Why Compare Physiological Sets?
A PBPK model represents the body through physiological inputs such as tissue volumes and blood flows. Comparing two age-specific input sets makes their differences explicit and helps reveal whether those differences are internally consistent. It is also important to distinguish a change in an input from a change in a drug concentration or exposure: the former is a model description, while the latter is a prediction that depends on the drug and the rest of the model.
This worked comparison uses a hypothetical young adult and hypothetical older adult. The values are chosen for teaching calculations; they are not reference values for either age group. The comparison is descriptive. It does not establish that age alone caused the differences, nor does it represent the variability among people of the same age.
Set Up a Consistent Comparison
The example compares adults of the same sex and similar height, but it does not attempt to hold body mass constant. That choice makes body composition changes visible rather than concealing them through an artificial matching constraint. Each body-composition category is treated as mutually exclusive, so the listed masses must add to total body mass. Organ volumes are listed separately and are not added to body mass. Tissue blood flows are defined as non-overlapping model compartments, so their sum must equal cardiac output.
The hypothetical body-composition inputs are:
| Input | Young adult | Older adult |
|---|---|---|
| Total body mass | 70 kg | 68 kg |
| Lean mass | 52 kg | 46 kg |
| Adipose mass | 14 kg | 18 kg |
| Other mass | 4 kg | 4 kg |
The mass totals are internally consistent: \(52+14+4=70\) kg for the young adult and \(46+18+4=68\) kg for the older adult. “Other mass” is included to make clear that lean and adipose mass need not exhaust total body mass under the stated model definitions.
Worked Example: Comparing Body Composition
Calculate the absolute and relative changes in lean and adipose mass. Then calculate each category’s fraction of total body mass in both sets.
Lean mass. The absolute change is \(46-52=-6\) kg. Relative to the young-adult value, the change is \(100(-6/52)=-11.54\%\). Thus, the older-adult set has 6 kg less lean mass, an 11.54% decrease relative to the young-adult set.
Adipose mass. The absolute change is \(18-14=4\) kg. The relative change is \(100(4/14)=28.57\%\). The older-adult set has 4 kg more adipose mass, a 28.57% increase relative to the young-adult set.
Fractions of total mass. For the young adult, lean mass is \(52/70=0.7429\), or 74.29% of body mass; adipose mass is \(14/70=0.2000\), or 20.00%. For the older adult, the corresponding fractions are \(46/68=0.6765\), or 67.65%, and \(18/68=0.2647\), or 26.47%.
The lean-mass fraction falls by \(67.65-74.29=-6.64\) percentage points, while the adipose fraction rises by \(26.47-20.00=6.47\) percentage points. These are percentage-point differences between fractions, not relative percentage changes in mass. The distinction matters because the denominators differ: total body mass is 70 kg in one set and 68 kg in the other.
Organ Volumes: Compare Like With Like
For organ volume, the same relative-change formula applies, with volume as the unit. The table below gives hypothetical values in litres. It includes several organs but is not intended to be a complete list of all PBPK compartments.
| Organ | Young adult (L) | Older adult (L) |
|---|---|---|
| Liver | 1.50 | 1.35 |
| Kidneys, combined | 0.30 | 0.25 |
| Brain | 1.40 | 1.35 |
| Heart | 0.30 | 0.28 |
Worked Example: Calculating Organ-Volume Changes
For each listed organ, calculate the absolute change in litres and the relative change from the young-adult value.
For the liver, the absolute change is \(1.35-1.50=-0.15\) L. The relative change is \(100(-0.15/1.50)=-10.00\%\).
For the combined kidneys, the absolute change is \(0.25-0.30=-0.05\) L, and the relative change is \(100(-0.05/0.30)=-16.67\%\).
For the brain, the absolute change is \(1.35-1.40=-0.05\) L, and the relative change is \(100(-0.05/1.40)=-3.571\%\).
For the heart, the absolute change is \(0.28-0.30=-0.02\) L, and the relative change is \(100(-0.02/0.30)=-6.667\%\).
In this illustrative set, the largest relative decrease among these organs is for the combined kidneys, although its absolute decrease is tied with the brain at 0.05 L. Reporting both quantities prevents a small organ from seeming unimportant merely because its absolute volume change is small, while also avoiding the implication that a large percentage change necessarily represents a large change in litres.
Tissue Perfusion and Flow Conservation
Tissue blood flow \(Q_i\) is the volume of blood delivered to compartment \(i\) per unit time. The compartment perfusion rate per unit tissue volume is \(Q_i/V_i\), where \(V_i\) is that compartment’s volume. These are distinct quantities: total flow can fall while flow per unit volume stays similar, or vice versa. For a consistent flow allocation, the modeled compartments must be defined without overlap and their flows must sum to cardiac output.
| Model compartment | Young flow (L/min) | Older flow (L/min) |
|---|---|---|
| Liver, total hepatic flow | 1.25 | 1.00 |
| Kidneys, combined | 0.90 | 0.72 |
| Brain | 0.70 | 0.68 |
| Skeletal muscle | 1.00 | 0.85 |
| Skin | 0.35 | 0.25 |
| Adipose tissue | 0.10 | 0.12 |
| Other modeled tissues | 0.70 | 0.88 |
| Total cardiac output | 5.00 | 4.50 |
“Liver, total hepatic flow” is one model flow here; it must not be added to separate portal and hepatic-arterial flows if those are already included in the total. This avoids double-counting. The table’s other compartments are likewise defined as non-overlapping for this allocation.
Worked Example: Checking Flow and Calculating Perfusion
First verify flow conservation. For the young adult, the compartment flows sum to \(1.25+0.90+0.70+1.00+0.35+0.10+0.70=5.00\) L/min, equal to the stated cardiac output. For the older adult, they sum to \(1.00+0.72+0.68+0.85+0.25+0.12+0.88=4.50\) L/min, also equal to the stated cardiac output. Both sets therefore satisfy the specified flow-balance check.
The older-adult cardiac output is \(4.50-5.00=-0.50\) L/min lower, a relative change of \(100(-0.50/5.00)=-10.00\%\). The kidney flow decreases by \(0.72-0.90=-0.18\) L/min, or \(100(-0.18/0.90)=-20.00\%\). Adipose-tissue flow instead increases by \(0.12-0.10=0.02\) L/min, or \(100(0.02/0.10)=20.00\%\).
For the liver, the perfusion rate per unit volume is \(1.25/1.50=0.8333\) L/min/L in the young-adult set and \(1.00/1.35=0.7407\) L/min/L in the older-adult set. Its absolute change in this rate is \(0.7407-0.8333=-0.09259\) L/min/L, and its relative change is \(100(-0.09259/0.8333)=-11.11\%\). The total hepatic flow and liver volume both decrease, but by different relative amounts; therefore, flow per unit volume also decreases.
Why the Ratios Change: A Short Justification
Perfusion per unit volume is a ratio, so its relative change is not generally equal to the relative change in flow or volume alone. Write the older-to-young ratio for perfusion as
For the liver, \(Q_{\text{older}}/Q_{\text{young}}=1.00/1.25=0.8000\), while \(V_{\text{older}}/V_{\text{young}}=1.35/1.50=0.9000\). Thus the perfusion ratio is \(0.8000/0.9000=0.8889\): older-adult perfusion per unit volume is 88.89% of the young-adult value, a decrease of 11.11%. This ratio calculation justifies why simply subtracting the percentage decrease in volume from the percentage decrease in flow would not give the exact percentage change in perfusion.
How to Interpret and Check the Comparison
A useful comparison separates three questions: how much an input changes in its original units, how large that change is relative to a stated reference, and whether the full set remains physiologically and mathematically coherent. A positive or negative difference alone does not establish whether a PBPK prediction will rise or fall. That prediction also depends on drug-specific properties, the model structure, and the interactions among physiological inputs.
The principal check here is flow conservation. A mismatch between the sum of compartment flows and cardiac output signals an inconsistency in the allocation, a unit error, or possible double-counting. Body-composition masses should also sum to total body mass under the chosen definitions. Units need attention: organ volumes are in L, flows in L/min, and perfusion per unit volume in L/min/L. A value of \(Q_i/V_i\) is not a total flow.
A second check is the denominator behind every percentage. Relative changes in mass use the young-adult mass as the reference. Changes in body-mass fractions, by contrast, are best described as percentage-point differences. If different reference values or definitions are used, apparently conflicting percentages may both be correct but answer different questions.
Common Pitfalls and Alternative Comparisons
- Attributing the contrast to aging alone. The example contains only two hypothetical sets. It cannot separate age effects from differences in body size, health, or other covariates. Describe it as a comparison, not a causal estimate.
- Confusing percentage change with percentage points. The lean-mass fraction changes from 74.29% to 67.65%, a difference of \(-6.64\) percentage points. That is not the same statement as an 11.54% decrease in lean mass.
- Checking flows without checking compartment definitions. A sum that matches cardiac output is necessary for this allocation but does not prove that the partition is physiologically appropriate. Overlapping compartments can produce a misleading total.
- Applying a change in total flow uniformly without justification. If every tissue flow is scaled by the same factor, tissue flow fractions remain unchanged. The example does not make that assumption: the allocations change unevenly, and the remaining flow is explicitly assigned to other modeled tissues.
Absolute differences are most useful when the practical question concerns units such as litres or kilograms. Relative changes help compare inputs of different magnitudes, but depend on the selected reference. A third approach is to compare flow fractions, \(Q_i/\mathrm{CO}\), which describes how cardiac output is distributed rather than the amount delivered per minute. None of these summaries replaces the others. For instance, an organ’s flow fraction may rise while its absolute flow falls if total cardiac output falls enough.
These calculations also have limitations. The values are deterministic inputs, not estimates with uncertainty intervals; they do not describe within-age variability. Organ volume does not identify functional capacity, and perfusion per unit volume does not by itself determine drug clearance. In addition, physiological definitions and measurement methods must be compatible before values from different sources are combined. A coherent arithmetic comparison is a useful model-building check, not proof that the inputs are clinically representative.
Check Your Understanding
Use the formulas and conventions from the comparison. The solutions are included so each calculation can be checked.
- A tissue volume changes from 0.40 L to 0.34 L. What are its absolute and relative changes?
- A model compartment receives 0.60 L/min in the young set and 0.54 L/min in the older set. Calculate the relative flow change.
- Cardiac output is 4.8 L/min and the listed non-overlapping compartment flows sum to 4.6 L/min. Does the allocation pass the flow-conservation check, and what is the discrepancy?
- A tissue’s flow decreases by 10%, while its volume decreases by 20%. Does perfusion per unit volume increase or decrease? Calculate the relative change in the ratio.
Solutions. (1) The absolute change is \(0.34-0.40=-0.06\) L. The relative change is \(100(-0.06/0.40)=-15.00\%\). (2) The relative change is \(100[(0.54-0.60)/0.60]=-10.00\%\). (3) No. The listed flows are \(4.6-4.8=-0.2\) L/min below cardiac output, so 0.2 L/min remains unallocated or the inputs are inconsistent. (4) The older-to-young ratio for flow is 0.90 and for volume is 0.80. The perfusion ratio is \(0.90/0.80=1.125\), so perfusion per unit volume increases by 12.50%.