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models.partition

models.partition

Body-composition partitioning: how much of a mass change is fat.

This is where the model earns its keep relative to the folk formula “3500 kcal = 1 lb”. That rule assumes every pound lost is pure fat. In reality a mass change is a mixture of adipose tissue at ~9441 kcal/kg and fat-free tissue at ~1816 kcal/kg – a factor of five apart – so the same energy deficit moves the scale by very different amounts depending on what is being mobilised.

Two effects set the mixture.

Forbes’ law. Fitting body composition across a wide range of adiposity, Forbes found fat-free mass to be logarithmic in fat mass,

\[ L = 10.4 \ln F + C . \]

Differentiating and rearranging gives a parameter-free prediction for the fat fraction of any mass change:

\[ p_{\text{Forbes}} = \frac{dF}{dW} = \frac{F}{F + 10.4} . \]

The fatter you are, the more of what you lose is fat. At 30 kg fat mass the prediction is 74 %; at 15 kg, 59 %. This is why the last stretch of a diet is genuinely harder than the first: not only is maintenance lower, but a larger share of each lost kilogram is lean tissue, which carries far less energy and so comes off faster per unit deficit while costing more that you would rather keep.

Protein intake. Adequate protein during an energy deficit shifts the mixture toward fat. Modelled as an additive shift on the fat fraction, proportional to protein intake relative to a reference of 1.6 g/kg/day.

Functions

Name	Description
effective_energy_density	Energy density of the tissue currently being gained or lost, kcal/kg.
fat_fraction	Fat fraction of the current tissue change, in [PARTITION_MIN, PARTITION_MAX].
forbes_fat_fraction	Fat fraction of a mass change predicted by Forbes’ law, dimensionless.

effective_energy_density

models.partition.effective_energy_density(x, u, p)

Energy density of the tissue currently being gained or lost, kcal/kg.

\[ \rho_{\text{eff}} = p \rho_F + (1 - p) \rho_L \]

Ranges from about 5630 kcal/kg at the lower partition bound to 9290 at the upper one. Since the rate of mass change is the energy imbalance divided by this quantity, a lean subject on low protein can lose scale weight almost twice as fast as a fat subject on high protein at an identical deficit – while losing considerably more muscle in the process.

fat_fraction

models.partition.fat_fraction(x, u, p)

Fat fraction of the current tissue change, in [PARTITION_MIN, PARTITION_MAX].

Combines the Forbes baseline with a protein-driven shift. Clipping the result keeps the effective energy density inside physical bounds; without it an unconstrained partition_kappa could drive the fraction outside [0, 1] and produce a negative energy density, which is not so much inaccurate as meaningless.

forbes_fat_fraction

models.partition.forbes_fat_fraction(fat_kg)

Fat fraction of a mass change predicted by Forbes’ law, dimensionless.