© 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.
2
• 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
3
• 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
4
Effect of Reducing PK Variability on PD Separation What we are trying to achieve
5
Effect of Reducing PK Variability on PD Separation What we are trying to achieve
6
• Assume prior 0 ~ N(B, 2)
• The posterior at step i then has mean and variance
Bayesian Dose Adjustment
7
Convergence Converges with quadratic speed
8
• 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
9
• 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
10
• 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
11
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
12
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
13
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.
14
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
15
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
16
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
17
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.
18
Comparison of Performance
Red = fixed design
Blue = Bayes adaptive
Green = D optimal
Bias SD
ED50
ED70
ED90
19
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
20
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
21
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
22
• 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)
23
Personalized Design is More Efficient Simulation results
24
• 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
25
• 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
