Particle-based mesoscale modelling of blood flow in
microvessels
Leonor García-Gutiérrez, under the supervision of Prof. Robert Kerr
Mathematics Institute, The University of Warwick
l.garcia-gutierrez@warwick.ac.uk
Model: MPC-AT
Poiseuille flow
MPC-AT: Multi Particle Collision Dynamics with Anderson Thermostat.
I Parabolic profile is reproduced.
I Based on a coarse-grain description of molecular collisions.
I A modification of Bird’s Direct Simulation Monte-Carlo.
I Alternative approach to solve Boltzmann equation by alternating
streaming and colliding steps.
Z
0.35
Y
0.40
0.36
0.32
0.28
0.24
0.20
0.16
0.12
0.08
0.04
0.00
0.04
Stream
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0.25
0.20
v_x
Motivation
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0.10
0.05
0.00
20
15
10
5
Y (cells)
0
I Partially filled cells at the boundary, but density is uniform across
the lumen.
Collide
Z
14.5
20
14.0
Stream
particle density
13.0
12.5
Y
I UK - more than £1 billion in drugs alone in 2006 (nice.org.uk)
I US - $93.5 billion in healthcare in 2010 (Heidenreich PA et al. Circulation. 2011;123:933-44)
I Pathophysiology of essential hypertension is not well understood.
13.5
12.0
Evolution is governed by Newton’s equations of motion:
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11.5
5
11.0
10.5
0
20
vi (t + ∆t) = vi (t) + g ∆t
Why microvessels?
ri (t + ∆t) = ri (t) + vi (t) ∆t
“(...), it should never be forgotten that the more obvious
phenomena of the circulation are but a means through
which the real object of maintaining an adequate capillary
flow is attained”.
Carl J. Wiggers
15
I Standard expression in literature, implicitly assuming an
acceleration-driven flow (g) with no additional forces.
I Forces perpendicular to the flow can be introduced using:
15
10
5
Y (cells)
0
I Slip is observed (with both SAM and CAM averaging methods).
Typically corrected using “virtual particles” in the collision step.
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0.30
vi (t + ∆t) = vi (t) + a ∆t
0.25
v_x
1
ri (t + ∆t) = ri (t) + vi (t) ∆t + a∆t 2
2
I Time-fractioning algorithm: A particle that would escape the
lumen in ∆t is moved back close to the impact point with the
boundary; the impact time α∆t is calculated; impact velocity is
reversed (bounce-back: v → −v); and the particle is let to stream
for the remaining fraction of ∆t. The fractioning repeats if it collides
with the boundary before (1 − α)∆t.
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0.10
Simulation (fit)
NS
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0.00
20
15
10
Y (cells)
5
0
Red Blood Cells in a stenosed capillary
Collide
Simulation box containing the system is coarsely divided into equal
cells. Random shift of the grid is applied to ensure Galilean invariance:
I Cross-sectional images (left) resemble experimental TEM images
(right).
Particle occupation, t = 19.750ms, x = 100µm
35
30
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25
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Normal
Hypertensive
Maximum capillary density in pregnancy. Image courtesy of D. Singer
20
Z
?
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15
?
I Capillary rarefaction is observed also in healthy individuals with a
genetic predisposition for high blood pressure.
I Treating functional blockages in blood capillaries may prevent or
delay the onset of hypertension.
I This requires a deep understanding blood flow at the capillary level
mathematical model + simulations.
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?
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5
0
0
5
10
15
20
25
30
35
−Y
Particle occupation, t = 141.750ms, x = 150µm
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30
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25
Mesoscale, particle-based methods are more suitable than NavierStokes equations because at microcirculation level blood is a complex
fluid with disparate time,length,energy scales.
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10
w
I Apply periodic boundary conditions (flow direction).
I Reset random component of velocity for all particles.
kB T
uj |x,y,z ∈ N 0,
mj
20
Z
Why mesoscale methods?
?
?
15
10
5
I Compute average mass, velocity and random velocity for each cell.
(i)
~vcell
=
1
X
(i)
mcell j ∈P (i)
0
0
5
10
15
20
25
30
35
−Y
TEM images: http://remf.dartmouth.edu/imagesindex.html
mj ~vj
I Clustering (Rouleaux formation and dynamics).
I Update total velocity of each particle j in cell i, ensuring momentum
conservation.
vj (t + ∆t)|post −coll =
(i)
vcell (t
+ ∆t) + uj −
1
X
(i)
mcell j ∈P (i)
mj u j
Solutes: Hybrid MPC-MD
Coupling with solutes is straightforward in MPC
I Coarse-grained model of the solute: particle + interactions.
I Stream particles at timesteps ∆τ (velocity Verlet).
I Solute and solvent particles are indistinguishable in the collision
step.
Cross section of capillary with red blood cells.
Credit: Tissues and Organs—Visuals Unlimited/Getty Images
100
Continuum dynamics
I Fåhræus Effect is reproduced: reduction of the hematocrit in
non-stenosed microvessels, due to vRBC > vplasma (velocities in
µm ms−1 ).
Time scale (s)
10−3
Mesoscale
Run 1
Run 2
Run 3
−6
10
−9
v¯1 pl
v¯2 pl
RBC0
RBC1
RBC2
RBC3
RBC4
0.577
0.582
0.566
0.560
0.588
0.577
0.851
0.857
0.942
0.931
0.888
0.887
0.992
0.989
1.013
0.912
0.977
1.008
0.937
0.998
1.086
Molecular Dynamics
10
Acknowledgements
Funding: “La Caixa” Foundation and Warwick DARO. Computing facilities: Centre for
Scientific Computing of the University of Warwick.
10−12
10−9
10−6
Length scale (m)
10−3
100