Simplified Resting Metabolic Rate—Predicting Formulas for Normal-Sized and Obese Individuals

Authors

Edward H. Livingston,

Corresponding author

VA North Texas Health Care System and the Division of Gastrointestinal and Endocrine Surgery, University of Texas Southwestern School of Medicine, Dallas, Texas

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Gastrointestinal and Endocrine Surgery, UT Southwestern Medical Center, 5323 Harry Hines Blvd., Room E7-126, Dallas, TX, 75390-9156. E-mail: edward.livingston@utsouthwestern.edu

Abstract

Objective: Resting metabolic rate (RMR) is known to be proportional to body weight and to follow allometric scaling principles. We hypothesized that RMR can be predicted from an allometric formula with weight alone as an independent variable.

Research Methods and Procedures: An allometric, power-law scaling model was fit to RMR measurements obtained from a cohort of patients being treated for weight loss. This, as well as many of the commonly used RMR-predicting formulas, was tested for RMR prediction ability against a large publicly available RMR database. Bland-Altman analysis was used to determine the efficacy of the various RMR-predicting formulas in obese and non-obese subjects.

Results: Power law modeling of the RMR—body weight relationship yielded the following RMR-predicting equations: RMR_{Women} = 248 × Weight^{0.4356} − (5.09 × Age) and RMR_{Men} = 293 × Weight^{0.4330} − (5.92 × Age). Partial correlation analysis revealed that age significantly contributed to RMR variance and was necessary to include in RMR prediction formulas. The James, allometric, and Harris-Benedict formulas all yielded reasonable RMR predictions for normal sized and obese subjects.

Discussion: A simple power formula relating RMR to body weight can be a reasonable RMR estimator for normal-sized and obese individuals but still requires an age term and separate formulas for men and women for the best possible RMR estimates. The apparent performance of RMR-predicting formulas is highly dependent on the methodology employed to compare the various formulas.

Resting metabolic rate (RMR)^{1} is frequently estimated in clinical medicine to determine minimal energy requirements for patients receiving nutritional therapy. RMR estimates are also used to counsel patients regarding caloric intake in weight loss programs. RMR measurement by indirect calorimetry requires quantification of oxygen consumption and carbon dioxide production while a patient is resting. Metabolic activity above resting levels must be avoided to ensure that the measured metabolic rate as closely approximates the basal state as possible. These measurements are cumbersome to perform and not practical for routine clinical use. For this reason, in clinical practice, RMR is estimated from mathematical formulas.

The RMR estimating formula most commonly used was derived by Harris and Benedict in 1919 (1). In deriving their formula, Harris and Benedict empirically fit a number of anthropomorphic measurements into a regression equation relating them to RMR (2). RMR scales to body size allometrically such that it increases to the 3/4 power of an animal's weight (3). Although numerous modifications of the original Harris-Benedict equation have been proposed, none have used this well-established relationship. Theoret-ically, an RMR-predicting formula that follows a mathematical construct explaining the physiological basis for metabolism—size interaction will result in more accurate predictions. Because RMR is proportional to size, it is important to include obese patients in any cohort used for development of RMR-predicting equations, and obese individuals were included in only two prior studies of RMR prediction (4,5). We hypothesized that an RMR-predicting formula based on power law mathematics would provide superior estimates of RMR for a wide range of body weights. For this reason, we used power law equations to develop an RMR-predicting formula from a database of the RMR measurements we obtained that included obese individuals that was combined with the original Harris-Benedict and Owen datasets. We used the Institute of Medicine (IOM) database of metabolism measurements to test the performance of the newly derived RMR-predicting formula.

Because we hypothesized that RMR could be predicted from body weight alone, we performed a partial correlation analysis of the contribution of body weight, height, and patient age to RMR variance to determine their relative importance in RMR predicting formulas.

Research Methods and Procedures

Data Used for Allometric Modeling

Nonlinear regression provides the best estimates of an equation's coefficients when there are many data points with a very large numerical spread. For this reason, we obtained as many RMR measurements as possible to use in the allometric modeling study. We acquired the original Harris-Benedict and Owen databases. We also performed our own RMR measurements in a group of patients that included many obese individuals.

Harris-Benedict RMR Data

Harris and Benedict published extensive tables containing detailed information regarding body size and RMR (1). Data for 239 individuals were digitized, and information regarding age, weight, and RMR was entered into a Microsoft Excel database (Microsoft, Redmond, WA).

Owen RMR Data

Owen et al. published two papers that contained measured RMR data that were included in our analysis. RMR was measured by indirect calorimetry using a Beckman Metabolic Cart, and the results for 44 women (6) and 60 men (7) were digitized into our databases.

RMR Measurement

RMR was measured as part of a comprehensive medical evaluation for 327 patients presenting to a university-based weight management program involving a physician-supervised very low calorie diet. Consecutive patients who met the following criteria were included: 1) complete medical evaluation showing that patients were free of significant disease that potentially could alter RMR measurements; 2) metabolic panel showing that the patients did not have thyroid conditions; and 3) chest X-ray and medical exam verifying that respiratory illness was not present. Patients taking medications were not excluded, although if a patient was on a medication known to alter RMR, the test was not performed. For eligible patients, RMR was measured before entry into any dietary program. Patients were instructed to fast overnight before their arrival in the clinic the following day.

RMR was measured using a metabolic cart (SensorMedics Corp., Yorba Linda, CA) by one of two certified exercise physiologists in a standardized room. RMR was measured in the early morning after an overnight fast, and the patients were instructed to avoid exercise during the fasting period. Patients were instructed on pretest preparations and arrived at the clinic in the morning of the test day. They initially rested in a reclining chair, in a quiet room, for 20 minutes before the examination. They were placed in a reclining chair such that they were lying down. Measurements were repeated, and interest variability was limited to a predetermined range. The test was conducted twice, with the higher of the two RMRs selected for analysis. Patients were asked whether they had the subjective observation that they were not at rest and, if so, the results were rejected. Data used for this study were disassociated from the patient's identifiers and clinic chart to ensure anonymity.

Data Used for Testing RMR-predicting Equations and the Relative Contribution of Weight, Height, and Age to RMR Variance

IOM Macronutrient Publication-RMR Measurements

The IOM published a series of metabolic rate measurements compiled from various laboratories. Results reported as the basal energy expenditure observed as part of doubly labeled water total energy expenditure studies (8) were assumed equivalent to RMR for our purposes. Information was downloaded from the National Academy of Science's website (http:www.nap.eduopenbook0309085373). The tables were digitized into Microsoft Excel (Microsoft Corp.) file format using Omni-Pro (ScanSoft, Peabody, MA). These were used in the Bland-Altman analysis to test the accuracy of the RMR-predicting formulas we derived.

Nonlinear Regression

The weight—RMR relationship was assessed by nonlinear regression with the allometric power formula:

where a is a coefficient with units of kilocalories per 24 hours per kilogram and b is a non-dimensional power coefficient. Nonlinear regression was performed with the SigmaPlot Graphing package (SPSS, Chicago, IL). For assessment of the effect age had on RMR prediction, a second equation was solved:

where the age was in years and c is another non-dimensional coefficient. This regression was performed on the database created by combining the RMR measurements we obtained with those previously published by Harris and Benedict (1) and Owen et al. (6,7).

Multiple Regression

RMR prediction with the Harris-Benedict type of formula follows the general form of RMR = constant + A_{1}Weight + A_{2}Height + A_{3}Age.

Where constant, A_{1}, A_{2}, and A_{3} were solved by multiple regression using the SAS program (SAS Institute, Cary, NC). Significance of the fitted coefficients was assumed when p < 0.05. The proportion of RMR variance accounted for by each coefficient was determined by semipartial correlation analysis using type II sum of squares (9). The IOM database (8) was used not only for validation of the various RMR-predicting formulas effectiveness but also for determining the contribution of individual variables variance to the explained RMR using this multiple regression equation.

Calculations

RMR was estimated using several of the commonly referenced RMR-predicting formulas (1,4,6,10,11,12,13,14). Many of the formulas are presented elsewhere (15).

Predictive formula performance relative to measured RMR values was assessed by the Bland-Altman method (16,17). Additionally, we calculated how far RMR estimates deviated from the actual RMR. The percentage deviation was calculated as (RMR_{estimated} − RMR_{actual})/RMR_{actual.} The proportion of those falling within 10% of the actual RMR was reported.

Each of the tests used to assess RMR-predicting formula effectiveness measures a different aspect of their perfor-mance. The formula providing the greatest number of RMR estimates close to the measured value is identified by counting the number of estimates falling within 10% of measured RMR. This does not assess systematic deviations that may occur as a function of RMR, which are best evaluated by Bland-Altman slope determination. Bland-Altman means provide a measure of an RMR-predicting formula's systematic deviation above or below measured RMR but provides little information regarding the number of estimates that are close to the measured values or RMR-related systematic deviations. We simultaneously assessed these features by developing a scoring system. The proportion of RMR estimations falling within 10% of the actual value was subtracted by 10 times the absolute value of the Bland-Altman slope and also 1/10 of the Bland-Altman mean. In this way, we downgraded equations that appeared good based on providing large numbers of RMR estimates falling within 10% of the measured RMR but displayed RMR-related systematic deviation of RMR estimation or systematic over- or underestimation of RMR.

Statistical significance of the null hypothesis was determined by t testing with α < 0.05, using the Statistical Analysis Software package V.8.01 (SAS).

Results

Characteristics of the populations used for these studies are presented in Table 1. The IOM database had, on average, older and smaller individuals than the combined database consisting of the original Harris-Benedict and Owen data sets and measurements we obtained. However, both databases had wide ranges of age and weight, facilitating parameter estimation for nonlinear regression. The wide range also enabled the simultaneous assessment of the RMR-predicting equations performance for small and large patients.

Table 1. . Characteristics of the populations used for this study

IOM women

IOM men

Measured women

Measured men

IOM refers to the Institute of Medicine population of individuals with RMRs observed as reported in the doubly labeled water total energy expenditure tables. This data set was used to derive the power formula for RMR prediction. Columns denoted as measured refer to measurements we obtained in our laboratories that were combined with the original Harris-Benedict and Owen databases.

n

433

334

356

299

Age range (years)

20 to 96

20 to 96

18 to 77

18 to 95

Age mean± SD (years)

48 ± 20

51 ± 20

39 ± 13

36 ± 15

Weight range (kg)

43 to 164

54 to 216

36 to 261

33 to 278

Weight mean± SD (kg)

69 ± 16

82 ± 18

90 ± 37

96 ± 45

RMR range (kcal/24 h)

910 to 2576

1133 to 3035

870 to 2740

997 to 3762

RMR mean± SD (kcal/24 h)

1366 ± 212

1729 ± 273

1535 ± 334

1855 ± 416

RMR-predicting equations were derived from basal metabolic rates we measured combined with the Harris-Benedict and Owen datasets (8). Nonlinear regression of the power equation yielded:

where units of RMR were kilocalories per 24 hours, Weight is the body weight in kilograms, and R^{2} is the nonlinear regression goodness of fit statistic. Like with the correlation coefficient for linear regression, this statistic provides an estimate of the proportion of observed variance that is explained by the regression equation. We also calculated a regression equation with the addition of an age parameter:

Because the power exponents were very similar for men and women, we calculated RMR predicting equations that were not stratified for sex:

We performed multiple regression analyses on the IOM data set to determine the relative contribution of weight, height, and age on the explained RMR variance. These results are presented in Table 2. For women, weight explained 52% of the RMR variance, height explained 19%, and age explained 12% when these were assessed individually by simple linear regression. When all three were simultaneously entered into a multiple regression equation, the amount of variance explained by weight and age remained approximately the same. The amount of explained variance attributable to height fell from 19% to 3%. Correlation analysis between weight and height yielded a correlation coefficient of 0.32. For weight and age, it was 0.07. Thus, there was a strong correlation between weight and height. These results show that there was significant overlap between weight and height in terms of their contribution to the explained RMR variance. Their covariance resulted in the height's contribution to the explained variance being almost completely eliminated when it was assessed in conjunction with weight in multiple regression analysis. Weight and age were more weakly correlated such that when these were simultaneously assessed in multiple regression, they both retained their contribution to variance. Results for the analysis on men were very similar to those for women.

Table 2. . Regression analysis of the IOM database assessing the relative contribution of weight, height, and age to the RMR variance

Weight

Height

Age

Weight/ height/age

The row All represents the r^{2} value for the regression equation. For the weight, height, and age columns, simple linear regression was used. For the Weight/height/age column, multiple regression with weight, height, and age as independent variables was performed. The rows denoted Ptl represent the partial correlation coefficients for weight, height, and age.

Women

All

0.52

0.19

0.12

0.62

Ptl Weight

0.51

Ptl Height

0.03

Ptl Age

0.13

Men

All

0.53

0.23

0.13

0.64

Ptl Weight

0.51

Ptl Height

0.02

Ptl Age

0.15

Table 3 summarizes the effectiveness of various RMR-predicting equations as tested against the IOM database. The left columns summarize the proportion of predicted RMRs falling within 10% of the measured RMR sorted in descending order. The middle columns present the slopes of the Bland-Altman plots [RMR_{predicted} − RMR_{observed} on the y-axis and (RMR_{predicted} + RMR_{observed})/2 on the x-axis] in descending order. The right columns report the means of the subtracted values (RMR_{predicted} − RMR_{observed}) in descending order. No single measure presented in Table 3 can be exclusively applied to determine the best RMR-predicting equation. For example, in men, the James equation performed best, with 82% of predicted RMR falling within 10% of the actual measured RMR. A slope of zero with Bland-Altman analysis suggests no systematic deviation is estimates as a function of RMR, but the overall mean of −65 suggests that there is an overall tendency to underestimate RMR. The Harris-Benedict equation estimated only 69% of RMRs within 10% of the actual value. Although it had a Bland-Altman mean of zero, the slope was −0.1317, suggesting that the equation overestimates low RMRs and underestimates high RMRs.

Table 3. . Effectiveness of RMR-predicting equations

10%

Slope

Mean

The equations are presented in descending order based on their performance by various measures. The first column represents the fraction of predicted RMRs falling within 10% of the measured RMR. The second column is the slope of the Bland-Altman plot, and the third column is the mean of the differences between the predicted and measured RMRs.

WHO, World Health Organization; NLR, non-linear regression.

Men

James

82

Cunningham

0.6199

Bernstein

350

Mifflin

75

Owen

0.4560

WHO

339

NLR-age

73

NLR-age

0.2708

NLR-age

68

Cunningham

70

Mifflin

0.1694

Mifflin

55

Harris-Benedict

69

Bernstein

0.0000

Owen

19

Owen

66

James

0.0000

Harris-Benedict

0

WHO

52

WHO

0.0000

Cunningham

−10

Schofield

45

Harris-Benedict

−0.1317

James

−56

Bernstein

13

Schofield

−1.6300

Schofield

−635

Women

WHO

77

WHO

0.1760

Bernstein

221

James

76

Schofield

0.1735

Owen

78

NLR-age

69

James

0.1640

NLR-age

54

Harris-Benedict

68

NLR-age

0.1401

Mifflin

53

Schofield

67

Harris-Benedict

0.1032

Harris-Benedict

−24

Mifflin

66

Owen

0.0000

Schofield

−27

Cunningham

62

Bernstein

0.0000

James

−36

Owen

58

Cunningham

0.0000

WHO

−47

Bernstein

25

Mifflin

−0.0351

Cunningham

−53

When tested, we found that the inclusion of an age parameter to the nonlinear power equation substantially increased its predictive abilities. Thus, we did not use the power equation with only the weight term in our analysis comparing the various RMR predicting equations. Similarly, the use of a single equation for men and women also substantially degraded the RMR predicting equations performance resulting in our rejection of this equation as a viable RMR predictor.

When the scoring system was applied to the RMR predicting equations, the James equation was the most effective for both men and women (Table 4). The next most reliable equation that performed equally well in men and women was the Harris-Benedict equation followed by the power equation we derived (Equations 3 and 4).

Table 4. . Effectiveness score for RMR-predicting equations

Men

Women

The score was derived by subtracting 10 times the Bland-Altman slope and 1/10 the mean from the proportion of RMR predictions that were within 10% of the actual RMR.

WHO, World Health Organization; NLR, non-linear regression.

James

76.4

James

70.8

Mifflin

67.8

WHO

70.5

Harris-Benedict

67.7

Harris-Benedict

64.6

NLR-age

63.5

Schofield

62.6

Cunningham

62.8

NLR-age

62.2

Owen

59.5

Mifflin

60.7

WHO

18.1

Cunningham

50.6

Bernstein

−22.0

Owen

43.6

Schofield

−34.8

Bernstein

−1.8

Discussion

We assessed the efficacy of an equation based on physiological scaling principals relating RMR to body size in normal-sized and obese individuals. We found that simply relating RMR to body weight with a power exponent equation resulted in reasonable RMR estimates. For men, the equation accounted for 73% of the RMR variance, and for women, it was 67%. Regression analysis showed that age significantly influences RMR. When we added a linear age term to our power exponent equation, the RMR variance accounted for by the equation increased to 77% for men and 71% for women, resulting in much improved RMR estimates. Although the coefficients and exponents for these equations were similar for men and women, when we attempted to derive a single equation for men and women, the RMR prediction ability was diminished. Nonetheless, the new equation we derived based on the physiological basis for body size influence on metabolic rate was not necessarily better than empirically derived formulas previously applied to RMR prediction. Additionally, body weight alone could not reliably predict RMR. Age and sex had to be added to the RMR-predicting formula to improve its reliability.

We also found that the apparent RMR predicting power for an equation is reliant on the methodology used to assess its performance. Equations that seemed to be effective whenthe proportion of RMR predictions falling within 10% of the actual value proved less so when tested with the Bland-Altman method. There are many characteristics to consider when selecting an RMR-predicting equation, including its ability to generate RMR values close to the actual ones, its tendency to systematically under- or overestimate RMR, and systematic errors in RMR prediction occurring as a function of the patient's size or RMR. For example, in men, the James equation had the highest proportion, 82%, of RMR estimates, falling within 10% of the actual value. The Bland-Altman slope was zero, showing that the RMR estimate was independent of the RMR values itself. The Bland-Altman mean was −56, such that there was an overall tendency to underestimate RMR at all RMR values. In contrast, the Owen equation resulted in 66% of estimates, falling to within 10% of the actual value. Its Bland-Altman slope was 0.4560, suggesting that it tended to overestimate RMRs for those patient's having high RMRs. The Bland-Altman mean value of 19 suggests that, on average, the equation overestimates RMR for those with high values and underestimates those with low values. This can be shown by graphical analysis of the residual difference between predicted and actual RMR plotted against the mean of these two values.

We developed a scoring system to simultaneously account for these RMR predicting equation characteristics. Ideally, an equation will result in a large proportion of predicted RMRs falling within 10% of the actual values. Those that are greatly influenced by the RMR value itself have Bland-Altman slopes that significantly deviate from zero. The greater the deviation, the greater the dependence on the RMR value. For this reason, we subtracted 10 times the absolute value of the Bland-Altman slope from the proportion calculation. We multiplied by 10 to approximate the scale of these slope values to that of the proportion data. We subtracted 1/10 the absolute value of the Bland-Altman mean values from the result obtained above. Similar to the slope data, we needed to account for scale differences and also wanted to reduce the final score if the equation caused a significant, systematic deviation in its RMR estimates.

Using this scoring system, we found that the James equation provided the most consistent RMR estimates. The Harris-Benedict equation was the next most effective, followed by our power equation. The World Health Organization equation performed well in women but poorly in men, as did the Schofield equation. The Bernstein equation did not perform well for either sex.

Most prior efforts to develop an RMR-predicting formula have used some form of the Harris-Benedict formula relating RMR to weight. Harris and Benedict derived their formula empirically, assuming that RMR would be proportional to height, weight, sex, and age (1). Each of the subsequently derived RMR-predicting formulas have used the same basic form of the Harris-Benedict formula, including all of the parameters (4,10,13,18), weight and age (12,14), or weight alone (6,7). Numerous biological attributes scale to some power of an animal's weight (19). The relationships are described by power equations in the form of Y = aWeight^{b}, where Y is a biological property (e.g., metabolic rate, bone diameter, heart rate), a is a constant, and b is the scaling exponent. Extensive studies regarding the allometric relationship between metabolism and body weight have concluded that the power exponent is 3/4 when comparing metabolic rates of different species of animals of various sizes (3). The underlying explanation relates to nutrient delivery to tissues (20,21). Although it is well established that, with increasing size, metabolic rate increases allometrically, prior efforts at developing RMR-predicting formulas have not incorporated an allometric power equation relating RMR to body weight. We found that when a power formula was used, RMR could be predicted from weight and sex alone as well as it could from weight, height, age, and sex in a Harris-Benedict type of formula.

Most of the RMR-predicting formulas that have been proposed are linear formulas with body weight terms. We and others (22) have found that, of the possible variables that contribute to RMR prediction, weight explains the largest amount of the observed variance. When applied to obese cohorts, the Harris-Benedict formula has been shown to overestimate RMR (23,24). Several RMR predicting formulas have been developed using linear formulas with a body surface area (BSA) term (25,26,27,28). Studies have evaluated the performance of these formulas in predicting RMR. Of 12 assessed, the Robertson and Reid and Fleisch formulas correlated best to the measured RMR in obese cohorts (15). It was hypothesized that this occurred because the BSA-predicting formulas used in these studies contained the nonlinear term Weight^{425}. Robertson and Reid and Fleisch used this term because it was thought that RMR was proportional to BSA. One of the original BSA-predicting formulas was derived by DuBois, and it contained a Weight^{425} term (29,30,31). In retrospect, the weight term exponent was very similar to that we derived in the current study by power modeling of the RMR—body weight relationship. These prior studies combined with our current findings support the concept that RMR-predicting formulas should have a nonlinear term raising weight ∼0.5 power.

In summary we found that a power equation RMR-predicting formula based on body weight and age predicted RMR approximately as well as more complex models. Height contributes little to the explained RMR variance such that an equation relying exclusively on a nonlinear weight term combined with a linear age term performs as well as one that includes height and age in addition to weight. We also found that the assessment of an RMR-predicting formula is highly dependent on the methodology employed for comparing predicted to measured RMR values.

Acknowledgement

This work was supported, in part, by the Hudsen-Penn Endowment Fund.

Footnotes

^{1}

Nonstandard abbreviations: RMR, resting metabolic rate; IOM, Institute of Medicine; BSA, body surface area.