Dose-to-Dose Variability Versus Continuous Diffusion

Two Different Sources of Pharmacokinetic Variability

An amount may be uncertain because the administered dose is not fully or consistently available, or because the amount experiences random perturbations throughout time. These mechanisms can both produce variable concentrations, but they do not describe the same process. A random bioavailability multiplier acts at a dosing event; continuous diffusion acts throughout an interval, including times when no dose is given.

The distinction matters when specifying a model. If variability is introduced only at administration, adding a diffusion term can incorrectly spread that uncertainty across time. Conversely, random dose multipliers cannot represent ongoing process noise between administrations. A model may include both, but their roles and timing should remain explicit.

The multiplier may represent variable bioavailability, a random delivered-dose fraction, or another source of administration-level variation. It is important to specify whether \(F_j\) is independently redrawn at every dose or is shared across doses for one individual. A shared patient-level multiplier is not dose-to-dose variability: it creates persistent between-person differences. Also, for an oral dose entering an absorption depot, the jump belongs in the depot amount, not automatically in the central amount.

Dose Multipliers: Moments and Their Timing

First consider a single dose into an amount state, followed by deterministic first-order elimination. Suppose the post-dose amount is \(D F\), where \(D\) is fixed, \(F\) has finite mean \(\mu_F\) and variance \(v_F\), and between doses \(dA(t)=-kA(t)\,dt\) for \(k\geq 0\). At elapsed time \(s\), the amount is \(A(s)=DF e^{-ks}\). Directly taking its first two moments gives

The random variation is already present immediately after dosing: \(E[A(0)]=D\mu_F\) and \(\operatorname{Var}(A(0))=D^2v_F\). Elimination then scales that variation down. There is no new random input between doses in this model.

For repeated doses, independent multipliers add independent contributions, while a multiplier shared across doses makes those contributions correlated. The distinction can be expressed exactly in a simple elimination model.

Proof. Deterministic first-order elimination over the interval \(\Delta_j\) maps the preceding amount to \(e^{-k\Delta_j}X_{j-1}\). Adding the dose jump gives the stated recursion. Taking expectations gives the mean formula by linearity. For the variance, independence implies \(\operatorname{Cov}(X_{j-1},F_j)=0\). Therefore \[ \operatorname{Var}(X_j)=e^{-2k\Delta_j}\operatorname{Var}(X_{j-1})+D_j^2\operatorname{Var}(F_j), \] which is the required expression. \(\square\)

Continuous Diffusion: Variability Accumulates Between Doses

In a continuous diffusion model, the Brownian increment \(dW(t)\) supplies random variation over time. For example, a proportional diffusion term \(g(A,t)=\sigma A\) makes the instantaneous random perturbation proportional to the current amount. To define a standard solution, one may assume the drift and diffusion coefficients are globally Lipschitz with at most linear growth; for the linear model below, these conditions hold. The result also assumes a deterministic initial amount and a Brownian motion with respect to the model filtration.

Derivation. The explicit solution is \[ A(t)=A_0\exp\bigl((-k-\tfrac12\sigma^2)t+\sigma W(t)\bigr). \] Since \(W(t)\sim N(0,t)\), the normal moment-generating function gives \[ E[e^{\sigma W(t)}]=e^{\sigma^2t/2}, \] and therefore \(E[A(t)]=A_0e^{-kt}\). Squaring the solution and using \(E[e^{2\sigma W(t)}]=e^{2\sigma^2t}\) gives \[ E[A(t)^2]=A_0^2e^{-2kt-\sigma^2t}e^{2\sigma^2t} =A_0^2e^{-2kt+\sigma^2t}. \] Subtracting the squared mean yields \[ \operatorname{Var}(A(t)) =E[A(t)^2]-E[A(t)]^2 =A_0^2e^{-2kt}(e^{\sigma^2t}-1). \] These moments are finite at every finite \(t\), as the displayed expressions verify. \(\square\)

Model Choice, Diagnostics, and Common Pitfalls

A useful specification question is: when, physically or empirically, does the randomness enter? If replicate administrations vary while the amount between administrations follows a deterministic trajectory, a dose multiplier may be appropriate. If variability develops during an interval even with no dosing event, a continuous diffusion may be needed. If both patterns are supported, combine them:

For this hybrid specification, the diffusion describes within-interval process noise and the multiplier describes administration-level variation. A random multiplier may be independent of the Brownian motion and of earlier multipliers, but those are modeling assumptions, not automatic properties. Dependence can be included when the scientific mechanism requires it.

Diagnostics should examine variability by timing and replication. Variation appearing as an abrupt change at dose times supports an event-level source; variability that grows or changes during dose-free intervals supports a continuous process component. Repeated measurements alone may not identify the source: observation error, patient-level random effects, uncertain doses, and process noise can all contribute to observed spread. The model should therefore be checked against dosing records, replicate patterns, and the time course of residual variability.

• Common mistake: putting dose uncertainty into the diffusion term. This treats random administration as if new noise were injected continuously. Represent the administration mechanism with a dose-time multiplier or jump.
• Common mistake: redrawing a patient effect at every dose. A persistent bioavailability difference across one patient's doses requires a shared effect, which induces dependence between dose contributions.
• Common mistake: assuming equal means imply equivalent models. As the worked comparison shows, two models can have the same mean trajectory but different variance and different timing of uncertainty.

The alternatives also have limits. A dose multiplier cannot describe random evolution between doses on its own; a diffusion does not automatically represent discrete administration variability. A hybrid model is more flexible, but it adds parameters and may be difficult to identify from sparse observations. Finally, the additive jump shown here assumes the dose enters the chosen state directly. For oral absorption, the dose is commonly added to a depot state, after which absorption transfers amount to the central compartment.