Why Organ Size and Blood Flow Must Be Considered Together
Age-related physiological change is not a single scaling rule. Some organs may become smaller with age, while the change in blood flow to an organ may be smaller, larger, or otherwise different from the change in its tissue mass. In a physiologically based pharmacokinetic model, these distinctions matter: organ volume contributes to the amount of tissue available for distribution, while blood flow describes a rate of delivery and removal.
The key comparison is therefore not just between an older and a younger organ’s volume or total flow. It is also between their flow per unit tissue. An organ can receive less blood in total but more blood per gram if its mass falls by a still larger proportion. Conversely, an organ can shrink while its blood flow is preserved, in which case tissue-specific flow rises.
Age-related changes vary across organs and individuals. The equations below define a way to represent and check those changes; they do not assert that every organ atrophies or that a particular direction or magnitude of change applies universally. Values in the worked examples are hypothetical.
Definitions and the Relative-Change Identity
If tissue mass is not measured directly, volume may be used to derive mass when a suitable tissue density \(\rho_i\) is specified: \(M_i(a)=\rho_i V_i(a)\). Thus, flow per unit volume, \(Q_i/V_i\), and flow per unit mass, \(Q_i/M_i\), are not interchangeable unless density is accounted for. Keep the denominator and its units explicit.
For a reference age \(a_0\) and a comparison age \(a_1\), define the flow and mass ratios as \(R_Q=Q_i(a_1)/Q_i(a_0)\) and \(R_M=M_i(a_1)/M_i(a_0)\). Dividing the definition of tissue-specific flow at the two ages gives a useful identity.
This identity is exact, provided the quantities refer to the same organ and compatible flow and mass definitions at both ages. It shows why subtracting percentage changes is only an approximation: the ratio of flow per tissue depends on the ratio of the two relative changes.
Worked Examples: Calculating Tissue-Specific Flow
Worked Example: Flow Falls Less Than Organ Mass
Suppose a hypothetical reference adult has a liver tissue mass of 1.50 kg and liver blood flow of 1.20 L/min. In a hypothetical older-adult model, the corresponding values are 1.20 kg and 1.08 L/min. Calculate absolute changes and flow per unit mass.
Convert masses to grams and flow to millilitres per minute. At the reference age, \(M=1{,}500\) g and \(Q=1{,}200\) mL/min, so
At the older age, \(M=1{,}200\) g and \(Q=1{,}080\) mL/min. Therefore,
Mass has fallen by \(300/1{,}500=20.0\%\), while absolute flow has fallen by \(120/1{,}200=10.0\%\). Tissue-specific flow has risen by \((0.900/0.800)-1=12.5\%\). The organ receives less blood per minute overall, but more blood per gram of remaining tissue. This result is a calculation from the hypothetical values, not a general claim about aging liver.
Worked Example: Absolute Flow and Cardiac-Output Fraction
Consider a hypothetical kidney with tissue mass 300 g, absolute flow 1.00 L/min, and cardiac output 5.00 L/min at the reference age. For an older-age model, suppose mass is 240 g, absolute flow is 0.840 L/min, and cardiac output is 4.20 L/min. Calculate the flow fraction and tissue-specific flow at both ages.
The reference kidney receives \(1.00/5.00=0.200\), or 20.0%, of cardiac output. Its tissue-specific flow is
At the older age, its flow fraction is \(0.840/4.20=0.200\), also 20.0%. Its tissue-specific flow is
Absolute kidney flow has fallen by 16.0%, the same proportion as cardiac output; the kidney’s share of cardiac output is unchanged. Kidney mass has fallen by 20.0%, so tissue-specific flow has risen by \(3.500/3.333-1=5.00\%\). The three statements describe different quantities and are all consistent.
Preserving Flow Accounting Across Organs
In a systemic model, organ flows should be compatible with the cardiac output represented by the model. For a set of non-overlapping systemic tissue beds, their flows should sum to the total flow being allocated, subject to the model’s explicit treatment of any omitted or separately represented pathways. The flow fraction for organ \(i\) is \(Q_i/Q_{\mathrm{CO}}\), where \(Q_{\mathrm{CO}}\) is cardiac output. A flow fraction is not absolute flow: it can stay constant while both organ flow and cardiac output decline.
For organs with multiple inflows, be clear about the quantity being compared. For example, if a model represents more than one vascular supply to an organ, total organ inflow is the sum of those supplies when calculating total organ flow per tissue mass. Do not compare one component of inflow at one age with total inflow at another.
Worked Example: Checking an Age-Specific Flow Allocation
A hypothetical reference model has cardiac output 5.00 L/min, allocated as 1.20 L/min to organ A, 0.800 L/min to organ B, and 3.00 L/min to all other represented tissues. An older-age model has cardiac output 4.50 L/min, allocated as 0.900, 0.720, and 2.88 L/min, respectively. Check the totals and compare flow fractions.
The reference flows sum to \(1.20+0.800+3.00=5.00\) L/min. The older-age flows sum to \(0.900+0.720+2.88=4.50\) L/min, so both allocations match their stated cardiac outputs.
Organ A’s reference flow fraction is \(1.20/5.00=0.240\), or 24.0%; its older-age fraction is \(0.900/4.50=0.200\), or 20.0%. Its absolute flow falls by \((0.900/1.20)-1=-25.0\%\). Organ B’s fraction is \(0.800/5.00=0.160\), or 16.0%, at the reference age and \(0.720/4.50=0.160\), or 16.0%, at the older age. Its absolute flow falls by 10.0%, matching the decline in cardiac output.
This allocation illustrates that organ flow fractions can change differently even when total flows remain correctly balanced. It does not identify the biological cause of the changes; it only checks their arithmetic and their relation to the stated cardiac outputs.
Encoding Age-Associated Changes and Checking Assumptions
A practical PBPK representation starts by defining the reference physiology, the ages or age range of interest, and the source or rationale for each age-dependent quantity. Organ volume, tissue mass, cardiac output, and organ flow should not be treated as interchangeable inputs. If a model specifies volume but the tissue-specific flow calculation uses mass, the density conversion must be explicit. If the model uses a directly specified age-specific \(Q_i\), do not also apply a flow-scaling factor unless that second adjustment has a distinct, justified meaning.
The calculation assumes that the compared quantities use compatible anatomical boundaries and measurement conventions. It also assumes that \(Q_i\) is flow to the same tissue mass represented by \(M_i\). If an apparent volume difference reflects a different organ definition, unmeasured tissue, or a change in the amount of non-parenchymal material, interpreting \(Q_i/M_i\) as perfusion of equivalent tissue may be misleading. The ratio is still mathematically calculable, but its physiological meaning depends on the denominator.
Age can be represented by age-specific values or by functions of age. Either approach requires evidence or a clearly identified modeling assumption. A single comparison between two age groups does not establish a smooth trajectory across all ages. Likewise, an age-associated difference in observational data should not automatically be described as an effect caused by aging; cohort composition and other physiological differences may contribute. The earlier material on age-related changes in body composition helps distinguish body-composition scaling from organ-specific changes, but neither alone supplies a universal organ-flow rule.
Name the reference and comparison ages or groups, and state whether each quantity is an observed value, a population summary, or a model assumption.
Check that organ boundaries, tissue mass or volume, and flow refer to compatible quantities. Convert units before calculating \(Q_i/M_i\).
Report the ratios or percentage changes in organ size, absolute flow, and tissue-specific flow rather than using one as a substitute for another.
Compare each organ flow with cardiac output and verify that the modeled flow allocation sums appropriately under the model’s stated accounting.
Diagnostics, Alternatives, and Limitations
A useful diagnostic is to compute the relative-change identity independently of the direct tissue-specific-flow calculation. If mass falls to 80% of reference and flow falls to 90%, the implied tissue-specific-flow ratio is \(0.90/0.80=1.125\). A direct calculation should give the same ratio, apart from rounding. Disagreement suggests a unit conversion, denominator, or transcription error.
Other warning signs include a flow total that exceeds the stated cardiac output without an explicit explanation; a flow fraction calculated using a different cardiac-output value than the one used for the organ flows; or an apparent increase in tissue-specific flow that is reported as an increase in absolute organ flow. A mathematically positive \(Q_i/M_i\) does not by itself establish physiological plausibility. Compare inputs with appropriate age-specific evidence and check that the model’s definitions support the interpretation.
There are several modeling choices, each with different assumptions. Keeping adult organ size and flow fixed across ages is simple, but assumes no age-related change in those parameters. Scaling organ mass while holding absolute flow fixed implies that flow per unit mass changes inversely with mass. Scaling flow in direct proportion to mass holds flow per unit mass constant, an assumption rather than a consequence of atrophy. Specifying age-specific mass and flow separately allows both to change independently, but requires support for both inputs and careful flow accounting. Reporting all three quantities makes these assumptions visible.
Finally, organ-level flow per gram is an average over the modeled tissue. It does not describe local heterogeneity, microvascular distribution, or every cellular region. Nor does a change in organ flow alone determine drug exposure: exposure also depends on other model processes and parameters. Treat \(Q_i/M_i\) as a useful physiological summary and model input, not as a complete prediction of pharmacokinetics.
Common Mistake
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A hypothetical organ’s mass falls from 500 g to 400 g, while its flow falls from 1.00 L/min to 0.900 L/min. What happens to flow per gram?
- Cardiac output and an organ’s absolute flow both fall by 12%. What happens to the organ’s fraction of cardiac output?
- Why is flow per unit volume not necessarily the same as flow per unit mass?
- A model scales organ flow in direct proportion to organ mass. What assumption does this make about flow per unit mass?
- Name two checks that can identify an inconsistent age-specific flow input.
Solutions. (1) Reference flow per mass is \(1{,}000/500=2.00\) mL/min/g. The later value is \(900/400=2.25\) mL/min/g, an increase of 12.5%, even though absolute flow fell by 10.0%. (2) The fraction is unchanged because the numerator and denominator are multiplied by the same ratio: \(0.88Q_i/(0.88Q_{\mathrm{CO}})=Q_i/Q_{\mathrm{CO}}\). (3) Mass and volume differ by tissue density, so \(Q/V\) and \(Q/M\) have different units and are equal only under the appropriate density conversion. (4) It assumes flow per unit mass remains constant, since proportional changes cancel in \(Q/M\). (5) For example, verify that organ flows sum to the stated cardiac output under the model’s accounting, and independently compare direct \(Q/M\) calculations with the ratio identity \(R_Q/R_M\).