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

Abstract

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

Methods: 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.

Findings: 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 [corrected]. 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.

Interpretation: 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.

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

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)

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.

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

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.

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) Data are mean; error bars show SD. Data were reported by Wells and colleagues.

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)

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.

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) 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.

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)

Model simulations of both obese and healthy weight children were begun at age 5 years at a healthy bodyweight and composition. After age 11 years, energy intake was decreased in obese boys and girls to maintain the same weight until age 16 years. Data were reported by Wells and colleagues. Datapoints show mean and error bars show SD.