Application of optimal model-based

design in dose ranging

Sergei Leonov

April 23, 2014

Durham, NC

R&D | Biometrics & Information Sciences

Trends and Innovat ions in

Clinical Trial Stat ist ics

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Outline

• What “optimality” means • Examples of optimal designs - Adaptive dose-ranging - Measuring efficacy and toxicity responses - PK/PD studies

• Treatment vs experimentation

• References on optimal model-based design - Classic monographs: Fedorov (1972), Silvey (1980), Pázman (1986), Pilz (1991), Atkinson

and Donev (1992), Pukelsheim (1993), Fedorov and Hackl (1997) - Recent books: Atkinson et al. (2007), Berger and Wong (2009), Morris (2010), Goos

and Jones (2011), Rasch et al. (2011), Pronzato and Pázman (2013), Fedorov and Leonov (2013) FL2013

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Optimal model-based design: components

• Model • Controls and its admissible values • Utility function • Cost/penalty • Estimation method • Optimality criteria • Computing optimal designs • Sensitivity analysis

• Key for design/estimation: individual information matrix µ(x,θ) of a predictor x

- dose or sequence of doses in dose finding - sequence of sampling times in PK studies

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Optimal Designs

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Optimality criteria

Optimality criterion

(utility): • Single parameter:

size of confidence

interval

• Several parameters:

“size” of confidence

region

D-criterion: |D| = λ1 ∙λ2 = (OA ∙OB)2 ; area (V) = π (λ1 ∙λ2 )1/2 ~ ( det )1/2

A-criterion: tr D = λ1 + λ2 = (OC)2 = D11 + D22

E-criterion: λ1 = (OA)2

FL2013, Ch. 2.2

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Cost-based (penalized) designs

FL2013, Ch. 4.1

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Engineering locally optimal designs

DESIGN ENGINE

Model, prior information, optimality criterion, individual

information matrix, constraints, cost, penalty, …

Optimal design, comparison of different designs, sample size,

robustness, scenario analysis, benchmarking, …

FL2013, Ch.3

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1. Adaptive design: Alzheimer’s disease study

• First-Time-in-Human, patient population • Biomarker

- Reduction expected after a single dose - Goal: find a dose with 90% reduction - Inhibition = Percentage decrease post-dose wrt baseline: 100*(Bpre-dose - Bpost-dose) / Bpre-dose

FL2013, Ch. 8.1

0

20

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0.000 0.001 0.010 0.100 1.000 10.000 100.000

Dose

% I

nh

ibit

ion

at

day 2

1mean

Emax

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Model: Emax

Measurements: Yi = η(xi , γ) + εi , i = 1,…, N {xi} - doses, N - number of patients, η – response function

Response parameters γ = (Emax , ED50 , β): – Emax : maximal response

– ED50: dose at which the response is half of Emax

– β : slope parameter

Variance model: Var εi = σ2A + σ2

M ηi (Emax – ηi ) (largest in the middle near ED50)

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• Measurement/statistical model (Emax) • Long half-life: each new dose given to a new cohort • Cohort-dependent design region: limit on dose-escalation

− 10-fold for “small” PD effect − 5-fold once “larger” PD effect is observed

• Subject numbers: safety is a priority (FTIH)

− All cohorts: 2 subjects on placebo − First cohorts: “fixed” dose escalation

• Criterion: find doses that are “best” for estimation

− All model parameters → D-criterion − Dose ED90 → c-criterion, Var(ED90)

Adaptive optimal design with constraints

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Adaptive optimal design

Addresses the problem of limited information at the start of the study: “locally” optimal designs, depend on model parameters

1. Collect some data using an initial design 2. Estimate model parameters using current data 3. Find optimal design (doses/weights) for current parameter estimates 4. Collect more data using this new design

Repeat steps 2-4 given available resources (# of subjects)

Box, Hunter (1965)

Size of pre-adaptation stage is critical!

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Simulation stand-alone tool (Matlab)

Parameter

panel

Simulation

settings

Variance

model

Criterion

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Single simulation: estimated curves Estimated curves for first cohorts: far

from “true” (doses too low to

properly estimate parameters)

After higher doses are tested, curves

start to approach the true curve

After first 4 cohorts (fixed doses) After 8 cohorts

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Multiple simulations Explore empirical distribution of adaptive designs

• Cohort 8: “cluster” weights close to locally optimal (0.3, 0.3, 0.4)

optimal doses

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The background normal distribution naturally incorporates correlation between responses

FL2013, Ch.6.5

2. Dose-ranging, efficacy and toxicity

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Utility functions and optimality criteria

• Utility functions - Location of “best” dose x* and value of p10(x*)

- Knowledge of all model parameters

• Optimality criteria - Precision of estimators for maximum of the utility function and its location

Var {estimated p10(x*)} + Var {estimated location of max[p10(x)]}

- Function of the variance-covariance matrix of estimated parameters

FL2013, Ch. 2.2, 9.2.1

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Penalties and constraints

Restricted design region: no observations beyond

X = { x: Pr{efficacy} > QE , Pr{toxicity} < QT }

FL2013, Ch.6.4.2

Pr{efficacy w/o toxicity} 1 - Pr{toxicity}

QE QT

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To learn or to treat?

To treat:

• Increase CE and CT

• Increase QE and decrease QT

DESIGN ENGINE

To learn:

• Decrease CE and CT

• Decrease QE and increase QT

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From locally optimal to other designs

• Locally optimal designs: guessing values of unknown parameters - "To guess is cheap, to guess wrongly is expensive“

• Bayesian designs: incorporating uncertainty of our prior knowledge

• Multistage/adaptive designs

- Size/design of initial stage is important (earlier/later examples)

• Fully adaptive (many adaptations)

- Complicated logistics, blinding, potential operational bias

- Little use when enrollment is fast and treatment results are not readily

available

- Can be inferior to two-three stage designs (Fedorov, Wu, Zhang (2012))

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3. Design of population PK/PD studies

• What we optimize/control • When to take samples

• Number of sampling times per patient

• Goal: find the most “informative” sampling schedule

• Key: individual information matrix µ(x,θ) of a k-dimensional

predictor x (sequence of sampling times)

• Population Optimum Design of Experiments (PODE)

• Annual workshop, nonlinear mixed effects models, started in 2006

• Population optimal design tools: PFIM (INSERM, Université Paris 7, France),

PkStaMp (started at GlaxoSmithKline), PopDes (CAPKR, Univ. of Manchester),

PopED (Uppsala Univ., Sweden), WinPOPT (Univ. of Otago, New Zealand)

• 2014 PODE meeting: Basel, September 11

• 2015 PODE meeting: Cambridge, INI, July

• Software comparison: Nyberg et al. (2014, Brit. J. Clin. Pharm.)

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Nonlinear mixed models

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PkStaMp: input screen, 2-compartment model

FL2013, Ch. 7.5

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0

20

40

60

80

100

0.001 0.010 0.100 1.000 10.000 100.000

DoseR

esp

on

se

EDp

p

Adaptive designs,

two approaches • Optimal design, efficient

experimentation to gather information for future patients

Robbins, Lai (1979),

Bartroff, Lai (2010)

• Allocate next patient(s) to current “best” dose, eg. current estimate of MTD

CRM, O’Quigley et al (1990) Shu,O’Quigley (2008)

Treatment vs experimentation dilemma

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Treatment vs experimentation

(1) Original CRM: one-parameter models, appropriate for estimating a single target dose (Bayesian c-optimal design)

(2) Some extensions: two-parameter models for estimating a single target dose, examples referenced in Shu, O’Quigley (2008)

Results from optimal/adaptive control: designs of type (2)

- may converge to the “wrong” dose

- may not converge at all

Bozin, Zarrop (1991), Azriel et al. (2011)

Designs of type (2): “Best-Intention” designs (BI)

Fedorov et al. (2011)

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“Best intention” designs

BI example:

Robbins-Monro (1951)

- finding a dose d* that

gives target response f *

yn = f(dn,θ) + εn ,

monotone f , {ε} – noise

Next dose:

dn+1 = dn – an (yn - f *)

Why it works (“on average”): if dn > d*, then yn > f *→ dn < dn+1

if dn < d*, then yn < f *→ dn > dn+1

Simplest case – linear,

f = θ1 + θ2d

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BI: Robbins-Monro

Procedure

dn+1 = dn – an (yn - f *)

• “Small” an ~ 1/n (to avoid “big” jumps)

• Fastest convergence: an = 1/(θ2n)

Parameters unknown: how about using estimate θ*2,n?

Shu, O'Quigley (2008): “being optimal for anything other than the

best estimated treatment for the next patient, or group of

patients, to be included in the study is not acceptable''

Next plots: FL2013, Ch. 9.2

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Linear model, optimal doses

Left panel:

Robbins-Monro (BI)

Right panel:

penalized adaptive design (PAD)

Penalty φ(x) = (x-x*)2

# of iterations:

- left columns: 100

- right : 400

Initial design:

- top row: 2-point

- bottom : 8-point

convergence to wrong doses

BI PAD

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θ1

Linear model, parameter estimates

BI PAD

θ1 θ1 θ1

θ2

θ2

# of iterations:

- left column: 100

- right : 400

Initial design:

- top row: 2-point

- bottom : 8-point

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Quadratic model f(x,θ) = θ1 + θ2 x + θ3 x2

convergence to wrong doses

Goal: find maxx f(x,θ)

x* = -θ2 /(2θ3 )

BI PAD

Left panel: BI

Right panel: PAD

# of iterations:

- left columns: 100

- right : 400

Initial design:

- top row: 3-point

- bottom : 12-point

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Quadratic model, parameter estimates

BI PAD

θ2 θ2 θ2 θ2

θ3

θ3

# of iterations:

- left columns: 100

- right : 400

Initial design:

- top row: 3-point

- bottom : 12-point

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Summary

• Optimal design: finding most “informative” levels of controls

(doses in dose-finding, sampling times in PK/PD studies, etc.)

• Validating standard/alternative designs (benchmarking)

• Test robustness of optimal designs (sampling windows, PK/PD)

• Can incorporate costs/penalties

• Can accommodate various practical constraints

• Practical, user-friendly GUI-based applications

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References

• Azriel, D., Mandel, M., Rinott, Y. (2011). The treatment versus experimentation dilemma in dose finding studies. J. Statist. Plann. Inf., 141 (8), 2759–2768.

• Aliev A, Fedorov VV, Leonov S, McHugh B, Magee M (2012). PkStaMp library for constructing optimal population designs for PK/PD studies. Comm. Statist. Simul. Comp. 41 (6), 717–729.

• Atkinson, A.C., Donev, A. (1992). Optimum Experimental Design. Clarendon Press, Oxford. • Atkinson, A.C., Tobias, R., and Donev, A. Optimum Experimental Designs, with SAS. Oxford

University Press, Oxford. • Berger, M.P.F., Wong, W.K. (2009). An Introduction to Optimal Designs for Social and

Biomedical Research (Statistics in Practice). Wiley, Chichester. • Box, G.E.P., Hunter, W.G. (1965), Sequential design of experiments for nonlinear models. In:

Korth, J.J. (ed.), Proceedings of IBM Scientific Computing Symposium, IBM, White Plains, New York, pp. 113-137.

• Bozin, A., Zarrop, M. (1991). Self-tuning extremum optimizer – convergence and robustness properties. In: Proceedings of ECC 91, First European Control Conference, pp. 672–677.

• Elfving, G. (1952). Optimum allocation in linear regression theory, Ann. Math. Statist., 23, 255-262.

• Fedorov, V.V. (1972), Theory of Optimal Experiment. Academic Press, NY. • Fedorov, V.V., Flournoy, N., Wu, Y., Zhang, R. (2011). Best intention designs in dose–finding

studies. Preprint NI11065-DAE, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK. http://www.newton.ac.uk/preprints/NI11065.pdf.

• Fedorov, V.V., Gagnon, R., Leonov, S., Wu, Y. (2007). Optimal design of experiments in pharmaceutical applications. In: Dmitrienko A, Chuang-Stein C, D’Agostino R (Eds), Pharmaceutical Statistics Using SAS. SAS Press, Cary, NC, pp. 151-195.

• Fedorov, V.V., Hackl, P. (1997). Model-Oriented Design of Experiments. Springer, NY • Fedorov, V.V., Leonov, S. (2005), Response driven designs in drug development. In: Wong,

W.K., Berger, M.P.F. (Eds.), Applied Optimal Designs, Wiley, Chichester, pp. 103-136. • Fedorov, V.V., Leonov, S.L. (2013). Optimal Design for Nonlinear Response Models. Chapman

& Hall/ CRC Biostatistics Series, Boca Raton, FL.

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• Fedorov, V.V, Wu, Y., Zhang, R. (2012). Optimal dose-finding designs with correlated continuous and discrete responses. Statist. Med., 31 (3), 217–234.

• Goos, P., Jones, B. (2011). Optimal Design of Experiments: A Case Study Approach. Wiley, New York.

• Lai, T.L., Robbins, H. (1979). Adaptive design and stochastic approximation. Ann. Statist., 7, 1196–1221.

• Leonov, S., Miller, S. (2009), An adaptive optimal design for the Emax model and its application in clinical trials. J. Biopharm. Stat. 19 (2), 360-385.

• Lindstrom, M.J., Bates, D.M. (1990). Nonlinear mixed effects models for repeated measures data. Biometrics, 46, 673-687.

• Magnus, J.R., Neudecker, H. (1988). Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley, NY.

• Mentré, F., Mallet, A., Baccar, D. (1997). Optimal design in random-effects regression models, Biometrika, 84 (2), 429-442.

• Mentré, F., Nyberg, J., Ogungbenro, K., Leonov, S., Aliev, A., Duffull, S., Bazzoli, C., Hooker, A. (2011). Comparison of results of the dierent software for design evaluation in population pharmacokinetics and pharmacodynamics. In: Abstracts of the Annual Meeting of the Population Approach Group in Europe (PAGE). ISSN 1871-6032. http://www.page-meeting.org/default.asp?abstract=2066.

• Morris, M. (2010). Design of Experiments. An Introduction Based on Linear Models. Chapman & Hall/CRC, Boca Raton, FL.

• Neumann AU, Lam NP, Dahari H, Gretch DR, Wiley TE, Layden TJ, Perelson AS (1998). Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon therapy. Science, 282, 103-107.

References (cont.)

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• Nyberg. J., Bazzoli, C., Ogungbenro, K., Aliev, A., Leonov, S., Duffull, S., Hooker, A., Mentré, F. (2014). Methods and software tools for design evaluation in population pharmacokinetics-pharmacodynamics. Brit. J. Clin. Pharm., early view. http://onlinelibrary.wiley.com/doi/10.1111/bcp.12352/abstract

• O’Quigley, J., Conaway, M. (2010). Continual reassessment and related dose-finding designs. Statistical Science, 25 (2), 245–257.

• O’Quigley, J., Pepe, M., Fisher, L. (1990). Continual reassessment method: a practical design for phase I clinical trials in cancer. Biometrics, 46, 33–48.

• Pázman, A. (1986). Foundations of Optimum Experimental Design. Reidel, Dordrecht. • Pilz, J. (1991). Bayesian Estimation and Experimental Design in Linear Regression Models.

Wiley, New York. • Pronzato, L. and Pázman, A. (2013). Design of Experiments in Nonlinear Models: Asymptotic

Normality, Optimality Criteria and Small-Sample Properties. Lecture Notes in Statistics, Volume 212, Springer, New York.

• Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York. • Rasch, D., Pilz, J., Verdooren, L.R. and Gebhardt, A. (2011). Optimal Experimental Design

with R. Chapman & Hall/CRC, Boca Raton, FL. • Robbins, H., Monro, S. (1951). A stochastic approximation method. Ann. Math. Statist., 22,

400-407.

• Shu, J., O’Quigley, J. (2008). Dose-escalation designs in oncology: ADEPT and the CRM.

Statist. Med., 27, 5345–5353.

• Silvey, S.D. (1980). Optimal Design. Chapman & Hall, London.

• Wynn, H.P. (1970) The sequential generation of D-optimal experimental designs. Ann. Math.

Statist., 41, 1655-1664.

References (cont.)

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Back-up, Ex.1: Comparing designs, D-efficiency

1/Eff: by how much sample size has to be increased to get

same precision (information) compared to optimal design ξ*

Composite (multi-stage) design: uses placebo & fixed doses from

4 fixed cohorts

• Total 58 patients (as adaptive)

• ξ0 = {16 on placebo, 3 each on [0.001, 0.01], 6 each on [0.05, 0.25] }

• Remaining 24 patients: allocated wrt composite D-optimality,

complementing existing design (analog of Bayesian design)

Ratio of utility functions:

Efficiency: adaptive - 0.89 (st.dev. 0.02)

FL2013, Ch. 2.7

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Combination drug for treating chronic hepatitis C (HCV) infection Neumann et al. (1998), Mentré et al. (2011)

Back-up, Ex.3, complex models: HCV dynamics

FL2013, Ch. 7.5.7