Population Priors for Individual Pharmacokinetic Parameters

From a Population Model to an Individual Parameter Distribution

An NLME model describes how pharmacokinetic parameters vary between individuals. For a new individual, fixed-effect estimates and known covariates specify a typical parameter value, while the random-effects distribution describes between-individual variation around it. Before using that individual’s concentration observations, these model components therefore imply a distribution of plausible parameter values.

Here, “prior” means the distribution for an individual parameter before conditioning on that individual’s concentration data. It is conditional on the individual’s known covariates and on the population-model parameters. If the population parameters are treated as fixed at their estimates, this distribution does not include uncertainty in those estimates. It is also not, by itself, a distribution updated using the individual’s observed concentrations.

Definition: Let \(x_i\) denote the known covariates for individual \(i\), \(\theta\) the population fixed effects, and \(\eta_i\) that individual’s random effects. If the individual parameter vector is \(\phi_i=g(x_i,\theta,\eta_i)\), the model-implied population prior is the conditional distribution \(p(\phi_i\mid x_i,\theta,\Omega)\), induced by the specified distribution of \(\eta_i\), such as \(\eta_i\sim N(0,\Omega)\). It describes between-individual parameter variation conditional on the covariates and population-model parameters.

A common scalar parameter model is \(\phi_i=\theta_\phi\exp(\eta_i)\), with \(\eta_i\sim N(0,\omega^2)\). This guarantees \(\phi_i>0\). Covariates can change the typical value, for example \(\phi_i=\theta_\phi\exp\{\beta(x_i-x_0)\}\exp(\eta_i)\). In both cases, the random effect is normal on the log scale; the parameter itself is lognormally distributed.

Assumptions and the Lognormal Result

The interpretation and calculations below rely on several modeling conditions. The individual’s covariates are known and appropriately measured; the structural covariate relationship is specified on the chosen scale; and the random-effects distribution is adequate for the population being represented. For the scalar lognormal result, the random effect must be normally distributed with mean zero and variance \(\omega^2\), and the multiplier of \(\exp(\eta_i)\) must be positive. Treating \(\theta\) and \(\Omega\) as fixed also means that uncertainty in their estimation is being set aside.

Theorem: If \(\phi_i=m_i\exp(\eta_i)\), where \(m_i>0\) is determined by fixed effects and individual covariates and \(\eta_i\sim N(0,\omega^2)\), then \(\ln(\phi_i)\sim N(\ln(m_i),\omega^2)\). The parameter’s median is \(m_i\), its arithmetic mean is \(m_i\exp(\omega^2/2)\), and its \(q\)-quantile is \(m_i\exp\{\omega\Phi^{-1}(q)\}\), where \(\Phi\) is the standard normal cumulative distribution function.

The median and mean differ because exponentiation is nonlinear: individuals are symmetric around zero on the random-effect scale, but the parameter distribution is right-skewed. In particular, \(m_i\) is often called the typical value or the median under this model; it is not the arithmetic mean when \(\omega^2>0\).

Write the individual parameter model.
Collect its fixed-effect and covariate terms into the positive multiplier \(m_i\), so that \(\phi_i=m_i\exp(\eta_i)\).

Take logarithms.
\(\ln(\phi_i)=\ln(m_i)+\eta_i\). Since adding a constant to a normal random variable shifts its mean, \(\ln(\phi_i)\sim N(\ln(m_i),\omega^2)\).

Obtain the median and quantiles.
The median of a normal variable with mean \(\ln(m_i)\) is \(\ln(m_i)\), so exponentiating gives median \(m_i\). Applying the normal \(q\)-quantile and exponentiating gives \(m_i\exp\{\omega\Phi^{-1}(q)\}\).

Obtain the arithmetic mean.
For a normal random variable, \(E[\exp(\eta_i)]=\exp(\omega^2/2)\). Thus \(E[\phi_i]=m_iE[\exp(\eta_i)]=m_i\exp(\omega^2/2)\).

Worked Calculations

Worked Example: Covariates Set the Individual-Specific Typical Clearance

Hypothetical model. Suppose clearance in L/h is \(CL_i=6\exp\{-0.20(eGFR_i-90)/10\}\exp(\eta_i)\), where eGFR is measured in mL/min/1.73 m\(^2\) and \(\eta_i\sim N(0,0.30^2)\). For an individual with eGFR \(60\) mL/min/1.73 m\(^2\), calculate the model-implied median and mean clearance.

The covariate contribution to the exponent is \(-0.20(60-90)/10=0.60\), not \(-0.60\). Therefore the covariate-adjusted multiplier is \(m_i=6\exp(0.60)=10.93\) L/h. Under the stated model, this multiplier is the median clearance.

The arithmetic mean is \(m_i\exp(0.30^2/2)=10.9332\exp(0.045)=11.436\) L/h. The mean exceeds the median because clearance is lognormally distributed. These calculations depend on the assumed covariate direction and the zero-mean normal random effect; they do not establish a causal effect of eGFR on clearance.

Worked Example: A Lognormal Prior for Clearance

Hypothetical model. Suppose \(CL_i=5(WT_i/70)^{0.75}(0.80)\exp(\eta_i)\) L/h for an individual with weight \(WT_i=80\) kg and a covariate category whose multiplier is \(0.80\). Assume \(\eta_i\sim N(0,0.25^2)\). Calculate the median, arithmetic mean, and central 95% model-implied interval for clearance.

First calculate the multiplier without rounding intermediate values:

m_i=5(80/70)^{0.75}(0.80)=4.421337\ldots\ \mathrm{L/h}.

This is the median clearance. The arithmetic mean is

E(CL_i)=m_i\exp(0.25^2/2) =5(80/70)^{0.75}(0.80)\exp(0.03125) =4.5616858\ldots\ \mathrm{L/h} \approx 4.562\ \mathrm{L/h}.

For a central 95% interval, use the 2.5th and 97.5th normal quantiles, approximately \(-1.96\) and \(1.96\). On the clearance scale the endpoints are \(m_i\exp(-1.96(0.25))\) and \(m_i\exp(1.96(0.25))\):

[4.421337\exp(-0.49),\;4.421337\exp(0.49)] \approx [2.709,\;7.217]\ \mathrm{L/h}.

Under this model, approximately 95% of individuals with these covariates have clearance between 2.709 and 7.217 L/h. This is a model-implied population interval, not a confidence interval for the population mean and not an interval conditioned on observed concentrations for this individual.

Worked Example: Probability of Clearance Exceeding a Threshold

Use the hypothetical model in the previous example. Calculate the model-implied probability that clearance exceeds 5 L/h for an individual with the same weight and covariate category.

Taking logarithms gives \(\ln(CL_i)\sim N(\ln(4.421337),0.25^2)\). Standardizing the threshold,

P(CL_i>5) =P\left(Z>\frac{\ln(5)-\ln(4.421337)}{0.25}\right) =P(Z>0.492\text{ approximately}) \approx 0.311.

Thus, under the stated population model, the probability of clearance above 5 L/h is approximately 31.1% for individuals with these covariates. The result is conditional on the specified model and its parameters; it is not a probability that a population parameter is true.

Correlated Random Effects and Joint Distributions

A PK model usually contains several parameters, such as clearance and volume. Their random effects may be correlated. If \(\eta_i=(\eta_{CL,i},\eta_{V,i})^\mathsf{T}\sim N(0,\Omega)\), then the joint individual parameter distribution is induced by transforming this vector through the clearance and volume models. Its dependence structure matters when considering combinations of parameter values, even though each parameter’s marginal distribution is determined by its own random-effect variance.

Worked Example: Reading the Dependence in a Two-Parameter Prior

Hypothetical model. Let \(CL_i=5\exp(\eta_{CL,i})\) L/h and \(V_i=40\exp(\eta_{V,i})\) L. Suppose the random effects are jointly normal with standard deviations 0.20 and 0.30 and correlation \(-0.50\). Write their covariance matrix and calculate the covariance between the random effects.

The covariance is the correlation multiplied by the two standard deviations: \(-0.50(0.20)(0.30)=-0.030\). Hence

\Omega= \begin{pmatrix} 0.20^2 & -0.030\\ -0.030 & 0.30^2 \end{pmatrix} = \begin{pmatrix} 0.040 & -0.030\\ -0.030 & 0.090 \end{pmatrix}.

The marginal median clearances and volumes remain 5 L/h and 40 L, respectively. The negative correlation means that, in the specified joint distribution, higher-than-typical clearance random effects tend to occur with lower-than-typical volume random effects. It does not mean every individual follows that pattern, nor does it establish a mechanistic relationship. The joint prior must reflect a valid covariance matrix; here its determinant is \(0.040(0.090)-(-0.030)^2=0.0027>0\), and both diagonal variances are positive.

Diagnostics, Alternatives, and Limitations

A model-implied prior is useful only to the extent that its inputs and distributional assumptions are credible. Before interpreting it, check the following:

Covariate coding and units: confirm that the individual’s covariate values, reference values, transformations, and category indicators match the model definition. A unit mismatch can substantially change \(m_i\).
Random-effects scale: verify whether the model places a normal random effect on the log parameter, adds a normal effect on the original scale, or uses another structure. These imply different distributions and may imply different support.
Distributional adequacy: assess whether the fitted random-effects distribution and covariance structure plausibly represent the population. Strong skewness, multiple subpopulations, or outliers may not be well represented by a single multivariate normal distribution.
Parameter plausibility: inspect implied quantiles and tail probabilities, not only typical values. Implausible or extreme values can reveal a scale error, inappropriate extrapolation of a covariate relationship, or a poor distributional assumption.

A common mistake is to report the typical value \(m_i\) as the mean clearance. With lognormal random effects, that is wrong unless \(\omega^2=0\): the mean is \(m_i\exp(\omega^2/2)\), while the median is \(m_i\). Another mistake is to call the interval calculated from random effects a confidence interval for the individual’s clearance. It represents between-individual variation under the model, not uncertainty in an estimated population coefficient and not an individual-specific distribution updated with concentration data.

The parametric NLME distribution is not the only possible representation. A discrete empirical distribution of estimated individual parameters can avoid a normality assumption, but depends on which individuals were sampled and how their parameters were estimated. A mixture model can represent distinct subpopulations, but requires additional mixture assumptions and parameters. Either alternative changes the induced distribution; neither automatically accounts for uncertainty in estimated population parameters. The standard parametric prior is compact and covariate-conditional, but its tails and subgroup behavior may be sensitive to model misspecification and covariate extrapolation.

Key takeaway: Fixed effects and known covariates determine the individual-specific typical parameter value; the random-effects distribution determines variation around it. State clearly whether a reported quantity is a median, arithmetic mean, or population quantile, and identify the assumptions and uncertainty components included.

Check Your Understanding

Answer each question, then compare with the solutions below.

If \(CL_i=7\exp(\eta_i)\) L/h and \(\eta_i\sim N(0,0.20^2)\), what are the median and arithmetic mean clearance?
In a lognormal clearance model, does a central 90% interval describe uncertainty in the estimated fixed effect or variation among individuals conditional on the model?
If the random-effect correlation between clearance and volume is zero, does that alone prove the two parameters are independent under any possible model?
For the clearance model in Worked Example 2, what is the model-implied probability that clearance is below 4.421337 L/h?

Solutions. (1) The median is \(7\) L/h. The mean is \(7\exp(0.20^2/2)=7\exp(0.02)=7.141\) L/h. (2) It describes variation among individuals conditional on the specified model parameters and covariates; it does not quantify uncertainty in the estimated fixed effect. (3) No. Zero correlation implies independence for jointly normal random effects, but zero correlation alone does not imply independence for arbitrary distributions. (4) The threshold is the median, so the lognormal distribution assigns probability \(0.500\) below it.