Endpoint-Specific Exposure Effects And Shared PK

One Exposure History, Distinct Endpoint Effects

When several outcomes are measured in the same person, the drug exposure driving them is ordinarily not a separate quantity for each outcome. A person's PK model supplies one concentration history, while the pharmacodynamic relationship to that history can differ by endpoint. One endpoint may change substantially with concentration, another only modestly, and individuals may differ in those endpoint-specific responses. The central modeling task is to represent those distinctions without duplicating or contradicting the shared PK history.

This setup is useful when endpoints have different scales or outcome distributions, yet are observed under the same dosing and concentration profile. It also provides a clear interpretation of dependence: people with a steeper-than-average response for one endpoint may tend to have a steeper-than-average response for another. That tendency is a correlation between individual exposure-response slopes, not a claim that the endpoints have the same drug effect.

Model Specification and Interpretation

Let \(i\) index individuals, \(j=1,\ldots,J\) index endpoints, and \(t\) denote observation time. A population PK model, with parameters \(\theta_{\mathrm{PK}}\) and individual PK effects \(\eta_i\), generates \(C_i(t;\theta_{\mathrm{PK}},\eta_i)\). The same \(C_i(t)\) is supplied to every endpoint model for individual \(i\). To make exposure scale explicit, define \(x_i(t)=C_i(t)/C_{\mathrm{ref}}\), where \(C_{\mathrm{ref}}\) is a prespecified positive reference concentration in the same units as \(C_i(t)\). Thus \(x_i(t)\) is dimensionless; the reference concentration affects coefficient interpretation, not the underlying concentration history.

A general endpoint-specific model is

Here \(Y_{ij}(t)\) is the outcome for endpoint \(j\), \(g_j\) is its link function, \(\alpha_j\) is the endpoint-specific intercept, and \(\beta_j\) is its population exposure slope. The individual deviation \(b_{ij}\) allows that person's slope to differ from the population slope. The total individual slope on the link scale is \(\beta_j+b_{ij}\). Different endpoints can use different links and outcome distributions: for example, an identity link for a continuous outcome and a logit link for a binary outcome.

The individual slope deviations are modeled jointly:

The diagonal elements of \(\Omega\) are the between-individual slope variances, and its off-diagonal elements are covariances between endpoint-specific slopes. For two endpoints, the correlation is

A positive \(\rho_{12}\) means individuals with above-average slopes for endpoint 1 tend, on average, to have above-average slopes for endpoint 2. It does not mean the fixed effects \(\beta_1\) and \(\beta_2\) are equal, that the endpoints have the same units, or that a change in one endpoint causes a change in the other.

For this specification, assume the shared PK model adequately represents the concentration histories; endpoint records are aligned to the relevant individuals and times; and the selected endpoint distributions and links are appropriate. The multivariate normal assumption for \(\mathbf b_i\) is a modeling choice for the population distribution of slopes. If the endpoint outcomes are treated as conditionally independent given the PK history and individual slopes, any remaining dependence in their observation errors must be negligible or addressed separately. PK effects and PD slope deviations are taken to be independent here unless their joint distribution is explicitly modeled. These assumptions separate shared exposure from additional sources of dependence.

Why Correlated Slopes Create Dependence

A correlated-slope model induces a specific form of cross-endpoint dependence: individuals' deviations in response to exposure are related. For identity-link endpoints, the derivation is direct. At a fixed exposure value \(x_j\), write the conditional mean for endpoint \(j\) as \(\alpha_j+(\beta_j+b_{ij})x_j\). Across individuals, the fixed terms do not vary, so the covariance of the two conditional means is

The first equality after expansion follows because constants have zero covariance and covariance scales by the product of constants. Thus this component of dependence grows with both exposure values and is zero if either endpoint is evaluated at \(x=0\). This result concerns covariance in conditional means across individuals; it does not assert that the actual observations have no additional residual covariance. With nonlinear links, the same intuition applies, but the covariance of outcomes is not generally exactly \(x_1x_2\Omega_{12}\).

Mixed Endpoint Types: A Shared Input, Different Response Models

Consider a continuous endpoint with normally distributed errors and an identity link, alongside a binary endpoint with a Bernoulli distribution and a logit link. Their conditional models might be

These equations use one \(x_i(t)\), but they do not force the endpoints to have the same response curve. The first slope is measured in continuous-outcome units per exposure unit; the second is in log-odds units per exposure unit. For a binary endpoint, exponentiating a log-odds change gives an odds ratio, not a risk difference.

Checking the Model and Choosing an Alternative

The model is useful only if its shared input and endpoint-specific structure are credible. First check that concentration predictions are aligned with the endpoint observation times and correspond to the same individual dosing history. A mismatch in time alignment can look like an endpoint-specific slope difference. Then assess whether the endpoint response patterns are compatible with the chosen links and whether residual patterns suggest unmodeled time trends, nonlinear exposure response, or a poor outcome distribution. For the slope distribution, inspect estimated variances and covariance, uncertainty in the correlation, and whether individual slope predictions show systematic patterns against covariates or exposure range.

A covariance estimate near the boundary of its allowable range, a very imprecise correlation, or substantial sensitivity to a few individuals warns that the data may not support a rich random-slope structure. Such results do not by themselves prove the model is wrong. They indicate that the slope variation and correlation may be weakly informed, so simpler specifications and sensitivity checks are important. A shared-slope model sets individual slope variation to zero; a model with independent random slopes allows variation but fixes their covariance to zero. Each answers a different question and should not be chosen solely because it is easier to fit.

A shared concentration history is not the same as a shared effect parameter. Nor does it ensure the PK model is known without error. Treating estimated concentrations as exact can understate uncertainty in the endpoint effects. The model also does not automatically account for residual dependence after conditioning on exposure and slopes; a separate residual dependence structure may be needed if supported by the design and data. These distinctions are important when interpreting the fitted slope correlation.

Common Pitfall

Incorrect reasoning: “The endpoints use the same PK model, so they should have the same exposure slope; any observed association between endpoint values must therefore be their shared drug effect.”

Correction: Sharing the PK model means the endpoints use the same individual concentration history. Their fixed slopes, links, outcome scales, and individual slope deviations can still differ. Slope correlation describes how individual response tendencies co-vary; it is not equality of drug effects and does not by itself explain all observed endpoint association. State which source of dependence is represented before interpreting a joint model.