末梢血管血流FSI
Theory and Physics
Overview
Professor, why is it necessary to couple fluid and structure in peripheral blood flow simulation?
Peripheral arteries are thin, with diameters below a few millimeters and thin vessel walls, so deformation due to blood pressure is relatively large. If the wall bulges, the cross-sectional area increases and flow velocity changes; if flow velocity changes, wall shear stress and pressure change. Ignoring this bidirectional interaction significantly reduces prediction accuracy for pulse wave propagation speed and wall stress.
Specifically, what medical applications are there?
Predicting atherosclerosis progression, evaluating restenosis risk after stent placement, supporting vascular bypass surgery design, etc. It is known that regions with low Wall Shear Stress (WSS) have a high risk of plaque accumulation, so obtaining accurate WSS distribution via FSI is clinically important.
Governing Equations
What equations are used on the fluid and structure sides?
The fluid side uses the incompressible Navier-Stokes equations. Blood is often treated as a non-Newtonian fluid.
Here, $\mathbf{v}_m$ is the ALE (Arbitrary Lagrangian-Eulerian) mesh velocity. Blood viscosity is often described by the Carreau-Yasuo model.
How is the structure side described?
The vessel wall is modeled as a hyperelastic material. The Mooney-Rivlin and the anisotropic Holzapfel-Gasser-Ogden model are representative.
Collagen fiber orientation is represented by two fiber families to reproduce the anisotropy of the arterial wall.
Coupling Interface Conditions
Where do the fluid and structure connect?
Three conditions must be satisfied on the vessel lumen surface.
1. Kinematic Condition: $\mathbf{v}_f = \dot{\mathbf{d}}_s$ (Fluid velocity at interface = Structural displacement velocity)
2. Dynamic Condition: $\boldsymbol{\sigma}_f \cdot \mathbf{n} = \boldsymbol{\sigma}_s \cdot \mathbf{n}$ (Traction continuity)
3. Geometric Condition: Fluid domain shape follows structural deformation
Satisfying these three conditions strictly is strong coupling, which is essential for vascular FSI. Weak coupling becomes unstable due to the added mass effect.
What is the added mass effect?
When the density ratio $\rho_f/\rho_s$ between fluid and structure is close to 1 (which applies to blood and vessel walls), partitioned weak coupling introduces artificial energy and diverges. This is the added mass instability. Methods to avoid this problem, such as the Robin-Neumann splitting method or Quasi-Newton method, are being researched.
Blood is "Not a Newtonian Fluid" — The Story of Carreau Fluid
A pitfall for beginners in peripheral vascular flow analysis is the question, "Is it okay to treat blood as a Newtonian fluid (constant viscosity)?" It's generally acceptable for large vessels, but in peripheral vessels close to thin capillaries, the Carreau fluid characteristic—where viscosity sharply increases at low shear rates—cannot be ignored. Specifically, when shear rate falls below 1s⁻¹, viscosity increases nearly tenfold from about 0.004Pa·s to 0.04Pa·s. Ignoring this nonlinear effect causes significant deviation in wall shear stress distribution.
Physical Meaning of Each Term
- Structure-Thermal Coupling Term: Thermal expansion due to temperature change induces structural deformation, and deformation affects the temperature field. $\sigma = D(\varepsilon - \alpha \Delta T)$. 【Everyday Example】Railroad tracks in summer expand and gaps narrow—temperature rise → Thermal Expansion → stress generation is a typical example. Electronic circuit boards warping after soldering is also due to differences in thermal expansion coefficients between materials. Engine cylinder blocks develop thermal stress from temperature differences between hot and cold parts, potentially leading to cracks.
- Fluid-Structure Interaction (FSI) Term: Bidirectional interaction where fluid pressure/shear forces deform the structure, and structural deformation changes the fluid domain. 【Everyday Example】Suspension bridge cables vibrating in strong wind (Vortex-Induced Vibration)—wind force shakes the structure, the shaken structure alters airflow, further amplifying vibration. Heart blood flow and vascular wall elastic deformation, aircraft wing flutter (aeroelastic instability) are also typical FSI problems. One-way coupling may suffice in some cases, but bidirectional coupling is essential for large deformations.
- Electromagnetic-Thermal Coupling Term: Feedback loop where Joule heating $Q = J^2/\sigma$ causes temperature rise, and temperature change alters electrical resistance. 【Everyday Example】Nichrome wire in an electric stove heats up (Joule heat) and glows red when current flows—temperature rise changes resistance, altering current distribution. Eddy current heating in IH cooking heaters, increased sag in power lines due to temperature rise are also examples of this coupling.
- Data Transfer Term: Interpolation resolves mesh mismatch between different physical fields. 【Everyday Example】When calculating "feels-like temperature" by combining "temperature data" and "wind data" in weather forecasting, interpolation is needed if observation points differ—in CAE coupled analysis, structural mesh and CFD mesh generally don't match, so data transfer (Interpolation) accuracy at the interface directly affects result reliability.
Assumptions and Applicability Limits
- Weak Coupling Assumption (One-way coupling): Effective when one physical field affects the other but the reverse is negligible
- Cases Requiring Strong Coupling: Large deformation in FSI, strong temperature dependence in electromagnetic-thermal coupling
- Time Scale Separation: Sub-cycling can improve efficiency when characteristic times of each physical field differ significantly
- Interface Condition Consistency: Ensure energy/momentum conservation at the coupling interface is satisfied numerically
- Non-applicable Cases: When three or more physical fields are strongly coupled simultaneously, monolithic methods may be necessary
Dimensional Analysis and Unit System
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Thermal Expansion Coefficient $\alpha$ | 1/K | Steel: ~12×10⁻⁶, Aluminum: ~23×10⁻⁶ |
| Coupling Interface Force | N/m² (Pressure) or N (Concentrated Force) | Check force balance between fluid and structure sides |
| Data Transfer Error | Dimensionless (%) | Interpolation accuracy depends on mesh density ratio. Below 5% is a guideline |
Numerical Methods and Implementation
ALE Formulation
The mesh moves when the vessel deforms, right? How is this handled specifically?
The ALE method introduces mesh velocity $\mathbf{v}_m$ and modifies the fluid convection term. At the interface, the mesh follows the structure; inside, mesh quality is maintained via Laplacian smoothing or elastic body analogy.
$k$ is a coefficient that diffuses mesh displacement; the trick is to set it small (high stiffness) near the interface and large farther away.
Doesn't the mesh collapse under large deformation?
Exactly. In cases with high stenosis rates (over 70% occlusion), ALE mesh may fail. Remeshing or meshless methods like the Immersed Boundary method become options.
Immersed Boundary Method
How does the Immersed Boundary method work?
A method developed by Peskin in 1972 for heart valve simulation. The structure is embedded on a fixed fluid mesh, and forces and velocities are exchanged via delta functions.
It is robust against large deformation as mesh regeneration is unnecessary, but has the weakness that accuracy near the interface depends on the delta function width.
1D-3D Coupling Method
It's too heavy to compute all peripheral vessels in 3D, right?
Exactly. The region of interest (e.g., carotid artery bifurcation) is solved in detail with 3D FSI, while upstream/downstream is modeled with a 1D model for systemic circulation. The governing equations for the 1D model are cross-sectionally averaged conservation laws.
The pressure-area relationship is closed by a tube law.
How is the interface between 3D and 1D handled?
The cross-sectional average flow rate and average pressure from the 3D side are passed as boundary conditions to the 1D model. Applying an RCR Windkessel model based on the method of characteristics at the outlet boundary is standard. Ansys Fluent and SimVascular support this coupling.
Time Integration and Convergence
How is the strong coupling iteration converged?
Perform Dirichlet-Neumann iteration within each timestep. The basic approach is alternately applying Dirichlet conditions (displacement specified) to the structure and Neumann conditions (traction specified) to the fluid, but this diverges due to the added mass effect.
The IQN-ILS (Interface Quasi-Newton with Inverse Least Squares) method is very effective for accelerating FSI convergence. It constructs an approximate inverse Jacobian from past interface residuals and updates. The preCICE library implements this method.
| Method | Characteristics | Convergence |
|---|---|---|
| Fixed-Point Iteration | Simple but slow convergence | Prone to divergence in vascular FSI |
| Aitken Relaxation | Acceleration via dynamic relaxation | Moderate |
| IQN-ILS | Quasi-Newton method | Converges in 5~10 iterations |
| Anderson Acceleration | Mixed Method | Comparable to IQN-ILS |
Pulsatile Flow Boundary Conditions — How to Apply the Heartbeat Waveform
Surprisingly tricky in peripheral vascular FSI analysis is "how to set inlet boundary conditions." To reproduce pulsatile flow synchronized with the heartbeat, using velocity waveforms measured by Doppler ultrasound is ideal, but data varies per patient. In practice, the method "approximating periodic velocity profiles using the Womersley solution (analytical solution)" is widely used. Understanding that the profile becomes flat in large vessels with Womersley number (α) over 10, and parabolic in thin vessels below 3—reduces boundary condition mistakes after grasping this difference.
Monolithic Method
Solves all physical fields simultaneously as one system of equations. Stable for strong coupling but complex to implement and memory-intensive.
Partitioned Method (Partitioned Iterative Method)
Solves each physical field independently, exchanging data at the interface. Easy to implement, can utilize existing solvers. Suitable for weak coupling.
Interface Data Transfer
Nearest neighbor (simplest but low accuracy), projection (conservative), RBF interpolation (robust to mesh mismatch). Balance between conservation and accuracy is important.
Sub-iteration
Performs sufficient iterations within each coupling step to ensure interface condition consistency. Residual criteria are scaled based on typical values of each physical field.
Aitken Relaxation
Automatically adjusts coupling iteration relaxation factor. Adaptive technique that prevents divergence from over-relaxation and accelerates convergence.
Stability Condition
Beware of added mass effect (in fluid-structure coupling when structural density ≈ fluid density). Apply Robin-type interface conditions or IQN-ILS method if unstable.
Analogy for Aitken Relaxation
Aitken relaxation is like "balancing a seesaw." If one side pushes too hard, the other side bounces up, and the recoil causes pushing too hard again—Aitken relaxation automatically adjusts the pushing force to suppress this oscillation. It's an adaptive technique that automatically adjusts the next correction amount based on the previous correction when coupling iterations oscillate and fail to converge.
Practical Guide
The typical workflow is as follows.
1. Acquire Medical Images: Obtain vascular geometry from CT Angiography (CTA) or MRA
2. Segmentation: Extract vessel lumen and wall using ITK-SNAP, 3D Slicer, Mimics, etc.
3. Geometry Repair and CAD Conversion: Smooth surface mesh, repair defects with Geomagic Wrap, etc.
4. Volume Mesh Generation: Hexahedral elements for vessel wall (3+ layers), prism boundary layer + tetrahedra for lumen
5. Material / Boundary Condition Setting: Hyperelastic material, pulsatile inlet velocity, Windkessel outlet condition
6. FSI Calculation Execution: Strong coupling for 2~3 heartbeat cycles (to remove initial transients)
7. Postprocessing: Evaluate WSS, OSI (Oscillatory Shear Index), wall displacement
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