Biomedical Engineering Reference
In-Depth Information
train medical devices (see [21]); last but not least, the simulation of realistic ECGs
is a necessary step towards the development of patient-specific models from clinical
ECG data, since it would be meaningless to address an inverse problem with a model
unable to solve the direct one correctly.
As will be briefly shown in Sect. 4.2, the modelling of the cardiac electrical
activity is a very complex and broad subject. The monographs [13, 53, 58, 61]
give excellent introductions to the topic. The numerical simulation of ECGs using
a whole-heart reaction-diffusion model has been addressed in many publications
[7, 30, 33, 38, 50, 52, 67]. Among them, only a few [7, 50, 52] provide meaningful
simulations of the complete 12-lead ECG. In [50, 52], simulations rely on either a
monodomain approximation or a heart-torso decoupling approximation and a multi-
dipole cardiac source representation (see [38, Sect. 4.2.4] and [28]). In publication
[7], several simulations are based on a flexible coupled heart-torso model which al-
lows to compare different modelling assumptions. More details about this model will
be presented in Sect. 4.4.
In this presentation, we will focus on the numerical simulation of ECGs using
a three-dimensional mathematical model fully based on partial/ordinary differen-
tial equations (PDE/ODE). The main ingredients of this model are standard: phe-
nomenological cell dynamics, bidomain equations for the heart and a generalized
Laplace equation for the torso. They are briefly overviewed in Sect. 4.2. A result on
the existence of solution is presented in Sect. 4.3 for several classes of ionic mod-
els. In order to provide realistic ECG simulations, critical modelling aspects have
to be discussed: heart-torso transmission conditions, cell heterogeneity, His bundle
modelling, tissue anisotropy, etc. In Sect. 4.4, we will present different modelling
options and show how they have been combined to get realistic ECGs. To comple-
ment the study, the impact on ECGs of alternative modelling choices is presented.
Then, some decoupled time discretization schemes for the bidomain and heart-torso
systems are analyzed in Sect. 4.5. Finally, some preliminary results obtained with a
reduced-order model based on the Proper Orthogonal Decomposition (POD) method
are presented in Sect. 4.6. This reduced-order model is about ten times as fast as the
full-order model. Our results show that it is not always accurate and that further
work is necessary to make it more reliable. Nevertheless, we will show that it seems
to provide an interesting alternative to the full-order model in some particular cases,
for example long-time simulations.
4.2 Mathematical modelling
The mathematical modelling of the ECG is known as the
forward problem
of electro-
cardiography (see [38]). It relies on three main ingredients: a model for the electrical
activity of the heart, a model for the torso (extracardiac regions) and some specific
heart-torso coupling conditions.
