Special Section: Spotlight on the Future–Imaging and Biomechanical Engineering

Construction of Analysis-Suitable Vascular Models Using Axis-Aligned Polycubes

[+] Author and Article Information
Adam R. Updegrove

Mechanical Engineering Department,
University of California, Berkeley,
Berkeley, CA 94709
e-mail: updega2@berkeley.edu

Shawn C. Shadden

Mechanical Engineering Department,
University of California, Berkeley,
Berkeley, CA 94709
e-mail: shadden@berkeley.edu

Nathan M. Wilson

Open Source Medical Software Corporation,
Santa Monica, CA 90403
e-mail: nwilson@osmsc.com

1Corresponding author.

Manuscript received March 15, 2018; final manuscript received June 5, 2018; published online April 22, 2019. Assoc. Editor: Ching-Long Lin.

J Biomech Eng 141(6), 060906 (Apr 22, 2019) (9 pages) Paper No: BIO-18-1142; doi: 10.1115/1.4040773 History: Received March 15, 2018; Revised June 05, 2018

Image-based modeling is an active and growing area of biomedical research that utilizes medical imaging to create patient-specific simulations of physiological function. Under this paradigm, anatomical structures are segmented from a volumetric image, creating a geometric model that serves as a computational domain for physics-based modeling. A common application is the segmentation of cardiovascular structures to numerically model blood flow or tissue mechanics. The segmentation of medical image data typically results in a discrete boundary representation (surface mesh) of the segmented structure. However, it is often desirable to have an analytic representation of the model, which facilitates systematic manipulation. For example, the model then becomes easier to union with a medical device, or the geometry can be virtually altered to test or optimize a surgery. Furthermore, to employ increasingly popular isogeometric analysis (IGA) methods, the parameterization must be analysis suitable. Converting a discrete surface model to an analysis-suitable model remains a challenge, especially for complex branched structures commonly encountered in cardiovascular modeling. To address this challenge, we present a framework to convert discrete surface models of vascular geometries derived from medical image data into analysis-suitable nonuniform rational B-splines (NURBS) representation. This is achieved by decomposing the vascular geometry into a polycube structure that can be used to form a globally valid parameterization. We provide several practical examples and demonstrate the accuracy of the methods by quantifying the fidelity of the parameterization with respect to the input geometry.

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Grahic Jump Location
Fig. 1

Graph simplification procedure: (a) the triangulated surface mesh generated from segmentation of medical image data (far left) of an abdominal aortic aneurysm and iliac bifurcation, (b) the extracted centerline geometry and surface groups from VMTK, and (c) the corresponding graph simplification with examples of some of the basic information stored within the graph simplification

Grahic Jump Location
Fig. 2

At each bifurcation location, the two vertices where the three branches meet are defined as the critical vertices (vc0 and vc1, in red). The “edge lines” separating each of the branches are then trisected, which give two new “slice vertices” (vsi) per edge line. After the child branches are categorized as aligning and diverging, the slice vertices are used to form the new bifurcation patch. In this case, the green branch is the diverging branch and slice vertices vs0, vs1, vs4, and vs5 are used in forming the new bifurcation patch.

Grahic Jump Location
Fig. 3

From the graph simplification (upper right), the polycube structure (lower right) can be easily formed. However, an additional bifurcation cube is added at branch intersections to better unite parameterizations between branches. This bifurcation cube maps to a new bifurcation patch on surface mesh, as shown on the left.

Grahic Jump Location
Fig. 4

The ring of vertices separating the parent branch from the aligning branch is used to calculate a central vertex (c(P0), in white). From this vertex, the plane of the ring of vertices is projected normally in both directions to the slice vertices identified earlier. Slicing the triangulated surface mesh along planes P1 and P2 and extracting the inner region gives the new bifurcation patch, which is mapped to the bifurcation cube.

Grahic Jump Location
Fig. 5

Angles αij and βij used for the harmonic edge weights

Grahic Jump Location
Fig. 6

Mapping the parameterization. Each surface patch (upper left) is conformally mapped (mapping C) to the plane (lower left). This enables parameter grid lines (lower right) to then be mapped to the model surface (upper right) via the mapping C−1I.

Grahic Jump Location
Fig. 7

Comparison of original surface mesh (left) with NURBS representation (right)

Grahic Jump Location
Fig. 8

An aorta model with celiac and iliac branches are decomposed into the polycube representation and parameterized with discrete conformal mappings. The leftmost image is the original surface mesh and the rightmost image is the final NURBS model, which consists of 16 NURBS patches.

Grahic Jump Location
Fig. 9

Multiple branches of an iliac artery are decomposed into the polycube representation and parameterized with discrete conformal mappings. The leftmost image is the original surface mesh and the rightmost image is the final NURBS model, which consists of 7 NURBS patches.

Grahic Jump Location
Fig. 10

Pulmonary arteries are decomposed into the polycube representation and parameterized with discrete conformal mappings. The leftmost image is the original surface mesh and the rightmost image is the final NURBS model, which consists of 10 NURBS patches.

Grahic Jump Location
Fig. 11

A variety of aneurysms parameterized using the same polycube representation. By parameterizing the input surface meshes with the same polycube representation, it is possible to perform statistical shape modeling for a variety of applications.



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