\
`
`• tn •
`
`Volume 15 Number 15
`
`15 August 1996
`
`T. Colton
`A. L. Johnson
`D. Machin·
`
`SMEDDA~ 15(15) 1605-1712 (1996)
`ISSN 0277-6715
`
`INTELGENX 1036
`
`

`
`Statistics in Medicine
`
`Theodore Colton
`Boston University School of
`Public Health, 80 East
`Concord Street, Boston,
`Massachusetts 02118, U.S.A.
`
`EDITORS
`Tony Johnson
`MRC Biostatistics Unit,
`Institute of Public Health,
`University Forvie Site,
`Robinson Way,
`Cambridge CB2 2SR, U.K.
`
`Deputy Editors
`
`David Machin
`MRC Cancer Trials Office,
`5 Shaftesbury Road,
`Cambridge CB2 2BW, U.K.
`
`Robert Glynn
`Department of Medicine,
`Harvard Medical School,
`900 Commonwealth Avenue East,
`Boston, Massachusetts 02215, U.S.A.
`
`Joel Greenhouse
`Department ofStatistics,
`Carnegie Mellon University,
`Pittsburgh, Pennsylvania 15213,
`U.S.A.
`
`Biostatistical Tutorial
`Editor
`Ralph D' Agostino
`Department of Mathematics,
`Boston University,
`Ill Cummington Street,
`Boston, Massachusetts 02215,
`U.S.A.
`
`Statistics in the Medical
`Literature
`Douglas Altman
`ICRF Medical Statistics Group,
`Centre for Statistics in Medicine,
`Institute of Health Sciences,
`PO Box 777, Headington,
`Oxford OX3 7LF, U.K.
`
`Editorial Assistants
`
`Book Review Editor
`
`Niels Keiding
`Department of Biostatistics,
`University of Copenhagen,
`Blegdamsvej 3,
`DK-2200 Copenhagen N,
`Denmark
`
`Suzanne Thompson
`Boston University School
`of Public Health
`
`Margaret Molloy
`MRC Cancer Trials Office
`
`AIMS AND SCOPE
`The Journal will publish papers on practical applications of statistics and other quantitative methods to medicine and
`its applied sciences. It will embrace all aspects of the collection, analysis, presentation and interpretation of medical
`data. Specific areas will include clinical trials, diagnostic studies, quality control, laboratory experiments, epi(cid:173)
`demiology and health care research. The journal will emphasize the relevanc~ of numerical techniques and will aim to
`communicate statistical and quantitative ideas in a medical context. Examples of applications of statistics to specific
`projects, articles explaining new statistical methods and reviews of general topics will be published; papers containing
`extensive mathematical theory will be excluded. The main criteria for publication will be appropriateness of the
`statistical method to the particular medical problem and clarity of exposition. The ultimate goal of Statistics in
`Medicine is to enhance communication between statisticians, clinicians and medical researchers with the common
`purpose of advancing knowledge and understanding of quantitative aspects of medicine. It is intended that both the
`readers and authors of the Journal include statisticians, clinicians, epidemiologists, health researchers, mathematicians
`and computer scientists interested in medicine.
`
`Advertising: To advertise in this journal or to rent the subscription list contact: Caroline Melling, Non-subscription Sales Manager, John Wiley & Sons Ltd., Baffins Lane, Chichester.
`West Sussex, P019 IUD, UK. Telephone 44 (0)1243 770351 , Fax 44 (0)1243 770429, Email: info-assets@wiley.co.uk.
`Reprints: Bulk reprints of articles published in this journal are available to order. Please contact the above address.
`
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`
`To subscribe: Statistics in Medicine (ISSN 0277-6715/USPS 680-830) is published semi-monthly by John Wiley & Sons Limited, Baftins Lane, Chichester, Sussex. England, 1996
`subscription price: U.S.$1 ,195.00. Second class postage paid at Jamaica, N.Y. 11431. Air freight and mailing in the U.S.A. by Publications Expediting Services Inc., 200 Meacham
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`
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`Telephone: +44 (0)1243 843288, Fax: +44 (0)1243 843232.
`
`INTELGENX 1036
`
`

`
`STATISTICS IN MEDICINE, VOL. 15, 1605- 1618 (1996)
`
`IMPROVED DESIGNS FOR DOSE ESCALATION STUDIES
`USING PHARMACOKINETIC MEASUREMENTS
`
`STEVEN PlANT ADOS I
`Oncology Biostatistics, Johns Hopkins Oncology Center, Johns Hopkins University School of Medicine,
`550 North Broadway, Suite l103, Baltimore, MD 21205, U.S.A.
`
`AND
`
`GUANGHAN LIU
`Department of Biostatistics, Johns Hopkins University School of Hygiene and Public Health, Baltimore, MD 21205.
`
`U s.A·
`·
`
`·at
`SUMMARY
`.
`.
`.
`.
`. . al trt
`We_ descnb: a method_ for mcorporatm~ pharmacokinetic (PK) data in~o dose escalatiOn chntCed uses
`designs. D?mg so can Improve _the efficiency and accuracy of these studi~s. The method ~ropos effects:
`a parametnc dose respo~s~ function that models the probability of response m each person with two t and
`the dose of drug admmistered and an ancillary pharmacokinetic measurement. After trea_tn:ten ed to
`observation of each subject (or group of subiects) for response one calculates the dose to be admintster f the
`• ate o
`the next individual (or group) to yield the target probability of response from the current best estim
`. tical
`dose-response curve. This procedure is a variant of the continual reassesment method (CRM). St~U~ilitY
`simulations emp~oyi~g a logisti~ dose- response model (that is, we model the logit of the resp~mse pro rAVC)
`as a linear combmat10n o~ ~redictors), dose of drug, and the area under the time-conce_ntration curve waY to
`demonstrate that the addition of pharmacokinetic information to the CRM is a practical and useful
`improve both dose-response modelling and the design of dose escalation studies.
`
`'
`
`•
`
`J
`
`.
`1. INTRODUCTION
`drugs 10
`·
`·
`·
`·
`Phase I dose. esc~lati?n studies are the earliest studies performed m t.he .testi.ng of ne'":'
`. nation
`humans. Their obJectives are to observe and quantify the absorption, distnbutwn and ehmi
`cer
`of the drug and to estim~te an optimal drug dose. When d.evel~ping cyt?toxic. drugs for ~a:ave
`therapy, a common settmg for use of dose escalation designs IS when mvestigators als .
`tive
`interest in estimating the drug dose associated with serious but reversible toxicity. 1 This obJec "ic
`stems from the fact that, to have maximal anti-tumour effect, one must administer most cytoto
`· elY
`drugs at the highest dose that the patient can tolerate.
`Meeting the objective of quantifying drug absorption, distribution and elimination is rei aU" ad,
`straightforward because it does not depend on administering 'optimal' doses of drug. !nsterial
`nearly any clinically reasonable dose of the drug suffices, provided investigators obtatn ~e etic
`blood (or other tissue) samples for drug levels and use simple but reliable pharmacoktn pie,
`models to quantify the rates of drug transfer between physiologic compartments (for exarn
`blood to tissue and vice versa).
`.
`nds
`Estimating the optimal or 'maximal tolerated' dose (MTD) is more difficult because It depe ugh
`on reliable observation of at least a region of the dose toxicity function for the drug. AlthO
`
`CCC 0277- 6715/96/151605- 14
`© 1996 by John Wiley & Sons, Ltd.
`
`•[]995
`Received Aprz
`5
`199
`Revised October
`
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`

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`S. PIANTADOSI AND G. LIU
`
`several designs have been proposed to estimate the dose associated with a specified probability of
`response efficiently (here termed the 'target dose'), ethical issues and other clinical constraints
`make it difficult to carry out the study designs perfectly. Investigators must often compromise
`with an operational definition for the MTD that depends on the number of toxicities seen in small
`groups of patients treated at the same dose.
`Recently, the continual reassessment method (CRM) has been proposed and studied as an
`improved dose escalating strategy for phase I trials. 2 In theory, the CRM permits treatment of
`each patient at the best estimate of the correct target dose, although it is probably better
`employed in a constrained fashion to prevent the patient from receiving too high a dose of the
`study drug. 3 Although clinical investigators suspect that both pharmacokinetic information and
`dose are important in predicting response, there are relatively few methods to incorporate this
`information formally into dose-response studies.4 Neither the CRM nor other methods proposed
`for conducting dose escalation studies has explicitly incorporated PK measurements into the
`dose escalation scheme. The CRM offers the best opportunity to accomplish this because it uses
`a parametric model for the probability of response, a feature that explicitly describes the effect of
`dose on outcome, and it could, in principle, also describe the effects of other variables on
`outcome.
`In this paper, we describe an approach to incorporate quantitative effects for both dose of drug
`and PK parameters into the process of estimating the optimal dose of drug to administer. The
`approach uses a version of the CRM and employs a two-variable logistic dose-response model.
`Using simulations, we compare the two-variable procedure to a conventional approach based
`only on dose of drug. Use of a parametric model to describe the dose-response curve is an
`idealization, but one frequently employed because it is helpful. However, this investigation
`focuses on the benefit of adding PK information to an efficient dose escalation method and has
`less concern with making that scheme universally optimal. Our modelling approach requires the
`predictions of a pharmacokinetic model that we describe first. We then discuss the motivation for
`the logistic dose-response model, dose escalation methods, and simulation methods. In Section 3,
`we present the results of the simulations. Finally, we discuss limitations and possible extensions.
`
`2.1. Pharmacokinetic model
`
`2. METHODS
`
`A source of reliable and accurate PK data is necessary to implement this method. Most often,
`information about pharmacokinetics comes from a compartmental model, although this is not
`a requirement of the procedure. We use only the predictions and the model itself is not explicitly
`incorporated into the estimation process. Therefore, one can use any PK model. Here, we
`consider a relatively simple model and type of drug administration to illustrate the approach.
`Consider a drug administered at a constant rate by continuous intravenous infusion as one
`might do in a phase I clinical trial. We assume that the drug is transferred from blood to a tissue
`compartment (and vice versa) with first-order kinetics, and eliminated from the blood, also by
`first-order kinetics. We can describe this situation with a two-compartment linear system as
`outlined in the Appendix, equations (10)-(16). If we stop the infusion at time t 0 , equations (18) and
`(20) give the concentrations of drug in the blood and tissue. We obtain the area under the
`time-concentration curve (AUC) for blood by integrating (18) to yield (19) or
`
`AUC = g0 t 0 + X(O) =dose + X(O)
`y
`y
`
`(1)
`
`INTELGENX 1036
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`DOSE ESCALATION STUDIES USING PHARMACOKINETIC MEASUREMENTS
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`1607
`
`where g0 is the constant rate of infusion, t0 is the infusion time, and y is the elimination rate from
`the vascular compartment as shown in the Appendix. With no drug initially present, X(O) = 0
`and equation (1) shows that AUC is propo~nal to total dose, that is AUC = g0 t0 /y = dose/y.
`In reality, dose and estimated AUC (AUC) are not likely to be perfectly redundant. For
`example, in the presence of population variability in the elimination rate, y, we expect to see an
`imperfect correlation between dose and AUC. Also, the estimated AUC may be subject to
`random variability because of inaccuracies in the model, measurement error, or curve fitting in
`the estimating process so that
`
`...-....
`where e is a random error component. Thus, AUC could be affected by either population
`variability in y or random error, or both, so that it is not likely to be redun~ with dose in
`practical applications. Below, we make explicit the random components for AUC used in this
`investigation.
`
`2.2. Dose-response model
`
`Phase I investigators often focus on drug toxicity as the response of interest. Here, we use
`'response' in a way that includes toxicity as well as beneficial responses. There have been
`numerous parameteric forms suggested for dose-response models, perhaps the most widely used
`being the logit and pro bit curves. s-? We assume the logistic form because its qualitative
`behaviour is reasonable and its mathematical simplicity facilitates incorporation of more than
`one predictor variable. Generally, we do not expect the behaviour of the model away from the
`target dose to be important so that the exact choice of a model may not be critical as long as it
`behaves properly in the region of interest.
`Specifically, we assume that the true probability of response, p, satisfies
`/3 1 x dose -
`
`logit(p) = {30 -
`
`f3z x AAuc
`
`or
`
`(2)
`
`where {3 0 , {3 1 and {3 2 are constants and AAuc = AUC - dose/y. In other words, if the actual AUC
`for a fixed dose level exceeds that predicted, the probability of response increases. If the AUC is
`close to that expected (AAuc ~ 0), we gain no additional information about response beyond that
`contributed by dose. Using AUC or other PK data in this way can improve the estimate of {3 1 and
`can yield a more accurate update of the target dose. {30 is the log-odds of the background
`probability of response and satisfies {30 » 0. The probability of response increases with increasing
`dose or increasing AUC so that {3 1 > 0 and {3 2 > 0. Because of additivity on the logit scale, this
`model easily generalizes to effects for other PK parameters (for example, peak concentration),
`interactions, or more than two predictors.
`We further assume that values of AAuc in the population have a Gaussian distribution with
`mean 0 and variance a 2
`• Although this is an assumption of convenience, it is also a plausible way
`to capture biological variability with minimum complexity. If a 2 and {3 2 are large enough, some
`individuals will have a higher (or lower) probability of response than dose alone indicates. Model
`behaviour in this regard seems consistent with what we expect to observe in the real world.
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`INTELGENX 1036
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`S. PIANTADOSI AND G. LIU
`
`2.3. Dose escalation method
`
`The dose escalation begins with the investigators' choice of a starting dose thought to lie in the
`region of the target dose. One may base this starting value, in part, on preclinical studies. The first
`m1 patients receive this dose and their responses are noted. Also, we estimate the A UC using
`standard pharmacokinetic methods and we determine the deviation from its predicted value.
`Using these data and the assumed dose-response model, we estimate the parameters and
`calculate a new dose by inverting the model equation. The new dose obtained does not have to be
`one of a pre-specified set. The next individual (or set of m2 patients) receives the new dose and
`then we repeat the .process until either the new dose no longer changes or we have treated
`a specified total number of subjects.
`Apart from formal incorporation of PK data, this procedure differs slightly from that often
`used in clinical practice. First, in practice, one often uses a small set of doses from which one
`chooses all subsequent doses. A set of pre-selected doses can limit the size of jumps and help
`protect against administration of excessively large doses. We do not require a restricted set of
`dose to test the benefit of adding PK information to the model. Second, actual dose escalation
`trials often entail treatment of patients in groups. Grouping, for example by three, also tends to
`reduce the size of the dose adjustments and is often used in escalation algorithms including the
`CRM. Again, grouping is not required to test the benefit of adding PK information. In any case,
`one could adapt this escalation method for groups and/or fixed doses.
`We use equation (2) to calculate the best dose for the next experimental subject, based on the
`current parameter estimates. Suppose that n is the target probability of response of interest and
`that we have treated k patients. Denote the estimates of the parameters after k patients by Pob
`Plk and Plk· Then, from equation (2) we estimate the best dose of drug to administer to the next
`patient:
`
`Pok - logit(rr)
`dosek + 1 =
`p"'
`.
`1k
`Note that we do not need P2 k because we replace ~Auc with its expected value ( = 0) for the next
`patient. Also, we assume that f3ok is a known constant. After sufficient data have been gathered, we
`could estimate f3ok also. However, we have not pursued this approach because it is not essential to
`the question of benefit that can result from adding PK measurements to the dose escalation
`strategy. We discuss parameter estimation in the next section.
`
`(3)
`
`2.4. Statistical simulations
`
`The simulation follows a path that investigators could use to implement this scheme in reality. In
`the following, a subscript i denotes a value for the ith patient and the symbol - indicates the
`probability distribution sampled. To begin the simulation, we choose true values for {3 0 , /3 1 and /3 2
`and the first patient receives a specified starting dose. More generally, we calculate dosei from the
`existing data using equation (3). We obtain a value for ~Auc; by
`
`~AUC;"' N(O,a2
`
`).
`
`(4)
`
`Using dosei and ~Auc;, we calculate the probability of response for each patient from equation (2)
`as qi = P(dosej, ~AucJ. To calculate qi, we use the true values of the coefficients in equation (2).
`Then, we generate a binary variable, rj, indicating response as
`
`if Ui ~ qi
`1
`{
`ri = 0 otherwise
`
`(5)
`
`INTELGENX 1036
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`DOSE ESCALATION STUDIES USING PHARMACOKINETIC MEASUREMENTS
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`1609
`
`where
`
`ui "' Uniform(O, 1).
`
`2. 4.1. Bayesian parameter estimation
`
`For binary outcome data, the likelihood function is
`
`L({Jb {32) = n P({Jb {32, dosei, .1AucJ'i [1 - P(f3 b {32, dosei, .1AucJJ 1-r;
`
`n
`
`i= 1
`
`(6)
`
`where n is the number of study subjects, ri is the response indicator as in equation (5), and
`P(f3b{32,dosei,.1AucJ is the dose-response model (equation (2)) viewed also as a function of {31
`and {32. We form this likelihood function after obtaining the results for each study subject. The
`next step in the simulation is to use all the available data to update the estimates of {31 and {3 2 (we
`assume {3 0 is known) in the dose-response model. We employed a Bayesian procedure to
`accomplish this. In particular,
`
`11
`1
`
`_ s; s; P1f(PbP2)L(f31,f32)df31dP2
`- s;s; J(Pbf32)L(PbP2)df31dP2
`where f(Pb {32) is the joint prior probability density function for {31 and {3 2 and L({Jb {32) is the
`likelihood function viewed as a function of {31 and {3 2 • For dose-response models that use only
`dose,
`
`(7)
`
`/1* _ s; P1f(pbo)L(pbo)df31
`1 -
`J; f({JbO)L({JbO)d{31
`.
`For /12 , we use an equation similar to (7) with {3 2 replacing {J1 in the numerator.
`For simplicity, the prior joint probability density function for {J1 and {32 was chosen to be
`a bivariate uniform density
`
`(8)
`
`We performed the integration in equations (7) and (8) using a Gauss-Legendre quadrature rule. 8
`One needs only a relatively small number of points for this quadrature rule to have good
`accuracy, for example, we used 10 in our simulations. The double integral was evaluated by
`iteratively calling a routine for the inner integral while evaluating the outer one.
`The choice of prior distributions is often a concern in Bayesian estimation. The distributions
`for {31 and {J 2 are likely skewed right because we have constrained them to be positive. Therefore,
`we employed fairly broad uniform priors. In particular, we assumed u1 = u2 = 0 and 11 = 12 = 10
`in the simulations. We explored the use of other prior distributions, including the normal and
`log-normal, and found no practical impact on the estimation.
`One alternative to Bayesian estimation is to select values for {J1 and {J 2 that maximize equation
`(6), that is, maximum likelihood estimates (MLEs). Although MLEs have wide use in many
`contexts, they are unsuitable here because of the small sample sizes on which we must base our
`estimation, especially early in the dose finding. Consequently, we did not use MLEs routinely for
`the simulation.
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`S. PIANTADOSI AND G. LIU
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`2.4.2. Simulation summaries
`
`Following the estimation of {3 1 and {3 2 for the AUC model, we calculate the dose for the next
`subject from equation (3) using P1 • We repeat the process until we have taken a fixed (user(cid:173)
`specified) number of steps. Then, we record for later summary the current dose employed and the
`estimated target probability of response. We follow the same procedure for the dose-only model
`and equation (3) using Pi. In this case, we assume {3 2 is 0.
`The true dose associated with the target response probability is D, that is
`
`{30 -
`
`logit(n)
`f3t
`.
`
`D =
`
`(9)
`
`A similar equation using estimated parameters yields the final recommended dose. We denote
`the final dose recommended by the dose plus AUC model (with .L\Auc = 0) as D and by the
`dose-only model as D*. Similarly, the estimated probabilities of response are fi and fi*. The
`principal outputs of each simulation were .L\D = D - D,
`.L\1t = fi - n,
`.L\1) = D* - D, and
`.L\: = fi* - n, the biases, in addition to the estimates themselves. From these, we calculated the
`average biases L\D, L\1t, L\1), and .1: and the estimated standard deviations of D, fi, D*, and fi* over
`all simulations.
`To help explain the results of the simulations, we considered the coefficient of variation for the
`AUC,
`
`and an ad hoc variation ratio (VR)
`
`VR =
`
`{3 2 a
`{3 1 dose + f12 a
`These quantities express the relative extent to which fluctuations in the AUC influence the
`probability of response. Both increase as the variability in AUC increases relative to the dose
`administered.
`We conducted simulations under a variety of different values for the true parameters to test
`differences in the performance of the two dose-response models. In particular, we varied {3 1 and
`{3 2 along with a, the random components of AUC. Although, in theory, the assumed prior
`distributions for {3 1 and {3 2 affect the estimation process, we found this of little practical
`importance for comparing the performance of the two models. In all cases, we took the target
`probability of response to be between 0·3 and 0·5 and we stopped dose escalations after 20-30
`subjects, roughly in accord with actual studies.
`
`2.5. Computer methods
`
`For speed and flexibility, we performed all computations and simulations using a custom
`C-language microcomputer program. All parameters required to control the simulations and
`parameter estimation are input into the program from a single data entry screen. The program
`can perform a single simulation with display of all dose steps, or multiple simulations with display
`of summary statistics upon completion. The program and source code are available from the
`authors on request. Because it uses a proprietary third-party screen manager, users cannot
`modify the program without the screen manager. Nevertheless, it is capable of a wide variety of
`simulations in its present form.
`
`INTELGENX 1036
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`We conducted all simulations on an IBM PC compatible microcomputer with a Pentium
`microprocessor running at 100 megahertz. The processor was a newly manufactured model free of
`the floating point error so widely found and discussed in late 1994 and early 1995. For the dose
`plus AUC model, 1000 simulations required approximately 3-4 minutes. For the dose-only
`model, the same number of simulations required only 1-2 minutes. These timings are 10-100
`times faster than some older microprocessors such as the 80386 and 80486 with slower clock
`speeds.
`
`3. RESULTS
`
`We provide details for simulations that use 0·3 as the target probability and a fixed sample
`size of 30 subjects. All other simulations performed yielded similar results. Use of either model
`generally yielded unbiased estimates of both the dose associated with a specified target proba(cid:173)
`bility of response and the true probability of response at the final dose, that is, .1D, and .11t were
`near 0 (Table I). In some cases however, estimation based on dose alone yielded biased estimates
`for the dose. This seemed to happen when VR was relatively large (Table I). This effect is probably
`due to the fact that the two-variable model produced unbiased estimates of the parameters
`/31 and /3 2 , whereas the dose-only model mis-estimated /31 as a consequence of the assumption
`that /3 2 = 0.
`With the differences in the biases standardized, namely
`d d·r _ lbias2l -
`lbiastl
`st . 1 . - - - - - - -
`(s.d.1 + s.d.2)/2
`
`where the subscript 2 refers to the dose plus AUC model, we can see the generally better
`performance of the dose plus AUC model, particularly as the coefficient of variation of AUC
`increases (Figure 1). We see the same qualitative behaviour as VR increases (Figure 2). Presenta(cid:173)
`tion of the results in this format helps to emphasize the importance of the relative size of fJ. These
`comparisons assume that positive and negative biases have the same cost, which may not hold for
`diseases such as cancer where many of the drugs tested are ineffective. Thus, we might find
`a negative bias more acceptable than a positive one. However, a large relative bias is probably not
`desirable.
`When both /31 and VR are small, the dose plus AUC model appears to perform more poorly
`than the dose-only model (Table I, panel1). This could result from the need for more information
`to estimate the additional parameter. However, as either dose or AAuc becomes important in
`predicting response as measured by increasing /31 or {3 2 , the two-variable model performs at least
`as well as the dose-only model.
`Most of the variability in D, it, D* and it* is due to the binomial variability that arises from
`implementation of equation (5). Therefore, the precision in the averages of these estimates would
`only increase with use of a larger study. Although possible for simulations, this strategy is
`impractical for most real phase I studies; hence we did not pursue it.
`We explored using maximum likelihood estimates for /31 and /3 2 to replace the Bayesian
`estimates discussed above. Other than the problem that the likelihood degenerates for small
`samples when there are either no response or all responses, the MLEs appeared to be seriously
`biased unless the sample size exceeded 50-60 subjects (data not shown). The mechanics of
`calculating MLEs was faster than that for Bayes estimates which offered some slight advantage
`for the simulations. However, the likelihood function is flat, especially when the sample size is
`small. Thus, the MLE is more variable than the posterior mean.
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`INTELGENX 1036
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`S. PIANTADOSI AND G. LIU
`
`Table I. Bias and standard deviation of parameter estimates from the simulation study. Each line
`represents 1000 runs with 30 subjects each, a true probability of response= 0·30 and Po = 3·0
`
`Simulation parameters
`pl
`p2
`VR
`11
`(J
`AUC
`
`Estimate
`
`Dose+ AUC
`Bias
`Std
`
`Dose only
`Bias
`Std
`
`1·0
`
`2·0
`
`0·1
`
`0·05 0.09
`
`1·0
`
`2·0
`
`0·3
`
`0·14 0·22
`
`1·0
`
`2·0
`
`0·5
`
`0·23 0·32
`
`1·0
`
`2·0
`
`1·0 0-46 0-48
`
`2·0
`
`1·0
`
`0·1
`
`0·05 0·04
`
`2·0
`
`1·0
`
`0·3
`
`0·14 0·12
`
`2·0
`
`1·0
`
`0·5
`
`0·23 0·19
`
`2·0
`
`1·0
`
`1·0 0·46 0·32
`
`2·0
`
`2·0
`
`0·1
`
`0·05 0·09
`
`2·0
`
`2·0
`
`0·3
`
`0·14 0·22
`
`2·0
`
`2·0
`
`0·5
`
`0·23 0·32
`
`2·0
`
`2·0
`
`1·0 0·46 0·48
`
`2·0
`
`3·0
`
`0·1
`
`0·05 0·12
`
`2·0
`
`3·0
`
`0·3
`
`0·14 0·29
`
`2·0
`
`3·0
`
`0·5
`
`0·23 0-41
`
`2·0
`
`3·0
`
`1·0 0·46 0·58
`
`Dose
`Prob
`Dose
`Prob
`Dose
`Prob
`Dose
`Prob
`Dose
`Prob
`Dose
`Prob
`Dose
`Prob
`Dose
`Prob
`Dose
`Prob
`Dose
`Prob
`Dose
`Prob
`Dose
`Prob
`Dose
`Prob
`Dose
`Prob
`Dose
`Prob
`Dose
`Prob
`
`0·1516
`0·0547
`0·2051
`0·0654
`0·1741
`0·0592
`0·1252
`0·0490
`0·0152
`0·0172
`0·0245
`0·0210
`0·0198
`0·0187
`0·0240
`0·0210
`- 0·0041
`0·0094
`0·0301
`0·0232
`0·0061
`0·0141
`0·0112
`0·0187
`0·0086
`0·0143
`- 0·0011
`0·0111
`0·0114
`0·0164
`0·0126
`0·0206
`
`0·0257
`0·0054
`0·0256
`0·0054
`0·0259
`0·0054
`0·0264
`0·0055
`0·0084
`0·0034
`0·0084
`0·0034
`0·0082
`0·0034
`0·0088
`0·0036
`0·0083
`0·0034
`0·0085
`0·0035
`0·0087
`0·0035
`0·0098
`0·0040
`0·0083
`0·0034
`0·0087
`0·0035
`0·0089
`0·0036
`0·0111
`0·0046
`
`0·0943
`0.0435
`0·0296
`0·0309
`- 0·1428
`- 0·0011
`- 0·6553
`- 0·0872
`0·0030
`0·0120
`- 0·0074
`0·0079
`- 0·0237
`0·0021
`- 0·1108
`- 0·0297
`- 0·0176
`0·0044
`- 0·0467
`- 0·0068
`- 0·1207
`- 0·0332
`- 0·2754
`- 0·0833
`- 0·0119
`0·0063
`- 0·1056
`- 0·0282
`- 0·1986
`- 0·0597
`- 0-4149
`- 0·1236
`
`0·0265
`0.0055
`0·0263
`0·0055
`0·0272
`0·0055
`0·0274
`0·0050
`0·0082
`0·0033
`0·0083
`0·0033
`0·0086
`0·0034
`0·0095
`0·0036
`0·0085
`0·0034
`0·0088
`0·0035
`0·0093
`0·0035
`0·0103
`0·0036
`0·0084
`0·0034
`0·0091
`0·0035
`0·0100
`0·0036
`0·0104
`0·0034
`
`4. DISCUSSION
`
`Our study demonstrates the feasibility of improving dose escalation clinical trials by incorporat-
`ing pharmacokinetic measurements into an experimental design that ordinarily relies on response
`only. The simulation study we performed approximates methods used in real phase I clinical
`trials. Therefore, we expect that our findings have practical importance in carrying out these
`studies. However, we have not demonstrated universal applicability, nor was this our goal when
`the study began. We have assumed certain idealizations: a particular and simple form for the true
`dose-response function with a fixed intercept term; a simple two-compartment pharmacokinetic
`model of drug distribution; the ability to assay blood samples reliably and quickly; and the
`willingness to administer individualized and changeable doses of drug rather than pre-specified
`amounts. It would be useful to explore the robustness of our conclusions with respect to each of
`these factors. However, because the addition of PK measures can improve an already efficient
`scheme such as the CRM, it is likely to help in more complex situations also.
`
`INTELGENX 1036
`
`

`
`DOSE ESCALATION STUDIES USING PHARMACOKINETIC MEASUREMENTS
`
`1613
`
`20
`
`10
`
`0
`
`4)
`
`(.) c: e 4)
`~
`i::5 -10
`-c
`4)
`N
`~
`0 -c -20
`c:
`0
`Vi
`
`§ b
`
`\
`\
`\
`\
`\
`
`-30
`
`-40;-~~~~~~~~~~~~~--~
`...r
`...r
`...r
`...r
`IL')
`IL')
`IL')
`IL')
`I")
`I")
`I")
`I")
`~ ~ ~ ~
`oooo . . . . - - - - - . . -NNNN
`c ) c ) c ) c ) c ) c ) c )d c ) c ) c ) c )
`c) c) c) c)
`
`Coefficient of Variation of AUC
`
`Figure 1. Standardized differences of absolute bias between dose plus AUC and dose-only parameters versus CV
`
`20
`
`10
`
`~ c: e 4)
`~ -10
`-c
`4)
`N
`'"E
`0 -g -20
`~
`
`-30
`
`-40
`
`~
`~
`
`...r
`0
`c)
`
`0>
`0
`c)
`
`0>
`N
`N
`0
`c) c) d
`
`0>
`c)
`
`N
`N
`N
`N
`c) d
`
`....,
`.....,
`0> N
`N
`N
`c) c) d
`
`....,
`:;: ~ ~ co
`N
`LO
`c) c) c) c) c)
`
`Variation Ratio
`
`Figure 2. Standardized differences of absolute bias between dose plus AUC and dose-only models versus VR
`
`4.1. Escalation method
`
`One of the criticisms of the CRM has been that investigators who base dose levels on it will treat
`a larger fraction of patients at higher doses than they would using conventional designs. 9
`Although one could address this concern by modifying the CRM to make adjustments only after
`
`INTELGENX 1036
`
`

`
`1614
`
`S. PIANTADOSI AND G. LIU
`
`groups of three patients, 3 it is also possible that the approach outlined here is helpful in
`preventing large dose escalations because of the additional information the PK data provide.
`Additional simulation studies could resolve this issue. This could be important because dose
`adjustments after each patient might constitute a more efficient strategy than does use of
`a grouped design.
`We have not yet studied how varying the escalation strategy affects this new design. For
`example, should we limit the relative size of dose steps recommended by the model? The primary
`purpose of our investigation was to establish the potential of PK measures to improve the
`accura