Predicting Energy Intake in Adults who are Dieting and Exercising

Abstract

Background:

When a lifestyle intervention combines caloric restriction and increased physical activity energy expenditure (PAEE), there are two components of energy balance, energy intake (EI) and physical activity energy expenditure (PAEE), that are routinely misreported and expensive to measure. Energy balance models have successfully predicted EI if PAEE is known. Estimating EI from an energy balance model when PAEE is not known remains an open question.

Objective:

The objective was to evaluate the performance of an energy balance differential equation model to predict EI in an intervention that includes both calorie restriction and increases in PAEE.

Design:

The Antonetti energy balance model that predicts body weight trajectories during weight loss was solved and inverted to estimate EI during weight loss. Using data from a calorie restriction study that included interventions with and without prescribed PAEE, we tested the validity of the Antonetti weight predictions against measured weight and the Antonetti EI model against measured EI using the intake-balance method at 168 days. We then evaluated the predicted EI from the model against measured EI in a study that prescribed both calorie restriction and increased PAEE.

Results:

Compared with measured body weight at 168 days, the mean (±SD) model error was 1.30 ± 3.58 kg. Compared with measured EI at 168 days, the mean EI (±SD) model error in the intervention that prescribed calorie restriction and did not prescribe increased PAEE, was −84.9 ± 227.4 kcal/d. In the intervention that prescribed calorie restriction combined with increased PAEE, the mean (±SD) EI model error was −155.70±205.70 kcal/d.

Conclusion:

The validity of the newly developed EI model was supported by experimental observations and can be used to determine EI during weight loss.

INTRODUCTION

Weight change studies require accurate measurements of energy intake (EI) and total daily energy expenditure (TDEE). Two reasons requiring accurate objective knowledge of EI is to 1) track dietary adherence during weight loss interventions^{1, 2} 2) determine true intervention effect in post-hoc analysis. If participant adherence differs by individual, then actual EI that was acheived should be factored into determining the effect of an intervention. Because the relationship between weight and EI^{3, 4} is not linear, we are not simply able to determine adherence from body weights alone. Measurements of EI and physical activity using self-reported instruments are flawed; typically resulting in underreported EI ⁵ and over-reported physical activity ^{6, 7}. Energy balance models that predict EI have been validated in clinical trials that collected EI measured by the intake-balance method ^{3, 8}. Energy balance models have also been successfully used to foster adherence to prescribed EI in dietary interventions ².

For weight loss, the recommended initial treatment strategy is a structured lifestyle intervention which includes a prescription for caloric restriction alongside a recommended increase in physical activity ⁹. In the case of lifestyle interventions that include both caloric restriction and increased caloric expenditure through physical activity, two variables contribute to a change in energy balance: 1) the change in EI and 2) the change in physical activity energy expenditure (PAEE). When expressed in the form of the energy balance equation as,

$$ES=\mathit{E}\mathit{I}-(\mathit{P}\mathit{A}\mathit{E}\mathit{E}+TEF+RMR)$$

where ES represents the rate of energy stores, EI the rate of energy intake, PAEE is the rate of physical activity energy expenditure, TEF the rate of the thermic effect of feeding and RMR is the resting metabolic rate, all in kcal/d. A differential equation energy balance model formulates terms and derivatives of the terms for each of the expressions in the energy balance equation. To estimate EI, the model is then inverted to solve for EI as:

$$\mathit{E}\mathit{I}=ES+\left(\mathit{P}\mathit{A}\mathit{E}\mathit{E}+TEF+RMR\right)$$

If both EI and PAEE (expressed as the terms in bold) are unknown, the above equation is referred to as an undetermined system ¹⁰ and cannot be used to estimate EI without making a simplifying assumption about PAEE.

To estimate EI using the energy balance equation, we aim to simplify the model, by assuming that the increase in PAEE throughout the intervention is zero. Importantly, this assumption does not imply that PAEE is zero, but that the impact on ES due to any increase in PAEE from an intervention is negligible. This assumption is justified because when physical activity is combined with CR to treat obesity, the addition of the physical activity will result in only small increases in weight loss of ~1.1 kg compared to diet-only programs^{11, 12}. However, the impact of PAEE on ES is likely dependent on the dose of physical activity, with higher doses of physical activity resulting in greater weight loss. Nonetheless, even when adults are provided a high dose of physical activity (~286 kcal/d or 2000 kcal/week), in line with recommendations for weight loss and weight loss maintenance¹³, the majority of adults struggle with adopting and maintaining these levels long-term¹⁴. Thus, the assumption that changes in PAEE have a negligible impact on ES may be valid.

To test whether this assumption can yield reasonable estimates of EI during a weight loss intervention, we relied on a thermodynamic energy balance model developed to predict weight during calorie restriction (CR). We then inverted this model to predict EI from body weight. Next, we compared the predicted EI to measured EI in an intervention that included both calorie restriction and carefully prescribed and monitored exercise (CR+EX). If this comparison yields reasonable estimates, the EI formula can be used to estimate participant EI in CR+EX lifestyle interventions during the intervention or post-hoc and in a cost-effective reliable manner.

METHODS

Study Design

The overarching model developed is a formula that outputs EI after input of biological sex, height, baseline weight, post-intervention weight, and the total number of intervention days. The final EI formula was derived in three steps.

Step 1: Solve for weight trajectories as a mathematical expression:

The first step was to solve the differential equation developed by Vincent Antonetti ¹⁵ that predicts body weight after a change in intake, EI. The Antonetti model solution yields a formula for weight trajectories after input of EI, age (years), height (cm), baseline weight (kg), and biological sex (male=1, female=0). The solution is not a numerical approximation that requires computer programming or specialized software, but a nonlinear formula that generates weight loss trajectories as a function of the inputs.

Step 2: Validate the weight trajectories in CR data:

The second step was to evaluate the Antonetti model weight predictions with measured body weight data from a large diet intervention that prescribed a change in CR and did not prescribe a change in PAEE (see a detailed study description further in the Methods) ¹⁶. The purpose of this step is to verify the quality of the weight predictions, which is the original outcome variable of the Antonetti model.

Step 3: Invert the weight trajectories to obtain a formula for EI:

The third step is to algebraically invert the validated formula for the weight trajectory from Step 1 to express EI as a function of time-dependent weight. This results in a formula for EI that yields a value after input of age (years), height (cm), baseline weight (kg), biological sex (male=1, female=0), number of intervention days, and final weight.

Step 4: Validate the EI formula against measured EI in CR data:

The fourth step is to validate the formula for EI against measured EI using the intake balance method in an intervention that prescribed CR and did not prescribe changes in PAEE (see details of the study description further in the methods) ¹⁷. The purpose of this step is to check the quality of the predicted EI against measured EI values in data where the underlying model assumption of no added increase in PAEE was satisfied.

Step 5: Evaluate the EI formula against measured EI in CR+EX data:

The final step is to compare EI model predictions against EI measured by the intake balance method in a CR+EX intervention (includes both prescribed CR and a prescribed increase in PAEE) ¹⁸. Details of this study appear further in the Methods. In this step, we estimate how sensitive our simplifying assumption that PAEE will not change beyond variations accounted for by changes in body weight is on predictions to EI.

A diagram of the study design steps appears in Figure 1.

Figure 1:

Diagram illustrating the steps in the process of building the model for predicted EI.

The Antonetti Energy Balance Weight Prediction Model

The original Antonetti energy balance model was first published in 1971 in the American Journal of Clinical Nutrition ¹⁵. A modified version that uses the Mifflin St. Jeor equations for the resting metabolic rate ¹⁹ renders the Antonetti differential equation solvable ²⁰ without requiring a specialized numerical program for simulation of weight trajectories (see the derivation of the final weight trajectory formula from the differential equation in the Supplemental Materials). The final weight trajectory that arises from the Antonetti energy balance model is:

$$W\left(t\right)=-\frac{\beta }{\alpha }+C{e}^{\alpha t}$$ (1)

where

$$\alpha =-\frac{m+10}{7700},$$

$$\beta =\frac{0.9EI+\gamma }{7700},$$

and

$$C=W_0+\frac{\beta }{\alpha }$$

where W₀ represents baseline weight in kg.

The parameters m and γ originate from model development from the energy balance equation and are defined as

$$m=\frac{0.9TDE{E}_0-10W_0-6.25H+5A-166Sex+161}{W_0}$$

$$\gamma =-6.25H+5A-166Sex+161$$

where TDEE₀ (kcal/d) represents baseline energy requirements in kcal/d, H is height in cm, A is age in years, and Sex is 0 for females and 1 for males. For full model derivation from the energy balance model to Antonetti differential equation, see ¹⁵ or the Supplemental Online content. Full details to solve for the weight trajectory from the differential equation appears in the Supplemental Online content.

CR Study Data used to validate the Antonetti Weight Model and EI Formula

The Comprehensive Assessment of Long-term Effects of Reducing Intake of Energy Phase II study (CALERIE II) ²¹ was designed to test the hypothesis that two years of caloric restriction at 25% below baseline levels will slow aging. The study was conducted at three sites: the Pennington Biomedical Research Center in Baton Rouge, LA, Tufts University in Boston, MA, and Washington University St. Louis, MO ¹⁶. The study also performed measurements of TDEE at baseline, 6 months, 12 months, and 24 months using the gold standard method of doubly labeled water (DLW). In addition, the study performed simultaneous measurements of body composition at 6 months, 12 months, 18 months, and 24 months using dual energy X-ray absorptiometry (DXA). Using the intake-balance method ²², which sums the measured body energy stores and TDEE, an objective clinically measured estimate of EI was obtained. This measurement is required as an input for the Antonetti model to generate body weight trajectories. CALERIE II included a sample of 220 participants randomized into two groups: a calorie restriction (CR) group prescribed 25% CR below baseline requirements (N=135) and an ab libitum control group (N=85). While neither randomized group received an exercise prescription, exercise was permitted. Body weight data at 168 days from the CR group was used to validate the Antonetti body weight trajectories. Longer term weight data was not used because measured EI varied over time and cannot be approximated well by a single 24-month constant EI for input into the Antonetti model.

The CALERIE II CR study data was also used to input baseline and 6-month body weights into the inverted Antonetti EI model to generate predicted EI values from body weight.

All participants at all sites provided written informed consent.

CR+EX Study Data used to Test the Assumptions behind the Antonetti EI model

The Comprehensive Assessment of Long-term Effects of Reducing Intake of Energy Phase I Study (CALERIE I) tested the effects of CR on biomarkers of longevity ¹⁸. Data from subjects enrolled at the Pennington Biomedical Research Center were used. Twelve of the CALERIE I subjects were placed on a weight-maintenance diet, which began with a low-calorie diet (LCD) of 890 kcal/d until 15% of baseline body weight was lost, followed by weight maintenance. Twelve of the CALERIE phase I subjects were placed on a calorie-restricted (CR) diet of a restriction of 25% of their baseline energy requirements, and 12 were prescribed a combination of CR and exercise (CR+EX: 12.5% CR of baseline energy requirements plus 12.5% increase in energy expenditure by structured aerobic exercise). The mean (X±SD) target prescription was 403± 63 kcal per session for women and 569 ±118 kcal per session for men, targeting 5 sessions per week ¹⁸. TDEE was measured using DLW at baseline, three months, and six months. Compliance to exercise prescriptions was measured from oxygen cost ²³ and the measured increase to PAEE was 336.91± 106.31 kcal/d.

Body composition was simultaneously measured at the three time points using DXA. Measured EI over the 6-month study was then estimated using the intake-balance method from the CR+EX arm and used to evaluate the capacity of the Antonetti EI formula to estimate EI during a CR+EX intervention. CALERIE I was approved by the Pennington Biomedical Research Center Institutional Review Boards (Baton Rouge, LA). All participants gave written informed consent before enrollment.

EI Model Derived from the Antonetti Energy Balance Model

Given a weight on day t, the weight trajectory in Equation 1 can be inverted and algebraically solved for EI resulting in:

$$EI=\frac{77000\alpha \left(W\left(t\right)-W_0{e}^{\alpha t}\right)-10\gamma \left(e^{\alpha t}-1\right)}{9\left(e^{\alpha t}-1\right)}$$ (2)

To use the formula to calculate EI at day 84 of an intervention for a female age 50, height 163 cm, baseline weight 80 kg, and weight on day 84 of 75 kg, we first estimate baseline energy requirements using the calculator from ²⁴

$$TDE{E}_0=2342\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}/\mathrm{d}.$$

Then we calculate m

$$\begin{array}{l}m=\frac{0.9TDE{E}_0-10W_0-6.25H+5A-166Sex+161}{W_0}\\ =\frac{0.9\times 2342-10\times 80-6.25\times 163+5\times 50-166\times 0+161}{80}\\ =\mathrm{8.75.}\end{array}$$

We next calculate α and γ:

$$\alpha =-\frac{m+10}{7700}=-\frac{8.75+10}{7700}=-0.00244,$$

$$\gamma =-6.25H+5A-166Sex+161=-6.25\times 163+5\times 50-166\times 0+161=-607.75$$

And then input these values into the formula for EI:

$$\begin{gathered} EI=\frac{77000\alpha \left(W\left(t\right)-W_0{e}^{\alpha t}\right)-10\gamma \left(e^{\alpha t}-1\right)}{9\left(e^{\alpha t}-1\right)} \\ =\frac{77000\left(-0.00244\right)\left(75-80e^{\left(-0.00244\times 84\right)}\right)-10\left(-607.75\right)\left(e^{\left(-0.00244\times 84\right)}-1\right)}{9\left(e^{\left(-0.00244\times 84\right)}-1\right)} \\ =1779\frac{\text{kcal}}{\text{d}} \end{gathered}$$

Validation of the Antonetti Energy Balance Weight Prediction Model and EI model

A Bland Altman analysis was performed to compare 6-month actual body weights versus Antonetti predicted body weights. EI was calculated using the intake-balance method. We used the energy density conversion constants in ⁸ and the DXA measured change in fat mass (FM) in kg and change in fat free mass (FFM) in kg to calculate the difference quotient for $\frac{dFM}{dt}$ and $\frac{dFFM}{dt}$ as in ²⁵

$$E{I}_{measured}=9500\frac{\Delta FM}{\Delta t}+1020\frac{\Delta FFM}{\Delta t}+E{E}_F$$

where ΔFM and ΔFFM are the change in fat and fat free mass in kg over the 6 month period, EE_F is the DLW measured TDEE in kcal/d at 168 days, and Δt is 168 days. A correlation plot and bias plot were generated and the R², bias, and 95% confidence intervals were calculated.

A Bland Altman analysis was also performed against EI measured by the intake-balance method and EI predicted by Equation 2. Similar to the validation of body weights, a correlation plot and bias plot were generated and the R², bias, and 95% confidence intervals were calculated.

Evaluation of EI Model for CREX

The EI model predictions in Equation 2 were compared against EI measured through the intake-balance method from CALERIE Phase 1 study data using the CR+EX arm. A Bland Altman analysis was performed consisting of a correlation plot and bias plot. The R²², bias, and 95% confidence intervals were calculated.

Web-based RShiny App Development

A web-based RShiny application that outputs EI at any time during an intervention was developed using in the programming language R (R Core Team (2013)).

RESULTS

Validation of the Antonetti Weight Trajectories

Weight at 168 days for each individual participant of the CALERIE Phase II study was predicted from the Antonetti weight trajectory (Equation 1) after entering sex, age in years, height in cm, and baseline weight in kg. The model parameter, m, was calculated as described in the Supplemental Materials. Figure 2 Panel A is the correlation plot of measured weight versus predicted weight. Figure 2 Panel B is a plot of the difference between measured and predicted versus the average of the measured versus predicted for each individual participant. Figure 2 Panel B shows how the error (difference between measured and predicted EI) is distributed as a function of EI (average of measured and predicted EI). The 95% confidence intervals plotted on the figure depict an interval of agreement that 95% of the error falls within. The hope is that the error is randomly distributed around a mean error (bias) of zero and that the 95% confidence interval is acceptably small ²⁶. The mean (±SD) model error was 1.30 ± 3.58 kg. The coefficient of determination between measured weight and predicted weight was R²=0.87 with line of regression y = 0.88x + 9.23. The bias was 1.30 kg with 95% confidence intervals [−5.72, 8.33]. There was a small negative trend in the error, R²=0.03.

Figure 2:

Bland Altman plots for 6-month EI measured using the intake balance method versus the Antonetti predicted EI in the CR participants of CALERIE Phase 2. The left panel is the correlation plot of measured versus predicted and the right panel is the difference between measured and predicted versus the average between measured and predicted. The grey line at 0 kg would be the expected difference if our model accurately predicted measured weight. The right panel includes the bias (solid dark horizontal line) and the 95% confidence intervals (dashed lines).

Validation of the Antonetti EI predictions

EI for each individual participant of the CALERIE Phase II study was calculated from Equation 2 after entering sex, age in years, height in cm, baseline weight in kg, and weight in kg at 168 days. The model parameter, m, was calculated as described in the Supplemental Materials. Compared with measured EI at 168 days in the intervention, the mean (±SD) model errors were −84.86 ± 227.38 kcal/d with 95% confidence intervals [−530.51, 366.80]. Figure 2 Panel C is the correlation plot of EI measured using the intake-balance method versus predicted EI. Figure 2 Panel D is a plot of the difference between measured and predicted EI versus the average of the measured and predicted EI for each individual participant. The coefficient of determination between measured EI and predicted EI was R²=0.69 with line of regression y = 0.83x + 256.68. There was no trend in the error in the difference-mean plot with line of regression, y = −0.005x −75.209 and coefficient of determination, R² = 07E-5.

Evaluation of the Antonetti EI predictions in CR+EX data

EI for each individual participant of the CALERIE Phase I CR+EX study arm was calculated from Equation 2 after entering sex, age in years, height in cm, baseline weight in kg, and weight in kg at 168 days. The model parameter, m, was calculated as described in the Supplemental Materials. Compared with measured EI at 168 days, the mean (±SD) model errors were −155.70 ±205.70. Figure 3 Panel A is the correlation plot of EI measured using the intake-balance method versus predicted EI. Figure 3 Panel B is a plot of the difference between measured and predicted EI versus the average of the measured and predicted EI for each individual participant. The coefficient of determination between measured EI and predicted EI was R²=0.84 with line of regression y = 0.94x −10.34. The bias was −155.70 kcal/d with 95% confidence intervals [−558.86, 247.47]. There was no trend in the error in the difference-mean plot with line of regression, y = −0.028x −219.96 and coefficient of determination, R² = 0.0043. The two participants with the highest measured EI also had the greatest overpredicted error.

Figure 3:

Bland Altman plots for 6-month EI measured using the intake balance method versus the Antonetti predicted EI in the CREX participants of CALERIE Phase 1 study. The left panel is the correlation plot of measured versus predicted and the right panel is the difference between measured and predicted versus the average between measured and predicted. The grey line at 0 kcal would be the expected difference if our model accurately predicted measured EI. The right panel includes the bias (solid dark horizontal line) and the 95% confidence intervals (dashed lines).

Web-based RShiny application

The web-based application requires users to enter sex, baseline weight in kg or lb, post-intervention weight in kg or lb, and duration of intervention in days. The app outputs the Antonetti model predicted EI during the intervention in kcal/d and generates the predicted dynamic model weight trajectory (Equation 1). The app is freely accessible at https://diana-thomas.shinyapps.io/EnergyIntake/.

DISCUSSION

Weight change interventions that include both diet and exercise are changing two terms within energy balance: changes in EI and changes in PAEE. Thus, a dynamic energy balance model cannot estimate EI without making assumptions regarding the impact of the prescribed physical activity on TDEE. In idealized clinical-based interventions, compliance with an exercise component of an intervention is quantified in real-time with supervised exercise bouts informed by indirect calorimetry or stable isotopes. However in clinical settings, exercise prescriptions are often unsupervised, or if they are supervised, they do not involve measuring energy expenditure using indirect calorimetry or stable isotopes. Understanding the contribution of exercise to energy balance is important for understanding weight change and for guiding treatment recommendations. Self-reported physical activity combined with self-reported EI can lead to two non-objective quantifications of energy balance.

When we assumed that the amount of energy expended in exercise was negligble, we showed that this simplifying assumption had only a modest effect on predictions of EI in an intervention that increased PAEE by an average of 336.91 ± 106.31 kcal/d. We demonstrated under this assumption and under this exercise dose, that a simple dynamic energy balance model can accurately estimate EI within a few hundred kcal/d in a combined diet and exercise intervention. Therefore, the model estimated EI provides a reasonable objective quantity that clincians can rely on during combined diet and exercise interventions.

The large variation in the error (difference between measured and predicted EI) captured by the 95% confidence interval, [−558.86, 247.47] kcal/d, is expected. The difference between measured and predicted EI includes DLW measurement error, biological variation, and model error. While DLW is the gold standard for measuring TDEE in community dwelling participants, when DLW was compared to 24-hour metabolic chamber measurements in domiciled participants, the 95% confidence intervals Bland Altman plots comparing chamber and DLW measurements were around [−150, 210] kcal/d. Additionally, energy balance models like the Antonetti model ^{27, 28} include error from parameter estimations, the regression models used to develop terms, and the specific assumptions behind the model. The models also cannot capture all possible individual biological variation. Combined with additional simplifying assumption proposed here, the size of the 95% confidence interval is unsurprising.

Despite this concern, the EI model is objective and is much more reasonable and accurate than self-reported intake^5–7. We point out that the most effective use of models like the one proposed here is when information is triangulated with self-report and other EI measures such as eating sensors^{29, 30}. A recent randomized control trial found that using a model combined with self-reported intake and a wrist worn eating sensor resulted in higher adherence compared to intervention arms that included only type of EI measurement.

There are several strengths in this study. First, we demonstrate that an overaching assumption to neglect additional PAEE in a lifestyle intervention that combines CR and EX still results in reasonable and reliable predictions of EI. Another strength of the study is the application of the Antonetti model which is simple and one dimensional. The simplicity results in closed form solutions of weight trajectories. These solutions can be used to express EI as a formula depending on baseline and final time weight inputs to arrive at an EI estimate for any time point during the course of the intervention. This simplicity makes the model easy to transform and can be used within a spreadsheet to quickly generate objective EI estimates. While simple, the model still gave highly correlated predictions to measured EI derived by the intake-balance method.

There are several study limitations. First, the EI model under the assumption that increases in PAEE is neglible was validated on a small independent sample obtained from one arm of a weight loss intervention that prescribed reductions in caloric intake of 12.5% combined with increased PAEE of 12.5%. The validation data was used because it included all energy balance estimates (DLW measured TDEE and body composition with DXA) requried to compare actual EI to EI measured using the intake-balance method. In addition, the validation data is in line with the Physical Activity Guidelines³¹ that PAEE that is between 250–500 kcal per day. The average prescribed exercise dose in CALERIE Phase I CR + EX was 403 kcal/d for five days a week which averages to 288 kcal/d during the entire week.

The model needs to be validated in a larger, more representative sample of participants undergoing both restricted energy intake and increased physical activity. In such a sample, the same measurements that allow for EI to be calculated from the intake-balance method will be necessary. In addition, the EI model assumes that participants are weight stable at the start of the intervention. Thus, our model may not be valid in participants who are not considered weight stable at baseline.

The majority of exercise prescriptions tend to align with the Physical Activity Guidelines for American’s ³¹ which equates to a PAEE that is between 250–500 kcal per day. The actual achieved PAEE is much lower, as observed in a systematic review of exercise studies ¹¹ or when 20 kcal per kg of body weight is prescribed and supervised for compliance to the prescription, the resultant change to body weight is 50% of expected due to compensation supporting the simplifying assumption of negligible increase³². However, we caution that while the model is doing well within the confines of doses of exercise observed in the CALERIE Phase I CR+EX arm ³³, higher doses will probably result in increased overpredicted EI. This is already evidenced with negative bias calculated in the CR+EX data. Negative bias indicates that EI is overestimated by the model. Since PAEE is underestimated by the assumption that the increase in PAEE is neglible, the energy balance model will predict lower TDEE than what is truly achieved. As a result, EI will be overestimated by the model and hence, the difference between actual and predicted EI will have higher negative error for larger PAEE. The model should only be used in interventions that prescribe PAEE doses at or lower than those achieved by the validation data set of the CALERIE participants, which was 337 kcal/d. Above this prescribed dose, we recommend alternative strategies such as the Remote Food Photography Method ³⁴and/or eating sensors^{29, 30}.

In comparison to existing models, a more complex EI model based on the first law of thermodynamics yielded similar 168 day bias and Bland Altman 95% confidence intervals as the work presented here⁸. The Antonetti model also has a similar bias to other more complex EI models ³⁵.The validation of the Antonetti EI predictions where model assumptions were justified (CALERIE Phase II CR study), yields a root mean square error of 216 kcal/d at 168 days. This is comparable to what was found for the model in ³⁵ at 26 weeks of a root mean square error of 191 kcal/d. While the Antonetti model had slightly higher root mean square error, the model has the advantage that it is simple enough to solve in closed form for weight trajectories and an explicit formula can be generated for each participant’s EI. This makes the EI model easy to program into dashboards like the app developed here.

CONCLUSIONS

For a combined diet and exercise weight change intervention, a simple mathematical model that assumes exercise energy expenditure is neglible can provide objective reasonsable predictions of EI over the intervention course.

Table 1:

Participant characteristics for the CALERIE Phases I (CREX) and CALERIE Phase 2 (CR) studies used for EI model validation. Data are reported in ^{21, 33}.(mean (SD)).

	Female CALERIE I	Male CALERIE I	Female CALERIE II	Male CALERIE II
n	7	5	95	40
age (years)	34.42 (3.97)	38.63 (7.14)	37.08 (7.10)	40.90 (6.65)
TDEE (kcal/d)	2341.00 (310.09)	3098.00 (404.24)	2246.60 (277.10)	2816.08 (342.13)
BMI (kg/m2)	27.9 (1.9)	27.1 (1.4)	25.00 (1.78)	26.29 (1.57)

Abbreviations:

BMI

body mass index

CR

calorie restriction

EX

exercise

DXA

dual energy X-ray absorptiometry

DLW

doubly labeled water

TDEE

total daily energy expenditure

EI

energy intake

PAEE

physical activity energy expenditure

SD

standard deviation

DATA AVAILABILITY STATEMENT

We used the CALERIE data which is publically available at https://calerie.duke.edu/samples-data-access-and-analysis.