PROGRESS TOWARDS AN ELECTROMECHANICAL MODEL OF THE HEART
FOR CARDIAC IMAGE ANALYSIS
M.Sermesant,Y.Coudière,H.Delingette,N.Ayache
INRIA Sophia-Antipolis
Projet Epidaure
2004,route des Lucioles
06902Sophia-Antipolis,France
ABSTRACT
We present a3D numerical model of the heart ventricles
which couples electrical and biomechanical activities.We
have adopted a simpler formulation of previously published
models to achieve a trade-off between biological accuracy
and computational efficiency.To carry out computations,
the FitzHugh-Nagumo equations are solved along with a
reaction diffusionconstitutive law based on the Hill-Maxwell rheological law.
Ultimately,the parameters of this generic model will be ad-
justed from the actual patient’s ECG and from the cardiac
motion measured in4D cardiac images.By including bio-
logical and physical a priori knowledge,we expect to ex-
tract quantitative ventricular function parameters from time
sequences of cardiac images1.
1.INTRODUCTION
The clinical motivation of this work is the quantitative mea-
surement of important ventricular function parameters from
cardiac images,like the ejection fraction,the myocardium
thickness and the local strain and stress.Those parameters
are useful to detect ischemic zones,measure the pathology
extent and control the therapy effectiveness.
The key idea of this paper is to build a“Beating Heart
Model”which contracts under electrical excitation.Byfit-
ting this model to cardiac image sequences we intend to re-
cover those ventricular function parameters.The knowledge
of the heart function has greatly improved at the nanoscopic,
microscopic and mesoscopic scales during the last decades,
thus a global integrative work of this organ becomes con-
ceivable[2].Although this is a work in progress,we be-
lieve that the proposed framework should allow a better un-
derstanding of the heart behavior by measuring quantities
such as stress that are beyond the pure geometric descrip-
tion currently achieved and it should help interpret cardiac
images.
Our“Beating Heart Model”is based on mathematical
systems of(non-linear)partial differential equations,set on
In the deformable model framework,a model evolves under the influence of two energies:an External E
nergy which makes the modelfit the images and an Internal En-ergy which acts as a regularization term and can include a priori information(shape,physical properties,motion,...)
Fig.2.Electromechanical model in a4D ultrasound image.
In our approach,the computation of this External En-ergy at a surface vertex depends not only on the v
ertex loca-tion but also on its normal direction.Different type of forces may be applied depending on the image modality.We chose to combine intensity and gradient information with a region-based approach[7]applied to the intensity profile extracted at each vertex in its normal direction.It consists in defin-ing a region with a range of intensity values and thenfind-ing its boundary by looking at the voxels of high gradient value.The extent of the intensity profile is decreased in the coarse-to-fine process.Then,we apply a force which is proportional to the distance to the closest boundary point of the image from the considered point of the mesh surface.
The volumetric nature of our model strongly decreases the importance of the image outliers in the motion estima-tion since it strongly constrains the geometric(for instance the thickness of the myocardium wall)and physical behav-ior.
The Internal Energy corresponds to an electromechan-ical“Beating Heart Model”which is described in the next three sections:first,the anatomical data necessary to build the model,then the electrical model used to compute the action potential wave propagation andfinally the mechani-cal model used to compute the contraction triggered by the action potential wave.
3.ANATOMICAL MODEL
To build our model we need data regarding both the3D ven-tricular geometry and the musclefiber directions.Indeed, the anisotropy created by thosefibers intervenes in both the electrical wave propagation and the mechanical contraction. There are different ways to obtain thosefiber directions. We are currently using data from a dissected canine heart available from the Bioengineering Research Group2of the University of Auckland,New Zealand and from reduced-encoding MR diffusion tensor imaging(dtMRI)[8].
In order to complete our anatomical model we also need data about the electrical network:the Purkinje network lo-
as in[10],or as the result of some equilibrium equations that govern the conducting continuum,like in the so-called bidomain model[12].
The anisotropy of the ventricles is taken into account through the diffusion tensor:,in a local orthonormal basis where is parallel to the fiber.is a scalar conductivity and the anisotropy ratio between the transverse and the axial conductivities.
4.3.Results of the wave propagation
Simulated isochrones of activation are presented(Fig.4),af-ter a wave was initialized at the apex,using a crude approx-imation of the Purkinje network and a slightly anisotropic diffusion
tensor.
Fig.4.Isochrones of activation(computed with a slightly
anisotropic diffusion tensor:)
We can simulate different singularities that may corre-
spond to pathologies by changing the conduction parame-
ters(Fig.5),for instance introducing a strong conductivity
anisotropy.
Fig.5.Apparitions of activation singularities with a highly
anisotropic diffusion tensor.
This time-dependent computed potential can then be used
as an excitation entry to the system describing the mechan-
ical behavior of the myocardium.
5.MECHANICAL MODEL
The myocardium is composed of musclefiber bundles spi-
raling around the two ventricles.It is a nonlinear viscoelas-
tic anisotropic active material.The qualitative behavior of
the electromechanical coupling is a contraction for a pos-
itive action potential and an active relaxation for a nega-
tive one.Moreover,the action potential also modifies the
stiffness of the material.The model introduced in[13,14]
by Bestel,Clément and Sorine captures this behavior.The
global muscle model is based on the Hill-Maxwell rheolog-
ical law which includes a contractile element,a series
element and a parallel element,as shown on Fig.6
and is detailed in
[15].
Ep
Fig.6.(Left)Hill-Maxwell rheological model.(Right)
Simplified rheological model.
In[14],Bestel,Clément and Sorine model the contrac-
tile element,controlled by the action potential,as fol-
lows:
around10minutes whereas other existing electromechan-ical models take several hours of computation.We are in-vestigating optimization and a parallel implementation to reduce the computation time to allow an interactive use of this model in time series of cardiac images.
The long term goal is to use this model to extract quan-titative parameters of the ventricular function from cardiac images.We can also simulate electrical wave singularities that may correspond to pathologies or test different loca-tions of electrical onset,useful for the implantation of pace-makers,for instance.Moreover,we will be able to simulate the influence of those electrical differences on the contrac-tion.We believe that this beating heart model could help understand the consequences of local electrical or mechan-ical failures on the global motion.Additional images and videos are available on th
e web3.
Globally,recent measurements of the electrical activ-ity,fiber directions and motion reconstruction(from tagged MRI)on the same heart should help adjust the different pa-rameters of the model[16].We will work during the com-ing months on validating this method with an entire dataset coming from the same heart.
Finally,a more accurate model may be considered,and different numerical techniques exist:the complete simula-tion process can be improved,always keeping in mind the balance between the complexity of the models(more equa-tions but more realistic)and the efficiency of the numerical methods(accuracy and speed).
7.ACKNOWLEDGEMENTS
This work is a part of the multidisciplinary project ICEMA4 (standing for Images of the Cardiac ElectroMechanical Ac-tivity),which is a collaborative research action between dif-ferent INRIA projects and Philips Research France[17].
8.REFERENCES
[1]M.Sermesant,Y.Coudière,H.Delingette,N.Ayache,and
J.A.Désidéri,“An electro-mechanical model of the heart for cardiac image analysis,”in Medical Image Comput-ing and Computer-Assisted Intervention(MICCAI’01).2001, vol.2208of Lecture Notes in Computer Science(LNCS),pp.
224–231,Springer.
[2]  A.McCulloch,J.B.Bassingthwaighte,P.J.Hunter,D.Noble,
T.L.Blundell,and T.Pawson,“Computational biology of the heart:From structure to function,”Progress in Biophysics& Molecular Biology,vol.69,no.2/3,pp.151–559,1998. [3]  A.F.Frangi,W.J.Niessen,and M.A.Viergever,“Three-
dimensional modeling for functional analysis of cardiac im-ages:A review,” Medical Imaging,vol.1, no.20,pp.2–25,2001.
[4]X.Papademetris,A.J.Sinusas,D.P.Dione,and J.S.Dun-
can,“Estimation of3D left ventricle deformation from

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