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Oxazolidinones in vitro time-kill (Schmidt 2009)

Source: vignettes/articles/Schmidt_2009_oxazolidinones.Rmd
Schmidt_2009_oxazolidinones.Rmd

Model and source

Schmidt et al. (2009) developed a susceptibility-based mechanism-based PK/PD model for the in vitro time-kill activity of two oxazolidinones – the investigational drug RWJ-416457 and the FDA-approved first-in-class representative linezolid – against methicillin-resistant Staphylococcus aureus (MRSA strain OC2878). The same structural model and the same set of variance parameters were fit jointly to both drugs; only the half-maximal effect concentration (EC50) and the broth degradation rate differ between drugs. This is encoded as two model files that share every other parameter:

  • modellib("Schmidt_2009_rwj416457") – RWJ-416457 (EC50 0.41 ug/mL; ~10% degraded over 24 h)

  • modellib("Schmidt_2009_linezolid") – linezolid (EC50 1.39 ug/mL; stable over 24 h)

  • Article: https://doi.org/10.1128/AAC.00633-09

Study system

The fit used two parallel in vitro models:

  1. Static time-kill curves – 50 mL cell-culture flasks were filled with 20 mL Mueller-Hinton broth at an inoculum of ~5e5 CFU/mL of MRSA strain OC2878. After a 2 h preincubation each flask was spiked with antibiotic at a constant concentration ranging from 0.25 to 16x MIC, and viable counts were sampled over 24 h. The MIC for RWJ-416457 was 0.5 ug/mL and for linezolid 1.0 ug/mL.

  2. Dynamic (changing-concentration) time-kill curves – the same flask system coupled to a syringe that periodically replaced a small volume of antibiotic-containing broth with fresh antibiotic-free medium, simulating a first-order elimination half-life. RWJ-416457 syringes replaced 2.2 mL every 4 h to mimic the human elimination t1/2 of ~24 h; linezolid syringes replaced 4.8 mL every 2 h to mimic t1/2 ~ 5 h.

Each static and each dynamic experiment was run in triplicate per drug per concentration. The data are antibiotic-free growth-control counts plus 7 antibiotic-exposed cohorts (0.25, 0.5, 1, 2, 4, 8, 16x MIC), fit simultaneously across both drugs.

The same information is available programmatically via readModelDb("Schmidt_2009_rwj416457") and readModelDb("Schmidt_2009_linezolid") (the population metadata of either file).

Mechanism

A susceptibility-based two-subpopulation model (Schmidt 2009 Fig. 1) splits the bacterial population into:

  • bact_susceptible (Ns) – self-replicating, antibiotic-susceptible cells.
  • bact_persister (Np) – dormant, antibiotic-insusceptible cells.

Bacteria can transition between pools via the rate constants ksp (S -> P) and kps (P -> S; held fixed at 0 because the data did not identify a back-conversion rate). Both pools experience first-order natural death at rate kd. Only the susceptible pool grows (with a logistic carrying-capacity limit at nmax) and only the susceptible pool is killed by the drug (Emax kill with kmax and ec50). Two delay-onset terms (1 - exp(-dgs*t)) and (1 - exp(-dks*t)) turn growth and killing on smoothly from 0 to 1 over the first ~hours of each experiment.

The full system implemented in model():

dNsdt=ks(1−Ns+NpNmax)(1−e−dgst)Ns−kdNs−kspNs+kpsNp−kmaxCEC50+C(1−e−dkst)NsdNpdt=kspNs−kpsNp−kdNpdCdt=−kdegCNs(0)=FN0Np(0)=(1−F)N0 \begin{aligned} \dfrac{dN_s}{dt} &= k_s \left(1 - \dfrac{N_s+N_p}{N_{\max}}\right) (1 - e^{-d_{gs}t})\,N_s - k_d\,N_s - k_{sp}\,N_s + k_{ps}\,N_p - \dfrac{k_{\max}\,C}{EC_{50}+C}(1-e^{-d_{ks}t})\,N_s \\ \dfrac{dN_p}{dt} &= k_{sp}\,N_s - k_{ps}\,N_p - k_d\,N_p \\ \dfrac{dC}{dt} &= -k_{deg}\,C \\ N_s(0) &= F\,N_0 \qquad N_p(0) = (1-F)\,N_0 \end{aligned}

The observation is Cc = log10(Ns + Np). The residual model is additive on log10 CFU/mL (see Assumptions and deviations for the natural-log-vs-log10 conversion that produced addSd = 0.234).

Source trace

The per-parameter origin is recorded as an in-file comment next to each ini() entry in inst/modeldb/specificDrugs/Schmidt_2009_rwj416457.R and inst/modeldb/specificDrugs/Schmidt_2009_linezolid.R. The table below collects the structural and variance estimates in one place. All values are from Table 1 of the paper unless otherwise noted; the bootstrap column matches sqrt(omega^2) for each of the four IIVs and the residual SIGMA.

Equation / parameter Value Source location
lks log(1.19) Table 1, structural (ks)
lkmax log(1.65) Table 1, structural (kmax)
lec50 (RWJ-416457) log(0.41) Table 1, EC50 RWJ-416457
lec50 (linezolid) log(1.39) Table 1, EC50 linezolid
lnmax fixed(log(3.39e9)) Table 1, Nmax (FIXED)
ldgs fixed(log(0.24)) Table 1, dgs (FIXED from growth control)
ldks log(0.50) Table 1, dks
lksp log(0.004) Table 1, ksp
kps fixed(0) Table 1, kps (FIXED to 0; ref 22)
lkd fixed(log(0.015)) Table 1, kd (FIXED from growth control)
logitf logit(0.83) Table 1, F
kdeg RWJ-416457 fixed(0.00439) Results ‘Drug stability’ (~10% loss over 24 h)
kdeg linezolid fixed(0) Results ‘Drug stability’ (stable)
lninit fixed(log(5e5)) Methods ‘Organisms’
etalks omega^2 = 0.013 Table 1, eta(ks)
etalnmax omega^2 = 0.219 Table 1, eta(Nmax)
etaldgs omega^2 = 0.184 Table 1, eta(dgs)
etaldks omega^2 = 0.079 Table 1, eta(dks)
addSd 0.234 (log10 SD) Table 1, residual variability (0.29 nat-log variance; see Assumptions)
Equation for d/dt(bact_susceptible) Eq 2 (text-described) Materials and Methods ‘Mathematical modeling’
Equation for d/dt(bact_persister) Eq 3 (text-described) Materials and Methods ‘Mathematical modeling’
Equation for d/dt(central) first-order decay Materials and Methods ‘Mathematical modeling’ final paragraph

Setup 

# Run a static (constant-concentration) time-kill simulation: dose the
# initial antibiotic concentration into `central` at t = 0, then read out
# bacterial Cc on a dense grid for 24 h. zeroRe() removes between-
# experiment IIV so we plot typical-value (population mean) curves that
# can be compared directly against Schmidt 2009 Fig. 2 fits.
sim_one <- function(mod_name, dose_ug_per_ml, kdeg_override = NULL,
                    grid_h = seq(0, 24, by = 0.25)) {
  mod_typ <- rxode2::zeroRe(rxode2::rxode2(readModelDb(mod_name)))
  params <- if (is.null(kdeg_override)) NULL else c(kdeg = kdeg_override)
  ev <- rxode2::et(amt = dose_ug_per_ml, cmt = "central", time = 0) |>
    rxode2::et(grid_h)
  sim <- rxode2::rxSolve(mod_typ, ev, params = params)
  as.data.frame(sim)
}

# Schmidt 2009 used MIC multipliers in {0.25, 0.5, 1, 2, 4, 8, 16}; the
# growth control is no antibiotic.
mic_multipliers <- c(0.25, 0.5, 1, 2, 4, 8, 16)
mic_rwj         <- 0.5    # ug/mL (Results 'MICs')
mic_lzd         <- 1.0    # ug/mL (Results 'MICs')

# Dilution-equivalent first-order elimination rates for the dynamic
# (syringe-replacement) experiments. RWJ-416457 syringes replaced 2.2 mL
# every 4 h (fraction remaining 17.8/20 = 0.89 per step ->
# kelim_dyn = -log(0.89)/4 = 0.0291 1/h, t1/2 ~ 24 h matching the paper's
# stated human t1/2). Linezolid syringes replaced 4.8 mL every 2 h
# (fraction remaining 15.2/20 = 0.76 per step ->
# kelim_dyn = -log(0.76)/2 = 0.137 1/h, t1/2 ~ 5 h).
kdeg_static_rwj  <- 0.00439                 # paper-published degradation
kdeg_dynamic_rwj <- 0.00439 + 0.0291        # degradation + dilution
kdeg_static_lzd  <- 0                       # stable
kdeg_dynamic_lzd <- 0.137                   # dilution only

Validation 1 – antibiotic-free growth control reaches Nmax

In the absence of drug the model must approach the published carrying capacity Nmax = 3.39e9 (log10 9.53). This is the equivalent of the ‘steady-state hold’ validation pattern for endogenous models – the final plateau Cc is a structural target.

gc <- sim_one("Schmidt_2009_rwj416457", dose_ug_per_ml = 0)
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
plateau_target <- log10(3.39e9)
plateau_sim    <- tail(gc$Cc, 1L)
cat(sprintf("Growth-control plateau Cc at 24 h: simulated = %.3f, target = %.3f\n",
            plateau_sim, plateau_target))
#> Growth-control plateau Cc at 24 h: simulated = 9.524, target = 9.530

ggplot(gc, aes(time, Cc)) +
  geom_line(linewidth = 1) +
  geom_hline(yintercept = plateau_target, linetype = 2, colour = "red") +
  labs(x = "Time (h)", y = "log10 CFU/mL",
       title = "Growth control reaches Nmax (red dashed = log10(3.39e9))")

Validation 2 – replicate Schmidt 2009 Figure 2 (static time-kill)

Each of the 7 concentration cohorts is simulated at its initial concentration; the antibiotic state declines first-order with the drug-specific kdeg. The model curves are the typical-value predictions of the published joint fit.

panel_A <- bind_rows(lapply(mic_multipliers, function(m) {
  sim_one("Schmidt_2009_rwj416457", dose_ug_per_ml = m * mic_rwj) |>
    mutate(mic_multiple = m, drug = "RWJ-416457")
}))
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
panel_A_gc <- gc |> mutate(mic_multiple = 0, drug = "RWJ-416457")
panel_A_all <- bind_rows(panel_A, panel_A_gc)

ggplot(panel_A_all, aes(time, Cc, colour = factor(mic_multiple), group = mic_multiple)) +
  geom_line(linewidth = 0.9) +
  scale_colour_viridis_d(name = "x MIC") +
  labs(x = "Time (h)", y = "log10 CFU/mL",
       title = "Figure 2A -- static time-kill, RWJ-416457",
       subtitle = "Lines = packaged-model typical predictions (replicates Schmidt 2009 Fig. 2A model fits)")

panel_C <- bind_rows(lapply(mic_multipliers, function(m) {
  sim_one("Schmidt_2009_linezolid", dose_ug_per_ml = m * mic_lzd) |>
    mutate(mic_multiple = m, drug = "linezolid")
}))
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
panel_C_gc <- sim_one("Schmidt_2009_linezolid", dose_ug_per_ml = 0) |>
  mutate(mic_multiple = 0, drug = "linezolid")
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
panel_C_all <- bind_rows(panel_C, panel_C_gc)

ggplot(panel_C_all, aes(time, Cc, colour = factor(mic_multiple), group = mic_multiple)) +
  geom_line(linewidth = 0.9) +
  scale_colour_viridis_d(name = "x MIC") +
  labs(x = "Time (h)", y = "log10 CFU/mL",
       title = "Figure 2C -- static time-kill, linezolid",
       subtitle = "Lines = packaged-model typical predictions (replicates Schmidt 2009 Fig. 2C model fits)")

Validation 3 – replicate Schmidt 2009 Figure 2 (dynamic time-kill)

For dynamic experiments the user supplies the dilution-equivalent elimination rate via rxSolve(..., params = c(kdeg = ...)). The helper sim_one accepts kdeg_override for this purpose; the static sibling above uses the in-file kdeg.

panel_B <- bind_rows(lapply(mic_multipliers, function(m) {
  sim_one("Schmidt_2009_rwj416457", dose_ug_per_ml = m * mic_rwj,
          kdeg_override = kdeg_dynamic_rwj) |>
    mutate(mic_multiple = m, drug = "RWJ-416457 (dynamic)")
}))
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
panel_B_all <- bind_rows(panel_B,
                         panel_A_gc |> mutate(drug = "RWJ-416457 (dynamic)"))

ggplot(panel_B_all, aes(time, Cc, colour = factor(mic_multiple), group = mic_multiple)) +
  geom_line(linewidth = 0.9) +
  scale_colour_viridis_d(name = "x MIC") +
  labs(x = "Time (h)", y = "log10 CFU/mL",
       title = "Figure 2B -- dynamic time-kill, RWJ-416457 (t1/2 ~ 24 h)",
       subtitle = "Lines = packaged-model typical predictions; kdeg = paper degradation + syringe-dilution")

panel_D <- bind_rows(lapply(mic_multipliers, function(m) {
  sim_one("Schmidt_2009_linezolid", dose_ug_per_ml = m * mic_lzd,
          kdeg_override = kdeg_dynamic_lzd) |>
    mutate(mic_multiple = m, drug = "linezolid (dynamic)")
}))
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
#> ℹ parameter labels from comments will be replaced by 'label()'
#> ℹ omega/sigma items treated as zero: 'etalks', 'etalnmax', 'etaldgs', 'etaldks'
panel_D_all <- bind_rows(panel_D,
                         panel_C_gc |> mutate(drug = "linezolid (dynamic)"))

ggplot(panel_D_all, aes(time, Cc, colour = factor(mic_multiple), group = mic_multiple)) +
  geom_line(linewidth = 0.9) +
  scale_colour_viridis_d(name = "x MIC") +
  labs(x = "Time (h)", y = "log10 CFU/mL",
       title = "Figure 2D -- dynamic time-kill, linezolid (t1/2 ~ 5 h)",
       subtitle = "Lines = packaged-model typical predictions; kdeg = syringe-dilution only (linezolid stable)")

Comparison against published numbers

The paper reports two head-to-head numerical comparisons that the packaged model can be checked against.

Potency comparison (EC50). Schmidt 2009 Discussion: “RWJ-416457 (EC50, 0.41 ug/mL) is approximately 3.4-fold more potent than the first-in-class representative, linezolid (EC50, 1.39 ug/mL).” The encoded ratio is identical (the EC50 values are the only structural difference between the two model files):

ec50_rwj <- 0.41
ec50_lzd <- 1.39
cat(sprintf("Linezolid / RWJ-416457 EC50 ratio: %.2f (paper: 3.4)\n",
            ec50_lzd / ec50_rwj))
#> Linezolid / RWJ-416457 EC50 ratio: 3.39 (paper: 3.4)

Drug stability. Schmidt 2009 Results ‘Drug stability’: “approximately 10% of the RWJ-416457 degraded over the 24-h time course of the experiment” (linezolid was stable). At the in-file kdeg:

loss_rwj <- 1 - exp(-kdeg_static_rwj * 24)
loss_lzd <- 1 - exp(-kdeg_static_lzd * 24)
cat(sprintf("RWJ-416457 24 h fractional loss in static experiment: %.3f (paper: ~0.10)\n", loss_rwj))
#> RWJ-416457 24 h fractional loss in static experiment: 0.100 (paper: ~0.10)
cat(sprintf("Linezolid  24 h fractional loss in static experiment: %.3f (paper: 0)\n",     loss_lzd))
#> Linezolid  24 h fractional loss in static experiment: 0.000 (paper: 0)

Dynamic-experiment half-life. Schmidt 2009 Methods ‘Dynamic time-kill curves’ states syringe replacement rates equivalent to a human t1/2 of ~24 h (RWJ-416457) and ~5 h (linezolid):

t12_rwj <- log(2) / kdeg_dynamic_rwj
t12_lzd <- log(2) / kdeg_dynamic_lzd
cat(sprintf("RWJ-416457 dynamic t1/2: %.1f h (paper: ~24 h)\n", t12_rwj))
#> RWJ-416457 dynamic t1/2: 20.7 h (paper: ~24 h)
cat(sprintf("Linezolid  dynamic t1/2: %.1f h (paper: ~5 h)\n", t12_lzd))
#> Linezolid  dynamic t1/2: 5.1 h (paper: ~5 h)

Assumptions and deviations

  • F is the susceptible-pool fraction at drug-addition time. The paper text defines F only as “the initial fraction of bacteria in the susceptible or persister stage” without specifying which pool. The packaged model encodes bact_susceptible(0) = F * N0 and bact_persister(0) = (1 - F) * N0. With F = 0.83 (Table 1) and kps = 0 (Table 1, fixed), the persister pool can only deplete by the slow natural-death rate kd = 0.015 1/h, giving a structural lower bound on total Cc at 24 h of roughly log10(85000 * exp(-0.36)) = log10(59300) = 4.77. The Results paragraph ‘Static time-kill curves’ describes “an ~2- to 2.5-log reduction in bacterial counts … at concentrations greater than 8x MIC” – consistent with the susceptible-pool reduction of more than 5 logs but slightly larger than the structurally bounded total-count reduction (~1 log to the persister floor). The packaged model reproduces the published Table 1 parameter values exactly; any prose-vs-fit discrepancy was not tuned away.

  • Residual error log base. The paper’s Data-analysis paragraph states the simultaneous fit used a “log error model” on log-transformed bacterial counts but does not name the log base. NONMEM’s LOG() is natural log, so SIGMA = 0.29 (Table 1) is interpreted as a variance on the natural-log scale (sqrt(0.29) = 0.539 natural-log SD). The packaged model reports Cc in log10 CFU/mL for microbiology readability; the natural-log SD is rescaled by 1/log(10) to give the encoded addSd = 0.234 (log10 CFU/mL SD). If the published value is instead a variance on the log10 scale, the corresponding addSd would be sqrt(0.29) = 0.539 log10 – a factor of 2.3 larger.

  • kdeg is on the linear (non-log-transformed) scale. Standard nlmixr2lib convention is lkdeg (log-transformed), but linezolid’s published value is exactly 0 (linezolid was stable over 24 h) which has no finite log. To keep the two sibling files structurally identical, both use the linear-scale form kdeg <- fixed(<value>).

  • Dynamic-experiment elimination is supplied at simulation time. The paper handled drug elimination in dynamic time-kill experiments as an effective first-order rate matching the observed syringe-replacement schedule. The packaged model’s kdeg defaults to the static (degradation-only) rate; for dynamic experiments the user overrides via rxSolve(mod, ev, params = c(kdeg = <total>)), with the paper-published combined rates shown in the vignette setup chunk.

  • Initial inoculum is treated as a fixed model parameter. The paper describes the experimental inoculum as “approximately 5e5 CFU/mL” (Methods ‘Organisms’) without an estimated value. The packaged model encodes lninit <- fixed(log(5e5)) so the simulated time-zero count matches the experimental design. The user can override via the params argument if simulating an experiment with a different inoculum.

  • No in vivo PK model. The Discussion suggests combining the time-kill parameters with in vivo PK data to predict clinical outcomes. The packaged files contain only the in vitro PD; users wishing to drive the PD with patient-level PK should provide concentrations through the central state (e.g. via time-varying covariates or by adding their own PK ODEs in a derived rxode2 model).