Converting Ke0 to an Equilibration Half-Life

Why Convert a Rate Constant to a Half-Life?

In an effect-compartment model, \(k_{e0}\) describes how quickly the effect-site concentration responds to plasma concentration. Its units are inverse time, such as \(\mathrm{h}^{-1}\) or \(\mathrm{min}^{-1}\). A half-life expresses the same rate in units of time, which can make the implied delay easier to interpret and communicate. The conversion is exact for the first-order equilibration model, not a separate biological process.

The crucial distinction is that the equilibration half-life describes how quickly the gap between plasma and effect-site concentrations closes when plasma concentration is held fixed. It is not, in general, the time for plasma drug concentration to halve. Plasma elimination and effect-site equilibration are different processes, each with its own rate parameter.

Deriving the Conversion

Suppose plasma concentration changes to a constant level \(C^\ast\) and remains there. Under the model, the effect-site concentration approaches \(C^\ast\). Define the remaining gap as \(D(t)=C^\ast-C_e(t)\). Differentiating this gap and substituting the model equation gives

This differential equation has the solution \(D(t)=D(0)e^{-k_{e0}t}\). The fraction of the original gap remaining after time \(t\) is therefore \(D(t)/D(0)=e^{-k_{e0}t}\). At the equilibration half-life, this fraction is \(1/2\). Solving that condition gives the conversion:

Conversely, if the equilibration half-life is given, the rate constant is \(k_{e0}=\ln(2)/t_{1/2,e}\). Here \(\ln(2)\) is approximately 0.6931. The units follow directly: dividing a dimensionless number by a rate in \(\mathrm{h}^{-1}\) yields hours. Always express the time and rate in compatible units before calculating.

The time constant \(\tau_e\) is the time for the remaining gap to fall to \(e^{-1}\), or about 36.8%, of its initial value. Equivalently, about 63.2% of the change toward the fixed equilibrium level has occurred after one time constant. A half-life instead describes the time to close half the gap. Both are useful, but they are not numerically equal.

Other Useful Equilibration Times

A half-life does not mean that equilibrium is reached after one half-life. It means that half the original difference remains. For the fixed-plasma step described above, the fraction of the change completed by time \(t\) is \(1-e^{-k_{e0}t}\). More generally, the time to close a fraction \(q\) of the initial gap, where \(0<q<1\), is

For example, closing 90% of the gap requires \(t_{90}=\ln(10)/k_{e0}\), which is about 3.322 equilibration half-lives. Such a time is a model-based measure of approach to equilibrium; it does not imply that the effect-site concentration becomes exactly equal to plasma concentration at a finite time.

Equilibration Half-Life Is Not Plasma Elimination Half-Life

For a one-compartment plasma concentration declining by first-order elimination after input stops, the elimination model is \(dC_p/dt=-k_{el}C_p\). Its plasma elimination half-life is \(t_{1/2,el}=\ln(2)/k_{el}\). Although this formula has the same mathematical form as the effect-site conversion, \(k_{el}\) and \(k_{e0}\) refer to different processes. The plasma half-life concerns the decline of plasma concentration; the equilibration half-life concerns the closing of the plasma-to-effect-site concentration gap under fixed plasma concentration.

There is no general rule that one of these half-lives must be longer than the other. They may be similar by coincidence, but that does not make them interchangeable. In more complex pharmacokinetic models, plasma decline can also have multiple phases, so a stated plasma half-life may be specific to a particular model or terminal phase. The effect-site equilibration parameter still describes the assumed effect-compartment process.

Assumptions, Checks, and Limitations

The conversion is valid when the effect-site dynamics are represented by a constant, positive first-order rate \(k_{e0}\). For its half-life interpretation, plasma concentration is held at a fixed level so that there is a well-defined equilibrium value toward which \(C_e\) approaches. Concentration is assumed to be represented consistently in the effect-compartment equation, and time units must match the units used for \(k_{e0}\). If plasma concentration varies over time, the same parameter remains a characteristic rate of the specified model, but the effect-site trajectory need not have a single observed “halfway time” that equals \(t_{1/2,e}\).

A practical check is to convert any reported rate to an intuitive time before interpreting it. A value reported in \(\mathrm{min}^{-1}\) may imply a timescale of only minutes, whereas the same numerical value in \(\mathrm{h}^{-1}\) implies a timescale of hours. Inspect the original model specification and units rather than relying on a parameter label alone. Also check that \(k_{e0}>0\): zero would imply no movement toward equilibrium, and a negative rate would not describe stable first-order equilibration.

The three parameterizations are equivalent only within the same first-order model. A pure fixed delay, for instance, means no response-site change until a specified time has passed; first-order equilibration instead begins changing immediately and approaches its target gradually. More elaborate delay or turnover structures may have different timescales and cannot generally be reduced to one \(k_{e0}\) without additional assumptions.

When an inferred or reported half-life seems implausible, first check units and conversion direction: \(k_{e0}=\ln(2)/t_{1/2,e}\), not \(t_{1/2,e}/\ln(2)\). Then check whether the quoted half-life refers to the effect site, the plasma, or another process. Even with correct arithmetic, the simple conversion cannot validate whether a first-order effect-compartment model adequately represents the data; that requires assessing the model itself.