A weighted differential entropy based approach for dose-escalation trials

Transcription

1 A weighted differential entropy based approach for dose-escalation trials Pavel Mozgunov, Thomas Jaki Medical and Pharmaceutical Statistics Research Unit, Department of Mathematics and Statistics, Lancaster University, UK September 29, th Early Phase Adaptive Trials Workshop 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 / 28

2 Dose escalation Limited prior knowledge about toxicities in humans Range of m regimes (doses, combinations, schedules) n patients Goal: Find the maximum tolerated regime that corresponds to a controlled level of toxicity, usually γ (0.2, 0.35) in oncology trials Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 2 / 28

3 Single agent dose-escalation designs Model-based methods CRM EWOC Algorithm based methods 3+3 design Biased Coin Design Fundamental assumption: a monotonic dose-response relationship Cannot be applied to: Combination trials with many treatments Scheduling of drugs Non-monotonic dose-toxicity relations Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 3 / 28

4 Unknown ordering problem. Example (I) Let us consider drugs combination dose-escalation trial with 3 dose levels of drug A: A 1, A 2, A 3 3 dose levels of drug B: B 1, B 2, B 3 (A 1 ; B 3 ) (A 2 ; B 3 ) (A 3 ; B 3 ) (A 1 ; B 2 ) (A 2 ; B 2 ) (A 3 ; B 2 ) (A 1 ; B 1 ) (A 2 ; B 1 ) (A 3 ; B 1 ) Even assuming monotonicity one drug being fixed, we cannot order (A 1 ; B 2 ) and (A 2 ; B 1 ); (A 1 ; B 3 ) and (A 2 ; B 1 ); (A 1 ; B 3 ) and (A 3 ; B 1 ) and so on... Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 4 / 28

5 Unknown ordering problem. Example (II) Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 5 / 28

6 Unknown ordering problem. Example (III) Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 6 / 28

7 Method for drug combinations Six-parameter model (Thall P. et al, 2003) Up-and-down design (Ivanova A, Kim S., 2009) Using the T -statistic Copula regression (G.Yin, Y.Yuan, 2009) Parametrization of drug-drug interactive effect POCRM (N.Wages, M. Conoway, J. O Quigley, 2011) Choose several ordering and randomize between them during the trial General restrictions: Strong model assumptions are usually needed No diagonal switching is allowed Synergistic effect is usually assumed Two combinations might be considered only Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 7 / 28

8 Goal To propose an escalation procedure that does not require any parametric assumptions (including monotonicity between regimes). Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 8 / 28

9 Problem formulation Toxicity probabilities Z 1,..., Z m are random variables with Beta prior B(ν j + 1, β j ν j + 1), ν j > 0, β j > 0 n j patients assigned to the regime 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) Let 0 < α j < 1 be the unknown parameter in the neighbourhood of which the probability of toxicity is concentrated Target toxicity γ Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 9 / 28

10 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) avel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 10 / 28

11 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) is defined as h φn (f n ) = 1 0 φ n (p)f n (p)logf n (p)dp. (2) Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 10 / 28

12 Weight Function The Beta-form weight function 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 11 / 28

13 Escalation criteria The difference of informations 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 regime 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 avel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 12 / 28

14 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 13 / 28

15 Escalation design Let d j (i) be a regime d j recommended for cohort i. The procedure starts from ˆ (0) j l cohorts were already assigned The (l + 1) th cohort of patients will be assigned to regime k such that d j (l + 1) : ˆ (l) k = inf ˆ (l) i, l = 0, 1, 2,..., C. i=1,...,m We adopt regime d j (C + 1) as the final recommended regime. Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 14 / 28

16 Alternative angle One can consider ˆ (n) j = (ˆp(n) ˆp (n) j j γ) 2 (1 ˆp (n) j ) as a loss function for a parameter defined on (0, 1). Loss function penalize ˆp (n) j close to 0 to 1 and pushes the allocation away from bounds to the neighbourhood of γ Does not include any definition of safety safety constraint is needed Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 15 / 28

17 Safety constrain (I) Considers regime d j as safe if at the moment n its PDF satisfies where 1 γ f nj (p)dp θ n (6) γ is some threshold after which all regimes above are declared to have excessive risk, γ = γ θ n is the level of probability that controls the overdosing Note that this depends on n Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 16 / 28

18 Why is a time-varying SC is needed? If β = 1 and θ n = θ = 0.50 then regimes with prior mode 0.40 will never be considered since 1 f 0 (p x = 0)dp = > Requirements to the function θ n θ n is a decreasing function of n θ 0 = 1 θ N 0.3 θ n = 1 rn Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 17 / 28

19 Choice of SC parameters γ = 0.55 γ = 0.50 γ = 0.45 γ = 0.40 γ = 0.35 γ = 0.30 r Table: Top row: Proportion of no recommendations for toxic scenario. Bottom row: Proportion of correct recommendations simulations. Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 18 / 28

20 Simulations For simulations below the following parameters were chosen: The cohort size c = 1 Total sample size N = 20 Number of regimes m = 7 The target probability γ = 0.25 Safety constraint n, if n 0.7; θ n = 0.3, otherwise. Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 19 / 28

21 Investigated scenarios Figure: Considering response shapes. The TD is marked as triangle. Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 20 / 28

22 Specifying the prior Assumptions: Vague beliefs about toxicity risk Prior belief: regimes have been correctly ordered monotonically A escalation to be started from d 1 The prior for regime 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 regimes is defined ˆp = [0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55] T. Vague prior β j = β = 1 for j = 1,..., m. Is there a unique set of prior parameters that lead to the equivalent performance? = ν j β j. Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 21 / 28

23 Choice of prior Figure: Proportion of correct recommendations: β = number of patients and difference between the risk of toxicity on lowest and highest dose across six scenarios. Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 22 / 28

24 Alternative methods We have also investigated Continual reassessment method (CRM) Partial ordering continual reassessment method (POCRM) All correct orderings used in simulation are incorporated in the model. Escalation with overdose control (EWOC) A target 25 th percentile is used. Non-parametric optimal benchmark Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 23 / 28

25 Simulation results. Ordering is correctly specified Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 24 / 28

26 Simulation results. Ordering is wrongly specified. d 1 d 2 d 3 d 4 d 5 d 6 d 7 No TR N True WDE SC CRM SC POCRM SC EWOC SC d 1 d 2 d 3 d 4 d 5 d 6 d 7 No TR N True WDE SC CRM SC POCRM SC EWOC SC Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 25 / 28

27 Simulation results. Highly toxic scenarios. d 1 d 2 d 3 d 4 d 5 d 6 d 7 No TR N True WDE SC CRM SC POCRM SC EWOC SC True No WDE SC CRM SC POCRM SC EWOC SC Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 26 / 28

28 Conclusions The WDE-based method performs comparably to the model-based methods when the ordering is specified correctly scenarios outperform them in wrongly specified setting 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 27 / 28

29 Further development Phase II Generalized weight function Consistency conditions Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 28 / 28

30 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 Wages N., Conaway M., O Quigley J. (2011a). Continual reassessment method for partial ordering. Biometrics 67(4), Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 29 / 28