Intravenous Bolus and Infusion Inputs

From Intravenous Dose to Central-Compartment Input

Intravenous administration places drug directly into the systemic circulation, so the model does not need an absorption depot to represent delivery from the administration site. The input still has a time course: a bolus is treated as an instantaneous addition of amount, whereas an infusion delivers amount over time. In a PBPK model, that input enters the central compartment and is then acted on by distribution and elimination processes represented elsewhere in the model.

It is important to keep the route input separate from the rest of the pharmacokinetic model. The input function specifies when and how much drug is delivered; the central and tissue equations specify what happens to drug after delivery. For transparent concentration calculations below, a one-compartment model with first-order removal is used as an illustrative special case. Its concentration formulas are not a substitute for the full tissue-distribution equations in a PBPK model.

The central-compartment mass balance can be written in general form as

where \(G_c\) collects the non-input terms in the central-compartment balance. It may include elimination, transfer from the central compartment to tissues, and return from tissues. In a PBPK model, these terms depend on the particular organ and tissue model. The route-specific input functions derived here do not require a particular form for \(G_c\).

Three Intravenous Input Functions

Use time zero as the start of administration. Let \(D\) denote the delivered dose, \(R_0\) a constant infusion rate, and \(T\) the duration of a finite infusion. An indicator function is 1 when its condition is true and 0 otherwise. A bolus is represented mathematically by a Dirac delta input, whose integral over time is one.

The delta-function notation is a convenient way to express an instantaneous input in a differential equation. Its key property is that integrating the amount balance across time zero adds exactly \(D\) to the central amount. Equivalently, the bolus sets the initial condition \(A_c(0^+)=A_c(0^-)+D\). It is not a very large but finite infusion rate over an unspecified interval.

The constant-rate input is an idealization if it is allowed to continue indefinitely. It is useful for describing a maintained infusion or the portion of a longer infusion before it stops. A finite infusion is the constant-rate input switched off at \(T\). Since its delivered amount is the integral of the rate over the infusion interval,

This dose-conservation check is useful both when writing model code and when translating a prescribed dose into an infusion rate and duration.

Concentration Profiles in a One-Compartment Special Case

For explicit concentration-time calculations, suppose the central amount is the only modeled amount and is removed by a first-order process with rate constant \(k\), where \(k>0\) and units are inverse time. Assume \(V_c\) is constant, drug is instantaneously and completely available in the central compartment after a bolus, and the infusion rate is constant during administration. The mass balance is then \(dA_c/dt=R_{\mathrm{in}}(t)-kA_c\). These assumptions simplify the disposition model; they do not change the definitions of the intravenous inputs.

Between input events, multiplying the differential equation by the integrating factor \(e^{kt}\) gives

Integrating this expression from time zero to \(t\), with initial amount \(A_c(0)\), yields the general solution for this one-compartment case:

The integral shows how each small amount delivered at time \(\tau\) contributes to the amount remaining at time \(t\). An amount delivered earlier has more time to be removed. For a bolus, integration across time zero gives the initial amount \(D\), after which the amount declines exponentially. For an infusion, integrating the rate over the period when input is active gives the accumulation during dosing.

For a constant-rate infusion that starts at zero with no initial amount, integrate the input \(R_0\) from zero to \(t\). Dividing the resulting amount by \(V_c\) gives the concentration during the infusion. For a finite infusion, the same expression applies only until the infusion stops; afterwards, the amount declines under the removal process without further input.

The post-infusion expression follows by taking the concentration at the stopping time, \(C_c(T)=\{R_0/(V_ck)\}(1-e^{-kT})\), and allowing that amount to decline for an additional \(t-T\) hours. Concentration is continuous at the stopping time in this model: evaluating either expression at \(t=T\) gives the same value. The input rate changes abruptly at \(T\), but the amount and concentration do not jump because no instantaneous dose is given at that time.

Checks, Alternatives, and Common Failure Modes

In an implementation, verify that the input function has the intended units and integrates to the intended dose. A bolus is an amount, not a rate; an infusion rate has amount-per-time units. For a finite infusion, check \(R_0T=D\), confirm the input switches off at \(T\), and ensure that the concentration expression changes to the post-infusion form at the correct time. With nonzero initial amount, its contribution must also be included; the zero-initial-amount formulas above cannot be applied unchanged.

A common mistake is to use \(D/V_c\) as the concentration throughout an infusion because \(D\) is the total dose. That calculation is the immediate concentration increment for a bolus in the one-compartment model, not the concentration during an infusion: drug is being delivered gradually while removal occurs. The correction is to use the rate-and-duration input and integrate it through the model, as in \(C_c(t)=\{R_0/(V_ck)\}(1-e^{-kt})\) during infusion.

Another useful comparison is between the input-function approach and a boundary or initial-condition implementation. A bolus can be implemented by adding \(D\) to \(A_c\) at time zero; an equivalent delta input represents the same jump. An infusion is naturally implemented as a rate term in the differential equation. These are alternative numerical representations of the same administered amount, not different pharmacokinetic mechanisms. A full PBPK model may require numerical solution of coupled organ equations; its concentration profile need not follow the single exponential expressions shown here. The limitation of the illustrative formulas is precisely that they replace tissue distribution with one central amount and a single first-order removal process.

Diagnostics for a failed profile include a total delivered amount that differs from the prescribed dose, a concentration that jumps when a finite infusion stops, an infusion that continues unintentionally after \(T\), or a result with incompatible units. Also check the timing convention: if an infusion starts at a time other than zero, shift its active interval rather than applying the zero-start expressions without adjustment. These checks address input implementation; disagreement with observed concentrations may additionally reflect an inadequate disposition model or incorrect parameter values.