# Bending Energy Schemes for Discrete‐Spring‐Network Structural Modelling of Red Blood Cells

## TL;DR

This paper compares three methods for modeling red blood cell deformation and finds one method is more accurate for predicting complex cell shapes.

## Contribution

The study introduces and validates a new bending energy scheme (BES C) for improved accuracy in red blood cell structural modeling.

## Key findings

- BES C accurately predicts complex RBC shapes like stomatocytes and echinocytes.
- BES A and BES B underestimate bending deformation in soft membranes.
- BES C captures necking behavior critical for accurate shape prediction.

## Abstract

Red blood cells (RBCs) undergo large structural deformation, including bending, when passing through capillaries. They also exhibit a range of complex shapes such as stomatocytes, discocytes and echinocytes that form due to altered blood pH and salt levels, ingested drugs and adenosine triphosphate depletion. Discrete‐spring‐network structural models of RBCs employ different numerical treatments of the continuum bending energy. This affects bending accuracy and the prediction of accurate RBC shapes. This research compares three representations called bending energy scheme (BES) A, B and C to evaluate their accuracy in shape predictions. BES A, seen throughout the literature, is based on the formulations of Kantor and Nelson, while BES B and BES C are, respectively, spring‐based and node‐based curvature calculation methods based on the formulations of Jülicher. Flat and enclosed spring‐network membrane test cases are presented, and predictions using the schemes are compared. The flat membrane test cases explored the bending of stiff and soft membranes while the enclosed membrane test cases evaluated equilibrium vesicle and RBC shape prediction, including predictions of the stomatocyte‐to‐discocyte‐to‐echinocyte sequence. Predictions showed that BES A and BES B have limitations and can underestimate the true bending deformation. Additionally, BES A and BES B are also unable to capture the necking behaviour critical to the accurate prediction of complex RBC shapes. BES C on the other hand was seen to be accurate and robust and predicted shapes closely matched expected biological shapes. Based on this research, BES C is recommended for all future spring‐network RBC structural modelling.

This research investigated three bending energy schemes for discrete‐spring‐network structural modelling termed bending energy scheme (BES) A, B and C. Flat and enclosed membrane test cases were presented, and predictions using the schemes were compared. All schemes accurately predicted the bending deformation of stiff membranes; however, for soft membranes, like vesicles and red blood cells, BES C proved to be the most accurate and robust, while the commonly employed scheme BES A was found to underestimate the true bending deformation.