Introduction
The time course of action of a neuromuscular blocking agent (NMBA) is governed by its pharmacokinetic and pharmacodynamic properties [^{1-4} ]. It is well recognized that the duration of neuromuscular block and the rate of recovery of the muscle response are dependent on the rate of decay of the plasma concentration. Also, it has been demonstrated that the time to peak effect of a submaximal blocking dose (T _{peak} ) is governed by the rate of decay of the plasma concentration profile [^{5} ]. The predominant role of pharmacokinetics on the time course of action of NMBAs can be explained by pharmacokinetic–pharmacodynamic (PK-PD) modelling [^{6-8} ]. The transport from plasma to the receptor site (also denoted effect compartment or biophase; for NMBAs, these terms refer to the neuromuscular junction or synaptic cleft) is also an important factor determining T _{peak} . The role of this transport process can be inferred from the observed relationship between the rate constant characterizing this process (usually denoted k _{e0} ) and time to peak effect [^{9} ]. Pharmacokinetics also affects the dose needed to achieve a certain peak effect of neuromuscular block (NMB_{peak} ), e.g. ED_{90} , and thus affects the ‘apparent potency' of NMBAs [^{2,5,8} ].

It is less obvious whether or not the pharmacodynamic properties of a NMBA, i.e. the concentration–effect relationship (characterized by EC_{50} , or ‘intrinsic potency') play a significant role in the time course of action of NMBAs. A possible mechanism for an influence of receptor affinity is a process called ‘buffered diffusion' [^{10-12} ]. The presence of a very high concentration of acetylcholine receptors (AChR) in the synaptic cleft causes repeated binding and release of NMBA molecules to these receptors, thus slowing down the rate of the diffusion of NMBA between the synaptic cleft and plasma. In vitro iontophoretic studies at the postsynaptic membrane of frog skeletal muscle fibres demonstrated the involvement of buffered diffusion in the kinetics of the action of NMBAs [^{13,14} ]. However, these findings do not prove that buffered diffusion plays a role in the time course of action of NMBAs when administered in vivo .

The aim of the present study was to determine whether the essential characteristics of buffered diffusion are recognizable in the time course of action of NMBAs, in particular in the time to peak effect. If buffered diffusion does play a significant role, PK-PD modelling predicts that T _{peak} decreases with increasing dose, and that this effect of dose is more pronounced for potent NMBAs and less pronounced or absent for less potent NMBAs [^{7,8} ]. This can be explained as follows. After injection of the NMBA the drug diffuses from the plasma into the synaptic cleft. Part of the drug molecules in the synaptic cleft binds to the postsynaptic AChR. The driving force for diffusion of NMBA molecules between plasma and effect compartment is the unbound NMBA concentration gradient between plasma and effect compartment. The peak effect is reached if the concentrations of unbound NMBA in plasma and effect compartment are equal. Binding of NMBA molecules to AChR results in a lower unbound NMBA concentration in the effect compartment, and thus increases the time point at which the concentration in the effect compartment equals the concentration in plasma, i.e. increasing T _{peak} . If a lower dose is given, i.e. if NMB_{peak} is lower, this increase in T _{peak} will be more pronounced, because the number of free binding sites on AChR for NMBA is higher. Similarly, T _{peak} will decrease with increasing dose. This effect of dose on T _{peak} will be more pronounced for a potent NMBA than for a less potent NMBA. For a less potent NMBA, the binding of NMBA molecules to AChR hardly affects the unbound concentration, since the effective concentration needed to reach the same degree of NMB is higher, whereas, according to receptor theory, the number of drug molecules bound to AChR is likely to be independent of the potency of the NMBA for the same degree of block. On the other hand, for a potent NMBA, the number of drug molecules bound to AChR will be high compared to the number of unbound drug molecules, and the effect of dose on T _{peak} will be more pronounced.

To test the hypothesis that buffered diffusion plays a role in the time to peak effect, we investigated the influence of dose on T _{peak} of a submaximal blocking dose in the pig for four NMBAs which vary in potency. In addition, we performed PK-PD simulations to support the expected relationships between potency, dose, NMB_{peak} and T _{peak} .

Methods
Following approval of the Animal Care Committee of the University of Groningen, 21 male pigs (body weight 20–28 kg) were studied. After a fasting period of 16 h with free access to water, the pigs were anaesthetized with 500 mg ketamine followed by 15 mg midazolam, given intramuscularly. The animal was weighed, and ear veins were cannulated to allow infusions of pentobarbital (2 mg kg^{−1} h^{−1} ) and fentanyl (2–4 μg kg^{−1} h^{−1} ), adjusted if needed to maintain an adequate depth of anaesthesia. The trachea was intubated and the lungs were artificially ventilated with an air/oxygen mixture, using a Cameco UV 705 (Cameco, Sweden) respirator, with a frequency of 19 breaths per minute, and 20 cm water pressure. The end tidal carbon dioxide level was maintained between 4 and 5 kPa (Godart Capnograph Mark II; E. Jaeger, Wuerzburg, Germany). Heart rate and blood pressure were measured continuously (Pressure Transducer, HP 78342A, Hewlett Packard, Boeblingen, Germany). Rectal temperature was measured continuously and maintained at 38°C with the use of a heating blanket. The axillary vein was cannulated for infusion of glucose 2.5% in saline 0.45%, and for administration of the NMBAs. The axillary artery was catheterized for continuous blood pressure measurements and for sampling of blood. The left peroneal nerve was exposed and attached to two silver stimulation electrodes. To prevent repetitive backfiring, the nerve was ligated proximal to the electrodes. The overlying skin was closed and the nerve was stimulated supramaximally with square wave stimuli of 0.2 ms at a frequency of 0.1 Hz (Grass S88; Grass Instruments, Quincy, MA, USA). The response of the tibialis anterior muscle was registered mechanomyographically using a force transducer (LB 8000 25N, Maryland Instruments Corp., Baltimore, MD, USA) connected to a muscle relaxation monitor MK II and to a recorder (MT 9000, Astro-Med, West Warwick, RI, USA). Preload was measured continuously and kept constant at approximately 75 g. The muscle response was allowed to stabilize before administration of a NMBA. After the experiment the animals were terminated by an overdose of pentobarbital.

The initial dose in the first animal of each NMBA group (rocuronium, vecuronium, pipecuronium or d-tubocurarine) was equal to the ED_{90} (dose needed to achieve a peak effect of 90% block) estimated from earlier studies, i.e. 600 μg kg^{−1} for rocuronium bromide (molecular weight (MW) 609.69), 100 μg kg^{−1} for vecuronium bromide (MW 637.74), 77 μg kg^{−1} for pipecuronium (MW 762.71) and 30 μg kg^{−1} for d-tubocurarine (MW 681.85). From NMB_{peak} obtained after this dose, the dose needed to achieve 20%, 40%, 60%, 75% and 90% block were estimated, assuming a relationship between NMB_{peak} and dose as described under data analysis, using a Hill coefficient of 4. This initial dose was adjusted in the next animals treated with the same NMBA if NMB_{peak} of this initial dose was not close to 90%.

Subsequently, five different doses of the same NMBA were administered to each animal, i.e. the estimated doses aiming at 20%, 40%, 60%, 75% or 90% block, in a random order. Each dose was given 45 min after complete recovery of the twitch response had been obtained. Each NMBA was studied in 5 or 6 animals.

The time to peak effect (T _{peak} ), defined as the time interval between the time of injection and the peak effect, was recorded after each dose.

Data analysis and statistics
The relationship between dose and NMB_{peak} was analysed by an iterative two-stage Bayesian population fitting procedure [^{15,16} ], using the program MultiFit (developed by J. H. Proost) for each NMBA separately, according to the following equation:

where ED_{50} is the calculated dose producing 50% NMB_{peak} and γ is the exponent or Hill coefficient. Data were unweighted, i.e. it was assumed that the difference between the measured and calculated NMB_{peak} was independent of the degree of block.

The relationship between NMB_{peak} and T _{peak} was analysed by standard multiple linear regression analysis (Excel 1997, Microsoft Corp, Redmond, WA, USA), with NMB_{peak} , sequential dose number (between 1 and 5), and the individual animals as regressors, for each NMBA separately, according to the following equation:

where T _{peak(50)} is the intercept, i.e. the time to peak effect for 50% NMB_{peak} (individual value for each animal), slope_{NMB} is the slope of the relationship between T _{peak} and NMB_{peak} , and slope_{N} is the slope of the relationship between T _{peak} and sequential dose number. The values 50 and 3 refer to the midrange value of the regressors NMB_{peak} and N, respectively. The mean value and SD of T _{peak(50)} was obtained from the individual values.

The relationship between dose and T _{peak} was analysed by standard multiple linear regression analysis, with the logarithm of dose, sequential dose number, and the individual animals as regressors, for each NMBA separately, according to the following equation:

where slope_{dose} is the slope of the relationship between T _{peak} and the logarithm of the dose (in μmol kg^{−1} ). The values log_{10} (ED_{50} ) and 3 refer to the midrange values of the regressors log_{10} (dose) and N, respectively.

The first dose administered to each animal was not taken into account in the data analysis, unless stated otherwise. Differences between NMBAs were tested using the U -test; a P value <0.05 was considered significant. Data are presented as mean ± SD.

PK-PD simulations
To investigate the expected effect of buffering on T _{peak} , computer simulations were performed using the PK-PD model and data for rocuronium in pigs described by De Haes and colleagues [^{17,18} ]. In this model, buffering is taken into account by including the binding of NMBA to AChR in the effect compartment, similar to earlier models [^{7,8} ]. In these earlier models, the neuromuscular blocking effect was related to the unbound NMBA concentration in the effect compartment, which is questionable from a mechanistic point of view. In the Unbound Receptor model (URM) of De Haes and colleagues [^{17,18} ], the degree of neuromuscular block is related to the concentration of unbound AChR available for acetylcholine to bind, resulting in muscle contraction. The URM and the model parameters for rocuronium in pigs are described in the Appendix .

To investigate the influence of the dose and NMB_{peak} on T _{peak} , the following simulations were performed. First, the dose producing x % NMB_{peak} (ED_{x} ) was calculated for x varying between 1% and 99% by an iterative procedure, and T _{peak} was determined for each ED_{x} . Then, ED_{50} and γ (Eq. (1) ), T _{peak(50)} , slope_{NMB} and slope_{N} (Eq. (2) ), and slope_{dose} and slope_{N} (Eq. (3) ) were estimated from the calculated values for x = 20%, 40%, 60%, 75% and 90%, similar to the experimental data as describe above. The simulations were performed with a study design similar to the experiments in pigs, i.e. an ED_{90} dose, followed by five consecutive doses ED_{20} , ED_{40} , ED_{60} , ED_{75} and ED_{90} in a random order. This procedure was performed five times, simulating five animal studies. Similar to the experiments in pigs, the results following the first dose were not used in the data analysis.

In addition to rocuronium, two additional hypothetical NMBAs were investigated, i.e. a more potent NMBA simulated by a tenfold decrease of the dissociation constant K _{d} (0.04 μM), and a less potent NMBA simulated by a tenfold increase of K _{d} (4 μM).

Results
Totally, 21 experiments were performed (Table 1 ). In two experiments (one with rocuronium and one with vecuronium), the last scheduled dose could not be administered for technical reasons. In three cases, the first dose resulted in 100% block, but NMB_{peak} was less than 100% in all the subsequent doses used for the data analysis.

Table 1: Number of animals, number of doses, potency (ED50), Hill coefficient (γ) and time to peak effect of 50% block (Tpeak(50)).

The relationship between dose and NMB_{peak} was analysed using Eq. (1) , and the results have been summarized in Table 1 . The four NMBAs showed a 20-fold range in potencies, expressed in their molar ED_{50} values. The steepness of the dose–response relationship, expressed in the Hill coefficient γ, was not significantly different between the four compounds.

The relationship between NMB_{peak} and T _{peak} is depicted in Figure 1 for each of the four NMBAs, and was analysed using Eq. (2) . The results are summarized in Tables 1 (T _{peak(50)} ) and 2 . The time to peak effect at 50% block (T _{peak(50)} ) for d-tubocurarine was significantly longer than for the other NMBAs, but the differences between rocuronium, vecuronium and pipecuronium did not reach statistical significance. However, statistical analysis of the measured T _{peak} pooled for each NMBA revealed significant differences in T _{peak} between any of the four NMBAs (rocuronium < vecuronium < pipecuronium < d-tubocurarine). The mean values of the measured T _{peak} for all doses were close to the T _{peak(50)} values listed in Table 1 .

Figure 1.:
Relationship between NMBpeak and Tpeak for rocuronium, vecuronium, pipecuronium and d-tubocurarine. Each line represents the data of an individual animal (n = 6 for vecuronium and n = 5 for rocuronium, pipecuronium and d-tubocurarine).

For rocuronium and pipecuronium, a significant positive correlation between NMB_{peak} and T _{peak} was observed; for vecuronium and d-tubocurarine the slope of relationship between T _{peak} and NMB_{peak} was close to zero, and no significant correlation was found (Table 2 ). Similar results were obtained by plotting T _{peak} against the administered dose using Eq. (3) (Table 3 ).

Table 2: Slope of the relationship between NMBpeak and time to peak effect (slopeNMB), slope of the relationship between the sequential dose number and time to peak effect (slopeN), and the P values for the hypothesis that the slopes are zero.

Table 3: Slope of the relationship between the logarithm of the dose and time to peak effect (slopedose), slope of the relationship between the sequential dose number and time to peak effect (slopeN), and the P values for the hypothesis that the slopes are zero.

For vecuronium, pipecuronium and d-tubocurarine, a significant positive correlation between the sequential dose number and T _{peak} was observed, both for the analysis of NMB_{peak} (Table 2 ) and dose (Table 3 ).

The relationship between ED_{50} and T _{peak(50)} for the four NMBAs is shown in Figure 2 .

Figure 2.:
Relationship between ED50 and Tpeak(50) for rocuronium, vecuronium, pipecuronium and d-tubocurarine. The symbol represents the data of an individual animal (n = 6 for vecuronium and n = 5 for rocuronium, pipecuronium and d-tubocurarine).

PK-PD simulations
In a first series of simulations, T _{peak} of rocuronium was calculated for different doses producing a peak effect of 1–99%, and the resulting relationship between NMB_{peak} and T _{peak} is shown in Figure 3 .

Figure 3.:
Relationship between NMBpeak and Tpeak predicted by PK-PD simulation for three NMBAs with different potencies, i.e. Kd = 0.04, 0.4 (rocuronium) and 4 μM, respectively.

The predicted values of ED_{50} and γ were 0.385 μmol kg^{−1} and 2.81, respectively. These values agree well with the values obtained from the experimental data (Table 1 ). The predicted time to peak effect (T _{peak(50)} ) was 91 s, and was markedly higher than the observed value of 53 s. The predicted values for the slopes were slope_{NMB} = −0.21 s, slope_{N} = 0.06 s (Eq. (2) ), slope_{dose} = −28 s and slope_{N} = 0.07 s (Eq. (3) ).

In addition to the simulations for rocuronium, this relationship was investigated in two hypothetical NMBAs, i.e. a more potent NMBA simulated by a tenfold decrease of the dissociation constant K _{d} (0.04 μM), and a less potent NMBA simulated by a tenfold increase of K _{d} (4 μM) compared with rocuronium (K _{d} 0.4 μM). For the more potent NMBA, T _{peak} is markedly longer than for rocuronium, and is more affected by the degree of NMB (Fig. 3 ). For the less potent NMBA, T _{peak} is shorter than for rocuronium, and is hardly dependent on the degree of NMB.

Discussion
The experimental results demonstrate that the time to peak effect of a submaximal blocking dose of NMBAs is not inversely related to dose, independent of the potency of the NMBA (Fig. 1 ). The PK-PD simulations of buffered diffusion predict a decrease of T _{peak} with dose, and an increase of T _{peak} with increasing potency (Fig. 3 ). These findings suggest that, even for the most potent NMBA in this study, buffered diffusion does not play a dominant role in the time to peak effect. Therefore it is unlikely that the observed inverse relationship between potency and T _{peak} of NMBAs is due to buffered diffusion.

The PK-PD simulations were performed using a mechanism-based model taking into account buffered diffusion. This URM was developed to explain the changes in potency and time course of action of rocuronium in case of a decreased number of AChR, as observed in myasthenic patients and pigs [^{17,18} ]. This model is also suited to predict the changes in time course of action due to differences in potency, similar to the model proposed earlier by Donati and Meistelman [^{7} ] and Proost and colleagues [^{8} ]. Although the PK-PD model parameters were obtained from a different study, the predicted values of ED_{50} and γ were in good agreement with the values obtained from the experimental data. However, the predicted T _{peak} (91 s) was markedly higher than the observed value (53 s). The anaesthetic and surgical procedures in both studies were comparable. The PK-PD model parameters were obtained after an infusion of rocuronium over 4–8 min, whereas in the present experiments rocuronium was administered as a bolus dose. Shortly after a bolus administration, mixing within the vascular system is the predominant distribution process, and compartmental models do not account for this process and assume an instantaneous mixing [^{19,20} ]. For this reason, short-lasting infusions are preferred in PK-PD studies [^{21} ]. The observed difference between the simulated and experimental values of T _{peak} may be related to this phenomenon.

The relationship between dose and NMB_{peak} (Eq. (1) ) was analysed by an iterative two-stage Bayesian population fitting procedure for each NMBA separately [^{15,16} ]. The use of a population analysis is nowadays standard practice in PK and PK-PD analysis, because it allows a robust and reliable estimation of the mean and SD of the model parameters, as well as individual parameter values of ED_{50} and Hill coefficient (γ). For the relationships between NMB_{peak} and T _{peak} (Eq. (2) ), and dose and T _{peak} (Eq. (3) ), we used standard multiple regression analysis, allowing the estimation of a single value for slope_{NMB} and slope_{N} , and slope_{dose} and slope_{N} , respectively.

An initial dose of the estimated ED_{90} was administered to allow a better prediction of the individual doses needed to achieve the aimed levels of block. To avoid bias due to a possible difference between the first dose (aiming at 90% block) and subsequent doses given at 45 min after complete recovery of the twitch response (aiming at 20%, 40%, 60%, 75% or 90% block, in a random order) with respect to the degree of block or T _{peak} , we excluded the data of the first dose from the analysis, ensuring similar starting conditions for each measurement, i.e. 45 min after complete recovery of the twitch response from a previous dose.

A possible difference between the first dose and subsequent doses was investigated for all NMBAs simultaneously by the following exploratory analysis. The mean ± SD T _{peak} of the first dose was 79 ± 26 s (n = 18; in three animals T _{peak} could not be measured because complete block was reached). T _{peak} of the second dose in the same animals was significantly shorter: 70 ± 23 s (P = 0.012). This difference may be related to the observed dose effect on T _{peak} , because the first dose was close to ED_{90} , whereas the average of the second dose was close to the ED_{50} . The time to the highest dose (dose comparable to the first dose) was 86 ± 25 s (P = 0.14, compared to first dose), an increase corresponding to the observed increase of T _{peak} with subsequent doses. These results indicate that T _{peak} after the first dose is not markedly different from that of subsequent doses. In addition, we compared the observed NMB_{peak} after the first dose with the calculated NMB_{peak} from Eq. (1) , using the individual values of ED_{50} and γ obtained from the analysis of the data of the subsequent doses, as reported in Table 1 . The mean ± SD observed NMB_{peak} after the first dose was 90 ± 9%; the calculated NMB_{peak} (mean ± SD 86 ± 9%) was slightly, but significantly lower (P < 0.001). This indicates that the neuromuscular junction may be less sensitive to NMBA at the first dose than at subsequent doses. If residual receptor occupancy plays a role, one would expect that the neuromuscular junction becomes more sensitive after an initial dose. This observation suggests that the interval between the experiments (45 min after complete recovery) was sufficiently long to ensure that residual receptor occupancy does not unduly influence the results of our experiments.

Interestingly, we found a positive correlation between NMB_{peak} , or dose, and T _{peak} for rocuronium and pipecuronium, whereas no correlation was found for vecuronium and d-tubocurarine. This discrepancy does not seem to be related to potency. The ‘buffering hypothesis' predicts a negative correlation between NMB_{peak} , or dose, and T _{peak} for potent NMBAs, and a less negative or no correlation for less potent NMBAs. Therefore it can be concluded that buffered diffusion does not play a role in the time to peak effect of NMBAs in the studied potency range, which is comparable to the potency range of NMBAs in current or former clinical use, i.e. ED_{50} ranging from 0.02 μmol kg^{−1} for doxacurium to 2 μmol kg^{−1} for rapacuronium. In contrast, it has been published that the onset time of atracurium and vecuronium [^{22} ], and the highly potent doxacurium [^{23} ] decreases with dose, but these authors did not discriminate between submaximal peak effect and complete block. The latter situation is more likely with higher doses and results in a shorter onset time compared to T _{peak} after a submaximal blocking dose; however, this does not reflect the time of maximal concentration in the biophase, or maximal receptor occupancy, as in the case of a submaximal peak effect. Therefore these data are insufficient to demonstrate that T _{peak} decreases with dose.

If buffered diffusion can be excluded as a dominant factor, differences in pharmacokinetics are likely to be responsible for the observed inverse relationship between potency and T _{peak} of a submaximal blocking dose [^{24-26} ], and confirmed in our study (Fig. 2 ) [^{5,7-9} ]. This finding is in agreement with earlier studies in the isolated perfused tibialis model in the rat [^{27} ]. We found that the rates of recovery of rocuronium and pancuronium were similar, in spite of a fourfold difference in potency, i.e. in the concentration needed to reach a stable 90% block. It should be noted that differences in k _{e0} are also responsible for differences in T _{peak} [^{9} ]. An increase in k _{e0} results in a decrease of T _{peak} , and a simultaneous decrease of ED_{90} ; therefore differences in k _{e0} do not explain the observed inverse relationship between potency and T _{peak} [^{8} ].

We do not have an explanation for the apparent discrepancy between the positive values for slope_{NMB} of rocuronium and pipecuronium, and the values close to zero for vecuronium and d-tubocurarine. Bias due to a temporal effect was excluded by randomization of the sequence of the doses. Possibly rocuronium and pipecuronium affect the blood perfusion of the muscle by a local pharmacodynamic effect. It was demonstrated that several less potent NMBAs possess calcium channel blocking effects in vascular smooth muscle [^{28} ], possibly resulting in local vasodilatation, and thus a faster onset of block. Therefore this mechanism does not explain the observed increase of T _{peak} with dose. However, a possible role of blood perfusion, e.g. due to a dose-dependent local vasoconstriction, is still likely, as this is the assumed mechanism for the decrease of onset time of rocuronium by ephedrine, and the increase by esmolol [^{29} ]. In the latter study, ‘onset time' was the time to reach complete block (disappearance of all four twitches of a train-of-four; data on the influence of these drugs on T _{peak} of a submaximal blocking dose are not available.

The observed increase of T _{peak} with dose of rocuronium and pipecuronium would imply that the value of k _{e0} , the rate constant of transport between plasma and biophase, obtained from a PK-PD analysis, would decrease with dose. A decrease of k _{e0} (and increase of EC_{50} ) with increasing dose was found in man for cisatracurium, a NMBAs of intermediate potency [^{30} ], although this has been disputed [^{31} ]. We are not aware of other studies demonstrating any effect of dose on the PK-PD model parameters.

For vecuronium, pipecuronium and d-tubocurarine, a significant positive correlation between the sequential dose number and T _{peak} was observed, both for the analysis of NMB_{peak} (Table 2 ) and dose (Table 3 ). This implies that T _{peak} increased for subsequent doses; for pipecuronium and d-tubocurarine this increases was about 5 s for each dose. This effect of dose number seems to be uncorrelated to the effect of NMB_{peak} and dose on T _{peak} , since these effects do not occur to the same extent in the four NMBAs. A possible explanation of the observed increase of T _{peak} may be found in a decreasing muscle perfusion during the experimental period, e.g. due to decrease of cardiac output or local vasoconstriction.

In conclusion, time to peak effect of a submaximal blocking dose of NMBAs is not inversely related to dose. This finding suggests that buffered diffusion does not play a dominant role in the time to peak effect of NMBAs in the clinical range. Therefore it is unlikely that the observed inverse relationship between potency and time to peak effect of NMBAs is due to buffered diffusion.

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Appendix

Unbound receptor model
Usually, PK-PD models assume that the neuromuscular blocking effect is related to the (unbound) concentration of the drug in the effect compartment, i.e. the neuromuscular junction [^{6-8} ]. However, this assumption is flawed from a mechanistic point of view, because the effect is the result of binding of the NMBA to the receptor and not just of the presence of free drug in the biophase. Alternatively, one might relate the effect to the concentration of bound NMBA. This hardly affects the model in a qualitative way, since the unbound concentration and the bound concentration are strongly correlated.

We developed a mechanism-based PK-PD model to explain the changes in potency and time course of action of rocuronium in case of a decreased number of AChR, as observed in myasthenic patients and pigs [^{17,18} ]. This model is also suited to predict the changes in time course of action due to differences in potency, similar to the model proposed earlier by Donati and Meistelman [^{7} ] and Proost and colleagues [^{8} ].

NMBAs are antagonists of acetylcholine, which is in turn responsible for neuromuscular transmission. Therefore we postulated that the contractile force of a muscle after supramaximal stimulation, measured as twitch height (TH), is related to the free AChR concentration according to the sigmoid Emax model (Hill equation):

where TH_{max} is the maximum TH, i.e. TH in the case that the number of free receptors is infinitely high, R _{free} is the concentration of free receptors, R _{free50} is the concentration of free receptors at which TH is 50% of TH_{max} , and β is an exponential coefficient.

In the absence of NMBA, the concentration of free receptors equals the total receptor concentration; on substitution in Eq. (4) it follows:

where TH_{c} is the TH in the absence of NMBA (control) and R _{tot} is the total receptor concentration.

The neuromuscular blocking effect (E ) of a NMBA is defined as

Substituting Eqs (4) and (5) in Eq. (6) yields

Eq. (7) describes the neuromuscular blocking effect as a function of the free AChR concentration. Total AChR concentration (R _{tot} ), R _{free50} and β are system parameters, and are independent of the NMBA [^{32} ].

Binding of the NMBA to the AChR binding sites is characterized by the equilibrium dissociation constant of the drug–receptor complex (K _{d} ):

where Cu_{e} is the unbound concentration of NMBA in the effect compartment, R _{free} is the concentration of free binding sites of AChR and R _{bound} is the concentration of AChR receptor sites to which a NMBA molecule is bound.

Defining R _{tot} as the total concentration of AChR binding sites, it follows upon rearrangement:

The time course of the unbound concentration in the effect compartment (Cu_{e} ) can be evaluated as described earlier [^{8} ]. Eq. (16) of that paper was simplified, not taking into account plasma protein binding and non-specific binding in the effect compartment, resulting in (10)

where C is the concentration in the central compartment (plasma concentration) and k _{e0} is the transport rate constant between the central compartment and effect compartment.

We assume that binding is very fast compared to the kinetic processes, so at any time equilibrium is assumed, i.e. Eq. (8) is valid at any time.

The PK-PD parameters were obtained from control animals of an earlier study [^{18} ]. The system parameters were R _{tot} = 1.47 μM, R _{free50} = 0.19 μM and β = 2.84. The PD and PK-PD link parameters for rocuronium were K _{d} = 0.40 μM and k _{e0} = 0.93 min^{−1} , respectively. The pharmacokinetics of rocuronium was described by a two-compartment model with parameters clearance (CL) = 26 mL min^{−1} kg^{−1} , volume of central compartment (V _{1} ) = 53 mL kg^{−1} , distribution clearance (CL_{12} ) = 10.5 mL min^{−1} kg^{−1} and volume of peripheral compartment (V _{2} ) = 90 mL kg^{−1} .