A predictive model of rat calorie intake as a function of diet energy density

Abstract

Easy access to high-energy food has been linked to high rates of obesity in the world. Understanding the way that access to palatable (high fat or high calorie) food can lead to overconsumption is essential for both preventing and treating obesity. Although the body of studies focused on the effects of high-energy diets is growing, our understanding of how different factors contribute to food choices is not complete. In this study, we present a mathematical model that can predict rat calorie intake to a high-energy diet based on their ingestive behavior to a standard chow diet. Specifically, we propose an equation that describes the relation between the body weight (W), energy density (E), time elapsed from the start of diet (T), and daily calorie intake (C). We tested our model on two independent data sets. Our results show that the suggested model can predict the calorie intake patterns with high accuracy. Additionally, the only free parameter of our proposed equation (ρ), which is unique to each animal, has a strong correlation with their calorie intake.

INTRODUCTION

Through the course of history, the human body has evolved to survive times of food deprivation. Additionally, through innate and learned responses, we generally find energy-dense foods the most palatable, and as such have learned which cues indicate the presence of those foods (37). Changes in agriculture and the food industry during the past century have provided unprecedented access to cheap, energy-dense foods. It has been argued by many scholars that one of the major contributing factors to the worldwide obesity epidemic has been easy access to highly palatable, energy-dense foods (2, 3, 27). In such a situation, understanding the role of rewarding aspects of food intake, including its appetitive and consummatory aspects, can help us find more efficient ways to prevent and mitigate the obesity epidemic. Despite the robust correlation observed between the functionality of the food reward system (i.e., the neural pathways that play a role in food wanting, liking, and reinforcement) and obesity, our knowledge about the rewarding aspect of food is incomplete, mainly due to its complex nature.

Mathematical and computational approaches to model different contributing factors of food decision making are one way to understand the complexities of these components, which has recently received much attention. Although it is tough to control and study all of these contributing factors to food behaviors in human settings, animal models can minimize some of the complexities of human studies and thus provide valuable insights into the understanding of these interactions. Developing models that can describe food intake patterns in animals (without including the connection to the rewarding aspects) has been the subject of many works. For instance, Lusseau et al. (23) presented a mathematical model for describing the food intake patterns of mice that undergo calorie restrictions using a Markov model by setting the probabilities of transition between the states of the Markov model (representing shifting across different physiological states). In addition to modeling food intake patterns, other studies have presented mathematical models for studying energy metabolism in animals (5, 11, 20). For instance, Guo et al. (12, 13) proposed several models to predict changes in body weights and energy expenditure of mice in different conditions. Also, from a different point of view, Jacquier et al. (15–17) presented a series of differential equations to model the hormonal regulation of food intake in rats.

Although various types of models have been proposed for describing different aspects of food intake in animals, a model that links food intake patterns and energy density of food does not exist. High-energy foods—like those high in sugar or fat—are generally considered more palatable and more rewarding. In this paper, we present a mathematical model that can predict the amount of calorie intake in rats as a function of the energy density of the diet. Our proposed model consists of an equation that includes four main variables: body weight (W), energy density of the diet (E), elapsed time since the beginning of the current diet (T), and finally a parameter ρ (rho) peculiar to each rat.

MATERIALS AND METHODS

We have used several existing models of food intake as the basis of developing our model. For inferring the level of perceived reward (R) in rodents, one objective measure that is commonly used in other works is licking rate when a liquid meal is consumed. It has been shown by other scholars that licking rate is a good indication of animals’ preferences in the oral sensations (25). Herein although not assuming licking rate measures perceived award, we use the licking rate as a proxy for our reward model. It has been shown that licking rate for each individual animal follows a Weibull function in the form of y = Aexp[–(Bt)^C], where A, B, and C are the parameters unique to each animal, and t is the time passed after the food was presented (8, 28). Similar exponential equations have been shown to be capable of modeling the relationship between the cumulative amount of calorie intake (C) and the meal time (26). Based on these two sets of formulations, the relation between the cumulative calorie intake (C) and reward (R) will follow a form of decreasing exponential curve with a form that is schematically shown in Fig. 1. As eating commences (or more precisely shortly after the beginning), the amount of reward is at the highest point. As more food is consumed, the reward also gradually decreases until it reaches zero.

Fig. 1.

A simple representation of the relation between the amount of the perceived reward and calorie intake (C) in a meal. The reward curve follows a logarithmic function, which becomes zero at calories = C_f.

For C > 0, this relation can be simplified with a logarithmic function. More details about this approximation are provided in the appendix. Such a logarithmic relation between C and R can be shown in a function that has the form of:

$$R_C\cong -\log (\beta C)$$ (1)

where, R_C shows the amount of perceived reward after consuming C calories, and β >0 is a constant multiplier. Here, reward refers to the palatability of food as inferred by the licking rate. To be able to describe different calorie intake patterns for different individual animals using this equation, the multiplier has to be specific to each animal. To determine the factors that affect the value of β, we consider a set of major elements that are known to influence the rewarding aspects of food intake and therefore β. Many studies have reported a positive effect of body weight (W) on the amount of experienced reward (9, 10). We also know that the amount of perceived reward has a direct relation to the energy density of the diet (E) (6, 33), and an inverse relation with the time elapsed from the first time that a new diet was introduced (T) (32). Using the fact that −log(x) is a decreasing function, we incorporate the variables T, W, and E into Eq. 1 as follows:

$$R_C\cong -\log \left({\beta }^{\prime }\frac{T}{WE}C\right)$$ (2)

where β′ > 0 is a new constant multiplier specific to each individual animal. In this formulation, higher values of T would make the slope of the curve in Fig. 1 steeper, while higher values of W or E would have an opposite effect. More generally, W, E, and T can be replaced by f_W(W), f_E(E), and f_T(T), where f_W, f_E, and f_T, are strictly increasing functions. In the next section, we discuss the specific functions that were chosen based on an analysis of experimental data.

Based on our model, eating ceases when the value of perceived reward is equal to zero. Both homeostatic and hedonic factors affect this stopping point, in which the combination of the different factors makes calorie intake initially rewarding, but gradually less appealing until the reward value equals zero. This stopping point is shown in Fig. 1 by C_f (subscript f for final). As C = C_f is the point where R_C reaches zero, we have, from Eq. 2:

$$-\log \left({\beta }^{\prime }\frac{T{C}_f}{WE}\right)\cong 0\Rightarrow \frac{{\beta }^{\prime }T{C}_f}{WE}\cong 1\Rightarrow {\beta }^{\prime }T{C}_f\cong WE\Rightarrow C_f\cong \frac{WE}{{\beta }^{\prime }T}$$ (3)

Hence, the amount of calories that an animal consumes can be predicted based on the values of W, E, T, and β′. For simplicity, we replace 1/β′ by a new parameter ρ:

$$C_f\cong \rho \frac{WE}{T}$$ (4)

The process described so far was used to find a starting point for our search for finding a logical mathematical model that can describe the relation between calorie intake and energy density of food. To see whether a formula in this form can be applied to the actual cases, and also determine which functions f_W(W), f_E(E), and f_T(T) are appropriate, we have used two data sets from two independent groups of rats, which are briefly introduced in the following.

Data

Study 1.

The first data set was related to a group of (n = 16) male Sprague-Dawley (Harlan) rats ~275–300 g at the age of 60–70 postnatal days. Following 7 days of habituation to the laboratory environment and maintenance on a chow diet with an energy density of 3.1 kcal/g (2018 Teklad, Harlan), the animals were assigned to two groups of eight. One group was maintained on the chow diet, and the other group was switched to a high-energy (HE) diet with an energy-density of 4.73 kcal/g (D12451, Research Diets) for 42 consecutive days. At this point, the diet of the HE group was switched back to the original chow diet for an additional seven consecutive days of recovery. Additional details about this study are also provided in the original article (36).

Study 2.

The second data set was collected through another similar experiment using (n = 20) rats of the same type, and similar diets. This time, after 14 days of habituation with the chow diet (3.1 kcal/g), one-half of the rats received HE diet (4.3 kcal/g) for 14 days. All experimental procedures were approved by the Institutional Animal Care and Use Committee at The Johns Hopkins University School of Medicine.

We used the data from the rats that received HE diet to fit and test our model. In the case of study 1, the data from both the initial chow diet and the recovery period were used to fit our mathematical model, and in study 2, only the data from the initial chow diet was used for fitting. The reason for considering the recovery period for study 1 was that the existing data for the initial chow diet period in the first experiment was insufficient (only 4 days for each rat) to properly fit our model. Fitting was performed using nonlinear regression analysis (fitnlm and associated functions in MATLAB software) to determine ρ with the least-squares fit to the data. The model was fitted and tested separately for each individual rat. After fitting the model using the chow diet data, it was tested on the data from the HE diet period.

RESULTS

Before fitting our model on the rat data sets to determine the best value of ρ for each rat, we determined the best form of Eq. 4. A similar procedure for finding the best equation to analyze food intake patterns has been used by others (18). More details about this procedure are provided in the appendix. We tested a set of common mathematical functions that can be used for the variables W, E, and T in Eq. 4, with arbitrary (synthetic) values of ρ. Specifically, we tested a range of power functions [x^c (in which c is constant)] and exponential functions [e^x and log(x)], and found that the following equation generates patterns that are close to the available data:

$$C=\text{ρ}\frac{W\text{log}\left(E\right)}{\left(T^{\frac{1}{4}}\right)}$$ (5)

During this process, the best values of ρ were determined for each rat using only chow data. The fitted model was then tested by predicting the amount of calorie intake during the period of HE diet. Having the values for ρ, W, E, and T, we used Eq. 5 to predict calorie intakes (C). The results of testing our model are shown in Fig. 2 (study 1 rats) and Fig. 3 (study 2 rats). Figure 2 illustrates the comparison between the actual calorie intake of the 8 rats of study 1 and the amount of calorie intake predicted by our model. These amounts of calorie intake refer to the period of receiving HE diet (the testing period). Each chart shows the daily calorie intake of one rat for a period of 42 consecutive days (t = 1–42). During this period, the energy density (E) of the rats’ diet was equal to 4.3 kcal/g, and the body weights of rats (W) were also available from the data.

Fig. 2.

Predicted calorie intake by our model vs. actual daily calorie intake of 8 rats from study 1 during the 42 days of high-energy diet presentation. Animals’ body weights are also color coded. Rat ID and the value of ρ (rho) for each rat are shown at top.

Fig. 3.

Predicted calorie intake by our model vs. actual daily calorie intake of 10 rats from study 2 during the 14 days of high-energy diet presentation. Animals’ body weights are also color coded. Rat ID and the value of ρ (rho) for each rat are shown at top.

Similarly, Fig. 3 shows the comparisons between the actual calorie intakes of the 10 rats of study 2 during the 14 days of receiving HE diet and our model’s predictions.

To explore the extent that the value of ρ—which is specifically assigned to each rat using the fitting data—is capable of predicting the calorie intake and weight gain of each rat, we ran a set of linear regression analyses. As mentioned earlier, the ρ values were calculated (fitted) using the chow diet data, and not the HE diet. Figure 4, A and B, shows the association between ρ and the average amount of daily calorie intake during the period of receiving HE diet.

Fig. 4.

The association between the values of ρ (rho) for each rat and the average daily calorie intake during the period of high-energy diet. A: 8 rats of study 1. B: 10 rats of study 2.

We have also examined the ability of our model to predict the weight changes of study 2 rats during the recovery period. A modified version of Eq. 5 {W = CT^1/4[log(E)ρ]⁻¹} and the derivative of this formula {dW/dT = (1/4)C[T^3/4log(E)ρ]⁻¹} were used to model body weight changes. The results of this experiment are presented in the appendix.

DISCUSSION

Our model presents a simple equation that can predict the amount of calorie intake of rats, based on their body weight, the energy density of the food, and the length of the period that the food was presented. These variables have been reported in various studies as the major factors that affect the food intake of rodents. For instance, in a recent study, the set of the main covariates of food intake that were considered for modeling the effects of calorie restriction on the behavioral phenotype of a group of mice included several variations of the time elapsed from the beginning of each diet change and body weight of animals (23).

Our model meets the four criteria in the so-called “first principles modeling approach” that have been proposed for assessing the quality of candidate models of food intake behaviors (1, 26, 35). Specifically, our model has a self-consistent theoretical basis that was described earlier. Additionally, it is a simple model that is not overly simplistic (one of the criteria). In fact, a major strength of our proposed model is that it has only one free parameter that needs to be evaluated. This parsimony is generally considered to be a desirable property of mathematical models (24).

The results obtained from testing our model on the two independent data sets showed that our model could predict the calorie intake patterns of rats with relatively high accuracy. Figures 2 and 3 show the predicted HE calorie intakes using our model vs. the actual data for the study 1 and study 2 rats, respectively. In these charts, proximity to the identity line (y = x) indicates better performance of the model. The increasing weights of the rats are shown using color, with scales indicated under each chart. One pattern that can be observed in these charts occurs after the switch in diet. The body weights attain their highest values and the models show relatively accurate results in this period. This is in contrast to the frequent fluctuations in the data during the initial period immediately after the switch in diet shifts that can be partly explained by the confusion of the rats due to the dietary shift (36). We have performed additional regression analyses and equivalence tests and observed a significant relationship between modeled and actual data (see appendix). The model’s performance has also been evaluated using two other types of analysis, with results shown in the appendix. In one of these analyses, the differences between the actual and predicted caloric intake is computed using the root-mean-square deviation (rmsd) and normalized root-mean-square deviation (nrmsd); in the second analysis, the Bland-Altman procedure is used. To investigate which of the models’ variables have a more important role on the model’s accuracy, we have also compared the differences between the data and model results by eliminating each of the 3 factors (W, E, or T) and reporting the model fits. Our results (shown in the appendix), indicate that the time variable (T) in the case of study 1, and body weight variable (W) in the case of study 2, had the highest impact. Although this observation does not identify a unique pattern, the strong associations between caloric intake and time and body weight are reasonable.

Regression analysis was used to test if the values of ρ could significantly predict the average amount of calorie intake during the HE diet. The regression results indicate that the predictor explains 44% and 51% of the variance in two studies (R² = 0.44, P = 0.07; R² = 0.51, P = 0.02). A similar analysis was done to study the relation between the ρ values and the rate of weight gain by the end of HE diet (presented in the appendix). In this case, ρ values explained 26% and 6% of the variance, respectively (R² = 0.26, P = 0.19; R² = 0.06, P = 0.50), and therefore no significant relation was found here. These results show that the variation in the value of parameter ρ can predict the change in calorie intake levels during dietary shifts. We have also studied the association between ρ and the amount of calorie intake during the chow diet and discussed its implications in the appendix. Each rat’s value of ρ is unique, which is in line with other studies that reported the role of genetic background in establishing individual differences in the responses to the changes in diet composition and energy availability (22). An immediate extension of our work would be to apply our model to two genetically different groups of rats [e.g., rats with diet-induced obesity (DIO) and diet-resistant (DR) rats], and measure the differences between the ρ values of these two groups.

Food with higher energy densities, such as high-sugar and high-fat foods, are generally considered more palatable and more rewarding to humans and nonhuman animals. As our model is specifically designed to predict the calorie intake of animals when a diet with a different energy density is offered, it directly relates to the studies of the rewarding aspects of food intake. Among these are studies based on the principles of reinforcement learning. Specifically, the behavior and meaning of the parameter ρ in our model closely relates to the so-called “reward sensitivity” parameter in reinforcement learning models of food choice (19). We hypothesize that ρ associates with reward sensitivity; investigating this hypothesis would be the subject of another study. Reduction in reward sensitivity is possibly the closest behavioral equivalent to the notion of a reduction in consummatory pleasure. Multiple studies have linked body weight changes to reward sensitivity and learning rate of individuals (7, 10, 19, 34), which are the two standard measures used in developing models of reinforcement learning. These two parameters are commonly estimated using maximum likelihood methods on data sets collected in probabilistic reward tasks experiments (14). Because of the way that ρ is defined in our model, it encodes a similar meaning as the reward sensitivity parameter. As collecting data using standard methods — i.e., from iterative probabilistic reward tasks — is challenging, our model can potentially serve as a simpler way to calculate the reward sensitivity parameter.

When a new diet is presented, the animals require some time to adjust and regulate their food intake. When the presented diet has a higher energy density, a spike in the overall amount of food intake is observed likely due to the novelty and/or palatability of the diet (36). The exponent, 1/4, used in the variable T in our model results in similar behavior. Other relevant effects such as sensory-specific satiety, as sometimes measured by food variety (availability of different types of food) (29, 31), can also be considered for extending our model to more complex scenarios of food intake (4).

The generated patterns of food intake by our model match many of the patterns reported in other studies of food intake in rodents. One pattern that we observed in our simulated results was that after switching to a high-energy diet, the generated calorie intake patterns initially increased significantly, and then decreased across subsequent days. In addition to matching our actual data, this is also consistent with other studies, which suggest that increased positive signals from oral stimulation and dampened inhibitory feedback are most robust during the first few days after high-energy diet presentation (36). Moreover, through our simulated experiments on body weight changes of study 2 rats, we observed that as the calorie intake of the rats decreased, the body weights computed by our model also plateaued. These patterns match the same patterns reported by Levin et al. (21, 22). They observed that rats that gained weight during high-energy diet presentation return to a “set point” after re-presentation of a chow diet.

Perspectives and Significance

One of the driving factors of the obesity epidemic has been an easier access to high-energy foods. Considering the challenges that exist in collecting and analyzing food intakes of humans, animal models can help us understand the role of the rewarding aspects of foods, as reflected by the amount of food consumed. The mathematical model proposed in this work is one step toward developing such models. It can predict the changes in food intake during shifts in the energy density of the rats’ diet. Our model can also be tested on other types of rodents or animals. Since the same type of rats was used in the two data sets of our work, we plan to test our model on different rat types to investigate its generalizability. Although we could have used a single partitioned data set for calibrating and testing our model, the tests on an independent second data set resulted in increased model confidence. Currently, our model only considers the total daily food intake of rats. The model can be extended in the future to simulate meal patterns, with daily variations in meal size and frequency. Although the total body weight might serve as a good proxy for body fat, we did not have access to lean and adipose tissue data for the studied animals to investigate their effect. It is possible that including such more elaborate measures in addition to body weight would increase the accuracy of our model.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

R.B. and Y.T. conceived and designed research; R.B. performed experiments; R.B. analyzed data; R.B. interpreted results of experiments; R.B. prepared figures; R.B. drafted manuscript; R.B., Y.T., T.I., and T.H.M. edited and revised manuscript; R.B., Y.T., T.I., and T.H.M. approved final version of manuscript.

Glossary

R

Overall perceived hedonic reward from the food

W

Body weight (g)

E

Energy density of the diet (cal/g)

ρ

Constant parameter (specific to each animal)

T

No. of days elapsed after the start of current diet

C

Amount of calories consumed (cal)

APPENDIX

Reward and Calorie Intake Relation

As specified in the main text, the most practical behavioral method for measuring the level of perceived reward by rodents that has been used in the literature is the licking rate when a liquid meal is consumed. To obtain a mathematical relation between the reward (R) as measured by licking rate, and the cumulative amount of calorie intake (C), we can use the equations that have been suggested for describing the relation between R and time (t), and the relation between C and t. It has been suggested that the patterns of licking rates over time follows a Weibull function in the form of:

$$R=\text{α}{e}^{\left[–{\left(\beta t\right)}^{\text{γ}}\right]}$$ (A1)

where α, β, and γ are the parameters that are unique to each animal, and t is the time passed from the moment that the consumption of a meal starts (8, 28). Additionally, McCleery (26) suggested that a negative exponential formula with three parameters is the best choice for this purpose, by comparing three types of equations for describing C over time (t) (including a power function and a hyperbolic formula). This equation has the form of:

$$C=\text{ζ}-\text{η}{e}^{-\text{θ}t}$$ (A2)

while ζ, η, and θ are the parameters of the model. This equation has been used, and also extended by many scholars, including a recent model proposed by Thomas et al. (35) as an extension of this equation. From Eq. A2, we will have:

$$C-\text{ζ}=-\text{η}{e}^{-\text{θ}t}$$

$$\Rightarrow \frac{\text{ζ}-C}{\eta }=e^{-\text{θ}t}$$

$$\Rightarrow \ln \left(\frac{\zeta -C}{\eta }\right)=\ln e^{-\text{θ}t}$$

$$\Rightarrow \ln \left(\frac{\zeta -C}{\eta }\right)=-\text{θ}t$$

$$\Rightarrow t=\frac{-\ln \left(\frac{\zeta -C}{\text{η}}\right)}{\text{θ}}$$ (A3)

We can replace t in Eq. A1 with the right side of the Eq. A3. After replacing t, R would be simplified to a polynomial function of C with new constant parameters (the process is not shown here). We know that during a course of a meal, there is one point that the amount of perceived reward is maximized, and this peak point is close to the beginning of the meal (8). This means that only one of the maxima of the polynomial function would fall between the C = 0 and C = C_f (end of the meal consumption). Based on these properties, one candidate function for capturing similar patterns would be a logarithmic function with a shape similar to the function shown in Fig. 1 of the main text of this paper. Although the reward cannot be equal to infinity at the beginning of the meal (C = 0), the logarithmic function can still present a good approximation of the remaining part of the meal consumption (from the peak of the perceived reward to the end of the meal).

Adjusting the Proposed Equation in Our Model

The purpose of this step was to find the best functions f, in the equation

$$C_f\cong \text{ρ}\frac{f\left(W\right)f\left(E\right)}{f\left(T\right)}$$ (A4)

that was introduced in the main text, which can best match to the actual patterns. We used a program (implemented in Matlab software) to test a set of strictly increasing functions for f and a set of synthetic values between −1 and +1 for ρ (with 0.1 intervals for search). Here, W shows the weight (g), E shows the energy density of diet (kcal/g), and T refers to the time elapsed after the beginning of this new diet (days). Specifically, we tested power functions and exponential functions. The power functions were in the form of xⁿ, n = 0.05, 0.10, . . . , 5. Additionally, we used natural exponential function (in the form of e^x) and its inverse [ln(x)], where, in both power and exponential functions, x refers to any of the three variables (W, E, and T) in Eq. A4. In this exhaustive search for finding the best candidates for the f functions, we used least-squares errors to decide which functions are the best (have the minimum error value).

Additional Types of Analysis for Comparing Our Model Against Actual Data

To formally measure the distance between the actual data sets and predicted patterns by our model, root-mean-square deviation (rmsd) and normalized root-mean-square deviation (nrmsd) between these two data sets are shown in Fig. A1 and Fig. A2 for each rat. Both rmsd and nrmsd are common ways of measuring the accuracy of different models. The range of values of rmsd was between 6 and 19, showing that our model’s predictions were close to the actual calorie intake patterns. Additionally, the values for normalized rmsd measurement were between 0.08 and 0.57 (with an average of 0.15 for study 1 and 0.36 for study 2).

In addition to this analysis, and the analysis that has been shown in the main text, we have also performed a Bland-Altman type of analysis to estimate the differences between predicted and actual daily caloric intake. The results related to these analyses are shown in Fig. A3 and Fig. A4.

Analyzing the Relationship Between the Modeled and Actual Caloric Intake

To formally study whether there is a significant relationship between the modeled and actual data, we have performed a set regression analyses on these two data sets. Tables A1 and A2 show the results of our experiments.

In addition to regression analyses, we have also used a form of equivalence testing procedure called “two one-sided t-tests” (TOST). This procedure tests if two sample sets come from distributions with different means (H0), against the alternative hypothesis that the distribution means are the same (H1) (30). The procedure’s outputs are two P values (one per t-test) and a 90% confidence interval. The null hypothesis should be rejected if max(pval-1, pval-2) > 0.05, or if the confidence interval falls inside of the equivalence interval. More details can be found in the original reference. Tables A3 and A4 show the corresponding results. The interval used for analyzing the study 1 rats was [−15, 15] and for the study 2 rats was [−20, 20].

Modeling Body Weight Changes

The purpose of this part of our work was to examine the ability of our model to predict body weight changes of rats, as resulted from the changes in their diet. Several existing works have reported that after switching to a chow diet, the rats that had received high-energy diet and gained weight, lost weight and returned to a so-called “set point” body weight after re-presentation of a chow diet (21, 22). To see whether our model can also generate patterns that are consistent with these findings, we used the proposed equation in this paper to predict the body weights of rats based on the energy density of their diet (E), total amount of calorie intake (C), and the duration in which the current diet was given (T). This was done using an alternate representation of the proposed equation for our model: W = CT^1/4[log(E)ρ]⁻¹. As discussed in the main text, we used the data from the initial baseline period of the study 2 rats for training the best values of ρ. Then the data from the recovery period of the study 2 rats was used for testing this experiment. Study 1 rats were not studied in this experiment because the data from the recovery period of that study was already used in the training phase (hence, the same data could not be used for testing). Since the suggested equation for W does not model the rate of the changes in the variables (as in differential equations), the predicted body weights of rats based on the amount of their food intake might have unrealistic fluctuations across the simulation days. This is due to the natural alternations in the animals’ daily calorie intake that is used in our model to predict daily body weights (input to the above equation). Accordingly, instead of reporting daily predictions, we compare the average of the modeled and actual body weight values of the animals.

The average of modeled and actual body weights for each animal is shown in Table A5. In addition to the average values, using a similar approach, we have also calculated the rate of changes in the modeled body weights. The derivative of the previous formula was used for this purpose {dW/dT = (1/4)C[T^3/4log(E)ρ]⁻¹}, with the assumption that T is a continuous variable. For all of the rats, the value of this derivative reached to zero (less than one) after the initial days of recovery. This indicates that the changes in the body weights plateaued after switching back to the chow diet. Collectively, the obtained values for the average body weights, and for the derivative of body weights, show that our model was able to regenerate realistic patterns that were observed in other studies: after switching to chow diet from a high-energy diet, the changes in rats’ body weights decreased significantly (21, 22).

Association Between ρ and Weight Gain

Similar to the analysis that was presented in the main text (Fig. 4), the association between the percentage of body weight gain and the values of ρ are shown below. The body weight of animals at the beginning of experiments was considered as 100%, and the reported percentages in Fig. A5 show the total increase by the end of the HE diet period. For instance, 44% indicates that the overall body weight of an animal reached to 144% by the end of the HE diet period.

Comparing Variables’ Role in Model

To investigate which variable in the proposed model has a more important role on the model’s performance (measured by its accuracy of predictions), we have compared the differences between the data and model results without including variation in one of its variables: W, E, or T. The results of this experiment on study 1 and study 2 rats are shown in Tables A6 and A7, respectively. These results indicate that the time variable (T) in the case of study 1, and body weight variable (W) in the case of study 2, had the highest impact. Although this observation does not identify a unique pattern, the strong associations between caloric intake and time and body weight are reasonable.

Correlation Between ρ and Chow Food Intake

We have also analyzed the association between the ρ parameter and the average amount of energy intake during the chow diet for the study 2 data set (results presented in appendix). The study 1 data points were not used, since the number of chow data points were limited. As Fig. A6 shows, we observed that chow calorie intake was able to predict ~49% of variance (R² = 0.49, P = 0.02). While a strong association between ρ and the chow calorie intake was observed, the association between chow energy intake (EI) and high-energy (HE) EI (R² = 0.34, P = 0.04) were weaker than the association between ρ and HE EI (R² = 0.51, P = 0.02). This shows that ρ is a stronger predictor of the EI during the HE period.

Fig. A1.

Comparison between the actual daily calorie intake of 8 rats from the study 1 (circles) and the predicted calorie intake by our model (stars) during the 42 days of high-energy diet presentation. Rat ID and the value of ρ (rho) for each rat are shown at top. rmsd, root-mean-square deviation; nrmsd, normalized root-mean-square deviation.

Fig. A2.

Comparison between the actual calorie intake of the 10 rats of study 2 (circles) and the predicted ones by our model (stars) during the 14 days of receiving high-energy diet. Rat ID and ρ (rho) value for each rat are shown at top. rmsd, root-mean-square deviation; nrmsd, normalized root-mean-square deviation.

Fig. A3.

Bland-Altman analysis between the actual daily calorie intake of 8 rats of study 1 and the predicted calorie intake by our model during the 42 days of high-energy diet presentation. Rat ID and the value of ρ (rho) for each rat is shown at top.

Fig. A4.

Bland-Altman analysis between the actual daily calorie intake of 8 rats of study 2 and the predicted calorie intake by our model during the 14 days of high-energy diet presentation. Rat ID and the value of ρ (rho) for each rat is shown at top.

Fig. A5.

The association between the percentage of body weight gain by the end of the high-energy diet period and the values of ρ for each rat in study 1 (A) and in study 2 (B).

Fig. A6.

The association between the amount of chow food intake and the values of ρ for the rats in study 2. Study 1 rats were not analyzed, since the number of chow data points was small.

Table A1.

Association between modeled and actual calorie intake per individual animal of study 1 rats

Animal ID:	1	4	5	9	10	11	12	14
R2	0.9481	0.8470	0.8193	0.9102	0.9003	0.8541	0.7500	0.9261

Table A2.

Association between modeled and actual calorie intake per individual animal of study 2 rats

Animal ID:	1	2	4	5	7	9	10	12	17	20
R2	0.958	0.95	0.958	0.958	0.957	0.953	0.959	0.945	0.916	0.914

Table A3.

TOST procedure results: equivalence test between modeled and actual calorie intake per individual animal of study 1 rats

Animal ID:	1	4	5	9	10	11	12	14
P value 1	0.94	0.23	0.67	0.82	0.34	0.78	0.98	0.83
P value 2	0.06	0.34	0.33	0.18	0.46	0.27	0.02	0.32
CI	[−6.2, 10.6]	[−3.4, 11.3]	[−1.1, 16.4]	[−12.4, 13.5]	[−5.6, 8.3]	[−6.4, 13.1]	[−9.8, 7.7]	[−6.2, 6.8]

TOST, two one-sided t-tests; CI, confidence interval.

Table A4.

TOST procedure results: equivalence test between modeled and actual calorie intake per individual animal of study 2 rats

Animal ID:	1	2	4	5	7	9	10	12	17	20
P value 1	0.88	0.85	0.81	0.77	0.80	0.84	0.79	0.81	0.79	0.88
P value 2	0.12	0.15	0.19	0.23	0.20	0.16	0.21	0.19	0.21	0.12
CI	[−20.5, 15.1]	[−20.5, 16.6]	[−18.3, 19.4]	[−15.6, 19.7]	[−18.6, 19.3]	[−17.7, 16.9]	[−19.2, 19.8]	[−16.8, 17.9]	[−15.2, 18.7]	[−19.2, 15.1]

TOST, two one-sided t-tests; CI, confidence interval.

Table A5.

Average body weight of study 2 rats during the recovery period, as predicted by our model and reported in the actual data set

Rat ID:	1	2	4	5	7	9	10	12	17	20
Weight (model), g	451.5	384.1	438.3	452.9	450.1	397.3	397.3	469.5	439.4	420.2
Weight (data), g	436.6	407.5	427.9	448.9	457.4	401.6	428.9	437.5	412.1	440.3

Table A6.

Comparing the predictions of the modified versions of our model on study 1 rats: 1) model without body weight (W) variable, 2) model without energy density (E) variable, and 3) model without time (T) variable

Rat ID:	1	4	5	9	10	11	12	14	All Rats
Model w/o W	9.24 (9.0)	17.06 (16)	13.81 (12.5)	11.25 (11.6)	10.78 (7.4)	13.44 (14.7)	11.77 (11.8)	7.9 (6.2)	11.91
Model w/o E	4.68 (3.6)	9.27 (8.8)	7.62 (9.3)	5.85 (4.7)	7.17 (4.5)	8.82 (7.3)	8.52 (11.8)	7.72 (6.1)	7.45
Model w/o T	17.17 (15.6)	14.4 (13)	16.26 (15.2)	13.12 (11.7)	18.42 (14.1)	14.44 (17)	19.61 (18.7)	20.2 (14.7)	16.7

The average of the difference between the predicted HE calorie intakes and actual data are shown as means (SD).

Table A7.

Comparing the predictions of the modified versions of our model on study 2 rats: 1) model without body weight (W) variable, 2) model without energy density (E) variable, and 3) model without time (T) variable

Rat ID:	1	2	4	5	7	9	10	12	17	20	All Rats
Model w/o W	10.84 (7.1)	11.63 (7.8)	9.8 (10.3)	9.91 (7.8)	10.32 (7.6)	8.46 (8.8)	10.1 (8.1)	10.58 (8.3)	12.05 (10.9)	14.37 (13.4)	13.76
Model w/o E	16.35 (8.1)	14.28 (11.9)	15.54 (11)	10.76 (6.2)	13.51 (8.2)	11.06 (8.3)	13.17 (9.2)	12.19 (8.6)	11.65 (11.1)	19.05 (15.3)	11.26
Model w/o T	12.5 (7.6)	11.85 (9.7)	11.54 (10)	9.02 (7)	10.48 (7)	9.69 (8)	10.13 (8)	10.66 (7.5)	11.32 (10.9)	15.4 (14.5)	11.58

The average of the difference between the predicted HE calorie intakes and actual data are shown as means (SD).