Two-Dimensional Maximin Power Designs for Combination Experiments of Drugs

Abstract

The combined use of two drugs is a major treatment approach for complex diseases such as cancer and HIV due to its potential for efficacy at lower, less toxic doses and the need to reduce developmental time and cost. Experimental designs have been proposed in the literature to test whether there are synergistic or antagonistic actions between the combined drugs. The existing designs for synergy testing are primarily one-dimensional (1D), allocating the doses of one drug while keeping the dose of another, the mixing proportion, or the total dose of the two drugs fixed. This paper considers two-dimensional (2D) designs in which the doses of two drugs can be varied simultaneously over the entire dose region. Based on the premise that prior information about the single-drug experiments is already available, we propose a succinct dose-response model that encompasses a wide class of potential synergistic/antagonistic actions deviated from additivity. We show that the uniform design measure over the 2D dose region is optimal under the proposed model in the sense that it maximizes the minimum power in the F-test to detect drug synergy. Methods for sample size calculation and design generation for our 2D optimal design are given. We illustrate the use of the proposed design and demonstrate its advantages over the 1D optimal design via a combination study of two anticancer drugs.

Keywords

Design of experiment / Optimal design / Response-surface modeling / Synergy testing / Uniform design / 62K05

References

[1] Abdelbasit KM, Plackett RL. Experimental design for joint action. Biometrics, 1982, 38: 171-179

[2] Almohaimeed B, Donev AN. Experimental designs for drug combination studies. Comput. Stat. Data. An., 2014, 71: 1077-1087

[3] Berenbaum MC. What is synergy?. Pharmacol. Rev., 1989, 41: 93-141

[4] Chen JB, Han X, Lin DKJ, Yang L, Zhou YD. On ordering problems: a statistical approach. Stat. Sin., 2023, 33: 1903-1922

[5] Dawson KS, Carter WHJr, Gennings C. A statistical test for detecting and characterizing/departures from additivity in drug/chemical combinations. J. Agr. Biol. Envir. St., 2000, 5: 342-359

[6] Fang HB, Chen XR, Pei XY, Grant S, Tan M. Experimental design and statistical analysis for three-drug combination studies. Stat. Methods Med. Res., 2017, 26: 1261-1280

[7] Fang HB, Ross DD, Sausville E, Tan M. Experimental design and interaction analysis of combination studies of drugs with log-linear dose responses. Stat. Med., 2008, 27: 3071-3083

[8] Fang KT, Li R, Sudjianto ADesign and Modeling for Computer Experiments, 2006New YorkChapman & Hall/CRC

[9] Fang KT, Liu MQ, Qin H, Zhou YDTheory and Application of Uniform Experimental Designs, 2018SingaporeSpringer

[10] Fang KT, Liu MQ, Zhou YDDesign and Modeling of Experiments, 2011BeijingHigher Education Press

[11] Feala JD, Cortes J, Duxbury PM, Piermarocchi C, McCulloch AD, Paternostro G. Systems approaches and algorithms for discovery of combinatorial therapies. WIREs Syst. Biol. Med., 2010, 2: 181-193

[12] Fitzgerald JB, Schoeberl B, Nielsen UB, Sorger PK. Systems biology and combination therapy in the quest for clinical efficacy. Nat. Chem. Biol., 2006, 2: 458-466

[13] Franco, J., Dupuy, D., Roustant, O., Kiener, P., Damblin, G., Iooss, B.: DiceDesign: Designs of Computer Experiment. R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R$$\end{document} package version 1.8-1, URL, https://cran.r-project.org/web/packages/DiceDesign/index.html (2019)

[14] Gennings C, Carter WHJr, Campain J, Bae DS, Yang RSH. Statistical analysis of interactive cytotoxicity in human epidermal keratinocytes following exposure to a mixture of four metals. J. Agric. Biol. Environ. Stat., 2002, 7: 58-73

[15] Houghton PJ, Stewart CF, Cheshire PJ, Richmond LB, Kirstein MN, Poquette CA, Tan M, Friedman HS, Brent TP. Antitumour activity of temozolomide combined with irinotecan is partly independent of O6\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O^6$$\end{document}-methylguanine- DNA methyltransferase and mismatch repair phenotypes in xenograft models. Clin. Cancer Res., 2000, 6: 4110-4118

[16] Kong, M., Lee. J. J.: A generalized response surface model with varying relative potency for assessing drug interaction. Biometrics 62, 986–995 (2006)

[17] Laska EM, Meisner M, Siegel C. Simple designs and model-free tests for synergy. Biometrics, 1994, 50: 834-841

[18] Lin DKJ, Peng JY. The order-of-addition experiments: a review and some new thoughts (with discussion). Qual. Eng., 2019, 31: 49-59

[19] Ning S, Xu H, Al-Shyoukh I, Feng J, Sun R. An application of a Hill-based response surface model for a drug combination experiment on lung cancer. Stat. Med., 2014, 33: 4227-4236

[20] Peng JY, Mukerjee R, Lin DKJ. Design of order-of-addition experiments. Biometrika, 2019, 106: 683-694

[21] Ryall KA, Tan AC. Systems biology approaches for advancing the discovery of effective drug combinations. J. Cheminform., 2015, 7: 7

[22] Shiozawa K, Nakanishi T, Tan M, Fang HB, Wang WC, Edelman MJ, Carlton D, Gojo I, Sausville EA, Ross DD. Preclinical studies of vorinostat (suberoylanilide hydroxamic acid, SAHA) combined with cytosine arabinoside (ara-C) and etoposide for treatment of acute leukemias. Clin. Cancer Res., 2009, 15: 1698-1707

[23] Straetemans R, O’Brien T, Wouters L, van Dun J, Janicot M, Bijnens L, Burzykowski T, Aerts M. Design and analysis of drug combination experiments. Biometrical J., 2005, 47: 299-308

[24] Tallarida RJDrug Synergism and Dose-effect Data Analysis, 2000New YorkChapman & Hall/CRC

[25] Tallarida RJ, Stone DJ, Raffa RB. Efficient designs for studying synergistic drug combinations. Life Sci., 1997, 61: 417-425

[26] Tan M, Fang HB, Ross DDEvaluating Synergy: Statistical Design and Analysis of Drug Combination Studies, 2022New YorkWiley

[27] Tan M, Fang HB, Tian GL. Dose and sample size determination for multi-drug combination studies. Stat. Biopharm. Res., 2009, 1: 301-316

[28] Tan M, Fang HB, Tian GL, Houghton PJ. Experimental design and sample size determination for testing synergy in drug combination studies based on uniform measures. Stat. Med., 2003, 22: 2091-2100

[29] Tian GL, Fang HB, Tan M, Qin H, Tang ML. Uniform distribution in a class of convex polyhedrons with applications to drug combination studies. J. Multivariate Anal., 2009, 100: 1854-1865

[30] Voelkel JG. The design of order-of-addition experiments. J. Qual. Technol., 2019, 51: 230-241

[31] Wiens DP. Designs for approximately linear regression: two optimality properties of uniform designs. Stat. Probabil. Lett., 1991, 12: 217-221

[32] Wu CFJ, Hamada MExperiments: Planning, Analysis, and Optimization, 20213New YorkWiley

[33] Yang JF, Sun FS, Xu H. A component position model, analysis and design for order of addition experiments. Technometrics, 2021, 63: 212-224

[34] Yang LQ, Zhou YD, Liu MQ. Ordering factorial experiments. J. R. Stat. Soc. Ser. B Stat. Methodol., 2023, 85: 869-885

[35] Zhao YN, Li Z, Zhao SL. A new method of finding component orthogonal arrays for order-of-addition experiments. Metrika, 2021, 84: 805-824

[36] Zhao YN, Lin DKJ, Liu MQ. Designs for order of addition experiments. J. Appl. Stat., 2021, 48: 1475-1495

[37] Zhao YN, Lin DKJ, Liu MQ. Optimal designs for order-of-addition experiments. Comput. Stat. Data. An., 2022, 165 Article ID: 107320

Funding

National Natural Science Foundation of China(12131001)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature