A computational model to determine energy intake during weight loss

Abstract

Background: Energy intake (EI) during weight loss is difficult and costly to measure accurately.

Objective: The objective was to develop and validate a computational energy balance differential equation model to determine individual EI during weight loss.

Design: An algorithm was developed to quantify EI during weight loss based on a validated one-dimensional model for weight change. By using data from a 24-wk calorie-restriction study, we tested the validity of the EI model against 2 criterion measures: 1) EI quantified through food provision from weeks 0-4 and 4-12 and 2) EI quantified through changes in body energy stores [measured with dual-energy X-ray absorptiometry (DXA)] and energy expenditure [measured with doubly labeled water (DLW)] from weeks 4-12 and 12-24.

Results: Compared with food provision, the mean (±SD) model errors were 41 ± 118 kcal/d and -22 ± 230 kcal/d from weeks 0-4 and 4-12, respectively. Compared with EI measured with DXA and DLW, the model errors were -71 ± 272 kcal/d and -48 ± 226 kcal/d from weeks 4-12 and 12-24, respectively. In every comparison, the mean error was never significantly different from zero (P values > 0.10). Furthermore, Bland and Altman analysis indicated that error variance did not differ significantly over amounts of EI (P values > 0.26). Almost all individual participants' values were within CI limits.

Conclusion: The validity of the newly developed EI model was supported by experimental observations and can be used to determine an individual participant's EI during weight loss.

FIGURE 1

A: An example of measured body mass (circles, left axis) in kilograms, model estimates of energy intake (EI; dashed curve, right axis) in kilocalories per day, prescribed EI (solid horizontal line, right axis), and differential equation-generated (7) weight-loss curve (solid curve, left axis) in kilograms applying model estimates of EI in a low-calorie diet subject. B: A calorie-restricted subject from the CALERIE (Comprehensive Assessment of Long-term Effects of Reducing Intake of Energy) phase I study generated after determining EI by applying the shooting method (iterative) for solving a boundary value (limiting independent variable) problem outlined in the supplementary materials under “Supplemental data” in the online issue. As expected, body weight increased only when EI exceeded the target energy requirements. Comparison of model estimates of EI to dual-energy X-ray absorptiometry and doubly labeled water estimates of EI during weeks 12–24 resulted in an absolute error <17 kcal/d for both subjects, which supports the finding that the subjects’ actual EI increased by amounts predicted by the model.

FIGURE 2

Model estimates of energy intake (EI; kcal/d) compared with a corresponding reference method for each CALERIE (Comprehensive Assessment of Long-term Effects of Reducing Intake of Energy) phase I study subject. A: Model EI estimates compared with food provision EI from baseline to week 4 of the study (r = 0.99, P < 0.0001) and weeks 12–24 (r = 0.92, P < 0 0.0001). B: Model EI estimates compared with food provision EI during weeks 4–12. C: Model EI estimates compared with EI from the dual-energy X-ray absorptiometry and doubly labeled water (DXA/DLW) method during weeks 4–12 (r = 0.90, P < 0 0.0001). D: Model EI estimates compared with EI from the DXA/DLW method during weeks 12–24 (r = 0.90, P < 0.0001). The y = x diagonal (dashed) is provided for reference along with the data regression line (solid). The 4 panels show good correlation between the model estimates of EI and the criterion method.

FIGURE 3

The Bland-Altman approach (12) comparing energy intake (EI) model estimates to EI from reference methods. A: EI from food provision as the criterion from baseline to week 4 [² = 0.000, F(1,21) = 0.009, P = 0.93]. B: EI from food provision as the criterion from weeks 4–12 [² = 0.059, F(1,21) = 1.33, P = 0.26]. C: EI from dual-energy X-ray absorptiometry and doubly labeled water (DXA/DLW) as the criterion from weeks 4–12 [R² = 0.029, F(1,21) = 0.636, P = 0 0.43]. D: EI from DXA/DLW as the criterion from weeks 12–24 [R² = 0.017, F(1,21) = 0.354, P = 0.56] for the CALERIE (Comprehensive Assessment of Long-term Effects of Reducing Intake of Energy) phase I (9) subjects. The y-axis represents the difference between EI model estimates and EI from the criterion method (food provision or DXA/DLW), and the x-axis represents the mean EI derived from the model and criterion method. The Bland-Altman graphs indicate good agreement between model estimates of EI and criterion EI in all cases. KR, kitchen record.