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© Copyright 2015 Quintiles
Efficiency of Randomized Concentration-Controlled
Trials Relative to Randomized Dose-Controlled
Trials, and Application to Personalized Dosing Trials
Russell Reeve, PhD
Quintiles, Inc.
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• Randomized dose-controlled trial
› Patients are randomized to dose groups
› Drug from dose distributes into the bloodstream, yielding exposure (concentration)
› Drug in the bloodstream finds its way to the site of action
› Drug at site of action
• Randomized concentration-controlled trial
› Patients are randomized to concentration groups
› Drug dose is calculated from the concentration target
› Drug from dose distributes into the bloodstream, yielding exposure (concentration)
» Hopefully this is close to the target
› Drug in the bloodstream finds its way to the site of action
› Drug at site of action
• Concentration can be summarized by AUC, Cmax, Cmin, mean concentration,
etc.
RCCT vs RDCT Nomenclature and setup
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• C = concentration (or summary statistic)
• Based on power model, log C = 0 + 1 log d +
› is occasion-to-occasion (but within subject) variability, with Var = 2
› 0 and 1 may vary subject to subject, but constant within subject
› We will assume Var 0 = 2, and Var 1 = 0
› For ease of exposition, also assume dose proportionality (i.e., 1 = 1)
• Challenge is to estimate 0 for each patient
• Algorithm
› First dose chosen at random from doses that achieve expected exposure on average
› Observed exposure
› Use that to adjust dose to bring onto target
Assume linearity Based on power model
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Effect of Reducing PK Variability on PD Separation What we are trying to achieve
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Effect of Reducing PK Variability on PD Separation What we are trying to achieve
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• Assume prior 0 ~ N(B, 2)
• The posterior at step i then has mean and variance
Bayesian Dose Adjustment
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Convergence Converges with quadratic speed
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• Model for effect
› Hill model Y = a + dq, where q = {1 + exp[(0 + 1m)]}1
• Hypotheses:
› H0: E{Y} 0 vs HA: E{Y} follow Hill model
• Let test T1 require n1 to achieve power p, and test T2 require n2 to achieve the
same power p. Then we define efficiency to be eff(T1, T2) = n2/n1 (Lehmann
1986, p. 321)
Definitions
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• Let PK parameter x be defined as x = () + + form some function of the
dose , patient-specific intercept , and residual variability ~ N(0, 2)
› The variance 2 will vary depending on the design, with RDCT being less variable
than RCCT.
› We note that RCCT RDCT.
• Theorem 1 (Reeve 1994): Fundamental RCCT Theorem
› eff(RCCT, RDCT) 1 with equality if and only if RCCT = RDCT.
• Theorem 2: Consistency of Bayesian estimates (cf. Ghosal Theorem 4 quoting
Ghosal, Ghosal, Samanta 1994 and Ghosal, Ghosh, Samanta 1995)
› limt it = i
RCCT Theorem Underlies justification
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• Improvement is a function of variability of within to between subjects (/)
• As long as adjustment is dampened enough, then result is asymptotically true
› Danger is when you do not dampen enough; see Deming’s concern about adjustment
in quality control setting
› Bayesian adjustment dampens enough
• Adaptive designs have also been shown to be more efficient in this setting
Commentary
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Forms of Hill Model
• Hill model was first developed around 1903
• E = a + Emax dh/(ED50h + dh)
• E = + (Erange ½)/(1 + exp(h(log x log ED50)))
• Aliases
› Michaelis-Menten
› Emax
› 4-parameter Logistic
Many forms available, all equivalent
0 200 400 600 800 1000
0.0
0.2
0.4
0.6
0.8
Hill Model
Dose
Effe
ct
0 1 2 3 4 5 6 7
0.0
0.2
0.4
0.6
0.8
Hill Model (Log X)
Dose
Effe
ct
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Hill Model Forms All are equivalent functions of the dose d
h
d
h
h
Kd
dEEdE
max
0)(
h
d
h
h
Kd
dEEEdE
)()( 00
))/(1()( max
h
dKd
EEdE
))/(1(
11)( 0 h
dKdPEdE
]}[logexp{1)( max
0cdh
EEdE
))(1(
11)( 0 hdPEdE
2
1
]}[logexp{1
1)( max0
cdhEEdE
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D-Optimal Design
• Let = (E0, Emax, c, h) be the parameters
• Let y(d; ) be the Hill model
• Z = dy/d
• Determinant is D = |ZT Z|
• Want to find design that maximizes D.
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D22 Optimal
• Define () log D22 log Emax2
• Then () = 2 1 + 8 exp()/[1 + exp()] 4 = 0.
• This can be solved approximately as
opt 4(3 + 2e)/(3 + e)2.
• This implies v 0.23 or 0.77.
-4 -2 0 2 4
0.0
0.2
0.4
0.6
0.8
1.0
v1
1e
xp
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Robustness of Optimal Location
D22
0.5
00
.60
0.7
00
.80
0.9
0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
D22
Hill
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Adaptive Design
• Start with optimal design for first K subjects
• Update distributions of parameters
• Use D-Bayes design to pick next set of M subjects
• Repeat process
• Should beat fixed design
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Simulation Study
• Truth
› E0 = 40, Emax = -5, h = 1.5, = 1, c {1, 1.1, …, 1.9, 2.0}
› This model based on actual experience in clinical trial
› Sample size N = 300
• Fixed design
› Doses = 0, 20/3, 40/3, and 4 (equal allocation of patients)
• D-optimal design
› 4 doses (equal allocation of patients)
• D-Bayes optimal design
› Use uniform prior for
› K = 120, M = 20
• Look at ability to accurate and precisely estimate log ED50, log ED70, and log
ED90.
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Comparison of Performance
Red = fixed design
Blue = Bayes adaptive
Green = D optimal
Bias SD
ED50
ED70
ED90
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Within Patient Adapting
• Back to RA motivating example
• We have several options
• Fixed design
› Ad hoc
› D-optimal
• Bayesian adaptive design
• Within patient adaptive
• Endpoint is ACR20
› Binary
› If a patient achieve 20%
reduction in symptoms, then set to
1; otherwise set to 0
1 2 5 10 20 50 100 200 500
01
02
03
04
05
06
0
Dose
AC
R T
rea
tme
nt E
ffe
ct
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Personalized Design
• ACR20 is binary
• P(ACR20 = 1) follows Hill model
• Question: Can we estimate ED70 better
than with Bayesian design or fixed
design?
• Patients are seen at baseline, weeks 4,
8, 12, 16, and 24
• If ACR20 = 0 for 3 consecutive visits,
increase dose
• If ACR20 = 1 for 3 consecutive visits,
decrease dose
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
Hill Models for Personalized Trial
Dose
AC
R2
0 p
rop
ort
ion
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Simulation Results of Personalized Dosing
1.0 1.5 2.0 2.5 3.0
0.0
15
0.0
20
0.0
25
0.0
30
0.0
35
0.0
40
0.0
45
Bias
ED50 Possibility
Bia
s
Fixed
Bayes
Pesonalized
1.0 1.5 2.0 2.5 3.0
0.0
10
0.0
15
0.0
20
0.0
25
0.0
30
0.0
35
Standard Deviation
ED50 Possibility
Sta
nd
ard
De
via
tio
n
Fixed
Bayes
Pesonalized
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• Objective is to estimate minimum effective dose (MED)
› MED defined as dose that yield some p% response
• Consider binary endpoints (e.g., rheumatoid arthritis trial)
• Measure response repeatedly
• Assume P(therapeutic success) = 0 + 1 log dose
• Update predictors of 0 and 1 via Bayesian updates for each patient
• Update patient’s dose
› No TS for 3 observations, increase dose
› TS for 3 observations, decrease dose
› Otherwise, keep at current dose
More Challenging: Personalized Design on Binary
Endpoint
Extend to PD endpoints (Reeve and Ferguson 2013)
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Personalized Design is More Efficient Simulation results
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• We have discussed three seemingly disparate designs, but they are linked by
a common theme
› RCCT
› Bayesian adaptive
› Personalized
• All are using data (Bayesian updating in all 3) to correct for mis-specification of
our model
› RCCT corrects for misspecified exposure
› Bayesian adaptive and Personalized misspecified dose-response
» Could be even better if we correct for misspecified dose-response and exposure simultaneously
• By correcting for misspecifications, we achieve efficiency in design
Discussion of Themes Tie together the 3 types of designs
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• RCCT and other Bayesian adaptations can significantly improve the efficiency
of designs
• Encourages us to think in terms of exposure-response models
› These trials provide us with better estimates of the dose response
Conclusions and Wrapup
1 2 5 10 20 50 100 200 500
01
02
03
04
05
06
0
Dose
AC
R T
rea
tme
nt E
ffe
ct