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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
Lancet Diabetes Endocrinol. Author manuscript; available in PMC 2013 December 10.
Published in final edited form as:
PMCID: PMC3857695

Dynamics of childhood growth and obesity: development and validation of a quantitative mathematical model



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


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.


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 1976–80. 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.


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.


One of the most disconcerting aspects of the global obesity epidemic1 is the high prevalence of childhood obesity and the associated health2 and economic3 consequences. Tackling childhood obesity will necessitate a broad approach and a range of interventions, from clinical management to policy changes, that affect the entire population.4 To be effective, interventions should address the problem of food intake that is in excess of energy requirements for healthy growth and development.5 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.6,7

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.8 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 obese915 could 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.

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.



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.6,1618

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),1921 and modelled how physical activity decreases with age.2225

We used the reference childhood body composition data of Fomon and colleagues26 and Haschke27 to 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).28 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 ft 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.

Role of the funding source

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.


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).29 Similarly, model simulations compared favourably with body composition data from African-American and Hispanic children (data not shown).29 The small difference between daily intake and expenditure rates—the so-called energy imbalance gap6—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.

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 ...

Whereas Ellis and colleagues29 provided cross-sectional body composition data in children, Spadano and colleagues30 reported 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.

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 ...

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.31 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.

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).32 The model was initiated with the mean body composition of healthy weight children aged 5 years29 and the rate of energy intake was adjusted to arrive at the bodyweights reported by Wells and coworkers32 in 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).

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 ...

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.

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,14 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.

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)

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):

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 2003–0633 with those gathered in 1976–8034 to 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 late 1970s—ie, before the onset of the obesity epidemic.

Mean excess bodyweight and calculated excess energy intake in children, by age

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.

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,35 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.

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 ...

The second study (by Lazzer and colleagues36) 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.

Parents and clinicians often think that if weight is held constant, an obese child will outgrow their obesity. We simulated the necessary decrease in energy intake to maintain constant weight in obese children who were investigated by Wells and coworkers.32 Constant weight was achieved from age 11–16 years in the simulated obese boys and girls (figure 6A, 6B). The weight of boys obese at 11 years had almost normalised to that of healthy boys by age 16 years. By contrast, girls obese at 11 years still had substantially increased weight compared with healthy weight girls at 16 years. Simulated obese boys lost a large amount of body fat from 11–16 years and their body composition was almost normalised relative to healthy weight boys (figure 6C). The fact that weight remained constant in the obese boys during this period implies that fat-free mass increased substantially during concomitant loss of fat mass. By contrast, simulated obese girls lost much less body fat than did obese boys during the same period of weight stability (figure 6D).

Figure 6
Model simulations of changes in body composition during a successful weight maintenance intervention in boys (A) and girls (B), and of body fat mass in boys (C) and girls (D)


Quantification of the dynamics of energy balance during the development of childhood obesity is complex because growth has a substantial effect on energy expenditure and body composition. Furthermore, the weight gain associated with healthy growth presents a moving target in terms of the definition of excess weight and obesity. Therefore, a model of energy balance and body composition change during healthy growth is a prerequisite for proper understanding and management of childhood obesity.

Only one previous model has attempted to disentangle the dynamics of energy balance of excess weight versus healthy childhood growth (panel).14 Our results suggest that the previously calculated mean excess energy intake needed to generate overweight in adolescents was roughly 50% too high. This discrepancy is mainly because of the very high assumed inefficiency of energy deposition by Wang and colleagues,14 leading to an overestimation of the excess energy intake needed for weight gain.

The mathematical model by Butte and colleagues9 calculated the energy cost of weight gain noted in children over a period of 1 year. Our results generally accord with those of that model in terms of mean energy intake needed for various rates of weight gain (data not shown). However, in our model, we deemed healthy growth the appropriate background for calculation of excessive weight gain and energy intake—Butte and coworkers did not separately account for healthy growth.

Quantification of the intervention magnitude needed for treatment of obese children necessitates a model that correctly predicts how a known intervention results in changes of both energy expenditure and body composition. Unfortunately, few data are available for how these variables simultaneously change in response to controlled weight-management interventions in children. Although further validation is clearly needed, our model represents the best available method for prediction of how an intervention will affect childhood energy metabolism and changes in body composition. We validated our model with a range of datasets that were not used to build the model, including body composition and energy expenditure data in healthy-weight and obese children and changes in body composition in obese children during periods of weight loss and regain. We showed that our model accurately reproduced the noted changes in body composition during weight loss resulting from diet interventions in obese children. This finding was particularly noteworthy because our model was calibrated on the basis of body composition data during weight gain associated with healthy growth and no additional assumptions were made about the relative proportions of loss of body fat and fat-free mass during a weight loss intervention. Proper validation of our model's predictions of changes in energy expenditure during controlled weight loss interventions in obese children is crucial.

Another interesting simulation result was the possibility that obese children might outgrow obesity if a successful weight-maintenance intervention is instituted during a period of rapid growth. In this situation, the model predicted that the strong drive to accrete fat-free mass translates to a substantial loss of body fat despite no change in bodyweight, which would be very difficult to achieve in adults. The prospect of outgrowing childhood obesity was predicted to be much greater in boys than in girls for two main reasons. First, the simulated obese girls had already accumulated roughly 9 kg of additional body fat compared with the simulated obese boys by age 11 years, and normalisation of body composition for such high amounts of body fat could not be achieved without a decrease in bodyweight. Second, the intervention occurred during the period of maximum growth in boys, who accrete more fat-free mass than do girls in this period. Thus, weight-management interventions that occur during periods outside the maximum growth potential are predicted to lead to loss of substantially less body fat. This important prediction about the optimum timing of weight-management interventions in childhood should be experimentally investigated.

The simulated interventions in our study were limited to those that affect energy intake. However, the mathematical model also represents physical activity expenditure, and simulations of physical activity interventions alone or in combination with diet interventions are possible. Such simulations are beyond the scope of this Article but will be the topic of future work.

A limitation of our mathematical model is that some individual children might not be well represented by the model parameters that seem to adequately capture the group average dynamics exemplified in this study. For example, children who enter adolescence substantially earlier or later than average might not be well represented by the average model predictions during this period. Similarly, the rules that we derived might not apply to obese children who arrived at their bodyweight by a trajectory substantially different from the gradual weight gain assumed in the model. However, different weight gain trajectories can be simulated in our model and will be the subject of future studies.

This study presents the first mathematical model of childhood energy balance that accurately simulates healthy growth, the development of childhood obesity, and the effect of interventions on the body composition of obese children. The model represents a substantial step forward by providing a quantitative technique to accurately estimate the energy balance dynamics of childhood bodyweight. Importantly, once growth stops, our childhood model transforms into our previously validated adult model,6,16 thereby allowing for the simulation of bodyweight and energy balance dynamics during most of the human lifespan. It does not address the upstream causes of obesity development, such as changes in the food environment, but it can be used to better quantify the energy excess needed to produce a given level of overweight or obesity. The model can also be used to calculate the magnitude of intervention necessary to achieve a desired change in bodyweight. Policy makers and clinicians now have a quantitative technique for understanding childhood obesity and specifying energy balance benchmarks for interventions to address the childhood obesity epidemic.

Panel: Research in context

Systematic review

We searched PubMed up to June 25, 2013, without language restrictions, for the terms “child” and “weight” and (“energy balance” or “energy imbalance” or “energy gap”) and (“model” or “quantify” or “calculate” or “estimate”). Our search returned 51 citations, but the relation between energy balance and childhood bodyweight was quantified in only seven.915 Three of these previous studies10,11,13 calculated the energy gap underlying childhood weight gain, but did not properly account for how weight gain affects energy expenditure. Only four previous studies attempted to account for the energy cost of increased bodyweight during growth.9,12,14,15


Previous estimates of the energy gap underlying excess childhood weight gain were based on calculation of the energy content of bodyweight change during a specified period along with an assumption about the energy efficiency of weight gain.10,11,13 Such an energy efficiency factor does not correctly account for changes in energy expenditure during growth.37 More realistic models that account for energy expenditure changes during childhood have been developed,9,12,14,15 but none adequately distinguished healthy growth from excessive weight gain, has been validated relative to data not used to build the model, or was designed to predict how interventions affect body composition during childhood.

We describe the first validated mathematical model of childhood energy balance that accounts for healthy growth and development of obesity that makes quantitative predictions about weight-management interventions in childhood. Our model provides policy makers and clinicians with a new technique to accurately quantify energy balance benchmarks for interventions in childhood obesity. Future research needs to validate the model predictions for energy expenditure changes during controlled weight loss interventions in obese children.


This work was partly supported by the Intramural Research Program of the National Institutes of Health, National Institute of Diabetes and Digestive and Kidney Diseases. We thank Steve Gortmaker, Patty Mabry, and Gary Sacks for their insightful comments on the Article.


See Online for appendix

For the Berkeley Madonna see

Contributors: All authors participated in the design of the study. KDH and CCC created the mathematical model. KDH calibrated and validated the model. NFB and BAS provided access to data used to calibrate the model. All authors contributed to the writing of the paper and approved the final version.

Conflicts of interest: KDH reports a US patent application assigned to the National Institutes of Health related to the use of mathematical models of human metabolism for bodyweight management. All other authors declare that they have no conflicts of interest.

Contributor Information

Kevin D Hall, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD, USA.

Nancy F Butte, Baylor College of Medicine, Houston, TX, USA.

Prof Boyd A Swinburn, School of Population Health, University of Auckland, Auckland, New Zealand.

Carson C Chow, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD, USA.


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