1

1D-0D coupled algorithms for haemodynamical modeling

Sergey Simakov Timur Gamliov

Moscow Institute of Physics and Technology

Moscow, INM, 29.10.2014

6th Russian Workshop on Mathematical Models and Numerical Methods in Biomathematics and

4-th International workshop on the multiscale modeling and methods in biology and medicine

2

Review

1D+0D coupled models

Vessel wall elasticity modeling remark

Blood flow + Heart

Respiratory flow + alveolar volume

3

Multiscale approaches in haemodynamics

0D Electric circuits 3D flows1D-3D

4

Global blood flow + heart

5

1D vascular network

6

Global blood flow

0uSS

t x

02

020

0

2,16 ... ,

,2

S SP S Su u u S SS S 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 junctions3.1

3.2

Compatibility conditions discretisation along outgoing characteristics

3.31 1 1 1n n n n

k k k ku S 2 1N equations

2 1N

7

Boundary conditions at junctions

,k k kV S u

,k k kg

1k k

kk

k k

u SFA PV u

S

0ki k kiW A E

k kk

V F gt x

k k kki ki ki ki kdV V VW W W gdt t x

8

Boundary conditions at junctions

1, , , , 1

,2 ,2 ,2

1,1 ,1 ,2 ,11 1 1

,1 ,1 ,1

n n n nk M k M k M k MM M M

k k k kk

n n n nk k k k

k k k kk

V V V VW W g

h

V V V VW W g

h

1 1, , , 1 , , 1 ,

1 1,1 ,1,2 ,1,2 ,1

n n n nk M k M M k M M k M

n n n nk k k k

S u

S u

,k k kV S u

,k k kg

0( 1)iki k k kPu c SS

0

( 1)

kki

ik

PcSWS

9

Boundary conditions at junctions

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

10

Heart model

Isovolumetric contraction (0.08 s), Ejection (0.293 s), Isovolumetric relaxation (0.067 s), Ventricles filling (0.56 s)

2

2 ( ) ( ), 1...4j j j extj j j j

j

d V dV VI r p t P t jdt dt c

ijij j i

ij

Q p pr

1 51 51 14 14

2 62 23 23

3 37 37 23 23

4 48 14 14

V Q Q

V Q Q

V Q Q

V Q Q

Mass conservation

Volume averaged chamber motion Left auricle

Left ventricle

11

Boundary conditions at heart junctions

Arteries:

Veins:

,0 ,0 , , 5,1 , 6,2

i ij k ki i

k k ijij

p t p Su t S t Q t i j

r

, , , , 3,7 , 4,8

j jk k ij j

k k k k ijij

p S p tu t L S t L Q t i j

r

Discretisation of compatibility conditions

1 1 1 1n n n nk k k ku S

12

Next step with 1D

51 37 48 26, , ,y V Q Q Q Q

551 37 48 26 6 7 8, , , , , ,new new new newQ Q Q Q S S S S

5,6,7,8max

i i

new old

iS S

5 6 7 8, , ,S S S S

5 6 7 8, , , ,y A t y B S S S S

1.

2.

3.

4.

5.

6.

Boundary conditions at heart junctions

13

Respiratory flow + alveolar

volume

14

Thanks to Yura Ivanov

1D trachea-bronchial tree

15

1 ,...,

0, 1

,M

m mk k k k

k k k

node m mk m k k k k

u S

p t x p t R u S

Junction:

1 ,0 Tp t p t

Nasopharynx:

Gray’s Anatom

y

Gray’s Anatom

y

1D trachea-bronchial tree

Discretisation of compatibility conditions1 1 1 1n n n n

k k k ku S

16

, , alk k k k

k

dVu t L S t L

dt

Coupling with alveolar volume

1 1 1 1n n n nk k k ku S

2

2extalv alv alv

alvd V dV VI r p P tdt dt c

alv kp P S

17

1D+0D potential

1) Aneurisms modeling in global circulation: 1D + 0D + 1D

2) 1D – 3D junction: 1D + 0D + 3D + 0D + 1D

18

Vessel wall elasticity modeling

remark

19

Boundary conditions at junctions

1, , , , 1

,2 ,2 ,2

1,1 ,1 ,2 ,11 1 1

,1 ,1 ,1

n n n nk M k M k M k MM M M

k k k kk

n n n nk k k k

k k k kk

V V V VW W g

h

V V V VW W g

h

1 1, , , 1 , , 1 ,

1 1,1 ,1,2 ,1,2 ,1

n n n nk M k M M k M M k M

n n n nk k k k

S u

S u

,k k kV S u

,k k kg

0( 1)iki k k kPu c SS

0

( 1)

kki

ik

PcSWS

? ?PP SS

20

Elasticity: analytic approximation

exp 1 1, 1

ln , 1f S

2,extP S P t x c f S

Kholodov

Quarteroni, Formaggia 1f S

Olufsen, Peskin 1 1f S

Toro, Mueller , 0, 2,0m nf S m n

Favorskii, Mukhin f S

Pedley, Luo

32

1, 1

1 , 1f S

…

0S S

21

Elasticity: qualitative analysis and modeling

Pedley, Luo

Holzapfel, GasserMultilayer elasticity simulations -> S-like curve

Collapsible tubes study in lab -> S-like curve

Still not included: viscoelasticity, autoregulation …

22

Elasticity modeling

T

,f Ts

* * *

*

, ,

0,

T R R

R

1) Tension in deformable fiber

2) Density of elasticity force

3) Tansmural pressure

for collagen fibers

* 1R

* 1R otherwise

,p f n h

Peskin, Rosar 2001

Xs

Implemented by Vassilevski, Ivanov, Salamatova 2011

23

Elasticity modeling

Vassilevski, Ivanov, Salamatova

24

Thank You!