Scaling Organ Volumes Across Body Sizes

Why Organ-Volume Scaling Matters

A PBPK model represents the body as organs and tissues with volumes that determine the spaces available for drug distribution. When the modeled individual differs in body size from a reference individual of the same species, an organ volume may need to be adjusted. The key question is whether that volume stays approximately constant, changes in direct proportion to body mass, or follows another systematic relationship.

These choices are not interchangeable. If an organ grows less rapidly than body mass, assuming proportional growth will overpredict its volume in larger individuals. If it grows more rapidly, the same assumption will underpredict it. The relationship is an empirical and physiological modeling choice, not a universal rule for every organ.

The power law includes the first two forms as special cases: \(b_i=0\) gives a fixed volume, and \(b_i=1\) gives proportional volume. An exponent between zero and one means that volume increases with body mass, but less than proportionally. An exponent greater than one means it increases more than proportionally. These statements describe the fitted mathematical relationship over its intended range; they do not establish a biological mechanism.

Assumptions and a Reference-Anchored Form

Any scaling rule is conditional on its setting. The individuals being compared should be from the same species, and the organ should be defined and measured on a consistent basis. The relationship should apply to a reasonably comparable physiological population and state, rather than combining groups with systematically different conditions. Body mass must use the same units throughout, and organ volume must use the same units throughout. The rule also assumes that the chosen functional form is an adequate approximation over the body-mass range where it will be used.

A convenient way to use a known reference is to define \(M_{\mathrm{ref}}\) as reference body mass and \(V_{i,\mathrm{ref}}\) as the corresponding organ volume. For a power law, dividing the value at \(M\) by the value at the reference gives:

This form is useful because the mass ratio is dimensionless, and the reference volume supplies the volume units. It also makes the special cases transparent. If \(b_i=0\), the ratio is one and the volume does not change. If \(b_i=1\), the volume ratio equals the body-mass ratio, so the organ maintains a constant fraction of body mass on this volume-per-mass scale.

Deriving the Exponent From Two Observations

Suppose two hypothetical observations from the same species give organ volumes \(V_1\) and \(V_2\) at body masses \(M_1\) and \(M_2\). Under the power-law assumption, \(V_1=aM_1^b\) and \(V_2=aM_2^b\), where the same coefficient and exponent apply to both observations. Dividing cancels the unknown coefficient:

Taking logarithms of both sides and solving yields the exponent. The logarithms are of dimensionless ratios, so no logarithm of a quantity with units is required.

Once \(b\) is obtained, the scale coefficient follows from either observation: \(a=V_1/M_1^b\). With only two observations, this calculation forces a power curve through both points; it does not assess measurement variability or show that a power law is the best description. With more observations, the relationship should be checked against all of them rather than inferred from a selected pair alone.

Comparing Fixed, Proportional, and Nonlinear Scaling

The following hypothetical values illustrate how alternative assumptions produce different predictions. Suppose an organ volume is 0.350 L at a reference body mass of 20 kg, and consider a target body mass of 35 kg. For illustration, let a plausible fitted power-law exponent be \(b=0.8\).

In practice, a fixed value may be a reasonable approximation over a narrow range if there is little relevant growth or size-related change. Proportional scaling is simple and interpretable when the organ maintains a roughly constant fraction of body mass. A power law permits that fraction to change systematically with size. A table of measured volumes at several body sizes is another option when the data support discrete values, but interpolation and behavior between entries still need a defined rule. No approach should be selected merely because it is convenient.

Diagnostics, Limits, and Common Pitfalls

A scaling relationship should be checked against the data and the intended PBPK use. For a proportional model, inspect the organ-volume-to-body-mass ratio \(V_i/M\): if it changes systematically with body mass, a constant fraction may be inadequate. For a power law, plotting \(\log(V_i)\) against \(\log(M)\) should show an approximately linear pattern over the fitted range; systematic curvature suggests that one exponent may not describe the relationship well. Predictions should also remain positive and physiologically plausible, and should not imply an organ volume that is impossible in the context of the individual.

An unusual fitted exponent can signal a data or modeling problem, but should not be rejected automatically: the measurement range, organ definition, and population composition matter. Check for inconsistent units, mismatched measurement bases, transcription errors, influential observations, and an unnecessarily broad range of body sizes. If observations are variable, two points are not enough to establish a stable exponent. With more data, inspect residuals and uncertainty in the estimated relationship; a good-looking exponent alone is not a diagnostic.

The power law is most defensible within the range represented by the observations. Far outside that range, even a well-fitting curve may give unrealistic predictions. A relationship can also change across developmental stages or physiological states, so data from unlike groups should not be combined without justification. The within-species exponent described here is not a rule for translating organ volumes between species; that is a separate scaling problem.

Common mistake: A modeler observes that body mass doubles and assumes every organ volume must double. That reasoning silently imposes \(b=1\), which is a proportional-scaling assumption, not a consequence of body mass changing. The correction is to use evidence appropriate to the organ and species, state the assumed relationship, and examine whether the resulting predictions are plausible in the intended range.

Another frequent error is to treat a power-law coefficient as unitless. In \(V_i=aM^b\), the coefficient has units of volume times mass to the power \(-b\), such as \(\mathrm{L\,kg}^{-b}\). The reference-anchored equation avoids carrying those coefficient units through each prediction, but it still requires consistent volume and mass units.