*DRUG tolerance distribution*is the distribution among a number of subjects of the critical level of intensity (dose or concentration) at which the drug will just produce or just prevent a reaction in each subject. Experiments may be designed to characterize all or part of the tolerance distribution. Experimental methods for binary response variables exist for determination of the dose associated with the 50% point along the dose–response curve or tolerance distribution if the response of the patient can be observed immediately.

^{1}The earliest use by the anesthesia research community of one particular sequential design—the up-and-down method (UDM)—seems to have been in studies of the effective concentration for a 50% response (EC

_{50}or minimum alveolar concentration) of inhalation anesthestics.

^{2,3}Columb and Lyons

^{4}used the same design and reported the EC

_{50}—labeled by them the minimum local analgesic concentration (MLAC)—of epidural bupivacaine and lidocaine for pain relief during labor. The use of this experimental design is now commonly seen in studies of intrathecal local anesthestics,

^{5}intrathecal opioids,

^{6}intravenous opioids,

^{7}intravenous α

_{2}agonists,

^{8}intravenous anesthetics,

^{9}and inhalation anesthetics

^{10}—all studies determining the concentration or dose (EC

_{50}or ED

_{50}) of these drugs.

*g*) along the dose–response curve, (2) demonstrate the risk of certain statistical methods commonly used in literature reports, (3) allow the estimation of the concentration or dose associated with the chosen quantile

*g*(EC

_{g}or ED

_{g}) without the assumption of the symmetry of the tolerance distribution, and (4) set bounds on the probability of response at the EC

_{g}or ED

_{g}. This article will detail these developments, briefly survey current use of the UDM in anesthesia research, reanalyze published reports of MLAC for epidural relief of pain during labor to demonstrate newer methods, and discuss appropriate inferences from UDM studies.

#### Materials and Methods

##### Experimental Design

*K*ordered dose or concentration levels that are denoted by the set Ω

*x*= {

*x*

_{1}<

*x*

_{2}< …<

*xk*< … <

*xK*}; e.g., if five concentrations of bupivacaine (0.07, 0.08, 0.09, 0.10, 0.11) are used, then Ω

*x*= {0.07, 0.08, 0.09, 0.10, 0.11}. The purpose of the study is to determine the dose level that has a probability of effect Γ; most commonly the EC

_{50}or ED

_{50}is desired, and thus the assignment Γ = 0.5 is made so that a response is produced in 100 × Γ = 50% of the target population. This will be denoted the target dose.

*N*denotes the total sample size,

*Nk*and

*Rk*are the number of subjects and the number of responses at dose level

*xk*, and

*Rk/Nk*= p̂(

*xk*) = p̂

*k*denotes the observed response rate at the dose

*xk*; the observed response rate p̂

*k*is an estimator of the true probability of response

*p(xk)*=

*pk*at dose

*xk*. Let

*X*(

*i*) denote the dose level administered to the

*i*th patient. The target dose is designated μ—with σ̂ being the estimator of μ. The probability of effect at σ̂ is

*p*(σ̂); the estimator of

*p*(σ̂) will be denoted p̂(σ̂).

^{11}To perform the study at the quantile Γ = 0.5, the

*K*dose levels are chosen, the sample size of patients from a target population is selected, and the definition of a positive response is specified; the

*K*doses are chosen with a fixed, constant interval between levels. The first patient is exposed to one of the

*K*dose levels; at the discretion of the investigators, this may be chosen to be the lowest dose, may be the dose thought to be closest to μ, or may be selected arbitrarily. The response of the first patient determines the dose level for the second patient. If the first patient has a positive response at dose

*x*

_{k}, then the second patient is exposed to the next lower dose

*x*

_{k−1}. If the first patient has a negative response at dose

*x*

_{k}, then the second patient is exposed to the next higher dose

*x*

_{k+1}. Successive patients are assigned a dose similarly by the response of the previous patient. At the lowest and highest doses of Ω

_{k}, appropriate adjustments for the next dose keep all doses within the specified dose range. Although Dixon proposed a modified UDM for very small samples (

*N*≤ 6),

^{12}anesthesia trials using the UDM typically have 20–40 patients.

^{1}; often a nonsequential design will require two to three times as many patients to have the same MSE. The UDM has been generalized as the biased coin design (BCD) by Durham

*et al.*

^{13}to find ED

_{g}for other quantiles

*g*. For example, suppose the ED

_{95}is to be determined (Γ = 0.95); the probability

*B*= 0.05 to the next lower dose and with probability 1–

*B*= 0.95 to the same dose. If a quantile less than 0.5 is selected, the probability

*B*to the next higher dose and with probability 1 –

*B*to the same dose. Intervals between doses do not have to be equal in the BCD. The BCD is optimal among the general class of up-and-down designs in the sense that the distribution of administered doses is most peaked around μ. At Γ = 0.5, the BCD reduces to the Dixon and Mood UDM. No studies in the anesthesia literature have attempted to study other quantiles on the dose–response curve using a sequential design.

##### Graphical Display

^{11}included this graphical display. It simultaneously displays the doses/concentrations administered (Ω

_{x}), the sequencing of administered doses (

*X*(

*i*)), and the positive or negative response for each subject at their assigned dose/concentration. As all experimental data are displayed, a reanalysis is easily possible. (Reanalysis is also possible from a tabular plotting of dose, number of events, and number treated.) Figure 1 shows a replotting of data from two published UDM studies.

^{14,15}Each study reported the MLAC from two UDM experiments—one experiment performed for levobupivacaine and one experiment performed for ropivacaine. (If two drugs are being compared in the same study, patients should be randomly allocated to the one or the other drug.) Long sequences of steadily increasing or decreasing doses with repeated ineffective or effective responses are common. In such circumstances, some assigned doses or concentrations must vary considerably from the underlying true ED

_{g}or EC

_{g}.

##### Target Dose Estimation

*p*(σ̂) may differ appreciably from the desired quantile. Although it is not possible to know the true probability of effect

*p*(σ̂) at σ̂, an estimator p̂(σ̂) with a CI should be obtained.

*X*(

*i*),

*i*= 1, …,

*n*, are not independent variables because each dose is dependent on the response to the previous dose; (2) because UDM studies are intended to minimize sample size and generally choose smaller sample sizes, large sample size assumptions (asymptotic properties) may not apply to the estimators; and (3) although the assumption of dose–response curve symmetry of certain mathematical models (

*e.g.*, normal or logistic distribution) is typically made, this assumption is unverifiable, and estimators may have a large bias. Algorithms for the exact distribution of estimators have been discussed, but computational power for samples larger than

*N*= 15 is lacking for their calculation.

^{16}Thus the derivation of the definitive statistical properties of estimators is an intractable problem; properties of estimators have been established by simulation studies in most statistical methods articles.

##### Dixon and Mood Estimator (Nonparametric)

*X*(

*i*) cluster unimodally around μ, the first nonparametric estimator is the empirical average σ̂ = ∑

*Ni=1X(i)*. There are several variants of the simple average, including truncation of the first run of all assigned doses with similar (either positive or negative) observed responses

^{17}and the use of only the assigned doses with positive or negative responses as suggested by Dixon and Mood.

^{11,18}The truncated version has been shown to be superior.

^{17}Most anesthesia UDM studies do not specify which variant of the simple average is being used.

##### Turning Point Estimator (Nonparametric)

^{19}The values of pairs of successive assigned doses at which the observed response changed direction (pair reversals) are averaged. The turning point estimator is commonly used in anesthesia UDM studies and has been used in a methods article on UDM in the anesthesia literature.

^{20}Choi

^{19}demonstrated by simulation that the turning point estimator was superior to the Dixon and Mood simple average, but only for an assumed normal tolerance distribution.

##### Logistic/Probit Estimators (Parametric)

^{21}specifically reexamined and criticized the use of such analysis in several published anesthesia UDM studies. By simulation, they demonstrated that the parameter estimate of the regression slope is biased and the CIs of the ED

_{50}are unrealistically narrow. This seemed to be a consequence of the nonindependence of assigned dose values. A logistic regression estimation routine adapted to the nonindependence of up-and-down data has been developed,

^{22}but the requirement of tolerance distribution symmetry is necessary. In addition, the σ̂ estimator does not always exist for all UDM data sets.

##### Isotonic Regression Estimators (Nonparametric)

Table 1 Image Tools |
Fig. 2 Image Tools |

^{23}An isotonic point estimate may be constant over some range of doses. At each assigned dose, an adjusted response probability

*pk** is easily calculated by the pooled-adjacent-violators algorithm (PAVA) (appendix 1). As shown in figure 2, the PAVA-adjusted response rate is monotonic.

_{g}or ED

_{g}are possible with isotonic regression. These include the following:

*pk** is minimized, σ̂

_{2}= (

*xk*: |

*pk** −Γ| is minimum among

*xk*∈Ω

*x*); and

*pk** and

*p*

_{k+1}* bounding Γ,

*et al.*have demonstrated that the isotonic estimator σ̂

_{3}has a smaller bias and MSE compared with the Dixon and Mood estimators.

^{22}Compared with the logistic regression estimator adjusted for nonindependence of data, the isotonic σ̂

_{3}estimator requires no symmetry assumption, always exists, and has similar MSE.

^{22}Simulation studies have also demonstrated that of the three isotonic regression estimators, the isotonic σ̂

_{3}estimator will have the narrowest CIs.

^{24}While the calculation of the isotonic σ̂

_{3}estimator is simple, the CIs are obtained by computer numerical methods. Stylianou

^{16}developed a parametric bootstrap routine calculating CIs by a bias corrected percentile method. This bootstrapping approach allows the estimation of CIs of σ̂

_{3}for an experimental design with any Γ, not just the 50th quantile. Also, bootstrapping is implemented to obtain an estimate of the bias of p̂(σ̂) from Γ with CIs (appendix 2).

##### Stopping Rules

_{g}or ED

_{g}. Simulation studies suggest that including at least 20–40 patients will provide stable estimates of the target dose for most realistic scenarios.

^{16}Other sequential methods use computationally intensive methods that require reanalysis of all data after each patient.

^{25}

##### Survey of UDM Trials in the Anesthesia Literature

_{50}or ED

_{50}, (5) the use of probit or logit regression analysis, and (6) the desired inferences.

##### Reanalysis of Published Studies

*et al.*

^{14}and Polley

*et al.*

^{15}each reported the MLAC for levobupivacaine and ropivacaine when administered for epidural analgesia in parturients during labor. In each study, patients who requested epidural analgesia were randomly allocated to receive either levobupivacaine or ropivacaine. Within each study, two independent UDM trials were conducted—one for levobupivacaine and the other for ropivacaine. The concentrations (levobupivacaine and ropivacaine) used in the first patient of each sequence were 0.10% (Polley

*et al.*

^{15}) and 0.11% (Benhamou

*et al.*

^{14}). Each study published figures (Benhamou

*et al.*,

^{14}fig. 1; Polley

*et al.*,

^{15}figs. 1 and 2) showing the sequential assignment of local anesthetic concentrations and the success or failure of the block. Data were extracted from these figures. A secondary analysis of results using these data were run using the previously described software; 9,999 bootstrap replications were simulated; the first concentration in the resampling algorithm was set to the first concentration used in the study. Comparison of MLACs for each study used the method of overlapping confidence intervals. If 83% CIs are nonoverlapping, then the null hypothesis of equal MLACs is rejected at an α of approximately 0.05.

^{26}

#### Results

^{6,14,15,27–39}Only 1 report did not display the raw data in either graphical or tabular form.

^{31}In 5 reports, sample sizes were less than 20.

^{31–33,35,39}It seems that all studies used a fixed sample-size stopping rule. Among the 16 reports, the most commonly used estimators of ED

_{50}or EC

_{50}included the Dixon and Mood average,

^{6,14,15,28,30,34,36,37}the turning point estimator of pair reversals,

^{27,29,38,39}and the small sample Dixon estimate.

^{31,35}These empirical mean estimators were accompanied in 9 studies by a probit or logit regression analysis.

^{15,28,29,33–38}In addition, 2 studies calculated an ED

_{95}by probit or logit regression.

^{27,29}Most studies of regional anesthesia using the UDM classified some patients as “rejects” not to be evaluated and used the same dose or concentration for the next patient in sequence.

^{14,15,28,29,34–38}Two studies pursued a study of drug interaction. In each study, three sequential trials were performed: drug A, drug B, and drugs A plus B in a fixed ratio; on an isobologram, the three ED

_{50}s with confidence intervals were plotted; no statistical properties of this isobolographic analysis were provided. Most studies sought a sparing effect on the EC

_{50}or ED

_{50}of one drug by adding a second drug,

^{6,27,33,37,38}compared the potency of one drug in different patient groups or under different clinical conditions,

^{29,32,35,36,39}or compared the potency of two or more drugs.

^{14,15,28}Two studies explicitly recognized that an EC

_{50}or ED

_{50}is not a dose recommendation

^{29}or that the entire response surface of drug combinations across low to high quantiles of effect must be studied to map the interaction of two drugs.

^{31}

*et al.*

^{14}and Polley

*et al.*

^{15}are presented in table 2 along with MLACs obtained from isotonic regression reanalysis of the original data. The sample sizes are adequate for stable estimation of parameters. Some estimates of MLACs by empirical mean and isotonic regression are similar (

*e.g.*, levobupivacaine 0.087%

*vs.*0.090%), whereas others are different by 0.10% (

*e.g.*, ropivacaine 0.089%

*vs.*0.079%). The widths (upper bound minus lower bound) of the 95% CIs are the same or slightly larger for the isotonic regression MLACs. To compare the isotonic regression MLACs between levobupivacaine and ropivacaine within each study, 83% CIs were estimated (table 2). In the report by Benhamou, levobupivacaine was more potent than ropivacaine, but without statistical significance; using the 83% CIs from the bootstrap distribution, the difference between levobupivacaine and ropivacaine was statistically significant. In the report of Polley, there was no difference between levobupivacaine and ropivacaine.

_{50}(Γ = 0.5) was the goal of each study, the CIs on the nominal probability of effect (50%) were wide: levobupivacaine (Benhamou: 28–76%; Polley: 27–80%) and ropivacaine (Benhamou: 29–67%; Polley: 33–76%).

*et al.*

^{15}analyzed the responses of 35 patients; an additional 29 patients were excluded. A sensitivity analysis of all 64 patients, those included and those rejected, revealed that imputing values of no analgesia for rejected patients estimates a larger MLAC, 0.097% (0.092–0.102%)

*versus*0.090% (0.079–0.095%).

#### Discussion

*dose–response curve*. The interpretation of this curve is dissimilar for continuous and binary responses. For continuous responses, this curve usually depicts the averaged observed values at different doses. For binary responses, this curve has a probabilistic meaning. The points along the curve represent the tolerance distribution of the population exposed to the anesthetic

^{40}; thus the dose at which 50% of the population responds is considered the median effective dose—the effective dose for a response by 50% of patients exposed (ED

_{50}). It must be emphasized that a patient does not have an ED

_{50}; if the probability distribution is known, then the ED

_{50}is a parameter describing a population of patients. The determination of this dose–response curve is called

*quantal bioassay*.

_{50}and the estimated value obtained by experimentation. Variability or variance can be defined as the square of the deviations of observed sample values from the estimated ED

_{50}. In comparing various experimental designs and estimators for determination of the ED

_{50}, the accuracy of a particular estimator is defined by the MSE. The MSE is the sum of the squared bias and the variance. In comparing estimators, a smaller MSE is desired. Simulation studies have demonstrated that performing an experiment to characterize the ED

_{50}by obtaining a fully specified tolerance distribution (nonsequential design with preassigned doses at multiple quantiles from ED

_{1}to ED

_{99}) is almost always less efficient than a sequential design to estimate only the ED

_{50}.

^{17,41}For both sequential and nonsequential designs, MSE becomes smaller as sample size increases. But it may require two to three times as many experimental subjects in a nonsequential design to achieve the same lower MSE of a sequential design. Because clinical trials are always subject to patient availability and the economic constraints of minimizing the number of subjects, a sequential design is to be preferred.

_{3}estimator—as demonstrated by simulation studies—has very favorable properties compared with other UDM estimators for obtaining the target dose with a smaller bias and greater precision (tighter CIs). The precision of the estimate may be narrowed by increasing the sample size. Because in a UDM experiment the assigned doses are concentrated around the target dose for the 50th quantile, the relatively modest sample size prevents any precision in target dose estimators for the lower and upper tails of the tolerance curve. As noted by Dixon in discussing the estimation of small or large percentage points from a UDM experiment, “Nothing short of an extensive exploration of the distribution, involving perhaps thousands of observations, will suffice in that case.”

^{18}With the generalization of the UDM process to BCD experiments, clinical trials may be planned for other quantiles along the tolerance curve. Simulation studies of BCD trials for quantiles far from the midpoint,

*e.g.*, Γ = 0.1, have demonstrated that the isotonic σ̂

_{3}estimator will have low bias and variance.

^{16}Thus sufficient accuracy exists to justify BCD trials at high or low quantiles.

^{25}Implementation of a continual reassessment method design requires the specification of the mathematical model of the dose–response curve and previous parameter values before initiation of the study; after each patient’s response is observed, the totality of all patients’ responses are reanalyzed to choose the drug dose for the next patient; the full involvement of a statistician is necessary to design, perform, and analyze the study.

^{25}By contrast, the BCD sequential study avoids unverifiable assumptions about the mathematical model of the dose–response curve, requires minimal computation during the study to assign successive doses, and has favorable statistical properties of the estimators of the target dose. New developments in BCD methods also are appearing.

^{42}

_{50}or ED

_{50}) in specifically defined clinical circumstances. For example, assuming the parallelism of the dose–response curves of two different anesthetics, the EC

_{50}s or ED

_{50}s alone will allow a comparison of the relative potencies.

^{11}There are particular reasons for the BCD to be of interest to anesthesia researchers. Although anesthesiologists discuss potency of inhaled anesthetics in terms of the EC

_{50}, the potency dose of real interest is the EC

_{95}or EC

_{99}. A BCD study can be performed setting Γ = 0.95, permitting a direct estimation of EC

_{95}and avoiding unverifiable extrapolations from the EC

_{50}value.

_{50}, was given to all patients. Because of random variation in patient responses and assuming a binomial distribution with π = 0.5, a sample size of 20 would be chosen to have a 95% CI extending from 0.28 to 0.72; even a sample size of 40 would only narrow the 95% CI to a range of 0.35 to 0.65. To achieve a 95% CI extending from 0.45 to 0.55 would require a sample size of 385 patients. When interpreting a UDM study, one should be cautious about the probability of effect at the estimated ED

_{50}or EC

_{50}because there is little precision.

_{95/99}or EC

_{95/99}of a drug. For several reasons, the ED

_{50}or EC

_{50}determined in a UDM experiment cannot provide any reliable insight into the upper tail of the tolerance distribution. Although it is often assumed that the sigmoidal dose–response curves of all drugs are well fitted by a symmetric logistic curve, in fact that is an unverifiable assumption, especially when dealing with small sample sizes. Near the midpoint (median) of the tolerance distribution, multiple mathematical models (normal distribution, logistic distribution, Cauchy distribution, log, complementary log–log, and so on) fit the observed data equally well, but at the upper tail of the distribution, the estimates of ED

_{95}or EC

_{95}may be quite different under these alternative mathematical models. When two different drugs are compared at the ED

_{50}or EC

_{50}, the equality or inequality of these median values do not necessarily allow inferences about the potency relation at the ED

_{95}or EC

_{95}. The tolerance distribution functions may not be the same; even if the same, there may be a lack of parallelism. The tolerance distributions also may have a lack of symmetry outside the median point.

_{50}or EC

_{50}values for isobolographic analysis

^{30,31}should be approached cautiously because the statistical properties are unknown; a nonparametric approach using a bivariate isotonic estimator for two-dimensional dose finding should be considered.

^{43}

^{14,15}Lassie

*et al.*

^{44}posed the question whether the MLAC method and by extension UDM design is robust enough. Using the restrictions and changes suggested in the Discussion, UDM maintains its advantages of smaller sample size and reasonably simple study performance. Caution is appropriate in other uses.

##### Appendix 1: Use of PAVA

*p̂k*≤ p̂

_{k+1}. The weight for each response rate is the number of patients assigned to that dose,

*N*

*k*.

_{k}and p̂

_{k+1}weighted by the number of patients:

*pk** = p̂

_{k}at each dose k.

*pk** no matter what the starting point in the dose sequence is. Cited Here...

##### Appendix 2: Software for PAVA Estimators and Confidence Intervals

^{24}; it is compatible at least to GAUSS version 3.6. The algorithms were translated into the public domain R: A Language and Environment for Statistical Computing (R Foundation for Statistical Computing, Vienna, Austria) by N.L.P.; it is compatible at least to R version 2.4.1.‡ Cited Here...