Upload
others
View
5
Download
0
Embed Size (px)
Citation preview
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
D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 5
2015 -
Motivations for reduced-dimensional modeling
D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 6
• 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
D.CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 7
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
D.CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 7
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
D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 8
2015 -
Cylindrical symmetry assumption
D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 9
• 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
D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 10
• 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)
D.CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 11
(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)
D.CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 12
�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
D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 13
• 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
D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 14
2015 -
Spherical symmetry assumption
D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 15
• 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
D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 16
• 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
D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 16
• 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 -
Model calibration / insight
D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 18
• 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 -
Model calibration / insight
D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 18
• 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
D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 19
Adaptation to heart rate variations
2015 -
Conclusions
D. CHAPELLE & P. MOIREAUMaster Biomechanical Engineering Active and passive tissues 20
• 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