Bridging the computational gap between mesoscopic and continuum modeling of red blood cells for fully resolved blood flow

Abstract: We present a computational framework for the simulation of blood flow with fully resolved red blood cells (RBCs) using a modular approach that consists of a lattice Boltzmann solver for the blood plasma, a novel finite element based solver for the deformable bodies and an immersed boundary method for the fluid-solid interaction. For the RBCs, we propose a nodal projective FEM (npFEM) solver which has theoretical advantages over the more commonly used mass-spring systems (mesoscopic modeling), such as an unconditional stability, versatile material expressivity, and one set of parameters to fully describe the behavior of the body at any mesh resolution. At the same time, the method is substantially faster than other FEM solvers proposed in this field, and has an efficiency that is comparable to the one of mesoscopic models. At its core, the solver uses specially defined potential energies, and builds upon them a fast iterative procedure based on quasi-Newton techniques. For a known material, our solver has only one free parameter that demands tuning, related to the body viscoelasticity. In contrast, state-of-the-art solvers for deformable bodies have more free parameters, and the calibration of the models demands special assumptions regarding the mesh topology, which restrict their generality and mesh independence. We propose as well a modification to the potential energy proposed by Skalak et al. 1973 for the red blood cell membrane, which enhances the strain hardening behavior at higher deformations. Our viscoelastic model for the red blood cell, while simple enough and applicable to any kind of solver as a post-convergence step, can capture accurately the characteristic recovery time and tank-treading frequencies. The framework is validated using experimental data, and it proves to be scalable for multiple deformable bodies.

• The paper introduces the novel npFEM, offering unconditional stability and efficiency for simulating red blood cell deformations.
• It combines lattice Boltzmann, finite element, and immersed boundary methods to accurately capture complex RBC membrane dynamics under shear and large deformations.
• Validation against stretching, recovery, and tank-treading tests confirms its effectiveness for realistic, in vivo-scale blood flow simulations.

Bridging the Computational Gap Between Mesoscopic and Continuum Modeling of Red Blood Cells for Fully Resolved Blood Flow

The paper presents a comprehensive computational framework aimed at simulating blood flow with detailed resolution of red blood cells (RBCs), leveraging a novel approach that integrates various computational methodologies. This framework innovatively combines a lattice Boltzmann solver for modeling the blood plasma, a finite element (FE) solver specifically designed for deformable bodies, and an immersed boundary method for fluid-solid interactions.

Core Contributions

The principal innovation of this study is the introduction of a nodal projective finite element method (npFEM) for modeling RBCs. This method holds several advantages over traditional mass-spring systems, commonly used in mesoscopic modeling. Notably, npFEM offers:

1. Unconditional Stability: This feature allows for stable simulations under large deformations and various physical conditions, such as confined flows or high shear rates.
2. Versatile Material Expressivity: It provides a robust framework that can accurately simulate different material behaviors without being dependent on mesh topology.
3. Efficiency Comparable to Mesoscopic Models: Despite its advantages, npFEM demonstrates computational performance that matches mass-spring systems, making it feasible for large-scale simulations.

The paper advances the Skalak model for RBC membrane dynamics to enhance strain hardening at higher deformations, achieving better concordance with experimental data. This improvement is critical for practical applications, especially in modeling the complex viscoelastic behavior of RBCs during significant deformations typical in physiological conditions.

Numerical Validation and Performance

The framework's accuracy and efficiency were validated against several experimental benchmarks and computational scenarios:

• Stretching and Recovery Tests: Simulations of RBCs undergoing deformation and recovery closely align with experimental observations, confirming the framework's robustness in capturing essential mechanical behaviors.
• Tank-Treading and Wheeler Experiments: The npFEM approach accurately emulates RBC dynamics under shear flow, capturing both tumbling and tank-treading behaviors. This feature is particularly critical for understanding blood flow in microcirculatory conditions.
• Scalability: The framework's ability to handle multiple interacting RBCs demonstrates its potential to model realistic blood suspensions, a significant step towards in vivo applicability.

Implications and Future Perspectives

This work represents a significant advancement in the computational modeling of biological materials, particularly in the domain of hemodynamics. The npFEM not only bridges a vital gap between continuum and mesoscopic models but also sets a precedent for future studies exploring large-scale biological systems. Potential directions for future research include:

• Extension to Multi-Physiological Conditions: Incorporating additional biological factors such as blood-clotting and cellular interactions.
• GPU Implementation: To further improve computational efficiency, a GPU-accelerated version of the npFEM could be developed, potentially facilitating real-time simulations of complex systems.
• Integration with Medical Imaging: Enhanced model realism and accuracy might be achieved by integrating with patient-specific geometries derived from medical imaging modalities.

Thus, this framework not only holds promise for advancing computational hemodynamics but also opens avenues for its application in personalized medicine and the design of medical devices. Its capability to balance accuracy with computational efficiency makes it a practical tool for researchers and practitioners seeking to explore the intricate mechanics of blood flow and RBC behavior.