Red blood cell tests

Collision tests

Each of the following use a PHS (r⁷) RBF to construct the interpolant. A fictitious force is applied to each cell to drive them into/past one another. The cells are constructed using 1,600 data sites and 15,000 sample sites.

Vertical-vertical collision test

2 red blood cells, initially aligned vertically. The simulation terminated due to timestep restrictions at t ≈ 4.7 ms.

Horizontal-vertical collision test

2 red blood cells, one initially aligned horizontally and the other vertically. The simulation terminated due to timestep restrictions at t ≈ 1.0 ms. The video is very brief, so changing the playback speed to 0.25× might help.

Horizontal-horizontal collision test

2 red blood cells, initially aligned horizontally. The simulation terminated due to timestep restrictions at t ≈ 3.23 ms.

Sliding test

2 red blood cells, initially aligned obliquely. The simulation terminated due to timestep restrictions at t ≈ 2.5 ms.

Whole blood tests

Linear model + Skalak law

12 red blood cells in shear flow with shear rate ɣ̇ = 1000 s^-1. The red blood cells use a piecewise linear interpolant with 15,000 points and generate forces according to the Skalak law* with shear modulus E = 2.5×10^-3 dyn/cm and bulk modulus G = 2.5×10^-1 dyn/cm. I terminated the simulation manually at t = 40 ms.

* I was lazy and areas are approximately 3+𝒪(ΔÂ) larger than they should be. ΔÂ is the difference in reference area across edges.

IMQ RBF model + Skalak law + bending energy

12 red blood cells in shear flow with shear rate ɣ̇ = 1000 s^-1. The red blood cells use an IMQ RBF interpolant with 1,600 data sites and 15,000 sample sites and generate forces according to the Skalak law with shear modulus E = 2.5×10^-3 dyn/cm and bulk modulus G = 2.5×10^-1 dyn/cm and bending energy with bending modulus κ = 2×10^-12 erg. While not pictured, there is a bumpy endothelial layer below the cells, which is tethered in place. The simulation terminated due to timestep restrictions at t ≈ 10.3 ms.

PHS RBF model + Skalak law + bending energy

12 red blood cells in shear flow with shear rate ɣ̇ = 1000 s^-1. The red blood cells use a PHS (r⁷) RBF interpolant with 1,600 data sites and 15,000 sample sites and generate forces according to the Skalak law with shear modulus E = 2.5×10^-3 dyn/cm and bulk modulus G = 2.5×10^-1 dyn/cm and bending energy with bending modulus κ = 2×10^-12 erg. While not pictured, there is a bumpy endothelial layer below the cells, which is tethered in place. The simulation terminated due to timestep restrictions at t ≈ 9.5 ms.

Investigations in rough force smoothing

Let F be the force on the surface of the cell. Let 𝒮 be the spreading operator and its transpose, 𝒮^†, be the interpolation operator. The idea is to get the "rough" part of F by computing

F_rough = (I - 𝒮^†𝒮)F.

The video below plots the rough part of the force on the cell surface.

Wendland's compactly supported RBFs

Since r = ‖x-y‖, ∇r = r^-1 ∇(x-y). So for RBF φ,

∇φ = r^-1 φ'(r) ∇(x-y) ≔ 𝒟φ(r) ∇(x-y).

Wendland defines

ℐφ(r) = ∫¹_r tφ(t) dt,

so that 𝒟ℐφ = -φ. The Wendland RBFs are constructed by repeated application of ℐ:

φ_ℓk(r) = ℐ^k(1-r)₊^ℓ,

where (x)₊ = x if x > 0 and 0 otherwise. For an object embedded in ℝ^d, φ_ℓk is positive definite for ℓ ≥ ⌊d/2⌋+k+1. The parameter k controls the degree of smoothness of the RBF at r = 1. The smallest k I consider is therefore 2.

The images below depict the forces on 12 cells in a translated and rotated, but otherwise reference, configuration. The only forces present should be due to bending.

Skalak law forces of 12 RBCs reconstructed with 5th order spherical harmonics — (a) Skalak and (b) bending forces on 12 translated and rotated RBCs, otherwise in their reference configuration. The cells are reconstructed with 5^th order spherical harmonics, to represent the truth. Since the deformations are rigid body motions, the cells generate no tension.

Bending forces of 12 RBCs reconstructed with 5th order spherical harmonics — (a) Skalak and (b) bending forces on 12 translated and rotated RBCs, otherwise in their reference configuration. The cells are reconstructed with 5^th order spherical harmonics, to represent the truth. Since the deformations are rigid body motions, the cells generate no tension.

Skalak law forces of 12 RBCs reconstructed with Wendland-4,2 — (a) Skalak and (b) bending forces on 12 translated and rotated RBCs, otherwise in their reference configuration. The cells are reconstructed with the `φ`_4,2 Wendland RBF. Since the deformations are rigid body motions, the cells should generate no tension, but appear to do so.

Bending forces of 12 RBCs reconstructed with Wendland-4,2 — (a) Skalak and (b) bending forces on 12 translated and rotated RBCs, otherwise in their reference configuration. The cells are reconstructed with the `φ`_4,2 Wendland RBF. Since the deformations are rigid body motions, the cells should generate no tension, but appear to do so.

Skalak law forces of 12 RBCs reconstructed with Wendland-7,5 — (a) Skalak and (b) bending forces on 12 translated and rotated RBCs, otherwise in their reference configuration. The cells are reconstructed with the `φ`_7,5 Wendland RBF. Since the deformations are rigid body motions, the cells generate no tension.

Bending forces of 12 RBCs reconstructed with Wendland-7,5 — (a) Skalak and (b) bending forces on 12 translated and rotated RBCs, otherwise in their reference configuration. The cells are reconstructed with the `φ`_7,5 Wendland RBF. Since the deformations are rigid body motions, the cells generate no tension.

Skalak law forces of 12 RBCs reconstructed with the 7th order polyharmonic spline — (a) Skalak and (b) bending forces on 12 translated and rotated RBCs, otherwise in their reference configuration. The cells are reconstructed with the 7^th order polyharmonic spline. Since the deformations are rigid body motions, the cells should generate no tension.

Bending forces of 12 RBCs reconstructed with the 7th order polyharmonic spline — (a) Skalak and (b) bending forces on 12 translated and rotated RBCs, otherwise in their reference configuration. The cells are reconstructed with the 7^th order polyharmonic spline. Since the deformations are rigid body motions, the cells should generate no tension.

Reduced domain tests

2 RBCs in an 8μm × 8μm × 16μm domain.