## Background

Left ventricular (LV) volumes and ejection fraction derived from two-dimensional echocardiography are two measures of adverse LV remodeling, which predict survival in patients with systolic heart failure. However, the geometric assumptions and image foreshortening that can occur with two-dimensional echocardiography reduces measurement accuracy and thus predictive value. By its nature, three-dimensional (3D) echocardiography allows the entire LV shape to be studied, providing a methodology to examine LV remodeling through LV curvature on a global and regional scale. The aim of this study was to correlate changes in global and regional LV shape to LV ejection fraction.

## Methods

Full-volume, 3D transthoracic echocardiographic studies of the left ventricle were performed in 106 consecutive patients with either normal left ventricles ( *n *= 59) or cardiomyopathies ( *n *= 47). Customized software (QLAB) was used to extract segmented 3D LV endocardial shells at end-systole and end-diastole and to analyze these shells to determine global and regional LV shape analysis. Independent *t *tests were used for intergroup comparisons, and linear regression was used to correlate regional shape changes with systolic performance.

## Results

Derivation and analysis of the 3D LV shells was possible in all patients. Patients with dilated cardiomyopathy had significantly smaller curvature values, indicating rounder global LV shape throughout the cardiac cycle. Regional analysis identified a loss of septal and apical curvatures in these patients. Systolic apical mean curvature was well correlated with LV ejection fraction ( *r *= 0.89).

## Conclusions

This is the first study to demonstrate that regional remodeling measured by regional 3D LV curvature correlates well with LV function. As well, this methodology is independent of the geometric assumptions that limit the predictive value of two-dimensional echocardiographic measures of LV remodeling. Overall, this is a novel tool that may have applications in the assessment and prediction of outcomes of different forms of dilated cardiomyopathy.

After presenting with myocardial infarction or heart failure, patients’ outcomes are directly related to the adverse left ventricular (LV) remodeling process. In fact, changes in LV volume and LV ejection fraction (LVEF) directly affect patient survival. New pharmacologic therapies, resynchronization devices, and remodeling surgery are being developed with the aim of slowing, arresting, or reversing this adverse remodeling process. Common clinical practice still uses two-dimensional (2D) echocardiographic methods to assess LV chamber size and volumes to assess the success or failure of these therapies. In addition, 2D strain measurements have been shown to be predictive of outcomes in heart failure.

Unfortunately, image foreshortening of the left ventricle and geometric assumptions regarding the shape of the left ventricle confound many 2D calculations, while limiting reproducibility and predictive power. Also, currently used 2D volume computation methodologies compare unfavorably with newer three-dimensional (3D) imaging–based methods used to assess global LV geometry such as 3D sphericity. Sphericity is computed using the LV long-axis length as the diameter of a sphere whose volume is compared with actual LV volume. The sphericity computation depends heavily on determining the true LV volume and on a nonforeshortened, straight LV long-axis cavity length. Ultimately, sphericity does not isolate regional remodeling in a segmental way. Sphericity has been shown to correlate highly with LVEF and prognosis.

Three-dimensional echocardiography has been shown to provide more accurate and reproducible measurements of LV volume compared with conventional 2D methods. This is because an LV endocardial cast or mesh of the endocardial tissue–blood interface can be easily created from 3D echocardiographic images without the confounding effects of geometric assumptions or image foreshortening. With the use of curvature analysis, this LV endocardial cast can be used to examine LV remodeling through LV shape analysis on a global and regional scale. LV curvature analysis has been previously used in LV ventriculography, but these analyses represent a projection of the left ventricle, not its true 3D geometry.

In this report, we describe a 3D methodology based on differential geometry that can automatically display and quantify 3D global and regional changes in ventricular geometry. We tested the hypothesis that specific regional LV shape indices measured from transthoracic real-time 3D data correlate with LV function.

## Methods

## Study Population

This was a cross-sectional, observational study of patients scheduled to undergo clinically indicated transthoracic echocardiography. We studied 106 patients: 59 with normal left ventricles (33 men) and 47 with cardiomyopathy (CM; 24 men). We recruited the normal controls from consecutive patients presenting for outpatient echocardiography with normal LV size and function and no evidence of valvular heart disease. The 47 patients with CM were consecutive patients with LVEFs < 45% presenting for transthoracic echocardiography. Baseline characteristics are presented in Table 1 . The University of Chicago Institutional Review Board approved this study.

Variable | Controls ( n = 59) |
Patients with cardiomyopathy ( n = 47) |
---|---|---|

Men | 33 (56%) | 24 (51%) |

Age (y) | 44 ± 14 | 57 ± 17 |

Body surface area (m ^{2 }) |
1.9 ± 0.2 | 1.8 ± 0.2 |

LVEF (%) | 61 ± 6 | 21 ± 7 |

Heart failure etiology | ||

Ischemic | 24 | |

Idiopathic | 13 | |

Peripartum | 3 | |

Alcohol | 2 | |

Cocaine | 1 | |

Adriamycin | 1 | |

Familial | 1 | |

LV noncompaction | 1 | |

Amyloid | 1 |

## Three-Dimensional Echocardiography

Echocardiography was performed using an iE33 (Philips Medical Systems, Andover, MA) echocardiographic system with a transthoracic x3 transducer. Apical views were optimized to acquire the entire left ventricle within the scanning volume of acquisition. Initially, gain settings were adjusted using the narrow-angle acquisition mode, which allows imaging of a pyramidal volume of approximately 30° × 60° without the need for electrocardiographic gating. Subsequently, four electrocardiographically gated subvolumes were combined to form a full-volume data set. Data were stored digitally and exported offline in Digital Imaging and Communications in Medicine format for later analysis.

## Three-Dimensional LV Shape Analysis

The following methodology was used for each consecutive frame in the cardiac cycle to obtain dynamic analyses of LV shape for the entire cardiac cycle. Offline 3D quantitative analysis was performed in two stages: the first for 3D LV segmentation and the second for 3D LV shape analysis. LV segmentation was performed using customized QLAB 3DQ Advanced (Philips Medical Systems) as follows ( Figure 1 ). First, the user aligned the appropriate multiplanar view to maximize the LV cavity long and short axes. Next, four mitral annular and one apical point were placed on the left ventricle as landmarks. An arrow oriented toward the mid interventricular septum was used to align the 17-segment American Heart Association model. The 17-segment model was used because the LV apex plays an important role in LV remodeling and hence in shape changes. This manual initialization was performed on both end-diastolic and end-systolic frames in two orthogonal LV apical views. End-diastole was defined as the frame before aortic valve opening, and end-systole was defined as the frame before mitral valve opening. The LV endocardial cavity was segmented using an automated algorithm (QLAB) that consists of deformable modeling devoid of geometric assumptions. This resulted in an endocardial shell or mesh that accommodated the LV shape completely around the cavity. Finally, an experienced echocardiographer assessed the accuracy of the border placement, and the segmented LV endocardial meshes (with the American Heart Association 17-segment classification for the entire cardiac cycle) were exported using custom software for the second stage of the data analysis.

Novel custom software was used to analyze regional curvature. Curvature is the amount by which an object deviates from being flat. Numerically, curvature is the reciprocal of the radius of a circle that tangentially fits the curved region in question ( Figures 2 and 3 ). Thus, a large radius represents a small curvature such that the measured curved region is not extremely rounded. A small radius represents a larger curvature and so a very round structure. Thus, curvature allows the assessment of regions of the left ventricle, and the data can also be combined to derive a global assessment. This software was first validated using two types of computer-generated 3D phantoms. One phantom was a sphere and the second an ellipsoid. Using these two phantoms, 3D meshes of the sphere and ellipsoid were generated using the custom software. The regional curvature values calculated from the meshes were correlated with the true values. A least squares approach to fitting curvatures was used. To perform regional 3D shape analysis of the LV mesh on each mesh, 840 equally distributed nodes were allocated to cover the entire LV endocardial surface. Then, the LV mesh was treated as a curved 2D surface so that each node would have two principal curvatures, a minimum (cm ^{−1 }) and a maximum (cm ^{−1 }), which are always orthogonal to each other. Figure 2 displays the nature of principal curvatures. Figure 3 displays the concept of orienting a curvature to a specific myocardial location and orientation. In addition to the principal curvatures, at each node, we also computed the arithmetic mean (cm ^{−1 }), Gaussian (cm ^{−2 }), longitudinal (cm ^{−1 }), circumferential/short-axis (cm ^{−1 }), and difference (cm ^{−1 }) curvatures. The arithmetic mean curvature is the average of the minimum and maximum curvatures. The Gaussian curvature is the product of the minimum and maximum curvatures (see Appendix ). The difference curvature is a new curvature, which is calculated as the difference between the maximum and minimum curvatures. This curvature tends to zero for a sphere or a plane (i.e. when the two principal curvatures equal each other) but is nonzero for local “asymmetric” curvature. This parameter was created to emphasize nonspherical, nonplanar features. The remaining six curvature measurements (principal, mean, Gaussian, longitudinal, and circumferential) were chosen to characterize basic LV curvature changes. Each type of curvature was also averaged for all nodes within the total left ventricle and for all nodes within each segment of the 17-segment LV model. Parametric images as well as segmentation are depicted in Figure 4 . Because each LV segment contains multiple nodes, the mean and standard deviation for the distribution of curvatures for each of the regions were ultimately computed as shown in Figure 5 . Lastly, from the 3D LV mesh, we were also able to obtain global 3D sphericity and 3D-derived surface area. Because of the numerous parameters measured, from preliminary study results, we decided to focus this study on the differences noted in the apical and septal curvatures. Once these 3D shape parameters were calculated for the cardiac cycle, we correlated the end-diastolic and end-systolic values with ventricular size and function. LVEF was obtained from the 3D left ventricle.

## Statistical Analysis

All statistical pairwise comparisons were performed using independent *t *tests for groups. Comparisons of curvature parameters for both groups were done using *t *tests for independent groups with Bonferroni’s correction to adjust for multiple comparisons. *P *values < .0001 were considered significant. Linear regression was performed between curvature parameters and LVEF. Plots between groups were made using the center as the mean and the box bounded by 25th and 75th quartiles. All statistical analysis was done using Statistica version 10 (StatSoft, Tulsa, OK).

## Results

Three-dimensional echocardiographic data acquisition and subsequent data analysis were feasible in all patients.

## Global Remodeling Shape Analysis

Compared with normal subjects, global analysis demonstrated significant ( *P *< .0001) shape changes in patients with CM when quantified during both systole and diastole using 3D sphericity, 3D-derived surface area, and Gaussian, mean, maximum, minimum, circumferential, and difference curvatures ( Table 2 ).

Parameter | Diastole | Systole | ||||
---|---|---|---|---|---|---|

Normal | CM | P |
Normal | CM | P | |

Global shape parameters | ||||||

Sphericity | 28% | 38% | <.0001 | 19% | 33% | <.0001 |

Surface area (cm ^{2 }) |
94.5 | 161.5 | <.0001 | 52.3 | 133.2 | <.0001 |

Gaussian (cm ^{−2 }) |
0.143 | 0.096 | <.0001 | 0.23 | 0.11 | <.0001 |

Mean (cm ^{−1 }) |
0.390 | 0.310 | <.0001 | 0.53 | 0.34 | <.0001 |

Maximum (cm ^{−1 }) |
0.606 | 0.470 | <.0001 | 0.90 | 0.54 | <.0001 |

Minimum (cm ^{−1 }) |
0.173 | 0.151 | <.0001 | 0.17 | 0.15 | <.001 |

Longitudinal (cm ^{−1 }) |
0.227 | 0.212 | NS | 0.25 | 0.22 | <.05 |

Circumferential (cm ^{−1 }) |
0.552 | 0.409 | <.0001 | 0.82 | 0.47 | <.0001 |

Difference (cm ^{−1 }) |
0.433 | 0.319 | <.0001 | 0.72 | 0.39 | <.0001 |

Apical shape parameters | ||||||

Gaussian (cm ^{−2 }) |
0.136 | 0.079 | <.0001 | 0.219 | 0.099 | <.0001 |

Mean (cm ^{−1 }) |
0.437 | 0.331 | <.0001 | 0.620 | 0.372 | <.0001 |

Maximum (cm ^{−1 }) |
0.697 | 0.528 | <.0001 | 1.048 | 0.597 | <.0001 |

Minimum (cm ^{−1 }) |
0.178 | 0.134 | <.0001 | 0.193 | 0.147 | <.0001 |

Longitudinal (cm ^{−1 }) |
0.204 | 0.158 | <.0001 | 0.231 | 0.184 | <.0001 |

Circumferential (cm ^{−1 }) |
0.670 | 0.504 | <.0001 | 1.010 | 0.560 | <.0001 |

Difference (cm ^{−1 }) |
0.138 | 0.081 | <.0001 | 0.855 | 0.450 | <.0001 |

Septal shape parameters | ||||||

Gaussian (cm ^{−2 }) |
0.086 | 0.063 | <.0001 | 0.115 | 0.073 | <.0001 |

Mean (cm ^{−1 }) |
0.341 | 0.275 | <.0001 | 0.472 | 0.307 | <.0001 |

Maximum (cm ^{−1 }) |
0.560 | 0.429 | <.0001 | 0.847 | 0.494 | <.0001 |

Minimum (cm ^{−1 }) |
0.122 | 0.121 | NS | 0.097 | 0.119 | <.01 |

Longitudinal (cm ^{−1 }) |
0.184 | 0.193 | NS | 0.186 | 0.193 | NS |

Circumferential (cm ^{−1 }) |
0.498 | 0.357 | <.0001 | 0.757 | 0.420 | <.0001 |

Difference (cm ^{−1 }) |
0.437 | 0.308 | <.0001 | 0.750 | 0.375 | <.0001 |

## Regional Remodeling Shape Analysis

Compared with normal subjects, regional analysis demonstrated significant ( *P *< .0001) apical shape changes (segments 13–17) in patients with CM when quantified during both systole and diastole using Gaussian, mean, maximum, minimum, longitudinal, circumferential, and difference curvatures. The mean curvature measured at end-systole depicted the largest significant intergroup differences ( *t *= 14.5; Table 1 , Figure 6 ). Mean curvature was also significantly different during diastole ( Figure 6 ). Another parameter, based on the standard deviation of Gaussian curvatures, was also significant during both systole (0.09 and 0.21 in patients with CM and normal controls, respectively, *P *< .0001) and diastole (0.17 and 0.32 in patients with CM and normal controls, respectively, *P *< .0001), indicating more variance of curvature in normal controls than in patients with dilated CM. Systolic apical mean curvature was well correlated with LVEF ( *r *= 0.89; Figure 7 ).

In patients with CM, regional shape analysis demonstrated a significant ( *P *< .0001) reduction in septal curvature (segments 4, 10, and 15) in Gaussian, mean, maximum, circumferential, and difference curvatures ( Table 1 ). The differences in LV minimum curvature between the normal and dilated populations are shown in Figure 8 . In contrast, no differences were noted in diastolic minimum curvature (i.e., large radius of curvature; 0.12 and 0.12 in patients with CM and normal controls, respectively, *P *= .84) or diastolic longitudinal curvature (0.19 and 0.18 in patients with CM and normal controls, respectively, *P *= .29). Difference curvature of the mid and basal segments was well correlated with LVEF ( *r *= 0.88; Figure 9 ).

## Results

Three-dimensional echocardiographic data acquisition and subsequent data analysis were feasible in all patients.

## Global Remodeling Shape Analysis

Compared with normal subjects, global analysis demonstrated significant ( *P *< .0001) shape changes in patients with CM when quantified during both systole and diastole using 3D sphericity, 3D-derived surface area, and Gaussian, mean, maximum, minimum, circumferential, and difference curvatures ( Table 2 ).

Parameter | Diastole | Systole | ||||
---|---|---|---|---|---|---|

Normal | CM | P |
Normal | CM | P | |

Global shape parameters | ||||||

Sphericity | 28% | 38% | <.0001 | 19% | 33% | <.0001 |

Surface area (cm ^{2 }) |
94.5 | 161.5 | <.0001 | 52.3 | 133.2 | <.0001 |

Gaussian (cm ^{−2 }) |
0.143 | 0.096 | <.0001 | 0.23 | 0.11 | <.0001 |

Mean (cm ^{−1 }) |
0.390 | 0.310 | <.0001 | 0.53 | 0.34 | <.0001 |

Maximum (cm ^{−1 }) |
0.606 | 0.470 | <.0001 | 0.90 | 0.54 | <.0001 |

Minimum (cm ^{−1 }) |
0.173 | 0.151 | <.0001 | 0.17 | 0.15 | <.001 |

Longitudinal (cm ^{−1 }) |
0.227 | 0.212 | NS | 0.25 | 0.22 | <.05 |

Circumferential (cm ^{−1 }) |
0.552 | 0.409 | <.0001 | 0.82 | 0.47 | <.0001 |

Difference (cm ^{−1 }) |
0.433 | 0.319 | <.0001 | 0.72 | 0.39 | <.0001 |

Apical shape parameters | ||||||

Gaussian (cm ^{−2 }) |
0.136 | 0.079 | <.0001 | 0.219 | 0.099 | <.0001 |

Mean (cm ^{−1 }) |
0.437 | 0.331 | <.0001 | 0.620 | 0.372 | <.0001 |

Maximum (cm ^{−1 }) |
0.697 | 0.528 | <.0001 | 1.048 | 0.597 | <.0001 |

Minimum (cm ^{−1 }) |
0.178 | 0.134 | <.0001 | 0.193 | 0.147 | <.0001 |

Longitudinal (cm ^{−1 }) |
0.204 | 0.158 | <.0001 | 0.231 | 0.184 | <.0001 |

Circumferential (cm ^{−1 }) |
0.670 | 0.504 | <.0001 | 1.010 | 0.560 | <.0001 |

Difference (cm ^{−1 }) |
0.138 | 0.081 | <.0001 | 0.855 | 0.450 | <.0001 |

Septal shape parameters | ||||||

Gaussian (cm ^{−2 }) |
0.086 | 0.063 | <.0001 | 0.115 | 0.073 | <.0001 |

Mean (cm ^{−1 }) |
0.341 | 0.275 | <.0001 | 0.472 | 0.307 | <.0001 |

Maximum (cm ^{−1 }) |
0.560 | 0.429 | <.0001 | 0.847 | 0.494 | <.0001 |

Minimum (cm ^{−1 }) |
0.122 | 0.121 | NS | 0.097 | 0.119 | <.01 |

Longitudinal (cm ^{−1 }) |
0.184 | 0.193 | NS | 0.186 | 0.193 | NS |

Circumferential (cm ^{−1 }) |
0.498 | 0.357 | <.0001 | 0.757 | 0.420 | <.0001 |

Difference (cm ^{−1 }) |
0.437 | 0.308 | <.0001 | 0.750 | 0.375 | <.0001 |

## Regional Remodeling Shape Analysis

Compared with normal subjects, regional analysis demonstrated significant ( *P *< .0001) apical shape changes (segments 13–17) in patients with CM when quantified during both systole and diastole using Gaussian, mean, maximum, minimum, longitudinal, circumferential, and difference curvatures. The mean curvature measured at end-systole depicted the largest significant intergroup differences ( *t *= 14.5; Table 1 , Figure 6 ). Mean curvature was also significantly different during diastole ( Figure 6 ). Another parameter, based on the standard deviation of Gaussian curvatures, was also significant during both systole (0.09 and 0.21 in patients with CM and normal controls, respectively, *P *< .0001) and diastole (0.17 and 0.32 in patients with CM and normal controls, respectively, *P *< .0001), indicating more variance of curvature in normal controls than in patients with dilated CM. Systolic apical mean curvature was well correlated with LVEF ( *r *= 0.89; Figure 7 ).

In patients with CM, regional shape analysis demonstrated a significant ( *P *< .0001) reduction in septal curvature (segments 4, 10, and 15) in Gaussian, mean, maximum, circumferential, and difference curvatures ( Table 1 ). The differences in LV minimum curvature between the normal and dilated populations are shown in Figure 8 . In contrast, no differences were noted in diastolic minimum curvature (i.e., large radius of curvature; 0.12 and 0.12 in patients with CM and normal controls, respectively, *P *= .84) or diastolic longitudinal curvature (0.19 and 0.18 in patients with CM and normal controls, respectively, *P *= .29). Difference curvature of the mid and basal segments was well correlated with LVEF ( *r *= 0.88; Figure 9 ).

## Discussion

The aim of this study was to examine the differences in global and regional LV curvature shape parameters obtained from 3D echocardiography in patients with CM and to correlate these shape parameters with LV function. We found that throughout the cardiac cycle, the global LV shape of patients with CM was rounder compared with that of control patients. When we examined regional LV curvature in these patients, we found that the global changes were due to a loss of curvature in the septum and apex of the left ventricle in the hearts of patients with CM compared with normal hearts. When we correlated these regional changes in curvature with LV function, we found that the systolic apical mean curvature value and the difference curvature values of the mid and basal septal segments were all highly correlated with LVEF.

The normal heart is “bullet shaped” and thus has a small circumferential radius (i.e., a large circumferential curvature value) compared with the longitudinal direction, which is more “flat” except at the apex. This means that the circumferential curvature value is very different compared with the longitudinal curvature value in the normal heart. As hearts become dilated and more spherical, these curvatures will be closer in value. Difference curvatures go down as the local shape becomes more spherical. In a left ventricle with normal function and shape, circumferential curvature will tend toward the maximum value (i.e., large curvature values) and longitudinal curvature toward the minimum value (i.e., small curvature values), which is consistent with the bullet shape of the left ventricle. As the left ventricle dilates in CM, there is a loss of circumferential curvature (i.e., small curvature values). This is consistent with a dilating short-axis and cross-sectional diameter. This explains why a significant difference was seen in global circumferential curvature between patients with CM and normal controls, but not global longitudinal curvature. The dilation of the left ventricle also causes hearts with CM to have greater sphericity and surface area and smaller mean curvature.

When we examined regional changes in curvature, apical and septal remodeling were most responsible for the global changes in LV shape and curvature in patients with CM. As the LV dilates, the apex loses its curved-conical shape and becomes rounded. This is demonstrated by the reduction in mean and circumferential curvatures and the lower variation in apical curvature parameter values. This is logical, as we are measuring a form of regional apical sphericity.

As with apical remodeling, loss of septal curvature especially during systole was measured in patients with CM. Septal systolic shape depends on interventricular coupling and likely changes dependent on right ventricular and LV loading conditions that alter net septal transmural pressure. In patients with CM, LV loading conditions are larger than right ventricular conditions, so the septum loses its normal curvature during the cardiac cycle.

Unlike traditional diastolic sphericity, we found that regional shape parameters, especially in systole, correlated very well with LVEF. This indicates that this technique quantifies regional remodeling associated with declining LV function. This is consistent with our above finding of differences in LV shape between CM and normal patients. As the heart remodels and dilates, the amount of loss of apical and septal curvature reflects the worsening LV function.

Survival in heart failure is negatively correlated with the development of adverse LV remodeling. As the prevalence of heart failure increases, elucidation of the etiology, pathophysiology, and diagnostic and predictive markers of this process is critical for the development and evaluation of new therapies that could result in favorable LV remodeling and improved survival. In clinical practice, the most frequent measures used to assess remodeling include LV cavity lengths, volumes, and mass, which are often measured at end-diastole using 2D echocardiography.

However, our understanding of which 2D echocardiographic markers are predictive and which should be used for the identification of early adverse LV remodeling are at an early stage and limited by the inherent weaknesses of this imaging modality. Geometric assumptions of LV shape and image foreshortening in 2D echocardiography lead to lower accuracy and reproducibility. Three-dimensional echocardiographic methods, compared with 2D echocardiography, provide a more accurate and reproducible method of assessing LV volume because of their ability to avoid the previously mentioned limitations ( Figure 10 ). Importantly, 3D echocardiography allows the entire left ventricle to be assessed in a single image. This development allows LV shape parameters such as LV sphericity to be fully computed in 3D space rather than extrapolating from 2D data.

LV sphericity is a dimensionless index used to describe the deviation of the LV cavity from normal bullet shape to a spherical shape. It has been used to assess the severity of the remodeling process as the left ventricle becomes more globular. Three-dimensional echocardiography is crucial for the accurate computation of LV sphericity, because this index requires accurate volume and long-axis measurements, which can be obtained with 3D guidance. It has been shown that 3D echocardiography–derived sphericity performed within 2 to 10 days of an acute myocardial infarction can accurately predict the development of LV remodeling. However, 3D sphericity assesses overall LV change from the underlying bullet shape. This ignores disease processes whereby there is globally reduced LV function but only regional LV shape changes, such as LV apical ballooning. Thus, a more sensitive method for quantifying sphericity, such as regional sphericity, may have better predictive value.

Disease processes such as aortic stenosis, mitral regurgitation, and ischemic CM all result in changes in LV shape. However, these diseases all have different pathophysiologic pathways, suggesting that the processes by which the left ventricle remodels in each of these diseases may not be uniform. Regional shape analysis not only quantifies what clinicians have clinically observed on LV images but also provides a methodology by which mechanistic insights into regional LV changes can be measured. From this study, we found that regional shape throughout the cardiac cycle varies between patients with dilated CM and normal subjects. We have also recently found that end-systolic septal short-axis/circumferential curvature is a predictor of cardiac events in patients with nonischemic dilated CM with severely reduced LVEFs when other parameters, such as LVEF and volumes, are identical. Although further studies are required, regional shape may provide a more sensitive analysis of changes in LV function and better identify higher risk patients in various disease processes.

## Limitations

One limitation to this study is that the patients with CM included in this study had extremely dilated hearts and reduced LV function. The changes noted in this study may not necessarily be applicable to those with milder degrees of LV dysfunction. As well, the current methodology used to obtain regional curvature measurements is time intensive and not suitable in its current form for clinical use. Additionally, in patients with extremely dilated hearts, the entire left ventricle cannot be captured within the pyramidal sector for 3D transthoracic echocardiographic acquisitions, limiting its use. Last, regional shape analysis is very dependent on image quality. If the endocardial border is inaccurately tracked, the measured regional curvature may be invalid.

## Conclusions

In summary, to our knowledge, this is the first study to demonstrate that regional remodeling measured by regional 3D LV curvature correlates well with LV function. As well, patients with CM have larger more globular hearts, primarily because of a rounder apex and bulging of the septum into the right ventricle. Moreover, unlike 2D echocardiography–derived parameters, 3D shape parameters are not dependent on geometric assumptions or a definition of cavity length. This methodology may prove extremely useful in following global and regional LV changes during various disease process with regional LV effects, such as infarct expansion, apical ballooning syndrome, and dilation due to valvular or idiopathic causes. As well, this method may be applied to understanding the LV changes seen in volume versus pressure overload, as these two processes result in different LV geometric shapes. Although we used 3D echocardiography in this study, the methodology hereby described could be easily applicable to other cardiac imaging modalities, such as cardiac magnetic resonance or computed tomography. Ultimately, the development of a reliable tool to serially assess adverse and favorable LV remodeling would be clinically useful in assessing the efficacy of the myriad pharmacologic, interventional, and surgical therapies used for the treatment of ischemic, valvular, and idiopathic CM.

## Appendix

The methods used for analysis of curvature in this work come from differential geometry and analysis on manifolds. The LV mesh is a curved, compact, connected, orientable 2D manifold (i.e., a surface) embedded in 3D Euclidean space homeomorphic to the sphere *S *^{2 }(the open LV cavity is closed by the nonanalyzed mitral regional cap). Theoretically, at each node, a neighborhood surrounding the node is analyzed for changes in curvature and ultimately principal curvatures: a minimum ( *k *_{min }, *N *) and maximum ( *k *_{max }, *X *) curvature are computed. Curvature is defined as the reciprocal of the radius of curvature, *k *= 1/ *R *. Generally, circumferential (short-axis) curvature is greater (“more curvy”) than long-axis curvature. Because of the reciprocal nature of curvature versus radius, the circumferential radius of curvature is smaller than the longitudinal radius of curvature. It is a principle of geometry that principal curvatures are always oriented 90° to each other in this context. Analytically, curvature tensors can be computed using both the metric and the connection of the manifold. Before computing discrete curvatures, the ventricular meshes were conditioned for continuity to facilitate numerical stability.

The following formulas are used to compute curvatures *k *and other parameters (such as geodesics):