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Atezolizumab in pediatric and young adult patients (Shemesh 2019)

Source: vignettes/articles/Shemesh_2019_atezolizumab.Rmd
Shemesh_2019_atezolizumab.Rmd

Model

Shemesh et al. (2019) refitted the adult two-compartment intravenous atezolizumab population PK model (Stroh et al. 2017) to 431 serum concentrations from 87 pediatric and young adult patients in the phase I/II iMATRIX-atezolizumab study, keeping the adult structure and re-estimating every parameter. Patients < 18 years received 15 mg/kg q3w (maximum 1200 mg); patients >= 18 years received 1200 mg q3w.

mod <- rxode2::rxode(readModelDb("Shemesh_2019_atezolizumab"))
#> ℹ parameter labels from comments will be replaced by 'label()'
mod$description
#> [1] "Two-compartment population PK model with intravenous infusion input for atezolizumab (anti-PD-L1 IgG1) in pediatric and young adult patients (7 months to 29 years, 8.7-154 kg) with relapsed or refractory solid tumors or lymphoma from the iMATRIX-atezolizumab phase I/II study (Shemesh 2019, n = 87, 431 serum concentrations). The adult atezolizumab popPK structure was refitted to the pediatric data with every parameter re-estimated: power effects of body weight and albumin on CL and V1, of baseline tumor burden on CL, and a multiplicative factor for treatment-emergent ADA on CL. Sex effects were not evaluated and V2 and Q carry no covariates."

Population 

str(mod$population)
#> List of 15
#>  $ species         : chr "human"
#>  $ n_subjects      : num 87
#>  $ n_studies       : num 1
#>  $ n_observations  : num 431
#>  $ age_range       : chr "7 months to 29 years"
#>  $ age_median      : chr "12 years (69 patients < 18 years); 22 years (18 young adults)"
#>  $ weight_range    : chr "8.7-154 kg"
#>  $ weight_median   : chr "38.9 kg (< 18 years); 61.0 kg (>= 18 years)"
#>  $ sex_female_pct  : num 46
#>  $ disease_state   : chr "Relapsed or refractory pediatric solid tumors (Ewing sarcoma, neuroblastoma, osteosarcoma, rhabdomyosarcoma, no"| __truncated__
#>  $ dose_range      : chr "15 mg/kg IV q3w (maximum 1200 mg) for patients < 18 years; 1200 mg IV q3w flat for patients >= 18 years; 60-min"| __truncated__
#>  $ age_groups      : chr "2 infants < 2 y, 29 children 2 to < 12 y, 38 adolescents 12 to < 18 y, 18 young adults >= 18 y"
#>  $ ada_positive_pct: num 13
#>  $ regions         : chr "Multinational (iMATRIX-atezolizumab, NCT02541604)"
#>  $ notes           : chr "Shemesh 2019 Table 1 (baseline demographics by age group); 40 of 87 female."

Baseline characteristics by age group (Shemesh 2019 Table 1): 2 infants < 2 years (median 9.1 kg), 29 children 2 to < 12 years (22.5 kg), 38 adolescents 12 to < 18 years (51.1 kg) and 18 young adults >= 18 years (61.0 kg). Median albumin was 33-41 g/L and median tumor burden 55-120 mm across groups; 13% of patients were treatment-emergent ADA-positive.

Source trace 

tibble::tribble(
  ~Quantity, ~Value, ~`Source location`,
  "CL (L/day)", "0.217", "Table 2",
  "V1 (L)", "3.01", "Table 2",
  "V2 (L)", "1.36", "Table 2",
  "Q (L/day)", "0.183", "Table 2",
  "WT exponent on CL", "0.795", "Table 2",
  "ALB exponent on CL", "-1.18", "Table 2",
  "TUMSZ exponent on CL", "0.122", "Table 2",
  "ADA factor on CL (theta^ADA)", "1.23", "Table 2; Methods 'PopPK model'; Discussion '23% increase'",
  "WT exponent on V1", "0.766", "Table 2",
  "ALB exponent on V1", "-0.566", "Table 2",
  "Sex on V1 / V2", "not evaluated", "Table 2 (NE)",
  "BSV CL / V1 / V2 (variance)", "0.0458 / 0.0140 / 0.311", "Table 2",
  "Correlation CL-V1", "0.510 (covariance 0.012914)", "Table 2",
  "Proportional residual variance", "0.051 (SD 0.2258)", "Table 2",
  "Additive residual variance", "68.9 (SD 8.301 ug/mL)", "Table 2",
  "Reference WT / ALB / TUMSZ", "77 kg / 40 g/L / 63 mm", "Methods adult CL and V1 equations",
  "Two-compartment IV infusion structure", "-", "Methods 'PopPK model'; Results",
  "Dosing and infusion durations", "15 mg/kg (max 1200) or 1200 mg q3w; 60 min then 30 min", "Methods 'Study design'"
) |>
  knitr::kable()
Quantity Value Source location
CL (L/day) 0.217 Table 2
V1 (L) 3.01 Table 2
V2 (L) 1.36 Table 2
Q (L/day) 0.183 Table 2
WT exponent on CL 0.795 Table 2
ALB exponent on CL -1.18 Table 2
TUMSZ exponent on CL 0.122 Table 2
ADA factor on CL (theta^ADA) 1.23 Table 2; Methods ‘PopPK model’; Discussion ‘23% increase’
WT exponent on V1 0.766 Table 2
ALB exponent on V1 -0.566 Table 2
Sex on V1 / V2 not evaluated Table 2 (NE)
BSV CL / V1 / V2 (variance) 0.0458 / 0.0140 / 0.311 Table 2
Correlation CL-V1 0.510 (covariance 0.012914) Table 2
Proportional residual variance 0.051 (SD 0.2258) Table 2
Additive residual variance 68.9 (SD 8.301 ug/mL) Table 2
Reference WT / ALB / TUMSZ 77 kg / 40 g/L / 63 mm Methods adult CL and V1 equations
Two-compartment IV infusion structure - Methods ‘PopPK model’; Results
Dosing and infusion durations 15 mg/kg (max 1200) or 1200 mg q3w; 60 min then 30 min Methods ‘Study design’

Virtual cohort

Covariates are drawn per age group around the Table 1 medians and truncated to the Table 1 ranges (log-normal for weight and tumor burden, normal for albumin); ADA status is Bernoulli at the observed group rate.

set.seed(2019)
rxode2::rxSetSeed(2019)

groups <- tibble::tribble(
  ~agegrp, ~n, ~wt_med, ~wt_min, ~wt_max, ~alb_med, ~alb_min, ~alb_max, ~tum_med, ~tum_min, ~tum_max, ~ada,
  "< 2 y", 20, 9.1, 8.7, 9.5, 33, 30, 35, 59, 35, 83, 0,
  "2 to < 12 y", 150, 22.5, 12.0, 74.4, 41, 23, 46, 55, 15, 301, 0.17,
  "12 to < 18 y", 150, 51.1, 28.2, 105, 41, 29, 49, 78, 11, 208, 0.11,
  ">= 18 y", 150, 61.0, 46.2, 154, 39, 27, 47, 120, 10, 258, 0.11
)

rtrunc_lnorm <- function(n, med, lo, hi) {
  x <- exp(rnorm(n, log(med), (log(hi) - log(lo)) / 4))
  pmin(pmax(x, lo), hi)
}
rtrunc_norm <- function(n, med, lo, hi) {
  pmin(pmax(rnorm(n, med, (hi - lo) / 4), lo), hi)
}

cohort <- groups |>
  rowwise() |>
  reframe(
    agegrp = agegrp,
    WT = rtrunc_lnorm(n, wt_med, wt_min, wt_max),
    ALB = rtrunc_norm(n, alb_med, alb_min, alb_max),
    TUMSZ = rtrunc_lnorm(n, tum_med, tum_min, tum_max),
    ADA_POS = rbinom(n, 1, ada)
  ) |>
  mutate(
    id = row_number(),
    agegrp = factor(agegrp, levels = groups$agegrp),
    regimen = ifelse(agegrp == ">= 18 y", "1200 mg flat", "15 mg/kg"),
    dose = ifelse(regimen == "1200 mg flat", 1200, pmin(15 * WT, 1200)),
    wtband = case_when(
      regimen == "15 mg/kg" & WT < 30 ~ "< 18 y: < 30 kg",
      regimen == "15 mg/kg" & WT < 45 ~ "< 18 y: 30 to < 45 kg",
      regimen == "15 mg/kg" ~ "< 18 y: >= 45 kg",
      WT < 57 ~ ">= 18 y: < 57 kg",
      WT < 65 ~ ">= 18 y: 57 to < 65 kg",
      TRUE ~ ">= 18 y: >= 65 kg"
    )
  )

cohort |>
  group_by(agegrp) |>
  summarise(n = n(), WT = median(WT), ALB = median(ALB), TUMSZ = median(TUMSZ),
            `ADA+ (%)` = 100 * mean(ADA_POS)) |>
  knitr::kable(digits = 1)
agegrp n WT ALB TUMSZ ADA+ (%)
< 2 y 20 9.0 33.0 61.0 0.0
2 to < 12 y 150 21.0 40.6 59.8 20.7
12 to < 18 y 150 48.3 40.6 88.3 16.0
>= 18 y 150 59.3 38.1 118.8 12.0

Event table

Ten q3w cycles; 60-min infusion in cycle 1 and 30-min infusions afterwards (Shemesh 2019 Methods). Cycle 1 and cycle 10 (steady state, as in the paper) are sampled densely; the cycle trough is read at day 20.99 of the cycle, the last sampling time the paper used.

tau <- 21
dose_times <- tau * (0:9)
ss_start <- dose_times[10]
grid1 <- sort(unique(c(0, seq(0.01, 3, by = 0.05), seq(3.5, 20.5, by = 0.5), 20.99)))
obs_times <- c(grid1, ss_start + grid1)

doses <- cohort |>
  select(id, dose) |>
  tidyr::crossing(time = dose_times) |>
  mutate(
    dur = ifelse(time == 0, 1 / 24, 0.5 / 24),
    amt = dose, rate = amt / dur, evid = 1, cmt = "central"
  ) |>
  select(id, time, amt, rate, evid, cmt)

obs <- cohort |>
  select(id) |>
  tidyr::crossing(time = obs_times) |>
  mutate(amt = 0, rate = 0, evid = 0, cmt = "central")

events <- bind_rows(doses, obs) |>
  left_join(select(cohort, id, WT, ALB, TUMSZ, ADA_POS), by = "id") |>
  arrange(id, time, desc(evid))

Simulation 

sim <- rxode2::rxSolve(mod, events, returnType = "data.frame",
                       keep = c("WT", "ALB", "TUMSZ", "ADA_POS")) |>
  left_join(select(cohort, id, agegrp, regimen, wtband), by = "id")
sim |>
  mutate(cycle = ifelse(time < ss_start, "Cycle 1", "Cycle 10"),
         tcyc = ifelse(time < ss_start, time, time - ss_start)) |>
  group_by(agegrp, cycle, tcyc) |>
  summarise(med = median(Cc), lo = quantile(Cc, 0.05), hi = quantile(Cc, 0.95),
            .groups = "drop") |>
  # the pre-dose zero of cycle 1 cannot be drawn on a log axis
  filter(lo > 0) |>
  mutate(cycle = factor(cycle, levels = c("Cycle 1", "Cycle 10"))) |>
  ggplot(aes(tcyc, med, colour = agegrp, fill = agegrp)) +
  geom_ribbon(aes(ymin = lo, ymax = hi), alpha = 0.15, colour = NA) +
  geom_line() +
  geom_hline(yintercept = 6, linetype = "dotted") +
  facet_wrap(~cycle) +
  scale_y_log10() +
  labs(x = "Time after dose in cycle (day)", y = "Atezolizumab (ug/mL)",
       colour = "Age group", fill = "Age group")
Simulated cycle 1 and cycle 10 profiles (median and 5th-95th percentiles) by age group. The dotted line is the 6 ug/mL target trough (Shemesh 2019 Fig. 3).

Simulated cycle 1 and cycle 10 profiles (median and 5th-95th percentiles) by age group. The dotted line is the 6 ug/mL target trough (Shemesh 2019 Fig. 3).

Cycle 1 trough by age group

Shemesh 2019 Results report geometric-mean cycle 1 Cmin of 55.9 ug/mL in children 2 to < 12 years and 62.4 ug/mL in adolescents 12 to < 18 years.

geo_mean <- function(x) exp(mean(log(x)))

cmin_age <- sim |>
  filter(abs(time - 20.99) < 1e-6) |>
  group_by(agegrp) |>
  summarise(sim_gm = geo_mean(Cc), .groups = "drop") |>
  inner_join(tibble::tibble(agegrp = c("2 to < 12 y", "12 to < 18 y"),
                            published = c(55.9, 62.4)), by = "agegrp") |>
  mutate(pct_diff = 100 * (sim_gm - published) / published)
cmin_age |>
  dplyr::rename("Age group" = agegrp, "Simulated GM Cmin (ug/mL)" = sim_gm,
                "Published GM Cmin (ug/mL)" = published, "% diff" = pct_diff) |>
  knitr::kable(digits = 1)
Age group Simulated GM Cmin (ug/mL) Published GM Cmin (ug/mL) % diff
2 to < 12 y 49.8 55.9 -10.9
12 to < 18 y 62.7 62.4 0.5 
stopifnot(all(abs(cmin_age$pct_diff) < 25))

PKNCA validation

Cycle 1 (days 0-20.99) and cycle 10 (days 189-209.99) intervals. The PKNCA grouping is the Table 3 dose regimen / weight band.

conc_df <- sim |>
  filter(!is.na(Cc)) |>
  select(id, time, Cc, wtband)
dose_df <- doses |>
  left_join(select(cohort, id, wtband), by = "id") |>
  select(id, time, amt, wtband)

conc_obj <- PKNCA::PKNCAconc(conc_df, Cc ~ time | wtband + id)
dose_obj <- PKNCA::PKNCAdose(dose_df, amt ~ time | wtband + id)
intervals <- data.frame(
  start = c(0, ss_start), end = c(20.99, ss_start + 20.99),
  cmax = TRUE, cmin = TRUE, auclast = TRUE
)
nca <- PKNCA::pk.nca(PKNCA::PKNCAdata(conc_obj, dose_obj, intervals = intervals))

nca_res <- as.data.frame(nca) |>
  filter(PPTESTCD %in% c("cmax", "cmin", "auclast")) |>
  mutate(cycle = ifelse(start == 0, "Cycle 1", "Steady state"))
# Cycle 1 minimum over [0, 20.99] is the pre-dose zero; use the end-of-cycle
# trough instead, which is how the paper defines cycle 1 Cmin.
cmin_c1 <- sim |>
  filter(abs(time - 20.99) < 1e-6) |>
  transmute(wtband, id, cycle = "Cycle 1", PPTESTCD = "cmin", PPORRES = Cc)
sim_long <- nca_res |>
  filter(!(cycle == "Cycle 1" & PPTESTCD == "cmin")) |>
  select(wtband, id, cycle, PPTESTCD, PPORRES) |>
  bind_rows(cmin_c1)

Comparison against Shemesh 2019 Table 3

Table 3 reports medians of the individual post-hoc exposures by dose regimen and weight band; the simulated values are medians over the virtual cohort.

bands <- c("< 18 y: < 30 kg", "< 18 y: 30 to < 45 kg", "< 18 y: >= 45 kg",
           ">= 18 y: < 57 kg", ">= 18 y: 57 to < 65 kg", ">= 18 y: >= 65 kg")
published <- bind_rows(
  tibble::tibble(wtband = bands, cycle = "Cycle 1",
                 cmax = c(270, 330, 349, 492, 486, 326),
                 cmin = c(55.6, 65.0, 65.5, 97.3, 98.8, 57.6),
                 auclast = c(2085, 2757, 2988, 4268, 4330, 2733)),
  tibble::tibble(wtband = bands, cycle = "Steady state",
                 cmax = c(400, 463, 460, 664, 651, 404),
                 cmin = c(120, 125, 112, 171, 164, 88.1),
                 auclast = c(4045, 4781, 4510, 6692, 6574, 3861))
) |>
  pivot_longer(c(cmax, cmin, auclast), names_to = "PPTESTCD", values_to = "PPORRES")

cmp <- nlmixr2lib::ncaComparisonTable(
  simulated = sim_long,
  reference = published,
  by = c("wtband", "cycle"),
  units = c(cmax = "ug/mL", cmin = "ug/mL", auclast = "ug*day/mL"),
  tolerance_pct = 20
)
cmp |>
  dplyr::rename("Regimen / weight band" = wtband, "Cycle" = cycle) |>
  knitr::kable(caption = "Simulated medians vs Shemesh 2019 Table 3. * differs by more than 20%.")
Simulated medians vs Shemesh 2019 Table 3. * differs by more than 20%.
NCA parameter Regimen / weight band Cycle Reference Simulated % diff
Cmax (ug/mL) < 18 y: < 30 kg Cycle 1 270 267 -1.2%
Cmax (ug/mL) < 18 y: < 30 kg Steady state 400 380 -5.1%
Cmax (ug/mL) < 18 y: 30 to < 45 kg Cycle 1 330 319 -3.4%
Cmax (ug/mL) < 18 y: 30 to < 45 kg Steady state 463 448 -3.3%
Cmax (ug/mL) < 18 y: >= 45 kg Cycle 1 349 359 +3.0%
Cmax (ug/mL) < 18 y: >= 45 kg Steady state 460 479 +4.1%
Cmax (ug/mL) >= 18 y: < 57 kg Cycle 1 492 548 +11.3%
Cmax (ug/mL) >= 18 y: < 57 kg Steady state 664 718 +8.1%
Cmax (ug/mL) >= 18 y: 57 to < 65 kg Cycle 1 486 482 -0.8%
Cmax (ug/mL) >= 18 y: 57 to < 65 kg Steady state 651 624 -4.2%
Cmax (ug/mL) >= 18 y: >= 65 kg Cycle 1 326 363 +11.4%
Cmax (ug/mL) >= 18 y: >= 65 kg Steady state 404 463 +14.5%
Cmin (ug/mL) < 18 y: < 30 kg Cycle 1 55.6 47.6 -14.4%
Cmin (ug/mL) < 18 y: < 30 kg Steady state 120 107 -10.9%
Cmin (ug/mL) < 18 y: 30 to < 45 kg Cycle 1 65 59.1 -9.1%
Cmin (ug/mL) < 18 y: 30 to < 45 kg Steady state 125 112 -10.1%
Cmin (ug/mL) < 18 y: >= 45 kg Cycle 1 65.5 66.5 +1.5%
Cmin (ug/mL) < 18 y: >= 45 kg Steady state 112 115 +2.8%
Cmin (ug/mL) >= 18 y: < 57 kg Cycle 1 97.3 95.4 -2.0%
Cmin (ug/mL) >= 18 y: < 57 kg Steady state 171 164 -4.1%
Cmin (ug/mL) >= 18 y: 57 to < 65 kg Cycle 1 98.8 69.7 -29.5%*
Cmin (ug/mL) >= 18 y: 57 to < 65 kg Steady state 164 141 -14.0%
Cmin (ug/mL) >= 18 y: >= 65 kg Cycle 1 57.6 64.7 +12.4%
Cmin (ug/mL) >= 18 y: >= 65 kg Steady state 88.1 99.5 +13.0%
AUClast (ug*day/mL) < 18 y: < 30 kg Cycle 1 2080 1810 -13.0%
AUClast (ug*day/mL) < 18 y: < 30 kg Steady state 4040 3620 -10.6%
AUClast (ug*day/mL) < 18 y: 30 to < 45 kg Cycle 1 2760 2570 -6.8%
AUClast (ug*day/mL) < 18 y: 30 to < 45 kg Steady state 4780 4220 -11.8%
AUClast (ug*day/mL) < 18 y: >= 45 kg Cycle 1 2990 3050 +2.2%
AUClast (ug*day/mL) < 18 y: >= 45 kg Steady state 4510 4730 +4.8%
AUClast (ug*day/mL) >= 18 y: < 57 kg Cycle 1 4270 4390 +3.0%
AUClast (ug*day/mL) >= 18 y: < 57 kg Steady state 6690 6720 +0.3%
AUClast (ug*day/mL) >= 18 y: 57 to < 65 kg Cycle 1 4330 4030 -7.0%
AUClast (ug*day/mL) >= 18 y: 57 to < 65 kg Steady state 6570 5940 -9.7%
AUClast (ug*day/mL) >= 18 y: >= 65 kg Cycle 1 2730 3180 +16.5%
AUClast (ug*day/mL) >= 18 y: >= 65 kg Steady state 3860 4480 +15.9% 
pct <- sim_long |>
  group_by(wtband, cycle, PPTESTCD) |>
  summarise(sim = median(PPORRES), .groups = "drop") |>
  inner_join(published, by = c("wtband", "cycle", "PPTESTCD")) |>
  mutate(pct_diff = 100 * (sim - PPORRES) / PPORRES)
stopifnot(
  abs(median(pct$pct_diff)) < 10,
  quantile(abs(pct$pct_diff), 0.75) < 25
)

The pediatric weight bands (n = 21-25 patients each in the study) agree within about 15% for every metric. The largest differences are in the young-adult bands, each of which holds only 6 study patients, so their Table 3 medians are driven by individual covariate values that a Table 1-based virtual cohort does not reproduce (for example the cycle 1 Cmin of the 57 to < 65 kg band).

Typical-value half-life

Shemesh 2019 Results: terminal half-life of ~2-3 weeks in pediatric patients and young adults. The terminal half-life of the typical patient in each age group follows directly from the micro-constants.

thalf <- groups |>
  transmute(
    agegrp,
    cl = 0.217 * (wt_med / 77)^0.795 * (alb_med / 40)^-1.18 * (tum_med / 63)^0.122,
    vc = 3.01 * (wt_med / 77)^0.766 * (alb_med / 40)^-0.566,
    vp = 1.36, q = 0.183
  ) |>
  mutate(
    kel = cl / vc, k12 = q / vc, k21 = q / vp,
    beta = 0.5 * ((kel + k12 + k21) - sqrt((kel + k12 + k21)^2 - 4 * kel * k21)),
    t_half_day = log(2) / beta
  )
knitr::kable(select(thalf, agegrp, cl, vc, t_half_day), digits = 3)
agegrp cl vc t_half_day
< 2 y 0.049 0.654 31.894
2 to < 12 y 0.078 1.157 25.447
12 to < 18 y 0.156 2.168 18.068
>= 18 y 0.201 2.554 15.775 
# Deterministic: typical adolescents and young adults fall in the 2-3 week band.
stopifnot(all(thalf$t_half_day[thalf$agegrp %in% c("12 to < 18 y", ">= 18 y")] > 14 &
  thalf$t_half_day[thalf$agegrp %in% c("12 to < 18 y", ">= 18 y")] < 21))

The typical child (22.5 kg) and infant (9.1 kg) have longer terminal half-lives (about 3.6 and 4.6 weeks), because V2 and Q are not scaled by body weight in the pediatric model while CL and V1 are. The paper’s “~2-3 weeks” statement summarizes individual post-hoc estimates across all patients, most of whom are adolescents, and the authors note that infant results “have limited interpretation”.

Assumptions and deviations

  • Reference covariate values. The paper prints the normalization references (77 kg, 40 g/L, 63 mm) only for the adult equations; the pediatric model “utilized the same structure”, so the same references are used here.
  • ADA effect form. Per the Methods, the pediatric ADA effect is parameterized theta^ADA (1.23), unlike the adult (1 + theta) form.
  • Residual error. Table 2 reports variances; the model carries their square roots. The additive component is taken to be on the ug/mL scale, consistent with the adult model and the reported concentrations.
  • IIV. Exponential (log-normal) between-subject variability assumed on CL, V1 and V2 as in the adult model; only the CL-V1 correlation was estimated.
  • Sex was not evaluated (NE) in the pediatric model and is omitted; the adult-model sex effects are not carried over.
  • Adult model column of Table 2. The adult comparator (Stroh 2017) is not extracted here; its disposition parameters ship as the fixed IV layer of Chan_2025_atezolizumab.
  • Virtual cohort. Covariate distributions are approximated from Table 1 medians and ranges; the 2 infants of the study are represented by 20 simulated infants. ADA status is treated as time-constant.
  • Table 3 comparison. Table 3 is based on the study’s individual post-hoc parameters, so the comparison with a virtual cohort is approximate and is asserted on the median and interquartile envelope rather than every cell.