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Bayesian penalized D-optimality for dual-agent phase I oncology trials

Graham Wheeler1, Michael Sweeting2 and Adrian Mander1

1MRC Biostatistics Unit, Cambridge, UK

2Cardiovascular Epidemiology Unit, University of Cambridge, UK

Adaptive Designs WorkshopTuesday 28th January 2014

Outline

1 Phase I trials

2 Dual-agent phase I trials

3 Bayesian penalized D-optimality

4 Simulation study

5 Discussion and future work

Graham Wheeler (MRC Biostatistics Unit) Dual-Agent Phase I Oncology Trials Tuesday 28th January 2014 2 / 45

Outline

1 Phase I trials

4 Simulation study

Phase I Trials

A phase I trial is an exploratory study of a drug in humans for the first time.

In oncology, aim is to determine how safe certain doses of a new drug are in patients

Typical phase I trial for cytotoxic anti-cancer drug

Non-comparative, dose-escalation study, 15 − 50 patients (exhausted standard treatments)

Patients dosed sequentially (individuals or small groups)

Based on whether dose is deemed safe or not, change dose level for next patient/group

Aim is to find the Maximum Tolerated Dose (MTD).

Definition

MTD: The dose expected to produce some degree of medically unacceptable, dose-limiting toxicity (DLT)in a specified proportion of patients (e.g. 30%). [Babb and Rogatko, 2003]

Phase I Trials

Definition

Phase I Trials

Definition

Phase I Trials

Definition

Phase I Trials

Definition

Phase I Trials

Definition

Usually, a patient is said to have experienced a DLT if they have at least one grade 3 or higher toxicity,as defined by official guidelines [NCI, 2013].

DLT is usually recorded as a binary response Yi for patient i, where

1 if patient i has a DLT0 otherwise

For trials of cytotoxic anti-cancer drugs, we assume that as we increase the dose of a drug

the probability of a patient experiencing a DLT increases;

Therefore, our aim is to gradually increase the dose of a drug until we find a dose that we stronglybelieve will cause DLTs in, say, 30% of all patients that receive that dose.

Outline

1 Phase I trials

4 Simulation study

Dual-Agent Phase I Trials

In oncology, many chemotherapy regimens use combinations of drugs

We can conduct a dual-agent dose-escalation study to

explore the toxicity profile of two anti-cancer drugs given in combination;

identify one or more MTD combinations.

Regulators usually require single-agent phase I trials to have been conducted on each drug given as amonotherapy before a dual-agent trial can go ahead [FDA, 2010]

The most common approach (Standard Method) proceeds as follows:

Fix one agent at its MTD when used as monotherapy;

Use simple rule-based escalation (e.g. 3+3 method [Storer, 1989]) to escalate other drug.

Standard Method for Dual-Agent Phase I Trials

● ● ● ● ●

0 1 2 3 4

Drug A (dose level)

MTDBMTDB

MTDAMTDA

Figure 1: Example of the standard method for dual-agent phase I trials.

Fails to circumvent numerous pitfalls with the 3 + 3 escalation method:

Slow dose-escalation

Method does not target a specific probability of DLT [Lin and Shih, 2001]

Dose chosen as MTD often far less toxic than investigators expect it to be [Kang and Ahn, 2002]

Design has short-memory — only considers outcome of last cohort to guide dose escalation

Furthermore, fails to explore the surface of dose combinations [Harrington et al., 2013].

Fails to circumvent numerous pitfalls with the 3 + 3 escalation method:

Slow dose-escalation

Method does not target a specific probability of DLT [Lin and Shih, 2001]

Dose chosen as MTD often far less toxic than investigators expect it to be [Kang and Ahn, 2002]

Design has short-memory — only considers outcome of last cohort to guide dose escalation

Furthermore, fails to explore the surface of dose combinations [Harrington et al., 2013].

A far better approach to conducting a dual-agent phase I trial would

incorporate prior information about dose-toxicity relationships for each drug;

make dose-escalation decisions based on all available information;

explore surface of dose combinations;

choose one or more dose combinations for phase II testing from the MTD contour.

Dual-Agent Phase I TrialsA far better approach to conducting a dual-agent phase I trial would

Dose-Toxicity Surface

MTDContour

0 0.2 0.4 0.6 0.8 1Drug A (standardized dose)

Probabilityof DLT

Figure 2: Example of dose-toxicity surface for Drug A and Drug B (MTD Contour shown for target probability of 0.30).

incorporate prior information about dose-toxicity relationships;

These objectives can be achieved by modelling the dose-toxicity surface with a statistical model.

Specifically, model the probability of DLT and aim to solve the equation

Prob(DLT∣∣∣ dose) = 0.30.

However, dual-agent trials raise extra problems not present in single-agent trials:

How do we decide where to escalate to?

Furthermore, how do we decide where to de-escalate to?

How do we model the interactive behaviour between the two drugs?

Outline

1 Phase I trials

4 Simulation study

Dose-Toxicity Model

Let x = (x1, x2) be the dose combination of x1 units of drug A and x2 units of drug B (or standardizedunits).

We can model the probability of DLT at dose combination x by the six-parameter model

π((x1, x2), θ) =α1xβ1

1 + α2xβ22 + γ1(xβ1

1 xβ22 )γ2

1 + α1xβ11 + α2xβ2

2 + γ1(xβ11 xβ2

2 )γ2,

where θ = {α1, β1, α2, β2, γ1, γ2} are parameters > 0 [Thall et al., 2003].

When only one drug is given, this model reduces to a two-parameter logistic model, e.g.

π((x1, 0), θ) =α1xβ1

1 + α1xβ11

and π((0, x2), θ) =α2xβ2

1 + α2xβ22

Furthermore, this model is flexible enough to model many different interactions between the two drugs(synergistic, independent, antagonistic) [Gasparini, 2013].

Dose-Toxicity Model

π((x1, x2), θ) =α1xβ1

1 + α2xβ22 + γ1(xβ1

1 xβ22 )γ2

1 + α1xβ11 + α2xβ2

2 + γ1(xβ11 xβ2

2 )γ2,

π((x1, 0), θ) =α1xβ1

1 + α1xβ11

and π((0, x2), θ) =α2xβ2

1 + α2xβ22

Dose-Toxicity Model

π((x1, x2), θ) =α1xβ1

1 + α2xβ22 + γ1(xβ1

1 xβ22 )γ2

1 + α1xβ11 + α2xβ2

2 + γ1(xβ11 xβ2

2 )γ2,

π((x1, 0), θ) =α1xβ1

1 + α1xβ11

and π((0, x2), θ) =α2xβ2

1 + α2xβ22

Dose-Toxicity Model

π((x1, x2), θ) =α1xβ1

1 + α2xβ22 + γ1(xβ1

1 xβ22 )γ2

1 + α1xβ11 + α2xβ2

2 + γ1(xβ11 xβ2

2 )γ2,

π((x1, 0), θ) =α1xβ1

1 + α1xβ11

and π((0, x2), θ) =α2xβ2

1 + α2xβ22

Dose-Toxicity Model

π((x1, x2), θ) =α1xβ1

1 + α2xβ22 + γ1(xβ1

1 xβ22 )γ2

1 + α1xβ11 + α2xβ2

2 + γ1(xβ11 xβ2

2 )γ2,

π((x1, 0), θ) =α1xβ1

1 + α1xβ11

and π((0, x2), θ) =α2xβ2

1 + α2xβ22

Dose-Toxicity Model

From past single-agent trials and clinical opinion, construct informative priors for marginal parameters{α1, β1} and {α2, β2} [Bailey et al., 2009, Legedza and Ibrahim, 2001, Thall et al., 2003].

Assume vague prior beliefs for interaction parameters {γ1, γ2}.

Let Dn be the set of dose combinations and DLT responses for patients 1, . . . , n.

L(θ | Dn) =n∏

π(x(i), θ)yi (1 − π(x(i), θ))1−yi

Form posterior belief of the model parameters (and thus the dose-toxicity surface)

p(θ | Dn) =L(θ | Dn)p(θ)∫θ

L(θ | Dn)p(θ) dθ.

Given a posterior belief of the shape of the dose-toxicity surface and MTD contour, how do we choosewhich combinations to give to the next cohort of patients?

Dose-Toxicity Model

L(θ | Dn) =n∏

π(x(i), θ)yi (1 − π(x(i), θ))1−yi

Dose-Toxicity Model

L(θ | Dn) =n∏

π(x(i), θ)yi (1 − π(x(i), θ))1−yi

Dose-Toxicity Model

L(θ | Dn) =n∏

π(x(i), θ)yi (1 − π(x(i), θ))1−yi

Dose-Toxicity Model

L(θ | Dn) =n∏

π(x(i), θ)yi (1 − π(x(i), θ))1−yi

Dose-Toxicity Model

L(θ | Dn) =n∏

π(x(i), θ)yi (1 − π(x(i), θ))1−yi

Proposed Design

We need to consider the trade-off between

Patient gain — dose patients on the MTD contour

Population gain — maximize information about the shape of the dose-toxicity surface/MTD contour

However, most informative dose combinations may be well below (or worse, above) the MTD contour.

Aim for each cohort:

Dose the next cohort of patients at the set of most informative dose combinations, such that the dosecombinations chosen have estimated probability of DLT close to 0.30.

Proposed Design

Penalized D-optimal Design

Let ξc denote a set of c dose combinations x1, . . . , xc on the dose-toxicity surface.

The Cost of ξc quantifies how far dose combinations are from the estimated MTD contour. For example,

Cost of ξc = Ψ(ξc , θ) =c∑

(log-odds of DLT at xl − log-odds of DLT on MTD contour)2

(π(xl , θ)

1 − π(xl , θ)

)− log

1 − 0.30

(π(xl , θ)

1 − π(xl , θ)

)− log

1 − 0.30

(π(xl , θ)

1 − π(xl , θ)

)− log

1 − 0.30

Cost Function for Cohort of Two Patients

0.0 0.2 0.4 0.6 0.8 1.0

Probability of DLT for patient n+1

0.0 0.2 0.4 0.6 0.8 1.0

Figure 3: Cost function for cohort of two patients.

For past patients 1, . . . , n, denote the set of previous n dose combinations as ξn

Denote the Past Fisher information matrix as

M(ξn , θ) =n∑

I(x(i), θ)

Similarly, denote the Fisher information matrix of ξc by

M(ξc , θ) =c∑

I(xl , θ)

Given posterior parameter distribution p(θ | Dn), dose next c patients at ξ∗ such that

ξ∗ = arg maxξc

{log det

{M(ξc , θ) + M(ξn , θ)

}− λΨ(ξc , θ)

}p(θ | Dn) dθ

= arg maxξc

{Posterior Mean of

{Information at ξc given past data − λ × Cost of ξc

Penalized D-optimal DesignFor past patients 1, . . . , n, denote the set of previous n dose combinations as ξn

M(ξn , θ) =n∑

I(x(i), θ)

M(ξc , θ) =c∑

I(xl , θ)

ξ∗ = arg maxξc

{log det

{M(ξc , θ) + M(ξn , θ)

}− λΨ(ξc , θ)

}p(θ | Dn) dθ

= arg maxξc

{Posterior Mean of

M(ξn , θ) =n∑

I(x(i), θ)

M(ξc , θ) =c∑

I(xl , θ)

ξ∗ = arg maxξc

{log det

{M(ξc , θ) + M(ξn , θ)

}− λΨ(ξc , θ)

}p(θ | Dn) dθ

= arg maxξc

{Posterior Mean of

M(ξn , θ) =n∑

I(x(i), θ)

M(ξc , θ) =c∑

I(xl , θ)

ξ∗ = arg maxξc

{log det

{M(ξc , θ) + M(ξn , θ)

}− λΨ(ξc , θ)

}p(θ | Dn) dθ

= arg maxξc

{Posterior Mean of

M(ξn , θ) =n∑

I(x(i), θ)

M(ξc , θ) =c∑

I(xl , θ)

ξ∗ = arg maxξc

{log det

{M(ξc , θ) + M(ξn , θ)

}− λΨ(ξc , θ)

}p(θ | Dn) dθ

= arg maxξc

{Posterior Mean of

ξ∗ = arg maxξc

{Posterior Mean of

The value of λ weights the importance of minimizing cost with respect to maximizing information.

Given a cost function and cost limit C, choose λ ≥ 0 such that

(Cost of ξ∗ given λ) = C.

We therefore consider both patient-gain and population-gain in our dose-escalation decisions.

ξ∗ = arg maxξc

{Posterior Mean of

ξ∗ = arg maxξc

{Posterior Mean of

ξ∗ = arg maxξc

{Posterior Mean of

Summary of method

Require a certain amount of trial data before employing penalized D-optimality so determinant ofM(ξc , θ) + M(ξn , θ) is non-zero

Use start-up dose-escalation stage on diagonal of dose-toxicity surface until enough data acquired

Similar to forbidding dose-skipping of untested doses [Goodman et al., 1995]...

At start of trial, only have small amount of dose-toxicity surface open to experimentation

As trial continues, open more of dose-toxicity surface up for experimentation

1) Obtain prior belief of the dose-toxicity surface

2) Dose cohorts of patients up the diagonal within constrained region as close to estimated MTDcontour as possible

3) Once enough data acquired, choose λ for penalized D-optimality stage given cost function andcost limit C

4) Dose future cohorts of patients using penalized D-optimality, subject to constrained region(members of the same cohort may receive different combinations)

5) Continue until all patients dosed

6) Identify MTD contour and therefore set of MTD combinations to potentially test in phase II.

Summary of method

Example Trial (60 patients)

Gemcitabine (Standardized Dose)

0.0 0.2 0.4 0.6 0.8 1.0

●●● 1 2

Diagonal

MTD Contour

No DLT

Figure 4: Illustration of proposed design for Drug A and Drug B - Cohort 1

0.0 0.2 0.4 0.6 0.8 1.0

●●

Diagonal

MTD Contour

No DLT

0.0 0.2 0.4 0.6 0.8 1.0

●●

Diagonal

MTD Contour

No DLT

0.0 0.2 0.4 0.6 0.8 1.0

●●

Diagonal

MTD Contour

No DLT

0.0 0.2 0.4 0.6 0.8 1.0

●●

Diagonal

MTD Contour

No DLT

0.0 0.2 0.4 0.6 0.8 1.0

●●

Diagonal

MTD Contour

No DLT

0.0 0.2 0.4 0.6 0.8 1.0

●●

Diagonal

MTD Contour

No DLT

0.0 0.2 0.4 0.6 0.8 1.0

●●

Diagonal

MTD Contour

No DLT

0.0 0.2 0.4 0.6 0.8 1.0

●●

●●●●

●●

●● ●●

●●

● ●●

●●●

●●

Diagonal

MTD Contour

No DLT

Figure 12: Illustration of proposed design for Drug A and Drug B - All patients.

Outline

1 Phase I trials

4 Simulation study

Simulation Study

Dual-agent phase I trial of gemcitabine and cyclophosphamide [Thall et al., 2003]

Use following prior distributions — all prior means equal truth

Distribution Parameter Shape Rate Mean Variance

Gemcitabine (marginal) Gammaα1 1.74 4.07 0.43 0.11β1 10.24 1.34 7.65 5.71

Cyclophosphamide (marginal) Gammaα2 2.32 5.42 0.43 0.08β2 15.24 1.95 7.80 3.99

Parameter µ σ Mean Variance

Interaction Log-normalγ1 -0.69 1.18 1.00 3.00γ2 -0.32 0.80 1.00 0.90

Table 1: Prior parameter distributions for simulation study.

Simulation StudyDual-agent phase I trial of gemcitabine and cyclophosphamide [Thall et al., 2003]

True Dose-Toxicity Surface

0.30.4

0.50.6MTD Contour

0 0.2 0.4 0.6 0.8 1Gemcitabine (standardized dose)

0.00.10.20.30.40.50.60.70.80.91.0

Probability DLT

Figure 13: True dose-toxicity surface (MTD contour marked in blue).

Simulation Study of Proposed DesignDual-agent phase I trial of gemcitabine and cyclophosphamide [Thall et al., 2003]Use following prior distributions — all prior means equal truth

Sample size of 60 patients, dosed in cohorts of two

Estimate λ after 8 patients dosed

Target probability of DLT is 0.30; Cost Limit of C = 0.50

Simulation Study of Proposed DesignDual-agent phase I trial of gemcitabine and cyclophosphamide [Thall et al., 2003]Use following prior distributions — all prior means equal truth

Sample size of 60 patients, dosed in cohorts of two

Estimate λ after 8 patients dosed

Target probability of DLT is 0.30; Cost Limit of C = 0.50

We consider three different methods that utilise D-optimality:

Bayesian Penalized D-Optimality (BPDO)

1) Exactly the approach discussed

Local Penalized D-Optimality (LPDO)

1) Exactly the approach discussed, but use plug-in estimates of the parameters for computations —computationally quicker, but no information on the uncertainty around the model parameters —similar to Dragalin et al. [2008], but fully sequential, non-composite design, with toxicity-onlyendpoint

Thall method Thall et al. [2003]

1) Doses 20 patients dosed on diagonal of dose-toxicity surface.

2) Estimates MTD contour and doses two patients at combination on the contour, but above thediagonal, that maximizes the information on parameters

3) Repeats procedure, but switches to portion of MTD contour below diagonal, then above, thenbelow...

500 trials simulated for each method using R and OpenBUGS

0.0 0.2 0.4 0.6 0.8 1.0

a) Thall Model

1.0 0.3

0.0 0.2 0.4 0.6 0.8 1.0

b) LPDO Model

1.0 0.3

0.0 0.2 0.4 0.6 0.8 1.0

c) BPDO Model

1.0 0.3

Figure 14: Density plot of experimentation for BPDO method (MTD contour marked in blue).

0.0 0.2 0.4 0.6 0.8 1.0

a) Thall Model

1.0 0.3

0.0 0.2 0.4 0.6 0.8 1.0

b) LPDO Model

1.0 0.3

0.0 0.2 0.4 0.6 0.8 1.0

c) BPDO Model

1.0 0.3

Figure 15: Density plot of experimentation for LPDO method (MTD contour marked in blue).

0.0 0.2 0.4 0.6 0.8 1.0

a) Thall Model

1.0 0.3

0.0 0.2 0.4 0.6 0.8 1.0

b) LPDO Model

1.0 0.3

0.0 0.2 0.4 0.6 0.8 1.0

c) BPDO Model

1.0 0.3

Figure 16: Density plot of experimentation for Thall’s method (MTD contour marked in blue).

Information and Cost

MethodTotal Relative RelativeCost Efficiency Information per Cost

Mean SE Mean SE Mean SEThall 5.92 0.006 1 - 1 -

LPDO 6.06 0.003 1.54 0.023 1.50 0.021BPDO 6.18 0.024 1.73 0.024 1.67 0.024

Table 3: Relative Efficiency and Relative Information per Cost for simulation study relative to method of Thall et al.[Thall et al., 2003].

Area Between True and Estimated MTD Contours

MethodMean Area Between Standard PercentilesTruth and Estimate Error 5% 50% 95%

Thall 0.0382 9.24 × 10−4 0.0133 0.0347 0.0773LPDO 0.0398 9.25 × 10−4 0.0139 0.0360 0.0793BPDO 0.0368 8.98 × 10−4 0.0127 0.0326 0.0761

Table 4: Differences between ABC distributions per method. Smaller figures imply better performance.

Outline

1 Phase I trials

4 Simulation study

Discussion

Penalized D-optimality is an ideal approach for dual-agent phase I dose-escalation studies

Considers patient gain and population gain simultaneously

Explores the surface of dose combinations rather than a forced linear escalation

Using Bayesian methods, can update information accrued during trial with ease and incorporateprior information from previous single-agent studies

Bayesian penalized D-optimality is better at estimating MTD contours, but doses more patients at doseswith higher probabilities of DLT than the Thall method.

Consider the design features with care, in particular:

How do we elicit a cost function and cost limit C from clinicians?

Number of patients in first-stage of dose-escalation?

Discussion

Future Work

Rather than estimating λ, restrict the dose-toxicity surface based on the cost function:

find dose combinations for the next cohort from the exclusive set of dose combinations that willsatisfy the cost constraint.

Bayesian Penalized D-optimality is very computationally intensive!

Need to produce faster code so more scenarios can be investigated thoroughly

Providing easy-to-use open-access code/software will allow for further exploration and improvement ofdesign features [Gonen, 2009, Jaki, 2013].

Future Work

Acknowledgements

Michael Sweeting and Adrian Mander(MRC Biostatistics Unit)

Hub for Trials Methodology Research(Medical Research Council)

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