Jean-Frédéric GerbeauProject-team REO

INRIA Paris-Rocquencourt & Laboratoire J-L. Lions, Paris 6 universityFrance

Computational Methods for Blood flow

Associated Team “Cardio”

Associated Team “Cardio”

Coordinator• Irène Vignon-Clementel

Members• Stanford (CVBRL, Dept Bioengineering & Mech Engng)

C. Taylor, A. Figueroa, N. Xiao, G. Troianowski, J Feinstein, A Marsden, S Shadden, J Kim, R. Raghu

• INRIA: - Project-team REO: J-F. Gerbeau, M. Fernández, I. Vignon-

Clementel, M. Astorino, C. Bertoglio, G. Troianowski- Project-team MACS: D. Chapelle, P. Moireau

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Website: https://idal-siege.inria.fr/cardio/

Associated Team “Cardio”

Similar goals and complementary approaches• Modeling the blood flow in large arteries• Interaction simulation / medical data

Collaboration themes• Boundary conditions (fluid & fluid-structure)• Advance post-processing techniques• Image-based fluid-structure interaction

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Outline

• Surgical planning• Fluid-Structure Interaction in blood flows• Medical Data assimilation / Inverse problems• Viscoelasticity (Rashmi Raghu, Stanford)

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Surgical planning

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Glenn-Fontan surgery• congenital heart disease• multi-step complex

procedure

Glenn FontanNumerical simulations• Patient-specific geometries• Forecast pressure drop, flow split, wall-shear stress

Troianowski, Taylor, Feinstein, Vignon-ClementelStanford/INRIA

http://www.americanheart.org

Internships of A. Birolleau, G. Troianowski

Geometry from 5 patients MRI data

Surgical planning

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Y-graft in general improves:• energy efficiency• hepatic flow distribution• SVC pressure under rest & exercise

Troianowski, Taylor, Feinstein, Vignon-Clementel

Fluid-Structure Interaction

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Aorta (CVBRL, Stanford)

Cardiac valves(Hôpital Laval)

u uσf

σf

u

u

Coupler

Fluid Solid

ρf∂u

∂t |x+ (u−w) ·∇u

− 2µdiv(u) +∇p = 0, in Ωf(t)

divu = 0, in Ωf(t)

ρs∂2d

∂t2− div

F (d)S(d)

= 0, in Ωs

• Many works about fluid boundary conditions... but not that many about wall !

C. Taylor et al. , StanfordAbdominal aorta

Blood flow in aortaExternal tissue support

Moireau, Xiao, Astorino, Figueroa, Chapelle, Taylor, JFG, (Biomech Model Mech 2011)

• Typical b.c. on the external part of the vessel :

σsn = p0n

• A simple and affordable way to model the external tissues :

σsn = −ksd− cs∂d

∂t

DataContours without supportContour with support

Legend:Solid

Fluid Pressure with supportPressure without supportFlow with supportFlow without support

DiastoleSystole

spinespine

spinesp

ine

spinesp

ine

!

!

!

1

2

3

!1

!2

!3

60

80

100

120

140

0.0 0.3 0.90.6

60

80

100

120

140

0.0 0.3 0.90.60

10

20

30

40

0

40

80

120

160

60

80

100

120

140

0.0 0.3 0.90.60

90

180

270

360

!0

!0

(mmHg)

(cc/s)

(mmHg)

(cc/s)

Blood flow in aortaComparison simulation / images

Moireau, Xiao, Astorino, Figueroa, Chapelle, Taylor, JFG, (Biomech Model Mech 2011)

Medical Data Assimilation

Data assimilation• Reduce model uncertainties using observations• Access to “hidden” quantities• Smooth the data

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INRIA CVBRL, Stanford

Data assimilation in FSI

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• Partial observations of X: Z = H(X)

• Uncertainties on the initial condition X0 and the parameters θ

• FSI dynamical system:

BX = A(X, θ) +RX(0) = X0

• Time discretization: Xn+1 = Fn+1(Xn, θ)

• State variable: X=[u, p,df ,d,v]

• Parameters: θ = [Young modulus, viscosity, boundary conditions, . . . ]

Data assimilation

• State estimation through “simple” feedback terms. For example, for velocity measurements:

Moireau, Chapelle, Le Tallec, 2008

• Parameter estimation : reduced Unscented Kalman filter (SEIK) Dinh Tuan Pham, 2001 Moireau, Chapelle, 2009

• With respect to a variational approach: no tangent, no adjoint• Counterpart : as many resolutions as parameters (“particles”)...• ... but easy to run them in parallel

MX + KX = f −γHT (HX − Yobs)

Summary

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• Prediction:

Xn+1− = Fn+1(Xn

+, θn+, Z

n+1,K)θn+1− = θn+

• Correction:

Xn+1+ = X

n+1− + K

n+1X (Zn+1 −H(Xn+1

− ))θn+1+ = θ

n+1− + K

n+1θ (Zn+1 −H(Xn+1

− ))

Luenberger filterfor the state

Kalman-like filter forfor the parameters

Data assimilation

Newton Loop 3d visco-hyperelastic in

large displacement

Newton Loop 3d visco-hyperelastic in

large displacement

Newton Loop 3d visco-hyperelastic in

large displacement

FSI master controlling the messages between

the codes

FSI master controlling the messages between

the codes

FSI master controlling the messages between

the codes

Implementation

Fluid

Observations: Full displacements, normal displacements or surfaces

Structure3D hyperelasticity

Stress on the interface

FSI coupler

interface velocity and displ.

Correction step of the identification process

Data assimilation: parametric estimation

solid disp and vel fluid disp and vel

observationsparameters

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Rp

Rd

Stiffness estimationParameter estimation

• Synthetic data with E1 = 0.5, E2 = 2, E3 = 4MPa• Initial guess: E = 2MPa in the three regions• Observations: wall velocity

• Parameter estimation: Young modulus E in 3 regions

Simulation : C.Bertoglio (INRIA)

• Similar experiment with 5 regions• With noise (10%) and resampling:

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0 0.2 0.4 0.6 0.80

0.5

1

1.5

2

Time [s]Yo

ungs

mod

ulus

[M

Pa]

0 0.2 0.4 0.6 0.8−2

−1.5

−1

−0.5

0

0.5

1

Time [s]

Velo

city

[cm

/s]

PerfectNoised

Simulation : C.Bertoglio (INRIA)

Data assimilation(state only)

Direct similationSimulation: N. Xiao (Stanford)

Example 2State estimation

Future

• Charley Taylor & col. are about to leave Stanford• ...but not the end of the collaboration !

With A. Figueroa & N. Xiao : - inverse problems in full-body-scale 3D vasculature

With A. Marsden (UC San Diego) : - Pulmonary arteries in the Transatlantic Network (Leducq

foundation)- Respiration modeling

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