A nonlinear rotation-free shell formulation with prestressing for vascular biomechanics

Abstract

We implement a nonlinear rotation-free shell formulation capable of handling large deformations for applications in vascular biomechanics. The formulation employs a previously reported shell element that calculates both the membrane and bending behavior via displacement degrees of freedom for a triangular element. The thickness stretch is statically condensed to enforce vessel wall incompressibility via a plane stress condition. Consequently, the formulation allows incorporation of appropriate 3D constitutive material models. We also incorporate external tissue support conditions to model the effect of surrounding tissue. We present theoretical and variational details of the formulation and verify our implementation against axisymmetric results and literature data. We also adapt a previously reported prestress methodology to identify the unloaded configuration corresponding to the medically imaged in vivo vessel geometry. We verify the prestress methodology in an idealized bifurcation model and demonstrate the significance of including prestress. Lastly, we demonstrate the robustness of our formulation via its application to mouse-specific models of arterial mechanics using an experimentally informed four-fiber constitutive model.

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

“…The filter is used to determine suitable parameter values in agreement with time-resolved provided patient data, such as localised pressure or flow measurements [ 55 , 30 , 12 ]. Panel D displays results of a volumetric mesh motion algorithm, which has been developed as a component of an Arbitrary Lagrangian Eulerian (ALE) based FSI framework utilizing a rotation-free nonlinear shell formulation [ 17 ]. This method will enable simulation of large vascular strains and motions, incorporating biologically-relevant constitutive models for the vessel wall (for example, nonlinear strain energy functions for soft tissue, including four fibre models), and involves a novel mathematical formulation that retains displacement-only degrees of freedom for the shell (i.e.…”

“…C: Time history of parameter convergence during Kalman filter data assimilation [ 12 ]. D: Arbitrary Lagrangian-Eulerian deformation of a vascular mesh between diastole and systole [ 17 ]. E: Support for arbitrary run-time Python code for modelling cardiovascular control systems and changes of state, by controlling component parameters during simulations [ 16 ].…”

“…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 ].…”

About prestretch in homogenized constrained mixture models simulating growth and remodeling in patient-specific aortic geometries

“…The elastic radial rollers also helped to reduce stress concentration at the proximal border in the patient-specific ATA simulations. Similar boundary conditions were previously used by Nama et al (2020) and Moireau et al (2012) to model the external tissue support of the aorta (e.g. the surrounding organs), but our study is the first one dedicated to G&R.…”

Rotation-Free Based Numerical Model for Nonlinear Analysis of Thin Shells

“…Symmetric element tangent stiffness matrices lead to computational efficiency and major computer storage saving. In the last forty years, few numerical models dealing with rotation-free formulations [11][12][13] have emerged, in which only three translations at each node are retained as the degrees of freedom. The advantage of such numerical models is that they significantly simplify the problem and are computationally highly efficient compared to these numerical models involving rotation.…”