Setting Up the One-Tissue Infusion Balance
A perfused tissue receives drug in arterial blood and returns drug in venous blood. During intravenous infusion, the central blood pool also receives drug directly. Following the control-volume and amount-state approach, the accounting question is precise: given the infusion rate, tissue blood flow, and concentrations at the tissue boundaries, how do the amounts in the tissue and central pool change?
Let \(A_b(t)\) be the drug amount in a well-mixed central blood pool, and let \(A_t(t)\) be the drug amount in one tissue. Let \(Q\) be the blood flow through that tissue, \(C_a(t)\) its arterial inlet concentration, and \(C_v(t)\) its venous outlet concentration. Let \(R_{\mathrm{in}}(t)\) be the intravenous infusion rate. Concentration is measured as amount per volume, so \(Q C\) has units of amount per time.
With these assumptions, outflow from the central pool to the tissue is \(Q C_a\), while tissue venous return to the central pool is \(Q C_v\). The amount balances are therefore:
The terms in each equation are rates, not amounts. A positive \(C_a-C_v\) means the tissue gains drug; a negative difference means the tissue returns more drug than it receives and its amount decreases. The central pool changes according to infusion minus the net amount transferred into the tissue.
Why the Total Amount Balance Is a Useful Check
Adding the two compartment equations cancels the exchange terms. This is the central check for this example: internal movement between blood and tissue can change each compartment’s amount, but cannot create or remove drug from their combined total.
Integrating from time \(0\) to time \(T\) gives the corresponding amount identity:
This identity follows from the model assumptions, regardless of whether the boundary concentrations are constant. It also provides an independent check on calculations: the change in the sum of the two amounts must equal the total amount infused. If it does not, a flow term, sign, unit conversion, or arithmetic step is inconsistent with the stated model.
For constant \(Q\), the tissue amount change can be calculated by integrating the boundary-concentration difference. The central amount change then follows either from its own balance or from the total amount identity.
Worked Example: Constant Boundary Concentrations
Suppose a hypothetical tissue is perfused at \(Q=0.75\ \mathrm{L/h}\). Throughout a \(4\ \mathrm{h}\) interval, its arterial concentration is \(C_a=4.0\ \mathrm{mg/L}\) and its venous concentration is \(C_v=2.8\ \mathrm{mg/L}\). The constant intravenous infusion rate is \(R_{\mathrm{in}}=0.90\ \mathrm{mg/h}\). Initially, \(A_b(0)=12.0\ \mathrm{mg}\) and \(A_t(0)=20.0\ \mathrm{mg}\). Find both amounts at \(4\ \mathrm{h}\).
Check the net tissue transfer rate. The concentration difference is \(4.0-2.8=1.2\ \mathrm{mg/L}\), so:
The rate is positive, so the tissue gains drug. Over \(4\ \mathrm{h}\), its amount increases by \((0.90\ \mathrm{mg/h})(4\ \mathrm{h})=3.6\ \mathrm{mg}\). Thus \(A_t(4)=20.0+3.6=23.6\ \mathrm{mg}\).
The central pool receives \(0.90\ \mathrm{mg/h}\) by infusion and transfers a net \(0.90\ \mathrm{mg/h}\) to the tissue. Its rate of change is \(0.90-0.90=0\ \mathrm{mg/h}\), so \(A_b(4)=12.0\ \mathrm{mg}\).
Check the total. The initial total is \(12.0+20.0=32.0\ \mathrm{mg}\). The infused amount is \((0.90\ \mathrm{mg/h})(4\ \mathrm{h})=3.6\ \mathrm{mg}\). The final total, \(12.0+23.6=35.6\ \mathrm{mg}\), equals \(32.0+3.6\ \mathrm{mg}\). The two compartment results are consistent with conservation.
When Boundary Concentrations Vary with Time
A constant infusion rate does not imply constant arterial or venous concentrations. When concentrations vary, the net tissue transfer over an interval depends on the area between the two concentration profiles, multiplied by flow. For time-varying quantities, rates must be integrated over time; using a single concentration difference is valid only if it represents the appropriate time average.
Worked Example: Varying Concentrations with a Consistent Infusion
For a hypothetical system, let \(Q=0.80\ \mathrm{L/h}\), \(V_b=2.0\ \mathrm{L}\), and, for \(0\leq t\leq2\ \mathrm{h}\), let \(C_a(t)=3.0+0.50t\ \mathrm{mg/L}\) and \(C_v(t)=1.0+0.25t\ \mathrm{mg/L}\), where \(t\) is in hours. The central pool is well mixed, so \(A_b(t)=V_bC_a(t)\). The infusion is \(R_{\mathrm{in}}(t)=2.6+0.20t\ \mathrm{mg/h}\). Initially \(A_t(0)=10.0\ \mathrm{mg}\). Calculate the amount changes and check consistency.
The arterial concentration implies \(A_b(t)=2.0(3.0+0.50t)=6.0+1.0t\ \mathrm{mg}\). Thus the central amount increases at \(1.0\ \mathrm{mg/h}\), from \(6.0\ \mathrm{mg}\) to \(8.0\ \mathrm{mg}\).
The concentration difference is \(C_a-C_v=2.0+0.25t\ \mathrm{mg/L}\). Therefore the tissue transfer rate is \(0.80(2.0+0.25t)=1.6+0.20t\ \mathrm{mg/h}\). Its change over two hours is:
The final tissue amount is \(10.0+3.6=13.6\ \mathrm{mg}\). To check the central balance instantaneously, subtract the net tissue transfer from the infusion:
This equals the central amount rate implied by \(A_b(t)\). The infused amount is \(\int_0^2(2.6+0.20t)\,dt=[2.6t+0.10t^2]_0^2=5.6\ \mathrm{mg}\). The combined amount rises from \(6.0+10.0=16.0\ \mathrm{mg}\) to \(8.0+13.6=21.6\ \mathrm{mg}\), also an increase of \(5.6\ \mathrm{mg}\). Both compartment equations and the total balance agree.
Reconstructing a Missing Amount from the Balance
The same equations can solve for an unknown infusion or amount change when the remaining quantities are known. The total identity is particularly convenient for this inverse check: in a closed two-state accounting system without elimination, the total amount infused must equal the change in the sum of the two states.
Worked Example: Inferring the Infusion from Amount Changes
Suppose a hypothetical system has \(Q=1.0\ \mathrm{L/h}\), with constant \(C_a=5.0\ \mathrm{mg/L}\) and \(C_v=3.0\ \mathrm{mg/L}\) over \(2.0\ \mathrm{h}\). The tissue amount increases from \(25.0\ \mathrm{mg}\) to \(29.0\ \mathrm{mg}\), while the central amount decreases from \(15.0\ \mathrm{mg}\) to \(14.0\ \mathrm{mg}\). Find the infusion rate that is consistent with these values.
The tissue transfer rate calculated from the boundaries is \((1.0\ \mathrm{L/h})(5.0-3.0\ \mathrm{mg/L})=2.0\ \mathrm{mg/h}\). Over two hours, this predicts a \(4.0\ \mathrm{mg}\) tissue gain, matching the stated change from \(25.0\) to \(29.0\ \mathrm{mg}\).
The central amount change is \(-1.0\ \mathrm{mg}\). From the central balance, the total infused amount must equal the central change plus the net transfer to the tissue: \(-1.0+4.0=3.0\ \mathrm{mg}\). If the infusion is constant, its rate is \(3.0\ \mathrm{mg}/2.0\ \mathrm{h}=1.5\ \mathrm{mg/h}\).
The total amount rises from \(15.0+25.0=40.0\ \mathrm{mg}\) to \(14.0+29.0=43.0\ \mathrm{mg}\). The \(3.0\ \mathrm{mg}\) increase agrees with the inferred total infusion. If instead the stated constant infusion rate were \(2.0\ \mathrm{mg/h}\), the infusion would add \(4.0\ \mathrm{mg}\), contradicting the stated compartment changes; those values could not all satisfy this model.
Model Checks, Alternatives, and Limitations
A practical check begins with units and directions. \(Q C_a\), \(Q C_v\), and \(R_{\mathrm{in}}\) must all be in amount per time. The tissue rate is inflow minus outflow, not the reverse; the central pool uses the opposite signs for tissue exchange. Once the individual rates are calculated, verify that the exchange terms cancel when the equations are added. Finally, compare the sum of the amount changes with the integrated infusion.
A common mistake is to add \(Q C_a\) as an input to both the tissue and central pool, or to subtract \(Q C_v\) from both. That double-counts or misdirects the transfer. The arterial outflow leaves the central pool as it enters the tissue; the venous return leaves the tissue as it enters the central pool. These are internal transfers, so they cancel in the combined balance.
An alternative is to track only the total amount \(A_b+A_t\). That is sufficient to check total conservation, but it cannot determine how much drug is in each state. The two-state balances retain that distribution of amounts and show how the specified concentration difference drives transfer. Conversely, the balances here do not supply a tissue constitutive relationship that predicts \(C_v\) from \(A_t\), nor do they establish a particular tissue distribution mechanism. Boundary concentrations must be supplied or modeled separately for that purpose.
The assumptions also limit the interpretation. If there is elimination, an additional loss term is required; if other organs or inputs are present, their flows and amounts must be included in the accounting. A changing flow requires using its time-dependent value in the transfer rate. In addition, measured or imposed boundary concentrations and a central amount treated as \(V_bC_a\) must be mutually compatible with the central balance, as checked in the varying-concentration example. An amount calculation can identify a mismatch, but cannot by itself determine which input or assumption is wrong.
Check Your Understanding
Use the two-state model above. Treat concentrations and flows as specified over the interval, and give units with your answers.
- A tissue has \(Q=0.50\ \mathrm{L/h}\), \(C_a=6.0\ \mathrm{mg/L}\), and \(C_v=4.0\ \mathrm{mg/L}\) for \(3.0\ \mathrm{h}\). What is its amount change?
- If the infusion rate is \(0.80\ \mathrm{mg/h}\) in Question 1, what is the central amount change over the interval?
- Suppose \(Q=1.2\ \mathrm{L/h}\), \(C_a=2.0\ \mathrm{mg/L}\), and \(C_v=2.5\ \mathrm{mg/L}\) for \(2.0\ \mathrm{h}\). Does the tissue gain or lose drug, and by how much?
- With no elimination, the central and tissue amounts together increase by \(7.0\ \mathrm{mg}\) over an interval. What total amount must have been infused?
- Why is a constant infusion rate alone insufficient to calculate the tissue amount change?
Solutions. (1) The transfer rate is \(0.50(6.0-4.0)=1.0\ \mathrm{mg/h}\), so the tissue gains \(1.0(3.0)=3.0\ \mathrm{mg}\). (2) The infused amount is \(0.80(3.0)=2.4\ \mathrm{mg}\). The central change is \(2.4-3.0=-0.6\ \mathrm{mg}\), a decrease of \(0.6\ \mathrm{mg}\). (3) The rate is \(1.2(2.0-2.5)=-0.60\ \mathrm{mg/h}\); over \(2.0\ \mathrm{h}\), the tissue loses \(1.2\ \mathrm{mg}\). (4) The total infused amount is \(7.0\ \mathrm{mg}\), by the combined amount identity. (5) The tissue transfer also depends on \(Q\) and the arterial–venous concentration difference over time. Without that boundary information, its amount change is not determined by the infusion rate alone.