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Half-Life Calculation

Bill Denney

Source: vignettes/v06-half-life-calculation.Rmd
v06-half-life-calculation.Rmd

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. Half-life may also be estimated with Tobit regression, which includes concentrations below the lower limit of quantification as left-censored values; see the “Half-life calculation with Tobit regression” vignette for that method.

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 TlastT_{last} and step back:
    • at least 3 points (customizable with PKNCA.options("min.hl.points"))
    • Not including TmaxT_{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 TlastT_{last} is 12 hours; and if TmaxT_{max} is 1 hour then the default point sets that would be fit are:

  1. 6, 8, and 12 hours;
  2. 4, 6, 8, and 12 hours;
  3. 3, 4, 6, 8, and 12 hours; and
  4. 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:

  1. λz>0\lambda_z > 0 and at the same time the selection statistic must be within a tolerance factor of the best value of that statistic.
    1. The statistic is the adjusted r2r^2 by default, and its tolerance is controlled by PKNCA.options("adj.r.squared.factor") which defaults to 10^{-4}.
    2. Selecting on the unadjusted r2r^2 instead needs only PKNCA.options("r.squared.factor") set to the tolerance: the two factors are alternatives, so setting either one takes the other out of use. Because the unadjusted r2r^2 does not reward larger windows, it usually selects fewer points than the adjusted r2r^2 does.
    3. These rules must be met simultaneously, so if the best value of the statistic is for a line with λz≤0\lambda_z \leq 0, the half-life may end up being unreportable.
  2. If fitting the log-linear concentration-time line fails, then it is not the best line.
  3. 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)

# 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

Curve stripping chooses the points automatically, and it is what happens when nothing is specified. Two optional columns given to PKNCAconc() change that: exclude_half.life drops specific points and then curve strips whatever remains, while include_half.life names the exact points to fit and switches curve stripping off entirely. The decision tree below shows how to choose between them and what each one changes.

Choosing a Point Selection Method

Decision tree for choosing between automatic curve stripping, exclude_half.life, and include_half.life. Automatic curve stripping is the default and needs neither column. exclude_half.life drops the points flagged TRUE and curve strips the rest. include_half.life keeps only the points flagged TRUE and fits them directly. In every case, BLQ and zero concentrations and points at or before the end of the last dose are dropped before fitting.

Two parts of the tree are easy to overlook:

  • Only include_half.life turns off automatic selection. exclude_half.life removes the flagged points and then runs the usual curve stripping on the rest, so min.hl.points, allow.tmax.in.half.life, adj.r.squared.factor, and r.squared.factor still apply to the points that remain.
  • The middle band of the tree applies to include_half.life as well. Naming a point does not force it into the fit; BLQ and zero concentrations, and points at or before the end of the last dose, are dropped first either way. Including eight points that span a four-hour infusion therefore yields a five-point fit, not an eight-point fit.

Column Requirements

Both columns must be logical (TRUE/FALSE/NA); a non-logical column (for example, character "yes") is an error. Whether a column applies is decided separately for each interval, using the rules below.

Situation for an interval Result
The column was not given to PKNCAconc() The method is not used
Every value is NA The method is not used; the interval is treated as though the column had not been given
Some values are NA The NA values are treated as FALSE
Any value is not NA, even if every value is FALSE The column counts as in use
Both columns are in use Error: “Cannot both include and exclude half-life points for the same interval”

Because an all-FALSE column still counts as in use, initialize these columns to NA (rather than FALSE) wherever the corresponding method should not apply.

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)

# 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)

# 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