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

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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

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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

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ou

tpu

t

left

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ht

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te

inp

ut

left

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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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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

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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

3

4

5

6

ArteriesAfter

surgery

BeforeAfter

V-P

Q-E

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Thank You!