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Rencontres GdR DYCOEC, Nice, 5-6 février 2008
Spatio-temporal Dynamics of Spatio-temporal Dynamics of Nonlinear Mechano-chemical Nonlinear Mechano-chemical
Processes Processes Driving Anisotropic Rhythmic Driving Anisotropic Rhythmic
ContractionContractionof Cardiac Myocytes of Cardiac Myocytes
Philippe TRACQUI
CNRS, Laboratoire TIMC-IMAG, Equipe DynacellInstitut de l’Ingénierie et de l’Information de Santé, In3S,
Grenoble, France
http://www-timc.imag.fr/dynacell
in collaboration with Jacques OHAYON
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
Mechanics is a key issue of heart function …
… but still remains largely over simplified in analyses and models of cardiac cells and cardiac tissues dynamics
Things are changing with the increasingly recognized importance of the transduction of mechanical signals (mechanotransduction) in cell signaling cascades
There is a real need for the development of the mechanobiology of cardiac cells and tissues, notably through the development of theoretical models as the cell and tissue levels
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
Slide from the Physiome project presentationSlide from the Physiome project presentation(R. McLeod & P. Hunter)(R. McLeod & P. Hunter)
Slide from the Physiome project presentationSlide from the Physiome project presentation(R. McLeod & P. Hunter)(R. McLeod & P. Hunter)
GenesGenes ProteinsProteins Biophysical modelsBiophysical models Constitutive lawsConstitutive laws Organ modelOrgan model Whole body modelWhole body model
GenomeGenome
ProteinProtein
PhysiologyPhysiology
StructuralStructural
Bioeng. MaterialsBioeng. Materials
ClinicalClinical
Modeling HierarchiesModeling Hierarchies DatabasesDatabases
Molecular Biology
Physiology
Bioengineering
Clinical medicine
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
Context: Analysis of excitation-contraction of isolated cardiac myocyte trough a multi-scale integrative approach of the structure and function
relationships
Aims:
an analysis of the cardiac performance based on a relevant description of the Ca2+ driven anosotropic and hyperelastic cardiomyocyte contraction
a modelling basis for theoretical and quantitative analysis of the mechano-regulation of cardiomyocyte contraction by mechanotransduction processes
Cellular levelTissue level
Intracellular level
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
Spontaneous contraction of Spontaneous contraction of rat cardiomyocyterat cardiomyocyte
size (110 x 20 m)
Spontaneous contraction of an isolated Spontaneous contraction of an isolated cardiomyocytecardiomyocyte
The sarcomere as the contractile unitThe sarcomere as the contractile unit
Mean experimental
contraction period
Mean experimental contraction
duration
17,0 5,8 s 1,5 0,4 s
Mean experimental contraction amplitude
~ 8 m ( 7%)
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
Associated calcium wave propagationAssociated calcium wave propagation
Visualisation of the propagation of an intracellular calcium wave using Ca labelling with the fluorescent Fluo3 probe
(t = 268ms between two successive images, cell length :110 m)
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
A theoretical model of the A theoretical model of the cardiomyocyte self-sustained cardiomyocyte self-sustained
contractioncontraction
expression of Ca2+ oscillations in a domain of the parametric space where travelling waves may exist
introduction of cytosolic Ca2+ variations in the formulation of an active stress tensor, taking into account cell architectural anisotropy
consideration of cardiomyocyte hyperelastic properties with appropriate passive stress-strain relationship
finite element simulation and experimental validation of the dynamical behaviour of the virtual cardiomyocyte in different contexts
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
Modelling calcium waves propagation Modelling calcium waves propagation in cells and tissuesin cells and tissues
Dupont et al. 96
(Means et al., 2006)
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
Goldbeter et al. (PNAS, 1990)
Z: Cytosolic Ca2+ concentration
Y: Ca2+ concentration in the sarcoplasmic reticulum
A simplified one calcium -pool modelA simplified one calcium -pool model
€
dZ
dt= v0 +v1.β − v2 +v3 + k f .Y −k.Z
€
dY
dt= v2 −v3 −k f .Y
nn
n
MZK
ZV +=
222 .ν
ppA
p
mmR
m
M ZK
Z
YK
YV
++= ..33ν
Autocatalytic process responsible for temporal Autocatalytic process responsible for temporal oscillations: Calcium-Induced-Calcium-Release oscillations: Calcium-Induced-Calcium-Release
(CICR)(CICR)
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
0
1
2
3
4
5
6
7
8
9
10
1,7 1,8 1,9 2 2,1 2,2 2,3 2,4 2,5
longueur sarcomère en micronsSarcomere length SL (m)
Tension (kPa)
Passive tension as a function of the Passive tension as a function of the sarcomere length sarcomere length ((Cazorla et al., 2003Cazorla et al., 2003))
20 20 mm
Elastic properties of the cardiomyocyteElastic properties of the cardiomyocyte
Uniaxial stretching of the Uniaxial stretching of the cardiomyocytecardiomyocyte
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
Constitutive stress-strain relationship (1): passive Constitutive stress-strain relationship (1): passive componentcomponent
•a1 , a2 , cellular material constants
• I1 is the first invariant of the right Cauchy-
Green strain tensor C (I1=Tr(C))
The cardiomyocyte is considered as an hyperelastic incompressible medium with passive strain energy function
21211 )3()3( −+−= IaIaWp
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
Active anisotropic Cauchy stress tensor given by: Active anisotropic Cauchy stress tensor given by:
ssactact ffCaSLT ⊗= + ),( 2τ 42 ICT actact =
(fs orientation of deformed fibers)
Constitutive stress-strain relationship (2): active componentConstitutive stress-strain relationship (2): active component
with:with: ( )),,( )( ),( 22 yxtCaSLTCaSLT Maxact
++ = γ
Tmax maximal tension
K(SL)=Ca2+50 half-maximal
value
nH Hill coefficient
[ ][ ]nHnH
nH
CaSLK
CaCa
+
++
+=
2
22
)()(γ
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
Interplay of calcium oscillations and cell Interplay of calcium oscillations and cell contractioncontraction
Calcium spatio-temporal Calcium spatio-temporal dynamics (waves)dynamics (waves)
Active and passive anisotropic mechanical
behaviour
Integrative mechano-biochemical model of the self-sustained cardiomyocyte contraction
Model Variables Model Variables
Z(Z(rr,t) , Y(,t) , Y(rr,t) and {u(,t) and {u(rr,t), v(,t), v(rr,t) },t) }
€
∂Z
∂t= v0 + v1.β − VM 2
Z n
K2 + Z n + VM 3
Z p
KAp + Z p
Y m
KRm +Y m
+ kf .Y − k.Z + ∇. D∇Z( )
∂Y
∂t= VM 2
Z n
K2 + Z n − VM 3
Z p
KAp + Z p
Y m
KRm +Y m
− kf .Y
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
DD diagonal diffusion diagonal diffusion tensortensor
5,0=xx
yy
DD
ssactTp ffTF
E
WFpI ⊗+
∂∂
+−=τ
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
Finite element simulation of the Finite element simulation of the cardiomyocyte spontaneous cardiomyocyte spontaneous
contractioncontraction
Geometry extracted from real cell image
Boundary conditions Stress free boundaries, localized zero displacement in the nucleus area
No calcium fluxes (Neuman conditions) on the cell boundaries
Permeability of the nucleus to cytosolic Ca
(Pustoc’h et al., Acta Biotheor. 2005)
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
Cardiomyocyte contraction driven by calcium waves originating Cardiomyocyte contraction driven by calcium waves originating from cell border (left) or from cell centre (right), as shown from cell border (left) or from cell centre (right), as shown
by videomicroscopy time-lapse observationsby videomicroscopy time-lapse observations
Saptio-temporal evolution of cytosolic calcium concentrations (Z(x,y,t))Saptio-temporal evolution of cytosolic calcium concentrations (Z(x,y,t))
Simulated self-sustained oscillating contraction of Simulated self-sustained oscillating contraction of an isolated cardiomyocytean isolated cardiomyocyte
Triggering calcium spark initiated on the left cell sideSoliton propagating from left to right in pace with cell shortening
Calcium spark initiated in the middle of the cellTwo solitary waves propagating in opposite directionsCell contracts at both ends simultaneously
Rencontres GdR DYCOEC, Nice, 5-6 février 2008
Conclusions and perspectivesConclusions and perspectives
A satisfactory and rather simple mechano-biochemical model of the isolated cardiomyocyte oscillatory contraction
Amenable to theoretical analysis (bifurcation analysis of model dynamics)
Exemplified mechanical aspects disregarded by reaction-diffusion models
A quantitative framework for analysing the effect of local mechanotransduction processes (titin, endothelin, ..)
A basis for elaborating of a 2D virtual myocardium in which the global tissue response (arrhythmia, contraction inefficiency, …) to localized perturbations (ischemia, …) can be studied
Acknowledgement: This work has been supported by a grant from the CNRS (ACI NIM “MOCEMY”)