Tutorial: Bayesian PK

Theory

See the Theory Guide for the MAP log-posterior derivation, MCMC posterior sampling, and Bayesian BE formulation.

This tutorial covers MAP individual PK estimation, full Bayesian posterior sampling, and Bayesian 2x2 crossover bioequivalence using OpenPKFlow.

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1. Setting up priors

PKPrior defines log-normal priors for a 1-compartment model. The defaults are weakly informative and suitable for many small-molecule drugs:

from openpkflow.bayes import PKPrior

# Default priors
prior = PKPrior()
print(f"CL prior: log-normal(mean={prior.log_cl_mean:.3f}, sd={prior.log_cl_sd})")
print(f"Vz prior: log-normal(mean={prior.log_v_mean:.3f}, sd={prior.log_v_sd})")

Tighten priors when you have population estimates from prior studies:

prior = PKPrior(
    log_cl_mean=1.791,   # log(6.0) --> prior CL ~ 6 L/h
    log_cl_sd=0.3,       # tighter SD
    log_v_mean=3.688,    # log(40.0) --> prior Vz ~ 40 L
    log_v_sd=0.3,
)

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2. MAP individual PK estimation

MAP estimation uses L-BFGS-B optimisation and works with the base install (no extras required). It is suitable for sparse-sampling scenarios with 3-5 data points:

from openpkflow.bayes import map_individual_pk

times = [0.5, 1.0, 2.0, 4.0, 8.0]
concentrations = [4.4, 9.8, 16.6, 16.1, 4.8]
dose = 100.0

result = map_individual_pk(
    times,
    concentrations,
    dose,
    route="oral",
    prior=prior,
    subject="S01",
)

print(result.summary())

The summary() output includes:

• MAP parameter estimates with standard errors
• Derived PK parameters (t1/2, AUCinf, Cmax, Tmax)
• Convergence diagnostics (gradient norm, Hessian condition)
• Any diagnostic warnings

Interpreting the output

if not result.converged:
    print("WARNING: Optimiser did not converge. Do not interpret as MAP.")

if not result.uncertainty_reliable:
    print("WARNING: Standard errors are unreliable (non-positive-definite Hessian).")

for warning in result.warnings:
    print(f"[!] {warning}")

print(f"CL_F = {result.CL_F:.2f} +/- {result.CL_F_se:.2f} L/h")
print(f"Vz_F = {result.Vz_F:.2f} +/- {result.Vz_F_se:.2f} L")
print(f"ka   = {result.ka:.3f} +/- {result.ka_se:.3f} 1/h")
print(f"t1/2 = {result.half_life:.2f} h")
print(f"AUCinf = {result.AUCinf:.2f} h*mcg/mL")

Plotting and reporting

result.plot(show=True)

result.report("map_report.html")
result.report("map_report.md", format="markdown")

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3. Full Bayesian posterior sampling

For full uncertainty quantification, use bayes_individual_pk. This requires PyMC (pip install openpkflow[bayes]):

from openpkflow.bayes import bayes_individual_pk

result = bayes_individual_pk(
    times,
    concentrations,
    dose,
    route="oral",
    prior=prior,
    n_samples=2000,
    tune=1000,
    chains=2,
    subject="S01",
)

print(result.summary())

The output includes posterior means with 95% credible intervals:

print(f"CL_F posterior mean = {result.cl_mean:.2f} [95% CrI: "
      f"{result.cl_95ci[0]:.2f}, {result.cl_95ci[1]:.2f}] L/h")
print(f"Vz_F posterior mean = {result.v_mean:.2f} [95% CrI: "
      f"{result.v_95ci[0]:.2f}, {result.v_95ci[1]:.2f}] L")

Shrinkage assessment

The shrinkage metric tells you how much the posterior is pulled toward the prior (0 = data-dominated, 1 = prior-dominated):

print(f"CL shrinkage toward prior: {result.shrinkage_cl:.1%}")

High shrinkage (> 50%) indicates sparse or noisy data.

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4. Convergence diagnostics

PyMC-based results compute effective sample size (ESS) and R-hat automatically. Diagnostic warnings are included in result.warnings:

for w in result.warnings:
    print(f"[!] {w}")

Guidelines:

• R-hat < 1.05: Chains have converged. Values > 1.05 indicate non-convergence; increase tune or n_samples.
• ESS > 100 * chains: Effective sample size is adequate. Low ESS means the chain is highly autocorrelated; increase n_samples or tune.

If you need direct access to posterior samples for custom diagnostics:

import numpy as np
cl_samples = result.cl_posterior
v_samples = result.v_posterior

print(f"CL CV: {np.std(cl_samples) / np.mean(cl_samples):.1%}")
print(f"CL-V correlation: {np.corrcoef(cl_samples, v_samples)[0, 1]:.3f}")

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5. Bayesian BE analysis

bayes_be fits a Bayesian 2x2 crossover model via PyMC NUTS and computes P(0.80 <= GMR <= 1.25) -- the posterior probability of bioequivalence.

First, prepare a DataFrame with the required columns:

import pandas as pd

data = pd.DataFrame({
    "subject": [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8,
                9, 9, 10, 10, 11, 11, 12, 12],
    "sequence": ["RT", "RT", "RT", "RT", "RT", "RT", "RT", "RT", "RT", "RT",
                 "TR", "TR", "TR", "TR", "TR", "TR", "TR", "TR", "TR", "TR",
                 "TR", "TR", "TR", "TR"],
    "period": [1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2,
               1, 2, 1, 2, 1, 2, 1, 2],
    "treatment": ["R", "T", "R", "T", "R", "T", "R", "T", "R", "T",
                  "T", "R", "T", "R", "T", "R", "T", "R", "T", "R",
                  "T", "R", "T", "R"],
    "value": [105, 110, 88, 92, 120, 115, 95, 100, 78, 85,
              118, 112, 102, 98, 85, 80, 130, 125, 92, 88,
              107, 103, 76, 72],
})

Run the analysis:

from openpkflow.bayes import bayes_be

result = bayes_be(
    data,
    metric="AUC",
    n_samples=2000,
    tune=1000,
    chains=2,
)

print(result.summary())

The summary displays side-by-side Bayesian and frequentist results:

print(f"Bayesian P(BE) = {result.p_be:.3f}")
print(f"  GMR = {result.gmr_mean:.4g} [95% CrI: "
      f"{result.gmr_95ci[0]:.4g}, {result.gmr_95ci[1]:.4g}]")
print(f"Frequentist GMR = {result.freq_gmr:.4g} [90% CI: "
      f"{result.freq_90ci[0]:.4g}, {result.freq_90ci[1]:.4g}]")

Bayesian BE reporting

result.report("bayes_be_report.html")
result.report("bayes_be_report.md", format="markdown")

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6. Choosing MAP vs full Bayesian

Consideration	MAP (map_individual_pk)	Full Bayesian (bayes_individual_pk)
Dependencies	Base install only	Requires openpkflow[bayes] (PyMC)
Runtime	< 1 second	30-120 seconds (MCMC)
Output	Point estimates + SEs	Full posterior distributions
Uncertainty	Quadratic approximation (Hessian)	Exact posterior via MCMC
Uses	Quick fit, diagnostics, reports	Full inference, credible intervals

MAP estimation is the recommended starting point. Move to full Bayesian sampling when you need posterior probabilities, asymmetric credible intervals, or full uncertainty propagation.

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Regulatory notes

• Bayesian PK results are exploratory. Regulatory submissions require validated population PK software (NONMEM, Monolix, Phoenix NLME).
• Bayesian BE is not accepted by most regulators as a standalone BE method. The frequentist 90% CI is provided alongside for comparison.
• MAP estimation is useful for study design, model diagnostics, and internal decision-making. Do not submit MAP-derived parameters in lieu of a full population PK analysis.