A Simple Model Predicting Individual Weight Change in Humans

2.1. History of Dynamic Weight Change Models

One approach to modeling human weight change is to incorporate all reasonable physiological factors in order to reflect reality. This type of model provides insight into how perturbations to any component involved during changes in energy intake ultimately effect weight change. Applying this perspective has led to the carefully developed system of compartmental equations with state variables tracking changes in energy derived from protein, fat and carbohydrates [18, 19]. The Hall model identifies detailed movement of energy from intake to expenditure providing important information into the mechanisms behind weight change.

The Hall model requires estimation of the amount of grams of protein, carbohydrates and fat a subject consumes per day along with numerous other parameters [18, 19]. In order to reduce the amount of initial data and parameter estimations, several one and two-dimensional models have been developed by applying simplifying assumptions [2, 4–6, 31, 32, 55]. These simplifications are derived by assuming glycogen stores can be modeled by a time averaged constant, resulting in elimination of the carbohydrate mass equation. Elimination of the protein equation is obtained by algebraically correlating fat mass to fat free mass (FFM).

Some models [2, 4–6, 31, 32, 55] either assume that FFM is a linear function of fat mass or assume FFM can be modeled as a time averaged constant. Chow and Hall explored a more sophisticated formulation by coupling the FFM model proposed by the Forbes equation through a two dimensional dynamic model [4]. Forbes conjectured that FFM and fat mass were companions; increases/decreases in fat mass will be followed by increases/decreases in FFM [12]. Based on this hypothesis, Forbes fit mean data for 167 women of similar stature to a log linear regression equation to arrive at what is now known as the Forbes curve:

where F represents fat mass in kg and D ≈ 2.55.

We develop here a one-dimensional model that incorporates a newly developed FFM − F relationship [54] along with several other terms based on recent experimental observations. We refer the reader to [55] for complete biological details on the full model development and present here a description of the essential components of the model, highlighting the modifications beginning with the energy balance principle.

2.2. The Energy Balance Principle

The energy balance principle discussed in physiology and nutrition literature is based on the application of the first law of thermodynamics to an open system [11]. The human body is considered an open system because energy can be added to the system by input of mass flow in the form of food. The human energy balance equation takes the form,

where R is the rate of energy stored/lost, I is the rate of energy intake, and E is the rate of energy expenditure. The quantities R, I, and E are typically measured in kcal/d [27, 41].

2.3. The rate of energy stored/lost: R

Energy that is not used by the body is stored in the form of glycogen, lipids and protein [24]. Glycogen is a storage molecule of the body's short-term energy resource, glucose [14]. Because of the regulatory mechanism that works to maintain the glucose/glycogen level in a narrow range, we can model glucose/glycogen energy measured in kcal by a time-averaged constant, G. We remark here that this assumption has been compared successfully to observed data [4].

Lipids (mainly in the form of fatty acids) are the nutrients containing the most energy per unit mass[24]. Unlike glucose, fat can be stored in large quantities for extended lengths of time in the form of triacylglycerols within the adipose tissue spread throughout the human body. Therefore, fat mass is the main long-term energy storage mechanism of the human body. We denote the total kg of fat mass of the body at time t by the function F(t).

Protein's building blocks, amino acids, can be broken down and transformed to glucose to be used for energy through amino acid metabolism [24]. Although there are small amounts of protein stored in the liver, the major portion is contained in the body's muscle tissue [14]. Protein is a component of FFM and so we track changes in protein energy through changes in FFM as discussed previously [4].

As mentioned in the introduction, the two-dimensional model [4] relies on the Forbes FFM function of fat mass (F ). Specifically, Forbes observed an algebraic relationship between fat mass and FFM [12] by fitting a log linear curve through mean data for 167 women of similar stature [12],

An analogous male Forbes model was developed using body composition data from approximately 200 men of average stature [54]:

where S ≈ 0.29.

The Forbes model has been established to predict the change in FFM in numerous weight change studies with a high degree of accuracy for group mean data[17, 54]. It may be tempting to think of the Forbes curve as a trajectory that body composition travels during weight loss, however, the model does not determine change in FFM by estimating the baseline and final time body composition [54]. The Forbes curve estimates for the change in FFM rest on the fact that individual data follow a parallel curve of identical slope and in this manner accurately determines change without identifying beginning and endpoints.

As a stand alone equation, the parameter D can be computed using baseline body composition data to determine the correct translate. It was shown [54] that D is a reflection of age, height,and race. In addition, the Forbes model has a range of positive fat mass that yields negative FFM values. Moreover, the intercept of the Forbes model is undefined.

The two dimensional model from [4] successfully incorporates the Forbes model by considering F and FFM as state variables and applying the p-ratio (proportion of body energy mobilized as protein during weight change) derived from the Forbes model [4]. Reduction of the two-dimensional model to a one-dimensional model (Equation 25 [4]) reduces the dimension of the model, but will require calculation of the parameter D.

In order to develop a universal FFM function that is representative of the current national population, has a biologically realistic intercept when F = 0, and can be immediately applied for each individual subject in a one dimensional energy balance model, a class of fourth order polynomials of FFM as a function of F that considers age, height, gender and race was formulated based on the recently released body composition data collected by the Centers for Disease Control under the The National Health and Nutrition Examination Survey (NHANES). NHANES is a program designed to assess the health and nutritional status of adults and children in the United States. NHANES performs a continuous, nationally representative health survey of the civilian, non-institutionalized United States population, collecting data on about 5000 persons each year from interviews, physical examinations, and medical tests including bone densitometry. In 1999 NHANES began performing dual energy X-ray absorptiometry (DXA) whole body measurements on survey subjects 8 years old and older in three mobile examination centers. DXA measurements are widely accepted as the gold standard for human body composition measurements. The DXA whole body data obtained from the mobile exam centers was compiled by the NHANES study group and released on the Center for Disease Control (CDC) website [29]. The FFM models developed earlier [54] provided comparable estimates for ΔFFM to the Forbes model after entering age, height, and gender as initial inputs.

The first major modification to our prior model [55] is the replacement of the linear FFM function of by the FFM functions based on NHANES data. These FFM model terms were statistically determined through a regression analysis [54],

Females:

Males:

where FFM(t) represents the kg of FFM on day t, F(t) is the kg of fat mass on day t, H denotes height in cm, and A₀ is baseline age.

Separating R from Equation 1 in the three different components, kg of glucose/glycogen, FFM(t), and F(t) we have

where c_l is the energy density of FFM, c_f is the energy density of fat mass, and the derivative of FFM is taken implicitly. The values of c_l and c_f were obtained from [26, 51] and appear in Table 2.

2.4. The rate of energy intake, I

The rate of energy intake, I has been historically difficult to measure [19]. Past dynamic models have assumed that I during weight change is constant and determined by target energy intake requirements [2, 5, 6, 18, 19, 32, 55? ]. We now describe the limitations behind this assumption and provide a novel method based on modern technology and measurements to assess I as a time varying function dependent on an individual subject.

First, baseline intake is determined by applying steady state (zero energy balance) at the conception of the study:

The gold standard for determining E is through doubly labeled water (DLW). The technique involves enriching the body water of a subject with an isotope of hydrogen and an isotope of oxygen, and then determining the washout kinetics of both isotopes as their concentrations decline exponentially toward natural abundance levels. It is not feasible to obtain DLW data for every study or individual so we applied individual DLW data from National Academy of Sciences/Institute of Medicine (NAS/IOM) database of over 200 subjects to fit a quadratic regression formula that defines E as a function of body mass, W. Specifically,

where the subscript F represents females and M represents males.

As stated earlier, when a study prescribes caloric restriction, it is tempting to model I as the prescription demands. For example, if we prescribe a change in baseline intake (ΔĨ) that is 25% below baseline intake, we would model I = Ĩ − ΔĨ where

and Ĩ represents baseline energy intake.

However, compliance to such dietary restrictions are rare. In an analysis of 181 participants placed on popular diets, only one subject in the study was 100% adherent by the end of one year [1]. Furthermore, it has been consistently observed that compliance is a decreasing function of time [1, 7, 25]. To determine I, we applied data from studies which calculated energy intake using DXA measurements of changes in energy stores and DLW measurments of energy expenditures during weight loss. Specific details on these calculations appear in the model validation simulations in Section 4.

2.5. The rate of energy expenditure, E

The rate of energy expenditure consists of four different quantities; the kcal/day used for dietary induced thermogenesis (DIT), volitional physical activity (PA), basal metabolic rate (RMR), and spontaneous physical activity (SPA) [15]:

DIT is the energy involved in processing food. This consists of digestion, absorption, metallization, storage and transport of ingested food and is estimated to account for 4-15% of total energy expenditure [15]. Although DIT was observed to be sensitive to amount of protein ingested, we do not examine this aspect of DIT [58]. In the case of an aggregated composition of food intake, we assume that DIT is a direct proportion of the rate of energy intake, I, with proportionality constant, β. In [20], the adjustment to calorie reduction on DIT was examined by multiplying by an adjustment factor. In fact, such an adjustment has also been observed in overfeeding studies by a factor of up to 19% [20, 37]. We model the adjustment by a multiplier that is greater than 1 for the overfeeding case and less than one in the calorie restriction case. As in [20] we absorb the multiplier into one constant β with ranges provided in Table 1.

Therefore, the energy expended for dietary induced thermogenesis is

By physical activity, PA, we specifically mean volitional exercise such as sports and fitness related activity. PA can accounts for 20-40% of total energy expenditure [15]. Strenuous exercise or starvation cause glycogen stores to deplete [14]. As a result the liver begins to produce ketone bodies as an alternative energy source. Because we are not modeling extreme cases such as starvation, we assume the physical activity is light to moderate and does not deplete glycogen stores. Some PA is what we refer to as weight bearing activity and some of PA is non-weight bearing. Weight bearing activities involve activities that require us to carry our body weight. For example, walking and running are examples of weight bearing activities. As in the two dimensional model [4], we assume that energy used for non-weight bearing activity is negligible.

Thus, the rate of weight bearing exercise is estimated as a direct proportion of body mass and so we model PA by,

where m is the proportionality constant in kcal/kg/day and W, total body mass, is the sum of FFM and F.

An individual's RMR is the rate of energy required to sustain life. RMR is measured in controlled conditions. The subject must be in a relaxed (preferably just having awakened), postabsorptive state (12 hours or more of fasting) [15]. Thus, the direct determination of RMR is not simple. There exists several simple statistical formulas that estimate RMR depending on sex, total body mass, height, and age. These formulas are based on extensive experimental data which was analyzed using predictive regression equations. Although these estimates work well in most situations, they tend not to be as reliable in extreme cases such as obesity [49].

Popular statistical formulas assume that RMR is a linear function of body mass, age, height, and gender [21, 42]. There is evidence that RMR is a nonlinear function of body mass of the form W^p where 0 < p < 1 [30, 60]. The Livingston-Kohlstadt RMR equations developed using the NAS/IOM database take the form of body mass to a power and provides accurate estimates of RMR when validated against other databases [38]. The Livingston-Kohlstadt formula is a function of a power of body mass and baseline age:

where a_M = 293, a_F = 248, p_M = 0.4330, p_F = 0.4356, y_M = 5.92, y_F = 5.09, W represents body mass, and A₀ represents baseline age.

We point out that the units of the coefficients are non-standard as the exponent of body mass is statistically determined. While analysis of data sets indicate that the power of body mass is fractional, the exact fraction is still debated and the physiological reasoning behind the fractional power is not fully understood at this time. There are several mathematical explanations, for example, the fractal based derivation [57], however, the science behind human RMR models is still in its infancy and the Livingston-Kohlstadt model provides the best known data driven fractional power formula to date.

It has been observed that calorie restriction results in RMR that is lower than expected [25, 39]. The expected basal metabolic rate based on body mass in our model is given by Equation (5). If a is the percent of metabolic adaptation where 0 ≤ a ≤ 1, then we model the contribution of RMR with adaptation to energy expenditure by adjusting RMR:

Spontaneous physical activity (SPA) is defined as the total energy expended in activities of daily living, change of posture and fidgeting [61]. During weight gain due to overfeeding, it was observed that the ratio of the change in SPA to change in total energy expenditures remained relatively constant [37]:

where Δ represents final time quantity minus baseline quantity.

SPA is difficult to measure even in highly controlled settings, however, by expanding E using (4) and applying the linearity of the Δ operation we arrive at:

Integration over time allows us to solve for SPA in terms of the remaining energy expenditure:

where C is a constant of integration. The constant C is determined by solving

By examining data measured in [34, 35], it was found that baseline SPA is approximately 32.6% of baseline energy expenditure determined by DLW; SPA(0) = 0.326E(0). As a result, we obtain a closed form expression for SPA in terms of the other components of energy expenditure. The closed form expression is an increasing function of weight supporting observations that SPA is found to change during both over-feeding experiments [33, 37] and caloric restriction in obese individuals [10, 46].

There are only a few overfeeding studies that provide individual DLW, RMR, and DIT data which presents challenges in locking down the precise value of s.

To estimate s, we analyzed several well known overfeeding studies with published individual data. Within these published studies, we carefully detailed which data sets were applicable. Individual subject data was either published or provided by the author in the overfeeding studies of Bandini [3], Diaz [9], Levine [37], Pasquet [43], and Siervo [52]. The Pasquet study examined the effects of overfeeding during Guru-Walla, a traditional Cameroon fattening season. The reported baseline energy expenditures of the Pasquet study were higher than expected when compared to the NAS/IOM regression formulas for E. This was due to increased physical activity of the subjects during the collection of baseline information. Subjects in the study were performing physically demanding agricultural work which raised their baseline energy expenditures and therefore moved them out of energy balance. Thus, calculations of the change in E over time (ΔE) were negative or close to zero for most of the subjects and could not be applied to determine s. The Siervo study was unusual as they overfed participants for a period of a few weeks and then allowed a relaxation of the requirements for a few weeks [52]. This process was repeated for three cycles with the amount overfed and the length of the periods increasing. Although baseline and final time energy expenditures were reported, computation of ΔE was confounded by the non-monotonic feeding pattern. Therefore, we removed the Siervo study from consideration.

Computing the value of s using individual data from Bandini, Levine, and Diaz studies yielded a value of s = 0.4, 0.67, and s = 0.6 respectively. The Bandini study overfed 13 adolescents for a period of 2 weeks, the Levine study overfed 16 subjects by 1000 kcal/d over baseline requirements for a period of 8 weeks, and the Diaz study overfed 9 male participants 50% over baseline for a period of 42 days. Subjects in the Diaz study conducted both cycling and stepping activity during baseline and overfeeding. Similar to the Pasquet study, baseline energy expenditures for most subjects was higher than expected when compared to the NAS/IOM predictions. The Diaz study, however, continued the stepping and cycling physical activity during the overfeeding study and thus, resulting in positive ΔE for most subjects. Explanations for the negative ΔE measurements were due to external circumstances and were not considered in study's group mean results [9]. Similarly, we restrict our calculation of s to the positive ΔE measurements. Based on the available current data, we estimate s as the average of the 0.4, 0.6, and 0.67, yielding s = 0.56.

Computations of s for the case of caloric restriction indicates that s = 0.67 [22, 44, 48, 50].

We now summarize the formulation for E:

1. RMR is modeled by the Livingston-Kohlstadt equation for RMR. Resting metabolic rate travels along the trajectory for weight determined by the differential equation. Age is continuously increased from initial age, A₀. RMR = a_i(W(t))^p − y_i(A₀ + t/365), where a_F, a_M, y_F, y_M are the Livingston Kohlstadt parameters subscripted for gender.

2. DIT is modeled as a direct proportion of time varying intake (ωI). If intake is being increased then there is an increase in the proportion of DIT which is reported from 14-19%. So we have DIT = βI where β is in the range of 0.075 ≤ β ≤ 0.086 (determined based on overfeeding or caloric restriction). Baseline DIT is estimated as 0.075I.

3. PA at baseline is determined by applying baseline SPA observed to be around 32.6% of baseline E [34, 35, 37]. Baseline PA is estimated then from TEE(0) − DIT(0) − SPA(0) − RMR(0). If this expression is less than zero, then there is no contribution of physical activity so PA(0) = 0. Otherwise PA(0) = TEE(0) − DIT(0) − SPA(0) ± RMR(0). We model the dynamic portion of PA by PA = mW where m = PA(0)/W(0) and W is time varying weight.

4. SPA is modeled by integrating

where s = 0.56 and solving for the constant of integration by applying baseline data.

Combining all expressions obtained for R, I and E and substituting into Equation (1) we obtain the final one dimensional model,

with initial condition

We remark that the state variable of Equation 6 is F(t) since FFM(t) is slaved to F(t) through the NHANES regression formulas. The derivative of FFM(t) on the left hand side of Equation 6 is taken implicitly.

3.1. Long-term dynamics

In the case where age is constant, and I is constant, solutions to (AE) converge monotonically to the steady state where intake equals expenditure, I = E [55]. Increases in age act as incremental decreases in energy expenditure which do not allow solutions to completely rest, but force a small pulling away from the steady state of (AE). To see this, we simulated the model with age held constant at A₀ alongside the non-autonomous model. One can see the phenomena described in Figure 1.

4.1. Caloric Restriction - CALERIE PHASE I study

The Phase I of the Comprehensive Assessment of Long-term Effects of Reducing Intake of Energy (CALERIE) trial tested the effects of calorie restriction on biomarkers of longevity [23]. Twelve of the CALERIE subjects were placed on a very low calorie diet of 890 kcal/d (LCD), twelve were placed on a low calorie diet of 25% below baseline energy requirements (CR), and twelve were prescribed a combination of caloric restriction (12.5% below baseline energy requirements) and exercise (physical activity increased to 12.5% above baseline total energy expenditure). Thirty of the subjects were overweight and six of the subjects had a baseline BMI classifying them as obese. The 24 subjects that were reducing their weight solely through caloric restriction (CR and LCD) were used to test the validity of our model.

The metabolic adaptation parameter a was determined by comparing the result of regression formulas for the resting metabolic rate to actual resting metabolic rate in the CALERIE Phase I study [39]. The value of a was not calibrated using the dynamical systems model presented here, nor was it computed using the Livingston-Kohlstadt model and is treated as a universal parameter.

Subject energy intake was determined using the energy balance equation. The value of R was estimated by,

where body composition changes were measured by DXA. E was measured using DLW and then I was estimated by computing R+E. DXA and DLW measurements were collected for all subjects at 4, 12, and 24 weeks. We restrict our attention to the 24 CR and LCD subjects and remark that the 12 combined caloric restriction and exercise subjects would require an extra measurement determining adherence to the exercise prescription.

In order to estimate actual intake as close as possible to known data, we set intake as a piecewise defined step function

where I₀ is initial intake and c₄, c₁₂, c₂₄ are the individual subjects measured percent decrease from baseline intake at 4, 12, and 24 weeks respectively. The piecewise definition of I will result in a continuous solution but not one that is differentiable everywhere. This is because we are guaranteed a unique nonnegative solution for each branch of I by our existence theorem. The continuation along the same trajectory (feeding the end values as initial data for the next leg) guarantees continuity, however, differentiability does not hold since all 19 subjects presented here varied their intake, thereby changing the slope.

For the sake of comparison, we refer to the model presented here as the Heymsfield model. At 24 weeks of calorie restriction, the Heymsfield model predicts the mean 24 week weight as 73.9 kg versus the actual mean observed weight of 72.6 kg (Table 1). Simulations of the one dimensional model in [4] yield a mean weight of 72.6 kg. Error between each subjects individual final predicted and observed weight at 24 weeks was computed by

where the subscript i denotes subject number, and W_actual,i represents the measured weight, and W_estimatei represents the model predicted weight for subject i. We refer to the mean absolute error as the mean of the set of ${\left\{{\psi }_i\right\}}_{i=1}^{19}$.

The mean absolute error in predicting individual final weight for the Heymsfield model was 1.8 ± 1.3 kg whereas the one dimensional model (Equation 25 [4]) was 4.5 ± 3.3 kg. The Heymsfield model predicted final body mass consistently with a maximum absolute error of 4.3 kg. Model reliability is also substantiated by the low standard deviation in the mean absolute error for the Heymsfield model. In terms of individual predictions, the one dimensional model (Equation 25 [4]) yielded a maximum absolute error of 12 kg and predictions for 9 subjects produced absolute error over the maximum absolute error held by the Heymsfield model. Individual subject final measured 24 week weight and model predictions for both the Heymsfield model and the one dimensional model (Equation 25 [4]) appear in Figure 2.

Subjects were provided with all food using an infeeding paradigm during the first 12 weeks of the study which appeared to affect the level of adherence to the prescribed restrictions in intake. Direct inspection of individual weight predictions against measured data at 4 weeks, 6 weeks, 12 weeks, and 24 weeks, revealed although the weight loss trend was modeled well by the adherence measurements, the initial drop of weight appeared higher than predicted by the model. Moreover, subjects appeared to adhere at the 24 week measured adherence percentages for a few weeks after infeeding was discontinued.

We surmised that subjects may have been highly compliant during the inception of the study with compliance decreasing over time. To test this assumption, we modeled intake by a piecewise defined step function applying the assumption that subjects were 100% compliant for the first 28 days (represented by the percentage c₀), followed by a decrease in intake to the observed adherence numbers c₄ for 28 ≤ t ≥ 60. We then assumed that participants attempted to match their infeeding behavior at home for a few weeks after infeeding was discontinued. This is reflected by a decrease in intake to by the measured percentage c₁₂ for 60 < t ≤ 100. Finally, after 100 days, intake decreased to the adherence percentages measured at 168 days:

Applying the intake step function, we discover a remarkable fit to the individual observed measurements. The mean absolute error in prediction for the Heymsfield model reduces to 1.5 ± 1.6 kg. The one dimensional Chow-Hall model predictions remain relatively similar to the previous case with a mean absolute error of 4.6 ± 2.7 kg.

A sample simulation for a single subject describing the impact of each definition of intake (full compliance, adherence data applied, full compliance is temporarily attained followed by the measured adherence values) is depicted in Figure 3.

4.2. The Racette study

To compare the effects of exercise and composition of reduced energy intake, twenty three obese women were assigned to four groups; control, exercise, low carbohydrate and low fat. A 12 week weight reduction phase followed a 5-week baseline measurement period. A third phase, where subjects maintained their weight reductions was also conducted, however, for the purposes of model validation we are solely interested in the 12-week weight reduction phase. Total daily energy expenditure was measured at baseline and twelve weeks using doubly labeled water and body composition was determined at baseline and at the end of the weight reduction period using under water weighing. From these two measurements, we estimate actual intake as R + E where R is determined by

and E is given by DLW measurement at 12 weeks. We restricted our analysis to the 13 subjects who were assigned to the high fat or low carbohydrate diets.

Parameters for model simulations are provided in Table 5 and individual subject results for the observed, Heymsfield model, and one dimensional Chow-Hall model (Equation 25 [4]) appear for each of the 13 subjects in Figure 4. Similar to the CALERIE study, the one dimensional Chow-Hall model predicted mean group results accurately with a mean final weight of 85.3 ± 9.3 kg, however the maximum mean absolute error was 11.1 kg as opposed to the Heymsfield model which yielded maximum absolute error of 4.3 kg. Mean absolute error in prediction for the Heymsfield model was 1.7 ± 1.2 kg and mean absolute error for the one dimensional Chow-Hall model was 2.9 ± 3.0. We again note the low standard deviation in mean absolute error for the Heymsfield model indicating low variance in the error for individual subject predictions.

4.3. Overfeeding

For model validation purposes in the case of overfeeding, we apply overfeeding data from 22 subjects who increased their caloric intake to 1000 kcal/d over baseline requirements for a period of 8 weeks[34, 35]. Similar to the caloric restriction studies, subject compliance was determined using the energy balance equation; namely R was measured by DXA, E was measured through DLW and then I was estimated by computing R + E. From these measurements, it was determined that subjects were on average 98% compliant to the 1000 kcal/d overfeeding prescription. Input parameters and baseline information appear in Table 6. Results of the simulations show that the Heymsfield model predicts with a mean absolute error of 2.5 ± 1.6 kg the actual weight at 8 weeks and the one dimensional Chow-Hall model predicts with a mean absolute error of 5.5 ± 2.1 kg.

In summary, the Heymsfield model predictions of weight during weight change had very good agreement with actual measured weights Figure 6. Figure 6 depicts the plot of individual actual final weight from all three studies considered here (y-axis) against model predictions.