<s>
The	O
forward	B-Algorithm
problem	I-Algorithm
of	I-Algorithm
electrocardiology	I-Algorithm
is	O
a	O
computational	O
and	O
mathematical	O
approach	O
to	O
study	O
the	O
electrical	O
activity	O
of	O
the	O
heart	O
through	O
the	O
body	O
surface	O
.	O
</s>
<s>
The	O
principal	O
aim	O
of	O
this	O
study	O
is	O
to	O
computationally	O
reproduce	O
an	O
electrocardiogram	B-Application
(	O
ECG	B-Application
)	O
,	O
which	O
has	O
important	O
clinical	O
relevance	O
to	O
define	O
cardiac	O
pathologies	O
such	O
as	O
ischemia	O
and	O
infarction	O
,	O
or	O
to	O
test	O
pharmaceutical	O
intervention	O
.	O
</s>
<s>
Given	O
their	O
important	O
functionalities	O
and	O
the	O
relative	O
small	O
invasiveness	O
,	O
the	O
electrocardiography	B-Application
techniques	O
are	O
used	O
quite	O
often	O
as	O
clinical	O
diagnostic	O
tests	O
.	O
</s>
<s>
Thus	O
,	O
it	O
is	O
natural	O
to	O
proceed	O
to	O
computationally	O
reproduce	O
an	O
ECG	B-Application
,	O
which	O
means	O
to	O
mathematically	O
model	O
the	O
cardiac	O
behaviour	O
inside	O
the	O
body	O
.	O
</s>
<s>
The	O
three	O
main	O
parts	O
of	O
a	O
forward	O
model	O
for	O
the	O
ECG	B-Application
are	O
:	O
</s>
<s>
Thus	O
,	O
to	O
obtain	O
an	O
ECG	B-Application
,	O
a	O
mathematical	O
electrical	O
cardiac	O
model	O
must	O
be	O
considered	O
,	O
coupled	O
with	O
a	O
diffusive	O
model	O
in	O
a	O
passive	O
conductor	O
that	O
describes	O
the	O
electrical	O
propagation	O
inside	O
the	O
torso	O
.	O
</s>
<s>
Such	O
model	O
is	O
typically	O
solved	O
by	O
means	O
of	O
finite	B-Application
element	I-Application
method	I-Application
for	O
the	O
solution	O
's	O
space	O
evolution	O
and	O
semi-implicit	O
numerical	O
schemes	O
involving	O
finite	B-Algorithm
differences	I-Algorithm
for	O
the	O
solution	O
's	O
time	O
evolution	O
.	O
</s>
<s>
Between	O
the	O
mathematical	O
model	O
on	O
the	O
macroscopic	O
level	O
,	O
Willem	O
Einthoven	O
and	O
Augustus	O
Waller	O
defined	O
the	O
ECG	B-Application
through	O
the	O
conceptual	O
model	O
of	O
a	O
dipole	O
rotating	O
around	O
a	O
fixed	O
point	O
,	O
whose	O
projection	O
on	O
the	O
lead	O
axis	O
determined	O
the	O
lead	O
recordings	O
.	O
</s>
<s>
Then	O
,	O
a	O
two-dimensional	O
reconstruction	O
of	O
the	O
heart	O
activity	O
in	O
the	O
frontal	O
plane	O
was	O
possible	O
using	O
the	O
Einthoven	B-Application
's	I-Application
limbs	I-Application
leads	I-Application
I	O
,	O
II	O
and	O
III	O
as	O
theoretical	O
basis	O
.	O
</s>
<s>
The	O
model	O
obtained	O
is	O
called	O
bidomain	B-Algorithm
model	I-Algorithm
,	O
which	O
is	O
often	O
replaced	O
by	O
its	O
simplification	O
,	O
the	O
monodomain	B-Algorithm
model	I-Algorithm
.	O
</s>
<s>
The	O
basic	O
assumption	O
of	O
the	O
bidomain	B-Algorithm
model	I-Algorithm
is	O
that	O
the	O
heart	O
tissue	O
can	O
be	O
divided	O
in	O
two	O
ohmic	O
conducting	O
continuous	O
media	O
,	O
connected	O
but	O
separated	O
through	O
the	O
cell	O
membrane	O
.	O
</s>
<s>
The	O
monodomain	B-Algorithm
model	I-Algorithm
is	O
a	O
simplification	O
of	O
the	O
bidomain	B-Algorithm
model	I-Algorithm
that	O
,	O
in	O
spite	O
of	O
some	O
unphysiological	O
assumptions	O
,	O
is	O
able	O
to	O
represent	O
realistic	O
electrophysiological	O
phenomena	O
at	O
least	O
for	O
what	O
concerns	O
the	O
transmembrane	O
potential	O
.	O
</s>
<s>
In	O
the	O
forward	O
problem	O
of	O
electrocardiography	B-Application
,	O
the	O
torso	O
is	O
seen	O
as	O
a	O
passive	O
conductor	O
and	O
its	O
model	O
can	O
be	O
derived	O
starting	O
from	O
the	O
Maxwell	O
's	O
equations	O
under	O
quasi-static	O
assumption	O
.	O
</s>
<s>
As	O
for	O
the	O
bidomain	B-Algorithm
model	I-Algorithm
,	O
the	O
torso	O
model	O
can	O
be	O
derived	O
from	O
the	O
Maxwell	O
's	O
equations	O
and	O
the	O
continuity	O
equation	O
after	O
some	O
assumptions	O
.	O
</s>
<s>
Using	O
these	O
boundary	O
conditions	O
,	O
it	O
is	O
possible	O
to	O
obtain	O
two	O
different	O
fully	O
coupled	O
heart-torso	O
models	O
,	O
considering	O
either	O
the	O
bidomain	B-Algorithm
or	O
the	O
monodomain	B-Algorithm
model	I-Algorithm
for	O
the	O
heart	O
electrical	O
activity	O
.	O
</s>
<s>
where	O
the	O
first	O
four	O
equations	O
are	O
the	O
partial	O
differential	O
equations	O
representing	O
the	O
bidomain	B-Algorithm
model	I-Algorithm
,	O
the	O
ionic	O
model	O
and	O
the	O
torso	O
model	O
,	O
while	O
the	O
remaining	O
ones	O
represent	O
the	O
boundary	O
conditions	O
for	O
the	O
bidomain	B-Algorithm
and	O
torso	O
models	O
and	O
the	O
coupling	O
conditions	O
between	O
them	O
.	O
</s>
<s>
The	O
fully	O
coupled	O
heart-torso	O
model	O
considering	O
the	O
monodomain	B-Algorithm
model	I-Algorithm
for	O
the	O
electrical	O
activity	O
of	O
the	O
heart	O
is	O
more	O
complicated	O
that	O
the	O
bidomain	B-Algorithm
problem	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
coupling	O
conditions	O
relate	O
the	O
torso	O
potential	O
with	O
the	O
extracellular	O
potential	O
,	O
which	O
is	O
not	O
computed	O
by	O
the	O
monodomain	B-Algorithm
model	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
it	O
is	O
necessary	O
to	O
use	O
also	O
the	O
second	O
equation	O
of	O
the	O
bidomain	B-Algorithm
model	I-Algorithm
(	O
under	O
the	O
same	O
assumptions	O
under	O
which	O
the	O
monodomain	B-Algorithm
model	I-Algorithm
is	O
derived	O
)	O
,	O
yielding	O
:	O
</s>
<s>
First	O
the	O
monodomain	B-Algorithm
model	I-Algorithm
with	O
its	O
usual	O
boundary	O
condition	O
must	O
be	O
solved	O
:	O
</s>
<s>
The	O
uncoupled	O
version	O
of	O
the	O
fully	O
coupled	O
heart-torso	O
model	O
that	O
uses	O
the	O
bidomain	B-Algorithm
to	O
represent	O
the	O
electrical	O
activity	O
of	O
the	O
heart	O
is	O
composed	O
of	O
two	O
separated	O
parts	O
:	O
</s>
<s>
As	O
in	O
the	O
case	O
of	O
the	O
fully	O
coupled	O
heart-torso	O
model	O
which	O
uses	O
the	O
monodomain	B-Algorithm
model	I-Algorithm
,	O
also	O
in	O
the	O
corresponding	O
uncoupled	O
model	O
extracellular	O
potential	O
needs	O
to	O
be	O
computed	O
.	O
</s>
<s>
The	O
monodomain	B-Algorithm
model	I-Algorithm
with	O
its	O
usual	O
boundary	O
condition	O
:	O
</s>
<s>
Usually	O
,	O
the	O
finite	B-Application
element	I-Application
method	I-Application
is	O
applied	O
for	O
the	O
space	O
discretization	O
and	O
semi-implicit	O
finite-difference	B-Algorithm
schemes	O
are	O
used	O
for	O
the	O
time	O
discretization	O
.	O
</s>
<s>
This	O
means	O
that	O
the	O
bidomain	B-Algorithm
and	O
monodomain	B-Algorithm
models	I-Algorithm
can	O
be	O
solved	O
for	O
example	O
with	O
a	O
backward	B-Algorithm
differentiation	I-Algorithm
formula	I-Algorithm
for	O
the	O
time	O
discretization	O
,	O
while	O
the	O
problems	O
to	O
compute	O
the	O
extracellular	O
potential	O
and	O
torso	O
potential	O
can	O
be	O
easily	O
solved	O
by	O
applying	O
only	O
the	O
finite	B-Application
element	I-Application
method	I-Application
because	O
they	O
are	O
time	O
independent	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
fully	O
heart-torso	O
model	O
that	O
uses	O
the	O
bidomain	B-Algorithm
model	I-Algorithm
for	O
the	O
electrical	O
simulation	O
of	O
the	O
cardiac	O
behaviour	O
can	O
be	O
solved	O
considering	O
domain	B-Algorithm
decomposition	I-Algorithm
techniques	O
,	O
such	O
as	O
a	O
Dirichlet-Neumann	O
domain	B-Algorithm
decomposition	I-Algorithm
.	O
</s>
<s>
To	O
simulate	O
and	O
electrocardiogram	B-Application
using	O
the	O
fully	O
coupled	O
or	O
uncoupled	O
models	O
,	O
a	O
three-dimensional	O
reconstruction	O
of	O
the	O
human	O
torso	O
is	O
needed	O
.	O
</s>
<s>
The	O
equations	O
can	O
be	O
easily	O
solved	O
with	O
classical	O
numerical	O
algorithms	O
like	O
Runge-Kutta	B-Algorithm
methods	I-Algorithm
for	O
ODEs	O
.	O
</s>
