How adaptations of substrate utilization regulate body composition

International Journal of Obesity (2007) 31, 1378–1383 & 2007 Nature Publishing Group All rights reserved 0307-0565/07 $30.00 www.nature.com/ijo ORIGINAL ARTICLE How adaptations of substrate utilization regulate body composition KD Hall, HL Bain and CC Chow Laboratory of Biological Modeling, National Institute of Diabetes & Digestive & Kidney Diseases, National Institutes of Health, Bethesda, MD, USA Objective: To elucidate the mathematical relationship between longitudinal changes of body composition and the adaptations of substrate utilization required to produce those changes. Design: We developed a simple mathematical model of macronutrient balance. By using an empirical relationship describing lean body mass as a function of fat mass, we derived a mathematical expression for how substrate utilization adapts to changes of diet, energy expenditure and body fat such that energy imbalances produced the required changes of body composition. Results: The general properties of our model implied that short-term changes of dietary fat alone had little impact on either fat or non-fat oxidation rates, in agreement with indirect calorimetry data. In contrast, changes of non-fat intake caused robust adaptations of both fat and non-fat oxidation rates. Without fitting any model parameters, the predicted body composition changes and oxidation rates agreed with experimental studies of overfeeding and underfeeding when the measured food intake, energy expenditure and initial body composition were used as model inputs. Conclusion: This is the first report to define the quantitative connection between longitudinal changes of body composition and the required relationship between substrate utilization, diet, energy expenditure and body fat mass. The mathematical model predictions are in good agreement with experimental data and provide the basis for future study of how changes of substrate utilization impact body composition regulation. International Journal of Obesity (2007) 31, 1378–1383; doi:10.1038/sj.ijo.0803608; published online 13 March 2007 Keywords: body composition; weight loss; weight gain; mathematical model; substrate utilization Introduction What determines the relative change of body fat and lean mass during weight loss or weight gain? Answering this question has important implications for treatment of obesity as the desired goal is to decrease body fat while preserving lean mass. Alternatively, other conditions of altered body composition, such as anorexia nervosa and cachexia, have the challenge of recovering lean body mass without excessive accumulation of fat. The concept of macronutrient balance is now accepted as the physiological basis for determining body composition changes.1–4 However, it remains to be elucidated exactly how longitudinal body composition changes are quantitatively related to the properties of macronutrient balance. Here, we asked the following question: how must substrate utilization Correspondence: Dr KD Hall, NIDDK/NIH, 12A South drive, Room 4007, Bethesda, MD 20892-5621, USA. E-mail: Received 14 October 2006; revised 31 January 2007; accepted 31 January 2007; published online 13 March 2007 quantitatively adapt to a given energy imbalance to produce the longitudinal body composition changes proposed by Forbes5? Addressing this question led to a simple equation that elucidates how interactions between diet, energy expenditure and substrate utilization are quantitatively connected to changes of body composition. Rather than develop explicit mathematical models of food intake and energy expenditure, our goal was to define the quantitative interactions between these variables, substrate utilization and the resulting changes of body composition. (For a more general model incorporating energy expenditure dynamics we refer the reader to a more complex model of macronutrient metabolism by Hall6). Thus, we only considered comparisons to experimental studies, where energy expenditure and food intake were measured and could be used as model inputs. The model then predicts changes of body composition and substrate utilization. This strategy allowed for an explicit mathematical connection to the Forbes body composition data and the resulting equations contained no free parameters thereby avoiding any model fitting procedures. Remarkably, our simple equations accurately predicted the changes of body composition and Theory of substrate utilization and body composition KD Hall et al 1379 substrate utilization rates during both experimental underfeeding and overfeeding when the measured food intake and total energy expenditure were provided as inputs to the model. Methods Forbes found the following cross-sectional curve of lean body mass (L) versus fat mass (F) and hypothesized that longitudinal changes of body composition during energy imbalance were described by movement along the curve: L ¼ 10:4 Loge F þ 14:2 ð1Þ 5 with L and F expressed in kg. The following differential relationship was thereby derived for infinitesimal weight changes: dL 10:4 ¼ dF F ð2Þ We began by considering the following macronutrient balance equations: dF ¼ IF  fF E dt ð3aÞ dL ¼ IL  ð1  fF ÞE dt ð3bÞ rF rL where E is the total energy expenditure rate, fF the fraction of the energy expenditure rate accounted for by fat oxidation, IF the metabolizable intake rate of fat, IL the sum of the metabolizable intake rates of protein and carbohydrate, and rF ¼ 9.4 kcal/g and rL ¼ 1.8 kcal/g are the energy densities of body fat and lean mass changes, respectively.6 Eq. 3a simply states that the rate of body fat mass change, dF/dt, results from differences of fat intake and oxidation rates. This assumes that de novo lipogenesis is negligible, which is typically true in humans.7 Eq. 3b is a similar equation for the combination of carbohydrate and protein balances and their impact on the rate of lean body mass change, dL/dt. Fortunately, combining carbohydrate and protein in this way does not introduce serious errors because both of these macronutrients have similar energy densities.8 and are associated with similar amounts of water.9 To connect the simplified macronutrient balance equations to Forbes’s theory, we divided Eq. 3b by 3a and used Eq. 2 to derive the following expression for the fat oxidation rate: FatOx  fF E ¼ ðC=FÞIF  IL þ E 1 þ C=F ð4Þ where C ¼ 10.4 rL/rF. Eq. 4 predicts how fat oxidation rates adapt to changes of fat intake, non-fat intake, energy expenditure and body fat mass. These adaptations form the physiological basis of Forbes’s hypothesis for longitudinal body composition regulation. To demonstrate the general properties of Eqs. 3 and 4, we considered an individual with an initial body weight of 65 kg with 31% body fat. The baseline intake rates of dietary fat and non-fat were 900 and 1700 kcal/day, respectively, and the energy expenditure of 2600 kcal/day was held constant. These intake rates were chosen so that the baseline body fat and l (...truncated)