RU2615286C1 - Noninvasive method for electrophysiological heart characteristics determination - Google Patents

Abstract

FIELD: medicine.

SUBSTANCE: based on the known detailed models, a stochastic model of the epicardium repolarization current is developed, and its parameters are determined based on epicardial potential values samples found in the solution of the inverse electrocardiographic task in the reference points of the patient's heart computer model.

EFFECT: method allows to extend the functionality of electrocardiographic examination, to determine the ion currents components based on the epicardium potential values in the reference points of the patient's heart computer model.

26 dwg

Description

The present invention relates to medicine, in particular to cardiology, and can be used as an electrocardiographic method for diagnosing the condition of the heart.

The basic electrophysiological characteristics of the heart are understood as the values of dipole moments, potential and ionic currents of the epicardium. The determination of these characteristics is based on the solution of the inverse problem of electrocardiography (ECG EC), and their knowledge expands the functionality of the standard electrocardiographic approach and increases the efficiency of the diagnosis of heart conditions. Modern methods for solving the ECG EC allow determining the dipole moments and potentials of the epicardium [2, 4, 5], restoring the electrical activity of the heart (EAS) at the reference points of the computer model of the heart, and constructing an “electric portrait” of the patient’s heart [1, 3].

A known method of non-invasive registration of the electrophysiological characteristics of the heart [4, 13], which consists in registering multiple electrocardiographic signals, measuring the potentials generated by the heart, adaptive spatial interpolation of the potentials of the heart by expanding in spatial spherical functions, calculating the epicardial distribution of the potential, determining the moment distributions electrophysiological characteristics on the surface of the heart, calculation electrophysiologically x characteristics of the heart and the display of the electrophysiological characteristics of the heart.

There is also a method of non-invasive electrophysiological study of the heart [1, 2, 3], which consists in registering and processing electrocardiograms (EX), interpolating EX on the surface of the body, restoring the potential of the epicardium.

The known method of non-invasive recording of the electrophysiological characteristics of the heart and the method of non-invasive electrophysiological study of the heart are based on the registration of multiple leads or the method of ECG mapping of the heart [7], which is one of the most informative methods for studying EAS. ECG mapping of the heart allows you to get the maximum information about the features of the electric field of the heart at any moment of depolarization and repolarization of the ventricles of the heart, however, it requires the participation of a highly qualified specialist in the diagnosis, the use of expensive equipment and significant time spent on one study. According to the authors of the present invention, the disadvantages of the known method of non-invasive registration of electrophysiological characteristics of the heart and the method of non-invasive electrophysiological study of the heart are:

- the inability to use in mass preventive examinations (screening) of the heart due to the registration of multiple electrocardiographic signals;

- the inability to determine the components of the ionic currents of the epicardium.

The closest to the achieved result to the present invention is a non-invasive method for determining the electrophysiological characteristics of the heart [5], which consists in recording the electrocardiogram (ECS) and measuring the potentials generated by the heart, recording the frontal and left-side fluorographic images of the patient’s heart, determining geometric parameters from the images the patient’s heart, synthesize a computer model of the patient’s heart, determine the potentials generated by the heart on the patient’s torso, determine they add dipole moments and epicardial potentials ϕ at the reference points of the computer model of the patient’s heart, model the propagation of the excitation wave in the epicardium, synthesize a model EX, compare the model EX with registered ECS, and adjust the calculated parameters of the model of propagation of the excitation wave in the epicardium by changing the parameters when determining the moment distributions of the main electrophysiological characteristics on the surface of the heart and visualize the electrophysiological characteristics of the heart on three-dimensional heart models by computer graphics in the most convenient way for perception.

The figure 1 shows a diagram of an algorithm that implements the known method of non-invasive determination of the electrophysiological characteristics of the heart.

The figure 2 illustrates the functionality of the known method of non-invasive determination of electrophysiological characteristics of the heart.

From the analysis of the description of the known method of non-invasive determination of the electrophysiological characteristics of the heart, it follows that its functionality consists only in determining the dipole moments and potentials of the epicardium at the reference points of the computer model of the patient’s heart and the impossibility of determining the components of the ionic currents of the epicardium at the reference points of the computer model of the patient’s heart.

Consider the determination of the electrophysiological characteristics of the heart in a known method of non-invasive determination of the electrophysiological characteristics of the heart. The physical essence of the known method of non-invasive determination of the electrophysiological characteristics of the heart lies in the relationship of dipole moments and epicardial potentials. Show it.

- вектор i-ой площадки поверхности эпикарда в опорной точке компьютерной модели сердца,

- расстояние между поверхностями двойного слоя биологического генератора.Figure 2, a illustration shows the structure of a biological generator in the form of a double layer of current sources of the i-th site of the surface of the epicardium at the reference point of the computer model of the heart. Figure 2, a illustrates the definition of the potential ϕ at the point M, where

is the vector of the i-th site of the surface of the epicardium at the reference point of the computer model of the heart,

- the distance between the surfaces of the double layer of the biological generator.

In the known method for determining the electrophysiological characteristics of the heart, a system of differential equations of Maxwell is used [12]. The task of electrocardiography is to determine the electrophysiological characteristics of the heart based on the solution of the Maxwell differential equation system for a medium in which the geometric relationships between the region of the heart-shaped current generators and surface points where the potential is measured are conditionally specified. The formula for calculating the potential in such a conducting structure has the form [27]:

- объемная плотность дипольного момента, т.е. вектор плотности тока, созданный полем сторонних сил в неоднородном участке проводника.where r is the distance from the observation point to the points of the region of the current generators, limited by the volume V;

is the bulk density of the dipole moment, i.e. vector of current density created by the field of external forces in an inhomogeneous section of the conductor.

,The bulk density of the dipole moment is equal to the product of the conductivity of the medium and the vectors of the field strength of external forces

,

Since the bulk density of the dipole moment is used to quantitatively characterize continuously distributed dipole current sources in the volume of space where there are biological generators, the product of this quantity by the surface area of the epicardium associated with the reference point of the computer model of the patient’s heart is equal to the value of the ion current passing through this epicardial surface area:

Дивергенция вектора, равного произведению скалярной функции радиуса на объемную плотность дипольного момента, равна сумме скалярного произведения вектора

на градиент функции радиуса и произведения функции радиуса на дивергенцию вектора

The divergence of a vector equal to the product of the scalar function of the radius and the bulk density of the dipole moment is equal to the sum of the scalar product of the vector

by the gradient of the radius function and the product of the radius function by the divergence of the vector

Integrating the left and right sides of expression (4) over the volume of the biological generator and applying the Ostrogradsky-Gauss theorem to its left side, we obtain an integral expression of the form:

), и, как следствие,Since there are no current generators on the surface S directly in contact with the epicardium, the bulk density vector of the dipole moment of the biological generator on this surface will be zero (

), and as a consequence,

the surface integral on the left side of expression (5) is also equal to zero. Then, having selected the integral factor of expression (1) from the right-hand side of expression (5), we obtain an expression of the form for calculating the electric field potential ϕ at point M:

принимается однородным. В этом случае в выражении (6) модуль вектора

можно вынести за знак интеграла. Для плоской площадки S_i эпикарда, связанной с опорной точкой, справедливо представление биологического генератора в виде двойного слоя [12, 13]. На фигуре 2,а приведена схема двойного слоя источников для определения потенциала ϕ в точке М, где

- вектор площадки поверхности эпикарда в опорной точке,

- расстояние между поверхностями двойного слоя источников. Тогда для потенциала ϕ в точке М выражение (6) примет вид:Obviously, to determine the potential ϕ directly on the surface of the epicardium, biological generators of only this reference point of the computer model of the patient’s heart are important. Within a separate site S _{i the} field of the vector of the bulk density of the dipole moment

accepted homogeneous. In this case, in the expression (6), the modulus of the vector

can be taken out of the sign of the integral. For a flat site S _{i of the} epicardium associated with the reference point, the representation of the biological generator in the form of a double layer is valid [12, 13]. Figure 2, a shows a diagram of a double layer of sources for determining the potential ϕ at point M, where

- the vector of the surface area of the epicardium at the reference point,

- the distance between the surfaces of the double layer of sources. Then for potential ϕ at point M, expression (6) takes the form:

, телесный угол Ω, остается постоянным. Тогда для потенциала точки, расположенной на внешней поверхности эпикарда, справедливо выражение вида:The surface integral in expression (7) is equal to the solid angle Ω, under which the area from point M is visible. For a point located on the outer surface of the epicardium, the solid angle, under which the surface of the reference point is visible, is equal to 2π (or half the solid angle of the closed space, equal to 4π). Provided

, the solid angle Ω, remains constant. Then, for the potential of a point located on the outer surface of the epicardium, an expression of the form is valid:

на толщину двойного слоя:

.where D is the value of the dipole moment of the pad ΔS associated with the reference pad ΔS: D = D _S ⋅ΔS; D _S - surface density of the dipole moment of the reference point, equal to the product of the bulk density of the dipole moment

on the thickness of the double layer:

.

The ion current I _ion passing through the site of the epicardium ΔS at the reference point of the computer model of the heart is equal to the product of the volume density of the dipole moment J by the area of the site ΔS of the epicardium:

Expression (9) establishes the relationship between the current, the dipole moment, and the epicardial potential for the reference point of the computer model of the heart. It follows from expression (9) that the potential of the epicardium, the dipole moment, and the ion current for a separate reference point of a computer model of the heart have the same form of functional dependence. Obviously, with the known function of the potential of the epicardium, the form of the function of the current through the surface of the epicardium is known.

Thus, in the known method for non-invasively determining the electrophysiological characteristics of the heart, it is possible to estimate the total current through the pad reference point based on the construction of the relationship of electrophysiological characteristics. Moreover, in the known method of non-invasive determination of the electrophysiological characteristics of the heart, the components of the ionic currents of the epicardium are not determined.

Further, in the known method of non-invasive determination of the electrophysiological characteristics of the heart, verification of their determination is based on the conceptual model of the electrical activity of the heart (EAS), proposed by R.R. Aliev [16]. A distinctive feature of the conceptual model of EAS is the simplicity of a formal mathematical description of the concept of the phenomenon, which makes it possible to highlight the qualitative properties of an object without revealing its internal structure. The use of conceptual models is aimed at describing the changes in an individual property of an object without describing its internal structure. This approach does not allow to reveal the causes of changes in electrophysiological characteristics. The main disadvantage of using the conceptual model in the known method for determining the electrophysiological characteristics of the heart is the lack of description of ionic currents, which does not allow to determine the components of the ionic currents of the epicardium. This drawback lies at the basis of constructing a conceptual model aimed at describing the nature of changes in individual electrophysiological characteristics of the epicardium, such as dipole moments and epicardial potentials, without describing the reasons for the changes in these characteristics.

In the known method for determining the electrophysiological characteristics of the heart [5], the Aliyev-Panfilov (AP) conceptual model is used to restore the spread of the transmembrane action potential (TMPD) over the epicardial surface and synthesize model EX. The electrophysiological characteristics of the heart used in the AP model (epicardial potentials and dipole moments on the surface of the epicardium) were obtained when solving the ECG EC. The two-component conceptual AP model is a system of two parabolic differential equations:

where u, v are the slow and fast variables of the AP model, respectively;

ε ₀ , k, а, μ ₁ , μ ₂ are the parameters of the AP model.

Despite the fact that the AP model (10) is convenient for describing the distribution of excitation over the myocardial surface, this model does not reflect the phase properties of the development of TMPD. Figure 2.6 shows an example of the restoration of TMPD using the AP model. One of the characteristic features of the AP model is that the decay rate of the leading and trailing edges of the TMPD are set by the same fast variable (v), which determines their interconnected change. However, the development processes of the leading and trailing edges of real TMPD are different and are associated with changes in various structural elements of the membrane [8, 11]. The leading edge of TMPD is determined by the properties of the channels for sodium ions, the opening and closing of which occurs within 3 milliseconds. The trailing edge of TMPD is characterized by a change in membrane conductivity for potassium ions. The rate of its change is due to the work of structural elements of a membrane of various nature, regulated by internal processes in a cardiomyocyte. The typical duration of the trailing edge of the TMPD is 30 ... 50 milliseconds, while the development of the leading edge occurs within 3 milliseconds. Obviously, conceptual models containing two variables are not suitable for describing ion currents for different phases of the development of TMPD.

In the known method for determining the electrophysiological characteristics of the heart [5] based on the decision of the ECG ECG does not contain information about the distribution of ionic currents of the epicardium. The solution of the OZ ECG for dipole moments and epicardial potentials at the reference points of the computer model of the heart also does not contain a separation of the phases of the development of TMPD and, especially, the distribution of currents to different phases of the development of TMPD.

Thus, despite the availability of information on ion currents, in the known method for non-invasive determination of the electrophysiological characteristics of the heart there is no model for describing ionic currents of the epicardium, which leads to the loss of important diagnostic information.

According to the authors of the present invention, the determination of the constituent ionic currents of the epicardium is an urgent need for cardiology and is necessary to increase the reliability of the diagnosis of heart disease and for the effective use of antiarrhythmic drugs.

The disadvantage of this method of non-invasive determination of the electrophysiological characteristics of the heart is the inability to determine the components of the ionic currents of the epicardium at the reference points of the computer model of the patient’s heart.

The aim of the invention is to expand the functionality of assessing the state of the heart by analyzing ion currents through the surface of the epicardium and isolating the component of the potassium ion current in the time interval of monotonous repolarization.

To do this, in a non-invasive method for determining the electrophysiological characteristics of the heart, which consists in registering an electrocardiogram (ECS) and measuring the potentials generated by the heart, registering the front and left-side fluorographic images of the patient’s heart, determining the patient’s heart heart parameters, synthesizing a computer model of the patient’s heart, determine the potentials generated by the heart on the torso of the patient, determine the dipole moments and potentials of the epicardium ϕ at the reference points of the computer model of the patient’s heart, model the propagation of the excitation wave in the epicardium, synthesize a model EX, compare the model EX with the registered EX, correct the calculated parameters of the model of the propagation of the excitation wave in the epicardium by changing the parameters to determine the moment distributions of the main electrophysiological characteristics on the surface of the heart and visualize the electrophysiological characteristics of the heart on a three-dimensional model of the heart by computer graphics in the most convenient way to reproduce acceptance form, characterized in that it further

- generate data for a stochastic model of epicardial repolarization current at reference points of a computer model of the heart by:

мера ИИК, ϕ_max, ϕ_min и N - максимальное, минимальное значение и число отсчетов в выборке значений потенциала эпикарда в опорных точках компьютерной модели сердца;determining the information-measuring quantum (IIC) for samples of the epicardial potential values at the reference points of the computer model of the heart for one cardiocycle: γ = Δϕ⋅Δt, where Δt is the time required to obtain one sample;

IIR measure, ϕ _max , ϕ _min and N - the maximum, minimum value and the number of samples in the sample of epicardial potential values at the reference points of the computer model of the heart;

, где ϕ_i - i-е значение потенциала в выборке {ф} одного кардиоцикла;for determining the number of IIC N _γ for a stochastic model of epicardial repolarization current

, where ϕ _i is the ith value of the potential in the sample {f} of one cardiocycle;

determining the number m of ranked values of t _pj , the minimum t _{min j} and the maximum t _{max j} boundaries for the time intervals of the IIR grouping

;

;

;

;

;

;

, где g(t_i)=1 если t_{min j}<t_j≤t_{max j}, иначе g(t_i)=0,

, where g (t _i ) = 1 if t _{min j} <t _j ≤t _{max j} , otherwise g (t _i ) = 0,

where j is the sequence number of the time interval grouping IIK;

и

в j-м интервале группирования ИИК для положительных и отрицательных значений потенциала эпикарда в опорных точках компьютерной модели сердца пациента:determining the probability distribution of IIC

and

in the jth interval of the IIR grouping for positive and negative values of the epicardial potential at the reference points of the computer model of the patient’s heart:

,

,

;

;

;determining the difference in the probability distributions of the IIR p _j in the jth grouping interval for potential values at the reference points of the epicardium of the computer model of the patient’s heart:

;

the formation of samples of time samples for positive and negative differences in the probability distributions of the IRC t ₊ = {t _j | t _j ∈ (p _j > 0)}, t _- = {t _j | t _j ∈ (p _j <0)};

allocation of the time interval of the initial fast epicardial repolarization phase at reference points of the patient’s computer model of the heart t _nbr = min {t _j | t _j ∈ (p _j <0)};

allocation of the time interval of monotonous repolarization of the epicardium, including monotonous repolarization of the plateau phase and the phase of the final rapid repolarization of the epicardium at the reference points of the computer model of the patient’s heart:

,

;

,

;

, где P_M - вероятность обнаружения ИИК во временном интервале монотонной реполяризации эпикарда

;normalization of the probability distribution of IIR for samples of the potential values of the time interval of the monotonous repolarization of the epicardium at the reference points of the computer model of the patient’s heart

, where P _M is the probability of detecting IIC in the time interval of monotonous repolarization of the epicardium

;

- determine the currents of potassium abnormal rectification at the reference points of a computer model of the heart by:

), где

стохастическая модель I_so тока калия быстрого задержанного выпрямления;

- стохастическая модель I_so тока калия медленного задержанного выпрямления;

- стохастическая модель тока калия аномального выпрямления;

,

,

- плотности распределений ИИК для тока калия быстрого и медленного задержанного выпрямления и для тока калия аномального выпрямления соответственно; K_so1, K_so2 и K_K1 - весовые коэффициенты стохастической модели токов калия быстрого и медленного задержанного и аномального выпрямления соответственно; β_so1, α_so1, μ_so1, β_so2, α_so2, μ_so2, β_K1, α_K1, μ_K1 - параметры распределений;the formation of a stochastic model of delayed and abnormal rectification potassium currents

), where

stochastic model I _so potassium current fast delayed rectification;

- stochastic model I _so of potassium current slow delayed rectification;

- stochastic model of potassium current of abnormal rectification;

,

,

- IIC distribution density for potassium current of fast and slow delayed rectification and for potassium current of anomalous rectification, respectively; K _so1 , K _so2 and K _K1 are the weights of the stochastic model of potassium currents of fast and slow delayed and anomalous rectification, respectively; β _so1 , α _so1 , μ _so1 , β _so2 , α _so2 , μ _so2 , β _K1 , α _K1 , μ _K1 - distribution parameters;

, выпрямления, где q_j=1, если

, иначе q_j=0;reference weighting coefficients of the stochastic model potassium currents Fast K _so1 = 0,5⋅ (1-K _K1) and slow K _so2 = 0,5⋅ (1-K _K1) delayed and abnormal

rectification, where q _j = 1 if

otherwise q _j = 0;

determination of parameters β _so1 , α _so1 , μ _so1 , β _so2 , α _so2 , μ _so2 , β _K1 , α _K1 , μ _K1 stochastic model of potassium currents of fast and slow delayed and anomalous rectification by minimizing the solution of the equation

- матрица стохастической модели токов калия быстрогоWhere

- matrix of a stochastic model of fast potassium currents

- матрица нормированных вероятностей ИИК;and slow delayed and abnormal straightening;

- matrix of normalized probabilities of IIC;

и медленного задержанного

и аномального выпрямления

на j-м интервале группирования ИИК;determining the probability distribution of IIC currents of potassium fast delayed

and slow detainee

and abnormal straightening

on the jth interval of the IIK grouping;

модели токов калия быстрого и медленного задержанного и аномального выпрямления в опорной точке компьютерной модели сердца пациента, где составляющие критерия вычисляют по формулам:determination of the adequacy criterion

models of potassium currents of fast and slow delayed and abnormal rectification at the reference point of the computer model of the patient’s heart, where the components of the criterion are calculated by the formulas:

,

,

,

,

,

,

where _{k e so1, κ so1, k} e so2, κ so2, k e K1, κ K1, _ coefficients and entropy kontrekstsessa symmetrical distributions for potassium current of fast and slow delayed and abnormal rectification;

comparing the criterion r ² ≤3⋅ln (α ^-2 ), where α is the significance level of decision making;

correction of the weight coefficients of the stochastic model of potassium currents fast K _so1 and slow K _so2 delayed and abnormal K _K1 rectification at the reference points of the computer model of the patient’s heart;

задержанного и аномального выпрямления определены корректно»;decision: “Parameters of the stochastic model of potassium current

delayed and abnormal rectification determined correctly ”;

тока калия аномального выпрямления по формуле:value definitions

potassium current abnormal rectification according to the formula:

опорных точках компьютерной модели сердца пациента путем:- determine the repolarization current of the epicardium

reference points of a computer model of the patient’s heart by:

реполяризации эпикарда

, где

- стохастическая модель переходного транзитного тока эпикарда;

- стохастическая модель медленного деполяризующего тока кальция; K_to, K_st и K_M - весовые коэффициенты стохастической модели тока реполяризации эпикарда;

и

- плотности распределений ИИК для переходного транзитного тока и медленного деполяризующего тока кальция соответственно; β_to, α_to, μ_to, β_st, α_st, μ_st - параметры распределений;forming a stochastic current model

epicardial repolarization

where

- stochastic model of transient transition current of the epicardium;

- stochastic model of slow depolarizing current of calcium; K _to , K _st, and K _M are the weights of the stochastic model of epicardial repolarization current;

and

- the density of the distribution of the IIC for the transient transit current and slow depolarizing current of calcium, respectively; β _to , α _to , μ _to , β _st , α _st , μ _st - distribution parameters;

реполяризации эпикарда путем минимизации решения уравненияdetermining the parameters of a stochastic current model

repolarization of the epicardium by minimizing the solution to the equation

,

,

- матрица стохастической модели тока реполяризации эпикарда; р=[р, р₂ … p_m] - матрица разностей распределений вероятностей ИИК;Where

- matrix of a stochastic model of epicardial repolarization current; p = [p, p ₂ ... p _m ] - matrix of differences in the probability distributions of the IIC;

реполяризации эпикарда в опорных точках компьютерной модели сердца пациентаdetermining current value

epicardial repolarization at reference points of the patient’s computer model of the heart

;

;

- restore the potential ϕ _mod at the reference points of the epicardium of the computer model of the patient’s heart by:

;determining the coefficient of reduction potential of the epicardium

;

;determine the coefficient of current reduction

;

;determining the values of the restored potential ϕ _{mod i} at the reference points of the epicardium of the computer model of the patient’s heart

;

- compare the restored ϕ _{mod the} potential of the epicardium with the potential of the epicardium ϕ determined on the EX

;

;

реполяризации эпикарда в опорных точках компьютерной модели сердца пациента;- adjust the distribution parameters β _to , α _to , μ _to , β _st , α _st , μ _{st of the} stochastic current model

epicardial repolarization at reference points of the patient’s computer model of the heart;

реполяризации эпикарда в опорных точках компьютерной модели сердца пациента определены корректно».- make a decision: “Current values

the repolarization of the epicardium at the reference points of the computer model of the patient’s heart was determined correctly. "

The figure 3 shows the timing diagrams of the currents forming the transmembrane action potential (TMPD) [17].

The figure 4 shows the relationship of the potential of the epicardium and TMPD of cardiomyocytes through ionic currents: a) during depolarization of the epicardium, b) during repolarization of the epicardium.

The figure 5 shows the timing diagrams of the currents forming the TMPD for the detailed model Iyer-Mazhari-Winslow (IMW) [35].

The figure 6 shows the timing diagrams of the currents forming TMPD for the detailed model Tusscher-Noble-Noble-Panfilov (TNNP) [34]: a) ventricular TMPD; b) calcium concentration; c) sodium current; d) transient transit current; e) potassium current of fast delayed rectification; e) potassium current slow delayed rectification; g) depolarizing current of calcium of L-type; h) potassium current of abnormal rectification.

The figure 7 shows a comparison of the time diagrams of the currents forming the TMPD for the detailed model Luo-Rudy (LRd) (a) and for the detailed model TNNP (b) [30].

Figure 8 compares the time diagrams of the currents forming the TMPD for the detailed Priebe-Beuckelmann (PB) model, for the detailed LRd model and for the detailed TNNP model [23]: a) ventricular TMPD; b) depolarizing current of calcium of L-type; c) potassium current of fast delayed rectification; d) potassium current slow delayed rectification.

The figure 9 shows the change in the shape of TMPD in violation of the interaction of ion channels in cardiomyocytes [18].

The figure 10 shows a table with a list of reasons for changing the duration of TMPD for various syndromes [14, 15, 17, 21, 22].

The figure 11 shows a table with a list of the reasons for the change in the form of TMPD in Bruged syndrome [14, 15, 17].

The figure 12 shows a diagram of an algorithm that implements the proposed method for non-invasive determination of electrophysiological characteristics of the heart.

The figure 13 shows a diagram of a data generation algorithm for a stochastic model of epicardial repolarization current at reference points of a computer model of the heart.

The figure 14 shows a diagram of an algorithm for determining the potassium current of abnormal rectification at reference points of a computer model of the patient's heart.

The figure 15 shows a diagram of an algorithm for determining the repolarization current of the epicardium at the reference points of a computer model of the patient’s heart.

The figure 16 shows a diagram of an algorithm for restoring the potential of the epicardium at the reference points of a computer model of the patient's heart.

The figure 17 shows a typical example of the potential of the epicardium obtained by restoring the electrocardiogram.

The figure 18 shows an illustration of the formation of an information-measuring quantum (IIC) from a sample of the epicardial potential for the reference point of a computer model of the patient’s heart.

The figure 19 shows the difference in the distributions of the IIK for positive and negative values of the potential of the epicardium.

The figure 20 shows a representation of stochastic models of potassium current delayed and abnormal rectification for the time interval of monotonous repolarization of the epicardium at the reference points of the computer model of the patient’s heart.

The figure 21 shows the sequence of formation of the probability distribution of the IIC symmetric random variable Z for a stochastic model of potassium current anomalous rectification, where it is indicated:

по временным интервалам группирования для стохастической модели тока калия аномального выпрямления;a) a histogram of the probability distribution of the IIC

by grouping time intervals for a stochastic model of potassium current of abnormal rectification;

для значений случайной безразмерной величины Y, полученной путем преобразования временных отсчетов ИИК для стохастической модели тока калия аномального выпрямления;b) the probability distribution of IIC

for values of a random dimensionless quantity Y obtained by converting time samples of the IIR for a stochastic model of potassium current of anomalous rectification;

для значений случайной безразмерной величины Z для стохастической модели тока калия аномального выпрямления, полученной путем симметричного отображения вероятностей для значений случайной величины Y;c) symmetric probability distribution of IIC

for values of a random dimensionless quantity Z for a stochastic model of potassium current of anomalous rectification obtained by symmetric mapping of probabilities for values of a random variable Y;

по интервалам группирования случайной величины Z для стохастической модели тока калия аномального выпрямления.d) a histogram of the probability distribution of the IIC

over grouping intervals of a random variable Z for a stochastic model of potassium current of anomalous rectification.

The figure 22 shows the sequence of formation of the probability distribution of the IIC symmetric random variable Z for a stochastic model of potassium current slow delayed rectification, where it is indicated:

по временным интервалам группирования для стохастической модели тока калия медленного задержанного выпрямления;a) a histogram of the probability distribution of the IIC

by grouping time intervals for a stochastic model of potassium current slow delayed rectification;

для значений случайной безразмерной величины Y, полученной путем преобразования временных отсчетов ИИК для стохастической модели тока калия медленного задержанного выпрямления;b) the probability distribution of IIC

for values of a random dimensionless quantity Y obtained by converting time samples of the IIR for a stochastic model of potassium current slow delayed rectification;

для значений случайной безразмерной величины Z для стохастической модели тока калия медленного задержанного выпрямления, полученной путем симметричного отображения вероятностей для значений случайной величины Y;c) symmetric probability distribution of IIC

for values of a random dimensionless quantity Z for a stochastic model of potassium current of slow delayed rectification, obtained by symmetric mapping of probabilities for values of a random variable Y;

по интервалам группирования случайной величины Z для стохастической модели тока калия медленного задержанного выпрямления.d) a histogram of the probability distribution of the IIC

over grouping intervals of a random variable Z for a stochastic model of potassium current slow delayed rectification.

The figure 23 shows the sequence of formation of the probability distribution of the IIC symmetric random variable Z for a stochastic model of potassium current fast delayed rectification, where it is indicated:

по временным интервалам группирования для стохастической модели тока калия быстрого задержанного выпрямления;a) a histogram of the probability distribution of the IIC

by grouping time intervals for a stochastic model of potassium current of fast delayed rectification;

для значений случайной безразмерной величины Y, полученной путем преобразования временных отсчетов ИИК для стохастической модели тока калия быстрого задержанного выпрямления;b) the probability distribution of IIC

for values of a random dimensionless quantity Y obtained by converting time samples of the IIR for a stochastic model of potassium current of fast delayed rectification;

для значений случайной безразмерной величины Z для стохастической модели тока калия быстрого задержанного выпрямления, полученной путем симметричного отображения вероятностей для значений случайной величины Y;c) symmetric probability distribution of IIC

for values of a random dimensionless quantity Z for a stochastic model of potassium current of fast delayed rectification, obtained by symmetric mapping of probabilities for values of a random variable Y;

по интервалам группирования случайной величины Z для стохастической модели тока калия быстрого задержанного выпрямления.d) a histogram of the probability distribution of the IIC

over grouping intervals of a random variable Z for a stochastic model of potassium current of fast delayed rectification.

The figure 24 shows the dependence of the adequacy criterion and the parameters of the model of potassium currents of delayed and abnormal rectification on the weight coefficient K _{1 of} the current model of abnormal rectification:

,

,

от весового коэффициента K₁ модели тока аномального выпрямления;a) the dependence of the criterion of the adequacy of the model of potassium current delayed and abnormal rectification g and its components

,

,

from the weight coefficient K _{1 of} the current model of abnormal rectification;

b) the dependence of the shape parameter γ of the potassium current model of anomalous rectification on the weight coefficient K _{1 of the} model of anomalous rectification current;

,

,

моделей составляющих тока калия быстрого задержанного, медленного задержанного и аномального выпрямления от весового коэффициента K₁ модели тока аномального выпрямления;c) the dependence of the scale parameters

,

,

models of the components of the potassium current of fast delayed, slow delayed and anomalous rectification from the weight coefficient K _{1 of} the current model of anomalous rectification;

,

,

моделей составляющих тока калия быстрого задержанного, медленного задержанного и аномального выпрямления от весового коэффициента K₁ модели тока аномального выпрямления.d) the dependence of the scale parameters

,

,

models of potassium current components of fast delayed, slow delayed and anomalous rectification from the weight coefficient K _{1 of} the anomalous rectification current model.

- ток реполяризации эпикарда;

- ток аномального выпрямления;

- ток быстрого задержанного выпрямления;

- медленного задержанного выпрямления;

- переходной транзитный ток эпикарда;

- замедленный деполяризующий ток кальция.The figure 25 shows the results of modeling the repolarization current of the epicardium, where the following notation is used:

- epicardial repolarization current;

- current abnormal rectification;

- current fast delayed rectification;

- slow delayed straightening;

- transient transit current of the epicardium;

- slow depolarizing current of calcium.

Figure 26 shows graphs epicardial potentials: φ - potential epicardium, resulting in solving the ECG OZ, in the known method, and φ _mod - epicardium capacity restored by the currents of the stochastic model of repolarization epicardium.

The introduced actions with their connections exhibit new properties that expand the functionality of the known method and allow to determine the components of the potassium current and establish the reasons for the change in the shape of the epicardial potential. The expansion of functionality in determining the electrophysiological characteristics of the heart is provided by:

- restoration using the proposed stochastic model of repolarization current of the epicardium of the epicardial potential;

- comparing the restored potential of the epicardium with the potential of the epicardium obtained as a result of solving the ECG EC;

- determination of the components of the potassium current in the case of equivalence of the compared potentials of the epicardium;

- correction of the stochastic model of the epicardial repolarization current to achieve equivalence of the compared epicardial potentials in the case of nonequivalence of the compared epicardial potentials.

The essence of the invention is to determine the potassium current of abnormal rectification due to the use of a stochastic model of the constituent ionic currents of epicardial repolarization during electrocardiographic examination.

It is known that EAS is due to the functioning of the ion channels of myocardial cells — cardiomyocytes [10]. Many heart diseases are associated with defective changes in the functioning of ion channels, which are manifested in the development of life-threatening arrhythmias. The figure 3 shows a timing diagram of the components of the currents of the epicardium [17], illustrating the relationship of ionic currents in the development of TMPD. All currents in a cardiomyocyte are divided into incoming, depolarizing and exiting, repolarizing currents. From the time diagrams of the currents shown in figure 3, it can be seen that the depolarization currents consist of two main components: a fast current of sodium I _Na and a slow depolarizing current of calcium I _Ca. The cardiomyocyte repolarization currents contain a whole complex of components: outgoing transit currents I _to , potassium currents of fast delayed I _Kr , slow delayed I _Ks and abnormal rectification I _K1 [17].

The authors of the invention are convinced that the knowledge of the features of ionic currents during various phases of the development of TMPD is an important diagnostic indicator. For example, to describe the currents of a detailed model, phase 2 of the TMPD plateau (see Figure 3) is divided into two parts. The authors of the invention consider it necessary to combine the phases 2 and 3 of the TMPD (see figure 3) to determine the currents of the delayed and abnormal rectifications (see figure 3), and thus to isolate the time interval of the monotonous repolarization of the TMPD (see figure 3). During the monotonic repolarization of TMPD, processes of controlling the concentration of calcium current occur, leading to a change in the duration of TMPD.

A distinctive feature of the proposed method in the sense of obtaining new diagnostic information is the possibility of analyzing the solution of the ECG ECG using a stochastic model of epicardial repolarization current, constructed taking into account the characteristic features of ion currents during monotonic repolarization of TMPD. The figure 4 shows the relationship of TMPD of the cardiomyocyte and the epicardial potential ϕ during the development of depolarization processes (see figure 4, a) and repolarization (see figure 4.6). During depolarization of the epicardium, a change in the sign and value of TMPD occurs, which is associated with a change in the excess charge of the internal environment of the cardiomyocyte. A charge change occurs when the TMPD propagates from the endocardium to the epicardium. The charge accumulated in the cardiomyocyte determines the excess charge of the internal environment of the cardiomyocyte. During repolarization, a change in the sign of TMPD occurs, which causes a process of recharging the internal environment of the cardiomyocyte. The processes of depolarization and repolarization are due to different ion currents, which causes the movement of ions in the direction of the surface of the epicardium due to changes in ion concentrations. As a result, an ion current is generated that passes through the epicardium and forms an ECS on the patient's torso.

Due to the fact that the same ions of the intercellular medium participate both in the formation of the current for the development of TMPD and in the formation of a current passing through the epicardium, these processes are interconnected. The presence of the interconnection of processes allows the use of current schemes of detailed models of the process of development of TMPD to describe the process of repolarization of the epicardium. Since the current through the epicardium is formed by the total current of depolarization and repolarization of the cardiomyocyte, its components are random in nature, which necessitates the use of stochastic models to describe the ionic current of epicardial repolarization and the formation of the epicardial potential <p.

To clarify the further discussion, we consider the detailed models of ion currents used to study the components of ion currents in the development of TMPD. (see figures 5-11). Modern detailed models of the electrical activity of the heart (EAS) were created by improving the well-known Hodgkin-Huxley formalism (model) [24], which is based on the processes of controlling the conductivity of membrane channel-forming proteins by changing the state of voltage-dependent particles. Moreover, the formal control of the behavior of an individual protein structure is given in the form of a system of differential equations simulating the state of voltage-dependent particles. The following detailed models are best known.

1. The Luo-Rudy (LRd) model is one of the first detailed models for TMPD containing the main structural elements for describing the conductivity through the membrane [28].

2. Priebe-Beuckelmann (PB) model [31]. This model is designed to study abnormal automatism and cellular electrophysiological consequences of cardiac arrest (contains 22 variables).

3. Model Tusscher-Noble-Noble-Panfilov (TNNP) [34]. This model, which uses the results obtained directly in the study of myocardial tissue (contains 17 state variables and 44 parameters) '

4. The Iyer-Mazhari-Winslow (IMW) model [26, 32]. This model describes ion flows in detail for individual membrane structures, includes 67 state variables and 94 parameters.

Detailed models of LRd, PB, TNNP, and IMW allow us to solve a direct problem: for known values of ion currents, determine the development of TMPD. When solving the inverse problem: determining the values of ion currents from the known form of TMPD, many solutions are possible depending on random influences. The use of detailed models of LRd, PB, TNNP, and IMW is also limited due to the large amount of calculations that are important, which is important when considering the problems of electric potential propagation and the occurrence of rientri in the myocardium.

5. A simplified model for TMPD [30]. This model is designed to study the spread of TMPD, the formation of rientri, the refractory properties of the myocardium and other processes of the development of arrhythmia due to changes in myocardial recovery. The model refers to a realistic physiological model and includes currents of all ion channels. The purpose of the model is to reproduce the morphological form of TMPD and curves of return to the resting potential based on the results of modeling more complex models.

We show the definition of the constituent ionic currents of the epicardium in detailed models of ionic currents. The timing diagram shown in figure 5 illustrates a detailed simulation of the components of the ion currents of the TMPD in the IMW model [23, 35], where depolarizing and repolarizing currents are indicated. The IMW model also takes into account additionally the currents I _{Na / Ca of the} membrane structure of the sodium-calcium exchanger. The figure 6 shows the timing diagrams of the main currents of the detailed TNNP model for ventricular tissues [34], from which it follows that to build a detailed TNNP model, the same set of constituent ion currents is required. To ensure the quality of the model, it is important to provide a form of the function of changes in ion currents. The diagrams of the constituents (components) of the currents of the detailed TNNP model shown in Figure 6 contain information about the form of the change in the components of the ionic currents over time. The figure 6 shows the following diagrams:

а - transmembrane action potential ϕ;

b - the transition process of calcium Cai;

in - sodium current I _Na ;

g - transient transition current I _t0 ;

d - potassium current fast delayed rectification I _Kr ;

e - potassium current slow delayed rectification I _Ks ;

g — Z-type I calcium current _CaL ;

s - potassium current of abnormal rectification I _K1 .

Due to the large volume of calculations, the use of detailed models of LRd, PB, TNNP, and IMW is difficult when studying the occurrence of rientri in the myocardium. To study the formation of rientri during the spread of TMPD, the refractory properties of the myocardium and other processes of the development of arrhythmia due to changes in myocardial recovery, a simplified model for TMPD has been developed [30]. The importance of constructing a simplified model is to reproduce the morphological form of TMPD and the curves of the return to the resting potential with the minimum number of ion currents making up the model. Saving the shape of the curves in a simplified model is achieved by summarizing the results of more complex detailed models.

In a simplified model, currents are divided according to their functional purpose. The nature of the current changes during myocardial activity, grouped according to their functional purpose, is given for the LRd models in figure 7, a, b and for the simplified model in figure 7, d, e [30, 28]. From the consideration of these figures it follows that in the simplified model all ion currents are divided into four groups in accordance with their functional purpose.

The first group consists of sodium currents of rapid depolarization directed inside the cardiomyocyte with the rapid termination of the process (rapid inactivation of the conduction channels).

The second group is the transit currents of the phase of fast initial repolarization of myocardial tissues. This group includes currents directed outward with a quick end to the process, from which currents of potassium and chlorine ions are isolated. The functional purpose of the currents is to transfer the TMPD to a balanced state in a short period of time. These currents approximate well the functions of transients.

The third group - currents of delayed depolarization directed into the cardiomyocyte with a slow end of the process. The functional purpose of the currents is to ensure the influx of calcium and maintaining its concentration inside the cytoplasm for the development of compression processes of cardiomyocytes. This group of currents includes a whole complex (components) of components of the depolarizing current: L-type calcium current, T-type calcium current.

The fourth group consists of outwardly directed potassium currents of repolarization, the functional purpose of which is to control the balanced state of TMPD and change its duration depending on the state of the cardiomyocyte: concentration of ATP and other ions.

Figure 8 contains the dependencies of the slow and fast ionic potassium currents of the delayed potential I _Ks and I _Kr for the L-type calcium ion current L _CaL where the graphs are based on the most well-known detailed models: PB, IMW, TNNP [23].

With the normal functioning of the EAS, a balance is maintained between the depolarizing and repolarizing currents due to the balanced interaction of the ion channels. All ion channels are in one of three states: activation, inactivation, and rest. During the transition of ion channels from a rest state to an activation state, an ion current forms. The impaired functioning of the channels is reflected in a change in the balance between the depolarization and repolarization currents of the cardiomyocyte and, as a result, in a change in the shape, primarily, the duration of the TMPD [18].

The figure 9 shows the most characteristic changes in the form of TMPD due to a violation of the interaction of ion channels in a cardiomyocyte. An important symptom of malfunctioning of ion channels is associated with an increase in the duration of TMPD. Among the main reasons for the increase in the duration of TMPD should be highlighted:

- the delay of repolarization as a result of a change in the plateau phase of the TMPD, provided that the repolarization time does not change from the plateau level to the level of complete repolarization;

- slowing down repolarization, in which the 3rd phase of TMPD is extended from the plateau level to the level of the resting potential while maintaining the duration of the "plateau" phase. With this transformation, TMPD takes on the shape of a triangle, which is characterized by the degree of triangle [18, 29].

According to research [18], an increase in the “triangularity” of TMPD, regardless of the causes of its appearance, indicates an increase in the likelihood of arrhythmias. A typical example of an increase in the "triangularity" of TMPD in case of a malfunction of the conduction channels is given in figure 9.6. The formation of a TMPD of a triangular shape can be due to the following reasons: suppression of the repolarizing current of delayed and abnormal rectification, amplification of the current of the Na-Ca exchanger, or suppression of the depolarizing current of calcium. The electrophysiological parameter "degree of triangularity" TMPD is calculated as the difference in the duration of TMPD at the level of 90% and 30% repolarization [18].

The appearance of oscillations in the duration of TMPD (figure 9, c) is also associated with disruption of the channels and increase the likelihood of developing cardiac arrhythmias. Oscillations in the duration of TMPD are due to the development of parametric resonance in the tissues of the cardiomyocyte as a result of a reduction in the interval of tissue repair. Figure 9, b shows an example of the formation of oscillations due to a decrease in the diastolic interval with an increase in heart rate. In this case, with normal TMPD, cardiomyocytes do not have time to recover within a short diastolic interval, which is manifested in impaired functioning of the channels and a change in the shape (duration) of TMPD and the appearance of TMPD of short duration. After a short TMPD, a long interval (CI) is formed during which the cardiomyocyte is completely restored. The normal functioning of the channels in the cardiomyocyte ensures the development of normal TMPD, after which the oscillation cycle will repeat [18].

The importance of determining the components of ionic currents during electrocardiography is also confirmed by the data shown in figures 8 and 9 on the symptoms of genetic disorders of various proteins of ion channels, leading to changes in the components of ionic currents and causing complex heart rhythm disturbances. Table 1 (see figure 10) [15] provides a list of genetic disorders of various ion channel proteins that affect potassium currents in slow and abnormal rectification. Genetic abnormalities in regulatory channel-forming proteins lead to improper functioning of the ion channels and cause the development of complex cardiac arrhythmias [14, 15, 21], which are manifested in lengthening or decreasing the QT interval of ECS. Genetic defects of Brugad syndrome (see Figure 11) also cause violation of the balance of ion currents, which is reflected in the change in the duration of the action potential [14, 15].

From the above it follows that to increase the reliability of the diagnosis of heart disease, it is necessary to determine the components of the ionic currents of the epicardium. According to the authors of the present invention, the use of modern detailed models of ion currents requires unreasonably large computational costs to determine the parameters of the model during an electrocardiographic examination.

The construction of modern detailed models is based on the Hodgkin-Huxley model, in which the probabilistic properties are laid down in the description of the behavior of individual protein membrane structures. Protein ion channels are controlled by the transmembrane action potential, the change of which in detailed models is assumed to be the same for all structures. For a cardiomyocyte, the potential changes along the membrane surface and is not related to the initial state of the potential of dependent particles. Since channel-forming membrane proteins that are located in different cavities of a cardiomyocyte participate in the formation of the total ion current through the epicardium, the effect of the electric field on such structures is random. The use of detailed models for determining the components of ionic currents passing through the epicardium is difficult due to the difference in the initial phases of the development of current in different structures of the cardiomyocyte and the difference in the state of the cardiomyocyte.

The authors of the present invention believe that it is more expedient to use probabilistic (stochastic) models directly to describe the components of the ion current passing through the epicardium. Due to the fact that the currents forming TMPD and passing through the epicardium are created by the same ions, the currents of detailed models were used to form stochastic current models. The use of detailed models for constructing stochastic models of ion currents is based on the similarity of the shape of the distribution density of the information-measuring quantum (IIC) for the stochastic model of ion currents to the form of changes in the ionic epicardial repolarization currents. Therefore, to solve the problem of analyzing the ionic currents of epicardial repolarization, we used the rules for choosing the most suitable (optimal) form of an approximating solution. Moreover, violations in the organization of channel-forming proteins are reflected in a change in the properties of the stochastic model.

The construction of a stochastic model of the distribution of the components of the ionic current of repolarization of the epicardium consists in choosing the appropriate form of the approximating function. To do this, preliminary, using detailed models, the form of the function of the component currents is estimated by calculating the asymmetry and excess of the distribution of the IIC of the function of the component currents, and a possible form of distribution is selected. Then, the asymmetry and kurtosis of the IIR distribution are calculated for the stochastic model of ion currents from the samples of the recorded ECS samples. A comparison of the shape parameters of the approximation function of stochastic models of ion currents with the shape parameters of ion currents of the detailed model allows us to estimate the components of the ionic currents of the epicardium. Monitoring changes in the parameters of the stochastic model of ion currents increases the reliability of assessing the state of the heart.

Consider the actions introduced in detail.

Data generation for a stochastic model of epicardial repolarization current at reference points of a computer model of the heart. The main goal of this action is to obtain a sample of samples based on the readings of one cardiocycle characterizing the probabilistic properties of the model of epicardial repolarization currents. The peculiarity of the solution consists in comparing the shape of the curves and the distribution of samples, which can be done if the solution is sought in the form of distribution functions whose areas are equal to unity. For data, the condition must also be satisfied that the total value used to model the values, multiplied by the time required to obtain one sample, must be equal to a unit area.

A diagram of a detailed algorithm for the action of data generation for a stochastic model of ionic currents of repolarization of the epicardium is shown in Fig. 13. To perform the data generation step, a sample of potential values ϕ of the reference points of the epicardium is used. The figure 17 gives a typical example of the distribution of time samples of the potential of the epicardium for the reference point of a computer model of the patient’s heart. From figure 13 it follows that the first distinctive effect of the proposed method for non-invasive determination of electrophysiological characteristics of the heart contains

- determination of the information - measuring quantum for a sample of potential values at the reference points of the epicardium. Data generation begins with the definition of information-measuring quantum (IIC) - the minimum mathematical formation that displays the essence of probabilistic physical processes, information about which is contained in a sample of ECS values of one cardiocycle.

The value of IIC γ is defined as the product of time Δt necessary to obtain one reference per measure of uncertainty Δϕ of the epicardial potential

The relationship of the IIR γ and the samples of the sample of the epicardial potentials obtained by solving the inverse problem for the reference point of the epicardium is illustrated in Figure 18. For a one-dimensional quantity, the measure of the IIR Δϕ is defined as the ratio of the difference between the maximum ϕ _max and minimum ϕ _min values of the sample potentials to the number of intervals for grouping data, which is equal to the square root of the number of N samples in the sample

The IIR measure characterizes the uncertainty in calculating the potential of the epicardium, which is limited by the interval of grouping data. The choice of the boundaries of the grouping of values is “sensitive” to rough readings. To exclude them, the condition is used that a value of more than 5⋅σ (ϕ) is considered a “rough” count. This approach allows you to immediately exclude from consideration sodium currents, which are determined by a unit time interval and a value two orders of magnitude higher than the standard deviation.

. Эта величина определяет меру кванта по оси потенциала.Thus, if for one cardiocycle the number of samples equal to 1000 is obtained, then the number of data grouping intervals will be equal

. This value determines the measure of the quantum along the axis of the potential.

- determination of the number of IIC for a stochastic model of epicardial repolarization current. This action of the algorithm in figure 13 is necessary to estimate the number of IIKs contained in the entire sample of samples of the epicardial potential.

The sample of samples contains the total number of IIC equal to the sum of the IIC of positive and negative samples. The number of quanta contained in one sample is equal to the ratio of the value ϕ _{i of the} sample to the measure of quantum uncertainty Δϕ:

Then the total number of quanta contained in the samples is equal to the ratio of the sum of the moduli of values for all samples in the sample to the measure of uncertainty Δϕ IIC:

Multiplying the numerator and denominator of expression (14) by the time of obtaining one sample Δt and taking into account expression (11), we obtain, for determining the number of information-measuring quanta N _γ , contained in the entire stochastic model of epicardial repolarization currents, a formula of the form

- determination of the number m, ranked values, boundaries for the time intervals of the grouping of IIK. The action, which consists in determining the number m of ranked values and boundaries for the time intervals for grouping the IRC, is necessary for the formation of time intervals for grouping quanta. For this, the ranked values of the time t _pi and the boundaries of all time intervals of the IIR grouping are determined using the expressions:

количества значений в выборке; t_{min j} и t_{max j} - минимальная и максимальная границы j-го интервала группирования данных.where m is the number of data grouping intervals equal to the square root of

the number of values in the sample; t _{min j} and t _{max j} are the minimum and maximum boundaries of the j-th data grouping interval.

и

в j-м интервале группирования ИИК для положительных и отрицательных значений потенциала эпикарда в опорных точках компьютерной модели сердца пациента: необходимо для выражения значений потенциала эпикарда в единицах меры ИИК. Цель действия состоит в количественном выражении значения результата в единицах меры неопределенности результата и организации процесса усреднения значений результатов путем оценки вероятностей распределения ИИК во временных интервалах группирования потенциалов эпикарда. Для сравнения результатов используется их приведенные оценки к количеству ИИК N_γ (числу), рассчитанного для всей выборки результатов. Отношение суммарного количества мер ИИК для положительных значений результатов, попавших в j-й интервал группирования, отнесенного к количеству мер ИИК, необходимых для оценки всей выборки результатов, представляет собой вероятность

обнаружения ИИК для положительных значений потенциала эпикарда в j-м интервале группирования данных. Аналогично, отношение суммарного количества мер ИИК для отрицательных значений результатов, попавших в j-й интервал группирования, отнесенного к количеству мер ИИК всей выборки результатов представляет собой вероятность

обнаружения ИИК для отрицательных значений потенциала эпикарда в j-м интервале группирования данных. Формулы для вычисления вероятностей распределения ИИК для положительных и отрицательных значений потенциала эпикарда имеют вид;- determination of the probabilities of IIC in jx grouping intervals for potential values. Determination of the probability distribution of IIC

and

in the jth interval of the IIR grouping for positive and negative values of the epicardial potential at the reference points of the computer model of the patient’s heart: it is necessary to express the values of the epicardial potential in units of the IIR measure. The purpose of the action is to quantify the value of the result in units of the measure of the uncertainty of the result and organize the process of averaging the values of the results by estimating the probabilities of the distribution of IIR in the time intervals of the grouping of the epicardial potentials. To compare the results, we use their reduced estimates for the number of NIK N _γ (number) calculated for the entire sample of results. The ratio of the total number of IIR measures for positive values of the results falling into the jth grouping interval, referred to the number of IIR measures necessary to evaluate the entire sample of results, is the probability

detecting IIC for positive values of the epicardial potential in the jth interval of data grouping. Similarly, the ratio of the total number of IIR measures for negative values of the results falling into the j-th grouping interval, referred to the number of IIR measures of the entire sample of results, is the probability

detecting IIC for negative values of the epicardial potential in the jth interval of data grouping. The formulas for calculating the probabilities of the distribution of IIC for positive and negative values of the potential of the epicardium have the form

и отрицательных

значений потенциала в интервалах группирования. Для расчета разности распределений вероятностей ИИК для положительных

и отрицательных

значений потенциала используется формула:- determination of the probability difference of the IIC in the grouping intervals for the potential values at the reference points of the computer model. The determination of the difference in the probability distributions of the IIR p _j in the jth interval of grouping the values of the potential at the reference points of the epicardium of the computer model of the patient’s heart is necessary to obtain an averaged value of the potential expressed in units of the IIR reduced to the total number of IIR in the sample of results. The sample of results contains positive or negative values. When quanta fall into one interval of grouping results, the quanta of positive and negative values are mutually destroyed. The shape of the curve of the useful signal reflects the probability difference between

and negative

potential values in grouping intervals. To calculate the difference in the probability distributions of IIC for positive

and negative

potential values the formula is used:

An example of the probability difference of the IIC for the reference point of the computer model is given in Figure 19. Using the difference in the probability distribution of the IIC over the grouping intervals allows you to highlight the characteristic time intervals of the cardiocycle.

- the formation of samples of time samples for positive and negative probability differences. The formation of samples of time samples for positive and negative differences in the probability distributions of the IIC is necessary to highlight the time section of the monotonous repolarization of the epicardium. For this, ranked samples of time values t _- and t ₊ are generated for positive p _j > 0 and negative p _j <0 probability differences according to the following rules:

- the allocation of the time interval of the phase of the initial rapid repolarization of the epicardium at reference points. Isolation of the time interval of the initial fast epicardial repolarization phase at the reference points of the patient’s computer model of the heart is obtained by evaluating the minimum time reference for selecting time values for negative p _j <0 probability differences

- allocation of the time interval of monotonous repolarization of the epicardium at reference points. Isolation of the time interval of monotonous repolarization of the epicardium, including the interval of monotonous repolarization of the plateau phase and the phase of final fast repolarization of the epicardium, is necessary to determine the parameters of the stochastic model of potassium current delayed and anomalous rectification. As a monotonic region of repolarization, a positive-sign IR region is selected, limited on the one hand by the maximum value of the time interval of the IIR grouping with a negative value, and on the other hand, by the maximum time value for a positive potential. To determine the boundaries of the intervals using the expression

Thus, the analysis of the change in sign of the data grouping intervals is an effective tool for isolating the various phases of TMPD of epicardial tissues.

Figure 19 shows an illustration for determining the time intervals of the initial fast repolarization phase ΔT _to and the monotonic repolarization interval ΔT _{K of the} epicardium, for which the difference in the probability of detection of the IIR is positive (i.e., greater than zero). The time interval ΔT _Ca with a negative difference in the probability of detecting IIC corresponds to the site of late depolarization of myocardial tissue due to uncompensated currents of calcium ions.

- normalization of the probability of the distribution of IIR for the potential of the time interval of monotonous repolarization of the epicardium. To establish the form of a stochastic model of delayed and anomalous rectification of potassium current according to the probability distribution of IIC, it is necessary to normalize the probability of the distribution of IIC for the potential of monotonous repolarization of the epicardium of the computer model of the patient’s heart. The formula for normalizing the probability of the jth data grouping interval is:

.where P _M is the probability of observing the IIR measure in the time interval of monotonous repolarization of the epicardium

.

The ranked time samples are independent and grouped in the form of a column vector t _m , i.e. matrices with dimension m × 1: [t _m ] ^T = [t _l , ..., t _j , ... t _m ]. Column Vector

.time samples t _m corresponds to the vector of normalized probabilities IIC p ⁿ for the monotonic repolarization time interval has the form:

.

Determination of potassium current of abnormal rectification at reference points of a computer model of the heart. The second distinctive effect of the proposed method for non-invasive determination of the electrophysiological characteristics of the heart is to determine the potassium current of abnormal rectification. Due to the fact that the normalized probabilities of the interval of monotonous repolarization of the epicardium contain information about the currents of abnormal rectification, it is possible to determine the current by isolating from the total number of IICs of the epicardial potential only those IICs whose appearance is due to the current of abnormal rectification.

For this, statistical distributions (normalized functions with a given shape) and the limits of variation of the parameters of these distributions for describing the currents of the monotonic repolarization interval are preliminarily selected based on detailed models. Then, the parameters of the stochastic model of potassium currents of delayed and anomalous rectification are determined by minimizing the difference between the model and normalized IR probabilities for samples of the potential values of the time interval of the monotonous repolarization of the epicardium.

The scheme of the detailed algorithm of action "Determination of the potassium current of abnormal rectification at the reference points of the computer model of the heart" is given in figure 14. From figure 14 it follows that the second distinctive effect of the proposed method of non-invasive determination of the electrophysiological characteristics of the heart contains:

- the formation of a stochastic model of potassium currents of delayed and abnormal rectification. In this case, the general form of the stochastic model of potassium currents of delayed and anomalous rectification has the form:

- стохастическая модель I_so тока калия быстрого задержанного выпрямления;

- стохастическая модель I_so тока калия медленного задержанного выпрямления;

- стохастическая модель тока калия аномального выпрямления;

,

и

) - плотности распределений ИИК для тока калия быстрого и медленного задержанного выпрямления и для тока калия аномального выпрямления соответственно; K_so1, K_so2 и K_K1 - весовые коэффициенты стохастической модели токов калия быстрого и медленного задержанного и аномального выпрямления соответственно; β_so1, α_so1, μ_so1, β_so2, α_so2, μ_so2, β_K1, α_K1, μ_K1 - параметры распределений;Where

 - stochastic model I_so potassium current fast delayed rectification;

 - stochastic model I_so potassium current slow delayed rectification;

 - stochastic model of potassium current of abnormal rectification;

,

 and

) are the distribution densities of the IRC for potassium current of fast and slow delayed rectification and for potassium current of anomalous rectification, respectively; K_so1, K_so2 and K_K1 - weighting coefficients of the stochastic model of potassium currents of fast and slow delayed and abnormal rectification, respectively; β_so1, α_so1, μ_so1β_so2, α_so2, μ_so2, β_K1, α_K1, μ_K1 - distribution parameters;

The determination of the components of ion currents is carried out according to the a priori known approximate model of the behavior of ion currents. For ion currents, information obtained using detailed mathematical models TNNP [34], LRd [28], IMW [26], and PB [31] is a priori known. Examples of simulation results of 1-type calcium ion currents, fast and slow potassium currents of delayed rectification using the well-known detailed models of PB, IMW, TNNP are given in figure 8 [23]. All the main currents of the detailed TNNP model are given in figure 6 [34].

и быстрого

токов задержанного выпрямления используются формы стохастических моделей, построенных на основе плотностей статистических распределений

,

и сохраняющих возможность изменения формы.The authors of the present invention propose to approximate the forms of ionic currents of the component forms of the ionic currents of the detailed models and to determine, thus, the components of the ionic currents of the epicardium. As an approximation of the normalized by area slow

and fast

delayed rectification currents, forms of stochastic models based on the densities of statistical distributions are used

,

and retaining the ability to change shape.

важно, чтобы временное значение положения максимума плотности распределения ИИК был больше временного значения положения для среднего значения тока калия задержанного выпрямления

, что следует непосредственно из форм зависимостей токов задержанного выпрямления (см. фигуру 8.в).When choosing an approximating distribution for constructing a stochastic model of delayed rectification potassium current

it is important that the temporary value of the position of the maximum density of the IIR distribution is greater than the temporary value of the position for the average value of the potassium current delayed

, which follows directly from the forms of dependences of the delayed rectification currents (see figure 8.c).

It should be noted that when constructing a stochastic model of delayed and abnormal rectification potassium currents based on the generated data (see figure 20), most of the delayed rectification currents are compensated by the L-type I _CaL depolarizing current of calcium. For this reason, the choice of the distribution form for constructing a stochastic model of delayed rectification currents is of a qualitative nature while maintaining the possibility of changing the shape over a wide range.

It was shown in [18] that, to approximate the potassium current of anomalous rectification, it is preferable to use the Weibull – Gnedenko distribution with a shape parameter of more than 2.4. In this case, the possibility of changing the shape of the model, which in the general case has an asymmetric form, remains.

- setting the weighting coefficients of the stochastic model of the currents of delayed and anomalous rectification at reference points. This stage is necessary for a preliminary assessment of the relationship between the statistical weight of the current model, from which a preliminary estimate of the weight coefficient of the stochastic model of currents of anomalous rectification follows. For this, the probability for the IIR in the second half of the time range of the monotonous repolarization of the epicardium is estimated, and the weight coefficient K _{K1 of the} stochastic model of anomalous rectification currents is compared. Weighting factors K _so1 and K _so2 for stochastic current model of fast and slow detainee and anomalous rectification defined by the expression:

K _so1 = K _so2 = 0.5 (1-K _K1 ).

получим систему нелинейных, плохо согласованных, уравнений вида:- determination of the parameters of the stochastic model of currents of delayed and abnormal rectification. To do this, a substitution is made into the mathematical model of the currents of delayed and anomalous rectification of the ranked values of time t _P j obtained for the intervals of grouping the IRC. The stochastic model is normalized by area and estimates the density of the distribution of IIC (uncertainties in the sample of samples) over time. By comparing the obtained expressions for the distribution densities with the probability ratios for the corresponding IIC grouping intervals, to the grouping time interval

we get a system of nonlinear, poorly coordinated, equations of the form:

where m is the number of intervals for grouping IIK.

вида:The column vector of time samples t _m corresponds to the matrix form of the column vector of the stochastic model

type:

.

.

The vector form of the system of equations (24) has the form:

The components of the stochastic model of potassium currents correspond to the probability of the distribution of IIR on the j-th time interval. Vector equation (25) is poorly conditioned, since the number of unknown parameters is much less than the number of equations. The solution of poorly conditioned equation (25) is obtained by minimizing the determinant:

As a result of the search for optimal parameters from expression (26), the parameters for the stochastic model of potassium currents (23) are obtained.

, полученная с помощью стохастической модели токов ионов калия. Из фигуры 20 наблюдается соответствие распределения и графика сглаживающей функции, полученное для интервала монотонной реполяризации. Там же даны временные зависимости и для составляющих стохастической модели тока задержанного

,

аномального

выпрямления. Для контроля формы стохастической модели токов задержанного выпрямления применена смесь двух распределений минимального значения. Составляющие тока калия быстрого и медленного задержанного выпрямления показаны пунктирными линиями

и

. Составляющим стохастической модели токов калия соответствуют вероятности распределения ИИК на j-м временном интервале группирования (см. фигура 20).- determination of the probabilities of IIC of potassium currents of delayed and abnormal rectification in the grouping intervals. The figure 20 shows a histogram for the density of the distribution of the IIK and approximation

obtained using a stochastic model of potassium ion currents. From figure 20 there is a correspondence between the distribution and the smoothing function graph obtained for the monotonic repolarization interval. The time dependences for the components of the stochastic model of the current delayed are also given there.

,

abnormal

straightening. To control the shape of the stochastic model of delayed rectification currents, a mixture of two distributions of the minimum value is used. The components of the potassium current of fast and slow delayed rectification are shown by dashed lines.

and

. The components of the stochastic model of potassium currents correspond to the probability of the distribution of IIR on the j-th grouping time interval (see figure 20).

The formulas for calculating the probabilities of the distribution of IIC at the time intervals of grouping, respectively, for stochastic models of potassium currents of fast and slow delayed rectification p _{so1 j} , p _{so2 j} and anomalous rectification p _{K1 j} have the form:

From the expression (15) it follows that the probability of the distribution of the IIR over the time intervals of the grouping is proportional to the normalized probability of the IIR and the ratio of the value of the component of the model to the value of the model at a given grouping interval.

- determination of the adequacy criterion r of the stochastic model of potassium currents of delayed and abnormal rectification at the reference point of the KMC epicardium.

,

и

. Для анализа параметров распределения с различной формой авторами применен энтропийно-параметрический критерий проверки адекватности сглаживающего распределения выборочным данным [19], в основе которого лежит преобразование несимметричных распределений к симметричным распределениям с последующим установлением соответствия в пространстве энтропии и контрэксцессса.The adequacy of the stochastic model of potassium currents is established from the condition of correspondence of the components of the stochastic model and the histograms of the distributions of the individual components of the potassium current obtained on its basis:

,

and

. To analyze the distribution parameters with different shapes, the authors applied the entropy-parametric criterion for checking the adequacy of the smoothing distribution to sample data [19], which is based on the transformation of asymmetric distributions to symmetric distributions with the subsequent establishment of correspondence in the space of entropy and counterexcess.

The formula for calculating the adequacy criterion for a stochastic model of potassium currents of fast and slow delayed and abnormal rectification at the reference point of a computer model of the patient’s heart looks like:

where r _so1 , r _so2 and r _{Kl are} independent components of the criterion calculated for symmetric

samples of random values of the probability of the distribution of the IIC of the potassium current delayed and anomalous rectification.

The components of the criterion are calculated by the formulas:

where k _{e so1} , κ _so1 , k _{e so2} , κ _so2 , k _{e K1} , κ _K1 , are the entropy and counterexcess coefficients of symmetrically distributed random variables Z _so1 , Z _so2 and Z _K1 for the potassium current of fast and slow delayed and anomalous rectification.

Application of the criterion is possible if a relation is established between a random variable [t] distributed over a separate component of the stochastic model and a symmetrically distributed random variable Z [20].

The formation of the probability distributions of the IIC of symmetric random variables Z _K1 , Z _so1 and Z _so2 for stochastic models of the components of the potassium current of anomalous rectification, slow and fast delayed rectification are given in figures 21, 22 and 23, respectively.

,

и

по временным интервалам группирования для стохастической модели тока калия аномального, быстрого и медленного задержанного выпрямления, которые приведены на фигурах 21а, 22а и 23а, соответственно. Для построения преобразования на основе гистограмм определяются значения элементов

вектора-столбца временных отсчетов t_s по формулеThe probability distributions of IIC of symmetric random variables Z _K1 , Z _so1 and Z _so2 are formed on the basis of histograms of the probability distribution of IIC

,

and

by grouping time intervals for a stochastic model of potassium current of anomalous, fast and slow delayed rectification, which are shown in figures 21a, 22a and 23a, respectively. To construct a transformation based on histograms, the values of the elements are determined

column vector of time samples t _s according to the formula

where m is the number of grouping intervals; n _s - the number of time samples of the vector in one interval grouping IIK; s is the serial number of the reference.

The probabilities of the distribution of the IIC of the delayed and anomalous rectification currents for the s-th time reference are calculated by the formulas:

вектора-столбца временных отсчетов t_s соответствует вероятность распределения ИИК для составляющих стохастической модели токов калия. Формально вероятность распределения ИИК для любого момента времени задана стохастической матрицей тока калия для 5-го отсчета времени

вида:The column vector of time samples t _{s is} necessary for constructing histograms of the probabilities of the distribution of IIC over the grouping intervals. To each s-th time sample

the column vector of the time samples t _s corresponds to the probability of the IIR distribution for the components of the stochastic model of potassium currents. Formally, the probability of the IIR distribution for any moment of time is given by a stochastic potassium current matrix for the 5th time reference

type:

.

.

The probabilities of the distribution of the IIR over random values of time samples are related to the exponential distribution of the random variable Y using the following relations. For stochastic models of fast and slow delayed rectification, the authors use a transformation to distribute the minimum value of a random variable t _j to a random variable y _{so j} having an exponential distribution. The conversion formula has the form:

where α _so , μ _so are the scale and displacement parameters of the distributions.

For the stochastic model of anomalous rectification, we use the Weibull-Gnedenko distribution of the form:

where α _K1 , μ _K1 , β _K1 are the parameters of the scale, displacement, and shape of the Weibull-Gnedenko distribution.

,

и

s-х значений случайных безразмерных величин Y_K1, Y_so1 и Y_so2 для стохастической модели тока калия аномального выпрямления, медленного и быстрого задержанного выпрямления приведены на фигурах 21б, 22б и 23б, соответственно.IIC probability distributions

,

and

s-values of random dimensionless quantities Y _K1 , Y _so1 and Y _so2 for a stochastic model of potassium current of abnormal rectification, slow and fast delayed rectification are shown in figures 21b, 22b and 23b, respectively.

,

и

для значений случайных безразмерных величин Z_K1, Z_so1 и Z_so2 для стохастической модели тока калия аномального выпрямления, быстрого и медленного задержанного выпрямления приведены на фигурах 21в, 22в и 23в, соответственно.The reflection of s-values of random variables Y _K1 , Y _so1 and Y _so2 relative to the origin allows you to create a matrix of values of symmetrically distributed random dimensionless variables Z _K1 , Z _so1 and Z _so2 . Since the number of matrix values increases by a factor of 2, the probabilities associated with symmetrically located values also decrease by a factor of two. Symmetric IIC probability distributions

,

and

for values of random dimensionless quantities Z _K1 , Z _so1 and Z _so2 for a stochastic model of potassium current of abnormal rectification, fast and slow delayed rectification are shown in figures 21c, 22c and 23c, respectively.

The calculation of the entropy coefficients and counterexcessions for symmetrically distributed random variables Z _K1 , Z _so1 and Z _{so2 is} carried out according to the histograms of the distribution of these quantities. To construct the histograms of the distribution of random variables Z _K1 , Z _so1 and Z _so2 , the maximum max [Z _so1 ], max [Z _so2 ], max [Z _K1 ] and minimum min [Z _so1 ], min [Z _so2 ], min [Z _K1 ] value for each random variable and the data grouping intervals are calculated

,

,

, минимальных

,

,

и максимальных

,

,

значений границ интервалов группирования данных.Grouping intervals are used to determine ranked values.

,

,

minimum

,

,

and maximum

,

,

the values of the boundaries of the data grouping intervals.

,

,

по интервалам группирования случайных величин Z_K1, Z_so1 и Z_so2 рассчитывается по формулам:IIC probability distribution

,

,

on the intervals of grouping random variables Z _K1 , Z _so1 and Z _{so2 is} calculated by the formulas:

,

,

по интервалам группирования случайных величин Z_K1, Z_so1 и Z_so2 для стохастических моделей тока калия аномального выпрямления, быстрого и медленного задержанного выпрямления приведены на фигурах 21г, 22г и 23г, соответственно.Histograms of probability distributions IIK

,

,

according to the grouping intervals of random variables Z _K1 , Z _so1 and Z _so2 for stochastic models of potassium current of abnormal rectification, fast and slow delayed rectification are shown in figures 21g, 22g and 23g, respectively.

The following situation is valid for the obtained distributions: if the distribution of the IIR over the time intervals of grouping corresponds to the models of the components of the potassium currents, then the random variables Z _K1 , Z _so1 and Z _{so2 are} distributed in accordance with the Laplace distribution. Then, in order to establish the correctness of the stochastic model of potassium currents of delayed and anomalous rectification, it is sufficient that the entropy and counterexcess distributions of the random variables Z _K1 , Z _so1 and Z _so2 correspond to the entropy coefficient and the counterexcess of the Laplace distribution. To calculate the entropy coefficients of symmetrically distributed random variables Z _K1 , Z _so1 and Z _so2 , the expressions are used:

,

,

- вторые центральные моменты распределений.Where,

,

,

- second central moments of the distributions.

To calculate the counterexcess of the distribution of random variables Z _K1 , Z _so1 and Z _so2 , the expressions are used:

,

,

,

,

,

,

- четвертые центральные моменты распределений.Where

,

,

- the fourth central moments of the distributions.

Converting to a symmetric distribution and calculating the parameters using expressions (36) and (37) allow us to evaluate the distribution histograms for the individual components of stochastic models of potassium current in a single space of the adequacy criterion for the stochastic model of delayed and anomalous rectification potassium currents.

. Модель принимается справедливой, если критерий адекватности r меньше некоторого максимального значения. r≤r_max - Comparison of the adequacy criterion with its maximum value. The choice of a fair model of the stochastic model of the delayed and abnormal rectification current is carried out on the basis of comparing the adequacy criterion r with the maximum value of the criterion specified using the decision significance level α:

. A model is accepted fair if the adequacy criterion r is less than a certain maximum value. r≤r _max

A change in the criterion for the stochastic model of currents and its components from the weight coefficient K _{K1 of the} stochastic model of potassium current of anomalous rectification is shown in Figure 24, and from where it can be seen that the most acceptable result was obtained with a coefficient value of 0.325, at which the criterion value is 1, 5. The figure 24, b shows the dependence of the shape parameter γ of the probability distribution of the IIR for potassium current of anomalous rectification on the weight coefficient of the model K _K1 , which implies that the shape parameter γ in the range of the weight coefficient of the model of potassium current 0.25 <K _K1 <0.45 monotonously decreases with increasing weight coefficient K _K1 . Figure 24 c and d show the dependences of the parameters of the stochastic model of the delayed and anomalous rectification potassium current on the weight coefficient K _K1 for the scale parameters λ _so1 , λ _so2 , λ _K1 and for the bias parameters μ _so , μ _so2 , μ _K1, respectively. From the above dependencies it follows that changes in the model parameters from the weight coefficient remain smooth in the region of changes in the model weight coefficient 0.25 <K _K1 <0.45.

- correction of the weight coefficients of the stochastic model of the potassium current delayed and abnormal rectification at reference points. If the criterion r is greater than the critical value, then the weighting coefficients of the stochastic model are corrected and the actions of the blocks included in the cycle of 4 actions are repeated:

1 - determination of the parameters of the stochastic model of the currents of delayed and abnormal rectification;

2 - determination of the probability of IIC of potassium currents of delayed and abnormal rectification in the grouping intervals;

3 - determination of the adequacy criterion for stochastic models of delayed and abnormal rectification currents;

4 - comparison of the criterion with the maximum value for choosing a stochastic model of delayed current and anomalous rectification.

- decision making: "The parameters of the stochastic model of the potassium current of delayed and anomalous rectification are determined correctly." If at the stage of comparison the criterion of model adequacy is less than a certain maximum solution, then the stochastic model of delayed and anomalous rectification currents is adequate to the probability distribution of the IIC. Making a decision: “The parameters of the stochastic model of anomalous rectification potassium current are determined correctly” is necessary to preserve the shape and scale parameters of the stochastic model of anomalous rectification current for further calculations. Since the distribution of the IIR for the stochastic model of potassium current of anomalous rectification is completely contained in the sample of the epicardial potential values, after isolating the distribution of the IIR of the potassium current of anomalous rectification from the entire data set and constructing a correct approximation of this sample, the distribution parameters should be kept unchanged during further calculations.

для всех значений вектора вектору временных отсчетов t_m.- determination of potassium current values of abnormal rectification. Further, a model of potassium current of abnormal rectification is considered known. To determine the parameters of the stochastic model of the epicardial repolarization current, the potassium current of abnormal rectification calculated using the formula

for all values of the vector, the vector of time samples t _m .

The next fourth distinguishing feature of the invention is the "Determination of the repolarization current of the epicardium at the reference points of the computer model of the patient’s heart." The scheme of the algorithm for determining the epicardial repolarization current at the reference points of the computer model of the patient’s heart is shown in Figure 15. The input to the algorithm is the difference in the probability of grouping the IIR for positive and negative samples.

- the formation of a stochastic model of epicardial repolarization currents. The stochastic model of epicardial repolarization currents includes the ^1st , ^2nd, and ^3rd phases of the development of the transmembrane action potential. The general form of the stochastic model of epicardial repolarization currents has the form

- стохастическая модель переходных транзитных токов эпикарда;

- стохастическая модель медленных деполяризующих токов кальция (отрицательный знак модели обусловлен ее деполяризующим действием); K_to, K_st и K_i - весовые коэффициенты стохастической модели токов реполяризации эпикарда;

и

- плотности распределений ИИК для переходных транзитных токов и медленных деполяризующих токов кальция соответственно; β_to, α_to, μ_to, β_st, α_st, μ_st - параметры распределений.Where

- stochastic model of transient transition currents of the epicardium;

- stochastic model of slow depolarizing currents of calcium (the negative sign of the model is due to its depolarizing effect); K _to , K _st, and K _i are the weights of the stochastic model of epicardial repolarization currents;

and

- IIC distribution densities for transient transit currents and slow depolarizing calcium currents, respectively; β _to , α _to , μ _to , β _st , α _st , μ _st - distribution parameters.

Using statistical functions with a controlled form of distributions for modeling the components of the repolarization current allows us to make the models flexible and take into account the change in the shape of ion currents. In this case, the ion currents of detailed mathematical models of potential development are used to estimate the initial values and boundaries of the shape parameters of stochastic models for the components of ion currents. It should be noted that detailed models contain information about the shape of currents of the same functional purpose. A priori information on currents is the starting point for constructing a stochastic model of currents.

When constructing a model of epicardial repolarization currents, the results of simulation of delayed and anomalous rectification potassium currents are used. In this case, the parameters of the stochastic model of potassium currents of abnormal rectification are assumed unchanged. The parameters of the stochastic model of delayed rectification potassium currents are taken as the initial simulation values. The fact is that the potassium current of the delayed rectification regulates the intake of calcium and, formally, for the most part they are compensated. The results of the experiment contain residual currents, the restoration of which is carried out according to the residual information.

получим систему нелинейных плохо согласованных уравнений вида:- determination of the parameters of the stochastic model of epicardial repolarization current by. To do this, we substitute the ranked values of time t _pj into the stochastic model of epicardial repolarization currents. Equating the obtained expressions for the distribution densities and the ratio of the probabilities for the corresponding intervals of the IIR grouping, to the duration of the grouping time intervals

we get a system of nonlinear poorly matched equations of the form:

In vector form, the system of equations (38) has the form:

- матрица стохастической модели тока реполяризации эпикарда; р=[p₁ р₂ … p_m] - матрица разностей вероятностей ИИК;Where

- matrix of a stochastic model of epicardial repolarization current; p = [p ₁ p ₂ ... p _m ] is the matrix of probability differences of IIC;

The ranked time samples are independent and represent a vector _{m of} dimension m. Vector equation (39) is poorly conditioned, since the number of unknown parameters is much less than the number of equations. We find the solution to the poorly conditioned equation (39) by minimizing the determinant.

As a result of optimizing expression (40), we obtain the parameters of the stochastic model of epicardial repolarization currents.

реполяризации эпикарда в опорных точках компьютерной модели сердца пациента в моменты времени вектора t_m используются для расчета коэффициента приведения тока при восстановлении модели. Формула для определения значений тока реполяризации эпикарда имеет вид- determination of the epicardial repolarization current at the reference points of the computer model of the patient’s heart. Current values

repolarization of the epicardium at the reference points of the computer model of the patient’s heart at time instants of the vector t _m are used to calculate the current reduction coefficient during model reconstruction. The formula for determining the epicardial repolarization current is

- ток реполяризации эпикарда;

- переходный транзитный ток эпикарда;

- медленный деполяризующий ток кальция;

,

- быстрая и медленная компоненты тока калия задержанного выпрямления;

- ток калия аномального выпрямления.The dependences of currents on time obtained using stochastic models are shown in figure 25, where the notation is given:

- epicardial repolarization current;

- transient transit current of the epicardium;

- slow depolarizing current of calcium;

,

- fast and slow components of the potassium current delayed rectification;

- potassium current of abnormal rectification.

The next fifth distinguishing feature is "Restoring potential at the reference points of the epicardium of the computer model of the patient’s heart." This action is necessary to compare the restored potential with the potential of the epicardium obtained as a result of processing EX of the known method. The main stages of the process of restoring the potential of the epicardium is illustrated by the algorithm in figure 16.

In accordance with Ohm's law, for any ith moment of time, the current through the surface of the epicardium is proportional to the potential of the epicardium:

where R _from is the resistance of body tissues.

Summing up the left and right sides of expression (42) for all counts of time and taking into account the invariance of the resistance of body tissues, we find that the sum of the potentials is equal to the sum of the currents:

Replacing the sum of the potentials on the left side and the sum of the currents on the right side of expression (43) through the epicardial potential reduction coefficient Bϕ and epicardial current reduction coefficient B ₁ , we find that the medium resistance is proportional to the ratio of the reduced coefficients for the current and potential.

Then, to restore the potentials of the epicardium from the values of the stochastic model of the repolarization current of the epicardium, it is necessary to determine the coefficients of reduction of the potential and current of the epicardium.

- determination of the coefficient of reduction of the potential of the epicardium. To restore the potential of the epicardium, the coefficient of reduction of the potential of the epicardium is determined, which is equal to the sum of the potentials at the reference point obtained as a result of processing the EX.

- determination of the coefficient of reduction of the current epicardium. The current reduction coefficient equal to the sum of the values calculated using the stochastic model of epicardial repolarization currents at times t _i according to the formula

- determination of the values of the restored potential at the reference points of the epicardium of the computer model of the heart. To determine the potential of the epicardium, it is sufficient to multiply the current of the epicardium of the ith time instant by the ratio of the reduced potential coefficient and the reduced current coefficient

The values of the restored potential are used to build an “electric portrait” of the heart. These potentials are then used to model the propagation of epicardial excitation.

The result of the restoration of the potential of the epicardium is shown in figure 25, where ϕ is indicated - the potential of the epicardium obtained by processing EX; ϕ _mod is the restored epicardial potential using a stochastic model of the probability distribution of the IIC of the epicardial repolarization currents. From the above dependencies follows their good coincidence.

Then the next, sixth distinguishing feature of the present invention is implemented: “Comparison of the restored and determined by the ECS potentials of the epicardium”. Comparisons of the potential of the epicardium determined as a result of processing the EX by a known method and the potential of the epicardium reconstructed using the stochastic model are necessary to assess the accuracy of the simulation.

The comparison process is specified using the formula for the maximum permissible reduced modeling error equal to the difference between the potentials of the epicardium reconstructed using the stochastic model and determined by the values of the ECS:

where ϕ _min and ϕ _max are the minimum and maximum values of the epicardial potential obtained during restoration using the stochastic model.

A comparison of the potential ϕ _mod reconstructed using the stochastic model of epicardial repolarization currents and the potential of the epicardium ϕ determined by the ECS in Figure 12 illustrates the conditional block “Comparison of the restored and determined by the ECS potentials of the epicardium”.

реполяризации эпикарда. Для этого изменяются начальные и граничные условия параметров при моделировании и повторяются три последних действия алгоритма: «Определение тока реполяризации эпикарда в опорных точках компьютерной модели сердца пациента», «Восстановление потенциала в опорных точках компьютерной модели сердца пациента» и «Сравнение восстановленного и определенного по ЭКС потенциала эпикарда».If, when comparing the given simulation error, δ _{ϕ is} greater than the permissible value δ _max , then the following sixth distinguishing feature of the invention is implemented: “Correction of the distribution parameters of the stochastic model of the epicardial repolarization current”. Correction is carried out by changing the distribution parameters β _to , α _to , μ _to , β _st , α _st , μ _{st of the} stochastic current model

repolarization of the epicardium. For this, the initial and boundary conditions of the parameters are changed during modeling and the last three actions of the algorithm are repeated: “Determination of the epicardial repolarization current at the reference points of the patient’s computer model of the heart”, “Restoration of potential at the reference points of the patient’s computer model of the heart” and “Comparison of the reconstructed and determined by ECS epicardial potential. "

Decision making: "The values of the epicardial repolarization current are determined correctly." If the value of the reduced error is δ _ϕ , the simulation is less than the accepted maximum value, a decision is made "The value of the epicardial repolarization current is determined correctly."

Thus, the identification of changes in the ionic currents of the epicardium contributes to the choice of antiarrhythmic drugs for the effective treatment of arrhythmias, their use with the lowest possible probability of arrhythmogenic effects, the development of a set of medical measures for the prevention of sudden cardiac death and the effective treatment of life-threatening arrhythmias [14, 18, 21].

In pathological conditions such as coronary heart disease, hypertensive heart disease, congenital and acquired heart defects, with myocardial hypertrophy, with electrolyte and hormonal disorders, there is a significant decrease in repolarizing currents of delayed I _Kr , I _Ks and abnormal I _K1 rectification, which, in its the turn is accompanied by an increased risk of complex rhythm and conduction disturbances. Thus, the control of potassium currents of abnormal rectification as a result of the use of the invention will allow the cardiologist to predict the occurrence of arrhythmias and quickly use high-tech cardiosurgical treatment methods.

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LIST OF ACCEPTED ABBREVIATIONS

AB Atrioventricular WHO World Health Organization KDS Computer Diagnostic System TMPD Transmembrane action potential ECG Electrocardiogram THE EX Electrocardiogram OZ ECG Inverse electrocardiography task Kmc Computer model of the heart LV Left ventricle MIS Medical Information System ICD International Classification of Diseases STS The cardiovascular system EAS Electrical activity of the heart EEGS Equivalent electric heart generator DTMPD duration of transmembrane action potential DI diastolic interval IIC information measuring quantum AP Model Aliev-Panfilov LRd Model Luo-Rudy RV Model Priebe-Beuckelmann TNNP Model Tusscher-Noble-Noble-Panfilov IMW Model Iyer-Mazhari-Winslow

Claims (46)

A non-invasive method for determining the electrophysiological characteristics of the heart, which consists in recording the electrocardiogram (EX) and measuring the potentials generated by the heart, recording the frontal and left-side fluorographic images of the patient’s heart, determining the patient’s heart heart parameters, synthesizing a computer model of the patient’s heart, determining potentials, generated by the heart on the patient’s torso, determine the dipole moments and epicardial potentials ϕ at the reference points of the computer model the patient’s hearts, model the propagation of the excitation wave in the epicardium, synthesize a model EX, compare the model EX with the registered EX, correct the calculated parameters of the model of propagation of the excitation wave in the epicardium by changing the parameters when determining the moment distributions of the main electrophysiological characteristics on the surface of the heart and visualize the electrophysiological characteristics of the heart on a three-dimensional heart models using computer graphics in the most convenient form for perception e, characterized in that it further - generate data for a stochastic model of epicardial repolarization current at reference points of a computer model of the heart by: determining the information-measuring quantum (IIC) γ for samples of the epicardial potential values at the reference points of the computer model of the heart for one cardiocycle: γ = Δϕ⋅Δt, where Δt is the time required to obtain one sample;

- the measure of IIC, ϕ _max , ϕ _min and N is the maximum, minimum value and the number of samples in the sample values of the epicardial potential at the reference points of the computer model of the heart; for determining the number of IIC N _γ for a stochastic model of epicardial repolarization current

, where ϕ _i - ie the value of the potential in the sample {ϕ} of one cardiocycle; determine the number m of ranked values

the minimum

and maximum

boundaries for the time intervals of grouping IIK

Where

if

otherwise

where j is the sequence number of the time interval grouping IIK; determining the probability distribution of IIC

and

in the jth interval of the IIR grouping for positive and negative values of the epicardial potential at the reference points of the computer model of the patient’s heart:

determining the difference in the probability distributions of the IIR p _j in the jth grouping interval for potential values at the reference points of the epicardium of the computer model of the patient’s heart:

the formation of samples of time samples for positive and negative differences in the probability distributions of IIC

allocation of the time interval of the phase of the initial rapid epicardial repolarization at the reference points of the computer model of the patient’s heart

the allocation of the time interval of monotonous repolarization of the epicardium, including monotonous repolarization of the plateau phase and the phase of the final rapid repolarization of the epicardium at the reference points of the computer model of the patient’s heart

normalization of the probability distribution of IIR for samples of the potential values of the time interval of the monotonous repolarization of the epicardium at the reference points of the computer model of the patient’s heart

, where P _M is the probability of detecting IIC in the time interval of monotonous repolarization of the epicardium

- determine the currents of potassium abnormal rectification at the reference points of a computer model of the heart by: the formation of a stochastic model of delayed and abnormal rectification potassium currents

Where

- stochastic model of potassium current fast delayed rectification;

- stochastic model of potassium current slow delayed rectification;

- stochastic model of potassium current of abnormal rectification;

- IIC distribution density for potassium current of fast and slow delayed rectification and for potassium current of anomalous rectification, respectively;

and

- weighting coefficients of the stochastic model of potassium currents of fast and slow delayed and abnormal rectification, respectively;

- distribution parameters; assignment of weight coefficients of a stochastic model of potassium currents fast

and slow

delayed and abnormal

rectification, where q _j = 1, if

otherwise q _j = 0; parameter definitions

a stochastic model of potassium currents of fast and slow delayed and abnormal rectification by minimizing the solution of the equation

Where

- matrix of a stochastic model of potassium currents of fast and slow delayed and abnormal rectification;

- matrix of normalized probabilities of IIC; determining the probability distribution of IIC currents of potassium fast delayed

and slow detainee

and abnormal straightening

on the jth interval of the IIK grouping; determination of the adequacy criterion

models of potassium currents of fast and slow delayed and abnormal rectification at the reference point of the computer model of the patient’s heart, where the components of the criterion are calculated by the formulas

Where

- coefficients of entropy and counterexcess of symmetric distributions for potassium current of fast and slow delayed and anomalous rectification; comparing the criterion r ² ≤3⋅ln (α ^-2 ), where α is the significance level of decision making; correction of weight coefficients of the stochastic model of potassium currents fast

and slow

delayed and abnormal

straightening at reference points of a computer model of the patient’s heart; decision: “Parameters of the stochastic model of potassium current

delayed and abnormal rectification determined correctly ”; value definitions

potassium current abnormal rectification according to the formula

- determine the repolarization current of the epicardium

at reference points of a computer model of the patient’s heart by: forming a stochastic current model

epicardial repolarization

where

- stochastic model of transient transition current of the epicardium;

- stochastic model of slow depolarizing current of calcium;

and

- weighting coefficients of the stochastic model of epicardial repolarization current;

and

- the density of the distribution of the IIC for the transient transit current and slow depolarizing current of calcium, respectively;

- distribution parameters; determining the parameters of a stochastic current model

repolarization of the epicardium by minimizing the solution to the equation

Where

- matrix of a stochastic model of epicardial repolarization current;

- matrix of differences in the probability distributions of the IIC; determining current value

epicardial repolarization at reference points of the patient’s computer model of the heart

- restore the potential ϕ _mod at the reference points of the epicardium of the computer model of the patient’s heart by: determining the coefficient of reduction potential of the epicardium

determine the coefficient of current reduction

determining the values of the restored potential ϕ _{mod i} , at the reference points of the epicardium of the computer model of the patient’s heart

- compare the restored ϕ _{mod the} potential of the epicardium with the potential of the epicardium ϕ determined on the EX

- adjust distribution parameters

stochastic current model

epicardial repolarization at reference points of the patient’s computer model of the heart; - make a decision: “Current values

the repolarization of the epicardium at the reference points of the computer model of the patient’s heart was determined correctly. "

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