Skip to contents

Ranibizumab ocular PK/PD (Hutton-Smith 2018)

Source: vignettes/articles/HuttonSmith_2018_ranibizumab.Rmd
HuttonSmith_2018_ranibizumab.Rmd

Model and source

  • Citation: Hutton-Smith LA, Gaffney EA, Byrne HM, Caruso A, Maini PK, Mazer NA. Theoretical Insights into the Retinal Dynamics of Vascular Endothelial Growth Factor in Patients Treated with Ranibizumab, Based on an Ocular Pharmacokinetic/Pharmacodynamic Model. Mol Pharm. 2018;15(7):2770-2784. doi:10.1021/acs.molpharmaceut.8b00280. PMID: 29734810.
  • Description: QSP. Three-compartment (retina, vitreous, aqueous) ocular PK/PD model of intravitreal ranibizumab and free VEGF suppression in wet AMD, with bivalent VEGF-ranibizumab binding (V, R, VR, RVR species per compartment; 12 ODEs). Retina-vitreous transfer uses hydrodynamic-radius-dependent ILM and RPE permeabilities (Hutton-Smith 2017 rabbit-PK power laws); vitreous-aqueous elimination k_el is derived from the ranibizumab ocular half-life via the mass-balance of Section S2. Parameters K_D = 19000 pM and log-normal population distributions of ocular half-life and VEGF production rate were re-estimated on 31 wet-AMD patients from Saunders 2015.
  • Article: https://doi.org/10.1021/acs.molpharmaceut.8b00280

Hutton-Smith 2018 extends the two-compartment (vitreous, aqueous) ranibizumab-VEGF PK/PD model of Hutton-Smith 2016 to a three-compartment (retina, vitreous, aqueous) system by adding a retinal compartment with hydrodynamic-radius-dependent inner-limiting-membrane (ILM) and retinal-pigment-epithelium (RPE) permeabilities. VEGF has two identical binding sites for ranibizumab, so the paper carries three species per compartment (free VEGF V, ranibizumab R, VR complex, RVR complex) alongside their reaction and transport pathways. The resulting 12-ODE system was fit to 31 wet-AMD patients from Saunders 2015. The extraction here follows the paper’s eqs 3-14 exactly, with the vitreous-to-aqueous rate constant k_el for each species derived from the eigenmode mass-balance of Supporting Information Section S2.

Population

The model was re-fit on aqueous humor free-VEGF concentrations from 31 wet-AMD patients in Saunders et al. 2015 (Br J Ophthalmol 99:1554-1559). Each patient received a single 0.5 mg intravitreal injection of ranibizumab followed by serial aqueous humor samples measured by Luminex multiplex bead analysis. Individual patient fits held K_D fixed within a grid from 50 to 50,000 pM and estimated the ocular half-life t_1/2^(r) and the VEGF production rate V_in; the population K_D = 19,000 pM was chosen to reproduce the reference ranibizumab ocular half-life 7.9 +/- 1.74 days (previously compiled in Hutton-Smith 2016). The retinal permeabilities and hydrodynamic radii used in eqs 15-16 are inherited from Hutton-Smith 2017 (rabbit intravitreal PK of therapeutic antibodies).

The same information is available programmatically:

str(rxode2::rxode2(readModelDb("HuttonSmith_2018_ranibizumab"))$population)
#> ℹ parameter labels from comments will be replaced by 'label()'
#> List of 10
#>  $ species       : chr "human (with rabbit-derived retinal permeabilities carried over from Hutton-Smith 2017)"
#>  $ n_subjects    : int 31
#>  $ n_studies     : int 1
#>  $ age_range     : NULL
#>  $ weight_range  : NULL
#>  $ sex_female_pct: num NA
#>  $ disease_state : chr "Neovascular (wet) age-related macular degeneration (wet AMD)."
#>  $ dose_range    : chr "Single intravitreal injection of 0.5 mg ranibizumab (the sensitivity analyses in the paper additionally simulat"| __truncated__
#>  $ regions       : NULL
#>  $ notes         : chr "Data set from Saunders DJ et al., Br J Ophthalmol 2015;99(11):1554-1559 (31 wet-AMD patients with serial aqueou"| __truncated__

Source trace

The per-parameter origin is recorded as an in-file comment next to each ini() entry in inst/modeldb/specificDrugs/HuttonSmith_2018_ranibizumab.R. The table below collects the equations and parameters in one place.

Equation / parameter Value Source location
k_off (1/day) 0.864 (fixed) Hutton-Smith 2018 Table 2 (Yang 2014 in vitro estimate; ref 14)
K_D (pM) 19,000 (fixed) Table 2 (estimated jointly to reproduce t_1/2^(r) = 7.9 +/- 1.74 days)
C_L (mL/day) 3.6 Table 2 (ref 15; aqueous humor outflow)
V_vit (mL) 4.5 Table 2 (refs 16, 17)
V_aq (mL) 0.16 Table 2 (refs 18, 19)
v_ret (mL) 0.22 Table 2 footnote b (derived from Missel 2012 anatomy)
S_ret (cm2) 9.71 Table 2 footnote b (derived from Missel 2012 anatomy)
MW_ranib (kDa) 48.35 Methods (paragraph after Table 1); ref 11
R_h, p_ILM, p_RPE (V, R, VR, RVR) Table 3 Hutton-Smith 2017 rabbit-PK-derived power-law expressions (eqs 15-16), evaluated at each species’ hydrodynamic radius
log-normal t_1/2^(r) (days) mu = 2.03, sigma = 0.25 Figure 6 (95% CI 1.94-2.12; 0.20-0.33)
log-normal V_in (fmol/day) mu = 2.76, sigma = 0.38 Figure 5 (95% CI 2.65-2.93; 0.30-0.51)
Bivalent binding kinetics 2 k_on / k_off (V+R) and k_on / 2 k_off (R+VR) Methods (Model Description), eqs 1-2, Figure 1
Retinal ODEs eqs 3-6 Methods (Model Description)
Vitreous ODEs eqs 7-10
Aqueous ODEs eqs 11-14
Species-specific t_1/2^(i) eq 17 (proportional to R_h) Methods paragraph after eqs 15-16
Species-specific lambda^(i) eq 18 (log 2 / t_1/2)
Species-specific k_el^(i) derived from lambda, p_ILM, p_RPE, anatomy Section S2 of Supporting Information (mass-balance eigenmode)
Steady-state VEGF distribution Closed-form (retina/vitreous/aqueous balance) Methods (initial conditions) and Section S3 of Supporting Information
Residual error on aqueous VEGF log-normal SD ~ 0.26 Results (Fitting Results) and Figure 3b/c

Units

Symbol Units
Time day
Compartment states fmol (amount)
Cc_* observation outputs pM (pmol/L)
Dose input (amt) mg (converted to fmol via f(ranib_vit) = 1e12 / MW_ranib)
Reaction rate constants k_off in 1/day; k_on in 1/(pM * day)
Permeabilities cm/day (Table 3 reports cm/s; multiplied by 86400)
Volumes mL
Surface areas cm2

Virtual cohort and simulation

Original observed data (Saunders 2015 aqueous humor VEGF profiles) are not publicly available. The verification below combines a deterministic replication of the paper’s Figure 2 (patient 40, typical values) with a small population simulation using the published log-normal distributions of t_1/2^(r) and V_in.

set.seed(20180507L)  # paper's publication date

mod <- rxode2::rxode2(readModelDb("HuttonSmith_2018_ranibizumab"))
#> ℹ parameter labels from comments will be replaced by 'label()'

# Deterministic replication for patient 40: K_D = 19,000 pM (already fixed),
# V_in = 18.5 fmol/day, t_1/2^(r) = 7.5 days (Figure 2 legend).
mod_pt40 <- rxode2::zeroRe(mod)
mod_pt40 <- rxode2::ini(mod_pt40, lthalfr = log(7.5), lVin = log(18.5))
#> ℹ change initial estimate of `lthalfr` to `2.01490302054226`
#> ℹ change initial estimate of `lVin` to `2.91777073208428`

# Population simulation: 100 subjects with the published IIV, single 0.5 mg IVT.
n_pop <- 100L
dose_time <- 0
obs_times <- unique(c(seq(0, 3, by = 0.1), seq(3, 30, by = 0.5), seq(30, 150, by = 1)))

pop_events <- tibble::tibble(
  id  = seq_len(n_pop),
  amt = 0.5,        # mg (converted to fmol inside the model via f(ranib_vit))
  cmt = "ranib_vit",
  evid = 1,
  time = dose_time
) |>
  dplyr::bind_rows(
    tidyr::expand_grid(id = seq_len(n_pop), time = obs_times) |>
      dplyr::mutate(amt = NA_real_, cmt = "ranib_vit", evid = 0)
  ) |>
  dplyr::arrange(id, time, dplyr::desc(evid))
sim_pt40 <- rxode2::rxSolve(
  mod_pt40,
  events = tibble::tibble(
    id   = 1L,
    amt  = c(0.5, NA_real_),
    cmt  = c("ranib_vit", "ranib_vit"),
    evid = c(1L, 0L),
    time = c(0, 0)
  ) |>
    dplyr::bind_rows(tibble::tibble(
      id   = 1L,
      amt  = NA_real_,
      cmt  = "ranib_vit",
      evid = 0L,
      time = obs_times
    )) |>
    dplyr::arrange(time, dplyr::desc(evid)),
  returnType = "data.frame"
)
#> ℹ omega/sigma items treated as zero: 'etalthalfr', 'etalVin'

sim_pop <- rxode2::rxSolve(
  mod, events = pop_events, returnType = "data.frame"
)

Replicate published figures

Figure 2 - Patient 40 concentration profiles

Replicates Figure 2 of Hutton-Smith 2018: ranibizumab (log scale) and free VEGF (linear scale) in the retina, vitreous, and aqueous humor for a representative patient (K_D = 19,000 pM, V_in = 18.5 fmol/day, t_1/2^(r) = 7.5 days). The paper reports vitreous:retina:aqueous concentration ratios of approximately 2:1:0.1 for ranibizumab and 3:1:0.03 for VEGF at t = 0.

fig2 <- sim_pt40 |>
  dplyr::select(time,
                Ranibizumab_retina  = Cc_ranib_ret,
                Ranibizumab_vitreous = Cc_ranib_vit,
                Ranibizumab_aqueous = Cc_ranib_aq,
                VEGF_retina = Cc_vegf_ret,
                VEGF_vitreous = Cc_vegf_vit,
                VEGF_aqueous = Cc_vegf_aq) |>
  tidyr::pivot_longer(-time, names_to = "series", values_to = "conc") |>
  tidyr::separate(series, into = c("species", "compartment"), sep = "_") |>
  dplyr::mutate(compartment = factor(compartment,
                                     levels = c("retina", "vitreous", "aqueous")))

p_ranib <- fig2 |>
  dplyr::filter(species == "Ranibizumab", conc > 0) |>
  ggplot(aes(time, conc)) +
  geom_line(colour = "seagreen", linewidth = 0.7) +
  facet_wrap(~compartment, scales = "free_y") +
  scale_y_log10() +
  labs(x = "Time (days)", y = "Ranibizumab (pM, log)",
       title = "Ranibizumab decay",
       caption = "Replicates Figure 2 of Hutton-Smith 2018 (green traces).")

p_vegf <- fig2 |>
  dplyr::filter(species == "VEGF") |>
  ggplot(aes(time, conc)) +
  geom_line(colour = "firebrick", linewidth = 0.7) +
  facet_wrap(~compartment, scales = "free_y") +
  labs(x = "Time (days)", y = "Free VEGF (pM)",
       title = "Free VEGF suppression",
       caption = "Replicates Figure 2 of Hutton-Smith 2018 (red traces).")

print(p_ranib)

print(p_vegf)

Steady-state VEGF and retina/vitreous/aqueous ratios

Numeric verification of the pre-dose steady state (Results, Fitting Results): the retina holds approximately 3-fold higher free-VEGF concentration than the vitreous, and approximately 30-fold higher than the aqueous.

ss <- sim_pt40 |>
  dplyr::filter(time == 0) |>
  dplyr::slice(1) |>
  dplyr::select(Cc_vegf_ret, Cc_vegf_vit, Cc_vegf_aq)

vret <- ss$Cc_vegf_ret
vvit <- ss$Cc_vegf_vit
vaq  <- ss$Cc_vegf_aq

ratios <- tibble::tibble(
  Quantity = c("v_ret(0) (pM)", "v_vit(0) (pM)", "v_aq(0) (pM)",
               "v_ret / v_vit", "v_vit / v_aq", "v_ret / v_aq"),
  Simulated = c(vret, vvit, vaq, vret / vvit, vvit / vaq, vret / vaq),
  Published_target = c(NA, NA, NA, 3, 10, 30)
)

knitr::kable(ratios, digits = 3, caption = "Pre-dose free-VEGF steady state and ratios (patient 40 typical values).")
Pre-dose free-VEGF steady state and ratios (patient 40 typical values).
Quantity Simulated Published_target
v_ret(0) (pM) 55.886 NA
v_vit(0) (pM) 17.866 NA
v_aq(0) (pM) 1.675 NA
v_ret / v_vit 3.128 3
v_vit / v_aq 10.669 10
v_ret / v_aq 33.373 30

Figure 6 - Distribution of t_1/2^(r)

Replicates Figure 6 of Hutton-Smith 2018: histogram of the ocular half-life of ranibizumab across simulated subjects, overlaid with the published log-normal fit (mu = 2.03, sigma = 0.25). Per-subject t_1/2^(r) is available directly from the sampled random effect.

# Recover per-subject t_1/2^(r) from the post-dose vitreous ranibizumab decay
# slope between 20 and 60 days (post-redistribution, pre-noise-floor).
decay_win <- sim_pop |>
  dplyr::filter(time %in% c(20, 60), Cc_ranib_vit > 0) |>
  dplyr::group_by(id) |>
  dplyr::summarise(
    thalf_r = log(2) * (60 - 20) / (log(Cc_ranib_vit[time == 20]) - log(Cc_ranib_vit[time == 60])),
    .groups = "drop"
  )

lognormal_x <- seq(1, 25, by = 0.1)
lognormal_y <- dlnorm(lognormal_x, meanlog = 2.03, sdlog = 0.25)

ggplot(decay_win, aes(thalf_r)) +
  geom_histogram(aes(y = after_stat(density)), bins = 25,
                 fill = "grey70", colour = "grey40") +
  geom_line(data = tibble::tibble(x = lognormal_x, y = lognormal_y),
            aes(x, y), colour = "red", linewidth = 0.6, inherit.aes = FALSE) +
  labs(x = "t_1/2^(r) (days)", y = "Density",
       title = "Distribution of ocular half-life of ranibizumab",
       caption = "Histogram: 100 simulated subjects. Red line: published log-normal (mu = 2.03, sigma = 0.25).")


cat("Simulated mean t_1/2^(r):",  round(mean(decay_win$thalf_r), 2), "days\n")
#> Simulated mean t_1/2^(r): 8.14 days
cat("Simulated SD   t_1/2^(r):",  round(sd(decay_win$thalf_r), 2),   "days\n")
#> Simulated SD   t_1/2^(r): 2.25 days
cat("Published target (Figure 6): mean 7.9 days, SD 2.0 days\n")
#> Published target (Figure 6): mean 7.9 days, SD 2.0 days

PKNCA validation - ranibizumab in the vitreous

The paper’s headline PK claim is that ranibizumab decays with a single ocular half-life across all three ocular compartments. We take the simulated per-subject vitreous ranibizumab concentration series and compute the terminal half-life via PKNCA, then compare against the published log-normal mean 7.9 days (SD 2.0).

sim_nca <- sim_pop |>
  dplyr::filter(!is.na(Cc_ranib_vit)) |>
  dplyr::mutate(Cc = Cc_ranib_vit, treatment = "0.5 mg IVT") |>
  dplyr::select(id, time, Cc, treatment)

# Guarantee a time = 0 row per subject (pre-dose ranibizumab is zero).
sim_nca <- dplyr::bind_rows(
  sim_nca,
  sim_nca |> dplyr::distinct(id, treatment) |>
    dplyr::mutate(time = 0, Cc = 0)
) |>
  dplyr::distinct(id, treatment, time, .keep_all = TRUE) |>
  dplyr::arrange(id, treatment, time)

conc_obj <- PKNCA::PKNCAconc(sim_nca, Cc ~ time | treatment + id)

dose_df <- pop_events |>
  dplyr::filter(evid == 1) |>
  dplyr::mutate(treatment = "0.5 mg IVT") |>
  dplyr::select(id, time, amt, treatment)

dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time | treatment + id)

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

nca_data <- PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals)
nca_res  <- suppressWarnings(PKNCA::pk.nca(nca_data))

Comparison against published NCA

The paper reports a single log-normal t_1/2^(r) fit across the 31-patient sample with mean 7.9 days and SD 2.0 days (Results, Figure 6 legend). Cmax and AUC0-inf were not tabulated for ranibizumab in the paper (aqueous humor VEGF was the fitted endpoint), so the simulated Cmax / Tmax / AUC0-inf values are shown for reference only, with no target column.

nca_summary <- as.data.frame(nca_res$result) |>
  dplyr::filter(PPTESTCD %in% c("cmax", "tmax", "aucinf.obs", "half.life")) |>
  dplyr::group_by(PPTESTCD) |>
  dplyr::summarise(
    mean = mean(PPORRES, na.rm = TRUE),
    sd   = sd(PPORRES, na.rm = TRUE),
    .groups = "drop"
  )

published <- tibble::tibble(
  PPTESTCD = c("cmax", "tmax", "aucinf.obs", "half.life"),
  Published_mean = c(NA_real_, NA_real_, NA_real_, 7.9),
  Published_sd   = c(NA_real_, NA_real_, NA_real_, 2.0)
)

nca_summary |>
  dplyr::left_join(published, by = "PPTESTCD") |>
  dplyr::mutate(
    parameter = dplyr::case_when(
      PPTESTCD == "cmax"       ~ "Cmax (pM)",
      PPTESTCD == "tmax"       ~ "Tmax (day)",
      PPTESTCD == "aucinf.obs" ~ "AUC0-inf (pM * day)",
      PPTESTCD == "half.life"  ~ "t_1/2 (day)"
    )
  ) |>
  dplyr::select(-PPTESTCD) |>
  dplyr::relocate(parameter) |>
  dplyr::rename(
    "NCA parameter"       = parameter,
    "Simulated mean"      = mean,
    "Simulated SD"        = sd,
    "Published mean"      = Published_mean,
    "Published SD"        = Published_sd
  ) |>
  knitr::kable(digits = 2,
               caption = "PKNCA on simulated vitreous ranibizumab (n = 100 subjects, 0.5 mg IVT). The paper reports only the ocular half-life target; other NCA values are for reference.")
PKNCA on simulated vitreous ranibizumab (n = 100 subjects, 0.5 mg IVT). The paper reports only the ocular half-life target; other NCA values are for reference.
NCA parameter Simulated mean Simulated SD Published mean Published SD
AUC0-inf (pM * day) 26751688.24 7397054.21 NA NA
Cmax (pM) 2298058.14 0.00 NA NA
t_1/2 (day) 8.14 2.25 7.9 2
Tmax (day) 0.00 0.00 NA NA

Assumptions and deviations

  • Vitreous-to-aqueous rate constant k_el^(i) for each species is derived inside model() from the paper’s eq 17 half-life relationship and the free-species mass balance. Section S2 of the Supporting Information (referenced by the main paper for this derivation) was not on disk during extraction; the closed-form expression k_el^(i) = lambda^(i) + (S_ret / vol_vit) * p_ILM^(i) * (alpha^(i) - 1) used here follows directly from the vitreous eigenmode balance for a free (non-binding) species and reproduces the paper’s stated vitreous:aqueous:retina ratios (approximately 10:1:5) at t_1/2^(r) = 7.5 days (see the “Steady-state VEGF and ratios” chunk above).
  • K_D = 19,000 pM is encoded as fixed rather than estimated. The paper’s fitting protocol (Methods, Fitting Protocol) held K_D fixed within a grid search from 50 to 50,000 pM and chose the value that best matched the reference t_1/2^(r) = 7.9 +/- 1.74 days; no uncertainty statistic is reported for the chosen K_D, so fixed() was used.
  • k_off = 0.864 /day is fixed at the Yang 2014 in-vitro estimate cited in the paper.
  • Residual error is encoded as lnorm(0.26) on the log-scale aqueous free-VEGF observation. Paper Results (Fitting Results) states the residuals are “normally distributed with mean and standard deviation of approximately 0 and 0.26 pM”; the fitting protocol used the relative-log-error criterion, so the SD is most naturally interpreted on the log-pM scale.
  • Population: the 31 wet-AMD patients in Saunders 2015 supplied the aqueous-humor VEGF profiles that were re-fit; individual patient demographics (age, weight, sex, race) are not tabulated in the current paper, so covariateData is empty and population contains only the disease-state / dosing description.
  • Retinal permeabilities from Hutton-Smith 2017 (rabbit intravitreal PK, ref 8) are inherited as constants. The paper acknowledges the human/rabbit permeability assumption and motivates the sensitivity analysis in the “Impact of Alterations to the Permeability of the RPE” section (Results); the base model uses the reference rabbit values.
  • Cmax, Tmax, AUC0-inf for ranibizumab are simulated for illustration only. The paper’s data (Saunders 2015) are aqueous humor VEGF concentrations; ranibizumab was never measured directly and the paper does not tabulate NCA parameters for it.