The Flow-Conservation Derivation for Organ Beds

Why Organ Flows Must Balance

A PBPK model assigns blood flow to organs because perfusion determines how quickly compounds can be delivered to and removed from tissues. Those assigned flows cannot be arbitrary: in a steady, closed circulation, the blood leaving the heart for the systemic circulation must be accounted for across the systemic destinations. This is a conservation constraint, not a claim that every organ receives the same flow or that each organ’s flow is constant in every physiological state.

The key is to distinguish flow through a bed from net accumulation within a bed. A bed can have a substantial flow entering and leaving while its blood volume remains unchanged. Calling its supplied flow a “net inflow” would confuse these two quantities and would make the balance incorrect.

Definition: Let \(Q_{\mathrm{CO}}\) denote cardiac output, the volume of blood pumped by the heart per unit time. Let \(Q_i\) denote the total flow supplied to (entering) systemic organ bed \(i\), counting all of that bed’s inflowing vessels once. It is not the difference between inflow and outflow. Let \(f_i=Q_i/Q_{\mathrm{CO}}\) denote the bed’s flow fraction when \(Q_{\mathrm{CO}}>0\).

The equality derived here applies to a steady-state, closed-flow accounting in which the modeled systemic beds collectively account for the heart’s systemic output. Each destination must be included once, and the flow values must refer to the same physiological condition and time basis. If a pathway is omitted or left unassigned, the listed organ flows alone need not add to cardiac output; the remainder must be made explicit.

Deriving the Flow-Conservation Equality

First consider blood within one organ bed. Let \(B_i\) be the amount of blood in bed \(i\), expressed as a volume. Conservation of blood gives the local balance: the rate of change in the bed’s blood volume equals total inflow minus total outflow. At steady state, the bed’s blood volume is not changing, so its total outflow equals its total inflow. This does not mean the flow through the bed is zero; it means the flow entering is balanced by the flow leaving.

\frac{dB_i}{dt}=Q_{i,\mathrm{in}}-Q_{i,\mathrm{out}}=0 \quad\Longrightarrow\quad Q_{i,\mathrm{in}}=Q_{i,\mathrm{out}}=Q_i

Now consider the systemic distribution of the heart’s output. At steady state, there is no continuing accumulation of blood in the distribution network. If the systemic destinations are indexed by \(i=1,\ldots,k\), and each destination is counted once, the total flow leaving the source for those destinations is the sum of the flows supplied to them. By definition, that total supplied flow is cardiac output. Therefore:

Q_{\mathrm{CO}}=\sum_{i=1}^{k}Q_i

This is the central conservation result. It is sometimes useful to express the same constraint in terms of flow fractions. Dividing both sides by \(Q_{\mathrm{CO}}\), which is positive for a functioning circulation, gives:

\sum_{i=1}^{k}\frac{Q_i}{Q_{\mathrm{CO}}}=1 \quad\Longrightarrow\quad \sum_{i=1}^{k}f_i=1

Conversely, if a set of fractions sums to one, multiplying each fraction by cardiac output gives absolute flows that sum to cardiac output:

Q_i=f_iQ_{\mathrm{CO}} \quad\Longrightarrow\quad \sum_{i=1}^{k}Q_i =Q_{\mathrm{CO}}\sum_{i=1}^{k}f_i =Q_{\mathrm{CO}}

Key result: For a complete, non-overlapping set of systemic destinations at steady state, the absolute flows sum to cardiac output, \(\sum_i Q_i=Q_{\mathrm{CO}}\), and the corresponding fractions sum to one, \(\sum_i f_i=1\). The result follows from conservation and the accounting boundary; it does not depend on the particular organ-flow values.

Worked Examples

Worked Example: Converting Fractions to Absolute Flows

Suppose a hypothetical steady-state flow table uses \(Q_{\mathrm{CO}}=5.20\ \mathrm{L/min}\) and gives five systemic-bed fractions: 0.18, 0.22, 0.27, 0.13, and 0.20. Check the fractions and calculate the absolute flows.

First check whether the fractions account for the whole output: \(0.18+0.22+0.27+0.13+0.20=1.00\). They satisfy the conservation condition.

For each bed, use \(Q_i=f_iQ_{\mathrm{CO}}\). The calculated flows are:

\begin{aligned} Q_1&=0.18(5.20)=0.936\ \mathrm{L/min},\\ Q_2&=0.22(5.20)=1.144\ \mathrm{L/min},\\ Q_3&=0.27(5.20)=1.404\ \mathrm{L/min},\\ Q_4&=0.13(5.20)=0.676\ \mathrm{L/min},\\ Q_5&=0.20(5.20)=1.040\ \mathrm{L/min}. \end{aligned}

Adding them gives \(0.936+1.144+1.404+0.676+1.040=5.200\ \mathrm{L/min}\), equal to the specified cardiac output. The fractions are dimensionless, while each calculated \(Q_i\) has units of volume per time.

Worked Example: Checking Absolute Flows and Recovering Fractions

Consider a hypothetical table with \(Q_{\mathrm{CO}}=5.00\ \mathrm{L/min}\) and five supplied bed flows of 0.90, 1.10, 1.40, 0.70, and 0.90 \(\mathrm{L/min}\). Determine whether the flows close to cardiac output, then calculate their fractions.

The flow sum is \(0.90+1.10+1.40+0.70+0.90=5.00\ \mathrm{L/min}\), so the listed flows satisfy the equality for a complete set of destinations. Calculate each fraction as \(f_i=Q_i/Q_{\mathrm{CO}}\):

(0.90/5.00,\ 1.10/5.00,\ 1.40/5.00,\ 0.70/5.00,\ 0.90/5.00) =(0.18,\ 0.22,\ 0.28,\ 0.14,\ 0.18)

Their sum is \(0.18+0.22+0.28+0.14+0.18=1.00\). Thus the absolute-flow and fraction checks agree. This agreement is an algebraic consequence of using the same cardiac output and the same set of flow entries in both calculations; it does not independently establish that the underlying measurements are accurate.

Worked Example: Making an Unallocated Remainder Explicit

A hypothetical table specifies \(Q_{\mathrm{CO}}=4.80\ \mathrm{L/min}\), but lists only four systemic destinations with flows 1.10, 0.90, 0.75, and 0.65 \(\mathrm{L/min}\). Find the flow not represented in the list and express it as a fraction of cardiac output.

The listed sum is \(1.10+0.90+0.75+0.65=3.40\ \mathrm{L/min}\). The unallocated remainder is therefore \(4.80-3.40=1.40\ \mathrm{L/min}\). Its fraction is \(1.40/4.80=0.2917\), or about \(29.2\%\). The listed fractions sum to \(3.40/4.80=0.7083\), or about \(70.8\%\); adding the remainder fraction gives \(0.7083+0.2917=1.0000\).

The \(1.40\ \mathrm{L/min}\) remainder is an accounting quantity, not a measurement of a particular organ. A model can retain it as an explicitly unassigned category while the flow table is incomplete, or assign it to specified destinations when their flows are known. It should not silently be treated as though it belonged to one listed bed.

Practical Checks and Common Failure Modes

A flow-conservation check is simple, but its interpretation depends on careful bookkeeping. A mismatch does not by itself identify which input is wrong. It may indicate an incomplete list, inconsistent units, rounding, overlapping categories, or values that do not describe the same condition. The check is most useful when the model boundary and the meaning of each entry are stated clearly.

Confirm the accounting boundary. Identify which systemic destinations are included and whether any flow pathway is intentionally outside the table. A complete sum requires complete coverage.
Count each flow once. If a bed has more than one inflowing vessel, combine those supplies into that bed’s \(Q_i\), or list the vessels separately—but do not do both. Otherwise, part of the flow is double-counted.
Use compatible units. Convert all absolute flows to a common unit before adding them. For example, \(1\ \mathrm{L/min}=1000\ \mathrm{mL/min}\).
Compare like with like. Cardiac output and organ flows must refer to the same physiological condition and time basis. Fractions from one condition cannot be combined with output from another and expected to close.
Account for rounding. Published fractions rounded to a limited number of digits may sum slightly above or below one. Check the unrounded source values if available; do not infer a physiological imbalance from a small display-rounding discrepancy alone.

One especially important language error is to define \(Q_i\) as a bed’s “net inflow.” In a steady-state volume balance, net inflow means inflow minus outflow, which equals the rate of blood-volume accumulation and is zero. The \(Q_i\) in the conservation equation is instead the total supplied flow entering the bed. It can be positive even when net accumulation is zero, because the same volume per unit time leaves the bed.

How the Accounting Formulation Changes for Incomplete or Aggregated Tables

The equality can be written in different but equivalent ways depending on how a table is organized. Absolute flows make the volume-per-time balance explicit. Fractions normalize each flow by cardiac output and make the total-one constraint easy to inspect. A table with an omitted category can instead show the known subtotal and a clearly labelled remainder. These are alternative presentations of conservation, not different physiological laws.

Presentation	Check	Best use
Absolute flows	\(\sum_i Q_i=Q_{\mathrm{CO}}\)	Inspecting or building a model with flow rates in volume per time
Flow fractions	\(\sum_i f_i=1\)	Comparing allocations after normalization to cardiac output
Known subtotal plus remainder	\(\sum_i Q_i+Q_{\mathrm{remainder}}=Q_{\mathrm{CO}}\)	Showing explicitly that the listed destinations do not yet cover all flow

Fractions are convenient, but they can conceal an incorrect denominator or hide the size of a flow in physical units. Absolute flows retain units and are often easier to compare directly with a specified \(Q_{\mathrm{CO}}\), but they can appear inconsistent if units differ or numbers have been rounded independently. Reporting both, when appropriate, allows a reader to see the allocation and the corresponding rates.

Scope and Limitations

This derivation assumes a steady state for the accounting interval, a closed circulation with no unmodeled source or sink of blood, and a complete, non-overlapping accounting of the modeled systemic destinations. “Steady state” here means that blood volume is not continuously accumulating in the relevant beds and distribution network. It does not require that every local flow be identical or that blood be motionless.

During a transient change, blood volume in a region or distribution segment can change, so its instantaneous inflow and outflow need not be equal. A flow table may also be incomplete by design, or may combine categories that overlap. In those cases, do not force the listed entries to equal cardiac output without first clarifying the boundary and accounting categories. The conservation identity remains useful, but the term “complete set” and the time condition matter.

Finally, satisfying the sum is necessary for a closed, steady-state allocation but is not sufficient to validate every individual flow. Several erroneous entries could still sum to the correct total. Conservation checks internal consistency; they do not establish the accuracy of the inputs or the suitability of a particular physiological state.

Key takeaway: Define each organ-bed flow as the total flow supplied to that bed. At steady state, a complete and non-overlapping systemic allocation obeys \(\sum_i Q_i=Q_{\mathrm{CO}}\), equivalently \(\sum_i f_i=1\). If the sum does not close, check coverage, units, rounding, and double counting before interpreting the difference.

Check Your Understanding

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

In a steady-state bed, inflow is \(0.60\ \mathrm{L/min}\). What is the outflow, and what is the net accumulation rate?
A complete hypothetical allocation has fractions 0.25, 0.30, and 0.45. If \(Q_{\mathrm{CO}}=4.00\ \mathrm{L/min}\), calculate the three absolute flows and verify their sum.
A table lists absolute flows totaling \(3.6\ \mathrm{L/min}\) when cardiac output is \(4.0\ \mathrm{L/min}\). What is the unallocated remainder and its fraction of cardiac output?
Why is it incorrect to define \(Q_i\) as “inflow minus outflow” in the steady-state flow-conservation equation?
Two entries in a flow table describe separate inflowing vessels to the same bed. What must be checked before including them in a sum of organ flows?

Solutions. (1) Steady-state volume balance gives outflow \(=0.60\ \mathrm{L/min}\), so net accumulation is \(0.60-0.60=0\ \mathrm{L/min}\). (2) The flows are \(0.25(4.00)=1.00\), \(0.30(4.00)=1.20\), and \(0.45(4.00)=1.80\ \mathrm{L/min}\); their sum is \(1.00+1.20+1.80=4.00\ \mathrm{L/min}\). (3) The remainder is \(4.0-3.6=0.4\ \mathrm{L/min}\), and its fraction is \(0.4/4.0=0.10\), or \(10\%\). (4) Inflow minus outflow is the rate of accumulation, which is zero at steady state; \(Q_i\) represents the positive total supplied flow, equal to the outflow under that condition. (5) Check that each vessel is counted once and that their flows are being combined into a single bed total rather than also being counted again as a separate bed entry.