The dynamics of human body weight change

Abstract

An imbalance between energy intake and energy expenditure will lead to a change in body weight (mass) and body composition (fat and lean masses). A quantitative understanding of the processes involved, which currently remains lacking, will be useful in determining the etiology and treatment of obesity and other conditions resulting from prolonged energy imbalance. Here, we show that a mathematical model of the macronutrient flux balances can capture the long-term dynamics of human weight change; all previous models are special cases of this model. We show that the generic dynamic behavior of body composition for a clamped diet can be divided into two classes. In the first class, the body composition and mass are determined uniquely. In the second class, the body composition can exist at an infinite number of possible states. Surprisingly, perturbations of dietary energy intake or energy expenditure can give identical responses in both model classes, and existing data are insufficient to distinguish between these two possibilities. Nevertheless, this distinction has important implications for the efficacy of clinical interventions that alter body composition and mass.

Figure 1. Possible trajectories (solid lines) for different initial conditions and nullclines (dotted lines) in the L–F phase plane for models with a stable fixed point (A), multi-stability with two stable fixed points separated by one unstable saddle point (B), an attracting invariant manifold (C), and a limit cycle attractor (D).

Figure 2. An example of a situation where the intake or energy expenditure rate is changed from one clamped value to another and then returned.

(A) Fixed point case. (B) Invariant manifold case. Dotted lines represent nullclines. In both cases, the body composition follows a fixed trajectory and returns to the original steady state (solid curves). However, if the body composition is perturbed directly (dashed-dot curves) then the body composition will flow to same point in (A) but to a different point in (B).

Figure 3. Time dependence of body mass and F vs. L trajectories.

In all the Figures, the solid line is for an intake reduction from 12 MJ/day to 10 J/day and the dashed line is for the same reduction but with a removal of 10 kg of fat at day 100. Time dependence of body mass for the fixed point model (A). Trajectories in the F vs. L phase plane for the fixed point model (B). Dotted lines are the nullclines. Time dependence (C) and phase plane (D) of the invariant manifold model for the same conditions.