A computational atlas of image-derived shapes refers to an alignment of maps that relate individual anatomical geometry and function to the distribution of biological variations across a population, which can be described at different scales from genotype to phenotype [1]. Atlas-based analyses of patients and healthy volunteers have recently been explored in several different medical areas. For that purpose, large imaging databases, which enable the construction of probabilistic shape atlases for specific organs or diseases, have been established. Atlas-based analysis of brain anatomy and pathology is well advanced (e.g. [2]), including analysis of occipitalization in children [3], and MRI-based probabilistic atlases of neuroanatomy [4]. In the heart, atlas-based analysis has recently shown the potential to reveal new measures of geometry and function [1]. For example, atlas-based methods have been used to quantify subtle differences in heart shape between individuals born prematurely compared with full term age matched controls [5]. In patients with cardiovascular disease, certain changes in heart shape over time, known as remodeling, are indicative of worse prognostic outcome [6]. After myocardial infarction, remodeling associated with an increase in heart size is a predictor of mortality, and remodeling associated with sphericalization of heart shape is linked with decreased survival [7]. However, standard clinical indices used to describe remodeling are typically simple measures of mass and volume, such as end-diastolic (ED) volume (largest volume), end-systolic (ES) volume (smallest volume) or left ventricular mass. These ignore much of the available shape information. We hypothesized that atlas-based analysis of patients with myocardial infarction would enable better quantification of remodeling features associated with myocardial infarction.

The Cardiac Atlas Project (CAP, http://www.cardiacatlas.org) is a world-wide web-accessible resource, comprising a population atlas of asymptomatic and pathological hearts [8]. The CAP facilitates large-scale data sharing of cardiac imaging studies and their corresponding derived analyses that describe the cardiac shape, structure and function across various population groups. The data has been contributed from several studies, including Defibrillators to Reduce Risk by Magnetic Resonance Imaging Evaluation (DETERMINE) [9], comprising patients with myocardial infarction, and the Multi Ethnic Study of Atherosclerosis (MESA) [10], comprising asymptomatic volunteers. We used cases from both studies to examine the principal components of shape variation between the two cohorts, and thereby characterize shape changes associated with myocardial infarction.

Cardiovascular magnetic resonance (CMR) imaging is a non-invasive modality, which provides detailed, quantitative data of the heart structure and function. Compared to other imaging modalities, CMR does not use ionizing radiation, and is not dependent on restricted views of the heart. As a result, many large research studies are using CMR to collect phenotypic data on cardiac disease. Model-based image analyses were developed in the last decades from the availability of large-scale data set of CMR images. This has led to the growing number of statistical analysis applications for cardiac shape and motion [1]. One particular shape representation is a finite element model, which provides an efficient and accurate representation of complex geometries [11]. This method has been shown to provide a compact and powerful representation of shape and function of the LV, and has been validated against ex-vivo LV mass, against manually-drawn contours in patients with regional wall motion abnormalities, and against cardiac output flow in healthy subjects [12], [13]. However, the statistical analysis of shape parameters has previously been limited by the lack of substantial sample size and/or bias between acquisition protocols.

In this study, we applied principal component analysis (PCA) to characterize the cardiac shape features in a large number of CMR cases. PCA is a widely used dimensionality reduction technique, which has been applied to heart shape analysis [14], motion analysis [15], 3D segmentation [16], and population analysis [17] [18]. After extracting shape features using PCA, we applied a logistic regression to analyze the differences between myocardial infarction patients and asymptomatic volunteers. We also compared the performance (goodness of fit) of the model with standard clinical indices, including LV mass and volume. We found that the shape indices derived from the principal components of the shape variation characterized remodeling better than standard LV mass and volume indices.

PCA was performed on the shape models at ED and ES as well as their combination (ED&ES). A scree plot [29] is given in Fig. 2 showing the cumulative variance explained by each mode. The shape variation described by each mode is shown in Fig. 3 for ED, Fig. 4 for ES, and Fig. 5 and Fig. 6 for ED&ES. Although most of the modes do not correspond with traditional shape measures, the first three modes in each case can be understood in terms of commonly used clinical measures of remodeling. Mode 1 explained 50% of the total variance at ED and 55% at ES. In both cases the first mode was primarily associated with the size of the LV. Mode 2 explained 10% of the total variance at ED and was primarily associated with the sphericity of the left ventricle. The third mode of ED was associated with mitral valve orientation. At ES the second mode accounting for 8% of the variance was associated with wall thickening. The third mode at ES was associated with sphericity. We retained 90% of the cumulative variance, which resulted in 13 modes at ED, 14 modes at ES and 20 modes for the ED&ES combination.

Five logistic regression models were studied using all available data. The coefficients (β), standard error, associated p-values, standardized coefficients and odds ratios (OR) were calculated for each model. A p-value of 0.05 or lower was considered significant. The first model included age, sex, height, weight, systolic blood pressure and diastolic blood pressure (Table 1). This was considered the baseline model (Table 2). The second model included the baseline model variables plus 14 ES PCA modes (ES PCA, Table 3). The third model consisted of the baseline variables plus 13 ED PCA modes (ED PCA, Table 4). The fourth model consisted of the baseline variables plus the modes obtained from the ED&ES combination of the ED and ES shape parameters (ED&ES PCA, Table 5). The fifth model included the baseline variables plus the ED volume (EDV), the ES volume (ESV) and the LV mass (MASSVOL, Table 6). The baseline model showed that the age, height and weight were statistically significant predictors of disease (Table 2). These were also significant in the baseline plus ES PCA model (Table 3), along with most of the PCA modes (except modes 5, 8, 11 and 13). The baseline plus ED modes (Table 4) also showed that most PCA modes were significantly associated with disease (except modes 4, 7, 10, 12 and 13). The baseline plus mass and volume model showed that EDV, ESV and MASS were all associated with disease (Table 6).

The standardized coefficients show which modes have greater effect on the probability that the case is a patient. Mode 1 and mode 4 have greater effect in the classification model at ES. Mode 1, mode 6 and mode 2 have greater effect in the classification model at ED. EDV, ESV and MASS are highly related with the disease, according to Table 6. The Odds ratios were relative measures of the effects of the shape indicators between the myocardial infarction patients and the normal people. Some shape indicators (OR>1), for instance, the mode 1 and mode 4 in the ES model (Table 3) and ED&ES model (Table 5) show higher odds of myocardial infarction than others. Some shape indicators (OR<1) show lower odds of the disease, for example, mode 1 and mode 6 in the ED model (Table 4). There are several modes whose OR is not significant as their confidence intervals overlap the null value (OR = 1).

The goodness of fit was compared between the five models with the indices of Deviance, AIC, BIC and AUC of each model, which are listed in Table 7. Overall, all the PCA mode models as well as the mass-volume model showed good performance. The ED&ES PCA model achieved the best performance in terms of Deviance, AIC, SC, and AUC values, followed closely by the ES PCA model and the ED PCA model. All PCA models better characterized patients from non-patients than traditional mass and volume measures. The ROC curves are shown in Fig. 7.

We have proposed an atlas-based disease analysis framework by means of shape parameters from LV finite element models with a large number of subjects. The framework consisted of three steps: (1) fitting a finite element model to the LV MR images, (2) principal component analysis of the aligned shape parameters, and (3) quantification of the association with disease using logistic regression. We hypothesized that patients with myocardial infarction have significant shape differences with respect to the normal population, due to cardiac remodeling. The results supported this hypothesis, with most modes significantly associated with disease. The PCA analysis also performed better than traditional indices of remodeling (mass and volume). This method can therefore be used as a clinical tool for the characterization of the patterns of change associated with remodeling. These methods can also be used to track individual patients over time, by quantifying the degree to which their shape modes conform to the remodeling spectrum. Patients who are moving toward the adverse side of the spectrum may benefit from more aggressive treatment regimes. Conversely, the reverse remodeling associated with treatment can also be quantified. Although in this study we applied the method to patients with myocardial infarction, this framework is generalizable to any disease group.

Note that we did not attempt to correct for colinearity between EDV, ESV and Mass in the MASSVOL model, or between SBP and DPB in the baseline model. EDV and ESV were strongly correlated (Pearson coefficient ρ = 0.911, p<0.05), as were ESV and Mass (Pearson coefficient ρ = 0.664, p<0.05), which would affect these coefficient estimates and odds ratios in the model. However, all three were input together in the MASSVOL baseline model to show the combined goodness-of-fit of traditional indicators, in order to assess the improvement given by the PCA modes. SBP, DBP (Pearson coefficient ρ = 0.604, p<0.05) and other baseline variables were included in all the regression models to control for any differences between the patient and asymptomatic groups (Table 1).

The finite-element method is a powerful representation of the LV model, which also provides traditional indicators such as volume and mass. This method has been used to characterize cardiac motion [30], [31] and deformation in a variety of disease groups [32]. Extensions to the right ventricle and atria have also been proposed [33], [34]. In this study, we have limited the application of these models to the description of shape; however, these models also have the capability of simulating the excitation, contraction and relaxation of cardiac mechanics [35].

PCA clusters the variability of the finite element models into orthogonal modes that can be interpreted from a global shape point of view. In [5], PCA was used to determine the shape differences between people born pre-term and people born full-term. PCA has the disadvantage that the modes are in general difficult to interpret from a clinical perspective. However, in the present study the first three modes were associated with well understood clinical indicators such as size or sphericity. Interestingly, both size and sphericity are associated with adverse outcomes after myocardial infarction [7], [36], [37]. In the current study, the ED_ES PCA regression model performed the best with an AUC of 0.9905. Adding ESV, EDV and MASS into this model did not improve this performance greatly (results not shown), indicating that the discriminatory information included in these mass and volume measures are already captured in the ED&ES PCA model. Adding stroke volume or ejection fraction to the MASSVOL model also did not improve the results greatly, since these are very dependent on the ESV and EDV already in the model.

Although heart size is known to be dependent on patient body habitus, we did not correct the shape vectors for height or weight, as done clinically using indexing methods. This was because the baseline model already included height and weight, so all PCA logistic regression analyses were automatically corrected for height and weight.

Further work is needed in several areas. Although PCA is one of the most common dimensionality reduction techniques, other techniques may be more appropriate, such as independent component analysis [18]. Secondly, logistic regression classification method is only one of many methods which can be used for the characterization of disease. For example, in [38] a three-dimensional cortical gray matter density map was established and validated using sparse multinomial logistic regression in the classification of schizophrenia. In [39], a maximum a posteriori classifier was used to distinguish brain tissue types. Expectation-maximization (EM) [40] and k-Nearest-Neighbor [41] classification have been successfully applied to evaluate brain tumors from MRI. Neural networks and support vector machines have been used to identify brain structures with MRI [42] and to predict wall motion scores [43]. Evaluation and comparison of these methods for the evaluation of cardiac disease should be performed. Thirdly, the transformation from GRE to SSFP models was learned using 40 normal volunteers. While [23] showed that this was sufficient to robustly characterize the transformation, more cases would provide a greater variation of heart shape and might improve the transformation parameters.