A cell-resolved, Lagrangian solver for modeling red blood cell dynamics in macroscale flows

Abstract

When red blood cells (RBCs) experience non-physiologically high stresses, e.g., in medical devices, they can rupture in a process called hemolysis. Directly simulating this process is computationally unaffordable given that the length scales of a medical device are several orders of magnitude larger than that of a RBC. To overcome this separation of scales, the present work introduces an affordable computational framework that accurately resolves the stress and deformation of a RBC in a spatially and temporally varying macroscale flow field such as those found in a typical medical device. The underlying idea of the present framework is to treat RBCs as one-way coupled tracers in the macroscale flow by capturing the effect of the flow on their dynamics but neglecting their effect on the flow at the macroscale. As a result, the RBC dynamics are simulated after those of the flow in a postprocessing step by receiving the fluid velocity gradient tensor measured along the RBC trajectory as the input. To resolve the fluid velocity in the immediate vicinity of the RBC as well as the motion of the membrane, we employ the boundary integral method coupled to a structural solver. The governing equations are discretized in space using spherical harmonics, yielding spectral integration accuracy. The predictions produced by this formulation are in good agreement with those obtained from simulations of spherical capsules in shear flows and optical tweezers experiments. The accuracy of the present method is evaluated using unbounded shear flow as a benchmark. Its computational cost grows proportional to p⁵, where p is the degree of the spherical harmonic. It also exhibits a fast convergence rate that is approximately $\mathcal{O}\left(p^6\right)$ for p ⪅ 20.

1. Introduction

The dynamics of isolated red blood cells (RBCs) have been the source of considerable study in recent years, with an increasing focus on using numerical tools to understand their behavior. One main driver for understanding the behavior of RBCs is hemolysis, the rupture of RBCs, which can occur when they are subjected to large stresses for extended periods of time [1]. This is particularly likely to occur in certain non-physiological conditions, such as in blood pumps or other ventricular assist devices [1]. Several methods have been introduced to predict the occurrence of hemolysis in an area of a flow. Some methods establish threshold values of shear stress and exposure time in the fluid, above which RBCs can undergo damage in a certain type of flow (e.g. [2]). Others use these fluid parameters in empirical equations to return the amount of hemoglobin released into the blood in certain areas of the flow (e.g. [3]). Lagrangian models also exist, such as the model by Arora et al. that uses the strain on ellipsoids to estimate RBC damage [4]. However, often these methods do not account for the actual behavior of individual RBCs and instead rely on flow properties such as shear stress and exposure time, or they use highly idealized measures of RBC strain. Only recently have new methods of tracking RBC strain been formulated in which the shape of the cell is resolved and tracked [5–7]. Such methods do provide a picture of RBC behavior in a flow, but the flow surrounding the RBCs is typically not resolved in conjunction with their deformation. This could be useful for the inclusion of, for example, cell-cell interactions or more complex ambient flow around RBCs. Accurate, well-resolved simulations of individual RBCs and their surrounding flow in the types of macroscale flow fields in which hemolysis often occurs may help predict when and where it will occur.

The use of computational tools has shed light on the behavior of RBCs in a variety of flow fields, with special attention given to simple shear flows [6,8–12] due to their simplicity and reproducibility. However, these types of idealized flows may not be representative of the types of flows RBCs experience in vivo. Unfortunately, such conditions are difficult to replicate numerically except in small vessels, as the separation of length scales between larger vessels and the RBCs, as well as their high volume fraction, makes fully-coupled, cell-resolved simulations prohibitively expensive.

As fully-resolved simulations of this behavior may still be somewhat distant, statistical descriptions of RBCs may be necessary to understand their behavior in such flows. The purpose of the present work is to introduce a solver that utilizes information from macroscale simulations, for example of large blood vessels, to cheaply and accurately obtain a statistical picture of the behavior of RBCs in that flow by resolving the flow in their immediate vicinity. Only a small statistical sample of the total number of cells is simulated, reducing the problem from one that is intractably large to one that can be solved on most modern computers.

The first step in this process is to simulate the macroscale fluid in a way that captures the effect of the individual RBCs, without simulating them directly. For example, although blood plasma is Newtonian, the particulate nature of whole blood results in a fluid that is shear-thinning and viscoelastic. With the effect of the RBCs on the fluid incorporated into the whole blood model, the effect of the fluid on the RBCs remains. Tracking the RBCs as Lagrangian particles then requires only one-way coupling, in which only the effect of the fluid on the RBCs must be solved. As such, the macroscale simulations need to be solved a single time, and the individual RBCs can be tracked along various pathlines as a post-processing step. As the RBCs are tracked along a pathline, their response to the flow is evaluated at each time step. The flow field surrounding a RBC is obtained from the macroscale simulation. This surrounding flow field is used in conjunction with the current configuration of the RBC to evaluate its motion. Additionally, by tracking the cell as it progresses through the flow, a history of the strain on the cell over time can be obtained. This is particularly relevant to hemolysis, as the residence time, in addition to the stress, is a key predictor of whether a cell will rupture [2,3,13].

Because of their size, the Reynolds numbers of the flows under consideration in this work are small, and, as such, the flow in the vicinity of RBCs is typically Stokesian. As such, the Stokes equations around the RBCs must be solved at each time step to evaluate the RBC’s response to the flow. This requires evaluating both the stress in the membrane and its resultant disturbance to the fluid. The RBC membrane is composed of a cytoskeleton primarily made up of a protein known as spectrin anchored to a lipid bilayer. Several methods exist to model the behavior of this membrane. Often, models of the membrane lump the characteristics of these individual components together, but multiscale simulations that capture the behavior of the individual components can be performed [14,15] at the expense of some computational cost. Particle-based discretizations are also common [16,17]. Alternatively the RBC membrane can be represented as a continuum that behaves according to some constitutive law. This approach is taken in the current work, as it can be relatively cheap and allows some flexibility in the choice of discretization.

For the solver outlined above, the most appropriate method of solving the fluid velocity is with the boundary integral method (BIM). Several other options are available, including grid-based fluid methods such as immersed boundary methods (e.g. [18,19]) or particle-based methods (e.g. [16,17]). However, BIMs provide several advantages over these other methods. First, the velocity is only explicitly solved on the surface of the RBC, as opposed to points in the surrounding fluid. Because these simulations are one-way coupled, the volumetric velocity field in the fluid domain is not needed. This reduces the overall cost. Additionally, in comparison to their size, the majority of cells in a large vessel will be far from the wall and it is assumed that their dynamics will not be influenced by the no-slip boundary. As such, the cells are in general treated as being in unbounded flow, a framework that presents no issues for BIMs. Due to this assumption, many features that are typically present in microcirculation, such as the development of an RBC-free layer near the walls [20], will be absent for the small percentage of cells that are closest to the wall. BIMs have been used for several applications, including drops [21,22], capsules [23,24], and RBCs [8,25].

As stated above, a primary goal of the introduced technique is computational affordability, such that the behavior of the cell can be tracked along pathlines many times longer than its radius. The discretization of the cell can be important in this regard. A natural choice for smooth objects of spherical topology is to use an expansion in terms of spherical harmonic basis functions. This type of spatial representation has been used previously, for drops [22], vesicles [26,27], and RBCs [25]. This representation is advantageous because the singular integrals that arise in the BIM can be evaluated with spectral accuracy with respect to p, the truncation of the spherical harmonic series [28,29]. The cost of the integration method outlined below scales as $\mathcal{O}\left(p^5\right)$ [26], but acceptable accuracy can be achieved at fairly low-order truncation where simulation remains feasible. While other methods have better asymptotic scaling [30], their cost is larger than the present technique at similar levels of accuracy when p ≈ 16 [29].

2. Methods

The motion of the RBC is solved in two major steps: First, the internal forces in the membrane are evaluated based on the membrane’s deformation. In general, there is a hydrodynamic traction jump over the membrane, and if the inertia of the membrane is assumed negligible, this traction must be in equilibrium with the membrane’s internal load [31,32,8]. Thus, the traction jump is related to the internal membrane load via

$$\Delta f=-p,$$ (1)

where Δf is the traction jump and p is the internal membrane load. As outlined below, this jump is a combination of the traction jumps resulting from in-plane tension Δf_t and out-of-plane bending Δf_b

$$\Delta f=\Delta f_t+\Delta f_b.$$ (2)

The next step is to solve for the fluid velocity resulting from the ambient flow field and the hydrodynamic traction on the surface of the cell with the boundary integral method. This is used to solve for the velocity on the surface of the membrane. With the velocity known, the RBC can be advanced in time. A schematic showing the discretized RBC as it traverses the flow field as well as the resulting velocity is displayed in Fig. 1. Several assumptions are made in solving for the motion of the RBC. First, the membrane is assumed to be a continuum. While finer methods like molecular dynamics or dissipative particle dynamics could be used, these methods are in general much more costly. Continuum methods cannot represent molecular-level behavior, but they can capture larger-scale events well [33], which are of interest to this work. Stokes flow is also assumed, a prerequisite to the boundary integral method used. At the scale of the cell, shear rates of $\mathcal{O}\left({10}^6\right){\text{\hspace{0.17em}s}}^{-1}$ would be required to reach $\text{Re}=\mathcal{O}(1)$, which is much higher than what is usually encountered in biological flows where, for example, in a highly stenosed artery, the maximum shear rate typically reaches $\mathcal{O}\left({10}^5\right)$ [34]. Cell-cell interactions, wall effects, and curvature of the surrounding velocity field are also not included in the below formulation. However, this formulation does not preclude the addition of these effects, which are potentially important to the behavior of the cell. Due to the large volume fraction of the cells in physiological flow, the cell-cell interactions are an important driver in the way cells behave, and could even influence the trajectory of the cell. These interactions could be included, while still keeping the solver cheap, by using an effective plasma viscosity that scales with the shear rate, which could qualitatively capture added stresses from the nearby cells. Using periodic boundary conditions with multiple cells is another avenue to capturing these effects. Although more expensive, this would be a more accurate solution. However, investigation of these types of effects is beyond the scope of this paper.

Fig. 1.

The problem setup of the fluid portion of the solver. The RBC traverses the flow, and the boundary integral method in conjunction with the hydrodynamic traction Δf and ambient velocity field u^∞(x, t) is used to calculate the motion of the RBC.

2.1. Membrane mechanics

The RBC membrane is primarily composed of two components, a lipid bilayer and an underlying spectrin cytoskeleton. Each is responsible for different behavior in the membrane [35,8,14,36,11]. The bilayer is primarily responsible for the membrane’s resistance to bending and area dilatation, and the cytoskeleton is primarily responsible for the membrane’s resistance to shear. Although these two components account for different behaviors of the membrane, they are lumped into a single, infinitely thin continuum that is used to represent the behaviors of both. The following sections detail how the shape of the membrane is parameterized and how the internal load is calculated in the membrane given a reference and current configuration.

2.1.1. Membrane shape definition

The surface of the membrane S is parameterized using spherical harmonic basis functions. The cell is assumed to be smooth and of spherical topology. The parameterization maps any square-integrable function f on the surface of the unit sphere ${\mathbb{S}}^2$, defined by the inclination angle θ ∈ [0, π] and the azimuthal angle ϕ ∈ [0, 2π), as

$$f(\theta ,\varphi )=\sum _{n=0}^{\infty }\sum _{m=-n}^n{\widehat{f}}_n^m{Y}_n^m(\theta ,\varphi ),$$ (3)

where $Y_n^m(\theta ,\varphi )$ are spherical harmonic functions defined as

$$Y_n^m(\theta ,\varphi )=\sqrt{\frac{2n+1}{4\pi }\frac{(n-m)!}{(n+m)!}}{P}_n^m(\text{cos\hspace{0.17em}}\theta )e^{im\varphi }$$ (4)

and ${\widehat{f}}_n^m$ are scalar coefficients defined by the inner product over the surface of a sphere,

$${\widehat{f}}_n^m=\left(f,Y_n^m\right)=\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}f(\theta ,\varphi )\overline{Y_n^m(\theta ,\varphi )}\text{\hspace{0.17em}sin\hspace{0.17em}}\theta \text{d}\theta \text{d}\varphi ,$$ (5)

where the overbar denotes the complex conjugate. $P_n^m$ are the associated Legendre polynomials. Eq. (5) is a result of taking the inner product of Eq. (3) with an arbitrary spherical harmonic function and applying the orthogonality of spherical harmonic functions, where

$$\left(Y_n^m,Y_{n^{\prime }}^{m^{\prime }}\right)=\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}{Y}_n^m(\theta ,\varphi )\overline{Y_{n^{\prime }}^{m^{\prime }}(\theta ,\varphi )}\text{\hspace{0.17em}sin\hspace{0.17em}}\theta \text{d}\theta \text{d}\varphi ={\delta }_{n{n}^{\prime }}{\delta }_{m{m}^{\prime }}.$$ (6)

δ in Eq. (6) is the Kronecker delta. The discrete counterpart to Eq. (5) involves truncating the series at some degree p, calculating the function f on a grid, and integrating numerically. Standard practice for this discretization [22,27,25,37,26] involves discretizing θ into p + 1 points according to the rule θ_i = cos⁻¹ z_i, where z_i are the Gauss-Legendre points of order p + 1. θ is discretized along cos⁻¹ z_i (as opposed to fitting a Gaussian grid along θ and not cosθ) because this discretization avoids clustering at the poles, which can cause a significant reduction in accuracy. ϕ is discretized into 2(p + 1) uniform points according to ϕ_j = π j/(p + 1). This grid is commonly termed a “p-grid”. The p-grid facilitates the following rule for numerical integration, which combines Gauss-Legendre quadrature in θ with the trapezoidal rule in ϕ

$$\underset{S}{\int }f(x)\text{d}{S}_x\approx \frac{\pi }{p+1}\sum _{i=0}^p\sum _{j=0}^{2p+1}{w}_i^{G}f\left({\theta }_i,{\varphi }_j\right)W\left({\theta }_i,{\varphi }_j\right)\frac{1}{\text{sin\hspace{0.17em}}{\theta }_i},$$ (7)

where $w_i^G$ are the Gauss-Legendre quadrature weights and W (θ_i, ϕ_j) is the infinitesimal area element. The term S_x implies the integral is evaluated with respect to x, a parameterization of the cell surface, which is itself a function of θ and ϕ. For a smooth function f on a C^∞ surface of spherical topology, this integration rule is superalgebraically convergent with p [26]. The inner product Eq. (5) can then be replaced with the discrete version

$${\widehat{f}}_n^m\approx {\left(f,Y_n^m\right)}_p=\frac{\pi }{p+1}\sum _{i=0}^p\sum _{j=0}^{2p+1}{w}_i^{G}f\left({\theta }_i,{\varphi }_j\right)\overline{Y_n^m\left({\theta }_i,{\varphi }_j\right)}.$$ (8)

For spherical harmonics of degree n, n′ ≤ p and |m| ≤ n, |m′| ≤ n′, discrete orthogonality of spherical harmonic functions is also ensured [22]

$${\left(Y_n^m,Y_{n^{\prime }}^{m^{\prime }}\right)}_p={\delta }_{n{n}^{\prime }}{\delta }_{m{m}^{\prime }}.$$ (9)

To represent the surface of the membrane, each Cartesian component of the position vector defining the surface is individually expanded as in Eq. (3) to yield

$$x(\theta ,\varphi )=\sum _{n=0}^p\sum _{m=-n}^n{\widehat{x}}_n^m{Y}_n^m(\theta ,\varphi ).$$ (10)

Note that θ and ϕ are values in parameter space, and do not correspond to angles in physical space.

Some differential geometry is required in the following sections, and the necessary notation is described here. Local covariant basis vectors are defined on the surface of the membrane by

$$\begin{gathered} a_1=\frac{\partial }{\partial \theta }x, \\ {a}_2=\frac{\partial }{\partial \varphi }x, \\ n=\frac{a_1\times a_2}{\left|a_1\times a_2\right|}. \end{gathered}$$ (11)

The curvature tensor is defined as

$$b_{\alpha \beta }=\frac{\partial a_{\alpha }}{\partial {\theta }_{\beta }}\cdot n,$$ (12)

where the Greek indices range over {1, 2}, and θ₁ = θ and θ₂ = ϕ.

The below formulation also requires surface derivatives up to fourth-order. These derivatives are applied directly to the spherical harmonic functions. For an arbitrary derivative, this results in the equation

$$\frac{{\partial }^k{\partial }^l}{\partial {\theta }^k\partial {\varphi }^l}x(\theta ,\varphi )=\sum _{n=0}^p\sum _{m=-n}^n{\widehat{x}}_n^m\frac{{\partial }^k{\partial }^l}{\partial {\theta }^k\partial {\varphi }^l}{Y}_n^m(\theta ,\varphi ).$$ (13)

Since up to fourth-order derivatives on the surface are required, ultimately fourth-order derivatives of spherical harmonic functions are required. This is accomplished through the following recursive relationship, built on the fact that a first order θ-derivative of a spherical harmonic function can be represented in terms of a spherical harmonic function of the same degree but different order

$$\frac{\partial }{\partial \theta }{Y}_n^m(\theta ,\varphi )=\frac{1}{2}\left[a_n^m{Y}_n^{m+1}(\theta ,\varphi )-b_n^m{Y}_n^{m-1}(\theta ,\varphi )\right],$$ (14)

where $a_n^m=\sqrt{(n-m)(n+m+1)}$ and $b_n^m=\sqrt{(n+m)(n-m+1)}$. Applying this to arbitrary degree yields

$$\frac{{\partial }^k}{\partial {\theta }^k}{Y}_n^m(\theta ,\varphi )=\frac{1}{2}\left[a_n^m\frac{{\partial }^{k-1}}{\partial {\theta }^{k-1}}{Y}_n^{m+1}(\theta ,\varphi )-b_n^m\frac{{\partial }^{k-1}}{\partial {\theta }^{k-1}}{Y}_n^{m-1}(\theta ,\varphi )\right],$$ (15)

where $Y_n^m=0$ for |m| > n. Derivatives in ϕ take the simpler form

$$\frac{{\partial }^l}{\partial {\varphi }^l}{Y}_n^m(\theta ,\varphi )={\left(im\right)}^l{Y}_n^m(\theta ,\varphi ).$$ (16)

2.1.2. Tension

An important step in modeling the mechanics of a RBC is the choice of constitutive model. One commonly used model to represent the tension is the Skalak model [38]. The Skalak model was developed specifically for RBCs and has been shown to model their behavior well (e.g. Refs. [39,12]). It is used in the present work. The Skalak model introduces the following surface strain energy function:

$$W=\frac{G}{4}\left[\left(I_1^2+2I_2^2-2I_2\right)+C{I}_2^2\right].$$ (17)

G is the shear elastic modulus, C is a non-dimensional parameter representing the membrane’s resistance to area dilatation, and I₁ and I₂ are two strain invariants, defined as

$$\begin{gathered} I_1={\lambda }_1^2+{\lambda }_2^2-2, \\ {I}_2={\lambda }_1^2{\lambda }_2^2-1. \end{gathered}$$ (18)

λ_1,2 are the principal strains. The principal strains can be calculated easily from the surface deformation gradient tensor [32]

$$F=a_{\alpha }{A}^{\alpha },$$ (19)

where repeated indices imply summation. A^α is the contravariant form of the basis vectors in Eq. (11) calculated in the reference configuration, and a_α is the covariant form in the current configuration. As such, Eq. (19) yields the identity tensor when a_α also corresponds to the reference configuration. The surface left Cauchy-Green deformation tensor is then defined as

$$V^2=F{F}^{\prime }.$$ (20)

${\lambda }_1^2$ and ${\lambda }_2^2$ are the eigenvalues of V². The principal surface tensions are calculated as

$$\begin{gathered} {\tau }_1^p=\frac{1}{{\lambda }_2}\frac{\partial W}{\partial {\lambda }_1}, \\ {\tau }_2^p=\frac{1}{{\lambda }_1}\frac{\partial W}{\partial {\lambda }_2}. \end{gathered}$$ (21)

After some manipulation, the surface tension tensor can be shown to be [32,25]

$$\mathit{\tau }=G\left[\frac{1}{{\lambda }_1{\lambda }_2}\left(I_1+1\right)V^2+{\lambda }_1{\lambda }_2\left(C{I}_2-1\right)(I-nn)\right].$$ (22)

I is the identity tensor. By performing a local force balance [40], the hydrodynamic traction jump arising from the tensions in the membrane is

$$\begin{gathered} \Delta f_t^{\beta }=-{\tau }_{\mid \alpha }^{\alpha \beta }, \\ \Delta f_t^3=-{\tau }^{\alpha \beta }{b}_{\alpha \beta }. \end{gathered}$$ (23)

The notation •|_α denotes the covariant derivative with respect to θ_α.

These tensions are evaluated with respect to some stress-free reference shape. However, the stress-free reference state of the membrane with respect to shear is a matter of some uncertainty. Although RBCs naturally assume a biconcave disk shape, experimental and numerical results imply the RBC is pre-stressed at this shape, and the shape is reached through an equilibrium between bending and tensile stresses. For example, Svoboda et al. [35] showed that the shape of the cytoskeleton, which is responsible for the cells resistance to shear [35], does not retain the biconcave disk shape after removal of the lipid bilayer, instead reaching a nearly spherical shape. Additionally, experimental results show that RBCs can maintain the biconcave disk shape during tank treading at low shear rates [41,42], which is typically only achieved in numerical experiments when a stress-free state other than the biconcave disk is chosen [9,11]. The actual stress-free state of the cytoskeleton is still somewhat uncertain. Several authors suggest, however, that it is somewhere between a sphere and the biconcave disk shape [10,9,11,43,44]. In light of this, an oblate spheroid is used as the stress-free state.

The biconcave shape is achieved by starting with an oblate spheroid with a reduced volume V_obl/V_sph = 0.98, where V_obl is the volume of the oblate spheroid, and V_sph is the volume of a sphere with the same surface area as the oblate spheroid. Similar to Cordasco et al. [9,45] and Peng et al. [14,11], a deflation process is used to achieve the biconcave disk shape. A small, inward normal velocity is applied to the oblate spheroid until it reaches the reduced volume of a RBC, V_RBC/V_sph = 0.644, at which point the cell is allowed to reach a stable resting shape, after which the RBC no longer deforms. The RBC returns to this shape in the absence of external forces. The biconcave shape results from the inward buckling of the oblate spheroid as the volume is reduced [44]. As shown by Lim et al. [43], the biconcave shape corresponds to one possible state in which the total energy in the membrane, which is the sum of the elastic and bending energies, is minimized. The target cross section is the commonly used parameterization from Evans and Fung [46] and adapted by Pozrikidis [8], where, for the xz-plane

$$\begin{gathered} z=a\frac{\alpha }{2}\left(0.207+2.003{\text{\hspace{0.17em}sin}}^2\chi -1.123{\text{\hspace{0.17em}sin}}^4\chi \right)\text{cos\hspace{0.17em}}\chi , \\ x=a\alpha \text{\hspace{0.17em}sin\hspace{0.17em}}\chi . \end{gathered}$$ (24)

This membrane is axisymmetric about the z-axis. χ is a parameter that varies from −π/2 to π/2 and α = 1.38581894 is the ratio between the maximum radius of the cell and a, the equivalent radius. a is defined as the radius of a sphere with the same volume as the RBC.

The resting shape after deflation depends on multiple parameters, namely the spontaneous curvature, the bending modulus, and the original reduced volume [9,11,43]. The spontaneous curvature c₀ is the curvature that minimizes the bending energy in the membrane, and the bending modulus E_B represents the membrane’s resistance to bending. Note that the shape that minimizes the bending energy would be a sphere of curvature c₀. Simulations across a variety of parameters were run in order to find combinations that resulted in a suitable resting shape, while attempting to remain in the physiological range for the bending modulus. The final resting shape compared to the shape given by Eq. (24) is displayed in Fig. 2. Although there are some slight discrepancies, excellent agreement has been reached overall.

Fig. 2.

A cross section of the shape of the RBC upon reaching a steady state after the deflation process plotted against the target cross section of Evans and Fung [46]. Excellent agreement has been reached overall.

A common issue with spectral methods is aliasing. Aliasing arises when high frequency basis functions (i.e. large values of n for spherical harmonics) cannot be suitably resolved on a given grid, causing those basis functions to artificially alter those of lower frequencies. This phenomenon is especially pronounced when calculating the tensions, which require many nonlinear operations that move energy to high frequency scales [25]. A common solution to aliasing is to perform the nonlinear operations on a finer grid, perform the spherical harmonic transform of the resulting quantities, and discard the higher wave number modes. This strategy is employed here for calculation of the traction jump. The traction jump, and by extension all quantities needed to calculate it, are calculated on a finer q-grid, analogous to the p-grid mentioned previously, where q > p. The spherical harmonic transform of these quantities is then performed, and the modes that are greater than p are discarded. These values are then interpolated back to a p-grid as needed using the inverse spherical harmonic transform for use in the flow solver below. Zhao et al. [25] showed that an upsampling rate of 2 (i.e. q = 2p) typically leads to stable simulations. This was also observed in the present work, so this value is used for the results below unless otherwise specified.

2.1.3. Bending

While bending stresses in the cell are typically small, they can contribute significantly to the dynamics of the cell [47,12]. The commonly used Helfrich bending model [48] was used to describe the RBC’s resistance to bending. In this model, the bending energy in the membrane is described by the equation

$$W_B=\frac{E_B}{2}\underset{S}{\int }{\left(2\kappa -c_0\right)}^2\text{d}S.$$ (25)

E_B is the bending modulus, κ is the mean curvature, and c₀ is the spontaneous curvature. Guckenberger et al. provide a comprehensive review on this equation and its use in Ref. [49]. By taking the first variation of Eq. (25),

$$\delta W_B=\underset{S}{\int }\Delta f_b\cdot \delta x\text{d}S,$$ (26)

the following equation for the hydrodynamic traction arising from bending of the membrane can be obtained [50]

$$\Delta f_b=E_B\left[2{\Delta }_s\kappa +\left(2\kappa +c_0\right)\left(2{\kappa }^2-{\kappa }_g-c_0\kappa \right)\right]n.$$ (27)

Δ_s is the Laplace-Beltrami operator, κ_g is the Gaussian curvature, and n is the normal vector. While other bending models are occasionally used to represent the RBCs resistance to bending [25,8,37], they are typically equivalent to the Helfrich model to first order [49].

2.2. Flow solver

The formulation of the boundary integral method has been well-covered in several publications, and will be only briefly covered here. This method is described in greater detail in Ref. [31], and the derivation of the relevant equations from this text is summarized here. Since typical Reynolds numbers associated with flow around RBCs are much less than one, the assumption of Stokes flow is utilized. The starting point is the Stokes equations as a function of location x and including a point force located at point x₀ of strength g

$$\begin{gathered} -\nabla P+\mu {\nabla }^{2}u+g\delta \left(x-x_0\right)=0, \\ \nabla \cdot u=0. \end{gathered}$$ (28)

P is the pressure, μ is the viscosity of the fluid, and u is the velocity. First, the velocity and stress Green’s functions for Eq. (28) are found. By using the reciprocal theorem and the divergence theorem to integrate these Green’s functions over the surface of a particle, the following equation for the flow in the exterior of a particle is obtained

$$u_j\left(x_0\right)-\frac{1}{8\pi }\underset{S}{\int }{T}_{ijk}\left(x,x_0\right)u_i(x)n_k(x)\text{d}{S}_x=u_j^{\infty }\left(x_0\right)-\frac{1}{8\pi \mu }\underset{S}{\int }{G}_{ij}\left(x,x_0\right)f_i^{+}(x)\text{d}{S}_x.$$ (29)

$u_j^{\infty }$ is the ambient flow velocity and $f_i^{+}$ is the surface traction of the fluid on the external side of the particle. n_k are the components of the outward normal vector. $u_j^{\infty }$ is obtained from the macroscale simulations. This represents the undisturbed flow field in the immediate vicinity of the RBC. Often, the higher order terms of this velocity field will be negligible at the scale of the RBC, such that the velocity field around a point x_c is given by u^∞(x) = u^∞(x_c) + (x − x_c) · ∇u^∞(x_c). Only the velocity gradient is retained, as it is assumed the rigid body velocity of the cell is equal to that of the surrounding fluid. The selection of x_c relative to the RBC is arbitrary as changing it only superposes a solid-body motion on the computed velocity of the RBC, leaving the end result unchanged. Such a formulation is used in the present work as the velocity gradient is almost universally available in the types of macroscale simulations under consideration, but it should be noted that the present formulation is not limited to such linear flow fields. G_ij is known as the Stokeslet, and in unbounded flow is

$$G_{ij}=\frac{{\delta }_{ij}}{r}+\frac{r_i{r}_j}{r^3}.$$ (30)

Here, r = x − x₀ and r = |r|. T_ijk is known as the stresslet, and in unbounded flow is

$$T_{ijk}=-6\frac{r_i{r}_j{r}_k}{r^5}.$$ (31)

The integrals in Eq. (29) are single and double layer integrals, and for simplicity are referred to as

$$\begin{gathered} S_{ij}[\psi ]\left(x_0\right)=\underset{S}{\int }{G}_{ij}\left(x,x_0\right)\psi (x)\text{d}{S}_x, \\ {D}_{ij}[\psi ]\left(x_0\right)=\underset{S}{\int }{T}_{ijk}\left(x,x_0\right)\psi (x)n_k(x)\text{d}{S}_x \end{gathered}$$ (32)

A similar equation to Eq. (29) can be obtained for the fluid in the interior of a particle. By applying the reciprocal theorem to Eq. (29) for a flow in the interior of the particle with viscosity λμ, where λ denotes the ratio of the viscosity of the fluid inside the cell to the viscosity of the exterior plasma, combining the result with Eq. (29), and letting the point x₀ approach the surface of the particle, the following equation representing the velocity on the surface of the particle is obtained [31]

$$u_j\left(x_0\right)-\frac{1-\lambda }{4\pi (1+\lambda )}{D}_{ij}\left[u_i\right]\left(x_0\right)=\frac{2}{1+\lambda }{u}_j^{\infty }\left(x_0\right)-\frac{1}{4\pi \mu (1+\lambda )}{S}_{ij}\left[\Delta f_i\right]\left(x_0\right).$$ (33)

Similar to the surface coordinates, the velocity is transformed into spectral space via Eq. (8) and truncated at some degree p. Performing this transformation, substituting into Eq. (33) and simplifying yields

$$\sum _{n=0}^p\sum _{m=-n}^n{\widehat{u}}_i^{nm}\left[4\pi {\delta }_{ij}{Y}_n^m\left(x_0\right)-\frac{1-\lambda }{1+\lambda }{D}_{ij}\left[Y_n^m\right]\left(x_0\right)\right]=\frac{8\pi }{1+\lambda }{u}_j^{\infty }\left(x_0\right)-\frac{1}{\mu (1+\lambda )}{S}_{ij}\left[\Delta f_i\right]\left(x_0\right).$$ (34)

To simplify Eq. (34), the following variables are introduced

$$\begin{gathered} A_{ij}^{nm}\left(x_0\right)=4\pi {\delta }_{ij}{Y}_n^m\left(x_0\right)-\frac{1-\lambda }{1+\lambda }{D}_{ij}\left[Y_n^m\right]\left(x_0\right), \\ {b}_j\left(x_0\right)=\frac{8\pi }{1+\lambda }{u}_j^{\infty }\left(x_0\right)-\frac{1}{\mu (1+\lambda )}{S}_{ij}\left[\Delta f_i\right]\left(x_0\right), \end{gathered}$$ (35)

which, substituted back into Eq. (34), yields

$$\sum _{n=0}^p\sum _{m=-n}^n{\widehat{u}}_i^{nm}{A}_{ij}^{nm}\left(x_0\right)=b_j\left(x_0\right).$$ (36)

Both $A_{ij}^{nm}$ and b_j are composed of known quantities and can be calculated at any point on the surface. The efficient construction of $A_{ij}^{nm}$ is described in Appendix C. The main difficulty lies in calculation of the single and double layer integrals, whose calculation is detailed in the following section. In order to discretize Eq. (36) and solve for ${\widehat{u}}_i^{nm}$, the Galerkin method is used with spherical harmonic test functions defined over the same range as the spherical harmonic basis functions

$$\sum _{n=0}^p\sum _{m=-n}^n{\widehat{u}}_i^{nm}{\left(A_{ij}^{nm}\left(x_0\right),Y_{n^{\prime }}^{m^{\prime }}\left(x_0\right)\right)}_p={\left(b_j\left(x_0\right),Y_{n^{\prime }}^{m^{\prime }}\left(x_0\right)\right)}_p.$$ (37)

The result of this discretization is a left hand side matrix of size 3(p + 1)² × 3(p + 1)² and a right hand side vector of length 3(p + 1)². Note that due to the real-valued nature of u_i, ${\widehat{u}}_i^{nm}={(-1)}^m{\overline{\widehat{u}}}_i^{n(-m)}$, which significantly reduces the number of unknowns. This system can be solved for ${\widehat{u}}_i^{nm}$ with any complex linear solver for dense matrices. Other discretization methods such as the collocation method are also appropriate for solving for these coefficients. As has been observed in other works [25–27], the highest degree terms (of degree p) are calculated with large error and are dropped.

The equation

$$\frac{\partial x}{\partial t}=u$$ (38)

is used to advance the surface of the membrane. This equation is solved with a first-order explicit time-stepping method, but higher-order and implicit methods are also applicable. Based on the property of orthogonality Eq. (9) the individual spectral coefficients of the position vector are advanced only with their respective velocity coefficients by

$${\left({\widehat{x}}_n^m\right)}_{t+1}={\left({\widehat{x}}_n^m\right)}_t+\Delta t{\left({\widehat{u}}_n^m\right)}_t.$$ (39)

Δt is the time step and t is the current time step.

2.2.1. Calculation of single and double layer integrals

The bulk of the time in calculating the spectral velocity coefficients in Eq. (37) is spent calculating the single and double layer integrals at the required integration points. Due to the 1/r terms in the kernels, singularities appear in these integrals when x approaches x₀ on the surface of the membrane. These singularities degrade the accuracy of integration rules such as Eq. (7) and require special attention.

One method for accurately computing these integrals involves rotating the reference frame of the parameter space such that the singularity lies at a pole. This approach was first used by Graham and Sloan for solving Helmholtz problems [28,29] and was later adapted to Stokes flow problems by Veerapaneni et al. [26]. The rule is as follows: for a point x₀ defined in parameter space at x(θ = 0, ϕ = 0)

$$\begin{gathered} S_{ij}[\psi ]\left(x_0\right)\approx \frac{\pi }{p+1}\sum _{k=0}^p\sum _{l=0}^{2p+1}{G}_{ij}\left(x\left({\theta }_k,{\varphi }_l\right),x_0\right)\psi \left({\theta }_k,{\varphi }_l\right)W\left({\theta }_k,{\varphi }_l\right)w_k^S\left({\theta }_k\right), \\ {w}_k^S\left({\theta }_k\right)=\frac{w_k^G\sum _{n=0}^p{P}_n\left(\text{cos\hspace{0.17em}}{\theta }_k\right)}{\text{cos}\left({\theta }_k/2\right)}. \end{gathered}$$ (40)

Assuming a smooth function ψ defined on a smooth surface, this rule is superalgebraically convergent to the single layer integral with p. The same p-grid is used as that in Eq. (7). However, the spherical harmonic representation of the surface must be rotated such that the pole coincides with the point x₀, a procedure described in Appendix B. Eq. (40) is also derived in Appendix A, but for a more rigorous derivation of this rule see Refs. [28,29,26].

To show the spectral accuracy of the above rule, the integral $I={\int }_S{G}_{ij}\left(x,x_0\right)\text{d}{S}_x$ is calculated on the surface of a RBC at x₀(θ = 1, ϕ = 1) for various values of p. The relative error from the p = 28 case ϵ_p = |I₂₈ − I_p|/I₂₈ is displayed in Fig. 3 using both the singular integration rule Eq. (40) and the regular integration rule Eq. (7). The regular integration rule converges slowly, whereas the singular integration rule converges superalgebraically.

Fig. 3.

The relative error in the calculation of the singular integral ${\int }_S{G}_{ij}\left(x,x_0\right)\text{d}{S}_x$ at various values of p for the singular and regular integration rules. The results show much faster convergence for the singular integration rule than the regular integration rule.

2.2.2. Volume conservation

Although this method is based on divergence-free equations, it is not explicitly volume-conservative [51,25] and it is possible there could be some volume drift over time. As such, a method similar to that used by Freud [51] for two-dimensional cells and Zhao et al. for three-dimensional cells [25] has been used here to enforce volume conservation. Each time step, the surface of the membrane x is adjusted to the volume-corrected surface x_VC

$$\begin{gathered} x_{VC}=x+z, \\ z=-\frac{V-V_0}{SA}n, \end{gathered}$$ (41)

where n is the outward normal vector, V is the volume of the cell before correction, V₀ is the original volume of the cell, and SA is the surface area of the cell before correction. This correction is accomplished by transforming the normal vector into spectral space, and employing the orthogonality of the spherical harmonic functions to add the resulting coefficients directly to the spectral surface coefficients. This correction is generally small; typically z is several orders of magnitude smaller than the characteristic radius of the cell. Note, however, that while the volume change in all simulations below is much less than 1%, there are no restrictions similar to conservation of mass on the surface area. The total surface area is expected to, and does, vary more significantly in actual simulations than the volume, although it is still limited according to the cell’s large resistance to change in surface area. This change in surface area is typically limited to approximately 5–10% in the most extreme cases.

2.3. Lagrangian particle tracking and one-way coupling

Because the cells are small, they are treated as passive tracers. The relaxation time for RBCs in plasma is $\mathcal{O}\left({10}^{-7}\right)\text{\hspace{0.17em}s}$. This is typically much smaller than characteristic flow times of the large scale flows, and thus the cell usually relaxes to the velocity of the flow almost immediately. As such, the RBCs assume the velocity of the fluid at their location at a given time step, and this velocity is used to advect the RBC simultaneously with the calculation of its deformation using the equation

$$\frac{\partial x_C}{\partial t}=u\left(x_C\right),$$ (42)

where x_c is the position of the cell.

2.4. Solver summary

The solver steps are summarized below.

1. The necessary values of the spherical harmonics are calculated on the p-grid and q-grid using Eq. (4). The values of the derivatives of the spherical harmonics are calculated on the q-grid using Eqs. (15) and (16).

2. The necessary geometric quantities for the reference state are stored.

3. The time step loop is started.

4. The surrounding undisturbed flow field u^∞ around the RBC center x_c is obtained via u^∞(x) = (x − x_c) · ∇u^∞(x_c).

5. Given the coefficients for the shape of the membrane, all geometric quantities needed for the traction jump are calculated on the q-grid.

6. The traction jump is calculated using Eqs. (23) and (27) and transformed into spectral space. To dealias, only modes up to degree p are retained.

7. $A_{ij}^{nm}\left(x_0\right)$ and b_j(x₀) in Eq. (35) are calculated on the p-grid.

   1. The single and double layer integrals are calculated for each point on the p-grid using Eq. (40). Each point requires its own rotated p-grid to calculate these integrals.

8. The inner product ${\left(A_{ij}^{nm}\left(x_0\right),Y_n^m\left(x_0\right)\right)}_p$ is performed to get a square matrix and ${\left(b_j\left(x_0\right),Y_n^m\left(x_0\right)\right)}_p$ to get a vector of equal size.

9. The spectral velocity coefficients ${\widehat{u}}_i^{nm}$ in Eq. (37) are solved for using a complex matrix solver, dropping the highest degree terms.

10. The membrane is advanced via Eq. (39) and the volume correction Eq. (41) is imposed. The cell is also advected along its pathline simultaneously.

11. The time step loop is restarted.

3. Results & discussion

3.1. Parameters

The above formulation depends on eight physical parameters, which can be collapsed into five non-dimensional parameters. The eight physical parameters are described in Table 1. The characteristic radius a is the radius of a sphere with the same volume as the RBC. The spontaneous curvature c₀ is often used as a free parameter to achieve a resting shape that agrees well with the typical resting RBC shape [55,9,14,11], which is done in this work as well to achieve the shape shown in Fig. 2. It is taken to be c₀ = −2/a. Note that most of these parameters are not used in the simulation directly. Instead, the non-dimensional properties listed in Table 2 are used.

Table 1.

Physical parameters and their approximate ranges for RBC flow.

Parameter	Symbol	Approximate range for RBC flow
Shear elastic modulus	G	2.4 − 2.75 × 10−6 N/m [39,52]
Bending modulus	E B	$\mathcal{O}\left({10}^{-19}\right)\text{\hspace{0.17em}J}$ [53,54,45]
Plasma viscosity	μ	1.2 cp
Cytoplasm viscosity	μ int	6 cp
Characteristic radius	a	2.82 μm
Spontaneous curvature	c 0	Problem dependent
Dilatation ratio	C	105 [47]
Characteristic flow time/velocity scale	${\dot{\gamma }}^{-1}$	Problem dependent

Table 2.

Non-dimensional parameters used in the solver.

Parameter	Definition	Values used
Ca	$\mu \dot{\gamma }a/G$	0–2.5
$E_B^{*}$	EB/a2 G	0.03
$c_0^{*}$	ac 0	−2
λ	μint/μ	5
C	-	100

The Capillary number Ca is a problem dependent parameter. It can alternatively be represented as the ratio of a characteristic membrane time scale to a characteristic fluid time scale, and represents the ratio of tensile forces in the membrane to viscous forces in the fluid. For the given value of $E_B^{*}$, values of G ≈ 2.5 × 10⁻⁶ N/m and E_B ≈ 6 × 10⁻¹⁹ J are acceptable, both of which fall within the ranges listed in Table 1. Although the dilatation ratio C in RBCs is approximately 10⁵ [47], this increases the stiffness of the problem and makes it infeasible to solve. It has been shown that the dynamics of the problem are unaffected for approximately C > 10 [56,12], which is partially confirmed for the current framework in Fig. 12. Even so, the value of C is taken to be C = 100.

Fig. 12.

The RMS differences between the simulation in Fig. 8 run at various values of C and a case run at C = 200. The results show fast convergence with respect to C.

3.2. Validation

3.2.1. Spherical capsule in shear flow

To validate the solver, simulation results were compared with a common benchmark: a spherical capsule suspended in a shear flow. For many cases, the capsule will elongate into a steady, nearly ellipsoidal shape with dimensions that are dependent on the properties of the capsule and the flow. The Taylor deformation parameter D_ij = (r_i − r_j)/(r_i + r_j) is displayed against Ca for two values of the viscosity ratio λ in Fig. 4. r_i are the axis lengths of an ellipsoid with the same inertia tensor as the capsule [57]. These approximate dimensions are displayed on a deformed capsule in Fig. 5. Bending resistance is neglected. The results show excellent agreement with previous simulations by Lac et al. [57].

Fig. 4.

The Taylor deformation parameter D₁₂ plotted against the Capillary number for two different values of the viscosity ratio λ. Results from the current simulation are plotted against results from Lac et al. [57], showing excellent agreement.

Fig. 5.

The spherical capsule after deformation showing the approximate lengths of r₁ and r₂.

3.2.2. Optical tweezers experiments

In order to test the effectiveness of the model for predicting RBC behavior, several simulations were run to replicate optical tweezers experiments, in which a RBC is stretched between two silica beads by a constant force [58]. A diagram of the simulation setup is shown in Fig. 6. To replicate this experiment, an additional traction b was added to the hydrodynamic traction Δf before the membrane motion was calculated from the fluid solver. At the beginning of the simulation, two diametrically opposed regions were selected on the sides of the cell, representing the contact area with the beads. This contact area is shown in Fig. 6a. The experiment run by Mills and other simulations [58,16,12,59] typically uses a contact diameter of d_c = 2 μm. Due to the relatively coarse grid facilitated by the spherical harmonic representation, a contact diameter of d_c = 2.178 μm was used, as this is the closest value to d_c = 2 μm possible. The additional traction b was added to any nodes on the fine q-grid that fall within this region and set such that a constant force is applied throughout the simulation.

Fig. 6.

The un-deformed RBC, displaying the nodes where the additional traction b is applied (a), and the shape and dimensions of the RBC after the application of a force F on either end (b).

Results are shown for the simulations, as well as the experiment by Mills [58], in Fig. 7. These results show the axial length D_a and transverse length D_t, shown in Fig. 6, as a function of the applied force. There is good agreement between the simulation results and the experimental results. It should be noted that the anti-aliasing procedure makes it somewhat difficult to apply forces at distinct locations, which could account for some error in D_a. The forces are applied at the q-grid, as this is where the hydrodynamic traction is calculated. However, the sum of the applied force and the calculated hydrodynamic traction is then interpolated via a spherical harmonic transform to the coarser p-grid for use, and the sharp rise in force can cause a Gibbs-like phenomenon on the p-grid. This manifests in some minor oscillations in the traction over the surface of the cell and a smooth transition between the contact region and the rest of the cell surface. As shown by Sigüenza et al. [59], D_a is sensitive to the size of the contact area, and this phenomenon may contribute to uncertainty in the size of this area. Ultimately, the experimental and computational results still match well, including D_t, which Sigüenza showed was not sensitive to the size of the contact area [59].

Fig. 7.

The dimensions D of the RBC in Fig. 6 plotted against the force F. The results from the current simulation are plotted against experimental results from Mills et al. [58], showing good agreement between the two.

3.3. Computational performance

The following sections display convergence and accuracy results for a number of different parameters. All results are based on the same flow: a RBC in shear flow. Simulations are run for Ca = 1, p = 16 and the above properties in Table 2 unless otherwise stated. Note that converting these parameters to those corresponding to physiological blood flow using Tables 1 and 2 would result in a shear rate of $\dot{\gamma }\approx 750{\text{s}}^{-1}$. Snapshots of the RBC at several points in time are displayed in Fig. 8. A material point initially in the center of the dimple is also tracked in this figure. The RBC exhibits a behavior somewhat between tumbling and tank-treading at these parameters. While the behavior is mainly characterized by tumbling, the material point does move somewhat tangentially along the surface of the membrane. In physical units, this RBC would complete one tumble in approximately 0.03 s. This type of behavior is consistent with previous work, in which the RBC would not be expected to exhibit pure tank-treading at this viscosity ratio [42,60,61,20]. However, keeping all other parameters the same and setting the viscosity ratio to λ = 1.5 causes the RBC to tank-tread with a non-dimensional frequency of $f=4\pi /\dot{\gamma }T=0.445$, where T is the time for a material point to complete one full revolution around the membrane. This value matches well with previous results [60]. In the following sections, the location of the point initially in the dimple is tracked as the RBC completes approximately one tumble, and the corresponding angle with respect to the z-axis is used to compare results across the different parameters. This angle is normalized by the rotation of a sphere in the same shear flow.

Fig. 8.

Snapshots of a RBC in a shear flow for p = 16 at Ca = 1 and λ = 5 at non-dimensional time $t\dot{\gamma }=(\text{a})0$, (b) 5, (c) 10, (d) 15, (e) 20, and (f) 25. The green dot follows a material point on the surface initially in the dimple. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

3.3.1. Mesh independence

Mesh independence is established by running the simulations in the above section while varying the value of p. All simulations were run with a time step of $\Delta t=2.5\times {10}^{-3}\dot{\gamma }$. The results are displayed in Fig. 9. The RMS differences between simulations run at various values of p and a p = 28 case are plotted in Fig. 10. Qualitatively, the results in Fig. 9 are similar at all values of p. However, it appears that there is between $\mathcal{O}\left(p^5\right)$ and $\mathcal{O}\left(p^6\right)$ convergence at the values of p plotted. The fact that there is algebraic convergence with respect to p can likely be explained by the fact that some surface quantities that are used in these calculations, such as the curvature [27], also converge algebraically with respect to p.

Fig. 9.

The angle of the material point in Fig. 8 at multiple values of p, normalized by the rotation of a sphere in the same shear flow. The simulation reaches mesh independence quickly with respect to p.

Fig. 10.

The RMS differences between the cases plotted in Fig. 9 and a p = 28 case. At low values of p, the convergence is between $\mathcal{O}\left(p^5\right)$ and $\mathcal{O}\left(p^6\right)$.

3.3.2. Time step convergence

Results for the accuracy with respect to the time step are displayed in Fig. 11. All simulations were run at a value of p = 16. These results display the RMS difference between several cases run at different time steps and a case run with $\Delta t=5\times {10}^{-4}\dot{\gamma }$. Since an explicit Euler time-stepping scheme is used in this solver, it is expected that there will be first-order accuracy with respect to time, and this is a result borne out by Fig. 11, where the RMS difference increases approximately linearly with respect to the time step.

Fig. 11.

The RMS differences between the simulation in Fig. 8 run at various time steps and a case run at $\Delta t=5\times {10}^{-4}\dot{\gamma }$. The results show the expected approximate first-order accuracy.

3.3.3. Dilatation ratio convergence

Convergence results with respect to the dilatation ratio are displayed in Fig. 12. These results display the RMS difference between several cases run at different values of C and a case run with C = 200.

The qualitative dynamics of the RBC are mostly unchanged as C is increased, but there are some significant quantitative differences, especially at lower values of C. However, there is fast convergence to a consistent solution at larger values of C. At lower values of C, the convergence is approximately $\mathcal{O}\left(C^{1.5}\right)$. Although previous works suggest convergence at values of C ≈ 10 [56,12], the results of Fig. 12 suggest that a greater value of C may be necessary.

3.3.4. Computational cost

The spectral accuracy of the integration method means that acceptable accuracy can be obtained at a fairly coarse discretization of the cell surface. As a result, inverting and solving for the velocity coefficients on the surface is relatively cheap, in spite of the fact the resulting matrix is dense. Instead, the main cost of the solver comes from the singular integration. The singular integration requires rotating the reference frame such that the singular point is at the north pole. Due to properties of spherical harmonics, this rotation can be achieved in $\mathcal{O}(p)$ operations for a single point [26]. There are $\mathcal{O}\left(p^2\right)$ points, and evaluating the integrals at one point requires rotating every point, which requires another $\mathcal{O}\left(p^2\right)$ operations per point. Integrals must be evaluated at every point, so in total the integration cost scales as $\mathcal{O}\left(p^5\right)$. Methods exist with better asymptotic scaling, but the break-even point for p is higher than what is used in this work (p ≈ 36 for a $\mathcal{O}\left(p^4\text{\hspace{0.17em}log\hspace{0.17em}}p\right)$ scheme and p > 100 for a $\mathcal{O}\left(p^4\right)$ scheme) [62]. The total cost to run a simulation with 10 time steps is displayed in Fig. 13, with the cost of several of the components displayed as well. The overall $\mathcal{O}\left(p^5\right)$ scaling can be seen in this figure. The “rotation/assembly” portion of this figure includes the actual rotation of all constants and the assembly procedure outlined in Appendix C, as the two are linked. As expected, this step is responsible for the bulk of the computational cost. Additionally, the actual rotation algorithm is comparable to other works. Gimbutas and Veerapaneni [62] found that the total cost of rotation in a time step, a total of six rotations at each discretization point, was 0.0312 seconds at p = 12. For the present work, the cost to perform six rotations at p = 12 for all points was 0.0892 seconds, which is comparable to the work of Gimbutas and Veerapaneni, although direct comparisons are difficult to make on account of potential differences in hardware and software.

Fig. 13.

The total cost to run 10 time steps of the simulation as a function of p, as well as the cost of the individual components. This plot shows the expected $\mathcal{O}\left(p^5\right)$ overall scaling of the method, with the cost of rotation being the most significant.

3.3.5. BT shunt simulations

To showcase the solver in a realistic scenario, it was used to track RBC deformation in Blalock-Taussig (BT) shunt simulations for infants. The geometry used for these simulations is displayed in Fig. 14. The blood vessel fluid simulations were performed by coupling the boundaries of the given geometry to a lumped-parameter network, the details of which are described in previous works [63,64]. Twenty RBCs were randomly seeded at the ascending aorta and the velocity gradient was extracted at their locations as they traveled through the domain. A computationally efficient algorithm was employed for extracting fluid quantities at the location of cells that optimally identifies the hosting cell in an unstructured grid [65]. These velocity gradients were used with the solver to compute the deformation of the RBCs. An example pathline of an RBC, with the deformation of the RBC shown at various points, is displayed in Fig. 15.

Fig. 14.

The mesh used in the BT shunt simulations.

Fig. 15.

The pathline of one RBC as it traverses through the shunt. The pathline is colored according to the cell’s velocity. The cell is shown at several points along the pathline and magnified in size for visualization purposes.

Note that the maximum stable time step was smaller for the RBC-resolved simulations than the blood vessel fluid simulations, so a quadratic interpolation was used when necessary to compute the velocity gradient between blood vessel time steps. The RBCs were run at a value of p = 12. Although many of the finer features on the surface of the cell are not resolved at this value of p, the large scale dynamics of the cell are. The results from these simulations are shown in Fig. 16. These figures show the maximum values of λ₁/λ₂, a representation of shear strain, and λ₁λ₂, a representation of areal strain, on the surface of the individual RBCs, as well as the average of these values across all RBCs. One may utilize these figures to estimate the average and maximum strain in an RBC as it travels through a domain. As expected, the shear strain is generally much larger than the areal strain, as RBCs in general are much more resistant to areal than shear deformation.

Fig. 16.

Individual (gray dashed lines) and mean (red solid line) results for 20 RBCs as they traverse the BT shunt. The plots show the maximum value on the surface of the RBC of (a) λ₁λ₂, representing areal strain, and (b) λ₁/λ₂, representing shear strain.

4. Conclusions

A computational framework has been presented that tracks RBCs as Lagrangian particles along pathlines and captures their deformation and stress by resolving the flow in the immediate vicinity of the cell at each time step. The cells were assumed to be capsules of fluid surrounded by a thin membrane with resistance to shear, dilatation, and bending. The flow around the cell was solved using the boundary integral method, in which a spherical harmonic representation of the cells was used to obtain spectral accuracy of the integrals. The computational framework uses the current configuration of the membrane to calculate the internal load, which is in turn used to solve for the velocity on the membrane surface. A first-order, explicit time stepping scheme was used to advance the membrane, but higher order schemes could be implemented as well.

The model for the RBC was validated both for droplets in shear flow and against optical tweezers experiments of RBCs, showing good agreement with both. An unbounded shear flow was used as a benchmark to evaluate the convergence and accuracy of the framework. The method showed approximately $\mathcal{O}\left(p^6\right)$ convergence at values of p ⪅ 20. The method also showed approximate first-order accuracy with regard to the time step size, a result of the explicit Euler scheme used in this work. In addition, the convergence with respect to the dilatation ratio, C was investigated. It was found that the method did converge with increasing C, but the convergence was somewhat slower than expected. This slow convergence could impact some results of interest, especially areal strain, as the cell’s resistance to areal strain is governed by C. One way to potentially address this would be to use a higher value of C that is closer to its physical value. The increased stiffness of the problem at higher C would then need to be addressed, potentially with unconditionally stable time-stepping methods.

The computational cost of the method was also analyzed. The majority of the computational time is taken up by the rotation of the spherical harmonics, which are required to achieve spectral accuracy of the integrals. Although the cost of this method scales as $\mathcal{O}\left(p^5\right)$, at low values of p, acceptable speed and accuracy can be obtained.

This study makes no attempts to address cell-cell interactions. Because the volume fraction of RBCs can reach approximately 50% in vivo, accurate simulations of RBCs in these contexts should address cell-cell interactions directly or indirectly. A possible indirect approach would be to model the effect of cell-cell interactions by adjusting the effective viscosity of the fluid. A more direct approach would be to simulate multiple cells at the desired volume fraction in a periodic box. The current formulation requires little modification to add more cells. In addition, this type of problem lends itself well to parallelization. Additionally, there are well-established methods of adding periodicity to boundary integral methods.

Appendix A. Derivation of integration rule Eq. (40)

The single and double layer integrals of Eq. (32) contain singularities that can greatly reduce the accuracy of the regular integration rule Eq. (7). The solution is to rotate the coordinate system such that the singularity is balanced by the infinitesimal area term W, which vanishes as it approaches the pole at the same rate that the singularity goes to infinity. The result is a smooth integrand. However, even after rotation, a singularity can still be present as a result of the Gaussian discretization along cosθ [26]. The following derivation details the solution to this problem and follows closely to Refs. [28,29,26]. First, consider the single layer integral

$$\underset{S}{\int }{G}_{ij}\left(x,x_0\right)f_j(x)d{S}_x.$$ (43)

x is a function of the spherical coordinates θ and ϕ, and x₀ is defined at some constant value (θ₀, ϕ₀). For the below derivation, the point (θ₀, ϕ₀) must correspond to the pole, (0, 0). This can be accomplished by a change of variables, θ′ = θ − θ₀ and ϕ′ = ϕ − ϕ₀. To accomplish this, another function x′ is required such that

$$\begin{gathered} x^{\prime }\left({\theta }^{\prime },{\varphi }^{\prime }\right)=x(\theta ,\varphi ), \\ {x}^{\prime }(0,0)=x\left({\theta }_0,{\varphi }_0\right)=x_0. \end{gathered}$$ (44)

Thus,

$$f_j(x(\theta ,\varphi ))=f_j\left(x^{\prime }\left(\theta -{\theta }_0,\varphi -{\varphi }_0\right)\right)=f_j\left(x^{\prime }\left({\theta }^{\prime },{\varphi }^{\prime }\right)\right).$$ (45)

This transformation is inexpensive because of the spherical harmonic representation of x. The spherical harmonic coefficients ${\widehat{x}}_n^m$ can be rotated such that the above holds with new constants ${\widehat{x}}_n^{\tilde{m}}$. This process is described in Appendix B. Writing out the integral in terms of θ′ and ϕ′ yields

$$\underset{S}{\int }{G}_{ij}\left(x^{\prime },x_0\right)f_j\left(x^{\prime }\right)\text{d}{S}_{{\chi }^{\prime }}=\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}{G}_{ij}\left(x^{\prime }\left({\theta }^{\prime },{\varphi }^{\prime }\right),x_0\right)f_j\left(x^{\prime }\left({\theta }^{\prime },{\varphi }^{\prime }\right)\right)W\left(x^{\prime }\left({\theta }^{\prime },{\varphi }^{\prime }\right)\right)\text{d}{\theta }^{\prime }\text{d}{\varphi }^{\prime }.$$ (46)

The change of variables to θ′ and ϕ′ causes no significant changes in the integral because it is a rigid body rotation, and thus the Jacobian of the transformation is equal to 1. Below, the implicit dependence of x′ on θ′ and ϕ′ is dropped. Next, the integrand is multiplied by sinθ′ / sinθ′, while also noting that the definition of an integral over the surface of a sphere ∂B in terms of θ and ϕ (or in the present case, θ′ and ϕ′ is

$$\underset{\partial B}{\int }f(x)\text{d}{S}_x=\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}f(x(\theta ,\varphi ))\text{\hspace{0.17em}sin\hspace{0.17em}}\theta \text{d}\theta \text{d}\varphi ,$$ (47)

resulting in

$$\begin{gathered} \underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}\left[\frac{G_{ij}\left({\theta }^{\prime },{\varphi }^{\prime },x_0\right)f_j\left({\theta }^{\prime },{\varphi }^{\prime }\right)W\left({\theta }^{\prime },{\varphi }^{\prime }\right)}{\text{sin\hspace{0.17em}}{\theta }^{\prime }}\right]\text{sin\hspace{0.17em}}{\theta }^{\prime }\text{d}{\theta }^{\prime }\text{d}{\varphi }^{\prime } \\ =\underset{\partial B}{\int }\frac{G_{ij}\left(x^{\prime },x_0\right)f_j\left(x^{\prime }\right)W\left(x^{\prime }\right)}{\text{sin\hspace{0.17em}}{\theta }^{\prime }\left(x^{\prime }\right)}\text{d}{S}_{x^{\prime }}. \end{gathered}$$ (48)

Now the Cartesian unit vectors in the parameter space are introduced, such that a given pair (θ, ϕ) corresponds to the unit vector $\widehat{z}=(\text{cos\hspace{0.17em}}\varphi \text{\hspace{0.17em}sin\hspace{0.17em}}\theta ,\text{sin\hspace{0.17em}}\varphi \text{\hspace{0.17em}sin\hspace{0.17em}}\theta ,\text{cos\hspace{0.17em}}\theta )$ and the pair (0, 0) corresponds to the unit vector $\widehat{n}=(0,0,1)$. For readability, the integrand in Eq. (48) is replaced with

$$\frac{G_{ij}\left(x^{\prime }\left(\widehat{z}\left({\theta }^{\prime },{\varphi }^{\prime }\right)\right),x_0\right)f_j\left(x^{\prime }\left(\widehat{z}\left({\theta }^{\prime },{\varphi }^{\prime }\right)\right)\right)W\left(x^{\prime }\left(\widehat{z}\left({\theta }^{\prime },{\varphi }^{\prime }\right)\right)\right)}{\text{sin\hspace{0.17em}}{\theta }^{\prime }}\left|\widehat{n}-\widehat{z}\left({\theta }^{\prime },{\varphi }^{\prime }\right)\right|=C_i\left(\widehat{z},x_0\right).$$ (49)

Note the $|\widehat{n}-\widehat{z}|$ factor in the above equation. The integral now has the form

$$\underset{\partial B}{\int }\frac{1}{\left|\widehat{n}-\widehat{z}\right|}{C}_i\left(\widehat{z},x_0\right)\text{d}{S}_{\widehat{z}}.$$ (50)

Now the discrete inner product Eq. (8) is taken with the spherical harmonic functions, and the following is noted

$$\begin{gathered} {\widehat{C}}_i^{nm}\left(x_0\right)\approx {\left(C_i\left(\widehat{z},x_0\right),Y_n^m(\widehat{z})\right)}_p, \\ {C}_i\left(\widehat{z},x_0\right)\approx \sum _{n=0}^p\sum _{m=-n}^n{\widehat{C}}_i^{nm}\left(x_0\right)Y_n^m(\widehat{z}), \\ {C}_i\left(\widehat{z},x_0\right)\approx \sum _{n=0}^p\sum _{m=-n}^n{\left(C_i\left(\widehat{z},x_0\right),Y_n^m(\widehat{z})\right)}_p{Y}_n^m(\widehat{z}). \end{gathered}$$ (51)

Note that the inner product in this equation is superalgebraically convergent in p. Eq. (51) can then be utilized to write

$$\begin{gathered} \underset{\partial B}{\int }\frac{1}{|\widehat{n}-\widehat{z}|}{C}_i\left(\widehat{z},x_0\right)\text{d}{S}_{\widehat{z}}\approx \underset{\partial B}{\int }\frac{1}{|\widehat{n}-\widehat{z}|}\sum _{n=0}^p\sum _{m=-n}^n{\left(C_i\left(\widehat{z},x_0\right),Y_n^m(\widehat{z})\right)}_p{Y}_n^m(\widehat{z})\text{d}{S}_{\widehat{z}} \\ \approx \sum _{n=0}^p\sum _{m=-n}^n{\widehat{C}}_i^{nm}\left(x_0\right)\underset{\partial B}{\int }\frac{1}{|\widehat{n}-\widehat{z}|}{Y}_n^m(\widehat{z})\text{d}{S}_{\widehat{z}}. \end{gathered}$$ (52)

Spherical harmonic functions are eigenfunctions of the final integral, such that

$$\underset{\partial B}{\int }\frac{1}{\left|\widehat{n}-\widehat{z}\right|}{Y}_n^m(\widehat{z})\text{d}{S}_{\widehat{z}}=\frac{4\pi }{2n+1}{Y}_n^m(\widehat{n}).$$ (53)

Thus, the final integral in Eq. (52) can be written as

$$\sum _{n=0}^p\sum _{m=-n}^n{\widehat{C}}_i^{nm}\left(x_0\right)\underset{\partial B}{\int }\frac{1}{\left|\widehat{n}-\widehat{z}\right|}{Y}_n^m(\widehat{z})\text{d}{S}_{\widehat{z}}=\sum _{n=0}^p\sum _{m=-n}^n{\widehat{C}}_i^{nm}\left(x_0\right)\frac{4\pi }{2n+1}{Y}_n^m(\widehat{n}).$$ (54)

$\widehat{n}$ is at the pole, where the spherical harmonic functions have simple, known values. Specifically

$$Y_n^m(\widehat{n})=\{\begin{array}{ll}\sqrt{\frac{2n+1}{4\pi }}\hfill & m=0,\hfill \\ 0\hfill & m\ne 0.\hfill \end{array}$$ (55)

This eliminates the second sum, leaving

$$\sum _{n=0}^p\sum _{m=-n}^n{\widehat{C}}_i^{nm}\left(x_0\right)\frac{4\pi }{2n+1}{Y}_n^m(\widehat{n})=\sum _{n=0}^p{\widehat{C}}_i^{n0}\left(x_0\right)\sqrt{\frac{4\pi }{2n+1}}.$$ (56)

Additionally, $Y_n^0(\widehat{z})$ can be expressed as

$$Y_n^0(\widehat{z})=\sqrt{\frac{2n+1}{4\pi }}{P}_n(\widehat{n}\cdot \widehat{z}),$$ (57)

where P_n are Legendre polynomials. The calculation of ${\widehat{C}}_i^{n0}\left(x_0\right)$ can then be expressed as

$$\begin{gathered} {\widehat{C}}_i^{n0}\left(x_0\right)\approx {\left(C_i\left(\widehat{z},x_0\right),Y_n^0(\widehat{z})\right)}_p \\ =\sqrt{\frac{2n+1}{4\pi }}{\left(C_i\left(\widehat{z},x_0\right),P_n(\widehat{n}\cdot \widehat{z})\right)}_p. \end{gathered}$$ (58)

Substituting these results into Eq. (50) and rearranging yields

$$\underset{\partial B}{\int }\frac{1}{\left|\widehat{n}-\widehat{z}\right|}{C}_i\left(\widehat{z},x_0\right)\text{d}{S}_{\widehat{z}}\approx {\left(C_i\left(\widehat{z},x_0\right),\sum _{n=0}^p{P}_n(\widehat{n}\cdot \widehat{z})\right)}_p.$$ (59)

From the definition of the discrete inner product Eq. (8):

$${\left(C_i\left(\widehat{z},x_0\right),\sum _{n=0}^p{P}_n(\widehat{n}\cdot \widehat{z})\right)}_p=\frac{\pi }{p+1}\sum _{k=0}^p\sum _{l=0}^{2p+1}{w}_i^G{C}_i\left(\widehat{z}\left({\theta }_k^{\prime },{\varphi }_l^{\prime }\right),x_0\right)\sum _{n=0}^p{P}_n\left(\text{cos\hspace{0.17em}}{\theta }_k^{\prime }\right).$$ (60)

Next, note the following properties:

$$\begin{gathered} \left|\widehat{n}-\widehat{z}\left({\theta }^{\prime },{\varphi }^{\prime }\right)\right|=2\text{\hspace{0.17em}sin}\left({\theta }^{\prime }/2\right), \\ \frac{1}{\text{cos}\left({\theta }^{\prime }/2\right)}=\frac{2\text{\hspace{0.17em}sin}\left({\theta }^{\prime }/2\right)}{\text{sin\hspace{0.17em}}{\theta }^{\prime }}. \end{gathered}$$ (61)

Substituting the above results into the single layer integral equation, Eq. (43), yields the following integration rule, Eq. (40), valid when the singularity is located at the north pole

$$\begin{gathered} \underset{S}{\int }{G}_{ij}\left(x,x_0\right)f_j(x)\text{d}{S}_x\approx \frac{\pi }{p+1}\sum _{k=0}^p\sum _{l=0}^{2p+1}{G}_{ij}\left(x^{\prime }\left({\theta }_k^{\prime },{\varphi }_l^{\prime }\right),x_0\right)f_j\left(x^{\prime }\left({\theta }_k^{\prime },{\varphi }_l^{\prime }\right)\right)W\left(x^{\prime }\left({\theta }_k^{\prime },{\varphi }_l^{\prime }\right)\right)w_k^S, \\ {w}_k^S=\frac{w_k^G{\sum }_{n=0}^p{P}_n\left(\text{cos\hspace{0.17em}}{\theta }_k^{\prime }\right)}{\text{cos}\left({\theta }_k^{\prime }/2\right)}. \end{gathered}$$ (62)

Note that θ′ and ϕ′ are on the normal p-grid, but the spherical harmonic functions representing the surface have been rotated. $w_k^S$ is calculated at the beginning of the simulation and stored. Also, note that the only approximation that was made was transforming $C_i(\widehat{z},\widehat{n})$ into spectral space, a superalgebraically convergent operation.

Appendix B. Rotation of spherical harmonics

The process to rotate functions written as a spherical harmonic series is described here. Specifically, when the double or single layer integrals are evaluated at a point x(θ₀, ϕ₀), the coordinate system must be rotated such that the evaluation point in the new coordinate system is situated at the pole $\left({\theta }_0^{\prime }=0,{\varphi }_0^{\prime }=0\right)$. The coordinate system is thus rotated three consecutive times: first by −ϕ₀, then by −θ₀ and finally by ϕ₀. A function f(θ, ϕ) represented by a spherical harmonic series with constants ${\widehat{f}}_n^m$ can be rotated to this coordinate system via the following rule [66]

$${\widehat{f}}_n^{\tilde{m}}=\sum _{\tilde{m}=-n}^n{d}_n^{m\tilde{m}}\left(-{\theta }_0\right){\widehat{f}}_n^m{e}^{i(m-\tilde{m}){\varphi }_0},$$ (63)

such that, in the new coordinate system (θ′, ϕ′)

$$\sum _{n=0}^p\sum _{\tilde{m}=-n}^n{\widehat{f}}_n^{\tilde{m}}{Y}_n^{\tilde{m}}\left({\theta }^{\prime },{\varphi }^{\prime }\right)=\sum _{n=0}^p\sum _{m=-n}^n{\widehat{f}}_n^m{Y}_n^m(\theta ,\varphi ).$$ (64)

One possible form of $d_n^{m\tilde{m}}(\alpha )$ is [22,66]

$$d_n^{m\tilde{m}}(\alpha )={(-1)}^{\tilde{m}-m}{[(n+\tilde{m})!(n-\tilde{m})!(n+m)!(n-m)!]}^{1/2}{S}_n^{m\tilde{m}}(\alpha ),$$ (65)

where

$$S_n^{m\tilde{m}}(\alpha )=\sum _{s=\text{max}(0,m-\tilde{m})}^{\text{min}(n+m,n-\tilde{m})}{(-1)}^s\frac{{\left(\text{cos}\frac{\alpha }{2}\right)}^{2(n-s)+m-\tilde{m}}{\left(\text{sin}\frac{\alpha }{2}\right)}^{2s-m+\tilde{m}}}{(n+m-s)!s!(\tilde{m}-m+s)!(n-\tilde{m}-s)!}$$ (66)

The values of $d_n^{m\tilde{m}}(-\theta )$ can be pre-computed and stored at the beginning of the simulation at each value of θ on the p-grid. Note that Eq. (63) can be accelerated via the discrete FFT [62].

Appendix C. $\mathcal{O}\left(p^5\right)$ evaluation of singular integrals

The $\mathcal{O}\left(p^5\right)$ evaluation of the singular integrals could require, naively, $\mathcal{O}\left(p^8\right)$ operations [29]. This can be reduced via properties of spherical harmonics and the observation that several components in the sums required for the integrals can be separated into independent components. The following process was outlined by Ganesh and Graham [29] and Graham and Sloan [28], and is adapted for the current work below.

To evaluate the Galerkin projection of $A_{ij}^{nm}\left(x_0\right)$, it must be evaluated at every point on the p-grid. To help with readability, the 3 × 3 components of $A_{ij}^{nm}\left(x_0\right)$ are stored in the matrix $A_n^m\left(x_0\right)$. Evaluated on the p-grid, this takes the form $A_n^m\left(x\left({\theta }_k,{\varphi }_l\right)\right)$. The most difficult portion of calculating this matrix is the calculation of the double layer integral, $D\left[Y_n^m\right]\left(x\left({\theta }_k,{\varphi }_l\right)\right)$. This term is evaluated using the integration rule of Eq. (40):

$$D\left[Y_n^m\right]\left(x\left({\theta }_k,{\varphi }_l\right)\right)=\frac{\pi }{p+1}\sum _{k^{\prime }=0}^p\sum _{l^{\prime }=0}^{2p+1}T\left(x_{kl}^{k^{\prime }{l}^{\prime }},x_{kl}\right)\cdot n\left(x_{kl}^{k^{\prime }{l}^{\prime }}\right)Y_n^m\left({\theta }_k^{k^{\prime }{l}^{\prime }},{\varphi }_l^{k^{\prime }{l}^{\prime }}\right)W\left(x_{kl}^{k^{\prime }{l}^{\prime }}\right)w_{k^{\prime }}^S,$$ (67)

where k and l denote the un-rotated p-grid on which $A_n^m\left(x\left({\theta }_k,{\varphi }_l\right)\right)$ is evaluated, and $x_{kl}^{k^{\prime }{l}^{\prime }}$ denotes the k′ and l′ components of the rotated p-grid about the point x_kl. The rotated grid is calculated for each value of x_kl via the rotation algorithm above, an $\mathcal{O}\left(p^5\right)$ operation total. Filling this matrix in this form takes $\mathcal{O}\left(p^6\right)$ operations, as the matrix is size $\mathcal{O}\left(p^4\right)$ and requires $\mathcal{O}\left(p^2\right)$ operations per entry. To remedy this, it is noted that, as above, the rotated spherical harmonics can be expressed as

$$\begin{gathered} Y_n^m\left({\theta }_k^{k^{\prime }{l}^{\prime }},{\varphi }_l^{k^{\prime }{l}^{\prime }}\right)=\sum _{\tilde{m}=-n}^n{d}_n^{\tilde{m}m}\left(-{\theta }_k\right)Y_n^{\tilde{m}}\left({\theta }_{k^{\prime }},{\varphi }_{l^{\prime }}\right)e^{i(m-\tilde{m}){\varphi }_l} \\ =\sum _{\tilde{m}=-n}^n{d}_n^{\tilde{m}m}\left(-{\theta }_k\right)c_n^{\tilde{m}}{P}_n^{\tilde{m}}\left(\text{cos\hspace{0.17em}}{\theta }_{k^{\prime }}\right)e^{i\tilde{m}{\varphi }_{l^{\prime }}}{e}^{i(m-\tilde{m}){\varphi }_l} \end{gathered}$$ (68)

where $c_n^{\tilde{m}}$ are the normalizing constants of Eq. (4) and $d_n^{\tilde{m}m}$ are the constants of Eq. (65). First, the function $\text{T}\left(x_{kl}^{k^{\prime }{l}^{\prime }},x_{kl}\right)\cdot n\left(x_{kl}^{k^{\prime }{l}^{\prime }}\right)W\left(x_{kl}^{k^{\prime }{l}^{\prime }}\right)w_{k^{\prime }}^S\pi /(p+1)$ is computed on the new grid, where $n\left(x_{kl}^{k^{\prime }{l}^{\prime }}\right)W\left(x_{kl}^{k^{\prime }{l}^{\prime }}\right)$ is obtained via rotation. This matrix is referred to as $B_{kl}^{k^{\prime }{l}^{\prime }}$. Substituting this and Eq. (68) into Eq. (67) yields

$$D\left[Y_n^m\right]\left(x\left({\theta }_k,{\varphi }_l\right)\right)=\sum _{k^{\prime }=0}^p\sum _{l^{\prime }=0}^{2p+1}\sum _{\tilde{m}=-n}^n{B}_{kl}^{k^{\prime }{l}^{\prime }}{d}_n^{\tilde{m}m}\left(-{\theta }_k\right)c_n^{\tilde{m}}{P}_n^{\tilde{m}}\left(\text{cos\hspace{0.17em}}{\theta }_{k^{\prime }}\right)e^{i\tilde{m}{\varphi }_{l^{\prime }}}{e}^{i(m-\tilde{m}){\varphi }_l}.$$ (69)

This sum is then split into three separate sums as

$$\begin{gathered} C_{kl}^{k^{\prime }\tilde{m}}=\sum _{l^{\prime }=0}^{2p+1}{B}_{kl}^{k^{\prime }{l}^{\prime }}{e}^{i\tilde{m}{\varphi }_{l^{\prime }}}, \\ {E}_{kl}^{n\tilde{m}}=\sum _{k^{\prime }=0}^p{C}_{kl}^{k^{\prime }\tilde{m}}{c}_n^{\tilde{m}}{P}_n^{\tilde{m}}\left(\text{cos\hspace{0.17em}}{\theta }_{k^{\prime }}\right), \\ D\left[Y_n^m\right]\left(x\left({\theta }_k,{\varphi }_l\right)\right)=\sum _{\tilde{m}=-n}^n{E}_{kl}^{n\tilde{m}}{d}_n^{\tilde{m}m}\left(-{\theta }_k\right)e^{i(m-\tilde{m}){\varphi }_l}, \end{gathered}$$ (70)

each of which requires $\mathcal{O}\left(p^5\right)$ total operations. Similar operations can be performed for the Galerkin projection.