Issues in the Design and Interpretation of Minimum Alveolar Anesthetic Concentration (MAC) Studies
2002; Lippincott Williams & Wilkins; Volume: 95; Issue: 3 Linguagem: Inglês
10.1097/00000539-200209000-00021
ISSN1526-7598
Autores Tópico(s)Neuroscience and Neuropharmacology Research
ResumoThe minimum alveolar anesthetic concentration (MAC) of anesthetic preventing movement in 50% of individuals in response to a noxious stimulus is used both experimentally (1) and clinically (2) to measure anesthetic potency. This article discusses the relationship of the two study designs for measuring MAC in sections I and II. Implications of results from MAC studies for anesthetic mechanisms are considered in sections III and IV. I. Experimental Design The first method used to measure MAC is the quantal study design. It is the method that is used in humans. Each individual is exposed to an anesthetic concentration for a defined time, a noxious stimulus (a skin incision in humans) is applied, and movement or lack of movement is noted. The resulting quantal (categorical; all-or-none) data are then fit to a logistic or sigmoid Emax equation. Fitting the data to these two equations gives nearly identical results (see Appendix 1). Quantal analysis gives the probability of nonmovement as a function of anesthetic dose. The dose at which the probability of nonmovement is one-half is the 50% effective dose (ED50), or MAC. An example of data produced with this study design is shown in Figure 1(3). These data are taken from a MAC study in mice. Six mice from each of 15 different inbred strains of mice were studied. Movement or lack of movement in response to a noxious stimulus at various desflurane concentrations was measured. In total, 370 move/no move determinations were made. These are plotted in Figure 1. The figure also shows the logistic curve that fits these data. Of note, with the quantal design, MAC is determined for a population. MAC values for an individual are not known. For an individual, it is only known whether there was movement or not to a noxious stimulus.Figure 1: This graph shows the response to a tail clamp of mice from different inbred strains inhaling desflurane. Movement is plotted as zero and nonmovement as one. Three hundred seventy points are plotted. The data were also fit by logistic regression; the resulting sigmoid curve gives the probability of nonmovement as a function of desflurane concentration. Minimum alveolar anesthetic concentration (MAC) is the 50% effective dose (ED50) of this curve. Desflurane concentration is measured as partial pressure in percent atmospheres (% atm). MAC is 8.28% atm for these data.The second method of measuring MAC is the bracketing study design. MAC for each individual is determined. This is often used in experiments in animals (4). An animal is exposed to an anesthetic as before, and movement or lack of movement is noted. Unlike the quantal design, additional measurements are then made. If the animal moves at the initial concentration of anesthetic studied, the anesthetic partial pressure is increased in steps, and the procedure is repeated until the animal does not move. If the animal did not move initially, the anesthetic concentration is decreased until movement occurs. With the bracketing study design, MAC in the individual animal is the average of the largest concentration permitting movement and the smallest concentration preventing movement. For a group of animals, MAC is the average of the MAC values for the individual animals. Do quantal and bracketing study designs give similar or different results? Because of the large number of animals studied within each mouse strain in the study in Figure 1 and the repeated measurements made on individual animals, the data for the strains in that study can be analyzed either by the quantal study design or the bracketing study design. Data for strains are used rather than individual animal data because all of the animals in an inbred strain are genetically identical and should have the same MAC. By studying several animals per strain, an accurate value for MAC for the strain is obtained. Figure 2 shows the results of doing both quantal and bracketing analyses on this data set. Both the quantal and bracketing study designs give the same MAC values.Figure 2: Minimum alveolar anesthetic concentration (MAC) for desflurane was determined in mice from 15 inbred strains using the quantal study design and the bracketing study design. Strain MACs are plotted. The two study designs give the same values for MAC: the slope and intercept of the regression line are not statistically significantly different than 1 and 0, respectively, and the correlation coefficient is 0.975.II. Measurement of Variability in MAC Variability in quantal study designs is measured by the Hill coefficient (5), which determines the sigmoidicity (slope) of the MAC dose-response curve. The SD of MAC values obtained from individual animals specifies variability in the bracketing study design. How are these two measurements of variability related? Studies in animals indicate that MAC values are normally distributed (3). Knowing the distribution of MAC values allows the variables of the quantal design to be derived from the bracketing design and vice versa as follows. The probability that an individual animal will not move in response to a painful stimulus at an arbitrary anesthetic concentration is the same as the fraction of the population that would not move in response to a painful stimulus if all individuals were exposed to that concentration of anesthetic. The fraction of the population that will not move is equal to the fraction of the population with a MAC less than the applied anesthetic concentration. This fraction is equal to the area under the normal distribution to the left of the applied concentration, i.e., by the cumulative normal distribution (see Figure 3). This analysis is approximate because it assumes that an animal will not move if its MAC is less than the applied concentration of anesthetic and will move if its MAC is more than this concentration. In fact, an animal may not move at anesthetic concentrations less than MAC or move at concentrations larger than MAC because of measurement error and other environmental factors that influence MAC. A more complex analysis that takes this into account is described in Appendix 2. That correction only slightly modifies the analysis.Figure 3: The value of the ordinate of the cumulative normal distribution corresponds to the area under the normal distribution curve. For example, in this plot, the value of the ordinate for the point denoted by the circle on the cumulative normal distribution corresponds to the area to the left of the vertical line on the normal distribution. (sd denotes standard deviation.)Thus, the cumulative normal distribution gives the relationship between probability of nonmovement in response to a painful stimulus and anesthetic dose. Both the sigmoid Emax and logistic equations are excellent approximations to this distribution, as shown in Figure 4, which justifies their use in MAC studies. Assuming the slopes of these curves are equal at their ED50 (see Appendix 1), then the Hill coefficient (n) can be calculated from the population mean (ED50) and sd obtained using a bracketing experimental design, or a sd can be calculated knowing the population mean and Hill coefficient in the quantal study design, via the following equation:Figure 4: Plot of probability of nonmovement against anesthetic concentration for the cumulative normal distribution, sigmoid Emax equation, and logistic equation. The curves were constructed to fit a large, simulated minimum alveolar anesthetic concentration (MAC) experiment. The MAC data were simulated in 100,000 animals by drawing 100,000 MAC values at random from a normal distribution of a mean of 1 and sd of 0.1, comparing those data with randomly selected applied anesthetic doses and assigning a probability of nonmovement of 0 when the applied anesthetic dose was less than the simulated MAC, and 1 otherwise. The 0's and 1's were then fit to the applied anesthetic doses using nonlinear regression to the sigmoid Emax equation and by logistic regression to the logistic equation. For comparison, the exact probability of movement of nonmovement, given by the cumulative normal distribution, is also plotted. Note the excellent fit of all three curves.Because logistic regression and nonlinear regression are designed to provide an optimal fit of all of the data rather than match the slope at the ED50, there is a slight bias associated with using either logistic regression or nonlinear regression to a sigmoid Emax equation to fit data from MAC studies. It can be shown (see Fig. 5) that the sigmoid Emax and logistic equation give slightly biased (larger) Hill coefficients than they should. However, this bias is small as seen in Figure 5.Figure 5: Hill coefficients calculated by fitting the sigmoid Emax or logistic equations to minimum alveolar anesthetic concentration (MAC) data are larger (give steeper slopes at the 50% effective dose [ED50]) than the underlying normally distributed data. The bias is small but consistently greater for the logistic equation compared with a sigmoid Emax equation. The data in this figure were produced by simulating a MAC study in 100,000 animals by drawing 100,000 MAC values at random from a normal distribution with a mean MAC of 1. This was repeated 21 times for distributions with sd ranging from 0.05 to 0.25 in increments of 0.01. The simulated MAC data were compared with randomly selected applied anesthetic doses and assigned a probability of nonmovement of 0 when the applied anesthetic dose was less than the simulated MAC, and 1 otherwise. Hill coefficients were calculated from the latter by nonlinear regression to the sigmoid Emax equation or by logistic regression to the logistic equation and compared with the true Hill coefficient based on the slope of the underlying normal distribution of MAC values at its ED50 using Equation (1) in the text. How well does Equation (1) fit the experimental data? For the population MAC study described previously (3), the Hill coefficient calculated from the bracketing analysis via Equation (1) is 13.9 ± 1.5 (mean ± se). The Hill coefficient calculated from the quantal analysis via logistic regression is 14.2 ± 1.5, whereas nonlinear regression to the sigmoid Emax equation gives a value of 13.8 ± 1.7. These are not significantly different from each other. This supports the relationship between the Hill coefficient, sd, and ED50 derived above. The quantal and bracketing study designs are related via equation (1). Three other points in regard to equation (1) bear emphasis. First, because the ED50 term in equation (1) is a scaling factor (the data could be normalized for analysis, in which case this factor would not appear), Hill coefficients are functions only of sd. Second, because Hill coefficients are functions only of sd, in population MAC studies, Hill coefficients are large because the population sd is small. Third, Hill coefficients in behavioral studies in populations bear no direct relation to Hill coefficients in molecular studies, where large Hill coefficients speak to issues of allosteric effects and cooperativity. In a population study, Hill coefficients characterize the behavior of animals, not molecules. III. Relationship of Small Variability to Genes Underlying MAC How does the small variability in MAC dose-response curves relate to the genes that underlie MAC? Because the variability (measured as variance, sd, or Hill coefficient) in MAC is the result of both genetic and nongenetic (termed environmental) factors, by itself the variability in MAC cannot be used to characterize the effect of genes on MAC. Before the effect of genes on the variability in MAC can be estimated, the contribution of genetic factors to MAC must be separated from environmental factors. The variance in MAC, which is the square of the sd and denoted by sd2MAC, can be partitioned into a genetic component sd2gene and an environmental component sd2environment(6) that includes factors such as experimental error, the effect of circadian rhythms, drugs, and other nongenetic influences on MAC: The environmental variance can be determined in an experiment in which the genetic variance is zero. This will be the case in inbred animals, because inbred animals are all genetically identical. For inbred mice (3), the variance for desflurane MAC for individual strains on average is approximately 0.29 (% atm)2: this is the environmental variance. For all strains considered as a population, the variance in desflurane MAC is 0.85 (% atm)2, yielding a genetic variance via Equation (2) of 0.56 (% atm)2. For comparison, MAC for desflurane in this population of strains is 8.28% atm. The question of how genes affect the variability in MAC can be rephrased as how do individual genes affect sd2gene? If there are "n" genes that influence MAC, and these genes are denoted by x1, x2, … xn, and if the function f(x1, x2, … xn) describes the relation between the genes and MAC (that is MACgene = f (x1, x2, … xn), where MACgene refers to the genetic component of MAC, then the genetic variance is described by the law of propagation of errors (7): where sd2i denotes the variance in MAC attributable to gene xi, and covij denotes the covariance in MAC between genes xi and gene xj. (Covariances are related to correlation coefficients: if two variables are correlated, then they have a nonzero covariance. Variances are nonnegative. Covariances, like correlation coefficients, can be either positive or negative (8).) Few of the terms in Equation (3) are known. The function f(x1, x2, … xn) is not known, nor are the individual variance and covariance terms on the right side of the equation. This makes definitive conclusions about the genetic variance impossible. However, some general observations can be made. The smallness in the genetic variance, sd2gene (or equivalently, the largeness of population Hill coefficients) could result from one of three scenarios: First, it may reflect small variances and covariances in the genes underlying MAC. Second, if the genes underlying MAC have large variances, or if there are many small variances that add up to a large value, the overall genetic variance could still be small if a large negative covariance cancelled the large positive variance. Last, if the partial derivatives of the function f(x1, x2, … xn) were small, they could moderate the effect of large variances. Some cautions apply in interpreting Equation (3). Much is known about anesthetic effects on isolated gene products in biochemical systems (9). It is tempting to explain MAC in animals in terms of these molecular mechanisms. Molecular studies in in vitro systems, however, do not speak to the question of what the terms in Equation (3) are. The variances and covariances above refer to the effects of genes in animals, not in biochemical systems. Also, the genes in the equation above will include not only those coding for direct targets on which anesthetics act, but also modulators of those targets. IV. What Does the Small Population Variability in MAC Mean? Despite the complexity of relating proximate genetic causes to MAC, the small population, or phenotypic, variability in MAC does have some general implications for the evolutionary causes of anesthesia. The variability in any phenotype, including MAC, is the result of an equilibrium between forces that increase and those that decrease variability. Mutation increases variability, whereas natural selection and genetic drift decrease variability (10). Because the capacity to be anesthetized seems to extend across animal phyla (although for other end-points than immobility to noxious stimuli, which cannot be measured in all animals), genetic drift is an implausible explanation of the small phenotypic variability. Likewise, a small mutation rate cannot explain the small variability because it would have to be selectively active on anesthesia determining genes for tens to hundreds of millions of years of animal evolution. Consequently, the small variability in MAC most likely reflects natural selection for the anesthetic state. Viewed another way, if the capacity for anesthesia were not selected for, mutation might over time have eliminated the capacity for anesthesia altogether. Because there are no endogenous or environmental anesthetics, selection must act on a trait correlated with the capacity for anesthesia. Why should there be selection for the anesthetic state or a trait correlated with it? Natural selection implies that the capacity for anesthesia, or the trait correlated with the capacity for anesthesia, confers a survival advantage (strictly speaking, a reproductive advantage) on the organism. In general evolutionary terms, the reason it is possible to anesthetize an organism is because the capacity for anesthesia correlates with (presumably, because anesthetics act on) a highly conserved trait that benefits the organism. Because responses to inhaled anesthetics have been reported for animals in different phyla, including chordates (11,12), arthropods (13), and nematodes (14), the theory of evolution by common descent (15) suggests that the capacity for anesthesia arose in a common ancestor of today's animals. The existence of anesthetic responses in organisms in different kingdoms of the domain eucarya, such as the plant mimosa(16) and the protist tetrahymena(17), raises the intriguing possibility that the capacity for anesthesia arose in unicellular organisms ancestral to today's eukaryotes.
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