British Journal of Anaesthesia 104 (6): 705–10 (2010)
doi:10.1093/bja/aeq094
Advance Access publication April 28, 2010
CLINICAL PRACTICE Determining the potency of neuromuscular blockers: are traditional methods flawed? A. F. Kopman 1*, C. A. Lien 1 and M. Naguib 2 1
Department of Anesthesiology, Weill Cornell Medical College, 70 East 10th Street, Apt. 17F, New York, NY 10003, USA. 2Department of Anesthesiology and Perioperative Medicine, The University of Texas, M. D. Anderson Cancer Center, Houston, TX, USA Background. Traditionally, the clinical potency of neuromuscular blocking drugs has been measured using linear regression analysis (LRA) after log dose and probit or logit data transformation. However, probit and logit analyses are meant to handle only quantal responses with binomial error distributions, not continuous data such as per cent of maximal response. Some statisticians now consider this approach outmoded and assert that non-linear regression (NLR) is the preferred way to analyse sigmoidal dose – response relationships. We were interested in the degree to which the method of regression analysis alters calculated ED50 and ED95 values. Methods. We analysed raw data for succinylcholine, rocuronium, rapacuronium, and cisatracurium from previously published studies using both LRA and NLR to determine the ED50 and ED95 values and the respective slopes of the dose – response relationships. We also estimated drug potency using the Hill equation (HE) using the slopes obtained from LRA and NLR. Results. ED50 values calculated by NLR, LRA, or the HE were interchangeable. LRA resulted in ED95 values that were 13 –18% lower than those determined by NLR. The 95% confidence limits (CL) for the ED50 did not exceed +8% of the estimated value no matter how it was calculated vs +20 –30% for the ED95. Conclusions. The ED50 is a very robust parameter. When comparing the potency of neuromuscular blockers, it is this value rather than the ED95 that should be used. The CL for the ED95, regardless of how it is calculated, are so wide that this parameter must be viewed, at best, as an approximation. Br J Anaesth 2010; 104: 705–10 Keywords: neuromuscular block; pharmacology, dose – response; potency, drug; potency, ED50 Accepted for publication: March 18, 2010
When plotted on arithmetic axes, the dose – response relationship for most drugs takes the form of a sigmoid curve. Because non-linear regression analysis (NLR) of these curves is complex, to determine a drug’s ED50 and ED95 (the doses required for 50% and 95% effect, respectively), investigators studying neuromuscular block have traditionally transformed their data to create a linear graph. They then apply linear regression analysis (LRA) to the transformed results, and then retransform the best-fit values of slope and intercept to find the parameters that are of import in characterizing a neuromuscular blocking agent. An early example of this approach for defining the potency of neuromuscular blocking drugs is described in a 1974 paper by Donlon and colleagues.1 In that study, the
authors plotted the dose (x-axis) vs the effect (y-axis) using log-probit paper. Three decades later, in June 2005, the 8th International Neuromuscular Meeting entitled ‘Frontiers in Neuromuscular Physiology and Pharmacology’ was held in Stockholm, Sweden. Participants, all of whom had a major interest and expertise in the clinical use of neuromuscular blockers, were invited to participate in a series of workshops at the end of the meeting. The result was a consensus paper published 2 yr later in Acta Anaesthesiol Scand,2 which currently represents the most authoritative guide for investigators interested in the clinical use of neuromuscular blocking drugs. The guidelines for dose – response studies state that data should be linearized using the log dose of the
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*Corresponding author. E-mail:
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Kopman et al.
Methods The present protocol was approved by the Human Subject Review Committee at Weill Cornell Medical College. All patients consented to participate in the earlier IRB approved studies from which the data were obtained. The methodology used in all four groups has been previously described in detail.5 – 7
Succinylcholine, rocuronium, and rapacuronium Forty-eight (rapacuronium), 50 (succinylcholine), and 41 (rocuronium) ASA Physical Status I – II, adult patients undergoing elective surgical procedures were included in
these studies. Anaesthesia was induced with alfentanil 40 mg kg21 and propofol 2.0–2.5 mg kg21 i.v. Tracheal intubation was accomplished without the use of neuromuscular blocking agents. Anaesthesia was maintained with nitrous oxide (65–70% inspired), a propofol infusion, and opioid supplements as needed. Ventilation was controlled, and end-tidal carbon dioxide was maintained at 4.5–5.3 kPa. The indirectly evoked integrated compound action potential of the first dorsal interosseous muscle to supramaximal stimulation of the ulnar nerve at the wrist was measured and recorded using an NMT 221 monitor (Datex, Tewksbury, MA, USA). Single stimuli at 0.10 Hz were administered during the period of observation, and twitch depression was continuously recorded. Control twitch height was established after a 15– 20 min period of baseline stabilization. Immediately after baseline calibration, a single dose of relaxant was administered. The first subject in each group received a bolus estimated from published peer-reviewed data to approximate an ED50. Using the Hill equation (HE) [with a postulated slope of 4.50 (rapacuronium) or 4.75 (succinylcholine, rocuronium)], the subject’s actual ED50 was then estimated. The second subject received a dose which equalled the calculated ED50 for Patient 1. Patients 3– 5 were given a dose that approximated the average estimated value of the ED50s of the subjects which preceded them. In a similar manner, Patients 6 –10 were administered a dose based on the results of the earlier subjects calculated to achieve 20% twitch depression. The next five subjects received doses which approximated the average estimated value of the ED80. In the remaining patients, doses of each blocking drug were selected to provide essentially equally distributed increments in the range estimated to span responses from 15% to 95% twitch depression. Complete twitch depression was plotted as an effect of 99.5%. We did not encounter any 0% responses.
Cisatracurium Fifty adult patients undergoing elective surgery consented to participate in the initial dose – response portion of this study. Anaesthesia was induced with midazolam, propofol, fentanyl, and 70% nitrous oxide in oxygen and was maintained with a continuous infusion of propofol and nitrous oxide (70% inspired) with incremental doses of fentanyl as needed. The trachea was intubated after induction without the aid of a neuromuscular blocking agent. Ventilation was adjusted to maintain end-tidal PCO2 at 4.3– 5.3 kPa. The ulnar nerve was stimulated with supramaximal train-of-four (TOF) stimuli every 12 s using a Myotest peripheral nerve stimulator (Biometer International, Odense, Denmark). The contraction of the adductor pollicis muscle was recorded using a force displacement transducer and neuromuscular function analyzer (Myograph 2000, Biometer International). The first response of the TOF was
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drug plotted against either the probit, logit, or arcsine transformation of the per cent effect. Thus, as recently as 2007, LRA after data transformation was confirmed as the generally accepted method for determining neuromuscular potency among clinical investigators interested in this area of pharmacology. However, probit and logit analysis is meant to handle only quantal responses with binomial error distributions.3 One should not use continuous data, such as per cent of maximal response, with probit or logit analysis as these data are likely to require regression methods that assume a different error distribution. The problem with data transformation methods is that they cause some assumptions of linear regression to be violated. For example, data transformation distorts the experimental error. Linear regression assumes that the scatter of points around the line follows a Gaussian distribution and that the standard deviation is the same at every value of X. These assumptions are rarely true after transforming data.4 Some statisticians now consider this approach outmoded and assert that NLR is the preferred way to analyse sigmoidal dose – response relationships (Personal communication: Dennis Fisher, MD ‘P less than’ Company, San Francisco, CA, USA). We became aware of the theoretical objections to the use of logit or probit data transformations for calculating dose – response relationships when a reviewer of a recent manuscript submission of ours raised this issue. He felt rather strongly that LRA was a flawed approach when estimating a blocking drug’s ED95. We found his comment troubling as it raises questions of some consequence. Should investigators use NLR rather than LRA in future dose – response studies? The extensive existing literature on the potency of neuromuscular blocking agents is based almost entirely on LRA following data transformation. Does NLR regression analysis of data result in different ED50 and ED95 values? If so, are the differences of clinical importance? In an attempt to provide provisional answers to these questions, we reanalysed raw data from our files of four neuromuscular blockers using both methods of determining the drug’s ED50 and ED95 values and compared the results.
Determining neuromuscular potency
considered the twitch height. Cisatracurium was administered after stabilization of the response to indirect muscle stimulation. All patients received a single bolus of cisatracurium, with 10 patients each given a predetermined dose of 20, 25, 30, 40, or 50 mg kg21. Because the highest dose produced 100% block in six out of 10 subjects, we did not include the 50 mg kg21 dose in the calculations that follow. Thus, only 40 cisatracurium data points were analysed.
Statistics
Linear regression analysis Log dose – logit transformation of the raw dose –response data for each drug was performed. The best-fit line of regression for each drug was determined after transformation using the method of least squares. Confidence limits (CL) for the slope and intercept of the best-fit line were also obtained. The ED50, ED95, and the slope of the line of regression then were calculated using the regression function from the Excelw Data Analysis Tool Pak from Excel X for Mac (Microsoft Corporation, Redmond, Washington, DC, USA). The 95% CL for the ED50 and ED95 were also calculated. In a similar manner, we also calculated the ED50, ED95, and the slope of the line of regression using log dose – probit transformations to confirm that logit and probit data analysis result in similar estimates of potency. Non-linear regression analysis A best-fit non-linear analysis was performed on the above data using GraphPad Prism version 5.02 for Windows (GraphPad Software, San Diego, CA, USA). The function used to calculate the ED50 was ‘log dose vs response— variable slope (four parameters)’. This function is based on the following equation: y¼bottomþ(top2bottom)/ (1þ10^((log ED502X)*Hillslope)). We set the upper constraint (top) as 100% and the bottom as zero. The function used to calculate the ED95 was ‘log dose vs response - find ED anything’. This function solves the following equation: log ED50¼log ED952(1/Hillslope)*log(95/(100295)). Hillslope is synonymous with the Hill coefficient or g.
Statistical analysis: log dose – logit vs non-linear regression The potency of a neuromuscular blocking drug can also be calculated from the HE if g (the Hill coefficient) is provided.6 We therefore calculated the ED50 and ED95 values for all four drugs by this method using the values for g determined using both linear and non-linear regression. An advantage of determining drug potency by this method is that the ED50 and ED95 values for each individual subject can be calculated. Once this information is available, the statistical significance of estimated differences in potency for any drug (linear vs non-linear analysis) can be determined by a simple paired (matched) Student’s t-test.
Results As expected, ED50 and ED95 values using either logit or probit analysis produced identical ED50 values and the estimated ED95s did not differ by more than +1.5%. We did not do any further analysis of our probit data since logit analysis has one significant advantage over probit results. The log dose –logit (but not probit) slope can be substituted for g in the HE. The calculated NLR and log dose – logit ED50 and ED95 values (Table 1) result in ED50 values which are essentially identical. However, ED95 values derived from NLR are 13– 18% higher than those estimated by LRA (P,0.001 in each case). The 95% CL expressed as per cent difference from best-fit estimate of the ED95 are significantly wider when NLR is used. The best-fit nonlinear dose-response regression curves for all four drugs are illustrated in Figure 1. The slopes (and its CL) of the dose – response relationship for all four drugs as estimated by both linear and nonlinear regression (Table 2) show that when the upper and lower 95% CL for slope are substituted for the best-fit value in the HE, large changes in slope have little effect on the ED50 but may significantly alter the estimated ED95.
Discussion When Donlon and colleagues1 in 1974 first described the application of log dose – probit analysis as a technique for estimating a drug’s ED50 and ED95 values, practical desktop computers and user-friendly software programs for desktop machines capable of doing non-linear regression were not available. Thus, when atracurium and vecuronium appeared in the mid-1980s, clinical investigators continued to use tools and methods that were readily available and with which they were familiar.
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Power analysis When the ED95 for all four drugs is calculated from the HE (see below) using the slope from the best-fit line of regression (log dose – logit) as g, the coefficient of variations for the mean ED95 varied from a low of 18.6% (cisatracurium) to a high of 22.3% (succinylcholine). Using a coefficient of variation of 22.5%, a power of 80%, an a of 0.05, and an a priori analysis (two tailed t-test for the difference between two dependent means), we determined the sample size required for determining that a difference of 15% between methods as 38 observations. All four of our groups exceeded this value in size.
The software program also provided the CL for the ED50, ED95, and g.
Kopman et al. Table 1 Neuromuscular potency as determined by LRA vs NLR. CL, confidence limits. aED50 and ED95 expressed in mg kg21. bPercentage deviation from the ED50 and ED95 as determined by the best-fit line of regression. †P,0.001 in each case Succinylcholine (n550)
Rocuronium (n541)
0.377 0.367 – 0.385 22.8 to þ2.0 0.733 0.678 – 0.820 27.5 to þ11.9
0.026 0.0255–0.0269 21.5 to þ0.9 0.040 0.0375–0.0457 27.2 to þ12.9
0.170 0.158 – 0.184 27.6 to þ8.2 0.367 0.280 – 0.480 223.7 to þ30.8
0.375 0.358 – 0.398 25.9 to þ6.2 0.854 0.714 – 1.020 216.4 to þ19.4
0.026 0.0244–0.0278 26.2 to þ6.7 0.045 0.0361–0.0555 219.8 to þ23.3
0.0 þ12.9
20.5 þ16.5
0.0 þ12.5
First twitch depression (% of control)
First twitch depression (% of control)
0.170 0.163 – 0.175 24.4 to þ2.7 0.325 0.291 – 0.394 210.5 to þ21.2
100 80 60 40 20 0 0.1
0.2
0.3
100 80 60 40 20 0
0.4
0.2
100 80 60 40 20 0 0.025
0.032
0.3
0.5
0.8
Rapacuronium (mg kg–1) First twitch depression (% of control)
First twitch depression (% of control)
Rocuronium (mg kg–1)
0.02
Cisatracurium (n540)
0.04
100 80 60 40 20 0 0.08 0.1 0.13 0.16 0.2 0.25
Cisatracurium (mg kg–1)
Succinylcholine (mg kg–1)
Fig 1 The best-fit lines of regression log dose vs per cent twitch depression as calculated using NLR.
However, as early as 1980, a theoretical objection to this approach was recognized. In a study of D-tubocurarine, Donlon and colleagues8 noted that log-probit plots are ‘used in pharmacology to describe dose – response
relationships that are based on all-none events’. He rationalized that since each myofibril responds in such a manner, probit transformation could be applied to the response of an entire muscle group. Unfortunately, the
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Linear regression analysis (log dose vs logit) EDa50 0.138 95% CL EDa50 0.136 –0.140 21.7 to þ1.2 D% CL ED50b 0.273 EDa95 95% CL EDa95 0.248 –0.316 D% CL EDb95 29.6 to þ15.8 Non-linear regression (log dose vs effect) EDa50 0.143 95% CL EDa50 0.135 –0.152 D% CL EDb50 25.7 to þ5.9 EDa95 0.323 95% CL EDa95 0.272 –0.384 b 215.8 to þ18.9 D% CL ED95 % difference non-linear vs linear regression analysis ED50 þ2.2 ED†95 þ18.3
Rapacuronium (n548)
0.318 (212.3% to þ23.6%) 0.369 (216.0% to þ51%) 0.846 (210.5% to þ21.0%) 0.0459 (213.5% to þ44.0%) 0.281 (29.3% to þ16.4%) 0.332 (210.5% to þ23.5%) 0.751 (26.8% to þ11.1% 0.0411 (27.3% to þ14.4%) Succinylcholine Rocuronium Rapacuronium Cisatracurium
4.32 (3.49 – 5.14) 4.55 (3.33 – 5.77) 4.43 (3.66 – 5.20) 6.85 (5.20 – 8.50)
3.61 (2.86 –4.36) 3.84 (2.44 –5.23) 3.58 (2.80 –4.36) 5.43 (3.36 –7.50)
0.142 (1.4%) 0.174 (22.9% to þ2.3%) 0.386 (23.4% to þ3.1%) 0.0268 (+0.4%)
0.141 (0.7%) 0.171 (22.3% to þ2.9%) 0.372 (23.8% to þ3.5%) 0.0267 (23.0% to þ0.4%)
NRA LRA LRA NRA LRA
NRA
ED95 (mg kg21) (95% CL) ED50 (mg kg21) (95% CL) Slope and 95% CL Drugs
response of the muscle as a whole must be viewed as nondichotomous or continuous data. Our analysis of data from four previous dose – response studies confirms that ED50 values calculated by NLR or LRA after log dose – logit data transformation may be used interchangeably. However, ED95 values estimated by NLR averaged about 15% higher. For reasons outlined above, the continued use of LRA appears to be based on historic rather than scientific grounds. Additionally, if one accepts the position that LRA after data transformation (at least theoretically) is fundamentally flawed as a method of estimating the dose – response relationships of neuromuscular blocking drugs, much of the data on the potency of commonly used relaxants may overestimate their potency. Non-linear regression offers one additional advantage over LRA. Responses of zero and 100% can be plotted, but this is not allowed with probit or logit data transformation. It is also difficult to defend a protocol (LRA after data transformation) which we recognize to be based on a potentially unsound principle. We should, however, note one caution regarding the use of non-linear regression. Even the statistical software program we used, which is reasonably user friendly, presents the investigator with many regression models to choose from and decisions to make regarding data fitting data, data constraints, data weighting, and other parameters. Many researchers should probably seek expert statistical help before pursuing this type of analysis. When in doubt, classic LRA may be the safer way to proceed for many clinicians. Thus, an obvious question arises. Should future studies of neuromuscular potency use NLR or should we continue to use LRA in order to make comparisons with older drugs more meaningful? As is apparent from our data, the estimated ED95 is not a precise number. The 95% CL for ED95 values may span a range of – 10% to þ20% (LRA) or – 20% to þ30% (NLR) from the estimated best-fit value (Table 1). Although the CL of the ED50 and ED95 appear to be narrower when potency is calculated from the HE rather than from linear or non-linear regression, it should be remembered that the estimated slope which the latter method uses has its own CL which constrain its precision. The implications of this are striking. Wide variations in slope (g) have almost no effect on the ED50 when calculated using the HE. In contrast, estimated ED95 values are highly dependent on the g utilized (Table 2). Variations .20% in the ED95 are possible using values for g that are still within its 95% CL. This finding has several implications. First, if one assumes that the dose – response curves of most neuromuscular blocking drugs are roughly parallel,5 a strong case can be made that potency comparisons between different relaxants should be based on their respective ED50 values rather than their ED95 values as the former appears to be independent of how it is calculated. Secondly, the CL of the ED50 are narrower as a percentage of their estimated value than those of the ED95.
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Table 2 The effect of changes in slope (g) on ED50 and ED95 when using the HE. Percentage changes are the deviation from the ED value obtained from the best-fit line of regression when the upper and lower CL for slope are substituted in the equation. Large changes in slope have little effect on the ED50 but may significantly affect the estimated ED95
Determining neuromuscular potency
Kopman et al.
Conflict of interest A.F.K. has done ad hoc consulting for Organon Pharmaceuticals (now Schering-Plough, a Division of
Merck & Co., Whitehouse Station, NJ, USA) and served on its speakers bureau. He also has been an ad hoc consultant to Avera Pharmaceuticals (San Diego, CA, USA). M.N. has participated in two studies of sugammadex which were funded for Organon Pharmaceuticals.
Funding The present study was not funded. The cited studies on rapacuronium, rocuronium, and succinylcholine (refs 5 and 6) were supported in part by funds from the Department of Anesthesiology, St Vincent’s Hospital Manhattan, New York, NY, USA. The cited study on cisatracurium (ref. 4) was supported by funds received from the College of Medicine Research Center, College of Medicine, King Saud University.
References 1 Donlon JV, Ali HH, Savarese JJ. A new approach to the study of four nondepolarizing relaxants in man. Anesth Analg 1974; 53: 934 –8 2 Fuchs-Buder T, Claudius C, Skovgaard LT, Eriksson LI, Mirakhur RK, Viby-Mogensen J. Good clinical research practice in pharmacodynamic studies of neuromuscular blocking agents II: the Stockholm revision. Acta Anaesthesiol Scand 2007; 51: 789 – 808 3 Finney DJ. Probit Analysis. A Statistical Treatment of the Sigmoid Curve, 1st Edn. Cambridge: Cambridge University Press, 1947 4 Motulsky H, Christopoulos A. Fitting Models to Biological Data Using Linear and Nonlinear Regression: A Practical Guide to Curve Fitting. New York: Oxford University Press, 2003; pages 16 and 19 5 Naguib M, Samarkandi AH, Amnar A, Al-Zahrani S. Comparative clinical pharmacology of rocuronium, cisatracurium, and their combination. Anesthesiology 1998; 89: 1116– 24 6 Kopman AF, Klewicka MM, Neuman GG. An alternate method for estimating the dose –response relationships of neuromuscular blocking drugs. Anesth Analg 2000; 90: 1191 – 7 7 Kopman AF, Klewicka MM, Ghori K, Flores F, Neuman GG. Dose– response and onset/offset characteristics of rapacuronium. Anesthesiology 2000; 93: 1017 – 21 8 Donlon JV, Savarese JJ, Ali HH, Teplik RS. Human dose – response curves for neuromuscular blocking drugs: a comparison of two methods of construction and analysis. Anesthesiology 1980; 53: 161 –6 9 Kopman AF. How low can you go? Lowest effective dose of neuromuscular blocking agent for tracheal intubation. Can J Anaesth 2009; 56: 473 – 7 10 Lepage JY, Malinovsky JM, Malinge M, et al. Pharmacodynamic dose– response and safety study of cisatracurium (51W89) in adult surgical patients during N2O-O2-opioid anesthesia. Anesth Analg 1996; 83: 823 – 9 11 Belmont MR, Lien CA, Quessy S, et al. The clinical neuromuscular pharmacology of 51W89 in patients receiving nitrous oxide/ opioid/barbiturate anesthesia. Anesthesiology 1995; 82: 1139– 45
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The concept that the ‘intubation dose’ is equal to the 2 the ED95 is an antiquated one.9 Thus, the ED95 is a less useful concept than it once was. In defence of the ED95, there is some evidence that not all drugs have the same dose – response slope. Our data suggest that the slope for cisatracurium is steeper than the other three drugs we studied (Table 2). Cisatracurium appears to be 6.5 times more potent than rocuronium as measured by their ED50 values, but 8.1 times more potent when their ED95 values are compared. However, the method of dose selection and data collection used in the cisatracurium group was quite different from that used in the other three groups. In addition, the log dose – logit slopes calculated from other published ED50 and ED95 values for cisatracurium (5.17)10 (5.84)11 are only modestly steeper than those we found for succinylcholine, rocuronium, and rapacuronium. In conclusion, our findings clearly demonstrate that the estimated ED95 is at best an approximation subject to very wide CL. In contrast, the ED50 is a robust parameter which is independent of the methodology used in defining it and is only minimally effected by large variations in the slope used in calculating it from the HE. When comparing the potency of neuromuscular blocking drugs, it is the ED50 rather than the ED95 that is the more reliable parameter. Either non-linear regression, traditional log dose – logit, or probit analysis may be used when calculating this parameter. On theoretical grounds, the investigator who wishes to define the ED95 may be best served by using non-linear regression. A clinical study of this issue would be of interest. One group of subjects could receive a single ED95 bolus of relaxant as calculated by LRA, another cohort a single ED95 dose as estimated by NLR and the results compared. However, even if experimental data were to substantiate that NLR provides a more reliable estimate of the ED95 than that obtained by LRA, a 15% error in the ED95 is probably not clinically important. Clinicians rarely administer a single ED95 dose. Rather the size of the initial dose administered is usually based on intubation studies and personal experience. We believe the implications of our analysis may be of importance to investigators interested in the dose – response relationship. Although an argument can be made that in theory non-linear regression is a sounder approach, we do not believe that the use of traditional LRA methods needs to be summarily abandoned. The enormous database of peer-reviewed information using these traditional methods for determining neuromuscular potency has not led us seriously astray.
