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- 1. ABSTRACT
- 2. INTRODUCTION
- 3. STATISTICAL METHODS
- 4. EXPLORATORY DATA ANALYSIS
- 5. DATA PREPROCESSING: OUTLIER REJECTION AND DATA STANDARDIZATION
- 6. RESULTS
- 7. SUMMARY
- 8. DISCUSSION AND PERSPECTIVE
- References

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Front Biosci (Elite Ed). Author manuscript; available in PMC 2010 November 5.

Published in final edited form as:

PMCID: PMC2974574

NIHMSID: NIHMS244728

Send correspondence to: J. Jack Lee, Department of Biostatistics, Unit 1411, The University of Texas M. D. Anderson Cancer Center, P. O. Box 301402, Houston, Texas 77230-1402, Tel: 713-794-4158, Fax: 713-563-4242, Email: gro.nosrednadm@eeljj

The publisher's final edited version of this article is available at Front Biosci (Elite Ed)

See other articles in PMC that cite the published article.

Applying the *E _{max}* model in a Lowe additivity model context, we analyze data from a combination study of trimetrexate (TMQ) and AG2034 (AG) in media of low and high concentrations of folic acid (FA). The

Due to complex disease pathways, combination treatments can be more effective and less toxic than treatments with a single drug regimen. Successful applications of combination therapy have improved treatment effectiveness for many diseases. For example, the combination of a non-nucleoside reverse transcriptase inhibitor or protease inhibitor with two nucleosides is considered a standard front-line therapy in the treatment of AIDS. Typically, a combination of three to four drugs is required to provide a durable response and reconstitution of the immune system (1). Another example is platinum-based doublet chemotherapy regimens as the standard of care for patients with advanced stage non–small-cell lung cancer (2). Combination treatments have also been shown to prevent and to overcome drug resistance in infectious diseases such as malaria and in complex diseases such as cancer (3, 4). Emerging developments in cancer therapy involve combining multiple targeted agents with or without chemotherapy, or combining multiple treatment modalities such as drugs, surgical procedures, and/or radiation therapy (5, 6).

How do we assess the effect of a combination therapy? It is a simple question, yet it requires a complex answer. A superficial way to answer the question is to determine that a combination therapy is working if its effect is greater than that produced by each single component given alone. The notion of classifying drug interaction as additive, synergistic, or antagonistic is logical and easily understood in a general sense, but can be confusing in specific application without consensus on a standard definition. Excellent reviews of drug synergism have been written by Berenbaum (7), Greco *et al* (8), Suhnel (9), Chou (10), and Tallarida (11), to name a few. In essence, to quantify the effect of combination therapy, we must first define drug synergy in terms of “additivity.” An effect produced by a combination of agents that is more (or less) than the additive effect of the single agents is considered synergistic (or antagonistic). Then, we must further assess drug interaction in a statistical sense. Under a more rigorous definition, synergy occurs when the combined drug effect is statistically significantly higher than the additive effect. Conversely, antagonism occurs when the combination effect is statistically significantly lower than the additive effect.

Despite the controversy arising from multiple definitions of additivity or no drug interaction, the Loewe additivity model is commonly accepted as the gold standard for quantifying drug interaction (7–11). The Loewe additivity model is defined as

$$\frac{{d}_{1}}{{D}_{y,1}}+\frac{{d}_{2}}{{D}_{y,2}}=1.$$

(E1)

Here *y* is the predicted additive effect at the combination dose (*d _{1}*,

$$\frac{{d}_{1}}{{{f}_{1}}^{-1}(y)}+\frac{{d}_{2}}{{{f}_{2}}^{-1}(y)}=1.$$

(E2)

Note that (*E*2) involves an unknown variable *y*. By solving equation (E2), the predicted additive effect *y _{add}* can be obtained under the Loewe additivity model. Denote that the observed mean effect is

Alternatively, to measure and quantify the magnitude of drug interaction, the interaction index (*II*) can be defined as

$$II=\frac{{d}_{1}}{{D}_{{y}_{\mathit{obs}},1}}+\frac{{d}_{2}}{{D}_{{y}_{\mathit{obs}},2}}$$

(E3)

Note that *II* < 1, *II* =1, and *II* >1 correspond to the drug interaction being synergistic, additive, and antagonistic, respectively. Chou and Talalay (12) proposed the following median effect equation (E4) to characterize the dose-effect relationship in combination studies:

$$E(d)=\frac{{(d/{ED}_{50})}^{m}}{1+{(d/{ED}_{50})}^{m}},$$

(E4)

where ED_{50} is the dose required to produce 50% of the maximum effect. Although the median effect equation can be applied in many settings, it assumes that when *m* is positive, *E*(*d*)=0 for *d*=0 and *E*(*d*)=1 for *d*=∞. On the other hand, when *m* is negative, *E*(*d*)=1 for *d*=0 and *E*(*d*)=0 for *d*=∞. If we assume that the data follow the median effect equation, a linear relationship can be found by plotting the logit transformation of the effect versus the logarithm transformed dose. A more detailed account of the interpretation and use of the interaction index can be found in a number of references (13–16). Several methods for constructing the confidence interval estimation of the interaction index were proposed by Lee and Kong (17).

To help advance research developing and comparing methods for analyzing data for combination studies, Dr. William R. Greco at the Roswell Park Cancer Institute has organized an effort and invited several groups to participate in an exercise to compare rival modern approaches to model data from two-agent concentration-effect studies. We describe the data and statistical methods, including the *E _{max}* model, and the calculation of the interaction index under the

Two data sets provided by Dr. Greco are used to examine the effect of the combination treatment of trimetrexate (TMQ) and AG2034 (AG) in HCT-8 human ileocecal adenocarcinoma cells. The cells were grown in a medium with two concentrations of folic acid: 2.3 μM (the first data set, called low FA) and 78 μM (the second data set, called high FA). Trimetrexate is a lipophilic inhibitor of the enzyme dihydrofolate reductase; and AG2034 is an inhibitor of the enzyme glycinamide ribonucleotide formyltransferase. The experiment was conducted on 96-well plates. The endpoint was cell growth measured by an absorbance value (ranging from 0 to 2) and recorded in an automated 96-well plate reader. Each 96-well plate included 8 wells as instrumental blanks (no cells); the remaining 88 wells received drug applications. The experiments were performed using the “ray design,” which maintains a fixed dose ratio between TMQ and AG in a series of 11 dose dilutions. With 88 wells in each plate, each 5-plate stack allowed for an assessment of the combination doses at 7 curves (i.e., design rays) plus a “curve” with all controls. Two stacks were used for studying 14 design rays: TMQ only, AG only, and twelve other design rays with a fixed dose ratio (TMQ:AG) for each ray. The fixed dose ratios in the low FA experiment were 1:250, 1:125, 1:50, 1:20, 1:10, 1:5 (2 sets), 2:5, 4:5, 2:1, 5:1, and 10:1. The fixed dose ratios in the high FA experiment were 1:2500, 1:1250, 1:500, 1:200, 1:100, 1:50 (2 sets), 1:25, 2:25, 1:5, 1:2, and 1:1. Data from each of the 16 curves (2 for controls, 2 for single agents, and 12 for combinations) are grouped together. Curves 1–8 were performed on the first stack with curve 8 serving as the “control” experiment while curves 9–16 were performed on the second stack with curve 16 serving as the “control” experiment. The assignment of different drug combinations to the cells in the wells was randomized across the plates. Five replicate plates were used for each set of two stacks, resulting in a total of 10 plates for each of the two medium conditions (low FA and high FA). The maximum number of treated wells per medium condition is 880 (16 curves × 11 dilutions × 5 replicates). Complete experimental details and mechanistic implications were reported by Faessel *et al* (18).

Due to a plateau of the measure of cell growth such that it does not reach zero at the maximum dose levels used in the experiments, the median effect equation (E4) does not fit the data. Instead, we take the *E _{max}* model (19) to fit the data at hand.

$$E(d)={E}_{0}-{E}_{\mathit{max}}+\frac{{E}_{\mathit{max}}}{1+{(d/{ED}_{50})}^{m}},$$

(E5)

where *E _{0}* is the base effect, corresponding to the measurement of cell growth when no drug is applied;

As when using the median effect model, the *E _{max}* model can be applied to fit the single-drug and combination drug dose-response curves, and then the interaction index can be calculated accordingly. Although equation (E5) allows for different values of

Hereafter, we assume the dose-response curve follows the *E _{max}* model given in (

$$E(d)=1-{E}_{\mathit{max}}+\frac{{E}_{\mathit{max}}}{1+{(d/{ED}_{50})}^{m}}.$$

(E6)

The experiments we analyzed studied the ability of the combination treatments to inhibit the growth of cancer cells. The measure of the treatment effect was cell growth corresponding to the number of cells observed. Hence, the height of the dose-effect curve decreases when the dose increases. In this case, we have *m* > 0. In addition, as *d* goes to infinity, the effect plateaus at 1−*E _{max}*. Hence,

In a study of two-drug combinations, we need to fit three curves using the *E _{max}* model: curve 1 for drug 1 alone, curve 2 for drug 2 alone, and curve

$$d(e)={ED}_{50}{\left(\frac{1-e}{e-1+{E}_{\mathit{max}}}\right)}^{1/m}$$

(E7)

Note that the dose for the combination treatment is simply the sum of the doses of the single agents. This approach works well for a ray design with constant or varying relative potency between the two drugs (12, 17). Without loss of generality, we can assume that *E _{max, 1}* >

$$\begin{array}{l}\widehat{II}=\frac{{\widehat{d}}_{c}(e)\times p/(1+p)}{{\widehat{D}}_{y,1}(e)}+\frac{{\widehat{d}}_{c}(e)/(1+p)}{{\widehat{D}}_{y,2}(e)}\phantom{\rule{0.16667em}{0ex}}\text{for}\phantom{\rule{0.16667em}{0ex}}1-{\widehat{E}}_{\mathit{max},2}<e<1,\phantom{\rule{0.38889em}{0ex}}\text{and}\\ \widehat{II}=\frac{{\widehat{d}}_{c}(e)\times p/(1+p)}{{\widehat{D}}_{y,1}(e)}\phantom{\rule{0.38889em}{0ex}}\text{for}\phantom{\rule{0.16667em}{0ex}}1-{\widehat{E}}_{\mathit{max},1}<e\le 1-{\widehat{E}}_{\mathit{max},2}.\end{array}$$

(E8)

For *e* ≤ 1 − *Ê _{max,}*

We can apply the delta method to calculate the (large sample) variance of the interaction index (23). From our previous work (17), we found that better estimation of the confidence interval for the interaction index can be achieved by working on the logarithmic transformation of the interaction index.

By applying the delta method,
$\mathit{Var}(\mathit{log}(\widehat{II}))=\frac{1}{{\widehat{II}}^{2}}\mathit{Var}(\widehat{II})$. When 1 − *Ê _{max,}*

$$\mathit{Var}(\widehat{II})={\left(\frac{{\widehat{d}}_{c}(e)\times p/(1+p)}{{\widehat{D}}_{y,1}(e)}\right)}^{2}{V}_{1}+{\left(\frac{{\widehat{d}}_{c}(e)/(1+p)}{{\widehat{D}}_{y,2}(e)}\right)}^{2}{V}_{2}+{\left(\frac{{\widehat{d}}_{c}(e)\times p/(1+p)}{{\widehat{D}}_{y,1}(e)}+\frac{{\widehat{d}}_{c}(e)/(1+p)}{{\widehat{D}}_{y,2}(e)}\right)}^{2}{V}_{c}$$

(E9)

if 1 − *Ê _{max,}*

$$\mathit{Var}(\widehat{II})={\left(\frac{{\widehat{d}}_{c}(e)\times p/(1+p)}{{\widehat{D}}_{y,1}(e)}\right)}^{2}({V}_{1}+{V}_{c}).$$

(E10)

where

$${V}_{i}=\left(\frac{-1}{{\widehat{m}}_{i}(e-1+{\widehat{E}}_{\mathit{max},i})}\frac{1}{{\widehat{ED}}_{50,i}}\phantom{\rule{0.16667em}{0ex}}\frac{1}{{\widehat{m}}_{i}^{2}}\mathit{log}\frac{1-e}{e-1+{\widehat{E}}_{\mathit{max},i}}\right)\mathit{Var}\left({\widehat{E}}_{\mathit{max},i},{\widehat{ED}}_{50,i},{\widehat{m}}_{i}\right)\left(\begin{array}{c}\frac{-1}{{\widehat{m}}_{i}(e-1+{\widehat{E}}_{\mathit{max},i})}\\ \frac{1}{{\widehat{ED}}_{50,i}}\\ -\frac{1}{{\widehat{m}}_{i}^{2}}\mathit{log}\frac{1-e}{e-1+{\widehat{E}}_{\mathit{max},i}}\end{array}\right)$$

for *i*=*1*, *2*, *c*.

Upon calculation of the variance for
$\mathit{log}(\widehat{II})$, the point-wise (1-α)100% confidence interval for the interaction index (*II*) for a specified effect can be constructed as

$$\left(\widehat{II}\phantom{\rule{0.16667em}{0ex}}\mathit{exp}(-{z}_{\alpha /2}\sqrt{\mathit{Var}(\mathit{log}(\widehat{II})}),\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\widehat{II}\phantom{\rule{0.16667em}{0ex}}\mathit{exp}({z}_{\alpha /2}\sqrt{\mathit{Var}(\mathit{log}(\widehat{II})})\right)$$

(E11)

where *z _{α/}*

The overall objective of the data analysis is to assess the synergistic effect of the combination of TMQ and AG in both low and high FA media. We apply the exploratory data analysis first, and then estimate the dose-response relationship using the *E _{max}* model. We evaluate the drug interaction by calculating the interaction index under the Loewe additivity model. We perform an exploratory data analysis in order to understand the data structure and patterns and to determine whether preprocessing of the data in terms of outlier rejection and standardization would be required prior to data modeling. We analyze the low FA and high FA experiments separately then compare the results. For each experiment, we apply the

As in all data analyses, we begin with an exploratory data analysis. For the low and high FA experiments, there are 871 and 879 readings, respectively. Only 9 and 1 observations, respectively, are missing out of a maximum of 880 readings in each experiment. The data include designated curve numbers ranging from 1 to 16 and data point numbers ranging from 1 to 176. Each curve number indicates a specific dose combination. We re-label the curves as A-P where A and B correspond to the control (no drug) curves; C and D correspond to the curves of TMQ and AG administered alone, and curves E through P correspond to the combination curves with fixed dose ratios in ascending order. Each point number indicates the readings at each specific dilution of each curve. Because five duplicated experiments were performed, there are up to five readings for each specific point number. There is, however, no designation of the plate number in the data received. Figure 2 shows the variable percentile plot of the distribution of the effect from the low FA and high FA experiments using the BLiP plot, with each segment corresponding to a five percent increment (25). The plot gives an overall assessment of the distribution of the outcome variable of cell growth without conditioning on experimental settings. The middle 20% of the data (40^{th} to 60^{th} percentiles) are shaded in a light orange color. This figure indicates that the data have a bimodal distribution with most data clustered around either a low value of 0.2 or a high value of 1.2. For the low FA experiment, the distribution of the effect ranges from 0.072 to 1.506 with the lower, middle, and upper quartiles being 0.149, 0.449, and 1.150, respectively. Similarly, for the high FA experiment, the effect ranges between 0.070 and 1.545. The three respective quartiles are 0.213, 0.990, and 1.1495. The median of the data from the low FA experiment is smaller than the median of the data from the high FA experiment. The bimodal distributions could result from steep dose-response curves. As a consequence, the slope may not be estimated well in certain cases.

Variable width percentile plot for the observed effect in experiments with low and high folic acid media. Each vertical bar indicates a five percent increment. The middle 20% of the data are shaded in a light orange color.

To help understand the pattern of the fixed ratio dose assignment in a ray design and the relationship between the fixed ratio doses and curve numbers, we plot the logarithm transformed dose of TMQ and AG in Figure 3 for both the low FA and high FA experiments. As can be seen, curves A and B are the controls with no drugs. Curves C and D correspond to the single drug study of TMQ and AG, respectively. Curves E through P are the various fixed ratio combination doses of TMQ and AG. Note that curves J and K have the same dose ratios. Within each curve, the 11 dilutions are marked by 11 circles. For the combination studies, the curves for different dose ratios are parallel to each other on the log dose scale. If the same plot is shown in the original scale, these lines will form “rays,” radiating out from the origin like sun rays. Hence, the term “ray design” is an appropriate name for this type of experiment. The corresponding dose ranges used for each drug alone are 5.47×10^{−6} to 0.56 μM for TMQ in both the low FA and high FA experiments, and 2.71 ×10^{−5} to 2.78 μM for AG2034 in the low FA experiment and 2.71×10^{−4} to 27.78 μM in the high FA experiment.

Experimental design showing the logarithmically transformed AG2034 (AG) dose versus the logarithmically transformed trimetrexate (TMQ) dose in the fixed ratio experiments. 16 curves are shown. Curves A and B are controls; no drugs applied. Curves C and **...**

Figures 4 and and55 show the raw data of the effect versus dose level by curve for the low FA and high FA experiments, respectively. Instead of using the actual dose, we plot the data using a sequentially assigned dose level to indicate each dilution within each curve such that the data can be shown clearly. In addition, the data points at each dilution for each curve are coded from 1 to 5 according to the order of the appearance in the data set. We assume that these numbers correspond to the replicate number for each design point (the well position in the stack of 5 plates). Because the plate number was not listed in the data, we are not certain that this is the case. From the plot, we can see that there are outliers in several dilution series. Of note, in Figure 4, the effects from plate (replicate) #1 in curves B, E, F, and K tend to be lower than all other replicates. There are also some unusually large values, for example, in replicate 2 in curve A, dose level (dilution series) 6; replicate 3 in curve L, dose level 4; and replicate 2 in curve M, dose level 1. Similar observations can be made for the high FA experiment: plate #1 seems to have some low values in curves B, C, H, I, and J, and plate #4 seems to have some low values in curves E, K, N, O, and P. These findings indicate that certain procedures need to be performed to remove the obvious outliers in order to improve the data quality before the data analysis.

Distribution of the effect versus dose level for curves A through P for the experiment in a low folic acid medium.

Distribution of the effect versus dose level for curves A through P for the experiment in a high folic acid medium.

Figure 6 shows the perspective plot, contour plot, and image plot for the low FA experiment. From the perspective plots in Figure 6.A (back view), B (front view), and C (side view), we can see that the effect starts at a high plane plateau at an effect level of about 1.2 when the doses of TMQ are AG are small. As the dose of each drug increases, the effect remains approximately constant for a while and then a sudden drop occurs. This steep downward slope can be found by taking the trajectory of any combination of the TMQ and AG doses; it is also evident in the dose-response curves shown in Figures 4 and and5.5. The steep drop of the effect can also be found in the contour plot and the image plot. Similar patterns in the dose-response relationship are shown in Figure 7 for the high FA experiment. The steep drop of the effect occurs at smaller doses in the low FA experiment and at larger doses in the high FA experiment.

Perspective plots (A, B, C), contour plots (D, E), and image plot (F) for the effect versus logarithm transformed doses of trimetrexate and AG2034 for the experiment in a low folic acid medium.

To address the concern that outliers may adversely affect the analysis outcome, we devise the following simple plan. For each of the 176 point numbers (16 curves × 11 dilutions), the five effect readings should be close to each other because they are from replicated experiments. However, because the plate number is not in the data set, we cannot assess the plate effect. Neither can we reject a certain replicate plate entirely should there be a plate with outlying data, nor apply a mixed effect model treating the plate effect as a random effect. For the four or five effect readings in each point number (only 9 point numbers in the low FA and 1 in the high FA experiments have 4 readings), we compute the median and the interquartile range. An effect reading is considered an outlier if the value is beyond the median ± 1.4529 times the interquartile range. If the data are normally distributed (i.e., follow a Gaussian distribution), the range expands to cover the middle 95% of the data. Hence, only about 5% of the data points (2.5% at each extreme) are considered outliers. The number 1.4529 is obtained by qnorm(.975)/(qnorm(.75) - qnorm(.25)) where qnorm(*x*) is a quantile function which returns the *x*^{th} percentiles from a normal distribution. Upon applying the above rule, 129 out of 871 (14.8%) effect readings in the low FA experiment and 126 out of 879 (14.3%) in the high FA experiment are considered outliers and are removed before proceeding to further analysis. The numbers of outliers in replicates 1 to 5 are 60, 28, 19, 14, and 8 for the low FA experiment and 35, 18, 21, 34, and 18 for the high FA experiment, indicating a non-random pattern of outliers that could be attributed to experimental conditions. Note that the outlier rejection algorithm is only applied “locally.” In other words, it only applies to the replicated readings up to five replicates in each of the 176 experimental conditions.

After outliers are removed from the data, we compute the mean of the control curves. The means for curves 8 and 16 are 1.1668 and 1.1534 for the low FA experiment and 1.1483 and 1.1477 for the high FA experiment, respectively. To apply the *E _{max}* model in equation (E6) with

The *E _{max}* model in equation (E6) is applied to fit all of the dose-response curves. For the low FA experiment, the parameter estimates, their corresponding standard errors, and the residual sum of squares are given in Table 1. The dose-response relationships showing the data and the fitted curves are displayed in Figure 8. Note that although model fitting is performed on the original dose scale, the dose is plotted on the logarithmically transformed scale to better show the dose-response relationship. The fitted marginal dose-response curves for TMQ (curve C) and AG (curve D) are shown in a blue dashed line and a red dotted line, respectively. From Table 1, we see that
${\widehat{ED}}_{50}$ is 0.00133 for TMQ and 0.00621 for AG, indicating that TMQ is about 4.7 times more potent than AG at the

Effect versus logarithmically transformed dose plot for the combination study of trimetrexate and AG2034 in a low folic acid medium. Raw data are shown in open circles. Blue dashed line and red dotted line indicate the fitted marginal dose-response curves **...**

In all dose-response curves, the standardized effect level starts to drop between dose levels (dilutions) 3 to 6. Once the effect starts to drop, it drops quickly and plateaus at the 1 − *Ê _{max}* level. There are ample data points at the effect levels around 1 (dose levels 1–4) and 1 −

Based on the fitted dose-response curve, the interaction index (II) can be calculated over the entire effect range and at specific dose combinations. Table 2 gives a detailed result of the estimated interaction index and its 95% point-wise confidence interval at each dose combination for each combination curve. The II is calculated at the predicted effect level from the combination curve and not at the observed effect level. The results are shown in a trellis plot in Figure 9 where the red lines represent the 95% point-wise confidence intervals at each specific effect level and the black dashed lines indicate the 95% simultaneous confidence bands of the II for the entire range. From the figure we find that the interaction index can be estimated with very good precision in all curves except at the two extremes when the effect is close to 1 or 1 − *Ê _{max}*. The trend and the pattern of the interaction index are clearly shown in these figures. For curves E through K, i.e., with a TMQ:AG dose ratio ranging from 0.004 to 0.2, synergy is observed in the effect range between 0.2 to 0.9. For curves L and M, which have TMQ:AG ratios of 0.4 and 0.8, we see that synergy is observed at the low effect level from 0.2 to about 0.5. Beyond 0.5 the combinations are generally additive. For curves N, O, and P, with TMQ:AG ratios of 2, 5, and 10, the synergistic effect is lost and we see additivity in all dose ranges.

Trellis plot of the estimated interaction index (solid line) and its point-wise 95% confidence interval (red solid lines) and the 95% simultaneous confidence band (dashed lines) for the low folic acid experiment. Estimates at the design points where experiments **...**

Table 3 lists the parameter estimate, corresponding standard error, and sum of squares for all the curves in the high FA experiment. Unlike in the experiment using low FA media, the model fitting for all curves in the high FA experiment converge when using the *E _{max}* model. The estimated

Effect versus logarithmically transformed dose plot for the combination study of trimetrexate and AG2034 in a high folic acid medium. Raw data are shown in open circles. Blue dashed line and red dotted line indicate the fitted marginal dose-response curves **...**

Trellis plot of the estimated interaction index (solid line) and its point-wise 95% confidence interval (red solid lines) and the 95% simultaneous confidence band (dashed lines) for the high folic acid experiment. Estimates at the design points where **...**

In both the low FA and high FA experiments, TMQ is more potent than AG. At low TMQ:AG ratios, i.e., when a small amount of the more potent drug (TMQ) is added to a larger amount of the less potent drug (AG), synergy is achieved. However, when the TMQ:AG ratio reaches 0.4 or larger for the low FA medium, or when the TMQ:AG ratio reaches 1 or larger for the high FA medium, synergy decreases, or the interaction becomes additive. In general, a synergistic effect in a drug combination dilution series is stronger at higher doses that produce stronger effects (effects closer to 1−*E _{max}*) than at lower dose levels that produce weaker effects (effects closer to 1). The two drugs in this study are more potent in the low FA medium compared to the high FA medium. The drug synergy, however, is stronger in the high FA medium.

The data supplied by Dr. Greco provide an excellent opportunity to apply and compare various approaches for studying the effects of combination drug treatments. For the median effect model, a linear relationship between the logit transformed effect and the log-dose makes the model fitting straightforward and easy. However, when measuring cell growth, as in the experiments we analyzed, if the maximum drug effect reaches a plateau and does not kill all the cancer cells, even at the highest experimental doses, the median effect model (12) does not apply. We used the *E _{max}* model (19), which provides an adequate fit for most data. Parameter estimation under the

Upon construction of the marginal and combination dose-response curves, we applied the Loewe additivity model to compute the interaction index. We note that a definition of drug interaction such as the interaction index is model dependent. Additionally, no matter which model is used, based on the definition of the interaction index (7,8), the dose levels used in calculating the interaction index must be translated back to the original units of dose measurement. Under the given model, we found that the drug interaction between TMQ and AG was largely synergistic. Synergy was more clear and evident in the high FA experiment than in the low FA experiment. In addition, synergy was more likely to be observed when a small dose of the more potent drug (TMQ) was added to a large dose of the less potent drug (AG). When a large amount of a more potent drug is present, adding the less potent drug does not show synergy because the effect is already largely achieved by the more potent drug. In addition, the interval estimation showed that the 95% confidence intervals were wider at the two extremes of the effect, which were closer to 1 or to 1−*E _{max}*. This result is consistent with that of many regression settings in which estimation achieves higher precision in the center of the data distribution but lower precision at the extremes.

We have provided a simple, yet useful approach for analyzing drug interaction for combination studies. The interaction index for each fixed dose ratio is computed and then displayed together using a trellis plot. This method works well for the ray design. Other methods have been proposed to model the entire response surface using the parametric approach (27) or the semiparametric approach (28). The results from applying the semiparametric model are reported in a companion article (29).

We thank Dr. William R. Greco at the Roswell Park Cancer Institute for organizing this project comparing rival modern approaches to analyzing combination studies, for supplying the data sets, and for the invitation to present this manuscript. The authors also thank Lee Ann Chastain for her editorial assistance. This work is supported in part by grants W81XWH-05-2-0027 and W81XWH-07-1-0306 from the Department of Defense, and grant CA16672 from the National Cancer Institute. J. Jack Lee’s research was supported in part by the John G. & Marie Stella Kenedy Foundation Chair in Cancer Research.

- AG
- AG2034, an inhibitor of the enzyme glycinamide ribonucleotide formyltransferase
- ED
_{50} - dose required to produce 50% of the maximum effect
- E
_{max} - maximum effect attributed to the drug
- FA
- folic acid
- II
- interaction index
- TMQ
- Trimetrexate, a lipophilic inhibitor of the enzyme dihydrofolate reductase

1. Rathbun R Chris, Lockhart Staci M, Stephens Johnny R. Current HIV treatment guidelines - an overview. Curr Pharm Des. 2006;12:1045–1063. [PubMed]

2. Molina Julian R, Adjei Alex A, Jett James R. Advances in chemotherapy of non-small cell lung cancer. Chest. 2006;130:1211–1219. [PubMed]

3. Nyunt Myaing M, Plowe Christopher V. Pharmacologic advances in the global control and treatment of malaria: combination therapy and resistance. Clin Pharmacol and Ther. 2007;82:601–605. [PubMed]

4. Sankhala Kamalesh K, Papadopoulos Kyriakos P. Future options for imatinib mesilate-resistant tumors. Expert Opinion Investig Drugs. 2007;16:1549–1560. [PubMed]

5. Bianco Roberto, Damiano Vincenzo, Gelardi Teresa, Daniele Gennaro, Ciardiello Fortunato, Tortora Giampaolo. Rational combination of targeted therapies as a strategy to overcome the mechanisms of resistance to inhibitors of EGFR signaling. Curr Pharm Des. 2007;13:3358–3367. [PubMed]

6. Sathornsumetee Sith, Reardon David A, Desjardins Annick, Quinn Jennifer A, Vredenburgh James J, Rich Jeremy N. Molecularly targeted therapy for malignant glioma. Cancer. 2007;110:12–24. [PubMed]

7. Morris C. Berenbaum. What is synergy? Pharmacol Rev. 1989;41:93–141. [PubMed]

8. Greco William R, Bravo Gregory, Parsons John C. The search of synergy: A critical review from a response surface perspective. Pharmacol Rev. 1995;47(2):331–385. [PubMed]

9. Suhnel Jurgen. Parallel dose-response curves in combination experiments. Bull Math Biol. 1998;60:197–213. [PubMed]

10. Chou Ting-Chao. Theoretical basis, experimental design, and computerized simulation of synergism and antagonism in drug combination studies. Pharmacol Rev. 2006;58:621–681. [PubMed]

11. Tallarida Ronald J. An overview of drug combination analysis with isobolograms. J Pharmacol Exp Ther. 2006;319:1–7. [PubMed]

12. Chou Ting-Chao, Talalay Paul. Quantitative analysis of dose effect relationships: the combined effects of multiple drugs or enzyme inhibitors. Adv Enzyme Reg. 1984;22:27–55. [PubMed]

13. Machado Stella G, Robinson Grant A. A direct, general approach based on isobolograms for assessing the joint action of drugs in pre-clinical experiments. Stat Med. 1994;13:2289–2309. [PubMed]

14. Gennings Chris. On testing for drug/chemical interactions: definitions and inference. J Biopharm Stat. 2000;10:457–467. [PubMed]

15. Dawson Kathryn S, Carter Walter H, Jr, Gennings Chris. A statistical test for detecting and characterizing departures from additivity in drug/chemical combinations. J Agric Biol Environ Stat. 2000;5:342–359.

16. Lee J Jack, Kong Maiying, Ayers Gregory D, Lotan Reuben. Interaction index and different methods for determining drug interaction in combination therapy. J Biopharm Stats. 2007;17:461–480. [PubMed]

17. Lee J Jack, Kong Maiying. A confidence interval for interaction index for assessing multiple drug interaction. Stat Biopharm Res. 2009;1:4–17. [PMC free article] [PubMed]

18. Faessel Helene M, Slocum Harry K, Jackson Robert C, Boritzki Theodore J, Rustum Youcef M, Nair MG, Greco William R. Super in vitro synergy between inhibitors of dihydrofolate reductase and inhibitors of other folate-requiring enzymes: The critical role of polyglutamylation. Cancer Res. 1998;58:3036–3050. [PubMed]

19. Ting Naitee. Dose finding in drug development. Springer; NY: 2006. pp. 127–145.

20. Chambers John M, Hastie Trevor J. Statistical models in S. Chapman & Hall/CRC Press; Boca Raton FL: 1992. pp. 450–452.

21. Huet Sylvie, Bouvier Anne, Poursat Marie-Anne, Jolivet Emmanuel. Statistical tools for nonlinear regression: a practical cuide with S-PLUS and R examples. 2. Springer; NY: 2003.

22. SAS/STAT 9.1 User’s guide. SAS Institute; Cary, NC: 2007.

23. Bickel Peter J, Doksum Kjell A. Mathematical statistics: basic ideas and selected topics. 2. I. Prentice Hall; New Jersey: 2000. pp. 306–314.

24. Cox Christopher, Ma Guangpeng. Asymptotic confidence bands for generalized nonlinear regression models. Biometrics. 1995;51:142–150. [PubMed]

25. Lee J Jack, Tu Z Nora. A versatile one-dimensional distribution plot: the BLiP plot. Amer Statistician. 1997;51(4):353–358.

26. Crawley Michael J. The R book. Wiley; NY: 2007.

27. Kong Maiying, Lee J Jack. A general response surface model with varying relative potency for assessing drug interactions. Biometrics. 2006;62(4):986–995. [PubMed]

28. Kong Maiying, Lee J Jack. A semiparametric response surface model for assessing drug interactions. Biometrics. 2008;64:396–405. [PubMed]

29. Kong Maiying, Jack Lee J. Applying *E*_{max} model and bivariate thin plate splines to assess drug interactions. Front Biosci. In Press. [PMC free article] [PubMed]

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