Density and Attachment Agnostic CT Pulmonary Nodule Segmentation with Competition-Diffusion and New Morphological Operators

Abstract

Pulmonary nodules are potential manifestations of lung cancer, and their detection and inspection are essential for screening and diagnosis of the disease. The growth of a nodule is considered one of the most important cues for assessing its malignancy. Hence, the ability to segment the nodules accurately and measure their growth over time is crucial for prognosis. Accurate nodule segmentation is also vital for drug therapy development. A segmentation that can provide a consistent, reproducible, and accurate volumetric measure of nodule shrinkage/growth is very critical for evaluating the effectiveness of drug treatments

Appendix A: Notations and Definitions

In this chapter, the following typefaces are used. A lower bold letter (x) is used for a vector and a 3D coordinate, an upper bold letter (X) is used for a 3D volume and a set, an upper italic letter is for a function and a matrix (H), upper Greek letter for a volume-to-volume transformation (Ξ), and a lower italic letter (x, μ) is used for a scalar. Without ambiguity, we also use an upper bold letter associated with a binary volume (e.g., segmentation) to indicate a set of voxels whose binary values are nonzero.

We use ∂ to denote a set of boundary voxels in a binary volume. For example, ∂L is a set of boundary voxels in L. Throughout, L represents the foreground, which are radiographically denser anatomical structures and possibly opacities. We call the boundary of L (i.e., ∂L) surface voxels. For each foreground voxel (say x), the closest surface voxel (say s) is called a contact voxel. Note that there can be multiple contact voxels of x. A set of foreground voxels which regard s as their contact voxel is called the evacuation set of s. Among voxels in the evacuation set of s, the one furthest away from s is called the core voxel of s. A set of core voxels form a medial axis of the foreground. The evacuation set forms a straight line segment. A trace of voxels from x to its surface voxel in the straight path is called an exit path of x. A core voxel has multiple exit paths. Figure 6.25 illustrates some of these definitions.

Fig. 6.25

An illustration of some definitions

To cast the nodule segmentation problem to a more abstract geometrical one, we consider the following scenario. Let L ₁ be a convex nodule of interest and L ₀ be other structures in the image. Then, the foreground, L, is the union of L ₁ and L ₀. Let S ₁ = ∂L ₁ ∩ ∂L and S ₀ = ∂L ₀ ∩ ∂L. S ₁ and S ₀ are sets of surface voxels that are not occluded by each other. The correct segmentation we seek is Ξ(S ₁) ∩ L where Ξ(A) is the convex hull of a set of nonzero voxels in A. This definition makes the solution unique given L ₁ and L and not dependent on L ₀ which is arbitrary inside L ₁ . In Fig. 6.25, L ₁ is modeled by an ellipsoid (shown as an ellipse) while L ₀ is modeled by a nonconvex shape. The correct segmentation is the area colored by a hatched pattern. Table 6.4 lists symbols used in this chapter and their descriptions.

Table 6.4 Symbols and their meanings