Predicting weight-loss using differential equations (PRELUDE).

The 3500 kcal rule(1) is a weight-loss prescription that approximates a 3500 kcal energy deficit to 1 lb of weight-loss(2). Energy intake (EI) and energy expenditure (EE) are considered independent variables and weight is assumed to decrease at a fixed rate driven purely by behaviour i.e., calorie restriction or physical activity. However, such static modelling significantly overestimates weight-loss by disregarding the presence of obligatory and adaptive perturbations in energy balance in response to underfeeding.(3). Adaptive thermogenesis refers to the underfeeding-associated decline in resting energy expenditure beyond that caused by obligatory changes in body-composition(4). Moreover, physiological mechanisms alter both EI and EE to maintain weight at a genetically determined set-point(5), resulting in less-than-expected weight-loss. 
Our research proposes a mathematical model of weight-loss that uses inputs of weight and calorie intake to convert energy deficit to weight-loss over time. EE is subdivided into resting energy expenditure (REE), physical activity energy expenditure (PAEE) and diet-induced thermogenesis (DIT). REE is modelled using predictor equations based on gender- and weight-specific FFM estimates obtained from the literature. PAEE is modelled as a product of REE and DIT is modelled as a product of calorie intake. The resulting ordinary differential equation accounts for obligatory and adaptive adjustments in all three compartments in response to underfeeding. 
Our model was tested using a large database from a commercial very-low calorie weight-loss programme with an observed mean weight-loss of 8.5%. While static modelling overestimated weight-loss by ~50% (12.5±3.6% vs. 8.5±4.5%), our refined model had a mean error of <1% (9.3±2.2% vs. 8.5±4.5%).
Our research supports the role of mathematical modelling in clinical weight management.  Reliance on simple inputs of weight and calorie intake makes our model applicable in a clinical setting.