Skeletal model-based analysis of the tricuspid valve in hypoplastic left heart syndrome

Abstract

Hypoplastic left heart syndrome is a congenital heart disease characterized by incomplete development of the left heart. Affected children undergo a series of operations which result in the tricuspid valve becoming the only functional atrioventricular valve. Many of these patients go on to develop complications associated with heart failure and death such as tricuspid valve regurgitation. Predicting which patients will develop regurgitation as well as planning for corrective procedures could be greatly enhanced through better understanding of the relationship between geometry and function of the tricuspid valve. Traditional analysis has relied on simple, global anatomical measures which often can not capture localized structural changes. Recently, statistical shape modeling has proven to be useful for analyzing the geometry of the tricuspid valve. We propose to use skeletal representations (s-reps) for modeling the leaflets of the tricuspid valve in these patients. S-reps are a more feature-rich representation than traditional boundary-based models and have been shown to have advantages for statistical analysis. Unfortunately, it is more difficult to fit s-reps to many geometries which limits the application of their powerful analysis techniques. We propose an extension to previous s-rep fitting approaches which yields improved models for difficult to fit objects such as the leaflets of the tricuspid valve. We incorporate application-specific anatomical landmarks and population information to improve correspondence. We use several traditional shape analysis techniques to compare the efficiency of s-reps with boundary representations created using SPHARM-PDM. We observe that principal component analysis produces a more compact shape space using s-reps, needing fewer modes to represent 90% of the population variation, while distance-weighted discrimination shows that s-reps provide more significant classification results between valves with less regurgitation and those with more. These results demonstrate the power of using s-reps for relating structure and function of the tricuspid valve.

1. Introduction

Hypoplastic left heart syndrome (HLHS) is a form of congenital heart disease characterized by incomplete development of the left heart, affecting over 1,000 infants in the US per year and proving uniformly fatal without surgical intervention [5]. Surgery allows the children to survive, but many will go on to develop tricuspid regurgitation (TR). Because the tricuspid valve (TV) is the only remaining functional atrioventricular valve, this condition is highly associated with heart failure, requiring surgical intervention in as many as 30% of HLHS patients [7, 11].

The exact relationship between TV structure and TR is not known. Both imaging and surgical inspection are used to examine the TV but are not without limitations [22]. Metrics such as annular area [3, 14], septolateral diameter [3, 4, 14], bending angle [12], anterior leaflet prolapse [4], and anterior papillary muscle location [3, 14] computed from 3D echocardiograms have been used to detect the presence of TR. While some of these measures are semi-local, none of them can capture more localized or subtle differences in tricuspid valve geometry.

Previous work [25] has shown that boundary-based shape models of the TV can be used as a basis for analyzing a population of valves and performing tasks such as discriminating between valves with lower and higher amounts of TR. However, there are inherent limitations to representing objects using purely boundary information. In this work, we instead explore the use of discrete skeletal representations (s-reps) for modeling the TV.

S-reps have the benefit of explicitly modeling the entire object, including its interior, unlike pure boundary models such as those produced by methods like spherical harmonic point distribution model (SPHARM-PDM) [21]. S-reps have been shown to be a powerful representation in a variety of tasks, including segmentation [23], classification [9] and hypothesis testing [19].

These advantages are tempered by the fact that it is often more difficult to fit s-reps to objects than it is simpler boundary-based models. We introduce changes to the traditional s-rep fitting pipeline to make it easier to create s-reps for objects like TV leaflets which cause difficulty for existing approaches. We produce s-reps which have improved correspondence across the population compared with previous s-rep fitting techniques. We then compare the power of s-reps and s-rep-based analysis of the TV with previous work using boundary based models and show that it provides advantages for both general population analysis and discrimination.

2. Materials

2.1. Subjects

Acquisition of transthoracic 3DE images of the TV is part of the standard clinical echo lab protocol for HLHS at Children’s Hospital of Philadelphia (CHOP). Patients were retrospectively identified who had HLHS with a Fontan circulation and in whom 3DE of the TV had been previously performed. Exclusion criteria included presence of significant artifacts and inability to segment the TV. This study was approved by an Institutional Review Board. We utilized 100 3DE scans with age range 2 to 30 years (mean 10.36 years). Images were acquired using sector narrowed Full Volume or 3D Zoom mode with a wide field of view. Electrocardiogram-gated acquisitions were obtained when patient cooperation allowed. Transthoracic probes (X7 or X5) were used with the Philips IE33 and EPIQ 7 ultrasound systems.

3. Methods

3.1. Image Segmentation and Model Creation

Images were exported to DICOM via Philips QLAB and imported into 3D Slicer [1] using the SlicerHeart [2, 3] Philips DICOM converter. A single mid-systolic frame was chosen for modeling the TV. TV segmentation was performed using the 3D Slicer Segmentation module.

3.2. Skeletal Representations

In previous work [25], corresponding boundary point distribution models (PDMs) of each leaflet in the tricuspid valve were created using SPHARM-PDM [21] with correspondence-correcting post-processing [25]. While this process produces high quality models suitable for statistical analysis, there are inherent limitations to representing objects using purely boundary information. This is particularly true for TV leaflets which are very thin and sheet-like as the relationship between points which are close in space but on opposite sides of the leaflet can be just as if not more important than those of nearby points on the same side. For this reason we choose to use s-reps to model the leaflets of the TV in this work. As seen in figure 2, an s-rep consists of a grid of points on the skeleton and a set of vectors emanating from the skeleton to the boundary called spokes.

Fig. 2:

An s-rep fit to a tricuspid valve anterior leaflet. The green lines are the mesh of the object’s skeleton. The cyan spokes point to the top side of the object, the magenta to the bottom, and the yellow to the crest.

The general process for fitting an s-rep to an object consists of several steps [16]. Because directly computing a medial or skeletal model of an arbitrary object is a non-trivial process, the objects are first deformed to a simpler object such as an ellipsoid. This deformation is currently done via mean curvature flow. From there an s-rep of the ellipsoid (the skeleton of an ellipsoid is actually medial) can be directly computed. Finally, thin-plate splines (TPS) are used to interpolate a space-filling reverse deformation from the ellipsoid back to the original object. Finally, a refinement step that corrects any deficiencies in the s-rep fit caused by TPS warping by ensuring that the spokes touch the original object boundary and are approximately orthogonal to it is done.

This process works well for objects which are mostly ellipsoidal in shape and have three dimensions with obviously different principal radii. However, for objects such as TV leaflets where these assumptions do not hold, there can be substantial differences in how ellipsoids are mapped to the leaflets across a population. This will in turn yield s-reps which are not in correspondence because the same anatomical regions, such as the coaptation surfaces, will be mapped to very different parts of the s-rep. S-reps generated using this approach are thus not suitable for statistical analysis. In this work we instead leverage SPHARM-PDM models previously fit to the leaflet boundaries. Because these fits are based on a decomposition of the object’s boundary using spherical harmonics, by choosing to reconstruct the object using only first degree polynomials we obtain an ellipsoid that best fits the object and is in correspondence with the original SPHARM-PDM mesh. We can then proceed as before, computing an s-rep directly from this ellipsoid and using a TPS warp based on the existing correspondence. Because correspondence of these models across the entire population was already established, this process yields a set of s-reps which share this correspondence.

The final problem with fitting s-reps to TV leaflets is the choice of which dimensions of the leaflets should correspond to the axes of the ellipsoid. We choose to let the commisure-to-commisure axis of the leaflet be the major axis of the ellipsoid. The second axis of the ellipsoid is chosen to be the axis between the centerpoint of the leaflet along the annulus to its point closest to the center of the valve. Note that, in cases with significant billow or tenting, this center point is not precisely the center of the valve but the point along the ridge of the valve either above or below the annuluar plane. Figure 3 shows an example of these axes on a leaflet with tenting.

Fig. 3:

The leaflet axes used to map the first two ellipsoid axes. On the left, the line between the midpoint along the annulus and lowest part of the valve maps to the next longest ellipsoid axis. On the right, the left-to-right axis between the two commisure points maps to the long ellipsoid axis.

3.3. Analysis of s-reps

Discrete s-reps are represented as a series of tuples: a tuple of points representing samples on the object skeleton, a tuple of unit vectors representing spokes pointing from each skeletal point toward the object boundary, and a tuple of distances representing the length of these vectors. Because this representation is highly non-Euclidean, traditional statistical analysis methods which make Euclidean assumptions cannot be applied directly. Techniques such as principal geodesic analysis (PGA) were designed to act as analogues to traditional Euclidean methods on general non-Euclidean spaces. These techniques have been previously applied to s-reps, but because we know what manifolds s-rep features live on we can use a more customized approach. We use a method called Principal Nested Spheres (PNS)[10] which has been shown highly effective in Euclideanizing the s-rep representation [15].

PNS works by performing a principal component analysis (PCA)-like analysis, but operating on a data which lives on a sphere rather than on Euclidean data. It starts with data on an d dimensional sphere and finds the d – 1 dimensional subsphere on that sphere that minimizes the projected distances of each data point to that subsphere. This subsphere can be represented by an axis and distance which we call a polar system. The data is then projected to this subsphere and the process repeats until a 0 dimensional datapoint, the mean, is found. In each step, the projected distance is a Euclideanized feature. An example is shown in figure 4. Because many of the non-Euclidean features of s-reps live on spheres, this technique can be used to analyze them.

Fig. 4:

An example PNS projection step. The point p₁ on a d-dimensional sphere is projected onto the d – 1-dimensional subsphere defined by the polar system A(w¹, ψ₁). Z_d,1 is the Euclideanized feature resulting from this projection. This process is repeated until projecting onto a 0-dimensional subsphere.

To create a fully Euclideanized s-rep, we must perform several separate PNS analyses: one high-dimensional PNS for the tuple of skeletal points (n skeletal points live on the sphere S³ⁿ⁻⁴ and a two-dimensional PNS for each individual unit spoke direction which live on S². The Euclideanized features from these analyses are combined with appropriately normalized spoke lengths to create a matrix of Euclidean features suitable for being used with traditional Euclidean statistical analysis methods. Full details of this process can be found in [15].

3.4. Normalization

The question of how to normalize valve scale in group of children of varying size is an open one that requires further study. A traditional way to normalize shapes is using gross geometric scale using Procrustes analysis. In previous work on heart valves, body surface area (BSA) has been shown to be a meaningful basis for rescaling [20]. In this work we compare results using these two approaches but we do not directly address the question of which is the better approach or if there is another approach which provides better results.

4. Evaluation

4.1. Principal Component Analysis

We use principal component analysis (PCA) on the composite matrix described in section 3.3 to find a mean and modes of variation of our population of s-reps. One major use of PCA is for dimensionality reduction by finding a lower-dimensional space which captures most of the variation in the population. Compared to results using a traditional PCA approach on the SPHARM-PDM models, s-reps provide a more compact representation, being able to capture 90% of the population variation in 11 modes rather than 16.

4.2. Distance Weighted Discrimination

We use distance-weighted discrimination (DWD)[13], an SVM-like binary classification algorithm, for learning to separate valves with lower and higher amounts of regurgitation. Our data is grouped into four classifications: trivial, mild, moderate, and severe TR, with 19, 46, 30, and 5 cases respectively. Because these gradings are qualitative and somewhat subjective, as well as the low sample sizes particularly in the trivial and severe classes, we group the trivial and mild cases into one class and moderate and severe cases into another for training DWD.

In figure 5, we compare DWD results on both SPHARM-PDM and s-rep models using both normalization strategies discussed in section 3.4. These plots are generated by projecting all cases onto the computed separating direction and using kernel density estimation to fit distributions to each class. Though separation results are generally worse using BSA-based normalization than standard Procrustes analysis, we can see that there tends to be better separation, particularly between the largest mild and moderate groups, using s-reps.

Fig. 5:

Comparison of DWD projection plots between s-reps (left) and SPHARM-PDM (right) using both Procrustes normalization (top) and BSA-based normalization (bottom). Generally, Procrustes normalization performed better than BSA and s-reps performed better than SPHARM.

To further validate this result we use 10 rounds of 5-fold cross-validation to assess the accuracy of classifiers trained on both models and using both normalization methods. The performance on the trivial and severe groups, the smallest and most extreme, is similar across all combinations. For mild vs moderate, while there is still significant overlap between the two groups, s-reps show generally better separation, correctly classifying approximately 65% of moderates and 90% of mild cases using Procrustes normalization and 55% of moderates and 85% of milds using BSA normalization. This compares to approximately 50%/95% and 45%/90% for SPHARM. Interestingly, s-reps show significantly better results for the moderate group but sometimes slightly worse results for mild cases.

5. Discussion

In this work we propose a method for creating discrete skeletal representations for performing modeling and analysis of the individual leaflets of the tricuspid valve. We introduce improvements to previous s-rep fitting approaches to leverage the ease of creating population-level correspondences via methods which are less computationally intensive methods than s-rep fitting but which in turn produce less powerful models. This approach leads to models with improved suitability for statistical analysis. We employ several standard analysis techniques and demonstrate that s-reps provide a richer geometric representation than previous work using boundary-based PDMs. In addition, S-reps seem particularly well-suited to representing valve leaflets as these thin structures lend themselves to being represented by skeletal models with a defining medial surface, unlike boundary models. In contrast, for modeling of more globular shapes, such as the right ventricle, SPHARM may have benefits as the boundary may be more relevent than the internal volume in that setting.

The methods presented in this work are distributed as part of SlicerSALT [24], an open-source, free, comprehensive software that will allow biomedical scientists to precisely locate shape changes in their imaging studies, and SlicerHeart [3, 2], a 3DSlicer extension containing tools for cardiac image import (3D/4D ultrasound, CT, MRI), quantification, and implant placement planning and assessment.

The time needed for initial SPHARM-PDM and s-rep mapping are relatively low ( 5 minutes per case on an AMD Ryzen 3950X in a mostly single-threaded implementation), however, the current implementation of s-rep refinement can be quite slow ( 30 minutes - 1 hour, depending on how much refinement is needed) and may require much manual parameter tuning. We are currently exploring deep learning-based approaches for generating s-reps of objects to reduce required computation time as well as lower the need for manual adjustment for each new setting.

As discussed in section 3.4, the question of how to normalize TV leaflets in a population of children of varying size is an open one. We continue to investigate additional normalization schemes, including normalizing by age-adjusted average BSA to attempt to separate normal size increases due to aging from disease-related changes.

Clinically, shape methods such as s-reps could be used in multiple ways. They could be used to augment rapid segmentation of valves via inclusion into atlas based [17] or machine learning [8] approaches to create valve models while reducing time-consuming manual work. Shape analysis can be used to characterize the population and compare any individual valve in the context of that population. In association with clinical outcome data such as degree of regurgitation or other patient outcome, such information could allow for risk stratification and inform development of strategies for image-derived patient specific repair [6].

A future area that we wish to explore is in the use of skeletal models as a method of extraction of shell models for the finite element modeling (FEM) of heart valves. We believe that s-reps provide a powerful basis for an open-source application for more easily creating these models compared to previous medial approaches [18]. The structure of s-reps could also lend themselves well to create both skeletal and volumetric meshes for FEM. In addition, the inherent parameterization of valve leaflets may be beneficial in solving inverse problems in FEM such as those to derive leaflet material properties from images themselves.

Fig. 1:

A, B and C. Segmentation of TV from ultrasound; D and E. Atrial views of the annulus (D) and TV (E) with landmarks. Of particular note in this work are the commissures: ASC (anterior/septal commisure), APC (anterior/posterior commisure), and PSC (posterior/septal commisure). F. Cutaway showing the coaptation surface where leaflets meet.)