Experimental Designs for Drug Combination Studies

Transcription

1 Experimental Designs for Drug Combination Studies B. Almohaimeed & A. N. Donev First version: 1 October 2011 Research Report No. 8, 2011, Probability and Statistics Group School of Mathematics, The University of Manchester

2 Experimental Designs for Drug Combination Studies B. Almohaimeed and A. N. Donev Abstract The interest in drug combinations is growing rapidly due to the opportunities they create to increase the therapeutic effect and to reduce the frequency or magnitude of undesirable side effects when single drugs fail to deliver satisfactory results. Considerable effort in studying the mechanisms leading to the benefits of the joint action of drugs has been matched by the development of relevant statistical methods and tools for statistical analysis of the data obtained in such studies that allow important statistical assumptions to be taken into account, i.e. the appropriate statistical model and the distribution of the response of interest (e.g. Gaussian, Binomial, Poisson). However, much less attention has been given to the choice of suitable experimental designs for such studies, while only high quality data can ensure that the objectives of the studies will be fulfilled. We propose methods for construction of such experimental designs which are economical and make most efficient use of the available resources. We also propose simple but flexible experimental designs, which we call ray-contour designs, which are particularly useful when the use of low or high doses is undesirable and hence a standard statistical analysis of the data is not possible. Keywords: additivity; antagonism; ray design; ray-contour design; synergy. 1 Introduction Combinations of drugs have been known to be highly successful in treating different diseases for many years [1]. They provide additional opportunities over single drug therapies to achieve sufficient therapeutic effect at a lower 1

3 and potentially safer dose. Extensive discussions of the scientific background of such studies, include [1, 2, 3, 4, 5]. There are many examples of successful combination studies in the literature, e.g. [6, 7, 8, 9, 10, 11]. Recently there has been substantial interest in the appropriate statistical methods for analysis of data obtained in combination studies. These publications, including [9, 12, 13, 14, 15, 16, 17, 18, 19], discuss typical methods for statistical analysis of combination studies. Some of these methods have been implemented in statistical packages. Software developed for design and analysis of data from bioassay is also useful. Packages recently developed using the free statistical software R [20] include drc, grofit, dosefinding [21, 22, 23, 24, 25, 26]. They are regularly improved and updated. Common feature of these packages is that they allow for a variety of statistical models to be estimated and different assumptions about the distribution of the response of interest (e.g. Gaussian, Binomial, Poisson) to be made. In Section 2 we summarize the common statistical models used in the statistical analysis of data collected in a bioassay and present ways of assessing useful statistical properties of the results that depend on the choice of the experimental design. Combination studies are a great deal more complex than studying the effect of individual drugs separately. They typically take up longer time and substantial resources compared to studies of single drugs. Therefore, it is important that the experimental designs for such studies make use of all relevant available information in order to ensure statistically viable results and most efficient use of the available resources. Frequently used experimental designs require to collect data for the response of interest at different combinations (rays) of doses of the studied drugs. These ray designs [27], are easy to implement and the results obtained using them are easy to interpret. We discuss how economical but efficient ray designs can be constructed using information obtained in previous studies of the individual components in Section 3. We propose an approach for design of combination studies that takes into account the statistical analysis that will be carried out, i.e. the type of the model that will be estimated as well as the distribution of the experimental errors: we consider the cases when these distribution can be Gaussian, inverse Gaussian, Binomial, Poisson and Gamma. Designs constructed this way can be useful for many experimental situations and are particularly useful for preclinical studies. Finally, in Section 4 we provide an extension of the ideas discussed previously and propose the use of raycontour designs that permit a variety of combinations studies to be designed in such a way that low or high doses are possible to avoid and the statistical 2

4 analysis can be simplified. For example, ray-contour designs are particulary useful for combination studies where small doses should be avoided due to ethical considerations, e.g. in animal studies and in early studies in men when the use of higher doses may not be desirable because of a concern for their safety. We illustrate some benefits of using ray-contour in an example based on the results of a pilot studied carried out at the Paterson Institute for Cancer Research, Manchester, UK. All discussions in the paper are limited to cases where the combination of 2 drugs is required. However, the presented methodology can be easily extended to more general situations. Computer programs implementing all methods proposed in this paper have been written using the computer language R and are available at adonev/combinations. 2 Statistical analysis of combination studies When a ray design has been used, the statistical analysis of the data involves the estimation of the model relating the response of interest Y ij and the doses x ij, δ γ Y ij = η(x ij, θ) = γ + ( ) (x ij α i,)β λi + ɛ ij, (1) i where i, i = 1,..., r, denotes the drug combination (ray), j, j = 1,..., c, denotes the number of dose level at which the response Y ij is measured, x ij is the dose, while θ = (α i β i γ δ λ i ), i = 1,..., r, is a vector of all model parameters. The dose x ij is obtained by combining the doses d ij1 and d ij2 of the studied drugs. We denote R i = d ij1 d ij2 the ratios of the drugs in the combinations and p i = d ij2 d ij1 +d ij2 the corresponding proportions of drug 2. They both are the same for all rays. The model parameters have a useful and easy interpretation: α i represents the logarithm to base 10 of the dose leading to a response which is the average of the minimum and maximum possible response γ and δ, respectively. This dose is also known as the logic50, or logec50. In (1) β i is related to the rate at which the response changes with the dose and it is called Hill s slope. If λ i = 1, the relationship between the response and the dose is symmetric around the dose α i. However, including λ i in the model, allows it to capture departure from such symmetry. For simplicity, in this 3

5 paper we do not consider the case when one of the drugs or any drug combinations are not able to achieve 100% inhibition, but the extension of our results in such cases is straightforward. In (1) ɛ ij is the experimental error which we assume to be additive. Assumptions about the distribution of ɛ ij are usually made based on the knowledge of the nature of the response variable. When the response is continuous a common assumption is that ɛ is normally distributed with zero mean and a variance that may depend on the response. When the experimental errors are expected to be asymmetric around zero, a more common assumption is that their distribution is inverse Gaussian or Gamma. When the response of interest is counts, more suitable assumption for ɛ ij is that it has a Poison distribution, while if it is binary ɛ ij has a Binomial distribution. Often existing knowledge about the studied response allows for simpler models to be used. When λ is assumed to be one, Hill model [28] is obtained. In many cases it is possible to make further assumptions that γ = 0, or that γ = 0 and δ = 1, resulting in simpler models. Thus, models obtained by simplifying (1) that have 2, 3, or 4 parameters may be used. Depending on the distributions of the experimental errors model (1) is nonlinear or generalized nonlinear model and the estimation of its parameters θ can be done using available software. More complicated analysis required when more factors affect the results is also possible but is not discussed in this paper. A case when it may impractical or not possible to collect data to fit model (1) or a 2, 3 or 4 parameter simplification, is considered in Section 4. We use Loewe s [29] definition for additivity. Two compounds are considered to have additive joint action when the combination index CI = d 1 + d 2 D y0,1 D y0,2 (2) is equal to 1, where D y0,1 and D y0,2 are doses of compounds 1 and 2, respectively, required to achieve a defined level of response y 0, or equivalently an inhibition level I, 0 I 100. In (2) d 1 and d 2 denote doses of the two compounds which when used in combination lead to the same level of the response. For example, y 0 can be chosen to correspond to the IC50 (or EC50) dose, when I = 50, i.e. the dose required to obtain 50% of the maximum possible effect. In a study looking for an improvement of efficacy, it is desirable for the two compounds to have synergistic effect. Then CI < 1. 4

6 Alternatively, when the response is related to the drugs safety (e.g. existence, number or severity of undesirable side effects) it is hoped that the joint action of the drugs will be antagonistic, hence CI > 1. Desirable effect may exist only for some drug combinations. The estimation of the combination index and the evaluation of its statistical and biological interpretation plays an important role in combination studies. The values of the parameters α j in (1) are used to calculate the combination indices (2) for all studied drug combinations. [30, 31] show that when ɛ N(0, σ 2 ), the distribution of the logarithm of the combination index has approximately a normal distribution. This makes the interpretations of the results easy: a test for the statistical hypothesis that the combination ratio for a particular drug combination is zero is equivalent to testing whether or not the joint action of the drugs in that combination is additive. The accuracy of the estimate of the combination index depends on the experimental design used to collect the data and on the variability in the study. The Fisher s information matrix for the model parameters holds useful information about the statistical properties of the estimated model. Firstorder approximation of this matrix is given by M(θ) = X T W X, (3) where X is the n p design matrix, n is the total number of observations and p is the number of the model parameters. The columns of M are given by the partial derivatives of the model with resect to the model parameters evaluated at the observations. Appendix B lists the formulas for these calculations when model (1) is used. The information matrix (3) depends on the matrix W. When the observations are independent, W is diagonal with diagonal elements the variances of the observations, i.e. W = diag ( w 1 w 2 w n ). (4) The ith diagonal element of the matrix W depends on the distribution of the experimental errors and is given by ( ) dµ w i = V 1 (µ), (5) dη where µ is the mean of Y evaluated for the model parameters values. Table 5 in Appendix A provides the formulas for these entries when the distribution of ɛ is assumed to be Gaussian, inverse Gaussian, Binomial, Poisson or 5

7 Gamma. When simpler models with 2, 3 and 4 parameters are needed, the corresponding elements of the matrix W are obtained by setting γ = 0, δ = 1 and λ = 1 as necessary. The approximate standardized prediction variance at a dose x ij is d(x ij, θ) = nw 1 x ij f T (x ij, θ)m 1 (θ) f(x ij, θ), (6) where f(x ij, θ) for model (1) is defined by (14) given in Appendix B. An experimental design that minimizes the maximum of (6) over the design region is called G-optimum. The volume of the confidence region for the estimates of the model parameters is proportional to the inverse of the determinant of the Fisher s information matrix (3). The criterion requiring minimization of this volume, and hence the maximization of the determinant of (3), is called D-optimality. However, as the elements of (3) depend on the true values of θ, the determinant can only be calculated for specific values of θ. The same extends also to d(x ij, θ). This creates difficulties for comparing and constructing experimental designs with respect to these useful criteria. We discuss how they can be overcome in the next section. For a general review of the theory of design optimality, see [32]. 3 Experimental designs We consider a number of typical experimental scenarios and present statistical methodology and tools for the construction of suitable designs for combination studies. However, the choice of experimental designs for combination studies is affected by many considerations. The final choice of a design may be influenced by various additional scientific and practical considerations. We divide the experimental designs in three groups, each suitable for specific experimental situations: serial dilutions ray designs, small efficient ray designs, ray-contour designs. Ray-contour designs are the subject of the next section. A serial dilution design is specified by the maximum dose (MD) that is used, the number 6

8 of doses (ND= c) and the dilution factor (DF). In the case of combination studies, the dose is the total of the doses of the ingredients of the drug combination. These designs ensure good coverage of the different doses of the studied combinations and excellent statistical properties when the parameters of the design are suitably chosen. [33] provide advice about how MD, ND and DF can be chosen so that the required resources are minimized. They also provide computer code for their construction. However, the use of such experimental designs for combination studies is expensive. Also, these designs lack the sophistication to fully take into account the relevant model assumptions (e.g. model and distribution of experimental errors) and therefore may use badly the experimental resources. We devote the rest of this section to the construction of small and efficient ray designs which can be obtained using a specific design criterion for their construction. Similar designs have been used in monotherapeutic studies. For example, [34] propose the use of an optimality criterion for the construction of such designs and the observations are normally distributed with homogenous variance. As it was shown in Section 2 the elements of the information matrix depend on the unknown parameter values θ. One possibility to construct a D-optimum design for a specific combination study is to use plausible values for the model parameters in order to calculate the design criterion. The resulting designs are called locally D-optimum experimental designs. Typically locally D-optimum designs for estimating nonlinear models are saturated, i.e. require as many distinct observations as the number of the estimated parameters. In the case of a combination study with r rays and c = p doses used for each ray, c r drug combinations need to be tested, including a suitable number of positive and negative controls, i.e. measurements of the response when no drug is used and when a maximum effect can be seen. Clearly such designs are very economical and at the same time they make most efficient use of the experimental resources. We extend this idea to the cases when generalized nonlinear models are needed and consider the specific challenges that the combinations studies pose. A combination study will typically follow a careful study of the individual affects of the components of the studied drug combinations. As [35] point out, the estimates of the parameter values α 1 and α 2 obtained at that stage can be used to calculate the values α i, i = 3,..., r, for all studied drug 7

9 combinations. If the joint action of the drugs is additive, ˆα i = α 1α 2 (1 + p r ) α 1 + p r α 2, i = 3,..., r. (7) In order to estimate the remaining corresponding parameters in the case of additivity a functional relationship has to be assumed about how they change with the ratio of the components. In our experience, the values for β i and α i, i = 3,..., r can be successfully estimated using a linear interpolation between the corresponding values for the individual drugs, while γ, δ and λ i can be chosen in the same way as in the study of the individual drugs. However, it is clear that the parameter values used to obtain locally D- optimum designs may not be accurate and therefore, the optimum will only be approximately optimum. An alternative approach to construct small efficient ray designs is to choose the doses in such a way that the designs are approximately D-optimum for ranges of parameter values, or for their prior distribution. Such designs are called pseudo-bayesian D-optimum experimental designs. In general they may require a larger number of doses than the local D-optimum designs but they are more robust against possible discrepancy between the assumed model parameter values and those observed in the experiment. Pseudo- Bayesian D-optimum experimental designs are usually also approximately D-optimum. They are generally considered more robust against misspecification of the model parameters. Whatever choice is made, i.e. to use locally or pseudo-bayesian D- optimum designs, the solution of the resulting optimization problem is helped by the result of the General Equivalent Theorem of Kiefer and Wolfowitz [36] as it provides a way to verify whether the constructed design is optimum with respect to the chosen criterion of optimality for each studied drug combination. The theorem states that when the design is D-optimum, the maximum of the standardized prediction variance (6) is equal to the number of the model parameters, p and it is attained at the tested doses. In this case the experimental design is both D- and G-optimum. 3.1 Example 1. Locally D- and G-optimum designs We illustrate the construction of locally and pseudo-bayesian D-optimum experimental designs for combination studies for a variety of statistical models and possible error distributions with examples. Table 2 lists the locally 8

10 D- and G-optimum designs for model 1 and the values of the model parameters as given in Table 1 under an assumption for additive joint action of two drugs when Gaussian, Poisson, Binomial, Gamma, and inverse Gaussian distribution for the experimental errors are assumed. The ratios of the drug combinations for seven rays are chosen to be (0, 0.2, 0.3, 0.5, 0.7, 0.8, 1), respectively. The dose range was chosen to include doses up to 10 units. The designs can be easily recalculated for other choices. In all cases the maximum available dose was included in the design. Therefore, max is used in the tables of results to indicate that the optimum design point is located at the maximum available dose. Figure 1 shows the standardized prediction variance of the response 6 over the design region. In order to save space, this is done only for one of the rays (Ray 4). The plots for the remaining rays are similar. The standardized prediction variance do not exceed the number of model parameters, and it achieves its maximum at the design points, indicated by the dots ( ). Therefore, based on the result of the General Equivalence Theorem it is clear that the designs are both locally D- and G-optimum. 3.2 Example 2. Pseudo Bayesian D- and G-optimum designs Similarly to Example 1, Table 3 lists the pseudo Bayesian D- and G-optimum designs for model (1). The values of the model parameters used to construct the designs are given in Table 1, while Figure 2, shows the corresponding standardized prediction variances of the response for Ray 4. For the chosen design regions, the pseudo-bayesian D-optimum designs are saturated, but this may not be the case or other design regions and model parameters values. 4 Ray-contour designs The aim of these designs is to allow for local synergy or antagonism to be detected; avoid the use of too small or too high doses if necessary; simplify the statistical analysis of the data. 9

11 Table 1: Parameters values for Example 1 and Example 2. Parameters α 1 α 2 β 1 β 2 γ δ λ σ 2 κ υ n Example p Example 2 [1.31,3.01] [1.12,2.81] [0.50,1.15] [0.30,1.11] p Standardized prediction variance Gaussian errors Standardized prediction variance Poisson errors Dose Dose Standardized prediction variance Binomial errors Standardized prediction variance Gamma errors Dose Dose Standardized prediction variance Inverse Gaussian errors Dose Figure 1: Standardized variance for local D- and G-optimum design for combination ray 4. 10

12 Table 2: Locally D- and G-optimum designs for combination studies (r = c = 7). Doses Distribution Ray (max) (max) (max) Gaussian (max) (max) (max) (max) (max) (max) (max) Poisson (max) (max) (max) (max) (max) (max) (max) Binomial (max) (max) (max) (max) (max) (max) (max) Gamma (max) (max) (max) (max) (max) (max) (max) Inverse Gaussian (max) (max) (max) (max)

13 Table 3: Bayesian D- and G-optimum designs for combination studies (r = 7). Doses Distribution Ray (max) (max) (max) Gaussian (max) (max) (max) (max) (max) (max) (max) Poisson (max) (max) (max) (max) (max) (max) (max) Binomial (max) (max) (max) (max) (max) (max) (max) Gamma (max) (max) (max) (max) (max) (max) (max) Inverse Gaussian (max) (max) (max) (max)

14 Standardized prediction variance Gaussian errors Standardized prediction variance Poisson errors Dose Dose Standardized prediction variance Binomial errors Standardized prediction variance Gamma errors Dose Dose Standardized prediction variance Inverse Gaussian errors Dose Figure 2: Standardized variance for Bayesian D- and G-optimum design for combination 4. 13

15 This is achieved by including in the experimental design doses in such a way that if the joint action of the drugs is additive, the expected response to the j th dose of each ray is the same and equal to a predetermined inhibition level I i %, i = 1,..., r. A similar suggestion and a discussion of the possible benefits of using such an experimental design is given in [37]. However, a simple situation is only considered. The doses of a ray-contour design can be calculated by first calculating the expected values of ˆα i, i = 1,......, r, using equation (7). Then, the combined dose for each drug combination can be calculated as IC(I j ) = (I j /(100 I j )) β j 10 α j, (8) where β i, i = 1,..., r, are estimated by interpolation between β 1 and β 2 obtained in previous studies. The individual doses are obtained using the chosen proportions p i of the drugs in the combinations. Therefore, the rays of a ray-contour design are specified by the proportions p i, i = 1,..., r, of the studied drug combinations, while the contours are specified by the selected inhibition levels. The choice of the inhibition levels I i %, i = 1,..., c, as well as their number, can be made by taking into account practical considerations and research objectives. For example, in an in vitro study, a ray contour design would allow the experimenter to see whether synergy or antagonism is exhibited only at specific dose ranges. Also, these designs will be useful in early animal combination studies where in the absence of reliable information about the suitable dose range, the experimenter may want to avoid using too high doses. This approach also provides an easy way to design dose escalating clinical trials for combination studies. In other clinical trials, for example when the drug-combination is an anaesthetic, the experimenter may prefer to avoid unethically low doses. In this case the contours can be chosen to correspond to relatively high inhibition levels. No existing experimental designs are suitable in these cases. The statistical analysis of data obtained using a ray-contour design is simple and does not require fitting model (1). As a result of the construction of the experimental design, when the joint action is additive the expectations of the observations of the response taken at each contour (i.e. inhibition level) should be the same for all studied drug combinations. Therefore, the analysis of variance of the data (ANOVA) can be used to the test for synergy or antagonism at each inhibition level. ANOVA is a method well described in 14

16 many textbooks; see for example [38]. It is also implemented by all statistical packages. Example 3. Cancer Study. The aim of the study was to investigate the combined efficacy of a conventional cytotoxic drug (drug 1) and a novel mechanism based agent (drug 2) in colorectal cancer cell lines, with the ultimate intension of combining the two agents in the clinic. The ratios of the doses of Drug 2 to Drug 1 used in the combinations were: 0.5, 1.0, 2.0, 4.1 and 8.2 (these rays were numbered in reverse order), while the percentages inhibition to be studied were chosen to be 20, 35, 50, 60, 70, and 80. Data were also collected for the individual drugs (Ray 1 and Ray 2). The response was measured also at positive and negative control. The ray-contour experimental design that was used is given in Table 4 and was obtained as described above using the results of previous studies where the IC50 and Hill slopes were estimated as 4.62 µm and 9.47 µm, and and -3.82, for Drug 1 and Drug 2, respectively. The design was replicated on 3 plates. A plot of the means of the observations is shown in Figure 3. The benefits of using a ray-contour design are obvious - one does not need to perform a complicated statistical analysis in order to see that antagonism is observed at Ray 3 where Drug 2 is used 8.2 time more than Drug 1 in the combination. The synergy reduces and then diminishes as the ratio reduces. Many research laboratories that use ray designs with just a single drug combination been defined by the ratio of the IC50 values of the two drugs would have missed this useful result. Clearly, the choice of the drug combinations is very important to the success of the study. Usually it is not known in advance what drug ratio may possess useful properties. Therefore studying as large as possible number of ratios appears to be desirable. However, the experimental resources are usually limited. A compromise could be reached by keeping the number of rays large, while reducing the number of contours (i.e. the doses on each ray). Note that the contour corresponding to 50% inhibition ensures largest sensitivity to detect non-additive joint drug action. We trust the ray-contour designs will find many useful traditional and new applications that will make them popular with experimenters. For example, they will be well suited to new research devoted to identifying interaction thresholds [39]. 15

17 Inhibition Ray Figure 3: Means with error bars at one standard deviation for the observations at different inhibition levels. 16

18 Table 4: Ray-contour design for Example 3. Ray Drug ratio Inhibition (%) Dose 1 Dose inf inf inf inf inf inf inf inf

19 Acknowledgements. The authors are grateful to Dr. Christopher Morrow, Paterson Institute for Cancer Research, Manchester, UK, for providing the data of Example 3. References [1] Chou, T.C. Theoretical Basis, Experimental Design, and Computerized Simulation of Synergism and Antagonism in Drug Combination Studies. Pharmacological Reviews 2006; 58: [2] Berenbaum MC. What is Synergy?. Pharmacological Reviews 1989; 41: [3] Chou, T. C., and D. C. Rideout. Synergism and antagonism in chemotherapy. Academic Press, New York, N.Y [4] Greco, W. R.,Bravo G. and Parsons, J. C. The Search for Synergy: A Critical Review from a Response Surface Perspective. Pharmacological Reviews 1995, 47: [5] Tallarida, R. J. Drug Synergism and Dose-Effect Data Analysis. Chapman, Hall/CRC, [6] Berenbaum MC. The Expected Effect of a Combination of Agents: The General Solution. Journal of Theoretical Biology 1985; 114: [7] Faessel,H.M., Slocum,H. K., Jackson, R. C., Boritzki,T. J., Rustum,Y. M., Nair, M. G., and W. R. Greco: Super in Vitro Synergy between Inhibitors of Dihydrofolate Reductase and Inhibitors of Other Folaterequiring Enzymes: The Critical Role of Polyglutamylation. Cancer Research : [8] Loewe, S. Antagonisms and Antagonists. Pharmacological Reviews 1957;9, [9] Straetemans R. and Bijnens L. Application and review of the separate ray model to investigate interaction effects. Front Biosci 2010;E2: [10] Fitzgerald, J. B., Schoeberl, B., Nielsen, U. B. and Sorger, P. Systems biology and combination therapy in the quest for clinical efficacy. Nature Chemical Biology 2006;2(9):

20 [11] Kong M, Lee JJ. A General Response Surface Model with Varying Relative Potency for Assessing Drug Interactions. Biometrics 2006;62(4): [12] Kong M. and Lee JJ. Applying Emax model and bivariate thin plate splines to assess drug interactions. Front Biosci 2010;E2: [13] Lee JJ, Lin H.Y, Liu DD and Kong M. Emax model and interaction index for assessing drug interaction in combination studies. Front Biosci 2010;E2: [14] Peterson JJ. A review of synergy concepts of nonlinear blending and dose-reduction profiles. Front Biosci 2010; S2: [15] Donev, A. N. Comparison of methods for statistical analysis of combination studies. Front Biosci (Elite Ed) 2010;2: [16] Brun YF. and Greco W.R. Characterization of a three-drug nonlinear mixture response model. Front Biosci 2010; S2: [17] Palomares, I.Rodea, A.L. Petre, K. Boltes, F. Legans, J.A. Perdign- Meln, R. Rosal, F. Fernndez-Pinas. Application of the combination index (CI)-isobologram equation to study the toxicological interactions of lipid regulators in two aquatic bioluminescent organisms. Water Research 2010;44(2): [18] Yan Han, Bo Zhang, Shao Li and Qianchuan Zhao. A formal model for analyzing drug combination effects and its application in TNF-α-induced NFkB pathway. BMC Syst Biol 2010;4:50. [19] Fujimoto Junya, Kong Maiying, Lee J. Jack, Hong Waun Ki, and Lotan Reuben. Validation of a Novel Statistical Model for Assessing the Synergy of Combined-Agent Cancer Chemoprevention. Cancer Prev Res 2010, / CAPR [20] R Development Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing 2011;Vienna, Austria URL: [21] Ritz, C. and Strebig, J. Bioassay Analysis using R. Journal of Statistical Software 2005; 5:12. URL: 19

21 [22] Ritz, C. and Strebig, J. R package, Analysis of dose-response curves. URL: [23] Kahm, M., Hasenbrink, G., Lichtenberg-Frate, H., Ludwig, J. and Kshischo, M. grofit : fitting biological growth curves with R. Journal of Statistical Software 2010; 33, 121. [24] Boik J.C., Narasimhan N. An R Package for Assessing Drug Synergism/Antagonism. Journal of Statistical Software 2010; 34(6), 118. [25] Bornkamp, B., Pinheiro, J., Bretz, F. MCPMod : An R Package for the Design and Analysis of Dose-Finding Studies. Journal of Statistical Software 2009; 29, 123. [26] Bornkamp, B., Pinheiro, J., Bretz, F. DoseFinding : Planning and Analyzing Dose Finding experiments Journal of Statistical Software 2011; URL: [27] Mantel, N. An Experimental Design in Combination Chemotherapy. Ann. New York Acad. Sci. 1958;76, [28] Hill, A. V. The Combinations of Haemoglobin with Oxygen and with Carbon Monoxide. Biochem 1913, 7: [29] Loewe, S. The problem of synergism and antagonism of combined drugs. Arzneim. Forsch 1953; 3: [30] Lee JJ, Kong M, Ayers GD, Lotan R. Interaction Index and Different Methods for Determining Drug Interaction in Combination Therapy. Journal of Biopharmaceutical Statistics 2007; 17: [31] Lee J.J. and Kong M. Confidence intervals of interaction index for assessing multiple drug interaction. Statistics in Biopharaceutical Research 2009;1(1): 417. [32] Atkinson, A.C., Donev, A.N. and Tobias, R.D. Optimum Experimental Designs, with SAS. Oxford University Press, [33] Donev, A. N. and Tobias, R. Optimal Serial Dilutions Designs For Drug Discovery Experiments. Journal of Biopharmaceutical Statistics 2011, 21:

22 [34] Abdelbasit, K.M. and Plackett, R. L. Experimental Design for Joint Action. Biometrics 1982; 38: [35] Straetemans, R., OBrien, T., Wouters, L., Dun, J. V., Janicot, M., Bijnens, L., Burzykowski, T., Aerts, M. Design and analysis of drug combination experiments. Biometrical Journal 2005;47(3): [36] Kiefer, J. and Wolfowitz, J. The equivalence of two extremum problems. Canadian Journal of Mathematics 1960, 12: [37] Gennings, C. An efficient experimental design for detecting departure from additivity in mixtures of many chemicals. Toxicology 1995; 105: [38] Clarke, G.M. and Cooke, D. A Basic Course in Statistics, fourth edition. Arnold, [39] Hamm, A. K., Carter Jr., W. H. and Gennings, C. Analysis of an interaction threshold in a mixture of drugs and/or chemicals. Statistics in Medicine 2005; 24:

23 Appendix A. Table 5: The i th,i = 1,..., n, diagonal element of the W matrix for model (1). Distribution Normal with mean µ and variance σ 2, Expression i.e. w i = σ 2 Y i N (µ, σ 2 ) Poisson with mean µ, Y i Pois (µ) w i = e η(x i,θ) Binomial with n trials, and p probability of success, i.e. Y i B (n, p) and the variance is Var (Y i ) = np (1 p) w i = ne η(x i,θ) Gamma with scale ϑ and shape κ parameters i.e. Y i Γ (κ, ϑ) and variance Var (Y i ) = κϑ 2 w i = κη(x i, θ) 2 Inverse-Gaussian with mean µ and shape parameter υ i.e. Y i IG (µ, υ) w i = υ 4 (η(x i, θ)) 3/2 and variance Var (Y i ) = µ3 υ 22

24 Appendix B. The derivatives of model (1) with respect to the parameters (α i, β i, γ, δ and λ i, respectively) are η k (x ij, θ) α i = (δ γ) λ i 10 (x ij α i )β i β i ln (10) ( (x ij α i )β i ) λi ( (x ij α i )β i ), (9) η k (x ij, θ) β i = (δ γ) λ i10 (xij αi)βi (x ij α i ) ln (10) ( (x ij α i )β i ) λi ( (x ij α i )β i ), (10) η k (x ij, θ) δ = ( ( (x ij α i )β i ) λi ) 1, (11) and η k (x ij, θ) γ η k (x ij, θ) λ i = 1 ( ( (x ij α i )β i ) λi ) 1 (12) = ( ) (δ γ) ln (x ij α i )β i ( ) (x ij α i )β λi. (13) i Therefore f T (x ij, θ) = ( ηk (x ij,θ) α i η k (x ij,θ) β i η k (x ij,θ) δ η k (x ij,θ) γ ) η k (x ij,θ) λ i. (14) 23