Numerical considerations for advection‐diffusion problems in cardiovascular hemodynamics

Abstract

Numerical simulations of cardiovascular mass transport pose significant challenges due to the wide range of Péclet numbers and backflow at Neumann boundaries. In this paper we present and discuss several numerical tools to address these challenges in the context of a stabilized finite element computational framework. To overcome numerical instabilities when backflow occurs at Neumann boundaries, we propose an approach based on the prescription of the total flux. In addition, we introduce a “consistent flux” outflow boundary condition and demonstrate its superior performance over the traditional zero diffusive flux boundary condition. Lastly, we discuss discontinuity capturing (DC) stabilization techniques to address the well‐known oscillatory behavior of the solution near the concentration front in advection‐dominated flows. We present numerical examples in both idealized and patient‐specific geometries to demonstrate the efficacy of the proposed procedures. The three contributions discussed in this paper successfully address commonly found challenges when simulating mass transport processes in cardiovascular flows.

CRIMSON: An open-source software framework for cardiovascular integrated modelling and simulation

“…Here, a single scalar with zero initial condition is considered, an advective flux is imposed at the inlet with a zero total flux condition on all walls and outlets. This allows simulation of transport under high Peclet numbers and interaction of scalar species, including proteins, drugs, contrast agents and heat, and will enable study of phenomena such as thrombogenesis [ 15 ]. CRIMSON will support arbitrary numbers of reacting species, with reaction terms specified by the end-user at run-time in Python.…”

“…F: Scalar reactionadvection-diffusion (RAD) problem with one species transported in the blood stream. Run-time specification of arbitrary reactions between tens of species is under current development [15]. https://doi.org/10.1371/journal.pcbi.1008881.g008…”

“…We have made significant additions, including: (1) the conversion of the codebase from Fortran 77 to Fortran 90; (2) the addition of a modern C++11 layer to manage extensions, including the powerful new arbitrary BC design and control system; (3) the associated Python interface for dynamically adjusting BC parameters during simulations of transitional physiology; and (4) the use of the Verdandi libraries for data assimilation [ 30 ]. Additional modifications include the integration of the Google Test framework [ 31 ], and our ongoing work towards simulation of non-linear vessel wall-blood flow interactions [ 32 , 17 ] and flexible pipelines for simulation of systems of reaction-advection-diffusion (RAD) transport [ 15 ].…”

CRIMSON: An Open-Source Software Framework for Cardiovascular Integrated Modelling and Simulation

“…We have made significant additions, including: (1) the conversion of the codebase from Fortran 77 to Fortran 90; (2) the addition of a modern C++11 layer to manage extensions, including the powerful new arbitrary BC design and control system; (3) the associated Python interface for dynamically adjusting BC parameters during simulations of transitional physiology; and (4) the use of the Verdandi libraries for data assimilation (Chapelle, et al, 2013). Additional modifications include the integration of the Google Test framework (Alphabet Inc., 2016), and our ongoing work towards simulation of non-linear vessel wall-blood flow interactions (Le Tallec, et al, 2001), (Nama, et al, Submitted) and flexible pipelines for simulation of systems of reaction-advection-diffusion (RAD) transport (Lynch, et al, 2020).…”

“…Here, a single scalar with zero initial condition is considered, an advective flux is imposed at the inlet with a zero total flux condition on all walls and outlets. This allows simulation of transport under high Peclet numbers and interaction of scalar species, including proteins, drugs, contrast agents and heat, and will enable study of phenomena such as thrombogenesis (Lynch, et al, 2020). CRIMSON will support arbitrary numbers of reacting species, with reaction terms specified by the end-user at run-time in Python.…”