Nonlinear Dynamics Method in the Impedance Signals Analysis of the

Eye Blood Flow of Patients with Glaucoma

Anna A. Kiseleva

, Petr V. Luzhnov

, Alexander P. Nikolaev

, Elena N. Iomdina

and

Olga A. Kiseleva

Bauman Moscow State Technical University, Moscow, Russian Federation

Moscow Helmholtz Research Institute of Eye Diseases, Moscow, Russian Federation

Keywords: Nonlinear Dynamics, Transpalpebral Rheophthalmography, Eye Blood Flow, Glaucoma.

Abstract: The article considers the possibility of analyzing the signals of transpalpebral reophthalmography with the

help of the method of nonlinear dynamics. In analyzing the signals, it was considered a method with a

reconstruction of the signal attractor. An optimal time delay was chosen for the reconstruction, which allowed

us to construct an attractor in the space of selected coordinates. Next, we analyzed the mass centers of the

reconstructed attractors, the position of the coordinates allowed us to formulate a decisive rule for comparing

and dividing signals into groups.Verification of the results carried out on the model signals. The application

of this technique was shown with the help of the examples of signal analysis in patients with primary open-

angle glaucoma.

1 INTRODUCTION

In the structure of ophthalmopathology, primary

open-angle glaucoma (POAG) occupies one of the

first places among eye diseases that inevitably end in

blindness (Quigley, H. A., Broman, A. T., 2006).

Currently, the pathogenesis of the development and

progression of POAG is associated with increased

intraocular pressure (IOP) (Cherecheanu A.P.,

Garhofer G., et al., 2013). Increased IOP damages the

nerve cells of the retina, from which the optic nerve

is formed. It is manifested by a gradual loss of vision

function. However, other factors can lead to the

development of POAG. So, a lot of researches have

described studies where, along with increased IOP,

one of the risk factors for the development of POAG

and the progressive deterioration of visual functions

in this diseaseis a decreasing level of the blood filling

in the brain vessels and eye vessels (Caprioli, J.,

Coleman, A. L., 2010; Venkataraman, S. T., Flanagan,

J. G., et al., 2010; Calvo, P. et al., 2012). Studying the

eye hemodynamics can provide the necessary

information about the clinical course of POAG.

Electrical impedance method is a technique for

obtaining an image in sections of an impedance

distribution body (resistance of various organs in

response to an electric current) by means of non-

invasive electrical sounding (Adler A., Gaburro R.,

2016). The current flowing through the body creates

a volume distribution of electrical potential (voltage).

The potential decreases along the current line with

distance from the active (injecting current) electrode.

The voltage drop per unit length (electric field

strength) is proportional to the magnitude of the

current and the resistance of the body in accordance

with Ohm’s law. By measuring the voltage drop and

knowing the amount of current, it is possible to

calculate the resistance value. The reconstruction

algorithm allows the use of voltages measured only

on the surface of the body, to calculate the spatial

distribution of resistivity (or electrical conductivity)

within it. This method is the basis of the method of

reophthalmography (ROG) (Lazarenko V.I.,

Komarovskikh E.N., 2004). ROG is the method for

assessing the state of the blood flow in the eye. ROG

is a method for studying the pulse blood filling in the

vessels of various organs and tissues, based on

recording changes in the total electrical resistance of

tissues. In the classical method of registration, the

electrodes are mounted directly on the surface of the

eye near the lens. It leads to necessity of anesthesia

for conducting researches.

To solve the limitations of the classical technique,

a new registration technique has been developed - the

method of transpalpebral reophthalmography (TP

ROG) (Luzhnov P.V., Shamaev D.M., et al., 2015;

Kiseleva, A., Luzhnov, P., Nikolaev, A., Iomdina, E. and Kiseleva, O.

Nonlinear Dynamics Method in the Impedance Signals Analysis of the Eye Blood Flow of Patients with Glaucoma.

DOI: 10.5220/0007554800750080

In Proceedings of the 12th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2019), pages 75-80

ISBN: 978-989-758-353-7

Luzhnov P.V., Shamaev D.M., et al., 2017; Shamaev

D. M., Luzhnov P. V., et al., 2016). In this version of

the study, the electrodes are positioned on the closed

eyelid.

Analysis of the received signals includes

qualitative and quantitative analysis. Qualitative

analysis of signals is the definition of the pulse curve

form. Evaluating the type of the pulse curve, usually

emit hypertonic, normotonic and hypotonic types.

Unfortunately, now the method of analyzing the type

of the pulse wave is not objectivized. Factors

affecting vascular tone include biophysical,

biomechanical and hydrodynamic factors. Together,

they affect the type of the pulse curve and further

determine the result of the qualitative analysis of signals.

From the point of view of modern biophysics, the

combination of these factors, namely, their mutual

influence on each other, can be described using the

nonlinear properties of the system (Strogatz H., S.,

2014). Dynamic processes play an important role in

the development and manifestation of the nonlinear

properties of systems, in particular dynamic chaos.

Chaos is characterized, first of all, by internal self-

sustaining fractal fluctuations (Betelin, V.B., Eskov,

V.M., et al. 2017). This effect is found in the analysis

of many biological signals - studies of the electrical

activity of the heart (Elhaj, F., Salim, N., et al., 2016),

neural activity of the brain (Akar S.A., Kara S., et al.,

2015), respiration (Kiseleva A., Luzhnov P., et

al.,2018), and blood filling of various organs. The

theoretical basis for describing chaos in the work

(Takens F., 1981) was proposed by F. Takens. In

accordance with the Takens theorem, using phase-

spatial representations of signals - attractors for

analysis, it becomes possible to conduct their

qualitative and quantitative comparison (Charlton

P.H., Bonnici T., et al. 2017).

Thus, the main idea of this work is to eva-

luate the possibility of types transpalpebral

reophthalmography signals comparing with the help

of nonlinear dynamics methods, assessing the main

comparison parameters and further verifying these

parameters for analyzing signals of a transpalpebral

reophthalmography in patients with primary open-

angle glaucoma.

2 MATERIALS AND METHODS

2.1 Problem Statement

Due to the fact that in the qualitative analysis of pulse

curves, the presence of the physiological relationship

between vascular tones is taken into account, the

signals are usually divided into three main types:

hypotonic, hypertonic and normotonic types. As an

object of study, model signals of these types were

considered. An example of a pulse curve waveform is

shown in Figure 1.

Currently, a contour analysis based on the method

described in work (Webber C. L., Sbilut J. P., 1994),

is used for the analysis of such type signals (Figure

1). In contour analysis, the most informative

indicators are measured: the amplitudes of the

systolic and diastolic waves, the amplitude of the

rheographic wave at the incisure level, the duration of

the anacrotic limb and the catacrotic limb.

This analysis underlies the classification of

signals currently used in clinical practice. A

significant drawback of this classification is a high

probability of errors during automatic separation of

the signals.

Figure 1: The shape of the pulse blood filling curve of

model signals with the release of the main signal regions.

To solve this problem, it has been proposed to use

the analysis of phase-spatial portraits of the pulse

curve, which allows to classify signals not only by

their quantitative characteristics (wave amplitude

value, length of sections), but also helps to divide

signals into groups with the help of analysing the

shape of the pulse curve itself. As a result, the

following stages of work have been formulated:

Stage 1 - The reconstruction of the pulse curve

attractor;

Stage 2 - The analysis of the obtained form of

the attractors;

Stage 3 - The formulation of the classification

algorithm;

Stage 4 - The verification of results.

2.2 Reconstruction of the Attractor

The phase portrait (signal attractor) can be

constructed in two ways. The first method (the delay

method) involves constructing a pseudo-phase

BIODEVICES 2019 - 12th International Conference on Biomedical Electronics and Devices

portrait by displaying the observed values of x (t) on

a subspace with coordinates x (t + d), where d is the

time delay. The second method is based on the

imaging the phase portrait on a subspace with

coordinates x (t), x * (t), where x * (t) is the estimate

of the first derivative of x (t) in the t point (Palit K.,

Mukherjee S., et al., 2013). Due to the fact that TP

ROG signals are complex in their structure, we have

considered the first method to simplify calculations

and reduce processing time.

According to the Takens theorem (Takens F.,

1981), attractor reconstruction process of a biological

system consists in building on a number {x

}

1,...,

∞

new (reconstructed) attractor with the same

parameters as the original one, which have been

received as a result of measurement at certain

intervals of the considered signal x(t) (Palit K.,

Mukherjee S., 2011). For this, m-dimensional vectors

are constructed {zj}

1,...,

∞

, where z

j,k

j+d(k–1)

, where

d - time delay. Then, for sufficiently large m, the

parameters of the reconstructed attractor in the m-

dimensional space Z coincide with the parameters of

the original attractor in the n-dimensional space X

(Webber C. L., Sbilut J. P., 1994). Selection and

calculation of dimension m-dimensional space is

described in (Luzhnov P.V., Shamaev D.M. et. al.,

2018).

The key role for calculating all parameters of the

obtained attractor belongs to the correct choice of

time delay (Webber C. L., Sbilut J. P., 1994).

Typically, the time delay is selected so that each

reconstructed next vector add the most information

about the attractor or less can be correlated with the

previous one (Gracia J., Seppa V-P., et. al., 2017).

When reconstructing biological attractors, stable

results in the selection time delay gives technique

proposed in (Maiorov O. Yu., Glukhov A. B., et al.,

2007), which is based on the form of the original

signal. To analyze the pulse curve, it is necessary to

distinguish in the received portrait the beginning of

the cardiac cycle, the maximum amplitude and

incisure. Based on the analysis of the duration of

these sections, values of 0.08 sec and 0.22 sec have

been chosen in accordance with the data described in

the work (Kiseleva A., Luzhnov P., et al.,2018).

Figure 2 shows an example of the resulting

attractor of the signal of the pulse blood filling with

the 60 seconds duration. The waveform of the signal

itself (Figure 2a) and the resulting signal attractor

(Figure 2b) are shown.

Figure 2(a): The analyzed signal of pulse blood filling (10

sec. section), X axis - time, s, Y axis - signal amplitude,

Ohm.

Figure 2(b): The resulting phase portrait of a signal with the

60 seconds duration, on the X axis: signal x(t), Ohm, on the

Y axis: signal x(t - 0.08), Ohm, on the Z axis: signal x(t -

0.22), Ohm.

2.3 Analysis of the Shape Obtained

Attractors

In (Luzhnov P. V., Shamaev D. M., et al., 2018;

Kiseleva A., Luzhnov P., et al., 2018), an analysis of

the obtained attractors shape based on the mass center

of the obtained figures was considered. In the

framework of these works, the projection of the

attractors on 3 axes (XY, YZ, XZ) was considered,

the resulting form of the attractor was approximated

by two triangles with one common base. In the works,

the ratio of heights drawn from the vertices of

polynomial triangles to a common base was analyzed.

In this work, it has been proposed to modify the

algorithm and consider not the attractors of the pulse

curve, but analyze the attractors of the derived signal.

A significant advantage of this method is the exact

separation of the cardiocycle into systolic and

diastolic components. Figure 3 shows the derivative

of the pulse curve signal and its reconstructed

attractor. Areas A and B correspond to the systolic

and diastolic signal components, respectively.

Nonlinear Dynamics Method in the Impedance Signals Analysis of the Eye Blood Flow of Patients with Glaucoma

Figure 3(a): Derivative waveform pulse volume,

highlighting the main areas of signal.

Figure 3(b): The attractor derivative signal projected on XY

axis. On the X axis - the signal x (t), Ohm, on the Y axis is

the signal x (t - d), Ohm, area A - the systolic part, area B -

the diastolic part.

2.4 Formulation of the Classification

Algorithm

Analyzing the shape of the attractor (Figure 3b), it is

obvious that the resulting shape can be approximated

by two geometric figures - ovals.

To analyze the obtained forms of the attractor

(Figure 3b), it has been proposed to consider the

distance between the mass centers of regions A and B

(Figure 4). For the introduction of automatic signals

separation, it was necessary to divide the mass centers

of the signals according to three main types:

hypotonic type, hypertonic and normotonic type.

We considered model signals of each type

(normotonic, hypotonic, hypertonic), the

requirements for the centers of mass location for each

signal type (Figure 5) were formulated based on the

Figure 4: The attractor derivative signal with directions the

mass centers of systolic and diastolic components.

analysis. Moreover, it was determined that the

coordinates of the mass center of the systolic

component are identical for signals of three types and

had following coordinates: 0.002 (X coordinate) and

0.002 (Y coordinate). The following visual results

have been obtained on the projection plane of the

phase space based on the model signals with the

condition of their splitting into normotonic,

hypotonic and hypertonic types (Figure 5). The

results with the obtained coordinates of diastolic

centers are presented in Table 1. The diameter of the

region around the mass center of each signal type is

0.001 and was chosen according to the average spread

of the mass centers of the signal around the obtained

point.

Table 1: Mass center coordinates.

Type signal

Distance

between

centres

Normotonic

-0.008

0.010

Hypotonic

0.001

0.0005

0.001

Hypertonic

-0.002

0.004

As a result, the decision rule for signal

classification have been formulated: if the mass

center of the systolic component of the signal falls

within the range of 0.002 (X) and 0.002 (Y), and the

center of the signals diastolic component lies in the

ranges presented in Table 1, the distance between

centers do not exceed the values shown in Table 1 for

each type of signal, then the algorithm will

automatically separate the signals.

BIODEVICES 2019 - 12th International Conference on Biomedical Electronics and Devices

Figure 5: The mass centers of the systolic component and

the diastolic component for the types of signal. The dashed

line shows the boundary separation between the types of

signals.

2.5 Using of the Method in Clinical

Practice

The developed method can be used to classify the

pulse blood filling signals. In our work, the

application of this method for the signals of pulse eye

blood filling was considered. The application of the

considered classification method becomes possible

due to the identity of the forms of the pulse curves.

To test the method on the TP ROG signals, 3 types

of signals have been considered: 1) without

ophthalmopathology; 2) stage II POAG; 3) stage III

POAG. The study was performed in the group of 10

subjects (average age 54.2 ± 15.4 years) without

ophthalmopathology and in the group of 10 patients

with POAG were divided into two parts: the 1-st part

- 6 patients with stage II of POAG (mean age 72.0 ±

8.2 years), the 2-nd part - 4 patients with stage III

POAG (mean age 69.4 ± 6.8 years). The study has

been conducted on each signals type with the 3

seconds duration. Table 2 shows the coordinates of

the obtained diastolic centers. The obtained

coordinates of the systolic center are 0.30 ± 0.30 (X

coordinate) and 0.25 ± 0.02 (Y coordinate).

Similar to Figure 5, in Figure 6 for the case of a

POAG on the plane of projections, the boundaries of

the separation according to the stage of the disease are

shown.

To specify the values of the mass centers, it is

necessary to normalize TP ROG signals amplitudes

and conduct a study on a larger sample of TP ROG

signals.

Table 2: Mass center coordinates for TP ROG signals.

Type

signal

X, mOhm

Y, mOhm

Distance

between

centers

Healthy

-1.5

-1.2

2.1

stage

POAG

-1.7

-1.8

2.1

stage

POAG

-3.0

-2.1

3.8

Figure 6: The centers of mass of the systolic component for

ophthalmic signals. The dashed line shows the boundary

separation between the types of signals.

3 RESULTS

As a result of this work, a new method for separating

signals based on nonlinear dynamics has been

proposed. To solve the problem, the following steps

have been carried out: 1) the methods of nonlinear

dynamics have been considered and the optimal

method for biological systems has been selected; 2)

using the selected method, phase portraits of the

considered biological signals have been

reconstructed; 3) a classification algorithm has been

developed and a decisive rule for signal separation

has been formulated; 4) verification passed.

In the future, it is planned to carry out the

separation of signals in real time, as well as the

possible application of this algorithm for TP ROG

signals analysing for pulse blood filling in eyes

researches.

CONFLICT OF INTEREST

The authors declare that they have no conflict of

interest. The paper was supported by a grant from

RFBR (No.18-08-01192).

Nonlinear Dynamics Method in the Impedance Signals Analysis of the Eye Blood Flow of Patients with Glaucoma

REFERENCES

Adler A., Gaburro R., WRB Lionheart, “Electrical

Impedance Tomography”, in Handbook of

Mathematical Methods in Imaging, 2nd ed O Scherzer

(Ed), Springer, 2016.

Akar S.A., Kara S., Agambayev S., Bilgic V. (2015)

Nonlinear analysis of EEG in major depression with

fractal dimensions. 37th Annual International

Conference of the IEEE on Engineering in Medicine

and Biology Society: 7410–7413. DOI:

10.1109/EMBC.2015.7320104.

Betelin, V.B., Eskov, V.M., Galkin, V.A. et al. (2017)

Stochastic volatility in the dynamics of complex

homeostatic systemsDokl. Math. 95: 92. DOI:

10.1134/S1064562417010240.

Caprioli, J., Coleman, A. L. (2010) Blood Pressure,

Perfusion Pressure, and Glaucoma. Am. J.

Ophthalmol. 149, 704–712.

Calvo, P. et al. (2012) Predictive value of retrobulbar

blood flow velocities in glaucoma suspects. Investig.

Ophthalmol. Vis. Sci. 53, 3875–3884.

Charlton P.H., Bonnici T., Tarassenko L., et al. (2017)

Extraction of respiratory signals from the

electrocardiogram and photoplethysmogram: Technical

and physiological determinants. Physiological

Measurement 38(5): 669-690. DOI: 10.1088/1361-

6579/aa670e.

Cherecheanu A.P., Garhofer G., Schmidl D., Werkmeister

R., Schmetterer L. Ocular perfusion pressure and ocular

blood flow in glaucoma. CurrOpinPharmacol. 2013;

13: 36-42. DOI: 10.1016/j.coph.2012. 09.003.

Elhaj, F., Salim, N., Harris, A., Swee, T. and Ahmed, T.

(2016). Arrhythmia recognition and classification using

combined linear and nonlinear features of ECG signals.

Computer Methods and Programs in Biomedicine, 127,

pp.52-63. DOI 10.1016/j.cmpb.2015.12.024.

Gracia J., Seppa V-P., Pelkonen A., Kotaniemi-Syrjanen A.

et al. (2017) Nonlinear Local Projection Filter for

Impedance Pneumography. European Medical

andBiological Engineering Conference 65: 306-309.

DOI: 10.1007/978-981-10-5122-7_77.

Kiseleva A., Luzhnov P., Dyachenko A. and Semenov Y.

(2018). Rheography and Spirography Signal Analysis

by Method of Nonlinear Dynamics.In Proceedings of

the 11th International Joint Conference on Biomedical

Engineering Systems and Technologies - Volume 1:

BIODEVICES, ISBN 978-989-758-277-6, pages 136-

140. DOI: 10.5220/0006579301360140.

Lazarenko V.I., Komarovskikh E.N. Results of the

examination of hemodynamics of the eye and brain in

patients with primary

open-angle glaucoma. Vestnik Oftal’mologii 120(1),

32–36 (2004).

Luzhnov P.V., Shamaev D.M., Iomdina E.N., Markosyan

G.A., Tarutta E.P., Sianosyan A.A. (2017) Using

quantitative parameters of ocular blood filling with

transpalpebralrheoophthalmography. In: Eskola H.,

Väisänen O., Viik J., Hyttinen J. (editors). EMBEC &

NBC 2017. IFMBE Proceedings, vol. 65; p. 37–40,

https://doi. org/10.1007/978-981-10-5122-7_10.

Luzhnov P.V., Shamaev D.M., Iomdina E.N., Tarutta E.P.,

Markosyan G.A., Shamkina L.A., Sianosyan A.A.

(2015) Transpalpebraltetrapolarreoophtalmography in

the assessment of parameters of the eye blood

circulatory system.

VestnikRossiiskoiakademiimeditsinskikhnauk; 70(3):

372–377, https://doi.org/10.15690/vramn.v70i3.1336.

Luzhnov P. V., Shamaev D. M., Kiseleva A. A., Iomdina E.

N. (2018), Analyzing rheoophthalmic signals in

glaucoma by nonlinear dynamics methods. IFMBE

Proceedings 68/2: pp.827-831. DOI: 10.1007/978-981-

10-9038-7_152.

Maiorov O. Yu., Glukhov A. B., Prognimak A. B.,

Fenchenko V. N. (2007) Realization of the method of

displacement by estimating the sizes of the axes of the

attractor of the dynamic brain system. - Proceedings of

the Institute of Cybernetics. - K., p. 153, pp. 5–12.

Palit K., Mukherjee S., Bhattacharya D.K. (2013) A high

dimensional delay selection for the reconstruction of

proper phase space with cross auto-

correlation.Neurocomputing 113: 49-

57.https://doi.org/10.1016/j.neucom.2013.01.034.

Palit K., Mukherjee S. (2011) Generalized autocorrelation

and its application in attractorreconstruction. Bull. Pure

Appl. Math. 5(2): 218–230.

Quigley, H. A., Broman, A. T. (2006) Number of people

with glaucoma worldwide. Br. J. Ophthalmol. 90, 262–

267.

Shamaev D. M., Luzhnov P. V., Pika T. O., Iomdina E. N.,

Kleyman A. P., Sianosyan A. A. (2016), Applying

transpalpebral rheoophthalmography to monitor

effectiveness of the treatment of patients with

glaucoma”. International Journal of Biomedicine 6(4):

pp.287–289. DOI: 10.21103/Article6(4)_OA8.

Strogatz H., S. (2014). Nonlinear Dynamics and Chaos.

With Applications to Physics, Biology, Chemistry, and

Engineering. 2nd ed. Boca Raton, p.532.

Takens F. Detecting strange attractors in turbulence / In

Rand D.A. and Young L.S. // Dynamical Systems and

Turbulence, Lecture Notes in Mathematics. 1981. V.

898. Р. 366–381.

Venkataraman, S. T., Flanagan, J. G. & Hudson, C.

(2010) Vascular Reactivity of Optic Nerve Head and

Retinal Blood Vessels in Glaucoma- A

Review. Microcirculation 17, 568–581.

Webber C. L., Sbilut J. P. Dynamic Assessment of

Physiological Systems and States Using Rrecurrence

Plot Strategies.// J. Appl. Physiol., 1994, 76, p.965–

993.

BIODEVICES 2019 - 12th International Conference on Biomedical Electronics and Devices