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Basics Moving particles Coarse-grained blood model

Application of the Lattice Boltzmann method with moving boundaries in a coarse-grained suspension model for hemodynamics

Florian Janoschek Jens Harting

Department of Applied Physics, Eindhoven University of Technology, The Netherlands Institute for Computational Physics, Stuttgart University, Germany

3rd June 2009

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Lattice Boltzmann method Mid-link bounce-back boundary condition

The lattice Boltzmann method

D3Q19 lattice, drawing [Hecht and Harting, 2008]

Definitions discrete velocities cr populations nr(x, t) Hydrodynamic quantities density ρ(x, t) =

• r

nr(x, t) velocity u(x, t) =

• r nr(x, t)cr

ρ(x, t)

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Lattice Boltzmann method Mid-link bounce-back boundary condition

The lattice Boltzmann method

D3Q19 lattice, drawing [Hecht and Harting, 2008]

Time evolution Collision n∗

r(x, t) =

nr(x, t) − nr(x, t) − neq

r (ρ(x, t), u(x, t))

τ (BGK) with equilibrium population neq

r

Advection nr(x, t) = n∗

r(x − cr, t − 1)

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Lattice Boltzmann method Mid-link bounce-back boundary condition

Mid-link bounce-back boundary condition

[Nguyen and Ladd, 2002]

Advection/boundary condition nr(x, t) =        n∗

¯ r(x, t − 1)

x − cr wall n∗

r(x − cr, t − 1)

• therwise

with c¯

r = −cr

noslip: u = 0 at boundary

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Moving boundary condition Free motion of particles Lubrication correction

Moving boundary condition

[Nguyen and Ladd, 2002]

Steady state: nr = nr(x, t) = n∗

r(x, t) = neq r (ρ, u)

Conventional bounce-back condition nr = n¯

r

consistent with fluid at rest (u = 0) Moving boundary condition must be consistent with fluid at speed u 0.

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Moving boundary condition Free motion of particles Lubrication correction

Moving boundary condition

[Nguyen and Ladd, 2002]

neq

r (ρ, u) neq ¯ r (ρ, u) for u 0

Equilibrium population [Chen et al., 2000] neq

r (ρ, u) =

ραr

• 1 + βcru
• changes sign for r → ¯

r

+1 2β2(cru)2 − 1 2βu2 + O(|u|3)

• Moving wall boundary condition

nr(x, t) = n∗

¯ r(x, t − 1) + 2ραrβcru

• first order correction term

consistent with fluid speed u 0

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Moving boundary condition Free motion of particles Lubrication correction

Freely moving particles

Particle configuration characterized by translational and rotational position ri and ˆ

• i

Analytical particle surface defined by function Γ(x − ri, ˆ

• i) ∈
• [0; 1[

x within particle i [1; ∞[ x elsewhere discretized on the lattice Forces Fi and torques Ti acting on each particle Integration of equations of motion like in typical Molecular Dynamics codes (here: leap-frog integrator)

Fi → vi → xi Ti → ωi → ˆ

• i

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Moving boundary condition Free motion of particles Lubrication correction

Momentum balance

[Nguyen and Ladd, 2002]

Velocity at boundary node xb vb(xb) = vi + ωi × (xb − ri) Momentum transfer per timestep ∆t = 1 ∆p = (2n¯

r + 2ραrβcrvb)c¯ r

results in

force Fb = ∆p/∆t = ∆p and torque Tb = (xb − ri) × Fb

• n moving particle.

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Moving boundary condition Free motion of particles Lubrication correction

Fluid extinction

Fluid nodes xf covered by the particle turn into wall nodes. Fluid momentum is incorporated by the particle.

Force Ff =

• r

nr(xf)cr Torque Tf = (xf − ri) × Ff

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Moving boundary condition Free motion of particles Lubrication correction

Fluid creation

Wall nodes xp released by the particle turn into fluid nodes. Equilibrium population neq

r (ρ, vp) for the

system’s initial density ρ and the particle’s surface velocity vp is created. Fluid momentum is taken from the particle.

Force Fp = −ρvp Torque Tp = (xp − ri) × Fp

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Moving boundary condition Free motion of particles Lubrication correction

Lubrication correction

The fluid between particles close to contact is not resolved due to the finite lattice resolution. Static effects → assumption of equilibrium distribution at particle nodes Dynamic effects → correction term for spherical particles [Ladd and Verberg, 2001] Fij = −6πρν(RiRj)2 (Ri + Rj)2 ˆ rij[ˆ rij(vi − vj)]

• 1

rij − Ri − Rj − 1 ∆c

• with cut-off separation ∆c = 2/3

More flexible method: [Ding and Aidun, 2003]

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Human blood

Applications for blood simulation study of transport phenomena support for surgery design of lab-on-a-chip devices . . .

100 101 102 103 10−2 10−1 100 101 102 103 µ/µ(c = 0) ˙ γ [s−1]

data [Chien, 1970]

Blood properties ρblood ≈ 1.06ρH2O ≈ 55 vol. % plasma ≈ 44 vol. % red blood cells µ const: shear thinning

[Evans and Fung, 1972] Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Existing blood models

Continuum models µ = const (Newtonian fluid) µ = µ( ˙ γ) (Casson, Carreau-Yasuda) [Boyd et al., 2007] Particulate models that resolve RBC deformability

[Noguchi and Gompper, 2005]

[Noguchi and Gompper, 2005], [Dupin et al., 2007] Drawback no resolution of particulate effects Drawback computational cost

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

New phenomenological model

Similar to existing models cell trajectories integrated by molecular dynamics code plasma modelled as Newtonian fluid by lattice Boltzmann New in this model interactions of deformable cells modelled solely by soft potentials (at the present completely repulsive)

cell-cell cell-wall

hydrodynamic coupling between cells and plasma The goal resolution of particulate effects capability to simulate small macroscopic systems in 3D (∼ 1 mm3)

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Cells and vessel walls: BPrH-Potential

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 ˜ φrH r

Repulsive Hooke potential ˜ φrH(r) =

• (1 − r)2

r < 1 r ≥ 1 Orientation-dependent energy and range parameters ε(ˆ

• i, ˆ
• j) and σ(ˆ
• i, ˆ
• j,ˆ

rij) [Berne and Pechukas, 1972] φBPrH(ˆ

• i, ˆ
• j, rij) = ε(ˆ
• i, ˆ
• j) ˜

φrH(rij/σ(ˆ

• i, ˆ
• j,ˆ

rij)) Surface σ(ˆ

• i, ˆ
• j,ˆ

rij) = rij closely resembles positions of two ellipsoids or ellipsoid and sphere that are in touch Size parameters σ⊥, σ, and σr Strength parameters ¯ ε and ¯ εr

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Blood plasma: lattice Boltzmann

Blood plasma LB3D (D3Q19, BGK) viscosity µ, density ρf Interaction with blood cells suspended rigid particles [Nguyen and Ladd, 2002] no lubrication corrections, touching and

• verlapping is part of the model

density ρp, sizes R⊥ and R Interaction with vessel walls mid-link bounce-back

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics [Nguyen and Ladd, 2002]

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Unit conversion I

Factors for conversion between the dimensionless quantities of the simulation and physical units dx: physical distance between to lattice nodes dt: physical length of one lattice Boltzmann timestep dm: physical mass at one lattice site populated with unit density

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Unit conversion II

Space Sufficient resolution of RBCs → choose dx = 2

3 µm.

Time Numerical instability and increasing deviations between input radii and effective hydrodynamic radii for τ 1 → choose τ = 1. Still fluid viscosity ν = (2τ − 1)/6 should match physical plasma viscosity ν dx2 dt = ν(p) = 1.09 · 10−6 m2 s . This fixes dt = 6.79 · 10−8 s. Mass Assume unit density ρf = 1 for the fluid. Given dx and known density of blood plasma results in dm = 3.05 · 10−16 kg.

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Plane shear flow: parameter search

c=cpf=45 %

Preliminary parameters ρp = 1.07 σ⊥ = 6, σ = 2 R⊥ = 5.5, R = 1.833 ¯ ε = 0.05 µ = F A ˙ γ

2 3 4 5 6 7 101 102 103 104 µ / µ(c = 0) ˙ γ [s−1] ¯ ε 100 101 100 101 102 103 µ / µ(c = 0) ˙ γ [s−1] model Chien (1970)

Consistency with literature where attractive forces are negligible

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Channels: qualitative demonstration

c=cpf=42 %

Wall node potential parameters size σr = 0.5 strength ¯ εr = ?

0.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 25 30 vz / maxr(vz) r [µm] maxr(vz) [cm/s] 1.52 5.53 · 10−4 2 4 6 8 10 12 14 16 5 10 15 20 25 30 vz [cm/s] r [µm] ¯ εr 5 · 10−3 5 · 101

Behavior consistent with literature (Goldsmith and Skalak, 1975)

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Junctions: qualitative demonstration

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Qi /

j Qj

t [s] not narrowed narrowed

Observations clogging visible effect of ¯ εr red cells choose faster branch as in literature [Fung, 1981]

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Macroscopic systems

System 0.32 mm3 10243 lattice sites 2.3 · 106 particles Resources 1024 processes on XC2 (SSC Karlsruhe) 1.9 h/1000 LB steps Per process 10242 lattice sites 2.2 · 103 particles

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Conclusion and outlook

Results Model reproduces shear viscosity at high shear rates for plausible choice of parameters. Important effects in channels and junctions are reproduced at least qualitatively. The code is able to simulate macroscopic systems. Further work Implementation of attractive forces Improvement of choice of parameters Quantitative investigation of behavior in channels and junctions Optimization for sparse systems Enlargement of LB timestep dt if possible

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Bibliography I

Berne, B. J. and Pechukas, P . (1972). Gaussian model potentials for molecular interactions.

• J. Chem. Phys., 56(8):4213–4216.

Boyd, J., Buick, J. M., and Green, S. (2007). Analysis of the Casson and Carreau-Yasuda non-Newtonian blood models in steady and oscillatory flows using the lattice Boltzmann method.

• Phys. Fluids, 19:093103.

Chen, H., Boghosian, B. M., Coveney, P . V., and Nekovee, M. (2000). A ternary lattice Boltzmann model for amphiphilic fluids.

• Proc. R. Soc. Lond. A, 456:2043–2057.

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Bibliography II

Chien, S. (1970). Shear dependence of effective cell volume as a determinant of blood viscosity. Science, 168(3934):977–979. Ding, E.-J. and Aidun, C. K. (2003). Extension of the lattice-Boltzmann method for direct simulation of suspended particles near contact.

• J. Stat. Phys., 112(3/4):685–708.

Dupin, M. M., Halliday, I., Care, C. M., Alboul, L., and Munn, L. L. (2007). Modeling the flow of dense suspensions of deformable particles in three dimensions.

• Phys. Rev. E, 75:066707.

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Bibliography III

Evans, E. and Fung, Y.-C. (1972). Improved measurements of the erythrocyte geometry. Microvascular Research, 4:335–347. Fung, Y. C. (1981).

• Biomechanics. Mechanical Properties of Living Tissues.

Springer, New York. Hecht, M. and Harting, J. (2008). General on-site velocity boundary conditions for lattice Boltzmann. http://arxiv.org/abs/0811.4593, submitted for publication. Ladd, A. J. C. and Verberg, R. (2001). Lattice-Boltzmann simulations of particle-fluid suspensions.

• J. Stat. Phys., 104(5/6):1191–1251.

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics

Basics Moving particles Coarse-grained blood model Motivation Model Application on blood

Bibliography IV

Nguyen, N.-Q. and Ladd, A. J. C. (2002). Lubrication corrections for lattice-Boltzmann simulations of particle suspensions.

• Phys. Rev. E, 66:046708.

Noguchi, H. and Gompper, G. (2005). Shape transitions of fluid vesicles and red blood cells in capillary flows. PNAS, 102(40):14159–14164.

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics