The 1st International Workshop within the SCOPES project
Computational modelling of blood flow in the tumor vasculature
Computer modelling of
angiogenesis and
blood microcirculation
Prof. Dr. Nenad Filipovic
University of Kragujevac
SCOPES 2013, Kragujevac
Jan 25th, 2013
Outline
 Computer modeling approaches
Continuum method, Finite Element
 Discrete method, Dissipative Particle
Dynamics
 Thrombosis modeling
 Nanodrugs modeling
 Orbital shaker modeling
 Chorioallantoic Membrane Modeling
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Two different approach for
bioprocess modeling
 Continuum-based methods
 Discrete modeling
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Modeling mass transport in arteries
by continuum-based methods
  2c  2c  2c 
c
c
c
c
 vx
 vy
 vz
 D 2  2  2 
t
x
y
z
y
z 
 x
The mass transport process is governed by convection-diffusion equation
cw vw  k
c
 Kcw
n
ShD 
qw Dv
Db  PO2in  PO2 ref

Oxygen wall fluxes are frequently expressed in
The conversion of the mass
terms of the local Sherwood number
among the LDL passing
through a semipermeable wall
  2 ci  2 ci  2 ci
ci
ci
ci
ci
 vx
 vy
 vz
 Di  2  2  2
t
x
y
z
y
z
 x
SCOPES 2013, Kragujevac
Jan 25th, 2013

  Si

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Continuum model with three species, 1)
normal-resting platelets; 2) active platelets;
and 3) chemical agonist
n
 v  n  Dn n  R(c )n
t
a
 v  a  R( c )n
t
c
 v  c  Dc c  AR(c )n
t
Kinetics of platelet activation
 0,   1.0 


k pa   

,


1.0
t

 act

 is the activation function which depends
on the concentration of the j-th agonist and
n is concentration of normal-resting platelets, tact is a characteristic time for platelet
v is blood flow velocity, Dn is diffusion
activation
coefficient for resting platelets, R(c) is the
conversion rate of the resting into activate
platelets, a is concentration of activated
platelets, c is concentration of ADP, Dc is
diffusion coefficient of signaling chemical ADP
and A is the rate of creation of ADP
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Finite element method of diffusion-transport equations
1
 t M v 





n 1
K vv  
i 1
n 1
K μv  
i 1
n 1
J vv
i 1
n 1
K (vpi 1)
K Tvp
0
K cvi 1
0
n 1



0


1
 i 1
 i 1

M c  n 1 K cc
 n 1 J cc
t

0
V i    n 1 Fvi 1 
 i    n 1 i 1 
 P    Fp 
 i    n 1 i 1 
 C   Fc 
( M v ) jjKJ    N K N J dV ,


( M c ) jjKJ   N K N J dV
V
n 1
n 1

n 1

n 1
K cc
K
( i 1)
( i 1)
cv
J
( i 1)
cc
J
( i 1)
vv




jjKJ
jjKJ
  DN K , j N J , j dV
V
N
n 1 ( i 1)
K
,j
c
N J dV
V
jjKJ
  N
n 1 ( i 1)
K
j
v
N J , j dV
V
jkKJ
   NK
n 1
Fc (i 1) 
V

n 1

n 1

n 1
K v
( i 1)
K
( i 1)
vv
K
( i 1)
vp



jjKJ
jjKJ
   N K , j N J , j dV
V
  N
n 1 ( i 1)
K
j
v
N J , j dV

n 1
1
M c  n 1 C( i 1)  n C 
t
n 1
( i 1) n 1 ( i 1)
K cv  V   n 1 K cc (i 1)  n 1C(i 1) 
n 1
Fq 
n 1
Fsc ( i 1) 
Fq    N K q B dV
K
V
n 1
Fsc (i 1)   DN K  n 1 c
S
V
jjKJ
   N K , j Nˆ J dV
V
n 1
v j ,k N J dV
V
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i 1
 ndS
Example 1: Modeling albumin transport in a large artery
  2r  2 
v(r )  2V0 1    
  L0  


p 
vz
0
z
Reynolds number Re=448
Pecklet number Pe = 934000 Pe  L0V0 / D
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Example 2: Modeling the LDL transport through a
straight artery with the filtration through the wall
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Example 3: Modeling platelet accumulation on collagen-coated
wall of a tube with narrowing (stenosis)
Stenosis geometry for 75% reduction
(lengths are in [mm]). The shaded
domain is modeled by axisymmetric
finite elements. The finite element mesh
is shown schematically
Straight artery with 75% stenosis. a)
Intensity of blood velocity field; b)
Streamline contours; c) Wall shear rate
along the wall; d) Platelet accumulaton
rate along the wall (j*=j(x)/c0, kt =
5x10-3 [cm/s];
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The second approach: Discrete platelet adhesion
and aggregation modeling
Schematic representation of the mechanisms of platelet adhesion
and aggregation in flowing blood
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Discrete particle dynamics–platelet
aggregation and adhesion
Schematics of platelet aggregation
and adhesion. Activated platelets in
the vicinity of a injured wall
epithelium and binding of platelets at
the walls using springs. Interaction
forces for two aggregated platelets
The domain of the interaction
between platelets is denoted by rmax.
(Filipovic et al. 2007)
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DPD Method – Differential equations of motion
mi vi   fij
j
micrometers
fij  f
Conservative
ij
fijC
i
fij
f
Dissipative
ij
f
Brownian
ij
fijDissipative   vij
j
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Distance
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Example 1: Platelet aggregation in blood flow
between two parallel plates
Filipovic, N., Ravnic, D.J. Kojic, M., Mentzer, S.J., Haber, S. Tsuda, A.,
Interactions of Blood Cell Constituents: Experimental investigation and
Computational Modeling by Discrete Particle Dynamics Algorithm, Microvascular
Research, 75, 279-284, 2008.
Filipovic, N., Haber, S., Kojic, M., Tsuda, A., Dissipative particle dynamics
simulation of flow generated by two rotating concentric cylinders: II. Lateral
dissipative and random forces, J. Phys. D: Appl. Phys. 41 035504 , 2008
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Example 2: DPD simulation of Karino’s
example blood flow through expanded tube
T. Karino, H.L. Goldsmith, Adhesion of human platelets
to collagen on the walls distal to a tubular
expansion, Miscrovascular Research 17, 238-269,
1977.
Filipovic, N., Kojic, M., Tsuda, A., Мodeling thrombosis
using dissipative particle dynamics method, Phil
Trans Royal, A 366(1879), 2008
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ANIMATION: BLOOD FLOW AROUND THROMBUS
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Integration: Multiscale Modeling
Division of the flow domain into:
a) GLOBAL DOMAIN - Domain modeled by a continuum model (Finite Element) only
b) LOCAL DOMAIN - Domain modeled by both discrete particles (DPD) and FE
D
Common boundary
ABCD
C
v particle  v FE
A
B
DPD+FE domain
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FE domain
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Coupling the DPD and FE models
1) Decomposition of particle velocities
2) FE nodal forces in terms of the particle interaction forces
Particle interaction force

fij  fij  fij  fij
C
VKy
FE node K
j Velocit
v
Forcefij y i
Particle i
FJy
vi
vi
FJx
FE node J
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VKx
MESOSCOPIC BRIDGING SCALE METHOD
Mathematical interpretation of the coupling between discrete
particle (DPD) and finite element (FE) models
Kinetic energy
Ek  Ek  Ek
1 T
v M Av
2
1
1
Ek  vT M A v  VT MV
2
2
Ek 
Ek 
Diff. Eqs. of Motion
Lagrangian Description
1 T
v M A v
2
DPD
FE
Fluctuating
kinetic energy
M A v  f ext  f int
DPD
MV  Fext  Fint
FE
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Diff. Eqs. of motion Lagrangian description
Navier-Stokes
FE eqs. of
motion
and Continuity
1

t t ( i 1)
(i )
M

K
K



V
vp
 t

 (i )  


K Tvp
0   P 

( i 1)
1

1

 t  t Fin(it1)   t t Fext(i 1)   M  t t K (i 1) K vp   t t V   M t V 


  t
 t  t ( i1)    t


0
0
T

 
 
K vp
0   P   0 

Nodal internal
forces
t  t
FKiint(i 1)   N K , j t t ij(i1) dV
V
Stress
tensor


1
σ  n  mi vˆ i  vˆ i   rij  fij 
2 i j i
i

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
t t (i 1)
ij
M A v  f ext  f int
MV  Fext  Fint
Shear
stresses
From DPD
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Example 3: Thrombosis modeling in large arteries
DPD
method
FE method
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Nanomedicine – smart drugs
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Orbital shaker, shear stress computer simulation
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Orbital shaker, shear stress computer simulation
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Wall shear stress distribution at the bottom wall and free surface
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Shear stress distribution
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Velocity distribution
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Chorioallantoic Membrane
Computer Modeling
•
•
•
•
•
•
Contribution of blood flow to vessel structure remains a fundamental question in
biology.
intravascular pillars in the chick chorioallantoic membrane.
3-dimensional computational flow simulations indicated that the intravascular pillars
were located in regions of low shear stress.
Both wide-angle and acute-angle models mapped the pillars to regions with shear
less than 1 dyn/cm2.
Further, flow modeling indicated that the pillars were spatially constrained by
regions of higher wall shear stress.
The shear maps indicated that the development of new pillars was limited to
regions of low shear stress.
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Problem description
• Geometry and blood flow measurements derived from intravital microscopy
imaging
• Mapping of the mechanical forces within the CAM (chorioallantoic
membrane) vessels
• Calculation of the wall shear stress and blood pressure using 3D
computational flow simulations.
• Pillar geometry suggested the spatial constraint of high wall shear stress.
Further, the development of new pillars was limited to
• regions with low shear stress.
• The result suggests both a limiting and permissive influence of wall shear
stress on pillar development in the CAM.
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Experimental setup
• For all experiments, a modified, ex ovo (shell-less) culture method was
used
• Briefly, the eggs were kept in an RCOM 20 digital incubator (GimHae,
Korea) at 37.5°C and 70% humidity with automatic turning for 3 days.
• On embryonic development day (EDD) 3, the eggs were sprayed with
70% ethanol, air-dried in a laminar flow hood and explanted into a 20 ×
100 mm Petri dish (Falcon, BD Biosciences, San Jose, CA).
• The ex ovo cultures were maintained in a humidified 2% CO2 incubator at
37.5°C.
• To optimize the selective examination of the 2nd and 3rd order conducting,
as well as facilitate intravital microscopy identification of the intravascular
pillars, intravital microscopy was performed on EDD 13-16
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The digital images were acquired at a single wavelength (ex 430 nm; em 510 nm). The recorded image stacks were
analyzed for flow velocity (B) and recombined into a composite time-series image (C). A,B) A line of selectable orientation and
width was drawn along the vessel axis. The distance-time plane (B) provided a longitudinal view over the selected length of the
vessel. Cells or particles were tracked through multiple planes of the stack permitting a visual correlation in each plane. The white
object (arrow) represents a fluorescent particle; the slope of the diagonal line represents the velocity of the particle in the flow
stream. Note the different slope of the background speckle pattern--an observation suggesting the particle is near to, or
interacting with, the vessel wall. A,C) The image stack was digitally recombined and pseudo colored for presentation as a timeseries image (C). The region within the vessel demonstrating no detectable fluorescence (arrow) was defined as an intravascular
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pillar.
Spatial distribution of pillars in a region of the CAM. Time-series images of 9 contiguous regions
of a CAM, previously injected with FITCdextran, were digitally reconstructed and stitched into a 3 × 3
montage (A). The vessels were thresholded, binarized and mapped to a 2D grid (B).
Morphometric analysis of the binarized image provided a relative measure of both vessel and pillar
area. The vessels comprised 31.8% of the total surface area of the CAM in 2D projection; the
intravascular pillars comprised 0.4% of the vessel area (inset).
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Time-series flow visualization of the CAM intravascular pillars using
fluorescence intravital videomicroscopy and intravascular
tracers. A-D) CAM vessels visualized with the plasma marker FITC-dextran
and low-density fluorescent particle tracers. E-H) CAM vessels visualized with
high-density particle tracers. Pillars (red asterisk) were identified as
intravascular areas with no plasma marker or particle tracer throughout the
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time-series.
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3D computational flow modeling of a wide angle bifurcation
(60 degrees) in the CAM.
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3D computational flow modeling of an acute angle
bifurcation (5 degrees) in the CAM
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Time-series visualization of a developing intravascular
pillar using fluorescence intravital videomicroscopy
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Discussion
• Because new pillars were not predictable from our shear maps, we suspect that
mechanical forces have a permissive role in pillar development; that is, regions
of low shear stress permit pillar development stimulated by other growth or
developmental signals
• A diffusible endothelial activation signal provides one explanation for the vessel
irregularities observed during pillar development.
• A competing hypothesis is that mechanical forces stimulate new pillar formation
and may even initiate the related process of intussusceptive angiogenesis.
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Conclusions
• Modeling of such biological phenomena, is still a huge issue, which has to be
addressed through mathematical and algorithmic approaches and validated by
experimental approaches.
• The goal of the workshops is to exchange experiences, results and ideas based
on actual research projects in order to foster the creation of new interactions
between young scientists in Serbia and European colleagues
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