`
`• 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.
`
`USA Contacts:
`Advertising: Advertising Sales Department, John Wiley & Sons Inc, 605 Third Avenue, New York, NY 10158-<Xll2, USA. Telephone: (212) 850 8832. Fax: (212) 850 8888.
`Reprints: The Reprint Department, John Wiley & Sons Inc, 605 Third Avenue, New York. NY 10158-(XH2, USA. Telephone: (212) 850 8776.
`
`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
`Avenue, Elmont, N.Y. 11003. Orders should be addressed to: Subscriptions Department, John Wiley & Sons Ltd., Baffins Lane, Chichester, Sussex P019 IUD, England. Details of
`personal subscriptions are available from the publisher. Copyright© 1996 by John Wiley & Sons Ltd. Photo Typeset by Macmillan India Ltd. Printed and bound in Great Britain by
`Page Bros, Norwich. Printed on acid-free paper. ·
`
`For further subscription information: please contact: Journals Administration Department, John Wiley & Sons. Ltd., 1 Oldlands Way, Bognor Regis, West Sussex, P022 9SA.
`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
`
`INTELGENX 1036
`
`
`
`1606
`
`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
`
`
`
`DOSE ESCALATION STUDIES USING PHARMACOKINETIC MEASUREMENTS
`
`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.
`
`INTELGENX 1036
`
`
`
`1608
`
`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
`
`
`
`DOSE ESCALATION STUDIES USING PHARMACOKINETIC MEASUREMENTS
`
`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.
`
`INTELGENX 1036
`
`
`
`1610
`
`S. PIANTADOSI AND G. LIU
`
`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
`
`
`
`DOSE ESCALATION STUDIES USING PHARMACOKINETIC MEASUREMENTS
`
`1611
`
`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.
`
`INTELGENX 1036
`
`
`
`1612
`
`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

Accessing this document will incur an additional charge of $.
After purchase, you can access this document again without charge.
Accept $ ChargeStill Working On It
This document is taking longer than usual to download. This can happen if we need to contact the court directly to obtain the document and their servers are running slowly.
Give it another minute or two to complete, and then try the refresh button.
A few More Minutes ... Still Working
It can take up to 5 minutes for us to download a document if the court servers are running slowly.
Thank you for your continued patience.

This document could not be displayed.
We could not find this document within its docket. Please go back to the docket page and check the link. If that does not work, go back to the docket and refresh it to pull the newest information.

Your account does not support viewing this document.
You need a Paid Account to view this document. Click here to change your account type.

Your account does not support viewing this document.
Set your membership
status to view this document.
With a Docket Alarm membership, you'll
get a whole lot more, including:
- Up-to-date information for this case.
- Email alerts whenever there is an update.
- Full text search for other cases.
- Get email alerts whenever a new case matches your search.

One Moment Please
The filing “” is large (MB) and is being downloaded.
Please refresh this page in a few minutes to see if the filing has been downloaded. The filing will also be emailed to you when the download completes.

Your document is on its way!
If you do not receive the document in five minutes, contact support at support@docketalarm.com.

Sealed Document
We are unable to display this document, it may be under a court ordered seal.
If you have proper credentials to access the file, you may proceed directly to the court's system using your government issued username and password.
Access Government Site