Bayesian Updating of Time-Varying PK Parameters

Why Treat an Individual’s PK Parameters as a State?

An individual’s clearance or absorption characteristics need not remain fixed throughout treatment. A model with one constant individual parameter can miss gradual changes, while estimating a completely unrelated parameter at every sampling occasion discards the fact that nearby occasions are likely to be similar. A state-space model represents the parameter as evolving over time: it allows change, but constrains that change through an explicit transition model.

The central task here is to update the distribution of the current latent PK state when a new, possibly sparse, concentration observation arrives. The state is not itself observed. Rather, the concentration depends on the state through a PK model and is also subject to residual measurement error. Keeping these two sources of variation distinct is essential.

Specify the State and the Two Models

For positive PK parameters, a log scale is often convenient because it keeps the implied parameter positive and makes additive Gaussian changes correspond to multiplicative changes on the original scale. For example, let \(x_t=\log(CL_t/5)\), where \(CL_t\) is clearance in L/h and 5 L/h is a reference value. Then \(CL_t=5\exp(x_t)\) L/h. The reference is a scaling constant, not a claim that clearance must equal 5 L/h.

A general linear Gaussian transition model is

Here \(m\) is the long-run or reference state, \(A\) controls persistence, and \(Q_t\) is the covariance of process variation over the time interval. In a scalar mean-reverting model, \(A=\rho\), with \(0\leq\rho<1\): deviations from \(m\) tend to persist but shrink with time. A random walk is a different choice, \(x_t=x_{t-1}+w_t\), which has no mean-reversion term. The transition variance should reflect the elapsed time; it is generally not appropriate to assign the same process variance to a one-day and a ten-day gap without justification.

Given data up to the preceding occasion, suppose the current state distribution before observing the new concentration is \(x_t\mid\text{past}\sim N(a_t,P_t)\). For an observation \(y_t\), such as a log concentration, write a linear observation model as

The predicted observation at the prior state mean is \(c_t+H_ta_t\), and the innovation, or prediction error, is \(r_t=y_t-(c_t+H_ta_t)\). \(H_t\) describes how the observation changes with the state; \(R_t\) is residual measurement variance. In nonlinear PK models, the concentration prediction is generally a nonlinear function \(g_t(x_t)\). A local linear approximation around \(a_t\) uses \(H_t\), the derivative of \(g_t\) at that point. That approximation is an assumption to check, not an identity.

Gaussian Updating and Its Justification

Under a Gaussian prior state, a linear observation model, and independent Gaussian process and residual errors, the current state and observation have a joint Gaussian distribution. The posterior mean and covariance after observing \(y_t\) are

Here \(S_t\) is the predictive covariance of the observation, and \(K_t\) is the gain that converts an observation discrepancy into a state adjustment. For a scalar state and observation, these become \(S_t=H_t^2P_t+R_t\) and \(K_t=P_tH_t/S_t\). The posterior covariance cannot increase in this exact linear Gaussian update: the subtracted term is positive semidefinite. This does not mean the model’s uncertainty about future states can never increase; later process variation adds uncertainty again.

The update follows from conditioning a joint normal distribution. In particular, the covariance between \(x_t\) and \(y_t\) is \(P_tH_t^\mathsf{T}\), and the variance of \(y_t\) is \(S_t\). The conditional mean of a jointly normal vector is its prior mean plus the cross-covariance times the inverse observation covariance times the observation deviation. Substituting those two covariances gives \(a_t+P_tH_t^\mathsf{T}S_t^{-1}r_t\), which is the stated mean formula. The conditional covariance subtracts \(P_tH_t^\mathsf{T}S_t^{-1}H_tP_t\), giving the stated covariance formula.

For a nonlinear concentration model, this Gaussian update is exact only if the observation relation is linear (or otherwise yields the same conditional distribution). Applying it after linearizing a nonlinear PK model is an approximation. When the likelihood is strongly nonlinear, asymmetric, censored, or multimodal, a more faithful numerical representation of the posterior may be needed; importance sampling is one approach introduced in the earlier material on individual Bayesian forecasts.

What Controls the Size of the Update?

The innovation determines whether the data pull the state upward or downward only in combination with the sensitivity \(H_t\) and the signs in \(K_t\). In the scalar case, \(K_t=P_tH_t/(H_t^2P_t+R_t)\). If \(H_t<0\), as for log concentration decreasing with log clearance under a simple elimination-dominated relationship, a negative innovation gives a positive state adjustment. If \(H_t>0\), the direction reverses.

The gain also expresses how much weight the update gives the new observation relative to the state prior. Larger prior uncertainty \(P_t\) generally permits more movement in response to data; larger residual variance \(R_t\) generally reduces it. This is meaningful only when the prior and residual variances are calibrated and the observation is informative about the state. A sample taken at a time when the concentration is nearly insensitive to a parameter may have little ability to identify that parameter, even if its residual error is small.

Assumptions, Diagnostics, and Alternatives

The Gaussian formulas require a defensible transition distribution, an adequate observation model, and correctly represented uncertainty. For the exact formulas, transition errors and residual errors must have the specified distributions and covariance structure; in particular, the derivation above assumes they are independent. A nonlinear PK model additionally requires that a local approximation, if used, represent the observation response over the relevant state range. Dose times, amounts, and other inputs used to calculate the concentration prediction must also be correctly specified.

Useful checks include comparing observed concentrations with their predictive distributions, examining whether innovations show systematic trends, and checking whether posterior states repeatedly reach implausible ranges. Repeated innovations with the same sign may indicate a missing trend, an inappropriate transition mean, or an observation model that systematically misses concentration changes. Innovations that are much larger than their predicted standard deviations may indicate outliers, residual variance misspecification, incorrect dose records, or a poor local approximation. These checks diagnose incompatibility; they do not identify its cause by themselves.

A practical alternative is to keep individual parameters constant and represent repeated-occasion variation with occasion-specific effects, as covered in the earlier material on occasion-specific effects in repeated individual PK data. That approach suits changes that are transient or specific to an occasion, rather than gradual evolution of an underlying state. A state-space model instead links states across time and assigns uncertainty to their evolution. At the other extreme, estimating a separate unconstrained state at every occasion can overfit sparse concentrations and ignore temporal continuity. The choice should reflect the biological time scale and the sampling design, not merely which model gives a more flexible fit.

A state-space model cannot create information absent from the observations. With sparse sampling, clearance may remain confounded with volume, bioavailability, absorption, adherence, or residual error, depending on the design and concentration model. A Gaussian posterior can also be misleading when the likelihood is asymmetric or has multiple plausible parameter combinations. If the local sensitivity changes sharply across the state distribution, or if a Gaussian update gives implausible values, inspect the nonlinear likelihood and consider a method that represents its shape more fully. The update describes uncertainty conditional on the model; it does not account automatically for structural model uncertainty.