A SASIML program for simulating pharmacokinetic data 41
Fig. 1 Basic pharmacokinetic models and corresponding parameters used in developing the SAS pharmacokinetics
simulation program.
In contrast with other simulation programs, we provide open source code so that users may modify
the program to meet their particular need. Thus, a user may wish to compare existing and novel ap-
proaches to the evaluation of bioequivalence while another user may be more focused on traditional
comparisons of drug formulations.
Based upon our evaluation of bioequivalence study datasets that have been submitted to
FDACVM, we have identified a number of condi- tions that can complicate the design and analysis
of these trials. Using SAS programming language, we have developed a tool for exploring the conse-
quences of these various challenges. This project reflects a first stage in our efforts to explore the
performance of alternative metrics for evaluating product bioequivalence.
2. Computational methods and theory
The simulation software program allows for the use of either a one- or two-compartment open
body model [25]
. In both cases, drug enters and leaves from the central blood compartment. More
complex pharmacokinetic scenarios, such as three- compartment mamillary models
[26] are not be-
ing considered since one- and two-compartmental models describe the clinically relevant portion of
the concentrationtime profile of most compounds. Both the one- and two-compartment open body
models contain a single input rate constant K
a
and a single output rate constant K
el
. Instan- taneous intravenous administration can be sim-
ulated by setting the mean K
a
value to an ex- tremely large number e.g., 1000, with zero vari-
ability. The fundamental difference between a one and two compartment open body model is that the
two-compartment model includes rate constants describing the partitioning of the drug between
blood central compartment and some hypothet- ical ‘‘peripheral’’ or tissue compartment. The in-
tercompartmental rate constants k
cp
and k
pc
de- scribe the movement of drug between blood and
tissue Fig. 1
. While rate constants have been described as
both lettered and numbered variables through- out the pharmacokinetic literature where K
a
= K
01
, K
el
= K
10
, k
cp
= k
12
and k
pc
= k
21
, only lettered nomenclature is used in this SAS program.
The user specifies the terms that drive the model. This includes values for parameter means,
variances, and parameter correlation matrices for the test and reference groups. Each of these model-
ing components are defined separately for the test and reference products, allowing for a comparison
not only of formulation effects but also of physi- ological variables that can affect such basic phar-
macokinetic parameters as volume and clearance. Thus, the definitions of treatments 1 and 2 need
not be restricted to the comparison of two formu- lations but can be expanded to compare drug ex-
posure characteristics across conditions that may alter patient physiology. This latter point can be
particularly important when comparing blood level profiles across target animal species, a comparison
often used to provide substantial evidence of effec- tiveness for product use in a minor animal species,
such as goats, sheep, and wildlife
[27] .
On the basis of user-defined pharmacokinetic parameter means, variances [expressed as coeffi-
cients of variation CV], and correlation matrices, a mean vector and covariance matrix is generated
for each treatment group. The program then gen- erates a random set of terms for each subject using
a standard procedure for generating multivariate normal random variables similar to a macro MVN
available from SAS. Drug concentrations are calcu- lated deterministically, conditional on the subject-
specific parameters. The equations describing the concentration for a given set of parameters at time
42 E. Russek-Cohen et al.
‘‘t’’ C
t
is defined as follows [25,28]
: C
t
= F × K
a
× amtdose1[j]
V
c
× [gamma 1[i] × exp
− ˛×t
+ gamma 2[i] ×
exp
− ˇ×t
+ gamma 3[i] × exp
− K
a
× t
] 1
where gamma 1[i] =
k
pc
− ˛
˛ − ˇ × ˛ − K
a
2 gamma 2[i] =
k
pc
− ˇ
˛ − ˇ × K
a
− ˇ
3 gamma 3[i] =
k
pc
− K
a
ˇ − K
a
× ˛ − K
a
4 F
is the percent of dose actually absorbed and V
c
is the dilution factor estimating the fluid vol- ume of the blood central compartment within
which the administered dose distributes. For a one-compartment model, V
c
represents the en- tire distribution volume of the body. For a two-
compartment model, V
c
represents the volume of distribution associated with the central compart-
ment. The total distribution volume is the summa- tion of central and peripheral compartments, and is
represented by the equilibrium volume parameter, volume of distribution at steady state
[28] .
˛ = 1
2 ×
k
cp
+ k
pc
+ K
el
+ k
cp
+ k
pc
+ K
el 2
− 4k
pc
× K
el
5
ˇ = 1
2 ×
k
cp
+ k
pc
+ K
el
− k
cp
+ k
pc
+ K
el 2
− 4k
pc
× K
el
6 When the user selects a one-compartment body
model, ˛ and k
pc
are automatically set to equal zero. By setting k
pc
and ˛ to zero, then gamma 1 = 0
gamma 2 = 0 − ˇ
0 − ˇ × K
a
− ˇ
= −
ˇ −
K
a
ˇ + ˇ
2
7 which simplifies to
1 K
a
− ˇ
8 gamma 3 =
0 − K
a
ˇ − K
a
× 0 − K
a
= −
K
a
K
a 2
− K
a
ˇ 9
which simplifies to −
1 K
a
− ˇ
10 Thus, with ˛ and k
pc
set to zero, the two- compartment model Eq.
1 can be written as:
C
t
= F × K
a
× D
V
c
× 0 +
1 K
a
− ˇ
× exp
− ˇ×t
+ −
1 K
a
− ˇ
× exp
− K
a
× t
11 where D = amtdose1[j].
Since, in a one-compartment model, ˇ is the same as K
el
and V
c
is the same as V, the equation for a one-compartment open body model can be
written as: C
t
= F × K
a
× D
V K
a
− K
el
× exp
− K
el
× t
− exp
− K
a
× t
12 It should be noted that k
cp
values are not specified by the user. This is because values of k
cp
are auto- matically set by the values of ˛, ˇ, k
pc
, and K
el
due to the relationship
[25] :
k
cp
= ˛ + ˇ − k
pc
− K
el
13 In future iterations of this program, the absorption
component will be modified to allow for the inclu- sion of an inverse Gaussian model to describe the
input function as a bell shaped curve that asymp- totically approaches zero
[29] .
We establish that the population covariance ma- trix for each treatment group is positive definite by
checking to see that its determinant is positive. The assumption of positive definite is essential for sub-
sequent statistical calculations. If the determinant is less than or equal to zero, an error message is
printed in the output and the simulation is termi- nated. Usually, the correlation matrix is the source
of this problem and the user will need to think about alternative values for this matrix. Correlations that
are too high may be a possible issue.
The determination of product bioequivalence is based upon the use of TOST or a two one-sided tests
procedure [30]
. This hypothesis can be stated as follows
[30] , where R is the reference product and
T is the test product:
H0.
T
−
R
≤ − 0.20
R
or
T
−
R
≥ 0.20
R
14
H1. −
0.20
R T
0.20
R
15 Similarly, when considering ln-transformed data,
the interval hypothesis can be stated as follows:
H0.
T R
≤ 0.80 or
T R
≥ 1.25
16
A SASIML program for simulating pharmacokinetic data 43
H1. 0.80
T R
1.25 17
Assuming that a parallel study design is used to compare the untransformed values associated with
the test T and reference R treatments, the con- fidence intervals about the difference in treatment
means are estimated as follows:
L = ¯
X
T
− ¯ X
R
− t
1−˛v
× S
1 n
T
+ 1
n
R
18
U = ¯
X
T
− ¯ X
R
+ t
1−˛v
× S
1 n
T
+ 1
n
R
19 where t
1−˛v
is the point that isolates probability ˛ in the upper tail of the Student’s t-distribution with
v degrees of freedom, where v = n
T
− 1 + n
R
− 1
for a parallel design study, and ˛ = 0.05 per tail. To express these intervals relative to the ref-
erence means, the lower limit L
u
and the up- per limit U
u
about the difference in the untrans- formed means of treatment pharmacokinetic pa-
rameter values e.g., AUC and C
max
are estimated as follows:
L
u
= L
¯ X
R
20 U
u
= U
¯ X
R
21 Two formulations are considered bioequivalent
when L
u
≥ − 0.20 and U
u
≤ 0.20. When the phar-
macokinetic parameter values are ln-transformed and we assume that in Eqs.
18 and
19 we have
used the means and standard deviations of the ln- transformed data, then the lower L
t
and upper U
t
confidence interval about the ratio of the treat- ment means is estimated as follows:
L
t
= exp
L
− 1
22 U
t
= exp
U
− 1
23 We consider parameters to be bioequivalent when
L
t
≥ − 0.20 and U
t
≤ 0.25.
When only one variable is compared treatment, the t-test can be used to estimate the standard er-
ror of the estimate of the difference between treat- ment means. Alternatively, the estimator, s
2
, which will be used in the calculation of the 90 confidence
interval, can be obtained from the ‘‘error’’ mean square term found associated with an analysis of
variance ANOVA.
3. Program description
