Numerical Simulation of Red Blood Cells:a “Stokesian Dynamics” Approach
S. Faure, S. Martin, B. Maury & T. Takahashi
Groupe de Travail “Methodes numeriques”Laboratoire Jacques Louis-Lions
November 9, 2009
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 1 / 20
Aim of the project
Aim: Individual and collective behaviour of Red Blood Cells:aggregates, lateral migration...
Numerical tool: C++ program for granular flows (accurate handlingof contacts between rigid spheres)
Modeling: Fluid-RBC interactions, from dilute to densesuspensions...
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 2 / 20
Aim of the project
Aim: Individual and collective behaviour of Red Blood Cells:aggregates, lateral migration...
Numerical tool: C++ program for granular flows (accurate handlingof contacts between rigid spheres)
Modeling: Fluid-RBC interactions, from dilute to densesuspensions...
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 2 / 20
Aim of the project
Aim: Individual and collective behaviour of Red Blood Cells:aggregates, lateral migration...
Numerical tool: C++ program for granular flows (accurate handlingof contacts between rigid spheres)
Modeling: Fluid-RBC interactions, from dilute to densesuspensions...
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 2 / 20
References
Fictitious domain methods:Glowinski et al.: J. Comp. Phys. (2001)Wang, Pan & Glowinski: (2008)
Lattice Boltzmann methods:Ding & Aidun: Proc. A.P.S. (2003)Binder et al.: J. Colloid Interface Sci. (2006)Sun & Munn: Comp. Math. Appl. (2008)Zhang, Johnson & Popel: J. Biomech. (2008)
Other methods :Bagchi, Johnson & Popel: J. Biomech. Eng. (2005)Liu & Liu: J. Comp. Phys. (2006)Secomb, Styp-Rekowska & Pries: Annals Biomed. Eng. (2007)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 3 / 20
References
Fictitious domain methods:Glowinski et al.: J. Comp. Phys. (2001)Wang, Pan & Glowinski: (2008)
Lattice Boltzmann methods:Ding & Aidun: Proc. A.P.S. (2003)Binder et al.: J. Colloid Interface Sci. (2006)Sun & Munn: Comp. Math. Appl. (2008)Zhang, Johnson & Popel: J. Biomech. (2008)
Other methods :Bagchi, Johnson & Popel: J. Biomech. Eng. (2005)Liu & Liu: J. Comp. Phys. (2006)Secomb, Styp-Rekowska & Pries: Annals Biomed. Eng. (2007)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 3 / 20
References
Fictitious domain methods:Glowinski et al.: J. Comp. Phys. (2001)Wang, Pan & Glowinski: (2008)
Lattice Boltzmann methods:Ding & Aidun: Proc. A.P.S. (2003)Binder et al.: J. Colloid Interface Sci. (2006)Sun & Munn: Comp. Math. Appl. (2008)Zhang, Johnson & Popel: J. Biomech. (2008)
Other methods :Bagchi, Johnson & Popel: J. Biomech. Eng. (2005)Liu & Liu: J. Comp. Phys. (2006)Secomb, Styp-Rekowska & Pries: Annals Biomed. Eng. (2007)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 3 / 20
Outline
1. A brief description of SCoPI
2. How to simulate one single Red Blood Cell ?
3. Fluid / Red Blood Cells interactions in dilute suspensions
4. Fluid / Red Blood Cells interactions in dense suspensions
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 4 / 20
A brief description of SCoPI
SCoPI (simulation of collections of particles in interaction):4 written by Aline Lefebvre4 accurate handling of contacts between rigid spheres
Langevin formulation:
M · dU
dt= Fext , in R3N
Algorithm:
4 Euler schemeM ·Un+1 = M ·Un + h Fn
ext
qn+1 = qn + h Un+1
4 handling of contacts by Uzawa technique
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 5 / 20
A brief description of SCoPI
SCoPI (simulation of collections of particles in interaction):4 written by Aline Lefebvre4 accurate handling of contacts between rigid spheres
Langevin formulation:
M · dU
dt= Fext , in R3N
Algorithm:
4 Euler schemeM ·Un+1 = M ·Un + h Fn
ext
qn+1 = qn + h Un+1
4 handling of contacts by Uzawa technique
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 5 / 20
A brief description of SCoPI
SCoPI (simulation of collections of particles in interaction):4 written by Aline Lefebvre4 accurate handling of contacts between rigid spheres
Langevin formulation:
M · dU
dt= Fext , in R3N
Algorithm:
4 Euler schemeM ·Un+1 = M ·Un + h Fn
ext
qn+1 = qn + h Un+1
4 handling of contacts by Uzawa technique
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 5 / 20
A brief description of SCoPI
Idea (Maury, 2006): projection onto the set of admissible velocities:
K (qn) ={
V ∈ R3N , Dij(qn) + h Gij(qn) · V ≥ 0, ∀i < j}
Constrained minimization problem: splitting scheme:
Un+1/2 = Un + h M−1 · Fnext
Un+1 = minV∈K(qn)
1
2
∣∣∣V −Un+1/2∣∣∣2
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 6 / 20
A brief description of SCoPI
Idea (Maury, 2006): projection onto the set of admissible velocities:
K (qn) ={
V ∈ R3N , Dij(qn) + h Gij(qn) · V ≥ 0, ∀i < j}
Constrained minimization problem: splitting scheme:
Un+1/2 = Un + h M−1 · Fnext
Un+1 = minV∈K(qn)
1
2
∣∣∣V −Un+1/2∣∣∣2
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 6 / 20
A brief description of SCoPI
Saddle-point problem: the minimization problem reads asFind (Un+1, µn+1) ∈W such that
L(Un+1, λ) ≤ L(Un+1, µn+1) ≤ L(V, µn+1), ∀(V, λ) ∈W,
with W = R3N × RN(N−1)/2+ and
L(V, λ) = F(V)−∑
1≤i<j≤N
λij (Dij(qn) + h Gij(qn) · V)
where F denotes the unconstrained functional...
Numerical computations by Uzawa algorithm.
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 7 / 20
How to simulate one single Red Blood Cell ?
Numerical modeling of a Red Blood Cell:
Macro-body made of 11 rigid spheres
Shape of the RBC: two- / three-body interactions between the spheres
“Fake” overlapping of the annular spheres
Film 1 − formation of 1 RBC (with VTK)
Film 2 − formation of 1 RBC (with post-processing: POVRAY)
Film 3 − formation of 30 RBC (with post-processing)
Film 4 − formation of 500 RBC (with post-processing)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 8 / 20
How to simulate one single Red Blood Cell ?
Numerical modeling of a Red Blood Cell:
Macro-body made of 11 rigid spheres
Shape of the RBC: two- / three-body interactions between the spheres
“Fake” overlapping of the annular spheres
Film 1 − formation of 1 RBC (with VTK)
Film 2 − formation of 1 RBC (with post-processing: POVRAY)
Film 3 − formation of 30 RBC (with post-processing)
Film 4 − formation of 500 RBC (with post-processing)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 8 / 20
How to simulate one single Red Blood Cell ?
Numerical modeling of a Red Blood Cell:
Macro-body made of 11 rigid spheres
Shape of the RBC: two- / three-body interactions between the spheres
“Fake” overlapping of the annular spheres
Film 1 − formation of 1 RBC (with VTK)
Film 2 − formation of 1 RBC (with post-processing: POVRAY)
Film 3 − formation of 30 RBC (with post-processing)
Film 4 − formation of 500 RBC (with post-processing)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 8 / 20
How to simulate one single Red Blood Cell ?
Formation of 1 RBC (initialization) Formation of 1 RBC (at fixed time)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 9 / 20
How to simulate one single Red Blood Cell ?
Formation of 1 RBC (initialization) Formation of 1 RBC (at fixed time)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 10 / 20
Fluid / Red Blood Cells interactions (dilute suspensions)
Faxen laws (Faxen, 1922):
4 rigid sphere (radius a, velocity U),4 Stokes flow (viscosity µ, velocity field v∞).
F = −6πµa(
U− v∞|x=0
)+ πµa3 ∇2v∞|x=0
Couette flow (Film 5):
Tank-treading motion (t) and tumbling motion (b)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 11 / 20
Fluid / Red Blood Cells interactions (dilute suspensions)
Faxen laws (Faxen, 1922):
4 rigid sphere (radius a, velocity U),4 Stokes flow (viscosity µ, velocity field v∞).
F = −6πµa(
U− v∞|x=0
)+ πµa3 ∇2v∞|x=0
Couette flow (Film 5):
Tank-treading motion (t) and tumbling motion (b)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 11 / 20
Fluid / Red Blood Cells interactions (dense suspensions)
Langevin formulation:
M · dU
dt= FH + FP , in R3N
4 FH : hydrodynamic forces4 FP : non hydrodynamic forces
SCoPI algorithm:4 Definition of the so-called “grand resistance” matrix:
FH = −R ·U + F4 Implicit procedure:
(M + h Rn) ·Un+1 = h (Fn + FnP),
qn+1 = qn + h Un+1
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 12 / 20
Fluid / Red Blood Cells interactions (dense suspensions)
Langevin formulation:
M · dU
dt= FH + FP , in R3N
4 FH : hydrodynamic forces4 FP : non hydrodynamic forces
SCoPI algorithm:4 Definition of the so-called “grand resistance” matrix:
FH = −R ·U + F4 Implicit procedure:
(M + h Rn) ·Un+1 = h (Fn + FnP),
qn+1 = qn + h Un+1
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 12 / 20
Fluid / Red Blood Cells interactions (dense suspensions)
Stokesian Dynamics (Brady & Bossis, 1988):
4 suspension of rigid, non-Brownian particles4 bulk macroscopic shear flow4 the Reynolds particle number is small
Relationship between hydrodynamic forces and velocities (Happel &Brenner, 1965 ; Brenner & O’Neill, 1972):
4 resistance formulation: FH = −R · (U− v∞) + Φ : E(v∞)4 mobility formulation: M = R−1
Assumption: pairwise additivity of the interactions
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 13 / 20
Fluid / Red Blood Cells interactions (dense suspensions)
Stokesian Dynamics (Brady & Bossis, 1988):
4 suspension of rigid, non-Brownian particles4 bulk macroscopic shear flow4 the Reynolds particle number is small
Relationship between hydrodynamic forces and velocities (Happel &Brenner, 1965 ; Brenner & O’Neill, 1972):
4 resistance formulation: FH = −R · (U− v∞) + Φ : E(v∞)4 mobility formulation: M = R−1
Assumption: pairwise additivity of the interactions
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 13 / 20
Fluid / Red Blood Cells interactions (dense suspensions)
Stokesian Dynamics (Brady & Bossis, 1988):
4 suspension of rigid, non-Brownian particles4 bulk macroscopic shear flow4 the Reynolds particle number is small
Relationship between hydrodynamic forces and velocities (Happel &Brenner, 1965 ; Brenner & O’Neill, 1972):
4 resistance formulation: FH = −R · (U− v∞) + Φ : E(v∞)4 mobility formulation: M = R−1
Assumption: pairwise additivity of the interactions
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 13 / 20
Fluid / Red Blood Cells interactions (dense suspensions)
For two spheres, all elements of the resistance / mobility matrix for allseparation are known exactly (Jeffrey & Onishi, 1984).
4 At large separations, far-field expressions (method of reflections,...)4 At small separations, analytical results from lubrication theory4 At intermediate separations, tabulated results
Resistance matrix...
4 Resistance / mobility matrix defined for two spheres in interaction.4 The grand resistance matrix is obtained by pairwise additivity...
R = (M∞)−1 +R2B −R∞2B
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 14 / 20
Fluid / Red Blood Cells interactions (dense suspensions)
For two spheres, all elements of the resistance / mobility matrix for allseparation are known exactly (Jeffrey & Onishi, 1984).
4 At large separations, far-field expressions (method of reflections,...)4 At small separations, analytical results from lubrication theory4 At intermediate separations, tabulated results
Resistance matrix...
4 Resistance / mobility matrix defined for two spheres in interaction.4 The grand resistance matrix is obtained by pairwise additivity...
R = (M∞)−1 +R2B −R∞2B
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 14 / 20
Fluid / Red Blood Cells interactions (dense suspensions)
Far-field forces (Smoluchowski, 1911 ; Luke, 1989 ; Guazzelli, 2003)
−U2 =1
6πµaF2 +
1
8πµ
(I
r+
r · rr3
)·F1 +
a2
4πµ
(I
3r3− r · r
r5
)·F1 + ...
Lubrication forces (Jeffrey & Onishi, 1984 ; Dance & Maxey, 2003)
Fj→i · ni→j = −3πµa2
2
(Ui −Uj) · ni→j
r+O (ln r)
Intermediate-field forces (Kim & Karrila, 1991)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 15 / 20
Fluid / Red Blood Cells interactions (dense suspensions)
Far-field forces (Smoluchowski, 1911 ; Luke, 1989 ; Guazzelli, 2003)
−U2 =1
6πµaF2 +
1
8πµ
(I
r+
r · rr3
)·F1 +
a2
4πµ
(I
3r3− r · r
r5
)·F1 + ...
Lubrication forces (Jeffrey & Onishi, 1984 ; Dance & Maxey, 2003)
Fj→i · ni→j = −3πµa2
2
(Ui −Uj) · ni→j
r+O (ln r)
Intermediate-field forces (Kim & Karrila, 1991)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 15 / 20
Fluid / Red Blood Cells interactions (dense suspensions)
Far-field forces (Smoluchowski, 1911 ; Luke, 1989 ; Guazzelli, 2003)
−U2 =1
6πµaF2 +
1
8πµ
(I
r+
r · rr3
)·F1 +
a2
4πµ
(I
3r3− r · r
r5
)·F1 + ...
Lubrication forces (Jeffrey & Onishi, 1984 ; Dance & Maxey, 2003)
Fj→i · ni→j = −3πµa2
2
(Ui −Uj) · ni→j
r+O (ln r)
Intermediate-field forces (Kim & Karrila, 1991)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 15 / 20
Fluid / Red Blood Cells interactions (dense suspensions)
Film 6 − without lubrication forces Film 7 − with lubrication forces
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 16 / 20
Conclusion and perspectives
To be improved:4 Far-field forces:
method of reflectionscut-off distance (Satoh, 2001)
4 Lubrication:
method of boxes (complexity in N log N instead of N2)tangential contributionswall-particle lubrication
To be implemented:4 Intermediate-field forces4 Linear systems solvers4 Periodic boundary conditions
To be discussed:4 Lateral migration4 Aggregates (Film 8)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 17 / 20
Conclusion and perspectives
To be improved:4 Far-field forces:
method of reflectionscut-off distance (Satoh, 2001)
4 Lubrication:
method of boxes (complexity in N log N instead of N2)tangential contributionswall-particle lubrication
To be implemented:4 Intermediate-field forces4 Linear systems solvers4 Periodic boundary conditions
To be discussed:4 Lateral migration4 Aggregates (Film 8)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 17 / 20
Conclusion and perspectives
To be improved:4 Far-field forces:
method of reflectionscut-off distance (Satoh, 2001)
4 Lubrication:
method of boxes (complexity in N log N instead of N2)tangential contributionswall-particle lubrication
To be implemented:4 Intermediate-field forces4 Linear systems solvers4 Periodic boundary conditions
To be discussed:4 Lateral migration4 Aggregates (Film 8)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 17 / 20
Conclusion and perspectives
Red blood cell aggregates in a shear flow
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 18 / 20
Conclusion and perspectives
Spheres in a Poiseuille flow
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 19 / 20