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1
Mathematical Assessment of Arterio-Venous Malformations
Embolisation
1Moscow Institute of Physics and Technology2Novosibirsk State University
3Institute of Hydrodynamics SD RAS
Simakov1 S.S., Gorodnova1 N.O.,Chupakhin2,3 A.P., Khe2,3 A.K.
Workshop on mathematical models and numerical methods in
biomathematics 2013
2
Cardiovascular Loop
3
Motivation
AVM embolisation:
• Mortality (1.5-3%)
• Disability (3.8-7%)
• Postoperative risk (5% during the next 5 years)Without treatment:
Stroke at the age of 30-40 years
4
Motivation
5
Motivation
6
1D Network Mathematical
model of circulation
7
Global blood flow
0
uSS
t x
02
020
0
2,
16 ... ,,2
S SP S Su u
u S SSS St x Sd
S S
1) Mass balance
2) Momentum balance
1 ,...,
0, 1M
M Mk k k k
k k k
u S
, , 0,node mk k k M k k k k k kp S x p R u S x L
3) Boundary conditions at junctions
3.1
3.2
Compatibility conditions along outgoing characteristics (finite difference discretisaztion)
3.31 1 1 1n n n n
k k k ku S 2 1N
equations
2 1N equation
s
Kholodov 2001,et. al.
8
Boundary conditions: vessels junction
1 1 1 1 11 1 1 1 1 1 1
1 1 1 1 12 2 2 2 2 2 21
1 1 1 1 1
( )... ...
n n n n n
n n n n nn
n n n n nN N N N N N N
S S P S
S S P SD
S S P S
F S R 0
1 1 1 1 1
, , , , , [1, ]N N N N N
j ii k ij ki j j k k
j i j i k i k ik j k j
D R R R i j k N
R R
N Equations set
9
Vessel wall elasticity
Pedley, Luo, 1998
2,extP S P t x c f S
0 0
0 0
exp 1 1,
ln ,
S S S Sf S
S S S S
Analytic approximation
10
Vessel elasticity modeling
11
AVM Haemodynamic
s with 1D
Approach
12
AVM
13
1D AVM scheme
14
Pressure distribution
inp
ut
ou
tpu
t
left
ro
ute
rig
ht
rou
te
inp
ut
left
ro
ute
ou
tpu
t
inp
ut
left
ro
ute
occ
lud
ed
occ
lud
ed
occ
lud
ed
15
Velocity distribution
rig
ht
rou
te
ou
tpu
t
inp
ut
left
ro
ute
occ
lud
ed
occ
lud
ed
occ
lud
ed
occ
lud
ed
occ
lud
ed
16
Relative measures of embolisation quality
, 53,54R refi i
P refi
P Pe i
P
, 53,54R refi i
U refi
U Ue i
U
Pressure embolisation quality
Velocity embolisation quality
17
1D AVM scheme
18
Pressure embolisation quality
19
Velocity embolisation quality
20
P-U diagrams before and after surgical embolisation
21
P-U diagrams before and after surgical embolisation
22
Q-E diagrams before and after surgical embolisation
23
Q-E diagrams before and after surgical embolisation
24
ArteriesBefore surgery
BeforeAfter
12
3
4
5
6
7
2 4 6 8
2
4
6
8
12
34
5 6
7
20 40 60 80 100 120 140
10
20
30
40
50
60
70
V-P
Q-E
25
1
234
2 0 4 0 6 0 8 0 1 0 0 1 2 0
2 0
4 0
6 0
1
23
4
1 2 3 4 5 6
1
2
3
4
5
6
ArteriesAfter
surgery
BeforeAfter
V-P
Q-E
26
Thank You!