A machine learning approach to investigate the relationship between shape features and numerically predicted risk of ascending aortic aneurysm

Abstract

Geometric features of the aorta are linked to patient risk of rupture in the clinical decision to electively repair an ascending aortic aneurysm (AsAA). Previous approaches have focused on relationship between intuitive geometric features (e.g., diameter and curvature) and wall stress. This work investigates the feasibility of a machine learning approach to establish the linkages between shape features and FEA-predicted AsAA rupture risk, and it may serve as a faster surrogate for FEA associated with long simulation time and numerical convergence issues. This method consists of four main steps: (1) constructing a statistical shape model (SSM) from clinical 3D CT images of AsAA patients; (2) generating a dataset of representative aneurysm shapes and obtaining FEA-predicted risk scores defined as systolic pressure divided by rupture pressure (rupture is determined by a threshold criterion); (3) establishing relationship between shape features and risk by using classifiers and regressors; and (4) evaluating such relationship in cross-validation. The results show that SSM parameters can be used as strong shape features to make predictions of risk scores consistent with FEA, which lead to an average risk classification accuracy of 95.58% by using support vector machine and an average regression error of 0.0332 by using support vector regression, while intuitive geometric features have relatively weak performance. Compared to FEA, this machine learning approach is magnitudes faster. In our future studies, material properties and inhomogeneous thickness will be incorporated into the models and learning algorithms, which may lead to a practical system for clinical applications.

Appendix

The surface remeshing method has three steps:

Step-1: Find the shortest path between a node on the left boundary and a node on the right boundary. Given a pair of nodes on the left and right boundaries, the geodesic path between them is recovered. The points on the geodesic path are on the 3D surface, but may not be the nodes of the mesh. Then a set of geodesic paths are obtained for every pair of boundary nodes, and the shortest path is selected as a cut-line. The surface mesh is cut open along the cut-line as shown in Fig. 3a, and it becomes topologically equivalent to a rectangle.

Step-2: Compute mesh-parameterization of the 3D surface mesh. The 3D surface mesh, which is cut along the cut-line, is mapped onto a 2D rectangular region, which is called mesh-parameterization. After the mapping, the 3D surface mesh is transformed to a 2D planar triangle mesh as shown in Fig. 3b.

Step-3: Divide the 2D rectangular region into a 2D quad mesh and transform it to 3D. The 2D rectangular region is discretized into a 2D mesh with rectangular elements (i.e., quad elements), as shown in Fig. 3c. Then the transform from the points of the 2D quad mesh to the 3D surface is determined by barycentric interpolation (Botsch et al. 2010) of the 2D triangle mesh. After transforming the 2D quad mesh to the 3D surface and sealing the transformed mesh along the cut-line, a 3D surface mesh with quad elements is obtained, as shown in Fig. 3d.

We utilized the exact geodesic path finding algorithm proposed by Surazhsky et al. (2005), for Step-1. Based on the work of Yoshizawa et al. (2004), we developed a stretch-minimizing-based algorithm for Step-2, and it has two stages:

Stage-1 of Step-2: Find an initialization mesh-parameterization based on barycentric mapping and mean value theorem (Botsch et al. 2010). Barycentric mapping is used to build a parameterization of the 3D triangle surface mesh, i.e., transforming the 3D surface mesh to a 2D planar triangle mesh. The boundary of the 2D planar mesh forms a rectangle. Each triangle ${\varvec{P}}_i =( {{\varvec{p}}_1 ,{\varvec{p}}_2 ,{\varvec{p}}_3 } )$ of the 3D surface mesh is mapped to a triangle ${\varvec{Q}}_i =({{\varvec{q}}_1 ,{\varvec{q}}_2 ,{\varvec{q}}_3 })$ of the 2D planar mesh. Each node ${\varvec{p}}_i =[{x_i ,y_i ,z_i }]$ of the 3D surface mesh is mapped to a node ${\varvec{q}}_i =[{u_i ,v_i } ]$ of the 2D planar mesh. Here $[{x_i ,y_i ,z_i }]$ denotes 3D coordinate, and $[{u_i ,v_i }]$ denotes 2D coordinate. Based on barycentric mapping, the node coordinates of the 2D planar mesh are determined by

$$\begin{aligned} \left\{ {{\begin{array}{l} {\mathop \sum \limits _{j=1}^M a_{i,j} u_j =-\mathop \sum \limits _{j=M+1}^N a_{i,j} u_j } \\ {\mathop \sum \limits _{j=1}^M a_{i,j} v_j =-\mathop \sum \limits _{j=M+1}^N a_{i,j} v_j } \\ \end{array} }} \right. \end{aligned}$$

(8)

where M is the number of interior nodes and N is the total number of nodes. By using the mean value theorem, each coefficient is determined by

$$\begin{aligned} a_{i,j} =\frac{1}{{\Vert \varvec{p}}_i -{\varvec{p}}_j\Vert }\left( {\tan \left( {\frac{\theta _{i,j} }{2}} \right) +\tan \left( {\frac{\delta _{i,j} }{2}} \right) } \right) \end{aligned}$$

(9)

where $a_{i,j} >0$ if ${\varvec{p}}_i$ and ${\varvec{p}}_j$ are connected by an edge, otherwise $a_{i,i} =-\mathop \sum \nolimits _{j\ne i} a_{i,j}$ and $a_{i,j} =0$. $\theta _{i,j}$ and $\delta _{i,j}$ are angles between the edge from ${\varvec{p}}_i$ to ${\varvec{p}}_j$ and its two adjacent edges, respectively. After the coefficients $\{ {a_{i,j} }\}$ are calculated, the node coordinates of the 2D planar mesh are obtained by solving Eq. (1).

Now, an inverse transform from a point on the 2D plane to the 3D surface can be obtained: let ${\varvec{q}}$ be a point inside ${\varvec{Q}}_i$; then, its corresponding point ${\varvec{p}}$ on the 3D surface is determined by an affine mapping, namely barycentric interpolation:

$$\begin{aligned} {\varvec{p}}= & {} ( \langle {{\varvec{q}},{\varvec{q}}_2 ,{\varvec{q}}_3 } \rangle {\varvec{p}}_1 +\langle {{\varvec{q}}, {\varvec{q}}_3 ,{\varvec{q}}_1 } \rangle {\varvec{p}}_2 \nonumber \\&+\,\langle {{\varvec{q}}, {\varvec{q}}_1 ,{\varvec{q}}_2 } \rangle {\varvec{p}}_3 )/\langle {{\varvec{q}}_1 , {\varvec{q}}_2 , {\varvec{q}}_3 } \rangle \end{aligned}$$

(10)

where $\langle {{\varvec{q}}_a , {\varvec{q}}_b ,{\varvec{q}}_c } \rangle$ is the area of the triangle defined by the three points.

Stage-2 of Step-2: Refine the mesh-parameterization based on stretch minimization. After Stage-1, the 3D surface mesh is mapped onto a 2D parametric plane, resulting a 2D planar mesh composed of the same number of nodes and triangle elements. The goal of this refinement stage is to change the node coordinates of the 2D planar mesh such that mesh distortion is minimized. Mesh distortion is measured by the average stretch $\upmu$, given by

$$\begin{aligned} \upmu =\sqrt{\mathop \sum \nolimits _i A({\varvec{P}}_i )\upmu _{P_i }^2 /\mathop \sum \nolimits _i A({{\varvec{P}}_i })} \end{aligned}$$

(11)

where $A({{\varvec{P}}_i })$ denote the area of the triangle ${\varvec{P}}_i ; \upmu _{P_i }$ is the local stretch associated with triangle ${\varvec{P}}_i$, and it is defined as

$$\begin{aligned} \upmu _{P_i } =\sqrt{{\Gamma }^{2}+{\Upsilon }^{2}} \end{aligned}$$

(12)

where ${\Gamma }$ is max eigenvalue and ${\Gamma }$ is the min eigenvalue of the deformation gradient tensor derived from the affine mapping (Eq. 3). We utilize the algorithm proposed by Yoshizawa et al. (2004) to find the optimal node coordinates such that the average stretch $\upmu$ is minimized. This algorithm has two iteration steps:

• (1) Update the node coordinates of the 2D triangle mesh by minimizing the local energy function

  $$\begin{aligned} E=\mathop \sum \limits _j a_{i,j} {\Vert \varvec{q}}_i -{\varvec{q}}_j\Vert ^{2} \end{aligned}$$

  (13)

  In this step, the coefficients $\left\{ {a_{i,j} } \right\}$ are fixed. The solution of this minimization problem is found by solving a set of linear equations.

• (2) Update each coefficient by using each local stretch

  $$\begin{aligned} a_{i,j} \leftarrow \frac{a_{i,j} }{\upmu _{P_j } } \end{aligned}$$

  (14)

  The initial values of the coefficients are obtained in Stage-1.

After a few iterations, the average stretch $\upmu$ will be reduced. Then the rectangle region is discretized to a 2D planar quad mesh as shown in Fig. 3c. Using the affine mapping (Eq. 3) each node of the 2D planar quad mesh is transformed to the 3D surface. As a result, the 3D surface is now represented by a quad mesh as shown in Fig. 3d.