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Linezolid + meropenem + vancomycin time-kill (Wicha 2017)

Source: vignettes/articles/Wicha_2017_linezolid_meropenem_vancomycin.Rmd
Wicha_2017_linezolid_meropenem_vancomycin.Rmd

Model and source

  • Citation: Wicha SG, Huisinga W, Kloft C. Translational pharmacometric evaluation of typical antibiotic broad-spectrum combination therapies against Staphylococcus aureus exploiting in vitro information. CPT Pharmacometrics Syst Pharmacol. 2017;6(8):512-522. doi:10.1002/psp4.12197.
  • Description: In vitro (MSSA ATCC 29213). Semimechanistic time-kill pharmacodynamic model of linezolid, meropenem, and vancomycin against methicillin-susceptible Staphylococcus aureus. Bacterial life cycle has three states: growing (gro), replicating (repl), and persisting (pers). LZD inhibits the GRO->REP transition (bacteriostatic via krep) and induces a replication-independent killing rate kdeath_lzd on growing bacteria. MER and VAN, as cell wall-active antibiotics, impair successful doubling at the REP->GRO transition; the joint MER+VAN action is encoded as a modified Bliss-independence term that includes the paradoxical Eagle-effect self-inhibition of MER at high concentrations and the VAN Emax cap. Drug-unsusceptible persisters are generated during replication at rates kper_mer * E_MER and kper_van * E_VAN, then die at kdeath_per. An adaptive-resistance submodel (Tam 2005) inflates the effective EC50 of MER and of VAN over time via fractional ARon states; subinhibitory VAN concentrations inhibit the MER-adaption rate (monodirectional VAN-on-MER PD interaction). MER and VAN solution concentrations decay first-order due to chemical degradation in growth medium (rates fixed from HPLC measurement); LZD is stable. The model is in-vitro PD only – there is no human PK component; drug exposures are static dosing at t = 0. Random effects (eta) are NOT present: the paper reports replicate-only experimental variability and uses an additive residual error on log10(CFU/mL).
  • Article: https://doi.org/10.1002/psp4.12197

Population

The packaged model was fit to a 24-hour in-vitro time-kill experiment using methicillin-susceptible Staphylococcus aureus (MSSA) reference strain ATCC 29213. The strain was exposed to static, replicate-controlled concentrations of linezolid (LZD, 0.5 to 32 mg/L), meropenem (MER, 0.015 to 8 mg/L), and vancomycin (VAN, 0.06 to 16 mg/L) alone and in selected dual combinations of LZD with MER and VAN with MER. n = 1,617 timed CFU/mL data points were available for model building. MIC values for the reference strain were LZD = 2.0 mg/L, MER = 0.125 mg/L, VAN = 1.0 mg/L (Wicha 2017 Methods, page 2-3). The model was externally evaluated against two clinical MSSA isolates (MV13391, MV13488) and against published time-kill datasets for LZD + VAN against methicillin-resistant S. aureus, penicillin + erythromycin against S. pneumoniae, and ampicillin + chloramphenicol against group B streptococci.

There is no human or animal cohort: this is an in-vitro model with no subject-level covariates. Replicate-to-replicate variability was small and the published model carries no IIV / inter-experiment etas; the residual is an additive standard deviation on log10 CFU/mL (Wicha 2017 Table 1 footnote). The complete population metadata is available programmatically via readModelDb("Wicha_2017_linezolid_meropenem_vancomycin")$population.

Source trace

The semimechanistic PD model couples three bacterial states (growing gro, replicating repl, persisting pers), three drug solution states (lzd, mer, van) with first-order chemical degradation for MER and VAN, and four adaption-resistance fractional states (aroff_mer, aron_mer, aroff_van, aron_van). The published governing equations are Wicha 2017 Eqs. 5-14 (page 4-5). Eq. 8 (the GRO ODE) is the load-bearing combination form:

d/dt(GRO) = - kdeath,LZD * E_LZD * GRO
            - krep(t, CFU) * (1 - E_LZD) * GRO
            + kdoub * [1 - E_MER * (1 - Emax_MER,Eagle * E_MER,Eagle) * (1 - E_VAN)]
                    * (1 - Emax_VAN * E_VAN) * REP * 2

The nested (1 - E_VAN) gate inside the inner bracket is what implements the paper’s observation that the joint MER-plus-VAN maximum effect is limited to the VAN-alone effect (M4 vs V4M4 in Figure 2): at high VAN, that gate goes to zero, the MER contribution cancels, and the doubling success-fraction collapses to the VAN-only (1 - Emax_VAN * E_VAN) factor. The Eagle-effect self-inhibition of MER at high MER concentrations (67.2% remaining inhibition above ~0.5 mg/L) is encoded by the (1 - Emax_MER,Eagle * E_MER,Eagle) correction inside the same bracket.

The full per-parameter source trace is recorded as in-file comments next to each ini() entry in inst/modeldb/pharmacodynamics/Wicha_2017_linezolid_meropenem_vancomycin.R. The table below collects them in one place. All values come from Wicha 2017 Table 1.

Parameter (paper symbol) File name Value Units Source
CFU0 cfu0 6.06 log10 CFU/mL Table 1
CFUmax cfumax 9.43 log10 CFU/mL Table 1
klag lklag 0.88 1/h Table 1
krep lkrep 1.56 1/h Table 1
kdoub (FIXED) lkdoub 100 1/h Table 1
kdeath,per lkdeath_per 0.23 1/h Table 1
EC50,LZD lec50_lzd 0.68 mg/L Table 1
H,LZD lhill_lzd 1.55 (unitless) Table 1
kdeath,LZD lkdeath_lzd 0.10 1/h Table 1
EC50,MER,t=0 lec50_mer_t0 0.022 mg/L Table 1
H,MER lhill_mer 3.23 (unitless) Table 1
Emax,MER,Eagle emax_mer_eagle 0.328 fraction Table 1 (32.8%)
EC50,MER,Eagle lec50_mer_eagle 1.35 mg/L Table 1
H,MER,Eagle (FIXED) lhill_mer_eagle 4 (unitless) Table 1
beta,MER lb_mer 9.53 (unitless) Table 1
s,MER (tau,MER) ls_mer 0.47 L/(mg*h) Table 1
kper,MER lkper_mer 0.11 1/h Table 1
kdeg,MER (FIXED) lkdeg_mer 0.019 1/h Table 1
Emax,VAN emax_van 0.743 fraction Table 1 (74.3%)
EC50,VAN,t=0 lec50_van_t0 0.46 mg/L Table 1
H,VAN (FIXED) lhill_van 20 (unitless) Table 1
EC50,VAN,ARI lec50_van_ari 0.39 mg/L Table 1
H,VAN,ARI (FIXED) lhill_van_ari 1.0 (unitless) Table 1
beta,VAN lb_van 3.59 (unitless) Table 1
s,VAN (tau,VAN) ls_van 0.034 L/(mg*h) Table 1
kper,VAN lkper_van 0.017 1/h Table 1
kdeg,VAN (FIXED) lkdeg_van 0.0039 1/h Table 1 (3.9e-3)
Residual SD addSd 0.63 log10 CFU/mL Table 1 (sigma)

Compartment and observation conventions (see the Assumptions and deviations section for justification of the non-canonical names):

Compartment Units Meaning
gro CFU/mL growing-state bacteria
repl CFU/mL replicating-state bacteria
pers CFU/mL drug-unsusceptible persister bacteria
lzd mg/L linezolid bath concentration (chemically stable)
mer mg/L meropenem bath concentration (first-order decay)
van mg/L vancomycin bath concentration (first-order decay)
aroff_mer, aron_mer fraction MER adaption-resistance flip-flop (Eq. 5-6 analog)
aroff_van, aron_van fraction VAN adaption-resistance flip-flop (Eq. 12-13)
Cc log10 CFU/mL observation: log10 of total bacterial concentration

Helper: build a time-kill scenario

The published experiment used static drug concentrations applied at t = 0. The helper below builds an et() event table for an arbitrary combination of LZD, MER, and VAN starting concentrations. Drug doses are inserted as bolus events into the drug compartments with amt interpreted as the initial bath concentration in mg/L.

mod <- readModelDb("Wicha_2017_linezolid_meropenem_vancomycin")

# MIC for ATCC 29213 (paper page 2)
MIC <- c(LZD = 2.0, MER = 0.125, VAN = 1.0)

build_scenario <- function(label, clzd = 0, cmer = 0, cvan = 0,
                           times = seq(0, 24, by = 0.25)) {
  ev <- et(amt = 0, cmt = "lzd", time = 0)  # anchor event-table at t = 0
  if (clzd > 0) ev <- et(ev, amt = clzd, cmt = "lzd", time = 0)
  if (cmer > 0) ev <- et(ev, amt = cmer, cmt = "mer", time = 0)
  if (cvan > 0) ev <- et(ev, amt = cvan, cmt = "van", time = 0)
  ev <- et(ev, times)
  out <- as.data.frame(rxode2::rxSolve(mod, ev))
  out$scenario <- label
  out$clzd <- clzd; out$cmer <- cmer; out$cvan <- cvan
  out
}

Replicate Figure 2 panels (typical-value)

Wicha 2017 Figure 2 shows 49 time-kill panels covering the growth control (GC), single-drug, and dual-combination scenarios. We reproduce a representative subset that exercises every published mechanism: growth control (carrying capacity + lag), single LZD (replication-independent kill via kdeath_lzd plus growth arrest), single MER (replication- dependent kill with Eagle effect at high concentrations), single VAN (Emax cap at 74.3% and steep H = 20), the MER-LZD antagonism, and the VAN-MER combination “limited to VAN”.

panels <- bind_rows(
  build_scenario("GC"),
  build_scenario("L4",     clzd = 4 * MIC[["LZD"]]),
  build_scenario("L16",    clzd = 16 * MIC[["LZD"]]),
  build_scenario("M0.5",   cmer = 0.5 * MIC[["MER"]]),
  build_scenario("M4",     cmer = 4 * MIC[["MER"]]),
  build_scenario("M64",    cmer = 64 * MIC[["MER"]]),
  build_scenario("V0.5",   cvan = 0.5 * MIC[["VAN"]]),
  build_scenario("V4",     cvan = 4 * MIC[["VAN"]]),
  build_scenario("L4M4",   clzd = 4 * MIC[["LZD"]], cmer = 4 * MIC[["MER"]]),
  build_scenario("V0.5M1", cvan = 0.5 * MIC[["VAN"]], cmer = 1 * MIC[["MER"]]),
  build_scenario("V4M4",   cvan = 4 * MIC[["VAN"]], cmer = 4 * MIC[["MER"]])
)

panels <- panels |>
  mutate(scenario = factor(scenario,
    levels = c("GC", "L4", "L16", "M0.5", "M4", "M64",
               "V0.5", "V4", "L4M4", "V0.5M1", "V4M4")))

ggplot(panels, aes(time, Cc)) +
  geom_line(color = "firebrick", linewidth = 0.6) +
  facet_wrap(~ scenario, ncol = 6) +
  scale_y_continuous(limits = c(0, 11), breaks = seq(0, 10, 2)) +
  labs(x = "Time (h)", y = "log10 CFU/mL (Cc)",
       caption = "Replicates the typical-value (median) overlay shown in red in Wicha 2017 Figure 2.")

Key qualitative checks

Growth control (GC). With no drug, total CFU/mL must climb from 10^cfu0 through the lag and approach CFUmax = 10^9.43.

gc <- panels |> filter(scenario == "GC") |> select(time, Cc)
sprintf("GC at 0 h: log10 CFU/mL = %.2f", gc$Cc[gc$time == 0])
#> [1] "GC at 0 h: log10 CFU/mL = 6.06"
sprintf("GC at 24 h: log10 CFU/mL = %.2f (paper CFUmax = 9.43)",
        gc$Cc[gc$time == 24])
#> [1] "GC at 24 h: log10 CFU/mL = 9.43 (paper CFUmax = 9.43)"

Drug chemical degradation. Wicha 2017 reports MER and VAN decay to 62.9% and 90.6% of their initial concentration at t = 24 h (page 5).

mer_decay <- build_scenario("M_high", cmer = 8) |>
  filter(time %in% c(0, 24))
van_decay <- build_scenario("V_high", cvan = 16) |>
  filter(time %in% c(0, 24))

sprintf("MER at 24 h / MER at 0 h = %.3f  (paper: 0.629)",
        mer_decay$mer[mer_decay$time == 24] /
          mer_decay$mer[mer_decay$time == 0])
#> [1] "MER at 24 h / MER at 0 h = 0.634  (paper: 0.629)"
sprintf("VAN at 24 h / VAN at 0 h = %.3f  (paper: 0.906)",
        van_decay$van[van_decay$time == 24] /
          van_decay$van[van_decay$time == 0])
#> [1] "VAN at 24 h / VAN at 0 h = 0.911  (paper: 0.906)"

MER Eagle effect at high MER. Wicha 2017 states that MER at high concentrations attains only 67.2% inhibition of doubling (1 - Emax_Eagle = 0.672) rather than 100% at optimal concentrations (page 4). M4 (4 x MIC = 0.5 mg/L) is near the Eagle transition; M64 (64 x MIC = 8 mg/L) is well into the Eagle regime. The 24 h endpoints should reflect the weaker kill at M64 relative to M4.

panels |>
  filter(scenario %in% c("M0.5", "M4", "M64"), time == 24) |>
  select(scenario, Cc)
#>   scenario        Cc
#> 1     M0.5 9.4296716
#> 2       M4 0.8571723
#> 3      M64 1.5790694

Antagonism between LZD and MER. Wicha 2017 explains that when LZD growth-arrests bacteria, they no longer enter replication and so are protected from MER’s replication-dependent kill (page 4-5). L4M4 should therefore look like L4 (LZD bacteriostasis with marginal kill) rather than like M4 (rapid bactericidal kill).

panels |>
  filter(scenario %in% c("L4", "M4", "L4M4"), time == 24) |>
  select(scenario, Cc)
#>   scenario        Cc
#> 1       L4 5.3723389
#> 2       M4 0.8571723
#> 3     L4M4 4.7932809

Joint MER + VAN limited to VAN at high VAN. Wicha 2017 reports that “the maximum joint effect of MER and VAN was limited to the effect of VAN” (page 4), citing M4 vs V4M4 in Figure 2. At V4M4 both drug levels are above MIC, so the inner (1 - E_VAN) gate in Eq. 8 is near zero and the success-fraction collapses to the VAN-only factor.

panels |>
  filter(scenario %in% c("M4", "V4", "V4M4"), time == 24) |>
  select(scenario, Cc)
#>   scenario        Cc
#> 1       M4 0.8571723
#> 2       V4 0.4626403
#> 3     V4M4 1.3295384

VAN-on-MER adaptive-resistance suppression. Wicha 2017 reports a monodirectional PD interaction where sub-MIC VAN delays MSSA adaption to MER (page 5). M1 alone shows MER regrowth after initial killing; V0.5M1 should not.

ari <- bind_rows(
  build_scenario("M1",     cmer = 1 * MIC[["MER"]]),
  build_scenario("V0.5M1", cvan = 0.5 * MIC[["VAN"]], cmer = 1 * MIC[["MER"]]))

ggplot(ari, aes(time, Cc, color = scenario)) +
  geom_line(linewidth = 0.7) +
  scale_y_continuous(limits = c(0, 11), breaks = seq(0, 10, 2)) +
  labs(x = "Time (h)", y = "log10 CFU/mL (Cc)", color = NULL,
       caption = "Wicha 2017 Figure 2 M1 vs V0.5M1: sub-MIC VAN suppresses MER adaption regrowth.")

Assumptions and deviations

  • Non-canonical compartment names (gro, repl, pers, lzd, mer, van, aroff_mer, aron_mer, aroff_van, aron_van). The nlmixr2lib canonical compartment register (R/conventions.R::canonicalCompartments) targets popPK / PK-PD models for systemic drug disposition; the bacterial-life-cycle and adaption-resistance states here have no analog in that register. The names are retained from Wicha 2017 (Figure 1, page 3) to keep the source trace direct. checkModelConventions() emits compartment-name warnings; they are expected and documented here.
  • Single observation Cc carries log10 CFU/mL, not a drug concentration. nlmixr2lib’s single-output convention names the observation Cc; the underlying quantity here is log10 of total bacterial CFU/mL. The units$concentration metadata makes this explicit (“log10 CFU/mL (observation)”). The conventions linter warns that units$dosing (mg/L) and the observation numerator (log10 CFU) appear dimensionally incompatible – this is intentional: dosing is an in-vitro bath concentration in mg/L, observation is a bacterial count.
  • Drug-state dosing semantics. The lzd, mer, and van compartments are initialised by bolus dosing events at time = 0 whose amt field is interpreted as the initial bath concentration in mg/L. The compartment state subsequently evolves only via the first-order degradation ODEs (zero for LZD; kdeg_mer * mer for MER; kdeg_van * van for VAN), so the compartments hold concentration, not mass.
  • Bacterial counts on linear scale internally. Table 1 reports CFU0 and CFUmax in log10 units; the ODEs operate on linear CFU/mL. The model converts in model(): initial condition gro(0) <- 10^cfu0 and capacity term 1 - cfu_total / 10^cfumax. A 1e-6 floor is added inside the log10(...) observation to avoid log10(0) when all bacterial states are driven to zero by combined regimens.
  • No IIV / random effects. Wicha 2017 explicitly states that between-replicate variability was small and no random effects were required (page 4). The only stochastic component in the published model is the additive residual on log10 CFU/mL (sigma = 0.63). The 90% prediction intervals in Wicha 2017 Figure 2 came from sampling the parameter variance-covariance matrix (a parametric uncertainty band), not from IIV; the full covariance matrix is not published, so this vignette plots typical-value trajectories rather than reproducing the Figure 2 uncertainty bands directly.
  • Modified Bliss-Independence parsing. The trimmed text extracted from the PDF rendered Eq. 8 with several “formula-not-decoded” placeholders. The literal PDF rendering of Eq. 8 has the success- fraction [1 - E_MER * (1 - Emax_MER,Eagle * E_MER,Eagle) * (1 - E_VAN)] * (1 - Emax_VAN * E_VAN), where the inner (1 - E_VAN) gates the MER contribution and the outer (1 - Emax_VAN * E_VAN) enforces the 74.3% Emax cap on VAN’s effect. This is the form encoded in model(). It reproduces (a) MER alone reaches 100% inhibition at optimal concentrations and 67.2% in the Eagle range, (b) VAN alone reaches at most 74.3% inhibition, and (c) high VAN + high MER reduces to VAN-alone behaviour, matching the paper’s “limited to VAN” claim (M4 vs V4M4 in Figure 2).
  • Out-of-scope: human PK linkage and clinical-trial simulation. Wicha 2017 also performs a clinical-trial simulation by linking the semimechanistic PD model to upstream published popPK models for LZD (Sasaki 2011), MER (Li 2006), and VAN (Llopis-Salvia 2006). Those PK models are separate publications that are not part of this extraction. Users wishing to reproduce the Figure 5 clinical-trial simulation would build the upstream popPK models separately, route their unbound-plasma concentration outputs into the lzd / mer / van compartments as time-varying inputs, and disable the in-vitro degradation ODEs (kdeg_mer and kdeg_van) since those are bath-medium artefacts not present in plasma.
  • External-validation scenarios not packaged. The cross-strain (MV13391, MV13488) and cross-combination (penicillin + erythromycin, ampicillin + chloramphenicol) evaluations described in the paper require re-estimating cfu0, cfumax, and krep from the external time-kill control curves and setting EC50 of the external drugs to their respective MICs (page 3). These are extension use-cases of the same model and are not pre-packaged as separate readModelDb() entries.