Bayesian adaptive designs, with particular reference to dose finding methods in oncology

Transcription

1 IV BIAS Annual Congress - Padova 27/09/2012 Advanced Methods in Clinical Trials: surrogate endpoints and adaptive designs Bayesian adaptive designs, with particular reference to dose finding methods in oncology Mauro Gasparini Dipartimento di Scienze Matematiche - Politecnico di Torino mauro.gasparini@polito.it 1

2 Overview Dose finding in drug development. Dose finding in oncology as a binary regression problem. Relationship with other adaptive methods. One-dimensional dose-finding. CRM and alternatives. Extensions of CRM. 2

3 Positioning of dose finding in drug development pharmacology toxicology drug formulation animal PK/PD dose finding treatment mechanism human PK/PD proof of concept dose selection phase II to phase III preclinical Phase I Phase II confirmatory comparative trials } Phase III clinical marketing approval 3 pharmacoepidemiology additional indications expanded safety revisions post-marketing (Phase IV)

4 Bayesian adaptive designs in dose finding Dose finding is very delicate step in drug development. Adaptivity is necessary to prevent excessive toxicity. Bayesian methods are appropriate since dose finding trials have to be short (a few tenths of patients) and all previous (preclinical) knowledge can be used. 4

5 Phase I single-compound dose finding in oncology and the like few candidate dose levels x 1... x J response is toxicity (binary) risk (probability) of toxicity at x j : π(x j θ) T = target highest tolerated prob. tox. MTD = maximum tolerated dose π 1 (T) adaptive dose-escalation group sequential designs probability of toxicity 0 T π(x j θ) 1 x 1 MTD x j x J 5 dose

6 Estimating a quantile Quantile estimation a well known problem in Statistics (bioassay, toxicology, engineering). Trace back to the stochastic approximation method of Robbins and Monroe (1951): x i+1 = x i Y i T ib where x i are successive converging iterations, Y i is the response variable corresponding to x i and b is an appropriately chosen constant. A stochastic variation on iterative methods for finding a root of a nonlinear equation. 6 This is not the dose finding problem, but a continuous version of it.

7 Proper nonparametric and parametric dose-finding methods Nonparametric (or algorithmic) methods: 3+3 traditional designs A+B up-and-down designs Storer s two-stage 7 Model-based approaches: continual reassessment method (CRM) escalation with overdose control (EWOC) curve-free method TIme-To-Event (TITE) CRM CRM for survival outcomes CRM for survival outcomes interval toxicity probabilities dose finding for combination trials

8 Recent books Ying Kuen Cheung (2011). Dose Finding by the Continual Reassessment Method Chapman and Hall/CRC. Five papers in a recent issue of Statistical Science 2010, Vol. 25, No. 2 Chevret S (editor) (2006). Statistical Methods for Dose-Finding Experiments. Wiley. 8 Ting N (editor) (2006). Dose Finding in Drug Development. Springer.

9 Traditional 3+3 designs Evaluate 3 patients at initial dose: if 0/3 have DLT, evaluate 3 patients at the higher dose. if 1/3 have DLT, evaluate 3 more patients at the same dose. if 1/6 have DLT, evaluate 3 patient at the higher dose. if 2+/6 have DLT, discontinue dose escalation. if 2+/3 have DLT, discontinue dose escalation. When the trial is stopped, the dose level below that at which excessive DLT was observed is the estimated MTD. 9

10 Property of 3+3 and generalizations The result of the algorithm is that the probability of toxicity at finally estimated MTD is less than 0.33, most often around 0.26 A+B up-and-down designs generalize the 3+3 design to accommodate for different cohort sizes ans thresholds (Storer 1989, Korn 1994, Lin and Shih 2001) Estimation of MTD at the completion of dose escalation is done preferably by isotonic regression 10 Generalizations include a first accelerated phase and maximum likelihood methods (Storer s two stage design)

11 The continual reassessment method The CRM (O Quigley et al ) instead is a parametric method, based on sounder statistical concepts (by constrast, 3+3 designs are algorithmic methods). Unlike 3+3 designs, CRM uses information from all patients to get a current estimate of the MTD, so that the next patients can be assigned to it. The CRM is better explained in Bayesian terms, although non-bayesian versions exists. Prior information can properly be incorporated (Zohar et al. 2011). 11

12 The original one-parameter CRM The probability of toxicity (or simply the risk) π(x j θ) is parameterized with a low-dimensional parameter θ. For example, a one-parameter logistic, a two-parameter logistic or the very basic one-parameter model π(x j θ) = p θ i which makes use of (known) fixed skeleton probabilities p 1,..., p J This is a working model in the original formulation by O Quigley. In Statistics, the resulting parametric family is also known as (discrete) Lehman alternatives. 12

13 The dose escalation algorithm of CRM Start with a Bayesian prior inputed by physicians; do Bayesian update (using Bayes theorem); estimate of MTD (inverse of tolerated toxicity rate T) after each patient; assign next patient to the available dose closest to such estimate; stop after a fixed number of patients have been observed. 13

14 An example: the ssht phase I trial A phase I dose-finding and pharmacokinetic study of subcutaneous semisynthetic homoharrintonine (sshht) in patients with advanced acute myeloid leukemia. British Journal of Cancer (2006) 95, See presentation by Sarah Zohar: crm sshht zohar.ppt 14

15 Comments on the CRM Pros of CRM: away from traditional designs; adaptive; can shoot for any quantile, not just.33; great name! Cons: low-dimension of CRM brings too aggressive dose escalation; based only on the the posterior expected of MTD; difficult to grasp for physicians. 15 There is a lot of debate going on CRM. Critics feel it should be more explicitely Bayesian and more care should be attached to the specification of the model.

16 Modifications of the original CRM, slide I use a two-parameter model, like a logistic model; use a higher dimensional parametric model; (curve-free CRM, Gasparini and Eisele 2000, Whitehead et al. 2010); do not allow dose escalation by more than one level (modified CRM, Goodman et al. 1995); use two-stage modificatons; 16

17 Modifications of the original CRM, slide II use full posterior distribution instead of mean only, more in line with Bayesian thinking; for example, use posterior interval probabilities rather than posterior mean only Neuenschwander et al. (2008)). more flexible and sophisticated stopping rules, such as sufficient posterior certainty ) 17

18 Escalation with overdose control (CRM EWOC) EWOC (Babb et al 1998) is Bayesian predictive oriented. Like the remedies above, it prevents aggressive dose escalation. It first gives a reparameterization of the two-parameter exponential in terms γ, the MTD, and ρ, the initial dose toxicity level π(x γ, ρ) = exp { ln [ ρ 1 ρ 1 + exp { ln [ ρ 1 ρ ] + ln [ T(1 ρ) ρ(1 T) ] xγ } ] + ln [ T(1 ρ) ρ(1 T) ] xγ } 18

19 The CRM EWOC algorithm The CRM EWOC dose escalation algorithm is as follows: do Bayesian update after each patient obtain the posterior distribution function Π γ (γ) on γ, the MTD assign next patient to the available dose closest to Π 1 γ (α), (approximately, due to availability of doses), the α-quantile of the posterior distribution of the MTD, where α controls the Bayesian feasibility. 19 Notice α < T.

20 Software for the CRM Stand-alone software: CRM and CRMsimulator by researchers in MD Anderson R libraries: dfcrm is a companion software for Cheung s book bcrm is a new software by researchers in MRC Biostats 20

21 Higher dimensional models: product of beta Gasparini and Eisele (2000): θ 1 = 1 π 1 θ 2 = 1 π 2 1 π 1. θ 1,..., θ J independent with θ J = 1 π J 1 π J 1 π j Beta(a j, b j ), j = 1,....J The distribution induced on π 1, π 2... π J is called a product of beta prior (PBP) and can be used as prior in the dose finding problem by setting 21 π(x j θ) = π j Totally opposite to one-dimensional parameters.

22 Generalizations: dose finding for combination therapies combinations of two compounds, or schedules few candidate dose levels {x 1..., x J } {y 1... y K } underlying dose-response surface π(x j, y k, θ) the risk at (x j, y k ) adaptive dose-escalation, but more combinations possible T = target probability to be tolerated MTD = maximum tolerated dose combinations, possibly not unique straightforward generalization to more than two compounds: (d 1, d 2,..., d l ) (D 1 D 2... D l ) 22

23 Interaction with combination therapies It is important, when modeling combination trials, to take interaction into account. Suppose p and q are the risk of toxicity when only the first or only the second compound are used (also called marginal risks). Then No-interaction: g (p, q) = 1 (1 p)(1 q) = p + q pq. Synergy: Antagonism: g(p, q, γ) > p + q pq. g(p, q, γ) < p + q pq. 23 More on Gasparini M. (2013).

24 The treatment versus esperimentation dilemma An interesting recent theoretical development by Azriel et. al. (2011). Conflicting principles: treatment: want to treat every patient at MTD experimentation: want to estimate the MTD reliably Result: it is not possible to achieve both. If the next patient is treated at the current estimate of MTD, then this estimator can not be strongly consistent. The result is asymptotic (long run) but... 24

25 25... In the long run, we re all dead (Keynes).

26 Conclusions CRM is a good example of Bayesian adaptive ideas in a very crucial step of drug development such as Phase I dose finding. It has had a great role in thinking deeper about the rationale for dose finding. It has been improved thanks to a more thoughtful modeling, but debate is continuing nowadays. 26

27 27 Azriel, D., Mandel, M. and Rinott, Y. (2011). The treatment versus esperimentation dilemma in dose finding studies. Journal of Statistical Planning and Inference 141, Babb, J., Rogatko, A. and Zacks, S. (1998). Cancer phase I clinical trials: efficient dose escalation with overdose control. Statistics in Medicine 17, Bailey, S., Neuenschwander, B., Laird, G. and Branson, M. (2009). A Bayesian case study in oncology phase I dose-finding using logistic regression with covariates. Journal of Biopharmaceutical Statistics 19, Chevret, S., ed. (2006). Statistical Methods for Dose-Finding Experiments. Wiley. Cheung, Y.K. (2010) Stochastic Approximation and Modern Model-Based Designs for Dose-Finding Clinical Trials Statistical Science, 2010, Vol. 25, No. 2, Gasparini, M. and Eisele, J. (2000). A curve-free method for phase I clinical trials. Biometrics, 56, 2, Correction in Biometrics, 57, 2, Gasparini M. (2013). General classes of multiple binary regression models in dose-finding problems for combination therapies. Applied Statistics (JRSS C). Gasparini, M. Bailey, S. and Neuenschwander, B (2010). Correspondance: Bayesian dose in oncology for drug combinations by copula regression. Applied Statistics (JRSS C), 59, 2, Goodman, S.N., Zahurak, M.L. and Piantadosi, S. (1995). Some practical improvements in the continual reassessment method for phase I studies. Statistics in Medicine 14, Levy, V., Zohar, S., Bardin, C., Verkhoff, A., Choui, D., Rio, B., Legrand, O., Sentenac, S., Rousselot, P., Raffoux, E., Chast, F., Chevret, S. and Marie, J.P. (2006). A phase I dose-finding and pharmacokinetic study of subcutaneous semisynthetic homoharrintonine

28 (sshht) in patients with advanced acute myeloid leukemia. British Journal of Cancer (2006) 95, 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 I clinical trials in cancer. Biometrics 46, O Quigley, J., and Shen, L.Z. (1996). Continual reassessment method: a likelihood approach. Biometrics 52, Robbins, H. and Monroe, S. (1951). A stochastic approximation method. Ann. Math. Statist. 22, Storer, B. (1989). Design and analysis of phase I clinical trials. Biometrics 45, Storer, B.E. and De Mets, D. (1987). Current phase I/II designs: are they adequate? Journal of Clinical Research and Drug Development, 1, Thall, P.F., Millikan, R.E., Mueller, P. and Lee, S.-J. (2003). Dose-finding with two agents in phase I oncology trials. Biometrics 59, Ting, N., ed. (2006). Dose Finding in Drug Development. Springer. Whitehead, J. (1997). Bayesian decision procedures with application to dose-finding studies. International Journal of Pharmaceutical Medicine 11, Whitehead, J., Thygesen, H. and Whitehead, A. (2010). A Bayesian dose-finding procedure for phase I clinical trials based only on the assumption of monotonicity. Statistics in Medicine 29, Zohar, S., Baldi I., Forni, G., Merletti, F., Masucci, G., Gregori D. (2011). Planning a Bayesian early-phase phase I/II study for human vaccines in HER2 carcinoma. Pharmaceutical Statistics 10(3),