Multi-objective dose-finding Thomas Jaki Medical and Pharmaceutical Statistics Research Unit, Department of Mathematics and Statistics, Lancaster University, UK December 7, 2018 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. Thomas Jaki (Lancaster University) Multi-objective dose-finding 1 / 56
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 Thomas Jaki (Lancaster University) Multi-objective dose-finding 2 / 56
TD20 p(d) = P(toxicity|dose d) p(d) 1 0.2 TD20 dose Assume that a 20% risk of toxicity is an acceptable risk to pay for a chance of benefit Thomas Jaki (Lancaster University) Multi-objective dose-finding 3 / 56
General (Bayesian) approach Make assumptions about the form of p ( d ) 1 Impose a prior distribution for the parameters that determine p ( d ) 2 Choose next dose to optimise some form of expected gain 3 Stop once target dose level can be estimated accurately enough 4 Thomas Jaki (Lancaster University) Multi-objective dose-finding 4 / 56
Bayesian continual reassessment method p ( d i ) = d exp( β ) 1 i β ∼ N (0 , 1 . 34) 2 d ∗ = min i E � ( p ( d i ) − γ ) 2 � 3 Stop after N patients have been recruited 4 Thomas Jaki (Lancaster University) Multi-objective dose-finding 5 / 56
Single agent dose-escalation designs Algorithm based methods Model-based methods ‘3+3‘ design CRM Biased Coin Design EWOC Fundamental assumption: a monotonic dose-response relationship Cannot be applied to: Combination trials with many treatments Scheduling of drugs Non-monotonic dose-toxicity relations Thomas Jaki (Lancaster University) Multi-objective dose-finding 6 / 56
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... Thomas Jaki (Lancaster University) Multi-objective dose-finding 7 / 56
Unknown ordering problem. Example (II) Thomas Jaki (Lancaster University) Multi-objective dose-finding 8 / 56
Unknown ordering problem. Example (III) Thomas Jaki (Lancaster University) Multi-objective dose-finding 9 / 56
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 Only two combinations only Thomas Jaki (Lancaster University) Multi-objective dose-finding 10 / 56
Goal To propose an escalation procedure that does not require any parametric assumptions (including monotonicity between regimes). Thomas Jaki (Lancaster University) Multi-objective dose-finding 11 / 56
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 n j 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 Thomas Jaki (Lancaster University) Multi-objective dose-finding 12 / 56
Information theory concepts 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 � 1 h ( f n ) = − f n ( p ) log f n ( p ) d p 0 with the convention 0 log 0 = 0. Thomas Jaki (Lancaster University) Multi-objective dose-finding 13 / 56
Information theory concepts 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 � 1 h ( f n ) = − f n ( p ) log f n ( p ) d p 0 with the convention 0 log 0 = 0. It shows the amount of information needed to answer the question What is the toxicity probability? Thomas Jaki (Lancaster University) Multi-objective dose-finding 13 / 56
Weighted information Consider a two-fold experiment: (i) what is the toxicity probability (ii) is the probability of toxicity close to a target, γ Thomas Jaki (Lancaster University) Multi-objective dose-finding 14 / 56
Weighted information Consider a two-fold experiment: (i) what is the toxicity probability (ii) is the probability of toxicity close to a target, γ A: The weighted Shannon information � h φ ( f ) = − φ ( z ) f ( z ) log f ( z ) d z . R Thomas Jaki (Lancaster University) Multi-objective dose-finding 14 / 56
Weight Function The Beta-form weight function φ n ( p ) = Λ( γ, x , n ) p γ √ n (1 − p ) (1 − γ ) √ n . Thomas Jaki (Lancaster University) Multi-objective dose-finding 15 / 56
Escalation criteria Theorem Let h ( f n ) and h φ n ( f n ) be the DE and WDE corresponding to PDF f n when x ∼ α n with the weight function φ n given in (15). Then = ( α − γ ) 2 � h φ n ( f n ) − h ( f n ) � lim 2 α (1 − α ) n →∞ Thomas Jaki (Lancaster University) Multi-objective dose-finding 16 / 56
Escalation criteria Theorem Let h ( f n ) and h φ n ( f n ) be the DE and WDE corresponding to PDF f n when x ∼ α n with the weight function φ n given in (15). Then = ( α − γ ) 2 � h φ n ( f n ) − h ( f n ) � lim 2 α (1 − α ) n →∞ Therefore, for a regimen d j , j = 1 , . . . , m , we obtained that ∆ j ≡ ( α j − γ ) 2 α j (1 − α j ) . Criteria: ∆ j = i =1 ,..., m ∆ i . inf Thomas Jaki (Lancaster University) Multi-objective dose-finding 16 / 56
Estimation Consider the mode of the posterior distribution f n j = x j + ν j p ( n ) ˆ . j n j + β j ∆ ( n ) Then the following ”plug-in” estimator ˆ may be used j p ( n ) − γ ) 2 (ˆ ∆ ( n ) j ˆ = . j p ( n ) p ( n ) ˆ (1 − ˆ ) j j Thomas Jaki (Lancaster University) Multi-objective dose-finding 17 / 56
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 ) ∆ ( l ) ˆ k = inf i , l = 0 , 1 , 2 , . . . , C . i =1 ,..., m We adopt regime d j ( C + 1) as the final recommended regime. Thomas Jaki (Lancaster University) Multi-objective dose-finding 18 / 56
Alternative angle One can consider p ( n ) − γ ) 2 (ˆ ∆ ( n ) j ˆ = j p ( n ) p ( n ) (1 − ˆ ˆ ) j j as a loss function for a parameter defined on (0 , 1). p ( n ) Loss function penalize ˆ close to 0 to 1 and ‘pushes‘ the allocation j away from bounds to the neighbourhood of γ Does not include any definition of safety → safety constraint is needed Thomas Jaki (Lancaster University) Multi-objective dose-finding 19 / 56
Safety constraint Considers regime d j as safe if at the moment n its PDF satisfies � 1 γ ∗ f n j ( p ) d p ≤ θ n P (regime is overly toxic) = where γ ∗ is some threshold after which all regimes above are declared to have excessive risk, γ ∗ = γ + 0 . 2 θ n is the level of probability that controls the overdosing Note that this depends on n Thomas Jaki (Lancaster University) Multi-objective dose-finding 20 / 56
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) d p = 0 . 5107 > 0 . 50 0 . 45 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 Thomas Jaki (Lancaster University) Multi-objective dose-finding 21 / 56
Choice of SC parameters 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 γ ∗ = 0 . 55 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 γ ∗ = 0 . 50 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 γ ∗ = 0 . 45 26.15 26.02 26.81 25.18 22.26 21.75 15.16 11.05 7.47 37.95 59.49 70.52 80.53 88.32 94.18 97.63 γ ∗ = 0 . 40 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 γ ∗ = 0 . 35 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 γ ∗ = 0 . 30 24.21 18.09 14.40 11.42 7.13 0.95 0.08 0.04 Table: Top row: Proportion of no recommendations for toxic scenario. Bottom row: Proportion of correct recommendations. 10 6 simulations. Thomas Jaki (Lancaster University) Multi-objective dose-finding 22 / 56
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