Biomedical Engineering Reference
In-Depth Information
Two-compartment model: an inner cylinder
surrounded by an outer cylinder of same
resistivity (
Fig. 4.1-8
bottom)
DG
G
¼
D
y
D
y
þ
y
A
þ
y
t
ð
exact
Þ
(4.1.5)
C
v
C
In the two-compartment model the two cylinders are
physically in parallel, and the conductance model is
preferred with
L ¼
const.
D
y
þ
y
A
is the volume of the
inner cylinder, y
t
is the volume of the outer cylinder and
considered constant (implying that both the inner tube
and outer tube swell when
D
y
>
0). Equation
(4.1.5)
shows that the sensitivity falls with a larger surrounding
volume
v
t
.
In plethysmography the measurement should
be confined as much as possible to the volume where the
volume change occurs. Thus the problem of high sensi-
tivity plethysmography poses the same problem as in
electrical impedance tomography (EIT): to selectively
measure immittance in a selected volume.
Figure 4.1-9 Tetrapolar electrode system and the effect of
a bolus of blood passing the measured volume.
will be dependent on the bolus length with respect to the
measured length.
To analyze the situation with a tetrapolar electrode
system in contact with, for example, a human body, we
must leave our simplified models and turn to lead field
theory. The total measured transfer impedance is the ratio
of recorded voltage to injected current. The impedance is
the sum of the impedance contributions from each small
volume
d
y in the measured volume. In each small volume
the resistance contribution is the resistivity multiplied by
the vector dot product of the space vectors J
reci
(the
current density field due to a unit reciprocal current ap-
plied to the recording electrodes) and J
0
cc
(the current
density field due to a unit current applied to the true
current carrying electrodes). With disk-formed surface
electrodes the constrictional resistance increase from the
proximal zone of the electrodes may reduce sensitivity
considerably. A prerequisite for two-electrode methods is
therefore large band electrodes with minimal current
constriction.
If the system is reciprocal the swapping of the re-
cording and current carrying electrode pairs shall give the
same transfer impedance. It is also possible to have the
electrode system situated
into the volume
of interest
(e.g. as needles or catheters). Such volume calculation
(e.g. of CO) is used in some implantable heart pace-
maker designs, cf. Section 4.1.11.
4.1.2.2 The effect of different
conductivities
In the two-compartment, constant length, parallel cyl-
inder model analyzed above the conductivities in the
inner and outer cylinders were considered equal. Dif-
ferent conductivities will of course also change
G
. With
a conductivity s
t
of tissue outer cylinder and s
b
of blood
in inner cylinder, and with constant geometry, the con-
ductance is found from
G ¼
s
A/L
:
G ¼ð
s
A
A
A
þ
s
t
A
t
Þ
1
ð
constant volume
Þ
(4.1.6)
L
Equation (
4.1.6
) is not really a plethysmographic
equation when the assumption is that no geometrical
change shall occur. However, it relates to flow systems
with a varying conductivity of the passing liquid, and flow
with time is volume.
Changes in conductivity
Conductivity may change as a function of time (e.g.
caused by flow). The special case of a changing conduc-
tivity with a general but constant geometry was analyzed
by Geselowitz (1971) who developed an expression for
DZ
based on the potential field. Lehr (1972) proposed to
use current density instead of potential in the devel-
opment. Putting E
¼
-
VF
and as J
¼
sE in isotropic
media, we have:
4.1.2.3 Models with any geometry
and conductivity distribution
Figure 4.1-9
shows a dynamic system in a vessel where,
for example, a blood bolus volume on its passage leads to
a temporal local volume increase during heart systole.
The measured zone in the inner cylinder is filled with
blood (inflow phase). Later during the diastole the blood
is transported further (outflow phase), but also returned
via the venous system.
Figure 4.1-9
shows a tetrapolar
electrode system for the measurement of
G
.
In general a
tetrapolar
system is preferable; it may
then be somewhat easier to confine the measured tissue
volume to the zone of volume increase. The sensitivity
for bolus detection with a tetrapolar electrode system
ððð
J
0
0
$
DZ ¼
D
s
s
0
J
0
D
s
dy
½U
(4.1.7)
Here the integration is in the volume of conductivity
change. The volume is homogeneous but with changing
conductance, at a given moment of integration s is
