Oxygen Mass Transport in a Compliant Carotid Bifurcation Model

The purpose of the present study was to investigate oxygen mass transfer in the human carotid bifurcation, focusing on the effects of the wall compliance and flow field on the temporal variation and spatial distribution of the oxygen wall flux. Details of unsteady convective-diffusive oxygen transport were examined numerically using a compliant model of the human carotid bifurcation and realistic blood flow waveforms. Results reveal that axial flow separation at the outer common-internal carotid wall can significantly alter the flow field, oxygen tension field, and oxygen wall flux distribution. At the outer wall of the sinus, the Sherwood number, Sh (non-dimensional oxygen wall flux), takes on significantly lower values than at other sites due to the attenuation of transport rates by convective flow away from wall. More specifically, the lowest value of Sh was Sh∼6 (in the sinus), which is much lower than the value of the non-dimensional oxygen consumption rate (Damkohler number, Da) in the reactive wall tissue (Da=29–39). At the inner wall of the sinus, Sh∼170 is far above the expected value of Da. This implies that flow separation on the outer wall of the sinus provides a very strong fluid mechanical barrier to oxygen transport; whereas at the inner wall of the sinus, the mechanism of transport is controlled by the wall consumption rate.

APPENDIX

The equations governing the fluid motion are the momentum equations and the equation of continuity:

$$ \rho \frac{{\partial u_i }}{{\partial t}} + \rho u_{i,j} (u_j - \bar u_j ) = \sigma _{ij,j} + \rho f_i $$

(A1)

$$ u_{j,j} = 0 $$

(A2)

where u _j is the velocity component in the x _j direction, comma j (, j) denotes the partial derivative of a function with respect to the independent variable x _j . Any terms in which the same index appears twice stands for the sum of all terms obtained by giving this index its complete range of values (j=1, 2, 3). In addition, ρ is the fluid density, σ _ij is the stress, f _i is the body force at time t per unit mass, and $\bar u$ is the mesh velocity at time t. The equations governing the structural domain are the momentum equations, the equilibrium conditions and the constitutive equations, respectively:

$$ \rho a_i = \sigma _{ij,j} + \rho f_i $$

(A3)

$$ \sigma _{ij} n_j = {}^st_i $$

(A4)

$$ \sigma _{ij} = D_{ijkl} \varepsilon _{kl} $$

(A5)

where a _i represents the acceleration of a material point (where displacement is defined as $d_i = x_i - x_i^0$, and $x_i^0$ is the stress-free position) at time t, n _j is the outward pointing normal vector on the structural boundary at time t, ^s t _i is the surface traction vector at time t, D _ijkl is the material elasticity tensor, and ɛ _ij is the infinitesimal strain tensor.

The grid system is adjusted to accommodate the deformation of the structural body to avoid distortion due to significant deformation of the computational domain.