# Cardiovascular Modeling

Figure 1: Cardiac fluid dynamics models of blood flow on patient-specific moving left atrial geometries.

Figure 2: Cardiac electrophysiology (EP) simulations on patient specific biatrial geometries.

CFD simulations involve solving the **Navier-Stokes equations** using finite element methods, such as the **Arbitrary Lagrangian-Eulerian (ALE) method** to handle moving boundaries (Figure 1). The Navier-Stokes equations, governing the fluid dynamics, are given by:

$$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = -\nabla p + \nu \nabla^2 \mathbf{u} + \mathbf{f},$$

$$ \nabla \cdot \mathbf{u} = 0, $$

where $ \mathbf{u} $ is the velocity field, $ p $ is the pressure, $ \nu $ is the kinematic viscosity, and $ \mathbf{f} $ represents body forces (e.g., gravity).

The **ALE formulation** introduces a mapping from a reference domain $ \Omega_0 $ to a time-dependent domain $ \Omega(t) $. This mapping can be described by:

$$ \mathbf{x} = \mathbf{x}(\boldsymbol{\xi}, t), $$

where $ \boldsymbol{\xi} $ are the coordinates in the reference domain $ \Omega_0 $ and $ \mathbf{x} $ are the corresponding coordinates in the physical domain $ \Omega(t) $. The mesh velocity $ \mathbf{w} $ is defined as:

$$ \mathbf{w} = \frac{\partial \mathbf{x}}{\partial t}, $$

and the velocity field in the ALE formulation becomes:

$$ \frac{d \mathbf{u}}{dt} = \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} - \mathbf{w}) \cdot \nabla \mathbf{u}. $$

This formulation ensures that the mesh deforms smoothly with the boundary motion, maintaining numerical stability in the simulation.

For cardiac electrophysiology (EP) simulations, the **monodomain anisotropic reaction-diffusion equation** is solved either using finite-element method (Figure 2) or the **Lattice Boltzmann Method (LBM)** (Figure 3). The monodomain equation is:

$$ \frac{\partial V}{\partial t} = \nabla \cdot (\mathbf{D} \nabla V) + I_{\text{ion}}(V, w), $$

$$ \frac{dw}{dt} = f(V, w), $$

where $ V $ is the transmembrane potential, $ \mathbf{D} $ is the anisotropic diffusion tensor, $ I_{\text{ion}} $ represents the ionic currents, and $ w $ denotes the state variables for ion channels.

The **LBM** discretizes these equations by evolving particle distribution functions $ f_i(\mathbf{x}, t) $ along characteristic directions:

$$ f_i(\mathbf{x} + \mathbf{c}_i \Delta t, t + \Delta t) - f_i(\mathbf{x}, t) = -\frac{1}{\tau} \left( f_i(\mathbf{x}, t) - f_i^{\text{eq}}(\mathbf{x}, t) \right), $$

where $ f_i^{\text{eq}} $ is the equilibrium distribution, $ \mathbf{c}_i $ are discrete velocities, and $ \tau $ is the relaxation time controlling diffusion. This method captures the anisotropic nature of cardiac tissue efficiently and facilitates large-scale simulations of cardiac dynamics through its compatibility with high performance parallel computing.

Figure 3: Cardiac electrophysiology (EP) simulations on patient-specific biatrial geometry using the Lattice Boltzmann method.