Optimal Design for Hill Model

Transcription

1 Optimal Design for Hill Moel DA 01 D-optimal; aaptive esign; personalize esign Russell Reeve, PhD Quintiles Durham, NC, USA Copyright 01 Quintiles

2 Outline Motivating problem, from rheumatoi arthritis Hill moel D-optimal esign for Hill moel D-Bayes optimal esign for Hill moel Aaptive esign base on D-Bayes esign Personalize esign for RA

3 Co-authors on work presente herein From North Carolina State Univesity > Wei Xiao > Braley Ferguson 3

4 Rheumatoi Arthritis Progressive isease Large proportion of the population Very large market (> $5 billion/yr) Clinical trials > npoints: ACR0, DAS8 Time moel can be moele as exponential > ACR0 = a(1 e bt ) Also have ose-response 4

5 ACR Treatment ffect Dose Response Multiplie on top of time-response moel Moele as Hill moel Dose 5

6 ffect ffect Forms of Hill Moel Many forms available, all equivalent Hill moel was first evelope aroun 1903 = a + max h /(D 50 h + h ) = + ( range ½)/(1 + exp(h(log x log D 50 ))) Aliases > Michaelis-Menten > max > 4-parameter Logistic Hill Moel Hill Moel (Log X) Dose Dose 6

7 7 Hill Moel Forms All are equivalent functions of the ose h h h K max 0 ) ( h h h K ) ( ) ( 0 0 ) ) / ( (1 ) ( max h K ) ) / ( (1 1 1 ) ( 0 h K P ]} [log exp{ 1 ) ( max 0 c h ) ) ( (1 1 1 ) ( 0 h P 1 ]} [log exp{ 1 1 ) ( max 0 c h

8 Parameter Interpretation Asymptote Parameters ( ) 0 max Slope an location 1 1 exp{ h[log c]} 1 0 D 50 h max h log c Define v = (1 + exp{ h[log c]}) 1 8

9 Uses of Moel Dose response xposure response Combinations use in other applications Higher oses toxic Pharmacological response with mechanisms 1e-01 1e+01 1e+03 1e+05 Dose 1e-01 1e+01 1e+03 1e+05 Dose 9

10 Binary Responses Logistic Regression uses a Hill moel (logit) with asymptotes of 0 an 1 P(Y=1) = a + /{1 + exp[b(log x c)]} Use in mycophenolate mofetil RCCT > x = log AUC > Y = organ rejection status > Patients ranomize to AUC levels > Doses ajuste to get patients onto target using Bayesian upating of exposureresponse relationship - AUC = 1 + ose - Prior for 1 was base on variability in patient population - Distribution of 1 was upate after each ose, an use to ajust the next ose 10

11 D-Optimal Design Let = ( 0, max, c, h) be the parameters Let y(; ) be the Hill moel Z = y/ Determinant is D = Z T Z Want to fin esign that maximizes D. 11

12 Maximize Determinants Break into inepenent problems Asymptote esign vi nvi D11 n vi v Maximize by setting v to either 0 or 1 Satisfie by setting to 0 or Slope an location esign D [ [ v (1 v max h vi (1 vi )] [ vi (1 vi )] i i i )] where i h (log i c) Because D is symmetric about 0, this simplifies to i D max 4h( v (1 v )) log v1 1 v 1 1

13 v 1 1 exp D Optimal Define () log D log max Then () = exp()/[1 + exp()] 4 = 0. This can be solve approximately as opt 4(3 + e)/(3 + e). This implies v 0.3 or

14 Robustness of Optimal Location D D Hill

15 Bayesian D-optimal Put prior on parameters D Bayes ( ( c)) D( c c0,( / h) ) c e ( c c0,( / h) ) c Difficult to get expressions to solve this Wiens up the location of the maximum, but not as much as you might think 15

16 Aaptive Design Start with optimal esign for first K subjects Upate istributions of parameters Use D-Bayes esign to pick next set of M subjects Repeat process Shoul beat fixe esign 16

17 Simulation Stuy Truth > 0 = 40, max = -5, h = 1.5, = 1, c {1, 1.1,, 1.9,.0} > This moel base on actual experience in clinical trial > Sample size N = 300 Fixe esign > Doses = 0, 0/3, 40/3, an 4 (equal allocation of patients) D-optimal esign > 4 oses (equal allocation of patients) D-Bayes optimal esign > Use uniform prior for > K = 10, M = 0 Look at ability to accurate an precisely estimate log D 50, log D 70, an log D

18 Comparison of Performance Bias SD D 50 D 70 Re = fixe esign Blue = Bayes aaptive Green = D optimal D 90 18

19 ACR Treatment ffect Within Patient Aapting Back to RA motivating example We have several options Fixe esign > A hoc > D-optimal Bayesian aaptive esign Within patient aaptive npoint is ACR0 > Binary > If a patient achieve 0% reuction in symptoms, then set to 1; otherwise set to Dose 19

20 ACR0 proportion Personalize Design ACR0 is binary P(ACR0 = 1) follows Hill moel Hill Moels for Personalize Trial Question: Can we estimate D70 better than with Bayesian esign or fixe esign? Patients are seen at baseline, weeks 4, 8, 1, 16, an 4 If ACR0 = 0 for 3 consecutive visits, increase ose If ACR0 = 1 for 3 consecutive visits, ecrease ose Dose 0

21 Bias Stanar Deviation Simulation Results Bias Stanar Deviation Fixe Bayes Pesonalize Fixe Bayes Pesonalize D50 Possibility D50 Possibility 1

22 Conclusion Develope expression for D-optimal esign for Hill moels Use D-Bayes to construct aaptive trial D-Bayes aaptive trial significantly outperforme fixe trial Personalize trial beat again D-Bayes aaptive trial Take away: Faster you can utilize information, the better