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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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