Hodge decomposition of wall shear stress vector fields characterizing biological flows

Abstract

A discrete boundary-sensitive Hodge decomposition is proposed as a central tool for the analysis of wall shear stress (WSS) vector fields in aortic blood flows. The method is based on novel results for the smooth and discrete Hodge-Morrey-Friedrichs decomposition on manifolds with boundary and subdivides the WSS vector field into five components: gradient (curl-free), co-gradient (divergence-free) and three harmonic fields induced from the boundary, which are called the centre, Neumann and Dirichlet fields. First, an analysis of WSS in several simulated simplified phantom geometries (duct and idealized aorta) was performed in order to understand the nature of the five components. It was shown that the decomposition is able to distinguish harmonic blood flow arising from the inlet from harmonic circulations induced by the interior topology of the geometry. Finally, a comparative analysis of 11 patients with coarctation of the aorta (CoA) before and after treatment as well as 10 control patients was done. The study shows a significant difference between the CoA patients before and after the treatment, and the healthy controls. This means a global difference between aortic shapes of diseased and healthy subjects, thus leading to a new type of WSS-based analysis and classification of pathological and physiological blood flow.

Keywords: Hodge decomposition; vector field analysis; wall shear stress.

Figure 1.

Example of a Hodge–Helmholtz decomposition of a PCVF on a torus into gradient, co-gradient and harmonic field.

Figure 2.

An HMF-decomposition of a perturbed WSS vector field on a cylinder into five components: gradient, co-gradient, centre, Neumann and Dirichlet vector field.

Figure 3.

Perturbation of a laminar WSS on a cylinder starting from 0° to 90°. An increase in angle deviation decreases the Dirichlet component and increases the co-gradient components.

Figure 4.

Linear deformation of a pre- to post-intervention of a patient stenosis. A constant input velocity profile is used for the simulation. An increase in the Dirichlet component and a reduction in the co-gradient field is observed within 10 frames of the deformation.

Figure 5.

First row: Plug versus MRI input profile encoded in the centre and Neumann components. Second row: invariance of the Dirichlet component under the change of input profile.

Figure 6.

Comparison of the WSS of 10 healthy patients with MRI versus plug inlet velocity profile. The noise produced from the MRI can be identified by a significant increase of centre and Neumann fields.

Figure 7.

Analysis of artificial aorta models with a varying number of outlets. The streamlines show the flow of the gradient field. The diagram shows that the number of branching outlets is closely related to the gradient and Dirichlet field.

Figure 8.

Evolution of the WSS HMF-decomposition in a CFD simulation with an unsteady flow. The diagram shows 20 time points of the simulation. The close-ups are five phases from the 20 time points of the unsteady flow simulation (green dots). The colours are relative to the min–max magnitude of each input vector field.

Figure 9.

Analysis pipeline of WSS vector fields extracted from a simulated model and analysed via Hodge decomposition.

Figure 10.

Segmented aorta reconstructed from MRI images and used for the CFD simulation.

Figure 11.

Comparison of the WSS of 11 patients before and after intervention, and 10 healthy patients. An improvement in gradient and Dirichlet together with a reduction in co-gradient is observed.

Figure 12.

Examples of a function φ ∈ S_h together with $\mathrm{\nabla }\phi$, $J\mathrm{\nabla }\phi$ (a) and ψ ∈ S_h^* with $\mathrm{\nabla }\psi$, and $J\mathrm{\nabla }\psi$ (b) defined over the triangle T.

Figure 13.

Computation of the curl and divergence of a vector field at a point on $\mathrm{\nabla }{S}_h$ and $J\mathrm{\nabla }{S}_h^{\ast }$.