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Forecasting With Uncertain Adherence and Dose Times

Learn to propagate uncertainty about dose adherence and administration times through patient-specific parameter estimation into future concentration forecasts.

Advanced 13 min read

What You'll Learn

  • Represent uncertain adherence and administration times as alternative dose histories.
  • Combine prior history probabilities with concentration likelihoods to update their probabilities.
  • Propagate dose-history uncertainty into patient-specific parameter estimates.
  • Construct future concentration forecasts as mixtures over plausible histories and parameters.
  • Recognize when a single assumed history creates misleading precision.

Why Dose-History Uncertainty Matters

A patient-specific forecast depends on more than the measured concentrations and the pharmacokinetic parameters. It also depends on what doses were actually taken, and when. A dose documented as scheduled is not necessarily a dose taken; a reported administration time may also be approximate. If a forecast treats an uncertain dose as certain, the model may attribute its concentration effect to the patient’s clearance or volume instead. The resulting parameter estimate and forecast can be misleadingly precise.

The appropriate response is to represent uncertainty about dose events explicitly. Each plausible history specifies which doses were taken and their administration times. The concentration data then provide evidence about those histories, as well as about the patient’s parameters. Forecasts retain the resulting uncertainty rather than simply selecting the history that appears most likely.

Definition: A dose history is a record of dose amounts and administration times over a specified period. When adherence or timing is uncertain, let \(H\) denote a possible dose history. The posterior distribution \(p(H,\theta\mid y)\) describes uncertainty about both that history and the patient-specific parameters \(\theta\), given concentration data \(y\).

Representing Alternative Histories

For a linear one-compartment model with instantaneous doses and first-order elimination, a dose \(D_j\) administered at time \(t_j\) contributes \(D_j/V\) to the concentration immediately after administration, then declines exponentially. At a later time \(t\), the predicted concentration is the sum of the contributions from all earlier doses:

$$ C(t\mid \theta,H)=\sum_{j:t_j\leq t}\frac{D_j}{V}\exp\{-k(t-t_j)\}, \qquad \theta=(V,k). $$

Here \(V\) is volume of distribution and \(k\) is the elimination rate constant. A missed dose can be represented by omitting that dose from the sum; an uncertain administration time is represented by using its possible time or a probability distribution over times. This simple model is used below to make the calculations transparent. For other pharmacokinetic models, the same history-based reasoning applies, but each history’s concentration prediction must come from the appropriate model.

Suppose the concentration observations are conditionally independent given the parameters and history, and have a specified residual-error distribution. If \(p(H)\) is the prior probability of a history and \(p(\theta\mid H)\) is the parameter prior under that history, Bayes’ rule gives:

$$ p(\theta,H\mid y) = \frac{p(y\mid\theta,H)p(\theta\mid H)p(H)} {\displaystyle\sum_{H'}\int p(y\mid\theta',H')p(\theta'\mid H')p(H')\,d\theta'}. $$

For continuous administration times, the sum over histories becomes an integral over the uncertain times, or a sum-and-integral if some dose events are discrete and others continuous. The formula assumes that the prior probabilities meaningfully represent the information available before the concentration data, and that the observation model is appropriate conditional on the proposed history and parameters. Information from medication records or patient reports can inform the prior probabilities; it should not be counted again as if it were an independent concentration observation.

Worked Example: Updating the Probability of a Missed Dose

Worked Example: Updating the Probability of a Missed Dose

Hypothetical data. A patient receives 100 mg doses, and \(V=20\) L and \(k=0.100\) h\(^{-1}\) are treated as known for this illustration. Doses were scheduled at 0 and 8 hours, and a concentration was measured at 12 hours. The first dose was taken. Let \(H_T\) mean the dose at 8 hours was taken and \(H_M\) mean it was missed. Before seeing the concentration, assign probabilities \(P(H_T)=0.800\) and \(P(H_M)=0.200\). Suppose the measured concentration is 3.100 mg/L with a normal residual error having standard deviation 0.800 mg/L.

Under \(H_T\), both doses contribute; under \(H_M\), only the dose at time 0 contributes. Thus:

$$ C_T=5\{\exp(-0.100\times12)+\exp(-0.100\times4)\} =5(0.3012+0.6703)=4.858\ \text{mg/L}, $$
$$ C_M=5\exp(-0.100\times12)=5(0.3012)=1.506\ \text{mg/L}. $$

The normal likelihood density for an observation \(y\) with mean \(C\) and standard deviation \(\sigma\) is \(\exp\{-(y-C)^2/(2\sigma^2)\}/(\sqrt{2\pi}\sigma)\). Substitution gives likelihood densities of approximately 0.044640 per mg/L under \(H_T\) and 0.068503 per mg/L under \(H_M\). Multiplying by the prior probabilities gives unnormalized weights \(0.800(0.044640)=0.035712\) and \(0.200(0.068503)\approx0.013701\). Therefore:

$$ P(H_T\mid y)=\frac{0.035712}{0.035712+0.013701}\approx0.7227, \qquad P(H_M\mid y)\approx0.2773. $$

The observation increases the probability assigned to a missed dose from 0.200 to about 0.277, but the taken-dose history remains more probable because its prior probability was higher. These probabilities describe uncertainty between the specified histories under this model; they do not establish what the patient actually did.

Propagating History Uncertainty Into a Forecast

A forecast must average over the histories and parameters that remain plausible after the observed data. If \(C_*\) is a future concentration, its predictive distribution is obtained by integrating its conditional distribution over the posterior uncertainty:

$$ p(C_*\mid y)=\sum_H\int p(C_*\mid\theta,H,y)\,p(\theta,H\mid y)\,d\theta. $$

The conditional distribution \(p(C_*\mid\theta,H,y)\) must include any future-dose assumptions and relevant observation error. If the forecast concerns a future measured concentration, residual or assay error belongs in that distribution; if it concerns the underlying model-predicted concentration, it does not. Future adherence is a separate uncertainty from past adherence: a forecast conditional on a planned future dose should say so, rather than treating that future dose as guaranteed without qualification.

For discrete histories, the forecast is a weighted mixture. Even if each history yields a narrow conditional forecast, uncertainty about which history occurred can make the overall forecast broader or multimodal. In particular, averaging the parameters first and then predicting is generally not equivalent to averaging predictions over the joint posterior.

Worked Example: An Uncertain Administration Time

Worked Example: An Uncertain Administration Time

Hypothetical data. A single 100 mg dose with \(V=20\) L and \(k=0.100\) h\(^{-1}\) was administered either at time 0 (\(H_0\)) or time 2 hours (\(H_2\)), with prior probabilities 0.500 each. A concentration of 3.000 mg/L was observed at hour 6. Assume normal residual error with standard deviation 0.400 mg/L. The predicted concentrations are:

$$ C_0=5\exp(-0.100\times6)=2.744\ \text{mg/L}, \qquad C_2=5\exp(-0.100\times4)=3.352\ \text{mg/L}. $$

The likelihoods share the same normalizing constant, so their relative weights are determined by the exponential terms. Substituting the residuals and \(\sigma=0.400\), these terms are \(\exp\{-(3.000-2.744)^2/(2\times0.400^2)\}\approx0.815\) and \(\exp\{-(3.000-3.352)^2/(2\times0.400^2)\}\approx0.680\). With equal priors, the posterior probabilities are \(P(H_0\mid y)=0.815/(0.815+0.680)\approx0.545\) and \(P(H_2\mid y)\approx0.455\).

Suppose the forecast is for a measured concentration at hour 10, with no additional dose before then and the same independent 0.400 mg/L observation-error standard deviation. The two conditional mean concentrations are \(5\exp(-1.000)=1.839\) mg/L and \(5\exp(-0.800)=2.247\) mg/L. The posterior mean of the future measured concentration is approximately \(0.545(1.839)+0.455(2.247)=2.025\) mg/L.

The uncertainty has more than one source. The conditional measurement-error variance is \(0.400^2=0.160\) (mg/L)\(^2\). The variance between the two history-specific means is approximately \(0.545(0.455)(2.247-1.839)^2=0.04128\) (mg/L)\(^2\). Their sum is about \(0.2013\) (mg/L)\(^2\), giving a predictive standard deviation of \(\sqrt{0.2013}=0.449\) mg/L. Treating the administration time as known would omit the between-history component.

History Uncertainty Can Change Parameter Estimates

When parameters are not known, the concentration data inform both the history and the parameters. A low concentration could be consistent with higher clearance, a missed dose, or some combination. The posterior therefore needs to preserve the dependence between \(H\) and \(\theta\). This is especially important when the data are sparse: different histories may fit nearly as well while implying different parameter values.

The following discrete example uses equal prior probabilities for two possible elimination rates and two possible histories. It is a calculation aid, not a claim that a real patient’s rate constant has only two possible values. The likelihood is based on the same normal error model for every parameter-history combination.

Worked Example: Jointly Updating a Rate Constant and Dose History

Hypothetical data. A patient has a 100 mg dose at time 0 and a scheduled 100 mg dose at time 8; \(V=20\) L is treated as known. The later dose was either taken or missed. A concentration of 3.000 mg/L was measured at time 12, with normal residual error standard deviation 0.500 mg/L. The candidate values \(k=0.0800\) and \(k=0.1200\) h\(^{-1}\) each have prior probability 0.500, and the two dose histories each have prior probability 0.500, independently.

For each combination, the prediction is \(5\exp(-12k)\) if the later dose was missed and \(5\{\exp(-12k)+\exp(-4k)\}\) if it was taken. The likelihood, apart from a common normalizing constant, is \(\exp\{-(3.000-C)^2/0.500\}\), since \(2\sigma^2=0.500\). At \(k=0.0800\), the two predictions are 1.914 and 5.545 mg/L, giving relative likelihoods about 0.0948 and 0.0000024. At \(k=0.1200\), the predictions are 1.185 and 4.279 mg/L, giving relative likelihoods about 0.00137 and 0.0380.

The equal prior factors cancel when these likelihoods are normalized. Summing the two likelihoods for each rate constant and using the unrounded likelihoods gives:

$$ P(k=0.1200\mid y) = \frac{ \exp\{-(3.000-5\exp(-0.1200\times12))^2/0.500\} + \exp\{-(3.000-5[\exp(-0.1200\times12)+\exp(-0.1200\times4)])^2/0.500\} }{ \displaystyle\sum_{k\in\{0.0800,\,0.1200\}} \left[ \exp\{-(3.000-5\exp(-12k))^2/0.500\} + \exp\{-(3.000-5[\exp(-12k)+\exp(-4k)])^2/0.500\} \right] } \approx0.2938. $$

Thus \(P(k=0.0800\mid y)\approx0.7062\), and the posterior mean rate constant is \(0.7062(0.0800)+0.2938(0.1200)\approx0.09175\) h\(^{-1}\). This posterior mean alone does not describe the full uncertainty: the rate constant and dose history remain associated in the posterior.

For illustration, suppose a new 100 mg dose is planned at time 12 and the forecast is for the model-predicted concentration at time 16. The four history-parameter combinations predict approximately 5.021, 7.657, 3.827, and 5.741 mg/L, respectively (in order: \(k=0.0800\) with the earlier dose missed or taken, then \(k=0.1200\) with it missed or taken). Weighting these predictions by their joint posterior probabilities gives a posterior predictive mean of approximately 5.21 mg/L. The full forecast remains a mixture of the four predictions, not a single concentration at the value obtained by substituting the posterior mean \(k\) into the model.

Practical Checks, Alternatives, and Limitations

A practical analysis should make the histories and their prior probabilities explicit, then check whether the model’s predictions under those histories are compatible with the data and the clinical record. Useful checks include whether the proposed administration-time range is plausible, whether dose records and patient reports have been used consistently, and whether small changes in the history probabilities produce large changes in the parameter estimates or forecasts. Strong sensitivity is a warning that the data do not distinguish the competing explanations reliably.

The main alternative is to select one history, often the most likely one, and proceed as if it were certain. This may be reasonable when independent records establish the history with high confidence, but it discards uncertainty when the evidence is ambiguous. A sensitivity analysis that repeats the fit under several fixed histories can reveal how consequential that choice is; unlike Bayesian averaging, however, it does not assign a combined probability distribution unless results are explicitly weighted. When administration times are uncertain over a continuous interval, integrating over the time distribution is more faithful than rounding every time to one chosen value.

These methods cannot make uninformative data informative. If one concentration is compatible with several combinations of clearance and dose history, the posterior may remain broad or depend strongly on the priors. A numerical posterior probability is conditional on the model, the candidate histories, and the prior probabilities; it is not an objective probability that a particular dose was taken. An omitted possibility, such as an unrecorded extra dose, cannot be recovered by averaging over histories that do not include it.

Common pitfall: “The history with the largest posterior probability is the true history, so I can fit parameters as if it were certain.” The most probable history is still uncertain unless its posterior probability is effectively one. Conditioning only on that history can understate parameter uncertainty and omit other plausible forecast outcomes. Retain and propagate the posterior weights, or report clearly that a forecast is conditional on a selected history.
Key takeaway: Treat uncertain adherence and administration times as uncertainty in the dosing model itself. Update plausible histories with the concentration data, preserve their joint relationship with patient-specific parameters, and average future predictions over the resulting posterior uncertainty.

Check Your Understanding

Try each problem before reading its solution.

  1. A dose may have been taken or missed. What two ingredients, beyond the concentration observation, are needed to calculate the posterior probabilities of those histories?
  2. Why can choosing the most probable dose history and treating it as certain understate forecast uncertainty?
  3. In the uncertain-time example, what is the posterior probability of administration at time 2 hours, to three decimal places?
  4. A future concentration is forecast conditional on a planned dose. What does that forecast assume about future adherence?

Solutions. (1) Specify the prior probabilities for the histories and the likelihood of the observed concentration under each history, including the parameter assumptions and residual-error model. (2) Because other histories may retain posterior probability and predict different concentrations or parameter values; excluding them discards between-history uncertainty. (3) The posterior probability is approximately 0.455, calculated as \(0.680/(0.815+0.680)\). (4) Unless future adherence is also modelled as uncertain, the forecast is conditional on the planned dose being administered as specified.

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