A Bayesian decision-theoretic approach to incorporating pre-clinical information into phase I clinical trials

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1 A Bayesian decision-theoretic approach to incorporating pre-clinical information into phase I clinical trials Haiyan Zheng, Lisa V. Hampson Medical and Pharmaceutical Statistics Research Unit Department of Mathematics and Statistics, Lancaster University Industry collaborator: Clinical advisor: Dr Alun Bedding (AstraZeneca) Dr Malcolm Mecleod (Edinburgh University) IDEAS Think Tank, Traunkirchen 23rd June 2016 This project has received funding from the European Union s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No (IDEAS - H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 1 / 25

2 Motivation FDA (2005), Sharma and McNeill (2009), Reigner and Blesch (2002) Q: How can we use pre-clinical toxicology and pharmacology data to improve the design, conduct and analysis of phase I dose-escalation trials? Current approaches use pre-clinical data to determine a maximum recommended starting dose (MRSD) using allometric scaling: Using toxicology data: Human dose (mg kg 1 )= NOAEL (W A /W H ) 0.33 Using PK data: Estimate human PK parameters using allometric scaling, e.g., Cl H = Cl A (W A /W H ) b Scale doses by a safety factor of 10 in case of size-independent differences. Simply allometry can produce inaccurate predictions of human doses (e.g., diazepam, warfarin) leading to conservative or toxic starting doses. May be uncertainty about the best choice of allometric exponent Likely to be data on several animal species - which species is most relevant? H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 2 / 25

3 Research objectives To develop a Bayesian model-based procedure that uses pre-clinical information based on concerns about its degree of agreement with dose-toxicity relationship in humans: Is the drug predicted more (or less) potent in humans than it actually is? H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 3 / 25

4 Pre-clinical toxicology data Commensurability issues H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 4 / 25

5 Phase I clinical trials Principal aim: to estimate the maximum tolerated dose (MTD) using cumulative toxicity data from an ongoing phase I trial MTD - subject to an acceptable risk of dose-limiting toxicity (DLT): π d - target: π d (0.20, 0.35) Assumption: toxicity increases monotonically with the dose Approaches/Designs Algorithmic 3+3 design and more sophisticated up-and-down schemes Model-based Continual reassessment method (CRM) with a one-parameter working model Bayesian logistic regression method (BLRM) with a two-parameter dose-toxicity model... H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 5 / 25

6 Bayesian logistic regression method (BLRM) Cumulative toxicity data At each dose d j - Number of patients: n j - Number of DLTs observed: t j BLRM t j n j Binomial(n j, p(d j )), The dose-toxicity model: log p(d j) 1 p(d j ) = θ 1 + exp(θ 2 ) log d j Prior Bivariate normal distribution: (θ1, θ 2 ) MVN(, ) Pseudo-data envisaged for the lowest and highest doses Dose # Patients # DLTs d 1 n 1 t 1 d 0 n 0 t 0 H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 6 / 25

7 Bayesian methods for dose-escalation studies Expressing the prior Consider two potential doses d i, i = { 1, 0} p(d i ) Beta(t i, u i ), where u i = n i t i Two beta distributions are independent of one another p(d i ) is written as p i for illustration purposes The joint prior density is therefore given by g 0 (p 1, p 0 ) = 0 i= 1 p t i 1 i (1 p i ) u i 1, B(t i, u i ) and that of θ 1 and θ 2 by 0 [exp( θ 1 exp(θ 2) log d i) + 1] t i [exp(θ 1 + exp(θ 2) log d i) + 1] u i h 0(θ 1, θ 2) = B(t i, u i) i= 1 ( ) d 1 exp(θ 2) log using the Jacobian transformation. (Tsutakawa, 1975; Whitehead and Williamson, 1998) d 0 H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 7 / 25

8 Bayesian methods for dose-escalation studies Expressing the prior Pseudo-data indicates both the position of the parameters (θ 1, θ 2 ) and the strength of such an opinion: as if n 1 patients were administered the lower dose, t 1 DLTs observed n0 patients were administered the higher dose, t 0 DLTs observed Strength n 1 = n 0 = 3 is deemed proper H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 8 / 25

9 Bayesian methods for dose-escalation studies Expressing the prior Pseudo-data indicates both the position of the parameters (θ 1, θ 2 ) and the strength of such an opinion: as if n 1 patients were administered the lower dose, t 1 DLTs observed n0 patients were administered the higher dose, t 0 DLTs observed Strength n 1 = n 0 = 3 is deemed proper H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 8 / 25

10 Bayesian methods for dose-escalation studies Expressing the prior Pseudo-data indicates both the position of the parameters (θ 1, θ 2 ) and the strength of such an opinion: as if n 1 patients were administered the lower dose, t 1 DLTs observed n0 patients were administered the higher dose, t 0 DLTs observed Strength n 1 = n 0 = 3 is deemed proper Comments: 1 These are operational priors, calibrated to ensure the dose-escalation scheme has favourable operating characteristics 2 Informative priors can be formed using pre-clinical toxicology data H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 8 / 25

11 Addressing the potential prior-data conflict Marginal distribution function for p j The joint prior density can be given as 0 [exp( θ 1 exp(θ 2) log d i) + 1] t i [exp(θ 1 + exp(θ 2) log d i) + 1] u i h 0(θ 1, θ 2) = B(t i, u i) i= 1 ( ) d 1 exp(θ 2) log d 0 We further express the prior as the joint pdf. of p j and θ 2 : [ ( ] ti [ ( 0 exp log exp log m 0 (p j, θ 2 ) = i= 1 p j ) + exp(θ 1 p 2 ) log d j + 1 j d i 1 p j (1 p j ) exp(θ 2) ( d 1 log d 0 ) B(t i, u i ) p j ) exp(θ 1 p 2 ) log d j + 1 j d i The marginal distribution for p j, j = { 1, 0, 1, 2,..., J}, is g(p j ) = m 0 (p j, θ 2 )dθ 2. Notation: p(d j) is written as p j for illustration purposes. H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 9 / 25 ] ui

12 Addressing the potential prior-data conflict 1 The considered future observations of DLT (ỹ = 1) or not (ỹ = 0) are predicted using the prior predictive distribution of ỹ, P{Y = ỹ} = p j f (ỹ p j)g(p j)dp j, where f ( ) is the link function with the DLT probability p j, and the pior g(p j) is formed based on pre-clinical studies. 2 Predictions are optimal in the sense of maximising the prior expected utility ū(η) = u(ỹ, η)p{y = ỹ}, ỹ where u(ỹ, η) is the utility function that rewards/penalises predictions of ỹ as η: 0, if η = 0 while ỹ = 1 u(ỹ, η) = c, if η = 1 while ỹ = 0. 1, η = ỹ {0, 1} Note that 0 < c < 1. H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 10 / 25

13 Addressing the potential prior-data conflict (Cont d) 3 The optimal prediction ˆη is therefore chosen out of the whole decision set H = {0, 1} by maximising the prior expected utility ū(η): ˆη = arg max u(ỹ, η)p{y = ỹ}. η H 4 A 2 2 contingency table for the actual versus predicted DLTs and no-dlts ỹ Rewards and Penalties Actual obs (y) No-DLT DLT Human counts Prior predictions (ˆη) No-DLT u 00 u 10 n 00 n 10 DLT u 01 u 11 n 01 n 11 5 The predictive utility is then calculated at dose d j as U j pred = 1 l=0 1 m=0 u lmn lm, and the predictive accuracy as a j = 1 l=0 U j pred 1 n. m=0 lm 6 The average ā = a j/k measures the commensurability of animal and human data. H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 11 / 25

14 Prior tuning ā [0, 1), computed at each interim analysis, quantifies the degree of agreement between animal and human toxicology data Mixture prior with a weakly informative component will be considered f (θ) = w g(θ) + (1 w) p(θ), }{{}}{{} pre-clinical data weakly-informative the weight w is a function of ā, allowing for a flexible borrowing especially when the human data is sparse w will determine how quickly pre-clinical information is discounted in the case of a prior-data conflict p(θ) constructed based on the operational priors (Whitehead and Williamson, 1998) works reasonably well H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 12 / 25

15 The pre-clinical data weight f (θ) = w g(θ) + (1 w) p(θ) The weight w is defined as a function of predictive accuracy ā in the relevance of the information time n/n (say, N = 24) It governs how influential the pre-clinical data are as the trial proceeds One possible function considered in our study is w = ā 1 log(n/n) With the accumulation of human data, ā, serving as a measurement of prior-data conflict, becomes more reliable ā is computed dynamically at each interim analysis H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 13 / 25

16 Dose recommendation Fully Bayesian patient gain criterion G 1 = (p(d i ) π d ) 2 f (p(d i ) x)dp(d i ), π d (0.20, 0.35) Fully Bayesian determinant gain criterion G 2 = (deti(θ)) 1 f (θ x)dθ, θ = (θ 1, θ 2 ) where x denote the human data accumulated from each patient cohort, and f ( x) the (mixture) posterior distribution. Safety constraint: Controlling the probability of excessive toxicity at level δ: 1 γ g(p(d j ))dp(d j ) δ, where γ is some threshold above which the risk of toxicity is considered excessively high. H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 14 / 25

17 Simulations Six doses to be assessed: 2.2, 3.1, 3.3, 3.7, 4.0, 4.9 mg/m 2 A total number of 24 patients, cohort size of 1 Targeted probability π d = 0.20; that is, TD 20 is sought For all simulations, the true dose-response curve was assumed to be log p(d j) = θ1 + exp(θ2) log dj, 1 p(d j) Prior distribution for the model parameters was constructed as a mixture prior f (θ) = w g(θ) + (1 w) p(θ), with given information on the lowest and highest doses: For instance, p(d 1 ) Beta(5, 95) and p(d 6 ) Beta(95, 5) are summarised from pre-clinical toxicology studies Stochastic optimisation approach technique is used to derive the closed form for the bivariate normal distributions g(θ) and p(θ) Full (mixture) posterior distributions are obtained via MCMC simulations Results are presented based on 2000 simulated trials per scenario H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 15 / 25

18 Prior calibration Prior1 approximated as: Prior2 approximated as: Prior3 approximated as: MVN : (µ 1, µ 2, σ 1, σ 2, ρ) = ( 8.20, 1.92, 0.075, 0.037, 0.98) MVN : (µ 1, µ 2, σ 1, σ 2, ρ) = ( 5.76, 1.25, 0.094, 0.029, 0.78) MVN : (µ 1, µ 2, σ 1, σ 2, ρ) = ( 3.28, 0.66, 0.120, 0.032, 0.75) H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 16 / 25

19 Prior calibration (Cont d) Prior4 approximated as: Prior5 approximated as: Operational priors approximated as: MVN : (µ 1, µ 2, σ 1, σ 2, ρ) = ( 29.78, 3.16, 0.082, 0.062, 0.28) MVN : (µ 1, µ 2, σ 1, σ 2, ρ) = ( 10.74, 2.11, 0.101, 0.025, 0.95) MVN : (µ 1, µ 2, σ 1, σ 2, ρ) = ( 3.73, 0.84, 0.645, 0.367, 0.17) H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 17 / 25

20 Simulation results Table: Results for incorporating pre-clinical toxicology data or not are presented based on 2000 simulated trials, each with size of 24, per scenario. TD20 was sought with fully Bayesian determinant gain criterion used for dose-escalation. Dose (mg/m 2 ) d 1 d 2 d 3 d 4 d 5 d true p(d j) DLT SC1: animal data predict excess of DLTs 5.1 HLInc SelMTD (%) avgno.pts SC2: animal data predict insufficient DLTs 5.8 HLInc SelMTD (%) avgno.pts SC3: animal data indicate a shallow curve 4.6 HLInc SelMTD (%) avgno.pts H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 18 / 25

21 Simulations results (Cont d) Table: Continued. Dose (mg/m 2 ) d 1 d 2 d 3 d 4 d 5 d true p(d j) DLT SC4: animal data indicate a steep curve 5.0 HLInc SelMTD (%) avgno.pts SC5: animal data are perfectly commensurate with human data 4.6 HLInc SelMTD (%) avgno.pts SC6: animal data are completely discarded 5.1 w = 0 SelMTD (%) avgno.pts H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 19 / 25

22 Simulations results (Cont d) H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 20 / 25

23 Simulations results (Cont d) H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 21 / 25

24 Summary Incorporating pre-clinical data when they are commensurate with human data will lead to more efficient dose-escalation decisions and greater estimation precision More patients are allocated to the target dose Dose recommendations are sensible and robust The pre-clinical data will essentially be discounted if they are in clear conflict with the dose-toxicity relationship in humans H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 22 / 25

25 References FDA. Guidance for Industry. Estimating the maximum safe starting dose in initial clinical trials for therapeutics in adult healthy volunteers Sharma V, McNeill JH. To scale or not to scale: the principles of dose extrapolation. British Journal of Pharmacology 2009; 157: Reigner BG, Blesch KS. Estimating the starting dose for entry into humans: principles and practice. Eur J Clin Pharmacology 2002; 57: Tsutakawa RK. Technical report 52. Mathematical Science, University of Missouri, Columbia, Missouri Whitehead J, Williamson D. Bayesian decision procedures based on logistic regression models for dose-finding studies. J Biopharmaceutical Medicine 1998; 8: H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 23 / 25

26 Mixture prior derivation f (θ) = w g(θ) + (1 w) p(θ) 1 Synthesise information from pre-clinical toxicology studies to describe probabilities of toxicity p(d) especially for the lowest and highest doses 2 Derive g(p(d j )), the distribution for p(d j ), at each candidate dose d j 3 Calibrate g(p(d j )) to find a prior g(θ) in a way such that its implied prior information on the p(d) scale is in good agreement with g(p(d j )) using stochastic optimisation methods to minimise the discrepancy between the specified quantiles Q and the quantiles Q arising from g(θ): C(Q, Q ) = max q jk q jk, j = 1,..., J, k = 1,..., K j,k H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 24 / 25

27 Mixture prior derivation f (θ) = w g(θ) + (1 w) p(θ) 4 The BVN p(θ) can also be approximated using this quantile-based approach 5 Specifically, prior information at a given dose is summarised with three quantiles q jk = {q j (0.025), q j (0.5), q j (0.975)}, which correspond to the median and 95% percent credible intervals 6 Introducing a weakly-informative component can potentially guarantee the dose-escalation procedure performs properly, even (in an extreme case) when the animal data are completely incommensurate with the human data H Zheng, LV Hampson (Lancs Uni) Incorporating pre-clinical info IDEAS Think Tank 25 / 25