Conference Agenda

State estimation in large arteries presents a significant challenge, requiring robust methods that integrate physical models with experimental measurements such as Doppler ultrasound and 4D-flow MRI. In this study, we focus on reconstructing velocity, viscosity, and pressure states in shear-thinning blood flows using a projection-based approach. We compare three model order reduction strategies: classical Singular Value Decomposition (SVD), randomized SVD (rSVD), and a novel parametric Dynamic Mode Decomposition (DMD) method. The latter not only extrapolates partial velocity data beyond the acquisition region but also provides a predictive framework for super-sampling MRI data, which typically suffers from low temporal resolution.

Our methodology is validated in a patient-specific internal carotid artery siphon, where the state estimation is conducted within a manifold learning framework. Benchmark evaluations are performed for clinically relevant hemodynamic quantities, including pressure drops, oscillatory shear index (OSI), wall shear stress (WSS), and vorticity dynamics. Numerical simulations for the non-intrusive model order reduction are performed using the finite element method (FEM) for a non-Newtonian variation of the incompressible Navier-Stokes equations. A Carreau-like model is employed to account for varying hematocrit levels, while a variational multi-scale stabilization strategy ensures numerical stability in low-order elements.

Furthermore, we explore methodological advancements in inter-patient reconstructions. Specifically, we introduce an embedding strategy that enables state recovery in new arterial geometries without requiring a new offline training phase. This is achieved through a geometric mapping via a large diffeomorphism metric matching approach, enhanced by a multi-grid residual network for performance optimization. To ensure mass conservation, we employ a Piola transformation, while a nearest-neighbor search mechanism selects the most suitable pre-trained reduced-order model (ROM).

Our findings demonstrate the potential of combining model order reduction with data assimilation to improve the accuracy and efficiency of hemodynamic state estimation in clinical applications. This approach paves the way for real-time patient-specific simulations and enhanced diagnostic capabilities in vascular medicine.

Computational hemodynamics can enhance image-based diagnostic and provide complementary insights to predict, understand, and monitor treatments. The high computational costs and the complexity associated with handling patient-specific settings still remain a major challenge towards clinical applications. In this work, we propose a robust shape registration method for aortic geometries and its application to surrogate hemodynamic models and data assimilation tasks.

The approach is based on ResNet-LDDMM trained with a dataset of synthetic shapes, generated via SSM from an initial cohort of aortic coarctation patients and healthy subjects.

The loss function for training the ResNet is tailored to surface meshes, considering a modified Chamfer distance that accounts for mesh boundaries and orientations.

Moreover, the optimization utilizes a multigrid strategy, refining mesh size over the training epochs, that allows to handle realistic mesh sizes. The registration allows to define geometric encodings of different blood flow solutions on a single reference shape, as well as to build projection-based reduced-order models.

We employ this geometrical encoding to improve the training of encode-process-decode graph neural networks (GNN) and present potential applications in data assimilation problems, combined with a generalized Parametrized-Background Data-Weak formulation.

In partiular, we address the reconstruction of velocity fields and wall shear stresses, as well as the estimation of pressure fields and pressure-related biomarkers, like the pressure drop, from low-resolution velocity observations. We show different numerical tests based on the synthetic data, comparing the proposed strategies with state of the art estimators.

Accurate patient-specific blood flow simulations require precise specification of material parameters and boundary conditions to be a useful tool in clinical applications. Although no-slip conditions are commonly applied, allowing for some degree of slip at the vessel wall might, in many cases, lead to better agreement with flow patterns observed in medical images. In an attempt to address this issue, we are solving an inverse problem to estimate the boundary condition parameters, specifically focusing on the inlet velocity profile and wall slip parameters, using measured MRI velocity data (4D PC-MRI) obtained in descending aorta over one cardiac cycle.

The implementation is done using the Firedrake framework, an automated system for solving partial differential equations using finite element method, together with Pyadjoint, a library that enables efficient computation of adjoint equations by automatically differentiating the forward problem. Pyadjoint also contains an interface to various parallel implementations of optimization algorithms such as L-BFGS. These tools provide a flexible and efficient way to handle optimization problems in computational fluid dynamics, making them convenient for our application. Since solving inverse problems is usually computationally expensive, the forward problem is solved by the Newton or Picard method with a direct solver and discretized using P1P1 finite elements with interior penalty stabilization to reduce the overall cost.

To assess the robustness of the approach before applying it to real MRI data, we first conduct tests using artificially generated velocity fields under various conditions. This includes varying the amount of noise and resolution of the artificially generated velocity data, different numerical settings of the forward problem, and various regularization strategies of the error functional, which was done first for steady flow and then for transient flow.

The approach is also applied to multiple sets of real MRI patient-specific data, consisting of a morphological image (3D SSFP MRI) and velocity data (4D PC-MRI). Since patient-specific simulations usually contain some amount of uncertainty regarding the vessel wall, multiple segmentation of the morphological image are tested to study the sensitivity of the approach to this known phenomenon. Additionally, we compare the impact of Navier’s slip boundary condition (both constant and spatially varying) against the traditional no-slip condition, analyzing its influence on the flow simulations. By refining boundary condition estimations, this method can enhance post-processing analyses such as wall shear stress (WSS) calculations and pressure reconstruction, both of which are critical for assessing vascular health and disease progression. Improved flow accuracy may also lead to better estimates of hemodynamic biomarkers, further bridging the gap between computational modelling and clinical decision-making.

Internal carotid artery (ICA) stenoses are a major source of stroke. Besides typical cardiovascular risk factors, plaque composition, carotid bifurcation geometry, and hemodynamic parameters such as wall shear stress (WSS) are recognized as critical contributors to the progression and rupture of ICA plaques [1].

4D flow MRI provides time-resolved 3D velocity data for patient-specific hemodynamic assessment. Integrating this with CFD improves accuracy, especially in carotid stenosis, supporting risk stratification and treatment planning [2]. This study uses patient-specific CFD simulations to assess hemodynamics at the carotid bifurcation. The approach helps identify biomarkers for risk evaluation, personalized treatment, and stroke prevention across varying stenosis types.

Flow data and vascular geometries were obtained from 120 patients with mild carotid stenosis using 4D flow MRI from a previously published cohort [3]. These inputs enabled patient-specific simulations to comprehensively analyze hemodynamic variations and disease progression. Vessel segmentation was carried out using a nnU-Net model trained on semi-manual annotations by an experienced neurologist in MeVisLab.

Using VMTK, a centerline is generated for each geometry, with boundary patches constructed orthogonally along it. From phase-contrast images, 3D velocity vectors are extracted at these locations across all time points to define patient-specific inlet conditions. Appropriate meshes are using blockMesh and SnappyHexMesh utilities from OpenFOAM. CFD simulations are then run with validated solver parameters, followed by automated post-processing to derive relevant hemodynamic parameters.

This study focused on stenosed cases featuring plaques specifically located along the outer wall of the ICA. Key hemodynamic parameters included the pressure gradient, defined as the difference between maximum and minimum plaque surface pressure normalized by the projected centerline distance and the maximum WSS. Results demonstrated a consistent relationship between pressure gradient and maximum WSS, with elevated WSS often corresponding to increased pressure gradients. The disturbed flow was assessed using the oscillatory shear index (OSI) and relative residence time (RRT), where higher values indicated turbulence, recirculation, and fluctuations in WSS, conditions associated with plaque development and progression.

Stenosis severity alone does not determine hemodynamic risk; plaque morphology, location, and arterial geometry significantly influence localized forces. Certain geometrical features can lead to the co-localization of high-pressure gradients and WSS, increasing rupture risk. A steep reduction in cross-sectional area from the common carotid artery (CCA) to the peak stenosis in the ICA amplifies pressure gradients, particularly in steeper plaques, while flatter plaques exhibit lower gradients. Plaques extending into the CCA experience higher pressure gradients than those confined to the ICA. These findings highlight the critical role of arterial anatomy and plaque morphology in assessing plaque vulnerability.