## Abstract

Basic fluid dynamic principles were used to derive a theoretical model of optimum cardiovascular allometry, the relationship between somatic and cardiovascular growth. The validity of the predicted models was then tested against the size of 22 cardiovascular structures measured echocardiographically in 496 normal children aged 1 day to 20 yr, including valves, pulmonary arteries, aorta and aortic branches, pulmonary veins, and left ventricular volume. Body surface area (BSA) was found to be a more important determinant of the size of each of the cardiovascular structures than age, height, or weight alone. The observed vascular and valvar dimensions were in agreement with values predicted from the theoretical models. Vascular and valve diameters related linearly to the square root of BSA, whereas valve and vascular areas related to BSA. The relationship between left ventricular volume and body size fit a complex model predicted by the nonlinear decrease of heart rate with growth. Overall, the relationship between cardiac output and body size is the fundamental driving factor in cardiovascular allometry.

- heart size
- cardiovascular growth
- left ventricular volume

quantitative assessment of cardiac and vascular dimensions is essential to evaluation and management of cardiovascular disorders (5, 6, 9, 42, 50, 54, 68, 71, 72). Critical to interpretation of these measurements is the establishment of normal standards. Despite the physiological and clinical importance of the issue, the best means by which to adjust for the impact of body size on the size of cardiovascular structures remains controversial. Numerous studies (14, 17, 24, 28, 31, 36, 50, 52, 55, 58, 68, 69) have addressed this issue but have reached different and therefore suspect conclusions. Nearly all have relied on simple regression analysis without a firm theoretical basis (1). Because intra- and interspecies studies have found body heat production and cardiac output (CO) to be linearly related to body surface area (BSA) over a broad range, the most commonly used method is a per-ratio standard approach, linearly adjusting cardiac and vascular dimensions to BSA. The critical test of whether this method fully accounts for the effect of body size has rarely if ever been applied; namely, the BSA-adjusted variables should have a distribution independent of body size. The fallacy of such “per-surface area standards” (75) has been amply demonstrated (21, 25, 28), but the correct method of adjusting for body size remains unclear.

This study reports the derivation of a theoretical, model-based prediction of the nature of the relationship between body size and the size of cardiovascular structures based on known control mechanisms, followed by empirical testing of the predictions in a large population of normal children aged 1 day to 20 yr old. Two primary assumptions underlie the formulation of the analysis: *1*) in nonstenotic vessels with normal flow velocity, volume of flow is assumed to be the primary determinate of vascular and valve size, and *2*) this relationship between flow and vessel caliber is assumed to result in a similar relationship between flow and orifice size in cardiovascular structures that carry a constant proportion of total CO. The basis for these assumptions is discussed, and their validity is examined relative to the measured data.

### Glossary

- 2DEcho
- Two-dimensional echocardiography
- 2DEDV
- End-diastolic volume from two-dimensional echocardiography
- AoIs
- Aortic isthmus
- AoRt
- Aortic root
- AVA
- Aortic valve annulus
- BSA
- Body surface area
- BSA
^{0.5} - Square root of body surface area
- BSA
_{Bhw} - Body surface area calculated according to Boyd height and weight formula
- BSA
_{Bw} - Body surface area calculated according to Boyd weight formula
- BSA
_{D} - Body surface area calculated according to Du Bois and Du Bois formula
- BSA
_{H} - Body surface area calculated according to Haycock formula
- BSA
_{W} - Body surface area calculated according to Dreyer and Rey formula
- CO
- Cardiac output
- EDD
- Left ventricular end-diastolic diameter
- EF
- Ejection fraction
- EDV
- Left ventricular end-diastolic volume
- HR
- Heart rate
- LLPV
- Left lower pulmonary vein
- LPA
- Left pulmonary artery
- LCA
- Left carotid artery
- LSCA
- Left subclavian artery
- LUPV
- Left upper pulmonary vein
- LVV
- Left ventricular volume
- MMEDV
_{c} - End-diastolic volume from m-mode echocardiography, cubic formula
- MMEDV
_{al} - End-diastolic volume from m-mode echocardiography, area-length algorithm
- MPA
- Main pulmonary artery
- MV
_{AP} - Anteroposterior dimension of mitral valve
- MV
_{L} - Lateral dimension of mitral valve
- MVA
- Mitral valve area
- PVA
- Pulmonary annulus diameter
- RPA
- Right pulmonary artery
- RLPV
- Right lower pulmonary vein
- RUPV
- Right upper pulmonary vein
- STJ
- Aortic sinotubular junction
- TPVCSA
- Total pulmonary vein cross-sectional area
- TrAo
- Transverse aortic diameter
- TV
_{AP} - Anteroposterior dimension of tricuspid valve
- TV
_{L} - Lateral dimension of tricuspid valve
- TVA
- Tricuspid valve area

## METHODS

#### Predictive model of optimum dimensions of the aorta and pulmonary artery.

The optimum design of biological structures, evolved through the environmental pressure exerted by natural selection, is an established principle of biology that explains the curvilinear relation of the weight of a tree and the size of its branches (49), the relation between the diameters of pulmonary bronchi and bronchial air flow (30, 82), and the shape of eggs (78). There is a large body of evidence indicating that, under normal conditions, the geometry of the vascular pathways develops in a fashion that is optimal to minimize the hemodynamic cost of providing the volume of blood flow needed to support a wide range of body activities. For the vascular pathways, the operative physical principle is the need to minimize the energy required to propel blood through the vascular system by optimizing the interrelation between vessel radius and flow rates. This concept has been labeled the principle of minimum work. The energy required to propel flow is comprised of viscous and inertial forces. The viscous energy requirement relates to sheer stress (10, 85) and is inversely related to vessel radius. The inertial energy requirement is related to the oscillatory nature of blood flow, with the associated need to accelerate and decelerate flow with every beat, an energy requirement that varies directly with vessel radius. The dimension at which the sum of these two energy demands is minimized is the optimal vascular dimension (Fig. 1). The theoretical aspects and details of these calculations have been previously presented (56, 63, 66), and the theoretical foundations of the principle of minimum work (47, 48) and theoretical studies of optimality of the vascular system (20, 32, 33, 56, 67, 79, 86) have been validated for the coronary (66) and cerebral (63) arterial systems.

We calculated the aortic and pulmonary artery dimensions that theoretically minimize flow-related energy cost. The viscous and inertial energy requirements were calculated for the aorta and the main pulmonary artery (MPA) over the range of BSA from 0.2 to 1.8 m^{2} and over the range of CO values corresponding to normal physical activities (from 3.5 to 17.5 l·min^{−1}·m^{−2}). Viscous energy loss per unit of length (*E*_{V}) was calculated by the Poiseuille equation, *E*_{V} = (8*q*^{2}*vl*)/(π*r*^{4}), where *q* is the blood flow rate (cm^{3}/s), *v* is blood viscosity (0.03 dyn·s·cm^{−2}), *r* is the radius of the vessel, and *l* is the length of the vessel (56, 66). The inertial energy content (*E*_{I}) of blood volume was calculated as *E*_{I} = *b*π*r*^{2}*l*, where *r* and *l* are as above and *b* is the energy coefficient of the blood volume (calculated in dyn·cm^{−2}·s^{−1}), obtained as the product of the mean arterial pressure times mean blood flow rate divided by the arterial vascular volume (66). Calculations assumed normal BSA-adjusted mean values of arterial pressure and CO and an arterial vascular volume of 4% of the total blood volume for the aorta and 0.7% for the MPA (45). The values for the energy coefficient *b* were similar to the values previously reported for coronary arteries (66). The optimum dimensions of the aorta and the MPA were then calculated as the dimensions for each that minimized the sum of *E*_{V} and *E*_{I}.

#### Predictive model of aortic branch size.

Branching is an important feature of the vascular tree, which adds to the complexity of estimating optimal vessel size. In almost every part of the arterial tree, the area of the daughter branch is smaller than the area of the original vessel. In a constant-flow system, the velocity and energy expenditure increase exponentially as the diameter of the vessel decreases. If the same total flow is subdivided between two branch arteries of total cross-sectional area similar to the original vessel, the ratio of circumference to area rises, and more flow is exposed to the sheer stress of the vessel wall. Consequently, viscous forces are maintained at a similar level only if the sum of the daughter vessel areas is greater than that of the parent artery. In a pulsatile flow system, this must be balanced against the oscillatory energy cost of the system, which increases in proportion to the rise in cross-sectional area. The relation between the caliber of parent and branch vessels was originally described by Cecil D. Murray (47, 48) and further explored in later studies (20, 63, 67). The optimal model for the adaptation of the arterial tree is based on a balance between the radius of the vessel and the cubic root of the flow through the vessel, such that the radii of parent (*r*_{0}) and branch (*r*_{1}, *r*_{2},… *r*_{n}) vessels are related as (*r*_{0}^{x} = *r*_{1}^{x} + *r*_{2}^{x} + … + *r*_{n}^{x}), where *x* = 3 for laminar flow and *x* = 2.33 for turbulent flow (56, 62, 63, 66, 67, 79). To verify whether the aortic branching system obeys this model for adaptation of the arterial tree in children, we examined the relation between the dimensions of transverse aorta and its branches over the range of BSA.

#### Testing of the model.

The derived models were tested in a study group comprised of 496 normal children and young adults seen in the noninvasive laboratory at Boston Children’s Hospital for echocardiographic evaluation during the years 1987 to 1998 who had no evidence of structural or functional heart disease. Acquired or congenital heart disease and other systemic disorders were excluded by a careful review of the medical history, electrocardiogram, chest X-ray, and echocardiogram. Specific exclusion criteria included acute or chronic systemic disorder, hypertension, a family history of hypertrophic or dilated cardiomyopathy, and height or weight percentile outside the range of normal. This study was conducted under a protocol approved by the Institutional Review Board in 1985.

#### Calculation of BSA.

Height and weight were measured in the laboratory. To evaluate the potential differences between methods for calculating BSA, BSA was calculated according to five published methods.

The formula of Du Bois and Du Bois (16): (1)

The method of Dreyer and Ray (15): (2)

The two methods published by Boyd (7): (3) (4)

The formula of Haycock et al. (27): (5)

#### Data collection.

Echocardiographic studies were performed with a phased-array sector scanner (Hewlett-Packard or Acuson) equipped with transducers appropriate for body size and were recorded on 0.5-in. videotape cassette at 30 frames/s. Subxiphoid, apical, parasternal long and short axis, and suprasternal notch views were obtained for each patient. When necessary, subjects less than 2 yr of age were sedated with chloral hydrate. All measurements were performed by one of two observers. Measurements were performed either online during the exam or secondarily offline from videotape. Online measurements were performed from video loops captured in zoom mode by use of the electronic calipers on the ultrasound machine. Offline measurement was performed either using electronic calipers on a video overlay system or from video page prints using a digitizing tablet. All valvar and vascular dimensions were measured from inner surface to inner surface at the moment of maximal expansion during the cardiac cycle. The dimensions of the aortic valve annulus (AVA), aortic root (AoRt), and sinotubular junction (STJ) were measured on parasternal long axis views (61). The measurements of the transverse aorta (TrAo), left subclavian (LSCA), and left carotid arteries (LCA) at their origin and of the aortic isthmus (AoIs) were obtained from high parasternal long axis or suprasternal notch views (26, 73). The lateral dimensions of mitral and tricuspid valves were measured on four chamber apical views (MV_{L} and TV_{L}, respectively) and the anteroposterior dimension from the left parasternal long axis view (MV_{AP} and TV_{AP}, respectively) at the proximal attachment of the leaflets at each side of the annulus (26, 34). The mitral and tricuspid valve areas (MVA, TVA) were calculated as an ellipse [π·(*d*_{1}/2)·(*d*_{2}/2), where *d*_{1} and *d*_{2} are the anteroposterior and lateral dimensions, respectively] in those subjects for whom both measurements were available. The dimensions of the pulmonary valve annulus (PVA), the MPA, and the proximal right and left pulmonary arteries (RPA, LPA) were measured from left parasternal, short axis views. The pulmonary veins were measured from subcostal, apical, high parasternal, or suprasternal notch short axis views at the point of connection to the left atrium. Total pulmonary venous cross-sectional area (TPVCSA) was calculated as the sum of the cross-sectional areas of the four individual pulmonary veins. Two-dimensional end-diastolic volume (2DEDV) was calculated from long and short axis views of the left ventricle obtained from apical and parasternal windows, respectively. Diastolic images were taken as the frame preceding mitral valve closure, and the endocardial and epicardial volumes were calculated by the biplane Simpson’s algorithm (84).

Previously published (11) left ventricular end-diastolic short axis dimensions (EDD) obtained from digitized m-mode recordings in 290 normal children aged 0 to 18 yr were reanalyzed to obtain measurements of end-diastolic volumes by use of other algorithms, permitting us to examine the impact of the method of calculating end-diastolic volume on the relationship to body size. Heart rate was calculated from the R-R interval on the electrocardiogram. M-mode-derived left ventricular end-diastolic volume (MMEDV) was calculated by two methods. For all 290 subjects, cubic-formula end-diastolic volume (MMEDV_{c}) was estimated as the cube of the short axis dimension = EDD^{3}. For those subjects in whom left ventricular long axis dimensions could be measured (*n* = 176 of 290), area-length end-diastolic volume (MMEDV_{al}) was also estimated by using the 5/6·area·length algorithm. The two-dimensional (2DEDV) and m-mode (MMEDV_{c}, MMEDV_{al}) volume measurements represent independent data sets and were therefore analyzed separately.

#### Statistical analyses.

Linear, nonlinear, and multiple regression analyses were used to examine the relationship between parameters of body size and each of the echocardiographic variables. On the basis of preliminary exploratory analysis and theoretical considerations as discussed above, an exponential growth model with a zero intercept (*Y* = *aX*^{b}, where *Y* = cardiovascular dimension and *X* = body size parameter) was used to test the relation of echocardiographic measurements to body height, body weight, and BSA calculated according to the five methods described above. Three-parameter models of the form *Y* = *aX*^{b} + *c* were examined to determine whether a statistically significantly improved [*P* < 0.05 by *F*-statistic as described by Zar (87)] description of the relationship between body size and echocardiographic measurements could be achieved through the use of higher order models. Nonlinear curve fits were calculated by the Levenberg-Marquardt least-square method (41). The model with the highest *R* value and the lowest residual mean square and the lowest square root of residual mean square (= standard error of the estimate) was considered to provide the “best fit.” After derivation of the exponents for the exponential growth model, the significance of the difference between the derived and the theoretically predicted exponents was tested. Proceeding from the assumption that the control mechanisms for vascular growth should be uniform across the vascular and valve structures included in the analysis, we further evaluated the theoretical model based on whether a similar trend was observed for most or all of the variables. To test for the absence of heteroscedasticity and skewness in prediction (1) after transformation of valvar and vascular measured dimensions by division (index = dimension/square root of BSA_{H}, where BSA_{H} is BSA derived from the Haycock formula) the relations between body size and the indexed variables were tested for skewness and were analyzed by regression analysis for linear trends.

## RESULTS

#### Prediction of optimum vascular dimensions on the basis of the principle of minimum work.

Observed diameters of aortic valve, ascending aorta, pulmonary valve, and central pulmonary artery were compared with the values predicted by the principle of minimum energy dissipation over the entire range of body size (0.2–1.8 m^{2} BSA).

Optimal aortic dimensions calculated according to the principle of minimal work are presented for CO ranging from normal resting levels (3.5 l·min^{−1}·m^{−2}) to CO associated with maximal activity (5 × 3.5 l·min^{−1}·m^{−2}) in Fig. 2. The observed diameters were found to be optimal in terms of energy dissipation for CO up to two times the resting CO in infants, increasing to values that were optimal for three to four times the resting CO for older children. Data obtained from athletes and hypertensive subjects indicate that cardiac structures adapt to peak or mean 24-h levels of pressure and volume demand (8, 12, 18, 19, 43, 65). Heart rate response is the primary determinant of CO increase with exertion, and infants have an approximate twofold magnitude of heart rate reserve, a value that increases to three- to fourfold in older children and young adults. Thus the difference between observed diameters that are optimal in terms of energy dissipation in infants vs. children corresponds to the expected difference in CO associated with age-appropriate physical activity and range of intensity of exertion.

#### Predictive model of aortic branch size.

The calibers of the transverse aorta and branch vessels (LCA, LSCA, and AoIs) were found to be related according to *r*_{0}^{x} = *r*_{1}^{x} + *r*_{2}^{x} + *r*_{3}^{x} with *x* = 2.33, as demonstrated by linear regression and by calculation of the indexed mean values of the radii (Fig. 3). The value *x* = 2.33 is the value expected for an optimal relation between the calibers of parent and daughter vessels in turbulent flow (56). Curves, branches, and projections impose directional change on the flow streamlines, resulting in nonlaminar flow in the horizontal aorta and its branches (4, 40, 44). Over the range of body size, the observed relation between the dimensions of the transverse aorta and its branches were therefore found to conform to the theoretical principle of optimum dimension relation.

#### Comparison of methods for calculating BSA.

The relationship between BSA_{H} vs. each of the other four methods of calculating BSA is illustrated in Fig. 4. It has been previously documented that the formula of Du Bois and Du Bois increasingly underestimates (27) BSA as values fall below 0.7 m^{2}, an observation confirmed in these data. BSA calculated according to each of the other three formulas demonstrated significantly skewed comparison at lower compared with higher BSA values, although for each of the other methods lower values of BSA were overestimated compared with the Haycock formula. For each of the cardiovascular parameters examined in this study, BSA_{H} was more closely correlated than any of the other four formulas (data not shown). For these reasons, BSA_{H} was considered the most appropriate method for calculating BSA, and only analyses based on BSA_{H} are presented.

#### Relation of valvar and vascular dimensions and areas to body size.

The range, the number of observations, the mean and the median values of cardiac, valvar and vascular diameters, areas, and volumes are reported along with descriptive statistics for age, height, weight, and BSA in Table 1. For each of the valvar and vascular diameters, indexing to BSA_{H} failed to adequately account for the variance with BSA. Because the results of this analysis were similar for all 19 variables, the details of this analysis are illustrated only for the AVA (Fig. 5). In Fig. 5*A*, the close linear relationship between AVA and BSA_{H} is apparent. Nevertheless, when AVA is “indexed” by simple division by BSA, AVA/BSA is found to be strongly inversely and nonlinearly dependent on BSA, as shown in Fig. 5*B*. Thus this method of adjusting for body size fails to fully account for the dependence of AVA on BSA. This approach to normalization failed in a similar fashion for all 19 variables. The principle of minimal work predicts a linear relation between vessel cross-sectional area and the volume of flow through the vessel. For the cardiac valves and central vessels, the volume of flow is either total CO or a fixed proportion thereof. Therefore, their area should relate to BSA and their diameter should be proportional to BSA^{0.5}. Although the regression of AVA vs. BSA^{0.5} is only marginally better than the regression vs. BSA (Fig. 5*C*), when AVA is indexed by division by BSA^{0.5}, AVA/BSA^{0.5} is found to manifest no significant residual dependence on BSA, as shown in Fig. 5*D*. Figure 5 also illustrates the problem of nonconstant variance (heteroscedasticity) because there appears to be a larger data spread in AVA for higher values of BSA and larger spread of indexed AVA for lower values of BSA. However, this degree of skewness in the residuals was not statistically significant, as discussed below.

We next evaluated whether the mathematical relation between observed vessel and valve diameter and body size was significantly different from the theoretical prediction. We examined whether height, weight, or BSA_{H} provided a better means of accounting for the variance in the valvar and vascular diameters. Comparisons of the relationship between each of the valvar and vascular diameters vs. height, weight, and BSA_{H} are presented in Table 2. For each regression, the constant (*a*) and exponent (*b*) fit to a function of the form *Y* = *aX*^{b} are shown, with the correlation coefficient and the residual mean square. In each instance, the correlation coefficient was larger and the residual mean square was lower for the regression against BSA_{H}. Three parameter models of the form *Y* = *aX*^{b} + *c* were also examined for each variable. In each instance, the addition of the third parameter did not significantly improve the accuracy of the prediction of the *Y* values, justifying reliance on the simpler two-parameter model.

Comparison of the exponents in the regressions across variables in Table 2 indicated that vascular and valvar dimensions correlated on average with BSA_{H} raised to the 0.50 power (range 0.42 to 0.58). On the basis of the observations that *1*) all valve and vascular dimensions related more closely to BSA_{H} than to height, weight, or BSA calculated by other methods, *2*) on average valve and vascular diameters relate most closely to BSA_{H}^{0.5}, and *3*) each of the valvar and vascular diameters carry either the total or a constant proportion of CO and should therefore demonstrate a similar exponential dependence on BSA_{H}, we next examined whether the derived exponent for any of these variables vs. BSA_{H} was significantly different from the average exponent of 0.5. Table 2 presents the results for the linear regressions of each of the valvar and vascular diameters to functions of the form *Y* = *a*(BSA_{H})^{0.5} and of the form *Y* = *a*(BSA_{H})^{0.5} + *c*. For each of the parameters, there was no significant change in residual mean square compared with the more general *Y* = *aX*^{b} model (Table 2), indicating that the simpler linear model using (BSA_{H})^{0.5} as the independent variable was equally successful in describing the relationship between BSA and the size of vessels and valves.

When we evaluated whether the inclusion of a nonzero intercept in the linear model significantly improved the explained variance, the only variables for which the intercept was distant more than 1 mm from 0 were mitral and tricuspid annulus diameters and pulmonary venous diameters. For each of these, the *P* values for the intercept were less than 0.10. To determine whether these significant intercept *P* values related to age-related differences in the shape of the atrioventricular valves or relative size of individual pulmonary vein diameters, we analyzed the relationship of MVA, TVA, and TPVCSA to BSA. The analysis for these variables, conducted in a fashion similar to that for the diameters, is presented in Table 3. As predicted, the areas related more closely to BSA_{H} than to height or weight, and the exponent *b* of the *Y* = *aX*^{b} model for each area vs. BSA_{H} averaged 0.99 (range of 0.974 to 1.023) and in no case was significantly different from 1. For the areas, each intercept *P* (the probability that the intercept is different from zero) was not significant (Table 3). Thus the nonzero intercept of the regressions for mitral, tricuspid, and pulmonary venous diameters vs. the zero intercept of the regressions of the mitral, tricuspid, and total pulmonary vein areas could be explained by BSA-dependent differences in relative size of the diameters and by the change in shape of those structures with growth.

#### Indexed valvar and vascular dimensions and areas.

The model that best described the data in a statistical sense was therefore a linear model with zero intercept (*Y* = *aX*) where *X* = (BSA_{H})^{0.5} for vascular and valvar dimensions and *X* = BSA_{H} for vascular and valvar areas. On the basis of this finding, the adequacy of adjusting vessel and valvar dimensions by simple division by (BSA_{H})^{0.5} and of adjusting vessel and valvar areas by simple division by BSA_{H} was tested. As discussed above, if the effect of body size has been fully accounted for, the adjusted variable should have a distribution independent of body size. Indexed dimensions (*D*) were calculated as *D*_{I} = *D*/(BSA_{H})^{0.5} and indexed areas (*A*) were calculated as *A*_{I} = *A*/BSA_{H}. For each indexed variable, the regression vs. BSA_{H} was evaluated. The correlation coefficients and *P* values for these regressions are presented in Tables 2 and 3. With the exception of the mitral, tricuspid, and pulmonary vein dimensions, in each case the correlation was not significant, indicating that this method of indexing adequately accounts for the variance of the dimensions and areas with respect to BSA_{H}. The residual variance apparent in the indexed mitral and tricuspid dimensions was related to a BSA_{H} dependence of the magnitude of noncircularity of the valve orifice, with larger individuals tending to have more elliptical valves, although this relationship did not achieve statistical significance. This effect was eliminated when valve area was considered because there was no significant residual variance of the indexed valve areas with respect to BSA_{H}. Similarly, the significant residual variance in the indexed pulmonary vein dimensions was related to a BSA_{H} dependent variation in the relative size of the pulmonary veins, with larger subjects tending to have larger upper pulmonary veins compared with the lower veins. Again, when indexed TPVCSA was examined, there was no significant relation to BSA_{H}. The superiority of areas in accounting for the effect of BSA_{H} is the anticipated outcome if it is correct that total flow is the primary determinant of vascular and valvar size.

We tested for nonconstant variance (heteroscedasticity) of the relationship between indexed values vs. BSA_{H} by testing for skewness in the square of the residuals. The significance of this skewness is presented for each regression in Tables 2 and 3 (variance P). With the exception of the mitral and tricuspid lateral diameters, the magnitude of skewness was not statistically significant, indicating that this method of indexing adequately accounts for the effects of body size from a statistical point of view. Again, when mitral and tricuspid areas rather than diameters were considered there was no significant skewness to the residuals (Table 3), reflecting the same pattern as was noted for the issues of a significantly nonzero intercept and significant residual variance of the normalized variable vs. BSA_{H}. As with normal data obtained by others (1, 17, 23, 34, 38), the scatter of data points about the regression line tended to be greater for larger subjects as illustrated in Fig. 5*B*, an effect that is reduced but not completely eliminated by this method of indexing (Fig. 5*D*). However, because the nonconstant variance did not achieve statistically significant levels, estimation of the regression parameters remains unbiased (46).

#### Relation of left ventricular dimension and volume to body size.

A predictive model of the relationship between left ventricular volume and body size was developed from the fact that CO is equal to the product of heart rate (HR), end-diastolic volume, and ejection fraction (EF), that is, CO = HR·EF·EDV. Combining the known linear relationship between CO and BSA [that is, CO ≈ BSA^{1.0} (22, 36)] with our findings that HR ≈ BSA^{−0.4} (Table 4) and that EF is independent of BSA, the predicted relationship is EDV ≈ BSA^{1.4}. This prediction was evaluated against the observed data by analyzing the relationship between age, height, weight, and BSA as independent variables against left ventricular diastolic dimension and volume derived from two-dimensional echo and from m-mode echo (EDD, 2DEDV, MMEDV_{c}, and MMEDV_{al}). The results of this analysis are presented in Table 4. As was the case for the vascular and valvar measurements, the best predictive equations with the highest *R* value and the lowest residual mean square were obtained with BSA_{H} compared with the other four methods of calculating BSA. The three-parameter model that included a nonzero intercept (*Y* = *aX*^{b} + *c*) did not significantly reduce explained variance, justifying use of the simpler two-parameter model (*Y* = *aX*^{b}). Left ventricular dimension related most closely to BSA^{b} with *b* = 0.43, whereas EDV, determined by the several different methods, related to BSA^{b} where mean *b* = 1.38 (range 1.343–1.398). This value is not significantly different from the predicted value of 1.4, confirming the validity of the predictive model.

## DISCUSSION

#### Vascular and valvar dimensions.

On the basis of statistical analysis, BSA was a more important determinant of the size of each of the 19 valvar and vascular dimensions studied in 496 normal children than either height or weight alone. In addition, of the several methods of calculating BSA that are in common use, the Haycock method was superior for describing the growth of cardiovascular structures. All vascular and valve diameters were linearly related to the square root of BSA whereas valvar and vascular areas were related to BSA. Some previous studies, as summarized in Table 5, reached similar conclusions. If this order dependence of the relationship between cardiovascular pathway dimension and body size dimension is ignored by simply indexing all variables to BSA, the indexed values have significant residual dependence on body size. More complex models such as polynomial regression, variable exponents, or variable intercepts do not describe the data more effectively. As in previous observations (25, 31, 34, 37, 50, 59, 68, 73), the intercepts of the regressions of vascular and valvar dimensions were zero or not significantly different from zero, allowing normal valvar and vascular diameters to be represented accurately by values indexed to BSA^{0.5}. Similarly, valvar and vascular areas were represented accurately by values indexed to BSA.

This analysis also illustrates how the method of calculation of BSA can affect the results, particularly in small children. The potential influence of the method of BSA calculation has been given insufficient attention, as reflected by the failure of most papers to even reference which method was used (Table 6). Caution is clearly needed when analyzing outcome on the basis of dimensions of a pathway if the normal values have been obtained by use of a different modality or measurement technique. For example, because the aortic and pulmonary artery areas vary by 9–12% (39) and TVA changes by a mean value of 26% during the cardiac cycle (53), use of autopsy values to derive a normal range for comparison of angiographic (2) or echocardiographic values is unlikely to be valid. This issue is emphasized by the large differences in the “normal” Nakata index (RPA + LPA area) (50) derived by different imaging techniques (Table 6). One of the strengths of this study is a uniform approach to the method of measurement and the inclusion of measurement of a large number of different structures obtained in the same laboratory. As a consequence, the variation in the derived exponent between the variables included in this study was significantly less than the variation between the studies summarized in Table 5.

We observed, as have others (1, 17, 23, 34, 38), that data spread tended to increase directly as a function of BSA. This nonconstant variance was reduced but not completely eliminated when dimensions were indexed by division by BSA^{0.5}, as seen in Fig. 5*D*. If it is assumed that the variance is constant with respect to BSA_{H}, the resulting prediction intervals result in an excess number of cases being incorrectly classified as above or below 2 standard deviations from the normal mean value. As discussed by Abbott and Gutgesell (1), it is therefore desirable to employ methods that further reduce the trend toward nonconstant variance when constructing confidence and prediction intervals. These methods include multiple regression techniques (3), variance stabilizing transformations (1, 3, 35, 46, 87), and weighted least squares analysis (77). Such statistical methods, although improving the accuracy of prediction intervals, are invariably ad hoc and lack a sound physiological basis on which to base the choice of one method over another. In the present analysis, we were primarily attempting to understand the relationship between body size and growth of cardiovascular structures. For this purpose, the observation that the nonconstant variance did not achieve statistical significance was sufficient to ensure unbiased estimates of the regression parameters (46).

The *Y* = *aX*^{b} model of vascular and valve development with growth was predicted theoretically and confirmed empirically. The findings support a causal relationship because the observed measurements and the cardiovascular dimensions adjusted for body size according to the proposed model are similar to the values predicted by the theoretical analysis based on the principle of minimum work. Besides the validation of the analysis of the cardiovascular pathway dimensions vs. body size relation, the data support the hypothesis that cardiac and vascular pathways adapt to peak levels of pressure and volume demand. The diameters observed are optimal in terms of energy dissipation for CO equivalent to two times resting CO in infants and three to four times resting CO in older children. These findings are similar to the comparison between observed and theoretically predicted coronary size on the basis of the minimal work principle (66).

#### Ventricular dimensions and volumes.

In accord with some previous studies (14), we found BSA to be a more important determinant of left ventricular dimension and volume in normal subjects than age, height, or weight alone. Unlike the linear relation of valve and vessel diameters to the square root of BSA, the effect of body size on left ventricular dimensions was not fully accounted for by adjustments for the square root of BSA. Similarly, contrary to the predicted relationship of volume relating to weight^{1.0} and to BSA^{1.5}, ventricular size has a more complex relation with body size. In contrast to vascular and vessel growth, which depend directly on CO, the relationship between ventricular size and CO also depends on heart rate such that ventricular diastolic volume would only be predicted to relate to BSA^{1.5} if heart rate were a function of BSA^{−0.5}. In this study, we found that the relationship between heart rate and BSA had an exponent of −0.4, that is HR = *a*BSA^{−0.4}. Similar observations (HR = *a*′BSA^{−0.38}) were made in series of different species of mammals with a 1,800- to 500,000-fold increase of body weight (29, 60). The increasing resistance to blood flow and the power output available from the heart muscle with increasing body size are the limiting factors of the frequency of the heart beat, as derived mathematically and shown by the measured data available (74). Because of the inverse relation of heart rate with BSA and of the linear relation between CO and BSA (22, 36), left ventricular volume was more closely related to BSA^{1.4} as observed in the subjects of this study and in the literature (Table 5) than to BSA^{1.5}. Similar to our findings, left ventricular diameters have been most closely related to BSA^{0.4–0.45} (Table 5). It is also clear that when factors other than BSA influence basal heart rate, the relationship between BSA and ventricular size can be expected to change. Two well-known examples of this are complete heart block and training-induced bradycardia, both of which are associated with ventricular volumes larger than would be predicted for BSA. Less well investigated and outside of the scope of this study is whether the normal age-related fall in heart rate in adults similarly influences the relationship between heart rate and ventricular size, creating a need for indexing ventricular size for both age and BSA.

#### Limitations of the study.

A potential limitation of this study is that the optimal dimensions derived from the principle of minimal work were based on the assumption that CO is linearly related to BSA. The results of various investigations into the variance between CO and BSA have varied from a first order linear relationship (22, 36) to a 0.6 power relationship (29), but in humans the relationship has been found to be linear. The hypothesis that the relation between CO and BSA is linear is also supported by the observation that the relation between the dimensions of transverse aorta and its branches, which are linearly related to BSA, obeys the theoretical principle of optimum dimension relation that underlies vascular tree structure, and is calculated independently of CO values.

In conclusion, in children, BSA is a more important determinant of the size of each cardiac or vascular structure than height or weight alone. The method of calculation of BSA is also important, with the Haycock formula providing the best estimate with regard to the growth of cardiovascular structures. Cardiac and vascular pathway diameters vary linearly with the square root of BSA and valve and vascular areas vary linearly with BSA. Because the intercept of the regression is zero, the valve and vascular diameters are represented accurately by the values indexed for the square root of BSA by division. The distribution around the mean value of the index allows for an accurate representation of normals without the bias of heteroscedasticity and skewness in predictions using linear regression and ordinary or weighted least squares (1). Residual analysis showed that the effect of body size has been fully accounted for by demonstrating the independence of the index of valve and vascular dimensions and of the residuals with body size. Values derived from the theoretical principles of minimal work and the principles underlying vascular tree structure are similar to the observed measurements and to the dimensions adjusted for body size according to the proposed model. Aside from the validation of the analysis of cardiovascular dimension-body size relation, it supports the concept that cardiac and vascular pathways adapt to peak or mean 24 h level of CO. In contrast, because of a nonlinear decrease of heart rate with growth, the effects of body size on left ventricular size were not fully accounted for by similar adjustment for proportional powers of BSA, with ventricular volumes proportional to weight^{0.9} and BSA^{1.4}. The underlying relationship between BSA and CO does indeed appear to be the fundamental control mechanism that influences the growth of vessel and chamber size, but the variability of heart rate with body size adds an additional level of complexity to the control of chamber growth.

## Footnotes

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