Biomedical Engineering Reference
In-Depth Information
4.6.2 Examples of POD simulations with different sets
of parameters
The POD method has a practical interest only if a POD basis generated for a set of
parameters can be used with other sets of parameters as well. In particular, if we have
in mind to solve an inverse problem of parameter identification using POD, the pa-
rameters to be identified will take different values. It is therefore necessary to check
that the accuracy of the POD basis is stable with respect to parameter perturbation.
In this section, we study this issue for some parameters of the model.
Let us first consider the perturbation of the parameter
close
which corresponds
to the characteristic closing time of ionic channels. As detailed in the beginning of
Sect. 4.4, this parameter takes four different constant values. We focus on two of
these values,
τ
epi
close
and
rv
close
. A POD basis of 80 vectors is first constructed with
τ
τ
epi
close
rv
close
(
τ
. This basis is sufficient to get an excellent accuracy if the
same
experiment is run with the reduced-order model. More interestingly, it still
gives quite good results when
,
τ
)=(
80
,
80
)
epi
close
,
τ
rv
(
τ
close
)
are significantly modified. For example,
epi
close
rv
close
if we take
, Fig. 4.10 (left) shows a comparison of the first
lead of the ECGs obtained with the full model and the reduced model. The QRS is in
excellent agreement, while the T-wave is slightly underestimated, which is not sur-
prising since
(
τ
,
τ
)=(
90
,
120
)
τ
close
mainly affects the repolarization phase. It is interesting to note that
for the values
epi
close
rv
close
(
τ
,
τ
)=(
,
)
, the T-wave of the ECG is negative whereas it
is positive with (90,120). It is therefore particularly satisfactory to obtain the correct
T-wave orientation with the reduced-order model after perturbing these coefficients.
80
80
rv
close
. These parameters were
chosen since it was observed in [7] that the ECGs are particularly sensitive to
them. We have run simulations taking
Similar tests have been done with
τ
in
,
C
m
,
A
m
and
τ
10
−
3
rv
120
and we have compared the ECGs obtained with the full-order model and with the
reduced-order model corresponding to the basis generated with
τ
in
=
0
.
8
,
C
m
=
,
A
m
=
200
,
τ
close
=
τ
in
=
1
.
5
,
C
m
=
10
−
3
rv
2
70. The results are reported in Fig. 4.10 (right). We
see that the use of POD induces a temporal shift and a difference of amplitudes. On
the first lead, the wave orientation is preserved, but for some other leads changes of
orientation have been noticed.
The results get even worse with other parameters, like for example those govern-
ing the initial activation: if we use a POD basis obtained from a reference simulation
with an initial activation in the septum to run a reduced order simulation with an ini-
tial activation at the apex, the results are totally wrong, as shown in Fig. 4.11.
The last two examples show that POD approximation has to be used very care-
fully. In particular, it is necessary to examine to what extent the parameters chosen
to compute the POD basis have to be close to the parameters used in the simulation.
This issue is particularly important to address the problem of parameter identifica-
tion. Practical techniques to improve these results will be proposed in a forthcoming
publication.
×
,
A
m
=
100
,
τ
close
=
