Twostage dose finding for cytostatic agents in phase I ...download.xuebalib.com/xuebalib.com.7566.pdf · may beneﬁt from cytostatic agents even without observing tumor shrinkage,

Research Article

Received 13 October 2011, Accepted 24 May 2012 Published online 1 August 2012 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/sim.5546

Two-stage dose finding for cytostaticagents in phase I oncology trialsGuosheng Yin,a,b*† Shurong Zhengc and Jiajing Xua

Conventional dose-finding methods in oncology are mainly developed for cytotoxic agents with the aim of findingthe maximum tolerated dose. In phase I clinical trials with cytostatic agents, such as targeted therapies, designswith toxicity endpoints alone may not work well. For cytostatic agents, the goal is often to find the most efficaciousdose that is still tolerable, although these agents are typically less toxic than cytotoxic agents and their efficacymay not monotonically increase with the dose. To effectively differentiate doses for cytostatic agents, we developa two-stage dose-finding procedure by first identifying the toxicity upper bound of the searching range throughdose escalation and then determining the most efficacious dose through dose de-escalation while toxicity is con-tinuously monitored. In oncology, treatment efficacy often takes a relatively long period to exhibit comparedwith toxicity. To accommodate such delayed response, we model the time to the efficacy event by redistributingthe mass of the censored observation to the right and compute the fractional contribution of the censored data.We evaluate the operating characteristics of the new dose-finding design for cytostatic agents and demonstrateits satisfactory performance through simulation studies. Copyright © 2012 John Wiley & Sons, Ltd.

Keywords: adaptive design; censored data; dose escalation; dose de-escalation; efficacy; Kaplan–Meierestimator; delayed response; time to event; toxicity

1. Introduction

In oncology studies with cytotoxic agents, the primary objective of a phase I clinical trial is to identify themaximum tolerated dose (MTD). The MTD is typically defined as the highest dose with an acceptablelevel of toxicity, and the target toxicity rate is often prespecified by investigators. Many authors have pro-posed phase I dose-finding methods in the literature [1, 2]. In particular, the 3C 3 design is widely usedin practice because of its ease of understanding and implementation [3]. The 3C 3 design strictly fol-lows a set of dose-assignment rules, which can be viewed as a model-free, curve-free, or nonparametricmethod. Another algorithm-based design is the biased coin design, which uses a family of random walkrules [4]. These dose-finding methods are typically robust because they do not explicitly impose anyparametric model structure. However, nonparametric approaches tend to be less efficient as no strengthor information is borrowed across different doses under investigation. For a more efficient dose-findingprocedure, parametric models can be adopted such as the continual reassessment method, which has beenextensively studied to improve its practical performance [5–8]. However, such model-based designs mayinduce biased estimation in the case of model misspecification.

Although various nonparametric, parametric, or hybrid methods have been developed for dose finding,most of them only work for the conventional cytotoxic agents [9]. For example, chemotherapy directlydamages or destroys rapidly growing cancer cells and leads to dose-dependent tumor shrinkage: ahigher dose results in more shrinkage of the tumor, such as carboplatin and paclitaxel in lung cancer.Nevertheless, in the development of personalized medicine, targeted agents are often cytostatic, actingon molecular targets to inhibit tumor growth or prevent the proliferation of cancer cells [10]. Patientsmay benefit from cytostatic agents even without observing tumor shrinkage, and more importantly, lower

aDepartment of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong KongbDepartment of Biostatistics, The University of Texas: M. D. Anderson Cancer Center, Houston, TX 77030, U.S.A.cSchool of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, China 130024*Correspondence to: Guosheng Yin, Department of Statistics and Actuarial Science, The University of Hong Kong, PokfulamRoad, Hong Kong.

†E-mail: [email protected]

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Copyright © 2012 John Wiley & Sons, Ltd. Statist. Med. 2013, 32 644–660

G. YIN, S. ZHENG AND J. XU

doses of such agents may be as effective or even more effective than higher doses. For example, amonoclonal antibody trastuzumab and a tyrosine kinase inhibitor lapatinib specifically target humanepidermal growth factor receptor 2 positive in breast cancer. On the other hand, gefitinib prevents lungcancer cells from growing and multiplying by targeting the epidermal growth factor receptor, which isoften characterized by the disruption of epidermal growth factor receptor signal transduction for cell divi-sion, apoptosis, and angiogenesis. In contrast to cytotoxic agents, cytostatic agents work on cancer cellsfrom very different perspectives. Cytostatic agents are generally less toxic than cytotoxic agents. Insteadof using toxicity as the endpoint, efficacy would be more relevant for cytostatic agents. Nevertheless, theefficacy effect may not necessarily increase with the dose. The goal of dose finding with cytostatic agentsis to search for the optimum biological dose (OBD) that has the highest effectiveness as well as tolerabletoxicity. When searching for the MTD with cytotoxic agents, the investigator specifies a toxicity target,say, the maximum toxicity rate cannot exceed 30%. But for the OBD, there is typically no such prespec-ified efficacy target, as the aim is simply to find the most efficacious dose that is still tolerable. Thesekey differences between the two types of agents call for very different trial designs. Toward this goal,Hunsberger et al. proposed two dose-finding approaches by mimicking the 3C 3 design for molecularlytargeted agents that have little or no toxicity in the therapeutic dose range [11].

For cytotoxic agents in oncology, treatment efficacy or any favorable responsive activity is examinedin phase II trials after the MTD is identified in phase I trials [12]. There has been increasing interest in thedevelopment of dose-finding methodologies by jointly evaluating toxicity and efficacy [13–17]. Seam-lessly combining phase I and phase II clinical trials has many advantages over separately conductingthese two phases: (i) speeding up the drug development process; (ii) improving the dose-finding proce-dure by increasing the drug’s efficacy as well as controlling its toxicity; and (iii) enlarging the samplesize by pooling patients in phase I and phase II trials to produce more reliable estimates of toxicity andefficacy than would be achieved in each separate trial. Most of the phase I/II designs assume parametricmodels for the relationship between the toxicity–efficacy outcomes and the doses. In particular, Thalland Cook studied the trade-off contour by considering the toxicity and efficacy outcomes simultane-ously [16]. Yin et al. proposed a single-arm Bayesian adaptive phase I/II design based on toxicity andefficacy odds ratio trade-offs [17]. However, these phase I/II dose-finding designs have been developedfor cytotoxic agents in single-arm trials. For cytostatic agents, phase I trials are still single-arm focusingon efficacy effects, whereas phase II trials are often randomized with multiple treatment arms [18, 19].As a result, combining phase I and phase II trials with cytostatic agents would not be straightforward.

Our research is motivated by an open-label, multicenter, phase I trial to evaluate safety and activity ofan antibody in patients with metastatic or locally advanced malignant tumors. This antibody is expectedto be tolerable at a wide range of doses, although toxicity still needs to be carefully monitored forextreme cases. Adverse events are graded according to the National Cancer Institute Common Termi-nology Criteria for Adverse Events, CTCAE v3.0. For patients treated with cytostatic agents, toxicityoutcomes, if any, can typically be observed shortly after the treatment. However, efficacy may take arelatively long period to exhibit, for example, inhibiting tumor growth or stable disease. In the targetedtherapy development, often some surrogate or biomarker measurements may be used to characterize theefficacy endpoint. Such endpoint should be relevant to disease progression and also must be able to char-acterize patient response convincingly. It is possible to use a pharmacodynamic measure (i.e., inhibitingthe target or not) as an endpoint. For example, with vascular endothelial growth factor inhibitors, bloodpressure can be an indicator of whether the drug is hitting the target. Indeed, many cytostatic agentshave some type of pharmacodynamic biomarker indicating whether the compound is hitting the desiredtarget. For a broad range of cytostatic agents, we propose a two-stage dose-finding method. Stage 1takes dose escalation to determine the upper dose-searching bound by solely monitoring toxicity, fromwhich point stage 2 starts dose de-escalation to find the OBD within the tolerable dose range. In stage 2,dose escalation or de-escalation is mainly based on efficacy outcomes, although toxicity is also con-tinuously monitored for extreme cases. Because of possible late-onset efficacy, we model efficacy as atime-to-event endpoint and fractionize the censored efficacy event by redistributing the point mass to theright. Such a redistribution scheme can resolve the issues caused by delayed response and thus facilitateimmediate dose assignment upon each new cohort’s arrival.

We organize the remainder of the article as follows. In Section 2, we propose the two-stage designfor dose finding with cytostatic agents, with a particular emphasis on delayed efficacy outcomes. InSection 3, we present extensive simulation studies to examine the operating characteristics of the newdesign and also describe the sensitivity analysis to further investigate the new design’s properties. Weconclude with a brief discussion in Section 4.


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2. Cytostatic two-stage design

2.1. Toxicity and efficacy endpoints

In the targeted therapy development, toxicity is typically not the primary concern because most of thecytostatic agents are tolerable. Thus, phase I designs with cytostatic agents often focus on finding theOBD instead of the MTD. For cytotoxic agents, the dose-finding trial has a specific toxicity rate toachieve so that the identified dose should have a toxicity probability closest to but not exceeding thisupper toxicity bound. In contrast, we do not have such a target rate of efficacy for cytostatic dose find-ing, although the goal is simply to find the dose that has the maximum efficacy level while still tolerable.For cytostatic agents, phase I trials examine efficacy in addition to toxicity, although these phase I trialscannot replace the subsequent phase II studies. Once the OBD is determined in a single-arm phase I trial,a randomized multi-arm phase II trial is typically carried out to provide a more objective comparison withthe standard treatment.

In phase I trials with cytostatic agents, we monitor toxicity throughout the dose-finding process,although this type of agents are in general tolerable. For j D 1; : : : ; J , let pj and qj denote the probabil-ities of toxicity and efficacy associated with dose level j , respectively. For toxicity, one usually assumesa monotonically increasing relationship between pj and the dose level; that is, p1 < � � � < pJ . In con-trast, efficacy could stay at the same level or even decrease as the dose increases, and thus there is nosuch a monotonic constraint imposed for qj . Let Xij denote the toxicity outcome for subject i treated atdose level j , and let Yij denote the efficacy outcome of the same subject,

Xij D

�1 with probability pj0 with probability 1� pj ;

Yij D

�1 with probability qj0 with probability 1� qj :

In oncology, toxicity outcomes can often be observed shortly after the treatment, whereas efficacy out-comes are expected to be observable after a relatively longer period of follow-up. Such delayed responsemay cause difficulty in dose assignment, because some of the currently treated patients may still be wait-ing for their potential responses when a new cohort of patients enters the trial. For efficacy assessment,physicians often specify an evaluation window .0; �/, in which patients are expected to respond if theywill. If a patient has responded in .0; �/, we take the efficacy outcome y D 1; if after time � , a patientstill has not responded, we take y D 0; and in the middle of .0; �/, if a patient has not responded, y is notobserved and the time to efficacy is censored. It may happen that a patient has progressive disease duringthe follow-up, and then the patient would be taken off the study, and we take y D 0 (no response). Duringthe trial conduct, a cohort of patients is accrued every time interval a. The efficacy assessment period �is typically longer than a, say, a is 1 month and � is 3 months. By the time a new cohort is ready fortreatment, some of the patients in the trial may have been partially followed, and their efficacy outcomeshave not yet been observed. That is, the information needed for dose assignment is not available or onlypartially available.

2.2. Redistribution to the right for censored efficacy data

Viewing the patient response as an event, we can model late-onset efficacy as the time-to-event data. Ina more rigorous derivation, let ti denote the time to response for subject i , and let ui .ui 6 �/ denote theactual follow-up time. If we observe an efficacy event before time � , then yi D 1; and if by ui D � , thepatient still has not responded, then yi D 0. The difficulty arises when the efficacy outcome is censoredfor patients who have not responded (ui < ti ) and have not been fully followed up to time � (ui < � ).Figure 1 can illustrate this phenomenon. Three patients are enrolled within each time interval a; by thetime the fourth patient enters the trial, patient 1 has achieved a response (y1 D 1), whereas patients 2 and3 are not yet fully followed, and thus their efficacy outcomes are censored. In fact, during the remainingfollow-up, patient 2 also responded (y2 D 1), but patient 3 did not respond (y3 D 0). For censoredefficacy data, we can redistribute the mass of the censored observation to the right by reformulatingthe usual Kaplan–Meier estimator of the survival function as a self-consistent estimator [20–22]. If weobserve a censored efficacy event, one part of the weight is assigned to the censored point, and the otherpart is assigned to infinity (or somewhere that is larger than � ). When we calculate the probability ofefficacy, the censored observation would contribute some fraction of 1 to the proportion of responses.In our case, only those weights assigned to the points within .0; �/ count and those fractions assigned

646



0 a 2a 3a

Patient 3

Patient 2

Patient 1

Patient 6

Patient 5

Patient 4

Efficacy evaluation window

Time

Figure 1. Late-onset efficacy: the horizontal line segment represents the follow-up time, along which the efficacyevent is indicated by a cross and censoring by a circle. Each cohort arrives every a time interval and is followed

for a period of � .

beyond � are disregarded. More specifically, if subject i is censored by the decision-making time, wetake his or her fractional contribution to the overall response as

Pr.ti < � jti > ui ; í /DPr.ui < ti < � jí /

Pr.ti > ui jí /;

where í is the dose that patient i has been treated with. The weight or the fraction has a meaningfulinterpretation as a conditional probability of achieving a response within .ui ; �/ given that the subjecthas not responded till time ui . We can estimate the fractional contribution for a censored efficacyoutcome with

Oyi DOS.ui jí /� OS.� jí /

OS.ui jí /; (1)

where OS.�jí / is the estimated survival function given the dose information í . In particular, suppose thatthere are n patients enrolled in the trial thus far. For subject i , i D 1; : : : ; n, the censoring indicator �itakes a value of 1 if we have observed the response, otherwise�i D 0. Following kernel-based nonpara-metric estimation for the survival function [23, 24], let K.�/ denote a kernel density function and let hndenote the bandwidth satisfying hn! 0 as n!1. We may take a local Kaplan–Meier estimator [25]

OS.t jí /D

nYjD1

�1�

Wj .í /PnkD1 I.uk > uj /Wk.í /

�Nj .t/; (2)

where Nj .t/D I.uj 6 t; �j D 1/ and

Wj .í /DKf.í � ´j /=hngPnkD1Kf.í � ´k/=hng

:

If Wj .í /D 1=n, (2) reduces to the usual Kaplan–Meier estimator. In the calculation of (2), we treatpatients without experiencing the event in .0; �/, that is, those with yi D 0, as censored subjects at � .The usual independent censoring assumption is automatically satisfied because of patients’ staggeredentry. The purpose of using the local Kaplan–Meier estimator is to incorporate the dose informationwithout imposing any parametric model structure, such that the trial design is robust to the underlyingtime-to-event distributions.

At each dose level, we can estimate the response probability by the sample proportion. Not only dowe need to count all the patients with observed responses but also those censored observations with


647


some fractions of 1. Suppose that nj patients have been treated at dose level j , then the estimate for thecorresponding response probability is given by

Oqj D

PnjiD1 Oyi.j /

nj;

where Oyi.j / D 1 for responders, Oyi.j / D 0 for nonresponders, and Oyi.j / takes the form of (1) for censoredpatients.

2.3. Dose-finding algorithm

The dose-finding procedure can be divided into two sequential stages, in which we collect both toxicityand efficacy data throughout. Stage 1 adopts the 3C 3 design to locate the MTD, which is completelybased on the toxicity data. Starting from the MTD, stage 2 mainly uses the efficacy data to search for themost efficacious dose within the set of admissible doses, A, which contains all the doses satisfying

Opj < �T and Oqj > �E ; j D 1; : : : ; J; (3)

where �T and �E are the physician-specified upper toxicity and lower efficacy limits, respectively. Doseescalation starts from the lowest dose level with a cohort size of 3. Suppose that the current dose level isj and three patients are treated, the two-stage dose-finding procedure proceeds as follows.

Stage 1

(1) If none of the three patients experiences the dose-limiting toxicity (DLT), escalate to dose levelj C 1.

(2) If one of the three patients develops the DLT, then three more patients will be treated at the samedose level j .- If one of the six patients experiences the DLT, escalate to dose level j C 1 provided that fewer

than six patients have been treated at that dose level.- If two of the six patients experience the DLT, then the trial is finished, and the dose at the next

lower level j � 1 is declared as the MTD.- If more than two patients experience the DLT, the current dose level j has exceeded the MTD,

and three more patients will be treated at dose level j � 1 provided that fewer than six patientshave been treated at that dose level.

(3) If two or three patients experience the DLT (the current dose level j has exceeded the MTD), threemore patients will be treated at dose level j � 1 provided that fewer than six patients have beentreated at that dose level.

The MTD is defined as the highest dose at which six patients are treated and 1 DLT has been observed,or three patients are treated and no one has experienced the DLT. Suppose that the MTD identified instage 1 is at dose level j �, which will be the upper bound of the admissible dose levels. From then on,we restrict that the doses used to treat patients must be chosen from the admissible set A as defined in(3). If A is empty, the trial should be stopped early without recommending any dose.

Stage 2

(1) Starting from dose level j �, we examine the efficacy effects of all the doses belonging to A andassign the next cohort of patients to the dose that has the highest efficacy rate. If several doses aretied in terms of the efficacy rate, we choose the highest dose.

(2) When the maximum sample size is exhausted, we recommend the most effective dose chosen fromthe admissible set A. If there is a tie in efficacy, we recommend the highest dose.

In summary, stage 1 escalates the dose forward and mainly sets the upper bound for the dose searchingrange by examining toxicity only, and stage 2 searches for the most efficacious dose backward withinthe admissible dose set. Through such adaptive dose movement, we can quickly find the optimal dose interms of efficacy, which is also tolerable.

3. Simulation studies

We conducted extensive simulation studies to examine the operating characteristics of the proposedtwo-stage dose-finding design. In a typical oncology trial, toxicity is expected to occur quickly aftertreatment, whereas efficacy effects may take a relatively long period to exhibit. If we denote �T and �E

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npe

rcen

tage

ofdo

sele

vel(

Sel.%

)N

o.of

No.

ofT

rial

Dos

e1

23

45

6N

one

toxi

city

effic

acy

dura

tion

Scen

ario

1(0

.02,

0.40

)(0

.03,

0.25

)(0

.05,

0.20

)(0

.06,

0.15

)(0

.07,

0.10

)(0

.07,

0.05

)C

enso

red:

Sel.%

64.8

16.8

9.2

4.4

1.9

0.5

2.4

2.2

16.5

23.8

.Oq>0:2/%

75.2

46.5

32.4

20.9

10.8

4.1

No.

ofpa

tient

s27

.010

.77.

85.

84.

53.

7C

ompl

ete:

Sel.%

65.3

18.6

9.2

3.3

1.1

0.3

2.2

2.2

16.3

48.3

.Oq>0:2/%

73.9

45.1

32.1

19.5

9.5

3.6

No.

ofpa

tient

s25

.611

.18.

46.

14.

63.

7

Scen

ario

2(0

.02,

0.15

)(0

.04,

0.45

)(0

.07,

0.30

)(0

.09,

0.25

)(0

.10,

0.16

)(0

.15,

0.10

)C

enso

red:

Sel.%

2.0

66.1

17.1

9.1

2.6

0.8

2.5

3.7

18.9

23.7

.Oq>0:2/%

27.0

79.0

54.8

40.2

19.8

8.8

No.

ofpa

tient

s5.

426

.411

.17.

84.

93.

7C

ompl

ete:

Sel.%

2.3

64.0

18.4

10.1

2.2

0.5

2.5

3.7

18.7

47.8

.Oq>0:2/%

27.3

78.2

52.5

39.9

20.4

8.0

No.

ofpa

tient

s4.

925

.411

.38.

75.

23.

8

Scen

ario

3(0

.01,

0.10

)(0

.02,

0.15

)(0

.03,

0.35

)(0

.04,

0.18

)(0

.05,

0.12

)(0

.06,

0.07

)C

enso

red:

Sel.%

2.0

6.8

64.7

12.9

4.9

1.6

7.1

2.0

13.3

23.4

.Oq>0:2/%

9.4

18.8

68.9

25.4

11.2

4.6

No.

ofpa

tient

s5.

47.

525

.59.

26.

44.

8C

ompl

ete:

Sel.%

2.3

6.6

67.6

11.9

3.9

0.9

6.8

2.0

13.6

48.0

.Oq>0:2/%

10.6

20.3

70.4

23.9

11.2

4.1

No.

ofpa

tient

s5.

06.

826

.69.

46.

44.

7

Scen

ario

4(0

.01,

0.05

)(0

.02,

0.15

)(0

.04,

0.30

)(0

.06,

0.45

)(0

.08,

0.35

)(0

.10,

0.30

)C

enso

red:

Sel.%

0.1

1.6

12.6

51.0

20.6

11.2

2.7

3.6

19.6

23.7

.Oq>0:2/%

9.0

29.7

58.1

76.1

58.9

44.5

No.

ofpa

tient

s3.

54.

99.

821

.011

.88.

3C

ompl

ete:

Sel.%

0.1

1.6

12.4

49.5

21.7

12.0

2.6

3.7

19.7

47.9

.Oq>0:2/%

9.4

30.2

58.9

75.6

57.9

45.5

No.

ofpa

tient

s3.

44.

58.

920

.512

.59.

5


651


Tabl

eII

.C

onti

nued

.

Sele

ctio

npe

rcen

tage

ofdo

sele

vel(

Sel.%

)N

o.of

No.

ofT

rial

Dos

e1

23

45

6N

one

toxi

city

effic

acy

dura

tion

Scen

ario

5(0

.01,

0.15

)(0

.02,

0.25

)(0

.03,

0.33

)(0

.05,

0.47

)(0

.06,

0.60

)(0

.07,

0.40

)C

enso

red:

Sel.%

0.5

2.3

5.5

21.1

58.3

11.3

0.9

2.9

27.0

23.9

.Oq>0:2/%

36.6

55.5

66.4

80.7

86.3

66.7

No.

ofpa

tient

s3.

95.

26.

612

.223

.28.

5C

ompl

ete:

Sel.%

0.4

2.0

5.4

20.2

58.1

12.8

1.1

3.0

27.2

48.6

.Oq>0:2/%

35.0

56.0

67.7

81.0

86.6

67.1

No.

ofpa

tient

s3.

64.

66.

211

.624

.19.

4

Scen

ario

6(0

.01,

0.05

)(0

.02,

0.15

)(0

.03,

0.30

)(0

.04,

0.35

)(0

.05,

0.40

)(0

.06,

0.50

)C

enso

red:

Sel.%

0.1

0.9

7.9

13.0

22.5

54.0

1.7

2.7

22.3

23.8

.Oq>0:2/%

10.6

33.5

60.4

66.7

71.5

77.6

No.

ofpa

tient

s3.

44.

37.

89.

812

.521

.7C

ompl

ete:

Sel.%

0.1

0.8

7.1

12.4

20.8

56.9

1.9

2.7

22.7

48.4

.Oq>0:2/%

10.4

33.0

61.2

68.1

72.2

77.8

No.

ofpa

tient

s3.

34.

07.

09.

012

.024

.1

Scen

ario

7(0

.05,

0.01

)(0

.10,

0.02

)(0

.16,

0.03

)(0

.35,

0.05

)(0

.40,

0.07

)(0

.50,

0.09

)C

enso

red:

Sel.%

0.1

0.0

0.0

0.1

0.0

0.0

99.8

4.2

0.8

10.2

.Oq>0:2/%

0.0

0.0

0.0

0.0

0.0

0.0

No.

ofpa

tient

s6.

56.

25.

74.

21.

60.

5C

ompl

ete:

Sel.%

0.0

0.0

0.0

0.0

0.0

0.0

99.9

4.3

0.7

15.5

.Oq>0:2/%

0.0

0.0

0.0

0.0

0.0

0.0

No.

ofpa

tient

s6.

56.

25.

74.

21.

70.

5

The

targ

etdo

seis

inbo

ldfa

ce,a

ndth

ebi

vari

ate

log-

norm

aldi

stri

butio

nfo

rtim

esto

toxi

city

and

effic

acy

has

the

mar

gina

lvar

ianc

e�2D1

and

corr

elat

ion

coef

ficie

nt�D0:2

.

652



as the expected time windows for toxicity and efficacy events to occur, respectively, �T would be muchsmaller than �E . In our simulation studies, we set �T D 1 and �E D 3 months. The patient accrual ratewas three patients per month. Every month, we expected the toxicity outcomes of treated patients to befully observed, whereas efficacy outcomes would not be completely determined as the efficacy evalua-tion window was much longer than 1 month. In other words, patient accrual was about at the same paceas toxicity evaluation but was relatively faster with respect to efficacy evaluation.

To simulate bivariate toxicity and efficacy outcomes, we first generate a bivariate normal vector.ZT ; ZE / with the same marginal variance �2 and covariance cov.ZT ; ZE /D ��2, where ZT is asso-ciated with toxicity and ZE with efficacy. At different dose levels, we take different normal means withE.ZTj / D �Tj and E.ZEj / D �Ej , for j D 1; : : : ; J . We then exponentiate these two variates toobtain the bivariate log-normal variates. Let FT .�j�Tj / and FE .�j�Ej / denote the cumulative distribu-tion functions for the two log-normal variables at dose level j . In each scenario, we first specify thetoxicity and efficacy probabilities f.p1; q1/ : : : ; .pJ ; qJ /g for all J doses and then calculate the meansof the bivariate normal distributions, �Tj and �Ej , such that FT .�T j�Tj /D pj and FE .�E j�Ej /D qj .Following this route, we can simulate the bivariate times to toxicity and efficacy from the log-normaldistributions, in which we took the correlation coefficient between toxicity and efficacy, � D 0:2 or 0.5,and the marginal variance, �2 D 1 or 2. We considered six dose levels, J D 6, with the standardizeddoses of f´1; : : : ; ´6g D f0:1; 0:2; 0:3; 0:4; 0:5; 0:6g. In the estimation of the local Kaplan–Meier estima-tor, we used the biquadratic kernel function in the form of K.x/D .15=16/.1� x2/2I.jxj6 1/. For thedefinition of the admissible dose set, we took �T D 1=3 and �E D 0:15, so that the toxicity probabilitiesof admissible doses cannot exceed 1=3, whereas there is more room to adjust in terms of efficacy. Thetotal sample size was 60, and we simulated 10,000 trials for each configuration.

Tables I and II summarize the operating characteristics of the proposed two-stage design using thefractional contribution of censored data, when the time-to-event data were simulated from bivariate log-normal distributions. We also implemented the trial design using the complete data with a full follow-uptime �E for each cohort. Although the full evaluation of efficacy may not be practical because of theresulting over-lengthy trial duration, it may serve as a benchmark for comparison. As shown in Figure 2,toxicity is typically negligible, but efficacy may have a decreasing, umbrella-shaped, or increasing pat-tern, which corresponds to the maximum efficacy at the first, in the middle, or at the last dose. Undereach scenario, the first row of the tables represents the true probabilities of toxicity and efficacy; thesecond row corresponds to the selection percentage using the censored-data two-stage design; the third

Dose level

Pro

babi

lity Efficacy

Toxicity0.1

0.2

0.3

0.4

0.5

0.01 2 3 4 5 6

Dose level

Pro

babi

lity

Efficacy

Toxicity0.1

0.2

0.3

0.4

0.5

0.01 2 3 4 5 6

Dose level

Pro

babi

lity

Efficacy

Toxicity0.1

0.2

0.3

0.4

0.5

0.01 2 3 4 5 6

Figure 2. Illustration of three practical scenarios of probabilities of toxicity and efficacy with six dose levels.Toxicity is typically low, whereas efficacy may have three different patterns: decreasing, umbrella shaped,

and increasing.


653


Tabl

eII

I.Si

mul

atio

nst

udy

with

prob

abili

ties

ofto

xici

tyan

def

ficac

y.pj;qj/;jD1;:::;6

,for

six

dose

leve

lsus

ing

the

prop

osed

two-

stag

edo

se-fi

ndin

gde

sign

with

cens

ored

data

and

that

with

com

plet

eda

ta.

Sele

ctio

npe

rcen

tage

ofdo

sele

vel(

Sel.%

)N

o.of

No.

ofT

rial

Dos

e1

23

45

6N

one

toxi

city

effic

acy

dura

tion

Scen

ario

1(0

.02,

0.40

)(0

.03,

0.25

)(0

.05,

0.20

)(0

.06,

0.15

)(0

.07,

0.10

)(0

.07,

0.05

)C

enso

red:

Sel.%

64.7

16.1

9.4

4.3

1.8

0.6

3.1

2.2

16.4

23.7

.Oq>0:2/%

74.7

44.8

31.8

18.9

8.7

3.2

No.

ofpa

tient

s26

.910

.67.

85.

84.

53.

7C

ompl

ete:

Sel.%

66.2

17.6

8.9

3.7

1.1

0.2

2.2

2.2

16.3

48.4

.Oq>0:2/%

74.1

43.5

30.7

17.6

8.4

3.1

No.

ofpa

tient

s26

.111

.18.

26.

04.

53.

6

Scen

ario

2(0

.02,

0.15

)(0

.04,

0.45

)(0

.07,

0.30

)(0

.09,

0.25

)(0

.10,

0.16

)(0

.15,

0.10

)C

enso

red:

Sel.%

2.2

66.8

16.0

9.2

2.7

0.7

2.4

3.7

18.9

23.7

.Oq>0:2/%

24.8

79.2

51.3

36.9

17.4

5.4

No.

ofpa

tient

s5.

626

.810

.77.

74.

93.

6C

ompl

ete:

Sel.%

2.5

64.5

18.6

9.3

2.2

0.4

2.4

3.7

18.8

47.9

.Oq>0:2/%

26.5

77.9

50.8

36.9

16.3

5.6

No.

ofpa

tient

s5.

025

.711

.48.

45.

13.

7

Scen

ario

3(0

.01,

0.10

)(0

.02,

0.15

)(0

.03,

0.35

)(0

.04,

0.18

)(0

.05,

0.12

)(0

.06,

0.07

)C

enso

red:

Sel.%

2.0

6.3

65.5

12.4

5.0

1.8

7.0

1.9

13.3

23.5

.Oq>0:2/%

9.2

17.8

69.2

24.4

11.4

3.8

No.

ofpa

tient

s5.

67.

525

.69.

06.

44.

7C

ompl

ete:

Sel.%

2.5

6.2

67.5

12.5

3.7

0.8

6.9

2.0

13.6

48.0

.Oq>0:2/%

10.8

19.3

70.1

23.8

10.3

3.4

No.

ofpa

tient

s5.

16.

826

.59.

66.

34.

6

Scen

ario

4(0

.01,

0.05

)(0

.02,

0.15

)(0

.04,

0.30

)(0

.06,

0.45

)(0

.08,

0.35

)(0

.10,

0.30

)C

enso

red:

Sel.%

0.1

1.6

13.4

50.6

20.6

11.3

2.4

3.5

19.6

23.8

.Oq>0:2/%

8.5

28.9

56.1

74.2

56.3

42.7

No.

ofpa

tient

s3.

55.

010

.320

.711

.78.

2C

ompl

ete:

Sel.%

0.2

1.8

12.2

50.0

22.0

10.9

2.9

3.6

19.6

48.0

.Oq>0:2/%

10.1

29.4

58.1

74.3

56.4

42.2

No.

ofpa

tient

s3.

44.

58.

920

.912

.59.

1

654



Scen

ario

5(0

.01,

0.15

)(0

.02,

0.25

)(0

.03,

0.33

)(0

.05,

0.47

)(0

.06,

0.60

)(0

.07,

0.40

)C

enso

red:

Sel.%

0.5

2.1

5.8

21.8

58.4

10.6

0.8

2.9

26.8

23.9

.Oq>0:2/%

33.9

53.8

66.1

80.2

85.4

65.4

No.

ofpa

tient

s3.

95.

26.

912

.722

.98.

2C

ompl

ete:

Sel.%

0.6

2.2

5.2

19.7

57.9

13.2

1.1

3.0

27.1

48.7

.Oq>0:2/%

34.6

53.8

67.3

80.5

85.8

65.2

No.

ofpa

tient

s3.

74.

76.

211

.524

.19.

5

Scen

ario

6(0

.01,

0.05

)(0

.02,

0.15

)(0

.03,

0.30

)(0

.04,

0.35

)(0

.05,

0.40

)(0

.06,

0.50

)C

enso

red:

Sel.%

0.1

0.9

7.7

13.2

23.2

53.4

1.5

2.7

22.2

23.8

.Oq>0:2/%

10.7

31.5

60.0

65.0

70.1

77.0

No.

ofpa

tient

s3.

44.

48.

09.

812

.821

.2C

ompl

ete:

Sel.%

0.1

1.0

7.6

12.4

20.8

56.4

1.7

2.7

22.6

48.5

.Oq>0:2/%

10.8

31.1

60.5

67.3

70.6

76.2

No.

ofpa

tient

s3.

34.

17.

19.

011

.924

.1

Scen

ario

7(0

.05,

0.01

)(0

.10,

0.02

)(0

.16,

0.03

)(0

.35,

0.05

)(0

.40,

0.07

)(0

.50,

0.09

)C

enso

red:

Sel.%

0.0

0.0

0.0

0.0

0.0

0.0

99.9

4.1

0.8

10.1

.Oq>0:2/%

0.0

0.0

0.0

0.0

0.0

0.0

No.

ofpa

tient

s6.

56.

25.

64.

11.

60.

5C

ompl

ete:

Sel.%

0.0

0.0

0.0

0.0

0.0

0.0

100.

04.

10.

715

.1.Oq>0:2/%

0.0

0.0

0.0

0.0

0.0

0.0

No.

ofpa

tient

s6.

56.

25.

64.

01.

50.

4

The

targ

etdo

seis

inbo

ldfa

ce,a

ndth

ebi

vari

ate

times

toto

xici

tyan

def

ficac

yar

esi

mul

ated

from

aC

layt

onco

pula

mod

elw

ithm

argi

nal

dist

ribu

tions

ofW

eibu

ll.2;�T/

and

Wei

bull.2;�E/

and

the

asso

ciat

ion

para

met

er5.


655


row shows . Oq > 0:2/%, which is the probability that the estimated efficacy rate is greater than a certainthreshold, say 0.2; the fourth row is the number of patients treated at each dose; and the remaining rows5–7 correspond to the results using the complete-data two-stage design. We also present the percentageof inconclusive trials (denoted as ‘None’), the number of toxicities, the number of responses, and thetrial duration in months averaged over 10,000 simulated trials.

In scenario 1, the toxicity probability increases slowly with the maximum of 0.07 at the last dose,whereas efficacy monotonically decreases over the doses. Both the designs with censored data and com-plete data selected the first dose with the highest percentages, although the censored design performedslightly inferior to the complete design. However, the trial duration using censored data was immenselyreduced, by more than a half, compared with that using complete data. In scenario 2, the second doseis the target with the highest efficacy probability of 0.45 and a tolerable toxicity probability of 0.04. Inthis case, both designs performed well, although the censored design led to a much shorter trial duration.Scenarios 3 and 4 have the target at dose levels 3 and 4, respectively; that is, the efficacy probabilityreaches the maximum somewhere in the middle of the dose range and then starts decreasing till the lastdose. In both scenarios, similar conclusions can be drawn as before. In scenario 5, the efficacy probabil-ity curve is again umbrella shaped; both designs selected the target dose in over 50% of cases. Scenario 6has the target dose at the last dose level, that is, both toxicity and efficacy monotonically increase overthe doses, although toxicity increases at a much slower rate and all of the doses are tolerable. Bothdesigns selected the last dose with the highest percentage and also treated most of the patients at thatdose. In scenario 7, all the doses are not effective, so that most of the trials were stopped early becauseof a lack of efficacy. For all the scenarios considered, toxicities of the investigational doses are not ofmajor concern, and the optimal doses are mainly determined by efficacy. The selection percentages ofthe target doses are similar between the proposed two-stage design using censored data and that usingcomplete data. The number of patients experiencing toxicity and the number of responders are also veryclose between the two designs. Nevertheless, our method dramatically shortens the trial duration fromapproximately 4 years to less than 2 years. The simulation results in Table II are based on different val-ues of the marginal variance and correlation of the log-normal distributions, and we can draw the sameconclusions as those in Table I.

We also considered Weibull distributions for the marginal distributions of times to toxicity and effi-cacy and linked the two marginal survival functions by the Clayton copula [26]. We fixed the shapeparameter of the marginal Weibull distribution at 2 and adjusted the scale parameter, �T or �E , to matchthe toxicity and efficacy probabilities evaluated at �T and �E with the prespecified .pj ; qj /, respectively.The association parameter in the Clayton copula was set as 5 to induce moderate correlation between thebivariate distributions of toxicity and efficacy. We provide the simulation results in Table III. Althoughthe underlying copula model is very different from that of the bivariate log-normal distribution, we keptthe toxicity and efficacy probabilities at each dose to be the same as those in Tables I and II. The selectionpercentage and the number of patients treated at each dose are very close among the three tables. Fromthis sensitivity analysis, we conclude that our design is quite robust, because the main design scheme

Data in both stages 1 and 2

Time (months)

Cen

sorin

g pe

rcen

tage

scenario 1scenario 2scenario 3scenario 4scenario 5scenario 6scenario 7

5 10 15 5 10 15

0.00

0.05

0.10

0.15

0.00

0.05

0.10

0.15

0.20

0.25

Data in stage 2 only

Time (months)

Cen

sorin

g pe

rcen

tage

scenario 1scenario 2scenario 3scenario 4scenario 5scenario 6scenario 7

Figure 3. Censoring percentage caused by delayed response at each decision-making time point under theClayton copula model, with time starting from the initiation of stage 2.

656



Tabl

eIV

.Si

mul

atio

nst

udy

with

prob

abili

ties

ofto

xici

tyan

def

ficac

y.pj;qj/;jD1;:::;6

,for

six

dose

leve

lsus

ing

the

mod

ified3C3

desi

gn,w

hich

dire

ctly

iden

tifies

the

mos

teff

ectiv

edo

seaf

ter

iden

tifyi

ngth

eM

TD

.

Sele

ctio

npe

rcen

tage

ofdo

sele

vel(

Sel.%

)N

o.of

No.

ofT

rial

Dos

e1

23

45

6N

one

toxi

city

effic

acy

dura

tion

Scen

ario

1(0

.02,

0.40

)(0

.03,

0.25

)(0

.05,

0.20

)(0

.06,

0.15

)(0

.07,

0.10

)(0

.07,

0.05

)3C3

Des

ign:

Sel.%

34.3

20.2

16.9

13.2

8.5

4.3

2.6

1.0

3.8

8.5

.Oq>0:2/%

77.1

55.5

42.7

29.8

17.8

8.1

No.

ofpa

tient

s3.

23.

33.

43.

43.

33.

1

Scen

ario

2(0

.02,

0.15

)(0

.04,

0.45

)(0

.07,

0.30

)(0

.09,

0.25

)(0

.10,

0.16

)(0

.15,

0.10

)3C3

Des

ign:

Sel.%

4.8

40.1

21.8

17.8

9.3

3.5

2.7

1.6

4.8

8.6

.Oq>0:2/%

36.2

80.2

55.9

43.3

22.8

8.7

No.

ofpa

tient

s3.

23.

43.

53.

53.

33.

1

Scen

ario

3(0

.01,

0.10

)(0

.02,

0.15

)(0

.03,

0.35

)(0

.04,

0.18

)(0

.05,

0.12

)(0

.06,

0.07

)3C3

Des

ign:

Sel.%

3.6

8.2

40.4

20.7

14.5

7.8

4.8

0.7

3.1

8.4

.Oq>0:2/%

26.4

36.2

69.9

40.1

24.5

12.6

No.

ofpa

tient

s3.

13.

23.

33.

33.

33.

2

Scen

ario

4(0

.01,

0.05

)(0

.02,

0.15

)(0

.04,

0.30

)(0

.06,

0.45

)(0

.08,

0.35

)(0

.10,

0.30

)3C3

Des

ign:

Sel.%

0.7

4.0

13.4

34.2

24.3

21.3

2.1

1.1

5.4

8.6

.Oq>0:2/%

13.2

36.2

61.9

76.4

61.0

47.0

No.

ofpa

tient

s3.

13.

23.

33.

43.

53.

3


657


Tabl

eIV

.C

onti

nued

.

Sele

ctio

npe

rcen

tage

ofdo

sele

vel(

Sel.%

)N

o.of

No.

ofT

rial

Dos

e1

23

45

6N

one

toxi

city

effic

acy

dura

tion

Scen

ario

5(0

.01,

0.15

)(0

.02,

0.25

)(0

.03,

0.33

)(0

.05,

0.47

)(0

.06,

0.60

)(0

.07,

0.40

)3C3

Des

ign:

Sel.%

1.7

4.1

7.6

17.6

45.6

22.5

0.9

0.8

7.3

8.5

.Oq>0:2/%

37.7

57.0

67.2

80.5

86.1

65.8

No.

ofpa

tient

s3.

13.

23.

33.

43.

33.

3

Scen

ario

6(0

.01,

0.05

)(0

.02,

0.15

)(0

.03,

0.30

)(0

.04,

0.35

)(0

.05,

0.40

)(0

.06,

0.50

)3C3

Des

ign:

Sel.%

0.5

2.4

8.8

14.0

22.3

50.3

1.5

0.7

5.7

8.4

.Oq>0:2/%

13.4

36.8

63.0

67.8

72.3

77.9

No.

ofpa

tient

s3.

13.

23.

23.

33.

33.

3

Scen

ario

7(0

.05,

0.01

)(0

.10,

0.02

)(0

.16,

0.03

)(0

.35,

0.05

)(0

.40,

0.07

)(0

.50,

0.09

)3C3

Des

ign:

Sel.%

2.5

4.9

6.2

2.0

0.9

0.1

83.3

3.2

0.5

6.0

.Oq>0:2/%

1.7

2.4

1.5

0.2

0.1

0.0

No.

ofpa

tient

s3.

53.

94.

23.

51.

40.

4

The

targ

etdo

seis

inbo

ldfa

ce,a

ndth

ebi

vari

ate

times

toto

xici

tyan

def

ficac

yar

esi

mul

ated

from

aC

layt

onco

pula

mod

elw

ithm

argi

nal

dist

ribu

tions

ofW

eibu

ll.2;�T/

and

Wei

bull.2;�E/

and

the

asso

ciat

ion

para

met

er5.

658



is based on nonparametric proportions and does not rely upon any specific model structure. Therefore,the proposed two-stage design yielded very similar results regardless of whether the underlying modelis a bivariate log-normal distribution or a Clayton copula with Weibull marginal distributions. Figure 3presents the censoring percentage of the efficacy data starting from stage 2, because all patients in stage1 are fully followed and there is no censoring. The censoring percentage initially increases up to 10–20%and then gradually decreases as the trial proceeds. If we only count the data in stage 2, the censoringpercentage would be higher. By the time the trial is finished, there is no censoring because every subjectwould have been fully followed up to time �E .

Furthermore, we made a comparison with a more straightforward design, which only contains stage 1with the 3C 3 design and then immediately selects the most effective dose from all the doses below theMTD. If there are multiple doses tied in terms of efficacy, we would recommend the higher dose. Forfair comparison, we also implemented the lower bound for efficacy, that is, Oqj > 0:15. As shown by thesimulation results in Table IV, such direct application of the 3C 3 design to cytostatic agents performedinferior to the two-stage procedure, although the required sample size became much smaller.

4. Conclusion

In the development of personalized medicine, cytostatic agents are commonly encountered in targetedtherapy oncology trials. This type of agents are fundamentally different from conventional cytotoxicagents and thus call for new statistical designs. To fulfill the practical needs for emerging cytostaticagents, we have proposed a two-stage dose-finding procedure, which is suitable for agents that are gen-erally tolerable and efficacy is of more concern. The new design has been shown to work well in thesimulation studies, although some modifications can be made as follows. In stage 1, the acceleratedtitration design may be used instead of the 3C3 design to locate the MTD, which would save the samplesize by treating one patient at each dose till observing the first DLT [27]. In both stages, we can imposeparametric models for dose–toxicity and dose–efficacy curves, which may improve trial efficiency butmay cause bias as a result of possible model misspecification.

In a typical trial setting, drug efficacy is evaluated within a fixed period, after which the patientoutcome does not count. The proposed design uses the fractional contributions of censored efficacyevents to facilitate the continual trial conduct. The data used to estimate efficacy are subject to censor-ing, which may change as the trial proceeds. More efficacy events would be observed when a longerfollow-up is taken. As a result, there is a possibility that the updated efficacy data might reverse theprevious dose-escalation or de-escalation decisions. In the simulation, the chance of such reversion istypically small: less than 1% if we only count the cases of complete reversion (i.e., supposed to escalatebut de-escalate the dose instead, or vice versa) and around 15% if we count all cases of mismatching(e.g., including those supposed to stay at the same dose but escalate or de-escalate the dose). For smallsample sizes, the kernel smoothing method in the local Kaplan–Meier estimator may not be stable. Alter-natively, the Cox proportional hazards model may be applied to model the dose information, howeverthe proportional hazards assumption needs to be examined [28]. If the accrual is fast and the evaluationof efficacy is slow, more censored observations would be generated. The evaluation window for efficacyis subjective, which depends on the disease type, status, and the testing agent. In particular, the efficacyevent may still possibly occur after the evaluation window. In other words, even though we view the fullyfollowed patients as ‘complete’ data, they are in fact truncated or censored by the length of the evaluationwindow. The trial would collect much more information if patients are followed continuously till diseaseprogression, the occurrence of severe adverse events, or the end of the study. In this circumstance, doseselection may be based on time-to-event endpoints instead of binary endpoints, which becomes moreinvolved and warrants further investigation.

Acknowledgements

We would like to thank three anonymous referees and the Associate Editor for the very insightful comments thatsubstantially improved this paper. Yin’s research was partially supported by a grant (Grant No. 784010) from theResearch Grants Council of Hong Kong.

References1. Chevret S. Statistical Methods for Dose-Finding Experiments. John Wiley & Sons: England, 2006.2. Ting N. Dose Finding in Drug Development. Springer: Cambridge, MA, 2006.


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Institute 2004; 96:977–978.11. Hunsberger S, Rubinstein VL, Dancey J, Korn LE. Dose escalation trial designs based on a molecularly targeted endpoint.

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