Phase 2 doseresponse design: optimal doses

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Phase 2 dose-response design optimal doses
sample-size selection
Patrick Johnson, Pfizer Global Research
Development, UK Acknowledgement Virginie
Herben, Thomas Kerbusch, Jacqui Spanton Byron
Jones (Pfizer) Jakob Ribbing (Uppsala
University, Sweden)
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Agenda

• Overview purpose
• Dose-response study problem
• Dose selection
• Best Model
• Partial derivatives
• Simulation for information
• OTD warning
• Sample size
• Trial simulation
• Conclusion

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Overview

• After POC/POM main aim of Phase 2 to characterise
  dose-response
• FDA
• 10 years ago Have you got the correct dose?
• Now Whats the dose-response?
• Important design features of dose-response
  studies
• Dose selection
• Sample size

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Overview

• Dose selection
• How many?
• Which doses?
• How do you determine sample size?
• Most dose-response studies include formal
  (pair-wise) comparison
• Efficient pair-wise ? efficient dose-response

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Overview

• Post POC usual to have dose-response model based
  on current knowledge (quantified best guess)
• Optimal design theory could aid in
  dose-selection 
• Sample size subsequently selected using
  simulation (/or ODT)

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Purpose

• Describe efficient process of selecting dose(s)
  and sample size using optimal design theory and
  simulation in example dose-response study
• Example in spirit with a current project
• Simplified for illustration
• Splus (code last slide)

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Dose-response problem
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Background

• Example (any indication)
• Endpoint change from baseline ()
• No placebo effect

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Background

• Phase 2a 10, 25 50 mg

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X
X
X
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Background

• Phase 2a 10, 25 50 mg
• Drug is efficacious and safe
• Not fully characterised dose-response
• Lower dose may give similar effect and less/no
  AEs

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Aim

• Design study that, in combination with P2a data,
  fully characterises dose-response
• Questions
• Dose selection?
• Sample size (no pair-wise comparisons)?

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Dose-selection
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Dose-selection strategy

• Dose-response model based on P2a data
• Using best model, apply optimal design theory
  to select lower doses that will augment P2a data
  and fully characterise dose-response

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Assumptions

• Model parameter estimate are correct
• Unlikely to be true
• Model based on current knowledge so best guess
• Better than gut feeling

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Dose-response best model
X
X
X
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Sigmoid Emax model

• As Emax model with additional parameter (n)
• Modifies steepness of response

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Sigmoid Emax model
n5
n2
n1
n0.5
Symmetrical about ED50
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Richards model

• As Sigmoid Emax with additional parameter
• Incorporates possible asymmetry (ASSY)

• Nesting
• If ASSY1, Sigmoid Emax
• If ASSY1 n1, Emax

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Richards model
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Example Dose-response study
4 parameter problem (EMAX, EDI, n
(sigmoid) ASSY)
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Dose-selection problem

• Need to select 2 new doses
• What are the optimal 2 doses that maximise
  information collected (parameter estimates)?

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Wheres the information?
ASSY? n (sigmoid)?
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Look at partial derivatives
Splus deriv()
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Partial derivatives (Emax)
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Partial derivatives (EDI)
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Partial derivatives (n)
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Partial derivatives (ASSY)
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Maybe use partial derivative peaks?
At peaks youre collecting information on other
parameters
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Mathematical compromise?
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Partial derivatives

• Doses at PD peaks probably not too far off
• Need to incorporate
• Mathematical compromise parameter covariance
• P2a information we have at higher doses (Emax
  sigmoid)

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Why not use available software?

• Good optimal design software available (e.g.
  PFIM, POPT, OPTDES)
• Specify model design characteristics
• Optimal dose-selection solution
• No account for previous data at higher doses
• Get the question correct
• What are the 2 optimal doses that when added to
  data already collected will best characterise the
  dose-response?
• Simulation for information

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Simulation framework

• Richards model
• Incorporate previous P2a data
• Constrained by 2 additional doses
• Construct grid of possible dosing permutations
• Simulate for each dosing permutation
• Combine simulated data with previous data and
  calculate information collected
• Select dosing permutation with most information

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Simulation

• Richards model

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Richards model
EMAX 100 EDI 4 n (sigmoid) 2 ASSY 0.1
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Simulation

• Richards model
• Realistic grid of dosing permutations

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Realistic grid of dosing permutations
Constraints D1 0.5-9.5 mg, by 0.5 D2 0.5-9.5
mg, by 0.5 Removed duplications Used
expand.grid()
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Example dosing permutations
171 dosing permutations
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Simulation

• Need a model
• Realistic grid of dosing permutations
• For each dosing permutation

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For example
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Simulation

• Need a model
• Realistic grid of dosing permutations
• For each dosing permutation
• Add previous P2a doses

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For example
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Simulation

• Need a model
• Realistic grid of dosing permutations
• For each dosing permutation
• Add previous doses
• Calculate derivative for each parameter at each
  dose (J)

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For example
ASSY
n
EMAX
EDI
1.0
2.0
J
10.0
25.0
50.0
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Simulation

• Need a model
• Realistic grid of dosing permutations
• For each dosing permutation
• Add previous doses
• Calculate derivative for each parameter at each
  dose (J)
• Generate Fisher Information Matrix (FIM)
  (JwJT)

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Simulation

• Need a model
• Realistic grid of dosing permutations
• For each dosing permutation
• Add previous doses
• Calculate derivative for each parameter at each
  dose (J)
• Generate Fisher Information Matrix (FIM)
  (JwJT)
• Calculate information collected (detFIM)

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Top 10
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Dose selection

• Select 2.5 4.0 mg as 2 new doses
• Not exact solution (depends on resolution)
• Other theoretical methods

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OTD Warning
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Assumption/constraints

• A priori, need to specify
• Model structure (Emax etc.,)
• Parameter values
• What, if I knew that I wouldnt need to do the
  experiment!
• Well, estimated parameter values
• Not unusual to have these after POC/P2a

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Select sample size
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Sample size

• 2 new doses
• Need to select justify sample size
• No formal comparison (no NQuery)
• Collecting information on dose-response
• How much information?
• Investigate using trial simulation

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Trial simulation

• For each potential sample size (e.g. 16, 24, 32,
  40 48 subjects)
• Simulate 2 new doses from your model
• Combine simulated data with existing data
• Update model
• Goal good PARAMETER ESTIMATION
• For this example used EDI (measure of location,
  similar to ED50)

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EDI expected-estimation error ()
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Dose-response design solution

• Dose selection
• Optimal doses 2.5 4.0 mg (not exact)
• May not be feasible
• Can discriminate between possible dosing options
• Sample size
• Sample size of 32 subjects offers good estimation
  (in all parameters)
• Additional subjects offer little benefit (point
  of diminishing returns)

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Conclusion

• For illustration used simplified example
• Optimal theory allows more efficient resource
  allocation
• Optimal theory can incorporate
• Residual error model
• Random effects
• Sample size (w/o need for simulation)
• Number of doses
• Get benefit by going through the process (asking
  right questions)

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Splus simulation code

• APlot richards model
• DOSElt-0100 EMAXlt-100 SLOPlt-2 ASSYlt-0.1
  EDIlt-4
• RICHlt-EMAX/((1ASSYexp(SLOPlog(EDI/DOSE)))(1/A
  SSY))
• plot(DOSE,RICH,type'l',ylimc(0,110),xlab'Dose',
  ylab'Change from baseline ()',lwd2)
• Partial derivatives
• pdlt-deriv((EMAX/((1ASSYexp(SLOPlog(EDI/DOSE)))
  (1/ASSY))),c("EMAX","ASSY","SLOP","EDI"),functio
  n(EMAX,ASSY,SLOP,EDI,DOSEc(030)) NULL,formalT)
• derlt-pd(EMAX,ASSY,SLOP,EDI)
• parderlt-attr(der,"gradient")
• plot(030,RICH131,lwd2,type"l",xlab"Dose
  (mg)",ylab"Change from baseline ()")
• lines(030,parder,190,lwd2,col2)
• lines(030,-parder,45,lwd2,col5)
• lines(030,parder,37,lwd2,col4)
• lines(030,parder,24,lwd2,col3)
• legend(23,75,c("EMAX","ASSY","SLOP","EDI"),type"b
  ",ltyc(1,1,1,1),colc(2,3,4,5))

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Back-up slides
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Non-parametric bootstrap characterising model
uncertainty
Response
Dose