From a Physiological Scenario to a Flow Table
A PBPK organ-flow table is useful only when its numbers describe a coherent physiological scenario. A resting table should not be combined casually with an exercise cardiac output; nor should regional fractions from one condition be multiplied by a cardiac output from another. The table needs to state the condition, the cardiac output assigned to that condition, the regions being represented, and how well the allocation accounts for the total.
The construction uses the relationship between cardiac output, regional flow fractions, and absolute flows. The earlier material on organ blood-flow fractions and absolute flows establishes this relationship; here the emphasis is on assembling and checking a complete table for a chosen scenario.
A fraction is not a flow: \(f_i=0.20\) means that 20% of the specified cardiac output is allocated to region \(i\). Multiplying by \(Q_{CO}=5.0\) L/min gives \(Q_i=1.0\) L/min. If a source gives a percentage, convert it to a fraction before multiplying: 20% is \(0.20\), not \(20\).
The conservation statements apply to rows that partition the same outgoing flow into mutually exclusive destinations. This qualification matters for organs with serial or overlapping supplies. For example, portal venous blood reaches the liver after passing through splanchnic tissues. It is not a second, independent share of cardiac output to add on top of the upstream splanchnic arterial allocation. A table can report such internal flows separately, but its cardiac-output conservation sum must not count the same blood twice.
A Practical Construction and Audit
Start by writing the scenario in words: for example, “healthy adult at rest” or “same hypothetical adult during moderate exercise.” Assign \(Q_{CO}\) for that condition and identify the source and meaning of every regional fraction. Then list regions that form a non-overlapping partition of the flow being allocated. If the listed fractions do not account for all flow, include a residual category or state explicitly that the table is partial.
For each row, calculate \(Q_i=f_iQ_{CO}\). Keep units visible in the calculation: a dimensionless fraction multiplied by L/min gives L/min. Finally, sum the fractions and the absolute flows. The two checks are linked: if \(Q_{CO}\) is common to every row, then
Thus, a fraction sum of 1 implies an absolute-flow sum equal to cardiac output, apart from rounding. If the sum differs, investigate whether a region was omitted, counted twice, or assigned a fraction on a different basis. Do not conceal a discrepancy by changing values without documenting the adjustment.
Record the physiological condition and the cardiac output assigned to it, with units.
Choose mutually exclusive destinations for the conservation sum; identify any internal or overlapping flows separately.
Convert percentages to fractions, calculate \(Q_i=f_iQ_{CO}\), and report the units.
Check \(\sum_i f_i\) and \(\sum_i Q_i\), account for rounding, and record assumptions or residual flow.
Worked Example: A Resting Systemic Allocation
Hypothetical data. A reference adult at rest is assigned \(Q_{CO}=5.0\) L/min. Suppose a deliberately simplified table allocates systemic arterial output to seven mutually exclusive destinations. The regions are defined here as accounting categories; “splanchnic arterial bed” does not include a second row for portal return to the liver.
| Destination | Fraction \(f_i\) | Calculation | Flow \(Q_i\) (L/min) |
|---|---|---|---|
| Brain | 0.14 | 0.14 × 5.0 | 0.70 |
| Myocardium | 0.04 | 0.04 × 5.0 | 0.20 |
| Kidneys | 0.20 | 0.20 × 5.0 | 1.00 |
| Skeletal muscle | 0.17 | 0.17 × 5.0 | 0.85 |
| Skin | 0.06 | 0.06 × 5.0 | 0.30 |
| Splanchnic arterial bed | 0.25 | 0.25 × 5.0 | 1.25 |
| Other systemic destinations | 0.14 | 0.14 × 5.0 | 0.70 |
The fractions sum to \(0.14+0.04+0.20+0.17+0.06+0.25+0.14=1.00\). The flows sum to \(0.70+0.20+1.00+0.85+0.30+1.25+0.70=5.00\) L/min, equal to the assigned cardiac output. The table therefore conserves the specified output. Its values are hypothetical and illustrate bookkeeping; the table alone does not establish that the fractions are appropriate for every adult or every measurement definition.
Changing the Scenario Changes the Table
For a new condition, revisit both the cardiac output and the regional fractions. Some scenarios may change cardiac output substantially; others may primarily redistribute flow, or do both. Do not infer that a region’s absolute flow is unchanged merely because its fraction rises: \(Q_i\) depends on both \(f_i\) and \(Q_{CO}\).
Worked Example: Exercise With Redistribution
Hypothetical data. During a specified exercise condition, assign \(Q_{CO}=15.0\) L/min and use the following mutually exclusive fractions: brain 0.04, myocardium 0.05, kidneys 0.05, skeletal muscle 0.60, skin 0.10, splanchnic arterial bed 0.10, and other systemic destinations 0.06. These fractions are stipulated for this example, not offered as universal exercise values.
The corresponding calculations are: brain, \(0.04(15.0)=0.600\) L/min; myocardium, \(0.05(15.0)=0.750\) L/min; kidneys, \(0.05(15.0)=0.750\) L/min; skeletal muscle, \(0.60(15.0)=9.00\) L/min; skin, \(0.10(15.0)=1.50\) L/min; splanchnic arterial bed, \(0.10(15.0)=1.50\) L/min; and other destinations, \(0.06(15.0)=0.900\) L/min.
The fractions sum to \(0.04+0.05+0.05+0.60+0.10+0.10+0.06=1.00\). The flows sum to \(0.600+0.750+0.750+9.00+1.50+1.50+0.900=15.00\) L/min. For comparison, the resting example assigned 0.85 L/min to skeletal muscle, whereas this exercise scenario assigns 9.00 L/min. Both the higher cardiac output and the larger muscle fraction contribute to that difference.
Worked Example: Low Cardiac Output With a Residual Category
Hypothetical data. Consider a low-output disease scenario with \(Q_{CO}=3.6\) L/min. Suppose the available regional fractions are brain 0.18, myocardium 0.06, kidneys 0.16, skeletal muscle 0.15, and skin 0.05. Their sum is \(0.60\), so the listed regions account for only 60% of the assigned cardiac output. To create a complete accounting table, define “unallocated remainder” as \(1.00-0.60=0.40\). This residual is bookkeeping, not a physiological claim about one specific organ.
The absolute flows are brain, \(0.18(3.6)=0.648\) L/min; myocardium, \(0.06(3.6)=0.216\) L/min; kidneys, \(0.16(3.6)=0.576\) L/min; skeletal muscle, \(0.15(3.6)=0.540\) L/min; skin, \(0.05(3.6)=0.180\) L/min; and remainder, \(0.40(3.6)=1.44\) L/min.
The fraction check is \(0.18+0.06+0.16+0.15+0.05+0.40=1.00\). The flow check is \(0.648+0.216+0.576+0.540+0.180+1.44=3.600\) L/min. The table conserves the assigned output, while the residual makes clear that the supplied regional information was incomplete. If the scientific question requires naming the remainder, more information is needed; it should not be assigned to a particular tissue without justification.
Checks, Alternatives, and Limits
A useful audit examines more than the final total. Confirm that all fractions refer to the same scenario and cardiac-output basis, that percentages were converted correctly, and that each row has a clear region and unit. A sum near 1 may still be misleading if overlapping destinations were included. Conversely, a sum such as 0.98 may reflect rounding in reported fractions rather than a true two-percent unaccounted flow. State which explanation applies when it is known.
There are two common ways to populate the table. A fraction-based allocation starts with \(Q_{CO}\) and \(f_i\), then calculates \(Q_i\). It makes the conservation relationship explicit and is convenient when the model needs a coherent complete allocation. A direct-flow table starts with measured or otherwise specified \(Q_i\) values and sums them. This can preserve the measurements as reported, but the total may not match an independently reported \(Q_{CO}\), owing to measurement uncertainty, timing, definitions, or incomplete coverage. The difference should be reported and investigated, not erased by an undocumented rescaling.
When fractions are rounded, the displayed sum can differ slightly from 1 even if the underlying values sum to 1. For example, calculations using unrounded fractions may conserve flow while a table printed to two decimal places appears not to. Keep adequate precision during calculation, show a sensible number of significant figures in the final table, and distinguish rounding discrepancies from substantive mismatch. If fractions are intentionally normalized for a model allocation, state that choice and retain the original values for traceability.
A flow table is a scenario-specific input, not a guarantee that regional perfusion is spatially uniform or that a single value describes every individual. Disease and posture can produce heterogeneous changes within a named region, and an average regional flow may hide that variation. In addition, conservation validates the accounting, not the physiological accuracy of the inputs. The source, population, measurement condition, and intended model use remain important.
Check Your Understanding
Use the definitions and accounting rules above. Answers are provided after the questions for self-checking.
- If \(Q_{CO}=4.8\) L/min and a region has fraction \(0.125\), what is its absolute flow?
- A table’s listed fractions sum to 0.92. What is one transparent way to make the accounting complete, and what does the residual mean?
- Why should portal venous flow not automatically be added as a separate cardiac-output destination on top of its upstream splanchnic supply?
- For a scenario with \(Q_{CO}=6.0\) L/min, fractions 0.10, 0.25, and 0.65 are assigned to three exclusive regions. Calculate the flows and check conservation.
- Two fraction-based tables have the same fraction for a region, but one scenario has twice the cardiac output. How do their absolute flows for that region compare?
Solutions. 1. \(Q_i=0.125(4.8)=0.600\) L/min. 2. Add a clearly named residual fraction of \(1.00-0.92=0.08\), or report the table as partial. The residual represents flow not allocated to the listed regions; it is not automatically a specific organ. 3. Portal flow is internal recirculation from upstream splanchnic tissues to the liver, so counting both it and the upstream allocation as independent output destinations double counts blood. 4. The flows are \(0.10(6.0)=0.600\), \(0.25(6.0)=1.50\), and \(0.65(6.0)=3.90\) L/min. Their fractions sum to 1.00 and their flows sum to 6.00 L/min. 5. The absolute flow in the second scenario is twice as large, because \(Q_i=f_iQ_{CO}\) and the fraction is unchanged while cardiac output doubles.