Red blood cells as elastic surfaces: Cassini ovals, Helfrich shape equation, and biophysical regimes

Red blood cells (RBCs) are classically modeled as thin elastic surfaces governed by the Helfrich–Canham energy. Within the widely used axisymmetric framework, we provide what appears to be the first complete mathematical proof that Cassini ovals, except for the limiting round sphere, do not satisfy the Helfrich shape equation, even though they can approximate biconcave profiles over experimentally relevant parameter ranges. The argument proceeds by direct substitution of the Cassini meridian into the third-order reduced shape equation and yields an over-determined algebraic system with no consistent solution for any nonzero Cassini eccentricity.

Beyond this structural result, we place the analysis in a broader biophysical and geometric context. We review typical bending and shear moduli, reduced-volume constraints, and the role of area and volume conservation; we explain the geometric meaning of the Helfrich parameters and their relation to the Willmore functional; we connect our formulation to bilayer-couple and area-difference-elasticity (ADE) models; and we outline practical routes to parameter inference from micropipette aspiration, flicker spectroscopy, and optical tweezers. We also discuss why the specific curvature structure of Cassini ovals is conceptually incompatible with Helfrich equilibria, indicate regimes in which Cassini profiles remain useful surrogates for geometric descriptors and for initializing PDE-based solvers, and summarize recent models based on constant bending-energy density. Finally, we identify extensions with spatially varying spontaneous curvature that may accommodate membrane heterogeneity and more complex RBC morphologies.