Download An Application of Continual Reassessment Method Using SAS System

An Application of Continual Reassessment Method Using SAS System
Naoki Ishizuka, National Cancer Center Research Institute, Tokyo, Japan
ABSTRACT
The Bayesian technique, Continual Reassessment Method
(CRM), introduced by O’Quigley et al. (1990) is a new design for
phase I trials in cancer. The goal of this design is to reduce the
number of patients who receive lower doses, which are not
expected to be effective, and to obtain a more accurate estimate
of the maximum tolerated dose (MTD). The CRM has a couple of
attractive characters, which can not be expected with the
traditional three-cohort design. The allocated dose level to be
administrated is determined when the dose-response model has
been updated with observed responses. So that, numerical
integration and graphical presentation for the probability of the
density function are important in practice. The combination usage
of QUAD call of SAS/IML and PROC GPLOT of SAS/GRAPH is a
powerful tool.
Dose level :
xi (i=1,...,k)
Response of jth patient :
1 if toxic response
Yj (j=1,...,n)
0 if no toxic response
Dose-response (toxicity) model :
exp( 3 axi )
~
O ( xi , a ) , e.g. E[Yj ] G O ( xi , a ) 1 exp( 3 axi )
Prior distribution for the parameter a for jth patient :
f (a , j ) where j
Introduction
The difficulty of phase I cancer clinical trials of cyto-toxic drugs is
well described in a review paper by Ratain et al. (1993) The main
purpose of phase I trial is to determine or estimate the Maximum
Tolerated Dose (MTD) and the Recommended Dose for following
phase II trials. There is usually a deviation between research
objectives and patients’ incentive who expect (even minimal)
efficacy of treatment. A recommended dose is often inaccurate
due to large variability of patients’ response 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 lifethreatening toxicity or even toxic deaths, which can be avoided by
using a cautious dose-escalation strategy.
In a traditional so-called ‘3 cohort’ design, the dose level is
escalated to the next higher level when a fixed number of
patients, usually three, do not show the Dose Limiting Toxicity
(DLT) defined 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 show the DLT in the original cohort with 3 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.
The above traditional design is widely used in many practical
situations.
To fill the gap between research objectives and patients’ incentive
explained above and making the decision-making process more
explicit, the Continual Reassessment Method (CRM) is introduced
5
by O’Quigley et al. in 1990. The CRM is a Bayesian approach
and its goal is to reduce the number of patients who receive lower
doses, which are not expected to be effective, and to obtain a
more accurate estimate of the MTD. The CRM has a couple of
attractive characters, which can not be expected with the
traditional cohort design. One is to give quantitative interpretation
for the probability of the DLT occurrence. The another one is the
possibility of explicit utilization of prior information from a various
sources for expressing the ambiguity of dose-toxicity relationship
before the trial.
In Section 2 of this paper, the CRM with mathematical notations is
introduced. Then, the usage of the QUAD call of SAS/IML for
numerical integration is presented in Section 3. In Section 4, we
come back to the numerical example of the original paper. The
graphical presentation for the probability of the density function by
PROC GPLOT of SAS/GRAPH is also discussed.
The CRM
We adopt the notation in O’Quigley et al. (1990) as follows:
and
0
y ,..., y 1
j 1
f ( a , j ) da 1 ( j 1,..., n )
Prior mean response probability for jth patient and ith dose:
G ij 0 O ( xi , a ) f ( a , j ) da (i 1,..., k )
Approximate mean response probability with the use of prior
mean of parameter a , ( j ) :
q ij
y xi , m ( j ) (i 1,..., k ), m ( j ) 0 af ( a , j ) da
The original CRM proceeds as follows:
(i)
A dose-response (toxicity) model O ( xi , a ) is assumed,
(ii)
(iii)
(iv)
where xi, i=1,...,k are pre-determined dose levels; a is a
model parameter
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.
Define the target probability of DLT, θ; 20-33% are usually
used in practice.
Assign the jth patient to dose level x(j). (Allocation rules
are discussed later.) Once the dichotomous response yj of
toxic or no toxic response is observed, the posterior
distribution of a is updated using a following Bayesian
calculation. Then, calculate a revised (prior) probability
density of the DLT occurrence at each dose level. This
process continues until a predetermined fixed sample size,
e.g. 25, or some other condition is satisfied.
f (a , W j +1 ) =
ò
¥
0
f (a, W j )B x( j ), y j , a f (u, W j )B x( j ), y j , u du
,
g (a)Õ l =1B x(l ), yl , a j
=
ò
¥
0
where
g (u )Õ l =1B x(l ), yl , u du
j
yj
(1 y j )
B x ( j ), y j , a O xi , a 1 O xi , a .
Allocation rules and Modifications
There are some controversial issues in the CRM since the original
article appeared. The original CRM tends to allocate a higher
dose compared with the traditional design and some modifications
have been suggested by Faries (1991,1994), Korn et al. (1994)
and Moller (1995). The main suggestions have been to start with
the lowest dose and not to allow jumping in dose escalation.
Goodman et al. (1995) also suggested an option of assigning two
or three patients at one time in addition to the above
modifications. Piantadosi et al. (1996) proposed to use
pharmacokinetic information. Chevret et al. (1993) investigated
the operational characteristics of the CRM by simulation and
explored the effect of selection of the intercept parameter of
logistic regression. O’Quigley and others (1991,1992,1996)
investigated the property of the CRM in depth and discussed the
estimation problem based on the likelihood approach.
As regard the dose selection scheme for the jth patient, the
original paper suggested to select the dose level which minimizes
the following three criteria:
(G ij ,G ) , (G ij ,G ) , ( x j ,O a1 m (G ))
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 infinite
integrals to be performed, thus reducing computational resource.
Goodman et al. (1995) 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. (1990) and
O’Quigley (1992) adopted the selection rule based on (G ij ,G )
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.
Numerical Integration and Graphical
Presentation of the Prior Distribution
The prior information is updated with the observed response.
Regardless of the allocation rule, numerical integration is
essential in the CRM. We have to think either
G ij 0 O ( xi , a ) f ( a , j ) da (i 1,..., k )
or
q ij
proc sort data=work;
by doselvl theta;
run;
y xi , m ( j ) (i 1,..., k ), m ( j ) 0 af ( a, j ) da .
SAS/IML has QUAD call of SAS/IML performs numerical
integration of scalar functions in one dimension over infinite,
connected semi-infinite, and connected finite intervals. Suppose
you want to integrate exp(-t) from 0 to infinity: Example listing as;
proc iml;
/* DEFINE THE INTEGRAND */
start fun(T);
v = exp(-t);
retuen(v);
finish;
/* CALL QUAD */
a
= { 0 .P };
call quad(z,"fun",a);
print z[format=e21.14];
The model parameter a is a purely instrumental one and difficult
to interpret, especially for clinical investigators. It is rational to
~
monitor the distribution of the DLT occurrence probability G
~
instead of a. The distribution of G is directly derived from that of a
by a simple variable-transformation as follows:
f
f a , j da .
G ~ f a , j ~
dG G log(tanh xi 1) log 2
* ~
y10=0; y20=0; y30=0; y40=0; y50=0; y60=0;
y11=0; y21=0; y31=0; y41=0; y51=0; y61=0;
array dose{6} x1-x6;
do i = 1 to 6;
doselvl = i;
do theta = 0 to 1 by 0.005;
if theta = 0 then a = 1;
else a = log (theta)
/ log ( (tanh(dose{i}) + 1) / 2 );
/* Likelihood */
like = exp( a*y11*log( (tanh(x1)+1) / 2)
+ y10*log( 1 - ( (tanh(x1)+1) / 2)**a)
+ a*y21*log( (tanh(x2)+1) / 2)
+ y20*log( 1 - ( (tanh(x2)+1) / 2)**a)
+ a*y31*log( (tanh(x3)+1) / 2)
+ y30*log( 1 - ( (tanh(x3)+1) / 2)**a)
+ a*y41*log( (tanh(x4)+1) / 2)
+ y40*log( 1 - ( (tanh(x4)+1) / 2)**a)
+ a*y51*log( (tanh(x5)+1) / 2)
+ y50*log( 1 - ( (tanh(x5)+1) / 2)**a)
+ a*y61*log( (tanh(x6)+1) / 2)
+ y60*log( 1 - ( (tanh(x6)+1) / 2)**a));
/* Integration can be omitted */
density = exp(-a)*like;
f = - density/( theta*log
((tanh(dose{i}) + 1) / 2));
output;
end;
end;
label doselvl = 'Dose Level';
run;
If the figure of the probability density function for the prior
distribution of the MTD is of interest, numerical integration is not
essential as follows,
data work;
/* DOSE LEVEL */
x1 = -1.47221949; x2 = -1.09861229;
x3 = -0.69314718; x4 = -0.42364893;
x5 = 0.0;
x6 = 0.42364893;
/* NUMBER OF PATIENNS at DOSE LEVEL i,
RESPONSE Yj */
axis1 label=('PDF') v=none major=none
minor=none w=2;
axis2 label=(f=cgreek h=2.0 'q')
v=(h=2.0 f=swiss)
order=0 to 1.0 by 0.1 minor=(n=1)
offset=(0cm,0cm) w=2 length=80pct;
legend1 position=(top right inside)
value=(h=2.0 f=swiss) mode=share
across=1 shape=symbol(10,1)
label=(f=centxi h=2.0);
title1 f=mincho h=3.0 "Prior distribution of
DLT response probability for the first
patient";
title2 f=swissl h=2.0 "O'Quigley et al(1990)
Biometrics46 p40, Table1";
proc gplot;
plot f*theta=doselvl
/ frame legend=legend1 vaxis=axis1
haxis=axis2;
symbol1 i=join v=none l=1 w=1.5;
symbol2 i=join v=none l=3 w=1.5;
symbol3 i=join v=none l=5 w=1.5;
symbol4 i=join v=none l=7 w=1.5;
symbol5 i=join v=none l=9 w=1.5;
symbol6 i=join v=none l=11 w=1.5;
run;
EXAMPLE
There are some advantages to use a mean probability of the DLT
occurrence G ij instead of its approximation G ij taking account of
a wide (often skewed) variability in prior distribution of the DLT
occurrence. The data in Table 1 is the reproduction an illustrative
example of O’Quigley et al. (1990), where the dose-response
(toxicity) relationship was modeled as
~
f * (G )
a
O ( xi , a ) (tanh xi 1) / 2
with a target response of 20% and g(a)=exp(-a) is assumed as a
non-informative prior for the parameter a.
Table 1. Original example of O’Quigley et al. (1990)
Sequential trial of 25 patients;
1/ 2
Pr(Y 1) (tanh xi 1) / 2
j µ(j) xi yi
1 1.00 x3 0
2 1.38 x4 0
3 1.68 x4 1
4 0.92 x3 0
5 1.07 x3 1
6 0.71 x2 1
7 0.49 x1 0
8 0.55 x1 0
9 0.60 x1 0
10 0.65 x2 0
11 0.69 x2 0
12 0.73 x2 0
13 0.77 x2 1
j µ(j) xi yi
14 0.64 x2 0
15 0.67 x2 0
16 0.69 x2 0
17 0.72 x2 0
18 0.74 x2 0
19 0.76 x2 1
20 0.67 x2 0
21 0.69 x2 0
22 0.71 x2 1
23 0.64 x2 0
24 0.65 x2 1
25 0.61 x1 1
(25+1) 0.56 x1
~
G
Figure 2. Probability density function of the prior
distribution of DLT occurrence at each dose in the
numerical example of O’Quigley et al. (1990)
.
As we have discussed, 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 next
patients. If the prior distribution for the parameter a is exponential,
the density functions become to the figures in Figure 2, which
coding has already listed. This figure is correspondent to G i 1 .
The prior mean of the DLT occurrence at each dose level is as
follows:
~
G
i
xi
G ¢i 1
G i1
1
-1.47
0.05
0.25
2
-1.1
0.1
0.30
3
-0.69
0.2
0.38
4
-0.42
0.3
0.45
5
0.0
0.5
0.59
6
0.42
0.7
0.74
If the allocation rule is based on G ij as the original example, the
starting dose level is third one dose since it is the closet to the
target, 20%. However, the lowest dose level must be selected if
the rule is based on G ij . Table 2 shows how it does impact the
Dose level x
Figure 1. Assumed dose-response model in the
numerical example of O’Quigley et al. (1990)
Figure 1 shows the assumed dose-response model with Bayesian
credibility intervals (50%, 90%). If the prior were not so vague,
intervals would be narrow. In this figure, the “mean” curve stands
for the curve of O ( xi ,1) . For the first patient,
¥
m (1) = ò a exp(- a)da = 1,
0
q i¢1 = y {xi , m (1)}= y {xi ,1} (i = 1,..., k )
.
It is important to keep in the mind that the “mean” dose-response
curve,
O ( xi ,1) = {(tanh xi + 1) / 2} ,
1
does not correspond to the real mean (expected) probability
G i1 = ò0 {(tanh xi + 1) / 2} exp( -a )da
¥
a
(i = 1,..., k )
selected dose level for the first 13 patient. Both mean of the DLT
occurrence can be calculated in the same manner even if the
actual rule is different. These figures and tables demonstrate that
higher doses are allocated at earlier stages with the CRM based
on G ij . However, both of them become similar.
The final result, which is the probability density function of the
posterior distribution for the DLT occurrence 25 patients, is shown
in Figure 4.
CONCLUSION
It is never difficult to calculate the CRM if QUAD call of SAS/IML
and SAS/GRAPH are available. However, we must need some
information, ex. Preclinical, foreign affiliated data. Then, we have
to model as the prior information is reflected. This is the most
important step rather than calculation. Then, we have to
determine the allocation rule. Because the operating
characteristics fully depend on the rule as we suggested. The rule
may cause toxic related death, otherwise not so effective. SAS is
also a powerful tool, which provide various functions to do
simulation study.
Table 2. Mean for the DLT occurrence at each dose level
Underlines suggest the dose level which is the closet to 20%
G 1¢ j , G 2¢ j , G 3¢ j , G 4¢ j , G 5¢ j , G 6¢ j
j
µ(j)
xi yi
1 j , 2 j , 3 j , 4 j , 5 j , 6 j
1
1.000
x3
0
2
1.383
x4
0
3
1.684
x4
1
4
0.932
x3
0
5
1.073
x3
1
6
0.722
x2
1
7
0.501
x1
0
8
0.556
x1
0
9
0.602
x1
0
10
0.642
x2
0
11
0.687
x2
0
12
0.728
x2
0
13
0.766
x2
1
0.05, 0.10, 0.20, 0.30, 0.50, 0.70
0.25, 0.30, 0.38, 0.45, 0.59, 0.74
0.02, 0.04, 0.11, 0.19, 0.38, 0.61
0.12, 0.16, 0.24, 0.31, 0.47, 0.65
0.01, 0.02, 0.07, 0.13, 0.31, 0.55
0.06, 0.10, 0.16, 0.23, 0.39, 0.59
0.06, 0.11, 0.22, 0.33, 0.52, 0.72
0.14, 0.20, 0.30, 0.39, 0.56, 0.73
0.04, 0.08, 0.18, 0.27, 0.48, 0.68
0.10, 0.15, 0.25, 0.33, 0.51, 0.70
0.12, 0.19, 0.31, 0.42, 0.61, 0.77
0.18, 0.25, 0.36, 0.46, 0.63, 0.78
0.22, 0.32, 0.45, 0.55, 0.71, 0.84
0.28, 0.36, 0.48, 0.57, 0.72, 0.84
0.19, 0.28, 0.41, 0.51, 0.68, 0.82
0.24, 0.32, 0.44, 0.54, 0.69, 0.82
0.16, 0.25, 0.38, 0.48, 0.66, 0.81
0.21, 0.29, 0.41, 0.51, 0.67, 0.81
0.15, 0.23, 0.36, 0.46, 0.64, 0.80
0.19, 0.27, 0.39, 0.48, 0.65, 0.80
0.13, 0.21, 0.33, 0.44, 0.62, 0.78
0.17, 0.24, 0.36, 0.46, 0.63, 0.79
0.11, 0.19, 0.31, 0.41, 0.60, 0.77
0.15, 0.22, 0.34, 0.44, 0.62, 0.77
0.10, 0.17, 0.29, 0.40, 0.59, 0.76
0.13, 0.20, 0.32, 0.42, 0.60, 0.77
~
f * (G )
method: A practical design for phase 1 clinical trials in cancer’.
Biometrics , 46, 33-48 (1990).
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phase I cancer clinical trials’. American Statistical Association
1991 Proceedings of the Biopharmaceutical Section, 269-273
(1991).
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method for phase I cancer clinical trials’, Journal of
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M. C. and Simon, R. ‘A comparison of two phase I trial designs.’
Statistics in Medicine , 13, 1799-1806 (1994).
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using a preliminary up-and-down design in a dose finding study
in cancer patients, in order to investigate a greater range of
doses’, Statistics in Medicine 14, 911-922 (1995).
Goodman, S. N., Zahurak M. L. and Piantadosi S. ‘Some practical
improvements in the continual reassessment method for phase
I studies’, Statistics in Medicine, 14, 1149-1161 (1995).
Piantadosi, S. and Liu, G. ‘Improved design for dose escalation
studies using pharmacokinetic measurements’. Statistics in
Medicine, 15, 1605-1618 (1996).
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I clinical trials: a simulation study’. Statistics in Medicine, 12,
1093-1108 (1993).
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O’Quigley, J. and Shen, L. Z. ‘Continual reassessment method : a
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CONTACT INFORMATION
Your comments and questions are valued and encouraged.
Contact the author at:
Naoki Ishizuka
Cancer Information and Epidemiology Division
National Cancer Center Research Institute
5-1-1 Tsukiji, Chuo-ku, Tokyo 104-0045
JAPAN
Work Phone : 81-3-3542-3373
Fax
: 81-3-3542-3374
Email
: [email protected]
~
G
Figure 4. Probability density function of the final
posterior distribution for the DLT occurrence
REFERENCES
Ratain, M.J., Mick, R., Schilsky, R.L. and Siegler, M. ‘Statistical
and ethical issues in the design and conduct of phase I and II
clinical trials of new anticancer agents’. Journal of the National
Cancer Institute, 85, 1637-1643 (1993).
O’Quigley, J., Pepe, M., and Fisher, L. ‘Continual reassessment