Forecasting Before Steady State

Why Dose History Matters Before Steady State

Before steady state, concentration reflects not only the most recent dose but also drug remaining from earlier doses. A forecast based only on the latest dose can therefore miss a substantial part of the concentration. The size of that error depends on the dosing schedule and on how quickly the individual eliminates the drug.

For a patient-specific forecast, the dose record and the individual PK parameters work together. The dose record determines what drug input to include and when it occurred; the parameters determine how much of each dose remains at the forecast time. If earlier doses are omitted from the record, a model may attribute their unaccounted-for concentration to different PK parameters. Thus, an incomplete history can affect both a forecast made with fixed parameters and the parameter estimates inferred from observed concentrations.

Definition: A pre-steady-state concentration forecast is a predicted concentration at a specified time during a dosing course, calculated from the patient's individual PK parameters and the doses administered up to that time. Under linear one-compartment IV bolus assumptions, the forecast is the sum of the residual contributions from all prior doses.

The Repeated-Dose Concentration Model

Consider a one-compartment model with instantaneous IV bolus doses, volume \(V\), and first-order elimination rate constant \(k=CL/V\). Between doses, concentration declines exponentially. Each bolus dose \(D_j\) at time \(t_j\) produces an immediate concentration increment \(D_j/V\); after that, its contribution declines as \(e^{-k(t-t_j)}\). By superposition, the total concentration at time \(t\) is the sum of the contributions from all doses already given.

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

Here, \(D_j\) is in mg, \(V\) is in L, and \(k\) is in h\(^{-1}\) when times are in hours, so each contribution and the total concentration are in mg/L. The sum includes every dose at or before \(t\), including doses given at unequal intervals or in unequal amounts. The formula assumes doses were administered as recorded, the individual parameters stayed constant over the interval, and the drug follows linear one-compartment disposition with first-order elimination. It does not directly describe an oral absorption phase, an infusion, time-varying clearance, or nonlinear elimination.

The formula follows by applying the single-dose decay model to each dose separately. A dose \(D_j\) contributes \(D_j/V\) immediately after administration, and its contribution at a later time \(t\) is \((D_j/V)e^{-k(t-t_j)}\). Under linear disposition, one dose does not change the disposition of another, so the contributions add. Summing them gives the repeated-dose expression. This superposition argument is the reason the complete history matters: a dose does not cease contributing merely because a later dose has been given.

Worked Example: Forecasting After Several Equal Doses

Suppose a hypothetical patient receives 100 mg IV boluses at 0, 12, and 24 h. Use individual parameters \(V=20\) L and \(k=0.0800\) h\(^{-1}\) to forecast the concentration at 34 h, which is 10 h after the third dose. Assume the doses and times are known exactly and the model assumptions above hold.

At 34 h, the elapsed times since the three doses are 34, 22, and 10 h. Each dose contributes \(100/20=5\) mg/L at its administration time, before elimination. Therefore,

\begin{aligned} C(34) &=5\left(e^{-0.0800(34)}+e^{-0.0800(22)}+e^{-0.0800(10)}\right)\\ &=5\left(e^{-2.72}+e^{-1.76}+e^{-0.800}\right)\\ &=5(0.06587+0.1720+0.4493)\\ &\approx 3.436\ \text{mg/L}. \end{aligned}

The latest dose contributes about 2.247 mg/L, but the two earlier doses together contribute about 1.190 mg/L. A calculation using only the most recent dose would predict 2.247 mg/L, underestimating the full-history forecast by about 1.190 mg/L. The difference is not a change in the patient's parameters; it is the residual concentration from doses that the one-dose calculation discarded.

Forecasting With Unequal Dose Amounts

The sum is not restricted to regular schedules or identical doses. When dose amounts or intervals differ, calculate each term using that dose's amount and its own elapsed time. This is often more transparent and less error-prone than trying to replace the history with an average dosing interval.

Worked Example: An Unequal Dose History

For a hypothetical patient, suppose IV boluses of 100, 80, 100, and 50 mg were administered at 0, 12, 24, and 36 h. Use \(V=25\) L and \(k=0.0600\) h\(^{-1}\) to forecast at 42 h. Assume all four doses were administered as recorded and the linear one-compartment model is appropriate.

The elapsed times are 42, 30, 18, and 6 h. The dose-to-volume ratios are 4, 3.2, 4, and 2 mg/L, respectively. Thus,

\begin{aligned} C(42) &=4e^{-0.0600(42)}+3.2e^{-0.0600(30)} +4e^{-0.0600(18)}+2e^{-0.0600(6)}\\ &=4e^{-2.52}+3.2e^{-1.80}+4e^{-1.08}+2e^{-0.360}\\ &=4(0.08046)+3.2(0.1653)+4(0.3396)+2(0.6977)\\ &\approx 3.605\ \text{mg/L}. \end{aligned}

Each term uses the actual dose amount and time. For example, the 50 mg dose contributes about \(2(0.6977)=1.395\) mg/L, while the oldest, 100 mg dose contributes about \(4(0.08046)=0.3218\) mg/L. The older dose contributes less because more time has elapsed, not because it is excluded from the calculation.

How an Incomplete History Changes Parameter Estimates

When concentrations are used to estimate individual parameters, the likelihood depends on the predicted concentrations, and those predictions depend on the dose history. To make this connection explicit, suppose a measured concentration \(y\) has an additive normal error with known standard deviation \(\sigma\), conditional on candidate parameters and the recorded dosing history:

Y\mid \theta,\text{history}\sim N\left(C(t;\theta,\text{history}),\sigma^2\right), \qquad L(\theta)\propto \exp\left\{-\frac{[y-C(t;\theta,\text{history})]^2}{2\sigma^2}\right\}.

The posterior distribution is proportional to this likelihood times the prior distribution for \(\theta\). If a prior history is left out, the predicted concentration in the likelihood changes. The posterior can then favor different parameters, even though the measured concentration itself has not changed. The following example uses two candidate parameter values and equal prior probabilities to isolate this effect.

Worked Example: Comparing Posterior Estimates With Full and Incomplete Histories

Suppose a hypothetical patient received 100 mg IV boluses at 0, 12, and 24 h, and the concentration measured at 34 h was \(y=3.000\) mg/L. Use \(V=20\) L for both candidate parameter values. Candidate A has \(k=0.0800\) h\(^{-1}\); candidate B has \(k=0.0500\) h\(^{-1}\). Assign each candidate prior probability 0.5 and use an additive normal observation error with \(\sigma=0.800\) mg/L. Assume the recorded dose times and amounts are correct and both candidates use the same observation-error model.

With the full history, candidate A predicts 3.436 mg/L, as calculated earlier. Candidate B predicts

C_B(34)=5\left(e^{-0.0500(34)}+e^{-0.0500(22)}+e^{-0.0500(10)}\right) =5(0.1827+0.3329+0.6065) \approx 5.610\ \text{mg/L}.

For two candidates with equal prior probabilities, the posterior odds for B versus A equal the likelihood ratio. Using the normal likelihood,

\begin{aligned} \log\frac{P(B\mid y)}{P(A\mid y)} &=\frac{(y-C_A)^2-(y-C_B)^2}{2\sigma^2}\\ &=\frac{(3.000-3.436)^2-(3.000-5.610)^2}{2(0.800)^2}\\ &\approx \frac{0.190-6.814}{1.280} \approx -5.175. \end{aligned}

Therefore, the posterior odds for B versus A are \(e^{-5.175}\approx 0.00565\), and the posterior probability of B is \(0.00565/(1+0.00565)\approx 0.00562\). With the full dose history, candidate A is strongly favored because its predicted concentration is closer to the observation.

Now suppose the analyst incorrectly records only the 24 h dose. The corresponding predictions become \(5e^{-0.800}=2.247\) mg/L for A and \(5e^{-0.500}=3.033\) mg/L for B. Under this incomplete-history likelihood,

\begin{aligned} \log\frac{P(B\mid y,\text{incomplete history})} {P(A\mid y,\text{incomplete history})} &=\frac{(3.000-2.247)^2-(3.000-3.033)^2}{1.280}\\ &\approx \frac{0.5676-0.001066}{1.280} \approx 0.4426. \end{aligned}

The posterior odds for B versus A are now \(e^{0.4426}\approx 1.557\), giving posterior probability \(1.557/(1+1.557)\approx 0.609\) for B. The favored candidate has changed. This does not show that B is the patient's true parameter value; it shows that the wrong history changes the likelihood and, consequently, the posterior under the stated model and prior.

For these two-point candidate distributions, the posterior mean of \(k\) is also easy to calculate. With the full history it is \(0.9944(0.0800)+0.00562(0.0500)\approx 0.07983\) h\(^{-1}\). With the incomplete history it is \(0.3911(0.0800)+0.6089(0.0500)\approx 0.06173\) h\(^{-1}\). The change arises from altered likelihood contributions, not from a change in the prior or observation.

Checks, Alternatives, and Limitations

Before trusting a forecast, check that every known prior dose is included once, with the correct amount and time. Verify that elapsed times are calculated relative to the forecast time, and that \(k\) and time use reciprocal units. A negative elapsed time for a dose that should already have been administered, an implausible jump in predicted concentration, or a large mismatch between observed and predicted concentrations can signal a timing, dose-entry, or model problem. These checks do not identify the cause by themselves: a wrong dose record and unsuitable PK assumptions can both produce poor agreement.

A useful comparison is between three calculations:

Approach	What it uses	Main concern
Full-history forecast	Every known prior dose and the individual parameters	Requires a sufficiently accurate dose record and a suitable disposition model
Latest-dose-only calculation	Only the most recent dose	Omits residual drug from earlier doses during accumulation
Steady-state approximation	A repeating-dose pattern treated as having reached its repeated-cycle behavior	May not represent a patient who is still accumulating or whose prior schedule differs from the assumed pattern

A latest-dose-only calculation can be appropriate when earlier contributions are negligible for the question at hand, but that should be justified from the elimination rate and elapsed times, not assumed. Likewise, a steady-state approximation is not a substitute for summing a known non-steady-state history. The full-history expression is exact only within its assumptions. For oral doses with absorption, for example, each dose requires an absorption model as well as an elimination model; the instantaneous bolus contribution used here is not appropriate.

Parameter estimation also has limits. A single concentration may not distinguish among combinations of \(CL\) and \(V\), especially when the sampling time provides little information about the elimination profile. A complete history improves the specification of the likelihood, but it cannot guarantee that the available concentration data identify the individual parameters precisely. Similarly, if the history itself is wrong, a precise numerical optimizer or a narrow-looking parameter summary cannot repair the model misspecification.

Common Pitfall

Common mistake: “The last dose was at 24 h, so the 34 h concentration is just the 10-hour post-dose concentration from that dose.” This calculation discards earlier dose contributions. Under the linear repeated-bolus model, each earlier dose still contributes \(D_j e^{-k(t-t_j)}/V\) at 34 h. Include all known prior doses; omit a contribution only when it is negligible for the intended precision and that approximation is stated.

Check Your Understanding

Use the assumptions and equations in this tutorial. The questions are followed by concise solutions.

A patient receives 50 mg IV boluses at 0 and 8 h. For \(V=10\) L and \(k=0.10\) h\(^{-1}\), calculate the concentration at 10 h.
In the first worked example, what is the concentration contribution at 34 h from the dose at 0 h? Give units.
Why can omitting prior doses change a Bayesian parameter estimate even when the measured concentration and prior distribution are unchanged?
Name two assumptions needed to use the repeated-dose formula given here, and give one situation in which the formula would not directly apply.

Solutions. (1) The two elapsed times are 10 and 2 h, so \(C(10)=5(e^{-1.00}+e^{-0.200})=5(0.3679+0.8187)\approx 5.933\) mg/L. (2) The dose at 0 h contributes \(5e^{-0.0800(34)}=5e^{-2.72}\approx 0.3294\) mg/L. (3) Omitting doses changes the model-predicted concentration and therefore changes the likelihood; multiplying that altered likelihood by the same prior can produce a different posterior. (4) Assumptions include instantaneous IV boluses, linear disposition, and constant individual parameters with first-order elimination. The formula does not directly apply to an oral dose with an absorption phase.