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ˆ
M
c
(
k
)=
l
(
k
)
∗
QRS(
k
)
(6.3)
,
...
where
l
(
i
)
T
c
are the coefficients of the FIR filter and
T
c
is the heartbeat
period expressed in number of samples. In order to express the dependence of the
heartbeat component on the respiratory cycle, the coefficients of the filter represent-
ing
i
= 1
M
c
are considered varying, and linearly dependent on the lung volume
V
(
k
):
l
(
k
)=
l
0
(
k
)+
l
1
(
k
)
V
(
k
)
.
(6.4)
The filter coefficients
l
0
and
l
1
are identified online using a recursive least-squares
(RLS) algorithm. The future heart motion is predicted using this model and the
periodicity of the lung volume ensured by the artificial ventilation. The expected
motion at sample
k
+
n
can be written as
ˆ
ˆ
ˆ
M
(
k
+
n
)=
M
r
(
k
+
n
)+
M
c
(
k
+
n
)
=
F
(
k
−
k
res p
+
n
)+
l
0
(
k
−
k
QRS
+
n
)+
l
1
(
k
−
k
QRS
+
n
)
V
(
k
−
T
r
+
n
)
(6.5)
where
k
QRS
is the sample number representing the last QRS occurrence.
6.3.2
Amplitude Modulation Method
In [13], the authors modeled the coupling between the two motion components in
the temporal domain. In this section we present an alternate method consisting in
taking into account this coupling in a simpler way. Our algorithm can be seen as a
generalization of the Fourier linear combiner (FLC) framework [41]. The proposed
method is mainly based on experimental observations. Indeed, if we zoom in the
spectral analysis of the heart motion we obtain plots comparable to Figure 6.5. The
peaks around the heartbeat harmonics are similar to those that can be obtained by
plotting the frequency content of an amplitude modulation of two periodic signals of
frequencies
f
c
(the frequency of the carrier) and
f
r
(the frequency of the modulating
signal).
Hence, we propose to write the heartbeat component of the heart motion as the
result of an amplitude modulation. Two experimental observations should however
be underlined. First, the number of harmonics in the modulating signal is lower than
the number of harmonics in the respiratory component. These harmonics also ex-
hibit different amplitudes. Second, only the first low frequency heartbeat harmonics
are modulated. Therefore, we propose to write the heart motion as
M
(
k
)=
M
r
(
k
)+
C
c
(
k
)+
C
c
1
(
k
)
C
r
(
k
)
.
(6.6)
M
c
(
k
)
M
r
is the respiratory component of the heart motion.
C
c
contains all the significant
heartbeat harmonics whereas
C
c
1
contains only the first low frequency harmonics (a
truncated part of
C
c
).
C
r
is the modulating respiratory component. It has the same
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