# Summary of SIAM talk

Last Monday I gave a plenary talk at the joint Life Sciences and Annual SIAM meeting.  My slides can be downloaded from a previous post. The talk summarized the work I’ve been doing on obesity and human body weight change for the past six years.  The main idea is that at the most basic level, the body can be modeled as a fuel tank.  You put food into the tank by eating and you use up energy to maintain bodily functions and do physical work.  The difference between food intake rate and energy expenditure rate is the rate of change of your body weight.  In calculating body weight you need to convert energy (e.g. Calories consumed) into mass (e.g. kilograms).  However, the difficulty in doing this is that your body is not homogeneous.  You are comprised of water, bones, minerals, fat, protein and carbohydrates in the form of glycogen.  Each of these quantities has its own energy density (e.g. Calories/kg).  So in order to figure out how much you’ll weigh you need to figure out how the body partitions energy into these different components.

The problem is made simpler because the energy density of water, bones and minerals is zero, the energy density of fat is about 38 kJ/g and that of protein and carbohydrates is about 17 KJ/kg.  Thus, we really only need to consider the flux of the three macronutrients fat, carbohydrates and protein.   This is further simplified because the amount of glycogen in the body is small, a few kilograms at most. (“Hitting the wall” in a marathon is usually attributed to running out of glycogen and forcing the body to switch to burning protein.  However, this actually may not be true as recent studies seem to suggest that there still is some glycogen left when runners hit the wall.)  Hence, on a time scale of weeks, you can consider the rate of change of carbohydrates to be minimal and this reduces the system to two equations, one for fat and one for protein.  Given that a change in protein is associated with a change in water, this leads to equations for fat mass and fat free or lean mass.  The equations have the form

$\rho_F \frac{dF}{dt} = I_F - f E(F,L)$

$\rho_L \frac{dL}{dt} = I_L - (1-f) E(F,L)$

where $F$ and $L$ are the fat and lean body masses, $\rho_F$ and $\rho_L$ are the energy densities of fat and lean tissue and can be measured, $I_F$ and $I_L$ are the  fat and lean food intake energy rates, $E(F,L)$ is the total energy expenditure rate (it is a function of the fat and lean masses), and $f = f(F,L)$ is the fraction of the energy expenditure coming from utilizing fat (it is also a function of the the fat and lean masses).

In steady state (body mass not changing), we have the conditions

$E(F,L) = I_F + I_L \equiv I$

$f(F,L) =\frac{I_F}{I}$

In other words, when you are in steady state, you are in both energy balance (you burn all the food you eat), and you are in macronutrient balance (the amount of fat you burn is the amount of fat you eat, and the same for protein and carbohydrates).  In two dimensions, it is well known that stable steady states are restricted to either a single fixed point (unique body composition), multiple fixed points ( multiple possible body compositions), an invariant manifold or line attractor (where a continuum of states is possible), a limit cycle or multiple limit cycles (where the body composition would oscillate even when food intake rate and energy expenditure were fixed).  In order to know which, we need to estimate E and f.

The energy expenditure rate can be measured using indirect calorimetry and it turns out to be a linear function (more technically an affine function) of $F$ and $L$.  However,  it is difficult to measure $f$ because when you are weight stable, $f$ just reflects what your diet is.  You need to estimate $f$ in a dynamic situation, which is difficult to do in a single individual but is possible if we make a homogeneity assumption in which we assume that a heavier person is like a lighter person who gained weight and vice versa.  Gilbert Forbes did precisely that and found that body composition obeys a simple law given by

$\frac{dL}{dF} = \frac{10.4}{F}$

We can then impose this law on our body composition equations and get an estimate for $f$ that has no adjustable parameters and this matched experimental data  [2].  Forcing the body to obey Forbes law then leads to what we call the energy partition model

$\rho_F \frac{dF}{dt} = (1-p)(I - E(F,L))$

$\rho_L \frac{dL}{dt} = p(I - E(F,L))$

where

$p=\frac{1}{1+\frac{\rho_F}{\rho_L}\frac{F}{10.4}}$.

The interesting thing about the energy partition model, which almost everyone who’s tried to model body composition before has used, is that the steady state is a line attractor given by the condition $E(F,L)=I$.  This implies that you could eat the same diet and live the same lifestyle but be stable at an infinite number of different weights.  In some cases you would be heavy and others you would be thin.  However, obeying the Forbes curve doesn’t necessarily imply you obey the energy partition model.  It could also mean that your unique fixed point follows the Forbes curve.   Interestingly, it is impossible to tell if you have a fixed point or a line attractor  for changes in just energy intake or expenditure rates since they can give the same predictions.  What you need to do is to perturb body composition by either strength training or liposuction.  We show this and other implications in Ref. [3] listed below.

Armed with the model, we can now make predictions about how body weight will change.  We can estimate all the parameters in the model and in fact if you stay on a single Forbes curve and assume small body weight changes you can reduce the energy partition model to a simple one dimensional equation for a leaky integrator

$\rho\frac{dM}{dt}= I - \epsilon M -b$

where $\rho$ and $\epsilon$ are parameters that can be estimated and also depend on fat.  The model also immediately shows why the commonly used rule of “3500 Calories is a pound” is a poor approximation at best.  This rule only gives the initial rate of change of body weight if you are perturbed from a steady state and assumes that your body weight won’t change to compensate.   A more accurate rule that can be obtained from the model is that “10 Calories a day is a pound”, which means that reducing your diet by 10 Calories a day (Calorie = kilocalorie) will lead to a new steady state that is one pound lower.  However, the time constant for reaching steady state is about a year so it can take a long time to reach steady state.  This is one of the reasons why most diets will fail, the feedback is very slow so by the time you notice that you’re off the diet, it’s too late.  To add insult to injury because of the fat dependence of $\rho$ and $\epsilon$, the amount of weight you gain and the time constant for a given amount of food both increase with body fat.

The model can also address a long standing paradox in human metabolism, which as stated by Eugene Dubois in 1927, is that  “There is no stranger phenomenon than the maintenance of a constant body weight under marked variation in bodily activity and food consumption.”  Generally, it is stated as “since 3500 Calories is a pound and we eat a million Calories a year then we must have very precise intake control to maintain weight within a few pounds.” This argument is specious because it doesn’t account for saturation due to body weight change.  However, a more correct statement of the paradox is that “if 10 Calories a day is a pound and a cookie is 200 Calories then how do we maintain weight stability?”  The resolution of the paradox is simply that the time constant is very long so we average our intake over a year and the standard error of the mean yearly intake is small.  You can show this explicitly using stochastic calculus and I give some examples in my talk.  I still need to write the paper for this result. The one thing that can cause weight to be unstable is temporal correlations in food intake.  In previous work [1], we showed that the more correlated your food intake is from day to day or week to week the more likely you will have a high BMI.

Finally, we can address the question of what caused the American obesity epidemic.  Up until the mid seventies, the average body weight of the US population was very stable.  It then  increased linearly in time after that. The fraction of obese, which is defined as a body  mass index greater than 30, has also increased dramatically.  In Ref [4] below, we applied our model to the US population to predict how much food we should be eating to match the weight gain and compare it to the food available.  What we found is that the food available per capita as estimated by the USDA that accounts for exports, imports, spoilage and waste is more than sufficient to account for the increase in weight.  In fact, there has been a progressive increase in waste over the past 35 years.  This supports the hypothesis that the obesity epidemic is due to the “push” of food onto the American public.  In fact, the population is doing it’s best to resist the extra food and the consequence is extra waste.

References

[1] V. Periwal and C.C. Chow, ‘Patterns in food intake correlate with body mass index’, American Journal of Physiology: Endocrinology and Metabolism, 291 929-936 (2006) [PDF]

[2] K.D. Hall, H.L. Bain and C.C. Chow, How adaptations of substrate utilization regulate body composition’, International Journal of Obesity, 31 , 1378-83 (2007). [PDF]

[3] C.C. Chow and K.D. Hall, The dynamics of human body weight change’, PLoS Computational Biology 4 , e1000045 (2008).

[4] K.D. Hall, M. Dore, J. Guo, and C.C. Chow, ‘The progressive increase of food waste in America’, PLoS ONE 4(11): e7940 (2009).

## 5 thoughts on “Summary of SIAM talk”

1. […] push hypothesis for obesity By Carson Chow My blog post on the summary of my SIAM talk on obesity was picked up by Reddit.com.  There is also a story by […]

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2. […] as the plenary talk I gave at the SIAM Annual and Life Sciences meeting in 2010, which I summarized here.  Science and math writer Barry Cipra also wrote a piece about the talk in SIAM News.   I find it […]

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3. […] in obesity modeling”.  The slides can be found here.  The talk is a shorter version of my SIAM talk pointing out some open problems in the field and how to model […]

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4. […] can also read detailed insights and a summary of his AN10 talk on Dr. Chow’s blog here. Posted in All News, Conference News, News, Research News | Tags: math modeling, […]

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