Mathematical modelling of blood flows

Mathematical modeling of blood flow is a critical area in biomedical engineering and applied mathematics, offering insights into the circulatory system's behavior under various physiological and pathological conditions. The complexity of blood flow in the human body arises from the non-Newtonian nature of blood, the elasticity of blood vessels, and the intricate network of arteries, veins, and capillaries.

3. Hemodynamics in Arteries and Veins

  • 1D Models: Simplified models that reduce the problem to one dimension, considering the flow in large arteries or veins where the diameter is much larger than the vessel length. These models often solve the Navier-Stokes equations in a cylindrical coordinate system.

  • 2D and 3D Models: These models are more detailed and involve solving the full Navier-Stokes equations numerically. They are used for specific regions, like the heart or bifurcations in arteries.

4. Boundary Conditions

  • Inflow/Outflow Conditions: Often prescribed as velocity profiles or pressure conditions.
  • Wall Conditions: Blood vessel walls can be rigid or elastic, with elasticity introducing additional complexity (e.g., Fluid-Structure Interaction (FSI) models).

5. Fluid-Structure Interaction (FSI)

FSI models account for the interaction between the blood flow and the vessel walls' deformation. The vessel wall's elasticity is modeled using either:

  • Linear Elastic Models: Assume small deformations and linear stress-strain relationships.

  • Nonlinear Hyperelastic Models: Used for large deformations, typically in the arterial walls.

6. Computational Techniques

Numerical methods such as:

  • Finite Element Method (FEM): Used for complex geometries and FSI problems.
  • Finite Volume Method (FVM): Often used for solving the Navier-Stokes equations.
  • Lattice Boltzmann Method (LBM): A mesoscopic approach gaining popularity for its parallelization efficiency.

7. Applications

  • Aneurysm Analysis: Modeling blood flow to assess stress on vessel walls.
  • Stenosis: Evaluating blood flow through narrowed vessels.
  • Heart Valves: Simulating the flow through and around artificial or diseased valves.

8. Challenges and Future Directions

  • Multiscale Modeling: Integrating models at different scales, from molecular to organ levels.
  • Personalized Medicine: Patient-specific models using imaging data.
  • Real-time Simulations: For surgical planning and intraoperative guidance.

Conclusion

Mathematical modeling of blood flows is a powerful tool in understanding cardiovascular health and disease. With advances in computational power and imaging technologies, these models continue to evolve, offering more accurate and personalized insights into blood flow dynamics.