Modeling Mitral Valve Leaﬂets from Three-Dimensional Ultrasound
Robert J. Schneider1 , William C. Burke1 , Gerald R. Marx3 , Pedro J. del Nido2 , and Robert D. Howe1
1

Harvard School of Engineering and Applied Sciences, Cambridge, MA, USA 2 Department of Cardiac Surgery, Children’s Hospital, Boston, MA, USA 3 Department of Cardiology, Children’s Hospital, Boston, MA, USA

Abstract. The geometry of the mitral leaﬂets, most commonly viewed using three-dimensional ultrasound, is an important input for mechanical models predicting valve closure. Current methods for leaﬂet modeling from ultrasound either require extensive user interaction, rely on inaccurate assumptions, or produce generic results. The presented method for modeling the mitral leaﬂets from three-dimensional ultrasound of an open mitral valve is automatic and able to capture patient speciﬁc geometry. The method requires as an input the location of the mitral annulus, which we generate semi-automatically using our previously designed method. No additional user interaction is required. The leaﬂet modeling algorithm operates by ﬁrst constructing an extended surface at the location of the leaﬂets which extends beyond the leaﬂet edges into the blood pool, and then trims the surface to the observed length of the leaﬂets. Preliminary results suggest the leaﬂet modeling method, combined with our previous method for annulus segmentation, generates accurate patient-speciﬁc mitral valve models.

1

Introduction

The mitral valve is a complex three-dimensional anatomic structure which controls the blood ﬂow between the left atrium and left ventricle in the heart. While it can be imaged using magnetic resonance or computed tomography imaging, the shape and motion of the mitral valve is most accurately captured using threedimensional ultrasound (3DUS). Ultrasound, in addition to being inexpensive, portable, and non-ionizing, is fast enough to capture the fast moving structures of the valve. The geometry of the mitral valve is used in several applications, including mechanical modeling to predict valve closure [1, 2]. While these models typically included only generalized leaﬂet geometry, more recent eﬀorts have been made to predict valve closure using patient speciﬁc geometry [2]. For this reason, several eﬀorts have been made to model the leaﬂets from 3DUS. The methods for modeling the leaﬂets from 3DUS can be categorized as either volumetric, mesh, or parametric. An example of a volumetric method for 3DUS was shown in [3], which used an intensity-based level set method to locate the leaﬂets. Unfortunately the method included not only the leaﬂets but anything

connected to the leaﬂets with similar intensity, such as the left ventricle wall, thus failing to isolate the leaﬂets. A method for modeling the valve as a mesh was presented in [4], which used a thin tissue detector and level sets to locate the leaﬂets and surrounding tissue. The method subsequently required manual intervention and the assumption of a planar annulus to construct an isolated leaﬂet mesh geometry. The method presented in [5] segmented and tracked the locations of several features of the mitral and aortic valve structures in 3DUS using machine learning techniques. It then ﬁt a parametric model to the segmented locations. However, in ﬁtting the model to only a few points, the model lacked patient-speciﬁc detail. To compensate for the highlighted insuﬃciencies of previous methods, we have developed an automatic method for modeling the leaﬂets. The method is designed for open mitral valves because we want to diﬀerentiate between anterior and posterior leaﬂets, and making this distinction for coapted leaﬂets in 3DUS is extremely diﬃcult even for a human observer. After ﬁnding the location of the annulus using our previously developed semi-automatic algorithm [6] and an automated optical ﬂow tracking method based on the work by Lucas & Kanade [7], the leaﬂet modeling method automatically ﬁnds the location and extent of the mitral leaﬂets and generates a mesh to represent the geometry with no additional user interaction. We chose to represent the leaﬂets as a mesh because this representation can be easily used to generate either a volumetric or parametric representation if needed, is a suitable input for mechanical modeling, and is representative of the the thin leaﬂet tissue. The method uses the location of the annulus as a means to enforce knowledge about the location and orientation of the leaﬂets. The method then uses a series of steps to estimate and reﬁne a search space coordinate system and the shape and location of the leaﬂets. A mesh at the location of the leaﬂets is then constructed as a natural consequence of the geometry of the search space. The details of the algorithm are described in Section 2. A study quantifying the performance of the method as compared to manual leaﬂet tracings is then shown in Section 3.

2
2.1

Methods and Materials
Algorithm Summary and Components

The mitral leaﬂet modeling algorithm is comprised of ﬁve main components organized in a two-pass strategy that drives the algorithm to accurately locate the leaﬂets (Figure 1). The components can be described as deﬁning the search space, redeﬁning the search space axis based on the availability of new information, estimating an extended leaﬂet surface using a graph cut method, reﬁning the extended leaﬂet surface using an active surface method, and ﬁnally trimming the extended leaﬂet surface using dimensional reduction and a level set method.

(Section 2.2) Mitral Valve Annulus

(Section 2.3)

Search Estimate E E Extended Space Surface Construction

Final Pass

(Section 2.6)

(Section 2.4) Trim

Reﬁne E Extended Surface

E Extended
Surface

T
Redeﬁne ' Search Axis (Section 2.5)

Initial Pass

c
Trim Extended Surface (Section 2.4)

c
Mitral Valve Leaﬂets

Fig. 1. Flow chart summary of the leaﬂet modeling algorithm. A description of each component appears in the corresponding section.

2.2

Constructing the Search Space

The location of the mitral leaﬂets in the heart is such that they attach to and rotate about the mitral annulus. Therefore, we search for the location of the mitral leaﬂets by constructing a search space that is comprised of an arc system that is centered about the annulus (Figure 2). The arcs are constructed in planes that are rotated about a deﬁned axis. Initially, little is known about the leaﬂets aside from their orientation, and so the axis is generically deﬁned as the direction of least variance as determined from a principal component analysis of the mitral annulus. For the ﬁnal pass, the axis location is reﬁned, as described in Section 2.5, to account for the estimated leaﬂet orientation and position. The search planes are evenly spaced at an angular oﬀset of ∆Θp . The arcs are evenly spaced at a radial oﬀset of ∆R, and points are sampled along the arcs at an angular oﬀset of ∆Θs . The search space is designed around the assumption that the leaﬂets will intersect the search arcs, at most, at a single location along each arc. This point
Θs
∆R

LA MA

Θp

Search Space Axis LA

R

¨ % ¨

r j r
∆Θp LV (a) (b) LV ∆Θs (c)

Fig. 2. Example of the location (relative to the mitral valve) and spacing of the search space axis, planes, arcs, and sample points. (a) Open mitral valve in a clinical 3DUS slice. (b) Atrial view of an open mitral valve showing search planes oriented about the search axis. (c) Cut-plane view of an open mitral valve showing the search arc and sample point conﬁgurations. (LA - left atrium; LV - left ventricle; MA - mitral annulus)

along each arc is found ﬁrst using a graph-cut method to estimate the leaﬂet location, and in the ﬁnal pass, the location is reﬁned to account for surface curvature and the image.

2.3

Estimating an Extended Leaﬂet Surface

The location of the leaﬂets is estimated by ﬁtting a surface to their believed location. The surface is found as the min-cut of a graph [8], where the source connects to all points at the minimum Θs and the sink to all points at the maximum. The undirected edges of the graph, Γ , connect neighboring points at Θp ± ∆Θp , R ± ∆R, and Θs ± ∆Θs , making Γ 6-connected. The edges of the graph have a capacity that is a function of a drive image, which is computed at each point. The drive image on the ﬁrst pass is a product of the original 3DUS intensity (I) and a thin tissue detector (TTD) [6], Idrv = (I) (TTD). For the ﬁnal pass, because we have an estimate for the leaﬂet location, a weighting function, W, is incorporated into the drive image, Idrv = (I) (TTD) (W), to encourage the min-cut to be found at the estimated leaﬂet location. The edge capacity between points i and j is then deﬁned as Eij = 1
2,

1 + ω (Idrv,i + Idrv,j )

(1)

where ω is a scalar weight. The min-cut of the graph is found using Kolmogorov’s implementation [9]. The point representing the leaﬂet surface location along each arc is herein referred to as a leaﬂet node.

2.4

Trimming an Extended Leaﬂet Surface

The extended leaﬂet surface should reside at the location of the leaﬂets and extend past the edges of the leaﬂets into the blood pool of the left ventricle. It is therefore necessary to trim the surface at the leaﬂet edge so that the remaining leaﬂet geometry accurately portrays the location and geometry of the mitral leaﬂets. The trimming operation is done by isolating the nodes that reside at the leaﬂets from those that reside in the blood pool. This is done by ﬁrst reducing the ultrasound intensity information at the surface nodes to a two-dimensional image, where the axes of the image are the search plane angle and the arc radial oﬀset (Figure 3). The leaﬂet nodes are then separated from the blood pool nodes using the RCAC level set method [3], where the level set function is initialized using a k-means algorithm. The annulus nodes are always made to be included in the leaﬂet group during the level set evolution, and the continuity between the ﬁrst and last search plane is maintained. Nodes that are not found to be a member of the leaﬂet group are considered to be part of the blood pool and removed from the surface.

E Θp
R

c

(a)

(b)

(c)

Fig. 3. (a) Example of an extended and reﬁned leaﬂet surface. The gray level of the triangles reﬂect the 3DUS intensity. (b) The intensity image formed from the ultrasound intensity at the surface nodes. Also shown is the computed trimming contour (dashed white line) found using a level set formulation. (c) The surface from (a) that is trimmed according to the contour shown in (b). The black contours in (a) and (c) indicate the location of the mitral annulus.

2.5

Redeﬁning the Search Space Axis

The search space axis is initially deﬁned with only the knowledge about the general orientation of the leaﬂets. The problem with this generic assignment is that information about the leaﬂet position is unknown, and so the axis could potentially pass through the leaﬂet tissue, thereby creating an inconsistency in the estimated leaﬂet location. However, as information about the estimated leaﬂet location becomes available, the axis can be better deﬁned to account for the position of the leaﬂets. This is done by making the axis the vector that passes through the mitral annulus center point and the center of the estimated leaﬂet opening. In doing so, the axis will not pass through any leaﬂet tissue, and will provide for a more appropriate search space in which to ﬁnd the leaﬂets.

2.6

Reﬁning an Extended Leaﬂet Surface

The min-cut algorithm is well suited for estimating the location of the extended leaﬂet surface, but oftentimes fails to capture ﬁner leaﬂet detail. Therefore, a surface reﬁnement method treats the surface as an active surface and drives the it toward the location of the leaﬂets in the image and also regulates surface curvature. In the reﬁnement process, the nodes are restricted to move along their respective arcs in the search space. An image energy, Eimg , which is the inverse of the ultrasound intensity, drives surface nodes to the leaﬂet location along the arc. An internal energy, Ecrv , regulates surface curvature by driving surface nodes to locations that will result in uniform edge length. The minimized surface energy equation is then Esurf = ωimg Eimg + ωcrv Ecrv , where ωimg and ωcrv are scalar weights.

2.7

Mesh Generation

The nodes of the trimmed leaﬂet surface are formed into a triangular mesh by ﬁrst connecting nodes to those neighbors that exist at Θp ±∆Θp and R±∆R. Diagonal edges are then deﬁned between nodes at (Θp ,R) and (Θp + ∆Θp , R + ∆R). The normals of the triangles are deﬁned such that the positive normal direction points toward the top (atrial) side of the mitral leaﬂets.

3

Performance and Validation

We assessed the performance of the mitral leaﬂet modeling algorithm on a group of four anonymized clinical pediatric cases. The mitral valves in these cases were imaged using transesophageal echocardiography (iE33 Echocardiography System with X7-2t transesophageal probe, Philips Healthcare, Andover, MA, USA). A frame showing an open valve was manually selected from each of the four cases. The annulus, which acted as the input to the algorithm, was found using our semi-automatic annulus segmentation method [6] and a tracking algorithm based on the Lucas & Kanade method [7]. Using the location of the tracked annulus in the manually selected frame with an open valve, the leaﬂets were then found using the presented method. The leaﬂets in the chosen frames were manually traced in cut planes taken every 10o about the search space axis and compared to the meshes generated by the algorithm (Figure 4). The intersection of the leaﬂet mesh with the cut planes generated contour segments in the plane. A histogram of the distances of points along these contours from the manually traced leaﬂet contours were then computed (Figure 5). As indicated in the histogram plot, on average 95% of the leaﬂet mesh points resided within 2.10mm of the manually traced leaﬂets, with the average distance being roughly 0.76 ± 0.65mm. These errors were on the order of the ultrasound volume resolutions (0.5-1.0mm/voxel) and leaﬂet thicknesses observed in the volumes (2-3mm).

(a) Valve 1

(b) Valve 2

(c) Valve 3

(d) Valve 4

Fig. 4. Comparison of the algorithm-generated leaﬂet model (shaded surfaces) to manual tracings (contours) made in cut planes about the search space axis.

18 16 14 Valve 1 Valve 2 Valve 3 Valve 4 Average

Percent (%)

12 10 8 6 4 2 0 0 0.5 1 1.5 2 2.5 3

95th Percentile

3.5

4

Distance (mm)

Fig. 5. Histogram of the distances of the algorithm mesh from the manually traced points in each cut plane for each valve in Figure 4 and for the group average. The 95th percentile is shown for the group average.

4

Discussion

This paper presents a method that automatically generates a model of the leaﬂets for an open mitral valve in a 3DUS volume given the location of the mitral annulus. The presented method diﬀers from previous leaﬂet modeling methods in that it does not require extensive user interaction and generates a detailed, patient-speciﬁc leaﬂet mesh. The method uses a two-pass strategy that deﬁnes and reﬁnes a search space coordinate system and the leaﬂet surface to generate a mesh at the location of the leaﬂets. We showed the accuracy of this method by comparing the mesh to manual tracings of the leaﬂets made in cut planes taken about a designated axis, where it was found that 95% of the points along the mesh were less than 2.10mm away from the manual tracings. The leaﬂet modeling is robust and accurate partly due to the fact that it exploits prior knowledge about where to ﬁnd the leaﬂets based on the location of the mitral annulus. Accurately locating the annulus is therefore important as its location directly aﬀects the leaﬂet model. To ﬁnd the annulus, we ﬁrst use our annulus segmentation algorithm to ﬁnd the annulus in a frame with a closed valve [6]. We then track that annulus to all other frames using a modiﬁed version of the Lucas & Kanade optical ﬂow method [7]. We compared the location of the tracked annulus to manual delineations made by three experts across 30 frames and found an average RMS error of 1.67 ± 0.63mm between the tracked annulus and the expert mean. The accuracy of both the annulus and leaﬂet modeling methods makes for one of the most accurate and robust mitral valve modeling methods capable of capturing detailed patient-speciﬁc valve geometry. A limitation of the presented leaﬂet modeling method is that the 3DUS images of the mitral valve need to show the leaﬂets as a thin structure in an open valve state. This suggests that short-axis views of the valve will not suﬃce, as the chordal structure that becomes visible in this approach gives the leaﬂets

a thick appearance. Also, the valve cannot be completely open (peak diastole) as the leaﬂets are pushed up against the left ventricle wall and are typically indistinguishable from the wall as seen in 3DUS. The recommended images are then those that use either an enface or apex view of the mitral valve and capture the valve during the process of opening or closing. In these views, the anterior and posterior leaﬂets should be diﬀerentiable. Future work includes using the deﬁned leaﬂet mesh as a geometric prior to be tracked to other frames in a 3DUS sequence to provide for a four-dimensional image-driven model of the leaﬂets. The beneﬁt in this approach is that anterior and posterior mitral leaﬂets will be diﬀerentiated throughout a sequence, and because leaﬂet geometry can be preserved and collisions resolved, a coaptation line and surface should be found.

5

Acknowledgments

This work is supported by the U.S. National Institutes of Health under grant NIH R01 HL073647-06.

References
1. Kunzelman, K., Einstein, D., Cochran, R.: Fluid–structure interaction models of the mitral valve: function in normal and pathological states. Philosophical Transactions of the Royal Society B: Biological Sciences 362 (2007) 1393 2. Hammer, P., Perrin, D., Pedro, J., Howe, R.: Image-based mass-spring model of mitral valve closure for surgical planning. In: Proc. SPIE - Medical Imaging. Volume 6918. (2008) 69180Q1–8 3. Shang, Y., Yang, X., Zhu, L., Deklerck, R., Nyssen, E.: Region competition based active contour for medical object extraction. Computerized Medical Imaging and Graphics 32 (2008) 109–117 4. Burlina, P., Sprouse, C., DeMenthon, D., Jorstad, A., Juang, R., Contijoch, F., Abraham, T., Yuh, D., McVeigh, E.: Patient-Speciﬁc Modeling and Analysis of the Mitral Valve Using 3D-TEE. In: Information Processing in Computer-Assisted Interventions. Volume 6135 of Lecture Notes in Computer Science. (Springer) 135– 146 5. Ionasec, R., Voigt, I., Georgescu, B., Wang, Y., Houle, H., Vega-Higuera, F., Navab, N., Comaniciu, D.: Patient-Speciﬁc Modeling and Quantiﬁcation of the Aortic and Mitral Valves From 4-D Cardiac CT and TEE. Medical Imaging, IEEE Transactions on 29 (2010) 1636–1651 6. Schneider, R., Perrin, D., Vasilyev, N., Marx, G., del Nido, P., Howe, R.: Mitral Annulus Segmentation from 3D Ultrasound Using Graph Cuts. IEEE Transactions on Medical Imaging 29 (2010) 1676–87 7. Lucas, B., Kanade, T.: An iterative image registration technique with an application to stereo vision. In: Proc. 7th International Joint Conference on Artiﬁcial Intelligence, Vancouver, B.C., Canada (1981) 674–679 8. Boykov, Y., Kolmogorov, V.: An experimental comparison of min-cut/max-ﬂow algorithms for energy minimization in vision. IEEE Transactions on Pattern Analysis and Machine Intelligence 26 (2004) 1124–1137 9. Kolmogorov, V.: Software (2004) http://www.cs.ucl.ac.uk/staﬀ/V.Kolmogorov/software.html. [Accessed: October 1, 2008].