# Sparse NCA Calculations

#### Bill Denney

#### 2023-04-29

# 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 similarly to how normal, dense PK sampling is setup with PKNCA. The only difference are that you give the `sparse` option to `PKNCAconc()`, and in your interval calculations, you will request the sparse variants of the parameters. As of the writing of this vignette, the only sparse parameter for calculation is `sparse_auclast`. Any of the non-sparse parameters will be calculated based on the mean profile of the animals in a 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)

# Calculate!

Setup PKNCA for calculations and then calculate!

    library(PKNCA)
    o_conc_sparse <- PKNCAconc(d_sparse, conc~time|id, sparse=TRUE)
    d_intervals <-
      data.frame(
        start=0,
        end=24,
        aucinf.obs=TRUE,
        cmax=TRUE,
        sparse_auclast=TRUE
      )
    o_data_sparse <- PKNCAdata(o_conc_sparse, intervals=d_intervals)
    o_nca <- pk.nca(o_data_sparse)

    ## No dose information provided, calculations requiring dose will return NA.

    ## Warning: Too few points for half-life calculation (min.hl.points=3 with only 2
    ## points)

    ## Warning: Cannot yet calculate sparse degrees of freedom for multiple samples
    ## per subject

# Results

As with any other PKNCA result, the data are available through the `summary()` function:

    summary(o_nca)

    ##  start end N      cmax sparse_auclast aucinf.obs
    ##      0  24 1 3.05 [NC]      39.5 [NC]         NC
    ##
    ## Caption: cmax, sparse_auclast, aucinf.obs: geometric mean and geometric coefficient of variation

or individual results are available through the `as.data.frame()` function:

    as.data.frame(o_nca)

    ## # A tibble: 16 × 5
    ##    start   end PPTESTCD            PPORRES exclude
    ##    <dbl> <dbl> <chr>                 <dbl> <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 few points for half-life calcula…
    ##  6     0    24 r.squared            NA     Too few points for half-life calcula…
    ##  7     0    24 adj.r.squared        NA     Too few points for half-life calcula…
    ##  8     0    24 lambda.z.time.first  NA     Too few points for half-life calcula…
    ##  9     0    24 lambda.z.n.points    NA     Too few points for half-life calcula…
    ## 10     0    24 clast.pred           NA     Too few points for half-life calcula…
    ## 11     0    24 half.life            NA     Too few points for half-life calcula…
    ## 12     0    24 span.ratio           NA     Too few points for half-life calcula…
    ## 13     0    24 aucinf.obs           NA     Too few points for half-life calcula…
    ## 14     0    24 sparse_auclast       39.5   <NA>
    ## 15     0    24 sparse_auc_se         7.31  <NA>
    ## 16     0    24 sparse_auc_df        NA     <NA>