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

This paper proposes a modified suspension-balance model that incorporates hydrodynamic lift forces to efficiently simulate blood flows in microvascular channels, successfully capturing key phenomena such as cell-free layer formation, hematocrit and velocity profiles, and the Fahraeus and Fahraeus-Lindqvist effects.

The Gist

Imagine a busy highway where the cars are actually tiny, squishy red blood cells, and the road is a microscopic blood vessel. In a normal highway, you might expect traffic to be spread out evenly. But in our bodies, these "cars" have a weird habit: they hate being near the walls. They prefer to huddle together in the middle of the road, leaving a clear, empty lane right next to the pavement.

This empty lane is called the Cell-Free Layer (CFL). It's a crucial feature of our blood flow that helps our blood move faster and with less friction.

The Problem: The Old Map Was Missing a Turn

Scientists have been trying to build computer models to simulate how blood flows for years. They use something called a "Suspension Balance Model" (SBM). Think of this model as a traffic simulation software.

The old version of this software was good at predicting that cars would move toward the center of the road because of how they bump into each other. However, it failed to explain why the cars were so eager to leave the walls. It couldn't create that empty "cell-free lane" near the edge. It was like a GPS that knew the cars were moving but didn't know they were actively avoiding the curb.

The Solution: A New "Push" Button

The authors of this paper, led by Hugo Castillo-Sánchez and Leonardo Liu, decided to fix the software. They realized that because red blood cells are squishy (deformable), they generate a special kind of invisible force when they get too close to a wall.

They call this the Lift Force.

• The Analogy: Imagine you are swimming near the side of a pool. As you move, the water pushes you slightly away from the wall. For red blood cells, this "push" is much stronger because they are squishy and change shape as they squeeze past the wall.
• The Fix: The team added this "Lift Force" to their computer model. They created a Modified Suspension Balance Model (MSBM). Now, the software doesn't just watch the cars; it actively pushes them away from the wall, just like the water pushes a swimmer.

What Happened When They Ran the Simulation?

When they turned on this new "Lift Force" in their computer, the results changed dramatically:

1. The Empty Lane Appeared: The simulation successfully created that clear zone near the wall (the CFL) that we see in real life.
2. The Traffic Jam in the Middle: The red blood cells piled up in the center, creating a dense core.
3. The Shape of the Flow: Because the cells were bunched in the middle and the edges were clear, the blood didn't flow in a smooth, curved arch (like a normal river). Instead, it flowed like a solid plug or a piston, with a flat top. This is exactly what happens in real micro-vessels.

Testing the New Model

The team didn't just guess; they tested their new model against real-world data and other complex simulations:

• Time Travel: They watched how the "empty lane" formed over time. It started with cells everywhere, and slowly, the "Lift Force" pushed them away from the walls until the lane was clear. This matched the speed and behavior seen in high-speed camera experiments.
• The "Fåhræus Effect": This is a fancy term for a simple observation: blood flows faster in tiny tubes than you'd expect, and the concentration of cells in the middle is different from the concentration at the exit. Their new model predicted this perfectly.
• The "Fåhræus-Lindqvist Effect": This is the observation that blood becomes "thinner" (less sticky) when it flows through very small tubes. Their model captured this too, showing that the empty lane near the wall reduces friction, making the blood flow easier.

The Bottom Line

The paper claims that by adding a simple "push" (the lift force) to their computer model, they can now accurately simulate how blood behaves in tiny vessels.

• What it does: It captures the formation of the cell-free layer, the plug-like flow, and the famous "Fåhræus" effects that make blood flow efficient in our bodies.
• What it doesn't do (yet): The authors admit that for very large tubes (over 40 micrometers), the model pushes the cells away a little too much. They suspect this is because their model doesn't yet account for how cells "shield" each other from the wall when they are crowded together. They plan to fix this in future work.

In short, they built a better digital twin for blood flow that understands that red blood cells are not just passive passengers; they are active swimmers that push themselves away from the walls to keep the highway clear.

Technical Summary

Technical Summary: A Modified Suspension Balance Model for Deformable Particle Suspensions

Problem Statement
Modeling the flow of red blood cells (RBCs) and other deformable particle suspensions in confined geometries presents significant challenges due to inherent nonlinearity, multi-body interactions, and non-Newtonian rheological behaviors. A critical phenomenon in microcirculation is the formation of a cell-free layer (CFL), a near-wall region depleted of RBCs. This layer is physiologically essential for maintaining low apparent viscosity and promoting platelet margination. While cellular-scale direct computational models can capture CFL formation, they are often computationally prohibitive for vascular network or organ-scale simulations. Conversely, existing continuum models, such as the conventional Suspension Balance Model (SBM), often fail to capture the CFL because they do not adequately account for the deformability-induced hydrodynamic lift forces that drive particles away from walls.

Methodology
The authors propose a Modified Suspension Balance Model (MSBM) that augments the conventional SBM framework with a specific lift forcing term to account for wall-RBC hydrodynamic interactions.

• Theoretical Framework: The model retains the standard mass and momentum conservation equations for the suspension bulk and particle phases. The core modification lies in the particle-phase momentum balance. The conventional SBM accounts for particle migration driven by the divergence of particle-phase stress ($\nabla \cdot \langle \Sigma \rangle_p$) and hindered settling drag. The MSBM introduces an additional volume-averaged lift force term, $\phi L_\perp$, derived from experimental observations of deformable vesicles and RBCs.
• Lift Force Formulation: The lift force is modeled as a function of the local shear rate ($\dot{\gamma}$), the distance from the wall ($h$), and a reduced volume parameter ($\nu$). The term takes the form $L_\perp \propto \frac{\dot{\gamma}}{(h+h_0)^\beta}$, where $\beta$ is a power-law coefficient adjusted to reflect particle dependence (specifically for RBCs), and $h_0$ is a geometric factor to avoid singularity at the wall.
• Numerical Implementation: The MSBM was implemented in OpenFOAM, extending the existing SbmFoam solver to create SbmBloodFoam. The solver utilizes a Finite-Volume-Method (FVM) approach. To validate the continuum model, the authors employed an experimentally validated multiscale blood flow solver based on the Lattice Boltzmann method (LB) and coarse-grained deformable capsule models for RBCs.

Key Results
The study demonstrates the MSBM's capability to simulate blood flows in microvascular channels and tubes, capturing transient and steady-state behaviors:

1. CFL Formation and Velocity Profiles: Unlike the conventional SBM, which predicts a parabolic velocity profile and non-zero concentration at the wall, the MSBM successfully captures the formation of a CFL and the resulting "plug-flow" like velocity profile characteristic of blood. The model shows that increasing the lift force parameter ($\beta$) increases CFL thickness.
2. Parametric Sensitivity:
   • Bulk Hematocrit ($\phi_b$): As $\phi_b$ increases, the CFL thickness decreases, consistent with literature.
   • Maximum Packing Concentration ($\phi_m$): Higher $\phi_m$ values (representing greater RBC aggregation or packing propensity) lead to thicker CFLs and higher central hematocrit peaks.
3. Temporal Evolution: The model captures the transient development of the CFL. Initially, the hematocrit is uniform; over time, RBCs migrate toward the center, forming a cell-depleted layer near the wall and a concentrated core, eventually reaching a steady state.
4. Validation against Theory and Experiments:
   • Simple Shear Flow: The model reproduces the linear scaling of the cubic CFL thickness ($\delta_{CFL}^3$) with dimensionless time ($t\dot{\gamma}$) at short time scales, matching the scaling laws observed in single-particle migration experiments.
   • Channel Flows: MSBM predictions for velocity and CFL thickness align well with experimental data from Losserand et al. and Salame across various channel heights and bulk hematocrits.
   • Tubular Flows: The model successfully captures the Fåhræus Effect (discharge hematocrit exceeding tube hematocrit) and the Fåhræus-Lindqvist Effect (decrease in apparent viscosity with decreasing vessel diameter). The predicted apparent viscosity and hematocrit ratios show good agreement with empirical correlations (e.g., Pries et al.) and experimental data, particularly for vessel diameters below 40 µm.

Significance and Claims
The paper establishes a novel continuum computational framework that efficiently captures the microstructural heterogeneity and non-Newtonian flow behavior of concentrated deformable particle suspensions under confinement. The authors claim that by incorporating a deformability-induced lift force, the MSBM bridges the gap between computationally expensive discrete particle simulations and less accurate continuum models.

The work is presented as a step toward efficient, at-scale biomedical and industrial applications. The authors modestly note that while the model performs well for smaller vessels, it slightly overpredicts the CFL and the Fåhræus-Lindqvist effect for larger tubes (>40 µm). They attribute this to the lack of a "particle-screening effect" in the current lift force formulation, which would account for reduced lift forces in particle-laden regions due to local particle-wall lubrication interactions. This limitation is identified as a subject for future work rather than a current failure of the model.