Compromise design for combination experiment of two drugs

https://doi.org/10.1016/j.csda.2020.107150Get rights and content

Abstract

Preclinical experiment on two-drug combination is a stepping stone to multi-drug combination studies. Experimental designs have been proposed in the literature to test the presence of synergism between the combined drugs. However, a design that is efficient for synergy testing is not necessarily desirable for dose–response modeling and the latter is important for future development on drug interaction analysis. This work proposes an experimental design, called a compromise design to meet the dual requirements on synergy testing and dose–response modeling. The key idea of the design is to spread the design points uniformly on a pair of design regions where synergy testing and dose–response modeling are respectively carried out. Simulations and two illustrative examples are given to demonstrate the usefulness of the compromise design. In the illustrative examples, the good balance of the proposed design is visualized by 2-D projections of the design points. The simulation results indicate that the compromise design performs satisfactorily in terms of both testing power and model prediction accuracy.

Introduction

Drug combination studies have been widely applied in disease treatment such as cancer and HIV due to their potential for improved effectiveness and inhibited drug resistance at lower, less toxic doses and the need to move new therapies rapidly into clinical trials (for examples, see, Lilenbaum et al. (2005), Chou (2006), Fang et al. (2008), Jaynes et al. (2013), Foucquier and Guedj (2015) and Tan et al. (2016)). A key issue for such a study is to find which combinations are additive, synergistic or antagonistic. The combination experiment of two drugs is a fundamental stage to this issue because joint actions of the drugs are usually determined based on pairwise combinations (for examples, see, Fitzgerald et al. (2006), Shiozawa et al. (2009), Feala et al. (2010), Ning et al. (2014), Ryall and Tan (2015) and Fang et al. (2017)). To test the presence of synergism between two drugs in preclinical studies, various experimental designs have been proposed in the literature (see Tan et al. (2012) and Fang et al. (2017) for comprehensive overviews). Most of the existing designs are fixed-ratio type designs, i.e., they allocate the doses of one drug only while keeping the ratio of the doses of two drugs fixed. The fixed-ratio type designs are easy to implement with multiple ratios, and the results obtained by using them are easy to interpret because each fixed ratio can be treated as a single drug. Fixed-ratio type designs for response belonging to exponential family of distributions have also been considered in the literature (Almohaimeed and Donev, 2014). However, the fixed-ratio type designs are designed for models with strong parametric assumptions and they are suboptimal as they reduce the 3-D problem into a 2-D problem by fixing one dimension at a constant level. Under a general response surface model applicable to both in vivo and in vitro studies, Tan et al. (2003) proposed to use the uniform design (Fang, 1980) for two drug combination experiments. Tan et al. (2003) and a series of subsequent papers (Fang et al., 2008, Tan et al., 2009, Tian et al., 2009, Tan et al., 2016, Fang et al., 2017) have proved that the uniform design is a globally optimal design in the sense that it is a 3-D design for the experiment and it maximizes the minimum power of the lack-of-fit test to detect synergism between the combined drugs. The general model framework developed by Tan et al. (2003) will be used in this paper. In the rest of this opening section, we will show that there are two design regions, where synergy testing and dose–response modeling are respectively carried out, should be jointly taken into account under the model framework. This will be pointed out in Section 1.1. As demonstrated by many researchers, dose–response modeling in preclinical experiment is important for future development on drug interaction analysis (for examples, see, Kong and Lee, 2006, Jaynes et al., 2013, Fang et al., 2008, Fang et al., 2017). However, a motivating example will be given in Section 1.2 to demonstrate that there is a conflict in meeting synergy testing and dose–response modeling from the viewpoint of experimental design.

Consider a combination with two drugs D1 and D2, containing doses x1 of D1 and x2 of D2. When planning combination studies, we usually have already had experiments on the dose–response curve of each single drug. Without loss of generality, they are given as follows: y(x1)=α1+β1logx1=β1(logx1μ1) and y(x2)=α2+β2logx2=β2(logx2μ2),where y() is the dose–effect scaled into viability (proportion of cells surviving) or a tumor volume (with some transformation). This log-linear type of dose–response occurs in a wide variety of systems such as antimetabolite and antibiotics. In particular, Hill models, sigmoid dose–response curves and simple exponential dose–response curves can be transformed into the log-linear curves (Berenbaum, 1989, Kong and Lee, 2006, Fang et al., 2008, Tian et al., 2009, Yu et al., 2018). In Appendix A, we illustrate how the 4-parameter log-logistic function (for example, see Ritz (2010)), which is the most commonly used function for describing dose–response data in toxicology, can be transformed into a log-linear relation. Note that the dose–response curves in (1) still hold if x1 and x2 are respectively replaced by some known functions x1=g1(x1) and x2=g2(x2) (Tan et al., 2003, Tan et al., 2009). Therefore, one can assume that β1=β2=β by letting x1=x1β1β2. Then, under Loewe’s additivity (Berenbaum, 1989, Greco et al., 1995), the additive action of D1 and D2 is given by y(z,π1)=β{(logzμ1)+log[(1ρ)π1+ρ]}where z is the total dose and π1 is the proportion of the dose of D1 in the total dose, i.e., z=x1+x2 and π1=x1z,and ρ=exp(μ1μ2) is the potency of D2 relative to D1. The derivation of Eq. (2) is given in Appendix B. Since how D1 interacts with D2 is typically unknown before the experiment starts, a non-parametric statistical model is assumed below to capture the potential synergism (Tan et al., 2003) y(z,π1)=β{(logzμ1)+log[(1ρ)π1+ρ]}+f(z,π1)+ϵ,where ϵN(0,σ2) is the error term and the parameters β,ρ and μ1 are assumed given and estimated from the single drug curves in (1). The function f(z,π1) is unspecified that represents the departure from the additive action. Hence, testing the additivity of the drugs is equivalent to testing the following hypothesis: H0:f=0 versus H1:f0.The derivation of a test statistic for hypothesis (5) is expounded in Appendix C. It is worth noting that the test should be powered at a given anticipated size of synergism (η>0), i.e., if Tf2(z,π1)dzdπ1η2 where T denotes the design region for (z,π1), then f is considered not equal to zero. Therefore, the alternative hypothesis in (5) allows the presence of local synergy or local antagonism when they are interspersed in different regions of drug combinations.

For brevity, let T and M denote the design regions for (z,π1) and (x1,x2), respectively. Tan et al., 2003, Tan et al., 2009 and Fang et al., 2008, Fang et al., 2017 proved that the experimental design that spreads its design points uniformly over T is the optimal design in that it is a 3-D design for the combination experiment and it maximizes the minimum power of the lack-of-fit test for the synergy testing in (5). On the other hand, if the null hypothesis is rejected, a dose–response model should be established on M to further analyze the synergism between the drugs (Kong and Lee, 2006, Fang et al., 2008, Fang et al., 2017). However, the shapes of T and M are different. According to (3), it is easy to see that T=(zL,zH)×(0,1) is a rectangle where zL (zH) is the lower (upper) bound of the total dose. By using the inverse transformation of (3), M turns out to be a trapezoid with the four vertices being (zL,0),(zH,0),(0,zH) and (0,zL). For this reason, the design points on the T region (we call it the testing region hereafter) are not necessarily in close agreement with the design points on the M region (we call it the modeling region hereafter), as shown in the motivating example.

Temozolomide (TMZ) is a methylating agent that has been approved for the treatment of astrocytoma and has entered various phases of clinical evaluation against other tumors. Irinotecan (CPT-11) has been demonstrated broad activity against both murine and human tumor xenograft models and clinically significant activity against many types of cancer. Since xenograft models have shown activity for both of the drugs individually, the combination of the two drugs is of interest. Based on Houghton et al. (2000) and Tan et al. (2003) we have the following models for individual drugs y(x1)=0.5332+0.1728logx1=0.1728(logx1+3.0856),y(x2)=0.4348+0.1728logx2=0.1728(logx2+2.5162), where x1 is the dose of CPT-11, x2 is the dose of TMZ and y is the relative decreasing rate of the tumor volume. Then, the potency of TMZ relative to CPT-11 is ρ=0.5659. The total dose range can be chosen such that the response y is from 0.2 to 0.8 for the dose of CPT-11, i.e., the relative decreasing rate of the tumor volume is from 20% to 80%. Then, the total dose range is [0.1454,4.6834] according to CPT-11. Therefore, the T region is the rectangle (0.1454,4.6834)×(0,1) (see Fig. 1(b)) whereas the M region is the trapezoid with its four vertices being (0.1454,0),(4.6834,0),(0,4.6834) and (0,0.1454) (see Fig. 1(a)).

Tan et al., 2003, Tan et al., 2009 and Fang et al., 2008, Fang et al., 2017 proposed to use the uniform design on the T region to maximize the power of synergy testing. The design points of such a design with 25-run are visualized in Fig. 1(b), and it is easily seen that they are uniformly scattered. However, by using the inverse transformation of (3), the corresponding design points on the M region are not so uniform (see Fig. 1(a)). In particular, the design points on the M region become sparser and sparser as they are approaching the bottom of the trapezoid. A dose–response model built on such a design may miss some important drug interactions when the dose of CPT-11 and/or the dose of TMZ are large since such areas are not well covered by the design points. On the other hand, a uniform design over the M region is beneficial for dose–response modeling due to its thorough exploration of the M region (see Fig. 1(c)). However, the corresponding design points on the T region may be under-powered in synergy testing because there exists a large empty area on the left part of the T region (see Fig. 1(d)). This illustration clearly shows that a design that is efficient for synergy testing is not necessarily desirable for dose–response modeling and vice versa. Figs. 1(e) and (f) present a design that has a good balance between synergy testing and dose–response modeling, i.e., its design points have good uniformity on both the M region and the T region.

For simplicity of presentation, we call a design like in Fig. 1(e) or (f) a compromise design. For economic reasons, such a design is desirable because synergy testing and dose–response modeling can be done in a single step or with few follow-up experiments. How to obtain compromise designs for combination experiments of two drugs is a central contribution of this paper. The remainder of this paper is organized as follows: Section 2 elaborates details surrounding the compromise design including the design criterion and design construction methods. In Section 3, we apply the proposed experimental design to two illustrative examples of drug combination experiments. Then, in Section 4, simulation studies show comparisons with some other conventional design types, for examples, the Monte Carlo design and factorial design. A discussion is given in Section 5 and some additional contents that the readers may be interested in are given in the Appendices.

Section snippets

The compromise design

This section will be divided into three parts. Sections 2.1 The uniformity criterion, 2.2 Construction of uniform design on will present the uniformity criterion for design construction and provide a method for constructing uniform design on the M region, respectively. These will be served as bases for the construction of the compromise design, which will be discussed in Section 2.3.

Illustrative examples

Example 1

This example provides more details on the motivating example in Section 1.2. As stated in Section 1.2, 25 design points are supposed to be used for the combination experiment of Irinotecan (CPT-11) with Temozolomide (TMZ). Fig. 1(a) corresponds to the design whose points are uniform on the T region, Fig. 1(c) corresponds to the design whose points are uniform on the M region, and Fig. 1(e) corresponds to the compromise design. In this example, we denote these three designs as P1,P2 and P3,

Simulation

To further evaluate the merits of the compromise design, we compare five designs by simulations: P1—the design whose points are uniform on the T region, P2—the design whose points are uniform on the M region, P3—the compromise design, P4—the Monte Carlo design (random samples generated from a uniform distribution), and P5—the factorial design (a simple rigid grid of design including the extreme points). Suppose that 25 points (5 equally spaced levels in each direction for P5) with 2 replicates

Conclusion

Combinations of drugs are the hallmark of therapies for complex diseases such as cancer, HIV and hypertension. Since the experimental units (for example, the mice) are usually expensive, experimental design should jointly consider different issues to avoid extensive follow-up experiments, and this point was often unappreciated by the existing designs. In this paper, we propose an experimental design, called the compromise design to meet the dual requirements on synergy testing and dose–response

Acknowledgments

The authors thank the editor, the associate editor and the reviewers for their valuable comments and suggestions. This research was supported in part by the National Natural Science Foundation of China (Grant Nos. 11701109, 11971204, 11801331 and 11861017), Natural Science Foundation of Guangxi Province of China (Grant Nos. 2018JJB110027 and 2018AD19235), and Natural Science Foundation of Jiangsu Province of China (Grant No. BK20200108).

References (37)

  • FangK.T. et al.

    Design and Modeling for Computer Experiments

    (2006)
  • FangK.T. et al.

    Theory and Application of Uniform Experimental Designs

    (2018)
  • FangK.T. et al.

    Design and Modeling of Experiments

    (2011)
  • FangH.B. et al.

    Experimental design and interaction analysis of combination studies of drugs with log-linear dose responses

    Stat. Med.

    (2008)
  • FealaJ.D. et al.

    Systems approaches and algorithms for discovery of combinatorial therapies

    WIREs Syst. Biol. Med.

    (2010)
  • FitzgeraldJ.B. et al.

    Systems biology and combination therapy in the quest for clinical efficacy

    Nat. Chem. Biol.

    (2006)
  • FoucquierJ. et al.

    Analysis of drug combinations: current methodological landscape

    Pharmacol. Res. Perspect.

    (2015)
  • FrancoJ. et al.

    DiceDesign: Designs of Computer Experiment. R package version 1.8-1

    (2019)
  • Cited by (5)

    • Optimal designs for semi-parametric dose-response models under random contamination

      2023, Computational Statistics and Data Analysis
      Citation Excerpt :

      Finding optimal clinical trial designs is gaining interest among data scientists and pharmacologists since it can assist investigators to achieve higher quality results while working with limited resources. Important works include but are not limited to Dette et al. (2008a, 2010); Feller et al. (2017); Chen et al. (2017); Atkinson (2015); Wang and Ai (2016); Yu et al. (2018); Sverdlov et al. (2019); Rosa (2020); Dutta and SahaRay (2021) and Huang and Chen (2021). When researchers are only interested in a few pre-specified dose levels, the corresponding optimal design problem has been studied by Atkinson (2015), Wang and Ai (2016) and Rosa (2020).

    • Robust designs for dose–response studies: Model and labelling robustness

      2021, Computational Statistics and Data Analysis
      Citation Excerpt :

      Woods et al. (2006) proposed to optimize a loss function after averaging over a finite set of competing models, differing with respect to, for instance, the assumed parameter values, or the assumed link. For some recent work in this area see as well Huang and Chen (2021), Holland-Letz and Kopp-Schneider (2015), Feller et al. (2017) and Xu and Sinha (2021). The mathematical framework, and asymptotic properties, of our methods are outlined in the next two sections of this article.

    View full text