ICODOE (Tian)

Estimation Efficiency in Continual ReassessmentMethod

— Optimal Design Theory in Dose-FindingProblems

Tian Tian Min Yang Lei Nie

ICODOE 2016

T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 1 / 43

Problem introduction

Under simple power model

Background of dose-finding studyNotations and definitionsCRM procedureOptimal designModels

Background

Target toxicity rate (p0) and the maximum tolerated dose(MTD)

Two widely used approaches for identifying MTD

3+3 approachContinual reassessment method (CRM)(O’Quigley et al., 1990)CRM is more efficient (simulation studies)





Motivation

Why is CRM more efficient?

Can we improve on CRM?





Description of the problem

Binary response Y at dose level x is modeled as:

Y ∼ Ber(φ(x, θ))

whereY =

{1 tox0 no-tox

Design question: How to assign dose levels to patients in trialsuch that we can identify MTD (x∗ : φ(x∗, θ) = p0) accurately?

Problem: Both φ and θ are unknown!





The CRM algorithm

(1) Assume model φ(x, θ) and a prior for θ based on preclinicalinformation;

(2) Calculate the mean of θ, and assign estimated MTD to thefirst entered patient(s);

(3) Collect data and update the posterior distribution of θ;(4) Calculate the posterior mean of θ and assign estimated MTD

to the next patient;(5) Repeat procedure (3) to (5) until some predetermined

stopping rule is met.

Key point: Assign each patient the dose level with correspondingtoxicity rate p0.





Optimal design in a clinical study

MTD = b(θ) = φ−1(θ, p0)

V (MTD) =(∂b(θ)∂θ

)(∑nj=1 Ij(θ, d)

)−1(∂b(θ)∂θ

)T

Choose a design ξ∗ that minimizes V (MTD)





Locally optimal design

Challenge: The variance-covariance matrix for a nonlinear modeldepends on the unknown parameter!

Solution: ”Locally” optimal design - based on the best guess ofthe unknown parameter.

Moreover, it fits the sequential design framework.





Models in CRM

Two models are dominated in CRM-based designs:

Simple power model:

p = φ(x, θ) = xexp(α)

Two-parameter logistic model:

p = φ(x, θ) = eα+βx

1 + eα+βx




Under two-parameter logistic model

Model setupOptimal designInterpret the resultSimulations

Model and information matrix

Dose-response model:

p = φ(x, α) = xexp(α), α ∈ R and 0 < x < 1. (1)

Let β = exp(α), under design ξ = {(xi, ωi), i = 1, ..., k}, theasymptotic variance for MTD is:

V =[(p

1/β0)(

ln p0)(− 1β2

)]2[ k∑i=1

ωixβi (log xi)2

1− xβi

]−1.

Maximize f(x) = xβ(log x)2

1−xβx=p1/β⇐===⇒ g(p) = p(log p)2

1−p






Optimal design for simple power model

TheoremUnder simple power model (1), for any parameter α ∈ R, regardlessof the target toxicity rate p0 set in the trial, the optimal designalways choose the next dose level with corresponding toxicity ratep, where p is the solution to equation log p− 2p+ 2 = 0.

Numerical approximation: p ≈ 0.203.






Interpret the result

Surprising facts:

(I). Theoretically optimal!

Most clinical studies set p0 at 0.2 – not only medicallyreasonable, but also (nearly) optimal from statistical pointof view.

(II). Counter-intuitive!

No matter what p0 is, optimal design always collect data atdose level with toxicity rate 0.203.






Relative efficiency under different target toxicity rate

Table: Relative efficiency under different target toxicity rate

p0 0.1 0.15 0.2 0.25 0.3 0.35Relative efficiency 0.910 0.981 0.999 0.990 0.960 0.916

Under the simple power model, as long as p0 is chosen from areasonable range (say, from 0.1 to 0.35), the standard CRMprocedure will generate an optimal design, or at least a nearlyoptimal design with negligible efficiency loss.






Simulation results

Table: Comparison of perforemance of standard CRM and optimal CRMwhen β = exp(α) = 1

p0 0.15 0.25 0.3standard CRM (0.533,0.181) (0.416,0.271) (0.408,0.309)optimal CRM (0.561,0.223) (0.434,0.228) (0.440,0.226)

Similar percentage of correctly identifying MTD

Lower percentage of toxicity occurrence for p0 > 0.2






Simulation results (Cont.)

Table: Comparison of perforemance of standard CRM and optimal CRMwhen β = exp(α) = 2









Simulation results (Cont.)

Table: Comparison of perforemance of standard CRM and optimal CRMwhen β = exp(α) = 0.5







Delayed response introduction

Model setupOptimal design

Model

Dose-response model:

p = φ(x, a) = eα+βx

1 + eα+βx (2)

whereβ > 0, α < log p0

1− p0and x > 0. (3)






Information matrix

Under design ξ = {(xi, ωi), i = 1, ..., k}, the information matrix forθ = (α, β)T is:

IY (θ) =k∑i=1

ωi(xi, θ)hT (xi, θ)

where

h(x, θ) =( e

α+βx2

1 + eα+βx ,xe

α+βx2

1 + eα+βx

)T.






MTD and c-optimality

We write MTD as a function of θ, i.e.,

MTDdef= η = b(θ) = 1

β

(log p0

1− p0− α

).

Optimality criterion: c-optimality.

Tool: Elfving’s geometric approach (1952).






Elfving set

-0.4 -0.2 0.2 0.4h1

-0.6

-0.4

-0.2

0.2

0.4

0.6

h2

AB

C

D

R

P QΘ1

Θ2

Figure: Elfving set of model (2) with parameter (α, β) = (−2, 2)T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 19 / 43





Optimal design for two-parameter logistic model

TheoremUnder logistic model (2) with the assumptions (3), for any0 < p0 < 0.5, the optimal design selects the next dose level withtarget toxicity rate p0.

Under two-parameter logistic model, when p0 is set in a reasonablerange (0,0.5), the standard CRM algorithm is exactly optimal!




Likelihood at each study point

Background and motivationNotations and definitionsModel the pseudo outcomes

A new problem

Before — Incorporate optimal design theory into dose-findingproblems under a standard setup.

Now — Problems encountered in real-life clinical trials.

e.g., Monitoring late-onset toxicities.






Real-life clinical studies

In radio-therapy trials, dose-limiting toxicities often occur longafter the treatment is finished (Coia et al., 1995; Cooper etal., 1995)In a trial treating patients with pancreatic cancer, the fullevaluation period was 9 weeks while the accrual rate was 1per week (Muler et al., 2004).In the area of molecularly targeted agents, among a total of445 patients in 36 trials, 57% of the grade 3 and 4 toxicitieswere late-onset (Postel-Vinay et al., 2011).

——— It’s taking too long...






Some current methods

Time-to-event CRM (Cheung and Chappell, 2000)The predicted-risk-of-toxicity method (Bekele et al., 2008)Escalation with overdose control design (Mauguen et al.,2011)Treated as missing data and use a modified EM algorithm(Yuan and Yin, 2011)Treated as missing data and use Bayesian data augmentation(Liu and Ning, 2013)

——– How to bring in some optimal-design ideas here?






Description of the problem

Entire evaluation time is T ;

K + 1 interim study time points, u0 = 0, u1, ..., uK−1,uK = T ;

Binary response Zj,k denotes the toxicity outcome for patientj by time uk.






Weight function

A weight function wk to depict the relationship between trueresponse by time uK = T and outcome by interim study time uk,k = 1, ...,K (Cheung and Chappell, 2000):

Pr(Tox by time uk) = wkPr(Tox by time uK)⇒wk = Pr(Tox by time uk|Tox by time uK)






Outcome by time uk

True responsesafter entire evaluation time uK = T

Outcomes after actual follow-up time uk Row totalNo-tox Tox

No-tox 1− φ(xj , θ) 0 1− φ(xj , θ)Tox (1− wk)φ(xj , θ) wkφ(xj , θ) φ(xj , θ)

Column total 1− wkφ(xj , θ) wkφ(xj , θ) 1

Table: Joint probabilities of outcomes by time uk and uK = T

Outcome for patient j by time uk follows the marginal distribution:

Pr(Zj,k = 1) = wkφ(xj , θ)Pr(Zj,k = 0) = 1− wkφ(xj , θ)






Probabilities for different events

Marginal qj,k = Pr(Zj,k = 0) = 1− wkφ(xj , θ)

Conditionalpj,k = Pr(Zj,k = 1|Zj,k−1 = 0) = wkφ(xj ,θ)−wk−1φ(xj ,θ)

1−wk−1φ(xj ,θ)

qj,k = Pr(Zj,k = 0|Zj,k−1 = 0) = 1−wkφ(xj ,θ)1−wk−1φ(xj ,θ)

Jointpj,k = Pr(Zj,k = 1, Zj,k−1 = 0) = wkφ(xj , θ)− wk−1φ(xj , θ)

qj,k = Pr(Zj,k = 0, Zj,k−1 = 0) = Pr(Zj,k = 0) = 1− wkφ(xj , θ)

Relationpj,k · qj,k−1 = pj,k

qj,k · qj,k−1 = qj,k

Table: Probabilities for marginal/conditional/joint events




Preliminary results

Categorize patientsInformation matrices

Patient categories

At a certain interim study time point, there are three kinds ofpatients involved in the experiment:

Already been fully evaluated X1;

Not been fully evaluated yet X2;

Newly enrolled with dose assignments to be decided X3.




Preliminary results


Fully evaluated patients

K + 1 groups: X 11 ,...,XK1 , and XK+1

1 .

(1). For xj ∈ X k1 , k = 1, ...,K

Observed event:

{Zj,k = 1, Zj,l = 0, l < k} = {Zj,k = 1, Zj,k−1 = 0};

Contribution to the likelihood:

pj,k = Pr(Zj,k = 1, Zj,k−1 = 0) = wkφ(xj , θ)− wk−1φ(xj , θ).




Preliminary results


Fully evaluated patients (Cont.)

(2). For xj ∈ XK+11

Observed event:

{Zj,l = 0, l ≤ K} = {Zj,K = 0};


qj,K = Pr(Zj,K = 0) = 1− wKφ(xj , θ)wK=1===== 1− φ(xj , θ).




Preliminary results


Patients with delayed responses

K − 1 groups: X 12 ,...,XK−1

2 .

For xk ∈ X k2 , the random outcome by the next interim study pointis either {Zj,k+1 = 1, Zj,l = 0, l < k + 1} = {Zj,k+1 = 1, Zj,k = 0}or {Zj,l = 0, l ≤ k + 1} = {Zj,k+1 = 0}.

Here we introduce a new ”conditional random variable”,

Zj,k := Zj,k|Zj,k−1 = 0,

to depict the random outcome by the next interim study point,given the observed outcome at the current time.We have Zj,k+1 ∼ Ber(pj,k+1).




Preliminary results


Patients with delayed responses (Cont.)


qj,k · pZj,k+1j,k+1 q

1−Zj,k+1j,k+1

which can be interpreted as follows, observed part: Zj,k = 0 contribution========⇒ qj,k

random part: Zj,k+1 ∼ Ber(pj,k+1) contribution========⇒ pZj,k+1j,k+1 q

1−Zj,k+1j,k+1 .




Preliminary results


Newly enrolled patients

D groups: X 12 ,...,XD2 .

For xj ∈ X d3 , d = 1, ..., D, the random outcome by the nextinterim study point Zj,1 ∼ Ber(pd,1); and its contribution to thelikelihood is given by

pZj,1d,1 q

1−Zj,1d,1 = (w1φ(d, α)− w0φ(d, α))Zj,1(1− w1φ(d, α))1−Zj,1

w0=0===== (w1φ(d, α))Zj,1(1− w1φ(d, α))1−Zj,1 .




Preliminary results


Information matrices for simple power model

Ip1 =∑K+1k=1

∑x∈Xk1

(log x)2xexp(α)

(1− xexp(α))2 exp2(α).

Ip2 =∑Kk=2

∑x∈Xk−1

2

wk(log x)2xexp(α)

(1− wk−1xexp(α))(1− wkxexp(α))

exp2(α).

Ip3 =∑Dd=1 ωd ×

w1(log xd)2xexp(α)d

1− w1xexp(α)d

exp2(α).

under design ξ = {(xd, ωd), d = 1, , , D}.




Preliminary results


Information matrices for logistic model

Let c = exp(α+ βx),

Il1 =∑K+1k=1

∑x∈Xk1

c(1 + c2)

(1 xx x2

)

Il2 =∑Kk=2

∑x∈Xk2

[c(1 + c)(wk − wk−1c) + wk−1wkc2]

(1 + (1− wk−1)c)(1 + (1− wk)c)(1 + c)2

(1 xx x2

)

Il3 =∑Dd=1 ωd ×

w1cd(1 + cd)2(1 + (1− w1)cd)

(1 xdxd x2

d

)under design ξ = {(xd, ωd), d = 1, , , D}.



Preliminary results

Ongoing work

Weight function estimationOn simple power modelOn two-parameter logistic model

Weight function estimation

Two methods:

Linear and time-weighted estimation (Cheung and Chappell,2000).

wk = ukT.

Bayesian estimation (Ji and Bekele, 2009).



Preliminary results

Ongoing work


Bayesian estimation for wk

Conjugacy ⇒ wk ∼ Beta(α0k +

∑kl=1 n

toxl , β0k + ntox −

∑kl=1 n

toxl)

Posterior mean:

wk =α0k +

k∑l=1

ntoxl

α0k + β0

k + ntox.



Preliminary results

Ongoing work


Optimal result on simple power model

TheoremUnder simple power model (1), for any parameter α ∈ R, theoptimal design chooses the next dose level with correspondingtoxicity rate p, where p is the solution to equationlog p− 2w1p+ 2 = 0.

Notice that the optimal dose level depends on the estimate of thefirst-stage weight function w1.



Preliminary results

Ongoing work


Analytical D-optimality result on logistic model

TheoremAt the first stage, under logistic model (2), for any given doserange c = α+ βx ∈ [A,B] ⊆ [−10,− log 2], the D-optimal doseassignment is ξlopt = {(c∗, 1/2), (B, 1/2)}; and the choice of c∗would be one of the following two cases:

When 1− d2(A) < w1 and B > A+ d1(A), c∗ ∈ (A,B) is thesolution to equation f l(c) = 0.Otherwise, c∗ = A.

Notice that the optimal dose level here also depends on theestimate of the first-stage weight function w1.



Preliminary results

Ongoing work


Analytical D-optimality result on logistic model (Cont.)

Specifically,

f l(c) = 2(1− w1)(c−B − 1) exp(2c)

+ (c−B − 2− 2(1− w1)) exp(c) +B − c− 2

d1(A) = 2(1 + exp(A))(1 + (1− w1) exp(A))(1− exp(A)− 2(1− w1) exp(2A))

d2(A) = (2− log 2−A) exp(A) + 2 + log 2 +A

2 exp(A)[(A+ log 2− 1) exp(A)− 1]


Preliminary results

Ongoing work

Thank you

Current algorithm structureFuture work

Current algorithm structure

As for the situations where analytical results are challenging, weget help from algorithms.

In particular, here we adopt the optimal weight exchangingalgorithm (OWEA; Yang, Biedermann, and Tang, 2013).

Different optimality criteriaMulti-stage designParameter estimation (MLE/posterior mean)Weight function estimationDifferent targets, i.e., b(θ).


Preliminary results

Ongoing work

Thank you

Current algorithm structureFuture work

Existing problem and future work

Existing problem:

Restrained to situations where the assumed model is close to thetrue model.

Future research:

Incorporate the idea of optimal design into dose-finding studieswhere there exist model selection/averaging problem.


Ongoing work

Thank You


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