Common Errors in Cardiac-Output Allocation

Why Allocation Errors Matter

In a PBPK model, blood flow helps determine how quickly drug is delivered to each perfused tissue. A table can look plausible while still being internally inconsistent: it may label a fraction as a flow, assign more flow to organs than the heart supplies, or combine values from different physiological conditions. Such errors change the model’s implied delivery of drug, even if no equation in the model is written incorrectly.

The earlier material on organ blood-flow fractions and absolute flows establishes the conversion between a fraction and a flow; the flow-conservation derivation establishes that flows must reconcile to the output supplying the beds. Here the practical task is to catch common allocation mistakes and correct them before using a scenario-specific table.

Definition: Let $Q_{\mathrm{CO}}$ denote cardiac output, in units such as L/min. For a systemic organ bed $i$, let $Q_i$ be its absolute blood flow, also in L/min, and let $f_i$ be its dimensionless fraction of systemic flow. When the listed beds exhaust the systemic circulation, $$ Q_i=f_iQ_{\mathrm{CO}}, \qquad \sum_i f_i=1, \qquad \sum_i Q_i=Q_{\mathrm{CO}}. $$ If the listed beds are not exhaustive, the unlisted beds must account for the remaining fraction and flow. A scenario is the set of physiological conditions to which the cardiac output, fractions, and flows all refer.

These equalities assume a coherent circulation at the stated condition, with each relevant systemic bed represented once and with flows measured on a compatible basis. They do not require each organ fraction to remain constant when conditions change. In particular, a fraction is not an absolute flow: its value alone cannot specify a flow unless the corresponding cardiac output is known.

Check the Units Before the Arithmetic

A reliable first check is dimensional. The product of a dimensionless fraction and a flow has units of flow, whereas adding a fraction directly to a flow is meaningless. For example, $0.20$ is not $0.20\ \mathrm{L/min}$; it means 20% of the relevant output. If $Q_{\mathrm{CO}}=5.00\ \mathrm{L/min}$, the flow associated with $f_i=0.20$ is $1.00\ \mathrm{L/min}$.

Blood volume is a different quantity. Let $V_i$ denote the blood volume in a region, measured in units such as L. Dividing volume by flow gives a time, not a flow. Under a simple steady-flow interpretation, $V_i/Q_i$ is a characteristic turnover time for that volume; it cannot be substituted for $Q_i$ in an organ-flow table.

Worked Example: Converting Fractions to Organ Flows

Consider a hypothetical resting systemic table with $Q_{\mathrm{CO}}=5.00\ \mathrm{L/min}$. The listed fractions are brain 0.15, kidneys 0.20, skeletal muscle 0.20, splanchnic beds 0.25, and other systemic beds 0.20. Find the absolute flow to each group and check the allocation.

For each group, use $Q_i=f_iQ_{\mathrm{CO}}$. The fractions sum to $0.15+0.20+0.20+0.25+0.20=1.00$, so the listed groups are exhaustive under the stated setup.

Brain: $0.15(5.00\ \mathrm{L/min})=0.750\ \mathrm{L/min}$. Kidneys: $0.20(5.00\ \mathrm{L/min})=1.00\ \mathrm{L/min}$. Skeletal muscle: $0.20(5.00\ \mathrm{L/min})=1.00\ \mathrm{L/min}$. Splanchnic beds: $0.25(5.00\ \mathrm{L/min})=1.25\ \mathrm{L/min}$. Other systemic beds: $0.20(5.00\ \mathrm{L/min})=1.00\ \mathrm{L/min}$.

The absolute flows sum to $0.750+1.00+1.00+1.25+1.00=5.00\ \mathrm{L/min}$, matching cardiac output. Reporting the kidney value as “0.20 L/min” would be an error: 0.20 is the fraction, and the corresponding flow is $1.00\ \mathrm{L/min}$.

Reconcile Changes With the Total

A second common error is changing one or more organ flows while leaving the total and all other flows untouched. If the cardiac output is fixed, a redistribution must preserve the total: an increase in one bed must be offset by a decrease in one or more other beds. If no such offset is intended, then the proposed values imply a changed total output and should be checked against the stated cardiac output.

The conservation check follows directly from the fraction definition. For exhaustive systemic beds,

\sum_i Q_i=\sum_i(f_iQ_{\mathrm{CO}}) =Q_{\mathrm{CO}}\sum_i f_i =Q_{\mathrm{CO}}.

Thus, a proposed table with $\sum_i Q_i\ne Q_{\mathrm{CO}}$ cannot simultaneously describe all the stated systemic beds and the stated output. The discrepancy is a signal to investigate—not a reason to silently rescale values. First establish whether the cardiac output, a measurement, a unit conversion, a missing bed, or an assumption about the scenario is responsible.

Worked Example: Correcting an Unreconciled Redistribution

Use the hypothetical resting table above, with total output $5.00\ \mathrm{L/min}$. A draft revised table raises kidney flow from $1.00$ to $1.40\ \mathrm{L/min}$ and lowers skeletal-muscle flow from $1.00$ to $0.70\ \mathrm{L/min}$, leaving the other three flows unchanged. Is the revised table consistent with the same cardiac output? If not, make a stated correction.

The revised sum is $0.750+1.40+0.70+1.25+1.00=5.10\ \mathrm{L/min}$. It exceeds the stated $5.00\ \mathrm{L/min}$ by $0.100\ \mathrm{L/min}$, so the draft is not a complete allocation at fixed output.

One possible correction, if the “other systemic beds” group is permitted to absorb the discrepancy, is to reduce that flow from $1.00$ to $0.900\ \mathrm{L/min}$. The corrected total is $0.750+1.40+0.70+1.25+0.900=5.00\ \mathrm{L/min}$. Its fractions are the corrected flows divided by $5.00\ \mathrm{L/min}$: 0.150, 0.280, 0.140, 0.250, and 0.180. They sum to 1.00.

This is an arithmetic correction, not evidence that the “other” beds truly change by that amount. The physiological allocation must be supported by the scenario or data. If the proposed changes were intended to raise total output instead, the table would imply $5.10\ \mathrm{L/min}$, and that changed output would need to be stated and justified.

Keep Blood Volume and Blood Flow Separate

Blood volume describes how much blood is present in a region; blood flow describes how much passes through it per unit time. Their units make the distinction explicit: L versus L/min. A larger blood volume does not by itself imply a larger flow. Nor can a volume value be placed in a flow column just because both quantities refer to the same organ.

Worked Example: Detecting a Volume-for-Flow Substitution

Suppose, hypothetically, that a tissue region contains $0.300\ \mathrm{L}$ of blood and receives $0.750\ \mathrm{L/min}$. A draft table lists the tissue’s flow as $0.300\ \mathrm{L/min}$, apparently using the volume value. Identify the error and calculate the volume-to-flow time.

The stated flow is $0.750\ \mathrm{L/min}$, not $0.300\ \mathrm{L/min}$. The quantity $0.300\ \mathrm{L}$ is a volume and has the wrong units to fill the flow entry.

The volume-to-flow ratio is $$ \frac{V_i}{Q_i} =\frac{0.300\ \mathrm{L}}{0.750\ \mathrm{L/min}} =0.400\ \mathrm{min} =24.0\ \mathrm{s}. $$

The result is a characteristic time under the stated interpretation, not an alternative flow estimate. The calculation does not establish that the tissue is well mixed or that every blood element remains there for exactly 24 seconds.

Do Not Mix Physiological Scenarios

Fractions and flows depend on the physiological condition. A resting fraction combined with exercise cardiac output, or a flow from one condition combined with the total from another, may create a table that describes no specified scenario. Values should be paired by condition before calculating or comparing allocations. This is especially important when cardiac output and the relative distribution both change.

Worked Example: Separating Resting and Exercise Values

For illustration, suppose a hypothetical resting scenario has $Q_{\mathrm{CO}}=5.00\ \mathrm{L/min}$ and a kidney fraction of 0.20. A hypothetical exercise scenario has $Q_{\mathrm{CO}}=12.0\ \mathrm{L/min}$ and a kidney fraction of 0.08. Calculate kidney flow in each condition and assess the tempting calculation that uses the resting fraction with exercise output.

At rest, kidney flow is $0.20(5.00\ \mathrm{L/min})=1.00\ \mathrm{L/min}$. During exercise, it is $0.08(12.0\ \mathrm{L/min})=0.960\ \mathrm{L/min}$.

Using the resting fraction with exercise output would give $0.20(12.0\ \mathrm{L/min})=2.40\ \mathrm{L/min}$. That calculation is arithmetically valid but mismatches the stated exercise allocation. It is correct only under an additional assumption that the kidney fraction remains 0.20 during exercise, contrary to the hypothetical exercise value supplied here.

Systemic organ flows also should not be added to pulmonary flow as though they were separate portions of cardiac output. In a circulation at steady state, the same total output passes through the pulmonary and systemic circuits in sequence; a systemic allocation table accounts for systemic beds, while pulmonary flow is a separate description of flow through the lungs. Counting both circuits as independent destinations would double-count the output.

A Practical Audit

Use a short sequence of checks before accepting a flow table. The checks can identify an inconsistency, but they cannot determine which input is physiologically correct without further evidence.

Label the scenario.
Record the condition and ensure that cardiac output, fractions, and absolute flows refer to that same condition.

Label each quantity and its units.
Mark fractions as dimensionless, flows in volume per time, and blood volumes in volume units. Do not interchange them.

Check completeness.
Identify whether the table includes all systemic beds. If not, name or quantify the remainder rather than assuming the listed fractions sum to one.

Check both totals.
For exhaustive beds, verify that fractions sum to 1 and absolute flows sum to the stated $Q_{\mathrm{CO}}$, allowing only justified rounding differences.

Investigate, then correct.
Trace a mismatch to its source—such as a missing bed, conversion error, changed output, or scenario mismatch—and document any correction rather than silently forcing agreement.

These checks are complementary to, not replacements for, physiological judgment. An internally consistent table can still be physiologically implausible or based on uncertain measurements. Conversely, small differences caused by rounding may be harmless if their size and source are clear. When measurements disagree, diagnose the discrepancy rather than changing values solely to make the sum match.

Common Pitfalls and Limits

Calling a fraction a flow. The mistaken reasoning is that a value such as 0.20 is an organ flow. The correction is to label it as a fraction and multiply by the matching cardiac output to obtain L/min.
Changing a bed without checking the total. A change may be plausible in isolation but leave the table over- or under-allocated. Recalculate the sum and state whether output changed or another bed compensates.
Treating blood volume as perfusion. Volume and flow have different units; their ratio is time. Do not substitute one for the other.
Combining conditions for convenience. A fraction from rest is not automatically valid with exercise output. Use matched scenario values or explicitly justify an assumption that the fraction is unchanged.
Forcing a total by arbitrary rescaling. Rescaling may hide the source of an error and alter every value. First establish the intended beds, condition, and output.

The conservation and unit checks are basic safeguards, not a complete description of circulation. They do not resolve heterogeneous perfusion within an organ, measurement uncertainty, or the physiological validity of a proposed redistribution. Those questions require evidence and a clearly stated scenario.

Check Your Understanding

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

A hypothetical $4.00\ \mathrm{L/min}$ output supplies a bed with fraction 0.125. What is the bed’s absolute flow?
Three listed systemic flows are 0.80, 1.10, and $1.50\ \mathrm{L/min}$, while stated cardiac output is $3.60\ \mathrm{L/min}$. Assuming these are the only beds, is the table reconciled?
Why is a blood volume of $0.40\ \mathrm{L}$ not a flow of $0.40\ \mathrm{L/min}$?
A resting fraction is multiplied by an exercise cardiac output. What must be checked before interpreting the resulting flow?
If exhaustive organ fractions sum to 0.92, give two possible explanations that should be investigated before changing the values.

Solutions. (1) $Q_i=0.125(4.00\ \mathrm{L/min})=0.500\ \mathrm{L/min}$. (2) The flows sum to $0.80+1.10+1.50=3.40\ \mathrm{L/min}$, which is $0.20\ \mathrm{L/min}$ below output, so the table is not reconciled under the stated assumption. (3) The volume has units L, whereas flow has units L/min; volume divided by flow would have units of time. (4) Check whether the fraction applies during exercise; otherwise the calculation mixes scenarios. (5) A bed may be omitted, or one or more fractions may refer to a different condition or contain a measurement or transcription error.)