shows our model simulations of weight change and body fat change along with experimental data from the CALERIE study80
that investigated 6 months of 25% caloric restriction, 12·5% caloric restriction plus exercise, and 3 months of a liquid diet of 3·7 MJ per day followed by a period of weight maintenance. The close agreement between the model predictions and the data provides some validation of the mathematical model, since these data were not used for model development. shows another validation study, in which two obese women were provided with liquid diets of about 3·3 MJ per day in an inpatient setting.81
The model accurately reproduced the different rates of weight loss resulting from the differing predicted energy expenditure rates in these women. Similarly, , F shows the simulated and measured weight changes over the course of a 30 day fast in obese men and women, respectively.83
Again, the simulated weight change dynamics agree reasonably well with the data thereby providing some confidence that the model accurately captures the quantitative physiology of weight loss.
Predicted bodyweight and fat-mass changes by use of a dynamic simulation model of human metabolism
Calculation of the energy deficit generated by a given diet requires knowledge of the energy needed to maintain the baseline bodyweight. Unfortunately, we cannot measure the initial energy requirements of a free-living individual with a precision better than about 5%.83
More typically, the initial uncertainty will be much greater than 5% unless the specialised and expensive doubly-labelled water method is used to measure energy expenditure before the intervention. The uncertainty of the baseline energy requirements translates to an expected inter-individual variability of weight loss even if adherence to the prescribed diet is perfect. This is a fundamental limitation on our ability to precisely calculate the predicted bodyweight time course of an individual.
shows the predicted bodyweight time course of a 100 kg (220 lb) sedentary man following a step reduction of energy intake by 2 MJ per day (480 kcal per day). This constant diet perturbation was predicted to result in a bodyweight plateau at about 75 kg (165 lb) over a 10-year simulation taking roughly 1 year to reach half of the maximum weight loss and reaching 95% of this value after about 3 years. The dashed curves on illustrate the ±4 kg weight-loss variability after several years and show that even seemingly small initial uncertainties can lead to large expected long-term inter-individual variability of weight change. This expected variability will be exacerbated by imperfect adherence to the intervention as well as any differences in physiological variables between individuals.
Predicted long-term bodyweight change trajectories
By contrast with our dynamic model simulations, the popular dieting rule3–6
predicts that the same 2 MJ per day reduction of energy intake will result in a linear decrease of bodyweight over time with 22 kg lost in the first year (not shown), which is about 100% greater weight loss than our model prediction. This result shows the magnitude of the error introduced by ignoring dynamic changes of energy expenditure with weight loss. Moreover, it might help explain why even the most diligent followers of diet programmes often fail to reach weight loss goals that were set by use of the static weight-loss rule. Although practitioners of the erroneous dieting rule generally acknowledge that weight loss will slow over time, they had no way to estimate the weight-loss time course.
The timescale to reach a new bodyweight steady state is mathematically given by the effective energy density of the change in body tissue divided by the slope of the relation between the total energy expenditure rate and weight change (webappendix pp 3–4).28
Both of these factors are influenced by the initial body composition of the individual, and, therefore the bodyweight time course also depends on the initial body composition.33
shows the predicted change in bodyweight for the same step reduction of daily energy intake of 2 MJ per day in both a 100 kg man and an 80 kg man that differ in their initial body composition. Although the weight lost over the first year is similar, the greater initial fat mass of the 100 kg man results in a larger proportion of weight loss from body fat versus lean tissue than that in the 80 kg man. Because the energetically expensive lean tissue mass is preserved, the 100 kg man achieves a greater eventual weight loss than the initially lighter man because of the relative preservation of energy expenditure. However, to reach half of the maximum weight change takes longer for the 100 kg man than it does for the 80 kg man. Conversely, increased daily energy intake will result in greater weight gain in the 100 kg man than in the 80 kg man and a greater fraction of the weight change will be body fat (not shown).
These different bodyweight predictions at steady state result from the non-linear relation between initial mass of body fat and the fraction of weight change accounted for by change in lean tissue mass.30
Thus, the person with more initial body fat has a greater fraction of their weight change attributable to changes of body fat versus changes of lean tissue than does a person with less initial body fat. Since body fat contributes less than lean tissue to overall energy expenditure,39
the person with higher initial body fat will lose a greater amount of weight to achieve a new state of energy balance.37
Physical activity increases energy expenditure and can therefore cause weight loss, assuming no compensatory changes in energy intake. But does an increase of physical activity necessarily lead to the same weight loss as an energy-equivalent decrease of food intake? compares a step change of physical activity (ie, running roughly an additional 20 km per week at a moderate pace with an initial energy cost of about 1·2 MJ per day) with an energy-equivalent decrease of energy intake in the simulated 100 kg man. Such a relatively modest increase of physical activity expenditure results in slightly more rapid and greater predicted weight loss compared with an energy-equivalent reduction of food intake (). However, as the magnitude of each intervention increases, there is a point when diet leads to greater weight loss than does physical activity. Increased physical activity versus an initially energy-equivalent reduction of food intake leads to differing predicted weight loss because the energy expenditure of added physical activity is proportional to bodyweight itself.71
Therefore, by contrast with the assumption that a calorie is a calorie with respect to physical activity versus diet, our model shows that energy-equivalent initial changes of physical activity versus food intake can lead to differences in weight change, but experimental confirmation of this result would be difficult.
We have not yet modelled the potential effect of exercise on change of body composition although this effect is likely to be modest for most aerobic exercises. More importantly, our model simulations assume that changing physical activity does not alter energy intake and vice versa. However, there is evidence that increased physical activity results in compensatory changes of energy intake that act to attenuate the energy imbalance84,85
and that the extent of compensation has high individual variability.86
Conversely, changes of energy intake might result in compensatory adaptation of non-exercise physical activity, which also has a high degree of individual variability.87
These compensatory feedback relations between energy intake and physical activity clearly warrant further investigation. Because our simulation model predicts the expected responses in the absence of these feedback mechanisms, we suggest that the difference between the measured and model-predicted weight change can be used to quantify the magnitude of compensation.