Pulmonary and Systemic Flow Through the Circulation

Two Circuits, One Flow in the Intact Steady-State Circulation

The blood travels through the pulmonary and systemic circuits in sequence: blood returning from the body enters the right heart, passes through the lungs, returns to the left heart, and is then pumped to the body. This arrangement has an important consequence for PBPK models. In an intact circulation without a shunt, the total flow through the lungs and the total flow through the systemic organs are equal at steady state. This does not mean that each organ receives the same flow, or that pulmonary and systemic flow are separate portions to be added as if the circuits were parallel.

The equality follows from conservation of blood volume. It applies when blood is not accumulating in either circuit, and when the pulmonary and systemic pathways are connected in series without a pathway that bypasses one of them. Temporary changes in stored blood volume or a shunt can make pulmonary and systemic flows unequal, even if the circulation as a whole is not losing blood.

Definition: Let \(Q_{pul}\) denote total blood flow through the pulmonary vascular bed, and let \(Q_{sys}\) denote total blood flow through the systemic vascular beds. Both are measured as volume per time, commonly in liters per minute. In an intact, unshunted circulation at steady state, \(Q_{pul}=Q_{sys}=Q_{CO}\), where \(Q_{CO}\) is the common cardiac output through the serial circuits.

Deriving the Steady-State Equality

For a vascular region, conservation of blood volume states that the rate of change in the volume stored in that region equals inflow minus outflow. Consider first the systemic circuit, bounded by the left-heart outflow and the return to the right heart. In the absence of a shunt, its inflow is \(Q_{sys}\) from the left heart and its outflow is the same blood returning to the right heart. For the pulmonary circuit, the corresponding inflow is right-heart output and the outflow is blood returning to the left heart.

At steady state, the volumes stored in the pulmonary and systemic circuits are not changing on average. Therefore, inflow must equal outflow for each circuit. Since the circuits are in series, the outflow from one is the inflow to the next. The flow equality follows directly:

\frac{dV_{pul}}{dt}=Q_{pul,in}-Q_{pul,out}=0 \qquad\text{and}\qquad \frac{dV_{sys}}{dt}=Q_{sys,in}-Q_{sys,out}=0

Q_{RV}=Q_{pul}=Q_{LV}=Q_{sys}=Q_{CO}

Here \(V_{pul}\) and \(V_{sys}\) are the blood volumes stored in the pulmonary and systemic circuits, \(Q_{RV}\) is right-heart output, and \(Q_{LV}\) is left-heart output. The equations refer to average flow over an appropriate interval, not necessarily the instantaneous flow at every moment of a heartbeat. The conclusion assumes a closed, intact serial pathway with no net accumulation of blood in either circuit.

Worked Example: Checking Flow Through an Intact Circulation

Suppose a hypothetical adult has a measured right-heart output of \(5.1\ \text{L/min}\). The circulation is intact and unshunted, and blood volume in each circuit is stable over the measurement interval. What pulmonary and systemic flows should be used?

The steady-state serial-circuit relationship is:

Q_{RV}=Q_{pul}=Q_{LV}=Q_{sys}

Substitute the measured output:

Q_{pul}=Q_{sys}=5.1\ \text{L/min}

Thus, pulmonary flow and total systemic flow are each \(5.1\ \text{L/min}\). They are not \(2(5.1)=10.2\ \text{L/min}\) of unique cardiac output: the same blood passes through one circuit and then the other.

Total Systemic Flow Is Not a Regional Allocation

Systemic flow is distributed among regional beds, such as the liver, kidneys, and muscle. In the intact steady-state case, those systemic flows add to the systemic total. The regional shares can differ substantially; equality of \(Q_{pul}\) and \(Q_{sys}\) places no requirement on any pair of organ flows being equal.

If \(Q_i\) denotes the flow to systemic region \(i\), and all systemic regions are included in the accounting, then:

Q_{sys}=\sum_i Q_i \qquad\text{and, in the intact steady state,}\qquad Q_{pul}=\sum_i Q_i=Q_{CO}

The pulmonary flow is not an additional systemic regional share. If a model represents the lung as a perfused tissue, its pulmonary perfusion is \(Q_{pul}\); the systemic organs together also receive \(Q_{sys}\). These are flows through successive parts of the circulation, not parallel fractions of a single total that must sum together. When the model accounts for systemic regions using fractions \(f_i\), those fractions are relative to systemic flow and sum to one if the included regions exhaust the systemic allocation.

Worked Example: Allocating Systemic Flow Without Double-Counting the Lung

In a hypothetical unshunted steady-state model, total cardiac output is \(5.2\ \text{L/min}\). Four systemic regions have flow fractions of 0.25, 0.20, 0.25, and 0.30. Calculate their flows and state the pulmonary flow.

For each systemic region, multiply its fraction by \(Q_{sys}=5.2\ \text{L/min}\):

\begin{aligned} Q_1&=0.25(5.2)=1.30\ \text{L/min},\\ Q_2&=0.20(5.2)=1.04\ \text{L/min},\\ Q_3&=0.25(5.2)=1.30\ \text{L/min},\\ Q_4&=0.30(5.2)=1.56\ \text{L/min}. \end{aligned}

The fractions sum to \(0.25+0.20+0.25+0.30=1.00\), and the flows sum to \(1.30+1.04+1.30+1.56=5.20\ \text{L/min}\). Thus these regional flows account for the systemic total. The pulmonary flow is also \(5.2\ \text{L/min}\), by the steady-state equality. Adding it to the systemic regional flows would give \(10.4\ \text{L/min}\) of circuit passages, not a new whole-body cardiac output.

Shunts and Temporary Changes in Stored Volume

A shunt is a pathway that allows blood to bypass part of the usual serial route. In a simple left-to-right shunt, some blood from the systemic arterial side is redirected to the pulmonary circulation. The pulmonary circuit then receives the usual systemic venous return plus the shunted flow. In this accounting, pulmonary flow can exceed systemic flow. The reverse direction of shunting can produce systemic flow greater than pulmonary flow because some blood reaches the systemic side without passing through the pulmonary vascular bed.

For a simple left-to-right shunt, let \(S\) be the shunted flow, \(Q_{sys}\) the systemic flow reaching the body’s systemic beds, and \(Q_{pul}\) the flow through the pulmonary bed. With no other pathways or net volume accumulation, the shunt recirculates flow through the lungs:

Q_{pul}=Q_{sys}+S

This relation is an accounting identity for that simplified pathway, not a formula for every possible shunt anatomy. In clinical descriptions, the ratio \(Q_{pul}/Q_{sys}\) is often useful: in this simple left-to-right example it is greater than one.

Worked Example: A Simple Left-to-Right Shunt

Suppose a hypothetical circulation has systemic flow of \(4.8\ \text{L/min}\) and a left-to-right shunt of \(1.2\ \text{L/min}\). Assume the simple pathway described above, stable blood volumes, and no other bypasses. Calculate pulmonary flow and the pulmonary-to-systemic flow ratio.

The accounting equation is:

Q_{pul}=Q_{sys}+S

Substituting the given values:

Q_{pul}=4.8+1.2=6.0\ \text{L/min}

The ratio is:

\frac{Q_{pul}}{Q_{sys}}=\frac{6.0}{4.8}=1.25

The pulmonary flow is \(6.0\ \text{L/min}\), and it is 1.25 times the systemic flow. The difference, \(1.2\ \text{L/min}\), is the stated shunt flow under the example’s assumptions. Treating both circuit flows as one common cardiac output would hide this recirculation.

Flows can also differ temporarily without a shunt. A brief mismatch between inflow and outflow changes the blood volume stored in a circuit. For example, if left-heart output momentarily exceeds the return from the systemic circuit, blood volume in that circuit rises during the mismatch. The imbalance cannot be treated as a sustained steady-state condition if blood volume remains bounded and there is no ongoing source or loss.

\frac{dV_{sys}}{dt}=Q_{LV}-Q_{RV}

For instance, a hypothetical \(0.4\ \text{L/min}\) difference sustained for 30 seconds corresponds to a temporary volume change of \(0.4\ \text{L/min}\times0.5\ \text{min}=0.20\ \text{L}\). This calculation describes a change in stored blood over that interval; it does not establish a new steady cardiac output. In practice, flow pulsatility, measurement timing, and changes in vascular filling can all contribute to short-lived discrepancies.

Choosing and Checking a PBPK Flow Representation

A single common flow is appropriate when the model represents an intact, unshunted circulation at steady state and the regional flow inputs are consistent with that assumption. A model that needs to represent shunts should instead distinguish pulmonary and systemic flows and include the bypass pathway explicitly. A transient model may also need to track changing blood volumes, rather than forcing all measured flows to be equal at every instant.

Representation	Appropriate use	Main caution
One common cardiac output	Intact, unshunted circulation at steady state	Does not represent a shunt or a temporary flow imbalance
Separate \(Q_{pul}\) and \(Q_{sys}\)	Circulation with a shunt or another modeled bypass	Specify the pathway and ensure flows obey its conservation equations
Time-varying circuit volumes and flows	Transient conditions where stored blood volume changes	A brief mismatch should not be interpreted as a persistent steady state

Several checks help identify a flow representation that is going wrong. In the unshunted steady-state case, compare right- and left-heart outputs and verify that the systemic regional flows sum to \(Q_{sys}\). Check that the systemic fractions use the systemic total as their reference. If a model includes lung perfusion as well as systemic regional flows, make sure the lung flow has not been mistakenly added to the systemic allocation as though the two circuits were parallel. If \(Q_{pul}\ne Q_{sys}\), ask whether a shunt, transient storage change, omitted pathway, or inconsistent measurement interval explains the difference before changing the flow inputs.

A common mistake is to reason that the lungs and body are two destinations for cardiac output and therefore their flows must share one total. That treats serial circuits as parallel beds. In an intact steady-state circulation, blood passes through both in sequence, so each circuit carries the common flow. The systemic organs divide the systemic flow among themselves; the lungs are not another share in that regional systemic division.

The equality is a conservation result, not a guarantee that any two reported measurements will match exactly. Real measurements may differ because of timing, precision, transient physiology, or an unrepresented shunt. The one-flow assumption is also limited to the conditions that justify it: it should not be imposed on a model intended to resolve bypass flow or short-term changes in blood storage. The introductory material on reference organ blood-flow values provides the context for regional flow inputs; the present circuit distinction determines how those inputs relate to pulmonary perfusion.

Check Your Understanding

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

In an intact, unshunted steady-state circulation, right-heart output is \(4.7\ \text{L/min}\). What are \(Q_{pul}\) and \(Q_{sys}\), and why are they not added to obtain cardiac output?
Systemic flow is \(6.0\ \text{L/min}\), allocated to all systemic regions using fractions that sum to 0.96. What does this indicate about the allocation?
In the simplified left-to-right shunt accounting, \(Q_{sys}=5.0\ \text{L/min}\) and \(S=0.8\ \text{L/min}\). Find \(Q_{pul}\) and \(Q_{pul}/Q_{sys}\).
For 20 seconds, left-heart output is \(0.3\ \text{L/min}\) greater than systemic return. Under the stated volume-balance relationship, how much does systemic stored blood volume change, and can this be called a new steady state?

Solutions. (1) Both flows are \(4.7\ \text{L/min}\) because the circuits are in series and neither is accumulating blood. They describe successive passages of the same flow, not parallel portions of one total. (2) The fractions account for only 96% of the stated systemic flow; 4% is unallocated, so the regional list is incomplete or its fractions need correction. (3) \(Q_{pul}=5.0+0.8=5.8\ \text{L/min}\), and \(Q_{pul}/Q_{sys}=5.8/5.0=1.16\). (4) The interval is \(20/60=1/3\) minute, so the volume change is \(0.3(1/3)=0.10\ \text{L}\) of accumulation. This describes a temporary change, not a new steady state.