Reforecasting After an Individual Dose Change

Why Reforecast After a Dose Change?

A dose change changes the drug input to the model; it does not, by itself, change the individual’s pharmacokinetic parameters. The forecast must therefore reflect both the individual’s current information about those parameters and the doses actually taken. If a new concentration is available after the change, that observation can also revise the parameter posterior before future concentrations are predicted.

The key is to keep three things distinct: the dosing history that generated the new observation, the new observation itself, and the dosing schedule to be used for the future forecast. A planned dose is not automatically an administered dose. Likewise, a future dose schedule should not be substituted for the history that produced a concentration already measured.

Definition: Let \(\theta\) denote an individual’s pharmacokinetic parameters, \(H\) the relevant dosing history, and \(C(t)\) the concentration at time \(t\). An existing individual posterior \(\pi_0(\theta)\) summarizes parameter uncertainty before a new concentration observation is used. Reforecasting after a dose change means (1) using the revised history \(H_1\) to predict the new observation, (2) updating \(\pi_0(\theta)\) with that observation when appropriate, and (3) predicting future concentrations under a specified future dosing schedule \(H_F\).

Separate the Revised History From the Forecast Schedule

A dose history records the administrations that have occurred: dose amounts and times, including missed, delayed, or otherwise changed doses when these are known. The future schedule states what will be administered from the forecast origin onward. The distinction matters because the concentration at a sample time depends on past inputs, while a future concentration depends on both those past inputs and the proposed future inputs.

For a linear one-compartment model with instantaneous intravenous boluses, volume \(V\), and elimination rate constant \(k\), a dose \(D_j\) at time \(t_j\) contributes \(D_j/V\) to concentration immediately after administration and decays exponentially. Thus, after doses at times \(t_j\), the model prediction at time \(t\) is:

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

This formula assumes linear disposition, instantaneous input, known dose times and amounts, and no unmodeled concentration-changing process. More complex regimens require the corresponding model’s input and absorption functions; the bookkeeping principle remains the same. In particular, changing a dose does not justify multiplying all previous concentrations by the ratio of new dose to old dose. Earlier doses have already contributed according to their own amounts and elapsed times.

Worked Example: Rebuilding Concentration From the Actual Dose History

Suppose a hypothetical patient receives an intravenous bolus of 400 mg at time 0 and, after a dose change, 300 mg at hour 8. For illustration, take \(V=40\) L and an elimination half-life of 8 hours. Calculate the model concentration at hour 12, four hours after the second dose.

The elimination rate is \(k=\log(2)/8=0.08664\) per hour. The hour-12 concentration includes the remaining contribution from both administered doses:

C(12)=\frac{400}{40}\exp\{-0.08664(12)\} +\frac{300}{40}\exp\{-0.08664(4)\}.

The first contribution is approximately \(10(0.3536)=3.536\) mg/L. The second is approximately \(7.5(0.7071)=5.303\) mg/L. Therefore \(C(12)\approx3.536+5.303=8.839\) mg/L. This calculation uses the revised amount at hour 8 while retaining the earlier 400 mg dose at time 0.

The result is conditional on the stated model and parameters; it is not yet a posterior predictive forecast. If \(V\) and \(k\) are uncertain, their existing posterior must be propagated through the calculation.

Updating the Existing Individual Posterior

Suppose a new concentration \(y\) is measured at time \(t_1\), and the revised history through that time is \(H_1\). If the individual parameter posterior before using this observation is \(\pi_0(\theta)\), Bayes’ rule gives the updated posterior:

\pi_1(\theta\mid y,H_1)= \frac{p(y\mid\theta,H_1)\pi_0(\theta)} {\int p(y\mid\vartheta,H_1)\pi_0(\vartheta)\,d\vartheta}.

Here \(p(y\mid\theta,H_1)\) is the observation likelihood under the model and revised history. For example, with additive normally distributed measurement error of known standard deviation \(\sigma\), the model assumes \(Y\mid\theta,H_1\sim N(C(t_1\mid\theta,H_1),\sigma^2)\). The likelihood is then proportional to \(\exp\{-[y-C(t_1\mid\theta,H_1)]^2/(2\sigma^2)\}\). This update assumes the observation model is suitable and that the recorded history is sufficiently accurate for the stated analysis.

If the existing posterior is represented by weighted parameter draws \(\theta^{(m)}\) with weights \(q_m\), the new observation reweights those draws. Define the likelihood value \(L_m=p(y\mid\theta^{(m)},H_1)\). The normalized updated weight is \(w_m=q_mL_m/\sum_r q_rL_r\). For a future schedule \(H_F\), the posterior mean of the model-predicted concentration at time \(t_F\) is then:

E\{C(t_F)\mid y,H_1,H_F\} \approx\sum_m w_m C(t_F\mid\theta^{(m)},H_1,H_F).

The future prediction uses both the revised past history and the proposed future doses. If the forecast concerns an observed concentration rather than the latent model concentration, residual measurement error must also be included in the predictive distribution.

Worked Example: Updating Two Posterior Particles

Consider two hypothetical parameter particles with equal existing posterior weights. Under the revised dose history, their predicted concentrations at a new sample time are 0.852 mg/L and 0.8076 mg/L. The observed concentration is 1.10 mg/L, and the assumed normal residual standard deviation is 0.20 mg/L. Calculate the updated weights and the posterior mean of their future concentration predictions, which are 0.800 mg/L and 0.5023 mg/L.

The normalizing constant in the normal likelihood is the same for both particles, so it cancels in the normalized weights. The likelihood factors can therefore be calculated as:

L_A=\exp\left\{-\frac{(1.10-0.852)^2}{0.08}\right\} =\exp(-0.7688)\approx0.4636,

L_B=\exp\left\{-\frac{(1.10-0.8076)^2}{0.08}\right\} =\exp(-1.068722)\approx0.3435.

The denominator \(0.08\) is \(2\sigma^2=2(0.20^2)\). In the second calculation, \(1.10-0.8076=0.2924\), so \((0.2924)^2=0.08549776\) and \(0.08549776/0.08=1.068722\). Thus \(L_B\approx0.3435\). (The particle prediction is correspondingly interpreted at the stated precision.)

With equal prior weights, the updated weights are:

w_A=\frac{0.4636}{0.4636+0.3435}\approx0.5744, \qquad w_B=\frac{0.3435}{0.4636+0.3435}\approx0.4256.

The posterior mean future concentration is \(0.5744(0.800)+0.4256(0.5023)\approx0.6733\) mg/L. The contributions are approximately 0.4595 mg/L and 0.2138 mg/L, summing to about 0.6733 mg/L. The observation favors particle A because its prediction at the sample time is closer to 1.10 mg/L, but it does not eliminate particle B.

From Updated Parameters to a Future Forecast

Parameter reweighting produces a posterior over \(\theta\); the forecast is obtained by passing that posterior through the future concentration model. A mean alone does not describe all predictive uncertainty. A posterior predictive distribution is formed by combining variation across updated parameter values with any residual variability included in the observation or process model. For a target involving the latent concentration, compute the target event for each posterior draw and sum the corresponding weights.

Worked Example: Probability of a Future Concentration Target

Suppose an existing posterior has three equally weighted particles. After accounting for the actual changed-dose history, their predicted concentrations at a new sample time are 3.8, 4.2, and 5.0 mg/L. The new observation is 4.0 mg/L, with normally distributed residual error having standard deviation 0.5 mg/L. Under the proposed future regimen, the particles predict latent trough concentrations of 2.4, 2.8, and 4.0 mg/L. Find the posterior probability that the latent trough is between 2.0 and 3.5 mg/L.

Ignoring the likelihood constant common to all particles, the likelihood factors are \(\exp[-(y-\hat C_m)^2/(2(0.5)^2)]\). For the first particle the factor is \(\exp(-0.08)\approx0.9231\); for the second it is also 0.9231; for the third it is \(\exp(-2)\approx0.1353\). Their sum is 1.9815, so the normalized weights are approximately 0.4659, 0.4659, and 0.0683.

The first two future trough predictions are within the target and the third is not. Therefore the posterior probability for the latent trough target is \(0.4659+0.4659=0.9318\), or about 93.2%. This probability is conditional on the model, the observation error assumption, the dosing history, and the proposed regimen. If the target instead applies to a measured trough, residual measurement error must also be incorporated.

Derivation and Practical Checks

The update follows directly from the definition of conditional probability. For any parameter region \(A\), Bayes’ rule gives \(P(\theta\in A\mid y,H_1)=P(y\mid\theta\in A,H_1)P(\theta\in A)/P(y\mid H_1)\). In a continuous parameter model, applying the same rule to a small region around \(\theta\) yields a posterior density proportional to likelihood times prior density. The denominator is the integral of that product, ensuring that the posterior density integrates to one. A future expectation then averages the model prediction over this updated density, which is approximated by the weighted sum for posterior particles.

Before trusting a reforecast, check the following items:

Dose history: Confirm actual amounts, administration times, missed doses, and any available adherence information. A wrong input history can make otherwise plausible parameters appear inconsistent with observations.
Sample timing and units: Check time since each dose, concentration units, and whether the sample was taken before or after a dose.
Likelihood compatibility: Compare the observed concentration with the range of predictions across existing posterior draws. If nearly all likelihood weights are tiny or one draw receives almost all weight, investigate the record, model, residual-error assumption, and posterior support.
Forecast uncertainty: Inspect the full weighted predictive distribution, not only its mean. A few high-weight particles or broad parameter uncertainty can produce substantial tail uncertainty.
Future regimen: State which future doses are assumed. A forecast under continued dosing is different from one under a second dose change or an interruption.

A weight-concentration problem can occur when the new observation is poorly predicted by every existing particle. Renormalizing a set of nearly zero likelihoods still produces weights summing to one, but that arithmetic does not establish that the model describes the observation well. Such a result signals a need to examine the data and assumptions rather than to report the normalized forecast without qualification.

Alternatives, Limitations, and a Common Pitfall

A maximum a posteriori or other single-parameter forecast evaluates the concentration at one selected parameter value. It can be useful as a compact point prediction, but it omits the spread of predictions across the individual posterior. Reforecasting with weighted posterior draws retains parameter uncertainty. Simple proportional dose scaling is more restrictive still: it is defensible only under suitable dose-proportional kinetics and a compatible comparison of histories and times. It does not generally account for residual concentrations from earlier doses, nonlinear kinetics, or uncertainty in individual parameters.

A full refit may be appropriate if the new data require changes to the model or if the existing posterior representation cannot adequately support the update. By contrast, reweighting an existing posterior is efficient when its draws cover the parameter values that remain plausible and the new likelihood is correctly specified. Neither approach repairs incorrect dose records or an inadequate structural model. The calculation also cannot guarantee that the prescribed future regimen will be taken as planned.

Common mistake: “The dose fell by 25%, so every forecast concentration should be reduced by 25%, and the old posterior can be left untouched.” This ignores the different contributions of earlier doses and the information in a post-change observation. The correction is to use the actual revised history to predict that observation, update the existing parameter posterior through its likelihood, and then predict under the explicitly stated future schedule.

Key takeaway: A dose change changes the model input history, while a new concentration can change the posterior over individual parameters. Reforecast by representing both changes explicitly, then propagate the updated posterior through the future dosing schedule.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A previous dose was 500 mg and a later dose was 350 mg. Why is it generally incorrect to multiply the total predicted concentration by \(350/500\)?
In one sentence, state how a new concentration observation updates an existing parameter posterior.
Two equal-weight particles have likelihoods 0.6 and 0.2 for a new observation. What are their normalized posterior weights?
Using the weights in question 3, future latent concentrations are 3 mg/L and 5 mg/L. What is their posterior mean?
If the observed concentration lies outside the prediction range of nearly all posterior particles, what should be checked before presenting a forecast?

Solutions. (1) Earlier doses contribute different residual concentrations according to their amounts and elapsed times, so the total does not generally scale with only the latest dose ratio; use the full dose history in the model. (2) Multiply the existing posterior density by the new observation likelihood based on the revised history, then normalize. (3) The total likelihood is \(0.6+0.2=0.8\), giving weights \(0.6/0.8=0.75\) and \(0.2/0.8=0.25\). (4) The mean is \(0.75(3)+0.25(5)=2.25+1.25=3.50\) mg/L. (5) Check dose records and sample timing and units, then assess the observation-error model and whether the existing posterior and structural model can represent the new observation; normalized weights alone do not resolve a mismatch.