Pharmacokinetic and Pharmacodynamic Modelling

When to use

Any question about what the body does to a drug or what the drug does to the body: deriving exposure metrics from concentration-time data, fitting a structural model, building or checking a population analysis, choosing a dose or a regimen, relating exposure to effect, comparing formulations, or scaling to a new population.

The three rules

1. Fix the exposure metric and the analysis population before computing anything. AUC(0-t), AUC(0-inf), AUC(0-tau) at steady state, and Cavg are different quantities and answer different questions. So do AUCinf based on observed versus predicted Clast. Choosing after seeing the numbers is how a negative study becomes positive.

2. Structural model, variability model, and covariate model are three separate decisions. They get conflated constantly — an extra compartment added to absorb what is really unmodelled between-occasion variability, a covariate added to fix what is really a misspecified absorption model. Diagnose which one is wrong before changing any of them.

3. Convergence is not identifiability. A fit that converges with 200% relative standard error on a parameter, or a correlation of 0.99 between two, has told you the data cannot separate them. Every fitting script here reports both and flags them, because the parameter table alone looks fine in exactly this situation.

Scope

This skill computes, diagnoses, and structures. It does not decide that a formulation is bioequivalent, select a dose for a trial, recommend a dose for a patient, conclude that a drug has no QT liability, or replace a qualified pharmacometrician, clinical pharmacologist, or the regulatory review. The scripts report; none of them concludes. tdm_bayes.py in particular is a modelling aid — any change to a patient's regimen is the treating clinician's decision.

Scripts

cd skills/pkpd-modeling/scripts

All take --format table|tsv|json. Data goes to stdout, provenance and findings to stderr, so > out.tsv keeps them separate. Exit code is 0 for no findings, 1 when findings were raised, 2 for bad input, so any of them can gate a workflow.

Two private modules carry the shared machinery: _models.py (analytical solutions for linear mammillary models, plus integrated Michaelis-Menten, TMDD and indirect-response structures) and _common.py (I/O and reporting). Import them rather than re-deriving a Bateman function.

Workflow

1. Non-compartmental analysis

python3 nca.py -i profile.csv --dose 100 --route extravascular --partial-auc 0-24

Four choices decide the answer and are usually left implicit. This script makes all four explicit: --auc-method (default linup-logdown), --blq-rule, --lambda-z-points or an explicit --lambda-z-window, and whether you report auc_inf_obs or auc_inf_pred.

Lambda_z selection uses the standard rule: start from the last three quantifiable points, extend backwards, keep the longer window only if adjusted r-squared improves by more than 0.0001. Plain r-squared can only rise as points are added, so it would always pick the longest window. Points at or before Tmax are never eligible — including Tmax fits the tail of absorption and biases half-life, Vz and AUCinf downward.

On a noiseless simulated one-compartment oral profile with CL/F = 5, V/F = 20, ka = 1.2:

id  cmax     tmax  auc_last  lambda_z  t_half   r2_adj  auc_inf_obs  pct_auc_extrap  cl_f     vz_f
1   3.29678  1.5   19.8737   0.25      2.77259  1       19.8739      0.000781037     5.03173  20.1269

The 0.6% overestimate of CL/F is the trapezoidal rule on a sparsely sampled absorption phase, not an error — it is the irreducible bias of NCA on that sampling schedule, and it is why NCA and compartmental estimates of clearance never agree exactly.

The findings are the point. A steady-state profile truncated at tau produces:

finding: subject A: 25.2% of AUCinf is extrapolated (above 20%); AUCinf is driven by the
         lambda_z fit, not by data
finding: subject A: lambda_z window spans 0.58 half-lives (below 2.0); the terminal phase may
         not have been reached

Both are correct and both are routinely ignored. At steady state the reportable exposure metric is AUC(0-tau), not AUCinf; the script computes AUCinf anyway and tells you not to trust it.

2. Compartmental fitting and model selection

python3 fit_compartmental.py -i profile.csv --dose 500 --route iv-bolus --compare 1cmt,2cmt,3cmt

Parameters are estimated on the log scale, so they cannot go negative and their confidence intervals come out asymmetric. Weighting defaults to 1/y2 (constant CV), which is the right default for PK and the wrong one for a homoscedastic PD endpoint.

Fitting simulated two-compartment data (CL 4, V1 12, Q 6, V2 40, 8% proportional error):

model  parameters  wssr       aic       bic       f_vs_simpler  f_p_value    compared_with
1cmt   2           3.13201    -19.4956  -18.0795  n/a           n/a          n/a
2cmt   4           0.0309579  -84.7477  -81.9155  550.936       9.38016e-12  1cmt
3cmt   6           0.0232859  -85.0193  -80.771   1.4826        0.27762      2cmt

AIC picks the three-compartment model. BIC and the F test both reject it. AIC's fixed penalty of 2 per parameter is weak at this sample size, and it selects the overparameterised model more often than practitioners expect. The parameter table settles it:

finding: fit: Q3 has 98% RSE - not estimable from these data at this model size
finding: fit: V3 has 71% RSE - not estimable from these data at this model size

The one-compartment fit meanwhile earns:

finding: fit: residual signs are not random (runs test p = 0.0036) - a structural
         misspecification, which no amount of reweighting will fix

That distinction — structural misspecification versus a wrong error model — is the one to get right. A residual-versus-time plot with runs of the same sign means the model shape is wrong. Heteroscedastic residuals with random signs mean the weighting is wrong. Reweighting the first case hides it without fixing it.

3. Population PK

Check the dataset before running anything. This is where the time actually goes.

python3 check_popk_dataset.py -i nmdata.csv --covariates WT,CRCL --time-varying WT

The defects that matter are the silent ones. NM-TRAN does not reject a non-numeric DV — it reads BLQ as zero and fits it as a genuine zero concentration. A blank covariate becomes 0, so a missing body weight becomes a 0 kg patient. ADDL without II places no additional doses. Records sharing a timestamp are applied in file order, so whether a level is pre- or post-dose depends on which row came first. None of these stop a run.

severity  check                            detail
error     non-numeric DV                   DV contains text... NM-TRAN reads them as 0
error     subject with no dose             1 subject(s) have observations but no dose: 2
error     TIME not sorted                  1 subject(s) have out-of-order TIME: 1
error     covariate WT missing             1 record(s) have no value...
warning   duplicate TIME within a subject  NONMEM applies them in file order...

For the estimation itself, this skill does not reimplement NLME — see references/population-pk.md for estimation methods, the BLQ M1-M7 methods, covariate model building, and the diagnostics that decide whether a model is acceptable, and references/software-ecosystem.md for which tool to reach for.

4. Simulation and regimen selection

python3 simulate_regimen.py --cl 5 --v 40 --dose 500 --interval 12 --n-doses 10 --steady-state
python3 simulate_regimen.py --cl 5 --v 40 --dose 500 --interval 12 --n-doses 10 \
    --simulate 2000 --omega-cl 0.35 --omega-v 0.25 --target-trough 4.0

Deterministic simulation answers "what does the typical patient look like", which is almost never the question:

metric             p5        p25      median   p75      p95      geo_mean
peak               11.861    14.6018  16.6702  19.1476  22.9339  16.6453
trough             0.863226  2.15191  3.64412  5.49655  9.21053  3.27035

target       fraction_attaining
trough >= 4  0.444

The typical trough is 3.6 and the target is 4, so 44% of the population attains it. A regimen tuned on the typical patient leaves about half the population on the wrong side of the target. Reported attainment is still optimistic here: this is between-subject variability only, with no residual or between-occasion component.

Linear models are solved analytically and superposed, which is exact. --nonlinear switches to integrated Michaelis-Menten elimination, where superposition is invalid and multiple-dose behaviour cannot be inferred from a single dose at all.

5. Exposure-response

python3 exposure_response.py --emax -i er.csv --sigmoid
python3 exposure_response.py --cqtc -i qt.csv --cmax 250

The Emax fit reports fraction_of_emax_reached and flags a fit whose plateau is outside the data. When the highest observed exposure reaches only a third of the estimated Emax, Emax and EC50 are extrapolations that are strongly correlated with each other; quoting them as independent estimates is not supportable, and a "linear" exposure-response is simply the low-concentration limb of the same curve.

--cqtc evaluates the upper bound of the two-sided 90% confidence interval of predicted placebo-corrected change-from-baseline QTc against the 10 ms threshold, which is the question ICH E14 actually asks. A point estimate, or a 95% interval, answers a different one. The bundled model is an ordinary linear regression for screening; a submission-grade C-QTc analysis needs a mixed model with random intercept and slope per subject.

Every mode carries the same caveat, because it is the one that gets forgotten: patients are randomised to dose, not to exposure. Exposure-response across quantiles is observational even inside a randomised trial, and can reflect the covariates that drive clearance.

6. Bioequivalence

python3 bioequivalence.py -i be.csv --design 2x2 --metric AUC
python3 bioequivalence.py -i be.csv --design replicate --metric Cmax --scaling both
python3 bioequivalence.py --power --cv 0.30 --gmr 0.95 --target-power 0.80

Three criteria share the word "bioequivalence" and are not interchangeable: average BE (90% CI inside 80.00-125.00%), EMA's ABEL (limits widened as a function of CVwR, capped at 69.84-143.19%, point estimate still within 80-125%), and FDA's RSABE (a scaled linearised bound via Hyslop's method, not an interval at all). --scaling refuses to run on a 2x2 design:

error: reference-scaling requires --design replicate. High observed variability in a 2x2 study
does not license widening: without replicated reference administrations there is no estimate of
within-subject reference variability to scale to.

Sample size reproduces the published tables exactly (CV 30%, GMR 0.95, 80% power → N = 40 for a 2x2). Power is computed by integrating over the sampling distribution of the estimated standard deviation rather than treating the standard error as known — the normal approximation overstates power at realistic sample sizes. Note that N is driven far more by the assumed GMR than by CV; assuming 1.00 instead of 0.95 roughly halves the calculated N and is the usual reason a BE study comes in underpowered.

7. Scaling, paediatrics, and first-in-human

python3 allometry_and_fih.py --scale --cl 5 --weight-from 70 --weight-to 6 --pma-weeks 44
python3 allometry_and_fih.py --fih --noael rat=50,dog=10 --safety-factor 10

Scaling by size alone below about 2 years of age overpredicts clearance, in a neonate by several fold, because clearance is limited by enzyme and renal maturation rather than by size. Supplying --pma-weeks adds the Anderson-Holford sigmoidal maturation term; omitting it below 20 kg raises a finding.

parameter  reference  exponent  size_scaled  maturation_factor  final
CL         5          0.75      0.792063     0.30634            0.242641
V          40         1         3.42857      1                  3.42857

Size alone would predict 0.79 L/h; with maturation at 44 weeks post-menstrual age it is 0.24 L/h, a 3.3-fold difference. Volume is not matured — maturation describes eliminating capacity, not distribution space.

--fih uses the body-surface-area conversion from FDA's 2005 maximum-safe-starting-dose guidance and always emits a finding that a NOAEL-derived MRSD is not sufficient on its own for agonist immunomodulators: compute MABEL with --mabel and take the lower value.

8. Drug interactions

python3 ddi_static.py --basic --ki 0.5 --imax 2.0 --fu 0.05 --dose 0.4
python3 ddi_static.py --msm --ki 0.5 --imax 2.0 --fu 0.05 --dose 0.4 --fm 0.9 --fg 0.7

ICH M12 basic models with their cut-offs (R1 ≥ 1.02 hepatic, ≥ 11 intestinal; R2 ≥ 1.25 for TDI; R3 ≤ 0.8 for induction; transporter cut-offs by site), plus the mechanistic static model. The basic models are deliberately conservative: a negative is meaningful, a positive is a trigger for further work, not a prediction of clinical magnitude.

The mechanistic static model reports the ceiling alongside the prediction:

note: With fm = 0.9, no inhibitor of this pathway can raise the victim AUC above 10.00-fold. If
the prediction approaches that ceiling, fm is doing more work than the inhibition constants.

fm and Fg dominate the answer far more than the inhibition constants, and are usually the least well established numbers in the calculation.

9. Therapeutic drug monitoring

python3 tdm_bayes.py --model vancomycin-adult --weight 80 --crcl 75 \
    --dose 1500 --interval 12 --level [email protected] --level 42@2 --target-auc24 500

MAP Bayesian estimation shrinks towards the population when the data are uninformative and follows the data when they are not, which is why it beats both a trough read against population parameters and log-linear regression on two points. A single level raises a finding: it cannot separate clearance from volume, and whichever parameter the sample is uninformative about has simply returned its prior.

The bundled vancomycin parameterisation is explicitly labelled illustrative. Substitute a model validated in your population before the output means anything.

Software ecosystem

Verified against live sources on 2026-07-27; see references/software-ecosystem.md for the full map and references/source-ledger.md for provenance.

• Pharmpy 2.1.1 (2026-05-19) is the practical Python entry point — model-agnostic, drives NONMEM/nlmixr2/rxode2, and ships 19 run_* tools including run_amd, run_modelsearch, run_covsearch, run_structsearch, run_pdsearch, run_modelrank, run_vpc and run_qa. Two breaking changes are recent enough to catch you out: 2.0.0 (2026-02-12) changed dataset row indices to start at 1, and 2.1.0 (2026-05-08) renamed add_placebo_model to set_placebo_model and now requires numpy ≥ 2.
• NONMEM 7.6 (user guides dated November 2025) remains the regulatory default. New since 7.5: ADVAN16 (RADAR5 implicit Runge-Kutta for stiff delay differential equations), ADVAN17 (stiff delay differential-algebraic), NUTS Bayesian sampling, and SAEM storage of individual samples.
• nlmixr2 (requires rxode2 ≥ 5.0.0) is the credible open-source NLME alternative; babelmixr2 and monolix2rx translate models between it, NONMEM and Monolix.
• PKPy (PeerJ, 2025) is a Python popPK framework but is GitHub-only — not on PyPI, so pip install pkpy fails. chi-drm (1.0.3) is on PyPI for Bayesian PKPD.
• Open Systems Pharmacology Suite v12 (PK-Sim/MoBi) is the open-source PBPK platform; Simcyp and GastroPlus are the commercial ones. ospsuite is R-only and needs .NET 8.

Python has no mature NCA or NLME package of regulatory standing. That gap is why this skill ships its own validated NCA and fitting implementations rather than wrapping one.

What this skill exists to prevent

1. Lambda_z chosen by plain r-squared, or fitted through Tmax.
2. AUCinf reported from a profile where 25% of it was extrapolated.
3. AIC allowed to select a compartment whose intercompartmental clearance has 98% RSE.
4. Reweighting used to fix non-random residuals, which are a structural problem.
5. BLQ left in a DV column, where NM-TRAN reads it as a real zero.
6. A regimen chosen on the typical patient, with no attainment estimate for the population.
7. Emax and EC50 quoted as independent estimates when the plateau was never observed.
8. Reference-scaled bioequivalence limits applied to a 2x2 study.
9. Allometric scaling to a neonate with no maturation term.
10. An MRSD from a NOAEL used as the starting dose for an agonist immunomodulator.

References

• references/nca-conventions.md — parameter definitions, lambda_z rules, BLQ handling, steady state
• references/structural-models.md — closed-form solutions, parameterisations, NONMEM ADVAN/TRANS map
• references/population-pk.md — NLME estimation, covariate building, BLQ M1-M7, diagnostics, VPC
• references/pd-and-exposure-response.md — Emax, indirect response, effect compartment, ER analysis
• references/tmdd-and-biologics.md — TMDD approximations, monoclonal antibody PK, immunogenicity
• references/pbpk.md — when PBPK earns its cost, platforms, and what verification requires
• references/bioequivalence.md — designs, ABE/ABEL/RSABE, ICH M13 series, highly variable drugs
• references/special-populations.md — paediatrics, renal and hepatic impairment, obesity, pregnancy
• references/dataset-standards.md — CDISC PC/PP and ADPC/ADPP, NONMEM data items, common defects
• references/ddi-and-qt.md — ICH M12 stepwise assessment, static models, ICH E14/S7B C-QTc
• references/antimicrobial-and-tdm.md — PK/PD indices, PTA/CFR, vancomycin AUC-guided dosing, MIPD
• references/software-ecosystem.md — every tool, what it is for, licensing, and verified versions
• references/regulatory-guidance.md — the guidance ledger with dates, status, and what each requires
• references/source-ledger.md — provenance and research dates for every claim in this skill

Assets

• assets/popk-analysis-plan.md — population analysis plan structure, with the decisions stated up front
• assets/nca-reporting-checklist.md — what an NCA report has to state for the numbers to be interpretable