Why the Input Event Matters
A constant input rate can describe more than one pharmacological event. Drug may leave an extravascular depot at a constant rate, with only a fraction reaching systemic circulation; alternatively, drug may be delivered directly into the circulation by intravenous (IV) infusion. In a linear disposition model, those events can generate identical concentration-time profiles if their systemic input rates are equal. That mathematical equivalence does not make the doses, bioavailability assumptions, or model events interchangeable.
The distinction is important when translating a dosing record into a model. An extravascular dose is not automatically the amount entering systemic circulation, whereas an IV infusion rate is already a direct systemic input. Confusing the two can produce an incorrect dose interpretation even when the fitted concentration curve appears plausible.
Model and Assumptions
Consider a one-compartment model with central amount \(A(t)\), concentration \(C(t)=A(t)/V\), apparent volume \(V\), and first-order elimination rate constant \(k\). Assume \(V\) and \(k\) are constant, elimination is linear, and the system starts with no drug in the central compartment. The input starts at time zero and stops at \(T\). For the depot model, assume the entire depot dose is released uniformly over \(T\), and a constant fraction \(F\) reaches systemic circulation. The IV infusion is assumed to enter the central compartment directly.
For the depot, \(D/T\) is the release rate from the extravascular depot, while \(F D/T\) is the rate entering systemic circulation. For the IV infusion, \(R_0\) itself is the systemic input rate. The central amount obeys the same differential equation during input if \(R_0=F D/T\), but the two rates have different meanings.
The piecewise input equations presume an instantaneous start and stop at the specified times. They do not include a lag time, a gradual change in release rate, or a separate absorption process after the stated input duration. Those additions would change the event model.
Concentration Profile and Its Derivation
For either event, write \(R\) for the systemic input rate during \(0\le t<T\). The amount equation is \(dA/dt=R-kA\), with \(A(0)=0\). Multiplying by the integrating factor \(e^{kt}\) gives \(d(Ae^{kt})/dt=Re^{kt}\). Integrating from 0 to \(t\), then dividing by \(e^{kt}\), yields the amount and concentration during input.
At the end of input, the concentration is \(C(T)=\frac{R}{Vk}(1-e^{-kT})\). Once input stops, the amount follows \(dA/dt=-kA\), so it declines exponentially from its value at \(T\). Thus, for \(t\ge T\), \(C(t)=C(T)e^{-k(t-T)}\). The same formulas apply to a depot when \(R=FD/T\), and to an IV infusion when \(R=R_0\).
Worked Example: Concentrations After a Zero-Order Depot Dose
Suppose a hypothetical extravascular dose is \(D=100\) mg, with \(F=0.80\), released over \(T=4\) h. Let \(V=20\) L and \(k=0.10\) h\(^{-1}\). Assume the constant bioavailability and complete, uniform release described above.
The depot releases \(D/T=100/4=25\) mg/h, but the systemic input rate is only \(F D/T=0.80(25)=20\) mg/h. At the end of input,
Two hours after input ends, at \(t=6\) h, the concentration has declined for \(6-4=2\) h:
The post-input factor uses time since the input stopped, not time since dosing began. The dose administered was 100 mg, while the model's total systemic input was \(FD=80\) mg.
Worked Comparisons: Infusion and Profile Equivalence
Worked Example: An IV Infusion With the Same Nominal Dose
Now consider a hypothetical IV infusion of 100 mg over 4 h, with the same \(V=20\) L and \(k=0.10\) h\(^{-1}\). The infusion rate is \(R_0=100/4=25\) mg/h. Since the infusion enters systemic circulation directly, no extravascular bioavailability multiplier is applied.
At the end of infusion,
At \(t=6\) h, the infusion has been off for 2 h:
The IV concentration is higher than the depot concentration in the preceding example because the IV event delivered 100 mg systemically, whereas the extravascular dose delivered \(80\) mg systemically under \(F=0.80\). The nominal administered amounts and durations matched, but the systemic inputs did not.
Worked Example: Different Events With the Same Profile
Suppose an extravascular depot has \(D=125\) mg, \(F=0.80\), and \(T=5\) h. Its systemic rate is \(FD/T=0.80(125)/5=20\) mg/h. Compare it with a hypothetical IV infusion at \(R_0=20\) mg/h for the same 5 h. For both, let \(V=25\) L and \(k=0.12\) h\(^{-1}\). These choices satisfy the assumptions of a constant input and shared linear disposition.
The concentration at the end of input is the same for both events:
Two hours later, each profile is multiplied by \(e^{-0.12(2)}\):
The matching profiles follow because the systemic input rates and event times match, not because the administered doses are the same. The depot administered 125 mg, of which 100 mg reaches systemic circulation under the assumed \(F\); the infusion delivers 100 mg systemically. Concentrations alone, under these conditions, do not distinguish these events.
What to Check When Building or Assessing the Model
Before interpreting an estimated rate or dose, state what the event represents. For an extravascular depot, the modeled administered amount is \(D\), while the systemic amount is \(FD\). For an IV infusion, \(R_0T\) is the systemic amount delivered during the infusion. If the model uses a rate parameter, confirm whether it represents release from the depot or entry into systemic circulation; these rates differ by \(F\) under the stated depot assumptions.
Useful checks include:
- Units: \(D\) is an amount, \(T\) is time, and both \(D/T\) and \(FD/T\) are amount per time. The expression \(R/(Vk)\) has concentration units.
- Input duration: the constant-rate phase should end at the event's specified \(T\). Afterward, the model has elimination only.
- Endpoint continuity: concentration is continuous at \(T\); the input rate changes, but the central amount does not jump.
- Observed shape: during input, concentration rises toward the steady-input level \(R/(Vk)\); after input stops, it declines with rate constant \(k\). A profile inconsistent with these phases may indicate that the constant-rate event or the assumed disposition is inadequate.
A common pitfall is to set the depot input rate equal to \(D/T\), then treat that as systemic input while also claiming \(F<1\). This counts the full administered dose as reaching systemic circulation and contradicts the bioavailability assumption. The correction is to distinguish the depot release rate \(D/T\) from the systemic rate \(FD/T\). For IV infusion, by contrast, \(R_0\) is already systemic under this model.
Comparison, Interpretation, and Limits
| Feature | Extravascular zero-order depot | IV infusion |
|---|---|---|
| Input event | Drug released from a depot over \(T\) | Drug delivered directly into systemic circulation over \(T\) |
| Administered amount | \(D\); systemic amount is \(FD\) | \(R_0T\), delivered systemically |
| Systemic rate | \(FD/T\), under constant \(F\) | \(R_0\) |
| Bioavailability | May be less than one | Direct systemic input; no extravascular \(F\) multiplier |
A direct IV infusion model is often appropriate when the dosing record specifies a systemic delivery rate. The depot model is appropriate when the event is extravascular release and bioavailability must be represented. A first-order absorption model, already defined by its declining absorption rate from a depot, is a different input assumption; it should not be substituted merely because it is another extravascular model. Choosing between these descriptions requires knowledge of the dosing event and evidence about the concentration-time pattern, not just a visually similar curve.
The equivalence result is conditional. It assumes linear disposition, constant \(F\), matching systemic rates and event timing, and no additional delay or input pathway. It does not establish that the actual events are interchangeable, nor does a matching fit independently identify \(F\), the administered dose, and the input event. If only systemic concentrations are available, a known dose, event record, or other information may be needed to distinguish these interpretations. In a mixed-effects analysis, the event definition should therefore be correct before interpreting individual or population parameter estimates.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A 60 mg extravascular dose has \(F=0.75\) and is released over 3 h. What are the depot release rate and systemic input rate?
- An IV infusion delivers 18 mg/h for 4 h. What systemic amount does it deliver, and should an extravascular bioavailability factor be applied?
- For a one-compartment model, explain why equal systemic input rates over the same time interval produce equal profiles even if the event types differ.
- If an extravascular dose and an IV infusion have the same nominal dose and duration, must their concentration profiles match? Explain the condition that determines the answer.
Solutions. (1) The depot release rate is \(60/3=20\) mg/h; the systemic rate is \(0.75(20)=15\) mg/h. (2) The systemic amount is \(18(4)=72\) mg. No extravascular bioavailability multiplier is applied to this direct IV input. (3) Both events obey \(dA/dt=R-kA\) during input and \(dA/dt=-kA\) afterward. With the same \(R\), initial condition, duration, and disposition parameters, their amount and concentration solutions are identical. (4) Not necessarily. They match under the stated assumptions only if their systemic rates and input times match. With constant \(F<1\), the extravascular systemic rate is \(FD/T\), while an IV infusion of the same nominal amount over that duration has rate \(D/T\); these differ unless \(F=1\).