Intramural Research Program of the National Institutes of Health, National Institute of Diabetes and Digestive and Kidney Diseases.

This model quantifies the energy excess underlying obesity and calculates the necessary intervention magnitude to achieve bodyweight change in children. Policy makers and clinicians now have a quantitative technique for understanding the childhood obesity epidemic and planning interventions to control it.

The model accurately simulated the changes in body composition and energy expenditure reported in reference data during healthy growth, and predicted increases in energy intake from ages 5–18 years of roughly 1200 kcal per day in boys and 900 kcal per day in girls. Development of childhood obesity necessitated a substantially greater excess energy intake than for development of adult obesity. Furthermore, excess energy intake in overweight and obese children calculated by the model greatly exceeded the typical energy balance calculated on the basis of growth charts. At the population level, the excess weight of US children in 2003–06 was associated with a mean increase in energy intake of roughly 200 kcal per day per child compared with similar children in 1971–74. The model also suggests that therapeutic windows when children can outgrow obesity without losing weight might exist, especially during periods of high growth potential in boys who are not severely obese.

We developed and validated a mathematical model of childhood energy balance that accounts for healthy growth and development of obesity, and that makes quantitative predictions about weight-management interventions. The model was calibrated to reference body composition data in healthy children and validated by comparing model predictions with data other than those used to build the model.

Clinicians and policy makers need the ability to predict quantitatively how childhood bodyweight will respond to obesity interventions.

We introduce the first mathematical model of childhood energy balance and bodyweight dynamics that accounts for healthy growth and development of obesity and makes quantitative predictions about weight-management interventions.

To address this problem and improve predictions, we previously developed and validated mathematical models of adult human metabolism, which provided accurate predictions of bodyweight dynamics resulting from interventions in both individuals and populations.Unfortunately, these adult models were not appropriate for the additional complexity of childhood growth. Previous models for estimation of energy imbalances in children who are of healthy weight, overweight, or obesecould not adequately distinguish healthy growth from excessive weight gain and were not validated with data other than those used to build the model. Furthermore, none of the previous models was designed to predict how interventions affect body composition during childhood.

Estimating the effects of energy imbalance on changes in body weight in children.

Comment on “Obesity and the environment: where do we go from here?”.

One of the most disconcerting aspects of the global obesity epidemicis the high prevalence of childhood obesity and the associated healthand economicconsequences. Tackling childhood obesity will necessitate a broad approach and a range of interventions, from clinical management to policy changes, that affect the entire population.To be effective, interventions should address the problem of food intake that is in excess of energy requirements for healthy growth and development.Clinicians and policy makers need to be adequately informed about the likely effect of an intervention affecting energy balance and downstream effects on childhood bodyweight dynamics. Calculation of these effects can be complex, and we have reported that the usual estimates for translation between energy balance and bodyweight are highly inaccurate and result in exaggerated predictions of weight loss.

Health and economic burden of the projected obesity trends in the USA and the UK.

The sponsors did not have roles in study design; data collection, analysis, or interpretation; or writing of the Article. The corresponding author had full access to all the data in the study and the final responsibility for the decision to submit for publication.

We used the reference childhood body composition data of Fomon and colleaguesand Haschketo calibrate the growth model and simulate the changing proportion of fat and fat-free mass deposition with age and the increasing energy density of fat-free mass ( appendix ).We validated the model by comparing the simulation results with data not used to build the model. We selected the validation data from studies of children and adolescents in which body composition at different ages was accurately measured; we tried to use studies with longitudinal cohorts and that included measurements of energy expenditure. Only the timecourse of energy intake and the initial conditions of the model were adjusted to make comparisons with the validation data. No other model parameters were fit to match these data. We used the Runge-Kutta 4 algorithm (implemented in Berkeley Madonna , version 8.3) to numerically integrate the model. We used a step size of 0·2 days for numerical integration, which is much smaller than the characteristic timescale of the model dynamics, thereby ensuring that the method accurately represented the solutions of the model differential equations.

Sex-specific growth was modelled as the combination of increasing energy intake with time and an age-dependent function representing the net effect of various complex physiological processes that stimulate accretion of fat-free mass while obeying both energy balance and macronutrient balance. We assumed that growth stopped in early adulthood. We accounted for the fact that resting metabolic rate per unit bodyweight is significantly higher in children than in adults (because of the relative increased contribution of high metabolic rate organs),and modelled how physical activity decreases with age.

Changes in physical activity patterns in the United States, by sex and cross-sectional age.

Metabolic rate and organ size during growth from infancy to maturity and during late gestation and early infancy.

The appendix contains a detailed description of the development and calibration of our mathematical model. Briefly, our model is a mathematical representation of the energy balance principle, whereby changes in bodyweight are dynamically calculated to result from the difference between the calories consumed in food and the energy expended by the body to maintain life and do physical work. It includes metabolic adaptations that occur during weight gain and loss and partitions energy imbalances between fat and fat-free masses. In this study, we incorporated growth into our previously published equations describing changes to metabolism and body composition that have been validated in adults.

Results

29 Ellis KJ

Shypailo RJ

Abrams SA

Wong WW The reference child and adolescent models of body composition. A contemporary comparison. 29 Ellis KJ

Shypailo RJ

Abrams SA

Wong WW The reference child and adolescent models of body composition. A contemporary comparison. 6 Hall KD

Sacks G

Chandramohan D

et al. Quantification of the effect of energy imbalance on bodyweight. Figure 1 Model-simulated rates of energy intake and energy expenditure in boys (A) and girls (B), model-simulated and cross-sectional data for fat mass and fat-free mass during healthy growth in white boys (C) and white girls (D), and model-simulated energy imbalance gap in boys (E) and girls (F) Show full caption 29 Ellis KJ

Shypailo RJ

Abrams SA

Wong WW The reference child and adolescent models of body composition. A contemporary comparison. Data are mean; SD shown by error bars in C and D. Dotted lines in A and B show the range of simulated energy intake and expenditure rates corresponding to a healthy range of body compositions similar to the cross-sectional data that were reported by Ellis and colleagues. To represent the normal increase in energy requirements during growth, we simulated a gradual increase in energy intake from ages 5–18 years of roughly 1200 kcal per day in boys and 900 kcal per day in girls ( figure 1A, 1B ). The resulting model predictions for accretion of childhood body fat and fat-free mass compared favourably with cross-sectional data from 292 healthy white boys and girls reported by Ellis and colleagues ( figure 1C, 1D ).Similarly, model simulations compared favourably with body composition data from African-American and Hispanic children (data not shown).The small difference between daily intake and expenditure rates—the so-called energy imbalance gap—was less than 100 kcal per day for the entire timecourse of healthy growth in both boys and girls ( figure 1E, 1F ). The energy imbalance gap was bimodal, with peaks at ages 10 years and 15 years in boys and 9 years and 13·5 years in girls.

29 Ellis KJ

Shypailo RJ

Abrams SA

Wong WW The reference child and adolescent models of body composition. A contemporary comparison. 30 Spadano JL

Bandini LG

Must A

Dallal GE

Dietz WH Longitudinal changes in energy expenditure in girls from late childhood through midadolescence. Figure 2 Longitudinal and model-simulated data for bodyweight and body fat mass (A) and energy expenditure (B) in healthily growing girls, and longitudinal and model-simulated data bodyweight and body fat mass (C) and energy expenditure (D) during development of obesity Show full caption 30 Spadano JL

Bandini LG

Must A

Dallal GE

Dietz WH Longitudinal changes in energy expenditure in girls from late childhood through midadolescence. 31 Salbe AD

Weyer C

Harper I

Lindsay RS

Ravussin E

Tataranni PA Assessing risk factors for obesity between childhood and adolescence: II. Energy metabolism and physical activity. Data are mean; error bars show SD. Data for (A) and (B) were reported by Spadano and colleagues;those for (C) and (D) were reported by Salbe and colleagues. Whereas Ellis and colleaguesprovided cross-sectional body composition data in children, Spadano and colleaguesreported unique longitudinal data for both body composition ( figure 2A ) and energy expenditure ( figure 2B ) during healthy growth in girls from ages 10–16 years. Our model accurately reproduced the body composition and energy expenditure data ( figure 2A, 2B ), and thus represents the energy balance and energy partitioning dynamics of healthy childhood growth.

31 Salbe AD

Weyer C

Harper I

Lindsay RS

Ravussin E

Tataranni PA Assessing risk factors for obesity between childhood and adolescence: II. Energy metabolism and physical activity. Definition of childhood obesity or excess weight gain is not straightforward because healthy growth is highly variable and the trajectories of excess weight gain can be non-linear. Various timecourses of energy intake could be simulated to generate a range of bodyweight trajectories. However, for simplicity, we simulated the development of childhood obesity by gradually increasing the rate of energy intake from age 5 years to generate an excess energy imbalance gap with time; all other parameters were identical to those in the model of healthy growth. Figures 2C and 2D show body composition and energy expenditure during the development of obesity between the ages of 5 years and 10 years measured in a longitudinal study by Salbe and colleagues.Compared with healthily growing children ( figure 2A, 2B ), the obese children had more than double the fat mass and their energy expenditure was roughly 300 kcal per day higher at 10 years old.

32 Wells JC

Fewtrell MS

Williams JE

Haroun D

Lawson MS

Cole TJ Body composition in normal weight, overweight and obese children: matched case-control analyses of total and regional tissue masses, and body composition trends in relation to relative weight. 29 Ellis KJ

Shypailo RJ

Abrams SA

Wong WW The reference child and adolescent models of body composition. A contemporary comparison. 32 Wells JC

Fewtrell MS

Williams JE

Haroun D

Lawson MS

Cole TJ Body composition in normal weight, overweight and obese children: matched case-control analyses of total and regional tissue masses, and body composition trends in relation to relative weight. Figure 3 Experimental and model-simulated data of bodyweight in obese and healthy weight boys (A) and girls (B), body composition phase plane showing the progression of body fat and fat-free mass between ages 5 years and 11 years in boys (C) and girls (D), and model-simulated energy intake during healthy growth and development of obesity in boys (E) and girls (F) Show full caption 32 Wells JC

Fewtrell MS

Williams JE

Haroun D

Lawson MS

Cole TJ Body composition in normal weight, overweight and obese children: matched case-control analyses of total and regional tissue masses, and body composition trends in relation to relative weight. Data are mean; error bars show SD. Data were reported by Wells and colleagues. We also calculated the energy balance dynamics underlying the body composition differences in pairs of age-matched and sex-matched healthy weight and obese children as measured by Wells and coworkers ( figure 3 ).The model was initiated with the mean body composition of healthy weight children aged 5 yearsand the rate of energy intake was adjusted to arrive at the bodyweights reported by Wells and coworkersin 11-year-old boys and girls ( figure 3A, 3B ). The simulated changes in body composition reproduced the data, showing that the model correctly simulated the reported tissue deposition during both healthy growth and the development of obesity ( figure 3C, 3D ).

From ages 5–11 years, the mean energy intake was roughly 750 kcal per day higher in obese than in healthy weight boys ( figure 3E ) and roughly 850 kcal per day higher in obese than in healthy weight girls ( figure 3F ). At the end of this simulated 6 year period, obese boys were predicted to be eating roughly 1100 kcal per day more than healthy weight boys and obese girls to be eating roughly 1300 kcal per day more than healthy weight girls.

14 Wang YC

Gortmaker SL

Sobol AM

Kuntz KM Estimating the energy gap among US children: a counterfactual approach. Figure 4 Predicted excess energy intake averaged since age 5 years to generate excess simulated bodyweight in boys (A) and girls (B), and corresponding predicted excess energy intake at the end of the simulation in boys (C) and girls (D) Show full caption 14 Wang YC

Gortmaker SL

Sobol AM

Kuntz KM Estimating the energy gap among US children: a counterfactual approach. 6 Hall KD

Sacks G

Chandramohan D

et al. Quantification of the effect of energy imbalance on bodyweight. 14 Wang YC

Gortmaker SL

Sobol AM

Kuntz KM Estimating the energy gap among US children: a counterfactual approach. Model simulations were compared with previously published calculations by Wang and colleagues.Their calculations refer to a 10 year period and correspond to our simulations for the period 5–15 years. The rule for sedentary adults is that every 22 kcal per day increase in energy intake will increase bodyweight by roughly 1 kg after several years.Error bars show the range of calculations under different energy efficiency assumptions of Wang and colleagues. To generalise our results, we simulated the mean energy intake in excess of healthy growth requirements to generate varying degrees of excess weight at different ages. The duration of excess energy intake increased with age because all simulations were initiated with healthy weight children at age 5 years. For a given mean excess energy intake during the simulation, greater excess weight accumulated as the child aged. For comparison, figure 4A, 4B plot the counterfactual analysis of Wang and colleagues,who calculated the mean excess energy intake during a 10 year period to produce the noted excess weight gains in adolescents. Their results should be compared with ours for ages 5–15 years; generally, results agreed for the mean excess weight gained. However, our calculated mean excess energy intake in overweight adolescents was roughly 250 kcal per day lower than that of Wang and colleagues, suggesting that less energy intake is needed to generate overweight and obese adolescents than was previously calculated.

Boys: kcal per day per kg = 68 - 2 · 5 × age

Girls: kcal per day per kg = 62 - 2 · 2 × age

To illustrate the practical use of these rules at the population level, we compared US bodyweight data gathered in 2003–06 33 McDowell MA

Fryar CD

Ogden CL

Flegal KM Anthropometric reference data for children and adults: United States, 2003–2006. 34 Ogden CL

Fryar CD

Carroll MD

Flegal KM Mean body weight, height, and body mass index, United States 1960–2002. Table Mean excess bodyweight and calculated excess energy intake in children, by age Boys Girls Excess bodyweight (kg) Excess energy intake (kcal per day) Excess bodyweight (kg) Excess energy intake (kcal per day) 7 years 1·7 85 3·0 139 8 years 5·0 238 3·2 142 9 years 3·0 135 4·7 198 10 years 5·9 251 8·6 343 11 years 8·0 320 8·0 301 12 years 6·8 255 6·2 220 13 years 7·9 276 5·6 186 14 years 6·8 220 4·2 130 15 years 9·9 295 4·3 124 16 years 9·2 251 4·7 125 17 years 6·4 158 6·5 159 18 years 2·9 64 9·4 209 33 McDowell MA

Fryar CD

Ogden CL

Flegal KM Anthropometric reference data for children and adults: United States, 2003–2006. 34 Ogden CL

Fryar CD

Carroll MD

Flegal KM Mean body weight, height, and body mass index, United States 1960–2002. We calculated these mean results by comparing mean US bodyweight data from 2003–06with that from 1971–74. Figure 4C and 4D plot the excess energy intake consumed at each age. Our model predicts that children need substantially greater excess energy intake to generate the same degree of excess weight compared with older and more sedentary adults. We derived the following sex-specific rule for excess energy intake per unit excess weight in childhood (ages 7–18 years):To illustrate the practical use of these rules at the population level, we compared US bodyweight data gathered in 2003–06with those gathered in 1971–74to estimate the mean excess energy intake in children aged 7–18 years ( table ). Averaged across all ages, mean bodyweight increased by 6·1 kg in boys and 5·7 kg in girls ( table ), which translates to a mean excess energy intake of 210 kcal per day in boys and 190 kcal per day in girls, and provides a quantitative estimate of the magnitude of intervention needed to prevent obesity in future generations. For example, reducing energy intake in a cohort of children by a mean of around 200 kcal compared with that in 2003–06 data will return the mean bodyweight to levels characteristic of the early 1970s—ie, before the onset of the obesity epidemic.

Another example contrasts our model predictions with the standard clinical practice of calculating excess positive energy balance. For a 10-year-old girl who is 10 kg overweight according to growth charts but 5 years ago was at the 50th percentile for bodyweight, the standard calculation assumes that each kilogram of excess weight is equivalent to roughly 7700 kcal of excess energy consumed, corresponding to a typical contribution of accretion of fat and fat-free masses. According to this calculation, she probably consumed around an extra 40 kcal per day over this period. By contrast, use of our rule implies that she is eating roughly 400 kcal per day in excess of a peer that remained at the 50th percentile from age 5–10 years. The marked differences in estimated kcal intake arise because the standard calculation focuses solely on the energy imbalance gap and does not account for the increase in energy expenditure resulting from the increased bodyweight. Our model calculation includes both energetic components and therefore estimates a substantially greater intake; compared with the standard calculation, the model presents a very different picture of the lifestyle changes needed for childhood obesity treatment.

35 Stallings VA

Archibald EH

Pencharz PB

Harrison JE

Bell LE One-year follow-up of weight, total body potassium, and total body nitrogen in obese adolescents treated with the protein-sparing modified fast. Figure 5 Experimental and model-simulated changes in body composition before and after a 3 month protein-sparing modified fast in obese children (A) and model-simulated changes in energy intake and expenditure (B); and experimental and model-simulated changes in body composition before and after an 8 month outpatient diet intervention in boys (C) and girls (D) Show full caption 35 Stallings VA

Archibald EH

Pencharz PB

Harrison JE

Bell LE One-year follow-up of weight, total body potassium, and total body nitrogen in obese adolescents treated with the protein-sparing modified fast. 36 Lazzer S

Boirie Y

Poissonnier C

et al. Longitudinal changes in activity patterns, physical capacities, energy expenditure, and body composition in severely obese adolescents during a multidisciplinary weight-reduction program. Data for (A) and (B) were reported by Stallings and colleagues;those for (C) and (D) were reported by Lazzer and colleagues.Datapoints show means and error bars show SD. We compared the outcomes of our model with the body composition changes measured during two weight loss intervention studies in children. The first study, by Stallings and colleagues,was unusual; children who were obese were put on an inpatient protein-sparing modified fast (880 kcal per day) for 3 months and subsequently followed up at 1 year. To simulate this study, we began the model with healthy body composition at age 5 years and overfed until age 15 years to generate the bodyweight and fat mass noted at the start of the intervention ( figure 5A ). Before the 880 kcal per day diet, the children were estimated to be consuming roughly 3000 kcal per day ( figure 5B ); we returned the simulated children to this energy intake after the 3 month diet. Our model correctly predicted the weight and fat mass changes recorded at the end of the diet intervention and after 1 year ( figure 5A ). Energy expenditure rate was predicted to fall substantially during the intervention and slowly recover during follow-up ( figure 5B ). Because the measured weight and fat mass at 1 year follow-up matched the model predictions when assuming a complete return to the original diet, we expect that these children received only a temporary benefit from the intervention and that they were on their way to obesity relapse.

36 Lazzer S

Boirie Y

Poissonnier C

et al. Longitudinal changes in activity patterns, physical capacities, energy expenditure, and body composition in severely obese adolescents during a multidisciplinary weight-reduction program. The second study (by Lazzer and colleagues) with which we compared the output of our model was a more moderate weight loss intervention lasting 8 months ( figure 5C, 5D ). The study comprised an outpatient diet intervention and thus food intake could not be rigorously assessed. The model estimated that energy intake was decreased by roughly 1000 kcal per day, resulting in the simulated changes of weight and fat mass that closely matched the data of Lazzer and colleagues ( figure 5C, 5D ). Thus, the model correctly predicted the recorded body composition changes during weight loss in obese children in this dataset.