Organ Blood-Flow Fractions and Absolute Flows

Why Express Organ Flow as a Fraction?

A PBPK model often needs organ blood flows as inputs, but those inputs may be reported in different forms. One source may give an absolute flow in litres per minute; another may report the share of cardiac output directed to an organ. Converting between these forms makes the values easier to compare and allows a flow table to be rescaled when cardiac output changes. The conversion is simple, but it is meaningful only when the organ flow and cardiac output refer to compatible physiological conditions and measurement bases.

The earlier material on cardiac output, stroke volume, and heart rate established cardiac output as the volume of blood pumped by the heart per unit time. Here, cardiac output is the reference flow against which an organ’s blood flow is expressed. The focus is the definition and conversion, not the detailed allocation of flow among all organs.

Definition: Let $Q_{\mathrm{CO}}$ denote cardiac output, measured as volume per unit time, and let $Q_i$ denote blood flow to organ $i$, measured on a compatible basis and over the same time interval. The organ blood-flow fraction $f_i$ is $$ f_i=\frac{Q_i}{Q_{\mathrm{CO}}}. $$ It is dimensionless. Its percentage form is $100f_i\%$. The equivalent conversion to absolute organ flow is $Q_i=f_iQ_{\mathrm{CO}}$.

For example, if $f_i=0.18$, the organ receives 18% of the reference cardiac output. The fraction itself has no flow units; $Q_i$ does. A percentage must be converted to its decimal fraction before multiplication: 18% is $0.18$, not 18.

Converting Between Fractions and Absolute Flows

To calculate absolute flow from a fraction, multiply the fraction by cardiac output. To calculate the fraction from absolute flow, divide the organ flow by cardiac output. In either direction, retain the units until the final check: flow divided by flow is dimensionless, while a dimensionless fraction multiplied by flow has units of flow.

Identify the denominator.
Use the cardiac output that matches the physiological state and population represented by the organ-flow value.

Put the fraction in decimal form.
For a percentage, divide by 100 before using it in a multiplication.

Calculate and check units.
Multiply $f_i$ by $Q_{\mathrm{CO}}$ to obtain $Q_i$, or divide $Q_i$ by $Q_{\mathrm{CO}}$ to obtain $f_i$. Check that the result has the expected units and scale.

Worked Example: Converting a Fraction to Absolute Flow

Suppose a hypothetical adult has cardiac output $Q_{\mathrm{CO}}=5.0\ \mathrm{L/min}$. A model input assigns 16% of this output to a specified organ bed. Find the bed’s absolute flow.

Convert the percentage to a decimal fraction: $f_i=16/100=0.16$. Then use $Q_i=f_iQ_{\mathrm{CO}}$:

Q_i=0.16\times 5.0\ \mathrm{L/min}=0.80\ \mathrm{L/min}.

The hypothetical organ-bed flow is therefore $0.80\ \mathrm{L/min}$. The units remain $\mathrm{L/min}$, because the fraction $0.16$ is dimensionless. As a scale check, 16% is less than one-fifth, so a result somewhat below one-fifth of $5.0\ \mathrm{L/min}$ is reasonable.

Worked Example: Recovering the Fraction From an Absolute Flow

In a separate hypothetical case, cardiac output is $4.8\ \mathrm{L/min}$, and the flow to a specified organ bed is $0.72\ \mathrm{L/min}$. Calculate the organ’s flow fraction and percentage.

Divide the organ flow by cardiac output:

f_i=\frac{0.72\ \mathrm{L/min}}{4.8\ \mathrm{L/min}}=0.15.

The units cancel, leaving a dimensionless fraction. As a percentage, this is $100(0.15)\%=15\%$. Thus, under the stated hypothetical conditions, the organ-bed flow is 15% of cardiac output. A check using the reverse conversion gives $0.15\times4.8\ \mathrm{L/min}=0.72\ \mathrm{L/min}$, as required.

Using Fractions in a Flow Table

A flow table can list fractions, absolute flows, or both. If cardiac output is fixed, either representation determines the other. A table that reports both provides a useful arithmetic check, but rounding can make the displayed values differ slightly from conversions performed with unrounded inputs.

For a set of organ beds, the sum of the displayed fractions has an interpretation only after the beds and denominator have been defined. If the entries are a complete set of mutually exclusive flows referenced to the same cardiac output, their fractions should sum to one, apart from rounding. If the table lists only some beds, the sum can be below one because flow to other beds is not listed. It is not appropriate to assume that a partial list sums to one.

The beds must also be defined so that the same flow is not counted twice. For instance, flow reported for a downstream organ that includes blood already counted in an upstream region cannot simply be added to the upstream value as if they were separate destinations. A fraction is only as interpretable as the flow definition in its numerator. For this tutorial’s conversions, treat each $Q_i$ as a clearly defined organ-bed flow on a consistent basis.

Worked Example: Converting a Partial Flow Table

Consider a hypothetical flow table with $Q_{\mathrm{CO}}=4.0\ \mathrm{L/min}$. It lists three non-overlapping organ beds with fractions $0.25$, $0.10$, and $0.30$. Convert each fraction to absolute flow and determine what the table does and does not establish.

For the first bed, $Q_1=0.25\times4.0=1.00\ \mathrm{L/min}$. For the second, $Q_2=0.10\times4.0=0.40\ \mathrm{L/min}$. For the third, $Q_3=0.30\times4.0=1.20\ \mathrm{L/min}$. Each calculation uses $Q_i=f_iQ_{\mathrm{CO}}$.

The listed fractions sum to $0.25+0.10+0.30=0.65$, and the listed absolute flows sum to $1.00+0.40+1.20=2.60\ \mathrm{L/min}$. These are consistent because $0.65\times4.0\ \mathrm{L/min}=2.60\ \mathrm{L/min}$. Since this is a partial table, it does not imply that the remaining $0.35$ of cardiac output is missing from the circulation. It indicates only that the listed beds account for 65% of the stated output; other beds are not represented in the table.

Why the Conversion Works

The fraction is defined as the ratio of two flows measured in the same units. Multiplying that ratio by the denominator recovers the numerator. This is an algebraic identity, rather than an additional physiological assumption:

f_i=\frac{Q_i}{Q_{\mathrm{CO}}} \quad\Longrightarrow\quad f_iQ_{\mathrm{CO}} =\frac{Q_i}{Q_{\mathrm{CO}}}Q_{\mathrm{CO}} =Q_i, \qquad Q_{\mathrm{CO}}\ne 0.

The condition $Q_{\mathrm{CO}}\ne0$ is required for the ratio to be defined. In ordinary applications, cardiac output is positive. The identity also shows why the denominator matters: changing the cardiac output used to calculate the fraction changes the fraction, even if the organ’s absolute flow is unchanged. Conversely, applying the same fraction to a different cardiac output produces a different absolute flow.

Checks, Alternatives, and Limitations

A reliable conversion begins with checking whether the two flow values are comparable. They should refer to the same state (for example, not one value at rest and another during a different condition), use compatible units, and have a consistent physiological definition. If one value is reported in $\mathrm{mL/min}$ and the other in $\mathrm{L/min}$, convert one before dividing. Since $1\ \mathrm{L}=1000\ \mathrm{mL}$, $600\ \mathrm{mL/min}=0.600\ \mathrm{L/min}$.

The fraction and absolute-flow representations serve different practical purposes. Fractions make relative allocation and rescaling transparent; absolute flows are required when the model input or calculation needs volume per unit time. Neither representation adds information when one is computed exactly from the other using a known cardiac output. If the organ flow and cardiac output come from separate measurements, however, the calculated fraction inherits their measurement and definition differences. The conversion does not resolve uncertainty or make incompatible measurements comparable.

Useful arithmetic diagnostics include confirming that a calculated fraction is dimensionless, that an ordinary positive organ flow gives a positive fraction, and that the fraction is on a plausible scale for the stated bed. If a purported fraction exceeds one, first check for a unit mismatch, a wrong denominator, or incompatible flow definitions; do not automatically treat the value as a valid share of total output. For a complete set of mutually exclusive beds, also check whether the fractions are close to one, allowing for stated rounding. That sum check does not apply to an incomplete table.

Using a percentage as if it were a decimal fraction. Multiplying by 16 rather than $0.16$ makes the computed flow 100 times too large.
Changing the denominator without noticing. A fraction is relative to a particular cardiac output, not an intrinsic fixed property of the organ.
Mixing units. Dividing a flow in $\mathrm{mL/min}$ by one in $\mathrm{L/min}$ without conversion creates a numerical error by a factor of 1000.
Forcing a partial table to sum to one. A sum below one can simply mean that some organ beds were not listed.
Adding overlapping flows. Fractions cannot be summed meaningfully if their numerator definitions count some of the same blood flow more than once.

Key takeaway: An organ’s blood-flow fraction is its defined flow divided by a compatible cardiac output. Multiply the fraction by that same cardiac output to recover absolute flow; divide the organ flow by cardiac output to recover the fraction. Always state the denominator, units, physiological basis, and whether the table is complete.

Check Your Understanding

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

A hypothetical cardiac output is $6.0\ \mathrm{L/min}$, and an organ-bed fraction is 12%. What is the absolute flow?
A hypothetical organ-bed flow is $450\ \mathrm{mL/min}$, while cardiac output is $5.0\ \mathrm{L/min}$. What is the flow fraction and percentage?
A table lists fractions $0.20$, $0.35$, and $0.25$. Must this table be incomplete? Explain what additional information is needed.
Why is it invalid to calculate a flow fraction by dividing an organ flow in $\mathrm{mL/min}$ directly by cardiac output in $\mathrm{L/min}$?
If $Q_i$ stays fixed while the cardiac output used as denominator increases, what happens to $f_i$?

Solutions. (1) Convert 12% to $0.12$: $Q_i=0.12\times6.0=0.72\ \mathrm{L/min}$. (2) Convert $450\ \mathrm{mL/min}$ to $0.450\ \mathrm{L/min}$; then $f_i=0.450/5.0=0.0900$, or 9.00%. (3) The listed fractions sum to $0.80$. The table could be incomplete, or the entries might not represent a complete, mutually exclusive set; the bed definitions and whether the list is complete are needed. (4) The units must match before division: otherwise the numerical result is off by a factor of 1000. (5) The fraction decreases, because $f_i=Q_i/Q_{\mathrm{CO}}$ with fixed numerator and a larger positive denominator.