© 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