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Modeling circadian rhythms of food intake by means of parametric deconvolution: results from studies of the night eating syndrome^1,2,3

¹ From the New Bolton Center, School of Veterinary Medicine, University of Pennsylvania, Kennett Square, PA (RCB and PJM), and the Center for Weight and Eating Disorders, Department of Psychiatry, University of Pennsylvania School of Medicine, Philadelphia, PA (KCA, JDL, and AJS)

² Supported by National Institutes of Health grants no. K12HD043459 (to KCA) and R01 DK056735 (to AJS).

³ Reprints not available. Address correspondence to RC Boston, New Bolton Center, 382 West Street Road, Kennett Square, PA 19348. E-mail: drrayboston{at}yahoo.com.

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS AND SUBJECTS RESULTS DISCUSSION REFERENCES

 
Background: Disordered temporal eating patterns are a feature of a number of eating disorders. There is currently no standard mathematical model to quantify temporal eating patterns.

Objective: We aimed to develop a simple model by which to describe the temporal eating patterns of adult humans. For this purpose, patients with the night eating syndrome (NES) and persons without an eating disorder were assessed.

Design: Data were obtained from 2 studies, each involving patients with NES and control subjects. Data were analyzed by means of a novel equation to describe the 24-h temporal eating patterns. The equation employed the integration over time of 3 Gaussian equations to describe the cumulative daily caloric intake.

Results: The new model accurately described and quantified the temporal eating patterns of the subjects in the 2 studies. The analyses showed differences in the temporal eating patterns and in the amount of intake of normal-weight and overweight subjects with NES.

Conclusions: This novel model can be used to accurately and objectively describe and quantify temporal food intake patterns. It can also be used to establish norms for various human populations.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS AND SUBJECTS RESULTS DISCUSSION REFERENCES

 
Our understanding of pathological eating behavior is limited by variability in the timing and size of meals during a 24-h period. Although there is general agreement that most people consume 3 meals during this period, little consensus exists as to what constitutes a normal temporal meal pattern. There is variability even in the definition of types of meals. For example, breakfast has been defined as the first meal after awakening, a meal consumed at a certain time of day, a certain type of food consumed, and whatever a person perceives as "breakfast" (1, 2).

Most efforts at studying 24-h food intake use tabulations of the amounts of foods consumed at various times (3, 4). This method and the use of arbitrary definitions of what constitutes specific meals may have inadvertently led to distortions in the interpretation of eating patterns and to a failure to detect important features of eating patterns. Such problems have afflicted studies of sleep-wake dynamics. Recently, however, mathematical models have been developed that describe these functions in terms of circadian rhythms (5).

We present here a mathematical model of circadian aspects of food intake that can facilitate the study of eating patterns, both pathological and "normal" (6). This model is independent of subjective definitions of what constitutes breakfast, lunch, and dinner. It is tested by comparing the eating patterns of persons manifesting the night eating syndrome [NES (7-9)] with the eating patterns of a control group. NES has been conceptualized as a delay in the circadian pattern of food intake, exemplified by evening hyperphagia and nighttime awakenings with ingestion (9). The definition of evening hyperphagia for NES has varied across studies, in part because of a lack of information regarding appropriate cutoff times for "evening" or "after-dinner" eating. This study is the first to identify objective patterns of meal intake for persons with NES and control participants by using a new mathematical model.

	METHODS AND SUBJECTS

TOP ABSTRACT INTRODUCTION METHODS AND SUBJECTS RESULTS DISCUSSION REFERENCES

 
Methods
In this investigation we describe energy intake, and, for this reason, we focus on calories. The methods described, however, could just as well be applied to dry matter (in g), protein (in g), or any other dietary constituent. The circadian pattern of food intake describes the food intakes occurring during 3 distinct meals. Our mechanistic model is based on the assumption that the average rate of food intake during each of these 3 meals can be described by a symmetric pulse (Figure 1).

View larger version (14K): [in this window] [in a new window]   FIGURE 1. A schematic depicting the new conceptual model (the Boston model) of the average rate of eating during a meal, peak intake rate, time of peak intake rate, and meal spread (ie, intake duration).

(1)

where P₁ (cal/h) represents the maximum rate of calorie consumption/h, and P₂ (h^–2) is a parameter related to the mean inverse of meal spread. Meal spread is defined as the length of time (in h) during which the rate of intake exceeds one-half the peak intake rate (Figure 1). The meal spread for breakfast (MS_B) is calculated by the following equation:

(2)

P₃ (h) is the time at which the maximum rate of caloric intake occurs (Figure 1). Similar equations with corresponding parameters can be used to describe the rate of caloric intake at lunch [EL(t)] and dinner [ED(t)], according to the following equations:

(3)

and

(4)

With the use of the results of equations 1, 3, and 4 as input rates, the average cumulative caloric intake CI(T) up to any time [T (h)] in a day can be estimated by deconvolution (10) as a sum of the integrals of the following equation:

(5)

where "dt" is the integration element.

Thus, from a practical point of view, evaluation of the individual integrals in equation 5 between 6 and 30 h provides estimates of the total caloric intakes for breakfast, lunch, and dinner. Alternatively, because EB(t) is a Gaussian curve, the total area under the curve (AUC) from t = – to t = (AUC_EB)—ie, the breakfast caloric intake—can, as was shown previously (11), be calculated according to the following equation:

(6)

In practice, in the analysis presented here, the meal AUCs in the time domains <6 h and after 30 h are negligible. Therefore, equation 6 provides a good approximation of the integrand of equation 1 from 6–30 h. Because equations 3 and 4 also are Gaussian equations, it follows that, by employing the appropriate parameters from equations 3 and 4, equations similar to equation 2 can be employed to calculate the meal spread of lunch (MS_L) and dinner (MS_D). Likewise, equations similar to equation 6 can also be used to calculate the lunch (AUC_EL) and dinner (AUC_ED) caloric intake.

Subjects
The data used to develop this model of food intake were derived from a study carried out at the University of Pennsylvania (12). That study presented graphs of the 24-h (0600 to 0559 the following day) average cumulative energy intakes (kJ) of a group of 10 persons with NES and a group of 10 control subjects. Subjects were classified as having NES if they initially reported that they consumed >50% of their daily intake between 2000 and 0600. The images of these graphs [see Figure 1 in Meier et al (10)] were electronically scanned, and the data were digitally extracted with the use of UN-SCAN-IT software (version 5.0; Silk Scientific Corporation, Orem, UT). The cumulative energy intake in kJ was converted to kcal, and equation 5 was then fitted to the data for NES and control subjects by means of nonlinear regression using WinSAAM software (version 3.0.7; Internet: http://www.winsaam.org) (13). Data in all regressions were weighted to the SD of the observations (14).

This modeling approach for analyzing cumulative energy intake data was then tested by using data previously collected from a second study of 148 night eaters (100 female) and 68 control subjects (51 female) (9, 15). Of the control subjects, 23 were normal-weight and 45 were overweight or obese. Of the night eaters, 19 were normal-weight and 129 were overweight or obese. All subjects completed 24-h food diaries for 7 d. For each stratum in this investigation, normal-weight and overweight or obese NES subjects and normal-weight and overweight or obese control subjects, average cumulative caloric intakes were calculated at hourly intervals from 0600 on day 1 to 0559 the following day and averaged across the 7 d and across subjects. Equation 5 was then fitted to these data. We used t tests to determine whether means for specific parameters were significantly different (P < 0.05) in the different strata (16).

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS AND SUBJECTS RESULTS DISCUSSION REFERENCES

 
It is shown in Figure 2 (panels C and D) how 3 Gaussian curves (the new conceptual model, here called the Boston model) can be used to describe the average rates of caloric intake during breakfast, lunch, and dinner for NES and control subjects from the first study. Panels A and B in Figure 2 show how the summed integrands of these Gaussian curves describe the average cumulative caloric intake of NES and control subjects from the first study. The same curves for the second study (9) are shown in Figure 3.

View larger version (12K): [in this window] [in a new window]   FIGURE 2. Use of the new (Boston) model to describe the average cumulative caloric intake of control subjects (A) and patients with night eating syndrome (NES) (B). •, data; —, the model predictions. The adjusted R2 for prediction equations shown in A and B were both >0.99, and the root mean square errors were 15 and 18 calories for A and B, respectively. C (control subjects) and D (NES subjects): the individual Gaussian curves that describe the average rate of eating during each of 3 separate meals. Data presented in A and B were extracted from study 1, and each data point represents a mean of 10 patients.

View larger version (12K): [in this window] [in a new window]   FIGURE 3. Use of the new (Boston) model to describe the average cumulative caloric intake of control subjects (A: means of 68 subjects) and subjects with night eating syndrome (NES) (B: means of 148 patients). •, data; —, the model predictions. The adjusted R2 for prediction equations shown in A and B were both >0.99, and the root mean square errors were 19 and 24 calories for A and B, respectively. C (control subjects) and D (NES subjects): the individual Gaussian curves that describe the average rate of eating during each of 3 separate meals. Data are from study 2.

Table 1

Table 2

In both the first and second studies, the mean cumulative 24-h caloric intakes were significantly higher in NES subjects than in control subjects, and the difference was specific by weight class in the second study. In the second study, the average total daily caloric intake of overweight or obese subjects was significantly (P < 0.05) greater than that of normal-weight subjects, whether they were NES or control subjects (Table 2). This excess consumption occurred primarily during the dinner meal. In NES subjects, the time of peak rate of eating during the dinner meal (P₂₃) was significantly (P < 0.05) later in normal-weight subjects (20.07 ± 0.10 h) than in overweight or obese subjects (18.21 ± 0.19 h). In normal-weight control subjects, on the other hand, P₂₃ occurred at 17.95 ± 0.08, which was slightly later than it occurred (17.83 ± 0.08 h) in the overweight or obese control subjects.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS AND SUBJECTS RESULTS DISCUSSION REFERENCES

 
The Boston model accounts for all features of the cumulative intake curves, as shown in Figures 2 and 3, and in no time domain is there any systematic departure of model predictions from the measured data. The very high R² values and relatively small RMSE values testify to the accuracy of the Boston model in describing intake patterns. This model is robust in that the same estimates for model parameters were obtained even when initial estimates for parameters were dissimilar to the final estimates.

Criteria for NES are currently defined as the consumption of 25% of daily caloric intake after the evening meal, 3 nighttime awakenings with ingestion/wk, or both (8). With respect to the data from the first study (16), the Gaussian curve for the evening meal of the NES subjects is significantly (P < 0.05) larger in every respect—ie, the maximum rate of eating, the time of the maximum rate of eating, and the SD of the duration of eating—than is the corresponding curve for that meal of the control subjects (Table 1). In fact, with respect to the evening meal, the NES subjects had a peak rate of eating (267 calories/h) at 21.23 h and the Gaussian curve describing their eating pattern extended from 1000 to 0600 the next day. This pattern is embodied in the first core criterion, consumption of 25% of daily caloric intake after the evening meal (Figures 2D and 3D).

A second feature of both the first and second studies was the relative magnitudes of the midday meal caloric AUC in NES and control subjects. In both studies, the lunch caloric AUC in the NES subjects was significantly (P < 0.05) less than that in the corresponding control subjects. This finding suggests that the morning anorexia of NES subjects extends into the lunch period.

There is one major difference between these 2 studies, for which we do not yet have an explanation. In the first study, the maximum rate of caloric intake at the evening meal (P₂₁) was significantly (P < 0.05) greater in the NES subjects than in control subjects, whereas, in the second study, P₂₁ was significantly (P < 0.05) greater in the control subjects than in the NES subjects.

The question arose as to whether the greater food intake of the overweight or obese subjects may be due to their greater weight. If this were the case, and if the data were first scaled on an individual patient basis by body weight, different outcomes may be obtained. We scaled the intake patterns in this way, but when the Boston model was fitted to the resulting mean data (results not shown), it did not significantly alter any of the outcomes.

The Gaussian curves depicting the rate of eating associated with each meal pulse provide a novel means of visualizing the distinct differences in meal patterns between the control and NES subjects. The major strengths of the Gaussian equation is that its parameters correspond to easily discernible and comprehensible features of a putative intake pulse (Figure 1), that initial estimates of these parameters are relatively easy to obtain, and that the parameters of Gaussian equations are readily estimable. We did consider other equation forms to represent the intake pulses. There are theoretical grounds for assuming that gamma functions may describe eating pulses more accurately than may Gaussian functions (11). However, our efforts with gamma functions were unsuccessful because their use was associated with unstable parameter estimation, especially in the case of the dinner pulse. Other candidate pulse functions that we considered but rejected included a sum of 2 exponentials, polynomials, and periodic functions such as sine and cosine curves. Cognizant of the need for parsimony with respect to the number of estimable parameters, we rejected a function composed of the sum of 2 exponentials, because that type of function would involve 4 parameters for each pulse. In contrast, the Gaussian equation requires the estimation of only 3 parameters. We rejected the possibility of using polynomials because these equations could predict negative (physiologically unfeasible) rates of eating, whereas the Gaussian function is always positive. Although periodic functions such as sine and cosine are suitable for modeling diurnal cyclic (repeating) patterns across days (6), we rejected using these types of equations, principally because they do not have the requisite shape to describe the nonrepeating pulses associated with rates of eating within a day. Moreover, the data presented here are means within a single day, not repeating data across days.

The parameters of the Boston model provide a means of objectively characterizing all of the features of eating patterns of different strata of a population. Furthermore, such quantification of the 3 main daily meals becomes an objective process unaffected by subjective and potentially confounding definitions of what constitute a particular meal (1, 2). For example, some ethnic groups may consume meals at times different from those of most white Americans.

A further strength of the Boston model is that it allows the Gaussian curves describing the 3 main meals to overlap temporally (Figures 2D and 3D). Thus, it is not necessary to define meals as occurring within a specified window of time, and there can be no inadvertent distortion of the estimation of meal caloric intakes for particular substrata of the population who may eat their meals at nonnormative times.

Because of these attributes, the Boston model could have applications in comparing eating patterns of different racial and ethnic groups who eat at different times, night shift workers and day shift workers, and groups who eat at different times during different seasons of the years, eg, summer and winter. The Boston model also circumvents the potentially confounding problem associated with snacking, which is common in night eaters (3). The Boston model should be particularly useful for analyzing measures of change in clinical trials, because it potentially shows more information than do the traditional measures of change (17). This model may help to assess the effects of bariatric surgical interventions and of the differences among them (18).

A limitation of the Boston model is that it may not be appropriate for the analysis of small numbers of persons, because mean cumulative intake may be unduly influenced by a few outliers. Despite this limitation, the Boston model provides a simple, objective, robust approach to quantifying circadian eating patterns in populations and in strata of populations.

Conclusions
Perhaps the most important contribution of this study is that it shows how parametric deconvolution can be used to derive explicit solutions for latent putative functions to describe the rates of input that influence the timing, magnitude, and duration of feed intake pulses that constitute specific meals. In this work, 3 separate Gaussian functions were found to be most suitable for describing the rate of intake during breakfast, lunch, and dinner. Integrating these functions over time, and combing the 3 integrands, results in a mathematical model that can describe the temporal pattern of cumulative caloric intake throughout the day. The parameters of each of the separate Gaussian equations can be used to quantify accurately, for human populations or population strata, mean rates of caloric intake during specific individual meals. In conclusion, in the study reported here, parametric deconvolution was successfully used to provide new insights into differences in eating patterns between populations without an eating disorder and those with NES.

	ACKNOWLEDGMENTS

 
The authors' responsibilities were as follows—KCA, JDL, and AJS: conducted the studies; RCB and PJM: analyzed data and wrote the manuscript; and all authors: contributed to critical revision of the manuscript. None of the authors had a personal or financial conflict of interest.

	REFERENCES

TOP ABSTRACT INTRODUCTION METHODS AND SUBJECTS RESULTS DISCUSSION REFERENCES

 

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