1. Introduction
Determining the type and parameters of the performed action is a device control systems key issue, which is based on bionic control. Such devices can be both medical devices (bioelectric prostheses, orthoses, exoskeletons, rehabilitation, surgical devices) and special-purpose devices.
The problems of develo** an anthropomorphic [
1] neuromuscular interface have been considered in many studies [
2,
3,
4]. The key point in the neuromuscular interfaces’ development is the definition of methods and biophysical principles for obtaining information about the performed movement, which should allow the determination of not only its type but also to carry out a numerical assessment of the movement force-torque characteristics [
1,
5].
Management based on surface electromyography is considered a classic. The method allows the non-invasive recording of the skin’s potential arising from muscle excitation. Analysis of a signal’s amplitude-frequency characteristics allows realizing discrete [
6], proportional [
6], and pattern recognition-based control [
7]. Alternative approaches are proposed to expand the functionality and overcome the typical limitations of this method [
7]. One of the methods is based on the electrical impedance (EI) analysis [
8,
9,
10].
EI studies are based on the passing a high-frequency probing current principle (with a frequency of 50–100 kHz and an amplitude of 1–10 mA) between current electrodes (CE) of the electrode system (ES) and recording the arising potential difference at the potential electrodes (PE) when they are located under the tetrapolar lead system [
10,
11,
12] (CE at the edges, PE in the middle) in the projection of the range of interest [
1]. The measured EI values carry information about the electrical biological tissues’ properties at the probing depths. EI studies are used in the application for predicting or detecting (diagnosing) several diseases of the neuromuscular apparatus [
13,
14], assessing the blood supply to organs, for detecting peripheral veins [
15] and venepuncture [
16], for evaluating and adjusting treatment [
17] or rehabilitation process [
9].
In this work, EI myography, which can be used to solve bionic control problems, which consists in assessing the change in the EI value during muscle contraction in real-time is considered [
1]. It is one of the earliest EI research [
18,
19] applications that has not lost its relevance today [
1,
10]. Despite the existing contradictory results obtained during attempts “ex vivo” [
18,
19] and “in vivo” [
8,
20], EI changes during isometric contractions and tetanic stimulation assessments, the concept of using EI for bionic control is attractive since it can provide new knowledge about biophysical processes occurring in muscle and surrounding tissues associated with the EI myography signals parameters formation.
The location of the ES in the EI of myography is important from the amplitude characteristics of the signal and reproducibility point of view when conducting experimental studies. So, in existing studies, as a rule, ES are used with electrodes located along the axis of the studied muscle groups [
21]. The performed literature review shows that the existing approaches do not always take into account the need to select an acceptable ES size (distance between electrodes) in EI myography, which is important in terms of the EI signal expressiveness and reproducibility [
8,
9,
10,
13].
Figure 1 shows that the individual mechanisms contribution for the EI signal conditioning described above may differ depending on the ES sizes, which sometimes leads to obtaining different results of EI studies. So, for example, at large interelectrode distances, the EI signal conditioning is mainly associated with the muscle’s resistance change and blood filling, and with an ES size comparable to the limb characteristic dimensions, with a change in the skin-fat layer thickness [
22].
In this work, we consider two main contributions that affect EI signal conditioning when the small size ES (
Figure 1, left) is located in the muscle projection area or forearm muscle group, which are active during the motion: a change in the skin-fat layer thickness and muscle resistivity. None of these signal conditioning mechanisms can be achieved separately, which is difficult in terms of evaluating the above contributions in practice [
20], as a result, in the present work, attempts to use simulation were made.
In the contractions process, the muscle body thickens, the skin-fat layer becomes thinner due to deformation from the pressed ES (
Figure 2a), and the muscle mutual position and the epicutaneous ES is changed [
20,
23]. Considering the muscles electrical properties [
10,
24,
25,
26] and their change during contraction (
Figure 2b) [
18,
19], including spatial dependence [
27,
28], during the action, a current density redistribution in the measurement area occurs [
23]. Everything noted is more related to superficial muscles; however, in the case of deep muscles consideration, which are covered by superficial ones on the forearm, then during their contraction, passive deformation of the latter occurs, which also affects the amplitude parameters of the EI measured and can lead to the interpretation of measurement results for superficial and deep muscles being different.
Thus, this work is devoted to determining the acceptable ES size for EI myography using the Pareto-optimality assessment method and the model for the EI signals conditioning in the forearm area, taking into account the change in the skin-fat layer and muscle groups electrophysical and geometric parameters when the brush acting. This study consists of two parts, including a preparatory part and a main part. In a preparatory part, using finite element modeling, an acceptable detailing of the investigated area of the forearm real geometric model was substantiated in terms of the absolute values of electrical impedance for different sizes of electrode systems. The results obtained made it possible to reasonably use the mathematical model when assessing the geometric dimensions of electrode systems for electrical impedance myography; this methodology is described in the main part.
3. Methodology for Selecting ES Geometric Dimensions for EI Myography
An important task in the control systems for technical devices based on EI design is to determine the ES geometric parameters. This ES provides the necessary measured EI signals sensitivity to determine the type of action performed [
10].
Figure 8 shows that on the forearm surface, a two-layer model justified above with the first layer thickness was used to solve this problem and calculate the EI (corresponding to the operator skin-fat layer), which can be determined individually based on the longitudinal forearm ultrasound investigation.
Papers [
12,
39] show an analytical expression that was obtained for the relationship between the EI for point electrodes (3) and model parameters such as the interelectrode distance a; AB, the distance between the current electrodes (3a for the considered ES); MN, the distance between the potential electrodes (a for the considered ES); the skin-fat layer ρ
1 and muscle tissues ρ
2 electrical resistances; and the skin-fat layer thickness h
1 for this model. Verification of the possibility of using an analytical relationship of electrical impedance for a two-layer model was performed using finite element modeling similar to
Appendix A.
The skin-fat layer thickness effect on the EI value can be demonstrated based on the apparent resistance (AR) calculation (4).
The effective ES probing depth ceases to capture the muscle tissue layer, and the AR increases as the ratio of the interelectrode distance to the skin-fat layer thickness
decreases. This means that the contribution to the change in the EI signal associated with the change in the muscle resistivity during contraction becomes smaller.
Figure 9 shows the AR relative change graph projection using the volunteer MRI section example with a forearm girth of 0.35 m and a skin-fat layer thickness of 0.01 m. In the case of using ES with an interelectrode distance of 0.01 m, the change in muscle resistivity during contraction will make a relative change in EI of no more than 25%. To increase this value, it is necessary to increase the interelectrode distance, which is limited by the requirements for the ES design and the measurement method. Taking these factors into account, it is necessary to use an individually-oriented approach to determine an acceptable interelectrode distance, which makes it possible to measure the muscle contraction parameters with higher sensitivity.
3.1. Acceptable ES Sizes Criteria
Solving the problem of determining the acceptable ES size for EI myography based on the papers [
12,
40] analysis and literature review [
18,
19,
41,
42], it is known that during contraction, the muscle electrical resistivity changes. It is possible to estimate the skin-fat layer thickness and its change in the process action using dynamic ultrasound and MRI investigation. Based on the studies carried out related to the assessment of morphological changes in the forearm, it was found that the thickness of the skin-fat layer becomes thinner in the process of performing the action. Considering that the electrical conductivity of muscles is several times higher than that of the skin-fat layer, it is expected that the apparent resistance will decrease with its thinning, and, accordingly, the EI will decrease, too.
Thus, four optimality criteria for choosing the ES size depending on the skin-fat layer thickness and electrical resistance were determined. So, for EI myography from the subsequent interpretation of signals point of view, it is important to determine the absolute and relative changes in signals as a result of muscle activity. In case of an analytical expression of EI, these changes can be represented as the values of the EI derivative for the studied parameter for the thickness of the skin-fat layer and the conductivity of muscle tissue, respectively. That is why the criteria were absolute (
,
) and relative (
,
) relationships between the partial derivatives of a two-layer model EI expressed by the skin-fat layer thickness (6) and the muscle tissues resistivity, respectively (7).
Figure 10 shows the EI change sensitivity depending on the thickness and skin-fat layer resistivity using analytical expressions for the optimality criteria and the MATLAB R2020b according to the assessment done. The relationsips were constructed for the range of thicknesses of the skin-fat layer from 2 to 10 mm and different specific electrical resistances of the skin-fat layer from 10 to 60 Ohm·m, which are considered acceptable for a frequency of 100 kHz [
32,
33,
34]. The range of values for the thickness of the skin-fat layer was selected based on the individual characteristic sizes of volunteers MRI and previous studies [
43]. If the ES size is small, the effective probing depth covers only the skin-fat layer, and if it is large enough, the probing depth corresponds to the muscle layer. The curves peaks correspond to the acceptable ES size, at which the ES is sensitive to changes in the medium parameters and the maximum change in the signal is observed for each presented criterion. The 70% threshold of the maximum value was chosen as the optimum for the relative change in the muscle tissues electrical resistivity criterion due to the ES design limitations.
Figure 11 shows the diagrams for selecting the sizes of the ES depending on the thickness and conductivity of the skin-fat layer, which were obtained for acceptable sizes. According to these relationships, it can be seen that the size of the ES weakly depends on the resistivity of the skin-fat layer, and it much more strongly depends on its thickness.
3.2. ES Size Selection by Pareto Optimality
To formalize the optimality problem, objective functions depending on variables with unknown values, which were varied in the optimization process to obtain an optimal solution, are used. When one optimality criterion is considered, the search usually boils down to obtaining the largest, as in the present case, or the smallest value of this criterion, that is, to solve the problems of maximization or minimization.
Since the EI change during muscle contraction is determined both by a change in the skin-fat layer thickness and by a change in the muscle tissue conductivity, the ES optimal sizes for these two processes are different. In the case when not one but several optimality criteria are set at once, it is possible to use an approach based on the Pareto principle.
The Pareto optimal solution’s main idea is such a feasible solution that cannot be improved by any of the available criteria without worsening by some other. The solution to the multiobjective optimization problem is the Pareto set of all Pareto-optimal feasible points.
Figure 12 shows the Pareto set corresponds to the Pareto frontier. In the Pareto set, points are not comparable with each other; i.e., all solutions to the problem are equivalent [
44].
In these studies, when constructing Pareto sets, the interelectrode distance a, which is varied in the range from 0 to 0.03 m, was used as a variable, and the EI sensitivity criteria were used as parametric functions.
Figure 13 shows the results.
Figure 14 shows the results of determination of the ES size based on the Pareto frontier representation set by dimensionless Pareto optimal solutions according to the criteria
and
since these criteria to a greater extent reflect the specifics, in fact, EI measurements.
4. Results and Discussion
In order to quantify the absolute changes in electrical impedance when performing an action, it is necessary to add to the calculations changes in the thickness of the skin-fat layer and the resistivity of muscle tissue, corresponding to real values. At the same time, we assume that there is no change in the resistivity of the skin-fat layer.
The change in the thickness of the skin-fat layer was assessed based on the analysis of the features of the morphological changes in the forearm during the performance of actions carried out in the framework of the authors’ previous studies using ultrasound [
45] and MRI [
43].
Figure 15 shows that using the means for graphic assessment of physiological parameters according to the images obtained, the thinning of the skin-fat layer was determined upon exposure, on average, by 0.5 mm for different volunteers without pressing.
In the framework of previous studies by the authors of [
45], the measured in vivo longitudinal electrical resistance of muscle tissue was comparable to the literature data obtained on the isolated tissues [
8,
18,
19,
20,
34]. Measurements in the projection of the center of the muscle body show that the resistivity of the muscle in the longitudinal direction increases by an amount of the order of 5%, which is associated with muscle contraction.
Thus, using the analytical solution for the two-layer model (3) and the presented changes in the thickness of the skin-fat layer and the resistivity of the muscle tissue, the absolute values of the electrical impedance were calculated. These data are presented in
Table 2 for electrode system sizes estimated using the Pareto optimality method described in the previous section. It is possible to trace how to change the contributions to the EI change for the two processes presented at ES different sizes. The table also shows the ratios
as EI changes due to these processes.
It is noted that it is necessary to use a larger ES to meet the criterion . The value corresponds to the selected acceptable interelectrode distance based on the criterion, while the value is based on the criterion. In this case, the acceptable size selection is made based on the range of values from to . When choosing the interelectrode distance, closer to , the ES will be more sensitive to changes in the skin-fat layer thickness; when closer to , it will be more sensitive to changes in muscle resistivity.
The approach analysis also shows that if the skin-fat layer conductivity is known and determined, for example, using the ES small size, its thickness and its change through ultrasound investigation, then the magnitude and change in the contracting muscle conductivity can be found as a result of the optimization procedure (8). The research development allows a more systematic approach to obtaining knowledge and data on the various muscle tissues during their contraction conductivity.
Pareto optimal solutions generally contain an infinite number of points. The optimization is carried out using optimization algorithms, as a result, the problem of multiobjective optimization is reformulated so that it can be solved. For this reason,
Figure 16 shows that the search for an acceptable ES size is limited to the case when the problem of maximizing the result of mathematical operations on objective functions—for example, sum, the sum of squares product—is considered.
The circles highlight the optimal solutions based on the objective functions’ sums corresponding to their maximum values.
Table 3 shows the acceptable ES sizes for the cases considered. Furthermore, this approach can be used to determine the acceptable ES size for different parts of the body individually.
5. Conclusions
In this paper, a method for determining the acceptable ES size for forearm muscle activity EI measurements using the method for assessing Pareto optimality according to criteria considering the skin-fat layer individual parameters is proposed for the first time. The sensitivity criteria were the EI absolute and relative changes in the skin-fat layer thickness and the muscle tissues resistivity. These mechanisms are the basis for the EI myography signals conditioning. In this case, the skin-fat layer thickness and its change can be determined, for example, using ultrasound investigation, and the skin-fat layer conductivity using small ES sizes.
To achieve the study results, based on the ultrasound investigation, MRI, and data on the tissues’ electrophysical properties, analytical and EI signal conditioning numerical models during muscle contraction were built. For the EI mathematical description relationships, a forearm two-layer model was reasonably chosen, which was represented by the skin-fat layer and muscles, which adequately describes the change in EI when hand actioning, provided that the ES size does not exceed the characteristic dimensions of the limb. At the same time, to simplify the mathematical model, the electrodes themselves were represented by points, which is acceptable when analyzing electrical impedance changes, considering the previous studies of the authors [
38].
Thus, the proposed mathematical model makes it possible to perform the calculations necessary to select the Pareto optimal sizes of the ES from the sensitivity of the EI measurements of muscle activity point of view, which were expressed by a change in the conductivity of muscle tissue and a change in the thickness of the skin-fat layer, considering individual parameters. These results can be used to develop systems for bionic control of the forearm based on EI. The study results allow considering the issues of solving inverse problems of limbs muscular activity electrical impedance probing in order to obtain new knowledge related to the muscle contraction biomechanical and morphological features and the EI myography signals conditioning mechanisms.