Age and ethnic dependence of BMI versus fat fraction relationship
All the body fat fraction measurements used here are based on 3C model measurements of body density and total body water. This decision was made not only because the 3C model has the largest database but because DEXA measurements of bone or fat density are problematical for morbidly obese subjects [
11‐
13]. The 3C method assumes a constant value of D
res
, the density of the non-fat and non-water body compartments (see eq.(
4)). One can estimate the error introduced by using the 3C model from the analysis of Silva et. al. [
10]. They found, for example, a value of D
res
of 1.555 ± 0.024 for men. Using extremes of D
res
of + and - one standard deviation (1.579, 1.531) yields a fat fraction range of 0.168 to 0.176 ((eq.(
4)). This error is small and probably within the range of the experimental errors in body density and total body water measurement. A major assumption of this analysis is that this constant value of D
res is valid for the morbidly obese subjects. There is no direct support for this assumption. Indirect support for this is provided by the observation that the plots of fat fraction versus either BMI using Model II (fig.
2) or density (fig.
5) do not show any obvious deviations at high fat fractions.
There is clear age dependence in the fat fraction versus BMI relation. For example, for males, the oldest age range subjects have about 60% more fat then the youngest for the same BMI (Table
3). For females the age dependence is less marked and more dependent on the BMI range. There is a roughly linear increase in fat fraction with age for women with a BMI < 26. However, for BMI in the range of 26 to 56 (average BMI of about 32), there was no significant age dependence (Table
4).
There was no significant difference in the relationship between fat fraction and BMI for Caucasians, Blacks and Hispanics for either males (Table
5) or females (Table
6) and these three ethnic groups were combined. This differs from previous studies that have found significant differences between Blacks and Caucasians [
5,
17‐
19]. In contrast, both Asians and Puerto Ricans have significantly greater fat fraction then Caucasians for the same BMI and age (Tables
5 and
6). This is a consistent observation for Asians [
5,
19]. The Puerto Rican result has not been previously reported.
Regression relations for predicting fat fraction from BMI
This analysis describes a new set of regression relations based on an analysis of 1356 subjects, the largest database currently reported. A comparison of the simple linear and the non-linear physiologically based regression models are shown in figs.
2,
3 and
4. Table
8 compares the mean square residual error (MSR) of the different models. The non-linear Models I and II deviate significantly from the linear fit only for values of BMI greater than about 50. Since all the men have BMI < 51 (figs.
2 and
3), there is no clear statistical advantage of Model I or II over the linear fit for men. The linear prediction in fig.
2 overestimates the fat fraction of the two men with the largest BMI (47.9, 50.3), suggesting that the non-linear regression may be superior for morbidly obese men. However, a larger data set will be required to prove this. For the women with BMI > 50, the predictions of Model I or II are clearly superior to that of the linear fit (figs.
2 and
4) and the MSR for Model II is about 25% smaller than the MSR for the linear fit to the same data (Table
8).
Table
8 compares the linear and Model I and II fits for subjects grouped in specific age ranges. For practical prediction of fat from BMI, it is preferable to use a general multi-regression relation that directly includes the age. Table
7 lists the parameters and MSR for the age dependent forms of the linear (eq.(
16)) and Model I (eq.(
9)) regressions for the different ethnic groups. The addition of this linear age dependence to Model I adds another adjustable parameter that weakens the physiological interpretation of the other parameters. The parameters listed in Table
7 for the age dependent Model I should be regarded simply as a set of empirical fitting parameters. The best estimate of the actual values of the physiological parameters is provided by the data listed in Table
8 which does not include the extra age dependent parameter in the data fitting.
The age dependent linear and non-linear Model I multi-regression relations both have 3 adjustable parameters. For women, the age dependent Model I regression has an MSR 15% less than the linear model (Table
7) and this non-linear model (eq.(
9)) represents the best choice for women, especially if highly obese subjects are included (see figs.
2 and
4). For men, there is no statistical difference in the accuracy of the linear and non-linear relations. However, since the non-linear relation has the correct limiting value at high fat fractions, it probably represents the best option for a general regression relation for men. In summary, this analysis suggests that the following age and BMI regression equation be used to predict body fat fraction in Caucasians, Hispanics and Blacks (see Table
7):
The corresponding regression parameters for Asians and Puerto Ricans are listed in Table
7.
The limitations of using the BMI to predict body fat fraction are well known [
4] and there can be large individual variations, depending primarily on body muscle mass. The mean square residual error (MSR) values of the current analysis provide one measure of the accuracy of the fat prediction (the "average" error is equal to the square root of the MSR). For Caucasian + Hispanic + Black ethnic males the MSR for the linear age dependent prediction is 0.00285, corresponding to an average error of about 0.053 in the value of the fat fraction predicted based just on the subject's age and BMI. For females the age dependent Model I fit is better, with an MSR of 0.00235, corresponding to an average error of about 0.048 in the value of fat fraction predicted from BMI and age for the entire BMI range (17 to 65).
Regression relations for predicting fat fraction from body density
Although the measurement of fat fraction using just the body density (2C model) is less accurate than the 3C method, it has the advantage of simplicity. It is particularly useful for following the time course of weight change (e.g. after bariatric surgery [
6]) where the frequent measurements complicate body water measurements. Table
9 lists the optimal least square regression parameters (a and b, eq.(
12)) for predicting fat from body density for Caucasian + Black + Hispanic males and females in different age classes. These predictions are described by the black lines in figs.
5,
6,
7.
Three different sets of regression coefficients (a and b) due to Siri [
14] and Brozek et. al. [
15] have been previously used to predict fat fraction from body density (eqs. (
13) – (15)). Figures
5 ,
6,
7 and Table
9 compare these predictions with the optimal predictions using the least square values of a and b. The MSR using these older parameters sets are from 40% to 300% larger then the MSR using the optimal values of a and b.
The Siri and Brozek relations were based on small data sets of relatively normal weight subjects. The new least square values of a and b listed in Table
9 for male and female subjects in different ages ranges should provide much more accurate relations for predicting fat fraction just from experimental measurements of body density. For men, there is a consistent age dependence in these parameters, with both a and b decreasing linearly with age. For females, the age dependence of the a and b parameters is small. The average MSR for males and females of about 0.0006 corresponds to an average error about 0.024 in the value of the fat fraction predicted using just the body density.
Physiological models for fat fraction as a function of BMI or body density
The same model of body composition was used to derive the relationship between fat fraction versus BMI and fat fraction versus body density. The basic assumption is that, as a subject with a given height gains weight, the extra weight has a constant, fixed composition with a fat fraction f
1 and a density d
1. This is clearly an approximation because one would expect that in severe obesity, as the subject becomes increasingly sedentary, there should be accompanying changes in both bone and muscle mass along with other pathological changes [
20]. However, the good fit of Model II for a BMI range of 17 to 65 (fig.
4) suggests that this is a roughly valid supposition.
The two models used to derive the fat fraction versus BMI differ in the body composition that is assumed for the lean "reference" subject to which the extra weight is added: for Model I, the reference subject is fat free, while, for Model II, the reference subject has a fat fraction f
0. For young males (fig.
3), Model II is clearly superior to Model I which underestimates the fat fraction at large BMI. The difference between the two models is smaller for older males (fig.
3) and females (fig.
4).
From the Model II fit to the fat fraction versus BMI data, the values of the fat fraction of the lean "reference" subject (f
0) and of the extra weight (f
1) can be determined. These values are listed in Table
9 for the different male and female age groups. For all males combined, the value of f
1 is 0.644. This is nearly identical to the value of 0.64 determined by Brozek et. al. [
15] from experimental weight gain or loss data in male subjects. For females the value of f
1 is larger, about 0.73.
The model for fat fraction versus body density relates the two regression parameters (a and b) to four physiological parameters (f
0, f
1, d
0, and d
1, eq.(
12)). Using the values of f
0 and f
1 determined from the fat fraction versus BMI regression (Table
8), these two equations for a and b can be solved for d
0 and d
1. Table
9 lists the assumed values of f
0 and f
1 and the corresponding values of d
0 and d
1 for the different male and female age dependent density regression relations.
With some additional assumptions, it is possible to use these 4 parameters (f
0, f
1, d
0, and d
1) to obtain a more detailed description of the tissue body composition. In Table
10, the body composition (as the fraction of total weight) is represented by muscle, bone and adipose with all the remaining tissues lumped into "other". The composition of both the lean "reference" subject and of the "extra" weight associated with the added fat are listed. It is assumed that the "other" tissue is essential functional tissue (gastrointestinal, nervous, heart, liver, etc.) that has a constant mass and, therefore, is not part of the additional "extra" weight that is added in obese subjects. It is assumed that "other" has the same weight fraction and density in the male and female "reference" subject. The experimental values of f
0, f
1, d
0, and d
1 place strong constraints on the body composition. For example, if each tissue's fat fraction and density are known then the values of f
1 and d
1 uniquely determine the adipose, muscle and bone composition of the "extra" tissue. Details about how the tissue weights were determined are described in the legend to Table
10.
Table 10
Body composition of standard male and female.
| | Wt. Fr. | Fat Fr. | Density | Wt. Fr. | Fat. Fr. | Density |
Male | Bone | 0.151 | 0 | 1.4 | 0.0427 | 0 | 1.4 |
| Muscle | 0.49 | 0 | 1.04 | 0.200 | 0 | 1.04 |
| Adipose | 0.136 | 0.8 | 0.92 | 0.758 | 0.85 | 0.915 |
| Other | 0.223 | 0 | 1.059 | 0 | 0 | 0 |
| Total | 1 | 0.109 | 1.0667 | 1 | 0.644 | 0.952 |
Female | Bone | 0.14 | 0 | 1.4 | 0.0367 | 0 | 1.4 |
| Muscle | 0.395 | 0 | 1.04 | 0.0974 | 0 | 1.04 |
| Adipose | 0.241 | 0.8 | 0.92 | 0.865 | 0.85 | 0.915 |
| Other | 0.223 | 0 | 1.059 | 0 | 0 | 0 |
| Total | 1 | 0.193 | 1.049 | 1 | 0.736 | 0.938 |
The most uncertain value used to derive the composition in Table
10 is the value of the fat fraction of adipose tissue. A number of studies have suggested that the fat fraction of adipose tissue increases in obese subjects, possibly as a result of an increase in the individual adipose cell volume [
21‐
23]. In Table
10 it has been assumed that adipose tissue is 80% fat in the lean references subject and 85% fat in the extra weight in obese subjects.
Table
10 provides a quantitative description of the composition of the extra weight that is gained (or lost) in obese subjects. This consists of the extra adipose tissue, plus the additional muscle and bone that is required to support this adipose tissue. In males, about 20% of this "extra" weight is muscle, while in females muscle is only about 10% of this extra weight. In both males and female, bone represents about 4% of the extra weight. This model, which uses a fixed composition of the "extra" tissue, provides a good description for the entire range of BMI (16 to 65) for females (see figs.
2 and
4). This suggests that the "extra" weight in morbidly obese subjects (BMI > 50) does not differ qualitatively from that of moderately obese subjects.
Table
10 lists the composition in terms of the weight fraction. This can be converted to absolute weight as follows. For a person with height H and weight W, the "reference" body weight = W
0 = BMI
0 × H
2(eq.(
6)) and the "extra" weight = W
1 = W - W
0, where BMI
0 for Model II is listed in Table
8. Multiplying the "reference" weight fraction by W
0 and the "extra" weight fraction by W
1 gives the absolute tissue weights.
By making additional assumptions about the extracellular (ECW) and intracellular (ICW) water fraction of the different tissues ([
24]), one can predict the changes in the corresponding water compartments in obesity. It is well recognized that the ratio ECW/ICW increases in obesity [
7,
25,
26]. This is because the "extra" weight in obese subjects is primarily adipose tissue (Table
10) whose water is almost all extracellular. In a comparison of the ECW and ICW compartments in obese versus matched non-obese controls, Waki et. al [
26] reported that this ratio increased from 0.63 in non-obese to 0.81 in obese females. This is similar to the predictions using the model data in Table
10 (assuming that the extracellular water weight fraction is 0.091 for muscle and 0.15 for the "extra" adipose tissue).
In addition to the model assumptions about body composition, another assumption in the derivation of eq. (
6) is that the body weight of the "reference" subjects scales as height
N, where N = 2. This is the basic assumption underlying the use of BMI as a parameter for obesity. A large, extensive review of the height versus weight relationship [
27] found an average value of N of 1.92 for males and 1.45 for females. However, this review analyzed the relation between height and total body weight while eq. (
6) assumes only that the "reference" weight scales as height
N. A more relevant test is to determine the height-weight relationship for the lean subjects in this current study, assuming that these lean subjects correspond to the "reference" subjects. Figure
8 shows a log-log plot of height versus weight for lean males (fat fraction < 0.15) and females (fat fraction < 0.24). For both males and females, the average value of N is close to 2 (1.96 for males and 1.95 for females).