Time-Dependent Simulations

Time-Dependent Simulations#

This section covers the time-dependent cardiac mechanics solvers in fenicsx-pulse. Unlike static problems where we solve for equilibrium at a single state, here we integrate the equations of motion over time to simulate a full cardiac cycle.

Mathematical Formulation#

The dynamic simulations solve the balance of linear momentum including inertia and damping effects. The governing equations in the reference configuration are:

\[ \rho \ddot{\mathbf{u}} - \nabla \cdot \mathbf{P} = \mathbf{0} \quad \text{in } \Omega_0 \]

subject to appropriate boundary conditions (Dirichlet, Neumann, or Robin).

  • \(\mathbf{u}\): Displacement field.

  • \(\mathbf{P}\): First Piola-Kirchhoff stress tensor.

  • \(\rho\): Mass density.

Time Integration#

To solve this system numerically, we discretize in time using the Generalized-\(\alpha\) method [EBB02]. This is an implicit, second-order accurate scheme that allows for control over high-frequency numerical dissipation. It solves for the displacement \(\mathbf{u}_{n+1}\), velocity \(\mathbf{v}_{n+1}\), and acceleration \(\mathbf{a}_{n+1}\) at each time step.

Benchmark Problems (Bestel Model)#

These examples implement the cardiac elastodynamics benchmarks described in [ArosticaNB+25]. They use a simplified analytical model (the Bestel model [BClementS01]) to drive the cavity pressure and active tension, focusing on the verification of the mechanical solver and the time integration scheme.

  • LV Benchmark: Simulates a beating Left Ventricle (LV) ellipsoid. It verifies the implementation of orthotropic passive material properties, time-dependent active stress, viscoelasticity, and dynamic Robin boundary conditions.

  • BiV Benchmark: Extends the benchmark to a Bi-Ventricular (BiV) geometry. This involves applying distinct pressure loads to the LV and RV cavities while handling the complex geometry of the septum and free walls.

Coupling to a circulation#

In these examples the 3D mechanics model takes the place of the ventricles in a 0D model of the circulation, and the two exchange cavity volumes and pressures. They differ in how the 0D side is advanced:

Style

How the 0D side is advanced

Demo

Pick it when

Five-phase cycle

pulse.cycle.CycleController, a Windkessel per cavity

complete_cycle

you want the classic phase-driven cycle without a closed loop

Split loop

any 0D code, called each step

land_circulation_biv

the circulation is an external solver

Monolithic

GotranxCirculation, 0D states inside Newton

monolithic_3d0d, monolithic_3d0d_biv

the circuit is a .ode file and you want one Newton system

Each style has a CLI template: pulse init --template complete_cycle, split_biv, monolithic_lv or monolithic_biv.

The closed loop in the last two is the circulation model of Regazzoni et al. [RSA+22]. Each demo also chooses its own activation: complete_cycle and the monolithic demos use the Bestel model [BClementS01], and land_circulation_biv runs the Land crossbridge model [LPHS+17] at every quadrature point. complete_cycle solves the dynamic problem, land_circulation_biv the quasi-static one, which needs no time integration scheme, and the monolithic demos can run either.

See also the Isometric Twitch Experiments & the Frank-Starling Mechanism section, which uses the same quasi-static formulation as land_circulation_biv but focuses on cellular-scale active tension models rather than a full circulation loop.

References#

[ArosticaNB+25]

Reidmen Aróstica, David Nolte, Aaron Brown, Amadeus Gebauer, Elias Karabelas, Javiera Jilberto, Matteo Salvador, Michele Bucelli, Roberto Piersanti, Kasra Osouli, and others. A software benchmark for cardiac elastodynamics. Computer Methods in Applied Mechanics and Engineering, 435:117485, 2025. doi:10.1016/j.cma.2024.117485.

[BClementS01] (1,2)

Julie Bestel, Frédérique Clément, and Michel Sorine. A biomechanical model of muscle contraction. In Medical Image Computing and Computer-Assisted Intervention–MICCAI 2001: 4th International Conference Utrecht, The Netherlands, October 14–17, 2001 Proceedings 4, 1159–1161. Springer, 2001. doi:10.1007/3-540-45468-3_143.

[EBB02]

Silvano Erlicher, Luca Bonaventura, and Oreste S Bursi. The analysis of the generalized-α method for non-linear dynamic problems. Computational mechanics, 28(2):83–104, 2002. doi:10.1007/s00466-001-0273-z.

[LPHS+17]

Sander Land, So-Jin Park-Holohan, Nicolas P Smith, Cristobal G Dos Remedios, Jonathan C Kentish, and Steven A Niederer. A model of cardiac contraction based on novel measurements of tension development in human cardiomyocytes. Journal of molecular and cellular cardiology, 106:68–83, 2017. doi:10.1016/j.yjmcc.2017.03.008.

[RSA+22]

Francesco Regazzoni, Matteo Salvador, Pasquale Claudio Africa, Marco Fedele, Luca Dedè, and Alfio Quarteroni. A cardiac electromechanical model coupled with a lumped-parameter model for closed-loop blood circulation. Journal of Computational Physics, 457:111083, 2022. doi:10.1016/j.jcp.2022.111083.