Sparse NCA Calculations
Sparse noncompartmental analysis (NCA) is performed when multiple individuals contribute to a single concentration-time profile due to the fact that there are only one or a subset of the full profile samples taken per animal. A typical example is when three mice have PK drawn per time point, but no animals have more than one sample drawn. Another typical example is when animals may have two or three samples during an interval, but no animal has the full profile.
Sparse NCA Setup
Sparse NCA is setup the same way as normal, dense PK sampling is
setup with PKNCA. The only difference is that you give the
sparse option to PKNCAconc(); the parameters
are requested by their usual names.
Two things happen behind that flag. For auclast and
aumclast, PKNCA uses the sparse estimators – the Bailer
point estimate with the Nedelman-Jia/Holder standard error – calculated
from the pooled samples of all the animals in a group, and reports the
standard error and degrees of freedom alongside the estimate as
auclast_se/auclast_df and
aumclast_se/aumclast_df. Every other parameter
is calculated from the arithmetic-mean profile of the animals in the
group.
The example below uses data extracted from Holder D. J., Hsuan F., Dixit R. and Soper K. (1999). A method for estimating and testing area under the curve in serial sacrifice, batch, and complete data designs. Journal of Biopharmaceutical Statistics, 9(3):451-464.
# Setup the data
d_sparse <-
data.frame(
id = c(1L, 2L, 3L, 1L, 2L, 3L, 1L, 2L, 3L, 4L, 5L, 6L, 4L, 5L, 6L, 7L, 8L, 9L, 7L, 8L, 9L),
conc = c(0, 0, 0, 1.75, 2.2, 1.58, 4.63, 2.99, 1.52, 3.03, 1.98, 2.22, 3.34, 1.3, 1.22, 3.54, 2.84, 2.55, 0.3, 0.0421, 0.231),
time = c(0, 0, 0, 1, 1, 1, 6, 6, 6, 2, 2, 2, 10, 10, 10, 4, 4, 4, 24, 24, 24),
dose = c(100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100)
)Look at your data. (This is not technically a required step, but it’s good practice.)
library(ggplot2)
ggplot(d_sparse, aes(x=time, y=conc, group=id)) +
geom_point() +
geom_line() +
scale_x_continuous(breaks=seq(0, 24, by=6))
Data Setup Note
Sparse NCA requires that subject numbers (or animal numbers) are given, even if each subject only contributes a single sample. The reason for this requirement is that which subject contributes to which time point changes the standard error calculation. If all individuals contribute a single sample, a simple way to handle this is by setting a column with sequential numbers and giving that as the subject identifier:
d_sparse$id <- 1:nrow(d_sparse)How Subjects Are Grouped for Sparse Calculations
With dense (normal) PK, every subject has a full concentration-time profile, so NCA parameters are calculated one subject at a time. Sparse parameters are different: they are calculated from the pooled samples of every subject that belongs to the same group. Knowing what defines a “group” is therefore important.
The groups are taken from the grouping variables on the right of the
| in the PKNCAconc() formula, with the
subject column removed. Every subject that shares the same
combination of the remaining (non-subject) grouping variables
contributes to a single pooled sparse concentration-time profile.
In the simple example above, the formula is
conc~time|id. Here id is the subject, and
removing it leaves no other grouping variables, so all of the data form
a single sparse group.
The behavior is easier to see with more grouping variables. Suppose the concentration and dose objects are created with the formulas below (illustrative code; not run here):
o_conc_sparse <- PKNCAconc(d_conc, conc~time|matrix+drug+usubjid/analyte, sparse=TRUE)
o_dose_sparse <- PKNCAdose(d_dose, dose~time|drug+usubjid)usubjid is the subject because, by default, the subject
is the last grouping variable before any / (or the last
grouping variable when there is no /). After dropping the
subject, the grouping variables that remain are matrix,
drug, and analyte. Sparse parameters are
therefore calculated by combining all subjects within each unique
combination of matrix, drug, and
analyte.
In other words, with that formula PKNCA does not
keep each usubjid separate, and it does
not group by matrix, drug, or
analyte alone. It pools subjects using the full set of
non-subject grouping variables together (matrix +
drug + analyte).
Because subjects are pooled within a group, all subjects in a group must share the same dosing. If subjects in the same group have different dosing information (for example, different dose amounts or dose times), PKNCA stops with an error identifying the inconsistent group.
Calculate!
Setup PKNCA for calculations and then calculate!
##
## Attaching package: 'PKNCA'## The following object is masked from 'package:stats':
##
## filter
o_conc_sparse <- PKNCAconc(d_sparse, conc~time|id, sparse=TRUE)
d_intervals <-
data.frame(
start=0,
end=24,
auclast=TRUE,
auclast_se=TRUE,
auclast_df=TRUE,
aucinf.obs=TRUE,
cmax=TRUE
)
o_data_sparse <- PKNCAdata(o_conc_sparse, intervals=d_intervals)
o_nca <- pk.nca(o_data_sparse)## The sparse estimators use the linear trapezoidal rule, so the auc.method option
## ("lin up/log down") does not apply to: auclast## Warning: Cannot yet calculate sparse degrees of freedom for multiple samples
## per subject## Warning: Too few points for half-life calculation (min.hl.points=3 with only 2
## points)pknca_interval_table() builds the same kind of
specification from a description of the study, and its
sparse_single_dose preset is the single-dose set with no
imputation (there is no individual profile to impute into):
pknca_interval_table(0, 24, preset="sparse_single_dose")Results
As with any other PKNCA result, the data are available through the
summary() function:
summary(o_nca)## start end auclast auclast_se auclast_df cmax aucinf.obs
## 0 24 39.5 7.31 NC 3.05 NC
##
## Caption: auclast, cmax, aucinf.obs: geometric mean and geometric coefficient of variation; auclast_se, auclast_df: arithmetic mean and standard deviation; NC: not calculatedor individual results are available through the
as.data.frame() function:
as.data.frame(o_nca)## # A tibble: 18 × 6
## start end PPTESTCD PPORRES PPANMETH exclude
## <dbl> <dbl> <chr> <dbl> <chr> <chr>
## 1 0 24 cmax 3.05 "" NA
## 2 0 24 tmax 6 "" NA
## 3 0 24 tlast 24 "" NA
## 4 0 24 clast.obs 0.191 "" NA
## 5 0 24 lambda.z NA "" Too fe…
## 6 0 24 r.squared NA "" Too fe…
## 7 0 24 adj.r.squared NA "" Too fe…
## 8 0 24 lambda.z.corrxy NA "" Too fe…
## 9 0 24 lambda.z.time.first NA "" Too fe…
## 10 0 24 lambda.z.time.last NA "" Too fe…
## 11 0 24 lambda.z.n.points NA "" Too fe…
## 12 0 24 clast.pred NA "" Too fe…
## 13 0 24 half.life NA "" Too fe…
## 14 0 24 span.ratio NA "" Too fe…
## 15 0 24 aucinf.obs NA "AUC: lin up/log down" Too fe…
## 16 0 24 auclast 39.5 "AUC: linear. Sparse: arithm… NA
## 17 0 24 auclast_se 7.31 "AUC: linear. Sparse: arithm… NA
## 18 0 24 auclast_df NA "AUC: linear. Sparse: arithm… NAauclast_se and auclast_df are reported
whether or not they were requested, because the sparse estimator returns
all three together. They are only calculable from sparse data:
requesting one for a dense PKNCAconc is an error.
Sparse AUC, AUMC, and Derived Parameters
Because the sparse estimate is stored under the standard parameter
name, every parameter derived from an AUC is available for sparse data
with no extra work. cl.last divides the dose by
auclast, mrt.last divides
aumclast by auclast, and so on down the usual
derived graph.
d_dose <- data.frame(id=unique(d_sparse$id), dose=100, time=0)
o_dose <- PKNCAdose(d_dose, dose~time|id, route="intravascular")
d_intervals_derived <-
data.frame(
start=0,
end=24,
auclast=TRUE,
aumclast=TRUE,
mrt.last=TRUE,
cl.last=TRUE,
kel.last=TRUE,
vss.last=TRUE,
vz.last=TRUE
)
o_data_derived <- PKNCAdata(o_conc_sparse, o_dose, intervals=d_intervals_derived)
o_nca_derived <- pk.nca(o_data_derived)## The sparse estimators use the linear trapezoidal rule, so the auc.method option
## ("lin up/log down") does not apply to: auclast, aumclast## Warning: Cannot yet calculate sparse degrees of freedom for multiple samples
## per subject## Warning: Cannot yet calculate sparse degrees of freedom for multiple samples
## per subject## Warning: Too few points for half-life calculation (min.hl.points=3 with only 2
## points)
as.data.frame(o_nca_derived)## # A tibble: 23 × 6
## start end PPTESTCD PPORRES PPANMETH exclude
## <dbl> <dbl> <chr> <dbl> <chr> <chr>
## 1 0 24 tmax 6 "" NA
## 2 0 24 tlast 24 "" NA
## 3 0 24 cl.last 2.53 "" NA
## 4 0 24 mrt.last 7.49 "" NA
## 5 0 24 vss.last 19.0 "" NA
## 6 0 24 lambda.z NA "" Too few points for half-lif…
## 7 0 24 r.squared NA "" Too few points for half-lif…
## 8 0 24 adj.r.squared NA "" Too few points for half-lif…
## 9 0 24 lambda.z.corrxy NA "" Too few points for half-lif…
## 10 0 24 lambda.z.time.first NA "" Too few points for half-lif…
## # ℹ 13 more rowsNote what vz.last does here. It is the clearance divided
by
,
the terminal rate constant fitted on the mean profile, which is the
standard toxicokinetic approach. These data have only two time points
after the mean profile’s peak, which is too few for a terminal fit, so
and therefore vz.last are NA. That is the
honest answer: the retired vz.sparse.last divided the
clearance by 1/MRT instead, which always produced a number
but made Vz numerically identical to Vss.
kel.last, like the rest of PKNCA’s kel
family, is 1/MRT, so it is unchanged from the retired
kel.sparse.last. Ask for lambda.z when you
want the terminal rate constant itself.
Deprecated Parameter Names
Sparse calculations were originally requested through a parallel set of names. All eleven still calculate and still give the values they always gave, but each now warns and will be an error in the next minor release of PKNCA:
| Deprecated | Use instead |
|---|---|
sparse_auclast |
auclast |
sparse_auc_se |
auclast_se |
sparse_auc_df |
auclast_df |
sparse_aumclast |
aumclast |
sparse_aumc_se |
aumclast_se |
sparse_aumc_df |
aumclast_df |
cl.sparse.last |
cl.last |
mrt.sparse.last |
mrt.last |
kel.sparse.last |
kel.last |
vss.sparse.last |
vss.last |
vz.sparse.last |
vz.last |
Every replacement but the last gives the same number as the name it
replaces. vz.last is
-based,
as described above, so it differs from vz.sparse.last by
design.
The functions behind the sparse estimators –
pk.calc.sparse_auc(),
pk.calc.sparse_auclast(),
pk.calc.sparse_aumc(),
pk.calc.sparse_aumclast(), as_sparse_pk(),
sparse_mean(), var_sparse_auc(), and
cov_holder() – are not deprecated. They are what the
unified estimators call, and they remain available for calculations
outside pk.nca().
Notes on Sparse Calculation Behavior
Degrees of freedom with multiple samples per subject
The degrees of freedom (auclast_df and
aumclast_df) can only be calculated when each subject
contributes a single sample to the profile (as in a serial sacrifice
design). When any subject contributes more than one sample, as in the
example data here, PKNCA warns that it “Cannot yet calculate sparse
degrees of freedom for multiple samples per subject”, and the degrees of
freedom are NA. That warning is the source of the warnings
in the results above. The point estimates and standard errors are still
calculated.
More than half of the measurements below the limit of quantification
When calculating the mean concentration at a time point (see
sparse_mean()), if strictly more than 50% of the
measurements at that time point are below the limit of quantification
(BLQ), the mean for that time point is set to zero. At exactly 50% BLQ,
the mean is calculated normally (including the BLQ values as zero).
Only the linear trapezoidal method is supported
The sparse variance theory is derived for a fixed-weight linear
combination of the per-time-point means, so the sparse estimators are
defined with the linear trapezoidal method only. Calling
pk.calc.sparse_auc() with any other method is
an error, and within pk.nca() the sparse estimators always
use the linear method: the auc.method option does not apply
to them, and pk.nca() says so once per run when the option
asks for anything else. Parameters that fall back to the mean profile,
such as aucinf.obs, do follow auc.method.