Effect-Site Dynamics During Infusion and Washout

Infusion Creates Two Linked Time Courses

During a constant-rate infusion, plasma concentration rises toward a plateau rather than jumping immediately to that level. The effect-site concentration follows with its own delay: it is driven by plasma concentration, but does not equilibrate instantaneously. When the infusion stops, plasma concentration declines, while effect-site concentration may initially continue to rise if it is still below plasma concentration. These trajectories matter when interpreting the timing of drug effects, even when the effect is determined by a concentration-response relationship already specified elsewhere in the model.

The calculations below use a one-compartment plasma model with first-order elimination and a first-order effect compartment. The infusion begins at time zero, and both plasma and effect-site concentrations are initially zero. The solution during infusion applies while the infusion rate remains constant; the washout solution begins when that infusion is stopped.

During a Constant-Rate Infusion

For a constant infusion beginning at zero concentration, the plasma balance is \(dC_p/dt=R_0/V-kC_p\). Its solution is \(C_p(t)=C_{ss}(1-e^{-kt})\): plasma concentration approaches \(C_{ss}\) asymptotically. The effect compartment is driven by this rising plasma concentration, so its trajectory is not simply the plasma curve multiplied by a fixed delay.

To obtain the effect-site solution, use the integrating factor \(e^{k_{e0}t}\) in the effect-compartment equation. Applying the initial condition gives the convolution integral below. Substituting the plasma infusion curve and integrating yields the closed form for unequal rate constants.

The result approaches \(C_{ss}\) as time increases, but initially \(C_e\) rises more slowly because it integrates the preceding plasma exposure. When \(k_{e0}=k\), the displayed expression has a zero denominator and must not be evaluated by direct substitution. Integrating the same convolution with equal rates gives its continuous limiting form.

After Infusion Cessation

Let \(T\) be the time the infusion stops, and write \(\tau=t-T\) for elapsed time since cessation. The plasma concentration at the stopping time is \(C_p(T)\). With no further input and first-order elimination, plasma concentration is \(C_p(T)e^{-k\tau}\). The effect compartment then responds to this declining plasma concentration while also retaining the concentration \(C_e(T)\) present at cessation.

Solving the effect-compartment equation from the initial value \(C_e(T)\), with \(C_p(T)e^{-k\tau}\) as its driving input, gives the following washout expression. The two terms have distinct roles: the first carries forward the existing effect-site concentration, and the second reflects continued equilibration with plasma after the infusion stops.

If the rates are equal, the corresponding limit is:

Interpreting the Trajectories and Checking the Model

During infusion, both concentrations approach \(C_{ss}\), but their rates of approach depend on different constants. A larger \(k\) means plasma approaches its infusion plateau more rapidly; a larger \(k_{e0}\) means the effect compartment equilibrates with the current plasma concentration more rapidly. These statements concern the modeled concentration trajectories, not a claim that the effect compartment is a directly measured anatomical space.

After cessation, plasma concentration must decline under the stated elimination model. The effect-site trajectory need not decline immediately: its instantaneous slope is \(k_{e0}[C_p-C_e]\). It rises when \(C_p>C_e\), is momentarily flat when \(C_p=C_e\), and declines when \(C_p<C_e\). This sign check is a useful diagnostic for detecting arithmetic or interpretation errors in a washout calculation.

The solution also has boundary checks. At \(t=0\), both infusion formulas give \(C_e(0)=0\). At long infusion times, \(e^{-kt}\) and \(e^{-k_{e0}t}\) vanish, so \(C_p\) and \(C_e\) both tend to \(C_{ss}\). At washout onset, setting \(\tau=0\) returns \(C_e(T)\); as \(\tau\) increases, both exponential terms ultimately vanish, so \(C_e\) tends to zero. A computed value that violates these checks warrants review of units, parameter substitution, or the selected equal-rate formula.

These closed forms are useful when the infusion rate and model parameters are constant over the interval being analyzed. They are not appropriate without modification if the plasma input follows a different time course, clearance is nonlinear or time-varying, the starting concentrations are not both zero, or the effect compartment feeds back on plasma pharmacokinetics. A general changing plasma profile requires solving the effect-compartment equation using that profile as its input. In practice, compare predicted plasma and effect-site trajectories with observations and inspect whether the assumed infusion timing, rate constants, and initial conditions are credible. A poor fit should not be repaired simply by interpreting an implausible \(k_{e0}\) as a physiological delay.

The main alternative is numerical integration of the differential equations. Numerical methods can accommodate more complicated inputs and models, but a closed-form solution is especially valuable for checking implementation and understanding parameter effects. For the constant-rate infusion and first-order washout considered here, the analytic formulas are direct and provide transparent checks on a numerical solution.