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.
Citation: Eugenio Aulisa, Magdalena Toda, Stone Fields, Erhan Güler. Red blood cells as elastic surfaces: Cassini ovals, Helfrich shape equation, and biophysical regimes[J]. Electronic Research Archive, 2026, 34(1): 31-47. doi: 10.3934/era.2026002
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.
| [1] |
W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch. C, 28 (1973), 693–703. https://doi.org/10.1515/znc-1973-11-1209 doi: 10.1515/znc-1973-11-1209
|
| [2] |
H. J. Deuling, W. Helfrich, Red blood cell shapes as explained on the basis of curvature elasticity, Biophys. J., 16 (1976), 861–869. https://doi.org/10.1016/S0006-3495(76)85736-0 doi: 10.1016/S0006-3495(76)85736-0
|
| [3] |
J. G. Hu, Z. C. Ou-Yang, Shape equations of axisymmetric vesicles, Phys. Rev. E, 47 (1993), 461–467. https://doi.org/10.1103/PhysRevE.47.461 doi: 10.1103/PhysRevE.47.461
|
| [4] |
B. Angelov, I. Mladenov, On the geometry of red blood cell, Geom. Integrability Quantization, 1 (2000), 27–46. https://doi.org/10.7546/giq-1-2000-27-46 doi: 10.7546/giq-1-2000-27-46
|
| [5] |
Q. Liu, H. Zhou, J. Liu, Z. C. Ou-Yang, Spheres and prolate and oblate ellipsoids from an analytical solution of the spontaneous-curvature fluid-membrane model, Phys. Rev. E, 60 (1999) 3227–3233. https://doi.org/10.1103/PhysRevE.60.3227 doi: 10.1103/PhysRevE.60.3227
|
| [6] | J. Hellmers, N. Riefler, T. Wriedt, Y. Eremin, Light scattering simulation by concave, peanut-shaped silver nanoparticles modeled on Cassini-ovals, in Proceedings of the International Conference on Electromagnetic Light Scattering (ICHMT), (2007), 53–56. https://doi.org/10.1615/ICHMT.2007.ConfElectromagLigScat.160 |
| [7] | G. Rodríguez-Lázaro, Red Blood Cell Mechanics: From Membrane Elasticity to Blood Rheology, Ph.D. Thesis, Universitat de Barcelona, 2014. |
| [8] |
B. Palmer, A. Pampano, Stability of membranes, J. Geom. Anal., 34 (2024), 328. https://doi.org/10.1007/s12220-024-01767-7 doi: 10.1007/s12220-024-01767-7
|
| [9] | E. Aulisa, S. Fields, M. Toda, Red blood cells as Helfrich surfaces with spherical topology, preprint, arXiv: 2507.21010. |
| [10] | E. Evans, R. Skalak, Mechanics and Thermodynamics of Biomembranes, CRC Press, 1980. |
| [11] |
U. Seifert, Configurations of fluid membranes and vesicles, Adv. Phys., 46 (1997), 13–137. https://doi.org/10.1080/00018739700101488 doi: 10.1080/00018739700101488
|
| [12] |
N. Mohandas, E. Evans, Mechanical properties of the red cell membrane in relation to molecular structure and genetic defects, Annu. Rev. Biophys. Biomol. Struct., 23 (1994), 787–818. https://doi.org/10.1146/annurev.bb.23.060194.004035 doi: 10.1146/annurev.bb.23.060194.004035
|
| [13] | A. Gruber, Curvature Functionals and p-Willmore Energy, Ph.D. Thesis, Texas Tech University, 2019. |
| [14] |
B. Athukorallage, G. Bornia, T. Paragoda, M. Toda, Willmore-type energies and Willmore-type surfaces in space forms, JP J. Geom. Topol. 18 (2015), 93–108. http://doi.org/10.17654/JPGTNov2015_093_108 doi: 10.17654/JPGTNov2015_093_108
|
| [15] | J. C. C. Nitsche, Lectures on Minimal Surfaces, Vol. 1, Cambridge University Press, 1989. |
| [16] |
A. Gruber, E. Aulisa, Computational p-Willmore Flow with Conformal Penalty, ACM Trans. Graphics, 39 (2020), 1–16. https://doi.org/10.1145/3369387 doi: 10.1145/3369387
|
| [17] | N. Vaidya, H. Huang, S. Takagi, Correct equilibrium shape equation of axisymmetric vesicles, in Integral Methods in Science and Engineering: Techniques and Applications, Birkhäuser Boston, (2008), 267–276. https://doi.org/10.1007/978-0-8176-4671-4_30 |
| [18] |
F. Jülicher, U. Seifert, Shape equations for axisymmetric vesicles: A clarification, Phys. Rev. E, 49 (1994), 4728–4731. https://doi.org/10.1103/PhysRevE.49.4728 doi: 10.1103/PhysRevE.49.4728
|
| [19] | K. Gaurav, Shape equations of the axisymmetric vesicles, (2022), 1–20. Available from: https://home.iitk.ac.in/kugaurav/projects/Shape_equations_of_the_axisymmetric_vesicles.pdf. |
| [20] |
R. C. Gonzalez, M. Toda, I. Mladenov, On the Shape of Red Blood Cell, J. Geom. Symmetry Phys., 72 (2025), 1–38. https://doi.org/10.7546/jgsp-72-2025-1-38 doi: 10.7546/jgsp-72-2025-1-38
|
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Eugenio Aulisa, Magdalena Toda, Stone Fields, Erhan Güler. Red blood cells as elastic surfaces: Cassini ovals, Helfrich shape equation, and biophysical regimes[J]. Electronic Research Archive, 2026, 34(1): 31-47. doi: 10.3934/era.2026002
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