Design for a combination of compounds: the balance between theory - - PowerPoint PPT Presentation

Design for a combination of compounds: the balance between theory and practice

Peter Lane & Yuehui Wu Research Statistics Unit

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Drug combination

Increasing interest in combinations of drugs Ever-pressing need to speed up development

– Will not wait to file individual components – Study dose-response in combination

Two new compounds, with just early-phase results

– Four doses of one (A, say): 1, 2, 4, 8 units – Three doses of the other (B): 1, 2, 4 units – Three levels of disease severity

Design Phase II trial(s) from which to choose doses for Phase III (maybe varying with severity)

– Propose factorial approach – Alternative is separate dose-ranging

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Mechanistic or empirical?

Mechanistic models appeal to underlying science

– Simple S-shaped curve – More complex model for PD effect

Empirical models are more robust computationally, given limitations of data

– Can approximate mechanistic shape – Mechanistic models may include parameters that can’t be estimated from the data – Empirical models can give unrealistic fit, complicating interpretation

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Mechanistic models – theory

Two-dimensional Gompertz for doses A and B of two drugs Y = µ + ν exp{ -exp[ α - βA - γB - δ√(A*B) ] } + ε Invariant under changes in scale for A and B Invariant under shift in dose scale if α not fixed Each drug alone has Gompertz response

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Gompertz model

10 20 30 40 50 B 100 200 300 400 A Y

• 0.1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

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Difficulties

When range of doses does not take response into left-hand part of sigmoid shape, there are problems of collinearity of ν and α

– Set α=0

Y = µ + ν exp{ –exp[ –βA – γB – δ√(A*B) ] } + ε Add additional terms such as severity of disease into µ

– Adding such terms into other parameters can lead to computational difficulties

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Empirical quadratic model – practice

Response is Y Continuous explanatories are A and B Categorical explanatory is Severity (i=1…3 ) Y = µi +αiA + βA2 + γiB + δB2 + θiAB + ε This is the simplest polynomial including curvature

– Interaction is only linear×linear – Quadratic effects constant over Severity levels

Has potential artefacts

– Peak and decline may appear because of model rather than because of data

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Example of quadratic model

Response Component A Component B

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To log or not to log? – theory

Effect of dose is often modelled on log scale

– Experience of effect tailing off as dose increases – Can be represented by logarithmic model

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To log or not to log? – practice

Quadratic model can be close to logarithmic

– Close enough for practical interpretation – Maybe problems if maximum is in range

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To log or not to log? – practice

Logarithmic model does not handle 0 dose

– So can’t use placebo data for model fitting

Quadratic model has no problem at dose 0 What behaviour is expected from the science?

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Strategy for Phase II study

Avoid designing on the basis of hypothesis tests Establish a required precision for estimates Design to give that precision across chosen range

• f doses

– E.g. with SD=1 unit, require maximum half-width of 95% CI for predicted means to be 0.18 units (i.e. CI is mean ±0.18)

Reduce number of doses for Component A to three (plus 0 dose to give B monotherapy)

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Optimal design

We are interested in constructing the design that reaches the target confidence interval width for the proposed model using the minimal sample size D-optimality criterion minimizes the generalized variance of unknown parameters |M-1(ξ,Θ)| where ξ={xi,λi}i=1…n, λi=Ni/N denotes the design

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D-optimal design

Locally D-optimal design for linear model doesn’t depend on the unknown parameters Once the model is defined, locally optimal design remains the same, e.g. the values of µi, αi etc. won’t affect the locally D-optimal design for the following model Y = µi +αiA + βA2 + γiB + δB2 + θiAB + ε

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Sensitivity function

At the optimal design support points, the sensitivity function for D-optimal design reaches the highest value compared to

• ther points within the design region

The design points are where one has the least information: so collect data there Sample size can be calculated based on the value of the sensitivity function at the optimal design support point

) , , ( )] , ( ˆ var[ ), ( ) , ( ) ( ) , , (

1

Θ ∝ Θ Θ = Θ

−

ξ ξ ξ x d x Y x f M x f x d

T

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Automated SAS program

We have a ready-to-use SAS program for constructing D- and A-optimal designs of linear and non-linear models (Fedorov, V.V., Gagnon, R.,

Leonov, S., Wu, Y., 2007)

Apply first-order exchange algorithm (references 1,2,3) for continuous design

– Forward step – Backward step

) , , ( min arg ) , , ( max arg Θ = Θ =

Χ ∈ − Χ ∈ + s x s s x s

x d x x d x ξ ξ

ξ

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Optimal design in practice

Theoretically, optimal design performs the best It may not be practical in real clinical trials

– Nonlinear model: the true values of parameters are unknown while locally optimal design depends on those values: go for adaptive design! – Linear model: weight of each support point is fixed whereas the investigator may have specific constraints

• n sample-size assignment or may want specific doses

in the trial

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Benchmarking

Locally D-optimal design should be built as the benchmark for the practical designs, N=724 to obtain the target CI width Relative G-efficiency (Geff) as the criterion for comparison, where ξ* denotes the locally D-

• ptimal design

This measure is proportional to sample size

) , , ( max arg ) , ( ) , ( ) , ( ) , ( * Θ = Θ Θ = Θ Θ =

Χ ∈

ξ ξ ξ ξ ξ x d d d m d d G

x eff

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Practical designs

Evenly spaced design: Geff=6/7.49=0.8 N=740

7.4 6.7

• Combination dose 2 (A)

and dose 2 (B) has large variance

• To obtain target CI-width

for all possible drug combinations within design region, 900 subjects are needed

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Force-point-in design

The investigators want to have certain drug combinations in the trial, regardless of whether they are optimal design points or not Force-point-in design is a modification of locally

• ptimal design that locates the “optimal” design

including the desired design points

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Force-point-in design (2)

Assuming the weights for the forced-in design points (denoted by ξ0) are known, the information matrix for ξ0 is D-optimal design problem becomes: find design ξm* such that Sensitivity function becomes

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Practical designs (2)

Force-point-in design: Geff=6/6.2=0.97 n=746

Must include in the design D-optimal design support points

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Outcome

Project team narrowly rejected the factorial approach:

• 1. Non-intuitive to study more patients at extreme

combinations

• 2. Complex logistics of factorial design
• 3. Lack of experience with going to FDA with factorial

design & response surface

• 4. Expectation that FDA would require pairwise

comparison for selection of doses

Instead: dose-range the two component drugs in separate studies

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Acknowledgements

Lynda Kellam and Caroline Goldfrad (Stats & Programming): project statisticians James Roger (Research Statistics Unit): two- dimensional Gompertz model

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References

1.

Atkinson, A.C. and Donev, A. Optimum Experimental Designs, 1992, Oxford: Clarendon Press.

2.

Fedorov, V.V. Theory of Optimal Experiments, 1972, New York: Academic Press.

3.

Fedorov, V.V. and Hackl, P. Model-Oriented Design of Experiments; Lecture Notes in Statistics 125; Springer- Verlag, New York, 1997.

4.

Fedorov, V.V., Gagnon, R., Leonov, S., Wu, Y. (2007), Optimal design of experiments in pharmaceutical

• applications. In: Dmitrienko, A., Chuang-Stein, C.,

D’Agostino, R. (Eds), Pharmaceutical Statistics, SAS Books by Users series, SAS Press, Cary, NC, pp. 151- 195.