Abstract
Correctly chosen doptimal designs provide efficient experimental schemes when the aim of the investigation is to obtain precise estimates of parameters. In the current work, estimates of parameters refer to the enzyme kinetic parameters V_{max} and K_{m}, but they also refer to the inhibition constant K_{i}. In general, this experimental approach is performed on a grid of values of the design variables. However, this approach may not be very efficient, in the sense that the parameter estimates (V_{max}, K_{m}, and K_{i}) have unnecessarily high variances. For good estimates of parameters, the most efficient designs consist of clusters of replicates of a few sets of experimental conditions. The current study compares the application of such doptimal designs with that of a conventional approach in assessing the competitive inhibitory potency of fluconazole and sertraline toward CYP2C9 and 2D6, respectively. In each instance, the parameter estimates, namely V_{max}, K_{m}, and K_{i}, were predicted well using the doptimal design compared with those measured using the rich data sets, for both inhibitors. We show that d optimality can provide more efficient designs for estimating the model parameters, including K_{i}. We also show that real cost savings can be made by carefully planning studies that use the theory of optimal experimental design.
Cytochrome P450 (P450) enzymes play a major role in drug metabolism, and CYP1A2, 2C9, 2C19, 2D6, and 3A4 are responsible for the majority of these reactions (Hasler et al., 1999). Altering the routes or rates of a metabolic reaction for a compound is particularly relevant with drugs that have a narrow therapeutic index, because small changes in the plasma concentration of the drug can potentially lead to an adverse effect. Such effects include reduction in efficacy or, even worse, toxicity. Inhibition of cytochromes P450 involved in the metabolic clearance of a drug, whether it is the drug itself or any coadministered drug, can potentially result in a metabolic drugdrug interaction (DDI). Inhibition of P450s results in the most frequently observed DDIs, and the associated mechanism of action can be classified as reversible, quasiirreversible, or irreversible. Reversible inhibitors, the mechanisms of action of which can be further subdivided into competitive, noncompetitive, or uncompetitive inhibition (Houston et al., 2003), pertain to the current work. Mixed inhibition has also been reported but is less common (Houston et al., 2003).
In a typical cytochrome P450 kinetic reaction, the enzyme binds substrate and metabolizes it into associated products. The binding step is reversible, whereas the catalytic step irreversible and is written as the following chemical model: where S, E, and P denote substrate, enzyme, and product, respectively. The reaction rate (v) is represented by the standard MichaelisMenten model (eq. 1): where V_{max} denotes the maximal velocity of the enzyme, [S] denotes the concentration of the substrate, and K_{m} denotes the MichaelisMenten constant, which represents [S] at which half of the maximal velocity is reached. The most prevalent mechanism of inhibition is competitive, where a compound binds reversibly to the enzyme and prevents the binding of substrate and vice versa. The competition between substrate and inhibitor for the enzyme is represented by the following equation: where K_{i} is the competitive inhibition constant. The initial rate of reaction v follows the mechanistic model (eq. 2): where [I] denotes the concentration of the inhibitor. For a fixed [I], the limit of this function is V_{max} when [S] becomes infinitely large. This result means that the same maximal velocity is obtained irrespective of the concentration of the inhibitor. However, in its presence, higher concentrations of the substrate are needed to come equally close to the asymptote of V_{max}. Hence, the apparent K_{m} increases with the concentration of the inhibitor, i.e., the reaction is slowed down.
The experimental procedure routinely used to determine these various parameters involves incubating the enzyme source (e.g., human liver microsomes) with a specific cytochrome P450 substrate over a range of different concentrations. The drug whose potency (i.e., K_{i}) is being determined is also studied at various concentrations under the aforementioned conditions, to see its effect on the amount of product being formed. From a pharmaceutical perspective, these P450 K_{i} experiments are typically performed for compounds in the developmental phase and require the study to be done on multiple occasions. K_{i} determinations are not limited to P450s, and there are a host of other screens (for example, target pharmacology potency screens) run within the pharmaceutical industry that generate K_{i} values.
Traditionally, this experimental paradigm is performed on a grid of values of the design variables. However, this approach may not be very efficient and could potentially generate superfluous data. Hence, application of doptimal designs allows for good estimates of parameters, where the most efficient designs consist of clusters of replicates of a few sets of experimental conditions. For linear models, the sets of conditions do not depend on unknown parameter values in the model. The design area is different for the model depicted by eq. 2, in which the parameters K_{m} and K_{i} enter nonlinearly. Then, the location of the clusters of design points depends on the K_{m} and K_{i} values, although not on the V_{max} value, which enters the model linearly.
The current study investigated whether there was scope to optimize the traditional design (for estimating the inhibition constant K_{i}) through the application of doptimal designs, and thus potentially affect efficiency by significantly reducing the sample numbers.
Materials and Methods
In Vitro Incubations.
All in vitro incubations were carried out using a MicroLabSTAR Autoload with 8 channels and a 96channel head (Hamilton Robotics, Bonaduz, Switzerland). The incubation mix consisted of the following (at their final concentrations): 50 mM KH_{2}PO_{4} buffer (pH 7.4), 5 mM MgCl_{2}, human liver microsomes (BD Bioscience, Woburn, MA), and 1 mM NADPH. Incubation times and protein concentration were selected such that they provided a linear reaction velocity with respect to product formation (as identified in preliminary experiments; data not shown). Incubation mixes were then supplemented with relevant inhibitors [fluconazole (Diflucan; CYP2C9) and sertraline (Zoloft; CYP2D6); Pfizer inhouse chemical store (Sandwich, England)] at up to eight concentrations. Reactions were subsequently initiated by the addition of specific P450 probes (Pfizer): dextromethorphan (CYP2D6) or diclofenac (Voltaren; CYP2C9) across 15 concentrations (0–50 μM), at each inhibitor concentration (0–60 μM). Reactions were terminated 1:2 in icecold acetonitrile (v/v) containing isotopically labeled dextrorphan or 4hydroxy diclofenac metabolites (50 ng/ml). Terminated reaction mixtures were analyzed directly by highperformance liquid chromatographymass spectrometry.
Samples were subsequently quantified using an Applied Biosystems/Sciex API 4000 QTRAP mass spectrometer (Applied Biosystems/MDS Sciex, Foster City, CA) in the positive ionization mode. A Phenomenex Synergi Fusion high pressure highperformance liquid chromatography column, 2.0 × 20.0 mm, 2.5μm particle size (Phenomenex, Torrance, CA) was used for chromatographic separation, at a flow rate 1 ml/min. A CTC auto sampler (CTC Analytics, AG, Zwingen, Switzerland) was used in conjunction with a Jasco XLC 3185PU highpressure, low dead volume, binary gradient pump, Jasco XLC 3067CO column oven and Jasco XLC 3080DG degasser (Jasco, Tokyo, Japan) (Youdim et al., 2008).
Data Analysis.
Conventional data were analyzed by nonlinear regression analysis using Grafit 4 (Erithacus Software Ltd., Horley, Surrey, UK) and applying models for MichaelisMenten kinetics with inhibition (eq. 2). The criteria used to select and check the most appropriate model included visual inspection of the residuals, together with tests of independence of the errors and the constancy of error variance, and F tests for the values of parameters.
Designing the Experiments.
From a pharmaceutical perspective, P450 K_{i} experiments are typically performed for compounds in the development phase and consist of taking measurements across several different concentration combinations of the substrate and inhibitor, with each combination normally repeated in triplicate. However, application of K_{i} studies for compounds during discovery, where numbers will exceed those in development, necessitates a reduced design. As such, single studies might be performed. However, a different setup is required if we want to optimize the experiment for parameter estimation. Then, the design usually consists of far fewer combinations of the concentrations, but each one is replicated several times. We denote such designs by ξ with a subscript N to indicate the total number of observations to be taken (i.e., eq. 3): where x_{j} denotes the design support points and represents the proportion of experimental effort at x_{j}, j = 1, … ,n. Note the following:
For the current study, the support points x_{j} specify n combinations of substrate and inhibitor concentrations that come from the design region Ω, i.e., x_{j} = ([S], [I]) and Ω = {[0,[S]_{max}] × [0,[I]_{max}]}, where [S]_{max} and [I]_{max} are the maximal allowable concentrations of the substrate and inhibitor, respectively. Although, in practice, setting such concentrations may be subject to error, in our experiments all x_{j} are determined with good precision, because the experiments were performed using an automated procedure.
In designing the current experiments, we assume that the errors in observing the rate of reaction v (eq. 2) are additive, independent, exhibit constant variance, and are approximately normally distributed. Analysis of conventional data supports these assumptions (data not shown). As such, the appropriate method of parameter estimation is nonlinear leastsquares. If the model were linear, the confidence region for the parameter estimates would be elliptical, or ellipsoidal with three or more parameters. For doptimal designs, we choose the x_{j} and r_{j} values to minimize this volume. If the parameter estimates are uncorrelated, this value is equivalent to finding a design that minimizes the variances of the estimates. However, the estimates are usually not independent, and d optimality minimizes the generalized variance of the estimates defined as the determinant of the matrix of variances and covariances of the estimates. This determinant is proportional to the square root of the volume of the confidence ellipsoid. A succinct summary of the optimal experimental design theory is given by Fedorov and Hackl (1997). For doptimal design for nonlinear models, see Atkinson et al., (2007).
Results and Discussion
The purpose of the current investigation was to compare a conventional approach with those that are d optimum. Our conventional designs had n (the number of support points) equal to 120, which consisted of a grid of 15 values of [S] and 8 values of [I] on a logarithmic scale, referred to as a “rich” data set. These rich data sets of substrate inhibitor pairings allowed us to estimate the parameters for both sertraline against CYP2D6 and fluconazole against CYP2C9 (Table 1). In each case, the inhibitory potencies obtained were consistent with those available in the literature for sertraline (Otton et al., 1993, 1996) and fluconazole (Youdim et al., 2008).
For many nonlinear models, the doptimal designs have to be found by numerical maximization. However, in our case, we were able to obtain the following analytical expressions for a doptimal design for the competitive model that has the form
The model contains three parameters, and, in this case, the doptimal design has an equal number of replicates at each of the three ([S], [I]) combinations. The values of the design variables s_{2}, s_{3}, and i_{3} are as follows: where K_{m}^{0} and K_{i}^{0} denote the prior values of the model parameters. Note also that in our model, [I]_{min} = 0. The substrate and inhibitor pairs and their replications identified for each study (based on parameter values estimated from the rich data sets) were as follows: for diclofenac∼fluconazole, for dextromethorphan∼sertraline,
In these displays, the pairs [such as (50, 0)] indicate the values of [S] and [I] to be applied to 10 or 7 samples, depending on the reaction being studied. As illustrated, all design support points were on the border of the design region for at least one of the substrate and inhibitor concentrations, and some were on the border for both concentrations. Given these doptimal designs, d efficiencies can be calculated for any other design (Atkinson et al., 2007). These efficiencies are such that a design with an efficiency of 50% requires twice as many trials as the doptimal design to provide parameter estimates of the same accuracy. The diclofenac∼fluconazole rich design had an efficiency of 0.254 on a per observation basis; the same efficiency for parameter estimation could be obtained with approximately 30 observations evenly split over the three support points of the doptimal design as that from the 120 observations from the rich design. For dextromethorphan∼sertraline, the d efficiency of the rich design is even less, 0.182; a design with seven observations at each of the points of the doptimal design provides greater efficiency than the rich design. The difference in the number of observations between the two studies arises from the dependence of the optimal design, as well as the efficiency of the design over a grid, on the prior values of the parameters. In theory, by running experiments that involve 30 and 21 observations, respectively, we should obtain parameter estimates that are as precise as those obtained from the 120 observations of the rich data sets. The results of Table 1 show that the experimental analysis supports our theory.
The conventional assay approach was subsequently repeated using these three support points with relevant replications of each. The estimates of the parameters determined from this experimental approach, using MATLAB (The MathWorks Ltd., Cambridge, UK) nonlinear leastsquares procedure nlinfit, are shown in Table 1. The parameter estimates, namely V_{max} and K_{m}, were predicted well using the doptimal design compared with those measured using the rich data sets. Estimates of the inhibition constants were of particular importance. For sertraline, the K_{i} toward CYP2D6 in the rich data set was estimated to be 2.6 μM, which agreed well with the value of 2.1 μM estimated using the doptimal design. The K_{i} for fluconazole toward CYP2C9 also compared well with an estimated value of 7.7 μM using the rich data set and 6.1 μM using the doptimal design. More importantly, the similarities between the S.E.s for each parameter, given the significant reduction in sample numbers compared with the richdata approach, demonstrate our cost saving implications using doptimal designs.
The aforementioned doptimal designs provide estimates of all three parameters with small variances. However, estimating the substrate/inhibitor pairings and relevant replicates requires retrospective analysis of the rich data set. The need to have this information up front, together with the availability of highthroughput IC_{50} screens providing an estimate of the K_{i} (Jones et al., 2009), limits the requirement to establish “bespoke optimized” K_{i} experimental designs. Depending on the mechanism, the K_{i} may reflect half the IC_{50} for competitive inhibitors or be equal to the IC_{50} for noncompetitive inhibitors. However, this approach may provide efficiency gains for compounds that progress during the developmental stage and at regulatory agencies that request more definitive measures of K_{i}. At this stage, there is likely to be sufficient information known about the parameter estimates, such as the K_{m} and V_{max} for substrates (probes) in the actual liver microsome matrix, against which the compound (inhibitor) is being tested against.
Given that K_{i} estimates from IC_{50} data could vary by 2fold (depending upon mechanism of inhibition), one could argue that there might be an advantage to having K_{i} determined earlier during discovery to better guide predictions of DDIs Obach et al., 2005, 2006). However, a strategy to replace the conventional IC_{50} assay must balance the need for data with cost effectiveness. The conventional approach for measuring K_{i} during early discovery clearly goes against this doctrine, as does the need to establish bespoke doptimal designs for every compound. Hence, the next steps will be to establish optimal designs that are not governed by discrete point estimates of inhibitory potency (i.e., IC_{50}) but rather designs that cover “regions” of potency, i.e., IC_{50} < 1 μM; 1 to 10 μM, >10 μM; a binning strategy often used by pharmaceutical companies as their firsttier approach to screen out compounds that pose a potential DDI risk. Such approaches are currently being investigated at the authors' institutions, using d_{S} optimality, which is an extension of d optimality.
Here, “s” indicates that interest is in a subset of the parameters in the model, i.e., K_{i}. In the estimation of just a single parameter (s = 1), a design is found for which the estimate has minimal variance. These designs differ from the doptimal designs shown above. In particular, the weights on the design points are often far from equal. In some cases, the designs are even singular, putting weight (as in our example) on less than three support points. Although not immediately useful, because not all model parameters can be estimated, such designs provide a reference against which the d_{S}efficiency of any other design can be calculated. Singular designs for nonlinear models have been reviewed previously (Atkinson et al., 2007). However, for illustrative purposes, the designs have been calculated for the current experimental data. Table 2 shows that the doptimal design has a d_{S} efficiency of 66.67% for dextromethorphan∼sertraline. This measure of efficiency was found by comparison with the d_{S}optimal design, which, for numerical reasons, was constrained to have a weight of at least 1 × 10^{−5} on the first support point. The rest of the weight is split equally between two points very close to those of the doptimal design. The last line of Table 3 shows the efficiencies for a design on these support points, with weights found to minimize the variance of the estimates of K_{i}. This design has a d_{S}efficiency of 100% and a d efficiency of only 4%. In between these extremes, a series of designs are presented in which the weights are in the ratio 1: r/r. The doptimal design corresponds to r = 1. Designs for r = 2, 3, and 4 are also presented. As the weights become less equal, the d efficiency decreases slowly and that for d_{S} increases. When r = 4 and the weights are one ninth, four ninths, and four ninths, the d and d_{S} efficiencies are 83.99 and 88.89. The results for diclofenac∼fluconazole (Table 3) are similar, where, as a result of r increasing, the designs become less balanced, resulting in increased d_{S} efficiency and decreased d efficiency. Values of 3 or 4 for r give designs that are not highly unbalanced and that have good efficiencies on both measures. It is not even necessary for r to be an integer. For example, with 30 measurements, the numbers at the three design points could be 4, 13, and 13, giving a value of 3.25 for r and a design with good d and d_{S} efficiencies. We have found that the optimal designs depend both on the prior estimates of the parameters K_{m} and K_{i} (although not on V_{max}) and on the assumed model. In practice, these parameter values will not be as well known as they are in our examples. However, if the value of the IC_{50} is known, only one design parameter remains unknown. Optimal designs and their efficiencies can then be calculated for a series of parameter values, and a design can be chosen with good efficiency over the range of values. If no such design can be found, d optimality can be extended by using the prior distribution of the parameters as weights in the calculation of a “Bayesian” design (Atkinson et al., 2007), which may require experiments at more than three combinations of concentrations. Likewise, compound optimality can be used (Atkinson et al., 2007) to find good designs when the mechanism of reaction is uncertain.
In conclusion, we have shown that d optimality can provide more efficient designs for estimating model parameters, including inhibitory K_{i}s. Such an approach may be of use for compounds that are in the later stage of drug development, where prior knowledge of potency can be used to guide these mathematical designs. Finally, because we have shown that doptimal designs can be applied successfully, this approach can be extended to include d_{s} optimality, where there is less reliance of prior knowledge of parameter estimates.
Acknowledgments.
We are grateful to Maurice Dickins and Barry Jones for their comments and guidance during the course of this work.
Footnotes
Article, publication date, and citation information can be found at http://dmd.aspetjournals.org.
doi:10.1124/dmd.110.033142.

ABBREVIATIONS:
 P450
 cytochrome P450
 DDI
 drugdrug interaction.
 Received March 11, 2010.
 Accepted April 16, 2010.
 Copyright © 2010 by The American Society for Pharmacology and Experimental Therapeutics