Deriving the Perfused-Tissue Amount Equation

From Flow Bookkeeping to an Amount Equation

A perfused tissue exchanges drug with blood: drug enters with arterial blood and leaves with venous blood. The difference between these two amount rates is not, by itself, necessarily the change in tissue drug amount. Drug may also be formed or removed locally. A mass balance puts these contributions together, with each term assigned a clear direction and unit.

The essential modeling choice here is to leave the venous concentration explicit. The equation can be derived without asserting that this concentration is equal to the arterial concentration, proportional to the tissue amount, or determined by a particular tissue-kinetic mechanism. Those relationships require additional model assumptions. The balance itself says how the boundary fluxes and local processes contribute to accumulation.

Definition: For tissue \(i\), let \(A_i(t)\) be the amount of drug in the tissue control volume at time \(t\). Let \(Q_i(t)\) be blood flow into the tissue, \(C_a(t)\) the drug concentration in incoming arterial blood, and \(C_{v,i}(t)\) the concentration in blood leaving the tissue. Let \(R_{\mathrm{form},i}(t)\) and \(R_{\mathrm{loss},i}(t)\) be the local rates of drug formation and loss, respectively. With one inlet and one outlet carrying the same flow, the perfused-tissue amount balance is

\frac{dA_i}{dt} = Q_i C_a - Q_i C_{v,i} + R_{\mathrm{form},i} - R_{\mathrm{loss},i}.

Here \(A_i\) is an amount, concentrations have units of amount per volume, \(Q_i\) has units of volume per time, and each \(R\) has units of amount per time.

The first product, \(Q_i C_a\), is the amount of drug entering per unit time. The second, \(Q_i C_{v,i}\), is the amount leaving per unit time. Formation adds drug to the control volume; loss removes it. Thus positive accumulation occurs when total entry plus formation exceeds total exit plus loss. For example, flow in litres per hour multiplied by concentration in milligrams per litre gives an amount rate in milligrams per hour.

Deriving the Balance

Consider a short time interval from \(t\) to \(t+\Delta t\). The amount change over that interval equals the amount that crosses into the control volume, minus the amount that crosses out, plus local formation, minus local loss. Dividing by the interval and taking its limit gives the instantaneous rate equation.

Identify the amount state.
The amount at the beginning and end of the interval is \(A_i(t)\) and \(A_i(t+\Delta t)\). Their difference is the net accumulation over the interval.

Count boundary transport.
Over a sufficiently short interval, inflow carries approximately \(Q_i(t)C_a(t)\Delta t\) of drug into the control volume, and outflow carries approximately \(Q_i(t)C_{v,i}(t)\Delta t\) out.

Add local processes.
Formation contributes approximately \(R_{\mathrm{form},i}(t)\Delta t\); loss contributes approximately \(-R_{\mathrm{loss},i}(t)\Delta t\).

Take the rate limit.
Sum the contributions, divide by \(\Delta t\), and let \(\Delta t\) approach zero to obtain the amount ODE.

In equations, the interval balance and its limiting form are

\begin{aligned} A_i(t+\Delta t)-A_i(t) &\approx \left[ Q_i(t)C_a(t)-Q_i(t)C_{v,i}(t) +R_{\mathrm{form},i}(t)-R_{\mathrm{loss},i}(t) \right]\Delta t,\\ \frac{dA_i}{dt} &= \lim_{\Delta t\to 0} \frac{A_i(t+\Delta t)-A_i(t)}{\Delta t}\\ &= Q_i C_a-Q_i C_{v,i} +R_{\mathrm{form},i}-R_{\mathrm{loss},i}. \end{aligned}

The approximation in the first line becomes exact in the limit when the rates are well-defined and sufficiently regular over the interval. If rates change substantially over a finite interval, the exact interval balance uses their integrals over time; replacing those integrals by a rate at one instant is not generally exact.

Assumptions for this form: The selected control volume has one blood inlet and one blood outlet; the flow entering and leaving is the same \(Q_i\), so there is no net accumulation of fluid in the control volume. Concentrations at both boundaries are defined as representative of the corresponding flowing blood. The listed local formation and loss terms account for local drug processes, and no other drug-transfer route crosses the control-volume boundary. If any of these conditions fails, the balance must be modified to include the relevant flow or transfer term.

Worked Calculations

Worked Example: Net Exchange With Local Formation and Loss

Suppose a hypothetical tissue has blood flow \(Q_i=1.80\ \mathrm{L/h}\). Its incoming arterial concentration is \(C_a=4.00\ \mathrm{mg/L}\), and its outgoing venous concentration is \(C_{v,i}=2.50\ \mathrm{mg/L}\). Local formation is \(0.300\ \mathrm{mg/h}\), and local loss is \(0.500\ \mathrm{mg/h}\). Find the instantaneous rate of change in tissue amount.

The one-inlet, one-outlet assumption gives the same \(1.80\ \mathrm{L/h}\) flow in both terms. Substituting in the amount balance,

\begin{aligned} \frac{dA_i}{dt} &=Q_iC_a-Q_iC_{v,i}+R_{\mathrm{form},i}-R_{\mathrm{loss},i}\\ &=(1.80)(4.00)-(1.80)(2.50)+0.300-0.500\\ &=7.20-4.50+0.300-0.500\\ &=2.50\ \mathrm{mg/h}. \end{aligned}

The tissue amount is increasing at \(2.50\ \mathrm{mg/h}\) at that instant. Blood contributes a net \(2.70\ \mathrm{mg/h}\), because the incoming amount rate exceeds the outgoing amount rate. Local processes together remove a net \(0.200\ \mathrm{mg/h}\), leaving positive accumulation. This calculation uses the stated venous concentration directly; it does not infer it from the tissue amount.

Worked Example: A First-Order Local Loss

Consider a hypothetical tissue with \(Q_i=0.750\ \mathrm{L/h}\), \(C_a=1.20\ \mathrm{mg/L}\), and \(C_{v,i}=0.800\ \mathrm{mg/L}\). There is no local formation. Suppose the specified local loss law is \(R_{\mathrm{loss},i}=kA_i\), with \(k=0.100\ \mathrm{h^{-1}}\), and the current amount is \(A_i=6.00\ \mathrm{mg}\). Calculate the amount rate.

First calculate the local loss rate using the given law:

R_{\mathrm{loss},i} =(0.100\ \mathrm{h^{-1}})(6.00\ \mathrm{mg}) =0.600\ \mathrm{mg/h}.

The two blood rates are \(Q_iC_a=(0.750)(1.20)=0.900\ \mathrm{mg/h}\) and \(Q_iC_{v,i}=(0.750)(0.800)=0.600\ \mathrm{mg/h}\). Therefore,

\frac{dA_i}{dt} =0.900-0.600+0-0.600 =-0.300\ \mathrm{mg/h}.

Although blood inflow exceeds blood outflow by \(0.300\ \mathrm{mg/h}\), local loss is \(0.600\ \mathrm{mg/h}\). The resulting amount is decreasing at \(0.300\ \mathrm{mg/h}\) at the stated instant. The first-order loss law is an additional assumption for this example, not a necessary part of the general mass balance.

Worked Example: Amount Change Over a Finite Interval

In a hypothetical calculation, let \(Q_i=2.00\ \mathrm{L/h}\), \(C_a=3.00\ \mathrm{mg/L}\), \(C_{v,i}=2.40\ \mathrm{mg/L}\), formation be \(0.100\ \mathrm{mg/h}\), and loss be \(0.400\ \mathrm{mg/h}\), all constant for \(0.500\ \mathrm{h}\). If the initial amount is \(10.0\ \mathrm{mg}\), find the amount at the end of the interval.

Because all rates are constant on this interval, the instantaneous balance is also the constant interval-average rate:

\begin{aligned} \frac{dA_i}{dt} &=(2.00)(3.00)-(2.00)(2.40)+0.100-0.400\\ &=6.00-4.80+0.100-0.400\\ &=0.900\ \mathrm{mg/h}. \end{aligned}

The amount change is rate multiplied by elapsed time:

\Delta A_i=(0.900\ \mathrm{mg/h})(0.500\ \mathrm{h})=0.450\ \mathrm{mg}, \qquad A_i(0.500\ \mathrm{h})=10.0+0.450=10.45\ \mathrm{mg}.

The result depends on the constancy of the rates over the interval. If they varied, the amount change would instead be obtained by integrating the net rate through time.

Interpretation, Diagnostics, and Alternatives

The equation is a conservation statement, not a complete description of tissue kinetics. It specifies how specified fluxes change the amount state. It does not, by itself, provide a rule for calculating \(C_{v,i}\) from \(A_i\), determine whether tissue and blood are at equilibrium, or prescribe the local formation and loss laws. Keeping these quantities separate makes the boundary assumptions visible and allows additional mechanisms to be introduced only when justified.

A useful dimensional check is that both flow-concentration products and both local rates must have units of amount per time. A useful sign check is to imagine one process at a time: with inflow alone the amount rises; with outflow alone it falls; formation adds; loss subtracts. For an interval with known constant rates, the calculated amount change should equal the net rate times the interval duration.

Unequal flows: If the control volume gains or loses fluid, the inlet and outlet flows need not match. Use separate flow rates in the two flux terms rather than forcing both to \(Q_i\).
Additional pathways: A lymphatic route, direct exchange with another compartment, or another boundary flux must be included with its own direction and rate. Leaving it out breaks the balance for the chosen control volume.
Double counting: If drug leaving in venous blood is already represented by \(Q_iC_{v,i}\), do not subtract the same vascular exit again as a local loss. Local loss denotes a distinct process occurring within the selected control volume.
Unjustified boundary substitution: Replacing \(C_{v,i}\) with \(C_a\) would set net blood exchange to zero. That is a substantive assumption, not an algebraic simplification implied by the mass balance.

The amount balance is often the clearest starting point because every term represents an amount rate and the signs follow directly from entry, exit, formation, and loss. An alternative is to write a balance for a concentration state, but that requires defining the volume associated with that concentration and accounting for any changes in it. Another alternative is to specify a direct net exchange rate instead of separate blood fluxes; that can be compact, but it hides the distinct inflow and outflow contributions. Neither alternative removes the need to define the boundary and local processes consistently.

Key takeaway: For a one-inlet, one-outlet perfused tissue with equal blood flow, the amount rate is inflow \(Q_iC_a\), minus outflow \(Q_iC_{v,i}\), plus local formation, minus local loss. The venous concentration remains an explicit boundary quantity until a separate model relation specifies it.

Check Your Understanding

Work each problem using the amount balance, then compare with the solutions below.

A tissue has \(Q_i=1.00\ \mathrm{L/h}\), \(C_a=2.00\ \mathrm{mg/L}\), and \(C_{v,i}=1.50\ \mathrm{mg/L}\), with no local formation or loss. What is \(dA_i/dt\), and what does its sign mean?
For the same tissue, a student writes \(Q_i(C_{v,i}-C_a)\) as blood contribution. Identify the sign error and give the correct blood contribution.
Suppose a second outflow pathway carries \(0.100\ \mathrm{mg/h}\) out of the control volume, in addition to venous blood flow. How should it enter the balance?
Why does the amount equation alone not determine the venous concentration?

Solutions. (1) The rate is \((1.00)(2.00)-(1.00)(1.50)=0.500\ \mathrm{mg/h}\), so amount is increasing at that instant. (2) Inflow must be positive and outflow negative, so the correct blood contribution is \(Q_i(C_a-C_{v,i})=(1.00)(2.00-1.50)=0.500\ \mathrm{mg/h}\). (3) Subtract the additional outward rate: include \(-0.100\ \mathrm{mg/h}\). (4) The balance relates the amount rate to the boundary concentrations and local rates, but supplies no independent relation that calculates \(C_{v,i}\) from \(A_i\); that relation must come from an additional model assumption or equation.

Deriving the Perfused-Tissue Amount Equation

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