using 2D and 3D full-field measurements

(1)

Identification of regional, nonlinear and anisotropic material properties in soft tissue biomechanics

using 2D and 3D full-field

measurements

(2)

Institut Mines-Télé[email protected]

PARIS

AUVERGNE RHONE-ALPES

MINES SAINT-ETIENNE

First Grande Ecole

outside Paris

Founded in 1816

(3)

Computational mechanics in the OR for vascular surgery?

www.predisurge.com

(4)

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

Arterial biomechanics and mechanobiology –

Position of

photomechanics

Stéphane Avril - 2018 March 21 - Toulouse Photomechanics 4

(5)

Adventitia Media Intima

Collagen

Endothelial Cells Layering

SMC

Elastin Collagen

EC

BL

Recruited SMC

EC BL

DEVELOPMENT

MATURITY Figure G&R1

Schematic representation of aortic structure

(6)

Basics of aortic mechanics

Stéphane Avril - 2018 March 21 - Toulouse Photomechanics Humphrey JD (2002) Cardiovascular Solid Mechanics: Cells, Tissues, and Organs, Springer-Verlag, NY

6

(7)

Functional biomechanical behavior

(8)

Material characterization and constitutive modeling

(9)

MEASUREMENT OF THE RESPONSE USING DIGITAL IMAGE CORRELATION

classical panoramic

(10)

Institut Mines-Télé[email protected]@emse.fr Stéphane Avril - 2018 March 21 - Toulouse Photomechanics

Bulge inflation test

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 Cristina Cavinato’s talk.

10

(11)

Davis et al. BMMB – 2015.

Davis et al. JMBBM – 2016

Zhao et al. Acta Biomaterialia - 2016

Identification of local material

properties

(12)

Understanding aneurysm growth using

mechanobiology and photomechanics

(13)

Prediction of risk of rupture and dissection

dissection

(14)

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!!

Stéphane Avril - 2018 March 21 - Toulouse Photomechanics

** 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

14

(15)

Illustrative

Clinical applications

TOWARDS ATAA GROWTH

PREDICTIONS

(16)

Altered mechanics induce biological responses, including gene expression, protein activation

and cell phenotype

(17)

APPROACH

1. Experiments

2. Material model

3. Inverse method

(18)

Study Design

Fibrillin 1 mgR/mgR Fibulin 4 SMC KO

Control

(19)

The pDIC technique

Posterior Anterior

(20)

pDIC measurements

Fibrillin 1 mgR/mgR Fibulin 4 SMC KO

dorsal

ventral inflation

6852 CS3 8

dorsal

ventral inflation

(21)

Measurement of bulk deformation fields by Digital Volume Correlation

on OCT images

OC T Las er

(22)

OCT-DVC applied to arterial mechanics

(23)

OCT-DVC validation – Rigid Motion

Digital Volume Correlation

50 mm

100 mm

150 mm

200 mm

250 mm

Original tif files*

50 mm

Mask Definition

Original Image

Final

Mask

(24)

Institut Mines-Télé[email protected]@emse.fr Stéphane Avril - 2018 March 21 - Toulouse Photomechanics 24

Image Processing Methodology – Rigid Motion

Title: REF

Width: 12.8125 inches (1230 pixels) Height: 5.3333 inches (512 pixels) Depth: 1.0417 inches (100 pixels) Size: 60MB

Resolution: 96 pixels per inch

Voxel size: 0.0104x0.0104x0.0104 inch^3 (2.44um)

Original

Image Parameters

50 mm

100 mm

150 mm

200 mm

250 mm

• Algorithmic Mask – Threshhold: 30

• Correlation window size [voxel]: 24 (8x8x8)

• Overlap : 75%

• Passes : 2

• Required valid voxel per window: 45%

(25)

Rigid Motion Results

Strains (xx, yy, xy) - Displacements

Infinitesimal Strain

Green Strain

(26)

OCT-DVC applied to arterial mechanics

(27)

OCT-DVC applied to arterial mechanics

(28)

OCT-DVC applied to arterial mechanics

(29)

Image Processing Methodology – Inflation Test

Title: REF

Width: 667 pixels Height: 640 pixels Depth: 100 pixels

Voxel size: 1x1x1 pixel^3 (2.38um)

Pressures: 20, 40, 60, 80, 100, 120, 140 Lambda (λ) 1 – 2 – 3

Original

Image Parameters *Original raw files

• Algorithmic Mask – Threshhold: 48

• Correlation window size [voxel]: 24 (8x8x8)

(30)

Comparison OCT-DVC with pDIC - Displacements

Inflation Test Results – Sample 81 – Lambda 2

20 – 40 mmHg

(31)

Comparison OCT-DVC with pDIC - Displacements

Inflation Test Results – Sample 81 – Lambda 2

20 – 140 mmHg

(32)

APPROACH

1. Experiments 2. Material model 3. Inverse method

(33)

CONSTITUTIVE MODEL

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

(34)

CONSTITUTIVE MODEL

FE implementation

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

(35)

PARAMETERS TO BE IDENTIFIED

(36)

APPROACH

1. Experiments 2. Material model 3. Inverse method

(37)

Inverse approach – traditional approach

Set of materials properties

Resolution of forward problem

Deviation between predictions and minimization

initialization

Identification of material properties

measurements

(38)

Inverse approach – FEMU approach

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

(39)

Inverse approach – FEMU approach

Set of materials properties

Resolution of forward problem

Deviation between predictions and minimization

initialization

Identification of material properties

J(µ)= 𝑇 𝑢 − 𝑇(𝑢 ^𝑒𝑥𝑝 ) ² + ^𝛼 𝐵(µ)

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

measurements

(40)

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 – FEMU approach

(41)

Alternative inverse approach:

the virtual fields method

Set of materials properties

3D estimation of

stresses

Deviation to the principle

of virtual minimization

initialization

Identification of material properties

Full-field strain

measurements

(42)

Set of materials properties

3D estimation of

stresses

Deviation to the principle

of virtual power minimization

initialization

Identification of material properties

Simple application of the constitutive model

for each element

Full-field stress reconstruction

Full-field strain

measurements

(43)

J =σ σ

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 Resolution:

Minimization of the equilibrium gap using the principle of virtual power

Bersi et al., J Biomech Eng, 2016

(44)

How to obtain 3D full-field strain measurements?

Set of materials properties

3D estimation of

stresses

Deviation to the principle

of virtual power minimization

Identification of material properties

Full-field strain measurements

Posterior Anterior ???

surface

3D

(45)

Tricubic Hermite geometric decomposition

𝑋(𝜌, 𝜃, ζ) = ෍

𝑖=1 4

෍

𝑗=1 4

෍

𝑘=1 4

𝐴 _𝑖𝑗𝑘 ψ _𝑖 (𝜌)ψ _𝑗 (𝜃)ψ _𝑘 (ζ)

where ψ _𝑖 is the Hermite cubic basis Reference

element

𝑌(𝜌, 𝜃, ζ) = ෍

𝑖=1 4

෍

𝑗=1 4

෍

𝑘=1 4

𝐵 _𝑖𝑗𝑘 ψ _𝑖 (𝜌)ψ _𝑗 (𝜃)ψ _𝑘 (ζ)

𝑍(𝜌, 𝜃, ζ) = ෍

𝑖=1 4

෍

𝑗=1 4

෍

𝑘=1 4

𝐶 _𝑖𝑗𝑘 ψ _𝑖 (𝜌)ψ _𝑗 (𝜃)ψ _𝑘 (ζ) ζ

𝜌

𝜃

(46)

3D reconstruction of inner and outer surface from OCT + thickness measurement

(47)

Alignment pDIC with OCT

pDIC points OCT points picked on outer surface

OCT points after

rigid registration

(48)

Tricubic Hermite geometric decomposition in the reference configuration

𝑋(𝜌, 𝜃, ζ) = ෍

𝑖=1 4

෍

𝑗=1 4

෍

𝑘=1 4

𝐴 _𝑖𝑗𝑘 ψ _𝑖 (𝜌)ψ _𝑗 (𝜃)ψ _𝑘 (ζ)

𝑌(𝜌, 𝜃, ζ) = ෍

𝑖=1 4

෍

𝑗=1 4

෍

𝑘=1 4

𝐵 _𝑖𝑗𝑘 ψ _𝑖 (𝜌)ψ _𝑗 (𝜃)ψ _𝑘 (ζ)

𝑍(𝜌, 𝜃, ζ) = ෍

𝑖=1 4

෍

𝑗=1 4

෍

𝑘=1 4

𝐶 _𝑖𝑗𝑘 ψ _𝑖 (𝜌)ψ _𝑗 (𝜃)ψ _𝑘 (ζ)

3D mesh at 80 mmHg

and in vivo axial stretch

(49)

Tricubic Hermite geometric decomposition at any pressure and in vivo axial stretch

𝑥(𝜌, 𝜃, 𝑧) = ෍

𝑖=1 4

෍

𝑗=1 4

෍

𝑘=1 4

𝑎 _𝑖𝑗𝑘 ψ _𝑖 (𝜌)ψ _𝑗 (𝜃)ψ _𝑘 (ζ)

𝑦(𝜌, 𝜃, 𝑧) = ෍

𝑖=1 4

෍

𝑗=1 4

෍

𝑘=1 4

𝑏 _𝑖𝑗𝑘 ψ _𝑖 (𝜌)ψ _𝑗 (𝜃)ψ _𝑘 (ζ)

𝑧(𝜌, 𝜃, 𝑧) = ෍

𝑖=1 4

෍

𝑗=1 4

෍

𝑘=1 4

𝑐 _𝑖𝑗𝑘 ψ _𝑖 (𝜌)ψ _𝑗 (𝜃)ψ _𝑘 (ζ) At any pressure

And any in vivo axial stretch

As the pDIC provided only surface

measurement, we cannot decompose x, y

(50)

We find the 𝜌 decomposition of x, y and z that provides the minimal elastic energy for an incompressible Neo-Hookean material

(51)

For all the pressures for which we have

pDIC measurements

(52)

3D estimation of

strains

3D estimation of

stress

Derivation of stress tensor using layer

specific constitutive behavior

(53)

Virtual power 1

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

After deriving virtual strains (infinitesimal) local internal virtual work is derived at every Gauss point and integrated across the volume

The virtual field is normalized such as the

(54)

Virtual power 2

After deriving virtual strains (infinitesimal) local internal virtual work is derived at every Gauss point and integrated across the volume

The virtual field is normalized such as the external virtual work equals F.

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

(55)

J =σ σ

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 Resolution:

Minimizing the equilibrium gap

(56)

Similar to material fitting at every position

P

Radius x x

x x x x

x

x x

x

F

P x x x

x x x x x x x

x x x

x

x x x x

x

x x

x x x

x x

x x x x

x

x x x x x

x x

Crosses represent external virtual work for every pressure and axial stretch Solid lines represent internal virtual work

The goodness of fit is evaluated with the R² value

(57)

Summary of the inverse approach

Set of materials properties

3D estimation of

stresses

Deviation to the principle

of virtual minimization

initialization

Identification of material properties

Full-field strain measurements

3D

(58)

Amendments to the inverse method for thick arterial walls

Set of materials properties

3D estimation of

stresses

Deviation to the principle

of virtual power minimization

initialization

Identification of material properties

Full-field strain

measurements

(59)

Set of materials properties

3D estimation of

stresses

Deviation to the principle

of virtual minimization

initialization

Material properties

Full-field strain measurements

In addition to the constitutive equations of the wall shown previously we

Identification of material properties

Thrombus

(60)

Measurement of bulk deformation fields by Digital Volume Correlation

on OCT images

OC T Las er

(61)

Amendments to the 3D tricubic decomposition

Full-field strain measurements

Set of materials properties

3D estimation of

stresses

Deviation to the principle

of virtual minimization

The information in 𝜌 cannot be 𝜌 ζ

𝜃

(62)

𝜌 ζ

𝜃

3D tricubic decomposition combining OCT-DVC and pDIC

After performing rigid registration of OCT against pDIC reconstructed geometries in the reference configuration,

The nodal values on the outer surface are derived from 3D bicubic

decomposition of the pDIC nodal positions

The remaining components of the 3D bicubic decomposition in 𝜌 and 𝜃 are derived by fitting the OCT-DVC voxel positions

The remaining ζ decomposition is

assumed to minimize cross section

warping

(63)

Stretch at p=20 mmHg

The reference configuration is set to:

P=80 mmHg Lambda in vivo

(64)

Stretch at p=140 mmHg

The reference configuration is set to:

P=80 mmHg Lambda in vivo

(65)

Results - Highlights

(66)

Full-Field Material Parameter

Estimation vs thickness distribution

(67)

Full-Field Material Parameter

Estimation vs local stress

(68)

Correlation with tissue µstructure

(69)

Vision

 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.

(70)

Some directions for photomechanics in

vascular mechanobiology

(71)

Non invasive reconstruction of in vivo stiffness distribution

Gated CT scan Stiffness map

(72)

Institut Mines-Télé[email protected]@emse.fr Stéphane Avril - 2018 March 21 - Toulouse Photomechanics

using 2D and 3D full-field measurements

Identification of regional, nonlinear and anisotropic material properties in soft tissue biomechanics