About the Model

A validated dynamic mathematical energy balance model that predicts weight change (1) was developed from the energy balance equation based on the first law of thermodynamics (2) which states that the rate of energy stored/lost, ES, is equal to the difference of rate of energy intake, EI, and the rate of energy expended, EE,

ES = EI - EE

The model considered the rate of energy stored/lost as the rate of change of fat free mass (FFM) energy and fat mass energy (FM). The energy densities of FFM and FM, derived from chemical tissue analysis, is estimated as 1020 kcal/kg and 9500 kcal/kg respectively (3,4)

ES = 1020 dFFM +9500 dFM
dt dt

EE was modeled as the sum of resting metabolic rate (RMR), voluntary physical activity (PA), dietary induced thermogenesis (DIT), and spontaneous physical activity (SPA)


The non-linear function of weight, gender, and age proposed by Livingston and Kohlstadt (5) was applied for the RMR term (Table 1):

RMR = ci Wpi -yiA

where ci ,pi ,yi are constants depending on gender: i = F,M. The Livingston-Kohlstadt model was developed using cross-sectional RMR subject data (N>600) and validated on over 700 subject data points (R2 > 0.71).

PA is modeled by a term that is directly proportional to weight:

PA = mW

and DIT is modeled as a direct proportion of energy intake (6) :


SPA was related to total energy expenditures using both overfeeding and underfeeding experimental conclusions. Specifically, it was observed that


during weight loss (7-9) and


during weight gain (10).

Combining all terms yields the full one dimensional differential equation energy balance model:


FFM-FM equations developed from NHANES data (N>10,000) (13).

FFM = -72.1+ 2.5FM - 0.04A + 0.7H - 0.002FM(A) - 0.01FM(H) - 0.04FM2 + 0.00003FM2A + 0.0000004FM4 + 0.0002FM3 + 0.0003FM2H - 0.000002FM3H
FFM = -71.7+ 3.6FM - 0.04A + 0.7H - 0.002FM(A) - 0.01FM(H) - 0.07FM2 + 0.00003FM2A - 0.000002FM4 + 0.0006FM3 + 0.0003FM2H - 0.000002FM3H

FFM is related to FM through a model derived from the recently released National Health and Nutrition Assessment Survey (NHANES) (13) which contained over 10,000 dual energy X-ray absorptiometry (DXA) measured body composition values, along with subject age, height race and gender (Table 1). We developed the FFM-FM relationship for specific use within the Heymsfield energy balance equation (15).

The model has been recently applied and validated as a tool to assess energy intake during weight loss (16).

  1. D. M. Thomas, C. K. Martin, S. B. Heymsfield, L. M. Redman, D. A. Schoeller, et al, A simple model predicting individual weight change in humans. J. Biol. Dyn., in press (2010).
  2. W.D. McArdle, F.J. Katch, V.L. Katch (eds). Exercise Physiology (Williams & Wilkins, Baltimore, MD, 2009) [seventh edition].
  3. S.B. Heymsfield, M. Waki, J. Kehayias, S. Lichtman S, F.A. Dilmanian, Y. Kamen, J. Wang J, R.N. Pierson Jr. Chemical and elemental analysis of humans in vivo using improved body composition models. Am. J. Physiol. 261, 191-198 (1991).
  4. Y. Schutz, Glossary of energy terms and factors used for calculations of energy metabolism in human studies. Human Energy Metabolism: Physical Activity and Energy Expenditure Measurements in Epidemiological Research Based Upon Direct and Indirect Calorimetry. A. J. H. van Es, Ed. (The Hague, The Netherlands: Koninklijke Bibliotheek, 1984). pp. 169-181.
  5. E.H. Livingston, I. Kohlstadt, Simplified resting metabolic rate—predicting formulas for normal-sized and obese individuals. Obes. Res. 13, 1255-1262 (2005).
  6. K.R. Westerterp, KR. Diet induced thermogenesis. Nutr. Metab. 1, 1-5 (2004).
  7. S.B Roberts, I. Rosenberg, Nutrition and aging:changes in the regulation of energy metabolism with aging. Physiol. Rev. 86, 651-667 (2005).
  8. L.G. Bandini, D.A. Schoeller, J. Edwards, V.R. Young, S.H. Oh, W.H. Dietz, Energy expenditure during carbohydrate overfeeding in obese and nonobese adolescents. Am. J. Clin. Nutr. 256, E357-E36 (1989).
  9. E.O. Diaz, A.M. Prentice, G.R. Goldberg, P.R. Murgatroyd, W.A. Coward, Metabolic response to experimental overfeeding in lean and overweight healthy volunteers. Am. J. Clin. Nutr. 56, 641-655 (1993).
  10. J.A Levine, L.M. Lanningham-Foster, S.K. McCrady, A.C. Krizan, L.R. Olson, P.H. Kane, M.D. Jensen, M.M. Clark, Interindividual variation in posture allocation: possible role in human obesity. Science 307, 530-531 (2005).
  11. L.K. Heilbronn, L. de Jonge, M.I. Frisard, J.P. DeLany, D.E. Larson-Meyer, J. Rood, T. Nguyen, C.K. Martin, J. Volaufova, M.M. Most, F.L Greenway, S.R. Smith, W.A. Deutsch, D.A. Williamson, E. Ravussin and Team, Pennington CALERIE. Effect of 6-Month Calorie Restriction on Biomarkers of Longevity, Metabolic Adaptation, and Oxidative Stress in Overweight Individuals. JAMA 295, 1539-48 (2006).
  12. S.B Racette, D.A. Schoeller, R.F. Kushner, K.M. Neil, K. Herling-Iaffaldano K. Effects of aerobicexercise and dietary carbohydrate on energy expenditure and body composition during weight reduction in obese women. Am. J. Clin. Nutr. 61, 486-494 (1995).
  13. G.L. Blackburn, National Health and Nutrition Examination Survey: where nutrition meets medicine for the benefit of health. Am. J. Clin. Nutr. 78, 197 – 198 (2003).
  14. D. Thomas, S. Das, J. Levine, C.K. Martin. L. Mayer, A. McDougall B.J. Strauss, S.B. Heymsfield, New fat free mass - fat mass model for use in physiological energy balance equations. Nutr. Metab. 7, 1-11 (2010).
  15. D. M. Thomas, C. K. Martin, S. B. Heymsfield, L. M. Redman, D. A. Schoeller, et al, A simple model predicting individual weight change in humans. J. Biol. Dyn., in press (2010).
  16. D. M. Thomas, D. A. Schoeller, L. A. Redman, C. K. Martin, J. A. Levine, et al, A computational model to determine energy intake during weight loss. Am. J. Clin. Nutr., doi:10.3945/ajcn.2010.29687 (2010).

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The java applet was made by Carl Bredlau and Steven Lettieri.