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Morphine (Franken 2015)

Source: vignettes/articles/Franken_2015_morphine.Rmd
Franken_2015_morphine.Rmd

Model and source

  • Citation: Franken LG, Masman AD, de Winter BCM, Koch BCP, Baar FPM, Tibboel D, van Gelder T, Mathot RAA. Pharmacokinetics of Morphine, Morphine-3-Glucuronide and Morphine-6-Glucuronide in Terminally Ill Adult Patients. Clin Pharmacokinet. 2016;55(6):697-709 (published online 29 December 2015). doi:10.1007/s40262-015-0345-4.
  • Description: Joint parent-metabolite population PK model for morphine and its two glucuronide metabolites (M3G, M6G) in 47 terminally ill adult palliative care patients (Franken 2015). Morphine: two-compartment disposition with three parallel first-order absorption routes (subcutaneous bolus, immediate-release oral liquid, controlled-release oral tablet) using route-specific fixed absorption rate constants; oral bioavailability F is estimated (SC F assumed 1). M3G and M6G are each one-compartment models fed by fixed-fraction transformation of morphine clearance (Fm1 = 0.55 for M3G, Fm2 = 0.10 for M6G, both fixed from literature). Morphine clearance decreases exponentially as time-to-death (TTD, days) approaches zero. Metabolite clearance depends on estimated glomerular filtration rate (eGFR, MDRD four-variable formula) and serum albumin via shared power-form covariate exponents. Residual variability was reported as additive error on the log-transformed observation (LTBS).
  • Article: https://doi.org/10.1007/s40262-015-0345-4

Population

The published analysis included 47 terminally ill adult palliative-care patients admitted to the Laurens Cadenza hospice in Rotterdam over a 2-year period. Median age 71 years (range 43-93), 55.3% female, 95.7% Caucasian and 4.3% Afro-Caribbean, with 95.7% having advanced malignancy as the primary diagnosis. Median duration of admittance was 33 days (range 7-457). Patients received morphine for pain and dyspnoea at daily doses of 15-540 mg administered as oral controlled-release tablets, oral immediate-release liquid, or subcutaneous bolus / infusion. Sparse plasma sampling (median 2 samples per subject, range 1-10) yielded 152 plasma concentrations for morphine, M3G, and M6G. Baseline blood chemistry medians at admission included albumin 26 g/L (range 14-39), creatinine 72 umol/L, and eGFR (four-variable MDRD) 96 mL/min/1.73 m^2 (range 27-239). See Franken 2015 Table 1 for the full baseline summary; the same information is available programmatically via readModelDb("Franken_2015_morphine")$population.

Source trace

The per-parameter origin is recorded as an in-file comment next to each ini() entry in inst/modeldb/specificDrugs/Franken_2015_morphine.R. The table below collects them in one place.

Equation / parameter Final value Source location
lka_sc (Ka SC bolus) 10 1/h (FIXED) Methods 3.1 (literature [26, 27])
lka_oral_ir (Ka IR liquid) 6 1/h (FIXED) Methods 3.1 (literature [26, 27])
lka_oral_cr (Ka CR tablet) 0.8 1/h (FIXED) Methods 3.1 (literature [26, 27])
lfdepot (F oral) 0.30 Table 2 Final
lcl (morphine CL asymptote) 47.5 L/h Table 2 Final
lvc (V1) 190 L Table 2 Final
lq (Q) 76.1 L/h Table 2 Final
lvp (V2) 243 L Table 2 Final
lcl_m3g (CL M3G) 1.44 L/h Table 2 Final
lvc_m3g (V M3G) 8.02 L Table 2 Final
lcl_m6g (CL M6G) 1.78 L/h Table 2 Final
lvc_m6g (V M6G) 8.24 L Table 2 Final
fm_m3g (Fm1) 0.55 (FIXED) Methods 2.3.1 (literature [21-23])
fm_m6g (Fm2) 0.10 (FIXED) Methods 2.3.1 (literature [21-23])
ttd_d (TTD peak drop in CL) 17.6 L/h Table 2 Final
ttd_rate (TTD first-order rate) 0.13 1/day Table 2 Final
e_crcl_cl_m3g = e_crcl_cl_m6g (eGFR exponent) 0.673 Table 2 Final (shared coefficient)
e_alb_cl_m3g = e_alb_cl_m6g (albumin exponent) 1.1 Table 2 Final (shared coefficient)
etalfdepot (BSV on F) 37.8% CV Table 2 Final
etalcl (BSV on morphine CL) 53.4% CV Table 2 Final
etalcl_m3g, etalcl_m6g 29.3% / 34.3% CV, rho = 1 Table 2 Final + Section 3.1
etalvc_m3g, etalvc_m6g 151.7% / 143.0% CV, rho = 1 Table 2 Final + Section 3.1
expSd (morphine residual) sqrt(0.432) Table 2 Final (LTBS variance)
expSd_m3g (M3G residual) sqrt(0.246) Table 2 Final (LTBS variance)
expSd_m6g (M6G residual) sqrt(0.265) Table 2 Final (LTBS variance)
Equation: morphine 2-cmt model with three parallel depots n/a Figure 2 schematic
Equation: TTD covariate first-order decay on CL n/a Eq. 3, Section 2.3.2
Equation: eGFR / albumin power covariates on metabolite CL n/a Eq. 1, Section 2.3.2

Virtual cohort

Original observed concentrations are not publicly available. The simulations below use typical-value virtual subjects whose covariates approximate the representative scenarios shown in Franken 2015 Figures 3-5. The model is exercised at the four eGFR levels of Figure 3, the three albumin levels of Figure 4, and the three time-before-death scenarios of Figure 5.

set.seed(20151229)

# Figure 3 cohort: vary eGFR at constant albumin = 25 g/L, TTD large
fig3 <- data.frame(
  id    = 1:4,
  CRCL  = c(10, 30, 50, 90),
  ALB   = 25,
  TTD   = 100,                       # "far from death" baseline
  scenario = factor(
    paste0("eGFR ", c(10, 30, 50, 90), " mL/min"),
    levels = paste0("eGFR ", c(10, 30, 50, 90), " mL/min")
  ),
  stringsAsFactors = FALSE
)

# Figure 4 cohort: vary albumin at eGFR = 78 mL/min, TTD large
fig4 <- data.frame(
  id    = 5:7,
  CRCL  = 78,
  ALB   = c(15, 25, 35),
  TTD   = 100,
  scenario = factor(
    paste0("albumin ", c(15, 25, 35), " g/L"),
    levels = paste0("albumin ", c(15, 25, 35), " g/L")
  ),
  stringsAsFactors = FALSE
)

# Figure 5 cohort: vary TTD at fixed eGFR / albumin (population medians)
fig5 <- data.frame(
  id    = 8:10,
  CRCL  = 96,
  ALB   = 26,
  TTD   = c(14, 7, 1),               # 2 weeks, 1 week, 1 day before death
  scenario = factor(
    paste0(c("2 wk", "1 wk", "1 day"), " before death"),
    levels = paste0(c("2 wk", "1 wk", "1 day"), " before death")
  ),
  stringsAsFactors = FALSE
)

Simulation

Figures 3 and 4 use a 30 mg subcutaneous bolus six times daily for ten days; Figure 5 uses a 50 mg subcutaneous bolus six times daily over the final week of life. Subcutaneous bioavailability is structurally 1 (so the SC depot’s f defaults to 1); oral routes (depot2 = IR liquid, depot3 = CR tablet) carry the estimated f_oral. All simulations use typical-value parameters (rxode2::zeroRe()) so the plots are mean-curve replications of the published figures rather than VPCs.

mod_typical <- readModelDb("Franken_2015_morphine") |> rxode2::zeroRe()

build_events <- function(cov_df, obs_times, dose_events) {
  events_list <- lapply(seq_len(nrow(cov_df)), function(i) {
    row <- cov_df[i, , drop = FALSE]
    dose_rows <- dose_events |>
      mutate(
        id       = row$id,
        CRCL     = row$CRCL,
        ALB      = row$ALB,
        TTD      = row$TTD,
        scenario = as.character(row$scenario)
      )
    obs_rows <- data.frame(
      id       = row$id,
      time     = obs_times,
      amt      = NA_real_,
      rate     = NA_real_,
      evid     = 0L,
      cmt      = "Cc",
      CRCL     = row$CRCL,
      ALB      = row$ALB,
      TTD      = row$TTD,
      scenario = as.character(row$scenario),
      stringsAsFactors = FALSE
    )
    dplyr::bind_rows(dose_rows, obs_rows)
  })
  dplyr::bind_rows(events_list) |>
    arrange(id, time, evid)
}

# 30 mg SC q4h x 10 days for Figures 3 and 4 (240 h total observation window)
dose_30mg_q4h <- data.frame(
  time = seq(0, 240 - 4, by = 4),
  amt  = 30,
  rate = 0,
  evid = 1L,
  cmt  = "depot",                    # SC bolus -> depot (depot1)
  stringsAsFactors = FALSE
)

obs_times_240 <- c(seq(0, 24, by = 0.25), seq(25, 240, by = 1))

events_fig3 <- build_events(fig3, obs_times_240, dose_30mg_q4h)
events_fig4 <- build_events(fig4, obs_times_240, dose_30mg_q4h)

sim_fig3 <- rxode2::rxSolve(
  mod_typical, events = events_fig3,
  keep = c("scenario", "CRCL", "ALB", "TTD")
) |> as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalfdepot', 'etalcl', 'etalcl_m3g', 'etalcl_m6g', 'etalvc_m3g', 'etalvc_m6g'
#> Warning: multi-subject simulation without without 'omega'
sim_fig4 <- rxode2::rxSolve(
  mod_typical, events = events_fig4,
  keep = c("scenario", "CRCL", "ALB", "TTD")
) |> as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalfdepot', 'etalcl', 'etalcl_m3g', 'etalcl_m6g', 'etalvc_m3g', 'etalvc_m6g'
#> Warning: multi-subject simulation without without 'omega'

# 50 mg SC q4h x 24 h for Figure 5 (TTD effect)
dose_50mg_q4h <- data.frame(
  time = seq(0, 24 - 4, by = 4),
  amt  = 50,
  rate = 0,
  evid = 1L,
  cmt  = "depot",
  stringsAsFactors = FALSE
)
obs_times_24 <- c(seq(0, 4, by = 0.1), seq(4.5, 24, by = 0.5))

events_fig5 <- build_events(fig5, obs_times_24, dose_50mg_q4h)
sim_fig5 <- rxode2::rxSolve(
  mod_typical, events = events_fig5,
  keep = c("scenario", "CRCL", "ALB", "TTD")
) |> as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalfdepot', 'etalcl', 'etalcl_m3g', 'etalcl_m6g', 'etalvc_m3g', 'etalvc_m6g'
#> Warning: multi-subject simulation without without 'omega'

Replicate published figures

Figure 3: effect of eGFR on M3G / M6G concentrations 

plot_fig3 <- sim_fig3 |>
  pivot_longer(cols = c(Cc, Cc_m3g, Cc_m6g),
               names_to = "species", values_to = "conc") |>
  mutate(species = factor(
    recode(species,
           "Cc"     = "Morphine",
           "Cc_m3g" = "M3G",
           "Cc_m6g" = "M6G"),
    levels = c("Morphine", "M3G", "M6G")
  ))

ggplot(plot_fig3, aes(x = time, y = conc, color = scenario)) +
  geom_line(linewidth = 0.8) +
  facet_wrap(~species, scales = "free_y", ncol = 3) +
  labs(x = "Time (h)",
       y = "Concentration (ug/L; morphine equivalents)",
       color = "eGFR scenario",
       title = "Replicates Figure 3 of Franken 2015",
       caption = paste("30 mg subcutaneous morphine every 4 h x 10 days,",
                       "albumin held at 25 g/L, TTD = 100 days.",
                       "Typical-value simulation (no IIV / no RUV)."))

Figure 4: effect of serum albumin on M3G / M6G concentrations 

plot_fig4 <- sim_fig4 |>
  pivot_longer(cols = c(Cc, Cc_m3g, Cc_m6g),
               names_to = "species", values_to = "conc") |>
  mutate(species = factor(
    recode(species,
           "Cc"     = "Morphine",
           "Cc_m3g" = "M3G",
           "Cc_m6g" = "M6G"),
    levels = c("Morphine", "M3G", "M6G")
  ))

ggplot(plot_fig4, aes(x = time, y = conc, color = scenario)) +
  geom_line(linewidth = 0.8) +
  facet_wrap(~species, scales = "free_y", ncol = 3) +
  labs(x = "Time (h)",
       y = "Concentration (ug/L; morphine equivalents)",
       color = "Albumin scenario",
       title = "Replicates Figure 4 of Franken 2015",
       caption = paste("30 mg subcutaneous morphine every 4 h x 10 days,",
                       "eGFR held at 78 mL/min/1.73 m^2, TTD = 100 days.",
                       "Typical-value simulation."))

Figure 5: effect of time-to-death on morphine concentrations 

plot_fig5 <- sim_fig5 |>
  filter(time > 0)

ggplot(plot_fig5, aes(x = time, y = Cc, color = scenario)) +
  geom_line(linewidth = 0.8) +
  labs(x = "Time after first dose (h)",
       y = "Morphine concentration (ug/L)",
       color = "Time before death",
       title = "Replicates Figure 5 of Franken 2015",
       caption = paste("50 mg subcutaneous morphine every 4 h x 24 h,",
                       "eGFR and albumin held at population medians.",
                       "Typical-value simulation (no IIV / no RUV)."))

Numeric checks against Section 3.3

Franken 2015 Section 3.3 quotes three numeric results from the final-model simulations:

  1. Morphine CL declines with TTD. Paper: 46.4 L/h three weeks before death (TTD = 21 d), 40.6 L/h one week before death (TTD = 7 d), and 29.9 L/h on the day of death (TTD = 0 d). The model encodes CL(TTD) = 47.5 - 17.6 * exp(-0.13 * TTD).
ttd_grid <- data.frame(TTD = c(21, 7, 0)) |>
  mutate(CL_predicted = 47.5 - 17.6 * exp(-0.13 * TTD),
         CL_paper     = c(46.4, 40.6, 29.9))
knitr::kable(ttd_grid, caption = "Morphine CL by TTD: predicted vs published Section 3.3.")
Morphine CL by TTD: predicted vs published Section 3.3.
TTD CL_predicted CL_paper
21 46.35214 46.4
7 40.41557 40.6
0 29.90000 29.9
  1. M3G CL decline with eGFR. Paper: at albumin = 25 g/L, M3G CL drops from ~1.6 L/h at eGFR = 90 mL/min to ~1.1 L/h at eGFR = 50 mL/min (~30% reduction). The packaged model uses references CRCL_ref = 96 (the Table 1 admission-time population median) and ALB_ref = 26, so the absolute typical-value M3G CL differs slightly from the paper’s narrative; the relative drop matches.
m3g_grid <- data.frame(CRCL = c(90, 50, 30, 10), ALB = 25) |>
  mutate(CL_m3g_predicted = 1.44 * (CRCL / 96)^0.673 * (ALB / 26)^1.1)
knitr::kable(m3g_grid, caption = "Typical-value M3G CL across eGFR at albumin = 25 g/L.")
Typical-value M3G CL across eGFR at albumin = 25 g/L.
CRCL ALB CL_m3g_predicted
90 25 1.3205732
50 25 0.8891276
30 25 0.6304634
10 25 0.3009936
  1. M3G CL decline with albumin. Paper: at eGFR = 78 mL/min, M3G CL drops from ~2.1 L/h at albumin = 35 g/L to ~1.4 L/h at albumin = 25 g/L (~31% reduction).
alb_grid <- data.frame(ALB = c(35, 25, 15), CRCL = 78) |>
  mutate(CL_m3g_predicted = 1.44 * (CRCL / 96)^0.673 * (ALB / 26)^1.1)
knitr::kable(alb_grid, caption = "Typical-value M3G CL across albumin at eGFR = 78 mL/min.")
Typical-value M3G CL across albumin at eGFR = 78 mL/min.
ALB CRCL CL_m3g_predicted
35 78 1.7365120
25 78 1.1993252
15 78 0.6837594

PKNCA validation

Single-dose NCA after the first 30 mg subcutaneous bolus of the Figure 3 cohort (window 0-4 h, before the second dose), computed separately for morphine and each metabolite. Treatment grouping uses the four eGFR scenarios so per-group summaries roll up.

sim_nca_window <- sim_fig3 |> filter(time <= 4)

dose_df_fig3 <- events_fig3 |>
  filter(evid == 1L, time == 0) |>
  select(id, time, amt, scenario)

intervals_sd <- data.frame(
  start    = 0,
  end      = 4,
  cmax     = TRUE,
  tmax     = TRUE,
  auclast  = TRUE,
  half.life = FALSE
)

Morphine 

sim_nca_morphine <- sim_nca_window |>
  filter(!is.na(Cc)) |>
  select(id, time, Cc, scenario)

conc_obj_morphine <- PKNCA::PKNCAconc(
  sim_nca_morphine, Cc ~ time | scenario + id,
  concu = "ug/L", timeu = "h"
)
dose_obj <- PKNCA::PKNCAdose(
  dose_df_fig3, amt ~ time | scenario + id, doseu = "mg"
)

nca_morphine <- PKNCA::pk.nca(
  PKNCA::PKNCAdata(conc_obj_morphine, dose_obj, intervals = intervals_sd)
)
knitr::kable(as.data.frame(summary(nca_morphine)),
             caption = "Morphine NCA over the 4 h after the first 30 mg SC bolus.")
Morphine NCA over the 4 h after the first 30 mg SC bolus.
Interval Start Interval End scenario N AUClast (h*ug/L) Cmax (ug/L) Tmax (h)
0 4 eGFR 10 mL/min 1 265 130 0.250
0 4 eGFR 30 mL/min 1 265 130 0.250
0 4 eGFR 50 mL/min 1 265 130 0.250
0 4 eGFR 90 mL/min 1 265 130 0.250

M3G 

sim_nca_m3g <- sim_nca_window |>
  filter(!is.na(Cc_m3g)) |>
  select(id, time, Cc_m3g, scenario) |>
  rename(Cc = Cc_m3g)

conc_obj_m3g <- PKNCA::PKNCAconc(
  sim_nca_m3g, Cc ~ time | scenario + id,
  concu = "ug/L", timeu = "h"
)
nca_m3g <- PKNCA::pk.nca(
  PKNCA::PKNCAdata(conc_obj_m3g, dose_obj, intervals = intervals_sd)
)
knitr::kable(as.data.frame(summary(nca_m3g)),
             caption = "M3G NCA over the 4 h after the first 30 mg SC morphine bolus.")
M3G NCA over the 4 h after the first 30 mg SC morphine bolus.
Interval Start Interval End scenario N AUClast (h*ug/L) Cmax (ug/L) Tmax (h)
0 4 eGFR 10 mL/min 1 2060 812 4.00
0 4 eGFR 30 mL/min 1 1940 737 4.00
0 4 eGFR 50 mL/min 1 1860 683 4.00
0 4 eGFR 90 mL/min 1 1730 605 4.00

M6G 

sim_nca_m6g <- sim_nca_window |>
  filter(!is.na(Cc_m6g)) |>
  select(id, time, Cc_m6g, scenario) |>
  rename(Cc = Cc_m6g)

conc_obj_m6g <- PKNCA::PKNCAconc(
  sim_nca_m6g, Cc ~ time | scenario + id,
  concu = "ug/L", timeu = "h"
)
nca_m6g <- PKNCA::pk.nca(
  PKNCA::PKNCAdata(conc_obj_m6g, dose_obj, intervals = intervals_sd)
)
knitr::kable(as.data.frame(summary(nca_m6g)),
             caption = "M6G NCA over the 4 h after the first 30 mg SC morphine bolus.")
M6G NCA over the 4 h after the first 30 mg SC morphine bolus.
Interval Start Interval End scenario N AUClast (h*ug/L) Cmax (ug/L) Tmax (h)
0 4 eGFR 10 mL/min 1 360 141 4.00
0 4 eGFR 30 mL/min 1 336 126 4.00
0 4 eGFR 50 mL/min 1 318 115 4.00
0 4 eGFR 90 mL/min 1 292 99.4 4.00

Assumptions and deviations

  • Residual-error interpretation. Franken 2015 Methods 2.3.1 reports residual variability as “additive error on the log scale” with Table 2 Final values 0.432 (morphine), 0.246 (M3G), 0.265 (M6G), no explicit units. The packaged model interprets these as NONMEM $SIGMA variance estimates on the log scale and supplies their square roots to rxode2’s lnorm() residual model so that log(Cobs) - log(Cpred) ~ N(0, expSd^2). If a future reading determines the values were already reported as log-scale SDs, the expSd* parameters should be set equal to the published numbers (without the square root).
  • Covariate reference values. The paper Methods 2.3.2 states “Continuous covariates were normalised to the population median values” without explicitly tabulating those medians per model. The packaged model uses the Table 1 admission medians eGFR = 96 mL/min/1.73 m^2 and albumin = 26 g/L as the reference. Section 3.3 simulation narratives imply a slightly different effective reference (the typical-value M3G CL at eGFR = 90 / albumin = 25 quoted as ~1.6 L/h is ~20% higher than the model evaluated at those covariates with admission-median references); this may reflect using the across-time median in the modeling dataset rather than the admission median in Table 1, but the exact reference is not stated. The relative effects of eGFR and albumin on metabolite CL match the paper’s reported ratios closely.
  • Shared covariate exponents on M3G and M6G clearance. Table 2’s “Covariate effect on M3G and M6G clearance” row reports a single eGFR coefficient (0.673) and a single albumin coefficient (1.1) applied to both metabolite clearances. The packaged model duplicates the coefficient onto the metabolite-suffixed names (e_crcl_cl_m3g = e_crcl_cl_m6g = 0.673; e_alb_cl_m3g = e_alb_cl_m6g = 1.1) so that the source-trace is explicit at the metabolite level. The nlmixr2lib convention does not currently have a syntactic form for “single estimate shared across metabolite suffixes”, so the duplication is the cleanest available encoding.
  • Fixed transformation fractions Fm1 = 0.55, Fm2 = 0.10. Per Methods 2.3.1, the fractions of morphine clearance that produce M3G and M6G could not be identified from the data (no mass-balance / urine measurements) and were fixed to literature values [21-23]. The metabolite clearance and metabolite volume estimates are therefore apparent values proportional to the assumed fractions; refitting under different Fm1 / Fm2 values would rescale CL_M3G and V_M3G (and similarly for M6G) by the same factor.
  • Three parallel depots for the three administration routes. The paper used route-specific fixed absorption rate constants (10 1/h SC, 6 1/h oral IR, 0.8 1/h oral CR) with a single estimated oral bioavailability F = 0.30 and an assumed SC bioavailability of 1. The packaged model encodes this as three parallel depot compartments (depot, depot2, depot3 for SC, oral IR, and oral CR respectively), with f(depot2) and f(depot3) set to the estimated f_oral and SC using the rxode2 default of 1. A dosing record selects the route by its cmt value.
  • TTD covariate is retrospective. The paper’s TTD covariate is the number of days remaining until the patient’s recorded time of death, which is known only after the fact. For prospective simulation of virtual cohorts where time of death is not known, set TTD to a large value (> 50 days) to recover the asymptotic typical-value morphine CL of 47.5 L/h. The packaged virtual cohorts in Figures 3-4 use TTD = 100 days for this purpose; Figure 5 deliberately varies TTD.
  • Dose unit is mg of morphine free base. The paper expressed M3G and M6G concentrations “in their morphine equivalents using the molecular weight” (Methods 2.3.1), so all internal state quantities are tracked in mg of morphine-equivalents and the observation equations convert mg/L to ug/L by a factor of 1000.
  • eGFR (CRCL canonical name). The paper used the four-variable Modification of Diet in Renal Disease (MDRD) formula; the canonical covariate CRCL register name covers MDRD-derived eGFR with the source alias eGFR. The covariateData[[CRCL]]$source_name field records the alias.