Method for Quantitative Assessment of the Eyes Pulse Blood Flow

with Linear Axisymmetric Model

Sergey B. Kazakov, Petr V. Luzhnov and Irina D. Davydova

Bauman Moscow State Technical University, Russian Federation

Keywords: Ophthalmoplethysmography, Oscillations Modelling, Diagnostic Algorithm, Eye, Blood Flow.

Abstract: The work examines a linear axisymmetric model of the eyes pulse blood flow. The main purpose of the

model is the determination of mathematical dependence of linear changes in the eye thickness of vascular

layer with the pulse. The main parameter is the volume of the input eye pulsation of blood flow. Ocular

hemodynamic parameters depend from various individual anatomical values of a patient's eye. In the paper

all assumptions of axisymmetric model are described. Calculations of the mathematical dependence

equation are based on geometric parameters of the eyeball. The analytical expression of volume changes in

the measuring ophthalmoplethysmography is presented. Accordance to our calculations, linear change in the

eyeball layers thickness varies within tens and units of microns after the input blood pulse volume.

Advanced technical methods that can register such a small range of length units are considered. Such a

registration and subsequent analysis of micro-displacement data will allow diagnosing different pathological

states of the eye hemodynamic in ophthalmology and neurology.

1 INTRODUCTION

Vascular factor is the essential and sometimes the

crucial aspect in the development of eye pathologies:

age-related macular degeneration (Mori F. et al.,

2001), diabetic retinopathy (Geyer O. et al., 1999;

Langham M.E. et al., 1989), glaucoma (Abegao

Pinto L. et al., 2016), circulatory disorders in the

retinal blood vessels (Tultseva S.N. et al., 2016). For

example, in the study of glaucoma, one theory has

been put forward that this pathology does not

contain only an increasing in intraocular pressure

(IOP). Moreover, it sometimes does not exclude a

violation of intraocular blood flow too. It was found

that at least 80% of people with increased IOP did

not develop optic nerve damage, while

approximately 30% of patients with glaucoma never

experienced increasing in IOP (Abegao Pinto L. et

al., 2016). These results allow us to understand that

one of the significant risk factors for such diseases is

a pathological change in the ocular hemodynamic

(Schmetterer L., Kiel J., 2012). Nowadays, current

technologies do not allow us to use reliable and

high-precision quantitative non-invasive

measurement of ocular pulse blood flow (Langham

M.E. et al., 1989).

Nevertheless, it is necessary to pay an attention

to the diagnostic calculations of eye blood

circulation due to using mathematical modelling in

order to improve its accuracy. New knowledge in the

field of eye hemodynamic will allow improving the

diagnostic algorithms of various eye diseases, to

create new therapeutic methods in ophthalmology

and neurology. Mathematical modelling of complex

ocular hemodynamics or its individual segments has

already been used to solve many different problems

(Shamaev D.M. et al., 2017). Even significantly

simplified models were useful for ophthalmologists.

We have already studied the hemodynamic

parameters of the eye in a normal, non-pathological

state in our previous work (Luzhnov P.V. et al.,

2020). Therefore, it is possible to form parameters

and technical characteristics of linear axisymmetric

model of the eyes pulse blood flow together with a

physical stand for ocular pulse modelling.

It is known that functional hemodynamic

changes are based on pathomorphological eye

conditions. The study of which can be useful in the

early diagnosis of eyes diseases. Thus, the purpose

of our work is the development of eye pulse

mathematical model. The main task of that model

will be the determination of mathematical

dependence of linear changes in the ocular

Kazakov, S., Luzhnov, P. and Davydova, I.

Method for Quantitative Assessment of the Eyes Pulse Blood Flow with Linear Axisymmetric Model.

DOI: 10.5220/0010385802390242

In Proceedings of the 14th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2021) - Volume 1: BIODEVICES, pages 239-242

ISBN: 978-989-758-490-9

239

geometrical sizes on the volume of the input eye

blood pulsation.

2 MATERIALS AND METHODS

The simplest linear axisymmetric model was chosen

for modelling the ocular hemodynamics. The main

assumptions of this model are presented below:

• The eyeball is considered as an object

isolated from surrounding biological tissues.

• The simplest model of the eye vascular layer

is an isotropic thin-walled spherical shell

with a thickness d, with a constant inner

radius R

and variable outer radius R

• The incompressible vitreous body is inside

the spherical shell.

• The eye anterior-posterior axis is D (23.0

mm). The initial value of the external radius

of the eye vascular layer is D/2 mm.

• The initial thickness of the eye vascular layer

d is 70.0 mm.

• The elasticity coefficient of a thin-walled

spherical shell must be ignored.

• The input volumes of eye blood flow

pulsation are set as constant data ∆Q. ∆Q is

the volume of input ocular blood flow

pulsation.

• The vascular layer of the eye is modelled as a

shaded shell that has an inner radius R

and

an outer radius R

In our model (see Fig.1) d

– the initial thickness

of the eye vascular, R

– the constant inner radius of

eye vascular layer, ∆d – the linear change in the

thickness of eye vascular layer, R

– the outer radius

of eye vascular layer, R

– the outer radius of eye

vascular layer after the volume of the input eye

pulsation of blood flow.

The main task of this model is the determination

of mathematical dependence of linear changes in the

eye thickness of vascular layer on the volume of the

input eye pulsation of blood flow.

The ophthalmoplethysmography method was

chosen as a method of recording and measuring the

variations in volume of the eyeball. Therefore, it was

necessary to determine the dependence of the

volume changes in the measuring chamber (i.e. in

the measuring eyecup) on the changes in the linear

size of our linear axisymmetric model.

Figure 1: Linear axisymmetric model of the eyeball.

Firstly, it is necessary to output an analytical

equation that defines the value of the outer radius of

eye vascular layer R

(see Fig.1). The initial volume

of the vascular layer V

is defined as

𝑉



𝜋∙(𝑅





− 𝑅





)

(1)

The volume of the vascular layer after the input

ocular pulsation of blood flow is defined as

𝑉



+∆𝑄=

𝜋∙(𝑅



− 𝑅





)

(2)

Due to this equation it is possible to determine

the outer radius of eye vascular layer:

𝑅





𝑅





3∙∆𝑄

4𝜋



(3)

Finally, due to previous calculations it is possible

to determine the linear change in the thickness of the

eye vascular layer, which is determined by the

following equation:

NDNSNT 2021 - Special Session on Non-invasive Diagnosis and Neuro-stimulation in Neurorehabilitation Tasks

240

𝑑=



𝑅





3∙∆𝑄

4𝜋



−𝑅



+𝑑



(4)

It is the value of the thickness of the vascular

layer after the input eye pulsation.

Secondly, it is necessary to output analytical

dependence of the change in the volume of the

measuring eyecup on the change in the linear size of

the thickness of the eye vascular layer. It can be

accomplished due to the equation for calculating the

volume of the ball segment

∆𝑉=𝜋∙ 𝑅





∙(



𝑅





3∙∆𝑄

4𝜋



− 𝑅



)

(5)

The volume of the measuring eyecup ∆V will

change as much as the volume of the segment of the

eyeball, which immersed in this measuring chamber.

3 RESULTS

Thus, by setting various individual values of a

patient:

• The value of the anterior-posterior axis of an

eye D (in mm).

• The volume of input ocular blood flow

pulsation ∆Q (in µl).

And, moreover, considering that d

is given as

the set of constant: d

, d

, it will be possible to

display on a three-dimensional graph the

dependence of the final thickness of the eye vascular

layer on the previous values (see Fig.2).

Figure 2: Graph of the dependence of linear changes in the

thickness of the eye vascular layer on the individual values

of a patient: the A-P axis of the eye, the value of the input

pulse volume, the initial thickness of the eye.

The necessary ranges of changes in values were

taken from the following literature sources. The

values of the eye anterior-posterior (A-P) axis are in

the range of 19.69 to 23.50 mm. The average value

is 21.60 mm. In the work (Tultseva S. N. et al.,

2017) the values of normal systolic increase in pulse

volume are ranged from 2.35 µl to 14.96 µl. The

average value is 7.27 µl. The values of constant are:

=0.065 µm, d

=0.070 µm, d

=0.075 µm.

Due to the equation (4), which defines the plane,

a three-dimensional graph was constructed for a

variable parameter d

(see Fig. 2).

At a large approximation three planes are visible

at different values of the Fig. 2 (see Fig. 3).

Figure 3: Approximate view of linear changes in the

thickness of the eye vascular layer.

Verification of our linear axisymmetric model

can be made by a technique which is described in

previous work (Kiseleva A.A. et al., 2020).

4 DISCUSSION AND

CONCLUSIONS

Accordance to our calculations, linear changes in the

eye thickness of vascular layer after the input pulse

volume varies within tens and units of µm.

Nowadays, there are several advanced technical

methods which can register such a range of length

units.

In the work (Zhu T. et al., 2012) the

development of fiber-optic sensors (FOS) is

considered. The vast majority FOS obtain an

external primary converter. Such a scheme works

when measured physical quantity (pressure,

temperature, acceleration, etc.) causes a mechanical

movement of a certain sensitive element (for

example, a membrane or an inertial mass), which in

Method for Quantitative Assessment of the Eyes Pulse Blood Flow with Linear Axisymmetric Model

241

turn leads to a modulation of the light intensity.

Movements in the FOS can be recorded by using an

interferometric measuring circuit. One of the

simplest devices of such a type can be considered a

fiber-optic end interferometer (FOEI). During using

quartz single-mode fiber and laser with emission

wavelength of 1.55 µm in FOEI, range of detected

linear motion of the mirror relative to the end face of

the optical fiber is in the range from 0.000025 µm to

640 µm, with an accuracy of ±0.000025 µm.

Moreover, there are methods for registering both

vibrations and displacement of surfaces using laser

radiation that directly probes a biological object. For

example, in work (Casaccia S. et al., 2015) describes

the technique of laser Doppler myography (LDM),

which is used as a non-contact method for

measuring the signal of mechanomyography (MMG)

from the biceps of a shoulder. The LDM signal was

measured by using a Polytec PDV100 laser Doppler

vibrometer, which uses a laser beam with a

wavelength of 633 nm, which corresponds to the

second class of the laser equipment (harmless to the

eye). Polytec PDV100 obtains the following

technical characteristics: a wide range of

measurement parameters of frequency fluctuations

from infra-low 0.05 Hz to ultrasonic 22 kHz,

measurement accuracy ±0.05 mm/s.

The methods described above and our linear

axisymmetric model will allow us to measure linear

displacements in the necessary range of length units.

CONFLICT OF INTEREST

The authors declare that they have no conflict of

interest.

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