=Paper=
{{Paper
|id=Vol-2267/359-363-paper-68
|storemode=property
|title=Visualisation of the quantum phase space of instantaneous heart rhythm
|pdfUrl=https://ceur-ws.org/Vol-2267/359-363-paper-68.pdf
|volume=Vol-2267
|authors=Sergey A. Mikheev,Viktor P. Tsvetkov,Ilya V. Tsvetkov
}}
==Visualisation of the quantum phase space of instantaneous heart rhythm==
https://ceur-ws.org/Vol-2267/359-363-paper-68.pdf
Proceedings of the VIII International Conference "Distributed Computing and Grid-technologies in Science and
Education" (GRID 2018), Dubna, Moscow region, Russia, September 10 - 14, 2018
VISUALISATION OF THE QUANTUM PHASE SPACE
OF INSTANTANEOUS HEART RHYTHM
S.A. Mikheev a, V.P. Tsvetkov b, I.V. Tsvetkov c
Tver State University, Tver, Russia
E-mail: a Mikheev.SA@tversu.ru, b Tsvetkov.VP@tversu.ru, c mancu@mail.ru
In this paper we offer an algorithm for quantization of the classical phase space of instantaneous heart
rhythm with a constant or in quantization increment of h. The process of quantization of the phase
space of instantaneous heart rhythm divides the phase space into cells with a finite size of h. The
primary objective of this approach is visualisation of the quantum phase space of instantaneous heart
rhythm. The phase cells of this space are color-coded with different colors depending on the values of
their occupation numbers. The estimates showed the sufficiency of use about 10 colors. At that, the
informativity level of visualisation of the quantum phase space of instantaneous heart rhythm remains
invariant. Color visualisation of the quantum phase space of instantaneous heart rhythm clearly
demonstrates that it can be used as a cardiovascular system state marker. We offer a method for
description of the quantum phase space of instantaneous heart rhythm which will allow to detect
statistical regularities of instantaneous heart rhythm chaos.
Keywords: quantization, quantization constant, phase space, instantaneous heart rhythm, visualisation.
© 2018 Sergey A. Mikheev, Viktor P. Tsvetkov, Ilya V. Tsvetkov
359
Proceedings of the VIII International Conference "Distributed Computing and Grid-technologies in Science and
Education" (GRID 2018), Dubna, Moscow region, Russia, September 10 - 14, 2018
1. Introduction
One of the major problems facing cardiology is representation of 24-hour Holter monitoring
(HM) data for analysis of RR intervals in the form combining simplicity and informativity [1-4]. It can
be done by means of visualisation of a data set obtained from RR intervals analysis based on the
quantum phase space (QPS) of instantaneous heart rhythm (IHR).
The classical phase space (PS) of IHR is a set of states in space R2 defined by the functions
y(t) (IHR) and v(t) (IHR change rate) [5-7]. The function v(t) is the difference derivation y(t) [6]. The
functions y(t) and v(t) are constructed according to the data from 24-hour Holter monitoring. To
perform IHR analysis, it is necessary to represent the RR intervals data from 24-hour Holter
monitoring in the form combining simplicity and informativity. We will image this data set based on
the QPS of IHR. Visualisation of the QPS of IHR is representation of digital IHR information in the
easy-to-analysis and -monitoring form.
2. Algorithm for Quantization of the Classical PS of IHR
Generally, quantization of the classical PS is understood to be a process of division of the PS
into cells with a finite size of h. The h parameter is known as either a constant or a quantization
increment. We perform the procedure of IHR PS quantization according to the algorithm:
yi = hyh 1 , vi = hvh 1 , (1)
i 1,2,...N (h),
where y, v – the coordinates of a point in the classical PS of IHR, [ ] – the “rounding to the nearest
whole number” operator. According to (1), yi, vi have the values which are multiples of a quantization
constant, that is h.
The set of points with the yi, vi coordinates shall be designated as “QPS of IHR”. At that, the
QPS of IHR is divided into unit cells with a volume of ΔΓ=|Δyi Δvi |= hD, since the PS of IHR creates
the fractal set which has dimensions of D [5].
The values of yi, vi determine the QPS cells, and their multiplicities determine the occupation
number, ni, of these states. At that, the range space {ni, yi, vi} contains full IHR information over the
whole time interval of interest.
The phase-space volume of QPS of IHR characterizes the IHR variability and is calculated by
the formula:
q = h D N (h). (2)
According to (2), the fractal dimension D of QPS of IHR is calculated by the formula:
N (h 1) (3)
D(h) = ln ln h .
N ( h)
Let the quantization constant h for all QPSs of IHR = 1. At a given value of h the structure of
QPS of IHR turns out to be informative enough. Hereafter the general cases for h≠1 will be
considered.
The critical task of our approach is visualisation of the set of states {ni, yi, vi}. For this purpose
we assign different colors to points yi, vi of the QPS of IHR depending on the values of occupation
numbers. Our estimates show that use of about 10 colors is quite enough.
Let nm=max{ni}. Let the IHR state {nm, ym, vm} is marked with "×". It corresponds to the
maximum probability of IHR being in this state during HM.
We select colors for the points of the QPS of IHR according to the following algorithm. We
divide the ni range space into J intervals:
j 1 J ni nm j J , j 1,2,..., J . (4)
Let us set the values of ni meeting the requirements stated in (4) as ni,j. Let us assume that
γ=1.6, J=10, and assign the following colors to points yi, vi of the QPS of IHR with the occupation
numbers ni,j:
360
Proceedings of the VIII International Conference "Distributed Computing and Grid-technologies in Science and
Education" (GRID 2018), Dubna, Moscow region, Russia, September 10 - 14, 2018
Then we have the following ni,j /nm ranges:
0 ni ,1 nm 0.0251, 0.0251 ni , 2 nm 0.0761,
0.0761 ni ,3 nm 0.1457 , 0.1457 ni , 4 nm 0.2308 ,
0.2308 ni ,5 nm 0.3299 , 0.3299 ni , 6 nm 0.4416 ,
0.4416 ni , 7 nm 0.5651, 0.5651 ni ,8 nm 0.6998 ,
0.6998 ni ,9 nm 0.8449 , 0.8449 ni ,10 nm 1.
The optimum choice of intervals and colors should be investigated further.
3. QPS of IHR of the Patients under Examination
Figures 1-4 represent the QPS of IHR of four patients of the Tver Regional Cardiology Health
Center (TvRCHC) constructed using the Maple program system according to the data from 24-hour
HM.
Figure 1. The QPS of IHR of the first patient, Figure 2. The QPS of IHR of the second patient,
× {nm=69, ym=-1, vm=1949} × {nm=69, ym=0, vm=2975}
Figure 3. The QPS of IHR of the third patient, Figure 4. The QPS of IHR of the fourth patient,
× {nm=54, ym=-1, vm=1864} × {nm=117, ym=-52, vm=2855} 361
Proceedings of the VIII International Conference "Distributed Computing and Grid-technologies in Science and
Education" (GRID 2018), Dubna, Moscow region, Russia, September 10 - 14, 2018
As Figures 1-4 show, the QPSs of IHR are divided into odd-shaped 10 colored bands. In all
cases the "×" cells are in the red-colored bands with the maximum probability of presence of phase
points in these bands. All the colored bands are clearly-worded geometric structures having defined
regularities of their configurations. The geometric structure of these colored bands reflects adequately
the cardiovascular system states of the patients under examination.
We represent the diagnoses of the patients under examination: Patient 1 - norm, Patient 2 -
Ryan's class 4а ventricular arrhythmia, Patient 3 - Ryan's class 4a ventricular arrhythmia, Patient 4 -
Ryan's class 5 ventricular arrhythmia.
The horizontal lines in Figures 1-4 divide the QPS of IHR into three areas; 1515. The first area corresponds to regular IHR, the second and the third areas – to IHR jumps
(catastrophes) [8].
For the patient with state “norm”, all color points of the QPS of IHR with the exception of the
first grey color j=1, reach the regular rhythm area. In case of Ryan's classes 4а and 5 ventricular
arrhythmias, the significant amount of cells of the QPS of IHR reaches the IHR jump area. The red-
colored areas with the peak values of occupation numbers of states, ni, in all four cases have the
infinitesimal phase-space volume Гq of the QPS of IHR: for the first patient – 7, for the second patient
– 6, for the third patient – 4, for the fourth patient – 15 dimensionless units. The total Гq will be as
follows: for the first patient – 11497, for the second patient – 19767, for the third patient – 13997, for
the fourth patient – 14570 dimensionless units. Note that the total Гq of the patient without cardiac
pathology is the minimum one among the examined patients.
4. Conclusion
In this paper, the algorithm for quantization of the classical PS of IHR with a constant or in
quantization increment of h was offered. In the process of quantization, the PS of IHR is divided into
cells with a finite size of h.
The prime advantage of our approach is color visualisation of the QPS of IHR. The phase cells
of this space are color-coded with different colors depending on the values of their occupation
numbers. Our estimates showed the sufficiency of use about 10 colors with the invariant informativity
level of visualisation of the QPS of IHR. Color visualisation of the QPS of IHR clearly demonstrated
its availability as a cardiovascular system state marker.
The proposed method of description of the QPS of IHR made it possible to detect statistical
regularities of IHR chaos.
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