On sensitivity in transfer impedance measurements

There are a number of parameters that are used to describe properties of a measurement system which consist of measurement instrumentation and measurand. One of these parameters has been given the name sensitivity, which may be misleading. The word sensitivity can be interpreted in at least two ways for a measurement system. Firstly, it specifies how much the output of a system, for example a transducer, changes as a function of a change in the input. This is the classical engineering point-of-view. When considering a 3D volume impedance system, such as a bioimpedance system, sensitivity takes another meaning as explained below.

A short repetition of some of the mathematics involved is useful in order to explain the problem. For those requiring more detailed information on the subject, the book Bioimpedance and Bioelectricity Basics [1] is recommended. If we look at an impedance measurement system with separated current carrying (CC) electrode pair and potential pick-up (PU) electrode pair, the measured transfer impedance is the potential on the PU electrodes divided by the current flowing in the CC electrode pair. To simplify the text, we are using the term impedance instead of the more proper term transfer impedance. The resulting current density of every point throughout the measurand can be quantified by vectors Jcc→$$\overrightarrow{{{J}_{cc}}}$$where CC reflects that this is the current density of the CC electrodes. The principle of reciprocity states that we can swap the CC and PU electrode pairs [2]. This would result in current density vectors JPU→$$\overrightarrow{{{J}_{PU}}}$$throughout the measurand. We can use the current vector field to calculate what we (currently) call sensitivity as shown inequation 1,

where S is sensitivity in all points throughout the measurand, I_CC is current in CC electrodes, and I_PU is the reciprocal current in the PU electrodes [1].

If the sensitivity is multiplied with the resistivity (ρ) as shown as shown in equation 2 we get the volume impedance density (z).

The volume impedance density can be integrated over all points in the measurand to get the measured impedance, Z, as shown in equation 3.

From equations 2 and 3, we see that both sensitivity and resistivity are required quantities when determining measured impedance which is the core of the critique: It is easy to be misled to believe that sensitivity as defined in equation 1 is telling everything there is to know about the sensitivity in a given point of the measurand. The sensitivity is giving important information, especially about which sign of the contribution of a given point, which can be negative if the two current density vectors have an angle of more than 90°, but the contribution to the measured impedance is also requiring knowledge of resistivity. Equation 2 defines volume impedance density, which is including both sensitivity and resistivity, which makes this parameter much more interesting since it tells us how much a point of a measurand is actually contributing to the measured impedance.

But still, the volume impedance density is only giving information of points of the measurand, and we still need a parameter that tell us something about how much measured impedance changes as a result of changes in the measurand. It must be mentioned that David B. Geselowitz touched the subject of reciprocity and sensitivity as defined later as perceptivity in 1971 [3].

The simulation results are shown in figures 1 through 3, and in table 1.

The current density vectors shown in figure 1 is the basis for calculating the sensitivity fields for the models as shown in figure 2. The sensitivity field for the center piece is more or less the same as it is for the surrounding regions for all models. This is misleading because the contribution from the center piece is much higher than it is for the surrounding regions. Figure 3 shows the volume impedance density field which gives information on how much each point contributes to the final measured (or simulated) impedance. The sensitivity field plots have correct sign, as do the volume impedance density field plots, but there stops the usability of sensitivity plots for this particular example.

Fig.2

Fig.3

In cases where the resistivity is constant throughout the model such as the model presented by Brown et. al. and Høyum et.al. [4, 5], the sensitivity field will give the similar information as the volume impedance density field. This is particularly true for graphical visualizations, but the sensitivity cannot be used to determine measured impedance by integration. So even for these cases, the volume impedance field is more useful.

From the volume impedance density field plots in figure 3, it is clear that all models have high z in center piece, and that there are some differences when it comes to the surrounding regions. One can see that model C is more focused on the center piece, but we have to look at the perceptivities to verify this. Table 2 shows that the perceptivities for all models are identical when using only three decimal places. This means that the change in impedance is the same for each model, which makes sense since the changes happens in the center piece with highest z. If we look at the relative perceptivities, on the other hand, these show that models A and B are very similar, and that model C has about 14 % higher value, which means that this electrode configuration is superior to the ones in model A and B. We have thus a tool that enables us to objectively compare electrode configurations.

The perceptivities is also useful to estimate requirements for the instrumentation. If, for example, we inspect the results for model C in table 2, we see that the perceptivity for geometrical changes is 3.6. This means that the maximum impedance will change by 3.6 Ω. So if we want to distinguish between n different levels, the accuracy must be better than 3.6⁄n Ω.

I have shown that volume impedance density is more useful to plot than sensitivity. A sensitivity plot may give a misleading impression of the contributions, and should thus be avoided. Calculation of relative perceptivities gives us tool to objectively compare how good a measurement setup is. It also let gives information regarding requirements for the impedance measurement instrumentation.