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RCCT vs RDCT Nomenclature an setup Ranomize ose-controlle trial Patients are ranomize to ose groups Drug from ose istributes into the bloostream, yieling exposure (concentration) Drug in the bloostream fins its way to the site of action Drug at site of action Ranomize concentration-controlle trial Patients are ranomize to concentration groups Drug ose is calculate from the concentration target Drug from ose istributes into the bloostream, yieling exposure (concentration)» Hopefully this is close to the target Drug in the bloostream fins its way to the site of action Drug at site of action Concentration can be summarize by AUC, C max, C min, mean concentration, etc. 2

Assume linearity Base on power moel C = concentration (or summary statistic) Base on power moel, log C = 0 + 1 log + is occasion-to-occasion (but within subject) variability, with Var = 2 0 an 1 may vary subject to subject, but constant within subject We will assume Var 0 = 2, an Var 1 = 0 For ease of exposition, also assume ose proportionality (i.e., 1 = 1) Challenge is to estimate 0 for each patient Algorithm First ose chosen at ranom from oses that achieve expecte exposure on average Observe exposure Use that to ajust ose to bring onto target 3

ffect of Reucing PK Variability on PD Separation What we are trying to achieve 4

ffect of Reucing PK Variability on PD Separation What we are trying to achieve 5

Bayesian Dose Ajustment Assume prior 0 ~ N(B, 2 ) The posterior at step i then has mean an variance 6

Convergence Converges with quaratic spee 7

Definitions Moel for effect Hill moel Y = a + q, where q = {1 + exp[( 0 + 1 m)]} 1 Hypotheses: H 0 : {Y} 0 vs H A : {Y} follow Hill moel Let test T 1 require n 1 to achieve power p, an test T 2 require n 2 to achieve the same power p. Then we efine efficiency to be eff(t 1, T 2 ) = n 2 /n 1 (Lehmann 1986, p. 321) 8

RCCT Theorem Unerlies justification Let PK parameter x be efine as x = () + + form some function of the ose, patient-specific intercept, an resiual variability ~ N(0, 2 ) The variance 2 will vary epening on the esign, with RDCT being less variable than RCCT. We note that RCCT RDCT. Theorem 1 (Reeve 1994): Funamental RCCT Theorem eff(rcct, RDCT) 1 with equality if an only if RCCT = RDCT. Theorem 2: Consistency of Bayesian estimates (cf. Ghosal Theorem 4 quoting Ghosal, Ghosal, Samanta 1994 an Ghosal, Ghosh, Samanta 1995) lim t it = i 9

Commentary Improvement is a function of variability of within to between subjects (/) As long as ajustment is ampene enough, then result is asymptotically true Danger is when you o not ampen enough; see Deming s concern about ajustment in quality control setting Bayesian ajustment ampens enough Aaptive esigns have also been shown to be more efficient in this setting 10

ffect 0.0 0.2 0.4 0.6 0.8 ffect 0.0 0.2 0.4 0.6 0.8 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) 0 200 400 600 800 1000 Dose 0 1 2 3 4 5 6 7 Dose 11

12 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 2 1 ]} [log exp{ 1 1 ) ( max 0 c h

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. 13

v 1 1 exp 0.0 0.2 0.4 0.6 0.8 1.0 D 22 Optimal Define () log D 22 log max 2 Then () = 2 1 + 8 exp()/[1 + exp()] 4 = 0. This can be solve approximately as opt 4(3 + 2e)/(3 + e) 2. This implies v 0.23 or 0.77. -4-2 0 2 4 14

Robustness of Optimal Location D 22 0.50 0.60 0.70 0.80 0.90 D22 Hill 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 15

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

Simulation Stuy Truth 0 = 40, max = -5, h = 1.5, = 1, c {1, 1.1,, 1.9, 2.0} This moel base on actual experience in clinical trial Sample size N = 300 Fixe esign Doses = 0, 20/3, 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 = 120, M = 20 Look at ability to accurate an precisely estimate log D 50, log D 70, an log D 90. 17

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

ACR Treatment ffect 0 10 20 30 40 50 60 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 ACR20 Binary If a patient achieve 20% reuction in symptoms, then set to 1; otherwise set to 0 1 2 5 10 20 50 100 200 500 Dose 19

ACR20 proportion 0.0 0.2 0.4 0.6 0.8 1.0 Personalize Design ACR20 is binary P(ACR20 = 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, 12, 16, an 24 If ACR20 = 0 for 3 consecutive visits, increase ose If ACR20 = 1 for 3 consecutive visits, ecrease ose 0 10 20 30 40 Dose 20

Bias 0.015 0.020 0.025 0.030 0.035 0.040 0.045 Stanar Deviation 0.010 0.015 0.020 0.025 0.030 0.035 Simulation Results of Personalize Dosing Bias Stanar Deviation Fixe Bayes Pesonalize Fixe Bayes Pesonalize 1.0 1.5 2.0 2.5 3.0 D50 Possibility 1.0 1.5 2.0 2.5 3.0 D50 Possibility 21

More Challenging: Personalize Design on Binary npoint xten to PD enpoints (Reeve an Ferguson 2013) Objective is to estimate minimum effective ose (MD) MD efine as ose that yiel some p% response Consier binary enpoints (e.g., rheumatoi arthritis trial) Measure response repeately Assume P(therapeutic success) = 0 + 1 log ose Upate preictors of 0 an 1 via Bayesian upates for each patient Upate patient s ose No TS for 3 observations, increase ose TS for 3 observations, ecrease ose Otherwise, keep at current ose 22

Personalize Design is More fficient Simulation results 23

Discussion of Themes Tie together the 3 types of esigns We have iscusse three seemingly isparate esigns, but they are linke by a common theme RCCT Bayesian aaptive Personalize All are using ata (Bayesian upating in all 3) to correct for mis-specification of our moel RCCT corrects for misspecifie exposure Bayesian aaptive an Personalize misspecifie ose-response» Coul be even better if we correct for misspecifie ose-response an exposure simultaneously By correcting for misspecifications, we achieve efficiency in esign 24

ACR Treatment ffect 0 10 20 30 40 50 60 Conclusions an Wrapup RCCT an other Bayesian aaptations can significantly improve the efficiency of esigns ncourages us to think in terms of exposure-response moels These trials provie us with better estimates of the ose response 1 2 5 10 20 50 100 200 500 Dose 25