Mathematical Assessment of Arterio-Venous Malformations Embolisation
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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
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Cardiovascular Loop
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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
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Motivation
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Motivation
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1D Network Mathematical
model of circulation
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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.
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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
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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
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Vessel elasticity modeling
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AVM Haemodynamic
s with 1D
Approach
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AVM
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1D AVM scheme
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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
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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
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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
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1D AVM scheme
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Pressure embolisation quality
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Velocity embolisation quality
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P-U diagrams before and after surgical embolisation
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P-U diagrams before and after surgical embolisation
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Q-E diagrams before and after surgical embolisation
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Q-E diagrams before and after surgical embolisation
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ArteriesBefore surgery
BeforeAfter
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2 4 6 8
2
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5 6
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20 40 60 80 100 120 140
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60
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V-P
Q-E
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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
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ArteriesAfter
surgery
BeforeAfter
V-P
Q-E
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Thank You!