Choosing Between Weight, BMI, BSA, and Fat-Free Mass

Why the Choice of Size Descriptor Matters

Body-size covariates are not interchangeable labels for the same quantity. Total weight measures the mass of the whole body; body-mass index (BMI) relates weight to height squared; body-surface area (BSA) combines weight and height; and fat-free mass (FFM) represents body mass excluding fat. Each descriptor therefore encodes a different view of size. If a population PK model uses one of them to scale clearance or volume, the predicted covariate effect depends both on the descriptor and on its fitted or fixed exponent.

The earlier material on power-function models and centering and referencing body-size covariates provides the basic model form. Here the emphasis is on what changes when the size descriptor changes, how to calculate and interpret the descriptors, and how to make a reasoned comparison without treating one as universally best.

Definition: Total body weight \(W\) is the measured mass of the whole body, usually in kilograms. BMI is \(W/H^2\), where height \(H\) is in metres, and is reported in \(\mathrm{kg/m^2}\). A commonly used BSA approximation is the Mosteller formula, \(\mathrm{BSA}=\sqrt{H_{\mathrm{cm}}W/3600}\), with height in centimetres and weight in kilograms; its result is in \(\mathrm{m^2}\). Fat-free mass (FFM) is body mass excluding fat mass, usually expressed in kilograms. FFM may be measured or estimated; the method used to obtain it must be stated.

How the Descriptors Scale

A typical reference-scaled PK relationship for a parameter such as clearance is

CL_i=CL_{\mathrm{ref}}\left(\frac{S_i}{S_{\mathrm{ref}}}\right)^{\theta}\exp(\eta_{CL,i}),

where \(S_i\) is the selected size descriptor for individual \(i\), \(S_{\mathrm{ref}}\) is its reference value, \(CL_{\mathrm{ref}}\) is clearance at that reference, \(\theta\) is the size exponent, and \(\eta_{CL,i}\) represents an individual deviation on the log scale. The ratio \(S_i/S_{\mathrm{ref}}\) is dimensionless. The same general form can be used for volume, with its own reference value and exponent. A particular exponent should not be assumed to apply equally to every descriptor or every PK parameter.

The algebra shows what information BMI and BSA add beyond weight. Let \(r_W=W_i/W_{\mathrm{ref}}\) and \(r_H=H_i/H_{\mathrm{ref}}\). Since BMI is weight divided by height squared, its ratio is \(r_W/r_H^2\). Under the Mosteller formula, the BSA ratio is \(\sqrt{r_Wr_H}\). FFM has no such fixed algebraic relationship to weight and height alone: its ratio depends on how much of each person's mass is fat-free.

\frac{\mathrm{BMI}_i}{\mathrm{BMI}_{\mathrm{ref}}} =\frac{r_W}{r_H^2}, \qquad \frac{\mathrm{BSA}_i}{\mathrm{BSA}_{\mathrm{ref}}} =\sqrt{r_Wr_H}, \qquad \frac{\mathrm{FFM}_i}{\mathrm{FFM}_{\mathrm{ref}}} =\frac{\mathrm{FFM}_i}{\mathrm{FFM}_{\mathrm{ref}}}.

The last ratio is written directly because FFM is a separately defined descriptor, not a formula that can be recovered from weight and height without further information. Taking logarithms makes the distinct height contributions especially clear: the log BMI ratio is \(\log(r_W)-2\log(r_H)\), while the log BSA ratio is one half of \(\log(r_W)+\log(r_H)\). Thus, at the same proportional increase in weight, a height change can reduce the BMI ratio or increase the BSA ratio.

Worked Example: Calculating and Comparing Descriptors

Consider a hypothetical reference individual weighing 70.0 kg and standing 1.70 m tall, with FFM of 52.0 kg. Compare a hypothetical individual weighing 80.0 kg and standing 1.75 m tall, with FFM of 60.0 kg. Use the Mosteller BSA formula.

For the reference, BMI is \(70.0/(1.70^2)=70.0/2.89=24.22\ \mathrm{kg/m^2}\). For the second individual, BMI is \(80.0/(1.75^2)=80.0/3.0625=26.12\ \mathrm{kg/m^2}\). The BMI ratio is \(26.12/24.22=1.078\) (using unrounded values, approximately 1.0785).

The reference BSA is \(\sqrt{170\times70.0/3600}=\sqrt{3.30556}=1.818\ \mathrm{m^2}\). The second individual's BSA is \(\sqrt{175\times80.0/3600}=\sqrt{3.88889}=1.972\ \mathrm{m^2}\). The BSA ratio is approximately \(1.972/1.818=1.085\). The weight ratio is \(80.0/70.0=1.143\), and the FFM ratio is \(60.0/52.0=1.154\).

The descriptors do not give identical ratios: the same individual is 14.3% heavier by total weight, but the BMI and BSA ratios are about 1.078 and 1.085, respectively, while the FFM ratio is 1.154. These differences arise from the height terms in BMI and BSA and the body-composition information in FFM. None of these calculations alone establishes which descriptor best predicts a PK parameter.

What a PK Model Says About Each Descriptor

For a given descriptor \(S\), the exponent \(\theta\) describes the modelled change in the PK parameter for a proportional change in \(S\), holding the model's other terms fixed. For example, if \(S_i/S_{\mathrm{ref}}=1.20\) and \(\theta=0.75\), the size component of the parameter ratio is \(1.20^{0.75}\), not 1.20. This interpretation is conditional on the chosen descriptor, the reference, the other model terms, and the model being an adequate description of the data.

If the model uses total weight, the exponent describes scaling per proportional change in total weight. With BMI, it describes scaling per proportional change in weight relative to height squared. With BSA, it describes scaling per proportional change in the combined weight-height measure. With FFM, it describes scaling per proportional change in estimated or measured fat-free mass. An exponent attached to one descriptor cannot be transferred unchanged to another and interpreted as the same biological relationship.

Worked Example: Translating a Weight-Based Clearance Model

Suppose a hypothetical model uses \(CL_{\mathrm{ref}}=4.00\ \mathrm{L/h}\) at \(W_{\mathrm{ref}}=70.0\ \mathrm{kg}\), with a fixed weight exponent of 0.75. Predict the typical clearance for an individual weighing 100 kg; set the individual deviation to zero for this calculation.

The model is \(CL_i=4.00(100/70.0)^{0.75}\ \mathrm{L/h}\). The weight ratio is \(1.42857\), and \((100/70)^{0.75}\approx1.3067\). Therefore \(CL_i\approx4.00(1.3067)=5.2268\ \mathrm{L/h}\) (approximately \(5.23\ \mathrm{L/h}\)).

This is the prediction from the stated hypothetical model, not a general claim that clearance must scale with weight to the 0.75 power. Replacing weight with BMI, BSA, or FFM would define a different covariate relationship. It would require its own reference and exponent, and the resulting prediction would need to be assessed in the relevant population.

Worked Example: Why Height Changes BMI and BSA Scaling

Two hypothetical individuals have the same height of 1.70 m, while weight increases from 60.0 kg to 90.0 kg. Compare the descriptor ratios and illustrate their consequences under a common exponent of 0.75. This common exponent is used only to make the arithmetic comparable; it does not imply the exponent should be the same in fitted models.

The weight ratio is \(90.0/60.0=1.500\). Because height is unchanged, the BMI ratio is also \(1.500\): dividing both weights by the same \(1.70^2\) does not change their ratio. The BSA ratio is \(\sqrt{1.500\times1}=\sqrt{1.500}=1.225\). Thus the size multipliers with exponent 0.75 are \(1.500^{0.75}=1.355\) for weight and BMI, and \(1.225^{0.75}\), equivalently \(1.500^{0.375}\), which is approximately \(1.164\) for BSA.

At fixed height, BMI and weight carry the same relative ordering and proportional change, although their numerical values and units differ. BSA changes less than weight in this example because the Mosteller formula takes the square root of the weight-height product. If height also varied, BMI and BSA ratios would no longer follow these same simple comparisons.

Choosing and Checking a Descriptor

Choosing a descriptor is a modelling decision, not a contest in which one measure is always correct. Consider the PK question, the likely biological relevance of the measure, how the measure was obtained, and whether the study data contain enough variation to distinguish its effect from correlated covariates. Weight is often straightforward to measure and interpret. BMI and BSA incorporate height in different ways. FFM may better distinguish individuals with similar total weights but different proportions of fat-free mass, but it depends on the validity and availability of its measurement or estimation method.

A careful comparison keeps the model interpretation clear. State the descriptor, its units, the reference value, and the exponent. Check that height units are consistent with the BMI or BSA formula; in particular, the Mosteller calculation uses centimetres and kilograms. Inspect the observed range and distribution of each candidate covariate, look for data-entry or calculation errors, and examine whether the fitted relationship is plausible across the data rather than only near the reference. If several descriptors are strongly related, their effects may be difficult to distinguish in the same model. A better fit from one candidate in a particular dataset is evidence about that setting, not proof that it is universally the right physiological measure.

No single descriptor removes the need to evaluate model adequacy. Useful checks include comparing observed covariate patterns with model predictions, checking residual or prediction trends against the descriptor, and assessing whether the estimated exponent is precise enough to support the interpretation. A poor fit at the extremes may signal an inadequate functional form or an unsuitable descriptor, but diagnosing extreme-body-size behavior requires attention to the population and is not settled by the formula alone.

Common Mistakes and Limitations

Calling the descriptors interchangeable. They use different information and have different scaling behavior. A model based on BMI is not simply a weight model with a new label.
Interpreting the exponent without naming the covariate. An exponent of 0.75 refers to a proportional change in the specific descriptor in the model, not to a generic increase in “size.”
Mixing units in the formula. BMI requires height in metres when reported in \(\mathrm{kg/m^2}\); the stated Mosteller formula uses height in centimetres and weight in kilograms.
Treating a calculated measure as error-free. BMI and BSA inherit errors in their input measurements. FFM may also have measurement or estimation error, and different FFM methods need not produce identical values.
Claiming biological superiority from a covariate association. A descriptor that predicts a PK parameter in a dataset does not by itself establish a causal mechanism or universal preference.

The formulas are compact summaries, not complete representations of body composition or physiology. BMI does not separate fat mass from fat-free mass; BSA combines height and weight according to a particular approximation; and FFM depends on how it is obtained. Their usefulness therefore depends on the target parameter, the population, the quality of the measurements, and the purpose of the model.

Key takeaway: Compare descriptors by the information they encode and by the covariate relationship they imply. Report the descriptor, units, reference, and exponent together; then assess whether the model is supported by the data and appropriate for its intended use.

Check Your Understanding

Answer each question, then check the worked solutions below.

Calculate the BMI for a hypothetical individual weighing 72.0 kg and standing 1.80 m tall.
A second hypothetical individual has 1.10 times the weight and 1.25 times the height of a reference individual. Calculate the BMI ratio and Mosteller BSA ratio.
A hypothetical weight-based clearance model has \(CL_{\mathrm{ref}}=4.00\ \mathrm{L/h}\) at 80.0 kg and exponent 0.75. Predict clearance at 100 kg, setting the individual deviation to zero.
In one or two sentences, explain why an exponent estimated for weight cannot automatically be used for FFM.

Solutions. (1) \(72.0/(1.80^2)=72.0/3.24=22.222\ldots\), so BMI is \(22.22\ \mathrm{kg/m^2}\). (2) The BMI ratio is \(1.10/(1.25)^2=1.10/1.5625=0.7040\). The BSA ratio is \(\sqrt{1.10\times1.25}=\sqrt{1.375}=1.1726\). (3) The weight ratio is \(100/80.0=1.25\), so \(4.00(1.25)^{0.75}=4.728708\ldots\ \mathrm{L/h}\), which rounds to \(4.729\ \mathrm{L/h}\) to four significant figures. (4) Weight and FFM are different descriptors: FFM excludes fat mass and its ratio is not determined by the weight ratio alone. The exponent describes scaling with the covariate actually used, so a weight exponent does not automatically describe scaling per proportional change in FFM.