Forecasting Across Planned Dose Changes

Why Planned Dose Changes Need Explicit Forecasts

An individual forecast after a dose change is not obtained by simply multiplying the current concentration by the ratio of the new dose to the old dose. The concentration at a future time can reflect several doses, administered at different times, and each contribution depends on how rapidly the individual eliminates drug. If the individual’s pharmacokinetic parameters are uncertain, that uncertainty must be propagated through the complete planned schedule.

The conditioning information includes the individual’s observed concentration data and the planned dose history. The posterior distribution for the individual’s parameters summarizes what is known about those parameters given the observations and model. Forecasting across a planned schedule means applying each plausible parameter value to the same known schedule, then summarizing the resulting distribution of future concentrations.

Definition: A posterior concentration forecast at time \(t^*\) is the distribution of a future concentration conditional on the observed data, the model, and the planned dosing schedule through \(t^*\). A latent-concentration forecast describes the model-predicted concentration without future measurement error. A posterior predictive forecast for a future measurement additionally includes the model’s residual variability.

Propagating the Dose Schedule Through a PK Model

Consider a one-compartment model with instantaneous intravenous bolus doses, first-order elimination, constant individual clearance \(CL\), and constant volume of distribution \(V\). Let dose \(D_j\) be administered at time \(t_j\), where \(t_j\leq t^*\). The elimination rate constant is \(k=CL/V\). Under linear elimination, each dose contributes to the concentration at \(t^*\), and the total concentration is the sum of these contributions:

C(t^*;\theta)=\sum_{j:t_j\leq t^*}\frac{D_j}{V} \exp\left[-\frac{CL}{V}(t^*-t_j)\right], \qquad \theta=(CL,V).

Here dose is measured in mg, volume in L, time in h, and concentration in mg/L. The elapsed time \(t^*-t_j\) must be calculated separately for every dose. A dose administered after \(t^*\) contributes nothing to this forecast. For a known schedule with intervals of different lengths, the elapsed times in the formula automatically reflect those intervals.

The formula assumes doses are known and given as scheduled, the individual parameters remain constant over the forecast, the one-compartment bolus model is adequate, and elimination is linear. If the actual model has absorption, infusion, multiple compartments, or time-varying covariates, the same forecasting principle applies, but the model’s corresponding state equations must be propagated instead of using this particular formula.

Justification: After a bolus dose \(D_j\), the model’s concentration contribution is \(D_j/V\) at \(t_j\) and decays as \((D_j/V)\exp[-k(t-t_j)]\). Linear elimination means the differential equation for total concentration is linear in concentration. Therefore the contributions from separate doses add, giving the displayed sum.

Let \(p(\theta\mid y)\) denote the individual’s posterior distribution of parameters given the observed data \(y\). For a latent forecast, the posterior distribution of concentration is obtained by applying the schedule-specific function \(C(t^*;\theta)\) to parameter values drawn from \(p(\theta\mid y)\). In integral form, its cumulative distribution is

P\{C(t^*;\theta)\leq c\mid y,\text{schedule}\} = \int I\{C(t^*;\theta)\leq c\}\,p(\theta\mid y)\,d\theta,

where \(I\{\cdot\}\) equals 1 when the condition in braces holds and 0 otherwise. In practice, posterior draws \(\theta^{(m)}\), \(m=1,\ldots,M\), are transformed through the same schedule to produce \(C^{(m)}(t^*)=C(t^*;\theta^{(m)})\). The distribution of these simulated concentrations approximates the latent posterior forecast distribution.

Fix the conditioning information.
Specify the observed data, forecast time, and every planned dose and administration time up to that time.

Draw individual parameters.
Draw \(\theta^{(m)}\) from the individual’s posterior distribution, retaining dependence among parameters.

Propagate each draw.
Evaluate the model at the forecast time under the complete known schedule, using that draw’s parameters.

Summarize the simulated forecasts.
Use their mean or median as appropriate and their selected quantiles for a posterior prediction interval.

Worked Forecasts Across Changing Doses

Worked Example: Summing Contributions Across Unequal Intervals

Suppose, hypothetically, an individual has \(V=20\) L and \(CL=2.0\) L/h, so \(k=CL/V=0.10\) h\(^{-1}\). The planned bolus doses are 100 mg at hour 0, 100 mg at hour 8, and 50 mg at hour 20. Forecast the latent concentration at hour 24.

The three elapsed times at hour 24 are 24 h, 16 h, and 4 h. Thus,

\begin{aligned} C(24) &=\frac{100}{20}e^{-0.10(24)} +\frac{100}{20}e^{-0.10(16)} +\frac{50}{20}e^{-0.10(4)}\\ &=5(0.090718)+5(0.201897)+2.5(0.670320)\\ &\approx 0.4536+1.0095+1.6758\\ &=3.1389\ {\rm mg/L}. \end{aligned}

The predicted concentration is approximately 3.139 mg/L. The lower dose at hour 20 contributes 1.676 mg/L at hour 24; it is not appropriate to omit the earlier doses, whose residual contributions total about 1.463 mg/L.

When parameters are uncertain, repeat that calculation across posterior draws rather than inserting a single parameter estimate. Every draw uses the same schedule; only the parameter values vary. The resulting quantiles describe uncertainty about the forecast under the posterior and model, not merely variation in the estimated mean.

Worked Example: Propagating Posterior Parameter Draws

Use the same hypothetical schedule as above and fix \(V=20\) L for illustration. Suppose three equally weighted posterior draws for \(CL\) are 1.6, 2.0, and 2.4 L/h. Their corresponding \(k\) values are 0.08, 0.10, and 0.12 h\(^{-1}\). Calculate the hour-24 forecast for each draw.

For \(k=0.08\) h\(^{-1}\):

C^{(1)}(24)=5e^{-1.92}+5e^{-1.28}+2.5e^{-0.32} \approx 0.7330+1.3902+1.8154 =3.9386\ {\rm mg/L}.

For \(k=0.10\) h\(^{-1}\), the calculation in the preceding example gives \(C^{(2)}(24)\approx3.1389\) mg/L. For \(k=0.12\) h\(^{-1}\):

C^{(3)}(24)=5e^{-2.88}+5e^{-1.92}+2.5e^{-0.48} \approx 0.2807+0.7330+1.5470 =2.5607\ {\rm mg/L}.

The three equally weighted latent forecast draws are therefore approximately 3.9386, 3.1389, and 2.5607 mg/L. Their average is \(9.6382/3=3.2127\) mg/L, and their median is 3.1389 mg/L. These summaries differ because the posterior forecast distribution need not be symmetric. In a real analysis, one would generally use many posterior draws and retain joint variation in \(CL\) and \(V\), rather than treating one parameter as fixed.

For this deliberately small discrete approximation, the equal-tail 80% quantiles under the inverse empirical-CDF convention are the smallest and largest draws, 2.5607 and 3.9386 mg/L. The interval contains all three simulated draw values because of the coarse approximation; interpolation conventions can give different endpoints. With a continuous posterior forecast distribution, its equal-tail 80% interval has 80% posterior probability for the latent future concentration, conditional on the data, model, and schedule. That posterior statement does not by itself assert 80% frequentist repeated-sampling coverage.

Latent Concentration and Future Measurements

The posterior distribution of latent concentration reflects parameter uncertainty, and any other uncertainty represented in the model’s latent dynamics. A future measured concentration can vary further because of residual error. For example, under an additive normal observation model, \(Y^*=C(t^*;\theta)+\epsilon^*\), where \(\epsilon^*\sim N(0,\sigma^2)\) and the future residual is conditionally independent of the parameter draw. To forecast a measurement, simulate a residual for each latent forecast draw and add it. Do not add residual error when the target is specifically the latent concentration.

Worked Example: Adding Residual Variability for a Measured Concentration

Continue the three-draw illustration, and suppose the residual error for a future measured concentration is additive with standard deviation \(\sigma=0.25\) mg/L. The three latent values are 2.5607, 3.1389, and 3.9386 mg/L, each with probability \(1/3\). Their posterior mean is 3.2127 mg/L. Their variance using probabilities \(1/3\) is approximately

\frac{(2.5607-3.2127)^2+(3.1389-3.2127)^2+(3.9386-3.2127)^2}{3} \approx 0.3192\ ({\rm mg/L})^2.

Under the stated independent, mean-zero residual model, the predictive mean for \(Y^*\) remains 3.2127 mg/L, while its variance is approximately

0.3192+(0.25)^2=0.3817\ ({\rm mg/L})^2.

The corresponding standard deviation is \(\sqrt{0.3817}\approx0.6178\) mg/L. This calculation illustrates why an interval for a future measurement is generally wider than one for latent concentration. It does not determine a particular quantile interval; that interval should be obtained from the simulated predictive distribution, including residual draws, rather than inferred from the standard deviation alone.

Diagnostics, Alternatives, and Limitations

A schedule-specific forecast is only as credible as its input schedule, posterior distribution, and structural model. Before interpreting the result, check that dose amounts and times use consistent units; the forecast time is later than the included administrations; and the time since each dose is calculated from its actual administration time. Verify that the posterior draws are plausible, preserve parameter dependence, and have adequate simulation precision for the quantiles being reported. For a numerical state model, compare the computed propagation against known special cases or a finer numerical time step.

A particularly consequential error is silently substituting the planned regimen for the historical regimen. Concentrations at the forecast time may still depend on doses administered before the change. Another error is forecasting from only the most recent dose when the model implies contributions from all prior doses. If actual dosing may differ from the plan, the schedule is no longer fully known; that uncertainty must be represented, for example by simulating adherence or dose timing, rather than hidden inside a parameter-only forecast.

The main alternatives answer different questions. A plug-in forecast evaluates the model at one point estimate of the individual parameters; it is simpler but does not represent parameter uncertainty. Posterior simulation propagates the full individual posterior through the schedule and is more suitable when the goal is a distribution of future concentrations. A steady-state formula may be convenient when repeated identical doses have reached steady state, but it is generally not a substitute for a transient forecast during or after a dose change. Direct numerical integration of state equations is needed when the analytic sum does not represent the chosen PK model.

Finally, the forecast is conditional on the assumed model and planned doses. It does not automatically account for future changes in clearance, unrecorded doses, uncertain administration times, or model misspecification. Posterior intervals quantify uncertainty represented in the model and the posterior; they cannot repair a schedule that is wrong or a model that poorly describes the individual.

Common pitfall: “The dose was halved, so the predicted concentration must be halved.” This reasoning ignores contributions from prior doses and the time elapsed since each administration. The correction is to propagate every dose at its actual time through the individual’s model, then summarize forecasts across the posterior parameter distribution.

Check Your Understanding

Use the model assumptions stated above unless a question specifies otherwise.

At a forecast time, why must the elapsed time be calculated separately for each past dose?
If a dose is planned for one hour after the forecast time, should it contribute to the forecast at that time? Explain.
Why can the posterior mean forecast differ from the concentration calculated at the posterior mean parameter values?
What additional simulation step is needed for an interval for a future measured concentration rather than latent concentration?

Solutions. (1) Each dose has a different administration time, so it has a different duration of elimination before the forecast time; its exponential contribution depends on that elapsed time. (2) No. The dose has not yet been administered at the forecast time, so it contributes nothing to concentration at that time. (3) The concentration function is nonlinear in parameters, so in general the average of \(C(t^*;\theta)\) over the posterior is not equal to \(C(t^*;\text{posterior mean of }\theta)\). (4) Simulate future residual error from the observation model and add it to each latent forecast draw before taking predictive quantiles.