Example: population toxicokinetics#

Part 5 · Stage 15 · 🌊 Basis Functions & Gaussian Processes · Lesson 125 of 144 · advanced

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Important

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A mechanistic model, fit hierarchically#

How does an inhaled solvent move through the body — absorbed, distributed among tissues, metabolised, exhaled? Gelman, Bois and Jiang answered this for tetrachloroethylene with a model that fuses three threads of this book: a mechanistic differential-equation model of physiology, a hierarchical treatment of person-to-person variation, and informative priors grounded in real biology.

The physiological model#

The body is divided into compartments — well-perfused tissues, poorly-perfused tissues, fat, and the liver where metabolism occurs — and the flow of the compound between them is governed by a system of ordinary differential equations. The parameters are physical quantities: blood flows, tissue volumes, partition coefficients, metabolic rates. The concentration over time is the ODE solution, and the likelihood compares it to measured blood or breath concentrations.

\[\frac{dC_{\text{tissue}}}{dt} = \frac{Q_{\text{tissue}}}{V_{\text{tissue}}} \Bigl(C_{\text{art}} - \frac{C_{\text{tissue}}}{\lambda_{\text{tissue}}}\Bigr) - (\text{metabolism}),\]

one such equation per compartment, coupled through the arterial concentration.

Hierarchy and physiological priors#

Each subject has their own physiological parameters, drawn from a population distribution — partial pooling, so a subject with sparse data borrows from the group. And because every parameter is a genuine physiological variable, informative priors are not a convenience but a duty: a tissue volume has a known plausible range, a blood flow a measured typical value. The priors carry decades of physiology.

import pymc as pm
with pm.Model():
    # population means of log physiological parameters, with literature-based priors
    mu = pm.Normal("mu", mu=prior_means, sigma=prior_sds, shape=n_params)
    tau = pm.HalfNormal("tau", 1, shape=n_params)
    theta = pm.Normal("theta", mu, tau, shape=(n_subj, n_params))   # per-subject, pooled
    C_pred = solve_pbpk_odes(theta, dosing)                         # ODE solution
    pm.Lognormal("y", mu=pm.math.log(C_pred), sigma=sigma, observed=concentration)

Why it is a landmark#

The analysis shows Bayesian inference automatically propagating the uncertainty in a large, mechanistic, hierarchical model — the many parameters, the ODE nonlinearity, the person-to-person spread, all carried into the predictions without approximation. Posterior predictive simulation checks the fit and its sensitivity to the priors, exactly the discipline of Stage 6. The lesson for Part V: nonlinearity here is scientific, dictated by physiology, not a flexible curve — and Bayesian methods let a genuine mechanistic model be fit with honest uncertainty. The lessons that follow relax that structure, replacing mechanism with flexible basis functions when the true form is unknown.

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