon entropy as measurement of drug effect, were 1.17 min (ranging from 0.4 to 2.0 min) and 4.0 vol% (ranging from 2.7 to 5.9 vol%), respectively. The estimates of T^1/2k^e0and EC⁵⁰for electroencephalographic Shannon entropy and previously used electroencephalographic parameters are summarized in figures 4 and 5. The values for k^e0and C⁵⁰for each patient were in good agreement between Shannon entropy and previously used electroencephalographic parameters.

In this study, we demonstrated a close correlation between Shannon entropy values and desflurane effect compartment concentrations (R²= 0.84 ± 0.08). Shannon entropy detected differences in electroencephalographic dynamics in the observed concentration range as well as approximate entropy, SEF 95, and BIS and better than electroencephalographic median frequency, total power and log total power. In addition, the use of the Shannon entropy as measurement of drug effect on the electroencephalogram yielded pharmacodynamic parameters comparable to those derived from pharmacodynamic modeling using established univariate descriptors of the electroencephalogram.

We are aware of the problems created by using the same electroencephalographic data as in a previous paper. 1However, because the idea for and the calculation of this first application of the Shannon entropy algorithm to electroencephalographic data was proposed after submitting the article about electroencephalographic approximate entropy, the Shannon entropy could not be included in the approximate entropy article.

Ideally, a univariate descriptor of drug effect on the electroencephalogram should require a minimal amount of data transformation and still identify the signal component most sensitive to changes of drug concentration. 9All commonly used univariate descriptors of the electroencephalographic require fast Fourier transform. Unequal voltage values at the beginning and end of an epoch cause artificially high power in the very-low- and the high-frequency components in the power spectrum. This necessitates the application of a window element before Fourier transform. This window element means a certain amount of data transformation and loss of data information. In contrast, the Shannon entropy is calculated directly from the digitized voltage signal, without any data transformation.

Furthermore, the mathematical method should be as simple as possible. Compared with the calculations necessary to obtain frequency-based parameters and the BIS, 4the Shannon entropy is by far the most straightforward algorithm. Because the algorithm takes into account only the frequency of observed amplitude values during an epoch and not their actual values, it is insensitive to infrequently occurring values, leading to a high robustness against artifacts.

For precise calculation of the Shannon entropy, the computed probability distribution of the amplitude values during one epoch must resemble the actual probability distribution as closely as possible. We varied the number of observed amplitude values by changing epoch length, and, in a separate step, we varied the number of different observed amplitude values by off-line downgrading of the amplitude resolution. In our study, 1,024 amplitude values sufficed for a precise calculation of Shannon entropy. Decreasing the number of amplitude values used for calculation of Shannon entropy decreased the correlation between Shannon entropy values and desflurane concentrations. Down to an amplitude resolution of 5 bit, the correlation between Shannon entropy values and desflurane concentrations remained remarkably stable, despite changing absolute Shannon entropy values and a decreasing number of different observed amplitude values.

However, we have to alert the reader to a peculiarity of the Shannon entropy, which is also shared by the canonical univariate parameter, 7,10,11total power, and log total power. In contrast to SEF 95, median frequency, or BIS, the Shannon entropy is not normalized to the total power. Therefore, the absolute value of the Shannon entropy in the absence of drug (baseline value) may vary between individuals because of interindividual differences of signal strength. It is not helpful to report parameters referring to absolute values of the Shannon entropy (E⁰and E^max). This precludes the use of the Shannon entropy as a clinical measure of anesthetic depth, for which a specific parameter value should correspond to a specific anesthetic depth regardless of the individual observed. Typical applications for the Shannon entropy include estimating potency and relative potency of drugs and their combinations.

Recently, a different measure of signal regularity has been successfully applied to the analysis of electroencephalographic data: approximate entropy. 1It is pivotal to understand that approximate entropy and Shannon entropy are two entirely different measures: approximate entropy measures the predictability of future amplitude values of the electroencephalogram based on the knowledge of the generally one or two previous amplitude values. Shannon entropy measures the predictability of future amplitude values of the electroencephalogram based on the probability distribution of amplitude values (1,024 in this study) already observed in the signal. With increasing desflurane concentrations, the electroencephalographic signal becomes more regular. The knowledge of the previous values allows a high prediction of the next value. Therefore, approximate entropy decreases with increasing desflurane concentrations. With increasing desflurane concentrations, electroencephalographic amplitude increases. The probability density function broadens and flattens. The knowledge of the probability distribution, without having a look at the previous values (e.g. , one or two), allows a low prediction of the next value. Therefore, Shannon entropy increases with increasing desflurane concentrations, whereas approximate entropy decreases with increasing desflurane concentrations.

We conclude that the Shannon entropy of the electroencephalographic amplitude values uniformly increases with increasing desflurane concentrations. The Shannon entropy seems to be a simple and robust electroencephalographic measure of anesthetic drug effect.