**5. Evaluation of limbs muscle asymmetry on the basis of the method of multifractal fluctuation analysis**

The study of signals of biological systems should be carried out with an account of such important factors as the existence of nonlinear interrelationships between different physiological indicators and between reports of organic biological signals. Their non-stationary nature is limited to the application of classical methods of analysis of biological signals.

Chaosticity of dynamic processes of biological systems and fractal properties of biosignals obtained from different structures of the organism require descriptions of the processes occurring in these systems with the use of nonlinear analogical dynamics, theories of dynamism. Based on the application of such methods of analysis, it is possible to obtain diagnostic information indicators, which come from each of the methods that can serve as an early predictor of disease of a particular organ.

In this article, the assessment of fluctuation of random processes using the method of fluctuation analysis of electromyographic signals is considered. The meaning of these signals is represented in the form of a dynamic time series to assess the asymmetry of the limbs of the human body [25–28].

There are not a large number of publications in science-based databases using multifractal approaches for the analysis of myographic signals. The author used fractal analysis to assess the morphological complexities of the surface of the connecting parts, scanned by a 3D scanner [29]. The possibility of predicting the asymmetry of electromyographic signals from the ends (arms and legs) was tested by the method of multifractal multiplicity [16]. The Higuchi fractal size was used to establish the imbalance of the jaw and the loss of muscle strength in the hands by helping the surface electromyography [17].

In the work are given examples of changes in the fluctuation functions of the electromyographic signal for different levels of load on a specific muscle [30].

Electromyographic signals are considered in this work, comparable to the ends: the right and left part of the hip muscle. In contrast to the classical approach to the assessment of asymmetry with the use of a single point (maximum) of the measured signal, we consider the multiplicity of the measured signals of the finite points as heterogeneous unevenly distributed fractional points. To assess the chaosticity of myographic signals, it is possible to use the characteristics of fractal multiplicity, such as the fractal size of Hausdorff, the indicator of Herst, generalized size, correlation, and information size [31–34].

#### **5.1 Materials and methods**

In the current period, in clinical conditions, the assessment of asymmetry is carried out at the maximum value of the amplitude of the muscles. This process includes in itself "maximum contraction of the muscle - the achievement of the peak in the maximum relaxation of the muscle", is the informative result and considered only the value of the peak. If you consider the whole process as a dynamic time series, then in order to make a decision it is necessary to include all the elements of the coincidence that creates this process. To examine all the data in a dynamic range and the minimum step size of a window in one report it is necessary to ensure the frequency of record lengths and window sizes. This segmentation is performed in two passages, performed in opposite directions.

The calculation algorithm consists of the following steps.

Initially from the series x(k), k = 0,1,2 … N allocate the total fluctuation (or fluctuation profile)

$$Y(i) = \sum\_{k=1}^{i} [\mathbf{x}(k) - \overline{\mathbf{x}}], \ i = 1, 2 \dots N,\tag{5}$$

where *x* - is the average arithmetic series х(к).

Divide the full interval ½ � **1**, *N* by *Ns* ¼ ½ � *N=s* segments, each of which contains s values. The elements of the new interval will be *x*ð Þ <sup>ν</sup>�**<sup>1</sup>** *<sup>s</sup>*þ**<sup>1</sup>**, … , *x*ν,*<sup>s</sup>*, ν = 1, … , Ns. It follows that in the case of *s* **>** *N=***4** the function of the deformed variance loses statistical informativeness due to the small number of *Ns* **< 4**, used in the medium. At the same time it is necessary to fulfill the inequality s **< 10** [35–37].

After changing the random variable *Y i*ð Þ we add *y*νð Þ*i* 6¼ **0** to find the polynomial for this function using the method of the least squares, and calculate the variance in the interval ν:

$$F^2(\nu, s) = \frac{1}{s} \sum\_{i=1}^{s} \left\{ \mathbf{y}[(\nu - 1)s + i] - \mathbf{y}\_{\nu}(i) \right\}^2 \tag{6}$$

for segments ν ¼ **1**, **2**, … *Ns*, in the case of fragmentation is performed in the direct direction, and for the reverse sequence ν ¼ *Ns* þ **1**, … , **2***Ns* we use the equation *Methods and Tools for Assessing Muscle Asymmetry in the Analysis of Electromyographic… DOI: http://dx.doi.org/10.5772/intechopen.103061*

$$F^2(\nu, s) = \frac{1}{s} \sum\_{i=1}^{s} \left\{ \mathbf{y}[N - (\nu - N\_s)\mathbf{s} + i] - \mathbf{y}\_\nu(i) \right\}^2 \tag{7}$$

Conduct the distribution of deformed dispersions at intervals

$$F\_q(\mathbf{s}) = \left\{ \frac{\mathbf{1}}{2\mathbf{N}\_s} \sum\_{\nu}^{2\mathbf{N}\_s} \left[ F^2(\nu, \mathbf{s}) \right]^{q/2} \right\}^{1/q} \tag{8}$$

Coefficient 2 in the signifier and in the upper limit of the sum used only for reflation of the algorithm with two passes.

At zero value of the order q this equilibrium contains indefiniteness, and then by definition

$$F\_0(\mathbf{s}) = \exp\left\{\frac{1}{4N\_s} \sum\_{\nu=1}^{2N\_s} \ln\left[F^2(\nu, \mathbf{s})\right]\right\} \tag{9}$$

To find the dependence *Fq*ð Þ*s* we change the time scale s with the fixed indicator *q* and represent it in binary logarithmic coordinates.

If the studied series corresponds to a similar number (s ! 0), then the scaling relationship is fulfilled

$$F\_q(\mathfrak{s}) \sim \mathfrak{s}^{h(q)},\tag{10}$$

where *h q*� � is a generalized index of Herst [35]. For stationary time series h (2) = H is the known index of the degree of Hearst, which with one side does not depend on *q*, and with the other variance is the same for all segments. In the positive/negative value q, *h q*� � indicates the scaling behavior of segments with large/small fluctuations [35].

For small values of s, *h q*� � we determine the linear regression [36].

$$F\_q(\mathfrak{s}) = h(q) \cdot \ln \mathfrak{s} \tag{11}$$

The standard representation of the scaling properties of the temporal series assumes a transition from the indicator Hearst *h q*� � to the mass indicator *τ q* � � and the spectral function f (α), the size of Rennie, which are in [37].

Mass index is calculated by the formula:

$$
\pi(q) = qh(q) - \mathbf{1} \tag{12}
$$

The spectral function has a connection with the mass indicator and the indicator Hearst:

$$a(q) = \frac{d\tau(q)}{dq} = h(q) + q\frac{dh}{dq}$$

$$f(a) = \mathbf{1} + q(a)[a - h(q(a))]\tag{13}$$

where *α* is an indicator of Gelder, which estimates the probability of the occurrence of the element of fractal multiplicity in the ν-th fragment.

The size of Rennie is determined by the equilibrium of spectral function and (12)

$$D\_q = \frac{qh(q) - 1}{q - 1},\tag{14}$$

Apparently the equation is not fulfilled when q = 1. For this value according to the rules of Lopital (14), we use the dependence:

$$D(\mathbf{1}) = h(\mathbf{1}) + \frac{\partial h}{\partial q}\bigg|\_{q=1} \tag{15}$$

#### **5.2 Discussion of results**

With the purpose of demonstration of the possibility of using this method for quantitative assessment of asymmetry of the patient's muscle. Conduct testing of the expressed method on EMG signals. Calculation is carried out separately for each multiplicity.

The experiments used muscle signals: Quadriceps femoris muscle - rectus femoris, Quadriceps femoris muscle - vastus lateralis, Quadriceps femoris muscle vastus medialis, Gastrocnemius muscle - lateral part, Gastrocnemius muscle medial part, and biceps femoris muscle.

In accordance with the above-mentioned algorithm for the computational experiment, medical records were used of the quadriceps femoris muscle-vastus lateralis, obtained with a 16-channel electromyograph ME6000 in lateral and right medial and medial lobes.

Signals are marked **х***l*ð Þ *k* - for the separation of the left part, **х***r*ð Þ *k* - for the separation of the right part. Each signal was broken down into 5 segments (*Ns* ¼ **5**). In the time series in both directions **х***l*ð Þ *k* s = 83 and in **х***r*ð Þ *k* s = 83.

Dependence *Fq*ð Þ*s* for differential values *q* ¼ **1**, **9** obtained after accounting trend in formulas (6) and (7) – **Figure 10**. When the values *q* <0 and *q* >4 are so different that it is possible to say that they are repeated (**Figure 10**) and with these values they lose their significance. It is found and in the values of the indicator Renée calculated using (12) (**Figure 11**).

The results of the indicator Hersta are shown in **Figure 12**. At *q*> **2** values it is possible to observe well-marked value of the indicator, which gives an opportunity to use it as an information indicator. Between the values of *h q* is determined by the relationship *h<sup>L</sup>* ð Þ*q* **>** *h<sup>R</sup>* ð Þ*<sup>q</sup>* and this is observed at positive values of *<sup>q</sup>*.

#### **Figure 10.**

*The dependence of the variance of the quadriceps femoris muscle-vastus lateralis on the size of the segment s ((a) - s = 83; (b) - s = 43) at different values of the deformation parameter q (q = 1–9).*

*Methods and Tools for Assessing Muscle Asymmetry in the Analysis of Electromyographic… DOI: http://dx.doi.org/10.5772/intechopen.103061*

**Figure 11.**

*Rennie's graphics for the right and left parts.*

#### **Figure 12.**

*Herst's indicator graphics are for the left and right parts.*

Defining *D*ð Þ *<sup>L</sup> <sup>q</sup> andD*ð Þ *<sup>R</sup> <sup>q</sup>* parameter *Dq* obtained from the left and right parts of the Quadriceps femoris muscle-vastus lateralis muscle of the patient. From **Figure 12** easily determines the relationship

$$D\_q^{(L)} > D\_q^{(R)}$$

This inequality is fulfilled for 0 < q < 4, and for other values we get repetitive values (it can be seen from the graphs), which lose their diagnostic value

$$D\_q^{(L)} = D\_q^{(\mathbb{R})}$$

The numerical value of the spectral function f (α) is shown in **Figure 13**.

As in the graphs of the indicator Hersta (**Figure 12**) here is also found good correlation in the values *q* >2. With the increase of the value of q in the signals of the left part of the mouse, the value of f(α) is increased in comparison with the value of the right part

$$f\_q^{(L)} > f\_q^{(\mathcal{R})}$$

#### **5.3 The conclusion**

The results show that the proposed method of acceptance and for the analysis of the detection of asymmetry of human extremities. It is possible to consider that the

**Figure 13.** *Graphs of the spectral function f(α) for the right and left parts.*

magnitudes that are used as the main characteristics of multifractals are informative signs that can be used to detect violations of the asymmetry of the ends. The results obtained show that in order to reveal the asymmetry of the finite points it is necessary to use the indicator Hearst and the value of the spectral function, as the informational range is wider than that of the indicator Reny.
