# Allometric scaling of left ventricular mass by body dimensions in males and females

Physiological variables must often be scaled for body size differences to permit meaningful comparisons between subjects or groups. This study aimed to determine the proper relationship between body dimensions and left ventricular mass (LVM) *via* allometric scaling (AS) in 142 subjects (78 males, 64 females; ages 18-40). A cubic formula was used to estimate LVM from wall thicknesses and left ventricular internal dimensions derived from M-mode echocardiography. Fat free mass (FFM) was predicted from anthropometry.“Best compromise” allometric equations (y = a·x^{b}) revealed a common body mass (BM) exponent of 0.78 (95% CI, 0.65-0.91). The widely adopted ratio scaling (RS) method assumes that the exponent *b*= 1. In this sample, use of RS would penalize heavier subjects by overcorrecting for BM. The equivalent mean FFM exponent of 1.07 was not different from unity (95% CI, 0.92-1.22). Hence, RS using BM would appear to penalize those subjects who are heavier owing to excess fat not excess FFM. Gender differences in LVM were 70, 44, and 18%, for absolute values per BM^{0.78} and per FFM^{1.07}, respectively (*P* < 0.05). This reveals quantitative differences in heart size independent of body dimensions. We conclude that sample specific AS permits meaningful intersubject or intergroup comparisons.

*Department of Exercise and Sport Science, the Manchester Metropolitan University, Crewe and Alsager Faculty, Alsager, ST7 2HL, ENGLAND, and School of Social Sciences, University of Teesside, Middlesbrough, TS1 3BA, ENGLAND*

Submitted for publication June 1995.

Accepted for publication June 1996.

Address for correspondence: Alan M. Batterham, Department of Exercise and Sport Science, the Manchester Metropolitan University, Crewe and Alsager Faculty, Alsager, ST7 2HL, ENGLAND. E-mail: A.BATTERHAM@mmu.ac.uk.

Current address for David R. Mullineaux: School of Social Sciences, University of Teesside, Middlesbrough, TS1 3BA, England.

**R**ecently there has been renewed interest in the importance of body size as a potentially confounding influence in studies of physiological function ^{(3)}. This paper will focus on how body dimension relates to cardiac dimension. First, the concept of scaling will be addressed, detailing various methods that have been adopted to relate body size variables to physiological variables. Second, experimental data will show the proper relationships between echocardiographically determined left ventricular mass and body dimension in a sample of males and females. Third, the general implications of these findings for scaling in sports echocardiography will be discussed.

## SCALING

A wide range of physiological variables are influenced by body dimensions, such that increments in body size result in an increase or decrease in the physiological response. To conduct meaningful interindividual or intergroup comparisons, it is often essential to partition out this confounding influence*via* the derivation of a relative index that is allegedly size independent. “Scaling” involves the normalization of a physiological variable (y) for differences in a body dimension variable(x).

Numerous approaches have been used to try to solve this problem. By far the most common approach is the use of ratio standards (RS) of the form y = b·x. Scaling *via* RS involves dividing the absolute physiological (dependent) variable by a body dimension (independent) variable such as body mass (BM), fat free mass (FFM), or body surface area (BSA). The theoretical and mathematical flaws in this method were recognized nearly fifty years ago ^{(22)}, and yet its use remains widespread. The RS approach assumes a linear relationship between the anthropometric and physiological variables with the line of identity passing through or close to the origin. Violation of these assumptions may lead to erroneous conclusions in research using the RS scaling method. An extension of RS involves the use of regression standards (RES) of the general form y = a + b·x + ∈, where *b* represents the gradient of the trendline, *a* the y-intercept, and ∈ the additive residual error term. Unfortunately, although often providing a better fit to the data with a reduction in residual error, positive intercepts are common, indicating that someone of zero body mass would exhibit a physiological response ^{(3)}. Therefore, extrapolation of the regression line beyond the actual range of data must be avoided.

## ALLOMETRY

Research in comparative physiology has consistently revealed that many physiological variables relate to body size in a log-linear, rather than linear fashion ^{(20)}. Plotting log-transformed data thus results in a straight-line fit. This relationship is best described by a power function or allometric equation of the following general form:Equation

where *y* is the physiological variable, *x* is the body size variable, and *a*, *b*, and ∈ represent the proportionality coefficient, the size exponent, and the multiplicative error term, respectively. The *a* and *b* values are derived from the log-log plot, with *b* representing the slope of the line and*a* the intercept at unity body mass (since the natural logarithm of 1= 0, that is, the origin on the log-log plot).

The most commonly adopted body size variable has been BM, thus enabling mass exponents (*b*) for several physiological variables to be identified. For example, it has been shown that maximum oxygen uptake is properly scaled by the power function ratio ˙VO_{2max}(absolute)/BM^{2/3} ^{(16)}. This suggests that smaller individuals have a higher ˙VO_{2max} per unit body mass than larger individuals because the mass exponent is less than one. Expressing˙VO_{2max} as a RS (ml·kg^{-1}·min^{-1}) to compare individuals or groups would thus penalize larger individuals by over-correcting for body size.

Allometric models have provided a better fit to the data with less residual error and thus arguably represent a more valid method for producing a dimensionless physiological variable ^{(16)}. An advantage of such power function approaches over regression standards models is the assumption of a multiplicative error term. Many physiological data sets indicate heteroscedasticity, a tendency for a greater spread in the data as body size increases, violating the assumption of constant error variance(homoscedasticity) in regression standards models. This further supports the statistical validity of allometric modeling ^{(3)}.

## RELATIONSHIP BETWEEN HEART SIZE AND BODY SIZE

There is little evidence in the literature of attempts to examine properly the relationship between left ventricular mass and body dimension indices in humans. The extant data indicate that cardiac dimensions may be closely correlated with fat free mass in particular, suggestive of a possible relationship between skeletal and cardiac muscularity^{(12)}. Animal studies ^{(18)} have revealed that the relationship between total heart mass (HM) and body mass(BM) is described best by the following equation:Equation

The exponent of 0.98 essentially equals unity. Hence, HM in a range of mammals represents a constant proportion of BM (0.6%). It has been assumed that a similar linear, proportional relationship exists in humans and that cardiac dimensions can therefore be properly scaled by RS methods constructing a LVM/BM ratio.

Use of RS scaling has been prevalent in echocardiographic studies in sports cardiology, particularly relating to the athletic heart^{(12)}. Attempts to document training specific adaptations in cardiac dimensions are clearly dependent upon the appropriate normalization of absolute heart size data for groups of disparate body size, e.g., weight lifters and endurance athletes. It has been suggested that use of a HM/BM index may be unsatisfactory as the mass of the organ is small relative to the mass of the body. Therefore, response of the organ weight may not be proportional to that of the whole body ^{(17)}. In addition, failure to meet the RS assumptions of linearity and zero intercept would result in inappropriate scaling.

The correct relationship between body size and echocardiographic indices of cardiac dimension remains to be established. The purpose of this study is to examine the proper allometric relationships between left ventricular mass and body mass and fat free mass in a human subject sample. In addition to informing scaling practice, allometry may raise interesting questions concerning gender differences in heart size-body size relationships and encourage a reevaluation of training specific changes in echocardiographic studies.

## METHODS

### Subjects

One hundred and forty two subjects volunteered for the study (78 male, 64 female, mean age 22 ± 1.5 yr, range 18-40). Subjects were screened medically with a standard laboratory questionnaire and all were found to be“apparently healthy,” asymptomatic and free from cardiovascular disease and major risk factors for coronary heart disease. Additional exclusion criteria included obesity and the chronic use of medications that may influence resting echocardiographic dimensions. Previous testing in the same laboratory had revealed no evidence of resting or exertional hypertension or electrocardiogram abnormalities.

The sample was drawn from a population of undergraduate sport and exercise science students who appeared relatively homogeneous with respect to habitual physical activity. A simple, “global” physical activity self-assessment tool was administered in a personal interview. The instrument was modified from that used in the Allied Dunbar National (England) Fitness Survey ^{(1)}, with the frequency and intensity of 20-min plus sessions in the previous 4 wk documented. An indication of lifetime physical activity participation was obtained from the proportion of years since age 14 that the subject had participated regularly in physical activity. All subjects were moderately to highly recreationally active, with 55% of males and 53% of females reporting an average of three or more 20-min sessions per week at a “vigorous” (7.5 kcal·min^{-1}) intensity. The remainder reported an equivalent frequency of “moderate” (5 kcal·min^{-1}) intensity activity. Group equivalence for physical activity indices was confirmed via the Kolmogorov-Smirnov two-sample test(*P* > 0.05) suggesting that males and females represent the same population with respect to the distribution of physical activity indices.

Procedures were in accordance with the American College of Sports Medicine policy statement regarding the use of human subjects and written informed consent ^{(2)}. A cross-sectional “snapshot” design was used, with subjects visiting the laboratory on one occasion.

### Procedures

**Echocardiography.** A Hewlett Packard (Andover, MA) Sonos 100 ultrasound imaging system (2.5 Mhz transducer) in sector 2-D mode was used to image a longitudinal axis view of the left ventricle from the parasternal window. M-mode recordings were derived from a cursor line crossing the left ventricle at the tips of the mitral valve leaflets. All echocardiograms were conducted and analyzed by a single experienced technician. The following measurements were made in centimeters according to the Penn convention^{(10)}: septal and posterior wall thicknesses at end diastole (ST and PWT) and left ventricular internal dimensions at end diastole(LVIDd). All readings were obtained at the peak of a simultaneous EKG R-wave, with subjects in the supine or left lateral decubitus position. Measurements represented an average of 3-5 heart cycles.

Left ventricular mass (LVM, g) was estimated by means of the regression corrected cubic formula of Devereux and Reichek ^{(10)}. This method involves the subtraction of internal LV volume from total LV volume, multiplied by an assumed constant for cardiac muscle density of 1.04. The major limitation of this approach lies in the calculation of LV volume by means of cubing obtained LV dimensions. Clearly, any measurement errors in ST, PWT, or LVIDd will inflate exponentially when estimating LVM. The 2.5 Mhz transducer represents a compromise between resolution and penetration, with optimal resolution of approximately 0.7-1.4 mm. A 1-mm error in PWT measurement, for example, could result in a 15% error in LVM estimation^{(17)}. Notwithstanding these limitations, Reichek and Devereux ^{(19)} reported a strong correlation between echocardiographic estimation of LVM and LVM determined at autopsy (r = 0.96). Moreover, the aim of the current study was to examine relationships between LVM and body dimensions using procedures commonly adopted in the extant literature ^{(12)} rather than criterion gold standard methods.

**Body composition.** Percent body fat was estimated by calculating the sum of bicep, tricep, subscapular, and suprailiac skinfolds^{(11)}. The mean skinfold of three rotations that agreed within 10% was used for subsequent analyses. Total body mass (BM ± 0.1 kg) and fat percent were used to partition body mass into its fat mass and fat free mass components.

**Data analysis.** Initially, it is essential to verify the inappropriateness of the RS approach before progressing to the allometry. Linearity checks were performed on BM against LVM and FFM against LVM. Scaling*via* RS can only be adopted if Tanner's “special circumstance” is satisfied ^{(22)}. The coefficient of variation for the body dimension variable (x) divided by the coefficient of variation for LVM (y) must equal the Pearson product moment correlation between the two variables: Vx/Vy = r x,y ^{(22)}. If this assumption is not met, RS scaling may lead to spurious conclusions.

**Allometry.** Prior to identifying a scaling index common to both genders, similarity of slopes of the relationship between body size and LVM must be confirmed. Significant gender differences found in the *b* exponents (LVM = a·BM^{b} (or FFM^{b})) would preclude intergroup comparison of scaled LVM, as it would indicate that the groups were qualitatively different. Commonality of *b* exponents was tested by including a gender × ln BM (or ln FFM) interaction term in a multiple log-linear regression model: Equation

The interaction term for both ln BM and ln FFM was not significant(*P* > 0.05) indicating that the *b* exponents were similar between groups. A common “best compromise” *b* exponent was then fitted by including “gender” as a predictor variable alongside BM or FFM in a multiple allometric regression model^{(16)}: Equation

To derive the power function exponents, gender was entered as a dummy variable (males coded 0, females coded 1) in a log-transformed model:Equation

The model provides a solution for a single *b* exponent, isolating the “gender independent” influence of body size on LVM. In addition, the equation allows for the derivation of adjusted values of the proportionality coefficient *a*, isolating the effects on LVM owing only to gender. With a common *b* exponent, *a* values can thus be compared to test for size-independent gender differences in LVM(alternatively, using the best compromise” *b* exponent, the power function ratios LVM/BM^{b} or LVM/FFM^{b} can be compared with exactly the same result).

All analyses were carried out using the SPSS 6.0 for Windows (SPSS Inc., Chicago, IL) statistical package, providing the power function equations and 95% confidence intervals for the *b* exponents. Gender comparisons of LVM were conducted with independent *t*-tests for absolute values and the body size corrected values (using power function ratios constructed from the common “best compromise” *b* exponents). The alpha level adopted for significance was *P* < 0.05.

## RESULTS

Table 1 shows the descriptive data for BM, FFM, and LVM. Males were on average 14.8 kg heavier than females, with 19.1 kg greater FFM. Absolute LVM was approximately 70% greater in males than in females.

**Linearity checks.** All checks revealed that the criteria for RS normalization of linearity and zero intercept had not been met. In all cases it was found that Vx/Vy did not equal r x,y. Use of RS scaling in the current study could thus distort the data.

**Allometry.** Kolmogorov-Smirnov one-sample tests revealed that the log-transformed dependent and independent variables, together with the allometric model residuals, were normally distributed (*P* > 0.1). In addition, no correlation was found (*P* > 0.05) between the absolute residual and the predictor variable (ln BM or ln FFM), indicating that the assumption of homoscedasticity for the log-linear allometric model was satisfied. The allometric power function equations are reported for BM against LVM and FFM against LVM. Kolmogorov-Smirnov two-sample tests revealed that males and females represented different populations with respect to the frequency distribution of left ventricular mass (*P* < 0.05). This further confirmed that multivariate allometry was warranted to identify common exponents. One test of the ability of the allometric model to correctly partition out the influence of body size is to correlate LVM/BM^{b} (or FFM^{b}) with BM (or FFM). The correlation should be close to zero if the power function ratio has properly scaled the data. That is, there should be no relationship between relative LVM and body size variables. Correlations between the ratio standards LVM/BM and BM, and LVM/FFM and FFM are presented for comparison. If the power function correlations are closer to zero than the RS, this represents superior scaling of the data.

**Body mass against left ventricular mass**Equation

The negative coefficient isolated for gender indicates the anticipated relationship-as gender tends towards zero (males) LVM increases. Proportionality constants (*a*) can be adjusted for the new common*b* exponent to compare the LVM of males and females independent of body mass: Equation

The *a* values reveal that, independent of body mass, males possess approximately 44% greater LVM (*P* < 0.05). The correlation checks reveal that the expression of LVM/BM^{0.78} did not penalize male or female subjects. For males, the power function ratio and the RS correctly partition out the influence of BM. For females, however, the correlation checks reveal that the RS approach results in a weak negative correlation, indicating that as body mass increases relative LVM decreases. The RS thus overcorrects for BM in females, penalizing heavier individuals in intersubject comparisons.

**Fat free mass against left ventricular mass**Equation

Similar to the results for BM, a negative gender coefficient was again revealed, indicating that independent of FFM males tend to have a higher LVM than females. The common *b* exponent of 1.07 results in the following adjustments to the proportionality constant *a*:Equation

The *a* values demonstrate that independent of FFM, males possess approximately 18% greater LVM than females (*P* < 0.05). The correlation checks reveal that the common power function correctly partitions out the influence of FFM for the female subjects. A weak positive correlation of 0.12 for the male sample suggests that the power function is slightly undercorrecting for FFM, thus exerting a minor penalty on smaller males in intersubject comparisons. Note, however, that the allometric scaling is still superior to ratio scaling in the same sample. For the females, both the allometric and the RS correctly account for FFM differences, with correlations not different from zero (*P* > 0.05).

## DISCUSSION

For a scaling technique to correctly partition out the influence of body size on a particular physiological variable, the scaled variable should be independent of body size. The findings demonstrate the statistical validity of the allometric scaling approach, with correlations between the scaled variable and the body dimension variable close to zero in all cases. In the female sample, ratio scaling of LVM resulted in a negative correlation with BM of-0.17. This overcorrection for BM would penalize larger female subjects in within-gender comparisons. The opposite effect occurred in the male sample for FFM. Ratio scaling of LVM resulted in a positive correlation with FFM of 0.16. This undercorrection would conversely penalize smaller subjects in interindividual comparisons. The findings demonstrate the utility of allometric scaling in this sample. However, Schmidt-Nielsen^{(20)} stated that the equations cannot be extrapolated beyond the range of data on which they are based. Therefore, it is recommended that sample specific allometry be conducted in all studies where scaling is required. In the current study, intersubject and intergroup comparisons are best conducted using power function ratios constructed from the common best compromise *b* exponents identified from the multiple regression model.

The best compromise mean *b* exponent for BM was found to be 0.78(95% confidence interval, 0.65-0.91). This exponent is different from unity(*P* < 0.05), contrary to the findings of Prothero for a range of mammals ^{(18)}. The mass exponent indicates that LVM increases with body mass at a lesser rate than that predicted from simple linear proportionality. As the exponent is less than one, use of a simple ratio standard would appear to overcorrect for BM and exert a penalty on larger subjects. The theory of geometric similarity ^{(20)} indicates that as body surface area (BSA) is proportional to the square of height and body volume (BV) is proportional to height cubed, it follows that BSA is proportional to BV^{2/3}. As body density is approximately equal to unity, BSA can be assumed to be proportional to BM^{2/3}. It can be seen that the mass exponent identified in the current study is not different from 2/3 (*P* > 0.05). It would appear that LVM is therefore proportional to BSA. This relationship has been documented in numerous studies^{(9,13)}. However, some authors^{(25)} have urged caution in scaling LVM using BSA, as the index may be confounded by differences in body composition. Clearly, any two individuals may present with similar BSA, yet differ widely in FFM. This may be of great importance given a proposed link between skeletal and cardiac muscularity ^{(12)}.

The scaling of LVM for differences in FFM offers additional insight. The“best compromise” mean *b* exponent of 1.07 (95% confidence interval, 0.92-1.22) is not different from unity, indicating that the relationship between LVM and FFM is close to constant proportion. Considering the BM and FFM exponents together, it would appear that ratio scaling of LVM per BM penalizes subjects who are heavier as a result of excess body fat, not excess FFM. Strikingly similar findings have been reported recently *via* allometric modeling of peak or maximal oxygen uptake in prepubertal children ^{(4)}, adult women^{(23)}, and older adults ^{(7)}. In these studies, the FFM exponent was not different from unity, whereas the BM exponent was significantly less than one. These findings, together with those of the current study, lend indirect support to the documented interrelationships between cardiac dimension, skeletal muscle mass, and functional capacity ^{(14)}. It appears that in echocardiographic studies requiring scaling body composition estimates must be secured. Correction of absolute cardiac dimensions by BSA or BM may be problematic because of variance in body fat percentage.

The gender difference in absolute LVM of 70% (*P* < 0.05) exceeded that reported in cross-sectional studies of 41-52%^{(5,9)}. This may be a result of disparities in sample characteristics (age, activity history, genetic factors) and/or specific methods employed (including different conventions adopted for LVM estimation). Allometric normalization of absolute LVM values for BM and FFM reduced the gender differences to 44 and 18%, respectively. These differences remained significant (*P* < 0.05). Independent of body size and composition then, males in this sample possessed larger left ventricular masses than females. Although Schmidt-Nielsen ^{(20)} issues a*caveat* that allometry is descriptive and does not represent biological laws, it is possible to briefly postulate mechanisms for these gender differences.

The proposed link between cardiac and skeletal muscularity suggests a possible dual role for testosterone. Testosterone has been used as a marker to reflect the general anabolic status of the body ^{(8)} and may positively influence cardiac growth. Gonadectomized male rats have been reported to have lower heart weights than controls ^{(15)}. This regression was reversed with testosterone replacement. Several authors have reported lower basal levels of testosterone in women compared with men. In addition, there may be a lack of post-exercise testosterone spiking in women ^{(24)} although contradictory evidence exists^{(6)}.

In addition to gender differences relating to testosterone, women have higher circulating levels of estrogens. Receptor sites for estrogen have been identified in cardiac myocytes, indicating that the heart may be a target organ for estradiol ^{(21)}. Estrogen may act as a testosterone antagonist in attenuating cardiac growth. This mechanism may be linked to the suggested cardioprotective influence of estrogen^{(12)}, based on epidemiological evidence of gender differences in the incidence of coronary heart disease.

## CONCLUSIONS

This study has examined the proper relationship between left ventricular mass and indices of body dimension in a human subject sample. Quantitative gender differences in cardiac dimension, independent of body size and composition, have been identified. The demonstration of the value of allometric scaling raises interesting questions regarding previous studies in sports cardiology. Echocardiographic data comparing groups of disparate gender, training status, and body size, may need to be reevaluated with more appropriate scaling techniques. It is hoped that our findings will promote discussion of this interesting problem and thus inform scaling practice.

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POWER FUNCTION ANALYSIS; ALLOMETRY; GENDER DIFFERENCES; ECHOCARDIOGRAPHY

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