Worked Prior Specification for a One-Compartment Model

From Individual Priors to Prior Predictions

A prior specification for a pharmacokinetic model is not just a list of distributions. It is a joint set of assumptions about plausible population parameters, between-subject differences, and observation error. The practical test is what those assumptions imply for concentration-time profiles under a specified dose and sampling schedule. This tutorial builds one such specification for a hypothetical oral study and examines simulated predictions from it.

Assume a single oral dose with known bioavailability $F=1$, first-order absorption, first-order elimination, and one central compartment. The example uses dose in milligrams, concentration in milligrams per litre, time in hours, clearance in litres per hour, volume in litres, and the absorption rate constant in inverse hours. These assumptions are deliberate simplifications; a different formulation, dosing history, or structural model would require a different prediction calculation.

Definition: For subject $i$, let $CL_i$, $V_i$, and $k_{a,i}$ denote clearance, volume of distribution, and the first-order absorption rate constant. With $k_i=CL_i/V_i$, the concentration after a single oral dose $D$ is $$ C_i(t)=\frac{F D k_{a,i}}{V_i(k_{a,i}-k_i)} \left\{\exp(-k_i t)-\exp(-k_{a,i}t)\right\},\qquad t>0,\quad k_{a,i}\ne k_i. $$ Here $C_i(t)$ is the model-predicted concentration without residual error. The limiting value is used if $k_{a,i}=k_i$, and the concentration at $t=0$ is zero for this model.

One justification for the concentration formula is to add the contributions of drug absorbed at each earlier time. At time $s$, the absorption rate is $Dk_a\exp(-k_a s)$, and the fraction remaining in the central compartment by time $t$ is $\exp[-k(t-s)]$. Thus

C(t)=\frac{Dk_a}{V}\int_0^t \exp(-k_a s)\exp[-k(t-s)]\,ds =\frac{Dk_a}{V}\frac{\exp(-kt)-\exp(-k_a t)}{k_a-k}.

The integral assumes complete systemic availability, no lag time, and the stated first-order processes. It also explains why the model requires care when $k_a$ is close to $k$: the displayed difference quotient can be numerically unstable even though its mathematical limit is finite.

A Coherent Prior Specification

Positive parameters are naturally specified through their logarithms. Let $TVCL$, $TVV$, and $TVk_a$ be the typical population values. For this worked specification, use independent log-scale normal priors with medians of 5 L/h, 50 L, and 1 h$^{-1}$, respectively. The values after the comma are log-scale standard deviations, not coefficients of variation.

\log(TVCL)\sim N(\log 5,0.25^2),\qquad \log(TVV)\sim N(\log 50,0.25^2),\qquad \log(TVk_a)\sim N(\log 1,0.35^2).

These distributions imply positive typical values. For example, the median of $TVCL$ is 5 L/h, while its mean is $5\exp(0.25^2/2)$, approximately 5.16 L/h. That distinction matters: the center specified on the log scale is the median on the original scale, not the arithmetic mean.

For the initial example, assume independent normal random effects:

\begin{pmatrix}\log CL_i\\ \log V_i\\ \log k_{a,i}\end{pmatrix} = \begin{pmatrix}\log TVCL\\ \log TVV\\ \log TVk_a\end{pmatrix} + \begin{pmatrix}\eta_{CL,i}\\ \eta_{V,i}\\ \eta_{k_a,i}\end{pmatrix}, \qquad \eta_{j,i}\mid\omega_j\sim N(0,\omega_j^2).

Here $j$ indexes clearance, volume, or absorption, and the three random effects are taken to be independent conditional on their standard deviations. Put half-normal priors on those nonnegative standard deviations:

\omega_{CL}\sim \operatorname{HalfNormal}(0,0.30^2),\qquad \omega_V\sim \operatorname{HalfNormal}(0,0.30^2),\qquad \omega_{k_a}\sim \operatorname{HalfNormal}(0,0.50^2).

The half-normal notation means the absolute value of a zero-mean normal draw with the stated scale. These priors place more mass on modest variability while allowing larger values. Since the random effects are on log scales, $\omega_j$ describes log-scale variability; it is not itself a percentage. Independence is a simplifying assumption, not a general pharmacokinetic fact. If prior knowledge supports correlated random effects, the joint random-effects distribution and its correlation prior must be specified accordingly.

For observations, use an additive-plus-proportional normal error model. Conditional on the individual prediction and error parameters, observations are independent with

Y_{ij}\mid C_i(t_{ij}),\sigma_{\rm add},\sigma_{\rm prop} \sim N\!\left(C_i(t_{ij}),\, \sigma_{\rm add}^2+\{\sigma_{\rm prop}C_i(t_{ij})\}^2\right).

For the hypothetical assay, let $\sigma_{\rm add}\sim\operatorname{HalfNormal}(0,0.05^2)$ mg/L and $\sigma_{\rm prop}\sim\operatorname{HalfNormal}(0,0.20^2)$. The additive term has concentration units; the proportional term is dimensionless. This is one explicit choice of priors, not a universal prescription. The units and scale should reflect the assay and the range of concentrations that the model is meant to describe.

Worked Example: Writing the Population and Variability Priors

Suppose the hypothetical study uses the model and scales above. Write the clearance prior and interpret it, including the role of its between-subject standard deviation.

The population prior is $\log(TVCL)\sim N(\log 5,0.25^2)$, so $TVCL$ is lognormal with median 5 L/h. Its log-scale prior standard deviation, 0.25, describes uncertainty about the population typical value. It is distinct from the subject-level random-effect standard deviation $\omega_{CL}$, whose prior is half-normal with scale 0.30.

For a particular value $\omega_{CL}=0.30$, a subject with $\eta_{CL,i}=0.30$ has clearance $CL_i=TVCL\exp(0.30)$. If $TVCL=5$ L/h, this is $5\exp(0.30)=6.749$ L/h. The individual clearance is positive, and the random effect changes it multiplicatively rather than adding 0.30 L/h. The prior on $\omega_{CL}$ allows uncertainty in the population degree of variability as well as variation among subjects.

Work Through a Concentration and Its Error

Prior prediction propagates uncertainty through the whole hierarchy. A simulated prediction first draws population parameters, then variability parameters, then subject random effects, and finally (for an observation prediction) residual error. The following calculations isolate one draw to make the model’s mechanics transparent.

Worked Example: Predicting Concentration and Residual Spread

Take a hypothetical prior draw with $TVCL=5$ L/h, $TVV=50$ L, $TVk_a=1$ h$^{-1}$, and all three individual random effects equal to zero. For a 100 mg oral dose, calculate the model concentration at 2 hours.

The elimination rate is $k=CL/V=5/50=0.10$ h$^{-1}$. Substituting $F=1$, $D=100$ mg, $V=50$ L, and $k_a=1$ h$^{-1}$ gives

C(2)=\frac{100(1)}{50(1-0.10)} \{\exp(-0.10\times2)-\exp(-1\times2)\} =2.2222(0.81873-0.13534) \approx 1.519\ {\rm mg/L}.

Now suppose a separate prediction has concentration $C=1.504$ mg/L, and the error parameters are $\sigma_{\rm add}=0.05$ mg/L and $\sigma_{\rm prop}=0.15$. Its residual standard deviation is

\sqrt{0.05^2+(1.504\times0.15)^2} =\sqrt{0.0025+0.05089536} =\sqrt{0.05339536} \approx 0.2311\ {\rm mg/L}.

Thus the conditional observation distribution is centered on the model prediction, with a standard deviation that combines an additive floor and a concentration-dependent component. It is not correct to add the two standard deviations directly; the model adds their variances.

Simulate and Inspect Prior Predictions

A prior predictive distribution is the distribution of possible observations implied by the prior and the model before using the study’s observed concentrations. Its role here is calibration: it can reveal that plausible-looking parameter priors combine to imply implausible concentration profiles, or that the error model is too wide or too narrow for the intended setting.

Draw population and variability parameters.
Draw $TVCL$, $TVV$, $TVk_a$, the $\omega_j$, and both residual error parameters from their stated priors.

Draw a hypothetical subject.
Draw the subject’s random effects conditional on the sampled $\omega_j$, then calculate $CL_i$, $V_i$, and $k_{a,i}$.

Calculate the profile and an observation.
Evaluate $C_i(t)$ at the planned times. For observation predictions, draw $Y_{ij}$ from the residual model; retain the noiseless $C_i(t)$ when checking structural profiles.

Inspect implications in context.
Plot profiles and summarize predictions at actual sampling times. Look for implausible magnitudes, shapes, variability, or residual spread, rather than judging a prior solely by its parameter-scale appearance.

Worked Example: Reading a Hypothetical Prior-Predictive Summary

Imagine 20,000 hypothetical prior simulations for a 100 mg dose and the priors specified above. The table gives illustrative 2.5th, 50th, and 97.5th percentiles of the simulated observations at three times. These numbers are an example of how to read a simulation summary, not results from a real study.

Time	2.5th percentile (mg/L)	Median (mg/L)	97.5th percentile (mg/L)
0.5 h	0.10	0.78	2.30
2 h	0.22	1.42	4.80
8 h	0.03	0.75	3.70

The median rises between 0.5 and 2 hours and falls by 8 hours, which is consistent with absorption followed by elimination for many draws. The broad intervals show that the priors permit substantial between-subject and parameter uncertainty. Whether the upper tail at 2 hours is acceptable cannot be decided from the table alone: it must be compared with plausible concentrations for the intended dose and population.

A useful extension is to compare predictions from parameter draws with observation draws. If the noiseless profiles appear reasonable but simulated observations are excessively dispersed, inspect the residual-error priors. If both are too dispersed, inspect the typical-value and random-effect priors. Simulations should also be checked at all planned sampling times; a plausible peak does not guarantee a plausible terminal profile.

Assumptions, Diagnostics, and Alternatives

This specification assumes a single dose, known $F=1$, a one-compartment disposition model, first-order absorption and elimination, lognormal individual parameters, independent random effects, and conditionally independent normal residuals with additive-plus-proportional variance. The prior predictive calculation assumes those components are combined exactly as stated. If bioavailability is not known, the product $F/V$ may be difficult to distinguish from concentration information alone; fixing $F=1$ is then a modeling convention, not evidence that actual bioavailability is one.

Inspect simulated profiles, not just marginal prior histograms. Warning signs include negative observation predictions at concentrations where negative values are impossible or unacceptable; peaks that occur outside a plausible time range; extremely high or near-zero concentrations over the study window; and large individual differences that overwhelm the population profile. With a normal residual model, negative observations can occur even when $C_i(t)>0$. Their frequency is especially informative when the mean concentration is small relative to the residual standard deviation.

The additive-plus-proportional model is not the only residual model. A proportional error model has variability that scales with concentration, while an additive model has constant absolute variability. A lognormal observation model can ensure positive observations, but it implies a different error structure and requires careful interpretation of its location and scale parameters. Similarly, correlated random effects can replace the independence assumption, and other structural models can replace the one-compartment model. These alternatives should be chosen to represent the study and scientific knowledge, not merely to improve the appearance of one prior-predictive plot.

Prior predictive calibration, covered in the earlier material on that subject, is an iterative check rather than a guarantee of a good posterior analysis. A prior can generate plausible predictions at a few times while still allowing unrealistic profiles elsewhere. Conversely, a broad prior predictive range may be appropriate when genuine uncertainty is large. Record the dose, schedule, simulation setup, and summaries used to make the judgment so the specification can be reviewed and reproduced.

Common pitfall: Treating the log-scale standard deviation of a typical-value prior as though it were the between-subject variability. These are different uncertainties: the former describes uncertainty about a population parameter, while $\omega_j$ describes variation among subjects conditional on that parameter. Specify both layers, simulate them in the correct order, and inspect the resulting profiles.

Check Your Understanding

Answer each question, then compare with the solutions below.

With $TVCL=6$ L/h and $TVV=60$ L, what is the elimination rate constant?
For $\sigma_{\rm add}=0.04$ mg/L, $\sigma_{\rm prop}=0.10$, and $C=2$ mg/L, calculate the residual standard deviation.
What is the difference between uncertainty in $TVCL$ and the subject-level random effect $\eta_{CL,i}$?
Name one diagnostic sign that the residual-error prior may be too wide, even if noiseless profiles look plausible.

Solutions. (1) $k=CL/V=6/60=0.10$ h$^{-1}$. (2) The variance is $0.04^2+(0.10\times2)^2=0.0016+0.0400=0.0416$ (mg/L)$^2$, so the standard deviation is $\sqrt{0.0416}=0.2040$ mg/L. (3) The prior for $TVCL$ describes uncertainty about the population typical clearance; $\eta_{CL,i}$ describes a subject’s log-scale deviation from that typical value, conditional on the population parameters and $\omega_{CL}$. (4) One sign is that observation simulations frequently produce negative concentrations or an implausibly broad spread around otherwise reasonable model predictions.