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The dynamics of human body weight change

Carson C. Chow, Kevin D. Hall

arXiv:0802.3234v1q-bio.TO

TL;DR

The p-ratio is p = p(F, L) = 1/(1 + α), and more fat is lost when initial body fat is higher. The paper treats two-dimensional autonomous lean- and fat-mass models as variants of the macronutrient partition model and classifies their fixed-point structures. The models include isolated fixed points and invariant manifolds, and an invariant-manifold model can behave indistinguishably from one with a fixed point.

  • Problem

    The p-ratio is p = p(F, L) = 1/(1 + α), and more fat is lost when initial body fat is higher.

    Source: §2 · p. 6

  • Method

    The paper treats two-dimensional autonomous lean- and fat-mass models as variants of the macronutrient partition model and classifies their fixed-point structures.

    Source: §III · pp. 12–13

  • Results

    The models include isolated fixed points and invariant manifolds, and an invariant-manifold model can behave indistinguishably from one with a fixed point.

    Source: §III · pp. 12–14

  • Takeaways & Limitations

    Allowing the partition parameter to depend weakly on body composition can produce dynamics that differ significantly from constant-p models.

    Source: §2 · p. 6

    Transient body-composition changes through procedures such as liposuction or temporary therapeutic administration are beyond the paper’s scope.

    Source: §III · pp. 13–14

Abstract

from arXiv · show

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 the long-term dynamics of human weight change can be captured by a mathematical model of the macronutrient flux balances and all previous models are special cases of this model. We show that the generic dynamical 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. However, this distinction is important for the efficacy of clinical interventions that alter body composition and mass.

I. INTRODUCTION

The paper frames human body-weight change as a problem of understanding how energy balance changes body mass and fat–lean composition, then analyzes macronutrient flux models as dynamical systems. For a clamped diet, these models can yield either one stable body composition or infinitely many steady-state compositions, and existing data cannot distinguish the classes.

I. INTRODUCTION

  • For a clamped diet and energy expenditure, models yield either one uniquely specified stable body composition or a continuum of steady-state compositions.The paper reports that existing data are insufficient to determine which class applies.Source: §I · p. 2
  • The paper proposes that experimental perturbations may distinguish the two classes, although existing data do not yet do so.The introduction identifies possible experimental approaches without specifying their outcomes.Source: §I · p. 2

A. General model of macronutrient and energy flux balance

  • The analysis represents body fat, glycogen, and protein through macronutrient flux balances linked to intake and energy-expenditure fractions.The three-compartment equations recover the energy-flux balance, with the expenditure fractions depending on body composition and intake.Source: A. General model of macronutrient and… · p. 3
  • Averaging removes fast intake and expenditure fluctuations to describe slow dynamics using moving averages over about a day.The averaged equations retain the flux-balance form at lowest order.Source: A. General model of macronutrient and… · p. 4
  • Hall’s three-compartment model approaches a unique steady state under clamped food intake, while limited glycogen storage supports an approximate two-dimensional reduction.The reduction permits analysis using dynamical-systems techniques.Source: A. General model of macronutrient and… · p. 4

1. Two compartment macronutrient partition model

The three-compartment macronutrient flux model can be reduced to a two-compartment model of fat and lean mass for long-term body-weight dynamics. The reduction treats glycogen as being in quasi-equilibrium, and the paper relates earlier two-dimensional models to this framework.

1. Two compartment macronutrient partition model

  • Assuming glycogen is in quasi-equilibrium reduces the three-compartment flux model to a two-compartment fat-and-lean model.The reduction uses dG/dt = 0 to substitute the carbohydrate balance into the other flux equations.Source: §1 · p. 4
  • A plausible alternative, with glycogen proportional to protein mass, leads to the same two-dimensional system.For the cited small proportionality constant, the resulting lean-mass energy density is approximately unchanged.Source: §1 · p. 5
  • Earlier two-dimensional models include linearized expenditure models and double-exponential weight-loss models equivalent to a zero-intake special case.The equivalence described for the latter uses energy expenditure linear in fat and lean mass.Source: §1 · p. 5

2. Energy partition model

The section derives energy partitioning from macronutrient partitioning and connects earlier models to this formulation. It describes how the p-ratio and fat–lean trajectories depend on body composition and model assumptions.

2. Energy partition model

  • Forbes’s relationship is similar for diet- and exercise-induced weight loss, but may differ with resistance exercise or substantial dietary protein changes; it does not describe infant growth well.The text identifies body composition data in growing infants as a basis for choosing α during the first two years.Source: §2 · p. 5
  • Hall, Bain and Chow’s two-compartment model using Forbes’s law matched a wide range of data without adjustable parameters.Their model used Eq. (15) with Forbes’s law.Source: §2 · p. 6
  • The energy partition model divides an energy imbalance between compartments by p(F, L), the fraction assigned to lean tissue, and follows fat–lean trajectories set by initial conditions.The model arises by substituting Eq. (15) into the macronutrient partition model; Eq. (12) describes a family of F-versus-L curves.Source: §2 · p. 6
  • Most earlier models are versions of the energy partition model, although their connection to macronutrient flux balance and dynamical-systems analysis was not noted.Some computational algorithms can also be translated into this model’s form.Source: §2 · p. 6
  • The p-ratio’s physiological interpretation remains obscure and it is difficult to measure directly, though it can be inferred indirectly from f measured by indirect calorimetry.The inference uses Eq. (15).Source: §2 · p. 6
  • Weight-loss data find that the fraction of body fat lost depends on body composition, with more fat lost when initial body fat is higher.A constant p-ratio may approximate the partitioning when α depends only weakly on body composition and changes are small.Source: §2 · p. 6

3. One Dimensional models

Energy-partition dynamics can be reduced to one dimension when fat and lean masses follow a unique relationship. The reduction holds along a fixed trajectory, but has limits when translating trajectories into mass dynamics or when body composition is directly altered.

3. One Dimensional models

  • Energy-partition dynamics follow fixed trajectories in the L–F plane, and a one-dimensional reduction is possible when F and L have a unique relationship.Continuity of α and its derivative provides sufficient, but not necessary, conditions for a unique solution over an interval.Source: §3 · pp. 6–7
  • With constant p, the trajectories are straight lines F = βL + C, with C specified by the initial body composition.The slope β depends on p and the fat- and lean-tissue energy densities.Source: §3 · p. 7
  • Forbes’s law also permits a one-dimensional trajectory reduction, but the resulting mass functions cannot be inverted in closed form to obtain an explicit mass differential equation.Mass dynamics can still be obtained using the fat- or lean-mass equation together with the relevant mass function.Source: §3 · p. 7
  • For large body-composition changes, the dynamics could differ significantly from those predicted by constant-p models.Source: §3 · p. 7
  • The one-dimensional model can represent time-varying intake along a fixed trajectory; a perturbation that directly alters body composition breaks the reduction while it acts.After the perturbation ceases, the one-dimensional model applies again with a new constant.Source: §3 · p. 7

C. Existence and stability of body weight fixed points

The two-dimensional macronutrient flux-balance model captures autonomous body-composition dynamics, whose fixed-diet outcomes depend on the structure and stability of its fixed points. A stable isolated fixed point yields a unique steady state, whereas an attracting invariant manifold permits a continuum of body compositions.

C. Existence and stability of body weight fixed points

  • The two-dimensional macronutrient partition model can express all autonomous two-dimensional and one-dimensional models of body-composition change.The analysis focuses on two dimensions because glycogen levels may average to an approximate constant after a few days.Source: C. Existence and stability of body… · p. 8
  • A single stable nullcline intersection uniquely determines steady-state body composition and mass, while collinear nullclines can permit infinitely many states.Multiple stable intersections can also make body composition non-unique.Source: C. Existence and stability of body… · p. 8
  • An isolated stable fixed point is guaranteed when the fat-utilization fraction increases with fat mass and decreases with lean mass.If one fixed-point condition automatically satisfies the other, a continuous curve of fixed points, or invariant manifold, results.Source: C. Existence and stability of body… · p. 9
  • The energy partition model has an invariant manifold because energy balance automatically satisfies its fat-fraction condition.Along the attracting manifold, the body can remain at any of infinitely many compositions on the energy-balance curve.Source: C. Existence and stability of body… · pp. 9–10
  • Multiple stable fixed points or a limit cycle require fat utilization to be nonmonotonic in lean mass and finely tuned.The paper therefore considers these outcomes less plausible than a single fixed point or an invariant manifold.Source: C. Existence and stability of body… · p. 10
  • The available data on energy expenditure and substrate use are consistent with an isolated stable fixed point, but fat-utilization dependence is not well established.The paper notes evidence that energy expenditure increases with fat and lean mass, while fat-utilization sensitivity depends on interrelated factors.Source: C. Existence and stability of body… · p. 10

D. Implications for body mass and composition change

For clamped intake or energy expenditure, fixed-point and invariant-manifold models can respond identically to those perturbations. Directly perturbing body composition can distinguish their outcomes, but existing evidence does not settle which class describes humans.

D. Implications for body mass and composition change

  • An invariant manifold allows a transient body-composition perturbation to produce a permanent change in body composition and mass.By contrast, in the fixed-point case the perturbation is corrected over time.Source: D. Implications for body mass and… · p. 10
  • Temporary changes in intake or energy expenditure move body composition along the same trajectory in both model classes and return it to the original steady state when restored.The paper says these conventional perturbations cannot reveal the distinction.Source: D. Implications for body mass and… · p. 10
  • A direct perturbation off the fixed trajectory leads to the same steady state in a fixed-point model but a different one in an invariant-manifold model.Possible perturbations include altering body composition directly or changing fat utilization.Source: D. Implications for body mass and… · p. 11
  • After 18 months of growth hormone administration, subjects returned very closely to original body composition; 18 months after therapy removal, lean had a slight (2%) but significant increase, offering some evidence for a fixed point but not conclusive evidence.Source: D. Implications for body mass and… · p. 11
  • Existing longitudinal weight-change data are insufficient to distinguish the classes because they altered weight through caloric intake alone.The paper notes that such data cannot rule out a fixed point.Source: D. Implications for body mass and… · p. 11

1. Numerical simulations

In numerical examples, reducing energy intake produced identical body-composition and mass time courses in fixed-point and invariant-manifold models. A direct fat-mass perturbation, however, permanently changed the final state only in the invariant-manifold model.

1. Numerical simulations

  • With ψ = 0, composition moves along a fixed trajectory; with nonzero ψ, an isolated fixed point can occur.The paper gives a corresponding fixed-point model for each invariant-manifold model so intake-perturbation trajectories match.Source: §1 · p. 12
  • Reducing intake from 12 MJ/day to 10 MJ/day produced identical body-composition and mass time courses in both model examples.Mass first decreased linearly and then approached a new stable fixed point.Source: §1 · p. 12
  • Removing 10 kg of fat at day 100 permanently altered final composition and mass in the invariant-manifold model but only transiently affected the fixed-point model.The examples used Forbes’s law and a p-ratio p = 2/(2 + F), with F measured in kg.Source: §1 · p. 12
  • The numerical simulations’ approach to the new fixed point has a time constant of several years for humans.The paper notes that this slow approach makes experiments distinguishing the models difficult.Source: §1 · p. 12
  • The time constant depends on energy-expenditure form, and steeper individual dependence on fat and lean mass may shorten it through adaptive thermogenesis.The authors state that the example should hold qualitatively if intake and activity variations are small relative to the induced change.Source: §1 · p. 12

III. DISCUSSION

The paper places autonomous two-compartment body-composition models into two broad classes: isolated fixed points and invariant manifolds. The classes can be difficult to distinguish from observed weight-change responses, although that distinction matters for clinical interventions that alter body composition.

III. DISCUSSION

  • The model classes are isolated fixed points, which specify body composition uniquely, and invariant manifolds, which permit infinitely many body compositions.The paper also notes that more exotic behaviors, including multistability and limit cycles, are possible but require fine tuning and are less plausible.Source: §III · pp. 12–13
  • The distinction matters for the potential permanence of body-composition changes from procedures such as liposuction or temporary therapeutic administration.The paper identifies the fraction of energy utilized as fat as a way to discriminate between the two possibilities.Source: §III · p. 13
  • The analysis covers slow dynamics under clamped average food intake, not hourly or daily fluctuations or slowly changing energy expenditure.Its stated time-scale fit is months to a few years in adults; feedback of body composition on food intake is outside scope.Source: §III · p. 14
  • An invariant manifold can behave indistinguishably from a fixed point, and decay toward a fixed point may take as long as a decade.Only experiments that perturb fat or lean compartments independently can distinguish the classes.Source: §III · p. 14

A. Method of Averaging

The averaging method removes fast fluctuations in intake, expenditure, and body compartments to describe slower body-mass and composition changes. It uses a fast time variable and expansions in a small parameter to derive averaged equations.

A. Method of Averaging

  • Averaging removes short-timescale fluctuations so the model describes slower changes in body mass and composition.The fast dynamics may be periodic or stochastic, provided their time average is of order ǫ or higher.Source: A. Method of Averaging · pp. 14–15
  • The method introduces a fast time variable τ = t/ǫ alongside slow time t, with ǫ representing the separation between timescales.For example, t in days and τ in hours gives ǫ approximately 1/24.Source: A. Method of Averaging · p. 14
  • Expansions of the body compartments and energy-expenditure fractions are substituted into the flux equations to derive averaged equations.The compartment expansions retain leading terms and first-order corrections in ǫ.Source: A. Method of Averaging · pp. 14–15
  • The resulting system represents lowest-order time-averaged macronutrient flux balances for long-term dynamics.The averaging can also extend over day-to-day intake fluctuations; choosing a different averaging interval changes the interpretation of the averaged quantities.Source: A. Method of Averaging · p. 15

B. Stability conditions for fixed points

The stability analysis links the behavior near fixed points and invariant manifolds to how the fat-utilization fraction changes with fat and lean mass. Under the stated conditions, unique stable fixed points are possible, while limit cycles and multistability require more specialized behavior.

B. Stability conditions for fixed points

  • If fat-utilization fraction f increases monotonically with fat mass F and decreases monotonically with lean mass L, the model has a unique stable fixed point.The analysis states that these monotonicity conditions guarantee stability.Source: B. Stability conditions for fixed points · p. 16
  • An invariant manifold has det J = 0; when energy expenditure increases monotonically in F and L, its trace is negative and the manifold attracts trajectories.The zero eigenvalue corresponds to marginal stability along the manifold.Source: B. Stability conditions for fixed points · p. 16
  • A limit cycle requires f to increase with L in some region but decrease at very small and large L, along with fine-tuned parameters.The paper links this possibility to a Hopf bifurcation.Source: B. Stability conditions for fixed points · p. 16
  • Multistability also requires nonmonotonicity and fine tuning.The paper presents it as less plausible than the generic behaviors.Source: B. Stability conditions for fixed points · p. 16
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