NONMEM Users Guide Part I - Users Basic Guide - Chapter F
In this chapter we return to theophylline plasma concentration vs time data, but where such data from 12 subjects, rather than from a single subject, are available. This is done to illustrate a regression with one-level nested random effects where the regression function is nonlinear. The fact that the regression function is nonlinear really does not introduce any new considerations regarding the inputs required by NONMEM. However, often with a nonlinear regression function and one-level nested random effects, a modeling approximation is necessary, and this is described here. Also, this example does bring together a number of concepts discussed in the earlier examples.
Each subject is given a single oral dose, the same dose for each subject. Each subject has a different weight. Often dose is expressed as the amount of drug administered per unit weight of the subject (weight-adjusted dose), and in this example the dose data item is the weight-adjusted dose. However, a weight data item is also included in the data record because it will be assumed that interindividual differences in plasma concentrations may be due to interindividual weight differences beyond those expressed through weight-adjusted dose. Also, 11 plasma concentrations are observed per individual at different times, and these times vary between individuals. (The clearance and rate constant measurements used in the examples of chapter E are obtained from this concentration vs time data and from similar data using an additional five doses per subject.)
A model for the jth observation from the ith individual might be given by
|
|
where
where 
,
, and
are the
(non-weight-adjusted) dose, time, and weight variables. Here
and
are not subscripted with a
j, indicating that the values of the dose and weight
variables do not vary within the individual. This model is
similar to one used in chapter C for data from a single
individual, but there are some notable differences. First,
the ith individual is regarded as having his own set of
pharmacokinetic parameters, these parameters are denoted by
,
, and
. Second, two of the
pharmacokinetic parameters are rate constant of absorption,
, and rate constant of
elimination, 
, as
previously, but the third basic parameter is clearance,
, rather than volume of
distribution. Third, these parameters are affected by random
interindividual affects, and thus random interindividual
variability is expressed in the model. Fourth, residual
error is an intraindividual effect. Note that an
individual’s clearance is linearly related to his
weight as in chapter E. The variance-covariance of the
random interindividual effects,
, is regarded as a full
matrix in this example.
As stated in section A.1, with the current version of NONMEM random effects must enter the model (for the observations) linearly. This requirement is not met in the above model; the random interindividual effects enter nonlinearly. One device that has been found useful under these circumstances is to approximate the above model, A, with another, B, obtained by expanding A with a first-term Taylor Series in the random effects about their mean values (0). In the case at hand B is given by
|
|
where
Written this way, the model is also displayed as
the NONMEM linear model schematic. Use of this first-order
approximation to the original model, along with use of the
ELS objective function, has been called the First-Order
Method for analyzing nonlinear mixed effects modeled
data. This method has been shown to be statistically
efficacious in particular situations (Sheiner and Beal,
1980, 1981, and 1983, and Beal 1984a). The first-order
approximation itself may be called the First-Order
Model. One practical problem with this method is that it
can require some nontrivial effort to obtain the partial
derivatives defining the g’s. Moreover, there is
little to be gained by examining these derivatives. Indeed,
rather than try to display explicit formulae for the
g’s in this example in this text, we refer the reader
to the PRED routine of Fig. 73 where code is given for these
formulae. Certain tools are available to help the user
obtain the first-order model. PREDPP is a package which can
be used with NONMEM and with pharmacokinetic data and which
automatically obtains the derivatives
, when, as in the example,
the effect of the 
is
through 
. PREDPP is
actually a very elaborate PRED subroutine. It then remains
for the user to supply code for the derivatives
; these are relatively
simple to obtain. Also, NM-TRAN, a computer program which
facilitates the problem of constructing inputs to NONMEM,
can be used to automatically obtain the derivatives
. (Both PREDPP and NM-TRAN
are distributed with NONMEM.)
Let I denote the number of individuals. Also, for
fixed i, let 
denote the
column vector of values of the
, let
denote the column vector of
values of the 
, let
denote the column vector of
values of the 
, let
denote the column vector of
values of the 
, and let
denote the column vector of
values of the 
. Then the
ELS objective function is given by
where
The last term in the expression for
is just a fancy way of
writing the diagonal matrix whose elements are all
. The matrix
is the variance-covariance
matrix of 
. The vector
is the vector of
weighted residuals from the observations
. As with previous
examples, it has the form residual (vector) divided by
standard deviation (matrix), and it is "squared"
in the expression for the objective function. The
weighted residuals are defined to be the weighted
residuals from all obervations
.
A code for PRED which implements the example is
given in Fig. 74. It is similar to that in Fig. 1. However,
the values returned in G are now very different, and a value
is also returned in H. The same rules for determining what
is returned in G and H, and that are given in chapter E,
apply here too. For clarity, code to compute the partial
derivatives that are returned in G is indented from the
other code. Note that in the expression for F the
weight-adjusted dose (DOSE) appears, rather than the
non-weight-adjusted dose, but that also THETA(3) occurs in
the denominator (E=THETA(3)*C) of that same expression, so
that weight itself need not enter this expression. On the
other hand, since 
adds to
mean clearance, weight does enter the expression for
G(3).
A control stream for this example is given in Fig. 75. The data set is embedded in it. Note that for readability and for the purpose of conveniently keying the data, the weight-adjusted dose and weight data items are blank for all data records of an individual record except the first data record. The PRED routine stores these data items in its local storage whenever the first data record of an individual record is passed to it (review the argument NEWIND described in section C.3.5.2).
The initial STRUCTURE record for the problem
specification has 1’s in fields 7 and 8, indicating
that 
is a full matrix, but
that 
is constrained to be
diagonal. (Again, since 
is
a scalar, it can be regarded as an unconstrained
matrix, but for the sake of
a more perspicuous problem summary, it is taken to be
diagonal.)
The final estimate, standard errors, and correlation matrix are shown in Figs. 76-78. It may interest the reader to see how remarkably well the final estimates in Figs. 66 and 76 agree for those parameters that occur in both the model in section E.4 and the model in section F.1. The final estimates of these parameters from both figures, their standard errors, and the ratios of standard error to estimate are given in Table F.2.2.i. Recall that the estimates in Fig. 76 are obtained using one-sixth the amount of data used to obtain the estimates in Fig. 66, since in the present example only the concentration data from one dose per individual are used, while in the previous example this same data, plus similar data from five additional doses per individual, are used.
Table F.2.2.i
Estimate Comparison
The first page of the requested table is shown in Fig. 79. Scatterplots of residual vs time and of weighted residual vs time, both separated by ID, are requested. The four scatterplots corresponding to individuals 4 and 5 are shown as examples in Figs. 80-83.
