A Complete System for Automatic Extraction of Left Ventricular Myocardium From CT Images Using Shape Segmentation and Contour Evolution

Abstract

The left ventricular myocardium plays a key role in the entire circulation system and an automatic delineation of the myocardium is a prerequisite for most of the subsequent functional analysis. In this paper, we present a complete system for an automatic segmentation of the left ventricular myocardium from cardiac computed tomography (CT) images using the shape information from images to be segmented. The system follows a coarse-to-fine strategy by first localizing the left ventricle and then deforming the myocardial surfaces of the left ventricle to refine the segmentation. In particular, the blood pool of a CT image is extracted and represented as a triangulated surface. Then, the left ventricle is localized as a salient component on this surface using geometric and anatomical characteristics. After that, the myocardial surfaces are initialized from the localization result and evolved by applying forces from the image intensities with a constraint based on the initial myocardial surface locations. The proposed framework has been validated on 34-human and 12-pig CT images, and the robustness and accuracy are demonstrated.

I. Introduction

Cardiovascular diseases are the leading causes of death in the world [1]. The diagnosis and treatment of these diseases may rely on different cardiac image modalities. No matter what modality is used, the clinical importance of delineating the myocardial boundaries is the highest [2]. Though this tedious task can be done manually in about 20 minutes with sophisticated interactive segmentation tools, the intra- and inter-observer variability is still inevitable [3]. The automatic segmentation methods with high accuracy are attractive because of the previous mentioned reasons. In this paper, we will focus on the segmentation of the left ventricular myocardium from computed tomography (CT) images.

A. Related Work

The main challenges in extracting the myocardium include large shape variability within cardiac cycles and between different patients, and weak edges between epicardium and heart fat or soft tissues. To get an accurate and robust segmentation, model-based methods have become dominant in this research [2]. Heart models are commonly used to represent the geometric or intensity features of the heart, and they are applied either explicitly or implicitly for segmentation. In the first type of methods, models created off-line are fitted to images for segmentation. For example, active shape models (ASMs) [4] build a statistical shape model from a set of aligned shapes by using the principal component analysis (PCA) technique, and have been used for left ventricle segmentation [5], [6]. Active appearance models (AAMs) extend this idea by incorporating gray level information [7] and have been used in segmenting the left and right ventricles from MR images [8]. The deformations allowed in the parametric models such as ASMs and AAMs are confined to the shape space where the heart models are embedded. A more sophisticated way of representing the shape space is by using 3D diffusion wavelets [9], which encode shape variations hierarchically. Using deformable models provides a flexible way to incorporate shape priors that are capable of adapting to local image content. For example, Ecabert et al. [10] modeled the whole heart as a multi-compartment, triangulated surface. The local adaptation was achieved by progressively optimizing the piecewise affine transformations of this model to match image boundaries. In [11], a mean shape of the heart was fitted to an image by estimating similarity transformations, which was then deformed to match image boundaries with the help of landmark points on the interventricular septum. Instead of deforming a pre-aligned model, atlas-based methods use shape information implicitly by directly registering each atlas image to a target image. Then, either the labels from multiple atlases are fused [12] or one single registered atlas is deformed [13] to extract the heart region. Model-free methods have also been widely used to explore the characteristics of heart geometry or intensity distribution from other perspectives. For example, the geometric and intensity features in the myocardial region were learned by using a random forests method for delineating the myocardium [14]. For a comprehensive literature review of heart segmentation, see [2], [3] and references therein.

Active contour models have been widely used in medical image segmentation because of their flexibility and robustness. In these models, energy functionals are commonly defined over image features such as edges [15], [16], region statistics [17], local characteristics [18], [19], and a combination of edges and regions [20], [21], which are optimized by using gradient descent techniques. Prior information can be incorporated as well to restrict the optimization space. For example, in [22], an active contour model was evolved in the shape space of the left ventricle obtained by applying the PCA to manually segmented images. Local variations may be captured by decomposing images into different regions using prior information for ventricles segmentation [23], [24] or by modeling a shape prior using pixel-wise stochastic level sets to extract the endocardium [25]. A shape constraint was also employed to control the search space of the myocardial contours between two consecutive image slices [26]. Coupled active contours have been proposed with distance constraints between contours for myocardium extraction [27], cortex segmentation [28], and cell tracking [29].

One important but less studied topic is how to locate the heart initially, especially for these methods using deformable models, which tend to get stuck in undesirable local extrema when started without a good initialization. Typically, the geometric features of the heart are used for localization. In [26], the endocardium was initialized by searching for a circular structure in a blood pool mask obtained via thresholding. Similar empirical rules were used to identify the left ventricle cavity [30]. To capture a more generic shape of the heart, the generalized Hough transform was utilized for heart detection [10]. In [11], the localization was achieved by searching for a similarity transformation in a hierarchical way. Atlas-based registration has also been used for coarse initialization [9], [13].

One fact that has been ignored in the literature for the localization is that the left ventricle is a salient component on the heart surface. This is where the shape decomposition/segmentation technique may be utilized to cluster the surface into meaningful components based on some given criteria as in computer graphics and geometric modeling [31], [32]. For example, a surface may be hierarchically decomposed into regions of deep concavities by using fuzzy clustering and graph partition techniques [33]. Prominent feature points [34] have also been used to cluster a surface into meaningful regions. Applications of shape segmentation in medical imaging can be found in heart modeling from images [35] and aneurysm neck detection on vessel surfaces [36]. Active contour models have been applied as well on surfaces to refine coarse segmentations [37] or extract objects of interest [38]. Among the few applications of the shape decomposition techniques to cardiac image segmentation, the narrowing of vessels around the left atrium was detected by merging local features based on given criteria to extract the left atrium [39]. As for the left ventricle localization, the region near the left ventricle is much more recognizable from the heart surface than from the volumetric data, which can be identified by a deep concave contour.

B. Method Overview and Our Contributions

In this work, we present a complete system for automatically extracting the myocardium from cardiac CT images without using training images. A coarse-to-fine strategy, consisting of global localization and local deformations, is used for the myocardium segmentation. The flowchart of the proposed method is shown in Fig. 1.

Fig. 1.

Flowchart of the proposed approach.

Before starting the localization step, the heart surface is approximated by the blood pool surface. Then, the apex point of the left ventricle on this surface is detected by using the relative orientation of ventricles with respect to the physical coordinate system of a CT image. The left ventricle is automatically detected by examining the distribution of the level sets starting from the apex point, which is further refined by performing the geometric active contour model on the blood pool surface. This contour decomposes the surface into two parts, and the one contains the apex point is chosen as the initial endocardial surface. Once the endcoardial surface is located, its corresponding mask is obtained via rasterization. Then, a variational region-growing method [40] is used to extract the initial epicardial surface based on the endocardium segmentation. Finally, these two surfaces are refined by employing an active contour model with a shape constraint, and the myocardium is obtained by extracting voxels between these surfaces.

The contributions of the proposed method are as follows:

• 1: we utilize the shape segmentation technique for localizing the left ventricle. Unlike other methods that only use low level information from voxels, our method captures a global geometric characteristic of the left ventricle that agrees with our visual perception. Hence, it is not sensitive to such issues as shape variability and changes of volume coverage. Note that, as an initialization step, the proposed method can be easily incorporated into other model-based frameworks.

• 2: we use a variational region-growing method to locate the epicardial surface given the segmentation of the endocardial surface. Then the localized endocardial and epicardial surfaces are employed as a constraint for the final segmentation. In this formulation, the shape variability is naturally handled and incorporated into our system without using training images. In addition, instead of simply imposing a constraint on the point-wise distance between two contours [28], the one used in our model is a surface-wise restriction that uses a distance field for guiding contour evolution process.

Therefore, the overall system is complete in that all the active contour models involved are initialized automatically and robustly, other than in those systems that active contours are either used as a single component or initialized manually.

The rest of this paper is organized as follows: Section II describes the details of left ventricle localization. Section III introduces the active contour model with a shape constraint obtained from the localization results. The robustness and accuracy of the proposed method are reported in Section IV. Finally, Section V concludes this paper.

II. Left Ventricle Localization

Assume that the orientation of a CT image is given and intensity contrast exists between blood pool and myocardium. The localization of the left ventricle is determined via searching for a deep concave boundary on the blood pool surface as follows.

A. Extract Blood Pool Surface

The extraction of the blood pool surface is carried out by a few mature techniques in the computer vision and graphics: since CT images have calibrated gray levels, the source image is thresholded to highlight the blood pool region. Then, the morphological opening operator is applied to remove noisy arteries and cut spines that may be residing in the same connected component of the heart. After that, the largest connected component is chosen and triangulated to get the blood pool surface.

B. Detect Apex Point

Suppose the coordinate system of the source image is Left-Posterior-Inferior (LPI) as shown in Fig. 2. In this system, the XYZ coordinates trace from left to right, posterior to anterior, and inferior to superior. Even though the long axis of pig and human heart has different orientations [41], the directions of left and right are clearly defined from the inferior view. The apex point is one salient feature that can be used to locate the left ventricle. Its location is determined as follows: 1) estimate the orientation of ventricles; 2) search for the left ventricle apex, which is the left tip point with respect to the estimated orientation.

Fig. 2.

Orientations of (a) human and (b) pig blood pool surfaces in the source image coordinate system. The reference directions are left(L), right(R), posterior(P), anterior(A), inferior(I), and superior(S).

To estimate the orientation of ventricles, the convex hull of the blood pool surface M_bp is first constructed. Let K(p̃) be the Gaussian curvature at each vertex p̃ of the convex hull. The vertices used for estimating the ventricle orientation are selected as

$$V_{ch}(\stackrel{\sim }{p})=\{\stackrel{\sim }{p}\mid K(\stackrel{\sim }{p})>{\mu }_K+{\sigma }_K\cap y(\stackrel{\sim }{p})>t_y\},$$ (1)

where μ_K and σ_K are the mean and standard deviation of K(p̃), and a threshold t_y defines the region of interest for the ventricles, which was empirically set as t_y = y_min + 0.5(y_max − y_min) to select points in the top half of the source image in the Y direction. Then, all points p̃ ∈ V_ch are translated as p̃_s = p̃ − μ_bp, where μ_bp is the centroid of M_bp. The PCA technique is utilized to find the principal component of these translated points p̃_s as the orientation of the ventricles, denoted by H. The positive direction is chosen so that H has negative component in the Y direction. A plane L_O passing through μ_bp with normal N = Z × H defines a reference plane such that the left ventricle points are mainly above the plane and otherwise for the right ventricle.

Let λ_max ∈ ℝ s.t. λ_max = max(p_s · H). A reference point is defined as p_rf = μ_bp + λ_max H. A constraint region-labeling process is employed to search for a neighborhood of the left ventricle apex. First, the corresponding points of V_ch on surface M_bp are sorted in an ascending order based on their distance to p_rf, which is denoted by {p_j}, j = 1 · · · n_ch, where n_ch is the cardinality of the set V_ch. Then, the distance field starting from p₁ is constructed using the fast marching method [42]. All points with their distances smaller than a threshold t_n are labeled as 1. After that, the unlabeled point with the smallest index in {p_j} is checked. If its distance to p_rf is smaller than the maximum distance between labeled points and p_rf, then the point is selected to start the labeling process again and the points within the range of t_n are labeled as 2. Let μ₁ and μ₂ be the centroids of these two labeled regions, respectively. The region with a larger projection (μ_i − μ_bp) · N, i = 1, 2, is selected as a neighborhood N_apx around the left ventricle apex. If only one labeled region is found, it is automatically selected as N_apx. Finally, the point in N_apx with the largest projection in N direction is selected as the apex point.

An illustration of the apex detection process is shown in Fig. 3.

Fig. 3.

Apex detection of human and pig hearts. Convex hull with high curvature points (green) for the human (a) and pig (c) hearts. Neighborhoods around apex points for human (b) and pig (d) hearts. The detected apex points are marked with red dots. The vector H represents the left ventricle orientation. The plane L_O with normal N identifies the directions of left and right.

C. Identify Cut Contour

A two-step segmentation strategy is used to identify the left ventricle by searching for a cut contour on M_bp.

1) Find an Initial Cut Contour

The initial cut contour, denoted by C₀, is determined based on the distance field starting from p_apx. Sampling this distance field evenly at an interval of 2 mm gives its isocontours/level-sets as shown in Fig. 4(a). The total length of each isocontour increases gradually and then drops slightly as it is traveling along the left ventricle. After that, it goes up first and then drops rapidly as it is propagating to the right ventricle and other regions (see Fig. 4(b)). Thus, the total length of an isocontour at distance d may be modeled by

Fig. 4.

Detection of the initial cut contour. (a) Distance field from the apex with isocontours. The initial cut contour C₀ is marked in red. (b) The length of isocontours vs. distance, and the determination of the optimal $d_j^{\ast }$.

$$h(d,\stackrel{\sim }{d})=\{\begin{array}{ll}{c}_0+c_{1}d+c_2{d}^2\hfill & \text{if}d\le \stackrel{\sim }{d}\hfill \\ {ae}^{-\frac{{(d-\mu )}^2}{2{\sigma }^2}}\hfill & \text{if}d>\stackrel{\sim }{d},\hfill \end{array}$$ (2)

where d̃ is serving as a turning point for these changes. Let (d_i, l_i), i = 1 · · · n, be a pair of a sampled distance and its corresponding isocontour length, where n is the number of samples over the distance field. Then, choose the isocontour at d_j^* as the initial cut contour, where j^* is determined by searching for the optimal turning point that minimizes the least-squares fitting error,

$$J^{\ast }=\underset{j}{argmin}\left(\underset{c_0,c_1,c_2}{argmin}\sum _{i=1}^j{(h(d_i,d_j)-l_i)}^2+\underset{a,\mu ,\sigma }{argmin}\sum _{i=j+1}^n{(logh(d_i,d_j)-log{l}_i)}^2\right)$$ (3)

Here, d_j is a sampling point serving as a trial turning point. The optimal turning point d_j^* is obtained by exhausting all elements in {d_j}, j = 1 · · · n. An illustration of the model fitting process described above is shown in Fig. 4.

2) Refine the Cut Contour

The geometric active contour model [15], [16] is utilized to refine the initial cut contour C₀. Suppose a contour on the surface M_bp is represented by the zero level set of a function U : M → ℝ with U(C(p, t)) = 0, where C(p, t) is a family of contours on M. Let g : M_bp → ℝ⁺ be a positive function that attracts an active contour to a conceptually desired boundary [37], defined as

$$g(p)=\frac{1}{1+{\left(\mid \kappa (p)\mid /S\right)}^2},$$ (4)

where κ(p) is the mean curvature at p, and S is a constant for a scaling to enhance the concave regions so that the values of g(p) for such regions are not overwhelmed by those of other regions. In our implementation, S = 0.01 was used in all the experiments. As in [37], κ is set to zero if it is positive. Then, the geometric active contour model on the surface M_bp is formulated in the level set framework as

$$E(U)={\int }_{M_{bp}}g(p)\delta (U(p))\mid {\nabla }_{M_{bp}}U(p)\mid dp,$$ (5)

where δ(U) is the Dirac delta function. The energy E(U) evaluates a weighted contour length. Similar as in ℝⁿ, the gradient descent flow of E(U) is

$$\{\begin{array}{l}{U}_t=\mid {\nabla }_{M_{bp}}U\mid {\nabla }_{M_{bp}}·\left(g(p)\frac{{\nabla }_{M_{bp}}U}{\mid {\nabla }_{M_{bp}}U\mid }\right)\hfill \\ \frac{\partial U}{\partial \stackrel{\rightharpoonup }{n}}{\mid }_{\partial M_{bp}}=0\hfill \\ U(0)=U(C_0),\hfill \end{array}$$ (6)

where ∂M_bp is the boundary of M_bp and n→ is the intrinsic outward normal of ∂M_bp. Here, ∇_M^bp is the del operator on M_bp. This flow drives a contour to segment desired boundaries while minimizing the weighted contour length. In particular, it moves a contour on M_bp by its geodesic curvature when g = 1, which produces the contour of the shortest length.

Numerically, a narrow band method is employed to solve Equation (6). The main steps are summarized as follows.

Algorithm.

Narrow band for Geometric Active Contour

Initialize the level set function U with C0. Construct a narrow band ΩMbp around the current contour on Mbp. Update U in ΩMbp, according to $$U(p,t+1)=U(p,t)+dt{\left(\mid {\nabla }_{M_{bp}}U\mid {\nabla }_{M_{bp}}·\left(g\frac{{\nabla }_{M_{bp}}U}{\mid {\nabla }_{M_{bp}}U\mid }\right)\right)}_{(p,t)},$$ (7) where dt is the time step in discretizing U. Find the new zero level set of U to update the contour C. Repeat steps 2–4 until it converges or reaches the maximum number of iterations.

In step 1, U is realized as the signed distance function from C₀, which decomposes M_bp into several regions. The sign of U is positive in the region that contains the apex point. The fast marching method is used to build Ω_M^bp from C₀ with a threshold ϕ_max to control the size of the narrow band.

Step 3 requires numerical approximations for gradient and divergence operators on a surface. The discretization schemes of [38] are adopted because they naturally capture the geometric properties of surfaces. The surface M_bp is represented by a triangle mesh as $M_{bp}=\{V={\{p_i\}}_{i=1}^N,T={\{T_k\}}_{k=1}^L\}$, where p_i ∈ ℝ³ is the ith vertex and T_k represents the kth triangle. The surface gradient and divergence operators at a vertex are approximated by taking the weighted average of the discretized operators over the first-ring neighbors of the vertex.

Let f = {f(p₁), f(p₂), f(p₃)} be a function and V = {V(p₁), V(p₂), V(p₃)} a vector field defined at each vertex. For any point p inside T_k, the values of f and V can be interpolated as

$$\{\begin{array}{l}p={\chi }^1(p_1-p_3)+{\chi }^2(p_2-p_3)+p_3\hfill \\ f(p)={\chi }^1(f(p_1)-f(p_3))+{\chi }^2(f(p_2)-f(p_3))+f(p_3)\hfill \\ \mathbf{V}(p)={\chi }^1(\mathbf{V}(p_1)-\mathbf{V}(p_3))+{\chi }^2(\mathbf{V}(p_2)-\mathbf{V}(p_3))+\mathbf{V}(p_3)\hfill \end{array}$$ (8)

where (χ¹, χ², 1 − χ¹ − χ²) is the barycentric coordinate of T_k. Here, χ = (χ¹, χ²) defines a local coordinate system for M_bp. Then the approximations of these two operators are

$${\nabla }_{M}f(p_i)=\frac{1}{{\sum }_l\mathit{Area}(T_l)}\sum _l\mathit{Area}(T_l){\nabla }_{T_l}f(p_i)$$ (9)

$${\nabla }_M·\mathbf{V}(p_i)=\frac{1}{{\sum }_l\mathit{Area}(T_l)}\sum _l\mathit{Area}(T_l){\nabla }_{T_l}·\mathbf{V}(p_i),$$ (10)

where l traces through all triangles in the first ring of the vertex p_i to average the discretized operators at each vertex (see Appendix for details). Typically, the maximum narrow band width is set as ϕ_max = min(10, 4l_max) mm, where l_max is the length of the longest edge on M_bp and 10 mm is about 10% of the maximum distance from the apex point to a turning point (see Section IV-B), such that there is enough support to compute the gradient and divergence operators. The time step of dt = 1 was used in our implementation.

In step 4, a new zero level set is obtained from U. The algorithm stops either when the contour stops evolution or the maximum number of iterations is reached.

To reduce the effect of local noise [see Fig. 5(b)], the contour evolution process described above are applied twice. In the first round, set g = 1 so that it shortens the initial contour by its geodesic curvature flow. Then, the feature function defined in Equation (4) is used to refine the contour so that it stops at locally concave locations. Finally, the endocardial surface, denoted by M_endo, is identified by the cut contour that separates the endocardial surface from other regions on M_bp. An illustration of the process for localizing the left ventricle is shown in Fig. 5.

Fig. 5.

Left ventricle localization. (a) Features on surface M_bp. (b) Initial contour C₀ and its narrow band Ω_M^bp. (c) Final contour C_cut. (d) Segmented endocardial surface.

III. Myocardium Wall Segmentation

The endocardial surface indicates the location of the left ventricle, which is rasterized to get its 3D mask Ĩ_endo for refinement. Instead of simply dilating Ĩ_endo for approximating the epicaridal mask Ĩ_epi, a variational region-growing model [40] is used by taking an outward neighborhood of Ĩ_endo as the seed region. After that, a localized region-based active contour model is utilized with a shape constraint imposed by these initial masks to refine the myocardial segmentation.

A. Initialize the Endocardial and Epicardial Masks

The surface M_endo is closed via triangulating the points along the cut contour C_cut. Here, M_endo is still used to denote the closed surface. Ĩ_endo is created by rasterizing M_endo with the same origin and resolution as the source image I. To remove noise and papillary muscles, the convex hull of M_endo is computed and set as a ROI for performing the morphological closing operation on Ĩ_endo. The size of the structure element for the closing operator depends on the radii of the papillary muscles, which was empirically set as 3 mm.

One way of initializing the epicardial mask is by dilating the endocardium to a given distance. This works well for the myocardial wall with a nearly uniform thickness. Here, we propose another way to initialize Ĩ_epi. Similar as in the dilation-based methods, a distance field from Ĩ_endo is computed and a small strip-region that lies between d_in and d_out away is chosen as the seed region for the epicardial mask. Typically, the values are set to d_in = 2 mm and d_out = 4 mm. The blood pool voxels are excluded from this region. Given the seed region of the epicardial mask, the robust-statistics-based energy functional [40] is defined as

$$E_{\text{RS}}(\mathrm{\Phi })={\int }_{\mathrm{\Omega }}p(\mathit{f}(x))H(\varphi (x))dx+{\lambda }_{\text{RS}}{\int }_{\mathrm{\Omega }}\delta (x)\mid \nabla \varphi (x)\mid dx,$$ (11)

where ϕ is the signed distance function from Ĩ_epi and H is the Heaviside function. Here, p(f(x)) is the probability density function of a feature vector f(x) that evaluates the intensity median, inter-quartile range, and median absolute deviation at point x, respectively, where p(f(x)) is learned from the seed region intensities. The first term in Equation (11) measures the intensity homogeneity inside the contour, and the second term is the length of the contour that controls the smoothness of the final result. In implementation, λ_RS was empirically set as λ_RS = 0.2.

The myocardial masks Ĩ_epi and Ĩ_end provide a good localization of the epicardium and endocardium, which impose a shape constraint for local refinement, especially on regions of low contrast or poor edges.

B. Evolve Active Contours With a Shape Constraint

Given a mask image Ĩ ∈ {Ĩ_endo, Ĩ_epi}, a feature function g̃ : ℝ → ℝ⁺ is defined over its signed distance function ϕ̃ as

$$\stackrel{\sim }{g}(\stackrel{\sim }{\varphi })=(d_{\mathit{max}}-d_{\mathit{min}}){\left(1+\mathit{exp}\left(-\frac{\mid \stackrel{\sim }{\varphi }\mid -\beta }{\alpha }\right)\right)}^{-1}+d_{\mathit{min}}.$$ (12)

Here, g̃(ϕ̃) is a modified sigmoid function, where α and β define the shape and width of the function, respectively, while d_max and d_min control the range of this function (see Section IV-B for details).

Then, the energy functional for refining the mask images is designed as

$$E(\varphi )={\int }_{{\mathrm{\Omega }}_x}\delta (\varphi (x)){\int }_{{\mathrm{\Omega }}_y}B(x,y)F(I(y),\varphi (y))\mathit{dydx}+{\lambda }_{\text{LG}}{\int }_{\mathrm{\Omega }}\stackrel{\sim }{g}(\stackrel{\sim }{\varphi }(x))\delta (x)\mid \nabla \varphi (x)\mid dx,$$ (13)

where B(x, y) is a ball of radius r centered at x, and F(I(y), ϕ(y)) is a generic internal energy term defined over Ω. In this work, the Chan-Vese energy [17] was used for F(I(y), ϕ(y)). The first term in Equation (13) is a localized region-based energy [21]. The second term is essentially the geometric active contour energy [15], [16] using the feature image g̃ to prevent the contour from evolving far from its initial location.

The sparse level set method [43] was used to implement the active contour model for its efficiency. In particular, the upwind scheme was used in discretizing |∇(·)| (see [44] for details). The refinement of the endocardial and epicardial masks were performed separately. In initializing the epicardial mask, a parameter d_w was used to control the maximum distance allowed in the region-growing process. This parameter is related to the average thickness of the myocardial wall, which typically ranges from 6 to 16 mm [3]. In our implementation, d_w was empirically set as d_w = 16 mm, and r = 4 mm was used for the radius of B(x, y). See Section IV-B for details.

C. Extracting the Myocardial Wall

The myocardial wall is defined as the volume between the endocaridal and epicardial masks. Note that the contour evolution process returns closed masks. To extract a complete myocardial wall, the voxels inside the blood pool need to be removed [seed Fig. 6(d)]. To this end, the wall is divided into two parts: one in which the myocardium can be completely determined by performing the XOR operation between the endocardial and epicardial masks, and the other formed by removing the voxels inside the blood pool from the epicardial mask. Let { $p_i^c$}, i = 1 · · · n_c be the points on C_cut. The unit normal of the plane that divides the wall is determined by

Fig. 6.

Segmentation of the myocardial wall. Initialize the endocardial surface (red) (a) before and (b) after removing papillary muscles. (c) Initialize the epicardial surface (yellow) from the initial endocarial surface (red) with a seed region (green). (d) Evolve the myocardial surfaces from the initial contours (yellow). (e) Extract the myocardial wall from endo- and epi- cardial masks by using the dividing plane (green). (f) The 3D visualization of the segmented myocardial wall.

$${\mathbf{N}}_L·(p_i-p_m)=0\mathrm{s}.\mathrm{t}.{\mathbf{N}}_L·(p_{\mathit{apx}}-p_m)<0,$$ (14)

where p_m is the centroid of { $p_i^c$}, and the constraint specifies the normal direction. The dividing plane is defined as

$${\mathbf{N}}_L·(x-p_L)=0,$$ (15)

where $p_L\in \{p_i^C\}$ and N_L · (p_i − p_L) ≤ 0, ∀i = 1 · · · n_c. That is, all points of C_cut lie on the same side of the dividing plane as p_apx does. The myocardial wall is the set of voxels that satisfy

$$V_{\mathit{Wall}}=\{\begin{array}{ll}{\stackrel{\sim }{V}}_{\mathit{Endo}}\oplus {\stackrel{\sim }{V}}_{\mathit{Epi}}\hfill & \text{if}{N}_L·(x-p_L)\le 0\hfill \\ {\stackrel{\sim }{V}}_{\mathit{Epi}}\cap {\overline{V}}_{BP}\hfill & \text{otherwise},\hfill \end{array}$$ (16)

where V̄_BP are the set of voxels outside of the blood pool, Ṽ_Endo and Ṽ_Epi are the volumes enclosed by the endocaridal and epicardial masks, respectively. An example of extracting the myocardial wall is given in Fig. 6. It is clear that, by using the procedure described above, a smooth endocardial mask is extracted in the presence of noise and papillary muscles. Further, the epicardial mask is separated from the background with soft tissues around the apex point and cut off right around the base area. The shape constraint makes the evolution process go smoothly while preserving the overall shape as well as adapting to the local intensity content.

IV. Experimental Results

This study was approved by the Institutional Review Board. We tested the robustness and accuracy of our method using cardiac CT images of 34 human and 12 pig hearts. The data include anomaly cases (hypertrophic cardiomyopathy and aneurysm) and volumes with different scanning quality.

A. Implementation

The overall framework was implemented in C++. Open source packages ITK [45] and [46] were used for basic image processing tasks, convex hull extraction and 3D visualization, respectively.

In extracting the blood pool, the source image was down-sampled to a voxel resolution of 2.0 × 2.0 × 2.0 mm³, where the heart shape is well preserved. The thresholds of 180 and 350 in Hounsfield units were used for human and pig images, respectively, so that the left and right ventricles can be separated after thresholding.

B. Parameter Determination and Robustness

To test the robustness of localizing the left ventricle described in Section II, sample points randomly selected from a neighborhood within 30 mm of the apex point were used as trial apex points to start the localization process. To quantify the errors, the distance between the cut contours obtained by using different sample points and the original contour was measured as

$$d(C,\stackrel{\sim }{C})=\frac{{\int }_{\stackrel{\sim }{C}(s)}D(C)ds}{{\int }_{\stackrel{\sim }{C}(s)}ds}.$$ (17)

Here, C is the original cut contour, D is the distance field starting from C, and C̃ is the cut contour obtained from a sample point. In implementation, D(C) was evaluated at triangle vertices. The initial and final errors d(C, C̃) for 20 randomly sampled points is shown in Fig. 7. All of the cut contours generated with sampled apex points converge to the original C with a tolerable numerical error. In addition, the average of the turning point d̃ are 94 ± 9 mm and 76 ± 6 mm for the human and pig data, respectively. This result shows that the feature used for identifying the cut contour is stable despite of the variability in heart shapes.

Fig. 7.

The initial and final errors for the cut contours generated from 20 randomly sampled apex points.

Similarly, the sensitivity of the scale factor S defined in Equation (4) was tested by computing the average distance between the original cut contour and the ones obtained with different scale factors. Experiments showed that the average distance was 0.08 mm as S ranging from 0.001 to 0.1.

The parameters in the feature function g̃(ϕ̃) were set as α = 1, β = 5, d_max = 1.0 mm, and d_mim = 0.02 mm, the width of which is slightly smaller than the average myocardium thickness. Example shapes of this function with varying α are shown in Fig. 8. It implies that larger values of α give a stricter constraint while minimizing Equation (13), and thus α = 1 was used.

Fig. 8.

Effects of α to g̃(ϕ̃) with β = 5, d_max = 1 mm and d_min = 0.02 mm.

The sensitivity of the epicardial initialization step to λ_RS and d_w were examined separately. Figure 9 shows the average point-to-surface distance for the example image when λ_RS varies from 0.1 to 0.9 and d_w from 8 to 24 mm, respectively. λ_LG is stable up to 0.8 for the epicardial initialization, after which the growing process stops before reaching the epdicardial boundary due to the high curvature constraint. Regarding d_w, the localization step is stable for d_w > 16 mm, which agrees with the thickness of a normal myocardial wall [3].

Fig. 9.

Sensitivity of the epicardial initialization with respect to λ_RS (Left) and d_w (Right).

The sensitivities of the parameters λ_LG and r were examined by varying one of them in a given range while keeping the other fixed. The coefficients of variation [47] are summarized in Table I. It shows that all these parameters are stable in the given range.

TABLE I.

Coefficient of Variation for Wall Refinement

Parameters	Range	Endo-Surface	Epi-Surface
λLG	[0.2, 0.8]	1.9%	3.4%
r	[2, 10]	1.2%	2.1%

Our method succeeded in localizing the left ventricle for all of the testing data with wide shape variations and volume coverages. Two examples of segmentation for the human data with completely different heart shapes are presented in Fig. 10. The results of pig data with different volume coverages are shown in Fig. 11. As can be seen from these results, the papillary muscles, pericardium, and soft tissues were successfully excluded from the myocardium.

Fig. 10.

Myocardium segmentation results of human data with completely different heart shapes. The first row shows segmentations in a diastole cycle. The second row shows the results of a patient with hypertrophic cardiomyopathy. From left to right are left ventricle localization, myocardial wall in axial, coronal, and sagittal views, respectively.

Fig. 11.

Myocardium segmentation results of pig data with different volume coverages. From left to right are left ventricle localization, myocardial wall in axial, coronal, and sagittal views, respectively.

C. Quantitative Analysis

To make fair comparisons, all testing data were resampled to the same resolution of the manual segmentations at 1.0 × 1.0 × 1.0 mm³. The mean and standard deviation of the point-to-surface errors for the human and pig data at the localization and refinement steps are shown in Table II. These results show that the localization step (LV localization) locates the endocardial and epicardial masks with high precision as compared to the manual segmentations, which are further refined after applying the active contour model with a shape constraint (AC refinement). Note that the accuracy of the endocardial segmentation is better than the epicardial segmentation as the former is better defined because of the high contrast between the blood pool and the myocardium, while in the latter case poor contrast and weak edges are present between the epicardium and background tissues.

TABLE II.

Mean and Standard Deviation of the Point-to-Surface Errors in Localization and Refinement (in mm)

Data	Stages	Endo-Surface	Epi-Surface
Human	LV localization	1.07 ± 1.03	1.27 ± 1.24
	AC refinement	0.88 ± 0.96	1.07 ± 1.16
Pig	LV localization	0.83 ± 0.89	1.05 ± 1.12
	AC refinement	0.72 ± 0.88	0.80 ± 0.99

The proposed method was compared to three types of training-based methods: an active contour method using localized PCA [23], [24], an active shape model (ASM) [4] method, and a standard multi-atlas method with a majority voting scheme [48] for segmenting the pig myocardium using the same dataset. For the latter two methods, the leave-one-out strategy was used due to the small number of images. In particular, the localization results from the proposed method were used to estimate the initial pose of the ASM model for segmentation. Regarding the multi-atlas-based method, affine registration was first applied to align training images to a testing image, which was refined by employing the B-Spline registration method [49]. Parameters were tuned to give the best performance for the three training-based methods. An example of segmentations from these methods is shown in Fig. 12, and the statistics of the surface-to-point errors of these methods are summarized in Table III. These results demonstrate the different characteristics of these four types of methods. The proposed method captured more local image content as compared to the other three methods. The PCA based active contour method also performed well but was less adaptive, since it relies on the sub-space learned from the training images. The ASM method failed to capture the finer details for both the endcardium and epicardium as it is essentially a PCA based method, which depends on how representative the training image are. The atlas-based method was not competitive with the other tested methods as it is mainly driven by global image information, and thus may not be able to compensate for large initialization errors especially when images have a wide volume coverage (see Fig. 11).

Fig. 12.

Comparison of myocardium segmentation. From left to right are results from the proposed method, localized-PCA, ASM, and Multi-Atlas methods, respectively.

TABLE III.

Comparison Between Proposed Method and Training-Based Methods for the Pig Data (in mm)

Method	Endo-Surface	Epi-Surface
Proposed method	0.72 ± 0.88	0.80 ± 0.99
Localized-PCA	1.08 ± 1.12	1.10 ± 1.40
ASM	1.12 ± 0.90	1.35 ± 1.37
Multi-Atlas	1.76 ± 1.97	1.48 ± 2.06

In addition, the proposed method was tested on a computer with Quad CPU 3GHz, 8G RAM. The average processing time is given in Table IV.

TABLE IV.

Average Computation Time (in Seconds)

	Human	Pig
LV Localization	25.2	15.6
AC Refinement	61.6	41.6
Total	89.0	58.1

Given the mask of a myocardium wall, we triangulated it and computed the volume enclosed by this triangulated surface using the same computational method as in [50]. The mass was computed as the product of the volume and density. The density ρ = 1.05g/mL was used in our experiments. For the human data, the average absolute difference between the masses from automatic and manual segmentations was 7.8 ± 5.0g, which is 5.5% ± 3.5% with respect to the mean mass. For the pig data, this difference was 4.2 ± 2.9g, which accounts for 6.4% ± 4.4% of the mean mass.

V. Discussion and Future Work

We have presented a complete system for automatically segmenting the myocardial wall from cardiac CT images. It follows the coarse-to-fine framework by first detecting the left ventricle, and then refining this result by employing contour evolution techniques with a shape constraint obtained on-line. Its performance has been evaluated by measuring the errors between automatic and manual segmentations. In these tests, our method achieved high accuracy as well as strong robustness for segmenting both the human and pig myocardium with large shape variabilities and different volume coverage.

Though the proposed method was specific for the segmentation of the left ventricular myocardium, it may be generalized in several possible ways for broader applications in cardiac image segmentations. It is straightforward to apply the shape segmentation technique to segment the right ventricle from CT images because of the similarity of ventricle structures. In addition, the proposed method may be applied to other image modalities as long as a smooth heart surface can be reconstructed. For example, in MR images, we may manually threshold an MR image to extract the blood pool and then generate the heart surface. Moreover, the proposed method can be easily integrated into user interactive segmentation frameworks, which are widely used in medical image segmentations. In particular, for ventricles with an arbitrary orientation, a user can effectively pinpoint the apex point to start the segmentation.

Therefore, in future work, we plan to extend the current method to segment ventricles both in CT and MR images and apply segmentation results to clinical applications such as evaluating the myocardial mass at risk caused by stenoses.

Biographies

Liangjia Zhu is a Post-Doctoral Fellow with the Department of Computer Science, Stony Brook University. He received the Ph.D. degree in electrical engineering from the Georgia Institute of Technology in 2013. His research interests include image processing, computer vision, and video analysis.

Yi Gao is an Assistant Professor with the Department of Electrical and Computer Engineering, University of Alabama at Birmingham. He is an Associate Scientist with the UAB Comprehensive Cancer Center. His research interests include computer vision, image, and shape computing.

Vikram Appia is currently with the Imaging Technology Laboratory in the Embedded Processing Research and Development Center, Texas Instruments, Dallas, USA. He received the Ph.D. degree from the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, USA, in 2012. He was a Research Intern with GE-Global Research, Bangalore. His research interests lie primarily in the fields of computer vision and image processing with a special focus in the areas of image segmentation, multicamera systems, multimodal data fusion, depth imaging, active contour models, PDE-based methods, and level set based methods.

Anthony Yezzi holds the position of Ken Byers Professor with the School of Electrical and Computer Engineering, Georgia Institute of Technology (Georgia Tech), where he directs the Laboratory for Computational Computer Vision. He has over 20 years of research experience in shape optimization via geometric partial differential equations. He received the Ph.D. degree in electrical engineering from the University of Minnesota in 1997. After completing a post-doctoral research appointment with the Massachusetts Institute of Technology, he joined the faculty of the Georgia Tech in 1999. His research lies primarily within the fields of image processing and computer vision with a particular emphasis on medical imaging and 3D surface reconstruction. He has consulted for a number of companies, including GE, 3M, MZA, Philips, Picker, and VTI. His work spans a wide range of image processing and vision problems, including image denoising, edgedetection, segmentation, shape analysis, multiframe stereo reconstruction, visual tracking, and registration. His research interests include curve and surface evolution, differential geometry, partial differential equations, and shape optimization.

Chesnal Arepalli is a Research Associate with the Department of Radiology, Emory University. He graduated from Gandhi and Osmania Medical Colleges, Hyderabad, India. He was a Clinical Fellow of cardiovascular radiology with AIIMS, New Delhi. He is interested in translating research and findings into clinical arena.

Tracy Faber has been a Professor with Emory University since 1992; previously she was on the faculty of the Southwestern Medical Center, Dallas. She was a member of the program faculty with the Joint Department of Biomedical Engineering, Emory University and Georgia Institute of Technology (Georgia Tech), and the Computing and Visualization Laboratory, Georgia Tech. Her research focused on computer aided diagnosis from medical images. This involves restoration, segmentation, labeling, registration, and 3D visualization.

Arthur Stillman is the William and Kay Casarella Professor of radiology and the Director of cardiothoracic imaging with Emory University. He is a Past President of the North American Society for Cardiovascular Imaging and is the Past Chair of the Cardiovascular Radiology and Intervention Council of the American Heart Association. He is a Distinguished Investigator of the Academy of Radiology Research. He has published over 139 scholarly papers mostly in areas of research involving cardiac MRI and CT. He is the Principle Investigator of a multicenter comparative effectiveness trial comparing coronary CT angiography to SPECT-MPI for guiding patients with symptoms of stable angina to optimized medical therapy (RESCUE: Randomized Evaluation of Patients with Stable Angina Comparing Utilization of Diagnostic Examinations). The findings are expected to result in validation of an evolving new standard of care for patients with stable angina that takes advantage of coronary CT angiography to more cost effectively drive appropriate care.

Allen Tannenbaum has held faculty positions in Israel, Canada, Switzerland, and the U.S. He is currently a Professor of computer science and applied mathematics/statistics with SUNY Stony Brook. He works in image processing, medical informatics, computer vision, and systems and control.

Appendix

Given the definition in Equation (8), the discretized gradient operator on T_k is computed as [38]

$$\begin{array}{l}{\nabla }_{T_k}f(p_1)=\sum _{i,j=1}^2{g}^{ij}\frac{\partial f}{\partial {\chi }^j}{\partial }_{{\chi }^i}\\ =(f(p_1)-f(p_3),f(p_2)-f(p_3))g^{-1}\left(\begin{array}{c}{\partial }_{{\chi }^1}\\ {\partial }_{{\chi }^2}\end{array}\right)\\ =(f(p_1)-f(p_3),f(p_2)-f(p_3))\times {\left(\begin{array}{cc}{\partial }_{{\chi }^1}·{\partial }_{{\chi }^1}& {\partial }_{{\chi }^1}·{\partial }_{{\chi }^2}\\ {\partial }_{{\chi }^2}·{\partial }_{{\chi }^1}& {\partial }_{{\chi }^2}·{\partial }_{{\chi }^2}\end{array}\right)}^{-1}\left(\begin{array}{c}{p}_1-p_3\\ p_2-p_3\end{array}\right)\end{array}$$ (18)

where ∂_χ¹ = p₁ − p₃ and ∂_χ² = p₂ − p₃ are the two tangent vectors at p₃ that span the tangent plane T_pM of a surface M. The matrix g⁻¹ = (g^{i, j}) is the inverse of the Riemannian metrics on M with

$$g=(g_{i,j})=\left(\begin{array}{cc}{\partial }_{{\chi }^1}·{\partial }_{{\chi }^1}& {\partial }_{{\chi }^1}·{\partial }_{{\chi }^2}\\ {\partial }_{{\chi }^2}·{\partial }_{{\chi }^1}& {\partial }_{{\chi }^2}·{\partial }_{{\chi }^2}\end{array}\right).$$ (19)

The vector field on T_k can be represented in the local coordinate as V = υ₁ ∂_χ¹+υ₂ ∂_χ². The discretization of the divergence operator on T_k is

$$\begin{array}{l}{\nabla }_{T_k}·\mathbf{V}(p_1)=\frac{1}{\sqrt{G}}\sum _{i=1}^2\frac{\partial }{\partial {\chi }^i}(\sqrt{G}{v}^i)\\ =\frac{\partial }{\partial {\chi }^1}(v_1)+\frac{\partial }{\partial {\chi }^2}(v_2)\\ =\left(g^{11}(\mathbf{V}(p_1)-\mathbf{V}(p_3))+g^{21}(\mathbf{V}(p_2)-\mathbf{V}(p_3))\right){\partial }_{{\chi }^1}+\left(g^{12}(\mathbf{V}(p_1)-\mathbf{V}(p_3))+g^{22}(\mathbf{V}(p_2)-\mathbf{V}(p_3))\right){\partial }_{{\chi }^2}.\end{array}$$ (20)