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.
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 [
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) [
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 = hyh1 , vi = hvh1 , (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 [
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).
According to (2), the fractal dimension D of QPS of IHR is calculated by the formula:
N (h 1) 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: (2) (3) 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.
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; 15<v<15, v<-1, and
v>15. The first area corresponds to regular IHR, the second and the third areas – to IHR jumps
(catastrophes) [
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 redcolored 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.
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.