Known Doses and Infusions in State Prediction

Known Inputs Change the Predicted Mean

A linear state prediction describes how the distribution of a latent pharmacokinetic state changes between observation times. A known dose or infusion contributes to that prediction: it changes the expected state, but, when its amount and timing are known exactly, it does not itself add uncertainty. The distinction is important. A bolus is an instantaneous state jump; a constant-rate infusion is a continuous input over its infusion interval. Treating either one as if it were the other can produce a plausible-looking but incorrectly timed prediction.

Let \(X(t)\) be a \(p\)-dimensional state, such as compartment amounts, and consider an interval \([t_i,t_{i+1}]\) on which the linear stochastic dynamics are \(dX(t)=(AX(t)+Bu(t))dt+L\,dW(t)\). Here \(A\) is a constant \(p\times p\) drift matrix, \(B\) maps the known input \(u(t)\) into the state, \(L\) is a constant diffusion matrix, and \(W(t)\) is a standard Brownian motion of dimension \(r\). Assume the initial state has finite second moments and is independent of future Brownian increments. The input is deterministic and integrable; piecewise-constant infusion rates satisfy this condition. A bolus is handled separately as a jump at its event time.

For a one-compartment model whose state is amount in the body, \(A=-k\), where \(k\) is the elimination rate constant. An intravenous bolus of amount \(D\) adds \(D\) to that state. An intravenous infusion at rate \(R\) adds \(R\) amount-units per unit time to the drift while it runs. If the state is concentration instead, the dose-to-state mapping must include the appropriate volume. The units of \(G\), \(B\), and \(u(t)\) must agree with the state units.

Deriving the Affine Prediction

Write \(\Delta=t_{i+1}-t_i\) and \(\Phi(\Delta)=e^{A\Delta}\). On an interval without bolus jumps, variation of constants gives \(X(t_{i+1})=\Phi(\Delta)X(t_i)+\int_{t_i}^{t_{i+1}}e^{A(t_{i+1}-s)}Bu(s)\,ds+\int_{t_i}^{t_{i+1}}e^{A(t_{i+1}-s)}L\,dW(s)\). The first integral is the deterministic, input-driven contribution; the second is the process-noise contribution.

To justify the transition, multiply the SDE by \(e^{-A(t-t_i)}\) and apply the product rule. Since this factor is deterministic and differentiable, the resulting differential is \(d(e^{-A(t-t_i)}X(t))=e^{-A(t-t_i)}Bu(t)\,dt+e^{-A(t-t_i)}L\,dW(t)\). Integrating from \(t_i\) to \(t_{i+1}\) and multiplying by \(e^{A\Delta}\) gives the stated representation. The deterministic integral is non-random. The stochastic integral has mean zero and, by the Itô isometry for deterministic integrands, covariance \(Q(\Delta)\). It is Gaussian because its integrand is deterministic and the driving process is Brownian motion. It is independent of \(X(t_i)\) by the assumed independence of future Brownian increments. Therefore the transition conditional on \(X(t_i)\) is Gaussian with the stated mean and covariance. Applying the mean and covariance rules to the affine transformation of the Gaussian \(X(t_i)\) gives the final formulas. \(\square\)

For a constant infusion rate \(u(s)=u_i\) across the whole interval, the input contribution becomes \(g_i=\int_0^\Delta e^{A(\Delta-r)}Bu_i\,dr\). If \(A\) is invertible, this can also be written \(A^{-1}(e^{A\Delta}-I)Bu_i\). The integral expression remains valid whether or not \(A\) is invertible. If the rate starts or stops inside the interval, split the integral at the change time, or evaluate the same integral with the appropriate piecewise value of \(u(s)\).

A useful second fact follows directly from the transition: adding a known bolus changes the mean prediction but not its covariance. Suppose a bolus \(Gd\) occurs at \(\tau\), with no other jump on \([t_i,t_{i+1}]\). Propagate first to \(\tau-\), add the dose, then propagate from \(\tau+\). The dose’s contribution at \(t_{i+1}\) is \(e^{A(t_{i+1}-\tau)}Gd\), while the covariance propagates through the two intervals and their process-noise terms. Because \(Gd\) is deterministic, it contributes zero variance and zero covariance with the state. This composition is exactly the same as applying the transition across the full interval and adding the appropriately propagated dose contribution. Thus event splitting is a valid way to implement the prediction, not an extra source of randomness. \(\square\)

Worked Predictions

Boluses at Observation Times

When a dose time coincides with an observation time, the observation convention must be explicit. The recorded observation may correspond to the state just before administration, \(X(\tau-)\), or just after it, \(X(\tau+)\). The prediction must target the same convention. If the observation is pre-dose, first predict to \(\tau-\), evaluate the observation there, and then apply the jump for subsequent propagation. If it is post-dose, apply the jump before evaluating the state at \(\tau+\). A continuous-time diffusion has no finite-duration evolution between \(\tau-\) and \(\tau+\); the difference is the known jump.

For several known events, order them by time. Propagate to the first event, apply any bolus jump, change the infusion rate if needed, and continue. Events at the same time require a stated convention for their ordering and for whether the observation is pre-event or post-event. In a linear model, additive known boluses at the same time can be combined into one jump, but the observation convention still matters.

Diagnostics, Alternatives, and Limitations

The continuous-time input integral and a discrete-time input matrix are two ways to express the same prediction. For a constant rate on an interval, one can define \(B_\Delta=\int_0^\Delta e^{A(\Delta-r)}B\,dr\) and write the input contribution as \(B_\Delta u_i\). This is convenient in a discrete-time implementation. Replacing \(B_\Delta\) with \(\Delta B\) is generally only an approximation: it ignores the effect of state evolution during the infusion. It may be adequate for a very short interval with slowly changing dynamics, but it is not the exact transition in general.

A practical check is to verify that a zero input gives \(g_i=0\), that a positive infusion increases the state in the intended direction, and that the infusion contribution tends to zero as the interval length tends to zero. For a bolus, check that the jump occurs exactly once and is propagated only over the time after administration. A dose applied at the start of an interval is attenuated over the full interval; a dose at its end is not attenuated before that endpoint. Check units as well: a rate multiplied by time has amount units, while a bolus is already an amount.

Common failures include treating a bolus as a short infusion without specifying its duration, applying a dose both in the initial state and again as an event, integrating an infusion after its stop time, and using a post-dose state when the observation is pre-dose. Another error is adding a known dose to the covariance. Under the stated assumptions, a known deterministic input contributes no uncertainty; uncertainty in dose amount, administration time, or adherence would require a different model.

These formulas rely on linear dynamics with known coefficients and known inputs over the prediction interval. If the input is uncertain or depends on the latent state, it cannot simply be treated as the deterministic \(g_i\) above. If the PK dynamics are nonlinear, the linear Gaussian prediction formulas do not apply directly. The covariance formula also depends on the specified process-noise model; adding a dose does not remove uncertainty already present from the initial state or process noise.