Diffusive interface approach to oxygen transport and metabolism under cellular flow dynamics in microcirculations

TL;DR

This work develops a diffusive interface, mixture-based framework for simulating oxygen transport in microcirculation under cellular flow. By rewriting the governing equations for oxygen concentration and HbO$_2$ saturation as a single mixture PDE using phase indicators, interfacial jump conditions are implicitly enforced on a fixed Cartesian mesh and coupled to RBC membrane dynamics via immersed boundary methods. The approach is validated against analytical spherical diffusion and demonstrated in straight capillaries and simple capillary networks, revealing how RBC distribution and the presence of the RBC membrane influence tissue oxygenation. The method offers a stable, scalable tool for exploring microvascular oxygen metabolism with potential extensions to tumor hypoxia, artificial oxygen carriers, and targeted drug delivery.

Abstract

The relationship between the spatiotemporal distribution of oxygen transport and cellular flow dynamics is of fundamental importance for understanding microcirculation systems. Three-dimensional (3D) modeling is indispensable for addressing complex oxygen transport and cellular behaviors in capillary networks; however, the computational approach is formidable for enforcing interface (or jump) conditions on largely moving and deforming interfaces. In this paper, we propose a diffusive interface approach for the oxygen transport using a mixture formulation. We formulate oxygen transport using an advection-diffusion-reaction equation and rewrite all governing equations in mixture forms using phase indicator functions, where all the interface conditions are included in the governing equations. This innovation avoids the complexity associated with discontinuities for largely moving interfaces in highly dense red blood cell (RBC) conditions. We model cellular flow as a fluid-membrane interaction problem using the immersed boundary method (IBM). The method allows the seamless calculation of coupling problems for cellular flows and oxygen transports in the cytoplasm (internal fluid) of the RBC, plasma (external fluid), and tissue regions using a fixed Cartesian coordinate mesh. The proposed method accurately captures the analytical solution for spherically symmetric diffusion, and successfully demonstrates oxygen transport in both straight capillaries and their networks.

Figures (11)

• Figure 1: (a) Schematic of the oxygen transport system in the present study. The contour colors represent the degree of oxygen concentration. (b) Definition of the computational regions $\Omega_i$ ($i \in [1,3]$), where $\Omega_1$ is the internal RBC (internal fluid) and $\Omega_2$ is plasma (external fluid), and their interfaces $\Gamma_{12}$ and $\Gamma_{23}$.
• Figure 2: (a) Schematic of an finite interface region $\tilde{\Gamma}_{12}$ (dashed area) representing the smoothed phase indicator function $\psi_1$ ($\in (0,1)$), and alternative phases: internal fluid $\tilde{\Omega}_1$ ($\psi_1 = 1$) and external fluid $\tilde{\Omega}_2$ ($\psi_1 = 0$). (b) Typical profile of the smoothed phase indicator function $\psi$ across the interface in the normal direction $r$.
• Figure 3: (a) Definition of the domains for a spherical diffusion problem: $\Omega_1$ is the internal RBC phase, $\Omega_2$ is the plasma phase, and $\Omega_3$ is the tissue phase. (b and c) Oxygen concentration $c$ as a function of the distance from the entrance of domain $r$ at $t = 0.01$ s and $0.1$ s, where solid black lines referent solutions. The symbols are numerical results with various spatial resolutions $\Delta x$.
• Figure 4: Numerical validation of oxygen transport with a moving (but non-deformable) interface. Instantaneous solutions of oxygen concentration $c$ for the fixed interface (a) and moving interface with a constant advection velocity (b). (c and d) Comparisons of axial profiles in both stream-wise $x$ and span-wise $z$ directions for $c$ and $s$ at $t=0.16$ s (after one period) for the fixed and moving interfaces. The results in panel (c) are shown as a function of the relative coordinate system based on the centroid of the spherical capsule, $x_c$.
• Figure 5: (a) Computational domains for oxygen transport with RBCs in a straight capillary, consisting of the internal RBC ($\Omega_1$), plasma ($\Omega_2$), and tissue ($\Omega_3$): ($left$) lateral view and ($right$) front view. The flow direction is from left to right. Snapshots of oxygen concentration $c$ (b) and saturation $s$ (c) on the lateral cross-sectional area ($z$ = 0) at ($left$) the initial state $t = 0$ s, ($middle$) $t = 0.05$ s, and ($right$) $t = 0.4$ s in $d = 9.66$$\mu$m. (d) Time history of $c_3$ averaged in the tissue phase $\Omega_3$.