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Anesth Analg 2002;95:609-614 © 2002 International Anesthesia Research Society
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Abstract |
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IMPLICATIONS: The two experimental designs (quantal and bracketing) used for population minimum alveolar anesthetic concentration studies give equivalent results. An expression relating variability in terms of Hill coefficients and SD is presented. Evolutionary implications of low population variability in anesthetic phenotypes is discussed.
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Introduction |
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The 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.
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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.
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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.
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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:
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equation
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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.
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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:
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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):
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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.
Appendix 1: Relationship Between the Logistic, Sigmoid Emax, and Cumulative Normal Distribution in MAC Studies
The three equations used to describe population MAC dose-response curves are the logistic equation, sigmoid Emax equation, and cumulative normal distribution:
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P(x) denotes the probability of nonmovement, x is the anesthetic dose, a and b are constants, n is the Hill coefficient, and 
is the SD.
Although these equations seem different, each describes a sigmoid function of one variable and two parameters that asymptotically approaches 0 at low values of the independent variable and 1 at high values. As Figure 4 shows, the three curves approximate each other reasonably well.
Equation (1) was obtained by equating ED50s and the slope at the ED50 for the three equations above. The slopes of the three equations will be equal at x = ED50 if their first derivatives are equal. The slope of the cumulative normal distribution at its ED50 is 1/(2
2)1/2. Equating this with the slope of the sigmoid Emax equation, n/(4ED50), yields the relationship between the Hill coefficient, n, and the population SD, :
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For the logistic equation P(x) = 0.5 at x = ED50, hence ED50 = -a/b. Using this relationship and equating the slope of the logistic and sigmoid Emax equations at their ED50s reveals that the coefficients in the logistic equation are related to the Hill coefficient and the population ED50:
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It is not surprising that the sigmoid Emax and logistic curves approximate each other. To demonstrate this approximation for normalized data (i.e., where ED50 = 1) and anesthetic concentrations in the range from 0–2 requires the series expansion ez = 1 + z + z2/2! + . . . Using the first two terms in this series gives approximately e(x-1) = x. (The difference between the sigmoid Emax and logistic equations will therefore be described by the higher order terms that are not included in this approximation.) Raising both sides to the -n power and adding 1 yields 1 + x-n = 1 + e-(nx-n). Taking reciprocals gives the final result,
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Although this is the result of an initial approximation and is subject to certain numerical constraints, as noted in Figure 4, the fit between the two curves is excellent over the range of normalized data of relevance to anesthesia (i.e., where 0 < x < 2).
Appendix 2: Probability of Nonmovement Versus Anesthetic Dose
Suppose that an individual is chosen at random from a population and exposed to an arbitrary dose of anesthetic. What determines whether the individual will move or not? The probability of nonmovement will depend on two factors.
The first factor relates to the distribution of the true MAC values in the population. True MAC values are defined here as those that would be achieved in the absence of environmental influences on MAC; they reflect only the genetic contribution to MAC. If sufficiently many measurements of MAC could be made in the same animal, the random environmental influences on MAC should average to zero, yielding the true MAC value. The distribution of true MAC values is significant because, in the absence of environmental influences on MAC, an animal will not move if exposed to a concentration of anesthetic more than its MAC, and it will move if exposed to a concentration less than its MAC.
Environmental influences on MAC are the second factor determining whether an animal will move at a given concentration of anesthetic. In the presence of environmental factors that influence MAC, if an animal has a true MAC close to but less than the applied concentration, it may move. Similarly, an individual with a true MAC slightly larger than the applied dose probably will move, but it may not.
The probability of nonmovement of a randomly chosen individual from the population at any given anesthetic concentration will depend on the product of the probability of each true MAC value in the population (i.e., factor one above), weighted by the probability of nonmovement at each true MAC (i.e., factor two above) summed over all true MAC values. To calculate the sum of the product of these factors requires integration of the product of the probability density functions describing the distribution of true MAC values and the environmental influences on MAC over all true MAC values and a range of anesthetic concentrations up to the applied concentration.
The true MAC values and the probability of nonmovement will be treated as normally distributed variables here, although the following approach could be applied for any distribution. For normally distributed true MAC values, let mean = MACpopulation and SD = SDgene. In section III, the genetic variance for desflurane MAC was SDgene = 0.75% atm. For the environmental influences, let SD = SDenvironment; for desflurane MAC, this is 0.56% atm. For comparison, the population SD = 0.92% atm for desflurane MAC. If we let N(x, SD, M) denote the value at anesthetic concentration x of the normal probability density function with mean, M, and standard deviation SD, and if we let Pnomove(k) = the probability of nonmovement at anesthetic concentration k, then
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The effect of this more detailed analysis compared to Equation (1) in the text is relatively small. The coefficient of variation in MAC in most studies is approximately 0.1 (3). As noted in the text and Figure 5, fitting normally distributed data with a SD = 0.1 by nonlinear regression to a sigmoid Emax equation gives a Hill coefficient that is slightly biased (in this case by 6% to 17.0) compared with the predicted Hill coefficient of the underlying data of 16.0 (from Equation 1). If instead, the genetic and environmental variances are partitioned in the ratio they contribute to desflurane MAC for a population with MAC = 1 and SD = 0.1, the above equation (B1) is evaluated numerically at several anesthetic concentrations and the resulting data were fit to a sigmoid Emax equation, the Hill number would be biased upward by 4% to 16.6.
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Acknowledgments |
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Supported, in part, by NIH grant 1P01GM47818.
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Footnotes |
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Presented, in part, at the Sixth International Conference on Molecular and Basic Mechanisms of Anesthesia, Bonn, Germany, June 28–30, 2001.
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