Inverse characterization of

(1)

Institut Mines-Télé[email protected]

Inverse characterization of regional, nonlinear and

anisotropic properties of arteries

Matthew R BERSI, Chiara BELLINI, Jay D HUMPHREY,

Stéphane AVRIL ([email protected])

(2)

Rupture risk estimation in thoracic aortic aneurysms

Stéphane AVRIL et al.

(3)

Institut Mines-Télé[email protected]@emse.fr

Where do I come from?

PARIS

AUVERGNE RHONE-ALPES MINES SAINT-ETIENNE

First Grande Ecole outside Paris Founded in 1816

euromech585 - 15 Feb 2017 3

Demanget et al., Perrin et al.

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Institut Mines-Télé[email protected]@emse.fr 5

What is an ATAA and why is serious?

euromech585 - 15 Feb 2017

dissection

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Epidemiology statistics

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Incidence : 10.4 per 100 000 persons

Elevated mortality without treatment for acute aortic dissection:

50% in 48 hours / 90% in 3 months BAV patients: ½ develops an ATAA

(7)

Risk management

 The International Registry of Acute Aortic Dissection (IRAD):

among 591 type A aortic dissection, 59% had a diameter <5.5 cm (Pape, 2007)

5.5 cm

Possible complication Possible

stability

Pape et al, Aortic Diameter ≥5.5 cm Is Not a Good Predictor of Type A Aortic Dissection Observations From the International Registry of Acute Aortic Dissection (IRAD), Circulation, 2007

Decision of surgical repair based on a measure if the Maximal Diameter

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Context

 More and more aneurysms are detected at an early stage (incidence >8% for males >65 years old).

 An intervention is recommended if the aneurysm grows more >1cm/year or it is >5.5cm. This represents >90000 interventions per year in Europe and USA

 BUT:

• 25% aneurysms <5.5cm rupture : 15000 deaths!

• 60% of aneurysms >5.5 cm never experience rupture!

 In summary: very high rate of inappropriate decisions and misprogramed surgical interventions!!

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Added value of biomechanics

 New insights on aneurysm rupture mechanisms

• Arterial wall mechanics

• How does it rupture ?

• When ?

 New patient-specific decision making tool

• Patient-specific

• From medical images

Vascops Scotti, 2007

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Recent developments in computational modeling and challenges

Index of Rupture?

Peak Wall Stress

Strength

Finite-element modeling

O. Trabelsi, et al, Patient specific stress and rupture analysis of ascending thoracic aneurysms, J. Biomech. (2015).

G. Martufi, et al, Is There a Role for Biomechanical Engineering in Helping to Elucidate the Risk Profile of the Thoracic Aorta?, Ann. Thorac. Surg. 101 (2016) 390–398.

S. Pasta et al., Constitutive modeling of ascending thoracic aortic aneurysms using microstructural parameters, Med. Eng. Phys. 38 (2016) 121–130.

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SAINT-ETIENNE PROTOCOL

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40 Patients with ATAA

Preoperative dynamic imaging

Dynamic CT Scanner

4D MRI

Collection of intraoperative aortic segment

Mechanical inflation tests

Histological Analysis

2014

2017

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Institut Mines-Télé[email protected]@emse.fr euromech585 - 15 Feb 2017 12

Collection of the samples

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PREPARATION

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Institut Mines-Télé[email protected]@emse.fr euromech585 - 15 Feb 2017

Bulge inflation test

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III) Inflation test device. Speckle pattern is made before starting the

inflation test Circumferential

A x ia l

Romo et al. Journal of Biomechanics -2014.

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Undeformed Deformed

x y

Full-field measurements using sDIC

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(16)

Davis et al. BMMB – 2015.

Davis et al. JMBBM – 2016

Zhao et al. Acta Biomaterialia - 2016

Identification of local material

properties

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Cristina Cavinato, Pierre Badel

Micromechanical quantification

using SHG microscopy

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Rupture profiles

50% of aortas r uptured wi th an angle θ equal to 90 °

Bl ood flow

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Local stress reconstruction

𝐴 ∙ 𝝈 = [𝐵]

𝑑𝑖𝑣 𝝈 + 𝑓 = 0

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(20)

Stress strain analysis in the bulge test

σ

_rup

: Rupture stress, λ

_rup

: Rupture stretch,

Physiological Stress

Stretch Stretch/Stress Curve

Rupture Stress

Physiological elastic modulus

Ca uchy Stre ss (MPa)

1 1.1 1.2 1.3

Physiological Stretch

1.4 1.5 1.6 1.7

Rupture Stretch

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Comparison with other studies

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0 1 2 3 4 5 6

0 10 20 30 40 50 60 70 80 90 100

σ _rup

Age

R.J. Okamoto et al,, Ann. Biomed. Eng. 30 (2002) 624–635.

D.A. Vorp et al,, Ann. Thorac. Surg. 75 (2003) 1210–1214.

C.M. García-Herrera et al., Med. Biol. Eng. Comput. 50 (2012) 559–566.

D.P. Sokolis et al, Med. Biol. Eng. Comput. 50 (2012) 1227–1237 C. Forsell et al, Ann. Thorac. Surg. 98 (2014) 65–71

S. Sugita. Card Eng Tech. 3:1 (2011) 41-51.

J.E. Pichamuthu et al., Ann. Thorac. Surg. 96 (2013) 2147–2154 A. Duprey at al. Acta Biomater 2016)

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Comparison with other studies

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1 1.2 1.4 1.6 1.8 2 2.2

0 10 20 30 40 50 60 70 80 90 100

λ _rup

Age

R.J. Okamoto et al,, Ann. Biomed. Eng. 30 (2002) 624–635.

D.A. Vorp et al,, Ann. Thorac. Surg. 75 (2003) 1210–1214.

C.M. García-Herrera et al., Med. Biol. Eng. Comput. 50 (2012) 559–566..

D.P. Sokolis et al, Med. Biol. Eng. Comput. 50 (2012) 1227–1237 C. Forsell et al, Ann. Thorac. Surg. 98 (2014) 65–71

S. Sugita. Card Eng Tech. 3:1 (2011) 41-51.

J.E. Pichamuthu et al., Ann. Thorac. Surg. 96 (2013) 2147–2154 A. Duprey at al. Acta Biomater 2016)

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Rupture risk estimation

Physiological Stress

Stretch Stretch/Stress Curve

Rupture Stress

E

_physio

= Physiological elastic modulus

Cauch y Stress (MP a)

1 1.1 1.2 1.3

Physiological Stretch

1.4 1.5 1.6 1.7

Rupture Stretch

γ _stress = Physiological stress Rupture stress γ _stretch = Physiological stretch

Rupture stretch

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Correlation between γ _stretch and E _physio

Duprey A, et al. Biaxial rupture properties of ascending thoracic aortic aneurysms. Acta Biomaterialia 2016.

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Distensibility measurements using the dynamic CT scan

 D = (A _max -A _min )/A _min /(P _max -P _min )

 E _physio = 2R/Dh

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Correlation between γ _stretch and E _physio

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R² = 0.7709 R² = 0.6959

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

0 1 2 3 4

in vivo in vitro

Linéaire (in vivo) Linéaire (in vitro)

g ^stretch

E ^physio (MPa)

Trabelsi et al., to be submitted 2017

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Summary

 2 ways of defining rupture:

 PWS – but unknown patient-specific strength

 γ _stretch correlated with in vivo distensibility

Higher distensibility  less risk because the aneurysm can more easily withstand volume variation

C. Martin et al., Acta Biomater. 9 (2013) 9392–9400

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Histological interpretation

 ATAA always manifests with elastin damage

 More and more collagen tends to be recruited in the physiological range

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Patient with

smallest γ _stretch Patient with

largest γ _stretch

elast coll

M.R. Hill et al,, J. Biomech. 45 (2012) 762–771

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State of the art

Disturbed

hemodynamics

Laminar flow

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Institut Mines-Télé[email protected]@emse.fr euromech585 - 15 Feb 2017

Vision

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 Our vision is that the evolution of the strength and of the wall stress of the aorta during the growth of an aneurysm can be predicted on a patient-specific basis by a

computational model.

 We plan to develop this computational model and to validate it on different cohorts

 On the basis of an MRI examination, our computational model, accessible by surgeons as an interactive intuitive user interface, would permit to predict when an aortic

aneurysm is going to reach a critical size or a critical

rupture risk.

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POINT OF VIEW OF MECHANOBIOLOGY

Altered mechanics induce biological responses, including gene

expression, protein activation and cell phenotype

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POINT OF VIEW OF BIOMECHANICS

Macroscopic manifestations ultimately dictate the mechanical functionality and structural integrity of the aortic wall

Martufi et al. Biomech. Model.

Mechanobio. pp. 917–928, 2014.

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CHALLENGES AND OBJECTIVES

Fundamentally understand local structure–function relationships in aortic aneurysms.

This requires an approach permitting:

1. To reconstruct the regional distribution of mechanical properties of the aorta during aneurysm growth.

2. To investigate their correlation with the underlying

microstructure and its evolutions.

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Illustrative

Identification of regional variations of material

properties in aortas

Application to mice models of aortic aneurysms

Clinical applications

Development of

mechanobiological models

Main milestones

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Inverse characterization of regional, nonlinear and

anisotropic properties of arteries

Matthew R BERSI, Chiara BELLINI, Jay D

HUMPHREY, Stéphane AVRIL ([email protected])

(36)

APPROACH

1. Experiments

2. Material model

3. Inverse method

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APPROACH

1. Experiments

2. Material model

3. Inverse method

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TENSION – INFLATION OF MICE AORTAS

Gleason, et al. J. Biomech. Eng.

126(6), p. 787, 2004.

Humphrey , Cardiovascular Solid Mechanics, 2002

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MEASUREMENT OF THE RESPONSE USING DIGITAL IMAGE CORRELATION

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classical panoramic

Genovese. Optics Lasers Eng, 47, p 995-1008, 2009.

Badel et al. CMBBE, 15, p 37-48, 2012.

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PANORAMIC DIGITAL IMAGE CORRELATION - pDIC

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Genovese et al, CMBBE, 14, p. 213- 237, 2011.

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Full-field strain measurement across the outer surface of the artery

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Posterior Posterior Anterior

ANT

POST

Circumferential Green Strain

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APPROACH

1. Experiments

2. Material model

3. Inverse method

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CONSTITUTIVE MODEL

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Bellini, et al., Ann. Biomed. Eng., 42(3), pp. 488–502, 2014

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CONSTITUTIVE MODEL

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FE implementation

Bellini, et al., Ann. Biomed. Eng., 42(3), pp. 488–502, 2014

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PARAMETERS TO BE IDENTIFIED

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(46)

APPROACH

1. Experiments

2. Material model

3. Inverse method

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Inverse approach – traditional approach

Set of materials properties

Resolution of forward problem

Deviation between predictions and measurements minimization

initialization

Identification of material properties

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(48)

Inverse approach – traditional approach

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Set of materials properties

Resolution of forward problem

Deviation between predictions and

measurements minimization

initialization

Identification of material properties 1. Write the weak form of the problem:

𝑅 𝑢, 𝑤, µ = 0 ∀𝑤

2. Discretize and represent as a linear

combination of piecewise bilinear functions 𝑢 ^ℎ = σ 𝑈 _𝑖 𝜑 _𝑖 (x)

𝑤 ^ℎ = σ 𝑊 _𝑖 𝜑 _𝑖 (x) µ ^ℎ = σ µ _𝑖 𝜑 _𝑖 (x)

3. Implement a finite-element implicit

resolution using the BFGS algorithm

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Inverse approach – traditional approach

Set of materials properties

Resolution of forward problem

Deviation between predictions and measurements minimization

initialization

Identification of material properties

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J(µ)= 𝑇 𝑢 − 𝑇(𝑢 ^𝑒𝑥𝑝 ) ² + ^𝛼

2 𝐵(µ)

Oberai et al., Inverse problems, 19, pp. 297-313, 2003

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Set of materials properties

Resolution of forward problem

Deviation between predictions and measurements minimization

initialization

Identification of material properties

1. Use a gradient based method (steepest descent or BFGS)

2. Need to derive the gradient of J with respect to µ at each iteration. With the adjoint method, this requires the resolution of 2 forward problems

3. Very unstable with hyperelastic models: many risks that the forward problems have a poor convergence

Inverse approach – traditional

approach

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Alternative inverse approach:

the virtual fields method

Set of materials properties

3D estimation of

stresses

Deviation to the principle

of virtual power minimization

initialization

Identification of material properties

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Bersi et al., J Biomech Eng, 2016

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A FE model of each artery is built Each artery is meshed with 5x40x25 hexahedrons

A quasi incompressible Neo-Hookean material is set

The measured displacements are assigned on the nodes of the outer surface as BCs

The Dirichlet problem is solved

3D estimation of

stresses

3D estimation of

strains

Alternative inverse approach:

the virtual fields method

(53)

Set of materials properties

3D estimation of

stresses

Deviation to the principle

of virtual power minimization

initialization

Identification of material properties

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Simple application of the constitutive model

for each element

Alternative inverse approach:

the virtual fields method

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We are looking for the material properties at a position x ₀ . 2 virtual fields are considered.

1. A local virtual radial “bulge”: u(x) = [f(x-x ₀ ) /r ² ] e _r

2. Local virtual axial extension: v(x) = f(x-x ₀ ) (z e _z –x/2 e _x –y/2 e _y ) Where f(x-x ₀ ) = exp(-||x-x ₀ || ² /d²)

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Deviation to the principle

of virtual power

Alternative inverse approach:

the virtual fields method

Avril et al. J. Biomech. 43(15), p 2978-2985, 2010 www.camfit.fr

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J =σ σ

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c

₃¹

,c

₃^2,3

min ,c

₃⁴

,𝛼,𝛽 min

c

^𝑒

,c

₂¹

,c

₂^2,3

,c

₂⁴

J u

𝐴 + J v 𝐵

minimization

2 p 

Linear least-squares

Genetic algorithm Resolution:

Alternative inverse approach:

the virtual fields method

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Summary of the inverse approach

Set of materials properties

3D estimation of

stresses

Deviation to the principle

of virtual power minimization

initialization

Identification of material properties

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(57)

RESULTS

1. Validation on control arteries

2. Application to ATAAs

3. Future work on dissected aneurysms

Con tro l

Dissect ion A T A A

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Local estimation of the quality of the minimization

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Examples of identified material parameters

Bersiet al., J BiomechEg, 2016

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Reconstruction of local strain energies

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Reconstruction of local linearized stiffnesses

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Validation against traditional stress strain analysis

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RESULTS

1. Validation on control arteries 2. Application to ATAAs

3. Future work on dissected aneurysms

Con tro l

Dissect ion A T A A

(64)

Results

Fibrillin 1 mgR/mgR Fibulin 4 SMC KO

Control

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(65)

pDIC measurements

Fibrillin 1 mgR/mgR Fibulin 4 SMC KO

dorsal

ventral inflation

6852 CS38

dorsal ventral

𝜆

_𝑧^𝑖𝑣

= 1.31 𝑂𝐷

_𝑚𝑎𝑥

= 3.62 mm

𝜆

_𝑧^𝑖𝑣

= 1.43 𝑂𝐷

_𝑚𝑎𝑥

= 2.93 mm

inflation

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(66)

Fibrillin 1 mgR/mgR Fibulin 4 SMC KO

Full-Field Thickness Estimation using OCT

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wt

(67)

Fibrillin 1 mgR/mgR Fibulin 4 SMC KO

Full-Field Thickness Estimation using OCT

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(68)

• Stored Energy (@ 𝜆 _𝑧 ^𝑖𝑣 and 80 mmHg)

Fibrillin 1 mgR/mgR Fibulin 4 SMC KO

Full-Field Material Parameter Estimation

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(69)

• Circumferential linearized stiffness (@ 𝜆 ^𝑖𝑣 _𝑧 and 100 mmHg)

Fibrillin 1 mgR/mgR Fibulin 4 SMC KO

Full-Field Material Parameter Estimation

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wt wt

(70)

• Axial linearized stiffness (@ 𝜆 _𝑧 ^𝑖𝑣 and 100 mmHg)

Fibrillin 1 mgR/mgR Fibulin 4 SMC KO

Full-Field Material Parameter Estimation

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wt wt

(71)

Fibrillin 1 mgR/mgR

Heterogeneity of the strain energy

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(72)

RESULTS

1. Validation on control arteries 2. Application to ATAAs

3. Future work on dissected aneurysms

Con tro l

Dissect ion A T A A

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Angiotensin-II Infusion Model of AAD

• Male ApoE ^-/- mice at 16-20 weeks of age surgically implanted with subcutaneous osmotic pump lateral to dorsal midline at mid-scapular region

• Maintained constant infusion of ANG-II at a rate of 1000 ng/kg/min

Anterior Posterior

Aortic Dissection

Dissecting Aortic Aneurysm

Suprarenal Descending Thoracic

Posterior Anterior

Intact Aorta

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(74)

Experimental Approach

 Angiotensin-II Infusion Model of AAD

 Timeline

 Peak in macrophage activity at day 4 with dissection occurring between days 1 and 4 followed by associated remodeling from days 4 to 10.

Day 0 4 6 8 10 12 14 21 28

ANG-II Infusion

Saraff et al. (2003)

IHC showing medial infiltration of macrophages after 48 hours

of treatment

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(75)

Material discretization

O

^W_AD

O

_AD^L

Wall Lumen

Thrombus

L

_AD

L

_FL

H

_AD

O

_AD^L

B A

Contours of model through planes A and B:

Segmented 3D model: Clipping planes:

Along A: Along B:

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(76)

Optical Coherence Tomography (OCT) data are available from the experiments

Digital volume correlation

0°

+90°

180°

-90°

OC T Las er

x y

(2D)

Male ApoE -/- 21d ANG-II

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Bulk strain measurement and

identification

(77)

Illustrative

Identification of regional variations of material

properties in aortas

Application to mice models of aortic aneurysms

Clinical applications

Development of

mechanobiological models

FUTURE WORK

(78)

Computer fluid dynamics

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(79)

Acknowledgements

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Funding:

ERC-2014-CoG BIOLOCHANICS

 Olfa Trabelsi

 Aaron Romo

 Jin Kim

 Pierre Badel

 Frances Davis

 Victor Acosta

 Jamal Mousavi

 Solmaz Farzeneh

 Francesca Condemi

 Cristina Cavinato

 Jérôme Molimard

 Baptiste Pierrat

 Miguel Angel Gutierrez

 Oscar Alberto Mendoza

 Ambroise Duprey

 Jean-Pierre Favre

 Jean-Noël Albertini

 Salvatore Campisi

 Chiara Bellini

 Matthew Bersi

 Jay Humphrey

 Katia Genovese

 Jia Lu

 Nele Famaey

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