ADAPTIVE EXPERIMENTAL DESIGNS. Maciej Patan and Barbara Bogacka. University of Zielona Góra, Poland and Queen Mary, University of London

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1 ADAPTIVE EXPERIMENTAL DESIGNS FOR SIMULTANEOUS PK AND DOSE-SELECTION STUDIES IN PHASE I CLINICAL TRIALS Maciej Patan and Barbara Bogacka University of Zielona Góra, Poland and Queen Mary, University of London

2 Introduction Clinical Trials Adaptive Designs Phase I Study Dose-Response Model PK Model Optimality Criteria Ethical Approach Efficiency of PK Estimation Optimization Problems Examples Approach First order kinetics Second order kinetics Conclusions Further Work

3 Introduction Clinical Trials Clinical trials are experiments on humans (or living animals) to explore a proposed treatment for a disease and to obtain a licence for the commercial use of the treatment on non-experimental patients. Phases of Clinical Trials: Phase I - first in human, small studies to understand PK parameters and to find a for further exploration in Phase II. Phase II - larger studies on patients to examine the evidence of a drug effect, compared to placebo. Phase III - very large studies on patients to further refine the selection. Phase IV - postmarketing clinical development.

4 Introduction Adaptive Designs The goal of adaptive designs is to learn from the accumulating data and to apply what is learnt as quickly as possible. In such trials changes are made by design and not on an ad hoc basis; therefore adaptation is a design feature aimed to enhance the trial, not as a remedy for inadequate planning. Gallo et al. (26) Main features of adaptive designs: The information is gathered sequentially. After each step of the trial the statistical model is updated. Next step of the trial depends on all the results up to date. The trial is stopped when the goal of the experiment is achieved (or the resources run out).

5 Introduction Phase I Study To establish PK parameters. To find a safe for further studies. concentration Efficacy Toxicity.4.2 c. t t max t time Dose

6 Introduction Phase I Study It is usually assumed that both efficacy and toxicity increase with. Then the Maximum Tolerated Dose is used for further studies in Phase II. Some Medicinal Products do not follow this pattern. For example, when the new treatment stimulates the immune system: large may decrease efficacy, -efficacy relationship may be unimodal.

7 Dose-Response Model After Zhang et al. (26) we consider three possible responses to a given : y - no efficacy and no severe toxicity ( neutral ), y - efficacy and no severe toxicity ( success ), y 2 - severe toxicity ( disaster ). We assign probabilities ψ i (x, ϑ ) to each of the responses, at a given x, such that ψ (x, ϑ ) - decreases with, ψ (x, ϑ ) - decreases or is unimodal or increases with, ψ 2 (x, ϑ ) - increases with and ψ (x, ϑ ) + ψ (x, ϑ ) + ψ 2 (x, ϑ ) =

8 Dose-Response Model The Continuation Ratio (CR) model assures such behaviour of the probabilities (Fan and Chaloner (24)) { } ψ (x; ϑ ) log = α + β x ψ (x; ϑ ) { } ψ2 (x; ϑ ) log = α 2 + β 2 x ψ 2 (x; ϑ ) where ϑ = (α, β, α 2, β 2 ) is a set of unknown parameters to be estimated.

9 Dose-Response Model Solving these two equations we obtain three nonlinear functions: ψ 2 (x; ϑ ) = exp(α 2 + β 2 x) + exp(α 2 + β 2 x) exp(α + β x) ψ (x; ϑ ) = [ + exp(α + β x)][ + exp(α 2 + β 2 x)] ψ (x; ϑ ) = [ + exp(α + β x)][ + exp(α 2 + β 2 x)]. Psi Psi Psi 2 Psi Psi Psi 2.8 Psi Psi Psi Dose Dose Dose

10 Dose-Response Model The approach we take here is more relevant when patients rather than healthy volunteers take part in the experiment. Questions: What should be recommended for further studies? What strategy to take to apply efficacious s as often as possible and get good estimates of the -response model parameters? How to incorporate the PK information obtained in the study?

11 PK Model Two compartments PK mechanistic model: d[b] dt = k a [A] λ k e [B] λ 2 d[a] dt = k a [A] λ + g(t x) with the initial concentrations [A] = and [B] =. ϑ 2 = (k a, k e ) is a vector of unknown rate of absorption and rate of elimination, respectively. (λ, λ 2 ) are known orders of the kinetic reaction. Time t is scaled in hours and g(t x) is the drug infusion rate: { cx t g(t x) = t > where c is some constant and x is a. We are interested in precise estimation of ϑ 2.

12 PK Model Question: What to apply and when to measure the concentration to optimally estimate the PK parameters?.5.5 concentration.5 concentration time time Design: { } t... t ξ(t x) = s w... w s

13 Optimality Criteria Biologically Optimum Dose Zhang et al. (26) propose the following decision functions for finding a BOD: δ (x; ϑ ) = I [ψ2 (x;ϑ )<π ], δ 2 (x; ϑ ) = ψ (x; ϑ ) λψ 2 (x; ϑ ), where I denotes an indicator function and λ [, ]. δ (x; ϑ ) = means that the toxicity at x is smaller than a pre-specified value π. x = arg max C(x) δ 2(x; ϑ ), C(x) = {x : δ (x; ϑ ) = } is the recommended BOD for next step of the trial.

14 Optimality Criteria Biologically Optimum Dose An example of a possible Adaptive Design: no reaction efficacy toxicity Response probability cohort.8.2.4

15 Optimality Criteria Biologically Optimum Dose At step k of the Adaptive Design: xk is applied to cohort k, responses are observed, parameters of the -response model estimated (Bayesian), x k is calculated, stopping rule is checked. Stopping rules: the last cohort of patients is treated, i.e., n = n max, the same x has been determined minimum m times,

16 Optimality Criteria Biologically Optimum Dose Some technicalities: the possible s were x = +.5i, i =,,..., 6, cohort size was q = 3 and maximum number of cohorts was n max = 5, the highest tolerable toxicity probability π was assumed to be.33 and λ =, it was allowed to skip only one during the escalation, no restrictions on de-escalations. uniform prior distributions for (α, β, α 2, β ) were taken over [ 3, 3], [, 4], [ 3, 7], [3, 9], respectively.

17 Optimality Criteria Efficiency of PK Estimation We use the D-optimality criterion for estimating PK parameters ϑ 2 = (k a, k e ): Φ{M(ξ)} = det M(ξ) where M(ξ) is the information matrix for θ 2 and efficiency of the estimation at a given x is defined as: E D (ξ (t x )) = p = 2, is the number of parameters. { det M(ξ (t x )) } /p max x det M(ξ (t x))

18 Approach BOD Constrained by Toxicity Level and by PK Efficiency Maximize (over x) δ 2 (x; ϑ ) = ψ (x; ϑ ) λψ 2 (x; ϑ ) subject to and ψ 2 (x; ϑ ) < π E D [ξ (t x)] η

19 Approach 2 PK Efficiency Constrained by Toxicity and Dose Efficacy Levels Maximize (over ξ) subject to and E D [ξ(t x )] ψ 2 (x ; ϑ ) < π δ 2 (x; ϑ ) max x δ 2 (x; ϑ ) ϱ where ϱ is the efficacy coefficient. Such a relative efficacy can be considered as an analogy to the efficiency coefficient E D.

20 Examples Approach : First Order Kinetics Probabilities of the responses y, y, y 2 and det M[ξ(t x)] at step k of the Adaptive Design for the parameter estimates ϑ and ϑ 2 obtained in step k : response probability π =.33 toxicity efficacy no reaction determinant

21 Examples Approach : First Order Kinetics D-optimum design for estimation of the kinetic parameters ϑ = (k a, k e ) does not depend on the : support coordinates weigths

22 Examples Approach : First Order Kinetics Examples of the trial runs for different levels of PK efficiency. (a) E D 4% (b)e D 8% (c)e D 9% BOD.2.2 BOD.2. BOD cohorts cohorts cohorts Dashed lines represent the adaptive bounds stemming from the E D constraint. The dot-dashed line represents the adaptive upper threshold for non-admissible toxicity.

23 Examples Approach : First Order Kinetics (a) E D 4% (b) E D 8% (c) E D 9% k e.3 k e.3 k e k a k a k a Final estimates of PK parameters from simulations - red square represents the values assumed for the simulations - green circle denotes the sample mean

24 Examples Approach : First Order Kinetics.8 (a) E D 4% (d) E D 8% (e) E D 9% efficacy efficacy efficacy efficiency efficiency efficiency.8.8 proportions of s.4 proportions of s.4 proportions of s Dose frequencies averaged over total number of cohorts in simulations. efficacy efficiency efficacy efficiency efficacy efficiency proportions of simulations proportions of simulations proportions of simulations Proportions of the final BOD frequencies in the simulations.

25 Examples Approach : Second Order Kinetics Probabilities of the responses y, y, y 2 and det M[ξ(t x)] at step k of the Adaptive Design for the parameter estimates ϑ and ϑ 2 obtained in step k :.8.7 toxicity efficacy no reaction.58 response probability π =.33 determinant Non-monotonous D-criterion as a function of restricts the optimum s to be in the middle of the range.

26 Examples Approach : Second Order Kinetics support coordinates weigths Design support points and weights for different s, for priors k a =.8 and k e =.3.

27 Examples Approach : Second Order Kinetics Examples of the trial runs for different levels of PK efficiency. (a) E D 75% (b)e D 9% (c) E D 98% BOD. BOD. BOD cohorts cohorts cohorts Dashed lines represent the adaptive bounds stemming from the E D constraint. The dot-dashed line represents the adaptive upper threshold for non-admissible toxicity.

28 Examples Approach : Second Order Kinetics (a) E D 75% (b) E D 9% (c) E D 98% k e.3 k e.3 k e k a k a k a Final estimates of PK parameters from simulations - red square represents the values assumed for the simulations - green circle denotes the sample mean

29 Examples Approach : Second Order Kinetics (a) E D 75% (b) E D 9% (c) E D 98% proportions of s.4 efficacy efficiency proportions of s.4 efficacy efficiency proportions of s.4 efficacy efficiency Dose frequencies averaged over total number of cohorts in simulations. proportions of simulations efficacy efficiency proportions of simulations efficacy efficiency proportions of simulations efficacy efficiency Proportions of the final BOD frequencies in the simulations.

30 Conclusions I Approach Incorporating the constraint on the efficiency of PK estimation into the optimization may seriously affect the selection design. Different orders of the kinetics will differently affect the selection design. A reasonable E D constraint may give high quality estimates of PK parameters as well as a good determined as BOD.

31 Conclusions II Approach 2 Here we maximize the efficiency of PK estimation with the constraint on the efficacy. The PK parameters are precisely estimated. The selected BODs are pooled towards the optimum region for PK estimation condition.

32 Conclusions III Both Approaches, and 2, are relevant when patients rather than healthy volunteers enter the trial. Patients are treated with possibly efficacious s. Some information on the response to BOD is already gathered in Phase I (prior for Phase II). Also: Approach : fewer potentially toxic s or non-efficacious s are applied, ensures good efficiency of estimation of the PK parameters, may be considered as a seamless phase I/II, leading to serious saving in resources, both financial and human. Approach 2: more relevant when toxicity is a minor issue, in case of first order kinetics, PK parameters are very precisely estimated.

33 Conclusions IV Defining appropriate criteria of optimality, suitable for the purpose of an experiment is paramount. Model-based simulation studies may be very useful for making decision about the criteria and foreseeing possible problems. Adding PK information, which is collected at Phase I anyway, may improve the optimization and lead to serious savings.

34 Further work to consider the kinetics order as unknown parameters, to include safety issues into stopping rules, to incorporate safety parameters into the optimization (time of injection, time between cohorts, toxicity probability) concentration.2.2 concentration time time The concentrations of A and B (solid lines) and their derivatives wrt reaction rates k a and k e

35 References Fan, S. and Chaloner, K. (24). Optimal desings and liminting optimal designs for a trinomial response, Journal of Statistical Planning and Inference 26: Gallo, P., Chuang-Stein, C., Dragalin, V., Gaydos, B., Krams, M. and Pinheiro, J. (26). Adaptive designs in clinical drug develoment - an executive summary of the phrma working group, Journal of Biopharmaceutical Statistics 6: Zhang, W., Sargent, D. and Mandrekar, S. (26). An adaptive -finding design incorporating both toxicity and efficacy, Statistics in Medicine 25: