Biomechanical Modeling of Active and Passive Biological ... · Active and Passive Biological...

Biomechanical Modeling of Active and Passive Biological Tissues — Application to Cardiac ModelingD. Chapelle, P. Moireau

Master Biomechanical Engineering Active and passive tissues

2015 -

Outline (of whole course)

D.CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 2

• Mechanical modeling of biological tissues

- passive behavior

- active behavior (muscles)

• Cardiac modeling

• Reduced-dimensional modeling for muscles and cardiac system

• Scientific computing

- spatial discretization and boundary condition

- time discretization and coupling

2015 -

Outline (of this course)

D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 3

• Motivations / context

• 1D reduced modeling

• 0D reduced modeling

• Conclusions

• Mechanical modeling of biological tissues

- passive behavior

- active behavior (muscles)

• Cardiac modeling

• Reduced-dimensional modeling for muscles and cardiac system

• Scientific computing

- spatial discretization and boundary condition

- time discretization and coupling

Lecture 4

Master Biomechanical Engineering Active and passive tissues

Motivations / context


2015 -

Motivations for reduced-dimensional modeling


• Sophisticated 3D multi-scale models are available

• However, difficult to calibrate and validate at the organ level

• Experimental data available at more reduced scales

- Tissue samples

- Myocytes

• Reduced-dimensional models intended to calibrate and validate at these scales while

retaining full biophysical complexity

• Much reduced computational cost ➫ longer time periods can be simulated (towards “real time”)

• Allow straightforward “mapping” to whole organ

2015 -

Summary of cardiac modeling


Principle of virtual work in total Lagrangian formulation (ref. configuration)

Constitutive behavior (2nd Piola-Kirchhoff):

with

Fiber-directed stress governed by active behavior (last lecture)

Main loading given by internal cavity pressure PV

�w � V(�0),

�

�0

�0� · w d� +

�

�0

� : dye · w d� = ��

�endo

PV � · F�1 · w J dS

� = �p + �1D(e1D, ec) � � �

e1D = � · e · � = 12 (I4 � 1)�1D

��

��

�p = �e + �v =�We

�e(J1, J4) +

�Wv

�e� pC�1

We = �1e�2(J1�3)2 + �3e�4(J4�1)2

W� =�

2 tr(e2) �J1�C

= I� 13

3 (1� 13 I1C�1)

�J4�C

= I� 13

3 (� � � � 13 I4C�1)

J1 = I1I� 13

3

J4 = I4I� 13

3

�Kc = �(|u| + � |ec|)Kc + n0K0 |u|+Tc = �(|u| + � |ec|)Tc + ecKc + n0T0 |u|+

2015 -

Summary of cardiac modeling


Principle of virtual work in total Lagrangian formulation (ref. configuration)

Constitutive behavior (2nd Piola-Kirchhoff):

with

Fiber-directed stress governed by active behavior (last lecture)

Main loading given by internal cavity pressure PV

�w � V(�0),

�

�0

�0� · w d� +

�

�0

� : dye · w d� = ��

�endo

PV � · F�1 · w J dS

� = �p + �1D(e1D, ec) � � �

e1D = � · e · � = 12 (I4 � 1)�1D

��

��

�p = �e + �v =�We

�e(J1, J4) +

�Wv

�e� pC�1

We = �1e�2(J1�3)2 + �3e�4(J4�1)2

W� =�

2 tr(e2)

Incompressibility assumptionLagrange multiplier (J=1)

�J1�C

= I� 13

3 (1� 13 I1C�1)

�J4�C

= I� 13

3 (� � � � 13 I4C�1)

J1 = I1I� 13

3

J4 = I4I� 13

3

�Kc = �(|u| + � |ec|)Kc + n0K0 |u|+Tc = �(|u| + � |ec|)Tc + ecKc + n0T0 |u|+

1D reduced modeling


2015 -

Cylindrical symmetry assumption


• Axisymmetry assumed on: material (transverse isotropy), loading, geometry

• Relevant for elongated tissue sample (along fiber)… or myocytes

C =

�

�C 0 00 C� 1

2 00 0 C� 1

2

�

�

6 M. Caruel et al.

Finally, the valve law (11g) can be expressed as

�V = 4⇡R2

0

⇣1 +

y

R

0

⌘2

y = f

�P

v

, P

ar

, P

at

�,

so that the initial system (11) finally leads to

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

⇢d

0

y +d

0

R

0

⇣1 +

y

R

0

⌘⌃

sph

= P

v

⇣1 +

y

R

0

⌘2

⌃

sph

= �

1D

+ 4�1� C

�3

�✓@W

e

@J

1

+ C

@W

e

@J

2

◆

+ 2@W

e

@J

4

+ 2⌘ C�1� 2C�6

�

�

1D

= E

s

e

1D

� e

c

(1 + 2ec

)2

(⌧c

+ µe

c

) = E

s

(e1D

� e

c

)(1 + 2e1D

)

(1 + 2ec

)3

k

c

= �(|u|+

+ w |u|� + ↵ |ec

|) kc

+ n

0

k

0

|u|+

⌧

c

= �(|u|+

+ w |u|� + ↵ |ec

|) ⌧c

+ n

0

�

0

|u|+

+ k

c

e

c

� V = 4⇡R2

0

�1 +

y

R

0

�2

y = f

�P

v

, P

ar

, P

at

�

C

p

P

ar

+ (Par

� P

d

)/Rp

= Q

C

d

P

d

+ (Pd

� P

ar

)/Rp

= (Psv

� P

d

)/Rd

.

[E]Add predictions of the ESPVR by computing the pres-sure in equilibrium with the maximum active stress

3.2 1D-formulation

ir

1

ir

2

R ix

L

Ftip

Fig. 2 Cylindrical model of a single papillary muscle

Geometry and kinematics This one-dimensional modelaims at reproducing the behavior of an elongated struc-ture made of myocardium, such as isolated muscle fibers,or even single myocytes, under uniaxial traction. As asimplified geometry we consider a circular cylinder ofradius R

0

and length L

0

in the reference configuration⌦

0

, see Fig.2, and we assume that material propertiesaccordingly enjoy cylindrical symmetry – namely, trans-verse isotropy – hence, the whole behavior has this samesymmetry. As an orthonormal basis we use a first vectori

x

oriented along the fiber – i.e. ⌧1

= i

x

– and we de-fine two arbitrary equivalent directions (i

r

1

, i

r

2

) in thecross section. An external force F

tip

is applied at theend of the fiber along the i

x

-direction, and we seek theresulting longitudinal displacement y(x) at each point

of the fiber. Due to the incompressibility condition, theCauchy-Green tensor takes the special form

C =

0

@C 0 0

0 C

� 1

2 0

0 0 C

� 1

2

1

A,

where C = (1 + y

0(x))2 is the strain in the ix

-direction.Therefore in the longitudinal direction we have�d

y

e · y⇤�xx

=�1 + y

0�(y⇤)0,

for a virtual displacement field y

⇤(x) = y

⇤(x) ix

.

Stress and equilibrium derivation Considering again thesmall thickness (diameter) of the fiber and the loadingin the axial direction, classical structural mechanics jus-tifies that the radial stresses ⌃

rr

are negligible. Like inthe 0D model reduction, this allows to compute the La-grange multiplier p, viz.

p = C

�1/2

�⌃

p

�rr

. (16)

The power of internal forces then reduces to

⌃ : dy

e · y⇤ = ⌃

xx

�1 + y

0�(y⇤)0,

with the axial stress given by

⌃

xx

= �

1D

+�⌃

p

�xx

� C

�3/2

�⌃

p

�rr

, (17)

with e = e

1D

= (C � 1)/2. In this case, we have for thehyperelastic part8>><

>>:

J

1

= C + 2C�1/2

J

2

= 2C1/2 + C

�1

J

4

= C

and8>>>>>>>><

>>>>>>>>:

@J

1

@C

= 1� 1

3

�C + 2C�1/2

�C

�1

@J

2

@C

=�C + 2C�1/2

�I � C � 2

3

�2C1/2 + C

�1

�C

�1

@J

4

@C

= i

x

⌦ i

x

� 1

3C C

�1

The derivative of the viscous pseudo-potential gives

@W

v

@e

=⌘

2C.

Then we can rewrite (17) as

⌃

xx

= �

1D

+ 2�1� C

�3/2

�✓@W

e

@J

1

+ C

�1/2

@W

e

@J

2

◆

+ 2@W

e

@J

4

+⌘

2C

�1 +

1

2C

� 9

4

�,

C =�1 + y �(x)

�2

� = ixFiber direction

y = y(x) ix + ... (transverse&sym.)

detC = 1

... +

�

�0

� : dye · w d�0 + ...��dye · w

�xx

=�1 + y�

�w� in

2015 -

1D model


• Uniaxial loading

• Invariants

• Viscoelasticity

• Principle of virtual work (1D): integrated over cross-section A0

∂Wv

∂e=

η

2C

��

��

�J1

�C= 1 � 1

3

�C + 2C� 1

2�C�1

�J4

�C= ix � ix � 1

3C C�1

�J1 = C + 2C� 1

2

J4 = C

� L0

0

�� y w + �xx

�1 + y�

�w�� dx =

FtipA0

w(L0)

� = �p + �1D � 1 � � 1 � p C�1, p = C� 12��p�

rr

�xx = �1D +��p�

xx� C� 3

2��p�

rr

�rr = 0

�xx = �1D + 2�1� C� 3

2��We

�J1+ 2�We

�J4+

�

2 C�1+12C

�3�

2015 -

Example: fiber experiments (vs. 1D model)


(4)

m1 m1

m2

m1

m2

(1) (2) (3)

Papillary muscles (laboratory rats)

Experimental data : Y. Lecarpentier (Institut du Coeur & Meaux hosp.)Paper: [Caruel et al. 2013]

Rest Preload Afterload Activation

2015 -

Model simulations vs. experiment (statics)


�0.5 0 0.5 1 1.5

0

1

2

3

1

23

e

Ftip/A

0(.10

4Pa)

Passive law

Active stress

Preloads

Afterloads

Passive stress

0 0.2 0.4

1

1.5

1

2 3 2

1

time (s)

Ftip/A

0(.10

4Pa)

(b)

0 0.2 0.40.7

0.8

0.9

1

1.11 2

3

2 1

time (s)

e

(c)

(a)

Stress/strain data

for initial loading (1) and max. shortening (3)

2015 -

Model calibration / insight


• Note: homogeneous activation ➫ homogeneous (ind. of x) stresses and strains

• Passive behavior (Points 1, i.e. extension under given preload)

• Active behavior (Points 3, i.e. maximum shortening under given afterload) Assuming large series stiffness Es we have

�1D = 0 � �xx = 2�1� C� 3

2��We

�J1+ 2�We

�J4

FtipA0

= �xx�1 + y�

�

�xx =n0(y�)T01+ y�

+ 2�1� C� 3

2��We

�J1+ 2�We

�J4

0D reduced modeling


2015 -

Spherical symmetry assumption


• Spherical symmetry assumed for: geometry, material (uniform distribution of fibers in all

tangential directions), loading (pressure)

• Relevant for: cardiac cavities (approximately), at least for left ventricle

R

d

Pv

i�1

�i�1

�i�2 i�2ir

detC = 1

y =�y + �(r � R0)

�ir

C =

�

�C�2 0 00 C 00 0 C

�

�

C = (1+ y/R0)2

... +

�

�0

� : dye · y� d�0 + ...

y = R � R0

efib =yR0

� (dye · w)�� = (1 + efib)(w/R0) in

2015 -

0D model


• Thin structure property

• Invariants

• Viscoelasticity

• Principle of virtual work (0D): integrated over spherical volume

∂Wv

∂e=

η

2C

�J1 = 2C + C�2

J4 = C

��

��

�J1

�C= 1 � 1

3

�2C + C�2

�C�1

�J4

�C= i�1

� i�1� 1

3C C�1

�sph = �1D + 4�1 � C�3� �We

�J1+ 2�We

�J4+ � C

�1 + 2C�6�

�rr = 0� = �p + �1D � 1 � � 1 � p C�1, p = C�2��p�

rr

�sph = ��1�1 + ��2�2 =��p�

�1�1+

��p�

�2�2+ �1D � 2C�3��p�

rr

� y w 4�R20d0 + �sph�1 + efib

� wR0

4�R20d0 = PVdVdy

w

2015 -

0D model


• Thin structure property

• Invariants

• Viscoelasticity

• Principle of virtual work (0D): integrated over spherical volume

∂Wv

∂e=

η

2C

�J1 = 2C + C�2

J4 = C

��

��

�J1

�C= 1 � 1

3

�2C + C�2

�C�1

�J4

�C= i�1

� i�1� 1

3C C�1

�sph = �1D + 4�1 � C�3� �We

�J1+ 2�We

�J4+ � C

�1 + 2C�6�

�rr = 0� = �p + �1D � 1 � � 1 � p C�1, p = C�2��p�

rr

�sph = ��1�1 + ��2�2 =��p�

�1�1+

��p�

�2�2+ �1D � 2C�3��p�

rr

� =d0R0

�d0 y + ��1 + efib

��sph =

�1 + efib � �

2 (1 + efib)�2�2�1 + �(1 + efib)

�3�PV

2015 -

Application: “real time” heartbeat simulations

Master Biomechanical Engineering Active and passive tissues 17D. CHAPELLE & P. MOIREAU

0 0.2 0.4 0.6 0.80.4

0.6

0.8

1

1.2

time (s)

V(·10

�4m

3)

0 0.2 0.4 0.6 0.8

�5

0

5

time (s)

Q(·10

�4m

3/s)

0 0.2 0.4 0.6 0.80

0.5

1

1.5

time (s)

Pv(·10

4Pa)

0.4 0.6 0.8 1 1.20

0.5

1

1.5

V (·10�4 m3)

Pv(·10

4Pa)

(a) (b)

(c) (d)

2015 -



• EDPVR (end-diastolic pressure-volume relationship): passive behavior / preload

• ESPVR (end-systolic pressure-volume relationship): active behavior / afterload (Es large)

0 50 100 150

0

1

2

3

V (mL)

P

v(·10

4Pa)

Passive behavior

ESPVR

Windkessel

Par = const.

�sph = 4�1 � C�3� �We

�J1+ 2�We

�J4

�1 + efib � �

2 (1 + efib)�2�2�1 + �(1 + efib)

�3�PV = ��1 + efib

��sph

�sph =n0(efib)T01 + efib

+ 4�1 � C�3� �We

�J1+ 2�We

�J4

2015 -



• EDPVR (end-diastolic pressure-volume relationship): passive behavior / preload

• ESPVR (end-systolic pressure-volume relationship): active behavior / afterload (Es large)

0 50 100 150

0

1

2

3

V (mL)

P

v(·10

4Pa)

Passive behavior

ESPVR

Windkessel

Par = const.

�50 0 50 100 150

0

1

2

3

4

Vd

V (mL)

Pv(·10

4Pa)

�0 = 1.8.105

�0 = 1.2.105

�0 = 0.6.105

diastolic fil.

�sph = 4�1 � C�3� �We

�J1+ 2�We

�J4

�1 + efib � �

2 (1 + efib)�2�2�1 + �(1 + efib)

�3�PV = ��1 + efib

��sph

�sph =n0(efib)T01 + efib

+ 4�1 � C�3� �We

�J1+ 2�We

�J4

2015 -

Valuable for “long-term” simulations


Adaptation to heart rate variations

2015 -

Conclusions


• Generic approach for dimensional reduction of cardiac models:

- 1D: tissue samples

- 0D: simplified cardiac cavities

• Can be adapted to virtually any cardiac model

• Application of 1D reduced model: detailed validation with experimental data

• Cross-validation with 0D/3D model (i.e. similar parameters) ➫ Hierarchy of compatible 0D/1D/3D models➫ Can be used for “in vitro to in vivo” mapping

• 0D model can be used for longer periods ➫ perspectives in real time monitoring

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