Mechanics of soft biological tissues

(1)

Institut Mines-Télécomavril@emse.fr

Mechanics of soft biological tissues

Prof Stéphane AVRIL

PARIS

AUVERGNE RHONE-ALPES

MINES SAINT-

ETIENNE

(2)

Institut Mines-Télécomavril@emse.fravril@emse.fr

OUTLINE

 PART I: computational modelling of soft biological tissues – Application to aortic aneurysm

 PART II: Experimental evaluation of regional variations in in the mechanical properties of soft tissues – Application to vascular mechanobiology

Stéphane Avril - 2020 Oct 12 - MecaBio

2

(3)

OUTLINE

 PART I: computational modelling – Application to aortic aneurysm

 PART II: Experimental evaluation of regional variations in in the mechanical properties of soft tissues – Application to vascular mechanobiology

3

(4)

Soft biological tissues: many challenges for continuum mechanics

Stéphane Avril - 2020 Oct 12 - MecaBio 4

(5)

5

0 0,2 0,4 0,6 0,8 1 Dé formation %

Entropie Ene rgie Inte rne

0 100 200 300 400 500 600 Dé formation %

Ene rgie Inte rne Entropie

enthalpic elasticity (cristal)

entropic elasticity

(biological tissue)

(6)

Essais de Treloar (1944)

0 10 20 30 40 50 60 70

0 1 2 3 4 5 6 7

Lambda-1

C on tr a int e no m ina le

Traction simple Equibiaxiale Traction plane

Treolar’s tests on rubber (1944)

Nom inal str ess F /S0 (MPa )

Stretch λ=L/L ₀

Biaxial tension

Planar tension Uniaxial tension

6 1 2 3 4 5 6 7

Hyperelasticity

(7)

Pseudo-hyperelasticity:

Irreversible effects are neglected…

7 Stretch λ=L/L ₀

1 2 3 4 5 6

No m inal str ess (MPa )

(8)

0 10 20 30 40 50 60 70

0 1 2 3 4 5 6 7

Co n tra in te n o min a le

Lambda-1

Nom inal str ess S =F/A 0 (MPa )

Stretch λ=L/L ₀

8 Stored energy per unit volume

1 𝐴 ₀ 𝐿 ₀ න 𝐹 ሶ𝐿𝑑𝑡 = න 𝑆 ሶ λ𝑑𝑡

1 2 3 4 5 6 7

Strain energy density

(9)

Deformation mapping in 3D

Deformation Gradient

1 2 3

( , , , )

i i i

y   x u x x x t

 

( )

or ⁱ _i _i _ij ⁱ _ij

j j j

y u

x u F

x x  x

    

  

    

  

y x u x F

x u(x)

e

₃

e

₁

e

₂

dx dy

u(x+dx)

Original Configuration

Deformed Configuration

i ik k

d d

dy F dx

 



y F x

9

(10)

e

₃

e

₁

e

₂

l

₀

Original Configuration

Deformed Configuration

m

l

1 1

( ) or ( )

2 2

T E ij F F ki kj  ij

    

E F F I

  ²

2 2

0 2 2

0 0 0

2 2

ij i j

l l l l

E m m

l l l

 

      

m E m

10 Green Lagrange strain

(11)

Nominal/ 1 ^st Piola-Kirchhoff

Material/2 ^nd Piola-Kirchhoff

1 1

ij ik kj

J ^ S JF ^ 

  

S F σ

1 T 1 1

ij ik kl jl

J ^ ^ JF ^  F ^

   

Σ F σ F 

e

₃

e

₁

e

₂

Original Configuration

Deformed Configuration

t

dA

₀

dA n

n

₀

x

u(x)

dP

⁽ⁿ⁾

dP

₀⁽ⁿ⁾

11 True / Cauchy

σ

Stress measures

π π

(12)

0 10 20 30 40 50 60 70

0 1 2 3 4 5 6 7

Co n tra in te n o min a le

Lambda-1

Nom inal str ess S =F/A 0 (MPa )

Stretch λ=L/L ₀

12 Stored energy per unit volume

න 𝑺: ሶ F𝑑𝑡 = න 𝝅: ሶ E𝑑𝑡 = Ψ(E)

1 2 3 4 5 6 7

Hyperelasticity

(13)

13 compressible hyperelastic behaviour

incompressible hyperelastic behaviour (𝐽=1)

 ?



Strain energy density:

𝛔 = 𝐽𝐅. 𝜕Ψ

𝜕𝐄 . 𝐅 ^𝑇

𝛔 = 𝐅. 𝜕Ψ

𝜕𝐄 . 𝐅 ^𝑇 + 𝑐𝐈

(14)

14     ^ el ^ ² ⁱ N

1 i i

N i

1 i

1 0 i 2

1 J 1

D 3 1

I C

J , I ,

I    

  



Polynomials of the first invariant

Initial shear modulus  ₀  2C ₁₀

Initial compressibility modulus

1 0

2 k  D

Ψ E = Ψ(ഥ 𝐼 ₁ , ഥ 𝐼 ₂ , 𝐽)

Ψ(ഥ 𝐼 ₁ , ഥ 𝐼 ₂ , 𝐽) ^𝐽

(15)

15 Particular case 1: Neo-Hookean behaviour (N=1)

    ^ _el ^ ²

1 1

10 2

1 J 1

D 3 1

I C

J , I ,

I    



    ^ el ^ ² ⁱ N

1 i i

N i

1 i

1 0 i 2

1 J 1

D 3 1

I C

J , I ,

I    

  



Polynomials of the first invariant Ψ E = Ψ(ഥ 𝐼 ₁ , ഥ 𝐼 ₂ , 𝐽)

Ψ(ഥ 𝐼 ₁ , ഥ 𝐼 ₂ , 𝐽) ^𝐽

(16)

16 Particular case 2: Yeoh behaviour (N=3)

    ^ el ^ ² ⁱ 3

1 i i

3 i

1 i

1 0 i 2

1 J 1

D 3 1

I C

J , I ,

I    

  



    ^ el ^ ² ⁱ N

1 i i

N i

1 i

1 0 i 2

1 J 1

D 3 1

I C

J , I ,

I    

  



Polynomials of the first invariant Ψ E = Ψ(ഥ 𝐼 ₁ , ഥ 𝐼 ₂ , 𝐽)

Ψ(ഥ 𝐼 ₁ , ഥ 𝐼 ₂ , 𝐽) ^𝐽

(17)

ARRUDA-BOYCE

  _{ }   ^ ^ ^{ } 



 



  



 





 

  el

2 el 5

1 i

i i 2 1

i 2 m

i 2

1 ln J

2 1 J

D 3 1

C I J

, I , I

673750

; 519 7050

; 19 1050

; 11 20

; 1 2 1

5 4

3 2

1  C  C  C  C 

C

Model with 8 chains

Statistical mechanics Gaussian chains





 ₀

k D 2

0 

 

 



  

 







 









 sinh

ln n n

Nk

_chain

0

 

 



  







^

and 3 3

I

₁ ₁ _chain

chain



17 Ψ E = Ψ(ഥ 𝐼 ₁ , ഥ 𝐼 ₂ , 𝐽) Other functions of the first invariant

(18)

18 OGDEN

    ^ el ^ ² ⁱ

N

1 i i

N

1 i

3 2

2 1 i

i 2

1 J 1

D 3 1

J 2 , I ,

I 

ⁱ

 

ⁱ

 

ⁱ

  



 

  





3 2

1 ,  , 

 Principal stretches  _i  J ^ ³ ¹  _i

 

 ^N

i

i 1

0 



1 0

2 k  D

Rivlin Mooney

; 2

N   ₁   ₂    

Hookean Neo

; 2

; 1

N   ₁   

Forms written with the principal stretches Ψ E = Ψ(ഥ 𝐼 ₁ , ഥ 𝐼 ₂ , 𝐽)

(19)

Anisotropy induced by collagen fibers

19

(20)

Anisotropy induced by collagen fibers

20

(21)

21 Anisotropic strain energy density

(Holzapfel et al., 2000)

21

(22)

Micro-fiber approach

22

(23)

Layer-specific constitutive model

Strain-energy function based on the constrained mixture theory

Humphrey & Rajagopal, Math Models Methods Appl Sci. (2002) ; Bellini et al, ABME (2014), Mousavi & Avril, BMMB (2017)

Deposition stretch of each constituent:

Collagen Elastin

SMC

Load

23

(24)

Damage and plasticity

24

(25)

Abaqus finite-element

implementation and verification

 FE software ABAQUS coupled with UMAT

 Hexahedral and tetrahedral elements

 Structural mesh (𝑟, 𝜃, 𝑧)

 Two different layers (media and adventitia)

Mousavi et al, IJNMBE, 2018

25

(26)

Time dependent material properties:

different approaches

26

(27)

Proteolytic injury and tissue adaptation

PROTEOLYTIC INJURY

Strain HEALING

Stress Stress

27

(28)

Growth and remodeling

28

(29)

29 𝐅 _tot ^𝑗 = 𝐅 _elc ^𝑗 𝐅 _gr ^𝑗 𝐅 _gr ^𝑗 = 𝐅 _r ^𝑗 𝐅 _g ^𝑗

𝐅 _r ^𝑗 and 𝐅 _g ^𝑗 should be updated if the artery is not in the homeostatic state

Elastic and inelastic decomposition of deformation gradient

Layer-specific constitutive model

Mousavi & Avril, BMMB (2017), Mousavi et al, BMMB (2019) Ghavamian et al, Front Bioeng Biotech (2020)

Stéphane Avril - 2020 Sept 16 - NUBioEng

29

(30)

Growth and Remodeling in

homogenized constrained mixture

Collagen mass production

ሶϱ ^𝑗 (t) = ϱ ^𝑗 (t)𝑘 _σ ^𝑗 𝜎 ^𝑗 𝑡 − 𝜎 _h ^𝑗

𝜎 _h ^𝑗 + ሶ𝜉 ^𝑗 (𝑡)

Inelastic deformation due to remodeling

Cyron et al, BMBB (2016), Braeu et al, BMMB (2017), Laubrie et al, IJNMBE (2019)

Collagen Elastin

SMC

Load

30

(31)

Simulation of aneurysm progression

Goal: Predict weakening in the aortic wall

31

(32)

32 Assumption

ATAAs are triggered by local proteolytic injury, which induce adaptation in the ascending thoracic aorta

Guzzardi et al, JACC (2014), Condemi et al, IEEE TBME (2019)

32

Stéphane Avril - 2019 July 9 – ESB Vienna

(33)

33 Patient-specific predictions

Growth and remodeling of a two–layer patient–specific human ATAAs due to elastin loss

Mousavi et al, BMMB (2019)

.

Localization function around the point of TAWSS max

33

(34)

34 Growth and remodeling of a two–layer patient–

specific human ATAAs due to elastin loss

Normalized Thickness Small growth parameter

Patient-specific predictions

34 Mousavi et al, BMMB (2019)

(35)

SUMMARY

 Patient-specific numerical model based on the constrained mixture theory including damage and G&R – coupling with CFD analyses, + active role of SMCs

 One of the major effect is mechanoregulation.

 Pressing need to get experimental insight about the regulation of tissue mechanics

35

(36)

OUTLINE

 PART I: computational modelling – Application to aortic aneurysm

 PART II: Experimental evaluation of regional variations in in the mechanical properties of soft tissues – Application to vascular mechanobiology

36

(37)

Institut Mines-Télécomavril@emse.fravril@emse.fr Stéphane Avril - 2020 Oct 12 - MecaBio Adventitia

Media Intima

Smooth Muscle Cells Fibroblasts

Collagen

Endothelial Cells

Elastic Lamina Layering

SMC

Elastin Collagen

EC

BL

Recruited SMC

EC BL

DEVELOPMENT

MATURITY Figure G&R1

What developmental biology tells us?

37

“Here and elsewhere we shall not obtain the best insights into things until we actually see them growing from the

beginning”

Aristotle (384-

322 B.C.)

(38)

APPROACH

1. Experiments 2. Material model 3. Inverse method

(39)

Functional biomechanical behavior

(40)

Study Design

Fibrillin 1 mgR/mgR Fibulin 4 SMC KO

Control

(41)

MEASUREMENT OF THE RESPONSE USING DIGITAL IMAGE CORRELATION

classical panoramic

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

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

41

(42)

DIC tend to be used for all kind of soft tissues

42

(43)

Undeformed Deformed

x y

Principle of DIC

43

(44)

The pDIC technique

Posterior Anterior

44

(45)

pDIC measurements

Fibrillin 1 mgR/mgR Fibulin 4 SMC KO

dorsal

ventral inflation

6852 CS3 8

dorsal

ventral inflation

(46)

OCT-DVC applied to arterial mechanics

(47)

Measurement of bulk deformation fields by Digital Volume Correlation

on OCT images

OC T Las er

(48)

Institut Mines-Télécomavril@emse.fravril@emse.fr Stéphane Avril - 2020 Oct 12 - MecaBio

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)

• Overlap : 75%

• Passes : 10

• Required valid voxel per window: 44%

48

(49)

APPROACH

1. Experiments 2. Material model 3. Inverse method

(50)

CONSTITUTIVE MODEL

Stéphane Avril - 2020 Oct 12 - MecaBio Bellini, et al., Ann. Biomed. Eng., 42(3), pp. 488–502, 2014

50

(51)

PARAMETERS TO BE IDENTIFIED

(52)

APPROACH

1. Experiments 2. Material model 3. Inverse method

(53)

Inverse approach – traditional approach

Set of materials properties

Resolution of forward problem

Deviation between predictions and measurements minimization

initialization

Identification of material properties

measurements

53

(54)

Inverse approach – FEMU approach

Set of materials properties

Resolution of forward problem

Deviation between predictions and measurements minimization

initialization

Identification of material properties

J(µ)= 𝑇 𝑢 − 𝑇(𝑢 ^𝑒𝑥𝑝 ) ² + ^𝛼

2 𝐵(µ)

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

measurements

54

(55)

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

55

(56)

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

Stéphane Avril - 2020 Oct 12 - MecaBio Bersi et al., J Biomech Eng, 2016

Full-field strain measurements

56

(57)

The principle of virtual work

57

(58)

Application to the identification of material parameters: the virtual fields method

Example in linear elasticity

58

(59)

Application to the identification of material parameters: the virtual fields method

Example in incompressible hyperelasticity

I c F

E . .

F ^t 





 





0 dS

u . T dV

: I c F

E . .

F ^*

V

* V

t     





 



 





 

  



59

(60)

Application to the identification of material parameters: the virtual fields method

Example for NeoHooke incompressible

I c F

. F C

2 ₁₀ ^t 





 ^F ^. ^F ^c ^I  ^: ^dV ^T ^. ^u ^dS

C

2 ^*

V

* V

t

10  









  ²  ^B ^c ^I  ^: ^dV

dS u

. T dV

: I c F

. F 2

dS u

. T

C _*

V

* V

t

* V

10   



 



 



60

(61)

Application in simple uniaxial tension

61

(62)

Application in simple uniaxial tension

FL dS

u . T ^*

V

 



 ^B ^B ^FL ^B  ^dV

C

V

33 22

11 10 ^  ^ ^

 

 



 

 







2 / z

2 / y

x u ^*

F

-F

L z

x

62

(63)

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

63

(64)

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

Genetic algorithm Resolution:

Minimization of the equilibrium gap using the principle of virtual power

Bersi et al., J Biomech Eng, 2016

64

(65)

3D estimation of

strains

3D estimation of

stress

Derivation of stress tensor using layer specific constitutive behavior

65

(66)

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 external virtual work equals P.

66

(67)

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 )

67

(68)

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

Genetic algorithm Resolution:

Minimizing the equilibrium gap

Bersi et al., J Biomech Eng, 2016

68

(69)

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

69

(70)

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

Full-field strain measurements

3D

Bersi et al., J Biomech Eng, 2016 Bersi et al, BMMB, 2018

70

(71)

Results - Highlights

(72)

Obtained linearized circumferential stiffness

(kPa) (kPa)

(73)

Correlation with tissue µstructure

Bersi et al, BMMB 2019

(74)

Cross sectional structure – function analysis

Bersi et al, Sci Rep 2020

(75)

Full-Field Material Parameter Estimation vs local stress

(76)

SUMMARY

76 - Inverse approach permitting to reconstruct the regional distribution of mechanical properties of the aorta.

- Towards correlations between mechanical properties and underlying microstructures during aneurysm growth.

Mechanics of soft biological tissues