Mathematical modeling of blood flow in
translational medicine
Simakov Sergey1,2
The work was supported by Russian Science Foundation
Grant No 14-31-00024 (New Laboratories)
Moscow Institute of Physics and Technology
(2) Institute of Numerical Mathematics
Workgroup on modelling blood flow and vascular diseases
(1)
7th Conference on Mathematical Modelling and
Computational Methods in Biomathematics
Moscow, INM RAS, 30.10.2015
1
1D Systemic Blood Flow
Model
2
Global blood flow
1) Mass balance
 S   uS 

 fS
t
x
2) Momentum balance
 u   u2 P  S  
2a)
  
  fu
t  x 2
 
 Q    Q2  S  P
Q
2b)
 
 K R  fQ

t  x S    x
S
3) Junctions
3.1

k  k1 ,..., kM
 kmuk Sk  0,  km  1
node
m
3.2a pk  Sk , xk   pm  k Rk uk Sk , xk  0, Lk
uk2  S k , xk  pk  S k , xk 
3.2b

 const
2

node
p
S
,
x

p


m
3.2c k k k
3
Vessel wall elasticity
Analytic approximation
P  S   Pext  t, x    c2 f  S 
exp  S S0  1  1, S  S0
f S   
ln  S S0  , S  S0
Pedley, Luo, 1998
Modelling

X 
T     


s

 
f 
 T  ,    
s
p  f ,n h
T   R ,   R ,
*
*
 *
  
0,   R*



 



Yu. Vassilevski, S Simakov, V. Salamatova, T. Dobroserdova, Blood Flow Simulation in Atherosclerotic Vascular
4
Network Using Fiber-Spring Representation of Diseased Wall, Math. Mod. In Nat. Phenom., 6(5):333-349, 2011
Physiological conditions:
autoregulation
Wall elasticity adaptation to average pressure
Mechanism:
endothelial layer permeability
NO production
shear stress

S
 
P   c  exp   1  1
 S0
 

2
cnew  Pnew 


cold  Pold 
1
2
Pold
Pnew
T
T
5
Gravity and Autoregulation
Head
S
Auotregulation
Collapsible tube
Leg
S
6
Biomedical
Applications:
 Angiosurgery
 Enchanced External
Counterpulsation
 Fractional Flow Reserve
 Cerebral Flow
7
Angiosurgery
(stenting)
8
Vascular surgery:
stenosis treatment
MRI/CT
Computational model:
• 1D vascular structure
• Functional parameters fitting
(elasticity, resistance)
• Simulations
Ultrasound dopplerography
9
Stenosis treatment: boundary
conditions and identification
Boundary conditions
Input (arteries):
Qin  Qheart (t ),
Output (veins):
Qout  Qheart (t ),
  0.21;
Parameters identification
• Ultrasound measurements (before surgery!)
• Angles between vessels at bifurcations
• Large vessels – rigid walls, small vessels – more elastic
• Occlusion – decreased lumen, high resistance
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Stenosis treatment:
1D vascular domain reconstruction
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Stenosis treatment
Peak blood velocity after treatment
Peak blood velocity before treatment
400
400
300
300
cm/s
cm/s
200
200
`
100
100
0
0
3
4
12
measured
5
7
simulated
9
3
4
12
measured
5
7
9
simulated
Patient-specific MRI and Doppler ultrasound data thanks to
I.M. Sechenov First Moscow State Medical University (Ph.Kopylov, et.al.)
S.Simakov, T.Gamilov, Yu.Vassilevskii, Yu.Ivanov, P.Kopylov, Patient specific haemodynamics modeling12
after occlusion treatment in leg, Mat. Mod. Nat. Phen, 2014
Enhanced External
Counterpulsation
(EECP)
13
EECP review
Indications
• Ischemia
• Arterial Hypertension
• Cardiovascular insufficiency
Risk factors
• Aneurisms rupture
• Atherosclerotic plaques rupture
• Varicose veins
Individual EECP strategy is required!!!
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EECP model
Wall state equation

S
 
P   c  exp   1  1  Padd
 S0
 

2
Padd :
Cardiac cycle
C
0
B
1
systole diastole
A
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Patient-specific EECP
treatment simulations
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EECP optimization:
sensitivity analysis
Impact of myocardial pressure
Impact of arterial stiffness
Impact of autoregulation response rate
T.Gamilov, S.Simakov, Modelling of coronary flow stimulation by enchanced external counterpulsation, Int.
17
J. Num. Met. Biomed. Eng (under review)
Virtual Fractional
Flow Reserve
Assesment
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Fractional Flow Reserve
Review
Clinical measurements require
• endovascular intervention
• expensive transducer
Solution: Virtual FFR assessment basing on
non-invasively collected data
(CT scans, systolic/diastolic pressure, heart rate …)
F.Yu. Kopylov, A.A. Bykova, Yu. V. Vassilevskii, S.S. Simakov, Role of measurement of fractional flow reserve in
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coronary artery atherosclerosis, Terapevticheskii Arkhiv, 87(9):106-113, 2015
Fractional Flow Reserve
1D vascular domain reconstruction
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Fractional Flow Reserve
T.Gamilov, Ph. Kopylov, R. Pryamonosov, S.Simakov, Virtual fractional flow reserve assessment in patient-specific
21
coronary networks by 1D haemodynamic model, Russ. J. Num. Anal. Math. Mod., 2015
Cerebral Flow:
Carotid Stenosis
Treatment
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Cerebral Flow
1D vascular domain reconstruction
Patient A
Patient B
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Cerebral Flow
Before treatment (with stenosis)
Common Carotid Art.
(No 26, 3)
Patient A
Internal Carotid Art.
(No 27, 86)
Common Carotid Art.
(No 13, 36)
Patient B
Internal Carotid Art.
(No 19, 28)
model
(cm/s)
Left
measured
(cm/s)
model
(cm/s)
Right
measured
(cm/s)
error
(%)
error
(%)
50
55
9
51
54
5,5
72
67
7
240
220
10
51
58
12
60
56
7
130
96
35
58
55
5
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Cerebral Flow
After treatment (stenting)
Vessel's index
Velocity, cm/s
Common Carotid Art.
Patient A
3, 26
Patient B
2, 13
Patient A
60
Patient B
59
Norm
50-104
Internal Carotid Art.
27, 86
19, 28
48
60
32-100
External Carotid Art.
74-75, 12-13
29-31, 14-16
60
90
37-105
42, 55
5, 10
50
35
20-61
54-52-50, 56-60-64
40, 3-4, 41-43-46
98
95
60-150
Vertebral Art.
Subcluvian Art.
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Conclusions
• Adequate tool for local and systemic blood flow
analysis is developed
• Successful patient-specific simulations cover




Femoral artery stenting
EECP impact to the blood flow
Virtual FFR assessment
Carotid artery stenting
• Full automatization of the presented algorithm
and validation with more clinical cases are
required to translate this research to the
bedside
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Thank You!
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