Improving a safety of the Continual Reassessment Method via a modified allocation rule
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1 Improving a safety of the Continual Reassessment Method via a modified allocation rule Pavel Mozgunov, Thomas Jaki Medical and Pharmaceutical Statistics Research Unit, Department of Mathematics and Statistics, Lancaster University, UK June 4, PSI Annual Conference, Amsterdam 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) Modified allocation rule for the CRM 1 / 17
2 Motivation Consider a Phase I clinical trial with binary responses and two doses: d 1, d 2 Goal is to find the maximum tolerated dose (MTD): γ = patients were assigned to each dose, 2 and 4 toxicities observed Q: Which dose should be administered to the next patient? avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 2 / 17
3 Motivation Consider a Phase I clinical trial with binary responses and two doses: d 1, d 2 Goal is to find the maximum tolerated dose (MTD): γ = patients were assigned to each dose, 2 and 4 toxicities observed Q: Which dose should be administered to the next patient? (ˆp i γ) 2 (1) The next patient can be assigned to either of doses, but one can argue that doses are not equal for two reasons. avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 2 / 17
4 Motivation Consider a Phase I clinical trial with binary responses and two doses: d 1, d 2 Goal is to find the maximum tolerated dose (MTD): γ = patients were assigned to each dose, 2 and 4 toxicities observed Q: Which dose should be administered to the next patient? (ˆp i γ) 2 (1) The next patient can be assigned to either of doses, but one can argue that doses are not equal for two reasons. 1 Criterion (1) ignores the randomness of the estimates. P (p 2 (0.25, 0.35)) > P (p 1 (0.25, 0.35)). avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 2 / 17
5 Motivation Consider a Phase I clinical trial with binary responses and two doses: d 1, d 2 Goal is to find the maximum tolerated dose (MTD): γ = patients were assigned to each dose, 2 and 4 toxicities observed Q: Which dose should be administered to the next patient? (ˆp i γ) 2 (1) The next patient can be assigned to either of doses, but one can argue that doses are not equal for two reasons. 1 Criterion (1) ignores the randomness of the estimates. P (p 2 (0.25, 0.35)) > P (p 1 (0.25, 0.35)). 2 ˆp 2 = 0.4 is an unacceptably high toxicity. avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 2 / 17
6 Motivation Consider a Phase I clinical trial with binary responses and two doses: d 1, d 2 Goal is to find the maximum tolerated dose (MTD): γ = patients were assigned to each dose, 2 and 4 toxicities observed Q: Which dose should be administered to the next patient? (ˆp i γ) 2 (1) The next patient can be assigned to either of doses, but one can argue that doses are not equal for two reasons. 1 Criterion (1) ignores the randomness of the estimates. P (p 2 (0.25, 0.35)) > P (p 1 (0.25, 0.35)). 2 ˆp 2 = 0.4 is an unacceptably high toxicity. It is usually of interest to balance two aims in a Phase I clinical trial avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 2 / 17
7 Current solutions Safety: Escalation with Overdose Control (EWOC) design by Babb et al. (1998): E ( α(γ P i ) + + (1 α)(p i γ) +) (2) + Low average number of DLTs Underestimation of the MTD Modifications: α n by Tighiouart et al. (2010) and Wheeler et al. (2017) avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 3 / 17
8 Current solutions Safety: Escalation with Overdose Control (EWOC) design by Babb et al. (1998): E ( α(γ P i ) + + (1 α)(p i γ) +) (2) + Low average number of DLTs Underestimation of the MTD Modifications: α n by Tighiouart et al. (2010) and Wheeler et al. (2017) Safety & Uncertainty Bayesian Logistic Regression Model (BLRM) by Neuenschwander et al. (2008). uses the whole distribution of the DLT probability and penalties for overly toxic intervals. For example, for γ = if p (0.00, 0.26); 0 if p (0.26, 0.41); L = 1 if p (0.41, 0.66); 2 if p (0.66, 1.00) avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 3 / 17
9 Goal We propose a new criterion for selecting doses in dose-escalation trials that accounts for 1 Uncertainty in the estimates 2 Ethical constraints and requires only one additional parameter to be specified. avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 4 / 17
10 Goal We propose a new criterion for selecting doses in dose-escalation trials that accounts for 1 Uncertainty in the estimates 2 Ethical constraints and requires only one additional parameter to be specified. We incorporate the proposed criterion to the one-parameter Bayesian continual reassessment method (O Quigley et al., 1990, CRM) avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 4 / 17
11 Novel Criterion The main object of estimation is the probability of DLT p i (0, 1) Squared distance is not a reliable measure for objects on the unit interval (Aitchison, 1992). avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 5 / 17
12 Novel Criterion The main object of estimation is the probability of DLT p i (0, 1) Squared distance is not a reliable measure for objects on the unit interval (Aitchison, 1992). Instead, we propose a distance satisfying the desirable properties δ(p, γ) = (p γ)2 p(1 p). (3) δ( ) = 0 at p = γ δ( ) as p 0 or p 1 The variance in denominator (Criterion 3 is a score statistic) avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 5 / 17
13 Novel Criterion The main object of estimation is the probability of DLT p i (0, 1) Squared distance is not a reliable measure for objects on the unit interval (Aitchison, 1992). Instead, we propose a distance satisfying the desirable properties δ(p, γ) = (p γ)2 p(1 p). (3) δ( ) = 0 at p = γ δ( ) as p 0 or p 1 The variance in denominator (Criterion 3 is a score statistic) In the illustration example above δ(ˆp 1 = 0.2, γ = 0.3) = 1/16 and δ(ˆp 2 = 0.4, γ = 0.3) = 1/24 (!) Single point estimate summarizes the information about uncertainty. avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 5 / 17
14 Introducing safety compound The target toxicity γ is always less than 0.5. Then for estimates ˆp 1 = γ θ and ˆp 2 = γ + θ, symmetric criterion favours ˆp 2. avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 6 / 17
15 Introducing safety compound The target toxicity γ is always less than 0.5. Then for estimates ˆp 1 = γ θ and ˆp 2 = γ + θ, symmetric criterion favours ˆp 2. We introduce an asymmetry parameter a: δ(p, γ) = (p γ)2 p a. (4) (1 p) 2 a 0 < a < 1 implies more severe penalty for more toxic doses. (!) Selection of under toxic doses remain to be undesirable as well. avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 6 / 17
16 Introducing safety compound The target toxicity γ is always less than 0.5. Then for estimates ˆp 1 = γ θ and ˆp 2 = γ + θ, symmetric criterion favours ˆp 2. We introduce an asymmetry parameter a: δ(p, γ) = (p γ)2 p a. (4) (1 p) 2 a 0 < a < 1 implies more severe penalty for more toxic doses. (!) Selection of under toxic doses remain to be undesirable as well. In the illustration example above, for a = 0.5 δ(ˆp 1 = 0.2, γ = 0.3, a = 0.5) < δ(ˆp 2 = 0.4, γ = 0.3, a = 0.5). avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 6 / 17
17 Asymmetry parameter (I) Parameter a balances the trade-off between ethical concerns and uncertainty How can one choose a meaningful value of a? avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 7 / 17
18 Asymmetry parameter (I) Parameter a balances the trade-off between ethical concerns and uncertainty How can one choose a meaningful value of a? Value a = 2γ leads to the same allocation as the squared distance a < 2γ leads to more conservative allocation of patients. avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 7 / 17
19 Asymmetry parameter (I) Parameter a balances the trade-off between ethical concerns and uncertainty How can one choose a meaningful value of a? Value a = 2γ leads to the same allocation as the squared distance a < 2γ leads to more conservative allocation of patients. Let (γ θ, γ + θ) be an interval such that among two estimates standing on the same squared distance from γ, the lower estimate would be preferred ( ( a = log γ θ ) ( / log 1 γ θ )) 1 γ + θ 1 γ + θ avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 7 / 17
20 Bayesian continual reassessment method DLT probability has the functional form ψ(d i, β) = d exp(β) i. f 0 (.) is prior distribution of β. After j patients have already been assigned to doses d(1),..., d(j) and binary responses Y j = [y 1,..., y j ] T were observed the posterior f j (β) is obtained. avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 8 / 17
21 Bayesian continual reassessment method DLT probability has the functional form ψ(d i, β) = d exp(β) i. f 0 (.) is prior distribution of β. After j patients have already been assigned to doses d(1),..., d(j) and binary responses Y j = [y 1,..., y j ] T were observed the posterior f j (β) is obtained. Then, the dose d k minimising ( ) (ψ(d i, β) γ) 2 E ψ(d i, β) a (1 ψ(d i, β)) 2 a (5) among all d 1,..., d m is recommended for the next group of patients avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 8 / 17
22 Bayesian continual reassessment method DLT probability has the functional form ψ(d i, β) = d exp(β) i. f 0 (.) is prior distribution of β. After j patients have already been assigned to doses d(1),..., d(j) and binary responses Y j = [y 1,..., y j ] T were observed the posterior f j (β) is obtained. Then, the dose d k minimising ( ) (ψ(d i, β) γ) 2 E ψ(d i, β) a (1 ψ(d i, β)) 2 a (5) among all d 1,..., d m is recommended for the next group of patients Convex Infinite Bounds Penalization with parameter a as CIBP(a). avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 8 / 17
23 Illustration (I) We revisit the Everolimus Trial in patients with HER2-overexpressing Metastatic Breast Cancer γ = 0.3. The study considers 3 regimens given together with Paclitaxel and Trastuzumab (PT): 1 Daily dosing of Everolimus 5mg plus PT (d 1 ) 2 Daily dosing of Everolimus 10mg plus PT (d 2 ) 3 Weekly dosing of Everolimus 30mg plus PT (d 3 ) Table : Aggregated data of the Everolimus trial Dose d 1 d 2 d 3 Number of Patients assigned Number of DLTs avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 9 / 17
24 Illustration (I) We revisit the Everolimus Trial in patients with HER2-overexpressing Metastatic Breast Cancer γ = 0.3. The study considers 3 regimens given together with Paclitaxel and Trastuzumab (PT): 1 Daily dosing of Everolimus 5mg plus PT (d 1 ) 2 Daily dosing of Everolimus 10mg plus PT (d 2 ) 3 Weekly dosing of Everolimus 30mg plus PT (d 3 ) Table : Aggregated data of the Everolimus trial Dose d 1 d 2 d 3 Number of Patients assigned Number of DLTs We compare original CRM and CIBP (0.3) using the same prior parameters. avel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 9 / 17
25 Illustration (II) Dose 1 Dose 2 Dose 3 Individual trial (CRM) Dose 1 Dose 2 Dose 3 Individual trial (CIBP) No Toxicity Toxicity cohort cohort Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 10 / 17
26 Illustration (II) Dose 1 Dose 2 Dose 3 Individual trial (CRM) Dose 1 Dose 2 Dose 3 Individual trial (CIBP) No Toxicity Toxicity cohort cohort Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 10 / 17
27 Illustration (II) Dose 1 Dose 2 Dose 3 Individual trial (CRM) Dose 1 Dose 2 Dose 3 Individual trial (CIBP) No Toxicity Toxicity cohort cohort Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 10 / 17
28 Illustration (II) Dose 1 Dose 2 Dose 3 Individual trial (CRM) Dose 1 Dose 2 Dose 3 Individual trial (CIBP) No Toxicity Toxicity cohort cohort Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 10 / 17
29 Illustration (II) Dose 1 Dose 2 Dose 3 Individual trial (CRM) Dose 1 Dose 2 Dose 3 Individual trial (CIBP) No Toxicity Toxicity cohort cohort Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 10 / 17
30 Illustration (II) Dose 1 Dose 2 Dose 3 Individual trial (CRM) Dose 1 Dose 2 Dose 3 Individual trial (CIBP) No Toxicity Toxicity cohort cohort Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 10 / 17
31 Illustration (II) Dose 1 Dose 2 Dose 3 Individual trial (CRM) Dose 1 Dose 2 Dose 3 Individual trial (CIBP) No Toxicity Toxicity cohort cohort Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 10 / 17
32 Numerical Study Setting by Wheeler et al. (2017). n = 40 patients; m = 6 doses; c = 1 cohort size; target γ = 0.33 β N (0, 1.34) a = {0.5, 0.25, 0.10}. Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 11 / 17
33 Numerical Study Setting by Wheeler et al. (2017). n = 40 patients; m = 6 doses; c = 1 cohort size; target γ = 0.33 β N (0, 1.34) a = {0.5, 0.25, 0.10}. We study the performance of designs in terms of (i) Accuracy m i=1 A = 1 m (p i γ) 2 π i m i=1 (p i γ) 2 (ii) mean number of toxic responses (DLTs) and focus on the mean performance. Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 11 / 17
34 Scenarios Toxicity Probability Scenario Scenario Scenario Scenario Scenario Dose Toxicity Probability Scenario Scenario Scenario Scenario Scenario Dose Dose Dose Dose Dose Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 12 / 17
35 Comparators We compare the performance of the proposed approach to EWOC TR design by Tighiouart et al. (2010) Toxicity-dependent feasibility bound (TDFB) by Wheeler et al. (2017) BLRM by Neuenschwander et al. (2008) We use the same prior distribution as Neuenschwander et al. (2008). Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 13 / 17
36 Results. Accuracy Accuracy Sc 1 Sc 2 Sc 3 Sc 4 Sc 5 Sc 6 Sc 7 Sc 8 Sc 9 Sc 10 Mean CIBP(0.5) CIBP(0.25) CIBP(0.1) TDFB EWOC TR BLRM Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 14 / 17
37 Results. Accuracy Accuracy Sc 1 Sc 2 Sc 3 Sc 4 Sc 5 Sc 6 Sc 7 Sc 8 Sc 9 Sc 10 Mean CIBP(0.5) CIBP(0.25) CIBP(0.1) TDFB EWOC TR BLRM Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 14 / 17
38 Results. Accuracy Accuracy Sc 1 Sc 2 Sc 3 Sc 4 Sc 5 Sc 6 Sc 7 Sc 8 Sc 9 Sc 10 Mean CIBP(0.5) CIBP(0.25) CIBP(0.1) TDFB EWOC TR BLRM Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 14 / 17
39 Results. Accuracy Accuracy Sc 1 Sc 2 Sc 3 Sc 4 Sc 5 Sc 6 Sc 7 Sc 8 Sc 9 Sc 10 Mean CIBP(0.5) CIBP(0.25) CIBP(0.1) TDFB EWOC TR BLRM Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 14 / 17
40 Results.DLTs Sc 1 Sc 4 Sc 7 Sc 10 Sc 2 Sc 5 Sc 8 Mean Average DLTs Sc 3 Sc 6 Sc 9 CIBP(0.5) CIBP(0.25) CIBP(0.1) TDFB EWOC TR BLRM Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 15 / 17
41 Results.DLTs Sc 1 Sc 4 Sc 7 Sc 10 Sc 2 Sc 5 Sc 8 Mean Average DLTs Sc 3 Sc 6 Sc 9 CIBP(0.5) CIBP(0.25) CIBP(0.1) TDFB EWOC TR BLRM Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 15 / 17
42 Results.DLTs Sc 1 Sc 4 Sc 7 Sc 10 Sc 2 Sc 5 Sc 8 Mean Average DLTs Sc 3 Sc 6 Sc 9 CIBP(0.5) CIBP(0.25) CIBP(0.1) TDFB EWOC TR BLRM Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 15 / 17
43 Results.DLTs Sc 1 Sc 4 Sc 7 Sc 10 Sc 2 Sc 5 Sc 8 Mean Average DLTs Sc 3 Sc 6 Sc 9 CIBP(0.5) CIBP(0.25) CIBP(0.1) TDFB EWOC TR BLRM Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 15 / 17
44 Conclusions The novel criterion requires one additional parameter only. The criterion incorporated into the one-parameter CRM method is found to result in 1 Similar accuracy, but fewer mean number of DLTS. 2 Greater accuracy, but similar mean number of DLTs. (!) The new criterion allows to make model-based design more ethical as it does not lead to any decrease in accuracy. Further work: Generalisation to dose-combination and dose-schedule trials including the case of delayed toxicity responses. Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 16 / 17
45 Aitchison, J. (1992) On criteria for measures of compositional difference. Mathematical Geology, 24, Babb, J., Rogatko, A. and Zacks, S. (1998) Cancer phase I clinical trials: efficient dose escalation with overdose control. Statistics in medicine, 17, Neuenschwander, B., Branson, M. and Gsponer, T. (2008) Critical aspects of the Bayesian approach to phase I cancer trials. Statistics in Medicine, 27, O Quigley, J., Pepe, M. and Fisher, L. (1990) Continual reassessment method: a practical design for phase 1 clinical trials in cancer. Biometrics, Tighiouart, M., Rogatko, A. et al. (2010) Dose finding with escalation with overdose control (ewoc) in cancer clinical trials. Statistical Science, 25, Wheeler, G. M., Sweeting, M. J. and Mander, A. P. (2017) Toxicity-dependent feasibility bounds for the escalation with overdose control approach in phase i cancer trials. Statistics in Medicine. Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 17 / 17
46 Asymmetry parameter (II) a θ Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 18 / 17
47 Comparators We compare the performance of the proposed approach to EWOC design using fixed α = 0.25 TR design by Tighiouart et al. (2010) using α 2 =... = α 9 = 0.25, α n = min (α n , 0.50). Toxicity-dependent feasibility bound (TDFB) by Wheeler et al. (2017) ( α n+1 = min 0.50, ( n 1 ) n i=1 y i BLRM by Neuenschwander et al. (2008) We use the same prior distribution as Neuenschwander et al. (2008). Pavel Mozgunov, Thomas Jaki (Lancaster University) Modified allocation rule for the CRM 19 / 17
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