Joint Constraints on Organ Volumes and Blood Flows

Why Organ Volumes and Flows Must Be Checked Together

A PBPK parameter set can contain individually plausible organ volumes and blood flows yet still describe no coherent physiology. Organ volumes must fit within the modeled body, and systemic organ perfusion must be consistent with cardiac output. In addition, a flow assigned to an organ should be considered in relation to that organ’s volume and to how its blood supply is represented. These are joint constraints: passing one check does not imply passing the others.

The audit is meaningful only when the inputs refer to the same population or individual, physiological state, and unit system. It also depends on a clear compartment definition. A PBPK tissue compartment usually represents the tissue as a whole, including its local vascular space; adding a separate whole-body blood volume to those tissue volumes can therefore double count volume. Model conventions differ, so determine what each volume includes before calculating a total.

The volume check assumes that compartments are nonoverlapping and collectively represent the declared body-volume target. The flow check assumes a closed circulation at steady state, with no unmodeled shunt, net accumulation, or mismatch in measurement state. These assumptions should be reviewed rather than silently presumed. If the parameter set represents only selected organs, the remainder must be identified; the listed organs alone need not sum to the whole body.

Writing the Joint Constraints

Let \(V_i\) denote the volume of modeled compartment \(i\), and let \(V_{\mathrm{body}}\) denote the body-volume target under the model’s convention. If the compartments form an exhaustive, nonoverlapping partition, the accounting condition is:

If only selected compartments are listed, their sum should not exceed the declared target, and the unrepresented volume should be assigned to a defined remainder rather than treated as unexplained slack. If the input is body mass rather than body volume, do not equate kilograms and liters without stating the density assumption or conversion convention.

For systemic perfusion, let \(Q_i\) be the flow assigned to each mutually exclusive systemic destination. Under steady-state circulation, conservation of flow gives:

The pulmonary circulation is not an additional systemic destination to add to this sum: in a closed circulation, pulmonary flow also equals cardiac output. Likewise, blood entering an organ through a secondary route must not be counted as a second independent contribution from the heart if that blood has already been counted upstream.

Volume and flow totals are necessary but not sufficient. For each organ with a defined perfusion, calculate its flow per unit volume:

Here \(q_i\) has units such as L/min/L. This ratio can expose a misassigned flow or volume that is hidden by correct totals. It is not a universal constant or a stand-alone pass/fail threshold: expected values depend on organ, species, age, physiological state, and the definition of perfusion used. Compare it with an appropriate source or model-specific reference, not with an assumed universal ratio.

A Short Derivation of the Flow Balance

At steady state, consider the systemic circulation as a control volume. Blood leaving the heart and entering systemic tissues is \(Q_{\mathrm{CO}}\). If there is no net blood accumulation in the circulation, total systemic outflow from those tissues must return to the heart at the same rate. With each systemic destination counted once, summing its inflow gives the cardiac-output constraint. The equality is therefore a conservation statement, not a claim that all organs receive the same flow.

This derivation also explains why the definition of an “organ flow” matters. For example, portal blood is first delivered to portal-drained tissues and then passes to the liver. It is a flow to the liver tissue, but it is not a second independent delivery from the heart. A consistent accounting scheme can sum primary systemic destinations, or use another explicitly defined scheme; it cannot mix the two by counting the portal flow both before and after its passage through the portal system.

Portal Flow and Other Accounting Traps

The portal system is a particularly important source of double counting. A model may report blood flow to the gut and also report total hepatic inflow, which includes portal venous return. Those values describe different stages of blood’s route, not two independent cardiac deliveries. When checking total systemic flow, use one internally consistent accounting convention and document which flows are included.

Balanced Totals Can Still Hide Implausible Assignments

Passing both sum checks does not validate the allocation among organs. One organ can be assigned too little flow while another receives too much, leaving cardiac output unchanged. Similarly, an organ volume can be too large while a remainder compartment is too small, preserving the whole-body total. A joint audit therefore has at least three stages: check definitions and coverage, check sums, and then inspect organ-specific relationships.

The following example uses an explicitly hypothetical project reference. It is not a universal physiological range. Its purpose is to show how to interpret a discrepancy and why an arithmetic adjustment alone does not establish the correct physiology.

Practical Audit Sequence and Failure Modes

A useful audit separates hard accounting identities from reference-dependent plausibility checks. The first two are usually exact within the declared model convention; organ-specific comparisons depend on suitable source data and matching definitions.

A common mistake is to regard a balanced sum as proof that the parameter set is plausible. The reasoning fails because summation can conceal compensating errors: one flow may be too low and another too high. The correction is to check both the totals and organ-specific flow-volume relationships. The converse mistake is to “fix” a sum by arbitrarily changing one organ. A numerical correction is not evidence of a correct physiological value.

Another failure mode is comparing values that use different boundaries—for example, a reported liver flow that includes portal inflow against a reference for hepatic arterial flow alone. A mismatch may then be definitional rather than physiological. Diagnostic checks should therefore record both the number and its meaning.

Choosing an Appropriate Comparison

Independent range checks are useful as an initial screen: they can quickly identify a negative volume, an implausible unit, or a flow exceeding the declared cardiac output. They do not enforce joint accounting. A constrained accounting audit adds the whole-body and cardiac-output equalities, while organ-specific comparisons test whether the allocation is credible. These approaches complement one another rather than compete.

When a set fails a joint equality, first check whether its components were defined consistently. If definitions are sound, the discrepancy may indicate missing compartments, incompatible data sources, or an incorrect input. A constrained fitting or reallocation can produce values that satisfy a sum, but it cannot determine which organ value is biologically correct without additional evidence. The audit identifies inconsistency; it does not automatically identify the true replacement value.