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

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 MODA Conference 8 June 2007 2 of 25

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 MODA Conference 8 June 2007 3 of 25

Mechanistic models – theory Two-dimensional Gompertz for doses A and B of two drugs Y = µ + ν exp{ -exp[ α - β A - γ B - 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 MODA Conference 8 June 2007 4 of 25

Gompertz model Y 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 50 400 40 300 30 200 B 20 A 100 10 0 0 MODA Conference 8 June 2007 5 of 25

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[ – β A – γ B – 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 MODA Conference 8 June 2007 6 of 25

Empirical quadratic model – practice Response is Y Continuous explanatories are A and B Categorical explanatory is Severity ( i =1…3 ) Y = µ i + α i A + β A 2 + γ i B + δ B 2 + θ i AB + 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 MODA Conference 8 June 2007 7 of 25

Example of quadratic model Response Component B Component A MODA Conference 8 June 2007 8 of 25

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 MODA Conference 8 June 2007 9 of 25

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 MODA Conference 8 June 2007 10 of 25

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? MODA Conference 8 June 2007 11 of 25

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 of 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) MODA Conference 8 June 2007 12 of 25

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 ={x i , λ i } i=1…n, λ i =N i /N denotes the design MODA Conference 8 June 2007 13 of 25

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 µ i, α i etc. won’t affect the locally D-optimal design for the following model Y = µ i + α i A + β A 2 + γ i B + δ B 2 + θ i AB + ε MODA Conference 8 June 2007 14 of 25

Sensitivity function − ξ Θ = ξ Θ T 1 d ( x , , ) f ( x ) M ( , ) f ( x ), Θ ∝ ξ Θ ˆ var[ Y ( x , )] d ( x , , ) At the optimal design support points, the sensitivity function for D-optimal design reaches the highest value compared to other 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 MODA Conference 8 June 2007 15 of 25

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 + = ξ Θ – Forward step x arg max d ( x , , ) s s ∈ Χ x − = ξ Θ x arg min d ( x , , ) – Backward step s s ∈ Χ x ξ MODA Conference 8 June 2007 16 of 25

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 on sample-size assignment or may want specific doses in the trial MODA Conference 8 June 2007 17 of 25

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 ( G eff ) as the criterion for ξ Θ ( * d , ) m comparison, where ξ * = = G ξ Θ ξ Θ eff denotes the locally D- d ( , ) d ( , ) optimal design ξ Θ = ξ Θ ( , ) arg max ( , , ) d d x This measure is ∈ Χ x proportional to sample size MODA Conference 8 June 2007 18 of 25

Practical designs Evenly spaced design: G eff =6/7.49=0.8 N=740 • Combination dose 2 (A) 7.4 and dose 2 (B) has large 6.7 variance • To obtain target CI-width for all possible drug combinations within design region, 900 subjects are needed MODA Conference 8 June 2007 19 of 25

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 optimal design that locates the “optimal” design including the desired design points MODA Conference 8 June 2007 20 of 25

Force-point-in design (2) Assuming the weights for the forced-in design points (denoted by ξ 0 ) are known, the information matrix for ξ 0 is D-optimal design problem becomes: find design ξ m * such that Sensitivity function becomes MODA Conference 8 June 2007 21 of 25

Practical designs (2) Force-point-in design: G eff =6/6.2=0.97 n=746 Must include in the design D-optimal design support points MODA Conference 8 June 2007 22 of 25

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 MODA Conference 8 June 2007 23 of 25

Acknowledgements Lynda Kellam and Caroline Goldfrad (Stats & Programming): project statisticians James Roger (Research Statistics Unit): two- dimensional Gompertz model MODA Conference 8 June 2007 24 of 25

References Atkinson, A.C. and Donev, A. Optimum Experimental 1. Designs , 1992, Oxford: Clarendon Press. Fedorov, V.V. Theory of Optimal Experiments , 1972, 2. New York: Academic Press. Fedorov, V.V. and Hackl, P. Model-Oriented Design of 3. Experiments ; Lecture Notes in Statistics 125; Springer- Verlag, New York, 1997. Fedorov, V.V., Gagnon, R., Leonov, S., Wu, Y. (2007), 4. 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. MODA Conference 8 June 2007 25 of 25