One-Compartment Constant-Rate Infusion

Why an Infusion Produces a Rising Concentration

A constant-rate intravenous infusion supplies drug continuously, while elimination removes drug continuously. At the start, the amount entering the body exceeds the amount eliminated, so concentration rises. As concentration increases, the elimination rate increases too. The rise slows as input and elimination approach balance. When the infusion stops, input becomes zero and concentration declines by elimination.

The equations below describe the structural concentration profile for one individual under a one-compartment model. They assume a well-mixed compartment, so the measured concentration is the amount in the compartment divided by its apparent volume. They do not, by themselves, describe between-person differences or measurement error.

The model assumes \(V>0\), \(CL>0\), and therefore \(k>0\), with these parameters constant over the period considered. Elimination is first-order: the elimination rate is proportional to the amount present. The drug enters directly into the compartment at a constant rate from time 0 to \(T\), and there is no additional input during that interval. Concentration is continuous when the infusion stops; only the input rate changes.

Deriving the Concentration During Infusion

Because \(A=VC\) and \(V\) is constant, dividing the amount-balance equation by \(V\) gives the concentration equation. Since \(k=CL/V\), the input term \(R_0/V\) can also be written \(kR_0/CL\).

For an infusion beginning with concentration \(C(0)=C_0\), multiply by the integrating factor \(e^{kt}\). The product rule gives \(d(e^{kt}C)/dt=(R_0/V)e^{kt}\). Integrating from 0 to \(t\), then solving for \(C(t)\), yields:

For an infusion into a compartment with no drug initially, \(C_0=0\), so the first term vanishes. The concentration rises toward \(R_0/CL\), but for finite \(t\) it remains below that value when the infusion begins at zero. At \(t=0\), the equation gives \(C(0)=0\), as required by the initial condition.

The units provide a useful check. If \(R_0\) is in mg/h and \(CL\) is in L/h, \(R_0/CL\) is in mg/L. The exponent \(kt\) is dimensionless when \(k\) is in h\(^{-1}\) and time is in hours.

After the Infusion Stops

At \(t=T\), the amount already in the compartment does not disappear when the infusion stops. For \(t>T\), set the input rate to zero and solve \(dC/dt=-kC\) using the concentration at the stop time as the initial value. This gives the post-infusion profile:

The concentration at infusion end is found from the during-infusion equation, using the initial concentration at \(t=0\). For an initially drug-free compartment, \(C(T)=C_{\mathrm{ss}}(1-e^{-kT})\). The complete profile is therefore the rising infusion curve up to \(T\), followed by exponential decline from that endpoint. The expression after stopping uses elapsed time since stopping, \(t-T\), not time since the infusion began.

Steady State, Duration, and Half-Life

The infusion rate and clearance determine the steady-state concentration, while the elimination rate constant determines how rapidly the profile approaches it. For an initially drug-free compartment, the fraction of steady state reached at time \(t\) is \(1-e^{-kt}\). The half-life is \(t_{1/2}=\ln(2)/k\). After one half-life of infusion, the concentration is half of \(C_{\mathrm{ss}}\); after each additional half-life, the remaining gap between the current concentration and \(C_{\mathrm{ss}}\) is halved.

Assumptions, Checks, and Alternatives

The formulas are exact for the stated model, but the assumptions matter in application. The infusion rate must be constant over the interval being modeled, and the relevant \(CL\) and \(V\) must be constant. The one-compartment description assumes rapid mixing relative to the time scale of interest. The exponential elimination equation assumes first-order elimination, rather than a rate that saturates as concentration rises. If there is drug present before infusion starts, use the general solution with the actual \(C_0\); do not use the drug-free formula.

Several checks can reveal arithmetic or model problems:

• During infusion from \(C_0=0\), concentration should rise monotonically toward \(R_0/CL\), because \(dC/dt=(R_0/V)e^{-kt}>0\).
• At \(t=0\), the drug-free formula gives zero; at infusion stop, the during- and post-infusion equations give the same value \(C(T)\).
• After stopping, concentration should decrease for \(k>0\), and after one half-life it should be half the value at the start of that interval.
• A negative exponent with time in hours requires \(k\) in h\(^{-1}\). A concentration unit error often signals that the units of infusion rate and clearance do not match.
• A persistent mismatch between observed profiles and the predicted shape may indicate that the constant-input, first-order, or one-compartment assumptions are inadequate; it is not fixed merely by changing the arithmetic.

The closed-form equation is the direct approach when the input is constant and the one-compartment assumptions are appropriate. For an infusion whose rate changes over time, one can solve the amount-balance equation separately over each constant-rate interval, carrying each interval's ending concentration forward as the next initial condition. More general time-varying inputs or more complex structural models can require numerical solution of differential equations. A bolus input is a different input pattern: it changes the initial amount rather than supplying drug continuously. These approaches are alternatives for different input schedules or model structures, not interchangeable formulas for the same dosing history.

The model has important limits. A good fit to this profile does not establish that the body is truly one well-mixed compartment, and a steady-state calculation is not a guarantee that a finite infusion will reach the target concentration. In practice, an infusion may stop before the asymptote is approached. The equations also describe the structural concentration, not assay variability or all sources of patient-to-patient variability.