Response Surface Model for Anesthetic Drug Interactions 论文

摘要

Click on the links below to access all the ArticlePlus for this article.Please note that ArticlePlus files may launch a viewer application outside of your web browser.DRUG interactions are the basis of anesthetic practice. For example, induction of anesthesia may consist of intravenous administration of a benzodiazepine before induction, a hypnotic to achieve loss of consciousness, and an opioid to blunt the response to noxious stimulation. Similarly, anesthesia often is maintained with a combination of a hypnotic (e.g. , propofol, isoflurane) and an analgesic (e.g. , fentanyl, nitrous oxide). Anesthetic drugs are often combined because they interact synergistically to create the anesthetized state.Pharmacodynamic drug interactions are typically described using mathematical models. The basic model is that of an isobole. Isoboles are iso-effect curves, curves that show dose combinations that result in equal effect. 1The combination of two doses (d1and d2) can be represented by a point on a graph, the axes of which are the dose axes of the individual drugs (fig. 1). The isobole connects isoeffective doses of the two drugs when administered alone, D1and D2. If the isobole is straight (fig. 1A), then the relation is additive. If the isobole bows toward the origin (fig. 1B), then smaller amounts of both drugs are needed to produce the drug effect when administered together, so the relation is supraadditive or synergistic. If the isobole bows away from the origin (fig. 1C), then greater amounts of both drugs are needed to produce the drug effect when administered together, so the relation is infraadditive. In table 1we propose a set of criteria that pharmacodynamic models of drug interactions should meet. In this article we propose an interaction model that meets these criteria, based on response-surface methodology. Response surfaces are a powerful statistical methodology for estimating and interpreting the response of a dependent variable to multiple inputs. 2Response-surface methodology is used for two principal purposes; to provide a description of the response pattern in the region of the observations studied and to assist in finding the region in which the optimal response occurs. Our model is a straightforward extension of the sigmoidal concentration–response relation for individual drugs. We test the proposed model using data from a study of the interaction of midazolam, propofol, and alfentanil with loss of consciousness. 3This article only considers pharmacodynamic interactions, the type of interaction most relevant to the practice of anesthesia. Pharmacokinetic interactions are entirely different and will not be considered. Appendix 1 (which can be found on the Anesthesiology Web site at www.anesthesiology.com) reviews several commonly used pharmacodynamic models of drug interactions and shows areas in which existing models fail to meet the criteria in table 1.The effects of individual drugs are often modeled by relating drug effect (E) to drug concentration (C) using a sigmoid model:where E0is the baseline effect when no drug is present, Emaxis the peak drug effect, C50is the concentration associated with 50% drug effect, and γ is a “sigmoidicity factor” that determines the steepness of the relation.This relation is shown graphically in figure 2. The concentration term often is defined as the concentration at the site of drug effect, but the model can be generalized to any measure of exposure (e.g. , dose, plasma concentration, or area under the curve). For models of probability, such as the probability of moving in response to surgical incision, E0is 0 and Emaxis the maximal probability (usually assumed to be 1). Dividing the numerator and denominator of equation 1by C50γ, we obtain an alternate form:In this model, concentration has been normalized to the concentration that results in 50% of maximal drug effect. This is a natural way to think about drug concentration—as a fraction of some measure of potency. For example, anesthesiologists are accustomed to thinking about volatile anesthetics in terms of minimum alveolar concentration (MAC), rather than in absolute concentration terms. This is precisely the concept of normalizing drug concentration to potency.The basic concept of our proposed interaction model is simple. Consider two drugs, each of which has a sigmoidal concentration–response relation. We will think of any given ratio (i.e. , B/(A + B), called θ herein) of the two drugs as behaving as a new drug. This new drug, which is actually a fixed ratio of the two drugs, has its own sigmoidal concentration–response relation, as shown in figure 3. This is the basic premise of our interaction model. The mathematics are simply an extension of the model for a single drug to a model that considers each ratio of two drugs as a drug in its own right. We will express the concentrations of drugs A and B as [A] and [B]. As suggested by equation 2, we must first normalize each drug to its potency, C50, and express the results in units (U) of potency. where UAis the normalized concentration of drug A, and UBis the normalized concentration of drug B. We can define a family of “drugs,” each being a unique ratio of UAand UB. Each drug will be defined in terms of θ, where θ is defined as By definition, θ ranges from 0 (drug A only) to 1 (drug B only). The “drug concentration” is simply UA+ UB. We can extend equation 2to describe the concentration–response relation for any ratio, θ, of the two drugs in combination:where θ is the ratio of the two drugs, the drug concentration is UA+ UB, γ(θ) is the steepness of the concentration–response relation at ratio θ, U50(θ) is the number of units (U) associated with 50% of maximum effect at ratio θ, and Emax(θ) is the maximum possible drug effect at ratio θ. Because Emax, C50, and γ in equation 2have been replaced by functions of θ, each ratio has the potential to have its own Emax, C50, and γ. This allows each ratio of drug A and drug B to behave as its own drug, with its own sigmoidal concentration–response relation, which is the basic premise of the model.The term “U50(θ)” is the potency of the drug combination at ratio θ relative to the normalized potency of each drug by itself. This requires careful explanation. Let us assume that only drug A is present, in a concentration of C50,A. In this case, the drug effect is half of the maximal effect, UA= 1, UB= 0, θ= 0, and the drug concentration is UA+ UB= 1. Because we have 50% of the maximum drug effect, and 1 unit of drug, then the number of units associated with 50% drug effect when only drug A is present, U50(0), must be 1. Similarly, let us assume that only drug B is present and the concentration of drug B is C50,B. In this case, the drug effect is half of the maximal effect, UA= 0, UB= 1, θ= 1, and the drug concentration is UA+ UB= 1. Because we have 50% of the maximum drug effect, and 1 unit of drug, then the number of units associated with 50% drug effect when only drug B is present, U50(1), must again be 1. By definition, if only drug A or drug B is present, U50(θ) = 1.Now, let us assume that drug A and drug B both are present, each in exactly half of the concentration that would cause 50% of the drug effect when administered alone. In this case, UA= 0.5, UB= 0.5, θ= 0.5, and the drug concentration is UA+ UB= 1. If this causes 50% of maximum effect, then the drugs are simply additive at θ= 0.5, and U50(0.5) = 1. However, if this combination produces more than a half-maximal effect, then 1 unit of this combination, at θ= 0.5, is more potent than 1 unit of either drug alone (i.e. , synergistic). In this case, U50(0.5) < 1. Conversely, if this combination produces less than a half-maximal effect, then 1 unit of this combination, at θ= 0.5, is less potent than either drug alone (i.e. , infraadditive). In this case, U50(0.5) > 1. Thus, U50(θ) is the potency of the combination compared with the potency of either drug alone, which is 1 by definition.Thus, the units of U50(θ) are not concentration units, but rather the number of units, at ratio θ, associated with 50% of maximal drug effect. U50(θ) is 1 for θ= 0 and θ= 1. For all values of θ between 0 and 1 (i.e. , all possible ratios of the two drugs), U50(θ) assumes a value determined by the data. If this value is 1, then the interaction is additive at θ. If the value is less than 1, then the drug effect is synergistic at θ. If the value is greater than 1, then the interaction is antagonistic at θ.Figure 4shows the relation between a three-dimensional response surface and a conventional two-dimensional isobolographic analysis. The two-dimensional isobologram is a cut through the three-dimensional surface, generally taken at the 50% response level. In this particular example, synergy is evident in the three-dimensional model as a bowing of the surface toward the reader. This bowing causes the conventional isobologram to deviate toward the origin from the straight line of additivity. Much pharmacodynamic literature supports the sigmoid relation in equation 1, equation 2, and equation 5. There is only modest information specifying the functions Emax(θ), U50(θ), and γ(θ). Our choice is to use functions that are capable of taking a variety of shapes, so that good approximations to the true relations can be determined empirically. To provide these flexible functions we chose fourth-order polynomials of the form where f(θ) is Emax(θ), U50(θ), or γ(θ). The coefficients (β0, β1, β2, β3, β4) are model parameters that are either constrained by the model or estimated from the data. Fortunately, two of these terms, β0and β1, can be replaced by other terms already defined.We already defined the values Emax(θ), U50(θ), and γ(θ) when only drug A is present, Emax,A, U50,A, and γA, respectively. Note in equation 6that when θ= 0 (only drug A is present), f(0) =β0. Therefore, when f(θ) is Emax(θ), U50(θ), or γ(θ), β0must be Emax,A, U50,A, and γA, respectively.Similarly, we also defined the values Emax(θ), U50(θ), and γ(θ) when only drug B is present, Emax,B, U50,B, and γB, respectively. Referring again to equation 6, when θ= 1 (only drug B is present), f(1) =β0+β1+β2+β3+β4. We can rearrange this as β1= f(1) −β0−β2−β3−β4. Thus, when f(θ) is Emax(θ), U50(θ), or γ(θ), β1must be Emax,B− Emax,A−β2,Emax−β3,Emax−β4,Emax, U50,B− U50,A−β2, U50−β3,U50−β4,U50, or γB−γA−β2,γ−β4,γ, respectively.This permits us to develop models that incorporate the individual drug parameters for Emax(θ), U50(θ), and γ(θ) as functions of θ. The equation for Emax(θ), using the substitutions previously mentioned for β0and β1, is U50,Aand U50,B, [equivalent to U50(θ) and U50(1)], are both 1 by definition. Thus, when f(θ) = U50(θ), the values of β0and β1in equation 6are 1 and −β2−β3−β4, respectively. Therefore, the equation for potency as a function of θ can be simplified to Many isobolograms have a simple inward or outward curvature, which can be readily encompassed with a simple quadratic form of equation 8with just one coefficient:If β2,U50is 0, then the value of U50(θ) will be 1 for all values of θ. This means that the interaction will be additive. If β2,U50is a positive number, then U50(θ) will be less than 1 for all values of θ between 0 and 1. The effect is to magnify the term in equation 5, making it appear that there is more drug present. This will produce a greater than additive effect, i.e. , synergy. If β2,U50is a negative number, then U50(θ) will be greater than 1 for all values of θ between 0 and 1. This reduces the term in equation 5, making it appear that there is less drug present. This will produce a less than additive effect. This assumes that drugs A and B have the same maximal effect. It is possible for some approaches to synergy analysis to show apparent synergy if the maximal effects of drugs A and B are not identical, even if U50(θ) = 1 for all values of θ.The model for the steepness term, γ(θ), can similarly be described from equation 6, with appropriate substitutions for β0and β1. The resulting equation is Equations 6–10describe straight lines (simple additivity) when the coefficients (i.e. , β2, β3, β4) are 0. They are the equations for parabolas if the respective β2coefficient is nonzero, and β3and β4are 0. More complex shapes are generated when β3and β4are nonzero.Figure 5shows Emax, U50, and γ as functions of θ for the synergistic interaction seen in figure 4. Emaxand γ are constant, and thus have no interaction. U50is necessarily 1 at θ= 0 and θ= 1, but is less than one between these extremes. This increases the potency of the drugs when administered in combination, resulting in the synergy seen in figure 4. The model can be readily expanded to show the interaction of more than two drugs. In the case of three drugs (A, B, and C) the proportion of each drug present can be expressed by θA, θB, and θC, where We can define the ratio of three drugs from just two of these ratios because θA+θB+θC=1. For our purposes here, we will use θBand θC. We again assume that for any fixed value of θBand θC, there is a sigmoidal relation between concentration and response. Therefore, if the three drugs could be administered to the effect site in an exactly fixed proportion, they would show a sigmoidal total concentration–response relation, where the “concentration” was the sum of the three normalized concentrations. This is precisely the notion that underlies the two-drug model. The equation for the model the model the parameters of the sigmoidal relation, Emax, and U50, are functions of θBand θC. The functions and are described in Appendix (which can be found on the Anesthesiology Web site at The point is that of a as in equations when three drugs are present, the parameters of the sigmoidal relation are surfaces for functions of θBand θC. The response-surface model was in the by the and as a for for also the model for of at 3This is on to the using and for the use of our response-surface model, we data previously by data are also the Anesthesiology Web site relations intravenous doses of midazolam, propofol, and alfentanil administered and in combination in for as to the to administration and or alfentanil to peak effect an intravenous the combination being midazolam, it was administered before the other drugs. The doses of midazolam, propofol, and alfentanil used and the proportion of for each dose are shown in table 2. This data set was because it three two-drug combinations that could be used to In the data are for the number of and the of the it the to a interaction model, the of the proposed response-surface model. We assumed in the model that all when no drug was 0 by definition. In we assumed that each drug was capable of if administered in a a of equation and can be constrained to 0 and 1 with no interaction the Thus, the probability of for any combination is where UB, and the doses of midazolam, propofol, and respectively. The units are of each dose to cause in 50% of the based on equation the data for the single and combination of and the data for the single and combination of and alfentanil the data for the single and combination of and alfentanil and the data set for the and combinations modeled parameters and estimated using by the for all the response of the either 0 to or 1 to and is the probability of response to for each dose be expressed in as the sum of the natural of the of response in the and in the The of the coefficients to the model was by the coefficients one at a by the model ratio and by of probability of for each dose and The response surfaces for the interactions and the surface was used to the synergistic combinations of the and the interactions based on the from the analysis of the data The intravenous doses to achieve probability of in this for each drug alone, for each combination, and for the To the application of the response-surface model with the parameters of , , and used to the for midazolam, propofol, and respectively. This was as the concentration at the of the respective then used to the of effect of in the these synergistic doses of midazolam, propofol, and administered alone and in The the and the from in of the to in of the The of effect was using for for by and shapes of the response surfaces generated by the equations are not readily We used three-dimensional to the response surface for a variety of interactions between two drugs, by the model parameters of equation 5. interactions and antagonistic interactions between two and interactions between and to the data for all in the in to the used by , the of the data for and for the analysis of the three drug interactions are shown in table 3. and for the analysis of the data set are shown in table 4. There no in the of the three drugs, there drug interactions the The for each drug to be the of the combination being modeled and This is an Because the data for single administration was in the interaction the entirely determined from each drug alone. the response surfaces for each of the drug interactions synergy. The synergy in the model was not Appendix on the Anesthesiology Web site at This when all three drugs are present, there is not synergy that from the interactions of all three drugs. The surface is shown in figure The maximum in values for the combinations are represented by the of the three of the as a = = = The maximum in for the combination is found at the point of the surface, which is at and This in figure and where the point as a on the 5shows the doses for the and combinations for maximum synergy associated with probability of a of based on the parameters shown in figure The between probability of to probability of no for each combination was based on the shown in figure the synergistic combination administration of one the dose, this results in a in the for to alone is the drug of choice when the point is a of A of the and of other such as and is to the drug combinations for other our response-surface model for an additive interaction a synergistic interaction and an interaction It also shows the interaction between a and a a and a and a and an way to describe our model for two drugs, A and B, is that the drug A and the drug B there are two sigmoid 1). Our proposed model connects these two sigmoid curves functions of θ interactions are then as coefficients of the polynomials that the to each each value of θ can have its own Emax, C50, and the model assumes that the sigmoidal is for all values of θ. This concept of a fixed ratio of two drugs own is not each drug ratio to be to a single response could be described by a single two-dimensional concentration–response that when using combinations of combination should be as a new with individual rather than the of the individual use of functions to the parameters of drug A to drug B assumes that the response surface is and the basic model the parameters estimated in sigmoid the polynomials that U50(θ), Emax(θ), and γ(θ) are more The use of response functions to complex response surfaces is in However, models with the variable present at than the are not often used because it to the proposed a flexible model. of the response surface, or application of the could result in of parameters that provide a but when more an for the are not unique to this model and can be by the one is in the terms can be from the model. The statistical of the terms should be the models using the ratio In to of model such as of and the pattern of the model should also be by the response surface, and by the individual model parameters as functions of θ (e.g. , equations to that the pharmacodynamic parameters not to For it must be that U50(θ) and γ(θ) are positive in the of 0 1. In the case of three drugs, the surface should be as shown in figure a mathematical that if and only if the isobolograms are straight For this to be be with to θ. The case of γ(θ) is more is with to θ, there is if Emax(θ) and U50(θ) are also not equal to γB, there is no way to use our interaction model to test for defined by 1). we with of the surface, the description of an interaction as or antagonistic may be For example, a drug combination can be synergistic in and antagonistic in our the on drug interactions can be to simple such as or the The interaction has the potential to be and than about which to the relation, the should be to the response the surface one can the combination to produce the model is It no about the of interaction between the drugs. However, we assume that the concentration–response relation for each of the drugs is described by a pharmacodynamic model. We have not to describe interactions between drugs that and that we are not of any that our could not be combined with the more of model for each drug not have to be the sigmoid 1). For example, the model could be a or response The model could also be a response as seen with some the model can be any so as it has parameters that the individual models. Thus, the only is that the interaction model reduces to the model for drug A when θ= 0 and to the model for drug B when θ= the of response-surface can For example, the of effects the of data in a to the model parameters for one of the drugs. to specifying a administered function for the concentration–response relation is to use a as described by the use of flexible functions that are to example, a can be constrained to an they in the of antagonistic interactions, it can be to model additive and synergistic drug used our response-surface model to the drug interactions for the hypnotic point between midazolam, propofol, and on this a maximum effect of for all three drugs described the data the dose of and alfentanil the response to in of the dose of the response to in only of results not that alfentanil will or in response to a surgical they that doses of will response to in of the of our response-surface model to potential interactions of two or three drugs at the effect site We that these concentrations are based on information parameters and not possible interactions between the three drugs. this we to that the synergistic dose ratio is not necessarily if a hypnotic effect is the the three-dimensional may in and different used to study drug interactions, it not information that be by a of two-dimensional such as the isobologram of the response surface, of three-dimensional the response to concentration of one drug in the of fixed concentration of the other drug to one and the response to fixed concentration ratios of the two any of these a three-dimensional of the response surface can be if are In the case of three drugs, it is no possible to the response surface because it is a surface, the model parameters can be in three (fig. of using axes to be in the of the interactions between three or more drugs. the study of drug interactions in anesthesia has used isobolographic analysis or multiple approaches have In the multiple is so with described in Appendix 1 in the web with pharmacodynamic that it should be The application of response-surface methodology to the study of drug interactions has the potential to the of these models. We proposed a flexible model for drug interactions, which the relation between the concentrations of two or three drugs and drug effect. We our new model using previously data and that this model can also describe of interaction between an a a and an of response-surface methodology permits of the concentration–response relation and can be used to develop for optimal drug