Numerical methods for polyline-to-point-cloud registration with applications to patient-specific stent reconstruction

Claire Yilin Lin, Alessandro Veneziani, Lars Ruthotto
2017 International Journal for Numerical Methods in Biomedical Engineering  
We present novel numerical methods for Polyline-to-Point-Cloud Registration and their application to patient-specific modeling of deployed coronary artery stents from image data. Patient-specific coronary stent reconstruction is an important challenge in computational hemodynamics and relevant to the design and improvement of the prostheses. It is an invaluable tool in large-scale clinical trials that computationally investigate the effect of new generations of stents on hemodynamics and
more » ... lly tissue remodeling. Given a point cloud of strut positions, which can be extracted from images, our stent reconstruction method aims at finding a geometrical transformation that aligns a model of the undeployed stent to the point cloud. Mathematically, we describe the undeployed stent as a polyline, which is a piecewise linear object defined by its vertices and edges. We formulate the nonlinear registration as an optimization problem whose objective function consists of a similarity measure, quantifying the distance between the polyline and the point cloud, and a regularization functional, penalizing undesired transformations. Using projections of points onto the polyline structure, we derive novel distance measures. Our formulation supports most commonly used transformation models including very flexible nonlinear deformations. We also propose two regularization approaches ensuring the smoothness of the estimated nonlinear transformation. We demonstrate the potential of our methods using an academic 2D example and a real-life 3D bioabsorbable stent reconstruction problem. Our results show that the registration problem can be solved to sufficient accuracy within seconds using only a few number of Gauss-Newton iterations. 2 C. Y. LIN ET AL. limited invasiveness. PCI implies the deployment of a prosthesis (called stent) [2], generally made of biocompatible materials with a metallic core, to open a coronary artery with severe occlusions. Next generation stents feature new bioabsorbable materials (generally absorbed within three years) and are targeted for acute pathologies in young patients [3] . The different mechanical properties of those materials require thicker struts to handle the pressure during and after the deployment. Thicker struts may interfere with the blood flow and eventually trigger biological processes and tissue remodeling with negative outcomes for the patient (reocclusion) [4, 5, 6, 7] . This is why an accurate assessment of the effect of the struts on the hemodynamics in patient-specific scenarios is of utmost importance. To this end the reconstruction of geometries for extensive fluid dynamics simulations based on clinical data and images is needed [8, 9, 10, 11, 12, 13, 14, 15 ]. An excellent overview of computational modeling of stented arteries, and comparison of imaging modalities used in this application is given in [16] . Computational Fluid Dynamics (CFD) is the tool of choice for this kind of investigations [17, 18, 19, 20, 21, 22], as it allows personalized quantitative analysis with a modest invasiveness for the patient. In particular, we target a fine analysis of the Wall Shear Stress (WSS -i.e. the tangential component of the normal stress) induced by the blood flow on the struts and the tissue [14] . The reliability of the results strongly depends on a precise patient-specific reconstruction of the stent and the lumen after deployment. Developing efficient (i.e., automatic or semi-automatic) methods for stent reconstruction is critical, for example, when processing a statistically significant number of patient datasets in large-scale clinical studies aiming at quantifying the effectiveness of the therapy [14, 23] . The accuracy and the efficiency of the reconstruction are challenged by the complexity of the sequence of steps and the large variability of cases in diverse patient-specific settings. We give a short description of the procedure currently developed in the Emory University Hospital in Sect. 6.1. To properly reduce patient variability and to improve the automation procedure, it is critical to guide the reconstruction with prior information available from the design of the stent. In fact, there are some practical limitations on patient-specific data. For instance, OCT images cannot resolve the entire vascular section (see Fig. 6 ) due to the shadow of the catheter. Therefore, a circular (sectiondependent) sector is missing in each image. To compensate the missing data, the information provided by the design of the stent provide a ground truth to guide the patient-specific reconstruction in an accurate and highly automated way. This requires to identify a map between the undeployed and the deployed geometries so that the missing information in the latter can be recovered by the mapping of the former one. This map can be calculated by a virtual deployment, i.e. a simulated operation mimicking the act of deployment. This can be done by a series of Boolean operations [24] or by mechanical simulation of the expansion [25, 26, 27] . While these approaches have great potential, the lack of knowledge of the mechanical properties of the wall to be used in the virtual deployment may be critical. In this work, we privilege a more data-driven approach, related to registration procedures. According to a similar guideline, in [28] the OCT-based stent-reconstruction is guided by an educated combination of a priori information on the stent design. In fact, the undeployed stent is registered to the point cloud of strut locations using a non-rigid point-to-point registration procedure [29] . The procedure is tested on one case of a metallic stent in a porcine artery. The key contributions of the present paper are to represent the stent efficiently as a polyline (defined in [30] as piecewise linear objects consisting of vertices and edges) and develop new numerical methods for registering polylines to point clouds. In the context of stent reconstruction, we assume that the elements of the point cloud (e.g., strut positions detected in OCT images) represent post-deployment points of the polyline (e.g., model of the undeployed stent), but the correspondence is unknown. Our goal is to establish the map by geometrically deforming the polyline object such that its distance to the point cloud is minimized. We exploit the polyline structure to compute the correspondence between the polyline and a given point, by projection onto the edges of the polyline. This assignment is (almost everywhere) differentiable with respect to the deformation and derivatives are easy to compute, thus, enabling fast optimization. Registration is known to be a challenging and ill-posed inverse problem and tailored approaches have been Here, Q k ∈ R np×d·np extracts the entries associated with the kth coordinate, e ∈ R np is a vector of all ones, squaring and square root are applied component-wise, and β > 0 is a conditioning This article is protected by copyright. All rights reserved. where the penalty functions ψ and ϕ are convex and chosen as, for example, in [43] . The volume penalty, ϕ, ensures invertibility of the optimal transformation as it satisfies ϕ(1) = 0, ϕ(z) = ∞ for
doi:10.1002/cnm.2934 pmid:29073332 fatcat:gvv64lcumbabbcnv7646yea5fe