Population PK Dosing Decisions from the Posterior (Concept Note)

Summary

Concept note, no raw source (Gelman, Bois & Jiang 1996 was not retrievable as an open copy). Once a population PK model is fit, a dosing rule is a decision problem: choose dose $D$ to maximise expected utility, or to maximise the probability of target attainment (PTA) that a patient’s exposure lies in a therapeutic window, integrating over between-patient variability and posterior parameter uncertainty. This is the “dosing decisions” half of example E64.

Overview

Two layers of uncertainty need separating: (i) between-patient variability (the population distribution ${\vartheta }_j\sim \text{logN}(\log {\vartheta }_{pop},\Omega )$) and (ii) parameter uncertainty about ${\vartheta }_{pop},\Omega ,\sigma$ (the posterior). For a new patient the predictive for exposure marginalizes both. For a known patient with observed concentrations, the individual posterior $p({\vartheta }_j\mid y_j,\text{population})$ supports individualized dosing (therapeutic drug monitoring).

Main Content

Exposure target and PTA

Let $E(D,\vartheta )$ be an exposure summary (e.g., average steady-state concentration $C_{ss,avg}=D/(\tau CL)$ for dose $D$ every $\tau$ hours; peak or trough concentration; AUC$=D/CL$ per dose). With window $[L,U]$,

$$\text{PTA}(D)=\mathrm{Pr}(L\le E(D,\vartheta )\le U)=\iint \mathbb{I}\{L\le E(D,\vartheta )\le U\}p(\vartheta \mid {\vartheta }_{pop},\Omega )p({\vartheta }_{pop},\Omega \mid y)d\vartheta d({\vartheta }_{pop},\Omega ).$$

Expected-utility dosing rule

$$D^{\ast }=\arg \underset{D}{\max }\mathbb{E}[u(E(D,\vartheta ))],$$

with a utility penalising sub-therapeutic exposure (failure) and toxic exposure asymmetrically. Posterior draws give the Monte Carlo estimate by averaging $u$ over simulated patients. Changing the loss changes the optimal dose; report the PTA curve in addition to $D^{\ast }$.

Algorithm (posterior simulation). For each posterior draw $s$: draw a new patient’s ${\vartheta }^{(s)}\sim \text{logN}(\log {\vartheta }_{pop}^{(s)},{\Omega }^{(s)})$, scale by body weight with the allometric rules, compute $E(D,{\vartheta }^{(s)})$ for each candidate $D$, evaluate $u$ or the window indicator. Average over $s$. In Stan this is a generated quantities block; in PyMC use pm.sample_posterior_predictive with a function of the posterior (Margossian 2022 - Population Two-Compartment Model).

Weight-based dosing. With allometric scaling $CL\propto w^{0.75}$, a fixed dose over-exposes light patients; per-kg dosing under-corrects because it scales as $w^1$ vs $w^{0.75}$. Compare PTA across weight strata to choose a rule.

Pharmacodynamic link. With PKPD (e.g. warfarin PK/PD in E64), the target is a response (e.g. INR in range, neutrophil nadir above a threshold) and requires the nonlinear ODE solves of Margossian 2022 - Nonlinear PKPD and ODE Solvers.

Examples

PTA for a q12h regimen (runs; verified). Clearance $C{L}_i=10e^{0.3z_i}$ L/h, target $C_{ss,avg}\in [5,15]$ mg/L, $C_{ss,avg}=D/(12C{L}_i)$:

import numpy as np
rng = np.random.default_rng(0); n = 20000; CL = 10.
CLi = CL*np.exp(0.3*rng.normal(size=n))
for D in (600, 900, 1200, 1800):
    css = D/12/CLi
    print(D, np.round([np.mean((css >= 5) & (css <= 15)), np.mean(css < 5), np.mean(css > 15)], 3))
# 600: [0.502 0.498 0.   ]   900: [0.902 0.087 0.011]   1200: [0.906 0.01 0.084]   1800: [0.498 0. 0.502]

Interpretation. PTA peaks at about 0.90 for 900-1200 mg q12h; 900 errs toward under-exposure (8.7% below, 1.1% above), 1200 toward over-exposure (1.0% below, 8.4% above). If toxicity is costlier than sub-therapeutic exposure, choose near 900; otherwise near 1200. The decision depends on utilities, not just the model. Adding posterior uncertainty about $C{L}_{pop}$ and $\omega$ widens the exposure distribution and lowers the maximum PTA.

Relevance to marketing measurement

The same structure turns an MMM posterior into a budget decision: simulate outcomes for each candidate budget over posterior draws (and geo-level variability), score with a utility, and report the probability of hitting a target. See Decision Analysis and Using a Fitted Model for Decision Analysis - Classification Competition.

Connections

Hierarchical Models, Posterior Predictive Checking, Margossian 2022 - Posterior Predictive Checks and LOO.

See Also

Margossian 2022 - Compartment Models and Event Schedules, Margossian 2022 - Bayesian Pharmacometrics - Overview.