Computer Methods in Applied Mechanics and Engineering
Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow☆
Introduction
Recently, patient-specific modeling was proposed as a new paradigm in simulation-based medical planning. Physicians, using computational tools, construct and evaluate combined anatomical/physiological models to predict the outcome of alternative treatment plans for an individual patient. A comprehensive framework has been developed to enable the conduct of computational vascular research [49], [6]. Blood flow simulations provide physicians with physical data to help them devise treatment plans.
Isogeometric analysis is a new computational technique that improves on and generalizes the standard finite element method. It was first introduced in [32], and expanded on in [20]. In an effort to instantiate the concept of isogeometric analysis, an analysis framework based on NURBS was built. Mathematical theory of this NURBS-based approach was put forth in [7]. NURBS is not the only possible basis for isogeometric analysis but it is certainly the most highly developed and widely utilized. For an introductory text on NURBS, see Rogers [41]. A more advanced treatment of the subject is given in Piegl and Tiller [38]. Other geometric modeling techniques that have potential as a basis for isogeometric analysis include: A-patches [4], T-splines [44], and subdivision [14]. These warrant further investigation.
Fig. 1 shows one such model, obtained from patient-specific imaging data. We have designed a set of procedures which allows us to create solid NURBS vascular models directly from patient-specific data. We have named this process the vascular modeling pipeline, which can be divided into four main steps:
- (1)
Preprocessing – in scanned computed tomography (CT) or magnetic resonance imaging (MRI) data, the intensity contrast may not be clear enough, noise exists, and sometimes the blood vessel boundary is blurred. Therefore, we use image processing techniques to improve the quality of CT/MRI data, such as contrast enhancement, filtering, classification, and segmentation.
- (2)
Path extraction – the goal is to find arterial paths. Vascular surface models can be constructed from the preprocessed imaging data via isocontouring. The skeleton is then extracted from the surface model using Voronoi and Delaunay diagrams. This skeletonization scheme is suitable for noisy input and creates one-dimensional clean skeletons for blood vessels.
- (3)
Control mesh construction – a skeleton-based sweeping method is developed to construct hexahedral NURBS control meshes by sweeping a templated quad mesh of a circle along the arterial path. Templates for various branching configurations are presented which decompose the geometry into mapped meshable patches using the extracted skeleton. Each patch can be meshed using one-to-one sweeping techniques. Some nodes in the control mesh lie on the surface, and some do not. We project nodes lying on the surface to the vascular surface. The blood vessel wall can be built by radially extending the surface outside 10–15% of the distance to the center line (see, e.g. [26]). It should be noted that the arterial wall is comprised of multiple layers, and may have significant regional variations in the thickness. Estimating the wall thickness correctly presents a significant challenge to the state-of-the-art imaging technology. In this work we make use of the MRI and CT data that contains no information on the wall thickness, and, as a result, we select the values that are consistent with what we found in the literature as well as private communication with vascular surgeons.
- (4)
NURBS construction and isogeometric analysis – after generating hexahedral control meshes, we construct solid NURBS geometric models and employ isogeometric analysis to simulate blood flow. Piecewise linear hex meshes can also be obtained. Three numerical examples, coronary, thoracic and abdominal arteries, are presented.
The remainder of this paper is organized as follows: Section 2 reviews related previous work. Section 3 describes the meshing pipeline and preprocessing. Section 4 discusses a skeletonization scheme. Section 5 talks about solid NURBS construction and isogeometric analysis. Section 6 explains the skeleton-based sweeping method and decomposition templates. Section 7 presents three numerical examples. Section 8 draws conclusions and outlines planned future work.
Section snippets
Sweeping method
Sweeping, or -D meshing, is one of the most robust techniques to generate semi-structured hexahedral meshes. One-to-one sweeping requires that the source and target surfaces have similar topology. The source surface is meshed with quadrilaterals [9], which are swept through the volume using linking surfaces as a guide [16].
However, few geometries satisfy the topological constraints required by one-to-one sweeping. In the CUBIT project [1] at Sandia National Labs, a lot of research has been
Meshing pipeline and preprocessing
The input images are often of poor quality which makes it difficult to generate quality meshes for regions of interest. To circumvent this problem we pass the raw imaging data through a preprocessing pipeline where the image quality is improved by enhancing the contrast, filtering noise, classifying, and segmenting regions of various materials. The surface model is then extracted from the processed imaging data, and the vessel path is obtained after skeletonizing the volume bounded by the
Skeletonization
The subsequent meshing process relies on generating a skeleton for the object to be meshed. Extracting the skeleton of a three dimensional object is a research problem that has drawn much attention for its wide applicability in graphics, solid modeling and in other diverse areas of science and engineering. Some of the techniques include, to name a few, topological thinning [11], distance field based methods [8], [12], [29], [62], potential field based methods [18], thinning via medial geodesic
Solid NURBS construction and isogeometric analysis
In a NURBS-based isogeometric analysis a physical domain in is defined as a union of patches. A patch, denoted by Ω, is an image under a NURBS mapping of a parametric domain (0,1)3whereIn the above, ’s are the rational basis functions, and ’ are the control points. In the definition of the
The skeleton-based sweeping method
Blood vessels are tubular objects, therefore we choose the sweeping method to construct hexahedral control meshes for NURBS-based isogeometric analysis.
Numerical examples
In this section we present applications of the meshing pipeline to three patient-specific vascular models: a model of a portion of the coronary tree, a model of the thoracic aorta, and a model of the abdominal aorta. Isogeometric analysis is then used to compute blood flow in the models. In all cases, time-dependent, viscous, incompressible Navier–Stokes equations were used as the blood model. The fluid density and dynamic viscosity were chosen to be representative of blood flow. The first
Conclusions and future work
We have developed a four-stage process to construct analysis suitable geometric models from patient-specific vascular imaging data with a goal of using them in isogeometric analysis of blood flow in arteries. We have focused on hexahedral solid NURBS modeling, and did not treat other geometrical modeling technologies, such as A-patches, T-splines, and subdivision. We would like to investigate these techniques in the future.
We use the sweeping method to construct hexahedral control meshes,
Acknowledgements
An early version of this paper appeared in 15th International Meshing Roundtable conference [61]. Y. Zhang was partially supported by the J.T. Oden ICES Postdoctoral Fellowship at the Institute for Computational Engineering and Sciences. This research of Y. Zhang, S. Goswami, and C. Bajaj was supported in part by NSF Grants EIA-0325550, CNS-0540033, and NIH Grants P20-RR020647, R01-GM074258, R01-GM073087. This support is gratefully acknowledged. We would also like to thank Fred Nugen, Bob
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