Optimal three-dimensional reconstruction for lung cancer tissues

2.1The application of fast fuzzy clustering to raise time resolution

One of the most effective algorithms for dealing with uncertain and inconsistent information is the fuzzy c-means (FCM) algorithm [10]. Consequently, different FCM derivative algorithms have been developed for different applications. Fast FCM (f-FCM) is one that can be used in the image processing field [11].

Suppose that a 2D image is composed of a set X of N pixels. After all pixels of the field have been classified into L gray levels, H⁢(l) represents the number of pixels partitioned to the l-th gray level. Then ui⁢l is the membership degree of the l-th gray level to the i-th cluster, where i= 1, 2, …, c. The objective function in the f-FCM algorithm is

where vi is the cluster prototype (representative of the gray level) of the i-th cluster, where i= 1, 2, …, C. The fuzziness exponent, m, is in the range [1.5, 2.5]. The algorithm assumes that the cluster number C is available. The prototype vector V and partition matrix U are updated according to

and

The terminal condition of this algorithm is set when a user-specified threshold value is reached according to the following formulas:

The fast FCM function is performed based on a one-dimensional attribute such as the gray-scale value. Thus, its computational complexity is O (ctL), where t is the total iteration time. In comparison, the computational complexity of the original FCM is O (ctdN), where d is the dimension of the investigated set. Usually, the number of pixels of any section is much larger than the number of gray levels (L= 256); thus, the f-FCM has the advantage of a shorter runtime as compared to the FCM algorithm.

2.2The application of validity index to raise spatial resolution

If the number of clusters is not known, it can be calculated using a proper function called the cluster validity index [12, 13]. The two most commonly used fuzzy validity indexes are the partitioning entropy (PE) index [14] and the Xie-Beni index [15]. The PE index is defined as

Where a is the base of the logarithm; P⁢E is the scalar function of the number of clusters (c) and is in the range [0, logca]; n is the number of pixels in set X. The optimal number of clusters, c*, can be obtained by minimizing Eq. (5).

On the other hand, Xie and Beni et al. [15] defined a validity index as

where the value of c is usually in the range [2, c𝑚𝑎𝑥], and c𝑚𝑎𝑥 is the upper bound of the number of candidate clusters, usually less than n. The range [2, c𝑚𝑎𝑥] must contain all possible numbers of clusters. Equation (6) selects the optimal cluster partition by using a trial-and-error method to find the optimal segmentation among all possible segmentations. The clustering algorithms must be performed repeatedly – (c𝑚𝑎𝑥-2) times; therefore, the value of c should be set as small as possible to ensure that the optimal segmentation is not lost.

The proposed 3D XCT imaging algorithm includes the following six steps for handling a series of 2D image sections:

Hereafter, the optimal 2D section-based 3D XCT reconstruction method will be referred to as OP-3D-XCT.

3.1Description of lung cancer types

Based on diverse histopathology characteristics and shape distributions, there are four common lung cancer types, as shown in Fig. 1.

Figure 1.

Pathological section of four types of typical lung cancer tissues (he, 50x).

3.2The optimal 2D sections

The 2D sections of musculoskeletal and viscera lung tissuesshown in Figs 2–4 were obtained by different numbers of clusters and the optimal segmentation identified by the f-FCM algorithm. Figures 2–4 clearly show that 2D segmentations under different numbers of clusters have various spatial resolutions, and that there is an optimal image for each type of tissue. These optimal images correspond to different numbers of clusters.

Figure 2.

CT image of musculoskeletal tissues in different number of clusters.

Figure 3.

CT image of Viscera tissues in different number of clusters.

Figure 4.

CT image of Lung tissue in different number of clusters.

Figure 5 shows the selection process of 2D optimal sections for four typical lung cancer tissues by the PE index function; Fig. 6 refers to the Xie-Beni index. The index function has obvious most value from the figure, the best image is inconsistent to the optimal value of the validity index by the Xie-Bine index and has a small different by the PE index. This further demonstrates that we can distinguish the optimal 2D image from a series of candidate images by using the proper validity indexes.

Figure 5.

Selection of the optimal 2D section for musculoskeletal tissues using PE index function in the OP-3D-XCT method.

Figure 6.

Selection of the optimal 2D section for viscera tissues usingXie-Beni index function in the OP-3D-XCT method.

Figures 2 through 4 show that different numbers of clusters for different 2D images correspond to fuzzy partitions of differing quality. Cluster validity indexes can identify the optimal number of clusters, and thus can provide the basis for the reconstruction of higher quality 3D images.

3.3The reconstructed 3D image

Figure 7 shows the reconstructed 3D image by the OP-3D-XCT method, clearly representing the structure of a human chest. Additionally, the shape distribution of the four types of typical lung cancer and the difference between surrounding tissues can be accurately observed. For comparison, the reconstructed images of the four types of typical lung cancer by direct use of 3D volume reconstruction without optimized 2D sections are shown in Fig. 8. These 3D images contain many irrelevant tissues and trail, so the region of interest cannot be observed clearly. Consequently, the reconstructed 3D images are not optimal images.

Figure 7.

Reconstructed 3D images of four types of typical lung cancer by OP-3D-XCT.

3.4Evaluation of the proposed method

The proposed method was evaluated in terms of time resolution and spatial resolution. The time resolution was evaluated by the absolute runtime of a frame of a 3D image; the runtime was measured by the “tic” and “toc” functions in the Matlab toolbox. The spatial resolution can be evaluated by the relative error of all K pixels in the reconstructed image as

where gj is the reference gray degree of the j-th pixel, that is, the real gray value; gj* is the gray value of the j-th pixel in the reconstructed image, where j= 1, 2, …, K; ξ is the average of all errors of K pixels.

To obtain the real gray values of all pixels, the anatomic parts were measured and recorded in the process of operating on the patient. However, only partial tissues could be observed; therefore, the comparison was performed on these observable tissues that correspond to partial pixels in each section.

Table 2 shows that the OP-3D-XCT method can improve time resolution and speed up the 3D reconstruction process by more than an order of magnitude. Please note that the computational complexity of f-FCM is independent of the number of clusters and is only related to the 256 gray levels. By comparison, the runtime of the FCM algorithm is directly proportional to the number of pixels; the more pixels the segmented images have, the longer the runtime of the segmenting process.

Figure 8.

Reconstructed 3D images of four types of typical lung cancer by the direct use of 3D volume reconstruction algorithms without optimal 2D possessing.

Table 3 shows that the OP-3D-XCT method can improve the spatial resolution, and offers advantages over the existing non-optimal method in four common types of lung cancer tissues. The average spatial resolution can be raised by an increment of over 8.6%.