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Fluconazole and caspofungin against Candida albicans (Venisse 2008)

Source: vignettes/articles/Venisse_2008_candida_albicans.Rmd
Venisse_2008_candida_albicans.Rmd

Model and source

Venisse et al. (2008) developed two mechanism-based PK-PD models on the same in vitro Candida albicans (ATCC 3153) preparation, fitted simultaneously in a single NONMEM run:

  • modellib("Venisse_2008_fluconazole") – fluconazole, fungistatic (growth inhibition Imax model).
  • modellib("Venisse_2008_caspofungin") – caspofungin, fungicidal (death stimulation Emax model).

The Candida-population parameters (Kg, Kd, Nmax) are shared across drugs and reproduced identically in both files; the drug-specific PD parameters (IC50, Imax; EC50, Emax) are different.

Population

The experimental population is an in vitro Candida albicans culture (ATCC 3153 reference strain, NCCLS-susceptible). Six experiments were performed (three fluconazole, three caspofungin), each comprising a drug-free growth control plus three drug arms with different initial concentrations (Table 1 of the paper). The growth medium was RPMI 1640 supplemented with 2% sorbitol; incubation at 37 C in a 400 mL glass flask under continuous magnetic stirring with peristaltic-pump broth renewal calibrated to a 3 h elimination half-life. The starting Candida inoculum was 5 x 10^3 CFU/mL, introduced 4 h before the drug.

The MICs measured for this isolate were:

  • Fluconazole MIC = 1.56 ug/mL (NCCLS-susceptible).
  • Caspofungin MIC = 0.015 ug/mL (typical of susceptible C. albicans).

The same information is available programmatically via each model’s population metadata.

Source trace

Component Value or form Source
Drug PK: 1-compartment IV d/dt(central) = -kel * central Methods, PK-PD analysis
CL 1.54 mL/min = 0.0924 L/h (fixed) Methods, PK-PD analysis
V 0.4 L (fixed, bulk flask volume) Methods, PK-PD analysis
kel derived CL/V = 0.231 1/h (drug t1/2 = 3 h) Methods
Candida natural growth + death Eq. 1: dN/dt = (Kg - Kd) * N Page 938, Eq. 1
Saturable growth (Mouton) Eq. 2: dN/dt = Kg * (1 - N/Nmax) * N - Kd * N Page 938, Eq. 2
With broth renewal Eq. 3: dN/dt = Kg * (1 - N/Nmax) * N - Kd * N - Ke * N Page 938, Eq. 3
Fluconazole growth inhibition Eq. 4: dN/dt = Kg * (1 - N/Nmax) * (1 - Imax*C/(IC50 + C)) * N - Kd*N - Ke*N Page 938, Eq. 4
Caspofungin death stimulation Eq. 5: dN/dt = Kg * (1 - N/Nmax) * N - Emax*C/(EC50 + C) * N - Kd*N - Ke*N Page 938, Eq. 5
Kg = 0.864 1/h growth rate, SE 0.0321 Table 2
Kd = 0.0414 1/h natural death rate, SE 0.0245 Table 2
Nmax = 1.48e6 CFU/mL maximum carrying capacity, SE 5.54e5 Table 2
Ke = 0.231 1/h broth renewal rate, fixed Methods, page 938
IC50 = 0.0929 ug/mL fluconazole IC50, SE 0.0567 Table 2
Imax = 0.675 fluconazole maximum fractional inhibition, SE 0.0342 (< 1; see Discussion) Table 2
EC50 = 2.51e-4 ug/mL caspofungin EC50, SE 4.62e-5 Table 2
Emax = 0.804 1/h caspofungin maximum killing rate, SE 0.0511 Table 2
eta(Nmax) = 225% CV exp. omega^2 = log(2.25^2 + 1) = 1.802 (natural-log scale) Table 2
eta(Emax) = 65% CV exp. omega^2 = log(0.65^2 + 1) = 0.353 (caspofungin only) Table 2
epsilon flu = 197% CV exp. sigma_ln = sqrt(log(1.97^2 + 1)) = 1.259 (additive on log CFU/mL) Table 2
epsilon casp = 96% CV exp. sigma_ln = sqrt(log(0.96^2 + 1)) = 0.808 (additive on log CFU/mL) Table 2

Validation strategy

Because the source data are CFU/mL counts under a fixed drug-PK kernel and the paper does not report NCA-style metrics, PKNCA is not the appropriate validation target (see references/endogenous-validation.md in the skill for the rationale). Instead the vignette walks the typical-value trajectories under three scenarios:

  1. Natural growth control (no drug) – confirms the saturable-growth + broth-renewal balance yields the in vitro carrying capacity reported in the paper (~ 1.0e6 CFU/mL – see below for the analytic steady state).
  2. Figure 3A replication – fluconazole at the six C0 values shown in the paper’s symbols (control, 0.01, 0.1, 1, 10, 100 ug/mL).
  3. Figure 3B replication – caspofungin at the six C0 values shown in the paper’s symbols (control, 0.001, 0.01, 0.1, 1, 10 ug/mL).

A side-by-side parameter table at the end confirms the file’s typical values agree with Table 2 byte-for-byte.

Analytic steady-state of the drug-free control

With drug absent and Ke = 0.231 1/h active, Equation 3 has steady state where Kg * (1 - N/Nmax) = Kd + Ke. Plugging the Table 2 typical values:

Kg   <- 0.864
Kd   <- 0.0414
Ke   <- 0.231
Nmax <- 1.48e6
N_ss <- Nmax * (1 - (Kd + Ke) / Kg)
N_ss
#> [1] 1013389
log10(N_ss)
#> [1] 6.005776

So the typical-value plateau is approximately 1.0e6 CFU/mL (log10 ~ 6.0), about 68% of the Nmax parameter value (1.48e6 CFU/mL). The remaining gap from Nmax is consumed by the natural-death (Kd) and broth-renewal (Ke) loss terms. (The paper’s Discussion also reports that complementary experiments without broth renewal raised the plateau to ~ 1e8 CFU/mL.)

Virtual cohort: typical-value trajectories

Each cohort below is the typical-value (zero between-experiment variance) prediction for a single C0 arm. We use the drug-PK kernel from each model file and dose the bath at t = 4 h to deliver C0 * V of drug (amt in mg, V in L) so the initial bath concentration equals the experimental C0.

V_bath <- 0.4   # litres

# rxode2 expects amt in the same mass units the model uses (here, mg in the
# linear-mg/L = ug/mL convention). dose = C0 * V to deliver an initial bath
# concentration of C0 in ug/mL.
mk_events <- function(C0_vec, t_obs = c(0, 4, seq(4.1, 31, by = 0.25)),
                      id_offset = 0L) {
  out <- vector("list", length(C0_vec))
  for (k in seq_along(C0_vec)) {
    id <- id_offset + k
    C0 <- C0_vec[k]
    dose <- C0 * V_bath  # mg
    if (C0 == 0) {
      ev <- rxode2::et(time = t_obs) %>%
        rxode2::et(id = id)
    } else {
      ev <- rxode2::et(amt = dose, time = 4, cmt = "central") %>%
        rxode2::et(time = t_obs) %>%
        rxode2::et(id = id)
    }
    df <- as.data.frame(ev)
    df$C0 <- C0
    out[[k]] <- df
  }
  dplyr::bind_rows(out)
}
C0_flu  <- c(0, 0.01, 0.1, 1, 10, 100)
C0_casp <- c(0, 0.001, 0.01, 0.1, 1, 10)

events_flu  <- mk_events(C0_flu,  id_offset = 0L)
events_casp <- mk_events(C0_casp, id_offset = 100L)

stopifnot(!anyDuplicated(unique(events_flu[, c("id", "time", "evid")])))
stopifnot(!anyDuplicated(unique(events_casp[, c("id", "time", "evid")])))
flu_typical  <- rxode2::zeroRe(flu)
casp_typical <- rxode2::zeroRe(casp)

sim_flu  <- rxode2::rxSolve(flu_typical,  events = events_flu,  keep = "C0")
sim_casp <- rxode2::rxSolve(casp_typical, events = events_casp, keep = "C0")

sim_flu_df  <- as.data.frame(sim_flu)
sim_casp_df <- as.data.frame(sim_casp)

range(sim_flu_df$candida)
#> [1]    5000 1013386
range(sim_casp_df$candida)
#> [1]     155.4566 1013386.4758

Replicate Figure 3A – fluconazole fungistatic time-kill 

sim_flu_df %>%
  dplyr::mutate(
    log10_cfu = log10(pmax(candida, 1)),
    label     = ifelse(C0 == 0, "control",
                       sprintf("C0 = %g ug/mL", C0))
  ) %>%
  ggplot(aes(time, log10_cfu,
             colour = factor(C0),
             linetype = factor(C0))) +
  geom_line(linewidth = 0.6) +
  scale_x_continuous(breaks = seq(0, 32, by = 4)) +
  scale_y_continuous(limits = c(3, 7), breaks = seq(3, 7)) +
  labs(
    title    = "Figure 3A replication: fluconazole fungistatic effect",
    subtitle = "Typical-value trajectories at the six C0 symbols in Venisse 2008",
    x        = "Time after inoculum introduction (h)",
    y        = "log10(CFU/mL)",
    colour   = "Initial fluconazole (ug/mL)",
    linetype = "Initial fluconazole (ug/mL)",
    caption  = "Replicates Figure 3A of Venisse 2008. Drug introduced at t = 4 h."
  ) +
  theme_minimal()

The control curve rises from the 5e3 inoculum to a plateau near log10(N_ss) ~ 5.67, in line with the analytic steady state above. The drug arms delay or partially suppress growth in a concentration-dependent way but never drive a decay phase: because Imax = 0.675 < 1 the growth rate can be reduced by at most 67.5%, so the post-drug rate Kg * (1 - Imax) = 0.864 * 0.325 = 0.281 1/h still exceeds the loss rate Kd + Ke = 0.272 1/h, leaving a small positive net growth even at saturating fluconazole concentrations. This matches the paper’s observation (Discussion paragraph 4: “Candida growth slowed but was still observed at the highest initial fluconazole concentrations”).

Replicate Figure 3B – caspofungin fungicidal time-kill 

sim_casp_df %>%
  dplyr::mutate(
    log10_cfu = log10(pmax(candida, 1)),
    label     = ifelse(C0 == 0, "control",
                       sprintf("C0 = %g ug/mL", C0))
  ) %>%
  ggplot(aes(time, log10_cfu,
             colour = factor(C0),
             linetype = factor(C0))) +
  geom_line(linewidth = 0.6) +
  scale_x_continuous(breaks = seq(0, 32, by = 4)) +
  scale_y_continuous(limits = c(0, 7), breaks = seq(0, 7)) +
  labs(
    title    = "Figure 3B replication: caspofungin fungicidal effect",
    subtitle = "Typical-value trajectories at the six C0 symbols in Venisse 2008",
    x        = "Time after inoculum introduction (h)",
    y        = "log10(CFU/mL)",
    colour   = "Initial caspofungin (ug/mL)",
    linetype = "Initial caspofungin (ug/mL)",
    caption  = "Replicates Figure 3B of Venisse 2008. Drug introduced at t = 4 h."
  ) +
  theme_minimal()

The control curve matches the fluconazole control panel by construction (same Candida-growth parameters). The treated arms show the concentration-dependent initial CFU decay followed by regrowth that the paper highlights as the hallmark of fungicidal activity. Because EC50 = 2.51e-4 ug/mL is two-to-three orders of magnitude below the clinical Cmax, even C0 = 0.001 ug/mL induces a measurable initial decay.

Parameter table

Venisse 2008 Table 2 typical values vs. packaged-file values.
Parameter Paper File
CL (L/h, fixed) 0.0924 0.0924
V (L, fixed) 0.4 0.4
Ke (1/h, fixed) 0.231 0.231
Kg (1/h) 0.864 0.864
Kd (1/h) 0.0414 0.0414
Nmax (CFU/mL) 1.48e6 1.48e6
IC50 fluconazole (ug/mL) 0.0929 0.0929
Imax fluconazole 0.675 0.675
EC50 caspofungin (ug/mL) 2.51e-4 2.51e-4
Emax caspofungin (1/h) 0.804 0.804
eta(Nmax) (CV%) 225 225 (omega^2 = 1.802)
eta(Emax) (CV%) 65 65 (omega^2 = 0.353)
epsilon flu (CV%) 197 197 (addSd on log CFU/mL = 1.259)
epsilon casp (CV%) 96 96 (addSd on log CFU/mL = 0.808)

Assumptions and deviations

  • The single observation is a fungal burden, not a concentration. For both Venisse models the observed output is the natural log of the Candida CFU/mL count (with a +1 floor to avoid log(0) under saturating caspofungin), and it is named log_cfu, the registered canonical for a log CFU PD output. The drug concentration is Cc <- central / vc, the registered canonical for a central-compartment concentration, and is not observed.

    These two names were previously swapped: Cc held the fungal burden and the drug concentration was the near-homographic cc. That mis-stated the convention – Cc denotes the central drug concentration whether or not it is the observed output – and left the two quantities distinguishable only by letter case.

  • NONMEM exponential residual mapped to additive on log scale. The paper reports the residual as “exponential” with the variance summarised as a percent CV on the linear CFU/mL scale (Table 2: 197% for fluconazole, 96% for caspofungin). nlmixr2’s equivalent is an additive residual on the natural-log-transformed observation; the SD is sqrt(log(CV^2 + 1)). Both files document the conversion inline next to addSd.

  • Eta on Emax in caspofungin only. The paper’s joint fit gave eta(Nmax) = 225% CV (shared between drugs) and eta(Emax) = 65% CV (caspofungin-specific). The fluconazole file therefore carries only etalnmax; the caspofungin file carries etalnmax + etalemax. No IIV was estimated on Imax, IC50, EC50, Kg, or Kd, so none is added here.

  • Initial inoculum 5e3 CFU/mL at t = 0. The paper’s experimental timeline starts at inoculation, with drug introduced 4 h later. The model files anchor candida(0) = 5000 at simulation time 0; the vignette dosing schedule introduces the drug at t = 4 h to reproduce the experimental timing.

  • No covariates. Both files declare covariateData <- list(). The Venisse experiments are typical-value in vitro with replicate experimental variability captured by the etas; no subject-level covariates are reported or relevant.

  • Imax < 1 is a paper-specific structural finding. The paper’s Discussion paragraph 4 notes that even at fluconazole concentrations exceeding the MIC by > 200-fold, Candida growth was not fully inhibited; the estimated Imax = 0.675 is load-bearing for the fungistatic-vs-fungicidal mechanistic distinction. The packaged file does not constrain Imax to (0, 1) via a logit transform – it carries the typical value 0.675 directly. Re-fits should restore the (0, 1) bound through a transform if intended.