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Russian Journal of Biomechanics
www.biomech.ac.ru
BIOMECHANICS OF CARDIAC ACTIVATION: THE SIMPLEST EQUATIONS AND MODELLING RESULTS
M.B. Hidirova
Department of Mechanics and Mathematics, Mirzo Ulugbek National University of Uzbekistan, Tashkent, 700095, Uzbekistan, e-mail: bahrom@vega.tashkent.su
Abstract. In this paper the questions of mathematical modelling of cardiac activation mechanisms are considered. On the example of the simplest functional equation, the importance of account of spatial-separated processes of wave origin and propagation on the heart surface is demonstrated using qualitative and quantitative analysis for definition of the solution behaviour regularities. Possibility of mathematical modelling of consequent activation of cardiac muscle cells using class of difference-differential equations is shown. They allow taking into account the temporary relations in cardiac regulatory system. Due to the complexity of created system of difference-differential equations, the methods for model system creation as reduced equations of cardiac activation are offered for model researches. The results of the qualitative analysis of developed equations of cardiac activation and using this equations for mathematical and computer modelling of the cardiovascular system are given.
Key words: cardiac activation, modelling, functional equations, difference-differential equations, model system, program software, cardiovascular system
Introduction
On the background of unremitting acuteness of cardiovascular diseases, the development of quantitative methods for investigation of circulation mechanisms is very actual. It becomes obvious that successful decision of given problem possible only at interconnected development of cardiovascular biology, biophysicists, biomechanics, biomathematics and biocybernetics. Herewith the role of cardiovascular biomechanics is basic due to combinations of theoretical statement of the problem with the enormous experience of using the exact methods and mechanical methods at the clarification of essential mechanisms of considered phenomena.
Quantitative researches of cardiovascular mechanisms suppose the analysis of regularities of cardiac activation. Results of experimental and model investigations [1-4] show the complex space-temporary variety of cardiac activities. In this paper the possible simplest equations of cardiac activity biomechanics are considered, in which the spatial-separated processes of activation origin and propagation are taken into account.
Method of functional equations
The electrical changes on the heart surface are typical manifestations of cardiac activity. Active (excitable) muscle fibre becomes negatively charged with respect to inactive (nonexcitable) muscle fibre. Thus, a difference of potentials appears always, when during
activation of cardiac tissue the pieces of heart fibres are excited, but other pieces of heart fibres are not excited [5]. Let common value of potential difference be some function fit) of time. Blood pressure in aorta at the moment of time t, basically, depends on amounts of excitable cells in ventricles and in auricles. Consequently, functional dependence exists between the blood pressure O(t) in aorta, and fit). If one takes into account that pressure of blood thrown by ventricles at a moment of time t depends not only on amounts of excited cells, but also on the value of blood pressure at preceding moments of time (from the beginning ventricles systole) and on the value of pressure of blood thrown by auricles in the ventricle during the systole, then in general case the dependence between Q(t) and fit) is very complex and requires using the equations allowing to take into account the spatial-separated processes of cardiac activity. Let us consider the following dependence
a(t)f(t) + b(t)f(t-h) = Q(t), (1)
where a(t), b(t) are continuous functions, which express the radial elasticity of auricles and ventricles. Note that if Qft) type on [0, qo) and fit) type on the initial interval [0, h] are
known, then it is sufficient to find /(0 on [h, co) under given a{t), b{t).
For analysis of the simplest particularities of solution (1) we will find fit) type under very strong admissions about ait), b(t) and h, let
a(t) = 6(0 = a~] = const * 0 and h be a small positive number. Equation (1) may be written
/(i+/»,)+ /(/-*,) = aQx{t), (2)
where
hx = hi 2 ; 0i(O - Q(t+ h/2).
We will consider that fit) e C2(0, T). Then
/(/ + hx) + fit- hx) = 2 fit) + h\f\t) + Oih2) or equation (2) may be approximately written (with account of initial conditions)
2/(0 + hx f'\t) = aQxit)';
/(0) = 0;/'(0)= /0.
Solution of this equation is [6]
/(0 = 4-/0 sin i^il + -A- )q] (X) sin - X)
;
dx.
yf2J° { 6, J 4lhx
Now we investigate the fit) behaviour at various suggestions relative to Q\it). 1. Let Q\it) = 0. Then
/(0 = ^/„sin(^Y
Since h\ is small, then we have oscillations with very small amplitudes. Consequently, there is such electrical change on the heart surfaces, under which pressure of blood flow is absent, i.e. there are such "excitable" impulses, which unable to excite heart cells. It means that there is so-called threshold value of impulses for cardiac cells. If impulse power is below the threshold value, then cardiac cells can not execute own contractile function. This fact in medicine is known [5]. If the heart will be stopped, it is possible to observe the rhythmical electrical changes on the heart surfaces for a long time.
2. Let Qx (t) = C = const. Then fit)
hx . f vr m
<j2t V h\ J
+
ac
1 - cos 4^-t «1
This is the oscillation near ac/2 with finite amplitudes. Sineeflt) > 0, then some pieces of the heart surfaces are contracting. This promotes continuous blood flow.
— CIt
3. Let Qx (t) = Ae ; a - const > 0; A = const > 0. Then
/(O = 4^/0sin
h f = ~4i sin
h
rV2
h
\
t + -
/
+
aA j
-at .
e sin
Jl \ h\
(t-x)
dx =
aA
2 + a h{
-at L . 2 t ah . .e ^2sin ——-i—j=rsin ,
2,2 J2h, ^
Let (j)(/2],0 is function consisting from items with h\\ §(hx,t) is small oscillation.
Then
/( 0 =
2 a A
n , 21 2 2 + a hx
-at . 2
e sm
4lh
+ <jKÄ1.0-
1 y
Here we have attenuation of oscillations. 4. In the case of practical modelling of cardiac activity it is very interesting when
Qx(t) = Ate~at; a = const > 0; A = const >0, (3)
(see real function of arterial pressure [5]).
Results of computer calculations have shown that in this case various regimes of cardiac activity depending on concrete values of parameters can be observed (Fig. 1).
The considered quantitative description of cardiac activity on the basis of equation (1) can be used for investigations of excitable cardiac tissue at limited problems. It is a simplified
UL_L
t
0
c d
Fig. 1. Some variants of (1) solutions by means of (3).
equation, but it reflects the essential sides of cardiac activity, it is necessary to note that it neglects dependence of cardiac activity from the blood pressure in arteries, existence of the left and right heart divisions, influence of organism condition on heart activity.
Methods of difference-differential equations
In works [7-9] the results of cardiac activity modelling based on the consequent quantitative description of cardiac fibre excitement, with regard to the above-mentioned sides of heart activity, are given. According to the given methodology, system of heart muscle fibres is considered as an excitable medium with the rhythm driver (pacemaker) in sinus node. Cardiac activation occurs due to propagation of activity waves on the heart surface. Pacemaker activity is regulated by means of the neuro-hormonal factors. Let x\ be the excitement value of muscle fibre at the moment to', xi be the excitement value of muscle fibre at the moment ¿o+x; ...; xB be the excitement value of muscle fibre at the moment tQ+(n-\). The excitement wave propagation on sinus node of cardiac muscle fibre can be described by the following difference-differential equations [7]
= aMxM(t -t) -b^(0 ; i = 2, ...,n, (4)
where a, is coefficient which expresses the velocity of excitement activity in /'+1 layer of fibre muscles; bt is coefficient which expresses the velocity of excitement abatement in /+1 layer of muscles fibre. Note that bt>a,+i and bra,t\ expresses the loss of power.
Then the excitement wave covers both auricles which have common musculature: the right auricle
<fyjit) dt
the left auricle
-- ßiJ^i-i(t - T) - c^iii) ; / = 2, ,.., m, (5)
¿fej (0
dt
$\xn(t-x)-c\zx(t)-
^P = ß;.zM(i-x)-c;zi.(0; i = 2,...,m, (6)
where y,{t), z,{t) (i = 1,2, ..., m) are amounts of excitement muscle fibres of the right and left auricles at the moment t. The ß(-, c;-, ß'y- c'j (i = 1, 2, ..., m-1) are coefficients like ones in equation (4).
Further excitement wave propagation on ventricle muscle fibres can be expressed by the following difference-differential equations: the right ventricle
= Yj©/_i (t - t) - «/,©-,(0 ; i = 2, ..., k, (7)
the left ventricle
dr\i(t)
dt
dT\i(t)
i\ -!)-</{ 11,(0;
dt
y'irii_l(t-T)-d'ii]i(t); / = 2,...,Jc, (8)
where ©,(/)> r|; (Y) (z'= 1, 2, ..., k) are amounts of excitement muscle fibre of the right and left ventricles at the moment t. The y,-, dt, y'j d'j (/ = 1,2,..., ri) coefficients are like ones in equation (4).
Let us consider the construction of equation of muscle fibre excitement in sinus node at the beginning the heart cycle. The muscles fibre excitement degree depends on power supply, functioning ventricles and auricles during previous stage of the heart cycle, the state of the heart and organism as a whole. Finally, above-mentioned parameters depend on the functioning muscles by means of which the whole organism is ensured by necessary resources. Then assuming that excitement nature depends on characteristics of excitable medium [3, 4, 10, 11] and taking into account the neuro-hormonal influence of organism on sinus node cells we can write
^ = Fs (®k (t), x\k (t))Ft (®k (0, tu (0)" Vi (0.
where Fs is stimulation function; Fl is inhibition function of considered muscle fibres excitement.
During study of similar processes, an inhibition function is usually considered as monotonous decreasing function of own arguments [11]. Taking into account modelling experience of regulatory closed system we can write
^ = Fs (®k (t), r\k (0) exp(-S,©* (0 - 62tu (/)) - Vi (0 •
Since heart functioning at norm expects synchronous ventricle activity and activity cessation of one of them leads to stopping heart activity as a whole, the stimulation function can be taken as homogeneous function of own arguments. Consequently, in the most simplified type for the excitement value of muscle fibres of sinus node on the initial stage of the heart cycle, we can write
^P = a©* (/)tu (0 exp(-5,©* (t) - 8^(0) - b0x, (/). (9)
Thus, we have obtained the closed system of n+2m+2k+\ nonlinear difference-differential equations (4)-(9). Values n, m, k can be determined from biological data. Solutions of the given system of equations can be obtained using the method of consequent integration under given initial conditions on time segments [10, 11].
Equations analysis and results
The system of equations (4)-(8) has zero initial conditions, since cardiac muscle fibres are in rest at the moment of new cycle beginning. For the equation (9), which expresses the excitement value of muscle fibres of sinus node, coefficients and initial conditions of the heart cycle are defined with account of blood pressure value of venous return. In this case the first equation of (4) is heterogeneous one
dx-y(t) , , .
£ =ai<p(0-¿i*2(0,
where cp(i) is known function of excitement value of muscle fibres of sinus node at the initial stage of the heart cycle. Solution of given equation defines heterogeneous part (at the length t) of the second equation of (4) and so on. Solutions of equations (4)-(8) and the right part of equation (9) are defined by means of this technique.
It is necessary to use corresponding reduced systems due to nonlinearity and greater amount of considered equations. In work [7] the main methods of reduced similar equations are stated in detail and the case of reduced equation of cardiac tissue excitement by means of this nonlinear difference-differential equation is considered
^p- = ax2(t-h) exp(2(1 - x(t - h)) - bx(t)).
This equation can be written in the following non-dimensional form
1) exp( 2(1 - x(t - 1)) - x(t) ,
1 dx (Q _ a_2( bh dt b K
where x{t) is the excitement value of muscle fibres at the moment t; a is parameter which expresses the average velocity of cardiac fibre excitement; b is parameter which expresses the average velocity of excitement suppression of cardiac fibre; h is parameter characterising the average time required for realisation of feedback in the system of cardiac activity. Values of a and b depend on level of blood supply of the heart muscle, value of h depends on organism state as a whole. It was shown that equation (10) has stable oscillatory solutions, expresses the functional-active regimes of heart activity and can be used for quantitative and qualitative analyses of the most common regularities of excitement of cardiac tissue as a whole [8, 9]. This equation was used (including Navier-Stokes equations for modelling of blood movement in arterial vessels and active and passive diffusion equation for description of blood transfer in organs and tissue of organism) for creation of minimal closed model of the cardiovascular system and for REGUS software [12]. Using REGUS for analysis of cardiovascular mechanisms at norm, arrhythmia origin, development and sudden cardiac death, we have shown that software can be used for quantitative analysis of cardiovascular functioning mechanism. Program REGUS was successfully used for solution of optimisation task at surgical treatment of portal hypertension under No 49-96 grant framework (with scientists from "Breast surgery" Institute of MHPUz under supervision of Prof. F.T. Adilova).
116.70
PRESS <H> TO CHANGE
heart activity - <h>
h=100.0
PRESS <P> TO CHANGE pressure in tissue - <p>
p=100.0
PRESS <F> TO FINISH
Fig. 2. The representation of the main cardiovascular characteristics during computing
experiments based on REGUS.
Figure 2 represents working state of REGUS display during quantitative analysis of the cardiovascular system at portal hypertension.
It is necessary to note that the reduced system of equations (derived from equations (4)-(9)) for description of cardiac tissue excitement with account of average values of excitable cells in pacemaker, auricles and ventricles can be used for analysis of possible anomalies in auricles, ventricles and pacemaker
= ax®{t - T0)ri(f - to^-W-^-M^o) _ blx(t);
^l = a2x(t-x1)-b2y(t);
M>=a3x(r-x2)-M0; (ID
where x(i), y{t), z(t), &(t), r\(t) are the variables, expressing levels of pacemaker, auricles and ventricles excitement, respectively; {a}, {b}, {5}, {x} are positive constants.
Results of qualitative analysis (11) show the presence of the trivial stable state. Existence of nontrivial equilibrium state depends on the solutions of equation
= 1, (12)
where
j = ; g a 4 § a3"5
bxb2...b5 ' 6264 b3b5
Analysis of (12) shows presence of two positive roots, when
A > 8e .
Qualitative and quantitative computer analyses of stability of nontrivial equilibrium (11) show the presence of unstable equilibrium and area with oscillatory solutions. These oscillations can be Poincare-type limit cycles (normal cardiac activity) and irregular fluctuations (various forms of arrhythmia). In some cases during computing experiments based on modified REGUS, there is the break-down of oscillations (effect of "black hole"), which denotes the sudden cardiac death and sharp fall in blood pressure in circulation system. Qualitative analysis of (11) in detail with determination of common regularities of solution behaviour in the phase space and construction of parametric portrait are subjects of further researches.
Conclusions
Depending on decided problems in the field of quantitative analysis of cardiac activity one of the following equations can be used: functional equation (1), difference-differential equation (10) and system of difference-differential equations (11). Their qualitative analysis showed the existence of stationary stable state, periodic solutions, irregular oscillations and effect of "black hole" under certain conditions. Moreover we can model concrete diseases of the cardiovascular system and offer the recommendations for choice of treatment tactics [13].
References
1. WITKOWSKI F X. Spatiotemporal evolution of ventricular fibrillation. Nature, 392: 78-82, 1998.
2. GOLDBERGER A.L. Cardiac chaos. Science, 243: 1419, 1989.
3. WINFREE A.T. Electrical turbulence in three-dimensional heart muscle. Science, 266: 1003-1006, 1994.
4. КРИНСКИЙ В.И., МИХАЙЛОВ Ф.С. Автоволны. Москва, Знание, 1984 (in Russian).
5. Физиология кровообращения. Физиология сердца. Ленинград, Наука, с. 72-82, 1980 (in Russian).
6. КАМКЕ Э. Справочник по обыкновенным дифференциальным уравнениям. Москва, ИЛ, 1951 (in Russian).
7. ХИДИРОВА М.Б. Об одной замкнутой модели сердечно-сосудистой системы (ССС): Механика возбуждения сердечной ткани. Проблемы механики, 2: 39-43, 1998 (in Russian).
8. HIDIROVA М.В. Cybernetic simulation of "heart-vessels" systems. Proceedings of the 10-th International Congress on Cybernetics and Systems. August, 26-31, Bucharest, 1996, Tome 2, p.
77.
9. ХИДИРОВА М.Б. Об одной замкнутой модели сердечно-сосудистой системы (ССС): Норма и аномалия сердечной деятельности. Проблемы механики, 3: 29-33, 1998 (in Russian).
10. ХИДИРОВА М.Б. Об одной замкнутой модели сердечно-сосудистой системы (ССС): Основные уравнения и результаты вычислительных экспериментов. Проблемы механики, 5: 26-30, 1998 (in Russian).
11. ХИДИРОВ Б.Н. Об одном методе исследования регуляторики живых систем. Вопросы кибернетики, 128: 41-46, 1984 (in Russian).
12. ХИДИРОВА М.Б. Программа для гибридного моделирования сердечно-сосудистой системы (REGUS). Свидетельство № 83. N ED GU 9700014. Зарегистрировано в Государственном реестре программ для ЭВМ 24.04.1997 (in Russian).
13. HIDIROVA М.В., ADILOVA F.T. Mathematical modelling of portal hypertension. Proceedings of the 9-th World Congress on Medical Informatics, August 18-22, Korea, Seoul. Edited by B. Cesnik, A.T. McGray Press. Amsterdam-Berlin-Oxford-Tokyo-Washington. DS: 399, 1998.
БИОМЕХАНИКА ВОЗБУЖДЕНИЯ СЕРДЕЧНОЙ ТКАНИ: ПРОСТЕЙШИЕ УРАВНЕНИЯ И РЕЗУЛЬТАТЫ МОДЕЛИРОВАНИЯ
М.Б. Хидирова (Ташкент, Узбекистан)
Успешный анализ закономерностей функционирования сердечно-сосудистой системы возможен при совместном развитии биологии, биофизики, биомеханики и биокибернетики кровообращения. Роль биомеханики кровообращения при этом является основополагающей вследствие сочетания ею теоретической постановки задачи с огромным опытом применения точных методов и средств механики при выяснении существенных механизмов рассматриваемых явлений.
Количественное исследование механизмов кровообращения предполагает анализ закономерностей возбуждения сердечной ткани. Результаты экспериментальных и модельных исследований показывают сложное пространственно-временное разнообразие процессов возбуждения сердечной ткани. В данной статье рассматриваются вопросы математического моделирования механизмов возбуждения сердечной ткани. На примере простейшего функционального уравнения путем качественного и количественного исследования закономерностей поведения его решений демонстрируется важность учета пространственной разделенности процессов возникновения и распространения возбуждения по сердечной ткани. Показана возможность математического моделирования последовательной активации мышечных клеток сердечной ткани путем привлечения класса дифференциально-разностных уравнений. Они позволяют учитывать временные взаимоотношения в системе сердечной регуляции. Вследствие сложности построенной системы дифференциально-разностных уравнений для модельных исследований предлагаются способы построения их модельных систем в виде редуцированных уравнений возбуждения сердечной ткани.
В статье приведены результаты качественного анализа разработанных уравнений возбуждения сердечной ткани и применения их для компьютерного моделирования системы кровообращения. Библ. 13.
Ключевые слова: возбуждение сердечной ткани, моделирование, функциональные уравнения, дифференциально-разностные уравнения, модельные системы, программное обеспечение, кровообращение
Received 21 April 2001
