Weighted differential entropy based approaches to dose-escalation in clinical trials

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1 Weighted differential entropy based approaches to dose-escalation in clinical trials Pavel Mozgunov, Thomas Jaki Medical and Pharmaceutical Statistics Research Unit, Department of Mathematics and Statistics, Lancaster University, UK June 23, 2016 ThinkTank, Traunkirchen, Austria Acknowledgement: This project has received funding from the European Union s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 1 / 22

2 Dose escalation Consider: Goal: First-in-men clinical trial Range of m doses, usually m (4, 8) N patients Find the maximum tolerated dose,(m)td, the dose that corresponds to a controlled level of toxicity, γ (0.2, 0.35) Challenges: Rough prior knowledge about toxicities of doses for humans N is small (usually less than 30) Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 2 / 22

3 Motivation Model-based methods CRM EWOC Algorithm based methods 3+3 design Biased Coin Design Fundamental assumption - a monotonic dose-response relation. Cannot be applied to: Non-monotonic dose-toxicity relations Combination trials with many treatments. Scheduling of drugs Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 3 / 22

4 Goal To propose a method of a dose-escalation which does not require a monotonicity assumption between doses. Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 4 / 22

5 Problem formulation Toxicity probabilities Z 1,..., Z m are independent random variables. Beta prior B(ν j + 1, β j ν j + 1), ν j > 0, β j > 0. n j patients assigned to the dose j and x j toxicities observed. Beta posterior f nj B(x j + ν j + 1, n j x j + β j ν j + 1) Random variable with Beta posterior PDF f nj : Z (n) j. Let 0 < α j < 1 be the unknown parameter in the neighbourhood of which the probability (risk) of toxicity is concentrated. Let γ be the target probability level. Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 5 / 22

6 Information theory concepts 1) A statistical experiment of estimation of a toxicity probability. The Shannon differential entropy (DE) h(f n ) of the PDF f n is defined as h(f n ) = with the convention 0log0 = f n (p)logf n (p)dp (1) Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 6 / 22

7 Information theory concepts 1) A statistical experiment of estimation of a toxicity probability. The Shannon differential entropy (DE) h(f n ) of the PDF f n is defined as h(f n ) = with the convention 0log0 = f n (p)logf n (p)dp (1) 2) A statistical experiment of a sensitive estimation. The weighted Shannon differential entropy (WDE), h φn (f n ), of the RV Z (n) with positive weight function φ n (p) φ n (p, α, γ, κ) is defined as h φn (f n ) = 1 where γ is the particular value of a special interest. 0 φ n (p)f n (p)logf n (p)dp (2) Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 6 / 22

8 Weight Function We consider the weight function of the form φ n (p) = Λ(γ, x, n)p γ n (1 p) (1 γ) n. (3) Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 7 / 22

9 Dose-escalation criteria The difference of information in two statistical experiments: Theorem Let h(f n ) and h φn (f n ) be the DE and WDE corresponding to PDF f n when x αn with the weight function φ n given in (3). Then ( lim h φ n (f n ) h(f n ) ) (α γ)2 =. (4) n 2α(1 α) Therefore, for a dose d j, j = 1,..., m, we obtained that j (α j γ) 2 2α j (1 α j ). Criteria: j = inf i. i=1,...,m Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 8 / 22

10 Estimation Consider the mode of the posterior distribution f nj ˆp (n) j = x j + ν j n j + β j. Then the following plug-in estimator ˆ (n) j may be used ˆ (n) j = (ˆp(n) ˆp (n) j j γ) 2 (1 ˆp (n) j ). (5) Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 9 / 22

11 Simulations For simulations below the following parameters were chosen: The cohort size c = 1 Total sample size N = 20 Number of doses m = 7 The target probability γ = 0.25 Number of simulation 10 6 Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 10 / 22

12 Investigated scenarios Figure : Considering dose-response shapes. The TD is marked as triangle. Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 11 / 22

13 Specifying the prior Assumptions: Rough beliefs about toxicity rates Prior belief: dose-response curve is monotonic A dose-escalation to be started from d 1 The prior for dose d j (1 j 7) is specified thought the mode ˆp (0) j Starting from the bottom: ˆp (0) 1 = γ. The vector of modes ˆp for all doses is defined ˆp = [0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55] T. Rough prior β j = β = 1 for j = 1,..., m. = ν j β j. Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 12 / 22

14 Alternative methods We have also investigated CRM EWOC (a target 25 th percentile is used) Non-parametric optimal benchmark Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 13 / 22

15 Pros and cons Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 14 / 22

16 Safety constrain (II) Requirements to the function θ n θ n is a decreasing function of n θ 0 = 1 θ N 0.3 We use the following form of θ n θ n = 1 rn where r > 0 is the rate of decreasing. Based on simulations in different scenarios we conjecture that reasonable trade-off is r = Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 15 / 22

17 Simulation results. Safety constrain (I) d 1 d 2 d 3 d 4 d 5 d 6 d 7 No TR true WDE (Plug-in) WDE SC (Plug-in) WDE SC (Bayesian) CRM SC EWOC SC true WDE (Plug-in) WDE SC (Plug-in) WDE SC (Bayesian) CRM SC EWOC SC Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 16 / 22

18 Simulation results. Safety constrain (II) d 1 d 2 d 3 d 4 d 5 d 6 d 7 No TR true WDE (Plug-in) WDE SC (Plug-in) WDE SC (Bayesian) CRM SC EWOC SC true WDE (Plug-in) WDE SC (Plug-in) WDE SC (Bayesian) CRM SC EWOC SC Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 17 / 22

19 Simulation results. Safety constrain (III) d 1 d 2 d 3 d 4 d 5 d 6 d 7 No TR True WDE (Plug-in) WDE SC (Plug-in) WDE SC (Bayesian) CRM SC EWOC SC True WDE (Plug-in) WDE SC (Plug-in) WDE SC (Bayesian) CRM SC EWOC SC Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 18 / 22

20 Intermediate results The WDE-based method performs comparably to the model-based methods in monotonic scenarios outperform them in non-monotonic settings However, WDE-based method experience problems in scenarios with no safe doses or with sharp jump in toxicity probability at the bottom. The time-varying safety constrain in the proposed form can overcome overdosing problems and increase the accuracy of the original method Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 19 / 22

21 Further development Phase II Generalized weight function Consistency conditions Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 20 / 22

22 References J. Babb, A. Rogatko, S. Zacks. Cancer phase I clinical trials: efficient dose escalation with overdose control. (1998). Statistics in Medicine, 17(10), M. Belis, S. Guiasu, A quantitative and qualitative measure of information in cybernetic systems (1968), IEEE Trans. Inf. Th.,14, Gasparini, M. and Eisele, J. (2000).A curve-free method for phase I clinical trials, Biometrics, 56, M. Kelbert, P. Mozgunov, Shannon s differential entropy asymptotic analysis in a Bayesian problem, Mathematical Communications Vol 20, 2015, N 2, J. O Quigley, M. Pepe, L. Fisher, Continual reassessment method: A practical design for phase I clinical trials in cancer, 1990, Biometrics O Quigley J, Paoletti X, MacCario J., Non-parametric optimal design in dose finding studies, (2002) Biostatistics; 3: M.K. Riviere, F. Dubois, S. Zohar, Competing designs for drug combination in phase I dose-finding clinical trials, Statistics in Medicine 2015, 34, 1-12 Shen L.Z., O Quigley J., Consistency of continual reassessment method under model misspecification, (1996) Biometrika; 83: L. N. Vandenberg et al Hormones and Endocrine-Disrupting Chemicals: Low-Dose Effects and Non-monotonic Dose Responses, Endocr Rev Jun; 33(3): Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 21 / 22

23 Choice of SC parameters γ = 0.55 γ = 0.50 γ = 0.45 γ = 0.40 γ = 0.35 γ = 0.30 r Table : Linear and an unsafe scenario for different parameters of the safety constraint. Results based on 10 6 simulations. Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 22 / 22