# New paper on childhood growth and obesity

Kevin D Hall, Nancy F Butte, Boyd A Swinburn, Carson C Chow. Dynamics of childhood growth and obesity: development and validation of a quantitative mathematical model. Lancet Diabetes and Endocrinology 2013 .

You can read the press release here.

In order to curb childhood obesity, we need a good measure of how much food kids should eat. Although people like Claire Wang have proposed quantitative models in the past that are plausible, Kevin Hall and I have insisted that this is a hard problem because we don’t fully understand childhood growth. Unlike adults, who are more or less in steady state, growing children are a moving target. After a few fits and starts we finally came up with a satisfactory model that modifies our two compartment adult body composition model to incorporate growth. That previous model partitioned excess energy intake into fat and lean compartments according to the Forbes rule, which basically says that the ratio of added fat to lean is proportional to how much fat you have so the more fat you have the more excess Calories go to fat. The odd consequence of that model is that the steady state body weight is not unique but falls on a one dimensional curve. Thus there is a whole continuum of possible body weights for a fixed diet and lifestyle. I actually don’t believe this and have a modification to fix it but that is a future story.

What puzzled me about childhood growth was how do we know how much more to eat as we grow? After some thought, I realized that what we could do is to eat enough to maintain the fraction of body fat at some level, using leptin as a signal perhaps, and then tap off the energy stored in fat when we needed to grow. So just like we know how much gasoline (petrol) to add by simply filling the tank when it’s empty, we simply eat to keep our fat reserves at some level. In terms of the model, this is a symmetry breaking term that transfers energy from the fat compartment to the lean compartment. In my original model, I made this term a constant and had food intake increase to maintain the fat to lean ratio and showed using singular perturbation theory that his would yield growth that was qualitatively similar to the real thing. This then sat languishing until Kevin had the brilliant idea to make the growth term time dependent and fit it to actual data that Nancy Butte and Boyd Swinburn had taken. We could then fit the model to normal weight and obese kids to quantify how much more obese kids eat, which is more than previously believed. Another nice thing is that when the child stops growing the model is automatically the adult model!

## 5 thoughts on “New paper on childhood growth and obesity”

1. this reminds me of standard math bio—eg you have malthusian growth which is exponential, and then you make the constant a function of population size (destiny (correct formulation) dependence or frequency dependent (jw curtsinger pnas 1984). i notice i have 5 or 6 setpoints tho it depends on lifestyle and diet (and those are not in dependent).
i wonder if this nonuniqueness is similar (isomorphic) to the SMD theorem in economics (s is for a former u chicago president) which shows if you do the crime, you do the time (where both crime and time are nonlinear functions of each other plus the rest of the universe (an error term like dark energy O(infinity))—eg my watch shows i’m always on time, even if its stopped. ‘ishi time’–bee hear now .

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2. Actually, it is nothing like saturated Malthusian growth, which is exponential. The growth here is secular and ends when the tap gets shut off.

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3. i glanced at the paper since i have a free subscription. the equations look like epidimeology, though they aren’t (nor langevin, since g(t) is not a stochastic force). looked at the press release—and i get some sort of second rate generic rock band so i can’t even read the article.
i think the issue is ‘junk food’ (which i was and still am slightly addicted to, except now i’m extremely allergic to it). its in schools, everywhere, and people don’t change. thats why one needs to deal with ‘frequency dependence’. and its not just junk food.

i don’t think ‘saturated malthusian growth’ is exponential except at first—it depends on the nonlinearity in the growth coefficient. see S Smale J Math Bio around 1976 april (smale had a few papers; eg computation over reals). In fact its universal computation.

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4. Consider the saturated exponential process given by the differential equation

dx/dt = x – a x^2

This is classic saturating Malthusian growth also known as logistic growth.

No matter how large the nonlinearity, close enough to x=0 will be a regime of exponential growth, which you have noted.

However, our childhood growth model is more like dx/dt ~ constant, so the dominant growth term is secular, which is very different from logistic growth.

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5. thanks for responding (tho you don’t need it or to); but i was thinking of dx/dt=a(x)x where a(x) can be any function of x (e.g. let a(x) be the universal polynomial, james p jones sept 1980 Bull Am Ma Assoc; of course that’s more variables but you can reduce them). (my fave BAMS article is George Mackey around 76 on group theory applied to everything; person who turned me onto it also almost flunked me cuz i didnt class)

one other point, as an unrecognized brilliant mathematician like ramanujan, i can show your equation can also be written as x(dot) = x(1-ax).
i think eliot montroll, feigenbaum, bruce west (reactionary global warming denialist and greedy warhawk) have written on this.

.(as an aside, when john wheeler (who i saw speak at u md—it for bit, boundary of a baoundary is zero) asked godel how to relate heisenberg uncertainty to logical incompleteness, godel through him out of his office at IAS. hit the road jack, and dont come back no more no more’.)
i wonder which growth model is applicable to, say, Cairo or giza (and i been there—-woke up sleeping on the beach with a bayonet in my faCE—no sleeping outside).
i think i ruptured my spleen running so i’m just killing time ; i may have to check out my doctor up the street—adis ababa (cool place)–police report says they got armed robbery last week.
peace out

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