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1D and 2D Mathematical Models for Blood Flow

1D Model

The 1D model is based on a cross-sectional integration of the Navier-Stokes equations under the assumption of incompressible flow in thin arteries. This model is particularly suitable for global studies of the arterial network, where the geometry is approximately linear or weakly curved.

Assumptions and Simplifications

  • The flow is considered incompressible.
  • The artery is modeled as a cylindrical tube with a cross-section varying with pressure.

Main Equations

The derived equations form a system of hyperbolic partial differential equations describing mass and momentum conservation:

  1. Mass conservation:

\[∂_t A + ∂_x Q = 0\]

  1. Momentum conservation:

\[∂_t Q + ∂_x \left( \frac{Q^2}{A} + \frac{1}{\rho} A P(A, x) \right) - \partial_x \left( 3\nu A \partial_x\left(\frac{Q}{A}\right) \right) = \frac{1}{\rho} P(A, x) ∂_x A - \frac{2\pi R K}{1-\frac{Rk}{4\nu}} \frac{Q}{A}\]

Energy and Entropy Relation of the 1D Model

The energy associated with the system is given by:

\[E(t, x) = \frac{A u_x^2}{2} + \frac{1}{\rho} A P(A, x) - \frac{\beta(x)}{3 \rho A_0(x)} A^{3/2}\]

The entropy relation verified by this energy is:

\[∂_t E + ∂_x \left( \left( E + \frac{\beta(x)}{3 \rho A_0(x)} A^{3/2} \right) u_x \right) = ∂_x \left( 3 \nu A ∂_x \left( \frac{Q}{A} \right) \right) u_x + \frac{2 \pi R k}{1 - R k / 4 \nu} u_x^2 ≤ 0\]

Under null boundary conditions:

\[∂_t \left( \int_0^L E \, dx \right) = - 3 \nu \int_0^L A (∂_x u_x)^2 \, dx - \frac{2 \pi R k}{1 - R k / 4 \nu} \int_0^L u_x^2 \, dx < 0\]

2D Model

The 2D model is derived from a radial integration of the Navier-Stokes equations, enabling better representation of local effects in complex geometric configurations, such as arterial bifurcations and severe aneurysms.

Assumptions and Simplifications

  • The flow is assumed incompressible.
  • The artery geometry is described using a curvilinear coordinate system (( s, \theta )).
  • The velocity profile is obtained without relying on a specific ansatz.

Main Equations

  1. Mass conservation:

\[∂_t A + ∂_θ \left( \frac{Q_{Rθ}}{A} \right) + ∂_s(Q_s) = 0\]

  1. Momentum conservation (radial and axial components):

\[∂_t (Q_{Rθ}) + ∂_θ \left( \frac{Q_{Rθ}^2}{2 A^2} + A P \right) + ∂_s \left( \frac{Q_{Rθ} Q_s}{A} \right) = \frac{2 R}{3} C \sin θ \frac{Q_s^2}{A} + \frac{2 R k Q_{Rθ}}{A} + P∂_θ (A)\]

\[∂_t (Q_s) + ∂_θ \left( \frac{Q_s Q_{Rθ}}{A^2} \right) + ∂_s \left( \frac{Q_s^2}{A} - \frac{Q_{Rθ}^2}{2 A^2} + A P \right) = - \frac{2 R}{3} C \sin θ \frac{Q_{Rθ} Q_s}{A^2} + \frac{k R Q_s}{A} + P∂_s (A)\]

Energy and Entropy Relation of the 2D Model

The energy associated with the system is given by:

\[E(t, θ, s) = A \left( \frac{9}{8} u_θ^2 + \frac{u_s^2}{2} + p \right) - \tilde{p}\]

The corresponding entropy relation is:

\[∂_t E + ∂_θ \left( \frac{3}{2} \frac{u_θ}{R} \left( E + \tilde{p} - \frac{9}{16} A u_θ^2 \right) \right) + ∂_s \left( u_s \left( E + \tilde{p} - \frac{9}{16} A u_θ^2 \right) \right) = \frac{9}{4} R k u_θ^2 + k R u_s^2 ≤ 0\]

This relation ensures that the energy locally decreases over time, guaranteeing the stability of the model.

Comparison of 1D and 2D Models

  • 1D Model:
    • Fast and efficient for global simulations of large arterial networks.
    • Well-suited for simple or weakly curved geometries.
    • Very low computational cost.
  • 2D Model:
    • More accurate for complex geometries (bifurcations, aneurysms).
    • Better captures local effects and fluid-structure interactions.
    • Moderate computational cost compared to three-dimensional models (3D NS-FSI).

The combined use of these two models provides an efficient alternative to 3D simulations, offering a good compromise between accuracy and computational cost.