Three-Compartment Amount-Balance Equations

From the Model Structure to Amount Changes

A three-compartment mammillary model has one central compartment connected to two peripheral compartments. Drug enters the system through the central compartment, can exchange between the central and either peripheral compartment, and is eliminated from the central compartment. The amount in any compartment changes according to a simple accounting rule: add the rates of all flows into it and subtract the rates of all flows out of it.

The equations describe how amounts change over time; they are not themselves concentration-time curves. The central concentration is related to central amount by \(C_c(t)=A_c(t)/V_c\), where \(V_c\) is the central volume. This tutorial focuses on the amount balances that determine the amounts.

With this notation, the three coupled amount-balance equations are

All rate constants are nonnegative. The equations assume that each exchange or elimination flow is first order in the amount in its source compartment. In particular, the flow from central to peripheral 1 is \(k_{12}A_c\), while the reverse flow is \(k_{21}A_1\). The model has no direct connection between the two peripheral compartments, and it includes no elimination from either peripheral compartment.

Deriving Each Balance

The central compartment receives external input at rate \(R(t)\), plus return flows from the two peripheral compartments. It loses drug through elimination and through both outward exchange pathways. Those flows are \(k_{10}A_c\), \(k_{12}A_c\), and \(k_{13}A_c\). Adding the incoming rates and subtracting the outgoing rates gives

Collecting the central-amount terms gives the first equation. Peripheral 1 receives \(k_{12}A_c\) from the central compartment and returns \(k_{21}A_1\) to it. Peripheral 2 receives \(k_{13}A_c\) and returns \(k_{31}A_2\). Since neither peripheral compartment has another incoming or outgoing pathway in this model, their balances contain just these two terms each.

Worked Equations in Context

Input and Initial Conditions

The input function and the initial conditions describe different features of the dosing situation. A constant-rate infusion over a time interval is represented by a constant \(R(t)\) during that interval, and \(R(t)=0\) outside it. An intravenous bolus is usually represented by an instantaneous increase in central amount: if the dose is \(D\), set \(A_c(0^+)=A_c(0^-)+D\). For a bolus into an initially drug-free system, this means \(A_c(0^+)=D\), \(A_1(0^+)=0\), and \(A_2(0^+)=0\). The ordinary differential equations govern the evolution after the jump; a bolus is not a constant infusion.

Total-Amount Balance as a Check

Let \(A_{\mathrm{tot}}=A_c+A_1+A_2\) be the total amount in the three modeled compartments. Adding the three differential equations makes each exchange term cancel: an exchange flow removes exactly the amount per unit time that it adds to the other compartment. The only remaining contributions are external input and elimination.

This identity is a useful derivation check. Exchange redistributes drug within the modeled system but cannot change the total amount. Only input adds drug to the total, and only the specified elimination removes it.

Assumptions, Diagnostics, and Alternatives

The equations apply when the compartment amounts are well-defined, exchange and elimination rates are proportional to source-compartment amounts, and the rate constants are fixed over the time interval being modeled. The external input is specified separately. For a physically meaningful amount model, initial amounts and input rates are nonnegative, and the rate constants are nonnegative. Under these conditions, the equations do not drive an amount below zero: when a compartment amount is zero, its outflow terms are zero, while its possible inflows are nonnegative.

Several checks can reveal an incorrectly written equation:

• Check source amounts. Every first-order flow must multiply the amount in the compartment it leaves. For example, central-to-peripheral-2 flow is \(k_{13}A_c\), not \(k_{13}A_2\).
• Check signs. An outgoing flow is subtracted from its source and added to its destination. The same exchange flow must therefore appear with opposite signs in the two relevant equations.
• Check units. A rate constant in h\(^{-1}\) multiplied by an amount in mg gives mg/h. It can be added to other terms in an amount-rate equation, but not directly to an amount.
• Check total balance. Adding the equations must leave \(R(t)-k_{10}A_c\). A remaining exchange term usually indicates a missing term, a wrong sign, or a wrong pathway.

The amount equations are often the clearest way to express the model’s flows. Concentration equations can instead be written by dividing each amount by its compartment volume, but the exchange terms then involve the relationships between amounts and concentrations as well as the volumes. The amount formulation makes conservation of drug during exchange especially transparent. It is also important to recognize what this model does not represent: direct exchange between the peripheral compartments, peripheral elimination, and nonlinear or time-varying rates are not included. If the intended system has one of those features, the corresponding balance equations must be changed rather than silently using these equations unchanged.