Skip to contents

Pazopanib mouse TGI (Ouerdani 2015)

Source: vignettes/articles/Ouerdani_2015_pazopanib_mouse.Rmd
Ouerdani_2015_pazopanib_mouse.Rmd

Model and source

  • Citation: Ouerdani A, Struemper H, Suttle AB, Ouellet D, Ribba B. Preclinical modeling of tumor growth and angiogenesis inhibition to describe pazopanib clinical effects in renal cell carcinoma. CPT Pharmacometrics Syst Pharmacol. 2015;4(11):660-668. doi:[10.1002/psp4.12001](https://doi.org/10.1002/psp4.12001).
  • The on-disk PDF is the corrected version of the article (revised online 2015-11-12); the paper’s first-page footnote states that Table 1 was replaced after the initial publication. All parameter values used here come from the corrected Table 1.

This vignette validates the preclinical (mouse, CAKI-2 xenograft) fit of the semi-mechanistic tumour-growth and angiogenesis-inhibition (TGI) model. The paired clinical fit lives in Ouerdani_2015_pazopanib and is covered by a separate vignette.

Population

Female CB-17 SCID mice, 8-10 weeks old, were inoculated subcutaneously with RCC CAKI-2 tumour cells. Once tumours had grown to 100-250 mm^3, mice were randomised into four dose groups of 8 animals each (vehicle, 10, 30, or 100 mg/kg pazopanib once daily by oral gavage for 24 days), giving 32 animals total. Tumour volumes were measured twice weekly by handheld caliper and calculated as (length * width^2) / 2. Source: Ouerdani 2015 Methods, preclinical-data section.

The same information is available programmatically via readModelDb("Ouerdani_2015_pazopanib_mouse")$population.

Source trace

Equations from Ouerdani 2015 Methods Equation 1 (preclinical fit with n = 1); parameter values from the corrected Table 1, preclinical column. The model has no PK ODE – drug exposure enters as the per-dose-group covariate AUC_PAZO, sourced from the FDA Pharmacology Review for pazopanib NDA 022465 (cited in the paper).

Equation / parameter Value Source location
d/dt(tumorSize) n/a Ouerdani 2015 Methods Equation 1
d/dt(carryingCapacity) n/a Ouerdani 2015 Methods Equation 1
a = a0 * AUC_PAZO^b_a n/a Ouerdani 2015 Methods Equation 2
c = c0 * AUC_PAZO^b_c n/a Ouerdani 2015 Methods Equation 3
lk_tumor log(0.166) Table 1 preclinical k = 0.166 (RSE 24%)
lk_cap (b) log(0.0183) Table 1 preclinical b = 0.0183 (RSE 58%); IIV fixed to 0
lk_aa0 (c0) log(0.007) Table 1 preclinical c = 0.007 (RSE 29%)
lk_cyto0 (a0) log(0.251) Table 1 preclinical a = 0.251 (RSE 13%)
lk_res (d) log(0.196) Table 1 preclinical d = 0.196 (RSE 26%)
lK0 log(543) Table 1 preclinical K0 = 543 (RSE 15%)
e_auc_pazo_k_aa (b_c) 0.332 Table 1 preclinical b_c = 0.332 (RSE 15%)
e_auc_pazo_k_cyto (b_a) fixed(0) Paper text: ba initially estimated at 0.0002 then fixed to 0
n fixed(1) Methods + Results: best preclinical fit at n = 1 (profile-tested 0.5-3)
etalk_tumor 0.2809 (= 0.53^2) Table 1 preclinical IIV on k = 53% (RSE 108%)
etalk_aa0 0.0361 (= 0.19^2) Table 1 preclinical IIV on c0 = 19% (RSE 127%)
etalk_cyto0 0.0576 (= 0.24^2) Table 1 preclinical IIV on a0 = 24% (RSE 96%)
etalk_res 0.1764 (= 0.42^2) Table 1 preclinical IIV on d = 42% (RSE 103%)
etalK0 0.1296 (= 0.36^2) Table 1 preclinical IIV on K0 = 36% (RSE 77%)
propSd 0.14 Table 1 preclinical e1 = 14% (RSE 23%)
addSd 3 Table 1 preclinical e2 = 3 mm^3 (RSE 17%)
AUC_PAZO for 10 mg/kg/day 220.2 ug*h/mL Ouerdani 2015 Methods, citing FDA Pharmacology Review NDA 022465
AUC_PAZO for 30 mg/kg/day 656.8 ug*h/mL Same
AUC_PAZO for 100 mg/kg/day 1140.8 ug*h/mL Same

Virtual cohort

The CAKI-2 xenograft study used 8 mice per group across four groups (vehicle, 10, 30, 100 mg/kg pazopanib). The virtual cohort below reproduces that allocation. Per-mouse baseline tumour volume (TUM_VOL) is drawn from a uniform distribution over the paper’s randomisation window (100-250 mm^3); drug exposure (AUC_PAZO) is held constant per dose group as reported in the paper.

set.seed(2015)

n_per_group <- 8L
dose_groups <- tibble::tribble(
  ~dose_mgkg, ~AUC_PAZO,
  0,             0,
  10,          220.2,
  30,          656.8,
  100,        1140.8
)

obs_times <- seq(0, 24, by = 0.5)  # twice-weekly measurements = ~every 3-4 days

make_cohort <- function(dose_mgkg, AUC_PAZO, id_offset) {
  ids <- id_offset + seq_len(n_per_group)
  per_subject <- tibble(
    id       = ids,
    TUM_VOL  = runif(n_per_group, min = 100, max = 250),
    AUC_PAZO = AUC_PAZO,
    dose_label = paste0(dose_mgkg, " mg/kg")
  )
  per_subject |>
    tidyr::crossing(time = obs_times) |>
    mutate(evid = 0L, amt = 0)
}

events_mouse <- bind_rows(
  make_cohort(0,    0,     id_offset =   0L),
  make_cohort(10,   220.2, id_offset =   n_per_group),
  make_cohort(30,   656.8, id_offset = 2*n_per_group),
  make_cohort(100,  1140.8, id_offset = 3*n_per_group)
) |>
  mutate(dose_label = factor(
    dose_label,
    levels = c("0 mg/kg", "10 mg/kg", "30 mg/kg", "100 mg/kg")
  ))

stopifnot(!anyDuplicated(unique(events_mouse[, c("id", "time", "evid")])))
nrow(events_mouse); n_distinct(events_mouse$id)
#> [1] 1568
#> [1] 32

Typical-value simulation (Figure 4 left panel)

A typical-value simulation (zeroRe(): no IIV, no residual error) reproduces the mechanistic shape Ouerdani 2015 highlights in Figure 4 (left panel): tumour volume rises monotonically in vehicle animals (pure logistic growth), declines monotonically at low pazopanib doses (cytotoxic effect dominates), and shows an initial decrease followed by limited regrowth at higher doses (the “dual-action” pattern as the antiangiogenic carrying-capacity reduction kicks in).

mod_mouse <- readModelDb("Ouerdani_2015_pazopanib_mouse")
mod_mouse_typ <- mod_mouse |> rxode2::zeroRe()
#> ℹ parameter labels from comments will be replaced by 'label()'

sim_typ <- rxode2::rxSolve(
  mod_mouse_typ,
  events = events_mouse,
  keep   = c("dose_label")
) |>
  as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalk_tumor', 'etalk_aa0', 'etalk_cyto0', 'etalk_res', 'etalK0'
#> Warning: multi-subject simulation without without 'omega'

# Population-typical median per dose group
typ_summary <- sim_typ |>
  group_by(dose_label, time) |>
  summarise(median_P = median(tumorSize),
            median_K = median(carryingCapacity),
            .groups = "drop")

ggplot(typ_summary, aes(time, median_P, colour = dose_label)) +
  geom_line(linewidth = 1) +
  labs(x = "Time (days)", y = "Tumour volume P (mm^3)",
       colour = "Dose group",
       title = "Typical mouse trajectory by dose group",
       caption = "Replicates Figure 4 (left panel) of Ouerdani 2015.") +
  theme_minimal()

Plotting the carrying capacity K alongside P shows the angiogenesis-inhibition mechanism: at zero exposure the carrying capacity grows unimpeded; under treatment, the antiangiogenic effect drives K below P and forces tumour shrinkage.

typ_long <- typ_summary |>
  pivot_longer(c(median_P, median_K), names_to = "state", values_to = "value") |>
  mutate(state = recode(state, median_P = "P (tumour)", median_K = "K (capacity)"))

ggplot(typ_long, aes(time, value, colour = state, linetype = state)) +
  geom_line(linewidth = 0.9) +
  facet_wrap(~ dose_label, ncol = 2, scales = "free_y") +
  labs(x = "Time (days)", y = "mm^3",
       colour = NULL, linetype = NULL,
       title = "Typical P and K trajectories by dose group",
       caption = "Replicates the mechanistic interpretation of Figure 4 of Ouerdani 2015.") +
  theme_minimal()

Stochastic simulation with IIV

Including IIV (etalk_tumor ~ 0.2809, etalk_aa0 ~ 0.0361, etalk_cyto0 ~ 0.0576, etalk_res ~ 0.1764, etalK0 ~ 0.1296) and the combined additive + proportional residual error gives a stochastic VPC analogous to Figure 2 of the paper (preclinical VPC stratified by dose).

sim_iiv <- rxode2::rxSolve(
  mod_mouse,
  events = events_mouse,
  keep   = c("dose_label")
) |>
  as.data.frame()
#> ℹ parameter labels from comments will be replaced by 'label()'

vpc_summary <- sim_iiv |>
  group_by(dose_label, time) |>
  summarise(
    Q05 = quantile(sim, 0.05, na.rm = TRUE),
    Q50 = quantile(sim, 0.50, na.rm = TRUE),
    Q95 = quantile(sim, 0.95, na.rm = TRUE),
    .groups = "drop"
  )

ggplot(vpc_summary, aes(time, Q50, colour = dose_label, fill = dose_label)) +
  geom_ribbon(aes(ymin = pmax(Q05, 0), ymax = Q95), alpha = 0.2, colour = NA) +
  geom_line(linewidth = 1) +
  facet_wrap(~ dose_label, ncol = 2, scales = "free_y") +
  labs(x = "Time (days)", y = "Tumour volume P (mm^3)",
       title = "Stochastic VPC: 5/50/95 percentile bands by dose group",
       caption = "Replicates Figure 2 of Ouerdani 2015 (preclinical VPC). Per-mouse residual error applied; 5th-percentile band clipped at 0.") +
  theme_minimal() +
  guides(colour = "none", fill = "none")

The 8 animals per group in the source dataset produce empirical 5th and 95th percentiles equal to the smallest and largest trajectory respectively; the paper itself notes that “VPC should be interpreted carefully, as the number of mice for each group is small.” The bands above use a 32-mouse virtual cohort (matching the source N) and so display the same wide spread.

Mechanistic sanity checks

This is a tumour-growth ODE model rather than a PK model, so PKNCA-based validation is not the right target (there is no drug-concentration / dose / AUC integration to perform). The mechanistic checks below catch the failure modes that would actually break this class of model.

1. Vehicle = pure logistic growth

At AUC_PAZO = 0, both drug-effect rates a and c collapse to zero via the model’s if (AUC_PAZO > 0) gate, and the ODE reduces to the classical Verhulst-Pearl logistic with an antiangiogenesis-driven capacity growth term. The simulated vehicle group should therefore show monotonic tumour expansion across the 24-day window.

veh_typ <- sim_typ |> filter(dose_label == "0 mg/kg")

# Monotonic non-decreasing tumour volume in vehicle (typical-value, no noise).
stopifnot(all(diff(veh_typ$median_P) > 0))

knitr::kable(
  veh_typ |> filter(time %in% c(0, 6, 12, 18, 24)),
  digits = 3,
  caption = "Typical vehicle-arm trajectory (no drug effect, pure logistic growth)."
)
Typical vehicle-arm trajectory (no drug effect, pure logistic growth).
id time k_tumor k_cap a0 c0 k_res K0 cyto_rate aa_rate ipredSim sim tumorSize carryingCapacity TUM_VOL AUC_PAZO dose_label
1 0 0.166 0.018 0.251 0.007 0.196 543 0 0 109.168 109.168 109.168 543.000 109.168 0 0 mg/kg
1 6 0.166 0.018 0.251 0.007 0.196 543 0 0 220.976 220.976 220.976 560.692 109.168 0 0 mg/kg
1 12 0.166 0.018 0.251 0.007 0.196 543 0 0 361.947 361.947 361.947 592.632 109.168 0 0 mg/kg
1 18 0.166 0.018 0.251 0.007 0.196 543 0 0 490.188 490.188 490.188 639.661 109.168 0 0 mg/kg
1 24 0.166 0.018 0.251 0.007 0.196 543 0 0 591.517 591.517 591.517 699.261 109.168 0 0 mg/kg
2 0 0.166 0.018 0.251 0.007 0.196 543 0 0 225.874 225.874 225.874 543.000 225.874 0 0 mg/kg
2 6 0.166 0.018 0.251 0.007 0.196 543 0 0 362.239 362.239 362.239 575.270 225.874 0 0 mg/kg
2 12 0.166 0.018 0.251 0.007 0.196 543 0 0 483.346 483.346 483.346 621.940 225.874 0 0 mg/kg
2 18 0.166 0.018 0.251 0.007 0.196 543 0 0 578.895 578.895 578.895 680.452 225.874 0 0 mg/kg
2 24 0.166 0.018 0.251 0.007 0.196 543 0 0 659.899 659.899 659.899 748.536 225.874 0 0 mg/kg
3 0 0.166 0.018 0.251 0.007 0.196 543 0 0 144.792 144.792 144.792 543.000 144.792 0 0 mg/kg
3 6 0.166 0.018 0.251 0.007 0.196 543 0 0 271.187 271.187 271.187 565.512 144.792 0 0 mg/kg
3 12 0.166 0.018 0.251 0.007 0.196 543 0 0 410.103 410.103 410.103 603.012 144.792 0 0 mg/kg
3 18 0.166 0.018 0.251 0.007 0.196 543 0 0 526.596 526.596 526.596 654.697 144.792 0 0 mg/kg
3 24 0.166 0.018 0.251 0.007 0.196 543 0 0 619.092 619.092 619.092 717.759 144.792 0 0 mg/kg
4 0 0.166 0.018 0.251 0.007 0.196 543 0 0 104.715 104.715 104.715 543.000 104.715 0 0 mg/kg
4 6 0.166 0.018 0.251 0.007 0.196 543 0 0 214.135 214.135 214.135 560.062 104.715 0 0 mg/kg
4 12 0.166 0.018 0.251 0.007 0.196 543 0 0 354.826 354.826 354.826 591.211 104.715 0 0 mg/kg
4 18 0.166 0.018 0.251 0.007 0.196 543 0 0 484.602 484.602 484.602 637.536 104.715 0 0 mg/kg
4 24 0.166 0.018 0.251 0.007 0.196 543 0 0 587.314 587.314 587.314 696.605 104.715 0 0 mg/kg
5 0 0.166 0.018 0.251 0.007 0.196 543 0 0 120.786 120.786 120.786 543.000 120.786 0 0 mg/kg
5 6 0.166 0.018 0.251 0.007 0.196 543 0 0 238.201 238.201 238.201 562.306 120.786 0 0 mg/kg
5 12 0.166 0.018 0.251 0.007 0.196 543 0 0 379.243 379.243 379.243 596.200 120.786 0 0 mg/kg
5 18 0.166 0.018 0.251 0.007 0.196 543 0 0 503.522 503.522 503.522 644.922 120.786 0 0 mg/kg
5 24 0.166 0.018 0.251 0.007 0.196 543 0 0 601.569 601.569 601.569 705.788 120.786 0 0 mg/kg
6 0 0.166 0.018 0.251 0.007 0.196 543 0 0 152.978 152.978 152.978 543.000 152.978 0 0 mg/kg
6 6 0.166 0.018 0.251 0.007 0.196 543 0 0 281.706 281.706 281.706 566.569 152.978 0 0 mg/kg
6 12 0.166 0.018 0.251 0.007 0.196 543 0 0 419.388 419.388 419.388 605.183 152.978 0 0 mg/kg
6 18 0.166 0.018 0.251 0.007 0.196 543 0 0 533.388 533.388 533.388 657.749 152.978 0 0 mg/kg
6 24 0.166 0.018 0.251 0.007 0.196 543 0 0 624.294 624.294 624.294 721.458 152.978 0 0 mg/kg
7 0 0.166 0.018 0.251 0.007 0.196 543 0 0 174.993 174.993 174.993 543.000 174.993 0 0 mg/kg
7 6 0.166 0.018 0.251 0.007 0.196 543 0 0 308.366 308.366 308.366 569.326 174.993 0 0 mg/kg
7 12 0.166 0.018 0.251 0.007 0.196 543 0 0 441.862 441.862 441.862 610.696 174.993 0 0 mg/kg
7 18 0.166 0.018 0.251 0.007 0.196 543 0 0 549.598 549.598 549.598 665.371 174.993 0 0 mg/kg
7 24 0.166 0.018 0.251 0.007 0.196 543 0 0 636.811 636.811 636.811 730.631 174.993 0 0 mg/kg
8 0 0.166 0.018 0.251 0.007 0.196 543 0 0 111.561 111.561 111.561 543.000 111.561 0 0 mg/kg
8 6 0.166 0.018 0.251 0.007 0.196 543 0 0 224.596 224.596 224.596 561.028 111.561 0 0 mg/kg
8 12 0.166 0.018 0.251 0.007 0.196 543 0 0 365.656 365.656 365.656 593.383 111.561 0 0 mg/kg
8 18 0.166 0.018 0.251 0.007 0.196 543 0 0 493.075 493.075 493.075 640.777 111.561 0 0 mg/kg
8 24 0.166 0.018 0.251 0.007 0.196 543 0 0 593.690 593.690 593.690 700.651 111.561 0 0 mg/kg

2. Drug-effect rates respect the AUC > 0 gate

For any individual with AUC_PAZO > 0 the per-individual rates cyto_rate and aa_rate are positive; for vehicle animals they are exactly zero. The gate is what keeps the cytotoxic rate a = a0 * AUC^0 = a0 from spuriously acting on vehicle animals in the mouse fit (where the exponent b_a is fixed to 0 and NONMEM-style 0^0 = 1 would otherwise leave a switched on).

gate_check <- sim_iiv |>
  group_by(dose_label) |>
  summarise(any_cyto = any(cyto_rate > 0),
            any_aa   = any(aa_rate > 0),
            .groups = "drop")

knitr::kable(
  gate_check,
  caption = "Drug-effect rates are zero in vehicle and positive in every drug arm."
)
Drug-effect rates are zero in vehicle and positive in every drug arm.
dose_label any_cyto any_aa
0 mg/kg FALSE FALSE
10 mg/kg TRUE TRUE
30 mg/kg TRUE TRUE
100 mg/kg TRUE TRUE 
stopifnot(identical(gate_check$any_cyto, c(FALSE, TRUE, TRUE, TRUE)))
stopifnot(identical(gate_check$any_aa,   c(FALSE, TRUE, TRUE, TRUE)))

3. Dimensional analysis of the ODE

Term Units Reduces to
k_tumor * tumorSize 1/day * mm^3 mm^3 / day
(1 - tumorSize / carryingCapacity) unitless unitless
cyto_rate * exp(-k_res*t) * tumorSize 1/day * unitless * mm^3 mm^3 / day
k_cap * tumorSize^n 1/day * (mm^3)^1 mm^3 / day (mouse n = 1)
aa_rate * carryingCapacity 1/day * mm^3 mm^3 / day

Both ODE right-hand sides come out as mm^3 / day, consistent with d/dt(state) where state is in mm^3 and t is in days. The mouse fit’s n = 1 makes k_cap’s units cleanly 1/day; in the paired clinical model n = 0.5 so k_cap’s units carry an implicit mm^(0.5) scaling that absorbs the source paper’s preclinical-to-clinical re-parameterisation.

Assumptions and deviations

  • checkModelConventions() deviations (intentional). Running nlmixr2lib::checkModelConventions("Ouerdani_2015_pazopanib_mouse") flags four warnings (no errors): (1) the tumorSize compartment is not on the canonical-compartment list; (2) the carryingCapacity compartment is not on the canonical-compartment list; (3) the single-output observation variable should be Cc; (4) units$concentration does not contain / (mass / volume) and units$dosing is n/a. All four are intrinsic to a tumour-volume-dynamics model with no drug-concentration / dosing ODE. The same deviations apply to every other tumour-size-dynamics model in the package (e.g., tgi_sat_logistic, oncology_xenograft_simeoni_2004, Zecchin_2016_tumorovarian). Tumour volume in mm^3 is not a drug concentration; Cc is reserved for plasma drug concentrations and would be misleading here. The deviations are intentional.
  • AUC_PAZO > 0 gate added to handle vehicle animals. Ouerdani 2015 fits the mouse model with b_a (AUC exponent on the cytotoxic effect) fixed to 0, which mathematically gives a = a0 at any positive AUC and, under the standard NONMEM convention 0^0 = 1, also a = a0 at AUC = 0. That keeps the cytotoxic rate switched on in vehicle animals, contradicting the paper’s vehicle-arm Figure 1 (pure logistic growth, no shrinkage). The nlmixr2lib model file gates both drug-effect rates on AUC_PAZO > 0 so that vehicle animals see a = 0 and c = 0, recovering the published vehicle-arm trajectory. The paper itself does not publish the source NONMEM control stream, so the exact form of the source author’s vehicle-handling cannot be confirmed; the gate is the simplest interpretation consistent with the reported Figure 1 vehicle trajectory and Equation 1 / Equations 2-3.
  • Per-mouse baseline tumour volume drawn uniformly from 100-250 mm^3. The paper sets P0 per-animal from the observed caliper measurement at randomisation but does not report individual values. The virtual cohort here draws TUM_VOL from Uniform(100, 250); switching to a triangular or empirical distribution would shift the per-group dispersion but not the typical-value trajectory.
  • AUC_PAZO per dose group is treated as time-fixed. The paper draws each dose group’s mean AUC from the FDA Pharmacology Review for pazopanib NDA 022465; per-animal AUC variability around the group mean is not retained in the model.
  • PKNCA validation is omitted. This is a tumour-volume-dynamics model with no drug-concentration / dosing ODE – there is nothing for PKNCA to integrate. The vignette substitutes the mechanistic-sanity checks above (vehicle = pure growth, gate respected, dimensional analysis), matching the validation pattern other tumour-volume models in the package use (e.g., Zecchin_2016_tumorovarian).