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Ciprofloxacin (Khan 2015)

Source: vignettes/articles/Khan_2015_ciprofloxacin.Rmd
Khan_2015_ciprofloxacin.Rmd

Model and source

  • Citation: Khan DD, Lagerback P, Cao S, Lustig U, Nielsen EI, Cars O, Hughes D, Andersson DI, Friberg LE. (2015). A mechanism-based pharmacokinetic/pharmacodynamic model allows prediction of antibiotic killing from MIC values for WT and mutants. J Antimicrob Chemother 70(11):3051-3060. doi:10.1093/jac/dkv233. DDMORE Foundation Model Repository: DDMODEL00000225.
  • Description: Mechanism-based six-compartment PK/PD model for ciprofloxacin against Escherichia coli K-12 wild-type and quinolone-resistant single-step mutants in static in vitro time-kill experiments (Khan et al. 2015 J Antimicrob Chemother). Two co-existing bacterial subpopulations (susceptible-growing, plus a small pre-existing less-susceptible subpopulation seeded at MUT*1e-6 of the inoculum) each carry growing, resting, and drug-induced non-colony-forming states; drug effect is a Hill-Emax term on bacterial kill with strain-specific EC50 (LM202 estimated; LM347, LM378, LM534, LM625, LM693, LM707 fixed at the published per-strain values), a separate EC502 for the resistant subpopulation, density-dependent active->resting flux, and a model-time gate that switches off the active->non-colony-forming transition after THETA(19) hours.
  • Article: https://doi.org/10.1093/jac/dkv233
  • DDMORE Foundation Model Repository entry: DDMODEL00000225

This model was extracted from the DDMORE Foundation Model Repository bundle for DDMODEL00000225 (scraped to dpastoor/ddmore_scraping/225/). The bundle contains:

  • executeable_cipro_pkpd.mod – the NONMEM control stream (ADVAN13, 6 compartments, 19 thetas, 1 omega FIX, 5 sigmas with one BLOCK(1) plus three BLOCK(1) SAME).
  • executeable_cipro.xml – MDL XML render of the same model.
  • output_simulated.lst – NONMEM listing from a refit on cipro_simulated.csv. The bundle does not ship an Output_real_*.lst from a re-fit on the original real-data cipro202.csv / cipro378.csv datasets. The simulated-refit reports MINIMIZATION SUCCESSFUL HOWEVER, PROBLEMS OCCURRED WITH THE MINIMIZATION (the standard “rounding errors but converged” status with NSIG = 5 reached).
  • cipro_simulated.csv – 9 in vitro tubes (one per CAB level for STR = 202) with 4 sample-position replicates per observation time; 207 rows, 148 observations.
  • cipro202.csv, cipro378.csv – the real-data datasets and their cipro202_VPC.mdl / cipro378_VPC.mdl companions.
  • Model_Accommodations.txt – notes that the MDL rendition drops L2, MTIME, B2, and M3 (“not supported in MDL”). The NONMEM .mod retains all four of those mechanisms.
  • DDMODEL00000225.rdf – RDF metadata (model-field-purpose: pkpd_0001024, model-implementation-source-discrepancies-freetext = "M3, B2, MTIME and L2 not included in mdl file.").
  • HO_Bacteria_PKPD _script.r, Command_target.txt, 225.json – helper scripts and scraper metadata.

The packaged model uses the .mod $THETA initial values as its parameter values, not the simulated-refit lst final estimates. That choice was confirmed by the operator (sidecar 025 q1 = mod_initials) on these grounds: the .mod initials appear to be the publication-derived point values that seeded the bundle’s simulated dataset – output_simulated.lst round-trip-recovers them within ~3 % for every theta – so the .mod initials are the closest available proxy for Khan 2015 Table 2 / Table 3, while no Output_real_*.lst is available. The output_simulated.lst final estimates differ materially only for SIGMA(1) (2.42 -> 1.78); the .mod initial 2.41597 is what the model file uses for addSd.

The Khan et al. 2015 publication itself (J Antimicrob Chemother 70(11):3051-3060) is paywalled and is not on disk in this worktree; PMID 26349518 has no PMC full-text mirror. A side-by-side comparison against the publication’s per-strain MIC table or per-tube time-kill figure is therefore out of scope here. Validation in this vignette is the F.3 mechanistic-sanity check from the extraction skill: the typical-value bacterial-count trajectory is qualitatively consistent with a 24-hour ciprofloxacin time-kill experiment on Escherichia coli LM202 wild-type at concentrations spanning 0-8 x MIC.

Population

Khan 2015 is a static in vitro time-kill study, not a clinical population-PK study: each tube is a single E. coli strain in Mueller-Hinton broth at a fixed ciprofloxacin concentration, with serial colony counts at 0, 1, 2, 4, 6, 9, 12, and 24 hours. The seven strains used are E. coli K-12 MG1655 (LM202 wild-type) and six gyrA / gyrB / parC / parE single-step mutants raised on the same MG1655 background (LM347, LM378, LM534, LM625, LM693, LM707). Per-strain MICs span four orders of magnitude (LM202 wild-type 0.047 mg/L -> LM707 highly resistant 92 mg/L), with the wild-type EC50 estimated and the six mutant EC50s fixed at the per-strain measured values in the published Table.

Because the experimental “subjects” are tubes rather than human participants, the model’s population metadata fields for n_subjects, age_range, weight_range, and sex_female_pct are intentionally NA. The disease_state, dose_range, and notes fields document the in vitro experimental design instead.

The DDMORE bundle’s cipro_simulated.csv ships only STR = 202 (wild-type) tubes spanning a 9-point CAB grid (0, 0.0029, 0.0059, 0.0118, 0.0235, 0.047, 0.094, 0.188, 0.376 mg/L; equivalent to 0-8 x wild-type MIC). The validation cohort below mirrors that covariate structure exactly.

Source trace

Per-parameter origin (also recorded as in-file comments next to each ini() entry of inst/modeldb/ddmore/Khan_2015_ciprofloxacin.R):

Equation / parameter Value Source location
lkgs log(1.70) executeable_cipro_pkpd.mod $THETA(1) = (1, 1.70) (init 1.70)
lemax log(5.24) .mod $THETA(2) = (0, 5.24)
lec50_347 fixed(log(0.037)) .mod $THETA(3) = (0, 0.037) FIX
lec50_202 log(0.057) .mod $THETA(4) = (0, 0.057) (only LM202 was estimated; the LM202 cell line is wild-type and the only strain shipped in cipro_simulated.csv)
lec50_378 fixed(log(0.65)) .mod $THETA(5) = (0, 0.65) FIX
lec50_534 fixed(log(0.30)) .mod $THETA(6) = (0, 0.30) FIX
lec50_625 fixed(log(1.0)) .mod $THETA(7) = (0, 1.0) FIX
lec50_693 fixed(log(31.0)) .mod $THETA(8) = (0, 31) FIX
lec50_707 fixed(log(92.0)) .mod $THETA(9) = (0, 92) FIX
lgam log(1.98) .mod $THETA(10) = (0.5, 1.98)
lpc_raw log(0.0186) .mod $THETA(11) = (0, 0.0186); multiplied by 1e-7 in model()
lkgs2 log(0.344) .mod $THETA(12) = (0.18, 0.344)
lec502 log(1.25) .mod $THETA(13) = (0, 1.25)
lmut_raw log(0.81) .mod $THETA(14) = (0, 0.81); multiplied by 1e-6 in model()
lksnc_max log(5.83) .mod $THETA(15) = (0, 5.83)
lknc_factor log(0.17) .mod $THETA(16) = (0, 0.17)
lgam_nc fixed(log(20)) .mod $THETA(17) = (0, 20) FIX
ltr50 log(0.24) .mod $THETA(18) = (0, 0.24)
ltmtime log(5.3158) .mod $THETA(19) = (2, 5.3158)
addSd 1.5544 = sqrt(2.41597) .mod $SIGMA 2.41597 (across-tube residual; per-position SAME blocks dropped – see Errata)
kk = 0.179 constant in model() n/a .mod $PK line 32 – hardcoded death rate, not estimated
Strain cascade (STR == X) * exp(lec50_X) sum n/a .mod $PK lines 39-45 IF(STR.EQ.X) cascade
ec50 selection by STR n/a dataset column STR in {347, 202, 378, 534, 625, 693, 707}
drugs = emax * CAB^gam / (ec50^gam + CAB^gam) n/a .mod $DES line 113 (EMAX equation, sensitive subpopulation)
drugs2 = emax * CAB^gam / (ec502^gam + CAB^gam) n/a .mod $DES line 114 (EMAX equation, pre-existing resistant subpop)
ksnc_1 = ksnc_max * (CAB/ec50)^gam_nc / ((CAB/ec50)^gam_nc + tr50^gam_nc) n/a .mod $PK line 82 (drug-induced active -> NC transition)
ksnc_2 analogous with ec502 n/a .mod $PK line 83
knc_s_1 = knc_factor * ec50 / (CAB + 1e-10) n/a .mod $PK line 85 (NC -> active back-reaction)
knc_s_2 = knc_factor * ec502 / (CAB + 1e-10) n/a .mod $PK line 86
flag = 1.0 * (t < tmtime) n/a .mod $PK lines 90-91 + $DES line 98 (MTIME(2) = THETA(19); FLAG = MPAST(1) - MPAST(2))
sr = pc * sum(bact_*) n/a .mod $DES line 102 (density-dependent active -> resting flux)
bact_s(0) = sbase * (1 - mut*1e-6) n/a .mod $PK line 72 (A_0(1) = SBASE * (1 - MUT*0.000001))
bact_spe(0) = sbase * mut * 1e-6 n/a .mod $PK line 74 (A_0(3) = MUT*0.000001*SBASE)
6-compartment ODE block n/a .mod $DES lines 117-123 (line-for-line image; see model file)
lnBact = log((bact_s + bact_r + bact_spe + bact_rpe) + 1e-8) n/a .mod $ERROR lines 130-132 (ATOT = A1+A2+A3+A5; IPRED = LOG(ATOT + 1e-8))

Virtual cohort 

set.seed(20260506L)

# Mirror cipro_simulated.csv: STR = 202 (LM202 wild-type, MIC = 0.047
# mg/L) at 9 CAB levels (0..8 x MIC). Time grid matches the
# experiment's 24-hour observation window.
mic202    <- 0.047
cab_grid  <- mic202 * c(0, 0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8)
obs_times <- c(0, 1, 2, 4, 6, 9, 12, 24)
base_log  <- 13                # ln(CFU/mL); the .mod default for a typical inoculum

events <- expand.grid(
  cab_level = seq_along(cab_grid),
  time      = obs_times,
  KEEP.OUT.ATTRS = FALSE,
  stringsAsFactors = FALSE
) |>
  mutate(
    id   = cab_level,
    CAB  = cab_grid[cab_level],
    STR  = 202L,
    BASE = base_log,
    evid = 0L,
    amt  = 0,
    fold_mic = CAB / mic202,
    treatment = sprintf("%.4g x MIC (CAB = %.4g mg/L)", fold_mic, CAB)
  ) |>
  arrange(id, time)

stopifnot(!anyDuplicated(unique(events[, c("id", "time", "evid")])))

Simulation 

mod <- rxode2::rxode2(readModelDb("Khan_2015_ciprofloxacin"))

# Typical-value (no IIV) trajectory: rxode2::zeroRe() zeros out the
# residual SD (the only random effect in this single-tube model).
mod_typical <- mod |> rxode2::zeroRe()
#> Warning: No omega parameters in the model

sim_typical <- rxode2::rxSolve(
  mod_typical,
  events = events,
  keep   = c("CAB", "STR", "BASE", "fold_mic", "treatment")
) |>
  as.data.frame()
#> Warning: multi-subject simulation without without 'omega'

Replicate published time-kill curves

The figure below replicates the qualitative shape of a Khan 2015 LM202 wild-type time-kill panel: at sub-MIC concentrations the bacterial count grows to a stationary plateau (capped by the density-dependent active -> resting flux); at 1 x MIC the count collapses sharply through ~24 hours; at 2-8 x MIC the count drops to near-eradication levels within the first 9 hours and stays low. The published Khan 2015 figures are not on disk for a side-by-side panel-by-panel comparison.

pal <- scales::viridis_pal(option = "C", end = 0.9)(length(unique(sim_typical$treatment)))
treatment_levels <- unique(events$treatment)

ggplot(sim_typical, aes(time, lnBact, colour = factor(treatment, levels = treatment_levels))) +
  geom_line(linewidth = 0.7) +
  geom_point(size = 1.2) +
  scale_colour_manual(values = pal, name = "ciprofloxacin (mg/L)") +
  labs(
    x = "Time (h)",
    y = "ln(CFU/mL)",
    title = "Khan 2015 time-kill: LM202 wild-type at 0-8 x MIC",
    caption = "Typical-value simulation; CAB = 0 is the no-drug control. MIC202 = 0.047 mg/L."
  ) +
  theme_minimal(base_size = 11) +
  theme(legend.position = "right",
        legend.text  = element_text(size = 9),
        legend.title = element_text(size = 9))

Mechanistic sanity checks

Three F.3 mechanistic-sanity checks for this PD model.

1 – No-drug control reproduces sigmoidal growth

In the absence of drug, the bacterial count must rise from the baseline inoculum and plateau (the density-dependent active -> resting flux is the only thing limiting unbounded growth).

no_drug <- subset(sim_typical, abs(CAB) < 1e-12)
end_count <- tail(no_drug, 1)
stopifnot(end_count$lnBact > base_log + 5)   # at least 5 ln-units of growth
stopifnot(end_count$lnBact < base_log + 12)  # but not unbounded
cat(sprintf("No-drug control: lnBact at t = 0 -> 24h: %.2f -> %.2f (delta = %.2f)\n",
            base_log, end_count$lnBact, end_count$lnBact - base_log))
#> No-drug control: lnBact at t = 0 -> 24h: 13.00 -> 20.54 (delta = 7.54)

2 – Static drug at MIC induces a >5 ln-unit drop by 24 h 

at_mic <- subset(sim_typical, abs(CAB - mic202) < 1e-12)
end_mic <- tail(at_mic, 1)
stopifnot(end_mic$lnBact < base_log - 5)     # at least 5-log kill
cat(sprintf("CAB = MIC: lnBact at t = 0 -> 24h: %.2f -> %.2f (delta = %.2f)\n",
            base_log, end_mic$lnBact, end_mic$lnBact - base_log))
#> CAB = MIC: lnBact at t = 0 -> 24h: 13.00 -> 2.79 (delta = -10.21)

3 – Drug effect saturates as CAB / EC50 -> infinity

At 8 x MIC the kill is near-complete; pushing CAB to ~100 x MIC should not produce a substantially deeper trough than 8 x MIC, since the Hill-Emax term saturates at emax.

ev_high <- data.frame(
  id = c(1L, 2L), time = rep(24, 2), evid = 0L, amt = 0,
  STR = 202L, BASE = base_log,
  CAB = c(8 * mic202, 100 * mic202)
)
sim_high <- rxode2::rxSolve(mod_typical, events = ev_high) |> as.data.frame()
#> Warning: multi-subject simulation without without 'omega'
delta <- diff(sim_high$lnBact)
cat(sprintf("CAB = 8 x MIC -> 100 x MIC moves lnBact(24h) by %.3f log-units\n", delta))
#> CAB = 8 x MIC -> 100 x MIC moves lnBact(24h) by -0.019 log-units
stopifnot(abs(delta) < 2)

Self-consistency vs the bundle’s simulated dataset

The DDMORE bundle ships cipro_simulated.csv (207 rows, STR = 202, 9 CAB levels, 4 sample-position replicates per observation time) as a smoke-test dataset. The chunk below overlays the bundle’s observations against the typical-value rxode2 trajectory. Per-subject exact matches are not expected – each NMTRAN subject in the bundle’s simulated dataset was generated with its own exp(ETA(1)*sqrt(SIGMA(1))) baseline-IIV draw and the four FLG2 position-EPS draws, neither of which is reproduced in this typical-value translation.

bundle_csv <- system.file(
  "extdata", "ddmore", "DDMODEL00000225_cipro_simulated.csv",
  package = "nlmixr2lib"
)

bundle_obs <- if (nzchar(bundle_csv) && file.exists(bundle_csv)) {
  read.csv(bundle_csv) |>
    dplyr::filter(EVID == 0, BOBS == 0) |>
    dplyr::transmute(
      id = ID, time = TIME, lnBact = LNDV,
      CAB = CAB, fold_mic = CAB / mic202,
      treatment = sprintf("%.4g x MIC (CAB = %.4g mg/L)", CAB / mic202, CAB),
      source = "DDMORE bundle (NONMEM simulation)"
    )
} else {
  NULL
}

p <- ggplot(sim_typical, aes(time, lnBact, colour = factor(round(fold_mic, 4)))) +
  geom_line(linewidth = 0.6, alpha = 0.9) +
  scale_colour_viridis_d(option = "C", end = 0.9, name = "fold MIC")

if (!is.null(bundle_obs)) {
  p <- p + geom_point(
    data = bundle_obs,
    aes(time, lnBact, colour = factor(round(fold_mic, 4))),
    shape = 21, fill = "white", size = 1.5, stroke = 0.5
  )
}

p + labs(
  x = "Time (h)",
  y = "ln(CFU/mL)",
  title = "Khan 2015 LM202 typical value vs DDMORE bundle simulated observations",
  caption = "Lines: rxode2 typical-value translation (this model). Points: bundle cipro_simulated.csv (when available)."
) +
  theme_minimal(base_size = 11) +
  theme(legend.position = "right")

PKNCA validation (not applicable)

PKNCA is not applicable to this model. Khan 2015 is a static in vitro bacterial PD study: the “concentration” being modelled is a log-bacterial count (CFU/mL), and the “dose” is a static drug concentration CAB (mg/L) held constant for the lifetime of each tube. There is no plasma-concentration time-course to take an NCA over and no clinical dose / clearance / volume to compare Cmax / AUC / half-life against. The validation strategy chosen by the operator (sidecar 025 q2 = verbatim) and the extraction skill’s references/ddmore-source.md Section “Validation strategy by model type” decision tree is the F.3 mechanistic-sanity recipe used above instead.

Assumptions and deviations

  • Source values are .mod $THETA initials, not lst final estimates. The DDMORE bundle does not ship an Output_real_*.lst from the published real-data fit; only output_simulated.lst (a refit on the bundled simulated dataset) is available. Per operator decision (sidecar 025 q1 = mod_initials), the model file uses the .mod $THETA initial values as the closest available proxy for the publication’s Table 2 / Table 3 final estimates. The output_simulated.lst values round-trip-recover the .mod initials within ~3 % for every theta but report SIGMA(1) = 1.78 instead of the .mod’s initial 2.41597; the model file uses addSd = sqrt(2.41597) = 1.5544 to remain consistent with the publication-derived initials.

  • The Khan 2015 publication itself is not on disk. PMID 26349518 / DOI 10.1093/jac/dkv233 is paywalled and has no PMC full-text mirror; a direct fetch returns only the abstract. As a result the per-strain MIC table, the per-strain EC50 table, the time-kill figure panels, and the residual-error structure description in the publication cannot be cross-checked here. The numerical values shown above match the bundle’s .mod; whether they match the publication’s printed tables exactly is a question this vignette cannot answer until full text is available.

  • Multi-level residual error collapsed to across-tube only. The .mod $ERROR block writes Y = IPRED + EPS(1) + SAM1*EPS(2) + SAM2*EPS(3) + SAM3*EPS(4) + SAM4*EPS(5) with SIGMA(1) = 2.41597 (across-tube random effect) plus four BLOCK(1) SAME SIGMA(2..5) slots gated on FLG2 (sample position 1..4) reflecting between-replicate variability. Per operator decision (sidecar 025 q3 = across_tube_only), only the across-tube SIGMA(1) is encoded as addSd = sqrt(SIGMA(1)); the four per-position residual slots are documented here but not reproduced. For typical-value time-kill trajectories this simplification has no effect; for stochastic VPC reproduction against a 4-replicate experiment a richer encoding would be required.

  • B2-method baseline IIV dropped from the typical-value encoding. The .mod $PK line 58 declares SBASE = exp(BASE) * exp(ETA(1) * sqrt(SIGMA(1))) – a NONMEM “B2 method” baseline-as-parameter encoding that ties the inoculum random effect to SIGMA(1) via an OMEGA = 1 FIX. This vignette and the packaged model use the deterministic sbase = exp(BASE) form (no ETA(1)); the B2-method baseline IIV is documented here. Reintroducing it would require an eta_sbase ~ fixed(1) IIV term and a multiplier of exp(eta_sbase * sqrt(addSd^2)) on the initial conditions.

  • NONMEM MTIME(2) translated to a step on t. The .mod uses MTIME(2) = THETA(19) and FLAG = MPAST(1) - MPAST(2) to switch off the active -> non-colony-forming transition once experimental time exceeds tmtime hours. Per operator decision (sidecar 025 q4 = if_t_in_model), the nlmixr2 translation is the literal flag <- 1.0 * (t < tmtime) step on the rxode2 model time variable. A smooth (logistic) replacement was rejected as a deviation from the publication’s intended discrete switch.

  • M3 BLQ method dropped. The .mod $ERROR lines 156-164 use NONMEM’s M3 method (F_FLAG = 1; Y = PHI((LLOQ - IPRED)/SIG) for FLG1 == 1, i.e., observations below the colony-counting limit of detection, LOQ = ln(10) ~= 2.303). nlmixr2’s standard residual-error pipeline does not take an M3 censoring branch via the ~ add(addSd) form, so the model file presents a single unconditional residual; users fitting to data containing BLQ observations should preprocess (e.g., set the M3-flagged rows to cens = 1 in the dataset and let nlmixr2’s M3 handling pick it up at fit time).

  • L2 (2nd-level OBS reset) dropped. The .mod $DATA line uses L2 = for an obscure NONMEM reset-on-occasion convention that does not have a clean rxode2 equivalent and is not relevant to the typical-value simulation.

  • Compartment names are paper-specific, not canonical. bact_s (sensitive growing), bact_r (sensitive resting), bact_spe (pre-existing resistant growing), bact_np (sensitive non-colony-forming under drug), bact_rpe (pre-existing resistant resting), bact_nppe (pre-existing resistant non-colony-forming). checkModelConventions() flags these as non-canonical (the canonical compartment register covers the human-PK universe of depot / central / peripheral1..2 / etc.); this is the same exemption pattern used by oncology_sdm_lobo_2002 (Lobo 2002 in vitro methotrexate on cancer cells), where tumorVol and transit1..4 are similarly paper-specific.

  • Observation variable is lnBact, not Cc. Following the conventions doc’s exemption for non-PK paper-named outputs (e.g., tumorSize, freeIgE, Cbrain_cerebellum), the colony count observation is named lnBact (natural-log of CFU/mL). checkModelConventions() flags this – it is the documented exemption that applies.

  • Covariates STR, CAB, BASE are in vitro experimental design columns, not population-PK covariates. They are documented in the model file’s covariateData entries (with source_name matching the dataset column name) but are not proposed for the canonical inst/references/covariate-columns.md register: the register’s remit is reusable cross-model human-PK covariates, and these three columns are tube-specific experimental-design parameters for a single in vitro publication.

  • Units field deliberately mixes mg/L (drug) and log CFU/mL (observation). The units list reports dosing = "mg/L" and concentration = "log CFU/mL". checkModelConventions() flags this as dimensionally incompatible – that is the intended reading: dosing here refers to the static drug concentration in the tube, while the observed quantity is a log-bacterial-count. The two are not the same physical quantity by design.

  • Bundle simulated dataset is a smoke-test cohort. The 9-tube, 207-row cipro_simulated.csv cohort is a regression-test artifact, not a recreation of the Khan 2015 trial. The virtual cohort built in this vignette mirrors the bundle’s covariate grid for the self-consistency overlay and is not an attempt to reproduce the actual published study.

  • output_simulated.lst minimization warning. The bundle’s simulated-refit lst reports MINIMIZATION SUCCESSFUL HOWEVER, PROBLEMS OCCURRED WITH THE MINIMIZATION – the standard NONMEM rounding-errors-but-converged status, with NO. OF SIG. DIGITS IN FINAL EST. = 5.2 and the requested NSIG = 5 reached. That warning is recorded here for completeness; it does not affect the packaged model because the packaged model uses .mod initials, not lst final estimates, per the q1 decision above.

  • “M3, B2, MTIME and L2 not included in mdl file.” The bundle’s Model_Accommodations.txt records that the MDL rendition (executeable_cipro.xml) drops these four NONMEM features because they are not supported in the MDL language. The NONMEM .mod itself retains all four. The packaged nlmixr2lib model retains MTIME (translated to a t-step) and drops M3, B2, and L2 as documented above; this is independently of the MDL-level discrepancies and is driven by the nlmixr2-vs-NONMEM mapping decisions, not by the MDL limitations.