Temporal linear mode complexity as a surrogate measure of the effect of remifentanil on the central nervous system in healthy volunteers

Abstract

AIMS

Previously, electroencephalographic approximate entropy (ApEn) effectively described both depression of central nervous system (CNS) activity and rebound during and after remifentanil infusion. ApEn is heavily dependent on the record length. Linear mode complexity, which is algorithmatically independent of the record length, was investigated to characterize the effect of remifentanil on the CNS using the combined effect and tolerance, feedback and sigmoid E_max models.

METHODS

The remifentanil blood concentrations and electroencephalographic data obtained in our previous study were used. With the recording of the electroencephalogram, remifentanil was infused at a rate of 1, 2, 3, 4, 5, 6, 7 or 8 µg kg⁻¹ min⁻¹ for 15–20 min. The areas below (AUC_effect) or above (AAC_rebound) the effect vs. time curve of temporal linear mode complexity (TLMC) and ApEn were calculated to quantitate the decrease of the CNS activity and rebound. The coefficients of variation (CV) of median baseline (E₀), maximal (E_max), and individual median E₀ minus E_maxvalues of TLMC were compared with those of ApEn. The concentration–TLMC relationship was characterized by population analysis using non-linear mixed effects modelling.

RESULTS

Median AUC_effectand AAC_reboundwere 1016 and 5.3 (TLMC), 787 and 4.5 (ApEn). The CVs of individual median E₀ minus E_max were 35.6, 32.5% (TLMC, ApEn). The combined effect and tolerance model demonstrated the lowest Akaike information criteria value and the highest positive predictive value of rebound in tolerance.

CONCLUSIONS

The combined effect and tolerance model effectively characterized the time course of TLMC as a surrogate measure of the effect of remifentanil on the CNS.

WHAT IS ALREADY KNOWN ABOUT THIS SUBJECT

• Remifentanil, an intravenous ultra short-acting opioid, depresses central nervous system activity with an increase in the delta band power, and causes beta activation after discontinuation, resulting in a rebound of the processed electroencephalographic parameters, including 95% spectral edge frequency, the canonical univariate parameter and electroencephalographic approximate entropy.

• A sigmoid Emax model, in which the highest predicted values of processed electroencephalographic parameters are restricted to the baseline value, cannot describe a rebound of these parameters.

• Electroencephalographic approximate entropy correlated well with the remifentanil blood concentration and demonstrated high baseline stability.

WHAT THIS STUDY ADDS

• A combined effect and tolerance model effectively characterized the time course of the remifentanil effect on the central nervous system, including the rebound which occurred during recovery from the remifentanil effect.

• Temporal linear mode complexity was comparable with approximate entropy as a univariate electroencephalographic descriptor of the effect of remifentanil on the central nervous system.

Introduction

Occasionally, the inhibition of a system may cause an excessive response above baseline upon cessation of inhibition, that is, a rebound effect [1]. Remifentanil, an intravenous ultra short-acting opioid, depresses central nervous system (CNS) activity with an increase in the delta band power, and causes beta activation after discontinuation, resulting in a rebound of processed electroencephalographic parameters, such as 95% spectral edge frequency, canonical univariate parameter and electroencephalographic approximate entropy (ApEn) [2, 3]. Conventionally, these EEG metrics have been used to characterize the time course of intravenous opioids [2, 4, 5]. In general, pharmacokinetic (one, two or three compartment mammillary models) and pharmacodynamic models (sigmoid E_max model), linked by an effect-compartment model, have been utilized to derive the blood–brain equilibration rate constant (k_e0). Modern computer-controlled drug delivery systems incorporate this rate constant to titrate intravenous hypnotics and opioids, which is called the target effect-site concentration-controlled infusion. However, there is no EEG metric showing rebound after termination of drug input which can be described by this PKPD modelling approach, because in a sigmoid E_max model the highest predicted values of the processed EEG parameters are restricted to a baseline value. Alternatively, a combined effect and tolerance model, with separate effect and tolerance compartments, may be a better modelling approach to this situation [6]. This model was used to describe the antinociceptive effect of morphine over time, including the development of tolerance and rebound.

We have demonstrated that depression of the CNS activity and rebound during and after remifentanil infusion was well characterized by ApEn and that its baseline stability was more stable [2, 3]. However, ApEn is heavily dependent on the record length, requiring time series of 100 or more points [7, 8]. Linear mode complexity (LMC), developed for the non-linear dynamical analysis of non-stationary biosignals, was used to evaluate the effect of an antiepileptic drug (valproate) on brain complexity and provide information about the spatiotemporal dynamics of the brain [9]. While spatial linear mode complexity (SLMC) quantifies the complexity of spatial linear modes in multichannel electroencephalograms (EEG), temporal linear mode complexity (TLMC), algorithmically identical to SLMC, measures the complexity of the temporal linear mode in a single channel EEG. TLMC focuses on the global structure of phase space reconstructed by a few trajectories of a short time series. In particular, TLMC is algorithmically independent of the record length (see Appendix 1).

The aims of this study were to characterize the time course of the effect of remifentanil on the CNS using a combined effect and tolerance model, and using TLMC as a surrogate measure of the effect of remifentanil.

Methods

The blood concentrations of remifentanil and raw EEG data obtained in our previous study were employed in the present study [2]. The patient characteristics are shown in Table 1.

Table 1.

Patient characteristics

	Young (≤40 years, n = 9)	Middle or elderly (>40 years, n = 19)
Weight (kg)	63.8 ± 11.3	59.1 ± 10.6
Age (years)	27.8 ± 5.7	57.7 ± 13.9
Height (cm)	167.9 ± 7.5	160.8 ± 9.6
Gender (M : F)	5:4	9:10
Rate of remifentanil Infusion (µg kg−1 min−1)	1–8*	3
Total amount infused (mg)	5.4 ± 2.8	2.7 ± 0.8
Infusion duration (min)	20.0*	15.3 ± 3.9

^*Volunteers were randomized to receive remifentanil for 15–20 min at a fixed rate of 1, 2, 3, 4, 5, 6, 7 or 8 µg kg⁻¹ min⁻¹. Data represent the mean ± SD.

Volunteer recruitment

After obtaining the approval of the institutional review board (Asan Medical Center, Seoul, Korea) and written informed consent, 28 volunteers aged 20–79 years who had no medical problems or abnormal laboratory test results were enrolled.

Drug administration and blood sampling

Drug administration and blood sampling are briefly described. Young volunteers (≤40 years of age, n = 9, M : F = 5:4) were randomly allocated to receive remifentanil by zero-order infusion of 1, 2, 3, 4, 5, 6, 7 or 8 µg kg⁻¹ min⁻¹ for 15–20 min. Middle-aged or elderly volunteers (>40 years of age, n = 19, M : F = 9:10) received remifentanil at 3 µg kg⁻¹ min⁻¹ until real-time 95% spectral edge frequencies showed no further changes after maximal suppression had been attained. Arterial blood samples (3 ml each) were taken at preset intervals as follows: (i) every 30 s during the first 5 min, every 1 min during the second 5 min and every 2 min during the third 10 min after the beginning of the remifentanil infusion; (ii) every 30 s during the first 5 min, every 1 min during the second 5 min and every 2 min during the third 10 min after discontinuation of the remifentanil infusion; and (iii) 25, 30, 35, 40, 45, 50, 60, 70, 80, 90, 100, 120, 150, 180, 210 and 240 min after discontinuation of the remifentanil infusion.

Electroencephalographic analysis

Each electroencephalogram was continuously recorded by QEEG-8 (Laxtha Inc., Daejeon, Korea) from frontoparietal montages (F3, F4, Cz, P3 and P4; the international 10–20 system) and digitized at 256 Hz. Data were stored on a hard disk for subsequent off-line calculation of TLMC and ApEn. Raw EEG signals were filtered between 0.5 Hz and 30 Hz and divided into epochs of 10 s, without overlap. For the calculation of ApEn, the length of the epoch (N) was 2560, the number of previous values (m) used to predict the subsequent values was 2, and the filtering level (r) was 10% of the SD of the amplitude values. No smoothing technique was applied to the calculation of TLMC and ApEn. Serious artefacts were excluded by checking the maximum amplitudes of each epoch. If an amplitude were greater than 200 µV, the epoch was excluded. The effectiveness of artefact rejection was manually confirmed. Artefact rejection and analysis of each EEG parameter was performed by a single experienced analyst.

The selection criteria for the EEG data points used in this study were as follows: (i) every 30 s during the first 5 min, every 1 min during the second 5 min and every 2 min during the third 10 min, after the beginning of the remifentanil infusion; (ii) every 30 s during the first 5 min, every 1 min during the second 5 min, and every 2 min during the third 10 min, after the termination of the remifentanil infusion; and (iii) thereafter, every 5–10 min until 170 min after the beginning of the remifentanil infusion [3].

Remifentanil exerts a more profound electroencephalographic effect on parietal montages than on frontal montages, as evidenced by the earlier observations that P3 and P4 data demonstrated consistently higher ratios of average maximal EEG effects compared with the inter-individual baseline variabilities in analyses employing ApEn, 95% spectral edge frequency and the canonical univariate parameter [2]. All of the volunteers in this study were right-handed. Hence, TLMC and ApEn derived from P4 montages were used, as suggested by inter-hemispheric asymmetries in EEG effects [10]. TLMC and ApEn data were normalized to individual baseline values, respectively.

Rebound in TLMC and ApEn

Rebound in TLMC and ApEn during recovery was defined as follows [3]. First, TLMC and ApEn values within 30 min after discontinuation of remifentanil infusion had to be >110% of individual baseline TLMC and ApEn, respectively. Second, any rebound of these had to last at least 30 s.

Areas above or below the effect vs. time curves of TLMC and ApEn during remifentanil infusion and during recovery from the remifentanil effect

The effect vs. time curves were drawn for both EEG metrics. Areas below the effect vs. time curves (AUC_effect), representing the decrease of the CNS activity during remifentanil infusion (pharmacological effect of remifentanil), were calculated. Areas above the effect vs. time curves (AAC_rebound), representing the rebound in each EEG parameter during recovery from a profound remifentanil effect (a physiological response) were also calculated (Figure 1). These areas were calculated by linear trapezoidal integration using WinNonlin Professional 5.2 (Pharsight Corporation, Mountain View, CA, USA).

Figure 1.

Schematic diagrams of areas above or below the effect vs. time curves of temporal linear mode complexity (TLMC, A) and approximate entropy (ApEn, B). This volunteer (ID14) received remifentanil for 15 min at the rate of 3 µg kg⁻¹ min⁻¹. Each effect was expressed as a value normalized to baseline (the dashed line). The solid lines are the rebound lines for TLMC and ApEn. Oblique striped regions indicate the areas below the effect vs. time curves of TLMC and ApEn (AUC_effect). The parallel striped regions represent areas above the effect vs. time curves of TLMC and ApEn (AAC_rebound)

Stability of TLMC and ApEn

The coefficients of variation (CVs) of the median baseline values (median E₀), median maximal values (median E_max) and differences between the individual median E₀ and median E_max values were calculated to determine the stability of TLMC and ApEn.

Pharmacodynamic modelling

Pharmacokinetic parameters from our previous study were employed in performing pharmacodynamic modelling [3]. As TLMC data normalized to individual baseline values were used, the baseline effects were set to 1.

The combined effect and tolerance model includes several submodels [11–14]. In a preliminary study, a model used by Ekblom et al. effectively described the data in this study [12]. One effect compartment and one tolerance compartment were linked to the central compartment to describe the delay of effect and rebound development as viewed from the central measureable compartment (Figure 2A). The rates of equilibration of remifentanil between blood and separate compartments are determined by the rate constants (k_e0 and k_t0) from the separate compartments. The effects in the effect and tolerance compartments were described by the sigmoid E_max model.

Figure 2.

Schematic illustration of combined effect and tolerance (A) and feedback (B) models. Combined effect and tolerance model: the effect (E) and tolerance (T) compartments are linked to a central compartment. The rate constants out of the effect and tolerance compartment are k_e0 and k_t0, respectively. Feedback model: H(t) is the inhibitory effect of remifentanil on the production of the response and (1 +M/M₅₀) is an amplifier of the loss of response. R and M denote the response and moderator, respectively. The k_in, k_out and k_tol are positive rate constants

The feedback and sigmoid E_max models were also fitted to the data as competing models (see Appendix). The feedback model describes a simplistic view of how the system behaves when opposing processes are operating simultaneously, and enables a prediction of the time course of drug action as well as the hypothetical modulator [1]. NONMEM® VII level 1 (ICON Development Solutions, Dublin, Ireland) was used to fit the combined effect and tolerance, feedback and sigmoid E_maxmodels to the data. A schematic illustration of combined effect and tolerance and feedback models is shown in Figure 2.

For a combined effect and tolerance model,

where the net effect is the calculated electroencephalographic effect of remifentanil and E_E is the electroencephalographic effect that is driven by the concentration of remifentanil in the effect compartment with a time delay determined by k_e0. The tolerance effect, E_T, is an opposing effect driven by the concentration of remifentanil in the tolerance compartment and the rate of the tolerance development is determined by k_t0. E_maxis the maximal possible effect. C_E and C_T are the remifentanil concentrations in the effect compartment and tolerance compartment, respectively. γ is the steepness of the concentration–response relation. C_E50 and C_T50 are the remifentanil concentrations in the effect and tolerance compartments associated with the half maximal effect and tolerance, respectively.

For a feedback model, the turnover of the response and moderator can be expressed as follows:

where H(t) is the remifentanil effect (inhibition) on the production of the response, and R and M denote the response and moderator, respectively. (1 +M/M₅₀) is an amplifier of the loss of response. E_max is the maximum fractional inhibition of k_in, IC₅₀ is the blood concentration associated with 50% maximal remifentanil effect, and C_b is the measured blood concentration of remifentanil. For simplicity, we assumed that M₅₀ was a unity [1]. The k_in, k_out, and k_tol are rate constants and γ is the steepness of the concentration-response relationship (see Appendix 2 for more detail).

For the sigmoid E_max model,

where effect is the electroencephalographic effect being measured (TLMC), E₀ is the baseline measurement when no drug is present, E_max is the maximum possible drug effect, C_E is the calculated effect site concentration of remifentanil, C_E50 is the effect site concentration associated with 50% maximal drug effect, and γ is the steepness of the concentration–response relation.

Model parameters were estimated using the ADVAN6 subroutine and the first-order conditional estimation (FOCE) with an interaction procedure. A diagonal matrix was estimated for the different distributions of η values. Inter-individual random variability was modeled using a log-normal model:

where P_i is the parameter value (k_out, IC₅₀, k_tol, E_max, γ, C_T50 or C_E50) of the ith volunteer, P_TV is the typical parameter population value, and η is a random variable with a mean of 0 and a variance of ω². Inter-individual random variability is reported as ω², the variance of η in the log domain. To evaluate the distribution of the η values relative to the normal distribution, the correlation of points in the normal probability plot was calculated and the Shapiro–Wilk test was employed to detect departures from normality [1, 15].

Residual random variability was modelled using an additive error model. Residual random variability is reported as σ², the variance in ε. Different models were evaluated using statistical and graphical methods.

The log likelihood ratio test was used to discriminate between hierarchical models [16]. A P-value of 0.05, representing a decrease in the objective function value of 3.84 points, was considered statistically significant (chi-squared distribution, one degree of freedom). The level of significance of a covariate was additionally assessed using a randomization test (fit4NM 3.1.8, Eun-Kyung Lee & Gyu-Jeong Noh, http://cran.r-project.org/web/packages/fit4NM/index.html, last accessed: 2010-10-28) [17]. Based on the null hypothesis that the covariate is not related to a pharmacodynamic parameter, data sets were generated by random permutation of the covariate empirical distribution. The final model was applied to these data sets. The distribution of the difference in objective function values between models with and without the covariate was obtained. The difference in objective function value at alpha = 0.05 and 1 degree of freedom was set as a critical value. If a change in objective function values between basic and covariate models fitted to an original data set was greater than the critical value, it was accepted as sufficient evidence to conclude that the covariate effect was statistically significant.

R software (version 2.10.1; R Foundation for Statistical Computing, Vienna, Austria) was used for graphical model diagnosis. Median weighted residuals and median absolute weighted residuals were calculated to examine the predictive qualities of the pharmacodynamic models in the study population.

The conditional weighted residuals of the pharmacodynamic models were calculated based on the first-order conditional estimation (FOCE) approximation, which reduces model misspecification by weighted residuals. The analysis is based on a first-order (FO) approximation [18].

Nonparametric bootstrap analysis was performed for internal model validation (fit4NM 3.1.8). Briefly, 2000 bootstrap replicates were generated by random sampling of the original data set. Final model parameter estimates were compared with median parameter values and the 2.5–97.5 percentiles of the nonparametric bootstrap replicates of the final model. Predictive checks were performed by simulating 2000 iterations and comparing simulated prediction intervals with the original data (fit4NM 3.1.8) [19].

Model selection criteria

In this study, model selection criteria for selecting one model out of the three models were determined as follows. First, the model selected had to exhibit the lowest Akaike information criteria. Second, it must have had the highest positive predictive value of rebound in TLMC.

where a true positive is rebound in observed and individually predicted TLMC. A false positive is rebound in only an individually predicted TLMC.

Statistical analysis

We compared AUC_effect and AAC_rebound by Mann–Whitney rank sum test using spss (version 13; SPSS Inc., Chicago, IL, USA). A P-value less than 0.05 was considered statistically significant. Prediction probability (P_K) was assessed as described by Smith and colleagues [20]. We calculated P_K values using Somers′ d cross-tabulation statistic on spss, which was then transformed from the −1 to 1 scale of Somers′ d to the 0 to 1 scale of P_K as P_K= 1 − (1 −∣Somers′ d∣) × 2⁻¹. TLMC was set as the dependent variable for the Somers′ d cross-tabulation statistic, and the responses calculated by A(4)/V4 in Appendix 3 for the feedback model and effect-site concentrations for the combined effect and tolerance as well as sigmoid E_max models were set as the independent variable, respectively. Prediction probabilities were calculated using the full measurement set. The SE of each P_K was calculated as (SE of Somers′ d) × 2⁻¹.

Results

Rebound in TLMC and ApEn

The number (%) of volunteers who exhibited beta activation and resultant rebound was 23 (82.1% of 28 volunteers), with 18 (64.3%) for TLMC and 17 (60.7%) for ApEn, respectively. All but one volunteer showed rebound in the observed ApEn as well as the observed TLMC. Neither age, gender nor total amount of remifentanil infused exerted any influence on the occurrence of rebound in TLMC (P > 0.05, Fisher's exact test for age and gender; P > 0.05, univariate logistic regression for the total amount of remifentanil infused).

Areas above or below the effect time curves of TLMC and ApEn during and after remifentanil infusion

The number of data points selected was 990 for TLMC and 986 for ApEn. The areas above or below the effect vs. time curves are shown in Figure 3. The AUC_effect and AAC_reboundof TLMC were slightly, but not significantly, higher than those of ApEn.

Figure 3.

Areas above or below the effect vs. time curves of temporal linear mode complexity (TLMC) and approximate entropy (ApEn). A) The areas below the effect vs. time curves of TLMC and ApEn (AUC_effect) represent a decrease in the central nervous system activity (a pharmacological effect of remifentanil). The central box covers the interquartile range with the median indicated by the line within the box. The whiskers extend to the 10^th percentile and 90^th percentile values. More extreme values (•) are plotted individually. The median (25%, 75%) values of AUC_effect were 1016 (836, 1319) for TLMC and 787 (631, 1201) for ApEn. B) The areas above the effect vs. time curves of TLMC and ApEn (AAC_rebound) represent the rebound of TLMC and ApEn during the recovery from the profound remifentanil effect. Median (25%, 75%) values of AAC_rebound were 5.3 (0.0, 74.2) for TLMC and 4.5 (0.0, 84.3) for ApEn. The solid lines indicate the median values

Stability of TLMC and ApEn

The median baseline (E₀), maximal (E_max) and individual median E₀ minus E_max values of TLMC and ApEn are shown in Table 2. ApEn was slightly more stable than TLMC.

Table 2.

Median baseline (E₀), maximal (E_max) and individual median E₀ minus E_maxvalues of temporal linear mode complexity (TLMC) and approximate entropy (ApEn)

	TLMC	ApEn
Median baseline value (E0)	0.85 ± 0.09 (10.08)	0.69 ± 0.04 (5.70)
Median maximal value (Emax)	0.51 ± 0.08 (15.91)	0.48 ± 0.05 (10.96)
Median E0 minus Emax*	0.34 ± 0.12 (35.64)	0.21 ± 0.06 (32.54)

Data represent the means ± SD (CV).

^*(individual median E₀ minus individual median E_max) value; CV, coefficient of variation = SD estimate⁻¹ × 100 (%).

Pharmacodynamic modelling

A total of 1520 TLMC data points were used to determine the pharmacodynamic characteristics. The performances of the three models are compared in Table 3 (see Appendix 3). Out of these models, the combined effect and tolerance model demonstrated the best performance and well described the rebound phenomenon. The equilibration half-lives for the delay of effect and tolerance development in a typical person were 1.1 min and 3.5 h, respectively (Table 4). The feedback and sigmoid E_max models were unable to capture rebound in TLMC. Predicted and observed TLMC values over time in two volunteers with the best and worst fits of the combined effect and tolerance model are shown in Figures 4 and 5, respectively. In the effect and tolerance model, E_T contributed to the development of rebound and hysteresis was collapsed. In the feedback model, the development of rebound is a complex function of the k_out, remifentanil blood concentration, response and moderator (k_in=k_out× 2, see Appendix).

Table 3.

Comparison between the combined effect and tolerance, feedback and sigmoid E_max models of remifentanil for temporal linear mode complexity (TLMC)

	Combined effect and tolerance model	Feedback model	Sigmoid Emax model
OFV	–6992	–5281	–5228
Number of parameters	13	10	10
AIC	–6966	–5261	–5208
Prediction probability	0.762	0.855	0.765
True/false positive (n)	7/0	0/0	0/0
True/false negative (n)	11/10	12/16	12/16
PPV (%)	100	0.0	0.0
NPV (%)	52.4	42.9	42.9

OFV, objective function value; AIC, Akaike Information Criterion = 2 ×N+ OFV, where N is the number of parameters. Prediction probability was calculated using Somers' d cross-tabulation statistics. True positive: rebound in observed and individually predicted TLMC; false positive: rebound only in individually predicted TLMC; true negative: no rebound in observed or individually predicted TLMC; false negative: rebound only in observed TLMC. PPV (positive predictive value) = true positive/(true positive + false positive) × 100 (%), NPV (negative predictive value) = true negative/(true negative + false negative) × 100 (%).

Table 4.

Population pharmacodynamic parameter estimates, inter-individual variability, and median parameter values (2.5–97.5%) of non-parametric bootstrap replicates of the combined effect and tolerance, feedback and sigmoid E_max models of remifentanil for temporal linear mode complexity (TLMC)

	Combined effect and tolerance model		Feedback model		Sigmoid Emax model
	Estimate (RSE, %CV)	(median, 2.5–97.5%)	Estimate (RSE, %CV)	(median, 2.5–97.5%)	Estimate (RSE, %CV)	(median, 2.5–97.5%)
Emax	0.48 (12.6, 27.3)	(0.46, 0.41–0.51)	0.52 (1.42, –)	(0.53, 0.46–0.58)	0.43 (3.76, 14.9)	(0.43, 0.39–0.46)
IC50or CE50 (ng ml−1)	CE50: 6.02 (43.7, 102)	(10.7, 5.1–19.1)	IC50=θ1– (AGE/44)θ2		CE50=θ1–θ2× (Age – 44)
θ1	–	–	15.0 (28.5, 82.1)	(16.7, 11.1–24.0)	18.6 (15.7, 63.6)	(18.5, 11.1–25.3)
θ2	–	–	3.81 (29.9, 82.1)	(4.26, 0.01–5.86)	0.39 (24.6, 63.6)	(0.38, 0.02–0.63)
CT50 (ng ml−1)	1.96 (30.56, 136)	(2.14, 1.01–15.4)	–	–	–	–
γ	3.72 (31.5, 74.0)	(4.09, 2.99–5.94)	3.35 (6.48, –)	(3.10, 2.43–4.43)	4.7 (10.6, 40.6)	(4.7, 3.69–6.05)
ke0(min−1)	0.62 (35.0, 119)	(0.69, 0.40–1.69)	–	–	1.03 (33.7, 140)	(1.00, 0.40–2.15)
kt0 (min−1)	0.0033 (45.2, 133)	(0.0027, 0.0005–0.019)	–	–	–	–
kin*(min−1)	–	–	0.74 (–, –)	–	–	–
ktol(min−1)	–	–	0.05 (2.33, 150)	(0.05, 0.01–0.11)	–	–
kout (min−1)	–	–	0.37 (29.2, 102)	(0.36, 0.22–0.59)	–	–
σ2	0.002	(0.002, 0.001–0.003)	0.01	(0.01, 0.004–0.018)	0.01	(0.01, 0.004–0.018)

TLMC data were normalized to individual baseline values. The baseline effects for all models were set to 1. The V₄ in the feedback model was also set to 1. Inter-individual and residual random variability were modelled using a log-normal model and an additive error model, respectively. Nonparametric bootstrap analysis was repeated 2000 times. Median parameter values (2.5–97.5%) of non-parametric bootstrap replicates are summarized. CV, coefficient of variation; σ², variance of residual random variability; RSE, relative standard error = SE estimate^–1× 100 (%).

^*The k_in was calculated from parameter estimate (k_out) with the following equation; k_in=k_out× 2.

Figure 4.

Predicted and observed TLMC values over time in a volunteer (ID14) with the best fit of the combined effect and tolerance model as determined by the lowest absolute value of the individual median of weighted residuals. A) combined effect and tolerance model, B) collapsed hysteresis loop in combined effect and tolerance model, C) feedback model, D) sigmoid E_max model. (A) , , . Weighted residual was calculated as (observed – predicted)/predicted, where population predictions were used as predicted. The absolute values of the individual median of weighted residuals were 6.9% (A), 8.2% (C) and 7.5% (D), respectively. (A) Observation (○); Population prediction of net effect (); Individual prediction of net effect (); Effect (——); Tolerance (- - - - -); (B) Blood (○); Effect-site (✗); (C) Observation (○); Population prediction of response (); Individual prediction of response (); Moderator (——); (D) Observation (○); Population prediction (); Individual prediction ()

Figure 5.

Predicted and observed TLMC values over time in a volunteer (ID8) with the worst fit of the combined effect and tolerance model as determined by the highest absolute value of the individual median of weighted residuals. A) combined effect and tolerance model, B) collapsed hysteresis loop in combined effect and tolerance model, C) feedback model, D) sigmoid E_max model. (A) , , . Weighted residual was calculated as (observed – predicted)/predicted, where population predictions were used as predicted. The absolute values of the individual median of weighted residuals were 56.7% (A), 93.5% (C) and 60.0% (D), respectively. (A) Observation (○); Population prediction of net effect (); Individual prediction of net effect (); Effect (——); Tolerance (- - - - -); (B) Blood (○); Effect-site (✗); (C) Observation (○); Population prediction of response (); Individual prediction of response (); Moderator (——); (D) Observation (○); Population prediction (); Individual prediction ()

Age was a significant covariate for IC₅₀ and C_E50 in the feedback model and sigmoid E_max model, respectively, resulting in an improvement in the objective function values (9.8 for the feedback model and 15.3 for the sigmoid E_max model, P < 0.01, degree of freedom = 1, respectively). The critical value of the randomization test was 3.4 for the feedback model and 2.3 for the sigmoid E_max model, respectively. However, age was not a significant covariate for C_E50 in the combined effect and tolerance model. With increasing age, the IC₅₀ and C_E50 of remifentanil decreased. These results coincide with our previous study [2].

The population pharmacodynamic parameter estimates, inter-individual variability and median parameter values (2.5–97.5%) of the nonparametric bootstrap replicates of the combined effect and tolerance, feedback and sigmoid E_max models are summarized in Table 4. The predictive checks of these models are shown in Figure 6.

Figure 6.

Predictive checks for combined effect and tolerance (A), feedback (B) and sigmoid E_max (C) models. The data distributed outside the 90% prediction intervals are 5.1% for the combined effect and tolerance model, 9.5% for the feedback model and 9.6% for sigmoid E_max model. The grey filled area represents the 90% prediction interval of each model. (+): observed TLMC

Discussion

A combined effect and tolerance model effectively described the rebound in TLMC during recovery from the remifentanil effect. TLMC was comparable with ApEn as a univariate EEG descriptor of the remifentanil effect.

Although drug effect and tolerance time courses have been described by several pharmacodynamics models [11, 21, 22], models capable of dealing with the rebound effect as well are limited. The effect and tolerance model and feedback model used in this study allow the inclusion of a rebound effect [1, 12]. In clinical settings, rebound phenomena are frequently observed. For example, rebound hypertension, hyperglycaemia and hyperalgesia are associated with clonidine, insulin and morphine discontinuation, respectively [12, 23–25]. A longer drug half-life or slower drug elimination lessens the rebound phenomenon [26]. Gabrielsson and colleagues showed a relationship between the time-gradient plot of the drug dose and rebound size. A rapid rise in the drug concentration tended to result in a large overshoot, while a rapid drop in drug dose caused a greater rebound [27]. Thus, the rapid metabolic clearance of remifentanil may result in rebound during recovery, as shown in our earlier study [3]. However, we observed a rebound of the processed EEG parameters after a high dose infusion (the total dose of remifentanil infused was approximately 2.9 mg) as well as after a small intravenous bolus dose (1 µg kg⁻¹) of remifentanil [2]. These findings suggest that the rebound phenomena of remifentanil may not be dose-dependent, as further supported by studies showing that differences in drug concentrations did not alter the rebound incidence or severity [26, 28].

In an earlier study, tolerance to antinociception developed after the long-term infusion of morphine (a period of several days) [12]. In this study, in which a short term infusion of remifentanil (less than 20 min) was applied, the blood-tolerance compartment equilibration half-life in a typical person was 3.5 h, and the duration of observation was up to 2.8 h after the beginning of remifentanil infusion, suggesting that tolerance was not likely to occur on average.

In general, β activation is associated with an increase in vigilance and awareness and plays an important role in the networks of inhibitory interneurons acting as pacemakers, and is gated by γ-aminobutyric acid type A (GABA_A) [29]. As can be seen in Figure 4, the combined effect and tolerance model successfully collapsed the hysteresis loop and accurately predicted rebound. Although describing rebound by the combined effect and tolerance model may not have a direct impact on the clinical decisions of anaesthesiologists, the results of this study may be used to characterize the concentration–response (i.e. the dynamic of the human brain) relationship of drugs characterized by the rebound phenomenon.

In a previous study, we used artificial neural network analysis to describe the rebound in approximate entropy following discontinuation of remifentanil [3]. However, ApEn lacks relative consistency and, if the ApEn of one dataset is higher than that of another, it should, but does not always remain higher for all of the conditions tested [7, 30]. Furthermore, artificial neural network analysis has several outstanding issues which need to be addressed. First, it provides little insight into the relative importance of the various input variables [31]. Second, it is impossible to ascertain whether predictors have positive or negative effects on output [31]. Third, the desirable value of most of the network design parameters can differ for each application and cannot be theoretically defined [18]. Fourth, the training of the network parameters tends to be problematic because of the difficulty to find the global optima in the complex and high-dimensional parameter spaces, which are needed for the overparameterization effects and error minimization in the training phase [18].

Conventional measures such as the correlation dimension and the Lyapunov exponent have all been used in non-linear dynamic analyses of biological time series [32]. For accurate analysis of biosignals using these measures, long record lengths are required and each time series should be stationary. However, these two requirements are contradictory to each other in terms of dynamic biosignals, because the preponderance of time series occurring in nature are non-stationary. Misleading results may be obtained by applying conventional algorithms to non-stationary time series. In particular, the EEG is not based on a stationary biosignal [33]. The EEG waveform may include a complex of regular sinusoidal waves, irregular spikes or spindles [33]. In most pathological conditions, such as an epileptic seizure, the EEG may show obvious non-stationarity. Linear mode complexity algorithmically decomposes multichannel time series data into uncorrelated modes by singular spectrum analyses [9]. Singular spectrum analysis was applied to detect EEG seizures and demonstrated a good performance [34]. Also, singular spectrum analysis extracts as much reliable information as possible from short and noisy time series without requiring any a priori knowledge about the underlying mechanism of the system [35]. In particular, TLMC was computationally more efficient than ApEn. The time needed to calculate TLMC offline from 61.4 MB of raw EEG data was 19 s, whereas it took 752 s to calculate the ApEn from the same data.

In summary, a combined effect and tolerance model better characterized the time course of the effect of remifentanil on the central nervous system, including rebound in TLMC. TLMC was comparable with ApEn as a surrogate measure to quantify the effect of remifentanil on the central nervous system.

Appendix 1: Temporal linear mode complexity and approximate entropy

1. Temporal linear mode complexity

The SLMC measure is proposed to quantify the complexity of spatial linear modes in multichannel EEGs, x_k(t_i) (k = 1,2, …, n; i = 1,2, …,N), and the TLMC is proposed to quantify the complexity of temporal linear modes in each channel of the EEG, where the index k represents the EEG of the k-th channel among the measured n channels, and the index i represents the i-th sampled value of the total sample number N. A normalized time series v_k(t_i) is formulated from raw EEG data in the form x_k(t_i) using the formula:

where is the mean of x_k(t_i).

For TLMC, phase space can be reconstructed employing a sequence of N=N_t−(n−1)d new vectors, , where d and n represent the delay number and embedding dimension. The covariance matrix, Ξ, is constructed from the time-averaged correlation between all element pairs in the n^th window.

The eigenvalues of the covariance matrix, σ_k(k = 1,2 …, n), are positive. Matrix diagonalization provides a set of principal axes for the n-dimensional space as eigenvectors with the corresponding eigenvalues σ_k. Linear mode complexity (LMC) for the temporal modes is defined by:

using the eigenvalues. LMC estimates how the eigenvalues are distributed. The normalization factor is determined from with σ₁=σ₂= … =σ_n=σ_t/n. That is, the LMC is normalized to a range between 1 and 0.

As the LMC increases, eigenvalues are more evenly distributed, which means that more modes are simultaneously participating in EEG dynamics. Note that the LMC is always 1 if the system is stochastic.

2. Approximate entropy (ApEn)

ApEn quantifies the predictability of subsequent amplitude values of the signal based on the knowledge of the previous amplitude values present in the time series [8].

(A1)

The normal procedure to calculate ApEn is the following. First, we start with the only data that we have, the discrete EEG time series, denoted by Equation A1, where T is the sample period and n, the number of samples of the EEG. The delays of the embedding vectors as usual are denoted by Equation A2.

(A2)

Here, L is the number of sampling intervals between successive components of an embedding vector, and j is the number of sampling intervals between the first components of multiple successive vectors. Then, the correlation sum is defined by ,

(A3)

where Θ is the Heaviside unit-step function, the norm ‘|| ||’ defines the distance between two vectors which is taken as the maximum distance between their components defined by Equation A4. In this study, the length of the epoch (N) was 2560, the number of previous values used for the prediction of the subsequent value (m) was 2, and the filtering level (r) was 10% of the SD of the amplitude values. The summation in this formula counts the number of pairs of vectors x(i) and x(j) for which ||x(i)−x(j)||₁ is less than the chosen distance r.

(A4)

The parameter given by Equation A5 is a simple normalization factor.

1 (A5)

ApEn is then defined by Equation A6, which can be considered as an approximation of the Kolmogorov-Sinai entropy [8].

(A6)

The EEG signals were processed with the following steps.

Step 1: A moving window of 10 s was applied to the five channels at the same time.

Step 2: The data inside the window were used for reconstruction of a phase space following Takens' delay theorem [36].

Step 3: The Grassberger & Procaccia algorithm was used for the point in the phase attractor to compute the correlation sum and then ApEn following the aforementioned algorithm [37].

Appendix 2: Derivation of a feedback model [1]

At steady state, and are equal to zero.

(A7)

(A8)

Rearranging Equation A2 and dividing by k_tol gives

(A9)

M_ss replaced by R_ss in Equation A1 gives

(A10)

When no drug is present in the body, by definition, R_ss=R₀ = 1 and H(t) = 1.

(A11)

To fit a feedback model to TLMC data, Equation A5 was used to calculate k_in from the estimate of k_out.

Appendix 3: Examples of the control streams used in pharmacodynamic modeling

1. Combined effect and tolerance model

$PROB RUN# 53033 (combined effect and tolerance model)

$input id oid time amt rate cmt dv mdv bwt ht age sex bsa lbm v1 k10 k12 k13 k21 k31

$DATA TLMC_tolerance_fit4NM.csv IGNORE = @

$SUBROUTINE ADVAN6 TOL = 6

$MODEL COMP (CENTRAL, DEFDOSE) COMP (PERIPH1) COMP (PERIPH2) COMP (EFFECT, DEFOBS) COMP (TOLER)

$PK

TH1 = THETA(1)

TH2 = THETA(2)

TH3 = THETA(3)

TH4 = THETA(4)

TH5 = THETA(5)

TH6 = THETA(6)

TEMAX = TH1

TCE50 = TH2

TGAM = TH3

TK41 = TH4

TCT50 = TH5

TK51 = TH6

MU_1 = LOG(TEMAX)

EMAX = EXP(MU_1 + ETA(1))

MU_2 = LOG(TCE50)

CE50 = EXP(MU_2 + ETA(2))

MU_3 = LOG(TGAM)

GAM = EXP(MU_3 + ETA(3))

MU_4 = LOG(TK41)

K41 = EXP(MU_4 + ETA(4))

MU_5 = LOG(TCT50)

CT50 = EXP(MU_5 + ETA(5))

MU_6 = LOG(TK51)

K51 = EXP(MU_6 + ETA(6))

V4 = 0.0001

K14 = V4 * K41/V1

V5 = 0.0001

K15 = V5 * K51/V1

$DES

DADT(1) = A(2)*K21 + A(3)*K31 + A(4)*K41 + A(5)*K51-A(1)*(K10 + K12 + K13 + K14 + K15)

DADT(2) = A(1)*K12-A(2)*K21

DADT(3) = A(1)*K13-A(3)*K31

DADT(4) = A(1)*K14-A(4)*K41

DADT(5) = A(1)*K15-A(5)*K51

$ERROR

CE = A(4)/V4

CT = A(5)/V5

IPRED = 1 – EMAX*CE**GAM/(CE**GAM + CE50**GAM) + EMAX*CT**GAM/(CT**GAM + CT50**GAM)

W = 1

IRES = DV – IPRED

IWRES = IRES/W

Y = IPRED + W * EPS(1)

$THETA; #6

(0.3, 0.5, 0.7); EMAX

(3, 5, 50); CE50

(1, 3, 10); GAM

(0, 0.5, 2); K41

(1, 2, 50); CT50

(0, 0.01, 1); K51

$OMEGA; #6

0.2; IIV_EMAX

0.2; IIV_CE50

0.2; IIV_GAM

0.2; IIV_K41

0.2; IIV_CT50

0.2; IIV_K51

$SIGMA; #1

3

$ESTIMATION NOTBT NOOBT NOSBT SIGL = 6 NSIG = 2 MAXEVAL = 9999 PRINT = 5 METHOD = 1 INTER MSFO = 53033.MSF NOABORT

$COVARIANCE PRINT = E MATRIX = S

2. Feedback model

$PROB RUN# 20112040 (feedback model)

$INPUT ID OID TIME AMT RATE CMT DV MDV BWT HT AGE SEX BSA LBM IV1 IK10 IK12 IK13 IK21 IK31

$DATA TLMC_RATIO_MICROGRAM_FIT4NM.CSV IGNORE = @

$SUBROUTINES ADVAN6 TOL = 6

$MODEL

COMP = (CENTRAL, DEFDOSE)

COMP = (PERI1)

COMP = (PERI2)

COMP = (RESP, DEFOBS)

COMP = (MODER)

$PK

TH1 = THETA(1)

TH2 = THETA(2)

TH3 = THETA(3)

TH4 = THETA(4)

TH5 = THETA(5)

TH6 = THETA(6)

TH7 = THETA(7)

TH8 = THETA(8)

V1 = IV1

K10 = IK10

K12 = IK12

K13 = IK13

K21 = IK21

K31 = IK31

TKOUT = TH1

TIC50 = TH2 – (AGE/44)**TH8

TNN = TH3

TKTOL = TH4

TEMAX = TH5

TRO = TH6

TV4 = TH7

MU_1 = LOG(TKOUT)

KOUT = EXP(MU_1 + ETA(1))

MU_2 = LOG(TIC50)

IC50 = EXP(MU_2 + ETA(2))

MU_3 = LOG(TNN)

NN = EXP(MU_3 + ETA(3))

MU_4 = LOG(TKTOL)

KTOL = EXP(MU_4 + ETA(4))

MU_5 = LOG(TEMAX)

EMAX = EXP(MU_5 + ETA(5))

MU_6 = LOG(TRO)

RO = EXP(MU_6 + ETA(6))

MU_7 = LOG(TV4)

V4 = EXP(MU_7 + ETA(7))

KIN = KOUT*(1 + RO)*RO

S1 = V1

S4 = V4

$DES

DRUG = 1 – EMAX*(A(1)/V1)**NN/(IC50**NN + (A(1)/V1)**NN)

DADT(1) =–K10*A(1) – K12*A(1) – K13*A(1) + K21*A(2) + K31*A(3)

DADT(2) = K12*A(1) – K21*A(2)

DADT(3) = K13*A(1) – K31*A(3)

DADT(4) = KIN*DRUG – KOUT*A(4)*(1 + A(5))

DADT(5) = KTOL*(A(4) – A(5))

$ERROR

IPRED = F

W = 1

IRES = DV – IPRED

IWRES = IRES/W

Y = IPRED + W * EPS(1)

$THETA;#6

(0, 0.3); KOUT_TH1

(0, 50); IC50_TH2

(0, 1.2); NN_TH3

(0, 0.06); KTOL_TH4

(0, 0.4, 0.6); EMAX_TH5

1 FIX; RO_TH6

1 FIX; V4_TH7

(0, 0.01); AGE_TH8

$OMEGA;#3

0.2; IIV_KOUT

0.2; IIV_IC50

0 FIX; IIV_NN

0.2; IIV_KTOL

0 FIX; IIV_EMAX

0 FIX; IIV_RO

0 FIX;IIV_V4

$SIGMA;#1

2

$ESTIMATION NOTBT NOOBT NOSBT SIGL = 6 NSIG = 2 MAXEVAL = 9999 PRINT = 5 METHOD = 1 INTER MSFO = 20112040.MSF NOABORT

$COVARIANCE PRINT = E MATRIX = S

3. Sigmoid E_max model

$PROB RUN# 20113 (Sigmoid Emax model)

$INPUT ID OID TIME AMT RATE DV MDV BWT HT AGE SEX BSA LBM V1 K10 K12 K13 K21 K31

$DATA TLMC_SigmoidEmax_fit4NM.csv IGNORE = @

$SUBROUTINE ADVAN6 TOL = 6

$MODEL COMP (CENTRAL,DEFDOSE) COMP (PERIPH1) COMP (PERIPH2) COMP (EFFECT, DEFOBS)

$PK

TH1 = THETA(1)

TH2 = THETA(2)

TH3 = THETA(3)

TH4 = THETA(4)

TH5 = THETA(5)

TEMAX = TH1

TCE50 = TH2 – TH5 *(AGE – 44)

TGAM = TH3

TK41 = TH4

MU_1 = LOG(TEMAX)

EMAX = EXP(MU_1 + ETA(1))

MU_2 = LOG(TCE50)

CE50 = EXP(MU_2 + ETA(2))

MU_3 = LOG(TGAM)

GAM = EXP(MU_3 + ETA(3))

MU_4 = LOG(TK41)

K41 = EXP(MU_4 + ETA(4))

V4 = 0.0001

K14 = V4 * K41/V1

$DES

DADT(1) = A(2) * K21 + A(3) * K31-A(1) * (K10 + K12 + K13)

DADT(2) = A(1) * K12-A(2) * K21

DADT(3) = A(1) * K13-A(3) * K31

DADT(4) = A(1) * K14-A(4) * K41

$ERROR

CE = A(4)/V4

IPRED = 1 – EMAX * (CE**GAM)/(CE50**GAM + CE**GAM)

W = 1

IRES = DV – IPRED

IWRES = IRES/W

Y = IPRED + W * EPS(1)

$THETA; #5

(0, 0.5); TH1_EMAX

(0, 5, 100); TH2_CE50

(1, 2, 10); TH3_GAM

(0, 0.2); TH4_K41

(0, 0.01); TH5_CE50_AGE

$OMEGA; #4

0.2; IIV_EMAX

0.2; IIV_CE50

0.2; IIV_GAM

0.2; IIV_K41

$SIGMA; #1

0.1

$ESTIMATION NOTBT NOOBT NOSBT SIGL = 6 NSIG = 2 MAXEVAL = 9999 PRINT = 5 METHOD = 1 INTER MSFO = 20113.MSF

$COVARIANCE PRINT = E

Competing Interests

There are no competing interests to declare.