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A1pi (Tortorici 2017)

Source: vignettes/articles/Tortorici_2017_a1pi.Rmd
Tortorici_2017_a1pi.Rmd

Model and source

  • Citation: Tortorici MA, Rogers JA, Vit O, Bexon M, Sandhaus RA, Burdon J, Chorostowska-Wynimko J, Thompson P, Stocks J, McElvaney NG, Chapman KR, Edelman JM. Quantitative disease progression model of alpha-1 proteinase inhibitor therapy on computed tomography lung density in patients with alpha-1 antitrypsin deficiency. Br J Clin Pharmacol 2017;83(11):2386-2397. doi:10.1111/bcp.13358.
  • Description: Empirical disease-progression model of intravenous alpha-1 proteinase inhibitor (A1-PI) augmentation therapy in alpha-1 antitrypsin deficiency (Tortorici 2017). Combines a sequential dose-exposure regression that predicts the per-subject median trough serum A1-PI from average dose rate (mg/day, supplied as DOSE covariate), baseline body weight, and baseline endogenous A1-PI, with a piecewise-linear-in-time exposure-response model that predicts CT lung density at total lung capacity from the dose-derived A1-PI exposure and baseline FEV1, with a slope transition at 720 days separating the RAPID-RCT and RAPID-OLE study phases.
  • Article: https://doi.org/10.1111/bcp.13358

Population

The model was fit to the combined RAPID-RCT (NCT00261833, n=180) and RAPID-OLE (NCT00670007, n=140) trials of intravenous human alpha-1 proteinase inhibitor (A1-PI; brand name Zemaira) augmentation therapy in adults with severe alpha-1 antitrypsin deficiency (AATD) and clinical emphysema. Enrolment criteria were a deficient genotype (typically PI*ZZ), serum A1-PI <= 11 umol/L, and post-bronchodilator FEV1 35-70% predicted. Patients were randomised to A1-PI 60 mg/kg/week intravenous infusion (n=93 in RAPID-RCT) or matching placebo (n=87) over 2 years; RAPID-OLE was a 2-year open-label continuation in 140 patients (76 Early-Start who continued A1-PI, 64 Delayed-Start who switched from placebo to active A1-PI). Samples for serum A1-PI were collected every 3 months at trough (day 7 post-infusion) and CT lung scans at baseline plus 3, 12, 21, 24, 36, and 48 months post-randomisation. The exposure-response analysis set modelled here comprises 134 subjects (61 placebo and 73 A1-PI) who had at least one post-baseline CT lung-density measurement; the dose-exposure analysis set was a slightly larger 308-subject set spanning both trials. Mean baseline characteristics in RAPID-RCT (Tortorici 2017 Table 1) were age 53 years, weight 77 kg, body-mass index 26 kg/m^2, FEV1 1.6 L, CT lung density at TLC 47 g/L, and antigenic A1-PI 6.4 umol/L (active arm) / 5.9 umol/L (placebo arm). Sex / race breakdown is not reported in the paper.

The same information is available programmatically via rxode2::rxode(readModelDb("Tortorici_2017_a1pi"))$population.

Source trace

The model is a sequential dose-exposure plus exposure-response disease progression model. The dose-exposure layer (Equation 6, parameter values from Table 2) predicts the per-subject median trough A1-PI exposure C from the average dose rate D, baseline weight WT, and baseline endogenous A1-PI (Cbase):

C=θ1⋅eη1⋅(Cbase/5.5)θ5+θ2⋅(WT/77)θ3⋅(Cbase/5.5)θ4⋅D C \;=\; \theta_1 \cdot e^{\eta_1} \cdot (C_\mathrm{base}/5.5)^{\theta_5} \;+\; \theta_2 \cdot (\mathrm{WT}/77)^{\theta_3} \cdot (C_\mathrm{base}/5.5)^{\theta_4} \cdot D

The exposure-response layer (Equations 7-10, parameter values from Table 3) predicts the CT lung density Y at total lung capacity from the dose-derived exposure and baseline FEV1, with a slope transition at 720 days separating the RAPID-RCT and RAPID-OLE study phases:

Y(t)=Int+DP1⋅min⁡(720,t)+DP2⋅max⁡(0,t−720) Y(t) \;=\; \mathrm{Int} + \mathrm{DP}_1 \cdot \min(720, t) \;+\; \mathrm{DP}_2 \cdot \max(0, t - 720)

where DP1 (RAPID-RCT phase decline rate) and DP2 (RAPID-OLE phase decline rate) are linear functions of A1-PI exposure (centred at the typical placebo-arm level) and FEV1 deviation from the cohort median; DP2 = DP1 plus a phase-2 increment.

Equation / parameter Value Source location
Dose-exposure structural form n/a Equation 6, page 2389
lc_pbo (typical placebo-arm A1-PI exposure, log-scale parameterisation of the linear value) 5.42 umol/L Table 2 theta1; the linear-scale interpretation is required to make Equation 6 dimensionally consistent with the placebo-arm Table 1 baseline (~6 umol/L) and the bootstrap predictions in Figure 2A (~16 umol/L typical post-treatment A1-PI for a 77 kg subject at 60 mg/kg/wk).
lcslope (slope of A1-PI exposure on dose rate) 0.02 (umol/L)/(mg/day) Table 2 theta2
e_wt_cslope (WT power exponent on slope) -0.85 Table 2 theta3; consistent with Kleiber’s-law allometric scaling of clearance per the paper’s Discussion
e_a1pi_cslope (Cbase power exponent on slope) -0.12 Table 2 theta4
e_a1pi_cpbo (Cbase power exponent on placebo intercept) 0.73 Table 2 theta5
Reference WT for power-form effect 77 kg paper Methods, dose-exposure section: “a reference individual with a weight of 77.0 kg”
Reference Cbase for power-form effect 5.5 umol/L paper Methods: “5.5, equates approximately to the median pre-treatment A1-PI concentration among those who were randomized to placebo in the RAPID-RCT study”
IIV omega1 (log-scale SD on placebo-arm exposure) 0.07 Table 2 omega1; encoded as variance 0.07^2 = 0.0049
Residual sigma (log-scale SD on Cc) 0.15 Table 2 sigma; encoded as proportional residual error
Exposure-response structural form n/a Equations 7-10, pages 2389-2390
bld (pre-treatment lung density intercept) 46.89 g/L Table 3 theta1
dpr_pbo (placebo-arm lung-density decline rate) -2.18 g/L/year Table 3 theta2
e_cc_dpr (A1-PI exposure effect on decline rate) 0.06 (g/L/year)/(umol/L) Table 3 theta3
dpr_phase2 (RAPID-OLE phase increment on decline rate) 0.20 g/L/year Table 3 theta4
e_fev1_dpr (FEV1 effect on decline rate) 0.56 (g/L/year)/L Table 3 theta5
Reference FEV1 for linear-deviation effect 1.6 L Table 1 mean FEV1 (1.6 L in both arms)
Knot day 720 Equations 7-10; “the slope transition from DPi1 to DPi2 at tijk = 720 days, when the Extension study begins” (paper page 2389)
Days-per-year unit conversion 365.25 Tables 2 and 3 report decline rates in g/L/year while equations 7-10 carry t in days; t is converted to years inside model() for dimensional consistency with the published slopes
3x3 IIV BLOCK on (Int, DPR, Cc-effect) – variances (15.28^2, 1.32^2, 0.09^2) = (233.48, 1.7424, 0.0081) Table 3 omega1..3 (SDs)
3x3 IIV BLOCK – correlations (omega12 = -0.270, omega13 = 0.26, omega23 = -0.75) Table 3
3x3 IIV BLOCK – lower-triangle covariances (-5.4476, 0.3576, -0.0891) derived from SDs and correlations above
Phase-2 IIV omega4 (additive SD on DP2 - DP1) 0.23 g/L/year Table 3 omega4 (SD); Table 3 footnote flags the 95% CI as 0-6644.52, “very large standard error”, retained on conservative-prediction grounds per the paper’s Model validity and limitations section
Residual sigma (additive SD on lung density) 2.60 g/L Table 3 sigma

Equation centring: the exposure-response model centres the per-subject A1-PI exposure at the typical placebo-arm level so that dpr_pbo + etadpr_pbo represents the natural rate of lung-density decline that would be observed in the absence of treatment. The paper Methods writes “The asterisks in the terms Ci1* and Ci2* indicate that exposure levels were centred at the median placebo levels observed in RAPID-RCT”. The model file uses exp(lc_pbo) (= 5.42 umol/L, the typical-value placebo exposure from the dose-exposure layer) as the centring reference, which is operationally equivalent to the paper’s Ci* construction at the typical individual.

Errata

No published erratum or corrigendum was found for this paper as of the model extraction date (2026-05-09); the PubMed erratum filter (PMID 28662542) returns no results, and the journal landing page lists no correction notice.

The published Supporting Information (Tables S1-S5 and Figures S1-S3) documents the bootstrap covariate-selection algorithm, predicted covariate-conditioned exposure / response tables, and goodness-of-fit plots. The supplement does not contain additional structural-model parameter values; Tables 2 and 3 in the main paper hold the complete final-model parameter set extracted into this model file. The supplement is referenced from the journal landing page at https://onlinelibrary.wiley.com/doi/10.1111/bcp.13358 (file bcp13358-sup-0001-supinfo.docx); a PMC mirror at https://pmc.ncbi.nlm.nih.gov/articles/PMC5651313/ lists the same file in its supplementary-data section but the direct download path was 404 at the time of extraction. Reviewers who need the bootstrap-table predictions to reproduce the paper’s covariate-conditioned summary figures (Figures 1 and 5) can fetch the supplement separately from Wiley after authentication.

The Table 2 column header for theta1 reads “Log A1-PI exposure for placebo (umol l-1)” while the underlying value 5.42 must be interpreted as a linear-scale (back-transformed) exposure for Equation 6 to be dimensionally consistent with the placebo-arm baseline (~6 umol/L) and the paper’s bootstrap predictions (~16 umol/L typical post-treatment A1-PI for the 60 mg/kg/wk reference individual). The model file encodes theta1 as lc_pbo <- log(5.42) and applies log-normal IIV via exp(lc_pbo + etalc_pbo), which back-transforms to the linear typical-value 5.42 and treats omega1 = 0.07 as a log-scale SD on this back-transformed value (equivalently, ~7% CV).

Virtual cohort

Original individual-level RAPID-RCT/RAPID-OLE data are not publicly available. The simulations below use a virtual cohort whose covariate distributions approximate the modelled population: weights drawn from a log-normal distribution centred on the observed median (77 kg, with between-subject SD reflecting Table 1’s pooled SD ~15 kg); baseline A1-PI drawn from a log-normal distribution centred on the placebo-arm median (~5.5 umol/L, SD ~2.5 umol/L truncated to the AATD enrolment range 1-11 umol/L); FEV1 drawn from a normal distribution centred on the cohort median (1.6 L, SD 0.5 L per Table 1).

set.seed(20260509)

n_subj <- 200

cohort <- tibble::tibble(
  id   = seq_len(n_subj),
  WT   = pmin(pmax(rlnorm(n_subj, log(77), 0.18), 50), 130),
  A1PI = pmin(pmax(rlnorm(n_subj, log(5.5), 0.30), 1), 11),
  FEV1 = pmin(pmax(rnorm(n_subj, 1.6, 0.5), 0.5), 3.0)
)

cat(sprintf(
  "Virtual cohort: n=%d. WT median %.1f kg [IQR %.1f-%.1f]; A1PI median %.2f umol/L [IQR %.2f-%.2f]; FEV1 median %.2f L [IQR %.2f-%.2f].\n",
  n_subj,
  median(cohort$WT),  quantile(cohort$WT,  0.25), quantile(cohort$WT,  0.75),
  median(cohort$A1PI),quantile(cohort$A1PI,0.25), quantile(cohort$A1PI,0.75),
  median(cohort$FEV1),quantile(cohort$FEV1,0.25), quantile(cohort$FEV1,0.75)
))
#> Virtual cohort: n=200. WT median 76.8 kg [IQR 67.9-87.8]; A1PI median 5.56 umol/L [IQR 4.26-6.79]; FEV1 median 1.62 L [IQR 1.33-1.95].

Replication: typical-value dose-exposure predictions (Figure 2A)

We first reproduce the paper’s Figure 2A: predicted A1-PI exposure level as a function of body weight, assuming proportional weight-based dosing at 60 mg/kg/week. The reference individual (77 kg, A1PI baseline 5.5 umol/L) should attain an exposure of approximately 16-18 umol/L; the figure shows weight-related variation across the observed range.

mod       <- readModelDb("Tortorici_2017_a1pi")
mod_typ   <- rxode2::zeroRe(mod)
#> ℹ parameter labels from comments will be replaced by 'label()'

wt_grid <- seq(50, 130, by = 1)

ev_dose_exp <- data.frame(
  id   = seq_along(wt_grid),
  time = 0,                # dose-exposure layer is time-independent
  amt  = 0,
  evid = 0L,
  WT   = wt_grid,
  A1PI = 5.5,
  FEV1 = 1.6,
  DOSE = 60 * wt_grid / 7  # mg/day for a 60 mg/kg/week regimen
)

sim_dose_exp <- rxode2::rxSolve(mod_typ, events = ev_dose_exp,
                                keep = c("WT", "DOSE")) |>
  as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalc_pbo', 'etabld', 'etadpr_pbo', 'etae_cc_dpr', 'etadpr_phase2'
#> Warning: multi-subject simulation without without 'omega'

ggplot(sim_dose_exp, aes(x = WT, y = Cc)) +
  geom_line(linewidth = 0.9, colour = "steelblue") +
  geom_hline(yintercept = 11, linetype = "dashed", colour = "grey50") +
  geom_vline(xintercept = 77, linetype = "dotted", colour = "grey50") +
  scale_x_continuous(breaks = c(50, 77, 100, 130)) +
  labs(
    x = "Body weight (kg)",
    y = "Predicted A1-PI exposure (umol/L)",
    title = "Typical-value A1-PI exposure vs body weight at 60 mg/kg/week",
    caption = "A1PI baseline = 5.5 umol/L (cohort median); dashed = 11 umol/L protective threshold."
  ) +
  theme_bw()
Replication of Figure 2A: predicted typical-value A1-PI exposure as a function of body weight at 60 mg/kg/week, with baseline A1-PI fixed at the cohort median 5.5 umol/L. The horizontal dashed line is the 11 umol/L putative protective threshold.

Replication of Figure 2A: predicted typical-value A1-PI exposure as a function of body weight at 60 mg/kg/week, with baseline A1-PI fixed at the cohort median 5.5 umol/L. The horizontal dashed line is the 11 umol/L putative protective threshold.

A spot-check at the 77 kg reference individual confirms the implementation matches the published equation:

ref_check <- data.frame(
  WT = c(60, 77, 100, 120),
  A1PI = 5.5,
  expected_Cc = 5.42 * (5.5/5.5)^0.73 +
                0.02 * (c(60, 77, 100, 120)/77)^(-0.85) *
                       (5.5/5.5)^(-0.12) * (60 * c(60, 77, 100, 120) / 7)
)

ev_check <- data.frame(
  id = seq_len(nrow(ref_check)),
  time = 0, amt = 0, evid = 0L,
  WT = ref_check$WT, A1PI = 5.5, FEV1 = 1.6,
  DOSE = 60 * ref_check$WT / 7
)

actual_Cc <- rxode2::rxSolve(mod_typ, events = ev_check) |>
  as.data.frame() |>
  dplyr::pull(Cc) |>
  unique()
#> ℹ omega/sigma items treated as zero: 'etalc_pbo', 'etabld', 'etadpr_pbo', 'etae_cc_dpr', 'etadpr_phase2'
#> Warning: multi-subject simulation without without 'omega'

knitr::kable(
  cbind(ref_check, actual_Cc = actual_Cc,
        diff = actual_Cc - ref_check$expected_Cc),
  digits  = 4,
  caption = "Typical-value Cc at canonical body weights: model output vs analytic Equation 6."
)
Typical-value Cc at canonical body weights: model output vs analytic Equation 6.
WT A1PI expected_Cc actual_Cc diff
60 5.5 18.1352 18.1352 0
77 5.5 18.6200 18.6200 0
100 5.5 19.1478 19.1478 0
120 5.5 19.5284 19.5284 0 

stopifnot(max(abs(actual_Cc - ref_check$expected_Cc)) < 1e-6)

Replication: typical-value dose-response (Figure 2B)

Figure 2B in the paper plots the predicted exposure distribution versus dose level (60, 90, 120 mg/kg/week), showing the median exposure rising approximately linearly with dose. The typical-value prediction at the 77 kg reference individual is reproduced below.

dose_levels <- c(0, 30, 60, 90, 120)  # mg/kg/week

ev_dr <- data.frame(
  id   = seq_along(dose_levels),
  time = 0, amt = 0, evid = 0L,
  WT   = 77, A1PI = 5.5, FEV1 = 1.6,
  DOSE = dose_levels * 77 / 7
)

sim_dr <- rxode2::rxSolve(mod_typ, events = ev_dr,
                          keep = c("DOSE")) |>
  as.data.frame() |>
  dplyr::mutate(dose_mgkgwk = dose_levels)
#> ℹ omega/sigma items treated as zero: 'etalc_pbo', 'etabld', 'etadpr_pbo', 'etae_cc_dpr', 'etadpr_phase2'
#> Warning: multi-subject simulation without without 'omega'

ggplot(sim_dr, aes(x = dose_mgkgwk, y = Cc)) +
  geom_line(linewidth = 0.9, colour = "steelblue") +
  geom_point(size = 2, colour = "steelblue") +
  geom_hline(yintercept = 11, linetype = "dashed", colour = "grey50") +
  scale_x_continuous(breaks = dose_levels) +
  labs(
    x = "Dose (mg/kg/week)",
    y = "Predicted A1-PI exposure (umol/L)",
    title = "Typical-value A1-PI exposure vs dose level at 77 kg / A1PI baseline 5.5 umol/L",
    caption = "Dashed line = 11 umol/L putative protective threshold."
  ) +
  theme_bw()
Replication of Figure 2B (typical-value form): predicted A1-PI exposure vs dose level for the 77 kg / A1PI baseline 5.5 umol/L reference individual.

Replication of Figure 2B (typical-value form): predicted A1-PI exposure vs dose level for the 77 kg / A1PI baseline 5.5 umol/L reference individual. 


knitr::kable(
  sim_dr |> dplyr::select(dose_mgkgwk, DOSE, Cc) |>
    dplyr::rename(`Dose (mg/kg/wk)` = dose_mgkgwk,
                  `Dose rate (mg/day)` = DOSE,
                  `Cc (umol/L)` = Cc),
  digits  = 2,
  caption = "Predicted A1-PI exposure across dose levels (typical-value)."
)
Predicted A1-PI exposure across dose levels (typical-value).
Dose (mg/kg/wk) Dose rate (mg/day) Cc (umol/L)
0 0 5.42
30 330 12.02
60 660 18.62
90 990 25.22
120 1320 31.82

The 60 mg/kg/week reference predicts ~18.6 umol/L (consistent with the ~16-18 umol/L the paper reports for 60 mg/kg/week dosing in Discussion); 30 mg/kg/week predicts ~12 umol/L (just above the 11 umol/L threshold); the 90 and 120 mg/kg/week extrapolations rise approximately linearly with no plateau, reproducing the paper’s qualitative finding that the dose-exposure relationship has no concentration-dependent saturation within the observed range.

Replication: typical-value exposure-response (Figure 3B)

Figure 3B in the paper plots the predicted lung-density decline rate as a function of post-baseline A1-PI exposure. We reproduce the typical-value prediction across the observed exposure range (~5-25 umol/L).

cc_grid <- seq(5, 30, by = 0.5)

# Set DOSE so that Cc takes the desired value at a 77 kg reference subject
# with A1PI baseline = 5.5: Cc = 5.42 + 0.02 * DOSE -> DOSE = (Cc - 5.42)/0.02
dose_grid <- pmax(0, (cc_grid - 5.42) / 0.02)

ev_er <- data.frame(
  id   = seq_along(cc_grid),
  time = 0, amt = 0, evid = 0L,
  WT   = 77, A1PI = 5.5, FEV1 = 1.6,
  DOSE = dose_grid
)

sim_er <- rxode2::rxSolve(mod_typ, events = ev_er) |>
  as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalc_pbo', 'etabld', 'etadpr_pbo', 'etae_cc_dpr', 'etadpr_phase2'
#> Warning: multi-subject simulation without without 'omega'

# DP1 = -2.18 + 0.06 * (Cc - 5.42) for the reference covariates
plot_er <- sim_er |>
  dplyr::mutate(
    decline_rate = -2.18 + 0.06 * (Cc - 5.42)
  )

ggplot(plot_er, aes(x = Cc, y = decline_rate)) +
  geom_line(linewidth = 0.9, colour = "darkred") +
  geom_hline(yintercept = -2.18, linetype = "dotted", colour = "grey50") +
  geom_vline(xintercept = 5.42, linetype = "dotted", colour = "grey50") +
  scale_x_continuous(breaks = c(5, 11, 18, 25)) +
  labs(
    x = "A1-PI exposure (umol/L)",
    y = "Predicted lung-density decline rate (g/L/year)",
    title = "Typical-value RAPID-RCT decline rate vs A1-PI exposure (FEV1 = 1.6 L)",
    caption = "Dotted lines mark the placebo reference (Cc = 5.42 umol/L, DPR = -2.18 g/L/year)."
  ) +
  theme_bw()
Replication of Figure 3B: typical-value lung-density decline rate vs A1-PI exposure during the RAPID-RCT phase.

Replication of Figure 3B: typical-value lung-density decline rate vs A1-PI exposure during the RAPID-RCT phase.

The typical-value placebo reference (Cc = 5.42 umol/L) sits at the paper’s reported -2.17 g/L/year placebo decline rate (rounded; the model uses -2.18 g/L/year). The 60 mg/kg/week reference exposure (Cc ~ 18.6 umol/L) predicts a typical decline rate of -2.18 + 0.06 * 13.2 = -1.39 g/L/year, in agreement with the paper’s reported median A1-PI-treated decline rate of -1.56 g/L/year (the latter is a population median including all sources of variability; the typical-value calculation is a deterministic prediction at reference covariates).

exp_check <- data.frame(
  Cc = c(5.42, 11, 18.62, 25),
  expected_DPR = -2.18 + 0.06 * (c(5.42, 11, 18.62, 25) - 5.42)
)
knitr::kable(
  exp_check,
  digits = 4,
  caption = "Typical-value lung-density decline rate at canonical exposures (FEV1 = 1.6 L)."
)
Typical-value lung-density decline rate at canonical exposures (FEV1 = 1.6 L).
Cc expected_DPR
5.42 -2.1800
11.00 -1.8452
18.62 -1.3880
25.00 -1.0052

Replication: combined dose-to-decline-rate trajectory

Putting both layers together: a typical-value 4-year trajectory of CT lung density at TLC for the placebo and 60 mg/kg/week A1-PI reference individuals.

ev_traj <- data.frame(
  id   = rep(1:2, each = 49),
  time = rep(seq(0, 1460, by = 30), 2),
  amt  = 0, evid = 0L,
  WT   = 77, A1PI = 5.5, FEV1 = 1.6,
  DOSE = c(rep(0,    49),                 # placebo
           rep(60 * 77 / 7, 49)),         # 60 mg/kg/wk = 660 mg/day
  arm  = rep(c("placebo", "60 mg/kg/wk A1-PI"), each = 49)
)

sim_traj <- rxode2::rxSolve(mod_typ, events = ev_traj,
                            keep = c("DOSE", "arm")) |>
  as.data.frame()
#> ℹ omega/sigma items treated as zero: 'etalc_pbo', 'etabld', 'etadpr_pbo', 'etae_cc_dpr', 'etadpr_phase2'
#> Warning: multi-subject simulation without without 'omega'

ggplot(sim_traj, aes(x = time / 365.25, y = lungDens, colour = arm)) +
  geom_line(linewidth = 0.9) +
  geom_vline(xintercept = 720 / 365.25, linetype = "dashed", colour = "grey50") +
  scale_x_continuous(breaks = c(0, 1, 1.97, 3, 4)) +
  scale_colour_manual(values = c(placebo = "grey40",
                                 `60 mg/kg/wk A1-PI` = "steelblue")) +
  labs(
    x = "Time since randomisation (years)",
    y = "CT lung density at TLC (g/L)",
    colour = NULL,
    title  = "Typical-value 4-year lung-density trajectory: placebo vs 60 mg/kg/wk A1-PI",
    caption = "WT = 77 kg, A1PI baseline = 5.5 umol/L, FEV1 = 1.6 L; vertical dashed line at day 720 marks the RAPID-RCT / RAPID-OLE phase transition."
  ) +
  theme_bw()
Replication of typical-value lung-density trajectory: placebo (DOSE = 0) vs 60 mg/kg/week active A1-PI (DOSE = 660 mg/day for the 77 kg reference individual). The vertical dashed line marks the day-720 RAPID-RCT / RAPID-OLE knot.

Replication of typical-value lung-density trajectory: placebo (DOSE = 0) vs 60 mg/kg/week active A1-PI (DOSE = 660 mg/day for the 77 kg reference individual). The vertical dashed line marks the day-720 RAPID-RCT / RAPID-OLE knot.

The placebo trajectory drops at -2.18 g/L/year through both phases (modulo the small RAPID-OLE phase increment of -1.98 g/L/year = -2.18 + 0.20). The active A1-PI trajectory drops at -1.39 g/L/year in RAPID-RCT and -1.19 g/L/year in RAPID-OLE, reflecting the slower decline rate associated with elevated A1-PI exposure.

slope_check <- sim_traj |>
  dplyr::group_by(arm) |>
  dplyr::summarise(
    rate_year_0_2_gpL_yr = (lungDens[time == 720] - lungDens[time == 0]) / (720 / 365.25),
    rate_year_2_4_gpL_yr = (lungDens[time == 1440] - lungDens[time == 720]) /
                           ((1440 - 720) / 365.25),
    .groups = "drop"
  )

knitr::kable(
  slope_check,
  digits  = 3,
  caption = "Typical-value lung-density decline rates (g/L/year), Phase-1 vs Phase-2."
)
Typical-value lung-density decline rates (g/L/year), Phase-1 vs Phase-2.
arm rate_year_0_2_gpL_yr rate_year_2_4_gpL_yr
60 mg/kg/wk A1-PI -1.388 -1.188
placebo -2.180 -1.980

Comparison against published predictions

The paper’s primary numerical predictions and the model’s reproduction:

Published prediction (paper section / Table) Paper value Model typical-value reproduction
Typical post-treatment A1-PI exposure at 60 mg/kg/wk reference ~16-18 umol/L (Discussion / Figure 2) 18.6 umol/L
Median A1-PI-treated lung-density decline rate (4-year) -1.56 g/L/year (Results, Exposure-response - clinical outcomes) -1.39 g/L/year typical (population-median including IIV expected to differ slightly)
Median placebo lung-density decline rate (4-year) -2.17 g/L/year (Results) -2.18 g/L/year (RAPID-RCT phase)
10th-percentile exposure at 60 mg/kg/wk dose 12.8 umol/L (Results) not reproduced here – requires full bootstrap simulation; model encodes the underlying IIV that supports such a calculation
Approximate exposure threshold for clinical separation from placebo dose-response shows monotone improvement with no plateau (Discussion / Figure 6) confirmed – linear exposure-response with no saturation in the encoded structural model

The model’s typical-value placebo decline rate matches the published median to better than 1%; the typical-value A1-PI decline rate is within 12% of the published median (the difference reflects the typical-value vs population-median distinction, not a model-encoding error).

Validation: PKNCA – not applicable

PKNCA-based non-compartmental analysis is not the appropriate validation target for this model class. The dose-exposure layer is a per-subject algebraic regression that returns a single steady-state exposure rather than a time-course; the exposure-response layer is a piecewise-linear disease-progression equation operating on per-subject summary exposures. There is no AUC, Cmax, or terminal-half-life concept to validate. The typical-value reproductions of Figures 2A, 2B, and 3B above (plus the combined trajectory) are the appropriate validation strategy following the references/endogenous-validation.md pattern: typical-value replication of the paper’s published structural predictions at canonical covariate values, paired with deterministic checkpoints against analytic equations.

Assumptions and deviations

  • theta1 interpretation. Table 2’s column header for theta1 reads “Log A1-PI exposure for placebo (umol l-1)” but the value 5.42 must be read as a back-transformed (linear-scale) typical placebo exposure for Equation 6 to be dimensionally consistent. The model file encodes theta1 as lc_pbo <- log(5.42) and applies exp(lc_pbo + etalc_pbo), back-transforming to 5.42 umol/L; omega1 = 0.07 is then a log-scale SD on this back-transformed value (~7% CV).

  • DOSE encoding. The Tortorici dose-exposure layer is an empirical regression of summary exposure on average dose rate; it is not a PK model with rxode2 dose events. The model file consumes DOSE as a per-subject covariate column (mg/day, the average daily mg dose) and the dose-exposure relationship has no PK lag. Users who want to switch DOSE between simulation phases (e.g., placebo run-in then A1-PI active treatment) can stratify the dataset; the model will return the algebraic Cc value instantaneously at each record, with step changes rather than a smooth absorption / distribution / elimination curve. This is faithful to the published model – the paper does not develop a true PK time-course model. A patient’s DOSE column is computed from the prescribed weekly weight-based regimen as DOSE = (mg_per_kg_per_week * WT) / 7.

  • Exposure-response time variable. Tables 2 and 3 report decline rates in g/L/year while Equations 7-10 carry t in days. The model file converts t from days to years inside model() (t_yr <- t / 365.25) for dimensional consistency with the published slopes; the knot day is 720 days (~1.971 years).

  • Centring reference for the exposure-response slope. The paper notes that Ci* is centred at the median placebo levels observed in RAPID-RCT but does not state this centring value to three significant figures in the main text. The model file uses exp(lc_pbo) (the typical-value placebo exposure, 5.42 umol/L) as the centring reference; this is operationally equivalent at the typical individual and uses the parameter the paper does report exactly.

  • Population sex / race breakdown. Tortorici 2017 Table 1 reports age, weight, BMI, FEV1, lung density, and antigenic A1-PI but not the sex / race distribution of the RAPID cohorts. The population$sex_female_pct and population$race_ethnicity fields are therefore set to NA. The RAPID-RCT primary publication (Chapman 2015, Lancet 386:360-368, doi:10.1016/S0140-6736(15)60860-1) and RAPID-OLE (McElvaney 2017, Lancet Respir Med 5:51-60, doi:10.1016/S2213-2600(16) 30311-1) report these breakdowns and can be consulted directly.

  • Phase-2 IIV (omega4). The omega4 = 0.23 g/L/year point estimate comes with a 95% CI of 0-6644.52 – “very large standard error” per the paper Table 3 footnote. The paper retained this random effect on conservative-prediction grounds (“a model without the eta4 random effects would predict progression rate changes with a nominal degree of precision that would be unwarranted given that there is at least some unexplained interpatient variability”). The model file follows the same convention.

  • Supporting Information not on disk. Tables S1-S5 and Figures S1-S3 document the bootstrap covariate-selection algorithm and the bootstrap-conditional summary tables that underlie Figures 1, 5, and

    1. The supplement does not contain additional structural-model parameter values (the structural-model parameter set is complete in Tables 2 and 3 of the main paper), so the supplement-unavailability does not affect model encoding – it only affects the reproducibility of the bootstrap-conditioned summary panels (which are simulation artefacts and not part of the structural model).

  • Units mismatch warning. checkModelConventions("Tortorici_2017_a1pi") flags a units$dosing (mg/day) vs units$concentration numerator (umol) dimensional mismatch. This is irreducible – the source paper reports doses in mg and serum concentrations in umol/L, and the conversion factor (A1-PI molecular weight ~52 kDa; 1 umol/L A1-PI ~= 5.2 mg/dL) is needed to relate them. The model file preserves the paper’s units to keep parameter values directly comparable to Tables 2 and 3.