Philippe TRACQUI

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


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


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


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


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%)


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)


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


Modelling calcium waves propagation Modelling calcium waves propagation in cells and tissuesin cells and tissues

Dupont et al. 96

(Means et al., 2006)


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)


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


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


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


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


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)


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


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”)

Philippe TRACQUI

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