The continual reassessment method and its applications: a Bayesian methodology for phase I cancer clinical trials

Transcription

1 STATISTICS IN MEDICINE Statist. Med. 2001; 20: (DOI: /sim.735) The continual reassessment method and its applications: a Bayesian methodology for phase I cancer clinical trials Naoki Ishizuka 1; and YasuoOhashi 2 1 Cancer Information and Epidemiology Division; National Cancer Center Research Institute; Tukiji; Chuo-ku; Tokyo , Japan 2 Biostatistics; School of Health Sciences and Nursing; University of Tokyo; Hongo 7-3-1; Bunkyo-ku; Tokyo; ; Japan SUMMARY We discuss the continual reassessment method (CRM) and its extension with practical applications in phase I and I=II cancer clinical trials. The CRM has been proposed as an alternative design of a traditional cohort design and its essential features are the sequential (continual) selection of a dose level for the next patients based on the dose toxicity relationship and the updating of the relationship based on patients response data using Bayesian calculation. The original CRM has been criticized because it often tends to allocate too toxic doses to many patients and our proposal for overcoming this practical problem is to monitor a posterior density function of the occurrence of the dose limiting toxicity (DLT) at each dose level. A simulation study shows that strategies based on our proposal allocate a smaller number of patients to doses higher than the maximum tolerated dose (MTD) compared with the original method while the mean squared error of the probability of the DLT occurrence at the MTD is not inated. We present a couple of extensions of the CRM with real prospective applications: (i) monitoring ecacy and toxicity simultaneously in a combination phase I=II trial; (ii) combining the idea of pharmacokinetically guided dose escalation (PKGDE) and utilization of animal toxicity data in determining the prior distribution. A stopping rule based on the idea of separation among the DLT density functions is discussed in the rst example and a strategy for determining the model parameter of the dose toxicity relationship is suggested in the second example. Copyright? 2001 John Wiley & Sons, Ltd. 1. INTRODUCTION The diculty of phase I cancer clinical trials of cytotoxic drugs is well described in a review paper by Ratain et al. [1]. The main purpose of phase I trials is to determine or estimate the maximum tolerated dose (MTD) and the recommended dose (RD) for following phase II trials. There is usually a deviation between research objectives and the incentive of patients Correspondence to: Naoki Ishizuka, Cancer Information and Epidemiology Division, National Cancer Center Research Institute, Tukiji, Chuo-ku, Tokyo , Japan. Contract/grant sponsor: Second Term Comprehensive 10-years Strategy for Cancer Control; contract/grant number: H10-Gan-027 and H12-Gan-012 Contract/grant sponsor: Ministry of Health, Labour and Welfare; contract/grant number: H-2 Copyright? 2001 John Wiley & Sons, Ltd.

2 2662 N. ISHIZUKA AND Y. OHASHI who expect (even minimal) ecacy of treatment. A recommended dose is often inaccurate due to the large variability of patients responses and there is a dilemma, that is, investigators want to increase the dose level quickly in order to give an adequate dose level to patients. On the other hand, a quick dose escalation may cause serious life-threatening toxicity or even toxic deaths that can be avoided by using a cautious dose escalation strategy. In a traditional so-called cohort design, the dose level is escalated to the next higher level when a xed number of patients, usually three, do not show the dose limiting toxicity (DLT) dened in the trial protocol. If one of three patients shows the DLT, then three more patients are added to the same dose level. If more than one (or two) patient do not show the DLT among six patients, then the dose level is increased. If more than one patient shows the DLT in the original cohort with three patients or more than two in the expanded cohort, then the dose escalation is terminated and the current dose level is estimated as the MTD. Many questions will arise in the mind of naive statisticians: (i) Are three patients enough to conrm the safety of a dose level? (ii) Are three or six patients enough to estimate the MTD with acceptable accuracy? Why is the information of patients response data at lower doses not utilized in the decision making process? (iii) Why should we wait for three patients response in the cohort when severe toxicity appears in the rst or rst two patients? (iv) Should de-escalation of doses be conducted in some cases? In practice, the above traditional design works well (surprisingly well from the authors viewpoint) in many situations. The main reason is that investigators are informally and implicitly utilizing the information of all available data including patients response data (of not only the DLT but also minor toxicity) at lower doses and their prior experience in trials of the same drug with dierent administration schedules or trials of similar drugs. Dose de-escalation is often done in real practice and decision making before observing three patients response is not so rare. In other words, the traditional cohort design can function in real practice with informal judgement of experienced investigators. For overcoming the dilemma between research objectives and patients incentive explained above and making the decision making process more explicit, several approaches have been proposed. Collins et al. [2] proposed the pharmacokinetically guided dose escalation (PKGDE) for accelerating the dose escalation based on the empirical fact that the area under the curve (AUC) of the plasma concentration of the compound or its active metabolite at the LD 10 in mice is near tothe AUC at the MTD in humans. Simon et al. [3] discussed accelerated strategies including an idea of intrapatient dose escalation using the information of mild toxicity (grade 2). Mick and Ratain [4] proposed the application of a linear model relating nadir white blood cell (WBC) count to dose and pretreatment WBC and tried to estimate the optimal dose for each patient. The continual reassessment method (CRM), introduced by O Quigley et al. [5], is a Bayesian approach and its goal is to reduce the number of patients who receive lower doses, which are not expected to be eective, and to obtain a more accurate estimate of the MTD. The CRM has a couple of attractive characteristics, which cannot be expected with the traditional cohort design. One is to give quantitative interpretation for the probability of the DLT occurrence. The other one is the possibility of explicit utilization of prior information from various sources for expressing the ambiguity of the dose toxicity relationship before the trial.

3 CONTINUAL REASSESSMENT METHOD AND ITS APPLICATIONS 2663 In Section 2, mathematical notations are introduced and the original and modied versions of the CRM are briey discussed. The original CRM has been criticized because it often tends to allocate too toxic doses to many patients and our proposal is to monitor a posterior density function of the DLT occurrence at each dose level. A stopping rule based on the idea of separation among the DLT density functions is discussed in the rst example of Section 4 and a strategy for determining the model parameter of the dose toxicity relationship is suggested in the second example. A simulation study in Section 3 shows that strategies based on our proposal allocate a smaller number of patients to doses higher than the MTD compared with the original method while the mean squared error of the DLT occurrence at the MTD is not inated. In Section 4 we present a couple of extensions of the CRM with real prospective applications: (i) monitoring ecacy and toxicity simultaneously in a combination phase I=II trial; (ii) combining the idea of PKGDE and utilization of animal toxicity data in determining the prior distribution. 2. CRM 2.1. The original CRM We adopt the notation in O Quigley et al. [5] as follows: Dose level: x i (i =1;:::;k). Response of jth patient: { 1 if toxic response Y j (j =1;:::;n) 0 if no toxic response Dose response (toxicity) model: (x i ;a), for example, E[Y j ]= = (x i ;a)= exp(3 + ax i) (1) 1 + exp(3 + ax i ) Prior distribution for the parameter a for jth patient: f(a; j ) where j = {y 1 ;:::;y j 1 } and f(a; 0 j ) da =1 (j =1;:::;n). Prior mean response probability for jth patient and ith dose: ij = 0 (x i ;a)f(a; j ) da (i =1;:::;k) (2) Approximate mean response probability with the use of prior mean of parameter a, (j): ij = {x i ;(j)} (i =1;:::;k); (j)= 0 af (a; j ) da (3) The original CRM proceeds as follows: (i) A dose response (toxicity) model (x i ;a) is assumed, where x i ; i=1;:::;k are predetermined dose levels; a is a model parameter. (ii) Assume a prior distribution for a. The unit exponential distribution g(a) = exp( a), a vague prior, has been used in most previously published examples. (iii) Dene the target probability of DLT, ; per cent are usually used in practice.

4 2664 N. ISHIZUKA AND Y. OHASHI (iv) Assign the jth patient todose level x(j). (Allocation rules are discussed later.) Once the dichotomous response y j of toxic or no toxic response is observed, the posterior distribution of a is updated using the following Bayesian calculation. Then, calculate a revised (prior) probability density of the DLT occurrence at each dose level. This process continues until a predetermined xed sample size, for example, 25, or some other condition is satised. j l=1 (x(l);y l;a) f(a; j )(x(j);y j ;a) f(a; j+1 )= f(u; 0 j )(x(j);y j ;u) du = g(a) g(u) j 0 l=1 (x(l);y l;u) du where (x(j);y j ;a)=[ (x(j);a)] yj [1 (x(j);a)] (1 yj) The modied CRM Some controversial issues in the CRM have been raised since the original article [5] appeared. With respect toconcerns by Ratain et al. [1] the original CRM tends to allocate a higher dose compared with the traditional design and some modications have been suggested by Faries [6; 7], Korn et al. [8] and Moller [9]. The main suggestions have been to start with the lowest dose and not to allow jumping in dose escalation. Goodman et al. [10] alsosuggested an option of assigning two or three patients at one time in addition to the above modications. Piantadosi and Lui [11] proposed the use of pharmacokinetic information. Chevret [12] investigated the operational characteristics of the CRM by simulation and explored the eect of selection of the intercept parameter of logistic regression. O Quigley and others [13 15] investigated the property of the CRM further and discussed the estimation problem based on the likelihood approach. As regards the dose selection for the jth patient, the original paper suggested selecting the dose level which minimizes the following three criteria: ( ij ;); ( ij;); (x j ; a=()) 1 where (a; b)= a b The authors emphasized the advantage of using the latter two of these criteria because of the reduction in the number of innite integrations to be performed, thus reducing computational resource. Goodman et al. [10] also mentioned that the calculation of the expected a produces essentially the same result as calculating the expected values of the probabilities themselves, with substantially less computation. O Quigley et al. [5] and O Quigley [14] adopted the selection rule based on ( ij;) in their illustrative examples and simulation studies, and the same rule has been adopted in all subsequent simulation studies by the other authors without critical examination. Only the original paper [5] mentioned the usage of ij but there has been no following discussion on the dierence in operating characteristics between selection rules based on ij and those on ij. Our proposal here is to use the mean probability of the DLT occurrence ij instead of its approximation ij, taking account of a wide (often skewed) variability in prior distribution of the DLT occurrence. The main aim of the proposal is to prevent dangerous dose escalation at an early stage of trials, as is illustrated as follows. The data in Table I are the reproduction of an illustrative example of O Quigley et al. [5], where the dose response (toxicity) relationship was modelled as (x i ;a)={(tanh x i +1)=2} a with a target response of 20 per cent and g(a) = exp( a) is assumed as a non-informative prior for the parameter a. Figure 1 shows the assumed dose response model with Bayesian (4)

5 CONTINUAL REASSESSMENT METHOD AND ITS APPLICATIONS 2665 Table I. Original example: hypothetical trial of 25 patients, Pr(Y =1) = {(tanh x i +1)=2} a, y j is response of the jth patient, x i is selected dose level, (j) is prior mean of the parameter a. j (j) x i y j x x x x x x x x x x x x x2 1 j (j) x i y j x x x x x x x x x x x x1 1 (25+1) 0.56 x1 Figure 1. Assumed dose response model in the numerical example of O Quigley et al. [5]. credibility intervals (50 per cent, 90 per cent). The mean curve stands for the curve of (x i ; 1) and it does not correspond to the mean (expected) probability ij. The prior mean of the DLT occurrence at each dose level is as follows: i x i 1:47 1:1 0:69 0: ij ij If we take account of the prior ambiguity of the dose response curve of Figure 1 and be cautious in giving too toxic doses to patients, the lowest dose will be assigned to the rst patient because ij, which reects the (skewed) variability of the prior distribution, is equal to

6 2666 N. ISHIZUKA AND Y. OHASHI Figure 2. Prior density function of DLT occurrence for the rst patient: (a) hyperbolic target function; (b) logistic function. the target toxicity. On the other hand, ij is determined only by the prior mean of a and does not reect any variability or the prior distribution. The two quantities ij and ij converge to each other as the information accumulates and the posterior distribution of a becomes more shrunken and symmetric. There is, however, a non-negligible dierence between the two quantities at early stages of the trial, resulting in dierent selection of dose levels. The model parameter a is a purely instrumental one and dicult to interpret, especially for clinical investigators. We propose monitoring the distribution of the DLT occurrence probability instead of a. The distribution of is directly derived from that of a by a simple variable-transformation as follows: [ ] f ( )= da f(a; j ) d f(a; j )= {log(tanh x i +1) log 2} (5) The (prior) distribution of the MTD, where the mean of the DLT occurrence probability is equal tothe target value, can alsobe derived in a similar way. Faries [7] suggested modelling the prior information for the MTD in the CRM. However, we have learned through discussion with clinical investigators that the density function of the DLT occurrence is intuitively easy to interpret for clinicians and useful in the decision making process of determining the dose for the next patients. If the prior distribution for the parameter a is exponential, the density functions become the gures shown in Figure 2(a) and the corresponding gures for the logistic regression model are shown in Figure 2(b). It is quite natural for these density functions to have a non-informative appearance and it is also reasonable to start from the lowest dose level if the probability that the DLT occurrence is over the dangerous level (say, per cent) is not negligible. In the example of Table I, the prior densities of the DLT occurrence are updated for the rst seven patients as shown in Figure 3; the nal ones with 25 patients are shown in Figure 4. These gures demonstrate that higher doses are allocated at earlier stages with the original CRM based on ij.

7 CONTINUAL REASSESSMENT METHOD AND ITS APPLICATIONS 2667 Figure 3. Updates of density function of DLT occurrence.

8 2668 N. ISHIZUKA AND Y. OHASHI Figure 4. Final posterior density function of DLT occurrence with 25 patients. In order to evaluate the speed of dose escalation, the selected sequence of dose levels is calculated for each of the escalation strategy in the case where there is no DLT observed: Original ij Modied ij Modied ij 2 patients=group ij Modied ij 2 patients=group In the above table, modied means the strategy which starts from the lowest dose and prohibits the jumping over one dose level and 2 patients means the strategy which requires a minimal two patients at a newly selected dose level for pharmacokinetic measurements. In this example, the allocation rules based on ij do not allow a dose escalation of more than one level higher than the previous. This hypothetical example also suggests that the stopping rule proposed by Korn et al. [8] and Goodman et al. [10], which asks for trial termination when the same dose level is selected in a consecutive series of six patients, prohibits the probably safe dose escalation over the fth level. In passing, a strategy of allocating two or more patients at one dose level should be preferably selected if information on linearity and variability of pharmacokinetic parameters is required. In summary: (i) Allocation rules based on ij should be used if the ambiguity of the (prior) distribution is taken intoaccount. (ii) The monitoring of the density function of provides intuitive interpretation of the current situation. It is possible to incorporate direct information derived from the density function of in the decision making of dose escalation or trial termination. One practical suggestion used in the examples of Section 4 is to prohibit the dose escalation when the probability that the DLT occurrence is over a prespecied level

9 CONTINUAL REASSESSMENT METHOD AND ITS APPLICATIONS 2669 Table II. Setting of simulation study. Number of patients in one trial 25 Number of trials 1000 Number of dose levels 6 Dose response model = (xi;a)= exp(3+ax i) 1+exp(3+ax i ) (i =1;:::;6) Prior distribution for a unit exponential Prior estimate i1 = (x i; 1) (i =1;:::;6) 0.05, 0.10, 0.25, 0.35, 0.50, 0.70 Target response probability 0.25 True toxic response Curve A: 0.05, 0.10, 0.25, 0.35, 0.50, 0.70 Curve B: 0.001, 0.01, 0.04, 0.09, 0.24, 0.49 Curve C: 0.30, 0.40, 0.52, 0.61, 0.76, 0.87 Allocation rule (1): original ij (1) : modied ij (2): ij (2) : modied ij (2) : modied ij 2 patients per group for rst 10 patients (say, per cent) is considerable (say, per cent). If L prespecied level, the above probability is calculated as follows: stands for a where Pr[ L ]= 1 L f ( ) d = al 0 f(a; j ) da a L = x 1 log L i ( L )= log(tanh x i +1) log 2) (6) (iii) As regards whether more than one patient is assigned to one dose level, there is no denite answer and it should be determined taking account of the balance between the precision of pharmacokinetic information (study objective) and the speed of dose escalation (patients incentive). It also depends on the condence in safety at lower dose levels. Some hybrid strategies, such as a combination of the PKGDE, the CRM and an idea of intrapatient escalation, are worth exploring. 3. SIMULATION 3.1. Method The design of the simulation study here is similar to those of Chevret [12] and Korn et al. [8] and its main objective is to see the dierence in operating characteristics among the allocation rules. The performance measures evaluated are the nal dose level with 25 patients, the posterior probability at the true target dose level, the number of experimentation (allocated patients) at each dose level and the number of patients who showed the DLT. The parameters used in the simulation are shown in Table II and ve allocation rules described in the previous section were examined. For each patient, one uniform random number

10 2670 N. ISHIZUKA AND Y. OHASHI Table III. Result of simulation study: curve A, 0.05, 0.10, 0.25, 0.35, 0.50, Allocation rule Per cent (%) True probability of dose level (1): original ij Recommendation Experimentation (1) : modied ij Recommendation Experimentation (2): ij Recommendation Experimentation (2) : modied ij Recommendation Experimentation (2) : modied ij 2 patients=group Recommendation for rst 10 patients Experimentation Allocation rule Posterior probability Number of toxic responses at true target dose Mean SD MSE Mean Maximum Minimum (1): original ij (1) : modied ij (2): ij (2) : modied ij (2) : modied ij 2 patients=group for rst 10 patients U was generated. A binomial response was dened as 1 (the occurrence of the DLT) if U was larger than or equal to the true toxic response probability at the administrated dose level for the patient and dened as 0 (no occurrence) otherwise Results The summary statistics of performance measures are tabulated in Tables III, IV and V. As is expected, the rules based on ij are generally superior to those based on ij. In Table III (curve A), the proportion of patients who received the 5th or 6th dose level, which is very toxic, is 15.5 per cent using the original CRM. Our modication reduces this proportion to 10.9 per cent, 8.5 per cent 5.4 per cent for allocation rules (2), (2) and (2), respectively, while the distribution of the estimated MTD (the nal dose level) is similar. Similar phenomena are observed in Table IV (curve B). In Table V (curve C), the proportions of patients who received the lowest (but still higher than the target) dose level is 66.9 per cent if the original CRM is used and this number increased to76.4 per cent, 76.8 per cent and 80.1 per cent with our modications. The target toxicity level or the nearest toxicity level to the target are 25 per cent (curve A), 24 per cent (curve B) and 30 per cent (curve C), respectively. There was small or no bias in the posterior mean of the DLT occurrence from the true values irrespective of allocation rules. However, regardless of the type of curves, the mean squared error (MSE) was smaller in the rules based on ij. The rules based on ij also gave fewer DLT responses compared with the rules based on ij.

11 CONTINUAL REASSESSMENT METHOD AND ITS APPLICATIONS 2671 Table IV. Result of simulation study: curve B, 0.001, 0.01, 0.04, 0.09, 0.24, Allocation rule Per cent (%) True probability of dose level (1): original ij Recommendation Experimentation (1) : modied ij Recommendation Experimentation (2): ij Recommendation Experimentation (2) : modied ij Recommendation Experimentation (2) : modied ij 2 patients=group Recommendation for rst 10 patients Experimentation Allocation rule Posterior probability Number of toxic responses at true target dose Mean SD MSE Mean Maximum Minimum (1): original ij (1) : modied ij (2): ij (2) : modied ij (2) : modied ij 2 patients=group for rst 10 patients Table V. Result of simulation study: curve C, 0.30, 0.40, 0.52, 0.61, 0.76, Allocation rule Per cent (%) True probability of dose level (1): original ij Recommendation Experimentation (1) : modied ij Recommendation Experimentation (2): ij Recommendation Experimentation (2) : modied ij Recommendation Experimentation (2) : modied ij 2 patients=group Recommendation for rst 10 patients Experimentation Allocation rule Posterior probability Number of toxic responses at true target dose Mean SD MSE Mean Maximum Minimum (1): original ij (1) : modied ij (2): ij (2) : modied ij (2) : modied ij 2 patients=group for rst 10 patients

12 2672 N. ISHIZUKA AND Y. OHASHI Table VI. Dose levels and corresponding prior estimate of toxic response occurrence of JCOG9512. Level x CPT-11 VP-16 CDDP Prior estimate mg=m 2 40 mg=m 2 60 mg=m mg=m 2 40 mg=m 2 60 mg=m mg=m 2 50 mg=m 2 60 mg=m mg=m 2 50 mg=m 2 60 mg=m mg=m 2 60 mg=m 2 60 mg=m EXAMPLES 4.1. JCOG9512 study Combination chemotherapy of two or more drugs is commonly used in treating cancer patients to achieve a higher response rate or cure. Although toxicity and ecacy of each drug at its recommended dose has already been examined in the previously conducted phase II trials, another trial, the so-called phase I=II trial, is required todetermine the MTD and=or the RD in a combination setting where overlapping toxicity often prevents the administration of each recommended dose. In contrast to phase I trials, investigators are interested in not only toxicity but also the ecacy of the new combination regimen and there is relatively ample prior knowledge as regards the ecacy and toxicity of test drugs. There are few research papers on the study design which focus on both toxicity and ef- cacy simultaneously in early phase cancer clinical trials. Thall and Russell [16] applied a Bayesian approach and used the proportional odds model for describing both toxicity and ecacy simultaneously. We have once tried to extend the CRM to deal with both toxicity and ecacy (tumour shrinkage or objective response) assuming a correlation structure between two bivariate variables (not published). Such a bivariate modelling, including those by Thall and Russell, is, however, often not practically feasible due to the delay in observing the objective response. Another diculty in bivariate modelling lies in an inherent complexity which exists in the relationship between toxicity and ecacy; positive correlation may occur due to the interpatient variability in pharmacokinetic variability while on the other hand negative correlation is also possible because some patients, who could expect tumour shrinkage after several courses of chemotherapy, may drop out due to the appearance of (not severe) toxicity. In spite of such diculty of modelling, the CRM has appealing features in phase I=II situations because it can provide an explicit strategy for dose escalation and also an objective termination rule of the phase II part of the trial, the objective of which is to estimate the objective response rate and possibly the toxicity prole with enough precision to judge the prospect for the conduct of phase III trials with the candidate combination regimen. Here we apply the CRM approach for marginal distributions of toxicity and ecacy while recognizing the future possibility of exible bivariate modelling. The Japan Clinical Oncology Group (JCOG) 9512 protocol [17] was designed to nd the appropriate dose levels of two drugs, CPT-11 and VP-16, with the xed dose level of CDDP for the three-drug combination chemotherapy for non-small-cell lung cancer patients. The dose levels selected are shown in Table VI with the prior estimates of the DLT occurrence. The

13 CONTINUAL REASSESSMENT METHOD AND ITS APPLICATIONS 2673 Figure 5. Assumed dose response model for JCOG9512 study. Figure 6. Prior density function of DLT occurrence at each dose level for JCOG9512. prior knowledge was reected in the determination of the dose response model: = (a; x)= exp(2:49 + ax) 1 + exp(2:49 + ax) and the normal prior distribution of the slope parameter a, whose mean was set to and standard deviation was set to 0:729=2:5; as is shown in Figures 5 and 6. The target toxicity level was dened as 33 per cent and the corresponding prior distribution of the MTD is shown in Figure 7. Actually, the mean of the prior distribution of a was determined from an initial guess based on the toxicity prole of two drug combinations of each of CPT-

14 2674 N. ISHIZUKA AND Y. OHASHI Figure 7. Prior density function of MTD: Less condent is the prior which was actually used for JCOG9512 study. 11 and VP-16 with CDDP and the standard deviation was determined after discussion with clinical investigators on the ambiguity (condence) of the initial guess of the MTD. Sensitivity analysis by changing the standard deviation parameter is possible and should often be useful in practice. The dose escalation strategy was started from the lowest dose level and based on our modied CRM described in the previous section. At least three patient were treated at each dose level because some ecacy was expected even at the lowest level and there was no ethical concern. The dose level was not escalated when Pr[ 50%] 1=3. The actual dose escalation history is shown in Figure 8; one patient at level 2 and two patients at level 4 experienced the DLT. As regards stopping rules of the CRM, Korn et al. [8] proposed terminating a trial if some prespecied number of patients, say 6, are administered the same dose level. O Quigley and Reiner [18] proposed stopping if the selected dose for the next patient is unchanged irrespective of the present patient s response and Heyd and Carlin [19] proposed stopping if the estimated MTD has a specied high precision. The attitude of Thall et al. [20] is against the specication of formal stopping rules. The proposal by Korn et al. seems to fail in some practical situations as is suggested in the previous section and we believe that the decision to terminate the trial or redene the dose levels should be based on the relative condence (narrowness) and separation of density functions of the DLT occurrence. That is: (i) We can terminate the trial if the (posterior) density functions among dose levels are well separated or we should redene a new dose level if the separation is good and any dose level does not cover the targeted toxicity level. (ii) We can terminate if the credibility interval of the DLT occurrence at the targeted dose level is narrow enough.

15 CONTINUAL REASSESSMENT METHOD AND ITS APPLICATIONS 2675 Figure 8. Dose escalation history and toxic response in JCOG9512 study. Figure 9. Posterior density function of DLT occurrence with 20 patients for JCOG9512 study. The above decision rule can be simply implemented by calculation and graphical presentation of the density functions of the DLT occurrence. In this example, the investigators and the advisory board approved the termination of the trial at the 20th patient; Figure 9 shows the posterior density functions at this point and good separation among densities was observed as is shown in Table VII. The nal recommended dose was determined as the level 4. As regards ecacy, ve patients experienced the objective response among ten patients whowere treated at level 4. The exact 90 per cent condence interval was per cent and this promising regimen is currently being examined in a randomized phase II trial in JCOG.

16 2676 N. ISHIZUKA AND Y. OHASHI Table VII. Posterior mean of distribution of toxic response occurrence at each dose level. Level Mean Pr[ 1=2] TOP-53 trial The design of this trial [21] is a hybrid combination of the CRM with the pharmacokinetic guided dose escalation (PKGDE) introduced in the Introduction. The TOP-53 is an analogue compound of VP-16, a widely used cancer drug. The pharmacokinetics of TOP-53 is linear in mice and dogs, and there is no active metabolite. Myelosuppression in the DLT in animal data and the strong correlation between the AUC and toxicity in animal data were observed. These phenomena were expected to hold in humans and these are prerequisites for conducting the PKGDE. The toxicity prole was similar to VP-16 in animal data, possibly alsoin human data and there were alsoavailable data from a phase I trial of etoposide. The starting dose was determined as 1=10th of mice LD 10 level (5:7 mg=m 2 =ln). We planned to use the PKGDE of 100 per cent increase of doses until 40 per cent of mice LD 10 level and a 33 per cent dose increase was planned to follow. The AUC of the next dose was estimated by using a posterior t-distribution under the assumption of a linear model relating the AUC to dose (without a constant term) and a vague prior for the slope parameter. After the 10th patient, the CRM calculation was started and the possibility of dose de-escalation was taken account. The target toxicity of the MTD was set to 33 per cent and Pr[ 0:33] was also monitored during the trial. For determining the prior distribution, animal toxicity data were utilized. The same toxicity experiments for determining the LD 10 or the toxic dose low (TDL) were conducted using both VP-16 and TOP-53 for three species: mice, rats and dogs. By comparing the LD 10 (for mice and rats) or the TDL (for dogs) levels, converting coecients or equitoxic dose of VP-16 to that of TOP-53 were estimated as 8.4 (95 per cent condence interval: 6 11) for mice, 3.5 (3 4) for rats, and 2.1 (condence interval: not available) for dogs. Phase I data of VP-16 for Japanese gave a guess that the MTD was around mg=m 2 (50 per cent, 30 per cent and 20 per cent at 360mg=m 2 ; 480mg=m 2 and 540mg=m 2, respectively). These twokinds of prior information were combined to provide a prior distribution of the MTD for TOP-53 shown in Figure 10 where each of three density functions for mice, rats and dogs corresponds to the estimated prior density function when the converting coecient of humans is the same as that of each species, respectively. We assumed the one-parameter logistic model with a gamma prior distribution for the slope part of the model: = (dose;a)= exp( 4:19 + a dose) 1 + exp( 4:19 + a dose) g(a)= 114:67 a 6 exp( 114:6a) (7) (11) (12)

17 CONTINUAL REASSESSMENT METHOD AND ITS APPLICATIONS 2677 Figure 10. Re-estimation of prior distribution of MTD density function with four patients using pharmacokinetic data. Figure 11. Assumed dose response model and credibility region (intervals) for TOP-53 study. where dose was measured in mg=m 2. The prior dose response (toxicity) relationship is shown in Figure 11 with credibility intervals and the gure of the corresponding prior density function of the MTD is shown on the left side of Figure 12. Here the prior distribution for the slope was selected so that the resulting distribution of the MTD could cover a range of expected distributions from animal data (especially from mice and rats). The prior distribution was slightly modied after observing four patients at the rst and the second levels by comparing the AUC level of humans and mice and it is superimposed in Figure 10.

18 2678 N. ISHIZUKA AND Y. OHASHI Figure 12. Updates of density function of MTD (left, initial prior; right, posterior with 24 patients). Figure 13. Eect of the xing intercept parameter on credibility region (intervals) of dose response curve: logit = a 0 + a (meta-meter). In this TOP-53 trial the starting dose was expected to be low enough not to show toxic response. The investigators agreed to set narrower credibility intervals at the lower dose levels while wider intervals were assumed at higher dose levels. This is a completely opposite situation to the previous example of the JCOG9512 trial where the toxicity at the higher doses was expected to be high enough with high credibility from prior experience. In the TOP-53 trail, it is impossible to reect this prior knowledge if the positive intercept, say 3, is used in the logistic model. Figure 13 shows the eect of selecting an intercept parameter on the credibility intervals of the dose response curve (x i ; â) when a unit exponential distribution is assumed for the slope parameter a. It is clear that spindle-shaped credibility regions converge to one xed point with a vertical axis value of logit 1 (a). If a positive value, say 3, is selected for a, only a region of toxicity probability (0; logit 1 (3)=0:9526) can be modelled irrespective of the determination of meta-meter x, a linear form of dose level.

19 CONTINUAL REASSESSMENT METHOD AND ITS APPLICATIONS 2679 Figure 14. Dose escalation history and toxic response in TOP-53 study. Figure 15. Updates of prior density function of DLT occurrence at each dose level (left, initial prior; right, posterior with 24 patients). If a negative value is selected, only a region (logit 1 (a); 1) can be modelled. Chevret [12] recommended using the value 3 for the intercept parameter from a simulation study. We, however, emphasize that the selection should be made taking account of the ambiguity in prior knowledge and considering the implicit restriction on the modelled shape of credibility regions. The dose escalation and de-escalation history of the TOP-53 trial are shown in Figure 14 and we could save a number of patients compared with the traditional cohort design. The study was terminated with 25 patients; one of them dropped out without receiving the test drugs and four patients showed the DLT responses. The estimated MTD was 18:8n and the corresponding nal density function of the MTD is shown on the right side of Figure 12. The posterior density functions of the DLT occurrence are shown in Figure 15; Pr[ 0:33]

20 2680 N. ISHIZUKA AND Y. OHASHI was estimated as 7 per cent and 70 per cent at 14:1n and 18:8n, respectively and a good separation between density functions was observed. 5. DISCUSSION We are now aware of many real trials where the CRM or its modication was used or stated tohave been used (for example, in Rinaldi et al. [22] and in many presentations in the annual meeting of the American Society of Clinical Oncology after 1997) but almost all presentations are lacking a description on methodological=statistical details such as selection of the dose response (toxicity) model or a determination of the prior distribution. It is a pity that we cannot know whether there are essential contributions and=or practical problems to be solved in the application of this probably innovative methodology, the CRM, from such presentations. The original CRM based on approximate estimates of the mean DLT probability, ij in our notation, has a tendency to allocate higher doses over the target. We have demonstrated that our modied version based on the directly estimated mean probability, ij in our notation, is theoretically superior to the original one in some operating characteristics evaluated in Section 3, although the simulation study presented in this paper is a limited one. We proposed utilizing the posterior probability that the DLT occurrence probability is over a specied toxic level in the decision making process. This is a useful measure both for safety monitoring and for judgement on the separation between density functions among dose levels, which was illustrated in two examples in Section 4. Some other useful measures derived from posterior densities may be created and utilized. The CRM is often justied from two aspects: its robustness against the selection of the prior distribution and its operating characteristics, which enable rapid dose escalation. In our personal view, the former is the other side of the same coin of ignorance and the latter is a (sometimes hazardous) consequence of neglecting the ambiguity of the prior information. We believe the main characteristics or potential of the CRM is in its exibility which enables the method to work as a communication tool between clinicians and statisticians and to work as a monitoring tool in early phase cancer clinical trials where complicated decision making is often required. The practical diculty in applying the CRM, which is also the key factor for fully utilizing the CRM, is in the selection of a dose response (toxicity) model and the determination of the prior distribution. In Section 4.2 we discussed an implicit restriction on the model, which is caused by the selection of the location parameter and has been ignored in previous research. In contrast to pure phase I trials of new compounds, we can utilize many information sources in phase I=II situations such as data on monotherapy. Even in phase I trials, we may be able to utilize foreign data, data from another administration schedule, data from mother compounds and animal toxicity data. We illustrated the utilization of the last two kinds of information in Section 4.2, although such an approach may be feasible only in the development of analogue compounds. In summary, much more eort should be made to determine the model and the prior distribution in applying the CRM. Such an eort should be based on a mutual communication between clinicians and statisticians and on scientic data. We believe such an attitude is necessary for creating a novel approach in this dicult eld with a dilemma between research and ethics.

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