Three-Dimensional Texture Feature Analysis of Pulmonary Nodules in CT Images: Lung Cancer Predictive Models Based on Support Vector Machine Classifier

Abstract

To extract texture features of pulmonary nodules from three-dimensional views and to assess if predictive models of lung CT images from a three-dimensional texture feature could improve assessments conducted by radiologists. Clinical and CT imaging data for three dimensions (axial, coronal, and sagittal) in pulmonary nodules in 285 patients were collected from multiple centers and the Cancer Imaging Archive after ethics committee approval. Three-dimensional texture feature values (contourlets), and clinical and computed tomography (CT) imaging data were built into support vector machine (SVM) models to predict lung cancer, using four evaluation methods (disjunctive, conjunctive, voting, and synthetic); sensitivity, specificity, the Youden index, discriminant power (DP), and F value were calculated to assess model effectiveness. Additionally, diagnostic accuracy (three-dimensional model, axial model, and radiologist assessment) was assessed using the area under the curves for receiver operating characteristic (ROC) curves. Cross-sectional data from 285 patients (median age, 62 [range, 45–83] years; 115 males [40.4%]) were evaluated. Integrating three-dimensional assessments, the voting method had relatively high effectiveness based on both sensitivity (0.98) and specificity (0.79), which could improve radiologist diagnosis (maximum sensitivity, 0.75; maximum specificity, 0.51) for 23% and 28% respectively. Furthermore, the three-dimensional texture feature model of the voting method has the best diagnosis of precision rate (95.4%). Of all three-dimensional texture feature methods, the result of the voting method was the best, maintaining both high sensitivity and specificity scores. Additionally, the three-dimensional texture feature models were superior to two-dimensional models and radiologist-based assessments.

Appendix

Contourlets

The LP is a multi-scale decomposition of the L²(R²) space into a series of increasing resolutions as given by Eq. 1.

$$ {L}^2\left({R}^2\right)={V}_{\mathrm{j}0}\oplus \left(\underset{\mathrm{j}=\mathrm{j}0}{\overset{-\infty }{\otimes {\mathrm{W}}_{\mathrm{j}}}}\right) $$

(1)

where V_j0 is an approximation sub-space at the scale 2^j0, whereas W_j contains the added details at the finer scale 2^j − 1. In the LP, each sub-space W_j is spanned by a frame {μ_{j, n}(t)}, n ∈ z² that assimilates a uniform grid on R² at intervals 2^j − 1 × 2^j − 1. For the directional filter bank, it can be shown that a l-level DFB generates a local directional basis for L²(Z²) that is composed of the impulse responses of the 2^l directional filters and their shifts (Eqs. 2 and 3, respectively).

$$ \left\{{g}_k^{(1)}\left[-{S}_k^{(1)}n\right]\right\},0\le k<{2}^1,\mathrm{n}\in {\mathrm{z}}^2 $$

(2)

$$ {S}_k^{(1)}={\displaystyle \begin{array}{c}\left[\begin{array}{cc}{2}^{l-1}& 0\\ {}0& 2\end{array}\right]\\ {}\left[\begin{array}{cc}2& 0\\ {}0& {2}^{l-1}\end{array}\right]\end{array}}{\displaystyle \begin{array}{c}0\le k<{2}^{l-1}\\ {}{2}^{l-1}\le k<{2}^l\end{array}} $$

(3)

In the contourlet transform, suppose that a l_j level DFB is applied to the detail sub-space W_j of the LP as in Eq. 4.

$$ {\mathrm{W}}_j=\underset{k=0}{\overset{2^{lj-1}}{\oplus }}{W}_{j,k}^{\left(1\mathrm{j}\right)} $$

(4)

Each sub-space ${W}_{j,k}^{\left({\mathrm{l}}_{\mathrm{j}}\right)}$ is spanned by a frame $\left\{{\rho}_{j,k,n}^{\left({\mathrm{l}}_{\mathrm{j}}\right)}\left(\mathrm{t}\right)\right\},\mathrm{n}\in {Z}^2$ with a redundancy ratio of 4:3, where $\left\{{\rho}_{j,k,n}^{\left({\mathrm{l}}_{\mathrm{j}}\right)}\left(\mathrm{t}\right)\right\}={\mathrm{g}}_{\mathrm{k}}^{\left({\mathrm{l}}_{\mathrm{j}}\right)}\left[m-{S}_k^{\left({\mathrm{l}}_{\mathrm{j}}\right)}n\right]{\mu}_{j,m}\left(\mathrm{t}\right)$. Furthermore, $\left\{{\rho}_{j,k,n}^{\left({\mathrm{l}}_{\mathrm{j}}\right)}\left(\mathrm{t}\right)\right\},\mathrm{n}\in {Z}^2$ is generated from a single prototype function and its shifts: $\left\{{\rho}_{j,k,n}^{\Big({1}_{\mathrm{j}}}\left(\mathrm{t}\right)\right\}={\rho}_{\mathrm{j},\mathrm{k}}^{\left(1\mathrm{j}\right)}\Big(\mathrm{t}-{2}^{\mathrm{j}-1}{\mathrm{S}}_{\mathrm{k}}^{\left(1\mathrm{j}\right)}\mathrm{n}\in {Z}^2$.

SVM

The objective of SVM is to find a mechanism to meet the classification requirements of the optimal separating hyperplane, such that the hyperplane can be maximized over a blank area on both sides of the plane while ensuring the classification accuracy.

In theory, the SVM can achieve the optimal linear separately regarding two types of data classification, for example, a given training set (x_i, y_j), i = 1, 2, ..., l, x ∈ {±1}, where the hyperplane is denoted as (w • x) + b = 0. To classify the face of all samples correctly and to classify the interval would require meeting the following constraints,y_i[(w • x_i) + b] ≥ 1, i = 2, 2, ...l. The classification interval can be calculated as 2/‖w‖. Therefore, the problem of optimal hyperplane structure is transformed for the sake of constraints under:

$$ \min \phi \left(\mathrm{w}\right)=\frac{1}{2}\left\Vert w\right\Vert -\alpha \left(y\left(\left(w\bullet x\right)+b\right)-1\right) $$

(5)

To address this constraint optimization problem, the Lagrange function is introduced:

$$ \mathrm{L}\left(\mathrm{w},\mathrm{b},\mathrm{a}\right)=\frac{1}{2}\left\Vert \mathrm{w}\right\Vert -\alpha \left(y\left(\left(w\bullet x\right)+b\right)-1\right) $$

(6)

where α_i > 0 is the Lagrange multiplier.

$$ \max \mathrm{Q}\left(\alpha \right)=\sum \limits_{j=1}^l{\alpha}_j-\frac{1}{2}\sum \limits_{\mathrm{j}=1}^l\sum \limits_{j=1}^l{\alpha}_{\mathrm{i}}{\alpha}_{\mathrm{j}}{y}_{\mathrm{i}}{y}_j\left({\mathrm{x}}_{\mathrm{i}}\bullet {\mathrm{x}}_{\mathrm{j}}\right) $$

(7)

$$ \mathrm{s}.\mathrm{t}.\sum \limits_{\mathrm{j}=1}^1{\alpha}_{\mathrm{j}}{y}_j=0,j=1,2,...,l,{\alpha}_j\ge 0,j=1,2,...,l $$

(8)

In the above dual problem, to avoid complex computing and high-dimensional inner products, it needs to be established if it is the objective function or decision-making functions that only relate to the product operation between the training samples in high-dimensional space.