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SPT-07A (D-borneol) whole-body PBPK in rats, dogs and humans (Zhu 2024)

Source: vignettes/articles/Zhu_2024_borneol_pbpk.Rmd
Zhu_2024_borneol_pbpk.Rmd

Model and source

Zhu 2024 builds a single whole-body PBPK structure and parameterises it three times – once per species – so the paper contributes three model files that share this vignette.

model_names <- c(
  rat   = "Zhu_2024_borneol_rat_pbpk",
  dog   = "Zhu_2024_borneol_dog_pbpk",
  human = "Zhu_2024_borneol_human_pbpk"
)
# readModelDb() returns the model FUNCTION; rxode2::rxode() resolves it to a ui.
uis <- lapply(model_names, function(nm) rxode2::rxode(readModelDb(nm)))
  • Citation: Zhu X, Kong W, Wang Z, Liu X, Liu L. (2024). Prediction of SPT-07A Pharmacokinetics in Rats, Dogs, and Humans Using a Physiologically-Based Pharmacokinetic Model and In Vitro Data. Pharmaceutics 16(12):1596. doi:10.3390/pharmaceutics16121596. PMCID PMC11676658. Model equations: Equations (7)-(16), pages 7-8. Rat physiology (volumes, blood flows, tissue:plasma partition coefficients, Rb, fu,p): Table 1, ‘Rat (0.25 kg)’ columns. Unbound whole-organ intrinsic clearances: Table 2, rat rows (RLMs / RKMs / RIMs). The adipose permeability-surface product PS is NOT reported anywhere in the paper; it is recovered by digitising the paper’s own rat adipose simulation (Figure 4J) – see the inline note on PS and the vignette Errata. No erratum or corrigendum was located for this article.
  • Article: https://doi.org/10.3390/pharmaceutics16121596
  • PMC full text: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11676658/

There is no supplementary material for this article, and no erratum or corrigendum was located.

SPT-07A is D-borneol, a bicyclic monoterpene in development in China for acute ischaemic stroke. The model is built bottom-up from in vitro metabolism data: SPT-07A is cleared almost entirely by UDP-glucuronosyltransferase (UGT) glucuronidation, and unusually the kidney and intestine carry a substantial share of it, which is what motivated the whole-body treatment. All tissues are perfusion-rate limited except adipose, which is permeability-limited because rat tissue-distribution work showed marked adipose accumulation.

rat – Preclinical (rat, Sprague-Dawley). PBPK (whole-body, Phoenix WinNonlin 8.3). SPT-07A (D-borneol), a bicyclic monoterpene under development in China for acute ischaemic stroke, after intravenous bolus dosing. Fourteen mass-balance compartments (lung, heart, brain, muscle, skin, adipose, kidney, spleen, stomach, liver, intestine, rest of body, plus arterial and venous blood) built entirely from in vitro and in silico data: every tissue except adipose is perfusion-rate limited, and adipose is permeability-limited (a vascular sub-compartment vp_adipose exchanging with the extravascular tissue adipose through a permeability-surface product PS), because rat tissue-distribution studies showed high adipose accumulation. Elimination occurs in three organs – liver, kidney and intestine – as unbound whole-organ intrinsic clearances measured in rat liver, kidney and intestinal microsomes and scaled by MPPGL / MPPGK / MPPGI; SPT-07A is cleared almost entirely by UDP-glucuronosyltransferase glucuronidation (hepatic UGT CLint,u 2060 vs CYP 5.14 uL/min/mg protein), with the liver, kidney and intestine contributing 62.2%, 32.6% and 5.2% of systemic clearance. Tissue:plasma partition coefficients were computed by the Schmitt tissue-composition method, not fitted. This is the species the model was BUILT and validated on before being scaled to dog and human (see modellib(‘Zhu_2024_borneol_dog_pbpk’) and modellib(‘Zhu_2024_borneol_human_pbpk’)); PS is the single parameter estimated from in vivo data (fitted to rat adipose concentrations) and the two other species inherit it by body-weight scaling. Deterministic typical-value simulator: the paper’s 5th-95th percentile bands come from 100 virtual subjects whose variability magnitude is never reported, so no IIV and no residual error are encoded (see the vignette Errata).

dog – Preclinical (beagle dog). PBPK (whole-body, Phoenix WinNonlin 8.3). SPT-07A (D-borneol), a bicyclic monoterpene under development in China for acute ischaemic stroke, after intravenous bolus dosing. Structurally IDENTICAL to the rat model that was built first (see modellib(‘Zhu_2024_borneol_rat_pbpk’)): fourteen mass-balance compartments, perfusion-rate-limited distribution everywhere except a permeability-limited adipose, and elimination in liver, kidney and intestine as unbound whole-organ intrinsic clearances. What changes is (a) beagle physiology (Table 1 dog column) and (b) dog-specific in vitro intrinsic clearances (Table 2 DLMs / DKMs). This is a pure FORWARD SCALE-UP, not a refit: nothing was estimated from dog in vivo data, and the one in-vivo-fitted parameter in the whole model – the adipose permeability-surface product PS – is inherited from the rat by linear body-weight scaling (Equation 13), which is what makes the dog plasma predictions a genuine cross-species test. NO intestinal elimination: glucuronidation of SPT-07A was not detectable in dog intestinal microsomes (Table 2 DIMs row is ‘-’), so CLintIntestine is fixed at zero and the dog clears the drug through liver (87.3%) and kidney (12.7%) only. Dog hepatic glucuronidation is by far the fastest of the three species (UGT CLint,u 12,200 uL/min/mg protein vs 2060 in rat and 745 in human). Deterministic typical-value simulator: no IIV and no residual error are encoded (see the vignette Errata).

human – PBPK (whole-body, Phoenix WinNonlin 8.3). SPT-07A (D-borneol), a bicyclic monoterpene under development in China for acute ischaemic stroke, in healthy adults after intravenous infusion. Structurally IDENTICAL to the rat model the framework was built on (see modellib(‘Zhu_2024_borneol_rat_pbpk’)): fourteen mass-balance compartments, perfusion-rate-limited distribution everywhere except a permeability-limited adipose, and elimination in liver, kidney and intestine as unbound whole-organ intrinsic clearances. What changes is (a) human physiology (Table 1 human column) and (b) human-specific in vitro intrinsic clearances (Table 2 HLMs / HKMs / HIMs). This is a pure BOTTOM-UP PREDICTION: no human in vivo data were used to build or fit it – the clinical data it is compared against are cited from a separate first-in-human study – and the one in-vivo-fitted parameter, the adipose permeability-surface product PS, is inherited from the rat by linear body-weight scaling (Equation 13). SPT-07A is cleared predominantly by UGT glucuronidation (hepatic UGT CLint,u 745 vs CYP 3.35 uL/min/mg protein), with liver, kidney and intestine contributing 76.5%, 23.1% and 0.4% of systemic clearance; UGT phenotyping identified UGT1A1 and UGT2B7 as the responsible isoforms, though a relative-activity-factor reconstruction recovers only ~36% of the measured hepatic activity, implying additional uncharacterised UGTs. The notably high renal contribution is consistent with the paper’s opening observation that the reported human clearance (1942 mL/min) exceeds hepatic blood flow (1450 mL/min). Deterministic typical-value simulator: no IIV and no residual error are encoded (see the vignette Errata).

Population

The three species were studied as follows (Methods 2.3.1-2.3.4). Rats (258 Sprague-Dawley animals across the kinetic and tissue-distribution studies) received 0.5, 1 and 2 mg/kg as a single IV bolus plus a 1 mg/kg qd multiple-dose arm, with plasma sampled to 240 min. Six beagle dogs received 0.25, 0.5 and 1 mg/kg as single IV boluses in a three-period cross-over, plus a multiple-dose arm, with plasma sampled to 300 min. The human data are not original to this paper – Methods 2.3.4 states they were cited from the paper’s reference [7]: 36 healthy volunteers in three cohorts of 12 received 10, 20 or 40 mg as a 1 h IV infusion, single dose on day 0 and then q12h for 7 days beginning at 48 h.

Two sampling details matter for the validation below: rats and dogs received an IV bolus with the first sample at 5 min, whereas humans received a 1 h infusion. That difference is the whole explanation for the Table 3 discrepancy discussed in the Errata.

str(uis$human$population)
#> List of 11
#>  $ species       : chr "human"
#>  $ n_subjects    : int 36
#>  $ n_studies     : int 1
#>  $ age_range     : chr NA
#>  $ weight_range  : chr "70 kg reference adult (Table 1)"
#>  $ sex_female_pct: num NA
#>  $ race_ethnicity: chr "Not reported (study conducted in China)"
#>  $ disease_state : chr "Healthy volunteers"
#>  $ dose_range    : chr "10, 20 and 40 mg as a 1 h IV infusion; single dose on day 0, then q12h for 7 days from 48 h"
#>  $ regions       : chr "China"
#>  $ notes         : chr "Methods 2.3.4: the clinical data are NOT original to this paper -- they are cited from the paper's reference [7"| __truncated__

Source trace

Every ini() entry carries an in-file comment naming its source location; the table below collects the model-level provenance in one place.

Equation / parameter block Source location
Perfusion-limited tissue ODE Equation (7), page 7
Arterial blood ODE Equation (8), page 7
Venous blood ODE Equation (9), page 7
Lung ODE Equation (10), page 7
Adipose vascular sub-compartment ODE (vp_adipose, paper’s C1) Equation (11), page 7
Adipose extravascular ODE (adipose, paper’s C2) Equation (12), page 7
Interspecies PS scaling PS_i = PS_rat * (W_i / W_rat) Equation (13), page 7
Liver ODE (eliminating, receives splanchnic inflow) Equation (14), page 7
Kidney ODE (eliminating) Equation (15), page 8
Intestine ODE (eliminating) Equation (16), page 8
Organ volumes, blood flows, Kt:pl, Rb, fu,p (all 3 species) Table 1
Adipose V1 / V2 split (all 3 species) Paper text below Equation (12) (PK-Sim 11.2)
Unbound whole-organ intrinsic clearances (all 3 species) Table 2, CL int, in vivo,u column
Well-stirred organ clearances used as a check below Results 3.1.3
Predicted vs observed Cmax / AUC / CL / t1/2 Table 3
PS (adipose permeability-surface product) Not reported anywhere – back-solved from Figure 4J; see Errata

Structural checks

Before comparing against published exposures, two checks confirm the physiology and the elimination algebra were transcribed correctly. Both are exact arithmetic identities, so they either hold or they do not.

Blood-flow mass balance

Every tissue flow must sum to the cardiac output printed on the Table 1 lung row, and the printed total hepatic inflow must equal the hepatic artery plus the three splanchnic organs that drain into the liver.

flow_check <- lapply(names(uis), function(sp) {
  th <- uis[[sp]]$theta
  tissue_sum <- sum(th[c("Qheart", "Qbrain", "Qmuscle", "Qadipose", "Qskin",
                         "Qkidney", "Qother", "Qhepart", "Qspleen", "Qstomach",
                         "Qintestine")])
  liver_in <- sum(th[c("Qhepart", "Qspleen", "Qstomach", "Qintestine")])
  data.frame(
    Species             = sp,
    `Sum of tissue flows` = tissue_sum,
    `Cardiac output (Table 1)` = unname(th["Qtotal"]),
    `Difference (%)`    = 100 * (tissue_sum - th["Qtotal"]) / th["Qtotal"],
    `Total hepatic inflow` = liver_in,
    check.names = FALSE
  )
}) |> dplyr::bind_rows()

knitr::kable(
  flow_check, digits = 3, row.names = FALSE,
  caption = paste(
    "Blood-flow mass balance. The rat closes exactly; the dog and human tables",
    "carry small rounding differences (0.13% and 0.006%) which are reproduced",
    "as printed rather than silently reconciled. Total hepatic inflow matches",
    "the Table 1 'Liver' blood-flow row exactly in all three species",
    "(rat 12.30, dog 323.33, human 1518.33 mL/min)."
  )
)
Blood-flow mass balance. The rat closes exactly; the dog and human tables carry small rounding differences (0.13% and 0.006%) which are reproduced as printed rather than silently reconciled. Total hepatic inflow matches the Table 1 ‘Liver’ blood-flow row exactly in all three species (rat 12.30, dog 323.33, human 1518.33 mL/min).
Species Sum of tissue flows Cardiac output (Table 1) Difference (%) Total hepatic inflow
rat 83.90 83.9 0.000 12.30
dog 1121.46 1120.0 0.130 323.33
human 5600.33 5600.0 0.006 1518.33

Well-stirred organ clearances

Equations (14)-(16) drive elimination with the plasma-equivalent concentration C_t / K_t:pl, so each eliminating organ behaves as a well-stirred compartment with blood clearance Q * fu,b * CLint / (Q + fu,b * CLint), where fu,b = fu,p / Rb. Reproducing the paper’s own Results 3.1.3 numbers from the packaged ini() values is a direct test that the elimination term was encoded correctly.

well_stirred <- function(ui, organ) {
  th <- ui$theta
  clint <- unname(th[paste0("CLint", organ)]) * unname(th["BW"])
  fub   <- unname(th["fup"]) / unname(th["Rb"])
  q <- switch(organ,
    Liver     = sum(th[c("Qhepart", "Qspleen", "Qstomach", "Qintestine")]),
    Kidney    = unname(th["Qkidney"]),
    Intestine = unname(th["Qintestine"])
  )
  unname(q * fub * clint / (q + fub * clint) / th["BW"])
}

# Results 3.1.3, mL/min/kg. Dog intestine is absent (no glucuronidation in DIMs).
published_cl <- tibble::tribble(
  ~Species, ~Organ,      ~Published,
  "rat",    "Liver",     46.6,
  "rat",    "Kidney",    24.4,
  "rat",    "Intestine", 3.90,
  "dog",    "Liver",     37.8,
  "dog",    "Kidney",    5.50,
  "human",  "Liver",     20.4,
  "human",  "Kidney",    6.14,
  "human",  "Intestine", 0.109
)

cl_check <- published_cl |>
  dplyr::mutate(
    Recomputed = mapply(function(s, o) well_stirred(uis[[s]], o), Species, Organ),
    `% diff`   = 100 * (Recomputed - Published) / Published
  )

knitr::kable(
  cl_check, digits = c(0, 0, 3, 3, 2),
  caption = paste(
    "Well-stirred organ blood clearances (mL/min/kg) recomputed from the",
    "packaged ini() values versus the values the paper reports in Results",
    "3.1.3. All eight agree to the printed precision."
  )
)
Well-stirred organ blood clearances (mL/min/kg) recomputed from the packaged ini() values versus the values the paper reports in Results 3.1.3. All eight agree to the printed precision.
Species Organ Published Recomputed % diff
rat Liver 46.600 46.607 0.02
rat Kidney 24.400 24.447 0.19
rat Intestine 3.900 3.905 0.13
dog Liver 37.800 37.837 0.10
dog Kidney 5.500 5.502 0.05
human Liver 20.400 20.363 -0.18
human Kidney 6.140 6.142 0.03
human Intestine 0.109 0.109 -0.12 
stopifnot(nrow(cl_check) == 8L, max(abs(cl_check$`% diff`)) < 1)

Simulation

The models are deterministic typical-value simulators: the paper never reports the magnitude of the parameter variability behind its 5th-95th percentile bands, so no IIV and no residual error are encoded. Every simulation below is therefore a single typical subject per dose arm (omega = NA is not passed – these models declare no etas).

solve_arm <- function(ui, dose_mg, times, infusion_min = 0, dose_times = 0) {
  rate_val <- if (infusion_min > 0) dose_mg / infusion_min else 0
  dosing <- data.frame(
    time = dose_times, amt = dose_mg, evid = 1L,
    cmt = "venous", rate = rate_val
  )
  obs <- data.frame(
    time = times, amt = NA_real_, evid = 0L,
    cmt = "venous", rate = 0            # ODE state name, never the observable "Cc"
  )
  ev <- dplyr::arrange(dplyr::bind_rows(dosing, obs), time, dplyr::desc(evid))
  rxode2::rxSolve(ui, ev, atol = 1e-10, rtol = 1e-10, returnType = "data.frame")
}

# A grid that resolves the post-bolus distribution phase finely enough for NCA.
grid_for <- function(tmax) {
  g <- c(
    seq(0, 5, by = 0.05), seq(5, 60, by = 0.25),
    seq(60, 600, by = 2), seq(600, max(tmax, 600), by = 10)
  )
  sort(unique(g[g <= tmax]))
}
arms <- tibble::tribble(
  ~species, ~dose_label,  ~dose_mg,        ~infusion_min,
  "rat",    "0.5 mg/kg",  0.5   * 0.25,    0,
  "rat",    "1 mg/kg",    1     * 0.25,    0,
  "rat",    "2 mg/kg",    2     * 0.25,    0,
  "dog",    "0.25 mg/kg", 0.25  * 8.5,     0,
  "dog",    "0.5 mg/kg",  0.5   * 8.5,     0,
  "dog",    "1 mg/kg",    1     * 8.5,     0,
  "human",  "10 mg",      10,              60,
  "human",  "20 mg",      20,              60,
  "human",  "40 mg",      40,              60
)

TMAX <- 6000  # min; >= 15 terminal half-lives in every species
sims <- arms |>
  dplyr::rowwise() |>
  dplyr::group_map(function(a, ...) {
    solve_arm(uis[[a$species]], a$dose_mg, grid_for(TMAX), a$infusion_min) |>
      dplyr::mutate(species = a$species, dose_label = a$dose_label,
                    dose_mg = a$dose_mg)
  }) |>
  dplyr::bind_rows() |>
  dplyr::mutate(arm = paste(species, dose_label))

Plasma concentration-time profiles (Figures 3, 5 and 6) 

sims |>
  dplyr::filter(!dplyr::near(time, 0), time <= 300, Cc > 1e-12) |>
  dplyr::mutate(species = factor(species, c("rat", "dog", "human"))) |>
  ggplot(aes(time, Cc * 1000, colour = dose_label)) +
  geom_line(linewidth = 0.7) +
  facet_wrap(~species, scales = "free_y") +
  scale_y_log10() +
  labs(x = "Time (min)", y = "Plasma SPT-07A (ng/mL)", colour = "Dose") +
  theme_bw() +
  theme(legend.position = "bottom")
Replicates Figure 3 (rats), Figure 5 (dogs) and Figure 6 (humans) of Zhu 2024: simulated typical-value plasma SPT-07A after single intravenous doses.

Replicates Figure 3 (rats), Figure 5 (dogs) and Figure 6 (humans) of Zhu 2024: simulated typical-value plasma SPT-07A after single intravenous doses.

Rat tissue distribution (Figure 4)

Figure 4 of the paper shows plasma plus ten tissues in rats after 2 mg/kg. The adipose panel (4J) is the one that carries information about PS: it is the only tissue whose simulated profile rises to a peak rather than declining monotonically, which is the signature of permeability-limited uptake.

rat2 <- solve_arm(uis$rat, 2 * 0.25, grid_for(300))
th_rat <- uis$rat$theta

tissue_conc <- rat2 |>
  dplyr::transmute(
    time,
    plasma    = Cc,
    heart     = heart     / (th_rat[["Vheart"]]     * 1e-3),
    liver     = liver     / (th_rat[["Vliver"]]     * 1e-3),
    spleen    = spleen    / (th_rat[["Vspleen"]]    * 1e-3),
    stomach   = stomach   / (th_rat[["Vstomach"]]   * 1e-3),
    brain     = brain     / (th_rat[["Vbrain"]]     * 1e-3),
    intestine = intestine / (th_rat[["Vintestine"]] * 1e-3),
    muscle    = muscle    / (th_rat[["Vmuscle"]]    * 1e-3),
    lung      = lung      / (th_rat[["Vlung"]]      * 1e-3),
    adipose   = adipose   / (th_rat[["Vadipose2"]]  * 1e-3),
    kidney    = kidney    / (th_rat[["Vkidney"]]    * 1e-3)
  ) |>
  tidyr::pivot_longer(-time, names_to = "tissue", values_to = "conc") |>
  dplyr::filter(!dplyr::near(time, 0), conc > 1e-12)

ggplot(tissue_conc, aes(time, conc * 1000)) +
  geom_line(linewidth = 0.6) +
  facet_wrap(~tissue, scales = "free_y", ncol = 4) +
  scale_y_log10() +
  labs(x = "Time (min)", y = "Concentration (ng/mL or ng/g)") +
  theme_bw()
Replicates Figure 4B-K of Zhu 2024: simulated typical-value tissue concentrations in the rat after a 2 mg/kg IV bolus.

Replicates Figure 4B-K of Zhu 2024: simulated typical-value tissue concentrations in the rat after a 2 mg/kg IV bolus.

Recovering PS from Figure 4J

PS is the single parameter in the whole model that was fitted to in vivo data, and its value is not reported. It was recovered by digitising the median line of Figure 4J and solving for the PS that reproduces it; the optimiser returns 0.499 mL/min, i.e. the authors’ round 0.5. The digitised points are reproduced below with their provenance so a reviewer can audit the back-solve.

# Digitised from Figure 4J (rat adipose, 2 mg/kg IV bolus), median line.
fig4j <- tibble::tibble(
  time = c(5, 30, 60, 90),
  conc = c(430, 410, 278, 205)     # ng/g
)

adipose_pred <- tissue_conc |>
  dplyr::filter(tissue == "adipose") |>
  dplyr::mutate(conc_ng = conc * 1000)

ps_cmp <- fig4j |>
  dplyr::mutate(
    Simulated = approx(adipose_pred$time, adipose_pred$conc_ng, xout = time)$y,
    `% diff`  = 100 * (Simulated - conc) / conc
  ) |>
  dplyr::rename("Time (min)" = time, "Digitised Fig 4J (ng/g)" = conc)

knitr::kable(ps_cmp, digits = c(0, 0, 1, 1),
             caption = "Back-solved PS = 0.5 mL/min against the digitised Figure 4J median.")
Back-solved PS = 0.5 mL/min against the digitised Figure 4J median.
Time (min) Digitised Fig 4J (ng/g) Simulated % diff
5 430 421.3 -2.0
30 410 413.7 0.9
60 278 305.8 10.0
90 205 215.4 5.1 

ggplot(adipose_pred, aes(time, conc_ng)) +
  geom_line(linewidth = 0.7) +
  geom_point(data = fig4j, aes(time, conc), colour = "firebrick", size = 2.5) +
  labs(x = "Time (min)", y = "Rat adipose SPT-07A (ng/g)",
       subtitle = "Line: packaged model. Points: digitised Figure 4J median.") +
  theme_bw()
Simulated rat adipose (extravascular) concentration at the back-solved PS = 0.5 mL/min versus the digitised median line of Figure 4J.

Simulated rat adipose (extravascular) concentration at the back-solved PS = 0.5 mL/min versus the digitised median line of Figure 4J. 

# The back-solve is only credible if it reproduces the digitised curve closely.
stopifnot(max(abs(ps_cmp$`% diff`)) < 15)

PKNCA validation against Table 3 

nca_input <- sims |>
  dplyr::filter(!is.na(Cc)) |>
  dplyr::mutate(id = arm)

dose_input <- arms |>
  dplyr::mutate(id = paste(species, dose_label), time = 0) |>
  dplyr::select(id, time, dose_mg, species, dose_label)

conc_obj <- PKNCA::PKNCAconc(nca_input, Cc ~ time | id)
dose_obj <- PKNCA::PKNCAdose(as.data.frame(dose_input), dose_mg ~ time | id)

intervals <- data.frame(
  start = 0, end = Inf,
  cmax = TRUE, aucinf.obs = TRUE
)

res <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals))

# PKNCA also returns the DEPENDENCIES of the requested parameters -- notably its
# own `half.life` / `lambda.z`, which aucinf.obs needs. Those extra rows must be
# dropped: a windowed half-life is appended below, and leaving PKNCA's
# asymptotic one in place would silently produce two half.life rows per arm,
# which ncaComparisonTable() would then average.
nca <- as.data.frame(res) |>
  dplyr::filter(PPTESTCD %in% c("cmax", "aucinf.obs")) |>
  dplyr::select(id, PPTESTCD, PPORRES) |>
  dplyr::left_join(dose_input, by = "id")

Terminal half-life is deliberately not taken from a 0-Inf PKNCA interval. This model has a slowly-equilibrating adipose compartment, so its true asymptotic terminal slope (rat 60, dog 139, human 217 min) is much shallower than anything a finite study could observe; comparing that against a published half-life would compare two different quantities.

Instead one uniform rule is applied to all three species – fit the terminal slope over the final half of that species’ sampling window – and the resulting windows are stated explicitly. Rats were sampled to 240 min and dogs to 300 min (Methods 2.3.1 and 2.3.3). The clinical sampling schedule is not given in this paper (the human data are cited from reference [7]), so a 12 h window is assumed, consistent with the q12h multiple-dose regimen. The rule is fixed in advance and applied identically to every species rather than tuned per arm.

# end of the sampling window (min); the slope is fitted over the final half
hl_end   <- c(rat = 240, dog = 300, human = 720)
hl_start <- hl_end / 2

half_life_windowed <- sims |>
  dplyr::group_by(species, dose_label) |>
  dplyr::group_modify(function(d, k) {
    sp <- k$species
    w <- d[d$time >= hl_start[[sp]] & d$time <= hl_end[[sp]] & d$Cc > 0, ]
    data.frame(PPTESTCD = "half.life",
               PPORRES  = unname(log(2) / -coef(lm(log(w$Cc) ~ w$time))[2]))
  }) |>
  dplyr::ungroup()

nca <- dplyr::bind_rows(nca, half_life_windowed)

# Guard against the duplicate-row trap above: ncaComparisonTable() takes the
# MEDIAN over repeated (group, parameter) rows, so a stray duplicate would be
# silently averaged rather than reported or flagged.
stopifnot(!any(duplicated(nca[, c("species", "dose_label", "PPTESTCD")])))

Table 3 reports predicted Cmax (humans only – the animal arms were IV boluses, for which the paper tabulates no Cmax), AUC and terminal half-life. The comparison below uses the paper’s predicted column, since reproducing the authors’ own simulation is what validates the packaged implementation.

# Zhu 2024 Table 3, "Pre" (predicted) columns.
# AUC in ug*min/mL == mg*min/L, which is the unit of Cc * time here.
published <- tibble::tribble(
  ~species, ~dose_label,  ~cmax,     ~aucinf.obs, ~half.life,
  "rat",    "0.5 mg/kg",  NA_real_,  3.78,        NA_real_,
  "rat",    "1 mg/kg",    NA_real_,  7.56,        58.2,
  "rat",    "2 mg/kg",    NA_real_,  15.1,        58.2,
  "dog",    "0.25 mg/kg", NA_real_,  3.60,        NA_real_,
  "dog",    "0.5 mg/kg",  NA_real_,  7.19,        76.7,
  "dog",    "1 mg/kg",    NA_real_,  14.4,        NA_real_,
  "human",  "10 mg",      0.0701,    5.76,        179,
  "human",  "20 mg",      0.1400,    11.5,        179,
  "human",  "40 mg",      0.2810,    23.0,        179
) |>
  tidyr::pivot_longer(c(cmax, aucinf.obs, half.life),
                      names_to = "PPTESTCD", values_to = "PPORRES") |>
  dplyr::filter(!is.na(PPORRES))

cmp <- nlmixr2lib::ncaComparisonTable(
  simulated     = dplyr::select(nca, species, dose_label, PPTESTCD, PPORRES),
  reference     = published,
  by            = c("species", "dose_label"),
  units         = c(cmax = "mg/L", aucinf.obs = "mg*min/L", half.life = "min"),
  tolerance_pct = 20
)

cmp |>
  dplyr::rename("Species" = species, "Dose" = dose_label) |>
  knitr::kable(
    caption = paste(
      "Simulated NCA from the packaged models versus the PREDICTED column of",
      "Zhu 2024 Table 3. * marks a difference above 20%. The human arms -- the",
      "only ones dosed as an infusion -- agree throughout; every starred row is",
      "a rat or dog AUC, for the reason set out in the Errata."
    ),
    align = c("l", "l", "l", "r", "r", "r")
  )
Simulated NCA from the packaged models versus the PREDICTED column of Zhu 2024 Table 3. * marks a difference above 20%. The human arms – the only ones dosed as an infusion – agree throughout; every starred row is a rat or dog AUC, for the reason set out in the Errata.
NCA parameter Species Dose Reference Simulated % diff
Cmax (mg/L) human 10 mg 0.0701 0.0702 +0.1%
Cmax (mg/L) human 20 mg 0.14 0.14 +0.3%
Cmax (mg/L) human 40 mg 0.281 0.281 -0.1%
AUC0-∞ (obs) (mg*min/L) rat 0.5 mg/kg 3.78 6.38 +68.8%*
AUC0-∞ (obs) (mg*min/L) rat 1 mg/kg 7.56 12.8 +68.8%*
AUC0-∞ (obs) (mg*min/L) rat 2 mg/kg 15.1 25.5 +69.1%*
AUC0-∞ (obs) (mg*min/L) dog 0.5 mg/kg 7.19 12.6 +75.1%*
AUC0-∞ (obs) (mg*min/L) dog 1 mg/kg 14.4 25.2 +74.9%*
AUC0-∞ (obs) (mg*min/L) dog 0.25 mg/kg 3.6 6.3 +74.9%*
AUC0-∞ (obs) (mg*min/L) human 10 mg 5.76 5.87 +1.9%
AUC0-∞ (obs) (mg*min/L) human 20 mg 11.5 11.7 +2.1%
AUC0-∞ (obs) (mg*min/L) human 40 mg 23 23.5 +2.1%
t½ (min) rat 1 mg/kg 58.2 59 +1.4%
t½ (min) rat 2 mg/kg 58.2 59 +1.4%
t½ (min) dog 0.5 mg/kg 76.7 75 -2.2%
t½ (min) human 10 mg 179 183 +2.5%
t½ (min) human 20 mg 179 183 +2.5%
t½ (min) human 40 mg 179 183 +2.5%

The human arms are the decisive test, because they are a pure bottom-up prediction: no human in vivo data entered the model at any point.

human_rows <- cmp[cmp$species == "human", ]
pct <- suppressWarnings(as.numeric(gsub("[^0-9.+-]", "", human_rows$`% diff`)))
# Tightened to the accuracy actually achieved (worst human row ~5%), so this
# gate fails on a regression rather than merely on a gross error.
stopifnot(nrow(human_rows) == 9L, !anyNA(pct), max(abs(pct)) < 8)

Dose proportionality

The model contains no saturable process, so exposure must be exactly dose-proportional within each species. This is a structural self-check rather than a comparison against the paper.

nca |>
  dplyr::filter(PPTESTCD == "aucinf.obs") |>
  dplyr::group_by(species) |>
  dplyr::summarise(
    `CL (mL/min/kg)` = paste(round(sort(unique(
      round(dose_mg / PPORRES / c(rat = 0.25, dog = 8.5, human = 70)[species[1]] * 1000, 2)
    ))), collapse = ", "),
    .groups = "drop"
  ) |>
  dplyr::rename("Species" = species) |>
  knitr::kable(caption = "Clearance is identical across dose levels within each species, as a linear model requires.")
Clearance is identical across dose levels within each species, as a linear model requires.
Species CL (mL/min/kg)
dog 40
human 24
rat 78

Multiple dosing (human, q12h)

For a linear system the steady-state AUC over a dosing interval equals the single-dose AUC to infinity. Table 3’s human multiple-dose predictions (5.87, 11.7 and 23.5) are indeed within ~2% of its single-dose predictions (5.76, 11.5 and 23.0), which is consistent with that identity; the simulation reproduces it.

md <- solve_arm(
  uis$human, 20, times = sort(unique(c(seq(0, 9 * 1440, by = 5)))),
  infusion_min = 60,
  dose_times = c(0, 2880 + seq(0, 13) * 720)   # single dose, then q12h x 7 days from 48 h
)

tau_window <- md |> dplyr::filter(time >= 2880 + 12 * 720, time <= 2880 + 13 * 720)
auc_tau <- with(tau_window, sum(diff(time) * (head(Cc, -1) + tail(Cc, -1)) / 2))
auc_sd  <- nca$PPORRES[nca$id == "human 20 mg" & nca$PPTESTCD == "aucinf.obs"]

knitr::kable(
  data.frame(
    Quantity = c("Steady-state AUC over tau (12 h), 20 mg q12h",
                 "Single-dose AUC to infinity, 20 mg",
                 "Zhu 2024 Table 3 predicted, 20 mg multiple dose",
                 "Zhu 2024 Table 3 predicted, 20 mg single dose"),
    `Value (mg*min/L)` = round(c(auc_tau, auc_sd, 11.7, 11.5), 2),
    check.names = FALSE
  ),
  caption = "Linear-system identity: steady-state AUC over a dosing interval equals the single-dose AUC to infinity."
)
Linear-system identity: steady-state AUC over a dosing interval equals the single-dose AUC to infinity.
Quantity Value (mg*min/L)
Steady-state AUC over tau (12 h), 20 mg q12h 11.72
Single-dose AUC to infinity, 20 mg 11.74
Zhu 2024 Table 3 predicted, 20 mg multiple dose 11.70
Zhu 2024 Table 3 predicted, 20 mg single dose 11.50 
md |>
  dplyr::filter(!dplyr::near(time, 0), Cc > 1e-12) |>
  ggplot(aes(time / 60, Cc * 1000)) +
  geom_line(linewidth = 0.5) +
  labs(x = "Time (h)", y = "Plasma SPT-07A (ng/mL)") +
  theme_bw()
Replicates the human multiple-dose panel of Figure 6: 20 mg 1 h infusion, single dose on day 0 then q12h from 48 h.

Replicates the human multiple-dose panel of Figure 6: 20 mg 1 h infusion, single dose on day 0 then q12h from 48 h. 

stopifnot(abs(auc_tau - auc_sd) / auc_sd < 0.02)

Assumptions and deviations

Errata and source inconsistencies

1. PS is not reported anywhere and was back-solved from a figure. Methods 2.5.1 states only that “PS is the permeability-surface product, which was estimated by fitting this value to the adipose concentration data in rats.” The fitted number appears in no table, figure caption or text, and the article has no supplementary material. It was recovered here by digitising the median line of Figure 4J and solving for the PS that reproduces it; the optimiser returns 0.499 mL/min against the authors’ evident round value of 0.5, and at PS = 0.5 the model matches the digitised curve to 5% on average and 10% at worst (table above). Because PS governs distribution and not elimination, AUC and clearance are mathematically independent of it – the back-solve affects the adipose profile and the terminal phase only, and none of the AUC or clearance validation above depends on it. The dog and human values (17.0 and 140 mL/min) follow from Equation (13), which as printed carries no exponent: the scaling is linear in body weight, not allometric.

2. Table 3’s rat and dog predicted AUC cannot be reconciled with the paper’s own equations, and the profiles show the equations are right. The packaged implementation reproduces the human Table 3 predictions to within about 2% on Cmax, AUC and clearance at all three doses, and reproduces all eight of the paper’s Results 3.1.3 organ clearances exactly. For rats and dogs, however, the simulated AUC to infinity is systematically higher than the Table 3 predicted column – by the same factor at every dose level within a species, though a different factor between the two species – even though the predicted terminal half-lives match. Every number quoted in this section is computed below rather than transcribed, so it cannot drift out of step with the packaged models.

trapz <- function(t, conc) sum(diff(t) * (head(conc, -1) + tail(conc, -1)) / 2)

published_auc <- published |>
  dplyr::filter(PPTESTCD == "aucinf.obs") |>
  dplyr::select(species, dose_label, `Table 3 predicted` = PPORRES)

auc_window <- sims |>
  dplyr::filter(species != "human") |>
  dplyr::group_by(species, dose_label) |>
  dplyr::summarise(
    `Model AUC(0-inf)`  = trapz(time, Cc),
    `Model AUC(5-inf)`  = trapz(time[time >= 5], Cc[time >= 5]),
    .groups = "drop"
  ) |>
  dplyr::inner_join(published_auc, by = c("species", "dose_label")) |>
  dplyr::mutate(
    `Model / Table 3` = `Model AUC(0-inf)` / `Table 3 predicted`,
    `Table 3 within window range` =
      `Table 3 predicted` > `Model AUC(5-inf)` &
      `Table 3 predicted` < `Model AUC(0-inf)`
  )

auc_window |>
  dplyr::rename("Species" = species, "Dose" = dose_label) |>
  knitr::kable(
    digits = c(0, 0, 3, 3, 2, 3, 0),
    caption = paste(
    "Zhu 2024 Table 3's predicted animal AUC against the packaged model's AUC",
    "over two windows (mg*min/L). The ratio is constant across doses within a",
      "species -- as a linear model requires -- and Table 3 falls between the",
      "model's AUC from 5 min onward and its AUC to infinity in every arm."
    )
  )
Zhu 2024 Table 3’s predicted animal AUC against the packaged model’s AUC over two windows (mg*min/L). The ratio is constant across doses within a species – as a linear model requires – and Table 3 falls between the model’s AUC from 5 min onward and its AUC to infinity in every arm.
Species Dose Model AUC(0-inf) Model AUC(5-inf) Table 3 predicted Model / Table 3 Table 3 within window range
dog 0.25 mg/kg 6.302 3.070 3.60 1.750 TRUE
dog 0.5 mg/kg 12.604 6.140 7.19 1.753 TRUE
dog 1 mg/kg 25.207 12.279 14.40 1.750 TRUE
rat 0.5 mg/kg 6.390 2.699 3.78 1.690 TRUE
rat 1 mg/kg 12.779 5.397 7.56 1.690 TRUE
rat 2 mg/kg 25.559 10.795 15.10 1.693 TRUE 
# The overprediction must be a pure scale factor within each species (linear
# model), and the sampling-window explanation must actually hold in every arm.
ratio_spread <- auc_window |>
  dplyr::group_by(species) |>
  dplyr::summarise(spread = diff(range(`Model / Table 3`)), .groups = "drop")
stopifnot(
  nrow(auc_window) == 6L,
  all(auc_window$`Table 3 within window range`),
  max(ratio_spread$spread) < 0.01
)

The factor is 1.69-fold in the rat and 1.75-fold in the dog, while the predicted rat terminal half-life matches (59.0 min simulated versus 58.2 reported). Two further observations locate the problem in the Table 3 summary rather than in the model:

  • The paper’s own stated systemic clearance for the rat – 75.0 mL/min/kg, Results 3.1.3 – agrees with the implemented equations, which give 78 mL/min/kg of plasma clearance (the small residual gap is the blood-to-plasma ratio plus sequential gut-to-liver extraction, neither of which the paper’s simple sum of organ clearances captures). Table 3’s predicted rat clearance of 129-132 mL/min/kg agrees with neither. The two numbers appear in the same paper and are mutually inconsistent, since AUC = Dose / CL.
  • The rat plasma profile of Figure 4A agrees with the packaged model, so the figures support the printed equations; only the Table 3 AUC/CL column for the animal species does not.
fig4a <- tibble::tibble(
  time = c(30, 60, 90),
  `Digitised Fig 4A (ng/mL)` = c(107, 51.3, 29.8)   # rat plasma, 2 mg/kg
) |>
  dplyr::mutate(
    `Model (ng/mL)` = approx(rat2$time, rat2$Cc * 1000, xout = time)$y,
    `% diff` = 100 * (`Model (ng/mL)` - `Digitised Fig 4A (ng/mL)`) /
      `Digitised Fig 4A (ng/mL)`
  ) |>
  dplyr::rename("Time (min)" = time)

knitr::kable(fig4a, digits = c(0, 1, 1, 1),
             caption = "Packaged rat model versus the digitised Figure 4A plasma profile (2 mg/kg IV bolus).")
Packaged rat model versus the digitised Figure 4A plasma profile (2 mg/kg IV bolus).
Time (min) Digitised Fig 4A (ng/mL) Model (ng/mL) % diff
30 107.0 106.7 -0.2
60 51.3 47.4 -7.5
90 29.8 29.1 -2.3 
stopifnot(max(abs(fig4a$`% diff`)) < 15)

The most likely mechanism is the sampling window. Rats and dogs received an IV bolus with the first sample at 5 min, so an NCA that starts at the first observed concentration misses the steep 0-5 min distribution phase entirely; humans received a 1 h infusion, which has no such spike, and their numbers reconcile. Consistent with this, the paper’s rat and dog predicted AUC values sit between the model’s AUC to infinity and its AUC computed from 5 min onward in every arm – which the table above checks explicitly. Following the standing policy that a printed equation outranks a conflicting summary, the equations as printed are what is implemented; the rat and dog AUC rows of the comparison table above are starred for this reason and should not be read as an implementation defect.

3. Dog multiple-dose regimen is stated inconsistently. Methods 2.3.3 says the dogs received 1 mg/kg qd for 7 days; Table 3 labels the dog multiple-dose row 0.5 mg/kg and reports the same predicted AUC (7.19) as its 0.5 mg/kg single-dose row. The Table 3 label is self-consistent with the reported exposure and is the one recorded in the model’s population metadata.

4. Blood-flow rounding. The dog tissue flows sum to 1121.46 mL/min against a printed cardiac output of 1120 (0.13%), and the human flows to 5600.33 against 5600 (0.006%). Both are reproduced as printed rather than reconciled, so the packaged values match Table 1 byte for byte. The rat closes exactly.

Assumptions

  • No IIV and no residual error. The paper’s 5th-95th percentile bands come from 100 virtual subjects, but the magnitude and distribution of the underlying parameter variability are never reported. Rather than invent variances, both are omitted: propSd is fixed(0) and no etas are declared. The packaged models are therefore deterministic typical-value simulators and cannot reproduce the prediction intervals in Figures 3-6, only the median lines.
  • Terminal half-life is window-dependent, and the window is assumed. The permeability-limited adipose compartment makes the model’s asymptotic terminal slope (rat 60, dog 139, human 217 min) much shallower than any observable half-life, so the comparison uses the final half of each species’ sampling window. Rat and dog windows come from Methods 2.3.1 and 2.3.3; the human window is assumed to be 12 h, since this paper does not report the clinical sampling schedule. The same rule is applied to all three species without per-arm tuning, and it recovers the paper’s predicted half-lives to within 5% in every species – but a reader who assumes a different clinical window will get a different human number, and none of the AUC, Cmax or clearance conclusions depends on this choice.
  • Observation compartment. The paper does not state which blood pool its simulated plasma concentration is drawn from. Venous plasma is used (Cc <- c_venous / Rb), matching the sampling site for the dog (forearm vein) and human (venous) studies; the rat study sampled the femoral artery. Arterial and venous concentrations differ materially only during the first few minutes after a bolus.
  • Blood versus plasma. Because Kt:pl is a tissue-to-plasma ratio while the perfusion terms carry Kt:pl / Rb, the arterial, venous and adipose vascular states hold blood concentrations; the reported plasma concentration is obtained by dividing by Rb. This follows directly from Equations (7)-(10) and is what makes the recomputed organ clearances match Results 3.1.3.
  • Body weight is not a covariate. Organ volumes and blood flows are the absolute Table 1 values for the reference animal (0.25 kg rat, 8.5 kg beagle, 70 kg adult) and are not weight-scaled. BW appears only in the per-kilogram intrinsic-clearance terms of Equations (14)-(16). Simulating a different body weight requires rescaling the volumes and flows as well, which the paper does not parameterise.
  • MPPGL / MPPGK / MPPGI are not model parameters. These microsomal-protein scalars appear in Table 1 but act upstream, converting the measured in vitro CLint,u into the whole-organ CLint,in vivo,u values of Table 2 that the ODEs actually use. Only the Table 2 outputs are packaged.
  • Compartment naming. The paper’s adipose sub-compartments C1 and C2 are mapped to the registered canonicals vp_adipose (vascular) and adipose (the tissue). The registered is_adipose was deliberately not used for C2, because the is_ prefix denotes interstitial fluid, whereas for a small lipophilic monoterpene with Kadipose:plasma of 1.72-2.84 the extravascular space is predominantly intracellular lipid. The paper’s “rest of body” is the registered other.