Introduction

Half-life is calculated by fitting the natural logarithm of concentration by time. The default calculation method is curve stripping (described in more detail below). Manual half-life points with no automated half-life selection can be performed, or specific points can be excluded while still performing curve stripping.

Curve Stripping Method

When automatic point selection is performed for curve stripping, the algorithm described below is used.

Select the Points

All sets of points that are applicable according to the current options are selected.

Drop all BLQ values, then
Drop all points at or before the end of the last dose administration, including infusion duration (if dosing information is provided), then
Choose all sets of points that start from the \(T_{last}\) and step back:
- at least 3 points (customizable with PKNCA.options("min.hl.points"))
- Not including \(T_{max}\) (customizable with PKNCA.options("allow.tmax.in.half.life"))

As a specific example, if measurements were at 0, 1, 2, 3, 4, 6, 8, 12, and 24 hours; if \(T_{last}\) is 12 hours; and if \(T_{max}\) is 1 hour then the default point sets that would be fit are:

6, 8, and 12 hours;
4, 6, 8, and 12 hours;
3, 4, 6, 8, and 12 hours; and
2, 3, 4, 6, 8, and 12 hours.

If PKNCA.options("min.hl.points") were set to 4, then the 6, 8, and 12 hour set would not be fit. If PKNCA.options("allow.tmax.in.half.life") were set to TRUE, then 1, 2, 3, 4, 6, 8, and 12 hours would be fit.

Select the Best Fit

After fitting all points, the best fit among the set of possible fit is selected by the following rules:

\(\lambda_z > 0\) and at the same time the maximum r-squared must be within an adjusted \(r^2\) factor of the best.
1. The adjusted \(r^2\) factor is controlled by PKNCA.options("adj.r.squared.factor") and it defaults to 10^{-4}.
2. These rules must be met simultaneously, so if the maximum adjusted \(r^2\) is for a line with \(\lambda_z \leq 0\), the half-life may end up being unreportable.
If fitting the log-linear concentration-time line fails, then it is not the best line.
If more than one fit still meets the criteria above, then choose the fit with the most points included.

Example

# Perform calculations for subject 1, only
data_conc <- as.data.frame(datasets::Theoph)[datasets::Theoph$Subject == 1, ]

# Keep all points
conc_obj <-
  PKNCAconc(
    data_conc,
    conc~Time|Subject
  )

# Only calculate half-life and parameters required for half-life
current_intervals <- data.frame(start=0, end=Inf, half.life=TRUE)
data_obj <- PKNCAdata(conc_obj, intervals=current_intervals)
result_obj <- pk.nca(data_obj)

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

# Extract the results for subject 1 
as.data.frame(result_obj)

## # A tibble: 12 × 7
##    Subject start   end PPTESTCD            PPORRES PPANMETH exclude
##    <ord>   <dbl> <dbl> <chr>                 <dbl> <chr>    <chr>  
##  1 1           0   Inf tmax                 1.12   ""       <NA>   
##  2 1           0   Inf tlast               24.4    ""       <NA>   
##  3 1           0   Inf lambda.z             0.0485 ""       <NA>   
##  4 1           0   Inf r.squared            1.000  ""       <NA>   
##  5 1           0   Inf adj.r.squared        1.000  ""       <NA>   
##  6 1           0   Inf lambda.z.corrxy     -1.000  ""       <NA>   
##  7 1           0   Inf lambda.z.time.first  9.05   ""       <NA>   
##  8 1           0   Inf lambda.z.time.last  24.4    ""       <NA>   
##  9 1           0   Inf lambda.z.n.points    3      ""       <NA>   
## 10 1           0   Inf clast.pred           3.28   ""       <NA>   
## 11 1           0   Inf half.life           14.3    ""       <NA>   
## 12 1           0   Inf span.ratio           1.07   ""       <NA>

Manual Point Selection

For both exclusion and inclusion methods below, the same NA handling rules apply on a per-interval basis. If all values are NA, then no inclusion or exclusion is applied (the interval is treated as-is, like the argument had not been given). If some values are NA for the interval, those are treated as FALSE.

A column counts as in use for an interval whenever it contains any non-NA value, even if every value is FALSE. Only one of exclude_half.life and include_half.life may be in use for the same interval; supplying both raises the error “Cannot both include and exclude half-life points for the same interval”. For this reason, initialize these columns to NA (rather than FALSE) when the corresponding method is not being used.

Exclusion of Specific Points with Curve Stripping

In some cases, specific points will be known outliers, or there may be another reason to exclude specific points. And, with those points excluded, the half-life should be calculated using the normal curve stripping methods described above.

To exclude specific points but otherwise use curve stripping, use the exclude_half.life option as the column name in the concentration dataset for PKNCAconc() as illustrated below.

data_conc$exclude_hl <- data_conc$Time == 12.12
# Confirm that we will be excluding exactly one point
stopifnot(sum(data_conc$exclude_hl) == 1)

# Drop one point
conc_obj_exclude1 <-
  PKNCAconc(
    data_conc,
    conc~Time|Subject,
    exclude_half.life="exclude_hl"
  )

data_obj_exclude1 <- PKNCAdata(conc_obj_exclude1, intervals=current_intervals)

# Perform the calculations
result_obj_exclude1 <- pk.nca(data_obj_exclude1)

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

# Results differ when excluding the 12-hour point for subject 1 (compare to
# example in the previous section)
as.data.frame(result_obj_exclude1)

## # A tibble: 12 × 7
##    Subject start   end PPTESTCD            PPORRES PPANMETH exclude
##    <ord>   <dbl> <dbl> <chr>                 <dbl> <chr>    <chr>  
##  1 1           0   Inf tmax                 1.12   ""       <NA>   
##  2 1           0   Inf tlast               24.4    ""       <NA>   
##  3 1           0   Inf lambda.z             0.0482 ""       <NA>   
##  4 1           0   Inf r.squared            1.000  ""       <NA>   
##  5 1           0   Inf adj.r.squared        0.999  ""       <NA>   
##  6 1           0   Inf lambda.z.corrxy     -1.000  ""       <NA>   
##  7 1           0   Inf lambda.z.time.first  5.1    ""       <NA>   
##  8 1           0   Inf lambda.z.time.last  24.4    ""       <NA>   
##  9 1           0   Inf lambda.z.n.points    4      ""       <NA>   
## 10 1           0   Inf clast.pred           3.28   ""       <NA>   
## 11 1           0   Inf half.life           14.4    ""       <NA>   
## 12 1           0   Inf span.ratio           1.34   ""       <NA>

Specification of the Exact Points for Analysis

In other cases, the exact points to use for half-life calculation are known, and automatic point selection with curve stripping should not be performed.

To use only specific points and bypass automatic curve stripping, use the include_half.life option as the column name in the concentration dataset for PKNCAconc() as illustrated below.

data_conc$include_hl <- data_conc$Time > 3
# Confirm that we will be including exactly six points
stopifnot(sum(data_conc$include_hl) == 6)

# Use only these points for the half-life
conc_obj_include6 <-
  PKNCAconc(
    data_conc,
    conc~Time|Subject,
    include_half.life="include_hl"
  )

data_obj_include6 <- PKNCAdata(conc_obj_include6, intervals=current_intervals)

# Perform the calculations
result_obj_include6 <- pk.nca(data_obj_include6)

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

# Results differ when including 6 points (compare to example in the previous
# section)
as.data.frame(result_obj_include6)

## # A tibble: 12 × 7
##    Subject start   end PPTESTCD            PPORRES PPANMETH              exclude
##    <ord>   <dbl> <dbl> <chr>                 <dbl> <chr>                 <chr>  
##  1 1           0   Inf tmax                 1.12   ""                    <NA>   
##  2 1           0   Inf tlast               24.4    ""                    <NA>   
##  3 1           0   Inf lambda.z             0.0475 "Lambda Z: Manual se… <NA>   
##  4 1           0   Inf r.squared            0.999  "Lambda Z: Manual se… <NA>   
##  5 1           0   Inf adj.r.squared        0.998  "Lambda Z: Manual se… <NA>   
##  6 1           0   Inf lambda.z.corrxy     -0.999  "Lambda Z: Manual se… <NA>   
##  7 1           0   Inf lambda.z.time.first  3.82   "Lambda Z: Manual se… <NA>   
##  8 1           0   Inf lambda.z.time.last  24.4    "Lambda Z: Manual se… <NA>   
##  9 1           0   Inf lambda.z.n.points    6      "Lambda Z: Manual se… <NA>   
## 10 1           0   Inf clast.pred           3.30   "Lambda Z: Manual se… <NA>   
## 11 1           0   Inf half.life           14.6    "Lambda Z: Manual se… <NA>   
## 12 1           0   Inf span.ratio           1.41   "Lambda Z: Manual se… <NA>

Half-Life Calculation