Why Dose Inputs Need Their Own Terms
A perfused-tissue model tracks drug amounts in specified blood and tissue compartments. Its internal transfer terms describe drug moving between those compartments; an intravenous dose is different. It is an external input to the modeled system, entering at a particular place and time. Keeping these two kinds of terms distinct makes the equations easier to interpret and helps prevent a dose from being omitted or counted twice.
Let \(A_j(t)\) denote the amount of drug in compartment \(j\), with units such as mg. The model’s internal processes determine how that amount changes through redistribution and any other processes included in the model. An intravenous input adds drug to the compartment where it is delivered. The input may be an instantaneous bolus or a rate maintained over an interval.
Adding an Input to an Amount-Based ODE
Write the amount equations for the model without a new dose as \(\frac{d\mathbf{A}}{dt}=\mathbf{f}_{\mathrm{int}}(\mathbf{A},t)\), where \(\mathbf{A}\) is the vector of compartment amounts and \(\mathbf{f}_{\mathrm{int}}\) contains the model’s internal processes. An infusion adds a rate to the equation for its receiving compartment. For a single input into compartment \(j\), the equations become:
Here \(u(t)\) is the amount entering per unit time, for example mg/h, and \(\mathbf{e}_j\) is a vector with a 1 at position \(j\) and 0 elsewhere. This vector notation says precisely where the dose enters: only the receiving compartment’s equation gets the input term. For an infusion delivering drug into arterial blood, for example, \(u(t)\) appears in the arterial-amount equation, not in every tissue equation.
This representation assumes that the modeled input is intravenous, that \(u(t)\) describes the amount actually entering the specified compartment per unit time, and that the compartment amounts are the model states. It does not itself add an absorption process or an unmodeled transit compartment. If the input rate changes over time, use that time-varying rate in \(u(t)\).
The units provide a direct check. An amount derivative has units of amount per time, so the infusion term must have those same units. A concentration such as mg/L cannot be added directly to an amount derivative. If the input is specified as a concentration rate or as a pump volume rate, convert it to an amount rate using the appropriate concentration or drug concentration in the infusate before inserting it into the amount equation.
Intravenous Bolus: An Instantaneous Amount Change
An idealized intravenous bolus is administered over a time interval short enough to be treated as instantaneous at the model’s time scale. If a bolus of amount \(D\) enters compartment \(j\) at time \(t_d\), the state changes at that time:
The superscripts \(t_d^-\) and \(t_d^+\) mean immediately before and after the dose. Amounts in other compartments do not jump from this dose unless the dose is explicitly delivered to those compartments. The ordinary differential equations govern the evolution between dose times. A bolus at the start of a simulation is therefore usually represented by setting the post-dose initial condition to the pre-dose amount plus \(D\).
A Dirac-delta input is another mathematical representation of an ideal bolus. It places a rate with units of amount per time into the ODE, whose time integral is the dose amount. In practical model setup, the jump condition is often clearer: it explicitly states the amount just before and after administration and avoids numerically integrating an infinitely narrow input.
Worked Example: Bolus Into Arterial Blood
Suppose a hypothetical model has arterial blood, one tissue, and venous blood as its amount states. Immediately before a dose at \(t=0\), their amounts are 10 mg, 20 mg, and 5 mg, respectively. An intravenous bolus of 100 mg enters arterial blood. Find the initial amounts used for integration after dosing.
The dose enters arterial blood, so apply the jump only to that compartment:
The other compartment amounts do not jump at the instant of this administration: \(A_t(0^+)=20\ \mathrm{mg}\) and \(A_v(0^+)=5\ \mathrm{mg}\). Thus the post-dose starting state is \((110,20,5)\) mg in the order arterial blood, tissue, venous blood. Subsequent changes in tissue and venous amounts arise from the model’s internal processes, not from adding the bolus to those equations as separate dose terms.
Intravenous Infusion: A Rate Over Time
For an infusion, drug enters over an interval rather than in a single jump. If a constant rate \(R_0\) starts at \(t_s\) and stops at \(t_e\), its input function is \(R_0\) during that interval and zero outside it. The cumulative amount delivered by time \(t\), for an infusion starting at \(t_s\), is:
When \(u(t)=R_0\) from \(t_s\) to \(t_e\), the delivered amount by the end of the infusion is \(R_0(t_e-t_s)\). The infusion term belongs in the equation for the receiving compartment throughout the infusion interval. When the infusion stops, the external input term returns to zero; drug already in the system remains part of the state and continues to move according to the model.
Worked Example: Infusion Into Arterial Blood
Consider a hypothetical three-state system with amounts \(A_a=30\) mg in arterial blood, \(A_t=40\) mg in tissue, and \(A_v=10\) mg in venous blood. For illustration, suppose the current internal transfer rates are 6 mg/h from arterial blood to tissue, 4 mg/h from tissue to venous blood, and 4 mg/h from venous blood to arterial blood. A constant infusion of 12 mg/h enters arterial blood. Calculate the three instantaneous amount derivatives.
Each internal transfer is a loss from its source and a gain to its destination. The external infusion adds to the arterial equation only:
The arterial amount is increasing at 10 mg/h at this instant, despite losing 6 mg/h to tissue, because it gains 4 mg/h from venous blood and 12 mg/h from the infusion. The tissue amount is increasing at 2 mg/h, and the venous amount has zero instantaneous change. The rates are hypothetical values chosen to illustrate the bookkeeping; the dose is still identifiable as the separate 12 mg/h external input.
Input Placement and the Amount Ledger
For a multi-compartment model, it is useful to make an input vector explicit. If the state order is \(\mathbf{A}=(A_a,A_t,A_v)^\mathsf{T}\) and the infusion enters arterial blood, then:
This is not a claim that drug remains in arterial blood. Internal perfusion and exchange terms may subsequently move it into tissue and venous blood. The vector only identifies the location where drug enters the modeled system. In the same way, an infusion into a different explicitly modeled compartment requires a different input vector.
A useful bookkeeping distinction is between administered amount and amount currently in a compartment. The former is the integral of the external input rate, plus any bolus amounts already administered. The latter is a state determined by both dosing and internal movement. They need not be equal: administered drug may be distributed among several compartments, and model processes may also remove drug if elimination is included.
Worked Example: Same Total Dose, Different Input Pattern
Compare two hypothetical regimens that each deliver 60 mg intravenously into the same compartment. Regimen A is an ideal bolus at \(t=0\). Regimen B is a constant infusion of 10 mg/h from \(t=0\) to \(t=6\) h. State how each is represented and verify the total amount delivered by 6 h.
For Regimen A, the receiving compartment’s initial amount is increased by 60 mg at \(t=0\):
For Regimen B, add 10 mg/h to the receiving compartment’s ODE during the six-hour interval. The total delivered is:
The regimens have the same total administered amount but different time courses of input: one has an instantaneous jump and the other supplies drug continuously. Equal total dose does not mean the model has equal compartment amounts at every time. Their internal redistribution proceeds while an infusion is being delivered, whereas a bolus supplies its full amount at once.
Deriving the Dose Term and Checking the Bookkeeping
The rate term follows directly from an amount balance over a short time interval. Suppose an infusion adds \(u(t)\) amount per time to compartment \(j\). Over an interval of length \(\Delta t\), its contribution to the amount is approximately \(u(t)\Delta t\) when the rate is nearly constant on that interval. Dividing by \(\Delta t\) and taking the limit gives the input contribution \(u(t)\) to \(dA_j/dt\):
For a bolus, take the interval across the administration time. The integrated input over that interval is the bolus amount \(D\). As the interval shrinks around \(t_d\), the amount immediately after the dose exceeds the amount immediately before by \(D\), giving the jump condition. Thus the rate and jump forms are consistent: one accumulates input over time, while the other records a single instantaneous accumulation.
This reasoning assumes the input amount is defined at the boundary of the modeled system. If a preparation, delivery device, or additional compartment is modeled explicitly, the input may first enter that component instead; the input location must match the chosen model boundary and state definitions.
Common Mistakes, Diagnostics, and Alternatives
The most common error is to add the dose to more than one compartment. A dose administered into arterial blood is not also a dose into each perfused tissue. Tissue gains drug through the internal fluxes already present in the tissue equation. Adding the administered amount again to the tissue equation would represent a different, usually unintended, administration.
Other practical checks help catch mistakes before interpreting simulated concentrations:
- Check dimensions. Infusion terms must have units of amount per time, and bolus amounts must have units of amount.
- Check the receiving state. Confirm that the nonzero entry of the input vector corresponds to where the dose enters.
- Check timing. A bolus should cause an amount jump at its stated administration time; an infusion should be active only over its delivery interval.
- Check for duplicate input. Do not represent one bolus both as a post-dose initial-condition increase and as an additional input during the same instant.
- Check the post-infusion behavior. Setting the input rate to zero after the infusion ends does not reset compartment amounts to zero.
A bolus initial-condition change is usually simplest when administration is treated as instantaneous at the simulation start. For a later bolus, a jump condition at that time is explicit; alternatively, integrate in separate intervals before and after the dose, using the updated state to start the second interval. A very short infusion can approximate a bolus when the actual administration has a nonzero duration or when that delivery profile matters. It should not be called an instantaneous bolus: its rate and duration are part of the model.
These representations have limitations. An ideal bolus ignores the finite time required to administer a real injection. A constant-rate infusion ignores changes in pump delivery unless those changes are included in \(u(t)\). Neither input representation alone specifies how drug distributes after entering the receiving compartment; that behavior comes from the internal model equations. The model also needs appropriate initial amounts and a clear definition of which compartments are included.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A 25 mg bolus is delivered into a modeled venous blood compartment at time zero. What changes in the initial state?
- An infusion enters compartment 2 at 8 mg/h for 3 h. What is the total amount delivered, and which ODE receives the input term?
- Why is it incorrect to add an arterial infusion rate to every tissue equation?
- A 40 mg bolus is represented as a 40 mg increase in the initial amount and also as an input term at time zero. What is the bookkeeping problem?
- An infusion ends at 5 h. What happens to the infusion term after 5 h, and what happens to drug already in the model?
Solutions. (1) Increase the venous amount by 25 mg; other amounts do not jump from that dose. (2) The delivered amount is \((8\ \mathrm{mg/h})(3\ \mathrm{h})=24\ \mathrm{mg}\), and the input term belongs in compartment 2’s equation. (3) Only the receiving compartment gets the external input; tissue amounts change through internal transfers unless the dose is explicitly administered there. (4) The same dose is counted twice, once in the initial amount and again as an input. (5) Set the infusion rate to zero after 5 h; amounts already present remain states and continue to evolve through the model’s internal processes.