Infusion Start, Stop, And Washout

Why infusion start and stop times matter

A constant-rate infusion does not make concentration jump to its eventual plateau. Concentration rises over time as drug enters the body while elimination removes drug. When the infusion stops, the input disappears, but drug already in the body remains and continues to be eliminated. The concentration therefore changes its pattern at the stop time: it rises toward a plateau during infusion and declines afterward.

The calculation is naturally piecewise: use one equation while drug is being infused and another after the input ends. The two equations must agree at the stop time. This tutorial derives those expressions for the one-compartment model and uses them to calculate post-infusion decline and washout.

Concentration while an infusion is running

Assume the infusion begins at time \(t_s\), continues at the constant rate \(R_0\), and the concentration at its start is \(C_s\). The model assumes instantaneous mixing in one compartment, constant \(V\) and \(CL\), linear first-order elimination, and a known infusion rate. There are no additional doses or changes in these parameters during the interval.

Since \(A=VC\), the amount-balance equation during infusion becomes \(V\,dC/dt=R_0-CL\,C\). Dividing by \(V\) gives \(dC/dt+kC=R_0/V\). The constant-input equilibrium concentration is \(R_0/CL\), because at equilibrium input equals elimination: \(R_0=CL\,C\). Solving the differential equation from the starting concentration \(C_s\) gives:

Here \(t_e\) is the stop time. The elapsed infusion time is \(t-t_s\), not necessarily the clock time \(t\). If the compartment is drug-free at the start, \(C_s=0\), and the equation reduces to \(C(t)=(R_0/CL)(1-e^{-k(t-t_s)})\). The concentration approaches \(R_0/CL\) asymptotically; it does not reach that value at a finite time under this model.

Stopping the infusion and joining the pieces

At \(t_e\), the infusion rate changes from \(R_0\) to zero. The model contains no instantaneous removal of drug at that moment, so concentration is continuous: the value just after stopping equals the value just before stopping. Denote this shared concentration by \(C_e=C(t_e)\). After the stop, the amount-balance equation is \(dA/dt=-CL\,C\), or \(dC/dt=-kC\). Its solution, initialized at \(C_e\), is:

A complete profile for a single infusion is therefore obtained by calculating \(C_e\) from the infusion equation and using it as the initial condition for the post-infusion equation:

The equations agree at the boundary: setting \(t=t_e\) in the second piece gives \(C_e e^0=C_e\), exactly the value calculated from the first piece. The slope can change at the stop because the input changes, even though concentration itself does not jump.

Washout to a specified concentration

Under first-order elimination, post-infusion concentration declines exponentially. For any positive threshold \(C_{\text{th}}\) below the concentration at stopping, solve \(C_{\text{th}}=C_e e^{-k\Delta t}\) for the time \(\Delta t\) after stopping:

This result requires \(C_e>C_{\text{th}}>0\). If the threshold is defined as a fraction \(f\) of the stop concentration, where \(0<f<1\), then \(C_{\text{th}}=fC_e\), so \(\Delta t=-\ln(f)/k\). For example, the time for concentration to fall to one half of its stop value is \(\ln(2)/k\), the elimination half-life. A positive exponential approaches zero indefinitely; in this mathematical model, it never becomes exactly zero at a finite time. “Washout” therefore needs a stated threshold or fraction to have a definite time.

Residual concentration and interrupted inputs

An infusion need not begin from a drug-free state. If \(C_s\) is already present at the start, the starting concentration contributes \(C_s e^{-k(t-t_s)}\), while the infusion contributes the remaining term. This follows directly from the solution above and prevents the common error of discarding residual drug when a new infusion begins.

Checks, comparisons, and limitations

A useful way to check a piecewise calculation is to verify the time origin, boundary value, direction of change, and units. During infusion from a concentration below \(R_0/CL\), concentration rises toward that equilibrium; after stopping, a positive concentration declines. At the stop, both pieces must give the same concentration. The exponent must be dimensionless, and all concentrations compared or added must use the same units.

The infusion profile differs from an IV bolus profile because infusion supplies drug over an interval rather than instantaneously. The constant-rate infusion equation describes the rising phase; after the input stops, the same first-order elimination process gives a simple exponential decline. A bolus-style decline equation is appropriate only after the relevant input has ended, and its starting concentration must be the concentration at that time. More complicated input schedules can be handled by applying the same logic over each interval with a defined input rate and carrying the boundary concentration forward.

These expressions are only as appropriate as their assumptions. A changing infusion rate requires a different input function or separate intervals; nonlinear elimination, changing clearance or volume, incomplete mixing, or an incorrect record of start and stop times can make the predicted curve misleading. Evidence of systematic disagreement between predictions and observed concentrations around the start or stop may indicate a timing problem, a misspecified input history, or inadequacy of the one-compartment assumptions. The equations alone cannot distinguish among those explanations.

Common mistake: Using the total time since the infusion began as the elapsed time in the post-infusion equation, or continuing to use the infusion equation after stopping. The correction is to calculate the stop concentration first, then use time since the stop: \(C(t)=C_e e^{-k(t-t_e)}\). Also, washout is not a finite time to exactly zero; report the threshold or fraction used.