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Right ventricular volumes revisited: A simple model and simple formula for echocardiographic determination
Right Ventricular Volumes Revisited: A Simple Model and Simple Formula for Echocardiographic Determination Stewart Denslow, PhD, and Henry B. Wiles, MD, Charleston, South Carolina
Our objective was to establish a crescentic model of the right ventricle as the basis of a reported 2⁄3 (Area)(Length) empirical formula for volume. This formula has been investigated by others without cognizance of its connection to a clear geometric model. The particular model, an ellipsoidal shell or difference of ellipsoids, has been investigated by several groups by using different volume formulas. Accordingly, we obtained echocardiographic images in 2 orthogonal planes from 7 patients and 4 volunteers. Specified area and length measurements from these images were used to calculate
In the management of congenital heart disease, the importance of monitoring the size of the right ventricle is often equal to or greater than that of the size of the left.1-3 Because of the low cost, portability, and noninvasive nature of echocardiography, this modality has been the subject of continuing interest in efforts to find an easily interpretable, geometric model–based method for right ventricular volume estimation.4-12 Although concurrent studies have focused on multiplane-based (Simpson’s rule) methods,13-15 such methods are generally more time consuming than methods that are based on models and require high-quality images at multiple image planes. Additionally, multiplane methods provide little conceptual framework for analysis of shape and shape changes. Despite the inherent advantages, however, geometric model application has been characterized by limited success. In large part this is caused by the relatively complex crescentic shape of the right ventricle16 and the difficulty of finding a simple-tounderstand model that describes this shape.7 Studied models have included single ellipsoid,8,12 combinations of ellipsoids,9 and crescentic shapes.4,5,10,11 Two significant investigations of an apparently From South Carolina Children’s Heart Center, Medical University of South Carolina. Reprint Requests: Stewart Denslow, PhD, South Carolina Children’s Heart Center, Medical University of South Carolina, 171 Ashley Ave, Charleston, SC 29425-0680. Copyright © 1998 by the American Society of Echocardiography. 0894-7317/98 $5.00 1 0 27/1/91976
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right ventricular volumes. These volumes were compared with values determined through multislice, magnetic resonance imaging with summation of lumen areas, a widely accepted standard. Obtained high correlations compared favorably with those of previous investigators who used equivalent but less well understood methods. We conclude that the ellipsoidal shell model of the right ventricle provides a simple arealength formula for the determination of lumen volume with echocardiography. (J Am Soc Echocardiogr 1998; 11:864-73.)
model-based approach are those of Levine et al7 and Gibson et al.6 Although the results of these studies indicated a high accuracy resulting from a simple formula for volume (2⁄3 3 Area 3 Length), the investigators were unable to determine a precise geometric shape corresponding to this formula. The objective of the present communication is to establish a clear geometric model as the basis of the simple area-length formula used by Levine et al7 and Gibson et al.6 This geometric model is that of an ellipsoidal shell or difference of ellipsoids. It has the advantages of a flattened, crescentic shape that includes the infundibulum and a simple formula for volume. This model has been used successfully, in some cases with the use of different but mathematically equivalent formulas, in the estimation of right ventricular volume from data including (1) dual-plane angiographic images,17 (2) dual-plane echocardiographic images,10 (3) singleplane and dual-plane magnetic resonance images,18-20 and (4) sonomicrometry data.21,22 Accordingly, we undertook a study comparing (1) volumes calculated with the above area-length formula with measurements from echocardiographic images and (2) volumes determined with Simpson’s rule with measurements from multiplane magnetic resonance images, a generally accepted “gold standard.”23-26 METHODS Subjects A retrospective search of divisional records was conducted for patients who had undergone a magnetic resonance
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imaging study that included a multislice, gradient echo study of the ventricles within 2 weeks of an echocardiographic study. A total of 7 patients (5 female, 2 male) (Table 1) were found. Four healthy adult volunteers (1 female, 3 male) who underwent the same imaging studies were also included. Informed consent was obtained when applicable. All studies were approved by the Institutional Review Board of our institution. Magnetic Resonance Imaging All studies were performed with the use of a Gyroscan T5 system (Philips Medical Systems of North America, Shelton, Conn) operating at 0.5 T. The imaging protocol followed established methods.23,25 Initial spin-echo localizer views, taken in sagittal, transverse, and coronal planes, were used for planning a multislice, spin-echo scan at right angles to the cardiac axis, with slices more than covering the ventricles from apex to base. After this scan, gradient-echo, cinescans of 1 to 3 slices (number of slices in each gradientecho scan was dependent on the subject’s heart rate) were planned that coincided in position and thickness with previous spin-echo slices that had indeed intersected the ventricles. At least 12 cardiac phases were obtained in each image plane to ensure adequate time resolution for location of end-systole. Acquisition parameters were slice thickness of 5 to 10 mm, flip angle 40 degrees, echo time 9 ms, and 4 signal averages. Resolution was 128 3 256 pixels interpolated to 256 3 256 pixels. Repetition time (364 to 632) varied with heart rate (45 to 150 bpm). Total number of contiguous slices was from 4 to 11. Data acquisition was triggered by the R waves of the electrocardiogram. Magnetic Resonance Image Analysis With the use of image measurement software (Gyroview, N.V. Philips, Eindhoven, The Netherlands) on a graphics workstation, areas from magnetic resonance images were determined from interactive videographic displays with a mouse-controlled cursor. Window width was set to the full range of pixel intensities in the image. Level was set at one-half the maximum intensity value (generally close to half the window setting). Window and level were kept constant during all analysis. In end-systolic and end-diastolic frames, measurements of cross-sectional lumen areas were made at the endocardial surface in the right ventricle. Papillary muscles and the moderator band were not included in the lumen area. End-diastole was taken as the first frame in each sequence, 8 to 16 ms after the R-wave trigger. This was also the largest lumen area in each sequence. End-systole was taken as the frame closest to the closing of the aortic valve that could be seen in slices near the base of each set. This frame also contained an area equal to the smallest area in each sequence. Volumes were calculated as the sum of all lumen areas times the slice thickness. Echocardiographic Imaging Two-dimensional echocardiographic studies were performed with either an Accuson 128 xp/10 (Mountain View, Calif) or an Interspec ATL (Ambler, Pa) with 5, 3.5,
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or 2 MHz transducers. Images were acquired from the apical 4-chamber view and either a parasternal short-axis or a subcostal-sagittal view that included tricuspid (inflow) and outflow portions of the right ventricle. Echocardiographic Image Analysis Areas and distances were obtained from echocardiographic images with interactive measurement systems installed on each of the echocardiography instruments. In the 4-chamber view, lumen area (A4ch) was obtained at end-diastole (time of tricuspid valve closure) and end-systole (frame immediately before tricuspid valve opening) by tracing the endocardial borders. Papillary muscles and the moderator band were not included in the lumen area. In the parasternal short-axis or subcostal sagittal view, we estimated the length of the model diameter (Figure 1, a, distance A-D) as the distance from the right ventricular free wall immediately bordering the tricuspid valve to the wall of the outflow tract on the opposite side of the pulmonary valve (Figure 2). All echocardiographic measurements were repeated on 6 separate cardiac cycles. Because the desired area and length were the anatomic maxima of these measurements in the chosen orientation, the maximum values of each measurement within a set was the final recorded value. Note that this often resulted in end-diastolic and end-systolic values originating from separate cardiac cycles. This protocol was followed to offset errors resulting from the imaging plane not cutting through the true maximum cross-sectional area or length. Geometric Model for Right Ventricle Volumes were calculated18-20 as Vol 5 2⁄3(A4ch)(DAD). This formula is derived from a difference-of-ellipsoids model of the right ventricle. As we have reported previously,18 this formula is mathematically exact for a range of crescent-shaped, regular, and irregular volumes that are quite similar in shape to the right ventricle, including its inflow, apical, and outflow regions (see Appendix). Figure 1, a, shows a basal area of the model superimposed on the atrioventricular junction of the right ventricle. Note that the superimposed area is composed of 2 partial crescents that match at their bases but are not of the same height (distance AE is not equal to distance ED). This situation, which helps in matching the model to the right ventricular anatomy, leaves the formula for volume unchanged (see Appendix). Figure 1, b and c, shows the degree of match between the model and the mid ventricle short axis and with the long axis. Figure 3 shows a 3-dimensional rendering of the model of the left ventricles and right ventricles. Note that the crescentic right ventricular lumen is modeled as largely wrapping around the left ventricular septal surface. However, the outflow tract is included in the model by specifying that the diameter to be measured should stretch beyond the epicardial surface of the left ventricle to the outer edge of the right ventricular outflow. Figure 4 shows the difference-of-ellipsoids model standing alone with indications of the area and length used for volume calculation. Although there exist two other combinations of area and length that can be used for volume estimation, this combi-
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nation was found to be most accurate with magnetic resonance images and was accordingly used in this study.18-20 An important point to note is that the 2⁄3(Area)(Length) formula also applies to convex (that is, noncrescentic) ellipsoidal shapes (see Appendix). The area to be measured is the same as before, whereas the distance to be measured becomes the diameter of the ellipsoid in the same plane as the previous distance between the “horns” of the ellipsoidal shell.18 This fact results in the formula for volume being applicable to a range of shapes from a rounded “left ventricle” type of volume to the concave “right ventricle” type of volume. Thus the existence of a septum that is flattened or bulging toward the left ventricle does not invalidate the use of this model and its formula. Statistical Analysis Volume estimates from the two methods were compared by using linear regression. Calculated estimates of parameters (slope and intercept) from different regressions were compared by using simultaneous confidence ellipses in parameter space.27 Comparison of calculated slopes with unity was done with a t test.27 Values of r were compared by using Fisher’s z transform to normality.27 Volume estimates were also compared by using analysis of residuals (differences of points from regression line)27 and analysis of agreement (plots of difference versus mean).28 Square root, logarithmic, and inverse transformations were performed to determine an appropriate method for elimination of dependence of variance on the volume.27,29
RESULTS Right ventricular volume calculated with the difference-of-ellipsoids formula correlated well with the magnetic resonance, Simpson’s rule– derived volumes (Figure 5 and Table 2). This correlation was equally good for separated end-diastolic, end-systolic, and stroke volumes (r2 values not significantly different) with half the sample size of the full set. End-diastolic and end-systolic slopes were not significantly different from each other at the 95% confidence level. Confidence ellipses of the end-diastolic
Figure 1 A, Section through base of ventricles with superimposed shape of base of proposed ellipsoidal shell model. Note that model is composed of 2 nonsymmetric sections that meet at line B-C. AV, Aortic valve; MV, mitral valve; PV, pulmonary valve; RVOT, right ventricular outflow tract; TV, tricuspid valve. B, Mid ventricular section through heart with superimposed cross section of proposed ellipsoidal model. LV, Left ventricle; RV, right ventricle. C, Four-chamber (long-axis) section through heart with superimposed cross section of proposed ellipsoidal model. LA, Left atrium; LV, left ventricle; RA, right atrium; RV, right ventricle.
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and end-systolic regression lines in parameter space (slope 3 intercept) showed no difference between these lines at the 95% level. The regression slope from the combined end-diastolic and end-systolic volumes was significantly below unity, as was the slope from the end-systolic volumes alone (P , .05). This would suggest that volumes from this method systematically overestimate the highly accurate magnetic resonance imaging volumes and that echocardiographic volumes would need to be corrected by a factor of .89, the slope of the regression line. The slopes from the stroke volumes and the end-diastolic volumes were not significantly different from unity. Because of the small range of ejection fraction values in the data from this study, the calculated 95% confidence intervals for the regression slope (2.42 to .96) and correlation coefficient (2.37 to .76) were extremely wide. Consequently, quantitative inferences concerning either accuracy or inaccuracy of ejection fraction estimates from the model used in this study could not be made with this data set.27 More data points would have to be obtained for any conclusion to be made. Analysis of residuals from regressions and analysis of agreement (Figure 6) both demonstrated a size dependence of the scatter, with increasing variance as volume increased. Rather than eliminating this dependence, both logarithmic and inverse transformations resulted in a decreasing variance at larger volumes. In contrast, plots of square-root– transformed data did not display any clear dependence of variance on volume (not shown). Accordingly, a square-root transformation was used to generate estimates of confidence intervals on the basis of 95% confidence intervals in the transformed domain (Figure 6). The confidence limits were almost identical between enddiastolic and end-systolic results. In both types of analysis, these lines appeared to somewhat overestimate the true magnitude of the confidence intervals. On the basis of the appearance of the plots, we estimated that the deviation of values of echo measurements from magnetic resonance imaging measurements is in the range of 20% to 30% of the mean volume.
DISCUSSION Search for Simple Model and Formula Over a span of 2 decades, the difference-of-ellipsoids model has been used by different investigators to estimate right ventricular volume.10,17,21,22 The lack of association of the difference-of-ellipsoids shape
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Figure 2 Echocardiographic image in subcostal sagittal view. Solid line shows measurement that estimates distance DAD discussed in text and illustrated in Figure 1, A, and Figure 3. It is measured from right ventricular free wall adjacent to tricuspid valve to wall of outflow tract on opposite side of pulmonary valve. RV, Right ventricle; RA, right atrium; PA, pulmonary artery; AO, aorta.
with the simple, area-length formula has been a significant limitation for the use of this approach despite the repeated good correlations reported in the literature.6,7,10,17,26 Without a specific shape as a model, it is unclear how much and what types of variation in shape can logically be tolerated while maintaining acceptable accuracy. As stated by Levine et al,7 the model should feature “a flattened, crescentic shape” that would taper both from the apex and the pulmonary valve. Our report describes a completely appropriate shape. This shape is that of an ellipsoidal shell or difference-of-ellipsoids model. The geometric volume of this model is found by a simple formula that has a previously demonstrated echocardiographic efficacy 6,7,26 and has been further validated here. Validation of Ellipsoidal Shell Model This study demonstrates that for right ventricles with relatively normal shape, (1) this model and its volume formula can be used explicitly to determine appropriate measurements for volume determination, and (2) that these appropriate measurements can be obtained echocardiographically. The results of our analysis of residuals suggest that differences in volume of 20% should be detectable and that anything above 30% will be clearly detectable. The analysis of agreement is more difficult to interpret except to confirm that the echocardiography-based volumes tend to overestimate the magnetic resonance imaging– based volumes. This interpretive ambiguity is to be expected because analysis of agreement was not developed for
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Figure 5 Linear regression results comparing echocardiographically derived, right ventricular volumes with volumes based on multislice, magnetic resonance images. Open circles indicate end-diastolic volumes; X, end-systolic volumes. Figure 3 Three-dimensional schematic of proposed model of right ventricle wrapped around “bullet-shaped” left ventricle. LV, Left ventricle; RV, right ventricle; RVOT, right ventricular outflow tract.
Figure 4 Schematic showing area (shaded) and length (DAD) used to calculate volume of ellipsoidal shell. Diameter DAD corresponds to distance A-D in Figures 1 and 2.
situations comparing, as in this case, new method results with results from a highly accurate method.28 Considering the retrospective nature of this data, with measurements being taken from images obtained for other purposes, these results are quite promising. It may be reasonably suggested that prospective collection of data would reduce the error. This conclusion is strengthened by independent past results with lower scatter6,7,10 collected prospectively and without reference to the connection between the shape and formula discussed here (Table 3).
Figure 6 Results of analysis of residuals (A) and analysis of agreement (B) comparing echocardiographically derived, right ventricular volumes with volumes based on multislice, magnetic resonance images. Closed circles indicate end-diastolic volumes; X, end-systolic volumes. Dependence of scatter on ventricular volume is clearly evident in both analyses. Confidence intervals are result of backtransformation from square-root–transformed domain, which displayed no noticeable variation in extent of scatter.
Comparison With Other Studies The results from this study compare well with those of previous echocardiographic studies6,7,10,26 that
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Table 1 Patients included in study of echocardiographic versus magnetic resonance volume estimation Patient
Age
BSA (m2)
Diagnosos
1 2 3 4 5 6 7
3 days 6 days 4 months 1 year 3 years 5 years 14 years
0.23 0.24 0.28 0.50 0.67 0.78 1.51
Supraventricular tachycardia Aortic coarctation, restrictive VSD Total AV Septal defect (unrepaired) Aortic coarctation, VSD repair; follow-up Aortic coarctation Pacemaker, interrupted IVC Syncope
BSA, body surface area; AV, atrioventricular; VSD, ventricular septal defect; IVC, inferior vena cava.
Table 2 Summary of regressions of echo versus MR-derived quantities: Echocardiographic quantities are dependent variables (MR 5 b1[Echo] 1 b0) Quantity
Slope
SD Slope
Y-intercept
SD Intercept
r
SEE
n
End-diastolic 1 end-systolic volumes End-diastolic volumes End-systolic volumes Stroke volumes Ejection fractions
0.8884 0.9049 0.7654 0.9433 0.2728
0.0527 0.0746 0.0978 0.1098 0.3048
20.1342 0.5602 2.3871 1.7783 0.3975
3.7106 6.5811 4.5393 4.7577 0.1478
.9666 .9707 .9337 .9441 .2858
11.2018 13.7852 9.5138 5.4113 0.1095
22 11 11 11 11
MR, Magnetic resonance.
Table 3
Summary of echocardiographic right ventricular volume results from literature
Reference
Subject
Modalities
Levine et al7 Gibson et al6 Tomita et al10 Helbing et al26
Human casts Human casts Human Human
Echo vs H2O displacement Echo vs angio Echo vs radio Echo vs MR
Formula
Regression equation
r
SEE
n
⁄ (Area)Length ⁄ (Area)Length Difference of volumes 2⁄3(Area)Length
Vecho 5 1.1VH2O 2 3.9 Vecho 5 1.0Vangio 1 4.6 Vecho 5 0.8Vradio 1 13.3 EDVecho 5 0.78VMR 1 21.5 ESVecho 5 0.64VMR 1 7.6
.95 .96 .935 .86 .82
5.2 6.8 15% ... ...
12 12 82 17 17
23 23
angio, Angiography; radio, radionuclide studies; MR, magnetic resonance; EDV, end-diastolic volume; ESV, end-systolic volume.
used either the 2⁄3(Area)(Length) formula or its equivalent (Table 3). The study of Helbing et al26 is particularly significant in that it compares results from several echocardiographic methods. The results from the 2⁄3(Area)(Length) method, which they describe as a “modified biplane pyramidal approximation,” gave the best correlation to the “gold standard” magnetic resonance– derived volumes. Their results show an identical regression slope to that obtained in our study (.89), although they found a greater y-intercept. This greater intercept may be caused by the lack of very small hearts in their sample resulting from the exclusion of children younger than 5 years. Interestingly, they found a good correlation for ejection fractions, an aspect not ascertainable from the present data. These authors cite limitations caused by the problems in obtaining a subcostal outflow (sagittal) view in all patients, a problem encountered with the adult volunteers in the present study. On the basis of the model and study described in this report, it could be speculated that this difficulty could be overcome
by (1) interchanging the subcostal view with a parasternal view and (2) reassessment of the quality of images in light of the fact that a distance (rather than an area) is required between landmarks that may be visible when the full outflow tract is still obscure. Appropriateness and Flexibility of Model As can be seen in Figures 1, 3, and 4, this ellipsoidal shell model looks like the right ventricle. The model has a concave septal surface and a planar tricuspid valve surface. Because the shell can be irregular without altering the mathematics (see Appendix), the outflow tract can be approximated by assuming one “horn” (octant) of the shell to be larger or more elongated than the other. It explicitly and implicitly takes into account the difficulties presented by the crescentic shape and separate infundibulum of the right ventricle. Mathematically, the 2⁄3(Area)(Length) formula applies to both concave and convex shapes as long as the right ventricle does not form the apex of the heart
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Figure 7 A, Illustration of fractions of ellipsoid for which volume can be determined by 2⁄3(Area)(Length) formula. B, Illustration of irregular composite ellipsoid for which volume equals 2⁄3(Area)(Length).
(see Appendix).18-20 This is the reason that a patient with atrioventricular septal defect can be reasonably included in this study. This defect often results in a flattening of the septum with equal pressures on the left and right, and the lumen shape becomes a simple quarter-ellipsoid. Use of the 2⁄3(Area)(Length) formula is in no way altered by this geometry; the measurements to be made are the same as for a concave-shaped right ventricle.
It might be speculated that the model that we used could be applied to more severely distorted right ventricles than those included in this study. Hypoplastic right ventricles might be modeled as shells wrapped around only a portion of the left ventricle for which the simple area-times-length formula could still be used with appropriately chosen dimensions. For greatly enlarged right ventricles, the two ventricles no longer share two common diameters and the
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formula for volume would revert to the more general difference-of-ellipsoids formula10,17,21,22: Volume 5 2⁄3Area (L ) 2 2⁄3Area (L ). The first term in this 1 1 2 2 equation contains measurements from the full biventricular unit minus the right ventricular free wall. The second term is the left ventricle plus myocardium. Ventricles in complex congenital malformations with inversion, dextrocardia, and superior-inferior relations could still be conceptualized as convex or concave ellipsoids. Thus the present framework for application of the 2⁄3(Area)(Length) formula remains a good candidate for estimation of volumes. Clearly, separate studies of these ventricular distortions would be necessary to demonstrate that application of this model could produce volumes of acceptable accuracy. Limitations As for any estimate based on echocardiographic data, the accuracy of volume estimates by the method used in this study depends on the quality of the images obtained and on the accuracy with which imaging planes cut through the precise anatomy required. Foreshortening is always a potential limitation, as is the existence of individuals with poor imaging windows. The use of 6 measurements in this study was intended to counteract the underestimation that results from foreshortening. The level of error indicated by analysis of residuals and analysis of agreement may be as high as 30%, although a value of 10% to 20% is suggested by the studies of others.6,7,10,27 While these values may seem high, they are comparable to what is obtained in routine clinical echocardiographic measurements. They are also sufficient to permit detection of the large changes in volume associated with many right ventricular abnormalities. This study included a relatively small number of individuals, unavoidably lowering the precision of analysis. This limitation is partly offset by the large amount of previous data obtained by others who were using either the ellipsoidal shell model or the 2⁄3(Area)(Length) formula.6,7,10,21,22,26 The population of patients included in the present study was heterogeneous in pathogenesis and age. None of the pathogeneses would be expected to cause a right ventricular shape outside of the shapes described by the formula used, as explained above. Additionally, the shapes and volumes were measured by magnetic resonance imaging, a method that makes no geometric assumptions and is generally agreed to be the most accurate. If any of the right ventricles were geometrically inappropriate for the use of the 2⁄3(Area)(Length) formula, the calculated, echocar-
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diography-based volumes should have deviated from the magnetic resonance volumes. For the clustered ejection fraction values, the precision was too low for any conclusion about regression slopes or intercepts to be made from the present data. However, the standard error of the data from this study (11%) does not show a statistically significant difference from that found in the study of Hiraishi et al12 (7%). The latter study, using a different echocardiographic method from the current study, examined twice as many subjects.22 When Helbing et al26 used the 2⁄3(Area)(Length) formula with a population that had a larger spread in ejection fractions, the standard error was only 4%. Thus both the comparison of the current ejection fraction results with those of independent studies and the accuracy of current absolute volume estimates provide support for usefulness of the model used in this study for ejection fraction determination.
CONCLUSIONS We have established that V52⁄3(Area)(Length), the previously validated, empirical formula for right ventricular volume, has a mathematical basis in a straightforward geometric model, the ellipsoidal shell. This crescentic model affords the opportunity for more knowledgeable utilization of this formula in echocardiographic monitoring of right ventricular size. Additionally, the current study, in combination with the cited previous studies, demonstrates that echocardiographic right ventricular volume estimation is possible and reliable. Although there are unavoidable limitations inherent in echocardiographic measurements, this modality offers portability and economy not found with the “gold standard” of magnetic resonance imaging. Appendix Mathematically, the volume of any regular or irregular ellipsoid is Volume 5
4 p~radius 1 ! ~radius 2 ! ~radius 3 ! 3
which can be easily manipulated algebraically to give Volume 5
2 ~Cross-sectional Area! 3 3 ~Perpendicular Diameter!
The volume of a half-ellipsoid, or bullet (Figure 7, A), will be half that of the original volume:
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Volume 5
~
2 Cross-sectional Area 3 2
!
REFERENCES
3 ~Perpendicular Diameter!
where the areas and lengths are those of the full ellipsoid. When the half-ellipsoid (sliced through its 2 smaller axes) is the model, it makes sense to simply use the existing “bullet” long-axis area in the formula: Volume 5
2 ~New Cross-section Area! 3 3 ~Perpendicular Diameter!
The appropriate diameter for the formula is measured across the planar surface of the half-ellipsoid and the area is the cross section of the shape along its long axis. The formula remains in the same form. Splitting this bullet along a second orthogonal plane halves the volume again, and again the 2⁄3(Area)(Diameter) form is valid for calculation when the areas and lengths are those of the quarter-ellipsoid. Similar reasoning applies to 8th-ellipsoid shapes as shown in Figure 7, A. An “irregular, composite ellipsoid” with 6 different radii can be constructed from octants that match along their faces as shown in Figure 7, B. The volume of this solid is the sum of the volumes of the octants. Because the radii match, this sum can be reduced algebraically to the 2⁄3(Area)(Diameter) form. Thus this formula does not require that the estimated volume be an ellipsoid of revolution or even that a cross section be a full ellipse. It only requires that each octant of the composite shape be an octant of a regular or irregular true ellipsoid. In this study, the right ventricular lumen is modeled as a quarter-ellipsoid (possibly composed of differing octants) from which a smaller quarter-ellipsoid has been subtracted (Figure 4). The smaller quarterellipsoid shares diameter DAD with the larger. The (long-axis) cross-sectional area of the quarter ellipsoid is A4ch, the lumen area in the 4-chamber view (shaded in figure), plus a remaining area, A. Volume 5
@
2 2 ~A 4ch 1 A!~D AD! 2 ~A!DAD 3 3
#
5
2 ~A4ch!DAD 3
Because of the shared diameter, DAD, the algebraic difference of 2 quarter-ellipsoids simplifies to 2⁄3(Area) (Length), the simple form that we seek.
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Right ventricular volume estimation using an ellipsoidal shell model and single-plane magnetic resonance imaging. Invest Radiol 1996;31:17-25. 21. Feneley MP, Elbeery JR, Gaynor JW, Gall SA, Davis JW, Rankin JS. Ellipsoidal shell subtraction model of right ventricular volume: comparison with regional free wall dimensions as indexes of right ventricular function. Circ Res 1990;67:1427-36. 22. Karunanithi MK, Michniewicz J, Copeland SE, Feneley MP. Right ventricular preload recruitable stroke work, end-systolic pressure-volume, and dP/dtmax-end-diastolic volume relations compared as indexes of right ventricular contractile performance in conscious dogs. Circ Res 1992;70:1169-79. 23. Higgins CB, Silverman NH, Kersting-Sommerhoff BA, Schmidt K. Assessment of cardiac function with cine magnetic resonance imaging. In: Higgins CB, Silverman NH, KerstingSommerhoff BA, Schmidt K, editors. Congenital heart disease: Echocardiography and magnetic resonance imaging. New York, NY: Raven Press, 1990:335-43.
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24. Longmore DB, Underwood SR, Bland C, et al. Dimensional accuracy of magnetic resonance in studies of the heart. Lancet 1985;1:1360-2. 25. Sechtem U, Pflugfelder PW, Gould RG, Cassidy MM, Higgins CB. Measurement of right and left ventricular volumes in healthy individuals with cine MR imaging. Radiology 1987; 163:697-702. 26. Helbing WA, Bosch HG, Maliepaard C, et al. Comparison of echocardiographic methods with magnetic resonance imaging for assessment of right ventricular function in children. Am J Cardiol 1995;76:589-94. 27. Neter J, Wasserman W, Kutner MH. Applied linear regression models. Homewood, Ill: RD Irwin, 1983:147-54. 28. Bland JM, Altman DG. Statistical methods for assessing agreement between two methods of clinical measurement. Lancet 1986;2:307-10. 29. Box GEP, Cox DR. An analysis of transformations. J R Stat Soc B 1964;26:211-20.
Our objective was to establish a crescentic model of the right ventricle as the basis of a reported 2⁄3 (Area)(Length) empirical formula for volume. This formula has been investigated by others without cognizance of its connection to a clear geometric model. The particular model, an ellipsoidal shell or difference of ellipsoids, has been investigated by several groups by using different volume formulas. Accordingly, we obtained echocardiographic images in 2 orthogonal planes from 7 patients and 4 volunteers. Specified area and length measurements from these images were used to calculate
In the management of congenital heart disease, the importance of monitoring the size of the right ventricle is often equal to or greater than that of the size of the left.1-3 Because of the low cost, portability, and noninvasive nature of echocardiography, this modality has been the subject of continuing interest in efforts to find an easily interpretable, geometric model–based method for right ventricular volume estimation.4-12 Although concurrent studies have focused on multiplane-based (Simpson’s rule) methods,13-15 such methods are generally more time consuming than methods that are based on models and require high-quality images at multiple image planes. Additionally, multiplane methods provide little conceptual framework for analysis of shape and shape changes. Despite the inherent advantages, however, geometric model application has been characterized by limited success. In large part this is caused by the relatively complex crescentic shape of the right ventricle16 and the difficulty of finding a simple-tounderstand model that describes this shape.7 Studied models have included single ellipsoid,8,12 combinations of ellipsoids,9 and crescentic shapes.4,5,10,11 Two significant investigations of an apparently From South Carolina Children’s Heart Center, Medical University of South Carolina. Reprint Requests: Stewart Denslow, PhD, South Carolina Children’s Heart Center, Medical University of South Carolina, 171 Ashley Ave, Charleston, SC 29425-0680. Copyright © 1998 by the American Society of Echocardiography. 0894-7317/98 $5.00 1 0 27/1/91976
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right ventricular volumes. These volumes were compared with values determined through multislice, magnetic resonance imaging with summation of lumen areas, a widely accepted standard. Obtained high correlations compared favorably with those of previous investigators who used equivalent but less well understood methods. We conclude that the ellipsoidal shell model of the right ventricle provides a simple arealength formula for the determination of lumen volume with echocardiography. (J Am Soc Echocardiogr 1998; 11:864-73.)
model-based approach are those of Levine et al7 and Gibson et al.6 Although the results of these studies indicated a high accuracy resulting from a simple formula for volume (2⁄3 3 Area 3 Length), the investigators were unable to determine a precise geometric shape corresponding to this formula. The objective of the present communication is to establish a clear geometric model as the basis of the simple area-length formula used by Levine et al7 and Gibson et al.6 This geometric model is that of an ellipsoidal shell or difference of ellipsoids. It has the advantages of a flattened, crescentic shape that includes the infundibulum and a simple formula for volume. This model has been used successfully, in some cases with the use of different but mathematically equivalent formulas, in the estimation of right ventricular volume from data including (1) dual-plane angiographic images,17 (2) dual-plane echocardiographic images,10 (3) singleplane and dual-plane magnetic resonance images,18-20 and (4) sonomicrometry data.21,22 Accordingly, we undertook a study comparing (1) volumes calculated with the above area-length formula with measurements from echocardiographic images and (2) volumes determined with Simpson’s rule with measurements from multiplane magnetic resonance images, a generally accepted “gold standard.”23-26 METHODS Subjects A retrospective search of divisional records was conducted for patients who had undergone a magnetic resonance
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imaging study that included a multislice, gradient echo study of the ventricles within 2 weeks of an echocardiographic study. A total of 7 patients (5 female, 2 male) (Table 1) were found. Four healthy adult volunteers (1 female, 3 male) who underwent the same imaging studies were also included. Informed consent was obtained when applicable. All studies were approved by the Institutional Review Board of our institution. Magnetic Resonance Imaging All studies were performed with the use of a Gyroscan T5 system (Philips Medical Systems of North America, Shelton, Conn) operating at 0.5 T. The imaging protocol followed established methods.23,25 Initial spin-echo localizer views, taken in sagittal, transverse, and coronal planes, were used for planning a multislice, spin-echo scan at right angles to the cardiac axis, with slices more than covering the ventricles from apex to base. After this scan, gradient-echo, cinescans of 1 to 3 slices (number of slices in each gradientecho scan was dependent on the subject’s heart rate) were planned that coincided in position and thickness with previous spin-echo slices that had indeed intersected the ventricles. At least 12 cardiac phases were obtained in each image plane to ensure adequate time resolution for location of end-systole. Acquisition parameters were slice thickness of 5 to 10 mm, flip angle 40 degrees, echo time 9 ms, and 4 signal averages. Resolution was 128 3 256 pixels interpolated to 256 3 256 pixels. Repetition time (364 to 632) varied with heart rate (45 to 150 bpm). Total number of contiguous slices was from 4 to 11. Data acquisition was triggered by the R waves of the electrocardiogram. Magnetic Resonance Image Analysis With the use of image measurement software (Gyroview, N.V. Philips, Eindhoven, The Netherlands) on a graphics workstation, areas from magnetic resonance images were determined from interactive videographic displays with a mouse-controlled cursor. Window width was set to the full range of pixel intensities in the image. Level was set at one-half the maximum intensity value (generally close to half the window setting). Window and level were kept constant during all analysis. In end-systolic and end-diastolic frames, measurements of cross-sectional lumen areas were made at the endocardial surface in the right ventricle. Papillary muscles and the moderator band were not included in the lumen area. End-diastole was taken as the first frame in each sequence, 8 to 16 ms after the R-wave trigger. This was also the largest lumen area in each sequence. End-systole was taken as the frame closest to the closing of the aortic valve that could be seen in slices near the base of each set. This frame also contained an area equal to the smallest area in each sequence. Volumes were calculated as the sum of all lumen areas times the slice thickness. Echocardiographic Imaging Two-dimensional echocardiographic studies were performed with either an Accuson 128 xp/10 (Mountain View, Calif) or an Interspec ATL (Ambler, Pa) with 5, 3.5,
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or 2 MHz transducers. Images were acquired from the apical 4-chamber view and either a parasternal short-axis or a subcostal-sagittal view that included tricuspid (inflow) and outflow portions of the right ventricle. Echocardiographic Image Analysis Areas and distances were obtained from echocardiographic images with interactive measurement systems installed on each of the echocardiography instruments. In the 4-chamber view, lumen area (A4ch) was obtained at end-diastole (time of tricuspid valve closure) and end-systole (frame immediately before tricuspid valve opening) by tracing the endocardial borders. Papillary muscles and the moderator band were not included in the lumen area. In the parasternal short-axis or subcostal sagittal view, we estimated the length of the model diameter (Figure 1, a, distance A-D) as the distance from the right ventricular free wall immediately bordering the tricuspid valve to the wall of the outflow tract on the opposite side of the pulmonary valve (Figure 2). All echocardiographic measurements were repeated on 6 separate cardiac cycles. Because the desired area and length were the anatomic maxima of these measurements in the chosen orientation, the maximum values of each measurement within a set was the final recorded value. Note that this often resulted in end-diastolic and end-systolic values originating from separate cardiac cycles. This protocol was followed to offset errors resulting from the imaging plane not cutting through the true maximum cross-sectional area or length. Geometric Model for Right Ventricle Volumes were calculated18-20 as Vol 5 2⁄3(A4ch)(DAD). This formula is derived from a difference-of-ellipsoids model of the right ventricle. As we have reported previously,18 this formula is mathematically exact for a range of crescent-shaped, regular, and irregular volumes that are quite similar in shape to the right ventricle, including its inflow, apical, and outflow regions (see Appendix). Figure 1, a, shows a basal area of the model superimposed on the atrioventricular junction of the right ventricle. Note that the superimposed area is composed of 2 partial crescents that match at their bases but are not of the same height (distance AE is not equal to distance ED). This situation, which helps in matching the model to the right ventricular anatomy, leaves the formula for volume unchanged (see Appendix). Figure 1, b and c, shows the degree of match between the model and the mid ventricle short axis and with the long axis. Figure 3 shows a 3-dimensional rendering of the model of the left ventricles and right ventricles. Note that the crescentic right ventricular lumen is modeled as largely wrapping around the left ventricular septal surface. However, the outflow tract is included in the model by specifying that the diameter to be measured should stretch beyond the epicardial surface of the left ventricle to the outer edge of the right ventricular outflow. Figure 4 shows the difference-of-ellipsoids model standing alone with indications of the area and length used for volume calculation. Although there exist two other combinations of area and length that can be used for volume estimation, this combi-
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nation was found to be most accurate with magnetic resonance images and was accordingly used in this study.18-20 An important point to note is that the 2⁄3(Area)(Length) formula also applies to convex (that is, noncrescentic) ellipsoidal shapes (see Appendix). The area to be measured is the same as before, whereas the distance to be measured becomes the diameter of the ellipsoid in the same plane as the previous distance between the “horns” of the ellipsoidal shell.18 This fact results in the formula for volume being applicable to a range of shapes from a rounded “left ventricle” type of volume to the concave “right ventricle” type of volume. Thus the existence of a septum that is flattened or bulging toward the left ventricle does not invalidate the use of this model and its formula. Statistical Analysis Volume estimates from the two methods were compared by using linear regression. Calculated estimates of parameters (slope and intercept) from different regressions were compared by using simultaneous confidence ellipses in parameter space.27 Comparison of calculated slopes with unity was done with a t test.27 Values of r were compared by using Fisher’s z transform to normality.27 Volume estimates were also compared by using analysis of residuals (differences of points from regression line)27 and analysis of agreement (plots of difference versus mean).28 Square root, logarithmic, and inverse transformations were performed to determine an appropriate method for elimination of dependence of variance on the volume.27,29
RESULTS Right ventricular volume calculated with the difference-of-ellipsoids formula correlated well with the magnetic resonance, Simpson’s rule– derived volumes (Figure 5 and Table 2). This correlation was equally good for separated end-diastolic, end-systolic, and stroke volumes (r2 values not significantly different) with half the sample size of the full set. End-diastolic and end-systolic slopes were not significantly different from each other at the 95% confidence level. Confidence ellipses of the end-diastolic
Figure 1 A, Section through base of ventricles with superimposed shape of base of proposed ellipsoidal shell model. Note that model is composed of 2 nonsymmetric sections that meet at line B-C. AV, Aortic valve; MV, mitral valve; PV, pulmonary valve; RVOT, right ventricular outflow tract; TV, tricuspid valve. B, Mid ventricular section through heart with superimposed cross section of proposed ellipsoidal model. LV, Left ventricle; RV, right ventricle. C, Four-chamber (long-axis) section through heart with superimposed cross section of proposed ellipsoidal model. LA, Left atrium; LV, left ventricle; RA, right atrium; RV, right ventricle.
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and end-systolic regression lines in parameter space (slope 3 intercept) showed no difference between these lines at the 95% level. The regression slope from the combined end-diastolic and end-systolic volumes was significantly below unity, as was the slope from the end-systolic volumes alone (P , .05). This would suggest that volumes from this method systematically overestimate the highly accurate magnetic resonance imaging volumes and that echocardiographic volumes would need to be corrected by a factor of .89, the slope of the regression line. The slopes from the stroke volumes and the end-diastolic volumes were not significantly different from unity. Because of the small range of ejection fraction values in the data from this study, the calculated 95% confidence intervals for the regression slope (2.42 to .96) and correlation coefficient (2.37 to .76) were extremely wide. Consequently, quantitative inferences concerning either accuracy or inaccuracy of ejection fraction estimates from the model used in this study could not be made with this data set.27 More data points would have to be obtained for any conclusion to be made. Analysis of residuals from regressions and analysis of agreement (Figure 6) both demonstrated a size dependence of the scatter, with increasing variance as volume increased. Rather than eliminating this dependence, both logarithmic and inverse transformations resulted in a decreasing variance at larger volumes. In contrast, plots of square-root– transformed data did not display any clear dependence of variance on volume (not shown). Accordingly, a square-root transformation was used to generate estimates of confidence intervals on the basis of 95% confidence intervals in the transformed domain (Figure 6). The confidence limits were almost identical between enddiastolic and end-systolic results. In both types of analysis, these lines appeared to somewhat overestimate the true magnitude of the confidence intervals. On the basis of the appearance of the plots, we estimated that the deviation of values of echo measurements from magnetic resonance imaging measurements is in the range of 20% to 30% of the mean volume.
DISCUSSION Search for Simple Model and Formula Over a span of 2 decades, the difference-of-ellipsoids model has been used by different investigators to estimate right ventricular volume.10,17,21,22 The lack of association of the difference-of-ellipsoids shape
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Figure 2 Echocardiographic image in subcostal sagittal view. Solid line shows measurement that estimates distance DAD discussed in text and illustrated in Figure 1, A, and Figure 3. It is measured from right ventricular free wall adjacent to tricuspid valve to wall of outflow tract on opposite side of pulmonary valve. RV, Right ventricle; RA, right atrium; PA, pulmonary artery; AO, aorta.
with the simple, area-length formula has been a significant limitation for the use of this approach despite the repeated good correlations reported in the literature.6,7,10,17,26 Without a specific shape as a model, it is unclear how much and what types of variation in shape can logically be tolerated while maintaining acceptable accuracy. As stated by Levine et al,7 the model should feature “a flattened, crescentic shape” that would taper both from the apex and the pulmonary valve. Our report describes a completely appropriate shape. This shape is that of an ellipsoidal shell or difference-of-ellipsoids model. The geometric volume of this model is found by a simple formula that has a previously demonstrated echocardiographic efficacy 6,7,26 and has been further validated here. Validation of Ellipsoidal Shell Model This study demonstrates that for right ventricles with relatively normal shape, (1) this model and its volume formula can be used explicitly to determine appropriate measurements for volume determination, and (2) that these appropriate measurements can be obtained echocardiographically. The results of our analysis of residuals suggest that differences in volume of 20% should be detectable and that anything above 30% will be clearly detectable. The analysis of agreement is more difficult to interpret except to confirm that the echocardiography-based volumes tend to overestimate the magnetic resonance imaging– based volumes. This interpretive ambiguity is to be expected because analysis of agreement was not developed for
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Figure 5 Linear regression results comparing echocardiographically derived, right ventricular volumes with volumes based on multislice, magnetic resonance images. Open circles indicate end-diastolic volumes; X, end-systolic volumes. Figure 3 Three-dimensional schematic of proposed model of right ventricle wrapped around “bullet-shaped” left ventricle. LV, Left ventricle; RV, right ventricle; RVOT, right ventricular outflow tract.
Figure 4 Schematic showing area (shaded) and length (DAD) used to calculate volume of ellipsoidal shell. Diameter DAD corresponds to distance A-D in Figures 1 and 2.
situations comparing, as in this case, new method results with results from a highly accurate method.28 Considering the retrospective nature of this data, with measurements being taken from images obtained for other purposes, these results are quite promising. It may be reasonably suggested that prospective collection of data would reduce the error. This conclusion is strengthened by independent past results with lower scatter6,7,10 collected prospectively and without reference to the connection between the shape and formula discussed here (Table 3).
Figure 6 Results of analysis of residuals (A) and analysis of agreement (B) comparing echocardiographically derived, right ventricular volumes with volumes based on multislice, magnetic resonance images. Closed circles indicate end-diastolic volumes; X, end-systolic volumes. Dependence of scatter on ventricular volume is clearly evident in both analyses. Confidence intervals are result of backtransformation from square-root–transformed domain, which displayed no noticeable variation in extent of scatter.
Comparison With Other Studies The results from this study compare well with those of previous echocardiographic studies6,7,10,26 that
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Table 1 Patients included in study of echocardiographic versus magnetic resonance volume estimation Patient
Age
BSA (m2)
Diagnosos
1 2 3 4 5 6 7
3 days 6 days 4 months 1 year 3 years 5 years 14 years
0.23 0.24 0.28 0.50 0.67 0.78 1.51
Supraventricular tachycardia Aortic coarctation, restrictive VSD Total AV Septal defect (unrepaired) Aortic coarctation, VSD repair; follow-up Aortic coarctation Pacemaker, interrupted IVC Syncope
BSA, body surface area; AV, atrioventricular; VSD, ventricular septal defect; IVC, inferior vena cava.
Table 2 Summary of regressions of echo versus MR-derived quantities: Echocardiographic quantities are dependent variables (MR 5 b1[Echo] 1 b0) Quantity
Slope
SD Slope
Y-intercept
SD Intercept
r
SEE
n
End-diastolic 1 end-systolic volumes End-diastolic volumes End-systolic volumes Stroke volumes Ejection fractions
0.8884 0.9049 0.7654 0.9433 0.2728
0.0527 0.0746 0.0978 0.1098 0.3048
20.1342 0.5602 2.3871 1.7783 0.3975
3.7106 6.5811 4.5393 4.7577 0.1478
.9666 .9707 .9337 .9441 .2858
11.2018 13.7852 9.5138 5.4113 0.1095
22 11 11 11 11
MR, Magnetic resonance.
Table 3
Summary of echocardiographic right ventricular volume results from literature
Reference
Subject
Modalities
Levine et al7 Gibson et al6 Tomita et al10 Helbing et al26
Human casts Human casts Human Human
Echo vs H2O displacement Echo vs angio Echo vs radio Echo vs MR
Formula
Regression equation
r
SEE
n
⁄ (Area)Length ⁄ (Area)Length Difference of volumes 2⁄3(Area)Length
Vecho 5 1.1VH2O 2 3.9 Vecho 5 1.0Vangio 1 4.6 Vecho 5 0.8Vradio 1 13.3 EDVecho 5 0.78VMR 1 21.5 ESVecho 5 0.64VMR 1 7.6
.95 .96 .935 .86 .82
5.2 6.8 15% ... ...
12 12 82 17 17
23 23
angio, Angiography; radio, radionuclide studies; MR, magnetic resonance; EDV, end-diastolic volume; ESV, end-systolic volume.
used either the 2⁄3(Area)(Length) formula or its equivalent (Table 3). The study of Helbing et al26 is particularly significant in that it compares results from several echocardiographic methods. The results from the 2⁄3(Area)(Length) method, which they describe as a “modified biplane pyramidal approximation,” gave the best correlation to the “gold standard” magnetic resonance– derived volumes. Their results show an identical regression slope to that obtained in our study (.89), although they found a greater y-intercept. This greater intercept may be caused by the lack of very small hearts in their sample resulting from the exclusion of children younger than 5 years. Interestingly, they found a good correlation for ejection fractions, an aspect not ascertainable from the present data. These authors cite limitations caused by the problems in obtaining a subcostal outflow (sagittal) view in all patients, a problem encountered with the adult volunteers in the present study. On the basis of the model and study described in this report, it could be speculated that this difficulty could be overcome
by (1) interchanging the subcostal view with a parasternal view and (2) reassessment of the quality of images in light of the fact that a distance (rather than an area) is required between landmarks that may be visible when the full outflow tract is still obscure. Appropriateness and Flexibility of Model As can be seen in Figures 1, 3, and 4, this ellipsoidal shell model looks like the right ventricle. The model has a concave septal surface and a planar tricuspid valve surface. Because the shell can be irregular without altering the mathematics (see Appendix), the outflow tract can be approximated by assuming one “horn” (octant) of the shell to be larger or more elongated than the other. It explicitly and implicitly takes into account the difficulties presented by the crescentic shape and separate infundibulum of the right ventricle. Mathematically, the 2⁄3(Area)(Length) formula applies to both concave and convex shapes as long as the right ventricle does not form the apex of the heart
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Figure 7 A, Illustration of fractions of ellipsoid for which volume can be determined by 2⁄3(Area)(Length) formula. B, Illustration of irregular composite ellipsoid for which volume equals 2⁄3(Area)(Length).
(see Appendix).18-20 This is the reason that a patient with atrioventricular septal defect can be reasonably included in this study. This defect often results in a flattening of the septum with equal pressures on the left and right, and the lumen shape becomes a simple quarter-ellipsoid. Use of the 2⁄3(Area)(Length) formula is in no way altered by this geometry; the measurements to be made are the same as for a concave-shaped right ventricle.
It might be speculated that the model that we used could be applied to more severely distorted right ventricles than those included in this study. Hypoplastic right ventricles might be modeled as shells wrapped around only a portion of the left ventricle for which the simple area-times-length formula could still be used with appropriately chosen dimensions. For greatly enlarged right ventricles, the two ventricles no longer share two common diameters and the
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formula for volume would revert to the more general difference-of-ellipsoids formula10,17,21,22: Volume 5 2⁄3Area (L ) 2 2⁄3Area (L ). The first term in this 1 1 2 2 equation contains measurements from the full biventricular unit minus the right ventricular free wall. The second term is the left ventricle plus myocardium. Ventricles in complex congenital malformations with inversion, dextrocardia, and superior-inferior relations could still be conceptualized as convex or concave ellipsoids. Thus the present framework for application of the 2⁄3(Area)(Length) formula remains a good candidate for estimation of volumes. Clearly, separate studies of these ventricular distortions would be necessary to demonstrate that application of this model could produce volumes of acceptable accuracy. Limitations As for any estimate based on echocardiographic data, the accuracy of volume estimates by the method used in this study depends on the quality of the images obtained and on the accuracy with which imaging planes cut through the precise anatomy required. Foreshortening is always a potential limitation, as is the existence of individuals with poor imaging windows. The use of 6 measurements in this study was intended to counteract the underestimation that results from foreshortening. The level of error indicated by analysis of residuals and analysis of agreement may be as high as 30%, although a value of 10% to 20% is suggested by the studies of others.6,7,10,27 While these values may seem high, they are comparable to what is obtained in routine clinical echocardiographic measurements. They are also sufficient to permit detection of the large changes in volume associated with many right ventricular abnormalities. This study included a relatively small number of individuals, unavoidably lowering the precision of analysis. This limitation is partly offset by the large amount of previous data obtained by others who were using either the ellipsoidal shell model or the 2⁄3(Area)(Length) formula.6,7,10,21,22,26 The population of patients included in the present study was heterogeneous in pathogenesis and age. None of the pathogeneses would be expected to cause a right ventricular shape outside of the shapes described by the formula used, as explained above. Additionally, the shapes and volumes were measured by magnetic resonance imaging, a method that makes no geometric assumptions and is generally agreed to be the most accurate. If any of the right ventricles were geometrically inappropriate for the use of the 2⁄3(Area)(Length) formula, the calculated, echocar-
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diography-based volumes should have deviated from the magnetic resonance volumes. For the clustered ejection fraction values, the precision was too low for any conclusion about regression slopes or intercepts to be made from the present data. However, the standard error of the data from this study (11%) does not show a statistically significant difference from that found in the study of Hiraishi et al12 (7%). The latter study, using a different echocardiographic method from the current study, examined twice as many subjects.22 When Helbing et al26 used the 2⁄3(Area)(Length) formula with a population that had a larger spread in ejection fractions, the standard error was only 4%. Thus both the comparison of the current ejection fraction results with those of independent studies and the accuracy of current absolute volume estimates provide support for usefulness of the model used in this study for ejection fraction determination.
CONCLUSIONS We have established that V52⁄3(Area)(Length), the previously validated, empirical formula for right ventricular volume, has a mathematical basis in a straightforward geometric model, the ellipsoidal shell. This crescentic model affords the opportunity for more knowledgeable utilization of this formula in echocardiographic monitoring of right ventricular size. Additionally, the current study, in combination with the cited previous studies, demonstrates that echocardiographic right ventricular volume estimation is possible and reliable. Although there are unavoidable limitations inherent in echocardiographic measurements, this modality offers portability and economy not found with the “gold standard” of magnetic resonance imaging. Appendix Mathematically, the volume of any regular or irregular ellipsoid is Volume 5
4 p~radius 1 ! ~radius 2 ! ~radius 3 ! 3
which can be easily manipulated algebraically to give Volume 5
2 ~Cross-sectional Area! 3 3 ~Perpendicular Diameter!
The volume of a half-ellipsoid, or bullet (Figure 7, A), will be half that of the original volume:
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Volume 5
~
2 Cross-sectional Area 3 2
!
REFERENCES
3 ~Perpendicular Diameter!
where the areas and lengths are those of the full ellipsoid. When the half-ellipsoid (sliced through its 2 smaller axes) is the model, it makes sense to simply use the existing “bullet” long-axis area in the formula: Volume 5
2 ~New Cross-section Area! 3 3 ~Perpendicular Diameter!
The appropriate diameter for the formula is measured across the planar surface of the half-ellipsoid and the area is the cross section of the shape along its long axis. The formula remains in the same form. Splitting this bullet along a second orthogonal plane halves the volume again, and again the 2⁄3(Area)(Diameter) form is valid for calculation when the areas and lengths are those of the quarter-ellipsoid. Similar reasoning applies to 8th-ellipsoid shapes as shown in Figure 7, A. An “irregular, composite ellipsoid” with 6 different radii can be constructed from octants that match along their faces as shown in Figure 7, B. The volume of this solid is the sum of the volumes of the octants. Because the radii match, this sum can be reduced algebraically to the 2⁄3(Area)(Diameter) form. Thus this formula does not require that the estimated volume be an ellipsoid of revolution or even that a cross section be a full ellipse. It only requires that each octant of the composite shape be an octant of a regular or irregular true ellipsoid. In this study, the right ventricular lumen is modeled as a quarter-ellipsoid (possibly composed of differing octants) from which a smaller quarter-ellipsoid has been subtracted (Figure 4). The smaller quarterellipsoid shares diameter DAD with the larger. The (long-axis) cross-sectional area of the quarter ellipsoid is A4ch, the lumen area in the 4-chamber view (shaded in figure), plus a remaining area, A. Volume 5
@
2 2 ~A 4ch 1 A!~D AD! 2 ~A!DAD 3 3
#
5
2 ~A4ch!DAD 3
Because of the shared diameter, DAD, the algebraic difference of 2 quarter-ellipsoids simplifies to 2⁄3(Area) (Length), the simple form that we seek.
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