A Modified Suspension-Balance Model for Deformable Particle Suspensions: Application to Blood Flows with Cell-Free Layer

TL;DR

The paper introduces a modified suspension balance model (MSBM) that adds a deformability-induced wall lift force to RBCs, enabling accurate prediction of cell-free layers in confined blood flows. By coupling this MSBM with a multiscale solver and OpenFOAM implementation, the authors demonstrate CFL formation, Fahraeus and Fahraeus-Lindqvist effects in channel and tubular geometries, and good agreement with experimental and DNS benchmarks. The approach yields a computationally efficient continuum framework that captures microstructural heterogeneity and non-Newtonian-like behavior in concentrated deformable particle suspensions under confinement. This framework has potential for scalable simulations of microcirculatory hemodynamics and related biomedical applications.

Abstract

We propose a modified suspension balance model (SBM) for the flow of red blood cells (RBCs) and other deformable particle suspensions in confined geometries. Specifically, the method includes the hydrodynamic lift force generated by deformable particles interacting with walls leading to a cell-free layer. The lift force is added to the SBM to drive RBCs migrating away from the wall. Using the modified SBM (MSBM), we simulate blood flows through microvascular channels and tubes. The method is able to capture the transient development of the cell-free layer (CFL) and the corresponding hematocrit and velocity profiles with the development of the CFL. The CFL thickness and hemorheological hallmarks in microcirculation, such as the Fahraeus Effect and the Fahraeus-Linqvist Effect, are captured and are in good agreement with existing experimental and direct numerical results of blood flows. This work establishes a novel continuum computational framework that can efficiently capture the microstructural heterogeneity and non-Newtonian flow behavior of concentrated deformable particle suspensions under confinement.

Figures (13)

• Figure 1: Flow geometry: two-dimensional channel flow. The snapshot that demonstrates the RBC suspension flows in a channel was obtained using our own multiscale blood flow solver (described in section \ref{['sec:multiscalesol']}) in a $42.7 \, \mu$m height channel with bulk hematocrit $\phi_b=0.2$.
• Figure 2: Comparison between the original suspension balance model (SBM) and our modified suspension balance model (MSBM). For both simulations, we use a channel height of $H=50\,\mu$m, the following parameters are used: $\mu_f/\rho= 1.30 \times 10^{-6}$ m$^2$/s, $\phi_m=0.5$, $\phi_b=0.2$, $a=2.82 \, \mu$m. For our MSBM simulations, we use $\alpha=4$, $f(1-\nu)=1.2$, $\beta=1.2$ and $h_0=1.0\times 10^{-12}$ m. On the left, we show the hematocrit vs channel coordinate profiles; on the right, we illustrate the velocity vs channel coordinate profiles. The solid red lines show the MSBM results, while the black dashed lines correspond to the SBM predictions.
• Figure 3: Temporal evolution of hematocrit (left) and velocity (right) profiles in a channel of $42.7 \, \mu$m height. The red solid line represents the snapshot for $t=1 \times 10^{-3}$ s, the black dashed line is for the case $t=1 \times 10^{-2}$ s, the green dashed-dotted line represents the case for $t=0.2$ s and the purple dashed line is the steady-state solution. The global and particle Reynolds numbers for these simulations are $Re=0.041$ and $Re_p= 1.26 \times 10^{-4}$, respectively.
• Figure 4: Comparison between the continuum model (MSBM, red lines) and our discrete simulation (black bars) results for the temporal evolution of the hematocrit in a $42.7 \, \mu$m height channel with average hematocrit value $\hat{\phi}=0.2$. These snapshots were taken at different $t$, and the MSBM parameter values used are $\mu_f/\rho= 1.30 \times 10^{-6}$ m$^2$/s, $\phi_m=0.50$, $\phi_b=0.2$, $\beta=1.20$, $\alpha=4$, $a=2.82 \, \mu$m, $f(1-\nu)=1.2$, $h_0=1.0\times 10^{-12}$ m and $\epsilon=a/H=0.13$.
• Figure 5: Hematocrit (left) and velocity (right) profiles obtained using our $\textit{SbmLiftFoam}$ solver for different values of the coefficient $\beta$. The blood parameters used in our simulations are: $\mu_f/\rho= 1.30 \times 10^{-6}$ m$^2$/s, $\phi_m=0.5$, $\phi_b=0.2$, $\alpha=4$, $a=2.82 \, \mu$m, $f(1-\nu)=1.2$ and $h_0=1.0\times 10^{-12}$ m. The solid red line corresponds to a value of $\beta=1.0$, the blue solid-dashed line is for $\beta=1.10$ and the black dashed line is for the case with $\beta=1.20$.