Apply Response Surface Analysis for Interaction of Dose Response Combine Treatment Drug Study

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1 Paper PO13 Apply Response Surface Analysis for Interaction of Dose Response Combine Treatment Drug Study Tung-Yi (Tony) Wu, Ph.D. Kos Pharmaceuticals Inc., Miami, FL ABSTRACT Combination treatments are widely used in medicine. When a clinical trial employs two (or more) drugs and is expensive, a matrix design with response surface analysis become an essential statistical method to obtain an overall profile, a minimum effective dose and/or maximum percentage change from baseline. In this paper, the author will demonstrate one example of a SAS program (using Proc RSREG and Proc G3D) to obtain response surface results. An example of low-cost response surface method design that does not require all possible dose-pair combination is also included. INTRODUCTION Combination drug treatments may involve synergy or antagonism. For the same combination of drugs, we might have both synergistic effects for dose-pairs in some regions and antagonistic effects in different dose-pair regions. Combination-dosages for prescription drugs for humans means that two (or more) drugs may be combined in a single-dosage form such that each component makes a contribution to the claimed effects of the combination and the dosage of each component is such that the combination is safe and effective for a relevant patient population. Such concurrent therapy is defined in the labeling for the drug [21 CFR ]. In order to obtain information for both effectiveness and safety, a matrix study design with response surface analysis is often used for combination drug treatment. Typically all combinations of doses of the two drugs are tested in humans. However, by looking at the expected matrix responses, a smaller design might be found. FIRST EXAMPLE The first example includes two drugs (A and B) and the response variable of Z (percentage change of response variable), obtained from a prior study sample data from environmental science are used to illustrate this method. The percentage change results (variable: Z) of this study were analyzed by estimating the response surface as a function of A and B. Such an analysis yields a response surface with the following overall structure: Z = ß 0 + ß A*A + ß B*B + B A2 *A 2 + ß B2*B 2 + ß AB*A*B

2 where: Z = percentage change of response variable, % A = A dosage B = B dosage AB = interaction term (combination of A and B) Example of SAS program using Proc RSREG and Proc G3D listed in Appendix I. The response surface analysis of the data was used to identify a percentage change pattern and identify the interaction conditions that can yield a "maximum" percentage change. In Table 1 the response surface analysis are presented (Raymond, 1971). These estimated parameters for the statistics indicated a wellestablished removal pattern which can be described by a response surface model with linear and quadratic terms (see partial F-test and F-test for overall [total] regression results in Table 1 and Appendix II). For this model variable A seems to be the most critical parameter (t-test; in Table 1), whereas the interaction term can be ignored. Canonical analysis of the results yielded negative eigenvalues (i.e and 10.58). This implies the existence of a maximum stationary point. The response surface analysis indicated a pattern which yielded a theoretical maximum percentage change of 88.9% at an A dosage value of 8.56 and B dosage value of 7.51 (i.e., at the stationary point; Figure 2). This finding illustrates the potential use of this procedure to have approximately 89% change of the measure response variable with a recommended A dosage close to 8.5 and, a B dosage above 7. SAS output of the Example I are listed in Appendix II. Table 1 RESPONSE SURFACE ANALYSIS RESULTS Parameters Estimates t-test Prob. ß ß A <0.001 ß B ß A <0.001 Partial F-Ratio test (Prob.) (<0.0001) 7.44 (0.0010) ß B ß AB (0.4456) R 2 = 0.77, F-value = [total regression], Prob. = <0.0001

3 Figure 1: The Response Surface Plot Figure 2: The Contour Plot

4 SECOND EXAMPLE The second example is an on going project, data are not available at this time; however, the study design is as follows: Study design: double-blind, parallel-group, muti-center, ten dose-pairs, escalating study design Study drug A dose: Low, Medium, Medium-High, High Study drug B dose: Low, High Combination of A and B: Low, Medium, Medium-High, High Study drug Placebo: Active Placebo Table 2 shows the study design of a possible efficient response surface clinical study design. This low-cost alternative response surface method design does not requires all possible dose-pair combinations. The partial matrix design of the response surface becomes a cost-effectiveness strategy for clinical studies. Table 2: Example of low-cost response surface method design STUDY DRUG Placebo (B) Low (B) Medium (B) High( B) Placebo(A) X X X Low(A) Medium(A) X X X Medium-High(A) X High (A) X X X CONCLUSIONS In the first example, the response surface design included all possible dosepairs; however, in the pharmaceutical industry the clinical trials are very expensive. Study designs become very important. Any additional dose-pairs will significantly increase the study cost. In the second example, the study design does not include all possible dose-pairs. The matrix design of response surface analysis using Proc RSREG and Proc G3D will be a cost-effectiveness method for statistical analysis. There are some limitations of this method of designing a partial matrix study. Prior data are needed. The example here benefited by having such data. However, other ways of getting prior information might include literature review, pilot study, or expectations for response for which this method might be generalized.

5 REFERENCE 1. Raymond H. Myers: Response Surface Methodology, SAS: SAS System for Regression, Cary, NC, SAS Institute, Inc. 3. SAS: SAS/GRAPH Software, version 6, Cary, NC, SAS Institute, Inc. 4. SAS: User s Guide, Cary, NC, SAS Institute, Inc. 5. Theodore Allen et al, Low-Cost Response Surface Methods From Simulation Optimization, Quality Reliability Engineering International 2002; 18:5-17 TRADEMARKS SAS is a registered trademark of SAS Institute Inc., in the USA and other countries. ACKNOWLEDGMENTS I would like to thanks Phillip Simmons and Caroline Malott for their review and provided great comments. CONTACT INFORMATION Your comments and questions are valued and encouraged. Contact the author at: Author Name: Tung-Yi (Tony) Wu Company KOS Pharmaceuticals Address: 2200 N. Commerce Parkway (suite 300), Weston, FL Work phone: (954) Fax: (954) twu@kospharm.com

6 APPENDIX I : THE RSREG EXAMPLE ******************************************************************* Program Name: rsreg1.sas Programmed by: Tony Wu Software: SAS 8 *******************************************************************; dm output 'clear;log;zoom off;clear'; libname SUG04 'c:\working_file\'; data PO13; set SUG04.po13; proc sort ; by dosagea dosagea; RUN; Proc rsreg data=po13 out=pr123; model pctchange=dosagea dosageb/lackfit predict residual; ridge max; data grid; do dosagea=1 to 10 by 0.2; do dosageb = 1 to 8 by 0.2; output; data PO13; set po13 grid; dosagea2=dosagea**2; dosageb2=dosageb**2; dosageab=dosagea*dosageb; proc reg data=po13; model pctchange=dosagea dosageb dosagea2 dosageb2 dosageab; output out=results p=yhat r=residual rstudent=rstudent h=hatvalue; id obs; data fit; set results; if pctchange=.; pctchange=yhat; proc g3d data=fit; plot dosagea*dosageb=pctchange / grid caxis=blue xticknum=7 yticknum=9 zticknum=6;;

7 *=== PLOT RESPONSE SURFACE BASE ON Proc RSREG results ===; data abc; a= ; b= ; c= ; d= ; e= ; f= ; do dosagea=1 to 10 by 0.5; do dosageb=1 to 8 by 0.5; output; output; data abcz; set abc ; do pctchange= a + b*dosagea + c*dosageb + d*dosagea*dosagea + e*dosageb*dosageb + f*dosagea*dosageb; output; proc g3d data=abcz; plot dosagea*dosageb=pctchange/ grid caxis=blue xticknum=8 yticknum=10 zticknum=10; format dosagea dosageb pct_change 5.0 ; proc gcontour data=abcz; plot dosagea*dosageb=pctchange / XTICKNUM=11 yticknum=8 levels= ; proc g3grid data=abcz out=gridnums; grid dosagea*dosageb=pctchange / spline smooth=0.05 axis1=1 to 10 by.25 axis2=1 to 8 by.25; run ; proc g3d data=gridnums ; plot dosagea*dosageb=pctchange / grid caxis=black xticknum=8 yticknum=10 zticknum=10; ******************************************************************** End of program. ********************************************************************;

8 APPENDIX II : THE RSREG EXAMPLE OUTPUT The RSREG Procedure Coding Coefficients for the Independent Variables Factor Subtracted off Divided by dosagea dosageb Response Surface for Variable pctchange Response Mean Root MSE R-Square Coefficient of Variation Type I Sum Regression DF of Squares R-Square F Value Pr > F Linear <.0001 Quadratic Crossproduct Total Model <.0001 Sum of Residual DF Squares Mean Square F Value Pr > F Lack of Fit Pure Error Total Error Parameter Estimate from Coded Parameter DF Estimate STD Error t Value Pr > t Data Intercept dosagea < dosageb dosagea*dosagea dosageb*dosagea dosageb*dosageb Sum of Factor DF Squares Mean Square F Value Pr > F dosagea <.0001 dosageb The RSREG Procedure Canonical Analysis of Response Surface Based on Coded Data Critical Value Factor Coded Uncoded dosagea dosageb Predicted value at stationary point: Eigenvectors Eigenvalues dosagea dosageb