Mathematical Biosciences and Engineering, 2009, 6(4): 873-887. doi: 10.3934/mbe.2009.6.873.

Primary: 58F15, 58F17; Secondary: 53C35.

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

A mathematical model of weight change with adaptation

1. Department of Mathematical Sciences, Montclair State University, Upper Montclair, NJ 07043
2. Department of Mathematical Sciences, Montclair State University, Montclair, NJ 07043
3. Department of Medicine, Endocrine Research Unit, Mayo Clinic and Mayo Foundation, Rochester, MN 55905
4. Pennington Biomedical Research Center, Ingestive Behavior Laboratory, Baton Rouge, LA 70808


As obesity and its related health problems grow around the world, efforts to control and manage weight is increasing in importance. It is well known that altering and maintaining weight is problematic and this has led to specific studies trying to determine the cause of the difficulty. Recent research has identified that the body reacts to forced weight change by adapting individual total energy expenditure. Key factors are an adaptation of resting metabolic rate, non-exercise activity thermogenesis and dietary induced thermogenesis. We develop a differential equation model based on the first law of thermodynamics that incorporates all three adjustments along with natural age related reduction of the resting metabolic rate. Forward time simulations of the model compare well with mean data in both overfeeding and calorie restriction studies.
  Article Metrics

Keywords metabolic adaptation; energy balance equation; non-exercise activity thermogenesis.

Citation: Diana M. Thomas, Ashley Ciesla, James A. Levine, John G. Stevens, Corby K. Martin. A mathematical model of weight change with adaptation. Mathematical Biosciences and Engineering, 2009, 6(4): 873-887. doi: 10.3934/mbe.2009.6.873


This article has been cited by

  • 1. Richard H. Singer, Mark Stoutenberg, Marc D. Gellman, Edward Archer, Sonia M. Davis, Nathan Gotman, David X. Marquez, Christina Buelna, Yu Deng, H. Dean Hosgood, Ruth E. Zambrana, Rebecca A Krukowski, Occupational Physical Activity and Body Mass Index: Results from the Hispanic Community Health Study / Study of Latinos, PLOS ONE, 2016, 11, 3, e0152339, 10.1371/journal.pone.0152339
  • 2. Corby K. Martin, Anastasia C. Miller, Diana M. Thomas, Catherine M. Champagne, Hongmei Han, Timothy Church, Efficacy of SmartLossSM, a smartphone-based weight loss intervention: Results from a randomized controlled trial, Obesity, 2015, 23, 5, 935, 10.1002/oby.21063
  • 3. Ruoyan Sun, Optimal weight based on energy imbalance and utility maximization, Physica A: Statistical Mechanics and its Applications, 2016, 442, 429, 10.1016/j.physa.2015.09.034
  • 4. James A. Levine, Sick of sitting, Diabetologia, 2015, 58, 8, 1751, 10.1007/s00125-015-3624-6
  • 5. Surabhi Bhutani, Eva Kahn, Esra Tasali, Dale A. Schoeller, Composition of two-week change in body weight under unrestricted free-living conditions, Physiological Reports, 2017, 5, 13, e13336, 10.14814/phy2.13336
  • 6. Kevin D. Hall, Metabolism of mice and men, Current Opinion in Clinical Nutrition and Metabolic Care, 2012, 15, 5, 418, 10.1097/MCO.0b013e3283561150
  • 7. Peter T. Katzmarzyk, Corby K. Martin, Robert L. Newton, John W. Apolzan, Connie L. Arnold, Terry C. Davis, Kara D. Denstel, Emily F. Mire, Tina K. Thethi, Phillip J. Brantley, William D. Johnson, Vivian Fonseca, Jonathan Gugel, Kathleen B. Kennedy, Carl J. Lavie, Eboni G. Price-Haywood, Daniel F. Sarpong, Benjamin Springgate, Promoting Successful Weight Loss in Primary Care in Louisiana (PROPEL): Rationale, design and baseline characteristics, Contemporary Clinical Trials, 2018, 67, 1, 10.1016/j.cct.2018.02.002
  • 8. Diana Thomas, Vincent Antonetti, , Advances in the Assessment of Dietary Intake, 2017, 211, 10.1201/9781315152288-13
  • 9. Penghong Guo, Daniel E. Rivera, Jennifer S. Savage, Danielle S. Downs, State Estimation Under Correlated Partial Measurement Losses: Implications for Weight Control Interventions * *Support for this work has been provided by the National Heart, Lung, and Blood Institute (NHLBI) through grant R01 HL119245. The opinions expressed in this article are the authors’ own and do not necessarily reflect the views of NHLBI. , IFAC-PapersOnLine, 2017, 50, 1, 13532, 10.1016/j.ifacol.2017.08.2347
  • 10. J.-Emeterio Navarro-Barrientos, Daniel E. Rivera, Linda M. Collins, A dynamical model for describing behavioural interventions for weight loss and body composition change, Mathematical and Computer Modelling of Dynamical Systems, 2011, 17, 2, 183, 10.1080/13873954.2010.520409
  • 11. Melinda M. Manore, Weight Management for Athletes and Active Individuals: A Brief Review, Sports Medicine, 2015, 45, S1, 83, 10.1007/s40279-015-0401-0
  • 12. Kevin D Hall, Gary Sacks, Dhruva Chandramohan, Carson C Chow, Y Claire Wang, Steven L Gortmaker, Boyd A Swinburn, Quantification of the effect of energy imbalance on bodyweight, The Lancet, 2011, 378, 9793, 826, 10.1016/S0140-6736(11)60812-X
  • 13. Paolo Nicola Barbieri, Modelling body weight, dieting and obesity traps, Physica A: Statistical Mechanics and its Applications, 2017, 468, 139, 10.1016/j.physa.2016.10.046
  • 14. Melinda Manore, D. Larson-Meyer, Anne Lindsay, Nobuko Hongu, Linda Houtkooper, Dynamic Energy Balance: An Integrated Framework for Discussing Diet and Physical Activity in Obesity Prevention—Is it More than Eating Less and Exercising More?, Nutrients, 2017, 9, 8, 905, 10.3390/nu9080905
  • 15. , , A Guide to Obesity and the Metabolic Syndrome, 2011, 283, 10.1201/b10790-15
  • 16. Nasim S. Sabounchi, Peter S. Hovmand, Nathaniel D. Osgood, Roland F. Dyck, Emily S. Jungheim, A Novel System Dynamics Model of Female Obesity and Fertility, American Journal of Public Health, 2014, 104, 7, 1240, 10.2105/AJPH.2014.301898
  • 17. Rachel A. H. Davis, Jacob E. Halbrooks, Emily E. Watkins, Gordon Fisher, Gary R. Hunter, Tim R. Nagy, Eric P. Plaisance, High-intensity interval training and calorie restriction promote remodeling of glucose and lipid metabolism in diet-induced obesity, American Journal of Physiology-Endocrinology and Metabolism, 2017, 313, 2, E243, 10.1152/ajpendo.00445.2016
  • 18. Leanne M Redman, L. Anne Gilmore, Jeffrey Breaux, Diana M Thomas, Karen Elkind-Hirsch, Tiffany Stewart, Daniel S Hsia, Jeffrey Burton, John W Apolzan, Loren E Cain, Abby D Altazan, Shelly Ragusa, Heather Brady, Allison Davis, J. Mick Tilford, Elizabeth F Sutton, Corby K Martin, Effectiveness of SmartMoms, a Novel eHealth Intervention for Management of Gestational Weight Gain: Randomized Controlled Pilot Trial, JMIR mHealth and uHealth, 2017, 5, 9, e133, 10.2196/mhealth.8228
  • 19. Kevin D. Hall, Modeling Metabolic Adaptations and Energy Regulation in Humans, Annual Review of Nutrition, 2012, 32, 1, 35, 10.1146/annurev-nutr-071811-150705
  • 20. K R Westerterp, Metabolic adaptations to over—and underfeeding—still a matter of debate?, European Journal of Clinical Nutrition, 2013, 67, 5, 443, 10.1038/ejcn.2012.187
  • 21. E J Dhurandhar, K A Kaiser, J A Dawson, A S Alcorn, K D Keating, D B Allison, Predicting adult weight change in the real world: a systematic review and meta-analysis accounting for compensatory changes in energy intake or expenditure, International Journal of Obesity, 2015, 39, 8, 1181, 10.1038/ijo.2014.184
  • 22. Peter Gennemark, Rasmus Jansson-Löfmark, Gina Hyberg, Maria Wigstrand, Dorota Kakol-Palm, Pernilla Håkansson, Daniel Hovdal, Peter Brodin, Maria Fritsch-Fredin, Madeleine Antonsson, Karolina Ploj, Johan Gabrielsson, A modeling approach for compounds affecting body composition, Journal of Pharmacokinetics and Pharmacodynamics, 2013, 40, 6, 651, 10.1007/s10928-013-9337-x
  • 23. Diana M. Thomas, M. Cristina Gonzalez, Andrea Z. Pereira, Leanne M. Redman, Steven B. Heymsfield, Time to Correctly Predict the Amount of Weight Loss with Dieting, Journal of the Academy of Nutrition and Dietetics, 2014, 114, 6, 857, 10.1016/j.jand.2014.02.003
  • 24. K.D. Hall, Mechanisms of Metabolic Fuel Selection: Modeling Human Metabolism and Body-Weight Change, IEEE Engineering in Medicine and Biology Magazine, 2010, 29, 1, 36, 10.1109/MEMB.2009.935465
  • 25. Diana M. Thomas, Corby K. Martin, Steven Heymsfield, Leanne M. Redman, Dale A. Schoeller, James A. Levine, A simple model predicting individual weight change in humans, Journal of Biological Dynamics, 2011, 5, 6, 579, 10.1080/17513758.2010.508541
  • 26. B. Jon Schoenfeld, A. Albert Aragon, J. W. Krieger, Effects of meal frequency on weight loss and body composition: a meta-analysis, Nutrition Reviews, 2015, 73, 2, 69, 10.1093/nutrit/nuu017
  • 27. Carl Pieper, Leanne Redman, Susan Racette, Susan Roberts, Manju Bhapkar, James Rochon, Corby Martin, William Kraus, Sai Das, Donald Williamson, Eric Ravussin, Development of adherence metrics for caloric restriction interventions, Clinical Trials: Journal of the Society for Clinical Trials, 2011, 8, 2, 155, 10.1177/1740774511398369
  • 28. Miguel A. Rojo-Tirado, Pedro J. Benito, David Atienza, Emiliano Rincón, Francisco J. Calderón, Effects of age, sex, and treatment on weight-loss dynamics in overweight people, Applied Physiology, Nutrition, and Metabolism, 2013, 38, 9, 967, 10.1139/apnm-2012-0441
  • 29. Emily J. Dhurandhar, Methodological Framework: Discovering Foods or Nutrients with a Causative Influence on Body Weight in Free-Living Settings, Frontiers in Nutrition, 2014, 1, 10.3389/fnut.2014.00017
  • 30. Edward Archer, Diana M. Thomas, Samantha M. McDonald, Gregory Pavela, Carl J. Lavie, James O. Hill, Steven N. Blair, The Validity of US Nutritional Surveillance: USDA’s Loss-Adjusted Food Availability Data Series 1971-2010, Current Problems in Cardiology, 2016, 41, 11-12, 268, 10.1016/j.cpcardiol.2016.10.007
  • 31. Bill I. Campbell, Danielle Aguilar, Laurin Conlin, Andres Vargas, Brad Jon Schoenfeld, Amey Corson, Chris Gai, Shiva Best, Elfego Galvan, Kaylee Couvillion, Effects of High Versus Low Protein Intake on Body Composition and Maximal Strength in Aspiring Female Physique Athletes Engaging in an 8-Week Resistance Training Program, International Journal of Sport Nutrition and Exercise Metabolism, 2018, 1, 10.1123/ijsnem.2017-0389
  • 32. Abby D. Altazan, Leanne M. Redman, Jeffrey H. Burton, Robbie A. Beyl, Loren E. Cain, Elizabeth F. Sutton, Corby K. Martin, Mood and quality of life changes in pregnancy and postpartum and the effect of a behavioral intervention targeting excess gestational weight gain in women with overweight and obesity: a parallel-arm randomized controlled pilot trial, BMC Pregnancy and Childbirth, 2019, 19, 1, 10.1186/s12884-019-2196-8
  • 33. Sunmi Yoo, Dynamic Energy Balance and Obesity Prevention, Journal of Obesity & Metabolic Syndrome, 2018, 27, 4, 203, 10.7570/jomes.2018.27.4.203
  • 34. Hazhir Rahmandad, Nasim S. Sabounchi, , Social Computing, Behavioral - Cultural Modeling and Prediction, 2012, Chapter 37, 306, 10.1007/978-3-642-29047-3_37
  • 35. Paolo Nicola Barbieri, Body Weight, Dieting and Obesity Traps, SSRN Electronic Journal, 2015, 10.2139/ssrn.2689501
  • 36. L. Anne Gilmore, Leanne M. Redman, Application of mathematical models in the management of obesity during pregnancy and the postpartum period in reproductive age women, Nutrition Research, 2019, 10.1016/j.nutres.2019.03.013
  • 37. Andrea Franco, Marco Correia, Jorge Cruz, , Artificial Intelligence in Medicine, 2015, Chapter 21, 166, 10.1007/978-3-319-19551-3_21
  • 38. Andrea Franco, Marco Correia, Jorge Cruz, , Progress in Artificial Intelligence, 2015, Chapter 5, 41, 10.1007/978-3-319-23485-4_5
  • 39. Anarina L. Murillo, Kathryn A. Kaiser, Daniel L. Smith, Courtney M. Peterson, Olivia Affuso, Hemant K. Tiwari, David B. Allison, A Systematic Scoping Review of Surgically Manipulated Adipose Tissue and the Regulation of Energetics and Body Fat in Animals, Obesity, 2019, 10.1002/oby.22511
  • 40. Ebenezer Bonyah, Kojo Ababio, Patience Pokuaa Gambrah, , Mathematical Modelling, Applied Analysis and Computation, 2019, Chapter 19, 275, 10.1007/978-981-13-9608-3_19
  • 41. Christoph Höchsmann, James L. Dorling, John W. Apolzan, Neil M. Johannsen, Daniel S. Hsia, Corby K. Martin, Baseline Habitual Physical Activity Predicts Weight Loss, Weight Compensation, and Energy Intake During Aerobic Exercise, Obesity, 2020, 10.1002/oby.22766
  • 42. Oladapo Babajide, Tawfik Hissam, Palczewska Anna, Gorbenko Anatoliy, Arne Astrup, J. Alfredo Martinez, Jean-Michel Oppert, Thorkild I. A. Sørensen, , Computational Science – ICCS 2020, 2020, Chapter 33, 441, 10.1007/978-3-030-50423-6_33

Reader Comments

your name: *   your email: *  

Copyright Info: 2009, , licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved