Why Low Cardiac Output Does Not Tell the Whole Flow Story
In low-output heart failure, the heart delivers less blood per unit time than it would in the reference state. This can reduce the absolute blood flow reaching organs. At the same time, the body may alter vascular tone in different regions, changing how the available systemic flow is distributed. These are related but distinct effects: cardiac output describes the total flow, while an organ-flow fraction describes that organ’s share of the total.
For PBPK calculations, separating these effects matters because organ perfusion is an input to the delivery of blood-borne compounds to tissues. A reduced cardiac output can lower flow to several organs even if their shares of total flow are unchanged. Conversely, a greater share directed to one organ does not by itself establish that its absolute flow has increased.
This relationship is an accounting identity once the output and fractions have been specified. It does not, by itself, predict what causes a fraction to change or establish that every organ responds in the same way. In heart failure, reduced pump performance can lower output; compensatory responses, including changes in sympathetic vascular tone, may alter regional allocation. The direction and size of organ-specific changes depend on the clinical state and cannot be inferred from the diagnosis alone.
Separating Total-Flow Change From Redistribution
A useful comparison specifies a reference state and a low-output state. In the reference state, organ \(i\) has flow \(Q_{i,\mathrm{ref}}=f_{i,\mathrm{ref}}Q_{\mathrm{CO,ref}}\). In the low-output state, its flow is \(Q_{i,\mathrm{low}}=f_{i,\mathrm{low}}Q_{\mathrm{CO,low}}\). These undivided equations remain valid even when an organ fraction is zero.
When the reference organ flow is positive, the ratio of low-output to reference flow can be written as a product of two ratios. The positivity condition matters: if \(Q_{i,\mathrm{ref}}=0\), the flow ratio is undefined, so the ratio form must not be used.
The first ratio on the right describes the change in total output; the second describes the change in the organ’s share. Their product gives the organ’s relative flow change. For example, if output falls to 60% of reference and the organ’s fraction rises by 25%, its flow becomes \(0.60\times1.25=0.75\) of reference: the fraction rose, but absolute flow fell by 25%.
If the fractions remain unchanged, every included organ’s absolute flow changes in proportion to cardiac output. This is a useful baseline calculation, not a claim that real heart failure preserves regional allocation. If fractions change, their sum must still equal 1 when the table covers all systemic beds. A higher fraction in one region therefore means a larger share, not necessarily a larger number of litres per minute.
Worked Example: Proportional Reduction With Unchanged Fractions
Suppose a hypothetical reference state has systemic cardiac output \(5.0\) L/min, and a kidney receives 20% of that output. In a hypothetical low-output state, output is \(3.0\) L/min and the kidney’s fraction remains 20%. Assume the fraction refers to the same defined systemic circulation in both states.
The kidney flow in the reference state is \(Q_{i,\mathrm{ref}}=f_{i,\mathrm{ref}}Q_{\mathrm{CO,ref}}=0.20\times5.0=1.0\) L/min.
In the low-output state it is \(Q_{i,\mathrm{low}}=f_{i,\mathrm{low}}Q_{\mathrm{CO,low}}=0.20\times3.0=0.60\) L/min.
The output has fallen by \(2.0\) L/min, or 40% of its reference value. Kidney flow has also fallen by 40%, from \(1.0\) to \(0.60\) L/min. This conclusion follows from the stated unchanged fraction; it is not a general prediction that kidney flow always changes proportionally in heart failure.
Worked Allocation With Redistribution
A practical way to avoid confusing shares with flows is to calculate both for each state. The following hypothetical table includes all systemic beds in this simplified example. The fractions sum to 1.00 in each state, so the flows should sum to the corresponding cardiac output.
Worked Example: A Larger Fraction but Lower Absolute Flow
Take a reference cardiac output of \(5.0\) L/min and a low-output cardiac output of \(3.0\) L/min. Suppose the illustrative fractions are as follows:
| Systemic bed | Reference fraction | Reference flow (L/min) | Low-output fraction | Low-output flow (L/min) |
|---|---|---|---|---|
| Kidneys | 0.20 | 1.00 | 0.25 | 0.75 |
| Brain | 0.14 | 0.70 | 0.18 | 0.54 |
| Coronary circulation | 0.05 | 0.25 | 0.07 | 0.21 |
| Splanchnic beds | 0.25 | 1.25 | 0.20 | 0.60 |
| Skeletal muscle | 0.18 | 0.90 | 0.12 | 0.36 |
| Skin | 0.06 | 0.30 | 0.05 | 0.15 |
| Other systemic beds | 0.12 | 0.60 | 0.13 | 0.39 |
| Total | 1.00 | 5.00 | 1.00 | 3.00 |
For the kidneys, the reference flow is \(0.20\times5.0=1.00\) L/min, while the low-output flow is \(0.25\times3.0=0.75\) L/min. The fraction increases from 0.20 to 0.25, but absolute kidney flow decreases by \(0.25\) L/min, or 25% relative to its reference flow. The increased share partly offsets the fall in total output; it does not overcome it.
The same pattern occurs for the brain and coronary circulation in this hypothetical allocation: their fractions rise, while their absolute flows fall. These numbers are chosen only to demonstrate the arithmetic and are not normal values or a prediction for a particular patient. The table is internally consistent because each row is calculated as fraction times output and the totals match the stated outputs.
The condition for an organ’s absolute flow to be preserved or increased can also be read from the ratio identity. Provided its reference flow is positive, low-output flow is at least as large as reference flow only when the relative increase in its fraction is large enough to offset the relative decrease in output. This is a mathematical threshold, not a claim that the body can or will achieve that redistribution.
Worked Example: How Much Fraction Increase Would Preserve Flow?
Suppose a hypothetical organ receives 20% of a reference output of \(5.0\) L/min, giving a reference flow of \(0.20\times5.0=1.0\) L/min. Output then falls to \(3.0\) L/min. What fraction would be required to keep this organ’s flow at \(1.0\) L/min?
Use \(Q_{i,\mathrm{low}}=f_{i,\mathrm{low}}Q_{\mathrm{CO,low}}\) and set the target flow equal to \(1.0\) L/min: \(1.0=f_{i,\mathrm{low}}\times3.0\). Therefore \(f_{i,\mathrm{low}}=1.0/3.0=0.333\), or about 33.3%.
The fraction would have to rise from 20% to about 33.3%, a relative increase of about \(0.333/0.20=1.67\), or 67%. A rise only to 25% would produce \(0.25\times3.0=0.75\) L/min, not preserve the original \(1.0\) L/min. In an actual allocation, increasing one fraction also constrains the fractions available to the other systemic beds.
Assumptions, Checks, and Limits
These calculations rely on a small set of explicit assumptions. Cardiac output and the fractions must refer to the same state and the same circulation boundary; the flow units must be consistent; and the listed beds must be defined without overlap. If the table is intended to cover all systemic output, its fractions must sum to 1 and its flows to cardiac output, allowing for rounding. A partial organ list need not sum to 1, but its missing share must not be mistaken for a flow-conservation error.
The calculation is an allocation model, not a full physiological explanation. It does not derive fractions from vascular resistance, pressure, autoregulation, or the severity and cause of heart failure. Nor does it establish that a given organ is adequately perfused: a calculated flow is not, by itself, a direct measure of oxygen delivery, tissue function, or drug concentration. Those questions require additional physiological information.
A useful diagnostic is to check three things before interpreting a result: whether the output values describe the stated conditions, whether fractions are on the same basis, and whether the calculated flows add up as expected. If a purported whole-body systemic table has fractions that sum far from 1, first look for omitted beds, overlapping categories, incompatible sources, or arithmetic errors. Rounding can explain small discrepancies, not a large mismatch.
The main alternative calculation is proportional scaling: hold all fractions fixed and multiply each reference flow by the cardiac-output ratio. It is simple and useful as a no-redistribution comparison, but it cannot represent organ-specific changes in allocation. A state-specific fraction table can represent redistribution, but requires justified fractions and careful flow-conservation checks. Neither approach alone explains the physiological mechanisms or the consequences for tissue function.
Check Your Understanding
Answer each question using the stated assumptions. Solutions are included below.
- Output falls from 4.0 to 2.0 L/min while an organ’s fraction remains 0.15. What are its flows in the two states?
- An organ’s fraction rises from 0.10 to 0.20 while output falls from 6.0 to 2.0 L/min. Does its absolute flow rise, fall, or stay the same? Calculate both flows.
- Why can the ratio \(Q_{i,\mathrm{low}}/Q_{i,\mathrm{ref}}\) not be used if the reference organ flow is zero?
- A table is described as covering all systemic beds, but its fractions sum to 0.82. Give two possible explanations to investigate before interpreting it.
Solutions. (1) Reference flow is \(0.15\times4.0=0.60\) L/min; low-output flow is \(0.15\times2.0=0.30\) L/min. (2) Reference flow is \(0.10\times6.0=0.60\) L/min; low-output flow is \(0.20\times2.0=0.40\) L/min, so absolute flow falls despite the doubled fraction. (3) A ratio with a zero denominator is undefined; use the undivided equations for the two states instead. (4) Possible causes include omitted systemic beds, overlapping or inconsistently defined categories, or a measurement or arithmetic problem. A sum of 0.82 is too far from 1.00 to attribute to ordinary rounding alone.