Assembling a Whole-Body Perfused-Tissue ODE System

From Individual Tissue Balances to a Whole-Body System

A perfused-tissue equation describes exchange at one organ, but a whole-body model must also track where the blood entering and leaving that organ goes. The key bookkeeping task is to connect every tissue to shared blood pools: arterial blood supplies the organs, and their venous outflows return to venous blood. When the equations are assembled consistently, each internal transfer leaves one modeled control volume and enters another.

Here the central circulation is represented by two well-mixed blood pools: an arterial pool and a venous pool. Each organ has its own tissue amount state. This structure makes the route of each flow explicit and distinguishes the concentration delivered to an organ from the concentration in blood leaving it.

Definition: For \(m\) perfused tissues, let \(A_a\) and \(A_v\) be the drug amounts in the arterial and venous blood pools, and let \(A_i\) be the amount associated with tissue \(i\), for \(i=1,\ldots,m\). Let \(V_a\) and \(V_v\) be the blood-pool volumes, with concentrations \(C_a=A_a/V_a\) and \(C_v=A_v/V_v\). Let \(Q_i\) be blood flow from the arterial pool through tissue \(i\), and let \(C_{v,i}\) be that tissue’s blood-outlet concentration. Total flow from the venous pool back to the arterial pool is \(Q_c\).

The concentration \(C_{v,i}\) is supplied by the model’s tissue-outlet relation. It may depend on the tissue state or on other states included in the tissue description. This tutorial does not prescribe a particular tissue-to-outlet relation; it shows how to couple a specified relation to the circulation balances. For the equations below to form a closed ODE system, each \(C_{v,i}\) must be evaluable from the model’s current states and parameters.

Assumptions and the Whole-Body Equations

The assembly below uses a single recirculating loop. Each tissue receives flow \(Q_i\) from the arterial pool and returns the same volume flow \(Q_i\) to the venous pool. All modeled cardiac output passes through the listed tissues, so \(Q_c=\sum_{i=1}^{m}Q_i\). There is no unmodeled shunt, dose input, elimination, or accumulation in a separate transit space in these balances. The arterial and venous pools are well mixed, and their volumes and the flows are treated as constant over the interval being modeled.

For any control volume, the amount rate of change is the rate entering minus the rate leaving. A tissue receives drug at rate \(Q_iC_a\) and returns drug at rate \(Q_iC_{v,i}\). The arterial pool receives venous blood at rate \(Q_cC_v\), while its total drug outflow to the tissues is \(\sum_i Q_iC_a\). The venous pool receives the tissue outflows and sends drug back to the arterial pool.

\frac{dA_i}{dt}=Q_iC_a-Q_iC_{v,i} =Q_i(C_a-C_{v,i}), \qquad i=1,\ldots,m

\frac{dA_a}{dt}=Q_cC_v-\sum_{i=1}^{m}Q_iC_a, \qquad \frac{dA_v}{dt}=\sum_{i=1}^{m}Q_iC_{v,i}-Q_cC_v, \qquad Q_c=\sum_{i=1}^{m}Q_i

These are amount balances, not concentration balances. If the blood-pool volumes are constant, concentrations can be recovered from amounts by \(C_a=A_a/V_a\) and \(C_v=A_v/V_v\). For example, dividing the arterial amount equation by \(V_a\) gives \(dC_a/dt=(1/V_a)dA_a/dt\). The factor \(1/V_a\) is needed because the equation’s right-hand side has units of amount per time, whereas a concentration rate has units of amount per volume per time.

Why the Internal Transfers Cancel

The tissue equations and blood equations must describe the same physical flows from opposite sides of their control-volume boundaries. For tissue \(i\), the arterial-to-tissue transfer \(Q_iC_a\) is an outflow from arterial blood and an inflow to the tissue. The tissue-to-venous transfer \(Q_iC_{v,i}\) is an outflow from the tissue and an inflow to venous blood. The return transfer \(Q_cC_v\) leaves the venous pool and enters the arterial pool.

Summing the right-hand sides shows the bookkeeping directly. Each transfer occurs once with a negative sign and once with a positive sign:

\begin{aligned} \frac{d}{dt}\left(A_a+A_v+\sum_{i=1}^{m}A_i\right) &=\left(Q_cC_v-\sum_{i=1}^{m}Q_iC_a\right) +\left(\sum_{i=1}^{m}Q_iC_{v,i}-Q_cC_v\right)\\ &\quad+\sum_{i=1}^{m}\left(Q_iC_a-Q_iC_{v,i}\right)\\ &=0. \end{aligned}

This identity is an assembly check under the stated closed-loop assumptions: internal circulation moves drug between modeled states but does not create or remove it. It does not say that any individual amount is constant. In general, drug can redistribute among blood and tissues while the sum of these amounts has zero derivative. If an external input, elimination process, or unmodeled route is added, the global balance must include its contribution.

Write one tissue balance per organ.
Use the arterial concentration for that organ’s inflow and its own outlet concentration for its outflow. Both use the same organ flow \(Q_i\).

Assemble the arterial balance.
Add return from the venous pool, \(Q_cC_v\), and subtract the sum of arterial drug flows to the tissues, \(\sum_i Q_iC_a\).

Assemble the venous balance.
Add each organ’s outlet drug flow, \(\sum_i Q_iC_{v,i}\), and subtract the return flow to the arterial pool, \(Q_cC_v\).

Audit the coupling.
Check \(Q_c=\sum_iQ_i\), confirm that every transfer appears with opposite signs in the two connected balances, and verify the units of every term.

Worked Examples

Worked Example: Assemble Two Tissue Equations and Both Blood Balances

Consider a hypothetical model with two tissues. Let \(Q_1=1.2\) L/min, \(Q_2=0.8\) L/min, \(C_a=4.0\) mg/L, \(C_{v,1}=2.5\) mg/L, \(C_{v,2}=3.0\) mg/L, and \(C_v=3.2\) mg/L. All values are hypothetical. The assumed total flow is \(Q_c=Q_1+Q_2=2.0\) L/min.

For tissue 1, the net rate is \(dA_1/dt=Q_1(C_a-C_{v,1})=(1.2\ \text{L/min})(4.0-2.5\ \text{mg/L})=1.8\) mg/min. The tissue amount is increasing at this instant because the arterial concentration exceeds its outlet concentration.

For tissue 2, \(dA_2/dt=Q_2(C_a-C_{v,2})=(0.8\ \text{L/min})(4.0-3.0\ \text{mg/L})=0.8\) mg/min.

The arterial amount rate is \(dA_a/dt=Q_cC_v-(Q_1+Q_2)C_a=(2.0)(3.2)-(1.2+0.8)(4.0)=6.4-8.0=-1.6\) mg/min. The arterial pool is losing drug at this instant.

The venous amount rate is \(dA_v/dt=Q_1C_{v,1}+Q_2C_{v,2}-Q_cC_v=(1.2)(2.5)+(0.8)(3.0)-(2.0)(3.2)=3.0+2.4-6.4=-1.0\) mg/min. The sum of the four rates is \(1.8+0.8-1.6-1.0=0\) mg/min, as required by the internal-flow bookkeeping.

Worked Example: Convert Blood Amounts to Concentrations Before Evaluating Rates

Suppose a hypothetical model has arterial amount \(A_a=12\) mg in \(V_a=3.0\) L and venous amount \(A_v=9.6\) mg in \(V_v=3.0\) L. Two tissue flows are \(Q_1=0.75\) L/min and \(Q_2=1.25\) L/min. Their outlet concentrations are \(C_{v,1}=2.8\) mg/L and \(C_{v,2}=3.4\) mg/L. Use the same closed-loop assumptions.

First calculate the pool concentrations: \(C_a=A_a/V_a=12/3.0=4.0\) mg/L and \(C_v=A_v/V_v=9.6/3.0=3.2\) mg/L. The total flow is \(Q_c=0.75+1.25=2.00\) L/min.

The tissue rates are \(dA_1/dt=(0.75)(4.0-2.8)=0.90\) mg/min and \(dA_2/dt=(1.25)(4.0-3.4)=0.75\) mg/min.

The arterial rate is \(dA_a/dt=(2.00)(3.2)-(2.00)(4.0)=6.40-8.00=-1.60\) mg/min. The venous rate is \(dA_v/dt=(0.75)(2.8)+(1.25)(3.4)-(2.00)(3.2)=2.10+4.25-6.40=-0.05\) mg/min. Together, \(0.90+0.75-1.60-0.05=0.00\) mg/min. Converting amounts to concentrations first avoids using a quantity with units of mg where the flow equations require mg/L.

Worked Example: Detect a Double-Counted Arterial Outflow

Return to the hypothetical values in the first example. The correct arterial outflow to both tissues is \((Q_1+Q_2)C_a=(2.0)(4.0)=8.0\) mg/min. Suppose an analyst subtracts this total and then subtracts the organ outflows a second time, writing \(dA_a/dt=Q_cC_v-2\sum_iQ_iC_a\).

That incorrect expression gives \(6.4-2(8.0)=-9.6\) mg/min instead of the correct \(-1.6\) mg/min. Keeping the correct tissue and venous equations from the first example, the sum of all rates would then be \(1.8+0.8-9.6-1.0=-8.0\) mg/min. The nonzero residual exposes the duplicated transfer: an amount flow has been removed from arterial blood twice but has only been assigned once to the tissues.

The correction is not to add another compensating term elsewhere. Write the arterial balance from its control-volume boundary once: inflow \(Q_cC_v\), outflow \(\sum_iQ_iC_a\). Then check that the tissue inflow terms are the matching positive terms.

Diagnostics, Alternatives, and Limits

Several quick checks help identify a faulty system before numerical simulation. First, each product \(QC\) must have units of amount per time: for example, \((\text{L/min})(\text{mg/L})=\text{mg/min}\). Second, a tissue’s net rate must be zero when its inlet and outlet concentrations are equal. Third, the arterial equation must contain the total arterial drug outflow to all modeled tissues, while the venous equation must contain their outlet drug flows. Finally, summing the model’s internal transfer terms should leave no residual.

Omitted organ: If tissue \(j\) has a tissue balance but its flow is missing from the blood balances, the circulation and tissue equations no longer describe the same set of transfers.
Flow mismatch: If \(Q_c\ne\sum_iQ_i\), the assumed circulation is incomplete or the flow values are inconsistent. Do not silently force the equations to balance; identify whether a shunt or another route must be represented.
Wrong outlet concentration: Using \(C_a\) instead of \(C_{v,i}\) for tissue return removes the tissue’s modeled concentration difference and changes the tissue and venous rates.
Concentration–amount confusion: A rate equation for an amount state needs amount-per-time terms. If writing a concentration rate, divide the amount rate by the corresponding constant volume.

An alternative is to combine arterial and venous blood into one central blood pool. That can be appropriate when the prediction target does not require distinguishing the two concentrations, but it removes the separate arterial supply and venous return states shown here. Likewise, pooling tissues replaces individual organ flows and outlet relations with aggregate quantities. These simpler structures can be useful, but they cannot represent organ-specific states unless the omitted distinctions are restored. The two-pool, organ-resolved system is also limited: it assumes well-mixed blood pools, specified tissue outlet relations, and the stated circulation topology. More detailed physiology requires additional states or routes, not extra copies of the same flow term.

Key takeaway: Assemble the whole-body system by following each blood flow across control-volume boundaries. For every tissue \(i\), use \(Q_iC_a\) as its inflow and \(Q_iC_{v,i}\) as its outflow; place the matching terms in the arterial and venous balances exactly once.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Two tissues have flows \(Q_1=0.6\) L/min and \(Q_2=1.4\) L/min. Under the stated closed-loop assumption, what is \(Q_c\)?
If \(Q_i=0.5\) L/min, \(C_a=3.0\) mg/L, and \(C_{v,i}=3.0\) mg/L, what is \(dA_i/dt\), and what does its sign indicate?
Why does the arterial balance subtract \(\sum_iQ_iC_a\) rather than subtracting only the flow through one tissue?
For constant arterial volume \(V_a\), how is \(dC_a/dt\) obtained from \(dA_a/dt\)?

Solutions: (1) \(Q_c=Q_1+Q_2=0.6+1.4=2.0\) L/min. (2) \(dA_i/dt=0.5(3.0-3.0)=0\) mg/min; equal inlet and outlet concentrations give no net tissue amount change at that instant. (3) The arterial pool supplies every modeled tissue, so its total drug outflow is the sum of their arterial drug flows, \(\sum_iQ_iC_a\). (4) Divide the amount rate by the constant arterial volume: \(dC_a/dt=(1/V_a)dA_a/dt\), with units of concentration per time.