**4. Discussion**

Computational engineering is considered a potentially powerful tool for biomedical sciences. Nevertheless, one of the most critical drawbacks is the time required to develop credible and accurate numerical solutions where fast patient-specific decision making is necessary, i.e., in the vascular biomechanics field. Advancing in this direction, the article presents a novel formulation that combines hierarchical level sets (from a 2D arterial segmentation) with an Immerse Boundary-Robin-based (IBR) formulation to obtain stress and strain distributions in arterial sections under physiological conditions of blood pressure.

The hierarchical level sets allow us to describe the arterial geometries (segmentations), including internal materials distributions (healthy tissue, lipid core, calcified core, among others) and their properties in a highly standardized format. The use of level sets also allows for using a single background mesh to simulate different patient-specific arterial segmentations, all with the same number of degrees of freedom, thus avoiding the preprocessing stages for developing a different conformal finite element mesh per geometry. Furthermore, having the same number of degrees of freedom allows developing so-called "a posteri" reduced-order models (ROMs) or dimensionality reduction methodologies toward fast (even real-time) simulations. It is worth mentioning that despite vascular tissues exhibiting nonlinear behavior, here, a simple linear elastic model is adopted to demonstrate the performance of the proposed approach in the benchmarks selected. Note that the proposed formulation is straightforwardly generalizable to any type of material model. As expected, the accuracy of the IBR depends on the size of the background mesh relative to the minimum size of the heterogeneity. One way to overcome this limitation is to consider an adaptive background mesh with a higher element density near the heterogeneities.

Using the elastic bed boundary conditions (instead of classical Dirichlet) allows us to effectively remove the rigid body motion without altering the natural deformation of the

arterial section due to the internal pressure. However, this boundary condition may also be used to account for the effect of the surrounding tissue on the artery in case the information is known. For instance, a uniform ballast coefficient, *α*, may represent an artery entirely surrounded by tissue as could be the case of the middle cerebral artery or a penetrating myocardial coronary artery. It could also be used to model a partially surrounded artery by specifying a non-uniform ballast coefficient along the external contour of the section. The results obtained with the proposed methodology in terms of displacement and stress fields were very similar compared to those imposing isostatic Dirichlet-type boundary conditions, which are widely accepted in the scientific community. To the best knowledge of the authors, this is the first attempt to propose Immersed Boundary methods with elastic bed boundary conditions for this type of simulation, showing the strong potential of the methodology for biomechanical applications.

**Author Contributions:** Conceptualization, P.D., J.F.R.M. and A.G.-G.; methodology, P.D., J.F.R.M. and A.G-.G.; software, S.G. and M.S.; validation, S.G. and M.S.; formal analysis, P.D.; investigation, J.F.R.M. and A.G.-G.; resources, P.D. and A.G.-G.; data curation, M.S. and S.G.; writing—original draft preparation, S.G.; writing—review and editing, M.S., P.D., J.F.R.M. and A.G.-G.; visualization, S.G. and J.F.R.M.; supervision, P.D., J.F.R.M. and A.G.-G.; project administration, P.D. and A.G.-G.; funding acquisition, P.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors acknowledge the financial support from the Ministerio de Ciencia e Innovación (MCIN/AEI/10.13039/501100011033) through the grants PID2020-113463RB-C33 and CEX2018-000797-S and the Italian Ministry of Education, University and Research (Grant number 1613 FISR2019\_03221, CECOMES).

**Acknowledgments:** The authors acknowledge Zhongzhao Teng of the University of Cambridge for providing the realistic 2D geometries used for this numerical development.

**Conflicts of Interest:** The authors declare no conflict of interest.
