Constructing Individual Likelihoods for Nonlinear PK Data

From a PK Prediction to a Likelihood

A nonlinear PK model maps a patient’s parameter values and dosing history to a predicted concentration at each sampling time. A likelihood adds a second ingredient: a model for how an observed concentration differs from that prediction. The prediction model alone does not say how surprising a measured value is, and a residual-error model alone does not say what concentration to expect. An individual likelihood joins the two.

For patient \(i\), let \(\boldsymbol{\theta}_i\) denote the patient-specific PK parameter vector, \(\mathcal{D}_i\) the recorded dosing history, \(t_{ij}\) the time of concentration measurement \(j\), and \(C_{ij}\) the observed concentration. Write the model prediction as \(C_{\mathrm{pred}}(t_{ij};\boldsymbol{\theta}_i,\mathcal{D}_i)\). The prediction may come from an algebraic formula or from numerically solving PK differential equations. In either case, it must use the relevant dose amounts, dose times, sample times, and model assumptions.

The likelihood depends on assumptions. The prediction model and dose history must be specified adequately; the chosen residual-error distribution must be appropriate for the data; and the dependence structure between observations must be stated. If observations are assumed conditionally independent given the PK parameters and dosing history, the joint likelihood is a product of the individual observation densities. That product is not justified merely because the measurements were taken at different times.

Choosing a Residual-Error Model

Let \(C_j\) be the concentration observed at time \(t_j\), and abbreviate its prediction as \(m_j=C_{\mathrm{pred}}(t_j;\boldsymbol{\theta},\mathcal{D})\). Under additive normal error, the observation is modeled as \(C_j=m_j+\epsilon_j\), where \(\epsilon_j\sim N(0,\sigma^2)\). This assumes errors have constant variance on the concentration scale and can take either positive or negative values. It is most plausible when variability is roughly constant in absolute concentration units and predictions are sufficiently far from zero that negative simulated observations are not a material concern.

A proportional normal-error model instead sets the residual standard deviation in proportion to the prediction: \(C_j\mid\boldsymbol{\theta},\mathcal{D}\sim N(m_j,(\sigma_p m_j)^2)\), where \(\sigma_p\) is a dimensionless proportional-error coefficient. This represents variability that grows with concentration. It still assigns some probability to negative values, and its variance approaches zero as the predicted concentration approaches zero; both features should be considered when deciding whether it is appropriate.

For positive concentrations with multiplicative variability, a lognormal model is one option. If \(\log C_j=\log m_j+\epsilon_j\), with \(\epsilon_j\sim N(0,\omega^2)\), then \(m_j\) is the conditional median, not the conditional arithmetic mean. The density on the original concentration scale includes a \(1/C_j\) factor because the logarithm changes the measurement scale.

A combined additive-plus-proportional model is also common: the conditional variance is \(\sigma_a^2+(\sigma_p m_j)^2\). It allows an absolute-error component to matter at low concentrations and a concentration-dependent component to matter at higher concentrations. The error structure should be selected from scientific understanding and data diagnostics, not simply because one formula is familiar.

Likelihoods for Several Sampling Times

Suppose \(J\) concentrations are observed for one patient. Under conditional independence, each with additive normal error and potentially time-specific standard deviation \(\sigma_j\), the joint likelihood is the product of the \(J\) normal densities. Taking logarithms converts the product into a sum, which is usually more stable numerically and easier to optimize. For a common \(\sigma\), the log likelihood is a constant minus a weighted sum of squared residuals. If \(\sigma\) is fixed, parameter values that produce smaller squared residuals have larger likelihood under this model.

The independence assumption concerns residuals after conditioning on the PK parameters and known design information. It may fail when samples share assay or handling effects, when there is serially correlated unexplained variability, or when an important time-varying process is omitted. In those cases, multiplying marginal densities can understate or misrepresent the information in the data; a joint error model is needed.

Why the Likelihood Construction Works

The product formula follows directly from the definition of conditional independence. For two observations, conditional independence means that their joint density equals the product of their conditional densities. Applying that definition repeatedly gives the product for \(J\) observations. For normal residuals, substituting the normal density into that product yields the displayed likelihood; taking a logarithm uses \(\log\prod_j a_j=\sum_j\log a_j\). This is a mathematical consequence of the stated model, not evidence that independence or normality is true for a particular data set.

For lognormal errors, the density also follows from a change of variable. Set \(Z_j=\log C_j\); the normal density describes \(Z_j\), but a density for \(C_j\) must account for \(dZ_j/dC_j=1/C_j\). Multiplying the normal density of \(\log C_j\) by this Jacobian produces the \(1/C_j\) term. Omitting it gives a density on the log scale, not the likelihood density for an observation recorded on the concentration scale.

Diagnostics, Alternatives, and Limitations

After calculating predictions, inspect residuals \(C_j-C_{\mathrm{pred},j}\) against sampling time, predicted concentration, and dose timing. A systematic trend over time can indicate a structural-model or dosing-history mismatch; residual spread that grows with prediction can indicate that constant additive variance is inadequate. A few unusually large residuals may reflect data-quality issues, an inappropriate error model, or genuine variability. Residual plots can reveal tension with assumptions, but do not by themselves identify the cause.

Before interpreting an apparent model mismatch, verify sample times, dose amounts and times, units, and whether the recorded concentration corresponds to the modeled analyte and sampling event. The earlier material on analytical uncertainty and contaminated samples is relevant when a measurement itself may be unreliable. Assumptions about dose history and interpretation also matter: an incorrect input can produce an apparently poor likelihood even if the PK equations are otherwise suitable.

The likelihood is conditional on the model and assumptions; it does not repair misspecified PK structure, uncertain dose records, or an unsuitable residual distribution. With sparse observations, several individual parameter combinations may yield similar predictions, so a likelihood can be broad or poorly identified. A likelihood value also cannot be interpreted in isolation as a probability that the model is correct, and raw likelihood values from different data scales or different observation sets are not directly comparable without an appropriate comparison framework.

This construction supplies the data component needed for individual PK inference. A prior distribution over parameters is a separate ingredient; the likelihood itself contains no prior information and is not a posterior distribution.