Analysis and modeling of equivalent schemes of biological tissue

Introduction.

By the nature of the electrical properties biological tissue is a heterogeneous environment. Organic substances (proteins, fats, carbohydrates, etc.) that make up the dense parts of tissues are dielectrics. However, all tissues and cells in the body contain fluids or are washed by them (blood, lymph, various tissue fluids), in addition to organic colloids, these fluids also contain electrolyte solutions, and therefore their resistivity to direct current is quite big.

The electrical conductivity of biological tissues (BT) is determined by the presence of free ions. The Ohm's law does not apply to BTs (due to polarization the current decreases in 2-3 ways/categories/). For the analysis of the effects of electrophysical properties of BTs on excitation processes, passive BTs are presented with equivalent electrical schemes (EESs) having alive BT impedance properties [1-6]. BT structural surveys are conducted in a wide range of frequencies (100 Hz... 10 KHz). For this purpose, the frequency capabilities of BTs are presented in the form of EESs that correspond to the regularity of the distribution of electricity in biological systems (the phenomenon of ionic conductivity and charge separation in BTs is caused by phase deviation of current and voltage). Tissue membranes have complicated structures, and according to Cole, they can be compared with capacitors [1]. The active ingredients of biological-electrical impedance (BEI) characterizing the flow of external and internal electrolytes (blood, lymph, interstitial fluid, etc.) are conditioned by the replacement of the amplifiers in the electric chain, and the capacitance components are characterized by separating the vacuum cleaners which is typical to the multidimensional BTs. At low frequencies (f < 100 Hz), the capability of the BT's is small, and the base deposit is an active component which is attributed to the upper layers of the skin. At high frequencies (f >10 kHz), the capacitance component of BTs decreases (decreases the impact of the discharging structures), and the active component strives for a constant value, characterizing the properties of high-definition BTs.

The nature of the problem and justification of the methodology. The construction of EES for BT is an important stage in the investigation of the given physical phenomenon. EES should not only reflect the properties of BTs in the frequency and temperature domains that are being investigated but should also anticipate their behaviour in larger areas. From this point of view, research, comparative analysis and modelling of the principles of construction of the BTs are a topical issue.

The results of the survey.

The number of BT model circuits is determined by the approximation of the tissue characteristics. You quency-dependent parameter allows evaluating the vitality and physiological state of the organism. The resistivity of biological tissues, determined for a given frequency of the current, can significantly change under the influence of physiological and pathophysiological factors. The non-affected BT characteristics have an exponential look, and dead BTs do not have a frequency dependency (the membrane that has the role of condensator is destroyed). The characteristic dependence of the BT impedance on the frequency, up to

Fig. 1. Frequency dependence of biotissue impedance

The cell is the basis of BT. The membrane is a per-the dielectric. Table 1 shows typical values of the elecmeable trical resistivity of the primary biological tissues for a components and is characterized by the properties of current frequency of 50 kHz barrier to the intracellular and extracellular.

Table 1. Typical values of electrical resistivity of biological tissues

The table shows that adipose and bone tissues have significantly lower electrical conductivity. Differences in resistivity can be explained, first of all, by different contents of fluid and electrolytes in organs and tissues. An important property of biological tissues is the dependence of their conductivity and relative dielectric constant on the frequency of the current.

Principles of construction of biological tissues 's equivalent circuits. The construction of equivalent circuits of biological tissues is an important step in its study. The analysis of existing schemes of existing BTs shows that the composition of the elements used is limited by the use of active and reactive elements. This allows formalizing the process of analyzing and synthesis of BT schemes.

For this purpose, the mathematical theory of the multitude theory can be used to formulate the problem.

Let's suppose that there is a multitude of a_; elements based on which it is possible to build an equivalent scheme. The quantity of the elements in the multitude is N.Suppose that K is the minimal collection of the required blocks. In that case the multitude of the main elements can be introduced in the following way:

If 1_; =| L_; |< N, it means that there exists another implementation for the construction of the i -the version, since but for the 1 main elements, there can also be C; = N -- 1_; additional elements that comprise C multitude. It is obvious that

Each element of the C multitude can be included or not included in the equivalent scheme. Consequently the number of implementations in i-th version will be C

n. = 2: The implementations of i -th version can be represented in the following way F = {fj}, j = l,n where f`- is the j-th implementation of the version i.

BT variants can be referred to the third type that are represented in the table 2.

Table 2. Topological variants of two, three and four element equivalent schemes of biological tissue

In the general case, the impedance Zx can contain both active - Rx and reactive Cx - capacitive resistances.

From Table 2 it can be seen that two and three elemental variants of biological tissues are realized in two sub-variants, and the four element variant is realized in three sub-variants.

The use of parallel and serial connection of active and reactive elements can be obtained various options for constructing equivalent electrical circuits of biological tissues. The resulting schemes can be considered as electrical circuits for the replacement of real biological tissue.

In the third table is given the scope of application of BTs, EESs and impedance.

Table 3. BT's known electrical equivalent schemes

The cell membrane is normally composed of a non-conductive lipid layer which is located between the two layers of the transmitter protein molecules. In low- frequency domain (<100 Hz) power/current/ does not flow through the cell membrane, it only passes through the extracellular liquid, and the intracellular fluid does not participate in the process. In the high-frequency range (> 100 Hz), the intracellular liquid also participates in the power transmission process, which extends across the extracellular and intracellular areas. Consequently, the parallel model of Table3 does not fully reflect the BEI capabilities of the BTs.

For this reason, Fricke-Morse's model (table 3) is used to investigate the frequency characteristics of the BEI, according to which the extracellular and intracellular fluids are transmitters and the cell membrane is dielectric that characterizes the electrical capacity. In low and high-frequency domains, the ability to inject in the BEI strives for zero, and, in the case of elevated, the membrane does not impede the transmission. Fricke- Morse's model can be replaced by the most suitable model for analysis (table 3).

As already noted, in the study of biological tissues an important task is to consider the nature of the dependences of the frequency characteristics of the im- pedance (full electrical resistance of an alternating current circuit) - biological tissues.

Table 4. The phase angle for different types of tissues

The absolute value (modulus) of the electrical impedance is determined by the expression:

In practice, the impedance value can be determined by measuring the amplitude (or effective) values of the voltage U0 and the current strength I0.

The phase angle 9 determines the ratio of the reactive and active components of the impedance

The values of the phase angle obtained at a frequency of 1 kHz for various biological tissues are given in Table 4 [10].

Table 4. The phase angle for different types of tissues

For qualitative and quantitative modelling of the electrical properties of individual parts of the human body, simple equivalent electrical circuits of biological tissues are widely used in Table 3, due to the presence of active and reactive components of the impedance. Therefore, it is possible to simulate the electrical properties of biological tissues using resistors, and capacitors - carriers of capacitive resistance. Due to this simulation, it is possible to evaluate the passive electrical properties of biological tissues, and the use of the MATLAB software environment to predict the behaviour of biological tissue based on the model's response.

These circuits were simulated using the MATLAB software at C = 8.5 ^F = 85 x 10^-7 F, R1 = 120 Q, and R2 = 100 Q. We calculated phase angle with <p=an- gle(Z) formula, where angle(Z) is taken from MATLAB function list. It returns the phase angles, in radians, for each element of complex array Z.

The results of the simulation are shown in Figure 2-9.

Pic. 2 Z1 phase angle dependence of frequency and active R resistance