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 633567. Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 1 / 22
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
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
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
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
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 = 0. 1 0 f n (p)logf n (p)dp (1) Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 6 / 22
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 = 0. 1 0 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
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
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
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
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
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
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
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
Pros and cons Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 14 / 22
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 = 0.035. Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 15 / 22
Simulation results. Safety constrain (I) d 1 d 2 d 3 d 4 d 5 d 6 d 7 No TR true 0.10 0.18 0.25 0.32 0.50 0.68 0.82 WDE (Plug-in) 8.77 18.52 27.17 27.50 13.81 3.33 0.90 29.18 WDE SC (Plug-in) 22.61 25.95 26.27 20.51 3.74 0.09 0.00 0.83 25.75 WDE SC (Bayesian) 16.78 26.43 29.54 22.51 3.76 0.11 0.00 0.87 26.18 CRM SC 13.88 23.80 29.70 22.90 7.78 0.62 0.02 1.22 24.66 EWOC SC 28.72 27.66 24.30 13.70 4.20 0.00 0.00 1.44 16.34 true 0.06 0.12 0.15 0.18 0.24 0.36 0.40 WDE (Plug-in) 4.14 10.20 19.09 22.65 26.37 12.60 4.95 17.82 WDE SC (Plug-in) 11.16 21.08 21.65 25.58 18.95 1.50 0.05 0.03 16.23 WDE SC (Bayesian) 7.03 14.72 23.33 30.11 23.34 1.39 0.05 0.03 16.85 CRM SC 2.30 5.90 13.10 20.60 27.20 18.90 11.60 0.32 22.05 EWOC SC 8.06 13.80 20.30 23.70 20.20 9.66 3.88 0.44 18.82 Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 16 / 22
Simulation results. Safety constrain (II) d 1 d 2 d 3 d 4 d 5 d 6 d 7 No TR true 0.15 0.20 0.50 0.55 0.60 0.65 0.70 WDE (Plug-in) 18.24 28.21 19.11 13.51 10.24 6.99 3.70 40.48 WDE SC (Plug-in) 42.26 41.18 6.09 3.33 1.40 0.27 0.03 5.44 28.47 WDE SC (Bayesian) 38.07 44.65 6.59 3.44 1.48 0.28 0.02 5.47 34.72 CRM SC 37.47 37.85 17.41 2.92 0.36 0.07 0.00 3.92 25.39 EWOC SC 51.00 26.11 11.01 0.88 0.13 0.00 0.00 10.87 18.03 true 0.50 0.55 0.60 0.65 0.70 0.75 0.80 WDE (Plug-in) 32.53 24.72 17.51 10.96 7.01 4.53 2.74 60.64 WDE SC (Plug-in) 13.46 5.57 2.50 0.93 0.30 0.05 0.01 77.18 18.39 WDE SC (Bayesian) 13.63 5.53 2.45 0.88 0.27 0.06 0.00 77.17 18.39 CRM SC 32.24 0.32 0.08 0.00 0.00 0.00 0.00 67.36 26.74 EWOC SC 16.17 0.00 0.12 0.00 0.00 0.00 0.00 83.71 15.37 Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 17 / 22
Simulation results. Safety constrain (III) d 1 d 2 d 3 d 4 d 5 d 6 d 7 No TR True 0.05 0.10 0.40 0.35 0.25 0.15 0.12 WDE (Plug-in) 3.74 7.62 17.53 20.34 28.87 15.80 6.07 24.30 WDE SC (Plug-in) 13.83 20.98 11.94 18.29 26.23 8.43 0.22 0.17 21.39 WDE SC (Bayesian) 14.11 19.13 11.77 18.27 27.90 8.50 0.23 0.15 21.34 CRM SC 4.26 19.90 17.70 6.31 2.84 3.00 46.10 0.31 16.06 EWOC SC 7.18 24.90 18.60 3.79 2.52 3.79 30.60 6.62 13.66 True 0.35 0.40 0.40 0.35 0.25 0.15 0.10 WDE (Plug-in) 9.87 10.87 14.20 17.69 27.44 15.27 4.65 32.48 WDE SC (Plug-in) 15.69 12.77 13.33 19.05 26.89 9.86 0.641 9.90 29.04 WDE SC (Bayesian) 15.57 12.65 13.31 18.27 27.92 8.90 0.58 9.96 29.06 CRM SC 47.41 2.51 0.97 0.48 0.72 0.40 30.10 27.30 21.00 EWOC SC 30.75 1.26 0.78 0.47 0.47 0.31 9.78 56.15 16.52 Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 18 / 22
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
Further development Phase II Generalized weight function Consistency conditions Pavel Mozgunov, Thomas Jaki (Lancaster University) WDE-based approaches to dose-escalation 20 / 22
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Choice of SC parameters γ = 0.55 γ = 0.50 γ = 0.45 γ = 0.40 γ = 0.35 γ = 0.30 r 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.00 0.32 4.32 18.47 36.15 49.06 61.49 75.70 26.47 26.65 26.40 26.05 26.85 25.03 24.10 20.23 0.15 2.50 17.76 38.75 52.74 63.06 74.94 87.22 26.27 26.22 26.53 27.24 25.46 23.30 19.35 17.10 1.13 12.72 35.72 56.49 67.16 77.55 86.53 93.49 26.15 26.02 26.81 25.18 22.26 18.75 15.16 11.05 7.47 37.95 59.49 70.52 80.53 88.32 94.18 97.63 26.04 25.91 24.90 21.98 17.66 14.47 8.05 3.51 33.98 58.22 74.42 84.14 90.52 94.86 97.90 99.20 25.65 24.54 20.45 15.55 13.77 7.21 3.25 0.70 55.51 77.02 87.21 92.99 96.50 98.55 99.37 99.83 24.21 18.09 14.40 11.42 7.13 0.95 0.08 0.04 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