Deriving Concentrations During and After Zero-Order Absorption

From Constant Input to a Concentration Profile

A constant absorption rate does not produce a constant concentration when the body is also eliminating drug. While drug enters, the amount in the central compartment accumulates only to the extent that input exceeds elimination. When absorption ends, input is zero, but drug remains in the body and its concentration declines. The resulting profile has one equation during input and another after input stops.

This tutorial considers a single dose absorbed at a constant rate over a finite interval. The earlier material on zero-order absorption establishes the constant-rate input assumption. Here, the goal is to solve the central-compartment concentration equation under that assumption and to interpret the two parts of the profile.

The model assumes linear, first-order elimination, constant \(CL\) and \(V\), and a constant systemic input rate from time zero through time \(T\). It also assumes a single dose, no lag before absorption begins, and an initial concentration of zero. The absorbed systemic amount is \(FD\). These assumptions define the concentration calculation; departures from them may require a different input function or disposition model.

Deriving the Concentration During Input

Write \(A(t)\) for the amount of drug in the central compartment. During absorption, the amount changes by input minus elimination. Elimination is \(CL\,C(t)\), where \(C(t)=A(t)/V\). Thus, for \(0\leq t\leq T\),

Since \(C(t)=A(t)/V\), dividing the amount equation by \(V\) gives a differential equation directly for concentration:

Multiplying by the integrating factor \(e^{kt}\) gives \(e^{kt}\frac{dC(t)}{dt}+ke^{kt}C(t)=\frac{R_{\mathrm{in}}}{V}e^{kt}\). The left side is the derivative of \(e^{kt}C(t)\). Integrating from zero to \(t\), and using \(C(0)=0\), yields the concentration during input:

Substituting \(R_{\mathrm{in}}=FD/T\) gives the version written in terms of dose and absorption duration. The factor \(R_{\mathrm{in}}/CL\) has units of concentration; it is the limiting concentration that would be approached if the same input rate continued indefinitely. With finite-duration absorption, this limit need not be reached.

After Absorption Stops

For times after \(T\), the input rate is zero. The amount equation becomes \(dA(t)/dt=-kA(t)\), or equivalently \(dC(t)/dt=-kC(t)\). Starting from the concentration reached at the end of input, the solution is exponential decline:

Substituting the during-input value at \(T\) gives a single expression for the post-input profile:

The piecewise equations agree at \(t=T\): the post-input expression evaluated at \(T\) is \(C(T)e^0=C(T)\). Concentration is therefore continuous when input stops. Its slope changes, however. Just before input stops, \(dC/dt=R_{\mathrm{in}}/V-kC(T)\); just after, \(dC/dt=-kC(T)\). The slope drops by \(R_{\mathrm{in}}/V\), reflecting the removal of the input term.

Interpreting the Profile and Checking the Model

During input, the rate of concentration change is \(R_{\mathrm{in}}/V-kC(t)\). At the beginning, \(C(0)=0\), so the initial slope is \(R_{\mathrm{in}}/V\). As concentration rises, elimination increases, reducing the net rate of accumulation. For a positive input and zero initial concentration, the concentration increases throughout the input interval: the during-input solution approaches \(R_{\mathrm{in}}/CL\) from below. Once input stops, concentration declines exponentially, with a half-life of \(\ln(2)/k\) under the model.

Several checks help detect an incorrect application:

• Units: \(R_{\mathrm{in}}/CL\) has units \((\text{amount}/\text{time})/(\text{volume}/\text{time})=\text{amount}/\text{volume}\). The exponent \(kt\) must be dimensionless.
• Boundary: The two equations must give the same concentration at \(T\).
• Direction: With positive input and zero initial concentration, concentration rises during input; after input stops, it falls.
• Input accounting: The rate is \(FD/T\), not \(D/T\), when only fraction \(F\) reaches systemic circulation.

The equations are useful because they distinguish the effects of input and disposition. A higher systemic input rate raises concentration during absorption; higher clearance lowers the input-related concentration scale \(R_{\mathrm{in}}/CL\). The value of \(k=CL/V\) controls how quickly elimination acts relative to the input interval. The same \(k\) determines the post-input decline in this one-compartment model.

The form also clarifies how this model differs from first-order absorption: here, systemic input is constant until a specified stopping time, whereas first-order absorption has an input rate that decreases over time. The mathematical concentration equation would also apply to a constant systemic infusion into the same disposition model. That mathematical resemblance alone does not identify the physical source of the input; the absorption model specifically represents drug entering systemically from an absorption process.

Common Mistakes and Limitations

• Using the full dose as systemic input. If \(F<1\), the systemic amount is \(FD\); omitting \(F\) overstates input and predicted concentrations.
• Continuing the input equation after \(T\). The constant-rate term is present only during absorption. After \(T\), use \(C(T)e^{-k(t-T)}\).
• Starting the post-input exponent at time zero. The concentration at \(T\) is already the starting value for decline, so the exponent is \(-k(t-T)\), not \(-kt\).
• Calling \(R_{\mathrm{in}}/CL\) the actual concentration at every time. It is the limiting value under indefinitely continued input, not generally the concentration reached by the end of finite input.

The profile relies on a single, well-mixed central compartment and linear elimination. A multicompartment disposition model, time-varying clearance, incomplete absorption within the specified interval, a nonzero initial concentration, or a nonconstant input changes the equations or their initial conditions. If the initial concentration is \(C_0\) rather than zero, for example, the during-input solution is \(C_0e^{-kt}+\frac{R_{\mathrm{in}}}{CL}(1-e^{-kt})\); the zero-initial-condition formula should not be used unchanged.