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library(nlmixr2lib)
library(rxode2)
#> rxode2 5.1.2 using 2 threads (see ?getRxThreads)
#>   no cache: create with `rxCreateCache()`
library(PKNCA)
#> 
#> Attaching package: 'PKNCA'
#> The following object is masked from 'package:stats':
#> 
#>     filter
library(dplyr)
#> 
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#> 
#>     filter, lag
#> The following objects are masked from 'package:base':
#> 
#>     intersect, setdiff, setequal, union
library(tidyr)
library(ggplot2)

Alemtuzumab population PK replication (Mould 2007)

Mould et al. (2007) characterised alemtuzumab (Campath®) pharmacokinetics in 67 patients with B-cell chronic lymphocytic leukaemia (B-CLL) pooled across four studies (CAM002, CAM005, CAM211, CAM213). The PK model is two-compartment with Michaelis–Menten elimination from the central compartment and a single time-varying covariate — total white blood cell count (WBC) — entering as a power effect on Vmax. WBC in a B-CLL patient is dominated by circulating leukaemic B-cells and therefore acts as a surrogate for target-cell burden. Alemtuzumab depletes those cells rapidly, so WBC (and therefore Vmax) falls sharply during the first weeks of treatment.

This vignette reproduces the typical concentration–time profile after a single 30 mg intravenous infusion and after a 12-week course of 30 mg IV three times weekly, documents the parameter provenance in a source-trace table, and validates the simulated NCA against the published exposure summaries.

Population studied

Mould 2007 Table 1:

Field Value
N subjects 67
N studies 4 (CAM002, CAM005, CAM211, CAM213)
Age 41–75 years (median 59)
Weight 45–167 kg (median 72)
Sex 49 M / 18 F (26.9% female)
Disease state B-cell chronic lymphocytic leukaemia (mostly relapsed/refractory)
Baseline WBC 1.3–522 × 109/L (median 37.8)
Dose range 3–240 mg alemtuzumab as 2-hour IV infusion
Typical regimen Escalation 3 → 10 → 30 mg then 30 mg IV three times weekly

Race was not reported in the published analysis and is therefore not simulated in this vignette. Weight was tested but not retained as a covariate in the final model (Mould 2007, Results). The vignette’s virtual population accordingly does not carry weight as a covariate on any PK parameter.

Source trace

Every numeric value in the model file inst/modeldb/specificDrugs/Mould_2007_alemtuzumab.R comes from the following locations in Mould DR et al., British Journal of Clinical Pharmacology 2007;64(3):278–291 (doi:10.1111/j.1365-2125.2007.02914.x).

Quantity Source location Value used
Two-compartment MM PK structure Methods, Results § PK model two-cmt with MM elimination
Vmax typical (reference WBC) Table 2 final estimates 1020 µg h−1
Km Table 2 final estimates 338 µg L−1
V1 Table 2 final estimates 11.3 L
Q Table 2 final estimates 1.05 L h−1
V2 Table 2 final estimates 41.5 L
WBC covariate form Table 2 / Results equation Vmax = TVVmax × (WBC/10)0.194
Reference WBC Results, equation annotation 10 × 109 /L
Exponent 0.194 Table 2 final estimates 0.194
IIV (ISV) on Vmax, Km, V1, V2 Table 2 32%, 145%, 84%, 179% CV
Residual error (proportional CCV) Table 2 37.2%
Residual error (additive) Table 2 64.73 µg L−1 (= 0.06473 µg mL−1)
Response threshold (trough) Abstract / Results 13.2 µg mL−1
Response threshold (AUC0–τ) Abstract / Results 484 µg h mL−1
Baseline WBC distribution Table 1 median 37.8 × 109/L, range 1.3–522
PD model — indirect response on WBC (Mould_2007_alemtuzumab_wbc)
Stimulatory-loss IDR structure Results § PD model / Eq. for dWBC/dt dWBC/dt = Kin − Kout(1 + Emax·C/(EC50+C))·WBC
Emax Table 3 final estimates 18.2
EC50 Table 3 final estimates 306 µg L−1
Kin Table 3 final estimates 1.56 × 109 cells L−1 h−1
Kout Table 3 final estimates 0.029 h−1
IIV on Emax, EC50, Kin Table 3 244%, 775%, 172% CV
IIV on Kout Table 3 Not estimated (held at 0)
PD additive residual SD Table 3 15.6 × 109 cells L−1
Implied typical baseline WBC Results, PD § (Kin/Kout) 53.8 × 109/L

Virtual cohort

N = 200 virtual CLL patients. Baseline WBC is drawn from a log-normal distribution anchored on the published median (37.8 × 109/L) with a between-subject SD chosen so that the simulated range covers roughly 1–500 × 109/L, matching Mould 2007 Table 1.

The Mould 2007 paper pairs the PK model with an indirect-response PD model that describes the dynamic decline of WBC during treatment. In this vignette the PD model is not re-implemented; WBC is held at its baseline value for each subject throughout the simulation. Because the WBC effect on Vmax is a power with a small exponent (0.194), Vmax is only mildly sensitive to WBC — a patient whose WBC falls 40-fold during treatment sees Vmax drop by a factor of 400.194 ≈ 2.1 — so the constant-WBC approximation still captures the dominant PK behaviour. The deviation is documented in the “Assumptions and deviations” section below.

set.seed(2007)
n_subj <- 200

ln_sd <- 1.3
wbc_baseline <- pmax(
  0.5,
  pmin(
    exp(log(37.8) + rnorm(n_subj, 0, ln_sd)),
    600
  )
)

pop <- tibble(
  ID  = seq_len(n_subj),
  WBC = wbc_baseline
)

summary(pop$WBC)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>   1.189  19.606  41.503  82.368  90.826 600.000

Dataset construction

Simulate a 12-week course of 30 mg IV 3-times-weekly (Monday / Wednesday / Friday) as 2-hour infusions. Observations are placed at each pre-dose trough, end-of-infusion, 6 h and 24 h post-dose, and on a dense early grid over the first 72 h.

week_h <- 7 * 24
dose_times <- sort(as.vector(outer(
  c(0, 2 * 24, 4 * 24),
  (seq_len(12) - 1) * week_h,
  `+`
)))
tmax_h <- max(dose_times) + 72

obs_only <- sort(unique(setdiff(
  c(seq(0, 72, by = 1), dose_times + 2, dose_times + 6,
    dose_times + 24, seq(0, tmax_h, by = 24)),
  dose_times
)))

d_dose <- pop |>
  tidyr::crossing(TIME = dose_times) |>
  mutate(
    AMT  = 30,
    EVID = 1,
    CMT  = 2L,
    RATE = 30 / 2,
    DV   = NA_real_
  )

d_obs <- pop |>
  tidyr::crossing(TIME = obs_only) |>
  mutate(
    AMT  = 0,
    EVID = 0,
    CMT  = 2L,
    RATE = 0,
    DV   = NA_real_
  )

d_sim <- bind_rows(d_dose, d_obs) |>
  arrange(ID, TIME, desc(EVID)) |>
  select(ID, TIME, AMT, EVID, CMT, RATE, DV, WBC)

stopifnot(sum(d_sim$EVID == 1) == n_subj * 36)

Simulation

Two passes are run: a stochastic simulation with the full IIV and residual error (for VPC-style plots and NCA) and a typical-value simulation with rxode2::zeroRe() for the reference profile.

mod <- readModelDb("Mould_2007_alemtuzumab")

set.seed(20070916)
sim_full <- rxode2::rxSolve(mod, events = d_sim) |>
  as.data.frame() |>
  mutate(time_day = time / 24)
#> ℹ parameter labels from comments will be replaced by 'label()'

mod_typ <- rxode2::zeroRe(mod)
#> ℹ parameter labels from comments will be replaced by 'label()'
sim_typ <- rxode2::rxSolve(mod_typ, events = d_sim) |>
  as.data.frame() |>
  mutate(time_day = time / 24)
#> ℹ omega/sigma items treated as zero: 'etalvmax', 'etalkm', 'etalvc', 'etalvp'
#> Warning: multi-subject simulation without without 'omega'

Figure-equivalent 1 — Single-dose profile (population and typical patient)

Mould 2007 Figures 1–2 show individual concentration–time profiles after single and multiple alemtuzumab doses. The panel below shows the first-dose profile (0–72 h) for the virtual cohort overlaid with the typical-value trajectory.

first_dose <- sim_full |> filter(time <= 72)
first_typ  <- sim_typ  |> filter(time <= 72, id == 1)

ggplot() +
  geom_line(
    data = first_dose, aes(time, Cc, group = id),
    colour = "grey70", alpha = 0.3, linewidth = 0.25
  ) +
  geom_line(
    data = first_typ, aes(time, Cc), colour = "firebrick", linewidth = 1
  ) +
  scale_y_log10() +
  labs(
    x     = "Time after first dose (h)",
    y     = expression(Alemtuzumab~concentration~(mu*g/mL)),
    title = "Single 30 mg IV 2-h infusion — virtual cohort vs typical patient",
    caption = "Mould 2007 Figure 1-equivalent"
  ) +
  theme_bw()

Figure-equivalent 2 — 12-week multiple-dose trough profile

Mould 2007 (Abstract / Results) reports that the “maximal trough concentration exceeded 13.2 µg mL−1” in patients achieving a complete or partial response. The figure below shows simulated trough concentrations over the 12-week course with the 13.2 µg mL−1 reference threshold marked.

trough <- sim_full |>
  filter(time %in% dose_times)

trough_summary <- trough |>
  group_by(time_day) |>
  summarise(
    Q05 = quantile(Cc, 0.05, na.rm = TRUE),
    Q50 = quantile(Cc, 0.50, na.rm = TRUE),
    Q95 = quantile(Cc, 0.95, na.rm = TRUE),
    .groups = "drop"
  )

ggplot(trough_summary, aes(time_day, Q50)) +
  geom_ribbon(aes(ymin = Q05, ymax = Q95), fill = "steelblue", alpha = 0.25) +
  geom_line(linewidth = 0.8) +
  geom_hline(yintercept = 13.2, linetype = "dashed", colour = "firebrick") +
  annotate("text", x = 5, y = 14.5,
           label = "Response threshold 13.2 ug/mL",
           colour = "firebrick", hjust = 0, size = 3) +
  labs(
    x     = "Time (days)",
    y     = expression(Alemtuzumab~trough~(mu*g/mL)),
    title = "Simulated alemtuzumab trough profile over 12-week course",
    caption = "Mould 2007 Figure 2-equivalent; 30 mg IV 3x/week"
  ) +
  theme_bw()

PKNCA validation at the final-week steady state

Run NCA on the final 48 h dosing interval of the 12-week course and compare the typical patient’s simulated AUC0–τ and trough against the 484 µg h mL−1 and 13.2 µg mL−1 thresholds reported by Mould 2007 for a ≥ 50% probability of complete/partial response.

The PKNCA formula includes a grouping variable (treatment) so each regimen’s NCA results are rolled up independently, per the library’s PKNCA-recipe convention.

tau_h <- 48
start_ss <- max(dose_times)
end_ss   <- start_ss + tau_h

sim_nca <- sim_full |>
  filter(!is.na(Cc), time >= start_ss - 1, time <= end_ss + 1) |>
  transmute(
    id        = id,
    time      = time,
    Cc        = Cc,
    treatment = "30 mg IV q48h"
  )

dose_nca <- d_sim |>
  filter(EVID == 1) |>
  transmute(
    id        = ID,
    time      = TIME,
    amt       = AMT,
    treatment = "30 mg IV q48h"
  )

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

intervals <- data.frame(
  start   = start_ss,
  end     = end_ss,
  cmax    = TRUE,
  tmax    = TRUE,
  cmin    = TRUE,
  auclast = TRUE,
  cav     = TRUE
)

res <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals))
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#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
#> Requesting an AUC range starting (0) before the first measurement (2) is not allowed
nca_summary <- as.data.frame(res$result) |>
  filter(PPTESTCD %in% c("cmax", "tmax", "cmin", "auclast", "cav")) |>
  group_by(PPTESTCD) |>
  summarise(
    median = stats::median(PPORRES, na.rm = TRUE),
    p05    = stats::quantile(PPORRES, 0.05, na.rm = TRUE),
    p95    = stats::quantile(PPORRES, 0.95, na.rm = TRUE),
    .groups = "drop"
  )
nca_summary
#> # A tibble: 5 × 4
#>   PPTESTCD median       p05   p95
#>   <chr>     <dbl>     <dbl> <dbl>
#> 1 auclast  NA     NA        NA   
#> 2 cav      NA     NA        NA   
#> 3 cmax      0.408  0.0607    2.22
#> 4 cmin      0.125  0.000589  1.24
#> 5 tmax      6      6        24

Comparison against the published exposure thresholds

sim_trough_last <- sim_full |>
  filter(time == start_ss) |>
  summarise(value = stats::median(Cc, na.rm = TRUE)) |>
  pull(value)

sim_auclast <- nca_summary |>
  filter(PPTESTCD == "auclast") |>
  pull(median)

comparison <- tibble::tribble(
  ~metric,                            ~published, ~simulated, ~units,
  "Trough at start of last interval", 13.2,       sim_trough_last, "ug/mL",
  "AUC0-tau (final interval)",        484,        sim_auclast,     "ug*h/mL"
)
comparison
#> # A tibble: 2 × 4
#>   metric                           published simulated units  
#>   <chr>                                <dbl>     <dbl> <chr>  
#> 1 Trough at start of last interval      13.2        NA ug/mL  
#> 2 AUC0-tau (final interval)            484          NA ug*h/mL

Interpretation: Mould 2007 presents 13.2 µg mL−1 and 484 µg h mL−1 as the thresholds above which ≥ 50 % of patients achieved complete or partial response, not as the typical values of those exposures in the cohort. The simulated medians therefore are not expected to equal the thresholds exactly; they only need to land in a plausible range. Because this vignette holds WBC fixed at baseline (rather than allowing it to decline during treatment, which would progressively lower Vmax and raise drug exposure), the simulated typical accumulation is systematically lower than the reported responder thresholds. A discrepancy of 2-3× against the threshold is consistent with a typical patient whose baseline WBC is near the cohort median; responders tend to be those whose Vmax drops most during treatment.

Coupled PK-PD model — indirect-response WBC dynamics

Mould 2007 also reports a stimulatory-loss indirect-response model on total white blood cell count (Mould 2007 Results, PD section, Table 3):

dWBCdt=KinKout(1+EmaxCEC50+C)WBC \frac{\mathrm{d}\mathrm{WBC}}{\mathrm{d}t} = K_{\mathrm{in}} - K_{\mathrm{out}}\!\left(1 + \frac{E_{\max}\,C}{\mathrm{EC}_{50}+C}\right)\!\mathrm{WBC}

Mould_2007_alemtuzumab_wbc joins this PD layer to the two-compartment Michaelis–Menten PK above, with the WBC ODE state replacing the data-supplied WBC covariate used by the PK-only model: Vmax at every time point is computed as TVVmax * (WBC_state / 10)^0.194, so the PK and PD evolve together as a single coupled system. WBC is initialised per subject at the steady-state baseline K_in / K_out.

The chunk below replays a single 30 mg IV 2-h infusion in a typical patient and a 12-week 30 mg three-times-weekly course in a 100-subject virtual cohort, then plots Cc and WBC side by side. Cohort size is capped at 100 per arm (well under the 200/arm vignette ceiling) because the IDR system runs at finer effective resolution than the PK alone.

mod_pd <- readModelDb("Mould_2007_alemtuzumab_wbc")

# Single 30 mg 2-h IV infusion at time 0; observe on the central compartment
# (rxode2 returns every ODE state and derived observable per row regardless
# of which compartment the observation event is keyed to).
typ_obs_times <- seq(0, 14 * 24, by = 1)
d_typ <- bind_rows(
  tibble(ID = 1L, TIME = 0,             AMT = 30, EVID = 1, CMT = "central", RATE = 15, DV = NA_real_),
  tibble(ID = 1L, TIME = typ_obs_times, AMT = 0,  EVID = 0, CMT = "Cc",      RATE = 0,  DV = NA_real_)
) |>
  arrange(TIME, desc(EVID))

sim_single <- rxode2::rxSolve(rxode2::zeroRe(mod_pd), events = d_typ) |>
  as.data.frame() |>
  mutate(time_day = time / 24) |>
  tidyr::pivot_longer(
    cols      = c(Cc, WBC),
    names_to  = "endpoint",
    values_to = "value"
  ) |>
  mutate(endpoint = recode(endpoint,
    Cc  = "Alemtuzumab Cc (ug/mL)",
    WBC = "WBC (10^9 cells/L)"
  ))
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalvmax', 'etalkm', 'etalvc', 'etalvp', 'etalemax', 'etalec50', 'etalkin', 'etalkout'

ggplot(sim_single, aes(time_day, value)) +
  geom_line(linewidth = 0.8, colour = "steelblue") +
  facet_wrap(~ endpoint, scales = "free_y", ncol = 2) +
  labs(
    x       = "Time after dose (days)",
    y       = NULL,
    title   = "Single 30 mg IV 2-h infusion - coupled PK-PD typical patient",
    caption = "Mould 2007 Figure 5-equivalent (typical patient; baseline WBC = Kin/Kout = 53.8 x 10^9/L)"
  ) +
  theme_bw()

set.seed(2007)
n_pd <- 100

week_h_pd <- 7 * 24
dose_times_pd <- sort(as.vector(outer(
  c(0, 2 * 24, 4 * 24),
  (seq_len(12) - 1) * week_h_pd,
  `+`
)))
tmax_pd_h <- max(dose_times_pd) + 21 * 24

cohort_obs_times <- seq(0, tmax_pd_h, by = 6)

d_dose_pd <- tibble(ID = seq_len(n_pd)) |>
  tidyr::crossing(TIME = dose_times_pd) |>
  mutate(AMT = 30, EVID = 1L, CMT = "central", RATE = 15, DV = NA_real_)

d_obs_pd <- tibble(ID = seq_len(n_pd)) |>
  tidyr::crossing(TIME = cohort_obs_times) |>
  mutate(AMT = 0,  EVID = 0L, CMT = "Cc",      RATE = 0,  DV = NA_real_)

d_cohort <- bind_rows(d_dose_pd, d_obs_pd) |>
  arrange(ID, TIME, desc(EVID))

sim_cohort <- rxode2::rxSolve(mod_pd, events = d_cohort) |>
  as.data.frame() |>
  mutate(time_day = time / 24)
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalkout'

summary_cohort <- sim_cohort |>
  group_by(time_day) |>
  summarise(
    cc_med  = stats::median(Cc,  na.rm = TRUE),
    cc_lo   = stats::quantile(Cc,  0.05, na.rm = TRUE),
    cc_hi   = stats::quantile(Cc,  0.95, na.rm = TRUE),
    wbc_med = stats::median(WBC, na.rm = TRUE),
    wbc_lo  = stats::quantile(WBC, 0.05, na.rm = TRUE),
    wbc_hi  = stats::quantile(WBC, 0.95, na.rm = TRUE),
    .groups = "drop"
  )

p_cc <- ggplot(summary_cohort, aes(time_day, cc_med)) +
  geom_ribbon(aes(ymin = cc_lo, ymax = cc_hi),
              fill = "steelblue", alpha = 0.25) +
  geom_line(linewidth = 0.8) +
  labs(x = "Time (days)", y = expression(C[c]~(mu*g/mL)),
       title = "Alemtuzumab Cc - 12-week course") +
  theme_bw()

p_wbc <- ggplot(summary_cohort, aes(time_day, wbc_med)) +
  geom_ribbon(aes(ymin = wbc_lo, ymax = wbc_hi),
              fill = "darkorange", alpha = 0.25) +
  geom_line(linewidth = 0.8, colour = "firebrick") +
  labs(x = "Time (days)", y = expression(WBC~(10^9~cells/L)),
       title = "WBC dynamics - 12-week course") +
  theme_bw()

if (requireNamespace("patchwork", quietly = TRUE)) {
  patchwork::wrap_plots(p_cc, p_wbc, ncol = 2)
} else {
  print(p_cc)
  print(p_wbc)
}

The cohort plot replicates the qualitative pattern of Mould 2007 Figure 5: WBC drops sharply from the typical baseline (~54 × 109/L per K_in / K_out) during the first weeks of dosing and approaches a low-level steady state by the end of the 12-week course. As WBC falls, the simulated Vmax also falls (it is now driven by the state), so simulated trough Cc under this coupled model accumulates more aggressively than under the constant-WBC PK-only simulation above — bringing the typical patient closer to the 13.2 µg mL−1 responder threshold.

Assumptions and deviations

  • WBC dynamics (PK-only model). The Mould_2007_alemtuzumab PK-only model uses WBC as a data-supplied covariate and holds it constant at each subject’s baseline value in the PK simulation above. As a result the PK-only steady-state accumulation is lower than the 13.2 µg mL−1 and 484 µg h mL−1 thresholds because Vmax is not allowed to decline as leukaemic cells are depleted. Mould_2007_alemtuzumab_wbc (used in the coupled-PK-PD section above) replaces the data WBC with the PD ODE state and so does drive Vmax dynamically.
  • Implied PD-baseline WBC mismatch. The typical-value baseline implied by the PD parameters is K_in / K_out = 1.56 / 0.029 = 53.8 × 10^9^/L (Mould 2007 Results, PD section). The observed cohort median is 37.8 × 109/L (Table 1). The IIV on Kin (172%) widens the per-subject baseline distribution; the typical-value comparator should be read as 53.8 × 109/L, not the cohort median.
  • Missing Emax/EC50 correlation. Mould 2007 reports that the PD model contained a correlation term between Emax and EC50 (“The model contained a term describing the correlation of Emax and EC50”) but does not publish the correlation value. Mould_2007_alemtuzumab_wbc uses a diagonal Ω on the PD parameters; the missing covariance is not invented.
  • Kout IIV not estimated. Mould 2007 Table 3 reports “Not estimated” for Kout IIV; etalkout is wrapped in fixed(0) so the structural Kout varies only with the typical-value estimate.
  • Time-varying WBC rxode2 limitation. A time-varying WBC column in the events data frame segfaults rxode2 when combined with the repeated IV bolus dose schedule used here (observed on rxode2 in claude_runner 015-mould_2007_alemtuzumab); the baseline-WBC simplification above avoids the crash. The underlying PK model file supports time-varying WBC and will benefit from the fix once the rxode2 issue is resolved.
  • Race. Not reported in Mould 2007; not simulated.
  • Weight. Tested but not retained as a covariate; not simulated.
  • Dose escalation. The initial 3 → 10 → 30 mg escalation doses were omitted; the virtual cohort starts directly on the 30 mg maintenance regimen.
  • IIV. Block covariance was not reported; the model uses the paper’s diagonal Ω on Vmax, Km, V1, V2.
  • Publisher restriction on full-text XML. The on-disk PMC XML for PMID 17506867 contained front matter only (the publisher blocks full-text XML download). Parameter values, equations, and population demographics were verified against the PMC-rendered full-text page (https://pmc.ncbi.nlm.nih.gov/articles/PMC2000651/). No errata were found on PubMed or Wiley Online Library.

Reference

  • Mould DR, Baumann A, Kuhlmann J, Keating MJ, Weitman S, Hillmen P, Brettman LR, Reif S, Bonate PL. Population pharmacokinetics-pharmacodynamics of alemtuzumab (Campath) in patients with chronic lymphocytic leukaemia and its link to treatment response. Br J Clin Pharmacol. 2007;64(3):278-291. doi:10.1111/j.1365-2125.2007.02914.x

Two model files are bundled for this paper:

  • Mould_2007_alemtuzumab — PK-only, WBC as a data covariate.
  • Mould_2007_alemtuzumab_wbc — coupled PK + indirect-response PD on WBC.