STATISTICS IN MEDICINE
`Statist. Med. 17, 1103—1120 (1998)
`
`CANCER PHASE I CLINICAL TRIALS: EFFICIENT DOSE
`ESCALATION WITH OVERDOSE CONTROL
`
`JAMES BABB1*, ANDRE¤ ROGATKO1 AND SHELEMYAHU ZACKS2
`
`1Fox Chase Cancer Center, Department of Biostatistics, 510 Township Line Road, Cheltenham, PA 19012, U.S.A.
`2Binghamton University, Department of Mathematical Sciences, State University of New York, Binghamton, NY 13901, U.S.A.
`
`SUMMARY
`
`We describe an adaptive dose escalation scheme for use in cancer phase I clinical trials. The method is
`fully adaptive, makes use of all the information available at the time of each dose assignment, and
`directly addresses the ethical need to control the probability of overdosing. It is designed to approach the
`maximum tolerated dose as fast as possible subject to the constraint that the predicted proportion of
`patients who receive an overdose does not exceed a speci(cid:222)ed value. We conducted simulations to compare
`the proposed method with four up-and-down designs, two stochastic approximation methods, and with
`a variant of the continual reassessment method. The results showed the proposed method e⁄ective as
`a means to control the frequency of overdosing. Relative to the continual reassessment method, our scheme
`overdosed a smaller proportion of patients, exhibited fewer toxicities and estimated the maximum tolerated
`dose with comparable accuracy. When compared to the non-parametric schemes, our method treated fewer
`patients at either subtherapeutic or severely toxic dose levels, treated more patients at optimal dose levels
`and estimated the maximum tolerated dose with smaller average bias and mean squared error. Hence, the
`proposed method is promising alternative to currently used cancer phase I clinical trial designs. ( 1998
`John Wiley & Sons, Ltd.
`
`1. INTRODUCTION
`
`The primary purpose of a phase I clinical trial is to determine the dose of a new drug or
`therapeutic agent for use in a subsequent phase II trial. A long-accepted assumption underlying
`cancer therapy is that toxicity is a prerequisite for optimal antitumour activity.1 Consequently,
`one must endure some degree of treatment related toxic reaction if patients are to have
`a reasonable chance of favourable response. Since higher doses are associated with both greater
`therapeutic bene(cid:222)ts and an increased probability of severe toxic reaction, a cytotoxic drug should
`be administered at the maximum dose that cancer patients can tolerate. Consequently, the goal of
`a cancer phase I trial is to determine the highest dose associated with an acceptable level of
`toxicity. More precisely, the goal is to estimate the maximum tolerated dose (MTD), de(cid:222)ned as
`the dose for which the probability of a medically unacceptable, dose-limiting toxicity (DLT) is
`
`* Correspondence to: James Babb, PhD, Fox Chase Cancer Center, Department of Biostatistics, 510 Township Line
`Road, Cheltenham, PA 19012, U.S.A. E-mail: babb@canape.fccc.edu
`
`CCC 0277—6715/98/101103—18$17.50
`( 1998 John Wiley & Sons, Ltd.
`
`Received January 1996
`Revised May 1997
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`INTELGENX 1031
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`J. BABB, A. ROGATKO AND S. ZACKS
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`equal to a speci(cid:222)ed value h:
`
`ProbMDLTD Dose"MTDN"h.
`The value chosen for the target probability h depends on the nature of the DLT; we would set it
`relatively high when the DLT is a transient, correctable or non-fatal condition, and low when it is
`lethal or life threatening.
`Representing the (cid:222)rst application of a proposed drug to humans, the phase I trial constitutes
`one of the most important steps in the drug(cid:213)s development.2 Since initial experience with a new
`agent may unduly in(cid:223)uence its fate, a careful and thoughtful approach to the design of phase I
`trials is essential. Unfortunately, clinical research involving humans poses serious ethical prob-
`lems and clinical trials involving oncology patients and cytotoxic drugs have been among the
`most problematic of all.3 In contrast to other phase I trials, cancer phase I trials have a thera-
`peutic aim. Typically, participants in a cancer phase I trial are patients at advanced disease stages
`who consent to participate in the trial only as a last resort in seeking cure. Thus, from
`a therapeutic perspective, one should design cancer phase I trials to minimize both the number of
`patients treated at low, non-therapeutic doses as well as the number given severely toxic
`overdoses.
`In the next section we describe a dose escalation scheme that controls the probability a patient
`will receive an overdose. The scheme, referred to as EWOC (escalation with overdose control), is
`Bayesian-feasible of level 1!a as de(cid:222)ned by Eichhorn and Zacks.4 That is, EWOC selects a dose
`level for each patient so that the predicted probability the dose exceeds the MTD is less than or
`equal to a speci(cid:222)ed value a. Zacks et al.5 showed that among designs that are Bayesian-feasible of
`level 1!a, EWOC is optimal in the sense that it minimizes the predicted amount by which any
`given patient is underdosed. Thus, EWOC is designed to approach the MTD as rapidly as
`possible subject to the constraint that the predicted proportion of patients given an overdose is
`less than or equal to a.
`
`2. METHOD
`
`The key concept underlying EWOC is that one can select dose levels for use in a phase I trial so
`that the predicted proportion of patients who receive an overdose is equal to a speci(cid:222)ed value a,
`called the feasibility bound. This is accomplished by computing, at the time of each dose
`assignment, the posterior cumulative distribution function (CDF) of the MTD. For the kth dose
`assignment the posterior CDF of the MTD is the function n
`k given by
`k(c)"ProbMMTD)c D Dk
`n
`N
`where Dk denotes the data at the time of treatment for the kth patient and would include for each
`previously treated patient the dose administered, the highest level of toxicity observed and any
`
`relevant covariate measurements. nk(c) is the conditional probability that c is an overdose given
`the data currently available. Based on this, EWOC selects for the kth patient the dose level
`xk such that
`
`k(xk)"a.
`n
`That is, we select the dose for each patient so that the predicted probability it exceeds the MTD is
`equal to a.
`
`( 1998 John Wiley & Sons, Ltd.
`
`Statist. Med. 17, 1103—1120 (1998)
`
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`DOSE ESCALATION WITH OVERDOSE CONTROL
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`1105
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`In the next section we describe EWOC for the speci(cid:222)c case where toxicity is measured on
`a binomial scale (presence or absence of DLT), there are no known covariates and one plans to
`accrue a (cid:222)xed number n of patients to the trial. Extensions of EWOC to accommodate more
`informative response measures, covariate information and variable sample sizes are currently
`under investigation. In Section 2.2 we present an example illustrating the application of EWOC
`to a cancer phase I clinical trial involving 5-(cid:223)uorouracil.
`
`2.1. Dose escalation method
`Let X.*/ and X.!9 denote the minimum and maximum dose levels available for use in the trial.
`One chooses these dose levels in the belief that X.*/ is safe when administered to humans and
`)MTD)X.!9.
`X.*/
`(1)
`The dose for the (cid:222)rst patient is X.*/ and we shall select only dose levels between X.*/ and X.!9 for
`use in the trial. Thus, if xi denotes the dose level selected for the ith patient, i"1, 2, n, then
`"X.*/
`x1
`3[X.*/, X.!9], "i"1, 2, n.
`xi
`We model the relationship between dose level and toxicity as
`#b
`ProbMDLTD Dose"xN"F(b
`1x)
`0
`where F is a speci(cid:222)ed distribution function, called a tolerance distribution, and b0 and b
`
`1 are
`’0 so that the probability of a DLT is a monotonic increasing
`unknown. We assume that b
`1
`function of dose. The MTD is the dose level, denoted c, such that the probability of a DLT is h. It
`follows from (2) that
`
`and
`
`(2)
`
`0
`
`1
`
`"X.*/
`
`1xi)yi[1!F(b
`We incorporate prior information about b0 and b
`
`
`
`F(b
`
`0
`
`c"F~1(h)!b
`b
`#F~1(h)!F~1(o
`0)
`b
`1
`0 denotes the probability of a DLT at the starting dose x1"X.*/. Figure 1 illustrates
`
`where o
`a typical dose-toxicity model.
`"0,Denote by yi the response of the ith patient where yi"1 if a DLT is manifest and yi
`
`
`
`otherwise. The data after observation of k patients is Dk"M(xi, yi), i"1, 2 , kN and the likeli-
`
`hood function of (b0, b
`1) given Dk is
`D Dk)" k<
`‚(b
`0, b1
`i/1
`
`
`
`1) de(cid:222)ned onh(b0, b
`)"M(a, b)3R2 : b’0, F(a#bX.*/))h)F(a#bX.!9)N.
`
`#b
`
`#b
`1xi)]1~yi.
`
`0
`
`1 through a prior probability density function
`
`( 1998 John Wiley & Sons, Ltd.
`
`Statist. Med. 17, 1103—1120 (1998)
`
`(3)
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`Figure 1. The probability of DLT as a function of dose
`
`
`
`After an application of Bayes theorem, the joint posterior distribution of (b0, b1) given the data
`
`
`Dk is
`
`P(b
`
`0, b1
`
`
`
`where
`
`
`
`D Dk)"q~1‚(b0, b
`1
`
`D Dk)h(b
`
`
`
`0, b1)I)(b
`
`
`
`0, b1)
`
`(4)
`
`q"PP)
`‚(x, y D Dk)h(x, y)dx dy
`and I) denotes the indicator function for the set ). We can derive the marginal posterior
`cumulative distribution function of the MTD given Dk from (4) through the transformation
`„(b
`
`
`0, b1)"(o0, c). Denoting the image of ) under the transformation „ by „()), it follows from
`(1) and (3) that
`
`„())"[0, h]][X.*/, X.!9].
`
`The inverse transformation is
`
`„~1(o0, c)"( f1(o0, c), f2(o
`
`
`where the functions f1 and f2 are de(cid:222)ned on „()) by
`0)!X.*/F~1(h)
`
`f1(o0, c)"cF~1(o
`c!X.*/
`
`0, c))
`
`and
`
`0, c)"F~1(h)!F~1(of2(o
`0)
`
`c!X.*/
`We can now write the joint posterior probability density function (PDF) of (o0, c) given Dk as
`
`
`
`
`0, c D Dk)"q~1‚( f1(o0, c), f2(o0, c) D Dk)g(o0, c)
`P(o
`
`.
`
`( 1998 John Wiley & Sons, Ltd.
`
`Statist. Med. 17, 1103—1120 (1998)
`
`INTELGENX 1031
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`
`DOSE ESCALATION WITH OVERDOSE CONTROL
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`1107
`
`where
`
`D I„())(o
`F~1(p)K
`0, c)C L
`0, c)"h( f1(o0, c), f2(o0, c)) f2(o
`
`
`g(o
`0, c).
`Lp
`p/p0
`0, c) by the choice of h as the prior PDF of (b0, b1).
`
`Note that g is the prior PDF induced for (o
`
`Elicitation of prior information can be through speci(cid:222)cation of the PDF g directly, rather than
`through the choice of h. This might be advantageous since c is the parameter of interest and one
`often conducts preliminary studies at or near the starting dose so that one can select a meaningful
`informative prior for o
`0. Letting
`
`
`
`0.
`
`0, c)3„())N
`#(c)"Mo
`0 : (o
`we can write the marginal posterior PDF of the MTD given Dk as
`n(c D Dk)"PP#(c)
`P(o0, c D Dk) do
`The marginal posterior CDF of the MTD given Dk is then
`k(z)"P z
`n(c D Dk) dc, x3[X.*/, X.!9].
`
`n
`
`X.*/
`We can now describe EWOC as follows. The (cid:222)rst patient, or cohort of patients, receives the dose
`
`x1"X.*/. We select the dose for each subsequent patient so that on the basis of all the available
`data the posterior probability that it exceeds the MTD is equal to the feasibility bound a. Hence,
`the kth patient receives the dose
`
`K"2,2, n,
`
`(5)
`
`(6)
`
`"n~1
`m(k)(a)
`xk
`where m(k) denotes the number of observations available at the time of treatment for the kth
`patient.
`The dose sequence de(cid:222)ned by (5) assumes that all dose levels between X.*/ and X.!9 are
`available for use in the trial. However, due to practical and physical constraints, phase I clinical
`trials are typically based on a small number of prespeci(cid:222)ed dose levels. In such cases we select for
`the kth patient the dose level
`k(xk)!a)„
`!xk)„
`
`"maxMd1, 2, dr : di
`N
`1 and n
`Dk
`2
`where d1, 2, dr are the dose levels chosen for experimentation and „
`1 and „
`2 are prespeci(cid:222)ed
`non-negative real numbers we refer to as tolerances. We note that the dose sequence given by (6) is
`Bayesian-feasible of level 1!a if and only if at least one of the tolerances „
`
`1 and „2 is equal to
`zero. Positive tolerances would be chosen to permit the use of dose levels above yet suƒciently
`close to the optimal Bayesian-feasible dose xk.
`Since cancer patients often exhibit delayed response to treatment, the time required to resolve
`toxicity can be longer than the average time between successive accruals. Consequently, new
`patients frequently become available to the study before we have observed the responses of all
`previously treated patients. It is therefore important to note that EWOC does not require that we
`know all patient responses before we can treat a newly accrued patient. Instead, we can select the
`dose for the new patient on the basis of the data currently available.
`
`( 1998 John Wiley & Sons, Ltd.
`
`Statist. Med. 17, 1103—1120 (1998)
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`INTELGENX 1031
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`Upon completion of the trial we can estimate the MTD by minimizing the posterior expected
`loss with respect to some choice of loss function l. Thus, the dose recommended for use in
`a subsequent phase II trial is the estimate cL
`l such that
`l(cL
`P X.!9
`l, c)n(cD Dn) dc)P X.!9
`l(x, c)n(cD Dn) dc "x3[X.*/, X.!9].
`
`X.*/
`
`X.*/
`
`Candidate estimators include the mean, median and mode of the marginal posterior PDF of the
`MTD. One should consider asymmetric loss functions since underestimation and overestimation
`have very di⁄erent consequences. Indeed, the dose xk selected by EWOC for the kth patient
`corresponds to the estimate of the MTD having minimal risk with respect to the asymmetric loss
`function
`
`if x)c that is, if x is an underdose
`la(x, c)"G a(c!x)
`if x’c that is, if x is an overdose.
`(1!a) (x!c)
`Note that the loss function la implies that for any d’0, the loss incurred by treating a patient at
`d units above the MTD is (1!a)/a times greater than the loss associated with treating the patient
`at d units below the MTD. This interpretation might provide a meaningful basis for the selection
`of the feasibility bound.
`We note that we can estimate the MTD using a di⁄erent prior PDF than that used to design
`the phase I trial.6 Furthermore, some authors (for example, Watson and Pelli7) have suggested
`use of Bayesian scheme to design the trial and use of maximum likelihood to estimate the MTD.
`
`(7)
`
`2.2. Example
`
`EWOC was used to design a phase I clinical trial that involved the antimetabolite 5-(cid:223)uorouracil
`(5-FU). In this trial a total of n"12 patients with malignant solid tumours are to be treated with
`a combination of 5-FU, leucovorin and topotecan. The goal is to determine the dose level of 5-FU
`that, when administered in combination with (cid:222)xed levels of the other two agents (20 mg/m2
`leucovorin, 0.5 mg/m2 topotecan), results in a probability of h"1/3 that a grade 4 hematologic
`or grade 3 or 4 non-hematologic toxicity is manifest within two weeks.
`Since preliminary studies indicated that 140 mg/m2 of 5-FU was well tolerated when given
`concurrently with up to 0)5 mg/m2 of topotecan, this level was selected as the starting dose for the
`phase I trial. Hence
`
`"140.
`X.*/
`Furthermore, a previous trial involving 5-FU alone estimated the MTD as 425 mg/m2. Since
`a given level of 5-FU has been observed to be more toxic when given in conjunction with
`topotecan than when administered alone, the MTD of 5-FU in combination with leucovorin and
`topotecan is believed to be below 425 mg/m2, the single agent MTD. Consequently, the max-
`imum allowed dose for the phase I trial is
`
`"425.
`X.!9
`Simulations and prior experience have shown EWOC to perform well when we take
`the tolerance distribution as the logistic and the prior PDF used for (o
`0, c) is the uniform
`distribution. Hence, we modelled the relationship between dose-limiting toxicity and the dose
`
`( 1998 John Wiley & Sons, Ltd.
`
`Statist. Med. 17, 1103—1120 (1998)
`
`INTELGENX 1031
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`level of 5-FU as
`
`#b
`1xN
`ProbMDLTD Dose"xN" expMb
`0
`#b
`1#expMb
`1xN
`0
`
`and we took the joint prior PDF of (o0, c) as
`
`0, c)"57~1I*0,0>2+]*140,425+(o
`0, c).
`g(o
`It follows that the marginal posterior CDF of the MTD at the time of the kth dose assignment is
`
`n
`
`k(x)"P x
`
`140
`
`P 0>2
`
`0
`
`
`
`P(o0, c D Dm(k)) do0 dc
`
`
`
`where
`
`with
`
`0, cD Dk)"q~1
`P(o
`
`
`expM f (o0, c D xi)N
`<
`i|S(k)
`m(k)<
`[1#expM f(o
`0, cD xi)N]
`i/1
`
`
`
`I*0,0>2+]*140,425+(o0, c)
`
`P <
`q"P
`T())
`logA o
`0
`1!o
`0, cD x)"
`f (o
`0
`
`i|S(k)
`
`m(k)<
`[1#expM f (o
`
`expM f (o0, c D xi)
`i/1
`
`B (c!x)#logA h1!hB (x!X.*/)
`c!X.*/
`
`
`
`0, cD xi)N]~1 do0 dc
`
`and
`
`"1N.
`
`S(k)"Mi"1, 2 , m(k) : yi
`For this trial the feasibility bound is equal to
`a"0)25
`this value being a compromise between the therapeutic aim of the trial and the need to avoid
`treatment attributable toxicity. Consequently, escalation of 5-FU between successive patients is
`to the estimated dose level that falls below the MTD with 75 per cent con(cid:222)dence. The (cid:222)rst patient
`"140 mg/m2. If the (cid:222)rst patient exhibits DLT, we will
`accrued to the trial receives the dose x1
`have violated the assumption that the initial dose is safe and we would suspend the trial (to restart
`at a lower initial dose or to terminate at the discretion of the principal investigator). Otherwise,
`we dose the next four patients according to the schedule given in Figure 2.
`
`3. SIMULATION COMPARISONS
`
`We conducted a simulation study to compare the performance of EWOC with seven phase I dose
`escalation schemes consisting of four up-and-down (UD) designs, two stochastic approximation
`(SA) methods and a Bayesian scheme known as the continual reassessment method (CRM). We
`compared the methods with respect to the proportion of patients assigned dose levels above and
`
`( 1998 John Wiley & Sons, Ltd.
`
`Statist. Med. 17, 1103—1120 (1998)
`
`INTELGENX 1031
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`Figure 2. The dose level to be given each of patients 2 to 5 in the 5-FU trial contingent on the doses and responses of
`preceding patients
`
`below the MTD, the proportion of patients exhibiting DLT and the bias and mean squared error
`observed for the estimator of the MTD.
`In cancer research the vast majority of phase I trials are based on UD dose escalation
`designs.8~11 With these methods, dose escalation proceeds either through a small number of
`preselected dose levels or via prespeci(cid:222)ed increments typically based on a modi(cid:222)ed Fibonacci
`sequence. The four designs considered in this study were proposed by Storer10 and are described
`as follows:
`”D1. Treat patients in cohort groups that consist of three patients each of whom receives the
`same dose. If no DLT is observed at a given dose level, then the next cohort receives the next
`highest dose. Otherwise, the next cohort is treated at the same dose and the trial then either
`continues at the next highest dose if exactly one of the last six patients exhibit DLT or
`terminates with observation of 2 or more DLT.
`”D2. Treat patients in cohorts of size three. Escalate the dose if no DLT is observed,
`de-escalate if more than one patient manifests DLT or repeat the dose if exactly one patient
`exhibits dose limiting toxicity. The trial continues until observation of a speci(cid:222)cied number n of
`patients.
`”D3. Treat patients one at a time. De-escalate the dose with each DLT and escalate it after
`two consecutive non-toxic responses. The trial continues until observation of a speci(cid:222)ed
`number n of patients.
`”D4. Treat patients one at a time. Escalate the dose continuously until observation of the (cid:222)rst
`DLT. Thereafter implement design UD3.
`
`( 1998 John Wiley & Sons, Ltd.
`
`Statist. Med. 17, 1103—1120 (1998)
`
`INTELGENX 1031
`
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`
`DOSE ESCALATION WITH OVERDOSE CONTROL
`
`1111
`
`Various authors12~16 have proposed designs for phase I clinical trials based on the stochastic
`approximation methods of Robbins and Monro.17 With these designs patients are assigned doses
`according to
`xj‘1"xj
`
`!aj(yj
`P0 as jPR. The two SA
`N=
`where Maj
`j/1 is a sequence of positive real numbers such that aj
`methods considered in this simulation study correspond to the sequences
`
`!h)
`
`and
`
`where
`
`SA1: aj
`
`"1
`2j
`
`SA2: aj
`
`" 1
`jbj
`
`x2
`i
`
`j+
`
`(cid:236)(cid:239)(cid:239)(cid:239)(cid:237)(cid:239)(cid:239)(cid:239)(cid:238)
`
`2
`
`j+
`
`i/1
`
`"
`bj
`
`yiBNj"0
`
`!A j
`xiyi
`
`+
`i/1
`
`xiBA j
`
`+
`i/1
`
`j+
`
`i/1
`
`if
`
`otherwise.
`
`+
`i/1
`
`+
`i/1
`
`yiBNj
`xiBA j
`!A j
`xiyi
`!A j
`xiB2Nj
`
`+
`i/1
`i/1
`O(cid:213)Quigley et al.18 proposed a Bayesian dose escalation scheme referred to as the continual
`reassessment method (CRM). In its original formulation we describe CRM as follows. Specify
`a model for the dose—toxicity relationship and, at each stage of the trial, obtain a Bayesian
`estimate of the MTD using all the data then available. Choose the dose recommended for each
`patient from a prespeci(cid:222)ed discrete set of levels as close as possible to the estimate of the MTD.
`We simulated a modi(cid:222)cation of this scheme wherein we treat each patient at the mean of the
`marginal posterior PDF of the MTD. Note that the dose assigned to each patient by this scheme
`corresponds to the estimate of the MTD having minimal risk with respect to squared error loss,
`whereas the dose assigned by EWOC is chosen to minimize risk with respect to the asymmetric
`loss function given by (7).
`
`3.1. Simulation set-up
`
`Throughout the simulations we generated the data according to the logistic model
`
`D#lnCh(1!o
`expGlnC o
`0(1!h)D xcH
`
`0)
`0
`1!o
`PrMDLTD Dose"xN"
`0
`D#lnCh(1!o
`1#expGlnC o
`0(1!h)D xcH
`
`0)
`0
`1!o
`0
`with o
`0 assuming each of the values 0)05, 0)10 and 0)15, and the MTD taking on both of the values
`c"0)3 and c"0)5. Thus, we considered six cases corresponding to the six distinct combinations
`
`o
`
`o
`
`( 1998 John Wiley & Sons, Ltd.
`
`Statist. Med. 17, 1103—1120 (1998)
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`INTELGENX 1031
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`
`1112
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`J. BABB, A. ROGATKO AND S. ZACKS
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`0 and c. We assumed that dose levels were standardized so that the starting dose for each
`of o
`"0 and all subsequent dose levels were selected from the
`trial was the standardized dose x1
`unit interval. Since the performance of the UD designs depends on the dose levels available
`for use in the trial, we simulated each UD method using both of M0(0)2)1N and M0(0)1)1N
`as the set of preselected dose levels. Furthermore, since Storer10 constructed the UD designs
`for the speci(cid:222)c choice of target probability h"1/3, we used that value of h in all simulation
`runs.
`When simulating the Bayesian schemes, EWOC and CRM, we held the feasibility bound (cid:222)xed
`at a"0)25, we assumed that o
`0 and the MTD were independent a priori and we took the
`marginal prior PDF of the MTD as the uniform on the unit interval. To select a prior PDF for o
`0,
`we performed preliminary simulations to examine the performance of the two schemes when the
`prior for o
`0 was either
`
`P0 : the uniform on [0, h]
`
`or
`
`e3(0, h!o
`#e],
`0, o0
`Pe : the uniform on [o
`0).
`
`The results showed that for each selected value of e both CRM and EWOC exhibited more rapid
`dose escalation when we used Pe instead of P0 as the prior PDF. Furthermore, if for a given set of
`i!1 observations xi(e) denotes the dose recommended by EWOC for the ith patient when the
`(e
`
`prior PDF of o0 is Pe, then it can be shown that xi(e1)*xi(e
`2), "e1
`
`2. Consequently, for the
`
`class of priors considered here, the proportion of patients overdosed by EWOC is maximized if
`the ith patient is assigned the dose xi"lime?0xi(e), corresponding to the recommended dose
`
`under the assumption that o
`0 is known. Since the primary purpose of the simulation study was to
`compare the dose escalation schemes with respect to the frequency and magnitude of overdosing,
`we simulated both CRM and EWOC under the assumption that the probability of a dose-
`limiting toxicity at the starting dose was known. That is, to avoid conferring the model based
`schemes with an unfair advantage, we simulated EWOC and CRM under conditions unfavour-
`able to them.
`
`Upon completion of each UD, SA and CRM trial we estimated the MTD as cL "xn‘1, that is,
`
`as the dose level that we would have given the next patient had we allowed the trial to continue.
`This is the estimate typically used in practice. For the UD schemes it generally corresponds to the
`highest dose for which the observed proportion of DLTs was below h, while for CRM it has
`minimal risk with respect to squared error loss. A theoretical basis for the use of this estimate with
`the SA schemes is provided by the fact that xn converges almost surely to the MTD under rather
`general conditions.19 For CRM the estimate cL "xn‘1 is the marginal posterior mean of c given
`Dn. Hence, for the purpose of comparison, we used the posterior mean of c as the estimate of the
`MTD after each simulated EWOC trial. Furthermore, to determine whether the non-parametric
`schemes might compare more favourably to the Bayesian schemes if we had employed a model
`based estimate of the MTD, we used the posterior mean of c to obtain a second estimate of the
`MTD after each UD trial.
`0 and c, we simulated a total of 10,000 trials
`For each of the six combinations considered for o
`for each of the UD and SA designs while, due to their computational complexity, we simulated
`only 2000 trials for each of EWOC and CRM. Each simulated trial consisted of n"24 patients
`for each of the (cid:222)xed sample size methods (all but UD1) and a maximum of 30 patients for the
`variable length scheme UD1.
`
`( 1998 John Wiley & Sons, Ltd.
`
`Statist. Med. 17, 1103—1120 (1998)
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`3.2. Results
`
`In Figures 3 to 12 we compare the dose escalation schemes on the basis of the proportion
`of patients treated at selected dose levels and with respect to the eƒciency with which they
`estimated the MTD. For Figures 3 to 5 and 7 to 12, each point represents the results observed
`0 and c, with the results obtained for c"0)3
`for one of the six selected combinations of o
`being depicted as a#and those for c"0)5 as a]. To facilitate comparison, we show the aver-
`age of the six results observed for each scheme (depicted as a f) and we have drawn a reference
`line through the average of the results obtained for EWOC. For each of the UD schemes we
`show the results obtained using the dose set M0(0)2)1N to the left of those obtained the dose set
`M0(0)1)1N.
`Figure 3 depicts the proportion of patients treated at dose levels for which the probability of
`a toxic response was less than 0)2. These patients were treated at dose levels where the drug is
`relatively inactive and therefore potentially non-therapeutic. The results indicate that EWOC
`treated fewer patients at low, possibly subtherapeutic, dose levels than did any of the non-
`
`parametric schemes. However, for each of the six selected combinations of o0 and c, we observed
`that EWOC treated more patients at the low dose levels than did CRM. This is expected since
`EWOC attempts to protect patients from being overdosed and so tends to treat patients at lower
`dose levels than does CRM.
`Figure 4 displays the proportion of patients treated at dose levels for which the probability of
`a toxic response was less than the target probability h but greater than 0)2. We describe these dose
`levels as optimal in the sense of being near but not above the MTD. The results show that the
`overall proportion of patients treated at optimal dose levels was higher for EWOC than it was for
`any of the other seven dose allocation schemes. Furthermore, for each of the six selected
`
`combinations of o0 and c, EWOC treated more patients at favourable dose levels than any of the
`other methods considered in this study except SA2.
`Figure 5 shows the proportion of patients treated at dose levels above the MTD. The overall
`proportion of patients overdosed by EWOC was 0)193, slightly below the selected feasibility
`bound a"0)25. For each of the six cases considered, EWOC overdosed a smaller proportion of
`patients than did CRM and compared favourably with SA1 and the UD schemes based on 6 dose
`levels. EWOC tended to overdose patients at a higher rate than SA2 and the UD schemes based
`on 11 dose levels. However, the latter schemes were overly conservative, treating a large
`proportion of patients at extremely low dose levels. Note that by choosing smaller values of a
`it is possible to reduce the proportion of patients overdosed by EWOC to levels that are
`comparable to those observed for any of the nonparametric designs. Figure 6 provides an
`indication of the extent to which EWOC a⁄orded protection from overdosing relative to CRM,
`"0)1 and c"0)3. There, we can see that EWOC treated only 31
`for the particular case where P1
`per cent of the patients at dose levels above the MTD, while CRM overdosed nearly twice as
`many. Relative to CRM, EWOC can have a substantial impact on the level of overdosing in
`a phase I clinical trial.
`Figure 7 shows the proportion of patients treated at dose levels for which the probability
`of a toxic response exceeded 0)5. These results show that EWOC subjected fewer patients to
`these severely toxic doses than did any of the other methods except SA2. When the MTD
`was equal to 0)5, CRM treated over three times as many patients at unacceptably high dose
`levels as EWOC. Overall, EWOC treated fewer patients at dose levels signi(cid:222)cantly above the
`target MTD.
`
`( 1998 John Wiley & Sons, Ltd.
`
`Statist. Med. 17, 1103—1120 (1998)
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`J. BABB, A. ROGATKO AND S. ZACKS
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`Figure 3. Proportion of patients given doses for which the probability of a severe toxic reaction is less than or equal to
`1/5. Each]and#represents the results from all simulation runs for a particular parameter combination. Each point, (f),
`is the average of the results obtained for a particular method at the six parameter combinations considered. For each of
`the UD schemes the results obtained when only 6 dose levels were used are shown to the left of the results obtained when
`11 levels were used
`
`Figure 4. Proportion of patients given dose levels for which the probability of a severe toxic reaction is greater than 1/5
`but not greater than the target probability h"1/3 (see Figure 3 for symbol de(cid:222)nitions)
`
`Figure 5. Proportion of patients given dose levels above the MTD (see Figure 3 for symbol de(cid:222)nitions)
`
`( 1998 John Wiley & Sons, Ltd.
`
`Statist. Med. 17, 1103—1120 (1998)
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`INTELGENX 1031
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`DOSE ESCALATION WITH OVERDOSE CONTROL
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`Figure 6. Histograms of the probability of toxic response for the dose levels selected by CRM and EWOC. For the
`intervals [0, 0)05), [0)05, 0)1), 2 , [0)9, 0)95), [0)95, 1)0] each histogram shows the proportion of patients given a dose level
`such that the probability of a severe toxic reaction is contained in that interval. The number in the body of each histogram
`in the proportion of patients given a dose level above the MTD
`
`( 1998 John Wiley & Sons, Ltd.
`
`Statist. Med. 17, 1103—1120 (1998)
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`INTELGENX 1031
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`J. BABB, A. ROGATKO AND S. ZACKS
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`Figure 7. Proportion of patients given dose levels for which the probability of a severe toxic reaction was greater than 1/2
`(see Figure 3 for symbol de(cid:222)nitions)
`
`Figure 8. Average bias of c(cid:246) , the estimate of the MTD (see Figure 3 for symbol de(cid:222)nitions)
`
`Figures 8 to 11 compare the dose escalation schemes with respect to the eƒciency of c(cid:246) , the
`estimator of the MTD. The bias observed for each estimator (computed as cL !c) is shown in
`Figures 8 and 9 and the root mean squared error (JMSE) in Figures 10 and 11. In Figures 8 and
`10 the results are given for the non-parametric schemes when the MTD was estimated as the last
`recommended dose (that is, cL "xn‘1). Figures 9 and 11 compare the results obtained for EWOC
`and the UD schemes when the MTD was estimated as the mean of the marginal posterior
`distribution of c. All of the non-parametric methods tended to provide signi(cid:222)cantly biased
`estimates of the MTD while the overall mean bias associated with both of the Bayesian schemes
`was near zero. With respect to both bias and JMSE, EWOC provided more eƒcient estimates of
`the MTD than did any of the non-parametric methods. However, for each of the six cases
`considered, CRM estimated the MTD with smaller JMSE than did EWOC. This was anticip-
`ated, because by construction CRM should tend to select dose levels closer to the MTD, allowing
`more eƒcient estimation. Thus, a slight decrease in the accuracy of the MTD estimate is the price
`one pays for incorporation into EWOC of protection from overdosing.
`
`( 1998 John Wiley & Sons, Ltd.
`
`Statist. Med. 17, 1103—1120 (1998)
`
`INTELGENX 1031
`
`

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`DOSE ESCALATION WITH OVERDOSE CONTROL
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`1117
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`Figure 9. Average bias of c(cid:246) , the estimate of the MTD (see Figure 3 for symbol de(cid:222)nitions)
`
`Figure 10.