A robust shape model for blood vessels analysis

TL;DR

A robust, low dimensional, patient specific vascular model is presented and used to study phenotype variability analysis of the thoracic aorta within a cohort of patients and an haemodynamics atlas of the aorta for the same cohort is built.

Abstract

The availability of digital twins for the cardiovascular system will enable insightful computational tools both for research and clinical practice. This, however, demands robust and well defined models and methods for the different steps involved in the process. We present a vessel coordinate system (VCS) that enables the unanbiguous definition of locations in a vessel section, by adapting the idea of cylindrical coordinates to the vessel geometry. Using the VCS model, point correspondence can be defined among different samples of a cohort, allowing data transfer, quantitative comparison, shape coregistration or population analysis. Furthermore, the VCS model allows for the generation of specific meshes (e.g. cylindrical grids, ogrids) necessary for an accurate reconstruction of the geometries used in fluid simulations. We provide the technical details for coordinates computation and discuss the assumptions taken to guarantee that they are well defined. The VCS model is tested in a series of applications. We present a robust, low dimensional, patient specific vascular model and use it to study phenotype variability analysis of the thoracic aorta within a cohort of patients. Point correspondence is exploited to build an haemodynamics atlas of the aorta for the same cohort. The atlas originates from fluid simulations (Navier-Stokes with Finite Volume Method) conducted using OpenFOAMv10. We finally present a relevant discussion on the VCS model, which covers its impact in important areas such as shape modeling and computer fluids dynamics (CFD).

Figures (9)

• Figure 1: Saggital view of the 30 thoracic aorta meshes used for the study.
• Figure 2: The Vessel Coordinate System is defined on a vessel section $V$, confined by two cross sections $A$, and $B$ (top). The longitudinal axis of the cylinder is represented by a curve, $\mathbf{c}$, on which a local reference frame is defined, $\left\{\mathbf{t}(t),\mathbf{v}_1(t),\mathbf{v}_2(t)\right\}$ (bottom figure, arrows in blue), with $\mathbf{t}$ the unitary tangent to $\mathbf{c}$. Given a point $\mathbf{x}$, its coordinates are its longitudinal location along the curve, $\tau$, and the polar coordinates of $\mathbf{x}$ in the plane orthogonal to the curve containing $\mathbf{x}$, $(\rho,\theta)$, represented in red in the figure.
• Figure 3: The representation of an aorta using the VCS. After computing the centerline of the vessel (in orange), the wall is approximated as a surface $\rho_w$ depending on coordinates $(\tau,\theta)$. This surface is represented, on the right, in VCS space. The color map corresponds to the distance of the wall points to the centerline, in millimeters, with the radius ranging, for this aorta, from $\sim$10mm to $\sim$17mm. Two points, $p_0$ (on the wall) and $p_1$ (inside the lumen) are represented in both Cartesian and VCS spaces, as well as two lines on the surface (in yellow), corresponding to $\theta=5\pi/4$ and $\tau=1/5$. The reference frame on the centerline is also shown in red in the Cartesian space for a point of the centerline.
• Figure 4: On the first row, from left to right, the input aorta, the patient specific B-Spline approximation using $L=9,R=15,K=19$, and the superimposition of both (the approximation light blue). On the second row, the distribution of residuals (in mm) on the input mesh, with detailed views of the sinuses of Valsalva and the aortic arch, where the approximation is less accurate.
• Figure 5: Plot of the sensitivity analysis on the influence of the parameters $L$, $R$, and $K$ over the estimation residuals of the patient specific vascular model. The top plot shows the cohort average of the mean residuals, and the bottom plot shows the cohort average of the 75$^{th}$ quantile of the residuals, in both cases expressed in millimeters. In both plots, the dots placed in a regular grid are color-coded to indicate the value of the residual at the corresponding values of $L$, $R$ and $K$.