Worked Case of Sparse Oral PK Bayesian Forecasting

A Sparse Oral PK Forecasting Case

Sparse concentration data provide limited information about an individual's pharmacokinetics. A population model supplies a principled starting point, but the forecast must reflect both what the samples say about this individual and the uncertainty that remains. The distinction between an estimated individual parameter and a future observed concentration is especially important: the latter includes uncertainty from both the parameter estimate and new observation-level variability.

This worked case uses a deliberately compact model. The population-derived oral PK model supplies a baseline concentration curve for the known dosing history. The individual is allowed to differ from that curve by a multiplicative exposure factor. This setup is useful for demonstrating the update and forecast calculations transparently; it is not a substitute for a richer model when the data can identify individual absorption or elimination differences.

Definition: Let $f_i$ be the positive concentration predicted by a population-derived oral PK model at sample time $i$, given the recorded dose history and sampling time. Let $\theta$ be the individual's log exposure multiplier relative to that curve. The observation model is $$ \log C_i = \log f_i + \theta + \epsilon_i, $$ where $C_i$ is the observed concentration and $\epsilon_i$ is a log-scale residual. Thus, $\exp(\theta)$ is the individual's exposure multiplier: values above 1 indicate concentrations scaled above the population curve, and values below 1 indicate concentrations scaled below it.

Model, Prior, and Assumptions

For this case, the prior is $\theta\sim N(0,0.35^2)$, and the residuals are independent with $\epsilon_i\sim N(0,0.20^2)$. The prior is centered at the population prediction, and its standard deviation represents uncertainty about the individual's log exposure multiplier before using that individual's samples. The values and all data below are hypothetical.

The calculation assumes that the population model's dosing history, dose times, sample times, and baseline predictions $f_i$ are correct; that concentrations are positive and measured on a consistent scale; and that the individual differs only by a constant exposure multiplier over the observations and forecast horizon. It also assumes that log residuals are unbiased, independent, normally distributed, and have the specified common standard deviation. These are substantive modeling assumptions, not consequences of having a normal prior.

Define $y_i=\log(C_i/f_i)$. Under this model, each $y_i$ is a noisy observation of $\theta$. The normal prior and normal residual model give a normal posterior that can be calculated directly. Let $\tau^2$ denote the prior variance and $\sigma^2$ the residual variance. Then

\theta\mid \mathbf{y}\sim N(m_n,v_n),\qquad v_n=\left(\frac{1}{\tau^2}+\frac{n}{\sigma^2}\right)^{-1},\qquad m_n=v_n\left(\frac{0}{\tau^2}+\frac{\sum_{i=1}^n y_i}{\sigma^2}\right).

The numerator includes the prior mean multiplied by its precision; here that mean is zero. More generally, if the prior mean is $m_0$, the numerator is $m_0/\tau^2+\sum_i y_i/\sigma^2$. In either form, the posterior mean is a precision-weighted combination of prior information and the observed log ratios. The posterior variance decreases as informative observations are added, provided the assumed model and residual variance are appropriate.

Worked Example: Updating From Three Sparse Samples

Suppose the population oral model predicts $f_i=(2.00, 1.50, 0.90)$ mg/L at three recorded sample times. Hypothetical measured concentrations are $C_i=(2.40,1.35,0.99)$ mg/L. The samples are sparse, and the times and dosing history are treated as known. Calculate the individual posterior.

First form the ratios and their logarithms:

\frac{C_i}{f_i}=(1.20,0.90,1.10),\qquad y_i=(\log 1.20,\log 0.90,\log 1.10) =(0.18232,-0.10536,0.09531).

Their sum is $0.17227$. The prior variance is $\tau^2=0.35^2=0.1225$, and the residual variance is $\sigma^2=0.20^2=0.04$. Therefore,

v_n=\left(\frac{1}{0.1225}+\frac{3}{0.04}\right)^{-1} =(8.1633+75)^{-1}=0.01202,

and

m_n=0.0120245\left(\frac{0.17227}{0.04}\right) =0.0120245(4.3068)=0.05179.

The posterior standard deviation is $\sqrt{0.0120245}=0.1097$ on the log scale. The posterior median exposure multiplier is $\exp(0.05179)\approx1.053$. This is a modest upward shift from the population curve, not evidence that all aspects of the individual's PK are 5.3% different.

From an Individual Posterior to a Future Concentration

For a future observation at a specified time, let $f_{\mathrm{new}}$ be the population-model prediction for the planned dose history at that time. Under the same observation model, the future log concentration is $\log C_{\mathrm{new}}=\log f_{\mathrm{new}}+\theta+\epsilon_{\mathrm{new}}$. Conditional on the samples, the posterior distribution of $\theta$ and the independent new residual together determine the predictive distribution:

\log C_{\mathrm{new}}\mid\mathbf{y}\sim N\left(\log f_{\mathrm{new}}+m_n,\;v_n+\sigma^2\right).

The variance includes $v_n$, the remaining uncertainty in the individual multiplier, and $\sigma^2$, the variability of a future observation around its model prediction. Omitting either term changes the question being answered. In particular, a prediction of the latent concentration curve without a new residual is not the same as a predictive distribution for a future measured concentration.

Worked Example: Predicting a Future Concentration

Continue the hypothetical case above. Suppose the population-derived model predicts $f_{\mathrm{new}}=2.40$ mg/L at a future sample time under the stated oral dosing history. Use the posterior from the three samples to calculate the predictive median, arithmetic mean, and approximate 95% predictive interval for a future observed concentration.

The predictive log mean and variance are

\log(2.40)+0.05179=0.9273,\qquad 0.0120245+0.0400=0.05202.

Thus the predictive log standard deviation is $\sqrt{0.05202}=0.2281$. Because a normal distribution is symmetric on the log scale, the predictive median on the concentration scale is

\exp(0.9273)\approx 2.528\ \text{mg/L}.

The predictive distribution is lognormal, so its arithmetic mean is not its median. The arithmetic mean is

\exp\left(0.9273+\frac{0.05202}{2}\right) =\exp(0.9533)\approx2.594\ \text{mg/L}.

An approximate central 95% predictive interval is obtained by exponentiating the normal interval on the log scale:

\left[ \exp(0.9273-1.96\sqrt{0.05202}), \exp(0.9273+1.96\sqrt{0.05202}) \right] \approx [1.616,3.952]\ \text{mg/L}.

Under the stated model, this interval describes the range containing 95% of future observed concentrations in repeated predictions made under the same conditions. It is not a statement that there is a 95% probability that a fixed unknown concentration lies in this particular interval.

Using the Forecast for a Target Range

A forecast can be summarized as a probability of falling within a clinically specified range, rather than only as a point estimate. That probability is conditional on the model, the data, the current dosing history, and the specified future sample time. It is not a guarantee about an individual outcome, and its clinical interpretation depends on how the target range was established.

Worked Example: Probability of Being Within a Target Range

For the future concentration in the preceding example, suppose a hypothetical target range is 2.00 to 4.00 mg/L. Calculate the posterior predictive probability that the future observed concentration falls in that range.

On the log scale, the predictive mean is 0.9273 and standard deviation is 0.2281. Standardizing the lower and upper limits gives

z_{\mathrm{low}}=\frac{\log(2.00)-0.9273}{0.2281}\approx-1.026,\qquad z_{\mathrm{high}}=\frac{\log(4.00)-0.9273}{0.2281}\approx2.012.

Therefore,

P(2.00<C_{\mathrm{new}}<4.00\mid\mathbf{y}) =\Phi(2.012)-\Phi(-1.026) \approx0.978-0.152=0.826.

The estimated probability is about 82.6% under this hypothetical model. It is a predictive probability for a future concentration, not the probability that the individual's parameter is in the target range. A point forecast alone would not provide this information.

Sequential Updating and Practical Checks

If another sample arrives and the same assumptions remain reasonable, its information can be added to the existing posterior. The posterior from the first samples becomes the prior for the next update; this is equivalent to updating once with all samples when the likelihood contributions are conditionally independent. The following calculation shows the effect of one additional hypothetical sample.

Worked Example: Updating After One More Sample

Suppose a fourth concentration is $1.30$ times the population-model prediction at its sample time. Its log ratio is $y_4=\log(1.30)=0.26236$. With the same residual variance, update from the three-sample posterior and recalculate the future forecast at $f_{\mathrm{new}}=2.40$ mg/L.

The updated precision is the previous posterior precision plus the new observation precision:

\frac{1}{v_4}=\frac{1}{0.0120245}+\frac{1}{0.04} =83.1633+25=108.1633, \qquad v_4\approx0.009245.

The updated mean is

m_4=v_4\left(\frac{0.05179}{0.0120245} +\frac{0.26236}{0.04}\right) \approx0.009245(4.3068+6.5591)=0.1005.

The future predictive log mean is $\log(2.40)+0.1005=0.9759$, and its variance is $0.009245+0.0400=0.04925$. The predictive median is $\exp(0.9759)\approx2.654$ mg/L. Its approximate 95% predictive interval is

\left[ \exp(0.9759-1.96\sqrt{0.04925}), \exp(0.9759+1.96\sqrt{0.04925}) \right] \approx[1.718,4.100]\ \text{mg/L}.

The new sample moves the forecast upward and modestly reduces uncertainty in the individual multiplier. Predictive uncertainty remains substantial because future residual variability is still included.

Several checks help determine whether the calculation is being used appropriately. Confirm that the sample times and doses used to obtain each $f_i$ match the recorded history, and that concentrations and predictions use the same units. Plot or inspect the log ratios $y_i$: a systematic trend over time can indicate that a constant exposure multiplier is inadequate, while an extreme ratio can indicate an input error, an unusual observation, or model misspecification. The assumed residual scale should be justified from the model and measurement context, not chosen merely to produce a narrow interval. Posterior predictive checks for individual PK fits and residual diagnostics for individual Bayesian forecasts provide more formal ways to examine fit and predictive behavior.

What This Case Does and Does Not Establish

The model is intentionally limited. It updates one stable multiplicative exposure parameter. It does not separately estimate clearance, volume, absorption rate, bioavailability, or adherence, and sparse concentrations may not distinguish those mechanisms. A time-varying exposure pattern, an incorrect dose history, nonlinear or saturable kinetics, or residuals with a different distribution could invalidate the likelihood. A population prior can stabilize sparse-data estimates, but it can also pull an individual toward the population prediction when the individual's data are weak or the prior is poorly matched.

The alternative of reporting only a maximum a posteriori (MAP) parameter and substituting it into the population curve produces a point forecast. That can be useful as a compact summary, but it does not represent the distribution of a future observation. A posterior predictive forecast propagates uncertainty in the parameter and includes future residual variability, so it is the appropriate object for prediction probabilities and predictive intervals. Conversely, including residual variability is not appropriate if the intended quantity is specifically the latent model-predicted concentration rather than a future measured concentration.

Common pitfall: Reporting $\exp(\log f_{\mathrm{new}}+m_n)$ as if it were a complete future forecast gives only the predictive median under this lognormal model. It does not quantify uncertainty. For a future observation, use variance $v_n+\sigma^2$, and state whether a reported summary is the median, arithmetic mean, interval, or target-range probability.

Key takeaway: Sparse oral samples update an individual estimate only through the structure and assumptions of the model. A useful forecast carries the posterior uncertainty forward, adds the variability of a future observation when predicting a measured concentration, and remains conditional on correct dose and sample histories.

Check Your Understanding

Use the model and assumptions stated above. Each answer is given after the question so the calculations can be checked.

If one sample has $C_i/f_i=0.80$, what is its log ratio, and what direction does it move the exposure estimate? Solution: $y_i=\log(0.80)\approx-0.2231$. It contributes evidence for an exposure multiplier below 1 and therefore moves the posterior mean downward, all else equal.
Why does the future observation variance include both $v_n$ and $\sigma^2$? Solution: The individual multiplier remains uncertain with posterior variance $v_n$, and a new concentration has residual variance $\sigma^2$ around the model prediction. The independent contributions add, giving $v_n+\sigma^2$ on the log scale.
In the first worked example, what is the posterior standard deviation of $\theta$, and what does it measure? Solution: $\sqrt{0.0120245}=0.1097$. It measures posterior uncertainty in the individual's log exposure multiplier, not the residual variability of a future concentration.
If the posterior predictive interval is wide despite several samples, name one model-based reason. Solution: The residual standard deviation may be large, leaving substantial future observation variability even if parameter uncertainty has decreased. A wide interval can also result from remaining posterior uncertainty or a forecast at a time with limited information.
Does a posterior median of 2.528 mg/L imply that the future concentration will be 2.528 mg/L? Solution: No. It is the median of the predictive distribution: under the model, half of future observations are below it and half above it. The predictive interval and target-range probability describe uncertainty more fully.