0092-86 15/97
Copyright 0 1997 Drug Information Association Inc.
D I U Information
~
Journal. Vol. 31. pp. 1243-1248, 1997
Rinkxi in the USA. All rights reserved.
THE UTILITY OF STEADY STATE
TROUGH CONCENTRATIONS IN
ASSESSING INTRASUBJECT VARIABILITY
ROBERTA. SMITH,PHD
Associate Director, Department of Biostatistics and Data Management
WEN-CHYISHYU,PHD
Principal Scientist, Department of Metabolism and Pharmacokinetics
WEI-CHILIAO,PHD, MD
Director, Department of Human Pharmacology
Bristol-Myers Squibb Pharmaceutical Research Institute, Princeton, New Jersey
During drug development, trough concentrations are usually monitoredfollowing repeated
dosing to ensure that steady state has been achieved. Further utility of this information
is explored in the estimation of intrasubject variability, Intrasubject variability is essential
to the determination of sample sizes for planning within-subject (ie, crossover) studies
of the effects of meals, concomitant drugs, or formulation changes (ie, bioequivalence).
The utility of steady state trough concentration was evaluated as an indicator of intrasubject variability compared to a conventional approach of using a fraction of intersubject
variability and to estimates obtained directly from crossover studies. Based on available
data, estimates of intrasubject variabiliv by the trough concentration approach were
comparable to those from the other two approaches.
Key Words: Intrasubject variability; Trough concentration; Steady state
INTR 0DUCTI 0N
between (inter-) subject variability of the rate
of
absorption and extent of absorption of the
UNDERSTANDING THE INTRASuBJE(T
variability of the pharmacokinetics of a drug
is necessary for optimizing a dosing paradigm. Intrasubject variability is also essential
to determine sample sizes for planning studies of drug formulations, food effects, or
drug-drug interactions during drug development. For a bioequivalence study, the statistical power depends on the within (intra-) and
Presented at the DIA “First International Taipei Symposium,” August 29-30, 1996. Taipei, Taiwan.
Reprint address: R. A. Smith, PhD, Department of
Biostatistics and Data Management, Bristol-Myers
Squibb Pharmaceutical Research Institute, P.O. Box
4O00, Princeton, NJ 085434000.
drug of interest. The intersubject variability
of a test formulation is often obtained from
the single or multiple dose pharmacokinetic
studies during the early drug development.
The approach to the assessment of intrasubject
variability, however, has not been consistent.
A replicate study design using a single
dose administration in up to five occasions
has been adopted but not widely used (1-6).
Utilizing complete pharmacokinetic profiles
obtained at the steady state has been proposed (7) but rarely performed. Recently, a
2 x 2 study design was recommended (8,9,10).
In a few cases, intrasubject variability was
obtained via trough plasma concentrations
I243
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I244
Robert A. Smith, Wen-Chyi Shyu, and Wei-Chi Liao
(1 1,12). Applications of the complete plasma
versus time profiles either via a replicate
study design or during steady state, or a 2 x
2 study design provide more rigorous evaluation. It is, however, a more costly and longer
process. Application of the trough concentrations at steady state can be more economical,
timely, and practical. This paper proposes to
examine the utility of steady state trough concentration as an indicator of intrasubject variability under different scenarios.
By rearranging the above two equations, one
can obtain a new expression for Css, min:
so that
log(Css,min) = log(AUC) + log f(k,, k, z).
PHARMACOKINETIC THEORY
where f(k,, k, z) depends only on the absorpSince the plasma profiles of most oral drugs tion rate, the elimination rate, and the dosing
exhibit monoexponential decline, a one-com- interval. Similar expressions may be derived
partment model is used to demonstrate the for Cmax and for other pharmacokinetic
theory of the authors' approaches. The one- models. Therefore, the intra- and inter-subcompartment pharmacokinetic model with ject variabilities of log(AUC) and log(Cmax)
first order absorption and elimination follow- may be related to the Cmin data.
ing a single dose administration is:
C(t) = {FDk$[V(k, - k)]}(e-k'- e-',')
where t is the time since dosing, F is the
absolute oral bioavailability, D is the dose
administered, V is the volume of distribution,
k, is the absorption rate constant, and k is
the elimination rate constant (13).
The AUC to infinity can be integrated as:
After repeated dosing to steady-state, the
plasma concentration can be expressed as:
where z is the time between doses and t is
the time since dosing (0 < t < 7).
The trough concentrations at steady state
(measured at t = z ) and the AUC over the
dosing interval can be expressed as:
STATISTICAL THEORY
Let Y denote a particular pharmacokinetic
variable of interest (eg, AUC, Cmax, or
Cmin), probably after log-transformation,
and let Y,,denote the j-th recorded value of
Y from the I-th subject in a study. One writes:
Y, = p, + Eij
where p, denotes the mean of all such measurements from that I-th subject and ej denotes the random intrusubject deviation of
the recorded value Yi, from its mean p,. These
within-subject deviations E are assumed to be
uncorrelated with zero mean and standard
Furthermore, let p denote
deviation oWiLin.
the mean of the p, from all possible subjects
and let Si denote the random infersubject
deviation of pi from the mean p. These between-subject deviations S are assumed to be
uncorrelated with zero mean and standard
deviation oBeIwmn
and to be independent of the
intrasubject deviations E. Thus,
Y, = p + si + Eij.
and
From this model, it follows that the total
variation of Y is the sum of between-subject
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1245
Steady State to Assess Intrasubject Variability
variation and within-subject variation:
to obtain an expression for the intrasubject
standard deviation
Furthermore, it follows that the inrrasubject
correlation p between any two observations
of Y from the same subject is:
oTadfrom an earlier study and an “educated
into which one can substitute the estimate of
That is, p is the proportion of the total variance of Y that is not due to intrasubject variability. For simplicity, the intrasubject deviations E have been assumed to be uncorrelated,
and this has led to an intrasubject correlation
p that is independent of the time between
Observations. Although it would seem preferable to have imposed some autocorrelation
structure (eg, an autoregressive error model),
most of the data sets that would be used had
only two or three replications, in which case
there would be no or little effect of autocorrelation.
guess” at p. In the authors’ experience, p has
usually been naively guessed to be 60%.
Cmin data from the typical pilot study
of safety and pharmacokinetics of multiple
doses, however, may be useful for predicting
p. If the ratio of the within-subject standard
deviations of log(Css,min) and log(AUC) is
similar to that of the between-subject standard deviations, then the intrasubject correlations of log(Css,min) and log (AUC) will
be approximately equal. In such a case, one
could use the intrasubject correlation of the
Cmin obtained as the educated guess for the
intrasubject correlation of the AUC or Cmax.
METHODS
Cmin data from multiple-dose studies of several drugs were available for use. In each
case an analysis of variance (ANOVA) was
performed on the log-transformed Cmin values using PROC GLM of SAS” Version 6.08
APPLICATION
(16) with subject and time point as factors.
In the process of modem drug development, The time points were restricted to the latest
studies of intrasubject effects of food, con- set of consecutive times for which differences
comitant drug administration, or formulation among them were not statistically significant
changes are often planned before estimates at the 5% level. The intrasubject standard
of within-subject variability of the relevant deviations oWirhin
were estimated by SWithmr
the
pharmacokinetic variables are available. For square root of the mean squared error, and the
example, to determine the number of subjects intersubject standard deviations okwcen
were
required for a bioequivalence study in which estimated by the square root of
the two one-sided test approach (14) is to be
used, one would ideally have an estimate
S’,,,,,, = [Mean Square for Subjects
of the within-subject standard deviations of
- ~ & ~ , , l / ( t hnumber
e
of time points).
log(AUC) and log(Cmax) with which to enter
the sample size tables prepared by Diletti,
Hauschke, and Steinijans (15). If only data The intrasubject correlations p of the Cmin
from between-subject studies or pilot studies were estimated by
of single or multiple doses are available, one
can rearrange the earlier expression for intrasubject correlation
In the few cases where some subjects had
incomplete data, the analysis was repeated
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1246
Robert A. Smith, Wen-Chyi Shyu, and Wei-Chi Liao
with SAS@PROC MIXED (17), but the esti- ments were assumed to have no effect on
mates changed very little.
either location or scale of the response. ProThe resulting estimates kinwere then jected intrasubject standard deviations and
combined with estimates of the total variabil- N B E obtained by assuming p = 60% were also
ity of log(AUC) and log(Cmax) to estimate calculated for comparison.
the corresponding intrasubject standard deviations, for example, for AUC,
RESULTS
These projected oWlhn
were then used in the
method of Diletti, Hauschke, and Steinijans
(1 5) to calculate the corresponding sample
sizes NBE required for bioequivalence trials
to have 80% power to meet Food and Drug
Administration(FDA) guidelines for bioequivalence (18) when the test and reference formulations were the same.
These projected intrasubject correlations,
intrasubject standard deviations, and N B E
were compared to more direct estimates obtained from crossover studies of the same
drugs. Either bioequivalence or food effect
studies were used. In order to use the same statistical methods on log(AUC) and log(Cmax)
that had been applied to log(Cmin), the treat-
Estimates of intrasubject standard deviations
of log(AUC) and log(Cmax) for seven drugs
studied in recent trials are presented in Table
1. The third column contains the indirect predictions of intrasubject variability obtained
from estimates of intersubject variability by
the proposed method of setting the intrasubject correlation p of the variable in question
equal to an estimate of intrasubject correlation among the Cmin. The fourth column
contains the indirect predictions similarly obtained by naively guessing p to be 60%. For
comparison, the last column contains corresponding estimates of intrasubject standard
deviations directly obtained from crossover
studies.
The proposed method performed no better
than the naive method based on guessing p
to be 60%. In nine of 14 cases, the naive
TABLE 1
Estimated Intrasubject Standard Deviations (Numbers in Parentheses
are Corresponding Sample Sizes Required tor 80% Power to
Demonstrate Bioequivaienceat 5% Level of Significance)
Drug
Avatriptan
Butorphanol
Cefprozil
Digoxin
Fosinopril
lfetroban
Irbesartan
Parameter
AUC
Cmax
AUC
Cmax
AUC
Cmax
AUC
Cmax
AUC
Cmax
AUC
Cmax
AUC
Crnax
Estimate from
Crnin
Estimate from
Intersubject
Variability
0.52 (94)
0.41 (59)
0.49 (84)
0.38(51)
0.17(12)
0.22 (18)
0.09 (5)
0.11 (6)
0.10(6)
0.08(5)
0.08(5)
0.08(5)
0.25(23)
0.32 (37)
0.06 (4)
0.08(5)
0.12 (7)
0.19 (14)
0.37(49)
0.15(10)
0.30 (32)
0.13 (8)
0.15 (10)
0.16 (11)
0.18(13)
Estimate from
Crossover Studies
0.10 (6)
0.24 (21)
0.22 (18)
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0.23 (20)
0.40 (56)
0.34 (41)
0.43 (65)
0.03 (4)
0.10(6)
0.11 (6)
0.13 (8)
0.17(12)
0.22 (18)
0.10 (6)
0.32 (37)
0.20 (15)
0.19 (14)
1247
Steady State to Assess Intrasubject Variabiliq
predictor was closer than the proposed predictor to the estimate obtained directly from
the crossover. For the proposed method, the
mean prediction error was -0.02, a slight
tendency, on average, to underestimate. For
the naive method, the mean prediction error
was 0.01, an even slighter tendency, on average, to overestimate. The root mean square
prediction errors were 0.12 and 0.08 for the
proposed and naive methods, respectively.
During these computations, a third approach was investigated. Since most pilot
studies of the safety and pharmacokinetics of
multiple doses collect plasma concentration
profiles after the first dose as well as after
the last dose, AUC and Cmax values from
these two time points and the methods described above can be used to generate estimates of intrasubject variability and correlation, provided the pharmacokinetics of the
drug do not significantly change upon repeated dosing. In such a case, AUC(k, after
the first dose and AUC(&T,,
and Cmax after
the last dose should be the same, as shown
above for the one-compartment model, while
AUC(&,, and Cmax after the first and last
doses should differ by some accumulation
index. For the available data on the seven
drugs listed in Table 1, this method had a
mean prediction error of 0.03, larger than for
either of the above two methods. The root
mean square prediction error, however, was
0.07, somewhat better than either of the
above two methods.
DISCUSSION
Three methods were explored for predicting
intrasubject variability from the limited data
normally obtained in early pilot studies of
pharmacokinetics of new drugs. Although
the method proposed in this paper for predicting intrasubject variability of the pharmacokinetic parameters AUC and Cmax did not, in
general, perform better than a naive approach
based on a guess at intrasubject correlation,
there may be some drugs where this method
of predicting the intrasubject correlation by
the intrasubject correlation of trough concentrations is superior. Estimates of intrasubject
variation based on AUC or Cmax values recorded only for the first and last doses may
prove to be the best method of all because
the data used in the calculations are derived
from richer data sets (ie, concentration profiles) rather than single time points (eg,
trough concentrations). Also, trough concentrations may be near the assay’s limit of quantification, adding another possible source of
error. More exploration of the factors influencing the potential use of trough concentrations for predicting intrasubject variability is
needed. Furthermore, the utility of trough
concentrations for other more clinical applications should be explored.
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