On the Complex Human Blood Flow Modeling

On the Complex Human Blood Flow Modeling
*Luboš Pirkl
**Tomáš Bodnár
* CFD support, s.r.o., Želkovická 823, 19014 Praha 9, Czech Republic
** Czech Technical University, Faculty of Mech. Engin., Technická 4, 166 07 Praha 6, Czech Republic
The paper presents a numerical simulation of blood coagulation process in the injured blood
vessel. There are three levels of modeling in this study. There are followed three major phenomenas: shear-thinning property of blood, viscoelasticity of blood and blood coagulation process.
Governing system is based on Navies-Stokes equations. The blood coagulation model consists of
interaction 28 chemical constituents described by advection-diffusion equations. All mentioned
models are implemented in OpenFOAM CFD toolbox.
Introduction
Coagulation is a complex process by which blood forms clots. It is an important part of hemostasis
(the cessation of blood loss from a damaged vessel), wherein a damaged blood vessel wall is covered
by a platelet and fibrin-containing clot to stop bleeding and begin repair of the damaged vessel. Disorders of coagulation can lead to an increased risk of bleeding (hemorrhage) or obstructive clotting
(thrombosis). A thrombus, sometimes called blood clot or simply clot, is the final product of the blood
coagulation step in hemostasis. It is achieved via the aggregation of platelets that form a platelet plug,
and the activation of the coagulation system (i.e. clotting factors). A clot is normal in cases of injury,
but pathologic in instances of thrombosis. In this study we perform a numerical simulation in injured
vessel watching the clot growth and clot lysis due to the blood chemical reactions. The original study
of the blood coagulation model was published in [1].
It is assumed the flow is laminar. Mathematical model is based on incompressible Navier-Stokes
equations which are generalized to take into account viscoelasticity and shear-thinning properties of
blood flow. The blood coagulation model consists of chemical reactions of 28 constituents described
by advection-diffusion equations. The model used to capture viscoelastic properties of the blood flow
is the generalized Oldroyd-B model, more details can be found in [2]. The model used to capture
shear-thinning properties of blood is Modified Cross Model, more details can be found in [4]. The
numerical method used for solution of the system of equations is based on the Finite Volume discretization. The computational test case is based on straight channel geometry imitating idealized
blood vessel.
Mathematical model
The governing system of equations is based on Navier-Stokes equations using Johnson-Segalman
model for stress tensor. The system of equations can be written in the following general form:
div u = 0
du
ρ
= divT − ∇p
dt
δD δT
= 2µ(γ̇) D + λ2
T + λ1
δt
δt
(1)
(2)
(3)
Here T is the stress tensor, D is symmetric part of the velocity gradient, γ̇ is the shear rate and λ1 and
λ2 denote the relaxation- resp. retardation time. Stress tensor T can be splitted into two parts:
T = Ts + Te
Ts = 2µs (γ̇)D
δT
Te + λ
= 2µe D
δt
(4)
(5)
(6)
Where Ts is solvent part of stress tensor that corresponds to Stokes law for Newtonian fluid. Te is
viscoelastic (extra stress) part of stress tensor. Both parts can be solved separately.
Viscoelasticity contribution
Viscoelastic part of stress tensor Te is a symmetric tensor of second order (as well as T and Ts ) therefore six components (in three dimensions) must be computed. Extra stress tensor can be evaluated
from the following equation:
∂Te
2µe
1
+ (u · ∇)Te =
D − Te + (WTe − Te W) − a(DTe + Te D)
∂t
λ
λ
(7)
Where model constants are: µe = 0.004 P a · s λ = 0.06s, a = 1.0, D & W are symmetric and
antisymmetric parts of velocity gradient. More details about extra stress equation can be found e.g.
in [3].
Shear-Thinning viscosity model contribution
For evaluation of the variable viscosity was used Modified Cross Model, where the viscosity decreases
from µ0 to µ∞ depending on shear-rate. Model parameters are obtained by fitting an experimental
data [4]. The Modified Cross Model is given by formula:
#
"
s
1 X 2
1 ∂ui ∂uj
1√
1
D:D=
di,j , D = (
+
)
, γ̇ =
µs (γ̇) = µ∞ + (µ0 − µ∞ )
m
a
[1 + (αγ̇) ]
2
2 i,j
2 ∂xj
∂xi
(8)
where: µ0 = 0.16 P a · s, µ∞ = 0.0036 P a · s, α = 3.736 s, m = 2.406, a = 0.254. More information
about blood viscosity models can be found e.g. in [4].
Blood coagulation model contribution
When the blood coagulation model is applied, the clot growth and clot lysis is modeled by a local
viscosity changes. The solvent viscosity can locally increase due to fibrin production to simulate the
clot. The time evolution of all 28 constituents are described by advection-diffusion:
dCi
= div (Di ∇Ci ) + Ri
dt
i = 1..28
(9)
Where Ri are source terms specific for all of 28 constituents, Di are diffusion coefficients.
The viscosity increase mimics the increase in fibrin concentration. There is a limiter applied on
viscosity growth, the viscosity can increase only to certain level e.g.:
C ibrin
µ limiter: µlocal = min(µlocal , Q · µ)
µlocal = Cffibrin
0
where increase factor Q is of the order of hundreds (still in developing phase). High viscosity at the
places with high fibrin concentrations simulate the clot (clot is, where the fibrin concentration exceeds
350 nM). Clotting surface is an area on the vessel boundary which is being damaged (simulating
injury). At the clotting surface non-Homogeneous Neumann boundary conditions for five selected
constituents are used:
∂IXa
∂x
∂IX
∂x
∂Xa
∂x
∂X
∂x
∂tP A
∂x
k7,9 · IX · T F V IIa L
K7,9M + IX
DIXa
k7,9 · IX · T F V IIa L
=
K7,9M + IX
DIX
k7,10 · X · T F V IIa L
= −
K7,9M + X
DXa
k7,9 · X · T F V IIa L
=
K7,9M + X
DX
= −
(10)
(11)
(12)
(13)
= −(kCtP A + kIIatP A · e−134.8·(t−T0 ) · IIa + kIatP A · Ia) · ENDO ·
L
DtP A
(14)
where L is characteristic length, surface concentration ENDO = 2.0 · 109 cells/m2 , Time evolution
of the surface concentration T F V IIa is based on an experimental data approximation, see [1].
Concentration T F V IIa is modeled using following formula:
2
2
T F V IIa = (kT F 7a · 10−15 ) · (93.93 · e(−((t−465.8)/123.4) ) + 58.66 · e(−((t−765.5)/352.2) ) )
Test case
The geometry of the test case is a straight channel or tube, to imitate an idealized blood vessel:
(a)
let/outlet
view
in-
(b) side view
(c) 3d channel
Figure 1: 3d tube, rectangular zone at the wall indicates the clot surface
(15)
(a) Fibrin concentration time development
(b) Fibrinogen concentration time development
(c) Thrombin concentration time development
(d) Prothrombin concentration time development
Figure 2: Selected concentrations at three selected points, 1 - geometrical center of the clot, 2 - point
in the middle of leading edge of the clot, 3 - point at the end of the clot, all three points are in one
single line
The cloting surface at the tube boundary is indicated in the Figure 1. It is a rectangular part of the
tube wall, asymmetrically placed closer to the inlet of the domain.
Boundary conditions are following: Five of twenty eight chemical constituents Ci have a special
treatment, at the clot surface, the time dependent non-homogeneous Neumann boundary conditions
are applied: ∂Ci /∂n = f(Ci ,t). For the rest of constituents homogeneous Neumann (∂Ci /∂n = 0 ) is
applied everywhere. Velocity U is zero at the walls and constant at the inlet. Pressure p is constant at
the outlet of the domain. Components of extra stress tensor Te are zero at the inlet.
The diameter of the tube is one centimeter, the length is 5.5 cm. The increase factor Q is 100. The
mean inlet velocity is 10 cm/s. The density of flowing blood is 1060 kg/m3. The computational grid
is of 25 000 cells.
Conclusion
Set of figures 2 shows time development of selected concentrations in three selected points of the
geometry. All three points are placed on the clotting surface according to Figure 1. The growth
and lysis of the clot were clearly observed. Set of Figures 4 shows time development of velocity
Figure 3: Time evolution of the clot volume in mm3
magnitude. On the right hand side of intersections one can observe the velocity decreases, which
clearly indicates the existence of the clot. Figure 3 shows the time development of the size of the clot.
The maximal clot size is at time t = 1200 s and clot disappears at time t = 2600 s, both times have
good agreement with [1].
The demands on the CPU power are quite high. In total, 38 differential equations are solved. The
model is still in experimental stage of development.
References
[1] M. Anand, K. Rajakopal, K. R. Rajakopal, : A model for the formation, growth, and lysis of clots in quiescent plasma. A comparison between the effects of antithrombin III deficiency and protein C deficiency,:
Journal of Theoretical Biology 253 (2008) 725-738.
[2] G. P. Galdi, R. Rannacher, A. M. Robertson, S. Turek, :Hemodynamical Flows - Modeling, Analysis and
Simulation, vol. 37 of Oberwolfach Seminars, Birkäuser, 2008.
[3] T. Bodnár , A. Sequeira, :Computational and Mathematical Methods in Medicine 9, 83 - 104 (2008).
[4] Cho, Y.I., Kensey, K.R., :Effects of Non-Newtonian Viscosity of Blood on Flows in Diseased Arterial
Vessel, Part 1: Steady Flows, vol.28 (1991), pp. 41-262.
(b) T = 0 s
(c) T = 200 s
(d) T = 300 s
(e) T = 400 s
(f) T = 500 s
(g) T = 600 s
(h) T = 700 s
(i) T = 800 s
(j) T = 900 s
(k) T = 1000 s
(l) T = 1100 s
(m) T = 1200 s
(n) T = 1400 s
(o) T = 1600 s
(p) T = 1800 s
(q) T = 2000 s
Figure 4: Time development of velocity magnitude in cross-section at point 1
Appendix
Constant name
Symbol
SI Unit
Value
h11L1
h11L1
[0 0 -1 0 -1 0 0]
216.6667;
h11A3
h9
h10
hTFPI
h2
hPC
hPLA
hPCI12a
halphaAP
h12A3
hPCI11a
h8
hC8
h5
hC5
h1
h12
hkalli
H1M
HC8M
HC5M
h11A3
h9
h10
hTFPI
h2
hPC
hPLA
hPCI12a
halphaAP
h12A3
hPCI11a
h8
hC8
h5
hC5
h1
h12
hkalli
H1M
HC8M
HC5M
[0 0 -1 0 -1 0 0]
[0 0 -1 0 -1 0 0]
[0 0 -1 0 -1 0 0]
[0 0 -1 0 -1 0 0]
[0 0 -1 0 -1 0 0]
[0 0 -1 0 -1 0 0]
[0 0 -1 0 -1 0 0]
[0 0 -1 0 -1 0 0]
[0 0 -1 0 -1 0 0]
[0 0 -1 0 -1 0 0]
[0 0 -1 0 -1 0 0]
[0 0 -1 0 0 0 0]
[0 0 -1 0 0 0 0]
[0 0 -1 0 0 0 0]
[0 0 -1 0 0 0 0]
[0 0 -1 0 0 0 0]
[0 0 -1 0 0 0 0]
[0 0 -1 0 0 0 0]
[0 0 0 0 1 0 0]
[0 0 0 0 1 0 0]
[0 0 0 0 1 0 0]
26666.6667;
270000;
5783333.33;
8000000;
11900000;
11;
1600000;
3666.6667;
183.3333;
0.0216667;
2300;
0.0037;
0.17;
0.002833333;
0.17;
25;
0.014166667;
0.01133333;
250000e-9;
14.6e-9;
14.6e-9;
Table 1: Summary of constituents, model parameters, part 2
Constituent name
Symbol
Diffusion coefficients Di
Initial value [M]
Fibrinogen
I
3.10e-11
7.00e-6
Fibrin
Ia
2.47e-11
7.00e-9
Prothrombin
II
5.21e-11
1.40e-6
Thrombin
IIa
6.47e-11
1.40e-9
Factor V
V
3.12e-11
2.00e-8
Factor Va
Va
3.82e-11
2.0e-11
Factor VIII
VIII
3.12e-11
0.70e-9
Factor VIIIa
VIIIa
3.92e-11
0.7e-12
Factor IX
IX
5.63e-11
9.00e-8
Factor IXa
IXa
6.25e-11
9.0e-11
Factor X
X
5.63e-11
1.70e-7
Factor Xa
Xa
7.37e-11
1.7e-10
Factor XI
XI
3.97e-11
3.00e-8
Factor XIa
XIa
5.00e-11
3.0e-11
Factor XII
XII
5.00e-11
5.00e-7
Factor XIIa
XIIa
2.93e-11
5.00e-9
Prekallikrein
PreK
4.92e-11
4.85e-7
Kallikrein
Kalli
4.92e-11
4.85e-9
Tissue Pathway Inhibitor
TFPI
6.30e-11
2.50e-9
Protein C
PC
5.44e-11
6.00e-8
Activated Protein C
APC
5.50e-11
6.0e-11
α1 Antitrypsin
α1 AT
5.82e-11
4.50e-5
α2 Antiplasmin
α2 AP
5.25e-11
1.05e-7
Plasminogen
PLA
4.93e-11
2.18e-9
Plasmin
PLS
4.81e-11
2.18e-6
Protein C1 Inhibitor
C1-INH
4.61e-11
2.41e-6
Antithrombin III
ATIII
5.57e-11
2.41e-6
tissue pathway Activator
tPA
5.28e-11
0.08e-9
Table 2: Summary of constituents, diffusion terms, initial values
Constituent name
Symbol
Chemical reaction source terms ßi
Fibrinogen
I
Fibrin
Ia
Prothrombin
II
Thrombin
IIa
Factor V
V
Factor Va
Va
Factor VIII
VIII
Factor VIIIa
VIIIa
Factor IX
IX
Factor IXa
IXa
Factor X
X
Factor Xa
Xa
Factor XI
XI
Factor XIa
XIa
Factor XII
XII
k1 ·IIa·I
K1M +I
LA·Ia
k1 ·IIa·I
− hH1 ·P
K1M +I
1M +Ia
−k2 ·V a·Xa·II
KdW ·(K2M +II)
k2 ·V a·Xa·II
−h2 · IIa · AT III
KdW ·(K2M +II)
−k5 ·IIa·V
C5 ·AP C·V a
− hH
K5M +V
C5M +V a
hC5 ·AP C·V a
k5 ·IIa·V
− HC5M +V a −h5 · V a
K5M +V
−k8 ·IIa·V III
K8M +V III
hC8 ·AP C·V IIIa
k8 ·IIa·V III
− HC8M +V IIIa −h8 · V IIIa
K8M +V III
−k9 ·XIa·IX
K9M +IX
k9 ·XIa·IX
−h9 · IXa · AT III
K9M +IX
−k10 ·V IIIa·IXa·X
KdZ ·(K10M +IX)
k10 ·V IIIa·IXa·X
−h10 · Xa · AT III − hT F P I · T F P I · Xa
KdZ ·(K10M +IX)
·XIIa·XI
−k11 ·IIa·XI
+ −kK12a
K11M +XI
12aM +XI
·XIa·AT III−h11L1 ·XIa·α1AT k11 ·IIa·XI
12a ·XIIa·XI
+ kK
+ −h11A3
−hC1Inh−11a ·XIa·C1IN H
K11M +XI
12aM +XI
Factor XIIa
XIIa
Prekallikrein
PreK
Kallikrein
Kalli
Tissue Pathway Inhibitor
TFPI
−hT F P I · T F P I · Xa
Protein C
PC
Activated Protein C
APC
kP C ·IIa·P C
KP CM +P C
kP C ·IIa·P C
−hP C · AP C · α1 AT
KP CM +P C
α1 Antitrypsin
α1 AT
−hP C · AP C · α1AT − h11L1 · XIa · α1 AT
α2 Antiplasmin
α2 AP
−(hP LA · P LA + hα2 AP · XIIa) · α2 AP
Plasminogen
PLA
Plasmin
PLS
Protein C1 Inhibitor
C1-INH
−(hP CI−12a · XIIa + hP CI−11a · XIa) · C1 − IN H
Antithrombin III
ATIII
−(h9 · IXa + h10 · Xa + h2 · IIa + h11A3 · XIa + hAT 3 · XIIa) · AT III
tissue pathway Activator
tPA
0
·Kalli·XII
−k12 ·XIIa·XI
+ −kKkalli
K12M +XII
kalliM +XII
−h12 ·XIIa−hC1Inh−12a ·XIIa·C1IN H ·Kalli·XII
k12 ·XII
+ kkalli
+ −h
K12M +XII
Kkalli +XII
αAP ·XIIa·α2AP −hAT 3 ·XIIa·AT III
−kP reKB ·XIIa·P reK
−kP reKA ·XIIa·P reK
+ KP reKBM +P reK −hkalli · Kalli
KP reKAM +P reK
kP reKA ·XIIa·P reK
P reKB ·XIIa·P reK
+ kK
KP reKAM +P reK
P reKBM +P reK
kP LA ·tP A·P LS
KP LAM +P LS
+
kP LA−12a ·XIIa·P LS
−hP LA · P LA · α2 AP
KP LA−12aM +P LS
−kP LA ·tP A·P LS
KP LAM +P LS
+
−kP LA−12a ·XIIa·P LS
KP LA−12aM +P LS
Table 3: Summary of constituents, reaction terms
Constant name
Symbol
SI Unit
Value
k12a
k12a
[0 0 -1 0 0 0 0] 0.000566667;
k11
k8
k9
k5
k10
k2
k1
kPLA
k12
kPreKA
kPreKB
kPC
kkalli
kPLA12a
K12aM
K11M
K9M
K8M
K5M
KdZ
K10M
KdW
K2M
KPLAM
K1M
KPLA12aM
K12M
KkalliM
KPreKAM
KPreKBM
KPCM
k79
k710
K79M
K710M
kCtPA
kIIatPA
kIatPA
kTF7a
k11
k8
k9
k5
k10
k2
k1
kPLA
k12
kPreKA
kPreKB
kPC
kkalli
kPLA12a
K12aM
K11M
K9M
K8M
K5M
KdZ
K10M
KdW
K2M
KPLAM
K1M
KPLA12aM
K12M
KkalliM
KPreKAM
KPreKBM
KPCM
k79
k710
K79M
K710M
kCtPA
kIIatPA
kIatPA
kTF7a
[0 0 -1 0 0 0 0]
0.00013;
[0 0 -1 0 0 0 0]
3.24;
[0 0 -1 0 0 0 0]
0.1883;
[0 0 -1 0 0 0 0]
0.45;
[0 0 -1 0 0 0 0]
39.85;
[0 0 -1 0 0 0 0]
22.4;
[0 0 -1 0 0 0 0]
59;
[0 0 -1 0 0 0 0]
0.2;
[0 0 -1 0 0 0 0]
0.033;
[0 0 -1 0 0 0 0]
3.6;
[0 0 -1 0 0 0 0]
40.0;
[0 0 -1 0 0 0 0]
0.65;
[0 0 -1 0 0 0 0]
7.25;
[0 0 -1 0 0 0 0]
0.0013;
[0 0 0 0 1 0 0]
2000e-9;
[0 0 0 0 1 0 0]
50e-9;
[0 0 0 0 1 0 0]
160e-9;
[0 0 0 0 1 0 0]
112000e-9;
[0 0 0 0 1 0 0]
140.5e-9;
[0 0 0 0 1 0 0]
0.56e-9;
[0 0 0 0 1 0 0]
160e-9;
[0 0 0 0 1 0 0]
0.1e-9;
[0 0 0 0 1 0 0]
1060e-9;
[0 0 0 0 1 0 0]
18e-9;
[0 0 0 0 1 0 0]
3160e-9;
[0 0 0 0 1 0 0]
270e-9;
[0 0 0 0 1 0 0]
7500e-9;
[0 0 0 0 1 0 0]
780e-9;
[0 0 0 0 1 0 0]
91e-9;
[0 0 0 0 1 0 0]
36000e-9;
[0 0 0 0 1 0 0]
3190e-9;
[0 0 -1 0 0 0 0]
0.54;
[0 0 -1 0 0 0 0] 1.716666667;
[0 0 0 0 1 0 0]
24.0e-9;
[0 0 0 0 1 0 0]
240.0e-9;
[0 2 -1 0 1 0 0]
1.087e-5;
[0 2 -1 0 0 0 0]
1.545e-13;
[0 2 -1 0 0 0 0] 8.4317e-20;
[0 0 0 0 0 0 0]
1.0e4;
Table 4: Summary of constituents, model parameters

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