First-Order Elimination

What First-Order Elimination Means

Elimination describes the removal of drug from the body. Under first-order elimination, the amount eliminated per unit time is proportional to the amount of drug present. The proportionality constant is the elimination rate constant, \(k\), with units of inverse time. This is a concentration-independent process in the sense that \(k\) does not change with concentration: doubling the amount doubles the elimination rate, while the fraction eliminated per unit time remains constant.

The distinction between \(k\) and \(CL\) matters. The rate constant \(k\) describes the fraction of the amount eliminated per unit time. Clearance \(CL\) describes the volume of plasma or blood from which drug is cleared per unit time. For example, \(k\) may be measured in \(\mathrm{h}^{-1}\), whereas \(CL\) may be measured in \(\mathrm{L/h}\). In a one-compartment model, the two are related through \(CL=kV\).

The simple exponential concentration curve requires more than first-order elimination alone. It also assumes a one-compartment disposition model, constant \(k\), and no ongoing input during the interval being modeled. If drug is still entering the body, concentration reflects both input and elimination. In a multicompartment model, elimination from the central compartment can be first order even though the observed concentration curve is not a single exponential.

Deriving Exponential Decline

After input has stopped, the amount in the one-compartment model changes only through elimination. Its rate of change is therefore the negative of the elimination rate:

Separating variables and integrating from time zero, when the amount is \(A_0\), to time \(t\), gives:

Exponentiating both sides yields \(A(t)=A_0e^{-kt}\). With constant \(V\), concentration is proportional to amount, so it follows that:

Thus, while the model’s assumptions hold, concentration declines exponentially and its natural logarithm declines linearly with time, with slope \(-k\). The fraction remaining after time \(t\) is \(e^{-kt}\); the fraction eliminated is \(1-e^{-kt}\). Neither fraction depends on the starting concentration.

The half-life expression follows by setting \(C(t_{1/2})=C_0/2\). Substitution into the exponential model gives \(1/2=e^{-kt_{1/2}}\). Taking natural logarithms gives \(-\ln(2)=-kt_{1/2}\), and hence \(t_{1/2}=\ln(2)/k\). A larger \(k\) therefore means a shorter half-life. In the one-compartment model, substituting \(k=CL/V\) gives the equivalent relationship \(t_{1/2}=\ln(2)V/CL\). This formula should not be applied as though it necessarily describes every observed phase in a multicompartment profile.

Clearance and Exposure

For first-order elimination in the one-compartment setting, the elimination rate can be written either as \(kA\) or as \(CL\,C\). Since \(A=VC\), equating these expressions gives \(kVC=CL\,C\), and therefore \(CL=kV\). This relationship connects the exponential rate of decline to the clearance and apparent volume that describe the model.

A useful exposure result follows from conservation of drug. Let \(R_{\mathrm{in}}(t)\) denote the rate at which drug enters the systemic circulation. With first-order elimination, the amount balance is \(dA/dt=R_{\mathrm{in}}(t)-CL\,C(t)\). For a single dose, integrate this balance over the full period from before dosing until all drug from that dose has been eliminated. If the initial and final amounts attributable to the dose are zero, the integrated change in amount is zero. The total systemic input is \(FD\), where \(D\) is the administered dose and \(F\) is the fraction reaching systemic circulation. Consequently:

For an IV dose, \(F=1\). For an extravascular dose, the expression assumes that \(F\) appropriately represents the fraction reaching systemic circulation and that the full concentration-time profile is captured. The relationship does not require instantaneous absorption: under linear, time-invariant clearance, changing the rate of input can change the shape and peak of the profile without changing this total exposure, provided the total systemic input remains \(FD\).

Ongoing Input and Observed Profiles

When drug is entering the body during the period of interest, the one-compartment amount balance is \(dA/dt=R_{\mathrm{in}}(t)-kA(t)\). The observed concentration can rise, remain nearly stable, or decline depending on the balance between input and elimination. The exponential decline equation describes the elimination-only interval, not every part of a concentration-time profile.

For example, after an oral dose, absorption may still contribute materially to concentration while elimination is occurring. Once input becomes negligible, the terminal slope reflects disposition. If absorption is slow and remains an important contributor to concentrations, the observed terminal slope can instead be absorption-limited, as in flip-flop kinetics. A straight line on a log-concentration plot is therefore not, by itself, proof that the observed slope represents the disposition elimination rate constant.

The one-compartment model is also distinct from first-order elimination in a multicompartment model. In the latter, elimination from the central compartment can be proportional to the amount there, but exchange between compartments also shapes plasma concentrations. The overall profile can contain multiple exponential phases. A terminal phase may be useful for describing late concentrations, but its slope is not automatically interchangeable with a one-compartment \(k=CL/V\) for the whole profile.

Diagnostics, Alternatives, and Limitations

A model should be checked against the design and the observed concentration-time data. A semilogarithmic plot of concentration against time can help assess whether a post-input interval is approximately linear, as expected under a single exponential. Curvature, systematic residual patterns, or different slopes across time may indicate that input is ongoing, distribution phases are present, the one-compartment structure is inadequate, or clearance changes with concentration or time. Sparse sampling can conceal these patterns, so apparent linearity is not conclusive evidence.

The first-order model is a natural choice when a constant fraction of the amount is removed per unit time and dose-normalized exposure is consistent with the assumed linear disposition. It contrasts with elimination models in which the removal rate is not proportional to amount across the relevant concentration range. Those alternatives may be needed when data show concentration-dependent or capacity-limited behavior. They should not be inferred from a single noisy curve or from a half-life calculated without confirming that the relevant phase is disposition-driven.

The most useful alternatives depend on what the data require. A multicompartment structural model can represent distribution and elimination phases when a single exponential fails. A model that allows nonlinearity may be appropriate when the elimination rate no longer scales proportionally with concentration. A time-varying clearance model may be considered when clearance changes over time. Each adds assumptions and parameters; the simplest model that adequately represents the data is preferable, but simplicity must not override clear evidence of misspecification.

First-order elimination is valuable because it yields transparent links among rate, half-life, clearance, volume, and exposure. These links support dose and concentration calculations, but they are conditional results, not universal properties of a drug. The assumptions should be stated whenever the equations are used.