A Two-Stage Response Surface Approach to Modeling Drug Interaction

Transcription

1 This article was downloaded by: [FDA Biosciences Library] On: 27 October 2012, At: 12:44 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Statistics in Biopharmaceutical Research Publication details, including instructions for authors and subscription information: A Two-Stage Response Surface Approach to Modeling Drug Interaction Wei Zhao a, Lanju Zhang a, Lingmin Zeng a & Harry Yang a a MedImmune, LLC, Gaithersburg, MD, Version of record first published: 01 Oct To cite this article: Wei Zhao, Lanju Zhang, Lingmin Zeng & Harry Yang (2012): A Two-Stage Response Surface Approach to Modeling Drug Interaction, Statistics in Biopharmaceutical Research, 4:4, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 A Two-Stage Response Surface Approach to Modeling Drug Interaction Wei ZHAO, Lanju ZHANG, Lingmin ZENG, and Harry YANG Studies of drug combinations have become increasingly important, especially in treating malignant cancers. Researchers are interested in identifying compounds that act synergistically when combined. Such synergy is usually measured through an interaction index. The existing statistical methods, in general, estimate the interaction index using pooled data from compounds administered individually and in combination. In this article, we propose a two-stage response surface approach. Parameters of monotherapy dose response curves are estimated and then incorporated in estimating the interaction index through a quadratic response surface model. Using multiple simulation studies, we demonstrate that the new method gives less biased estimates for both monotherapy dose response curves and interaction index. Also developed is a bootstrapping method that allows constructing a confidence interval for interaction index at any combination dose levels. An example is provided to illustrate the method. Key Words: Bootstrap; Drug combination; Loewe additivity; Nonlinear least squares; Simplex algorithm. 1. Introduction Advanced tumors are often resistant to single agents. There is an increasing trend to combine drugs to achieve better treatment effect and reduce safety issues (Ramaswamy 2007; Sun et al. 2011). It is desirable that the combination drugs are synergistic; that is, optimal treatment effect is achieved at lower dose levels when drugs are combined. The interaction effect of combination drugs can be synergistic, additive, or antagonistic. Commonly used models are Bliss independence (Bliss 1939) and Loewe additivity model (Loewe and Muischnek 1926). Greco, Bravo, and Parsons (1995) discussed these two reference models in detail. Generally, the Bliss independence is less desirable because it may incorrectly claim synergy or antagonism when two identical drugs are combined (sham experiment). Therefore, the reference model that we use in this article is Loewe additivity, which has been preferred by many authors (Chou and Talalay 1984; Greco, Bravo, and Parsons 1995; Kong and Lee 2006; Fang et al. 2008). Based on Loewe additivity, Berenbaum (1985) proposed to use interaction index to quantify the interaction between two drugs. The interaction index is defined as τ = d 1 + d 2, D y,1 D y,2 where τ is the interaction index at combination dose level (d 1, d 2 ) for drugs 1 and 2, and D y,i is the monotherapy dose level for drug i (i = 1, 2) that achieves treatment effect y. The ratio d i /D y,i can be thought intuitively to represent a standardized dose of drug i, and then τ can be interpreted as the standardized combination dose. τ<1 means that the same treatment effect can be achieved at a lower combination dose level; τ>1means that more drugs have to be given to achieve the same treatment effect; and τ = 1 means that the treatment effects are C American Statistical Association Statistics in Biopharmaceutical Research November 2012, Vol. 4, No. 4 DOI: /

3 Statistics in Biopharmaceutical Research: November 2012, Vol. 4, No. 4 additive and there is no advantage or disadvantage to combine them. More concisely, the three scenarios are summarized as < 1 synergy, τ = 1 additivity, > 1 antagonism. Many models have been proposed to estimate the interaction index. Some models use just one parameter to describe drug interaction across all combination dose levels. For example, methods proposed by Greco, Park, and Rustum (1990), Machado and Robinson (1994), and Plummer and Short (1990) are all in this category. On the other hand, saturated models (Lee and Kong 2009; Harbron 2010) calculate the interaction index separately for every combination. In this case, the number of parameters is as many as the number of combination doses. Other modeling approaches use fewer parameters than saturated models. For example, the response surface model of Kong and Lee (2006) includes six parameters to describe the interaction index. Harbron (2010) provided a unified framework that accommodates a variety of linear, nonlinear, and response surface models. Some of these models can be arranged in a hierarchical order so that a statistical model selection procedure can be performed. This is advantageous because, in practice, simple models tend to underfit the data and saturated models may use too many parameters and overfit the data. In existing models, monotherapy and drug combination data are pooled together for model fitting. Parameters describing monotherapy dose response curves and those modeling drug interaction indexes are estimated by fitting a single model. We refer to these as one-stage models. All existing models are one-stage models. In many situations, pooling data is a good statistical practice because it increases precision of parameter estimation. However, in the situation of drug combination, pooling data may compromise the accuracy of estimating drug interaction as will be demonstrated in this article. More specifically, we will show through simulation studies that the parameters estimated using one-stage models may significantly deviate from their true values, and thus potentially lead to false claims of synergy, additivity, and antagonism. In practice, effects of monotherapy treatments are usually evaluated before designing a drug combination study; the knowledge gained is used to optimize drug combination study designs. Fang et al. (2008) developed a method of interaction analysis befitting this two-stage paradigm. In their article, the monotherapy data are used to determine the drug response curve for each drug. Then an experiment for combination study is designed in some optimal sense, using the information from individual dose response curves. Finally, the interaction index is modeled using combination data only. However, the variability of individual dose response curves (or monotherapy data) is ignored in constructing a confidence interval for the interaction index. In this article, we propose a two-stage method to estimate the interaction index using a quadratic response surface model. The parameters in the interaction index model are estimated conditional on the estimates of monotherapy dose response parameters. The variances of model parameters are calculated using the bootstrap technique (Davison and Hinkley 2006) and their confidence intervals can be constructed at any combination dose levels. We also tackle a computational issue frequently encountered in drug combination analysis. Most models are nonlinear: monotherapy dose response models are nonlinear and interaction index models are nonlinear as well. Nonlinear least-square (NLS) method is commonly used to fit such models (Bates and Watts 1988; Lee et al. 2007; Harbron 2010). NLS is available in many existing statistical packages such as R and SAS and is easy to use once the models are correctly specified. However, NLS becomes unreliable for complicated nonlinear problems. We have experienced difficulties in parameter estimation using NLS, due to the singularity of Hessian matrices. Our investigation suggests that this issue is caused by the program s failing to calculate numerical derivatives. Since NLS uses a Newton Raphson type of procedure to perform parameter estimation, it relies heavily on the accuracy of numerical derivatives when the explicit form is not available. However, depending on the complexity of nonlinear equations, numerical derivatives may be difficult to get and NLS may stop working at any iteration. To improve the robustness of the model fitting, we develop a simplex method to search for parameters that maximize the likelihood function. Simplex method is a widely used direct search method to minimize or maximize an objective function (Zhao et al. 2004). The simplex algorithm, originally proposed by Nelder and Mead (1965), provides an efficient way to estimate parameters, especially when the parameter space is large. It is a direct search method for nonlinear unconstrained optimization. It attempts to minimize a scalar-valued nonlinear function using only function values, without any derivative information (explicit or implicit). The simplex algorithm uses a linear adjustment of the parameters until some convergence criterion is met. The term simplex arises because the feasible solutions for the parameters may be represented by a polytope figure called a simplex. The simplex is a line in one dimension, a triangle in two dimensions, and a tetrahedron in three dimensions. Since no division is required in the calculation, the divided by zero runtime error is also avoided. 376

4 Two-Stage Response Surface Approach The rest of the article is organized as follows. We describe the two-stage method in Section 2. In Section 3, we provide extensive simulation studies to demonstrate the performance characteristic of our method and show the consequence of pooling monotherapy and combination data. An example is provided in Section 4 and a discussion is given in Section Statistical Method In this section, parameter estimation and calculation of bootstrap variances are described in detail for the two-stage model. In Section 2.3, we use Greco s model to illustrate how data are fitted using the one-stage method. 2.1 First Stage In the first stage, the monotherapy dose effect is modeled using a median-effect (Greco, Bravo, and Parsons 1995) equation for each drug i. ( di ) mi D m,i y i = E i,max 1 + ( d i ) mi, D m,i where D m,i is median-effect dose of drug i, m i is a slope parameter, y i is the monotherapy drug effect at dose d i, and E i,max is the maximal effect of drug i. We assume E i,max = 1 in this article and the logistic regression model for the monotherapy dose effect data can be conveniently written in a linear regression form y i log = β 0,i + β 1,i log d i + ε i, (1) 1 y i where β 0,i = mlog D m,i and β 1,i = m i. In R, parameters can be easily estimated using simple linear regression function lm(). The distribution of ( ˆβ 0,i, ˆβ 1,i ) can be approximated by a bivariate normal distribution, N(( ˆβ 0,i ), ˆ ˆβ i ), where ˆ i is the estimated covariance 1,i matrix for ˆβ 0,i and ˆβ 1,i. 2.2 Second Stage In the second stage, interaction index τ is modeled using a quadratic response surface model τ = i=a,b d i exp ( β 0,i β 1,i )( y 1 y ) 1 β 1,i = exp ( γ 0 + γ 1 δ 1 + γ 2 δ 2 + γ 12 δ 1 δ 2 + γ 3 δ1 2 + γ 4δ2 2 ) = exp (XƔ), (2) where y is the response at the combination doses (d A, d B ), Ɣ ={γ 0,γ 1,γ 2,γ 12,γ 3,γ 4 }, δ i = log d i, and X is the design matrix. The response surface model we use in this article has six parameters so that it requires a dataset with at least six combination dose levels; otherwise, the model is not identifiable. If in fact there are fewer than six combination dose levels, one can remove some quadratic terms to make the model identifiable. If there is adequate number of combinations, higher-order terms can also be added. Conditional on ˆβ 0,i and ˆβ 1,i estimated in the first stage, Equation (2) defines an implicit function, f, between the expected logit response, y, and the combination doses, (d A, d B ). Since f is implicit, we solve y numerically using a bisection method. This relationship can be symbolically written as ( E log j=1 y 1 y ) ˆβ 0,i, ˆβ 1,i = f (Ɣ ˆβ 0,i, ˆβ 1,i, d A, d B ), d A d B 0. The model for combination dose response can be written as y log 1 y = f (Ɣ ˆβ 0,i, ˆβ 1,i, d A, d B ) + e. (3) Assuming e follows iid normal distribution N(0,σ 2 ), the log-likelihood function for the combination data is obtained as N log (2π) l = N log σ 2 ( ) 2 N log y j 1 y j f (Ɣ ˆβ 0,i, ˆβ 1,i, d 1, j, d 2, j ), 2σ 2 where N is the number of total combination data points. It is worth noting that this likelihood function is a conditional likelihood function, depending on ˆβ 0,i and ˆβ 1,i estimated in the first stage. The unknown interaction parameters, Ɣ, can be estimated by maximizing Equation (4). We propose to use the simplex method to estimate Ɣ. Conditional on ˆβ 0,i and ˆβ 1,i, ˆƔ approximately follows a multivariate normal distribution, N( ˆƔ, ˆ ˆβ 0,i, ˆβ 1,i ). Let j be the index of combination dose levels and F ={F j,k } be the Jacobian matrix, F j,k = f ( ˆƔ, d 1 j, d 2, j ˆβ 0,i, ˆβ 1,i ). ˆγ k Since f is implicit, its partial derivatives F j,k are also implicit and can only be calculated numerically. Then the estimated conditional asymptotic covariance matrix of ˆƔ can be written as ˆ = var( ˆƔ ˆβ 0,i, ˆβ 1,i ) = σ 2 (F F) 1. (5) (4) 377

5 Statistics in Biopharmaceutical Research: November 2012, Vol. 4, No. 4 By the theorem of conditional variance (Mood, Graybill, and Boes 1974), the covariance matrix of ˆƔ can be written as var ( ˆƔ ) = E ( var ( )) ˆƔ ˆβ 0,i, ˆβ 1,i + var ( E ( )) ˆƔ ˆβ 0,i, ˆβ 1,i. (6) Although an explicit form of var( ˆƔ) does not exist, it can still be calculated numerically. We propose a bootstrap method (Davison and Hinkley 2006) to calculate var( ˆƔ). First, a sample of ( ˆβ 0,i, ˆβ 1,i )isdrawnfrom N(( ˆβ 0,i ), ˆ ˆβ i ) distribution. Conditional on ( ˆβ 1,i 0,i, ˆβ 1,i ), interaction index parameters ˆƔ are calculated by maximizing Equation (4) using the simplex method and then conditional variance var( ˆƔ ˆβ 0,i, ˆβ 1,i ) is calculated using Equation (5). Repeat this exercise for a number of times and var( ˆƔ) can be approximated by replacing expectation and variance using sample mean and sample variance in Equation (6). In our experience, var( ˆƔ) becomes stabilized after 50 repeats. The 100(1 α)% confidence interval for interaction index τ at any combination dose (d A, d B ) can then be calculated using the following formula: ( ) CI = ˆτ exp ( z 1 α/2 x var( ˆƔ)x, )) ˆτ exp (z 1 α/2 x var( ˆƔ)x, where x = (1,δ 1,δ 2,δ 1 δ 2,δ 2 1,δ2 2 ) and z α/2 is the 100(α/2)th percentile of the standard normal distribution. The framework to calculate the interaction index is very general and other models such as Harbron s unified approach can be plugged in Equation (2) as well. 2.3 One-Stage Model There are many one-stage models in the literature; in this article, we use the one by Greco, Park, and Rustum (1990) as an example. The model is written as 1 = d 1 D m,1 ( y 1 y + ) 1 m 1 + d 2 D m,2 ( y 1 y ηd 1 d 2 ( D y ) 1 ( 2m m,1 1 y Dm,2 1 y 1 y ) 1 m2 ) 1 2m 2, where D m,i and m i are monotherapy parameters for drug i (i = 1, 2), the same as in Section 2.1; η is one single parameter to describe drug interaction at different dose levels. All the parameters are estimated using the pooled data. It is easily understood that the estimation of monotherapy parameters depends not only on how well the model fits the monotherapy data but also on the combination therapy data. On the other hand, the estimation of η depends on monotherapy data as well, which is not desirable since monotherapy and combination therapy effects are coupled and there is no way to quantify the interaction effect solely due to drug combination. 3. Simulation Study The simulation studies are designed to show that a one-stage model is not adequate when drug interactions are constant or varying across combination dose levels and also to demonstrate how the second-stage response surface model behaves when the first-stage model variances are increased. Six simulation studies are included. Logistic regression dose response curve is assumed for monotherapy drugs A and B. Two sets of parameters are taken for the monotherapy dose responses and three sets of parameters are taken for the combination dose response. The monotherapy models for drugs A and B are y A log = 2 + log d A + ε A, 1 y A y B log = logd B + ε B. 1 y B To make the simulation simple, var(ε A ) and var(ε B ) are assumed to be equal when generating the data, but they are estimated separately in the actual calculation. Two variances are used in the simulation, var(ε A ) = var(ε B ) = 0.2 and 0.4. Theoretical monotherapy dose response curves for drugs A and B are shown in Figure 1. Three response surface model setups for interaction indices are Setup 1: τ = exp( 0.7), Setup 2: τ = exp( δ 1 0.2δ 2 0.1δ 1 δ δ δ2 2 ), Setup 3: τ = exp( δ 1 0.5δ δ δ2 2 ). Six dose levels, 0.1, 0.5, 1, 2, 3, and 10, are assumed for each of the drugs. The total data points for each simulated data are 48, including 36 data points for combination doses and another 12 for monotherapy doses. We assume σ 2 = 0.3 as the error variance in Equation (3) in all six simulations. Schematic plots for the interaction index model setups 2 and 3 are shown in Figure 2. Tables 1 and 2 show the estimations of the secondstage response surface model and their corresponding square root mean square errors (RMSEs). The first row in each setup contains the true parameters used in that simulation. We ran 1000 repetitions for each simulation and recorded the estimated interaction index parameters. The initial values of the response surface parameters were derived from the least-square fit of the interaction index model using the combination data. The parameters 378

6 Two-Stage Response Surface Approach Figure 1. Theoretical monotherapy dose response curves for drugs A and B in simulation studies. Figure 2. Theoretical interaction index contours for simulation setups 2 and 3. White is for strong synergy and black is for antagonism. 379

7 Statistics in Biopharmaceutical Research: November 2012, Vol. 4, No. 4 Table 1. Estimated response surface parameters when var(ε A ) = var(ε B ) = 0.2. The first row in each setup is the true parameters used in the simulation; the second row is the mean of parameter estimations over 1000 repeats; and the third row is their corresponding square RMSEs γ 0 γ 1 γ 2 γ 12 γ 3 γ 4 Setup 1 True Mean RMSE Setup 2 True Mean RMSE Setup 3 True Mean RMSE converged to the true values in all simulations. When the variance in the monotherapy model increased from 0.2 to 0.4, the RMSE estimated parameters also increased. For example, RMSE of ν 0 in setup 1 increased from in Table 1 to in Table 2. Tables 3 and 4 demonstrate how much the estimated interaction indexes deviate from the true values. The first row at the top of each table lists six combination dose levels with equal component and the first row (τ) in each setup contains the corresponding true interaction index; ˆτ RS is the interaction index estimated using the two-stage response surface model and RMSE RS is the corresponding RMSE. Also, ˆτ GR and RMSE GR are for Greco s one-stage model and are given in rows 3 and 4 in each setup. For setup 1 with constant interaction index, the two-stage model is unbiased but Greco s one-stage model gives completely wrong calculation. For setups 2 and 3 Table 2. Estimated response surface parameters when var(ε A ) = var(ε B ) = 0.4. The first row in each setup is the true parameters used in the simulation; the second row is the mean of parameter estimations over 1000 repeats; and the third row is the corresponding square RMSEs γ 0 γ 1 γ 2 γ 12 γ 3 γ 4 Setup 1 True Mean RMSE Setup 2 True Mean RMSE Setup 3 True Mean RMSE Table 3. Estimated interaction index using two-stage response surface model and Greco s one-stage model at equal doses, with var(ε A ) = var(ε B ) = 0.2; ˆτ is the average of the estimated interaction index over 1000 repeats and RMSE is their corresponding rooted mean square error. Subscript RS is used to denote response surface model and GR is used to denote Greco s model (d A, d B ) (0.1, 0.1) (0.5, 0.5) (1, 1) (2, 2) (3, 3) (10, 10) Setup 1 τ ˆτ RS RMSE RS ˆτ GR RMSE GR Setup 2 τ ˆτ RS RMSE RS ˆτ GR RMSE GR Setup 3 τ ˆτ RS RMSE RS ˆτ GR RMSE GR with more complicated interaction index contours, it is easier to observe that the one-stage model is inadequate. When the variance in the first stage increases, the mean estimated interaction index remains unchanged for both Table 4. Estimated interaction index using two-stage response surface model and Greco s one-stage model at equal doses with var(ε A ) = var(ε B ) = 0.4; ˆτ is the average of the estimated interaction index over 1000 repeats and RMSE is their corresponding rooted mean square error. Subscript RS is used to denote response surface model and GR is used to denote Greco s model (d A, d B ) (0.1, 0.1) (0.5, 0.5) (1, 1) (2, 2) (3, 3) (10, 10) Setup 1 τ ˆτ RS RMSE RS ˆτ GR RMSE GR Setup 2 τ ˆτ RS RMSE RS ˆτ GR RMSE GR Setup 3 τ ˆτ RS RMSE RS ˆτ GR RMSE GR

8 Two-Stage Response Surface Approach Figure 3. Monotherapy dose response curves for drugs A and B fitted by the two-stage response surface model and Greco s one-stage model using Harbron s (2010) data. models, but RMSEs increase for the two-stage model and remain roughly unchanged for the one-stage model. 4. Example We use an example studied by Harbron (2010) to demonstrate how our method is performed in practice. Briefly, two drugs A and B were each studied under monotherapy dosing for nine dose levels with threefold spacing. They were studied in combinations in a factorial design for all of the lowest six doses, 36 combination doses in total (data are not shown). Table 5 gives parameter estimates and their corresponding estimated standard errors using the two-stage response surface Table 5. Estimated response surface parameters for Harbron s (2010) data γ 0 γ 1 γ 2 γ 12 γ 3 γ 4 Estimate Standard error model. If needed, information in the table can be used to do model selection. We compare the fitted monotherapy curves using the two-stage model with that using the one-stage model. Figure 3 shows the comparisons for both drugs A and B. The x-axis is dose level in log scale. Depending on drugs and dose levels, the monotherapy dose response fitting using one-stage models deviates from the two-stage model. Thus, the monotherapy doses may be overestimated (underestimated) and the interaction index will be underestimated (overestimated,) leading to false positive (false negative) claims. For drug A, the monotherapy doses using Greco s model may be underestimated by as much as 40% at high response levels; deviation from the twostage model can be 40% as well for drug B at low response levels. We present a contour plot of the upper 95% confidence interval for the interaction index in Figure 4. A region of combination doses can be claimed synergistic if all values in the region are less than one. It is easy to observe that the region in light gray consists of statistically significant synergistic drug combinations. The level of drug interaction depends primarily on drug A. Synergistic 381

9 Statistics in Biopharmaceutical Research: November 2012, Vol. 4, No. 4 Figure 4. Contour plot of upper 95% confidence interval of interaction index for Harbron s (2010) data. interaction is observed at high doses of drug A, while the upper bound of the confidence interval for the interaction index remains relatively constant at all doses of drug B. 5. Discussion Pooling data for analysis is a good statistical practice in many situations. However, blindly following this practice in other situations such as drug combination analysis may in fact lead to less accurate results and false conclusions. We propose a new method that estimates the drug combination interaction index through a two-stage approach using a response surface model. The method first fits monotherapy models and then carries the estimated parameters and their variances to the second stage of estimating the drug combination interaction index. For the two-stage model, the variance in the first stage is able to propagate to the second stage for estimating the interaction index, which may not be true for the one-stage model. This is because in general there are far more data points in the combination study than in the monotherapy study so that the variances from the monotherapy part are suffocated by the variances from the combination part. The first-stage monotherapy data serve as a foundation to evaluate drug interactions. Failure to properly incorporate the variance from the first stage to the second stage easily leads to false claims of drug synergy or antagonism. Using simulations and an example, we compare the performance of the proposed method with that of a onestage model. It is shown that the method produces more accurate results than the one-stage model. The monotherapy doses estimated using one-stage models deviate significantly from true values (data not listed), leading to biased estimation of interaction index and potential false claims. We also propose a simplex method to estimate the interaction index. It overcomes some of the pitfalls of the NLS procedures used in R and other statistical software packages. All calculations in the simulations and example converged to proper values and no error message was observed. Convergence means that the calculation converges both in parameters and in likelihood. Calculation is said to reach convergence when the difference between two iterations is less than a prespecified convergence limit. However, different software may have different ways to specify convergence limits. For example, fminsearch() function in MATLAB allows specifying convergence limits for both parameters and likelihood, but optim() function in R allows this only for likelihood. Nevertheless, convergence in likelihood is often equivalent to convergence in parameters. The default 382

10 Two-Stage Response Surface Approach tolerance limit in optim() is 1e-8 and we use 1e-20 in our calculation. To demonstrate, we use median response model as the monotherapy model to calculate monotherapy doses. This model can be easily replaced by other appropriate models. The response surface model we use to describe drug interaction is sufficient in most situations. However, this model can be replaced by other models if necessary. All models described by Harbron (2010) should apply. Using the parameter variances estimated from Equation (6), one can further do model selection to simplify the response surface model. Ideally, drug combination studies should be conducted in a stagewise fashion with insights gained with monotherapy experiments being fully used to optimize the design and analysis of drug combination experiments. In such a context, the two-stage analysis method proposed in this article becomes a natural method of choice to provide proper guidance and estimation of the effect of drug combination. [Received July Revised March 2012.] References Bates, D.M., and Watts, D.G. (1988), Nonlinear Regression Analysis and Its Applications, New York: Wiley. [376] Berenbaum, M.C. (1985), The Expected Effect of a Combination of Agents: The General Solution, Journal of Theoretical Biology,114, [375] Bliss, C.I. (1939), The Toxicity of Poisons Applied Jointly, The Annals of Applied Biology, 26, [375] Chou, T.C., and Talalay, P. (1984), Quantitative Analysis of Dose- Effect Relationships: The Combined Effects of Multiple Drugs or Enzyme Inhibitors, Advances in Enzyme Regulation, 22, [375] Davison, A.C., and Hinkley, D. (2006), Bootstrap Methods and Their Applications, New York: Cambridge University Press. [376,378] Fang, H.B., Ross, D.D., Sausville, E., and Tan, M. (2008), Experimental Design and Interaction Analysis of Combination Studies of Drugs With Log-Linear Dose Responses, Statistics in Medicine,27, [375,376] Greco, W.R., Bravo, G., and Parsons, J.C. (1995), The Search for Synergy: A Critical Review From a Response Surface Perspective, Pharmacological Reviews, 47, [375,377] Greco, W.R., Park, H.S., and Rustum, Y.M. (1990), Application of a New Approach for the Quantitation of Drug Synergism to the Combination of cis-diamminedichloroplatinum and 1-β-D-Arabinofuranosylcytosin, Cancer Research, 50, [376,378] Harbron, C. (2010), A Flexible Unified Approach to the Analysis of Pre-Clinical Combination Studies, Statistics in Medicine,29, [376,381,383] Kong, M., and Lee, J.J. (2006), A Generalized Response Surface Model With Varying Relative Potency for Assessing Drug Interaction, Biometrics, 62, [375,376] Lee, J.J., and Kong, M. (2009), Confidence Intervals of Interaction Index for Assessing Multiple Drug Interaction, Statistics in Biopharmaceutical Research, 1, [376] Lee, J.J., Kong, M., Ayers, G.D., and Lotan, R. (2007), Interaction Index and Different Methods for Determining Drug Interaction in Combination Therapy, Journal of Biopharmaceutical Statistics, 17, [376] Loewe, S., and Muischnek, H. (1926), Effect of Combinations: Mathematical Basis of Problem, Archives of Experimental Pathology and Pharmacology, 114, [375] Mood, A.M., Graybill, F.A., and Boes, D.C. (1974), Introduction to the Theory of Statistics, New York: McGraw-Hill Companies. [378] Nelder, J.A., and Mead, R. (1965), A Simplex Method for Function Minimization, The Computer Journal, 7, [376] Plummer, J.L., and Short, T.G. (1990), Statistical Modeling of the Effects of Drug Combinations, Journal of Pharmacological Methods, 23, [376] Ramaswamy, S. (2007), Rational Design of Cancer-Drug Combinations, The New England Journal of Medicine, 357, [375] Sun, T., Aceto, N., Meerbrey, K.L., Kessler, J.D., Zhou, C., Migliaccio, I., Nguyen, D.X., Pavlova, N.N., Botero, M., Huang, J., Bernardi, R.J., Schmitt, E., Hu, G., Li, M.Z., Dephoure, N., Gygi, S.P., Rao, M., Creighton, C.J., Hilsenbeck, S.G., Shaw, C.A., Muzny, D., Gibbs, R.A., Wheeler, D.A., Osborne, C.K., Schiff, R., Bentires-Alj, M., Elledge, S.J., and Westbrook, T.F. (2011), Activation of Multiple Proto-Oncogenic Tyrosine Kinases in Breast Cancer via Loss of the PTPN12 Phosphatase, Cell, 144, [375] Zhao, W., Wu, R.L., Ma, C.X., and Casella, G. (2004), A Fast Algorithm for Functional Mapping of Complex Traits, Genetics, 167, [376] About the Author Wei Zhao, Lanju Zhang, Lingmin Zeng, and Harry Yang, MedImmune, LLC, Gaithersburg, MD ( for correspondence: zhaow@medimmune.com). 383