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
where
If the sensitivity is multiplied with the resistivity (
The volume impedance density can be integrated over all points in the measurand to get the measured impedance,
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].
In engineering, the word sensitivity refers to how much the output of a system, typically a transducer, changes per change in input. This definition is only weakly related to the quantity defined in section 1, which is a problem since the engineering definition is what we actually need when evaluating a measurement system. To get around this problem, we use a synonym of the word sensitivity, perceptivity, to which we attach the engineering definition of sensitivity. Described in mathematical terms as
where Ψ is perceptivity, δ
Since a measured quantity often vary around an offset value, it can be useful to expand the perceptivity concept to what we call relative perceptivity defined as
where Ψ
Three examples are modelled in the Finite Element Model (FEM) tool COMSOL Multiphysics. For simplicity, the models are for stationary conditions. Only the three different material types shown in table 1 are used. The equations defining the model behavior is predefined in the COMSOL Multiphysics physics interface
Electrical properties of the materials used in FEM simulations.
Material | Conductivity (S/m) | Resistivity (Ωm) |
---|---|---|
lowConductivity | 0.1 | 10 |
mediumConductivity | 1 | 1 |
highConductivity | 10 | 0.1 |
Everything except the electrode placements are identical in the three models. This let us investigate three electrode placements with focus on sensitivity, volume impedance density, and perceptivity. Model figures are shown in figures 1 where current density vector colors can be used to identify CC and PU electrodes. Model A has side-by-side CC and PU electrodes placed far from the center region, model B has in-line CC and PU electrodes placed far from the center region, while model C has in-line CC and PU electrodes with CC electrodes placed far from the center region and PU electrodes placed near the center region. Four variants of the models are simulated:
Base model.
10 % increase of length of center section.
10 % increase of conductivity of center section.
10 % increase of both length and conductivity of center section.
The conducted research is not related to either human or animal use.
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.
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
Simulated impedances and associated perceptivities for the three models. Units for perceptivities are left out since we are considering change in phenomena which can be non-physical changes.
Parameter | Model A | Model B | Model C |
---|---|---|---|
Sim. 1 | 12.645 Ω | 12.645 Ω | 10.985 Ω |
Sim. 2 | 11.736 Ω | 11.736 Ω | 10.076 Ω |
Sim. 3 | 13.545 Ω | 13.545 Ω | 11.885 Ω |
Sim. 4 | 12.545 Ω | 12.545 Ω | 10.885 Ω |
Ψm | -3.636 | -3.636 | -3.636 |
Ψg | 3.600 | 3.600 | 3.600 |
Ψ | -0.400 | -0.400 | -0.400 |
Ψrm | -0.288 | -0.288 | -0.331 |
Ψrg | 0.266 | 0.266 | 0.303 |
Ψr | -0.032 | -0.032 | -0.036 |
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
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.