In what scenarios is FSI necessary for intravascular blood flow simulation?

It is necessary for assessing the risk of aneurysm rupture, predicting restenosis after stent placement, and designing coronary artery bypass grafts. The vessel wall is an elastic body that deforms radially by 5-10% with each pulse. This wall deformation affects the blood flow pattern, making FSI essential.

Is the incompressible Navier-Stokes equation used for the fluid side?

Yes, it is described in the Arbitrary Lagrangian-Eulerian (ALE) framework. $$ \rho_f \left( \frac{\partial \mathbf{u}}{\partial t}\bigg|_{ALE} + ((\mathbf{u} - \mathbf{w}) \cdot \nabla)\mathbf{u} \right) = -\nabla p + \nabla \cdot \boldsymbol{\tau} $$ Blood in large vessels is often approximated as a Newtonian fluid ($\mu \approx 3.5$ mPa·s). However, in low shear rate regions, non-Newtonian models such as the Carreau-Yasuda model are required. $$ \mu(\dot{\gamma}) = \mu_\infty + (\mu_0 - \mu_\infty)(1 + (\lambda \dot{\gamma})^a)^{(n-1)/a} $$

Hemodynamic Simulation

Q: How is the mechanics of the vessel wall modeled?

The vessel wall is treated as a nonlinear hyperelastic material. The Holzapfel-Gasser-Ogden model is widely used. $$ W = \frac{\mu}{2}(I_1 - 3) + \sum_{i=1}^{2} \frac{k_1}{2k_2} \left[ e^{k_2(\kappa I_1 + (1-3\kappa)I_4^{(i)} - 1)^2} - 1 \right] $$ Here, $\mu$ is the matrix stiffness, and $k_1, k_2$ are the stiffness parameters of the collagen fibers, with $I_4^{(i)}$ being the pseudo-invariant in the fiber direction.

Category: Analysis | Integrated 2026-04-06

CAE visualization for hemodynamics theory - technical simulation diagram

Hemodynamic Simulation

Hemodynamic: Theoretical Foundations

Overview of Hemodynamic FSI

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In what situations is FSI necessary for blood flow simulation inside blood vessels?

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For evaluating aneurysm rupture risk, predicting restenosis after stent placement, designing coronary artery bypass grafts, etc. Blood vessel walls are elastic bodies and deform radially by 5-10% with pulsation. This wall deformation affects the blood flow pattern, making FSI necessary.

Governing Equations

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How is the mechanics of the blood vessel wall modeled?

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The blood vessel wall is treated as a nonlinear hyperelastic material. The Holzapfel-Gasser-Ogden model is widely used.

$$ W = \frac{\mu}{2}(I_1 - 3) + \sum_{i=1}^{2} \frac{k_1}{2k_2} \left[ e^{k_2(\kappa I_1 + (1-3\kappa)I_4^{(i)} - 1)^2} - 1 \right] $$

Here, $\mu$ is the matrix stiffness, $k_1, k_2$ are collagen fiber stiffness parameters, and $I_4^{(i)}$ is the pseudo-invariant in the fiber direction.

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Is the fluid side the incompressible Navier-Stokes equations?

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That's correct. It is described in the ALE frame.

$$ \rho_f \left( \frac{\partial \mathbf{u}}{\partial t}\bigg|_{ALE} + ((\mathbf{u} - \mathbf{w}) \cdot \nabla)\mathbf{u} \right) = -\nabla p + \nabla \cdot \boldsymbol{\tau} $$

In large vessels, blood is often approximated as a Newtonian fluid ($\mu \approx 3.5$ mPa·s), but in low shear rate regions, non-Newtonian models like the Carreau-Yasuda model are necessary.

$$ \mu(\dot{\gamma}) = \mu_\infty + (\mu_0 - \mu_\infty)(1 + (\lambda \dot{\gamma})^a)^{(n-1)/a} $$

Coffee Break Casual Talk

Blood is a "Non-Newtonian Fluid" – A Mysterious Liquid Whose Viscosity Changes with Flow Velocity

The fluids first learned in engineering are "Newtonian fluids" like water and air, but blood is completely different. Blood is a non-Newtonian fluid whose viscosity changes with shear rate (the gradient of flow velocity); it has high viscosity at low flow rates and low viscosity at high flow rates. This is because red blood cells overlap each other at low flow rates (forming rouleaux) and disperse/align at high flow rates, reducing resistance. In capillaries (diameter ~8μm), red blood cells (diameter ~7μm) pass through in a single file, and blood behaves almost like a particle march rather than a viscous fluid. When calculating blood flow with CFD, the decision to use a non-Newtonian viscosity model like the Carreau-Yasuda model or simply use a Newtonian approximation (μ≈3.5 mPa·s) depends on the vascular site and flow rate region being analyzed.

Computational Methods for Hemodynamic

Stabilized Finite Element Method

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I heard SUPG/PSPG stabilization is needed for blood flow CFD. Why?

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Equal-order interpolation (P1-P1) does not satisfy the inf-sup condition, leading to spurious oscillations in pressure. Adding SUPG/PSPG stabilization yields stable solutions.

$$ \sum_e \int_{\Omega_e} \tau_{SUPG} (\mathbf{u} \cdot \nabla \mathbf{w}) \cdot \mathbf{R} \, d\Omega = 0 $$

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Local Reynolds numbers are high near vessel bifurcations and stents, so the design of stabilization parameters directly affects computational stability.

Coupling Algorithm

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Is weak coupling sufficient for vascular FSI?

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Blood vessel walls are thin and the added mass effect is significant, so weak coupling tends to diverge easily. Robin-Neumann splitting or Monolithic methods are recommended. Semi-implicit coupling methods using Generalized Robin conditions are gaining attention.

$$ \alpha_f \mathbf{u}_f + \boldsymbol{\sigma}_f \cdot \mathbf{n} = \alpha_f \dot{\mathbf{d}}_s + \boldsymbol{\sigma}_s \cdot \mathbf{n} $$

Image-Based Modeling Workflow

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Please explain the workflow from clinical images to mesh creation.

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1. Acquisition of CT angiography or MRA

2. Segmentation (Mimics, ITK-SNAP, 3D Slicer)

3. Surface smoothing and defect repair

4. Volume mesh generation (TetGen, VMTK, ICEM CFD)

5. Boundary layer mesh insertion (VMTK's boundary layer function)

6. Identification of inlet/outlet surfaces and addition of extension tubes

VMTK (Vascular Modeling Toolkit) is a dedicated OSS for blood vessels, capable of integrated processing from centerline extraction to boundary layer meshing.

Coffee Break Casual Talk

The Surprising Reason Why the Lattice Boltzmann Method (LBM) is Suitable for Blood Flow Analysis

The Lattice Boltzmann Method (LBM) has recently gained attention as a numerical method for blood flow analysis. It takes a completely different approach from conventional CFD (finite volume solution of Navier-Stokes equations), treating fluid as a "statistical distribution of virtual particles." Why is it suitable for blood flow? It excels at calculating suspension fluids with complex-shaped particles like red blood cells, and mesh generation is significantly easier compared to conventional methods. For capillary-level calculations, particle-resolved computation tracking individual red blood cells with LBM is possible. A group at Tohoku University performed the world's first LBM calculation of capillary networks (diameter 5-10μm) on the supercomputer "Fugaku," reproducing red blood cell flow in a model exceeding 500 million cells.

Hemodynamic in Practice

Boundary Condition Settings

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How are the inlet and outlet boundary conditions set?

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Flow rate waveforms obtained from 4D Flow MRI or Doppler echo are set at the inlet. The Womersley solution is often used for the initial profile.

$$ u(r,t) = \sum_{n=0}^{N} \hat{u}_n(r) e^{in\omega t} $$

The larger the Womersley number $\alpha = R\sqrt{\omega/\nu}$ (in the aorta, $\alpha \approx 15$), the flatter the profile becomes.

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How is the Windkessel model at the outlet set?

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Use a 3-element Windkessel, setting proximal resistance $R_p$, distal resistance $R_d$, and Compliance $C$. For multiple outlets, flow distribution is based on the cube of the diameter ratio according to Murray's law.

$$ \frac{Q_1}{Q_2} = \left( \frac{D_1}{D_2} \right)^3 $$

Wall Shear Stress Evaluation Metrics

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What kind of metrics are looked at in the results?

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Time-Averaged Wall Shear Stress (TAWSS) and Oscillatory Shear Index (OSI) are representative.

$$ \text{TAWSS} = \frac{1}{T} \int_0^T |\boldsymbol{\tau}_w| \, dt $$

$$ \text{OSI} = \frac{1}{2} \left( 1 - \frac{|\int_0^T \boldsymbol{\tau}_w \, dt|}{\int_0^T |\boldsymbol{\tau}_w| \, dt} \right) $$

Areas with low TAWSS (< 0.4 Pa) and high OSI (> 0.3) are considered high risk for atherosclerosis. FDA guidance also recommends these evaluations.

Coffee Break Casual Talk

Identifying "Dangerous Spots" in Cerebral Aneurysms with CFD – The Forefront of Clinical Application

Research is advancing to evaluate the rupture risk of cerebral aneurysms (a cause of subarachnoid hemorrhage) using blood flow simulation. Patient-specific 3D models are created from MRI/CT angiography, and wall shear stress (WSS) is calculated with CFD. In areas with extremely low WSS (below 0.4 Pa is a guideline), endothelial cells are prone to degeneration, increasing the risk of aneurysm wall thinning. In joint research with the University of Tokyo and Keio University Hospital, calculating the "area ratio of low WSS regions" with CFD reportedly enabled pre-discrimination between ruptured and unruptured aneurysms with about 80% accuracy. Blood flow analysis is beginning to provide quantitative evidence for difficult decisions like "should we operate or monitor?"

Hemodynamic: Software & Solver Comparison

Comparison of Hemodynamic Analysis Tools

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What tools are available for FSI analysis of hemodynamics?

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There is a wide range from dedicated tools to general-purpose solvers.

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Written by NovaSolver Contributors
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About the Authors

Tool	Type	FSI Support	Features
SimVascular	OSS (Stanford Univ.)	CMM Method	Dedicated to blood vessels. Integrated pipeline from image → Mesh → FSI.
CRIMSON	OSS (Univ. of Michigan)	Yes	Derived from SimVascular. Rich GUI.
ANSYS Fluent

Hemodynamic Simulation

Hemodynamic: Theoretical Foundations

Overview of Hemodynamic FSI

Governing Equations

Blood is a "Non-Newtonian Fluid" – A Mysterious Liquid Whose Viscosity Changes with Flow Velocity

Computational Methods for Hemodynamic

Stabilized Finite Element Method

Coupling Algorithm

Image-Based Modeling Workflow

The Surprising Reason Why the Lattice Boltzmann Method (LBM) is Suitable for Blood Flow Analysis

Hemodynamic in Practice

Boundary Condition Settings

Wall Shear Stress Evaluation Metrics

Identifying "Dangerous Spots" in Cerebral Aneurysms with CFD – The Forefront of Clinical Application

Hemodynamic: Software & Solver Comparison

Comparison of Hemodynamic Analysis Tools

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