Mechanical inhomogeneity of myocardium studied in paralleland serial cardiac muscle duplexes: experiments and models

Olga Solovyova *, Leonid Katsnelson, Slava Guriev, Larissa Nikitina,Yury Protsenko, Sergei Routkevitch, Vladimir Markhasin

Department of Molecular and Cellular Biomechanics, Ekaterinburg Branch of the Institute of Ecology and

Genetics of Microorganisms, Ural Division of Russian Academy of Sciences, 620219 Ekaterinburg, Russia

Abstract

We investigate, both experimentally and theoretically, contribution of the myocardium mechanical inhomogeneity

to the contractile function. We developed three approaches, named as Muscle Duplex Methods, to study the specific

effects and mechanisms of interaction in the simplest myocardial system consisting of two muscular units connected

either in parallel or in series. Our experimental approach is designed to study the interaction between two isolated

mechanically inhomogeneous cardiac muscles. The virtual duplex approach is based on a mathematical model of the

myocardium contraction. The hybrid duplex approach has been designed to support, in real time, interaction between a

natural muscle and its virtual counterpart. Using these approaches we showed the existence of a fine alignment between

mechanical characteristics of interacting inhomogeneous myocardial elements. Contractile properties of the elements

together with particular sequences and time delays in their stimulation specifically determine this alignment. We term as

‘‘tuning effects’’ all the phenomena concerning the interaction between inhomogeneous system’s elements. Within the

framework of the mathematical model we showed that the key mechanism underlying tuning effects is a feedback

between mechanical conditions and cooperative Ca2þ binding by troponin C. Thanks to the model analysis, we also

hypothesize that mechanical inhomogeneity of myocardium is apt to produce its electrical inhomogeneity. � 2002

Elsevier Science Ltd. All rights reserved.

1. Introduction

The term ‘mechanical inhomogeneity of myocardium’ (MIM) is used to address the differences between basic me-

chanical characteristics of adjacent or distant interacting contractile elements (CEs) in a cardiac muscle. These elements

may be small (e.g. separate sarcomeres) or large (strands of tissue), up to whole segments of the heart chambers. MIM

has been observed by many investigators. In particular, systolic and diastolic deformations of cardiac muscle are more

pronounced in sub-endocardium vs. sub-epicardium, as well as in the heart’s apex vs. its base [1]. Regional variations in

the left ventricular wall thickness and/or curvature [2], and of transmural pressure gradients [3] were found. It has been

shown that cardiomyocites in various regions within different heart chambers differ in their biochemical, metabolic,

electrophysiological and mechanical properties [1,4]. MIM is specifically prominent in pathological conditions like

myocardial ischaemia, which dramatically changes both the local and global kinetics of the ventricular walls [5]. MIM is

also widely expressed on the level of sarcomeres. For example, it was shown that sarcomere lengths differ in various

myocardial layers as a function of the ventricular volume [6]. Also, the onset and time courses of sarcomere activation

during a single beat vary in different layers of the ventricle [7]. Furthermore, MIM reveals itself on the molecular level

via variations in the ratio of myosin isoforms in cardiomyocytes lying in different myocardium layers [8,9]. Each

Chaos, Solitons and Fractals 13 (2002) 1685–1711www.elsevier.com/locate/chaos

*Corresponding author. Tel.: 7-3432-741316; fax: 7-3432-740070.

E-mail address: oas@cranium.uran.ru (O. Solovyova).

0960-0779/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0960-0779 (01 )00175-8

particular ratio corresponds to specific mechanic and energetic properties of a cardiomyocyte [10]. Besides, this ratio

may change due to variations in load or as a result of hormonal effects [11].

While MIM is a firmly established phenomenon, the basic properties of the interaction between elements of inho-

mogeneous myocardial structure have not been studied in detail yet. The identification of these properties would widen

the paradigm of cardiac biomechanics [10,12] and help to understand the contribution of MIM to cardiac regulation in

both norm and pathology.

Owing to the extremely complicated character of MIM in the whole heart, it is beneficial to use specific, simplified

approaches for an experimental and theoretical study of this phenomenon. We assume that inhomogeneous system may

be represented on any level (from the cell to the whole organ) as a composition of groups of basic inhomogeneous sub-

systems, each of which may be represented by a duplex, i.e. by two elements connected either in parallel or in series.

When creating the duplex method, we hypothesized that mechanical interaction between contractile structures should

produce, by itself, some new specific effects. This hypothesis follows from two points: (1) evidence that partners of an

inhomogeneous duplex continuously produce specific changes in mechanical conditions for each other, and (2) the

existence of a feedback between mechanical conditions during muscle contraction and the process of thin filament Ca2þ

activation being reported by many authors [13].

We have developed three approaches based on the concept of a duplex of cardiac muscles. We have studied duplexes

of the following three different types:

• Muscle duplex composed of two natural heart muscles connected in parallel.

• Parallel and serial virtual duplexes composed of two virtual muscles, each of which is represented by a mathematical

model of homogeneous myocardium.

• Hybrid duplex composed of a natural and a virtual muscle, which interact in real time simulating contraction of a

duplex of natural muscles connected in parallel.

All these approaches complement each other but we pay special attention to the method of mathematical modelling

because based on this method we can explain the mechanisms underlying inhomogeneity phenomena.

2. Mathematical model of homogeneous myocardium contraction (virtual muscle)

2.1. Brief description

First of all we will demonstrate the simulation capability of a mathematical model of homogeneous myocardium

contraction embodying virtual elements in both virtual and hybrid duplexes in our studies.

The full list of the model equations is given in Appendix A. In part we published this model of homogeneous

myocardium elsewhere [14–17]. Here we reproduce a brief description of the main model postulates and specific features.

Mechanical block of the model. A classical three-element rheological scheme is used to represent the homogeneous

muscle (see Fig. 1(A) for the scheme). It consists of a CE (sarcomere), plus series and parallel elastic elements (SE and

PE). We consider tensions of the two latter elements as exponential functions of their deformations (see Eqs. (A.3) and

(A.4)).

We assume that force generated by the sarcomere is determined as follows:

FCEðtÞ ¼ k � f ðvðtÞÞ � NðtÞ; ð1Þ

where t is the time, k is a scaling factor, f ðvÞ is an explicit parametrical function describing dependence of the averageforce f of one cross-bridge (XB) on the velocity v of sarcomere shortening or lengthening and N is a number of force-

generating XBs in the overlap zone of thin and thick filaments.

The fraction N of force-generating XBs depends on the number Nact of active sites on thin filaments available forinteraction with XBs and on the probability n for XBs to be attached to the actin:

N ¼ Nact � n: ð2Þ

The probability n ¼ n1 � n2 is determined as a product of the probability n1 for an XB to find an active site on thethin filament and the conditional probability n2 to interact with the bound site. We consider that the probability n1depends on the lattice spacing and increases with sarcomere length (Eq. (A.12)). This consideration takes its stand on a

number of experimental data [18,19]. The probability n2 depends on the velocity of sarcomere shortening or lengtheningand it is determined by the velocity dependent kinetics of XBs attachment–detachment (Eq. (A.11)).

Thus, force generated by the sarcomere nonlinearly depends on the sarcomere length, velocity of sarcomere

shortening or lengthening and Ca2þ activation of thin filament.

1686 O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711

Cooperative mechanisms of Ca2þ activation. Kinetics of Ca2þ complexes with specific troponin C (TnC) is one of thecentral points regulating the process of Ca2þ activation of thin filament and CE force development during contraction/

relaxation. In particular, we guess and prove by means of the model that peculiarities of this kinetics are responsible for

the experimentally observed phenomena of myocardium inactivation [13,20–23].

In general, Ca2þ binding with TnC is described by the following scheme:

Ca2þ þ TnC $aonaoff ðNA ;AÞ

A;

where aon; aoff are the on/off rate constants of Ca–Tnc association/dissociation; A denotes the concentration of Ca–TnCcomplexes and NA is defined below.

We took into account three types of cooperativity being relevant to Ca–TnC kinetics. All of them have been founded

experimentally and reported elsewhere [24–29].

Type 1. Affinity of TnC for Ca2þ increases with the average number of strongly bound XBs around each Ca–TnC

complex (this number is denoted as NA). We consider the Ca–TnC off-rate to be decreasing with increase in NA

(Eqs. (A.21) and (A.32)). Because of the NA depends implicitly on the sarcomere length (Eqs. (A.6), (A.7) and (A.31)),

off-rate for Ca–TnC is length-dependent as well. The latter circumstance produces, in turn, the length dependence of

myocardium Ca2þ activation.

Type 2. Affinity of TnC for Ca2þ increases with the Ca–TnC concentration resulting from conformation of the

troponin–tropomyosin complex due to Ca2þ binding to TnC. Adjusting, we consider the Ca–TnC off-rate to be de-

creasing with increase in A (Eq. (A.21)).

Type 3. Fraction Nact of active sites on the thin filament in the overlap zone increases cooperatively with Ca–TnCconcentration as a result of an end-to-end interaction between adjacent tropomyosins:

Nact ¼Al

Al þ KlA

� Loz; ð3Þ

where Loz is the length of the overlap zone; KA is the concentration of Ca–TnC for the half-maximum fraction of active

sites and l is the coefficient of cooperativity.

Fig. 1. Schemes of the mechanical and Ca blocks of the model. (A) Rheological scheme. Shown are the contractile element (CE), series

(SE) and parallel (PE) elastic elements. l1; l2 are the CE and PE lengthening over the rest length Lr; muscle length L is proportional toLPE. FPE; FSE – PE and SE tensions (see also Eqs. (A.3) and (A.4));muscle force F is proportional to FSE þ FPE. L¼ const under isometricconditions; F ¼ const under isotonic conditions. (B) Ca2þ handling scheme. The scheme demonstrates Ca2þ recirculation in cardio-myocyte during a beat. CaC – free Ca

2þ concentration in cytosol; Fin is the trans-sarcolemmal Ca2þ influx; FSRrel – flux of Ca

2þ released

from terminal cisterns (TC) of the SR; A – concentration of calcium associated with specific TnC; B1 and B2 – concentrations ofcalcium bound to fast and slow buffer ligands as a total; FSRpump – flux of Ca

2þ-uptake by SR-pumps; Fout – total flow of Ca2þ removed

out of the cell; FSRflow is a flow of Ca2þ between longitudinal SR (LR) and the TC; C – concentration of Ca–calsequestrin complexes;

CaLR;CaTC – free Ca2þ concentration in the LR and in the TC.

O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711 1687

All these types of cooperativity were firmly established in particular biochemical experiments and were discussed in

detail in our previous works [15,16].

Ca2þ handling modelling. We used a typical scheme of Ca2þ handling (Fig. 1(B)) including a simplified description of

the Ca2þ exchange between cytosol and external space, Ca2þ binding by TnC and other cytosolic ligands. Cytosolic

Ca2þ buffering was described in the model as Ca2þ binding by a generalized Ca2þ-buffer [30]. Ca2þ exchange with

sarcoplasmic reticulum (SR) includes: Ca2þ release from SR, Ca2þ pumping from cytosol to SR, Ca2þ flow between SR’

compartments and Ca2þ binding by calsequestrin within SR’s release compartment. All Ca-block equations of the

model are listed in Appendix A as (A.18)–(A.30). Particularly, we considered the mechanism of back-inhibition of the

SR-pump in our model (see right multiplier expð�kinh CaLRÞ in Eq. (A.28)) [16]. Owing to this mechanism velocity of SRCa2þ-pumping decreases with an increase in SR lumenal Ca2þ [31,32]. This effect may be important in the regulation of

muscle relaxation.

2.2. Simulation of myocardium mechanical behaviour by means of a homogeneous virtual muscle

Further on we demonstrate the main results of modelling obtained within the framework of the model presented

here. Some of them were earlier simulated by means of simpler versions of the model and were partially published

elsewhere [14–17].

Steady-state Ca2þ–force relationships at different lengths of a virtual muscle. We have adequately simulated Ca2þ–force relationships and the typical change in Ca2þ sensitivity with muscle length decrease.

First, we modelled pure isometric conditions for a virtual muscle implemented when considering an absolutely stiff

series elastic element (SE) (see Fig. 1(A) for the rheological scheme of a contractile unit). In this case there is no internal

sarcomere shortening, which means that sarcomere length is permanently equal to its initial length. When the muscle

length is fixed, force generated by the muscle at pure isometric conditions is determined only by the Ca2þ concentration.

Fig. 2. Model simulation of the effect of muscle length on Ca2þ–force relationships. Different pCa–force curves ðpCa ¼ � lgð½Ca2þÞÞcorrespond to three initial sarcomere lengths of 1.9, 2.0 and 2.2 lm (these lengths are indicated in the panels). Top: A virtual musclewith absolutely stiff SE (pure isometric conditions). Bottom: The same but with compliant SE (see Eqs. (A.3)). (A), (C) The unit of force

corresponds to the maximum force plateau obtained at an initial sarcomere length of 2.2 lm. Values of pCa1=2 of Ca2þ concentration

for half-maximum force are marked on the curves. The lower pCa1=2 values correspond to the higher Ca2þ concentrations and the

lower Ca2þ sensitivity of force response. (B), (D) Force is normalized by the maximum force corresponding to the given muscle length.

Hill coefficients of Ca2þ–force dependences are given in the panels (see Eq. (A.36)).

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The following changes in ‘pCa–force’ ðpCa ¼ � lgð½Ca2þÞÞ dependence due to muscle length decrease were observed innumerical experiments (Fig. 2(A)):

• decrease in the magnitude of the force plateau;

• decrease in pCa1=2 (pCa1=2 ¼ pCa for the half �maximum force). Corresponding pCa1=2 values are indicated on the

curves in Fig. 2(A);

• decrease in the slope of pCa–force curve where force is plotted in fractions of the maximum force plateau corre-

sponding to the largest muscle length.

At the same time we found out that the slope of the ‘pCa–normalized force’ curve, where force was normalized by the

maximum force corresponding to the given muscle length (Fig. 2(B)), did not depend on virtual muscle’s length. As a

result, Hill coefficients of ‘Ca2þ–force’ dependences (see Eqs. (A.35) and (A.36)), were also the same at different muscle

lengths (values of Hill coefficients are shown in Fig. 2(B)).

We also showed the influence of SE compliance on Ca2þ–force dependencies. Unlike the pure isometric conditions,

the compliance of SE resulted in a different internal sarcomere shortening depending on the Ca2þ level: the higher the

Ca2þ concentration the deeper the sarcomere shortening, and thus the more intense the inactivation of contractile

proteins. This inactivation revealed itself in the following effects (Fig. 2(C)): force plateau, pCa1=2, and the slope of pCa–

force curves decreased due to SE compliance. The main qualitative difference of the results obtained with accounting for

SE compliance as compared to pure isometric simulation is as follows. The Hill coefficient of Ca2þ–force relationship

increased with muscle length decrease (Fig. 2(D)). All the above results, the latter included, are in good concordance

with experimental data [33–35].

Isometric contraction–relaxation cycles. We have simulated the length dependence of isometric twitches and have

reproduced the following effects caused by a decrease in the muscle length (Fig. 3):

• peak force decrease (Figs. 3(A) and (C));

• insignificant decrease in time-to-peak force (TPF);

• essential decrease in relaxation time (T50) to 50% of peak force (TPF and T50 are marked on the curves in Figs. 3(B)and (D)).

These results of the numerical experiments are in good conformity with physiological data [15].

Fig. 3. Model simulation of the effect of muscle length on the isometric contraction–relaxation cycle. Time courses of isometric force

correspond to different initial sarcomere lengths which ranged from 1.9 up to 2.2 lm (boundary lengths are indicated in the panels).Top: A virtual muscle with absolutely stiff SE (pure isometric conditions). Bottom: The same but with compliant SE. In panels (A), (C)

force is normalized by the peak force obtained at muscle length Lmax when initial sarcomere length being equal to 2.2 lm. In panels (B),(D) force is normalized by the peak force corresponding to the given muscle length. Values of the time to peak force and relaxation

time to 50% of peak force at the shortest and the largest muscle lengths are marked on the corresponding curves.

O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711 1689

All the listed simulations proved to be qualitatively similar in both cases of pure isometric conditions for the virtual

muscle (Figs. 3(A) and (B)) and when SE compliance was taken into account (Figs. 3(C) and (D)). In the latter case time

to peak forces at all muscle lengths exceeded those observed during pure isometric conditions, but relaxation times

turned out to be much shorter. In other words, muscle’s capability for force generation decreased as a result of the

inactivation due to internal sarcomere shortening produced by the series element compliance.

Afterloaded contractions. Afterloaded muscle contractions at different constant afterloads are adequately simulated

in the model as well (Fig. 4). The modelling of sarcomere shortening–lengthening during afterloaded twitches

(Fig. 4(C)), in particular, two phases of sarcomere lengthening – fast and slow is also very similar to experimental data

obtained by laser diffraction [36]. Based on a set of afterloaded contractions we obtained the basic mechanical

Fig. 4. Afterloaded twitches of a virtual muscle. Shown are time courses of the below-listed model variables during afterloaded twitches

at constant afterloads (from 0 to 1 in step of 0.1 of the peak isometric force Fmax at muscle length Lmax). Thick lines are isometric twitchcurves. (A) Muscle force F is expressed in fractions of Fmax. (B) Muscle length is expressed in fractions of Lmax. (C) Sarcomere length.(D) Concentration of Ca–TnC complexes. (E) Concentration CaC of cytosolic Ca

2þ. (F) Upper traces represent the concentrations

CaLR of Ca2þ within longitudinal reticulum, lower traces represent the concentrations CaTC of Ca

2þ within terminal cisterns of the SR.

Time courses of all the variables depend essentially on the load applied. The lower the afterload the bigger the shortening and the

higher the shortening–lengthening velocity of both muscle and sarcomere and the lower the peak and the faster the decay of Ca–TnC.

Length dependent decrease in Ca–TnC leads to an increase in and prolongation of Ca2þ-transient corresponding to lower afterloads.

Increase in CaC is followed by an increase in peak CaLR as compared with that during isometric twitch.

1690 O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711

characteristics of active muscle such as length–force, force–velocity relationships, and load-dependent power and work

(Fig. 5). We also showed that a particular shape of all the above relationships, force–velocity included, depended on the

elastic properties of series and parallel elements (Fig. 5(B)).

We have simulated the principally important phenomenon of load dependence of relaxation (LDR) [4,15–17,23,37].

The essence of this phenomenon is as follows (see Fig. 6, left panels, for the model simulation):

• the lower the afterload the higher the velocity of muscle lengthening during the isotonic phase of relaxation

(Figs. 6(B) and (C));

• the lower the afterload the lesser the ratio T1=T2, where T1 indicates the duration of isotonic phase of the twitch underthe given afterload and T2 denotes the duration of such part of the isometric twitch where the force remains to behigher than this afterload (T1; T2 are shown schematically in Fig. 6(A)).We presume that the main factor underlying the load dependence of relaxation is the length-dependent inactivation

due to the length-dependent cooperativity of the first type (see the above item ‘cooperative mechanisms of Ca2þ ac-

tivation’). Within the framework of our model we obtained a faster Ca–TnC decay due to deeper sarcomere shortening

during afterloaded twitch (Fig. 4(D)). Feedback influence of mechanical conditions of contraction–relaxation on the

time course of Ca–TnC kinetics resulted, in turn, in a load-dependent difference between Ca2þ-transients during the

isometric cycle and lower-loaded twitches (Fig. 4(E)). The last result is also in good concordance with experiments [38].

By the way, this essential distinction in Ca2þ-transients led to a difference in the time courses of both free Ca2þ within

SR compartments (Fig. 4(F)), and calcium bound with cytosolic Ca2þ-buffers and with calsequestrin.

Fig. 5. Basic mechanical characteristics of active muscle. All dependencies presented are obtained using a set of afterloaded contrac-

tions at constant afterloads (from 0 to 1 in step of 0.1 of the peak isometric force Fmax at muscle length Lmax). (A) Length (right scale)–force curve is a plot of end-systolic muscle length L normalized by Lmax vs. normalized afterload value F =Fmax;Work (left scale)–forcecurve represents the relationship between load and muscle work calculated as the product of end-systolic force on end-systolic

shortening and normalized by the maximum value; Power (left scale)–force curve represents dependence between load and muscle

power calculated as the product of end-systolic force on maximum velocity of afterloaded shortening and normalized by the maximum

value. (B) Force–velocity curve represents the maximum velocity of afterloaded shortening plotted vs. the afterload value. Influence of

stiffness of series (SE) and parallel (PE) elastic elements on F–V dependence is shown. Control curve corresponds to the model with the

basic set of parameters (see table in Appendix A), and other curves were obtained at changed parameters of FSE and FPE functions (seeEqs. A.3 and A.4: more compliant SE was modelled by setting a1 ¼ 10 vs. 14.6 and b1 ¼ 0:1 vs 0.56 in the control); stiffer PE wasmodelled by setting a2 ¼ 18 vs. 14.6 in the control.

O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711 1691

Attenuation of load dependence of relaxation. Fig. 6 demonstrates the attenuation of LDR simulated by means of the

model. This effect was experimentally observed in different cases such as in severely hypertrophied myocardium, in the

myocardium of new-born animals, at high temperature, low ½Ca2þ0 level, etc. [39–43]. Mechanisms underlying the LDRattenuation were specific in each case. Using the model we classified the possible mechanisms, and in particular showed

that amplification of back-inhibition of SR-pump affected LDR in the model.

Perturbations in muscle load or length. Virtual muscle’s responses to load perturbation during the isotonic cycle are

shown in Figs. 7(A) and (B). They are in good concordance with experimental data [4,23]. For example, if a muscle

contracted with a load higher than a control one and then this load was abruptly reduced down to the baseline,

subsequent relaxation continued longer than that in control twitch (Fig. 7(A), curve 3).

Fig. 6.Modelling of load dependence of relaxation (LDR) of normal myocardium and LDR attenuation. Time courses of muscle force F

(normalized by the peak isometric force Fmax at muscle length Lmax), length L (normalized by Lmax) and velocity of muscle’s shortening(negative values) – lengthening (positive values) during afterloaded twitches are shown. (A)–(C) Simulation of the load-dependent

myocardium (see table in Appendix A for the basic values of the model parameters). (D)–(F) Particular example of simulation of

myocardium with attenuated LDR (the fraction of the SR in cell volume was decreased 1.5 times and maximum velocity kpump of SR-pumping was decreased 1.2 times vs. baselines). T1; T2 shown in panel (A) denote, respectively, duration of the isotonic phase at thegiven afterload and duration of that part of isometric cycle where force exceeds the afterload value. Index T1=T2 is used to quantify loaddependence of relaxation. This index is about 0.5–0.7 at middle afterloads in normal myocardium and it is close to 1 in myocardium

with attenuated LDR (hypertrophied, embryonic, new-born myocardium, etc).

1692 O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711

Figs. 7(C) and (D) demonstrate the effects of cyclic short-time deformations on isometric twitch. In concordance

with experimental data [15] we obtained that a short length change during the isometric cycle was followed by the

muscle’s force inactivation.

Fig. 7. Effects of load or length perturbations on a virtual muscle. (A), (B) Influence of load perturbations applied during muscle

shortening (panel A) or lengthening (panel B) on muscle relaxation. Time course of muscle force (upper plots) and length (lower plots)

during the twitch under the control afterload (curves 1, thick lines) are compared with test cycles (thin lines) after switching from/to

lower (curves 2) or higher (curves 3) afterloads. Time course of muscle relaxation depends essentially on the mode of muscle loading.

Shortening at lower-than-control afterload and following an abrupt increase in the afterload level (curve 2 in panel A) lead to ad-

ditional inactivation and earlier relaxation. Increase in the afterload level during lengthening phase (curve 3 in panel B) is also followed

by muscle inactivation and earlier relaxation. (C), (D) Influence of short-time cyclic deformations on isometric twitches. Thick lines

show time courses of isometric force (lower traces) at muscle length Lmax (upper traces). Thin lines show change in the muscle force dueto 5% muscle shortening/lengthening at different time moments during isometric cycle. Abrupt change in muscle length is followed by

force inactivation. First two deformations shown in panels C and D were applied at the same level of the muscle force, but first

deformation was applied earlier than peak force was reached and second one after that. Deformation applied in the latter case resulted

in more prominent inactivation because of decreased instantaneous level of Ca2þ activation reached during relaxation at the moment

when deformation was applied, as compared to the first case.

O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711 1693

Summarizing this part of the article, we should stress that our model of homogeneous muscle contraction reproduces

a very wide range of experimental data. They are as follows:

• steady-state Ca2þ–force relationships at different initial sarcomere lengths;

• time courses of force, muscle length and sarcomere length changes during isometric and afterloaded twitches at dif-

ferent initial muscle lengths;

• basic mechanical characteristics: length–force, force–velocity, force–work, force–power relationships;

• different aspects of load dependence of relaxation;

• effects of length or load perturbations on time courses of isometric or isotonic muscle twitches;

• Ca2þ-transients in the cytosol during muscle twitches.

Moreover, the model allows us to analyse mechanisms underlying the above-listed phenomena. We think that the

key factor is a feedback influence of mechanical conditions on cooperative activation–inactivation processes.

Based on the results obtained we think that our model is fit to serve as a virtual muscle in our experiments dealing

with the inhomogeneous system.

3. Muscle duplexes

We have been considering the muscle duplex as an elementary inhomogeneous system [44–46]. We composed du-

plexes of two elements mechanically connected either in parallel or in series (Fig. 8). In parallel connection duplex

shortening is equal to each element’s shortening and duplex force is equal to the sum of the element’s forces. In series

connection duplex shortening is equal to the sum of the element’s shortenings and the duplex force is equal to each

element’s force.

We developed three kinds of duplex methods: Natural Duplex, Virtual Duplex, and Hybrid Duplex. Below in this

chapter we briefly describe the methodological approaches specific for experiments with both natural duplex and

hybrid ones. Therein we also present the results of these experiments. As for the Virtual Duplex, it is performed as a

mathematical model of the muscle duplex composed either in series or in parallel. The latter model, in turn, was

created on the basis of the above-presented homogeneous virtual muscle. Particular mathematical formulas used to

combine the equations of two virtual muscles in the virtual duplex are given in Eqs. (B.1)–(B.4). The results of the

numerical experiments with both parallel and serial virtual duplexes are described below in the respective sections of

this chapter.

Fig. 8.Muscle duplexes. Scheme of contractile units’ connection within a duplex. (A) Parallel connection. Duplex shortening DL is equalto each element’s shortening DL1;2; duplex force F is equal to the sum of the element’s forces F1 þ F2. (B) Connection in series. Duplexshortening DL is equal to the sum of the element’s shortenings DL1 þ DL2 and the duplex force F is equal to each element’s force F1;2.Either a natural muscle or a virtual muscle can represent each contractile unit within a duplex. We consider a natural muscle duplex,

which consists of two natural muscles connected in parallel, a hybrid duplex composed of a natural and a virtual muscle connected in

parallel and parallel and serial virtual duplexes composed of two virtual muscles.

1694 O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711

3.1. Natural muscle duplex

Experimental method. We have created and used an experimental device for studying parallel inhomogeneous du-

plexes, composed of two muscles [44,45] (see Fig. 8(A) for the scheme of muscles’ parallel connection). In short, two

papillary muscles (£ < 0:7 mm) or trabeculae are excised from rat or rabbit right ventricles and placed in isolated

perfusion chambers with separate stimulating electrodes, preparation survival control, pre- and afterloading, force

recording. One end of each muscle is attached to the common lever of a linear servomotor and the other to an individual

force transducer. The servomotor is able to supply a range of velocities up to 200 lm/ms (while real velocities ofmyocardium shortening are 6 5 lm/ms). Experiments are controlled by a computer that also serves for both datagathering and processing.

At the beginning of every experiment basic mechanical characteristics (such as ‘force–velocity’, ‘length–force’, etc.)

are obtained separately for each of the two muscles. Subsequently, these muscles are mechanically connected in parallel

and the above characteristics are obtained for each of the interacting muscles and the whole duplex. Differences between

the characteristics of isolated and duplexed muscles allow one to assess the effects of interaction, both in control

conditions and during varying mechanical, chemical, pharmacological interventions, and/or during changes in bath

temperature, and/or during variations in time delay between the muscles’ stimulation (asynchronism).

Afterloaded contractions of a duplex. Fig. 9 demonstrates afterloaded twitches of a parallel duplex composed of two

thin papillary muscles excised from a right ventricle of rabbit. One of these muscles was cooled from 30 to 28 �C, sothat it contracted more slowly. In this way we obtained an inhomogeneous duplex consisting of a fast and a slow

element. Both elements of the parallel duplex contracted auxotonically under varying loads while the duplex contracted

isotonically. In this case, force decay in one muscle is compensated by the force increase in the second muscle so that

the sum of the forces keeps a constant value equal to the afterload applied to the duplex. In some cases complex

oscillations of the elements’ force were observed (Fig. 9). At the bottom part in Fig. 9 we show the results of the

corresponding numerical experiment with a virtual duplex. Virtual duplexes are considered in detail in the following

section.

Fig. 9. Afterloaded contractions of a parallel duplex. (A)–(D) Experimental record of contractions of a natural muscle duplex composed

of two thin papillary muscles excised from a rabbit right ventricle. (E)–(H) Results of a corresponding numerical experiment with a

parallel virtual duplex. Time courses of duplex shortening (panels A, E), duplex force (panels B, F) and force of each muscle interacting

within the duplex (panels C–D, G–H) during twitches at different afterloads applied to the duplex are shown. Muscles within the

duplexes contract auxotonically under varying loads and their forces reveal complex oscillations during the shown experiments. Force

decay in one of the elements is compensated by the force increase in the second element so that the sum of the forces is a constant value

equal to an afterload applied to the duplex.

O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711 1695

Length–force (L–F) relationships. Fig. 10 demonstrates L–F relationships between end-systolic shortening and end-

systolic force in a sequence of afterloaded contractions for duplex elements being separate or interacting, and L–F

relationships for the whole duplex as well. We also considered the ‘‘formal sum’’ of L–F curves in isolation obtained by

plotting given end-systolic shortening vs. sum of the two forces, where each force corresponds to the shortening of the

respective muscle in isolation.

When the muscles interacted their L–F dependencies slightly shifted in opposite directions as compared to their L–F

curves in isolation (Fig. 10(A)) it means that one muscle was additionally activated and the other was inactivated due to

their interaction within the duplex. Moreover, the ‘‘formal sum’’ did not coincide with the duplex length–force curve.

Thus, interaction revealed itself in the non-additive influence on the muscles’ L–F characteristics.

There was an influence of asynchronous stimulation of duplex elements on their length–force dependencies

(Fig. 10(B)). Delay in one muscle stimulation led to an increase in the above-mentioned shift of the L–F curve of each

interacting muscle as compared to such a curve in isolation. The mismatch between the ‘‘formal sum’’ and the duplex’s

L–F curve became more prominent as well.

At the bottom part of Fig. 10 we demonstrate corresponding data of numerical experiments with a virtual duplex.

Force–velocity (F–V) relationships. In Fig. 11 we show experimental F–V dependencies obtained for a fast and a slow

papillary muscle either being separate or interacting with each other within the duplex. In the same panels we plot F–V

curves for the duplex as a whole. For each muscle in isolation and for the whole duplex we use a common procedure,

when plotting F–V curves, i.e. we plot maximum velocity of the afterloaded shortening vs. corresponding value of the

afterload. F–V curves for each interacting duplex muscle are obtained by plotting the maximum velocity of the muscle’s

shortening (under every afterload applied to the duplex) vs. force developed by the muscle, when this velocities being

reached.

Investigating F–V relationships for duplex elements we found that in some cases their F–V curves became closer to

each other, thanks to the interaction between the muscles (Fig. 11(A)). It means that at each given load the fast muscle

Fig. 10. Effect of delayed stimulation of one element in parallel duplexes (both natural and virtual ones) on length–force (L–F) rela-

tionships. Relationships between end-systolic shortening and end-systolic force obtained in series of afterloaded contractions are

shown. Different curves represent L–F dependencies for duplex elements in both isolation (solid lines) and interaction (dashed lines), as

well as for the whole duplex (top dashed lines). ‘‘Formal sum’’ (top solid lines) of L–F curves in isolation is obtained by plotting the

given muscle shortening vs. sum of two forces, where each force corresponds to this end-systolic shortening for the respective muscle in

isolation. Experimental results (panels A–B) are compared with modelling (panels C–D). (A), (C) Muscle duplex with simultaneous

stimulation of the elements. (B) Parallel natural duplex with 80 ms delay in one element stimulation. (D) Parallel virtual duplex with 55

ms delay in one element stimulation.

1696 O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711

within the duplex shortened slower than in isolation, and the slow muscle contracted faster. Furthermore, we found in

some cases such a delay in stimulation of one duplex element where F–V relationships of both elements became very

close (almost coincided – see Fig. 11(B)).

3.2. Parallel virtual duplex

In order to analyse more precisely the effects found we used a virtual duplex which consisted of a fast and a slow

virtual element connected in parallel (see Fig. 8(A) for the scheme of parallel connection of duplex muscles). Time

courses of force and length of the duplex and its elements during afterloaded contractions are shown in Fig. 12.

Fig. 11. Effect of delayed stimulation of one element in a natural parallel duplex on force–velocity relationships (F–V). Shown are the

experimental F–V dependencies obtained for a fast and a slow papillary muscle either in isolation (solid lines) or when interacting

within the duplex (dashed lines) and for the duplex as a whole (thick lines). F–V curves are obtained by plotting the maximum velocity

of shortening vs. the force developed at the time when this velocity is reached. The duplex’s force is normalized by its maximum (i.e.

isometric peak force) value, whereas the muscle’s forces are normalized by their own isometric peak values. (A) Duplex with simul-

taneous stimulation of the elements. (B) Duplex with 30 ms delay in one element stimulation.

O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711 1697

Force–velocity dependencies. Fig. 13 demonstrates the effect of delayed stimulation of one duplex element on F–V

dependencies of both interacting muscles and their duplex. First we delayed stimulation of the fast muscle (by 30 and 50

ms). We observed F–V curves of both muscles drawing close to each other – up to their coincidence at 50 ms stimulation

Fig. 13. Effect of delayed stimulation of one element in the parallel virtual duplex on force–velocity relationships. F–V dependencies,

obtained as described above (see Fig. 10), for the fast and the slow virtual muscles (thin lines), interacting within the parallel virtual

duplex, and for the duplex as a whole (thick lines) are shown. The duplex’s force is normalized by its maximum (i.e. isometric peak

force) value, whereas the muscle’s forces are normalized by their own isometric peak forces at one and the same muscle length Lmax. (A)The duplex with simultaneous stimulation of the elements. (B), (C) The duplex with 30 or 50 ms delay in the fast muscle stimulation.

(D) The duplex with 50 ms delay in the slow muscle stimulation.

Fig. 12. Afterloaded contractions of a parallel virtual duplex composed of a ‘‘fast’’ and a ‘‘slow’’ muscle. Shown are time courses of

duplex force (panel A), duplex length (panel B) and force of each muscle interacting within the duplex (panels C–D) during twitches at

different afterloads applied to the duplex. Duplex elements are stimulated simultaneously. Isometric peak forces are almost equal in

both duplex elements. Time to peak force and relaxation time of the ‘‘fast’’ muscle (panel C) are shorter than those of ‘‘slow’’ muscle

(panel D). The lengths of the muscles are equal both to each other and to the duplex length. Forces are normalized by the maximum

force of the duplex Fmax reached during the duplex isometric contraction at length Lmax. Respectively, lengths are normalized by Lmax.

1698 O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711

delay applied to the fast muscle (Figs. 13(B) and (C)). After this we delayed the stimulation of the slow muscle. As a

result, two F–V curves of the interacting muscles proved to be even more distant than they were in isolation

(Fig. 13(D)).

Length–force dependencies. In Fig. 14 we show the influence of the above stimulation delays on L–F dependencies.

In the case of the 50 ms delay in the fast muscle stimulation F–L curves of interacting muscles became markedly

closer to each other (Fig. 14(B)), just as the corresponding F–V curves did (Fig. 13(C)). Contrariwise, when the slow

muscle stimulation was delayed by 50 ms, the L–F curves were even more diverged than without delay (Fig. 14(C)).

Summarizing, we can say that, for 50 ms delay in stimulation of the fast muscle, the basic mechanical character-

istics (L–F, F–V dependencies) of interacting inhomogeneous muscles became almost identical. In other words, the

muscles interacting within the inhomogeneous and asynchronous system behaved as functionally homogeneous

ones.

Ca2þ activation of duplex elements. In order to analyse factors determining this effect of functional tuning of in-teracting inhomogeneous muscles, we studied how Ca2þ activation of each element within the virtual duplex changed

due to the asynchronous stimulation. Fig. 15 demonstrates time courses of Ca–TnC concentration of both the fast and

slow muscles at different stimulation delays: simultaneous stimulation; stimulation of the fast muscle delayed for 50 ms;

and stimulation of the slow muscle delayed for 50 ms. When the fast muscle stimulation was delayed by 50 ms, the peak

of Ca–TnC concentration became lower in the fast muscle and higher in the slow muscle as compared to their si-

multaneous stimulation in the duplex. This means that the fast muscle was inactivated whereas the slow muscle was

additionally activated due to their asynchronous stimulation. When we changed the sequence of excitation and delayed

the slow muscle stimulation, Ca–TnC concentration slightly increased in the fast muscle as it essentially fell down in the

slow muscle, unlike the previous case.

We found some premises of equalization or, vice versa, divergence of the F–V curves for interacting muscles. Fig.

16 presents dependencies between normalized afterloads applied to the duplex and time to peak Ca–TnC concen-

trations of both interacting muscles. The curves are plotted for different stimulation delays of one element of the

duplex. As we noted above F–V curves of interacting muscles coincided in the case of 50 ms delayed stimulation of

the fast muscle. Exactly this delay induced close convergence of time to peak Ca–TnC concentrations in both in-

teracting muscles (Fig. 16(B)). In contrast, when stimulation of the slow muscle was delayed by 50 ms, time to peak

Ca–TnC concentrations diverged even more strongly than in the case of simultaneous stimulation of duplex muscles

(Fig. 16(C)).

Fig. 14. Effect of delayed stimulation of one element in the parallel virtual duplex on length–force relationships. Relationships between

end-systolic length and end-systolic force of the fast and the slow virtual muscle interacting within the duplex during afterloaded

contractions are shown. The muscle forces are normalized by their own isometric peak forces Fmax at one and the same muscle lengthLmax. Length is normalized by Lmax. (A) The duplex with simultaneous stimulation of the elements. (B) The duplex with 50 ms delay inthe fast muscle stimulation. (C) The duplex with 50 ms delay in the slow muscle stimulation.

O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711 1699

Fig. 17 demonstrates that during afterloaded twitches, Ca2þ-transients of interacting muscles in the virtual duplex

changed depending on stimulation delays, and these changes occurred in opposite directions as compared to Ca–TnC

concentration. In particular, the upsurge of Ca2þ concentration corresponded with the fall of Ca–TnC concentration.

Fig. 16. Effect of delayed stimulation of one element in a parallel virtual duplex on the Time to Peak Ca–TnC concentration in interacting

virtual muscles. Shown are dependencies between times to peak Ca–TnC concentration in the fast and the slow virtual muscles during

afterloaded twitches of the virtual duplex and the afterload values (normalized by isometric peak force of the duplex, Fmax). (A) Duplexwith simultaneous stimulation of the elements. (B) Duplex with 50 ms delay in the fast muscle stimulation. (C) The duplex with 50 ms

delay in the slow muscle stimulation.

Fig. 15. Effect of delayed stimulation of one element in the parallel virtual duplex on Ca–TnC kinetics in interacting virtual muscles. Time

courses of Ca–TnC concentration in the fast (left) and the slow (right) muscle during afterloaded twitches of the virtual duplex are

shown. (A) The duplex with 50 ms delay in the slow muscle stimulation. (B) The duplex with simultaneous stimulation of the elements.

(C) The duplex with 50 ms delay in the fast muscle stimulation.

1700 O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711

3.3. Parallel hybrid duplex

As we mentioned above, one of the methods we used to study the inhomogeneity phenomena is that of Hybrid

Duplex. We used the above-described virtual muscle as a counterpart for a natural muscle within a hybrid duplex (see

Fig. 8 for the scheme of a parallel duplex).

In more detail, the concept of the hybrid duplex method is as follows. A natural papillary muscle (or trabecula)

interacts mechanically – in real time – with a virtual one. In-parallel or in-series connection between natural and virtual

muscles is implemented by means of computer-controlled A/D & D/A input/output functions. These transform the

digital signal from the virtual muscle (tension calculated in the model at each current step – i.e. once per 100 ls; themodel calculation at every step takes about 50 ls) to an analogue output (instant load, applied to the natural muscle viaa servomotor). Vice versa, the program transforms the analogue signal from the natural muscle (its real instant

shortening/lengthening) to a digital reading to set the current length of the virtual counterpart. From this input, the

model calculates the next instant force of the virtual counterpart, which is applied to affect the next instant load on the

natural one, etc.

The Hybrid Duplex concept allows us:

• to create groups of standard virtual counterparts for interactions with natural muscles including representations of

normal, hypertrophied, and hypoxic muscles (parameter identification [47,48] of the above-described mathematical

model of homogeneous myocardium allows us to create these groups);

• to study the response of one and the same natural muscle to the interaction with either normal, hypertrophied or

hypoxic virtual muscle;

• to use the virtual muscle of any hybrid duplex as a transparent one, i.e. to observe in it a sequence of intracellular

events resulting from the interaction with the natural counterpart in the duplex;

• to change properties of virtual counterparts via parameters’ variation and to study how both the natural muscle and

the hybrid duplex (as a whole) respond (i.e. to study the sensitivity of an inhomogeneous system to changes in

intracellular mechanisms of its part).

We studied the afterloaded twitches of parallel hybrid duplexes composed of one and the same natural muscle with

two different virtual counterparts, one with strong load dependence of relaxation, and the other without load depen-

dence at all. The latter takes place, for instance, in severely hypertrophied myocardium. Fig. 18 shows afterloaded

twitches of a rabbit trabecula (natural muscle) and two virtual muscles both normal and ‘hypertrophied’ ones.

Fig. 17. Effect of delayed stimulation of one element in the parallel virtual duplex on Ca2þ-transients in the interacting virtual muscles.Time courses of cytosolic Ca2þ concentration in the fast (left) and the slow (right) muscle during afterloaded twitches of the virtual

duplex are shown. (A) The duplex with simultaneous stimulation of the elements. (B) Duplex with 50 ms delay in the fast muscle

stimulation. (C) Duplex with 50 ms delay in the slow muscle stimulation.

O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711 1701

Afterloaded twitches of the corresponding hybrid duplexes are also presented in the same figure. Hybrid duplex

properties, as well as the response of one and the same natural muscle to interaction with different virtual counterparts

within the duplex (Fig. 19), differed strongly, depending on particular properties of these counterparts.

We also studied the influence of asynchronous stimulation of hybrid duplex elements on their force–velocity rela-

tionships and found such delays in stimulation where F–V curves approached together closely (Fig. 20). We noted that

this closeness was achieved mostly by shifting the F–V curve of the natural muscle, and this peculiarity of the hybrid

duplex has not yet been clear for us.

3.4. Serial virtual duplex

At the present time we have almost finished creating a physiological setup for carrying out experiments with du-

plexes composed of two natural muscles connected in series. Meantime we have already developed a mathematical

model of such a virtual duplex (see Fig. 8(B) for the scheme of muscle connection within a serial duplex) and carried out

some numerical experiments as follows.

Isometric twitches of serial duplexes. In comparison with parallel connection elements of a serial duplex actually

interact during isometric mode of the whole duplex contraction, where the lengths of the elements redistribute whilst the

duplex length remains constant. Fig. 21(A) shows isometric twitches of a serial duplex composed of mechanically

homogeneous elements, both when they were stimulated without delay and when stimulation of one element was de-

layed by 50 ms. Peak force of the duplex with asynchronous elements decreased but the characteristic time of relaxation

did not change. Isometric twitches of the duplex composed of a fast and a slow muscle are shown in Figs. 21(B) and (C).

When stimulation of the fast muscle was delayed the duplex peak force was over that of the synchronous duplex and

relaxation became faster (Fig. 21(B)). On the contrary, if stimulation of the slow muscle was delayed the peak of the

duplex force decreased and relaxation time increased as compared to the synchronous duplex (Fig. 21(C)).

These results show that interaction between asynchronous inhomogeneous elements can affect duplex force quan-

titatively in different ways.

Characteristics of serial duplex relaxation. Fig. 22(A) shows that interaction between inhomogeneous fast and slow

muscles affected sensitivity of the relaxationof a serial duplex to its length.The slope of the normalized length–T30 curve (T30

Fig. 18. Afterloaded twitches of hybrid duplexes. Shown are time courses of force (upper traces) and length (lower traces) during af-

terloaded contractions of (A) a rabbit trabecula in isolation; (B) ‘‘normal’’ virtual muscle in isolation; (C) A+B hybrid duplex; (D)

‘‘hypertrophied’’ virtual muscle in isolation; (E) A+D hybrid duplex. Presented virtual muscles differ, in particular, in their load

dependence of relaxation (LDR). ‘‘Normal’’ virtual muscle simulates a natural muscle with strong LDR and ‘‘hypertrophied’’ virtual

muscle is characterized mechanically by attenuation of LDR. Mechanical behaviour of the hybrid duplexes composed of one and the

same natural muscle with different virtual counterparts appears to depend strongly on mechanical properties of these counterparts.

1702 O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711

is relaxation time to 30% of peak force) for the inhomogeneous duplex was much steeper as compared to those for ho-

mogeneous duplexes, each one being composed of two identical copies of either the fast or the slow element of the in-

homogeneous duplex. The slope of the duplex length–T30 curve was almost twofold steeper in the inhomogeneous duplex.Fig. 22(B) shows the effect of delayed stimulation of one duplex element on relaxation time T30 of a serial duplex. A

few such duplexes composed of different virtual pairs are represented. An increase in the delay in the fast muscle

stimulation in inhomogeneous duplex caused a decrease in the duplex relaxation time, but an increase in the delay in the

slow muscle stimulation vice versa resulted in an increase in the duplex relaxation time.

Thus, mechanical inhomogeneity as such essentially affected the mechanical behaviour of the contractile system.

Moreover, different order of asynchronous stimulation of inhomogeneous contractile elements significantly modulated

the mechanical response of the system.

4. Discussion

Conclusions. Summarizing all the above results we conclude that the methods of muscle duplexes allowed us to

describe and analyse a new class of biomechanical phenomena caused by myocardium element interaction. We showed

that owing to mechanical interaction between inhomogeneous duplex elements their basic mechanical characteristics

Fig. 19. Auxotonic twitches of a rabbit trabecula interacting with different virtual muscles during afterloaded contractions of hybrid

duplexes. (A), (D) Isometric twitches of a rabbit trabecula (see ‘‘N-muscle’’ curve in the panel), its virtual counterpart (‘‘V-muscle’’

curve) and the corresponding hybrid duplex (‘‘Duplex’’ curve). (B), (E) Force of the natural muscle contracting within the corre-

sponding hybrid duplex. (C), (F) Force of the virtual muscle interacting with the above natural one within the hybrid duplex. Virtual

muscles differ by both their time to peak force and relaxation time (left virtual muscle is faster than right one) as well as by their load

dependence of relaxation (right virtual muscle is not load dependent).

O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711 1703

changed, adapting to each other. This was observed for force–length, force–velocity dependencies. We found out that

the approaching of mechanical characteristics of both duplex elements closer to each other crucially depended on

stimulation asynchronism and the sequence of stimulations of these elements. In particular, there appeared to be such a

delay in stimulation of the faster element in the inhomogeneous duplex where length–force and force–velocity de-

pendencies of the interacting elements became practically identical. In other words, there were such conditions of

contraction for originally inhomogeneous systems where their elements became functionally homogeneous. We found

out that this functional tuning of the duplex elements was closely connected with and probably controlled by opposite

changes in Ca2þ activation of thin filaments of interacting contractile units so as to reach the closing of time to peak Ca–

TnC concentrations in these units.

Analyzing Ca2þ kinetics in interacting contractile units by means of the mathematical model, we found out, in

particular, that Ca2þ-transients of interacting elements changed also in opposite directions depending on the sequence

of asynchronous stimulations of the elements. This result, being compared with electrophysiological experimental data,

suggests the following hypothesis: mechanical inhomogeneity can induce or modulate electrical inhomogeneity of

myocardium. Following experimental data justify the validity of this hypothesis. It has been shown that in case of

isotonic shortening of the cardiac muscle the action potential duration (APD) increases [49]. The increase seems to be

induced by the length-dependent increase in the calcium transient which is also revealed in experiments [38]. Conversely,

the muscle lengthening results in both a decrease in the calcium transient and a reduction of the AP duration [49]. These

data point out that mechanical inhomogeneity of excitable environs may breed by itself electrical inhomogeneity and

cause perturbations during electrical wave propagation. This problem has not been properly studied yet.

In summary we can conclude that mechanical inhomogeneity proves to be an important factor regulating or modu-

lating heart muscle function. Interaction between inhomogeneous elements can affect myocardium biomechanical

properties not only quantitatively but also qualitatively. From the physiological point of view it seems important that

Fig. 20. Effect of delayed stimulation of one element in a hybrid duplex on force–velocity relationships. Shown are F–V dependencies,

obtained as described above (see Fig. 10), for a rabbit trabecula (N-muscle) and its virtual counterpart (V-muscle) interacting within

the hybrid duplex and for their duplex also. The duplex’s force is normalized by its maximum value (i.e. isometric peak force), whereas

the muscle’s forces are normalized by their own isometric peak forces at one and the same muscle length Lmax. (A) The duplex withsimultaneous stimulation of the elements. (B), (C) The duplex with 30 or 50 ms delay in the natural (slower) muscle stimulation.

(D), (E) The duplex with 30 or 50 ms delay in the virtual (faster) muscle stimulation.

1704 O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711

interaction between probably a great number of inhomogeneous cardiomyocytes should arrange specifically their be-

haviour within the whole system. It means that inhomogeneous cardiomyocytes have to be in tune with each other ac-

cording to their ownmechanical properties and time-space arrangement within myocardium.Wemean that depending on

the specific spatial arrangement of mechanical properties in an inhomogeneous cardiac tissue, there should exist a cor-

responding time arrangement of excitation that leads to an in-tune interaction between these tissue’s contractile elements.

Thus, the investigated biomechanical effects of interaction between inhomogeneous myocardial elements should be

considered as a new separate class of biomechanical phenomena, for which we propose the term ‘‘tuning effects’’.

Perspectives. We consider several directions for studying further inhomogeneity phenomena based on duplex ap-

proaches. They are as follows:

• to complete elaboration of an experimental device and computer software for both natural duplex and hybrid duplex

composed of muscles connected in series;

• to assess the effects of mechanical inhomogeneity of myocardium on the contractile function in pathology (local

hypoxia or hypertrophy);

• to verify the effects of mechanical inhomogeneity on cardiac pharmacological reactivity;

• to create mathematical models for complex mechanically inhomogeneous contractile systems:

1-D models of muscle chains composed of a number of virtual elements connected either in parallel or in series,

2-D and 3-D models of mechanically inhomogeneous tissue composed of several virtual muscles (quartet, octet,

etc.) connected partially in parallel and partially in series;

Fig. 21. Influence of delayed stimulation of one element in a serial virtual duplex on the duplex isometric twitch. Compared are time

courses of a virtual muscle forces and lengths during duplex isometric twitch in case of simultaneous stimulation of duplex elements

and of 50 ms delayed stimulation of one duplex element. Force is normalized by the peak force of synchronous duplex, length is

normalized by the same initial length in both muscles. (A) The duplex composed of two identical virtual muscles. (B) The duplex

composed of a fast and a slow muscle where stimulation of the fast muscle is delayed. (C) The duplex composed of the above muscles

where stimulation of the slow muscle is delayed.

O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711 1705

• to study the inhomogeneity phenomena on the molecular level by means of a mathematical model describing slow

and fast isomyosins presented in various ratios within the cardiomyocyte;

• to elucidate the interrelation between mechanical and electrical inhomogeneities in myocardium by means of an im-

proved mathematical model of homogeneous muscle contraction including simulation of mechano-electrical feed-

back influence on excitation–contraction coupling processes.

Acknowledgements

We are grateful to Dr. Peter Kohl from the University Laboratory of Physiology, Parks Road, Oxford OX1 3PT,

UK, for his helpful comments during the discussion of the manuscript. This is also our pleasure to acknowledge the role

of the Wellcome Trust Foundation supporting our study of inhomogeneous duplexes (grant # 061115/Z/00/Z), and

both RFBR (grant # 00-04-48323) and NATO Cooperative Linkage Grant (#LST.CLG. 975785) supporting our

mathematical modelling of homogeneous cardiac muscle.

Appendix A. Mathematical model of homogeneous myocardium contractions

Justification of all the model equations and choice of the basic parameter values are given elsewhere [14–17,45].

A.1. Mechanical block of the model (see 1(A) for the scheme)

t (ms) – time;

LCE ðlmÞ – length of the CE;LSE ðlmÞ – length of the SE;LPE ðlmÞ – length of the PE;Lr ¼ 1:78 ðlmÞ – the length of CE and PE in absolutely relaxed and non-stretched states.

Fig. 22. Influence of mechanical inhomogeneity on characteristic time of relaxation ðT30Þ in a virtual serial duplex during isometric twitch.T30 is relaxation time to 30% of isometric peak force. (A) Relationships between T30 in duplex isometric twitches and correspondingduplex length in serial duplexes composed of either homogeneous or inhomogeneous elements. Duplex length L is normalized by

duplex length Lmax at which initial sarcomere lengths of both duplex elements are equal to 2.23 lm. (B) Effect of delayed stimulation ofone element on T30 in isometric twitches of duplex composed of either homogeneous or inhomogeneous elements.

1706 O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711

l1 ¼ LCE � Lr; ðA:1Þ

l2 ¼ LPE � Lr: ðA:2Þ

From equality LSE þ LCE ¼ LPE follows that LSE ¼ l2 � l1.FCE (mN) – force of CE;FSE (mN) – force of SE;FPE (mN) – force of PE.

FSE ¼ b1 ðexpða1 ðl2 � l1ÞÞ � 1Þ; ðA:3Þ

FPE ¼ b2 ðexpða2 l2Þ � 1Þ; ðA:4Þ

FCE ¼ kfdl1dt

� �N ; ðA:5Þ

N ¼ Nact n; ðA:6Þ

n ¼ n1ðl1Þn2; ðA:7Þ

Nact ¼Al

Al þ KlA

Lozðl1Þ: ðA:8Þ

Mechanical block of the model equations contains ordinary differential equations for l1; l2 and n2. They are asfollows.

The following equation results from the identity FCE�tFSE and Eqs. (A.3) and (A.5).

fdl1dt

� �¼ b1 ½expða1 ðl2 � l1ÞÞ � 1

k ðAl=ðAl þ KlAÞÞn2 n1ðl1ÞLozðl1Þ

: ðA:9Þ

Required dl1=dt was obtained by calculating the inverse function v ¼ gðf Þ which was described in the explicit form.The following equation results from the identity l02 � 0 during isometric mode or from the identity ðFSE þ FPEÞ

during isotonic mode of muscle contraction.

dl2dt

¼0; isometric mode;

a1 b1 ðdl1=dtÞ expða1 ðl2�l1ÞÞa2 b2 expða2 l2Þþa1 b1 expða1 ðl2�l1ÞÞ

; isotonic mode;

�ðA:10Þ

dn2dt

¼ qndl1dt

� �G

dl1dt

� �0:9

�� n2

�: ðA:11Þ

Functions used in the above equations are listed below:

n1ðl1Þ ¼ 0:6 l1 þ 0:5; ðA:12Þ

Lozðl1Þ ¼ l1 þ 1:14; ðA:13Þ

f ðvÞ ¼ P ðvÞGðvÞ ; ðA:14Þ

GðvÞ ¼0:6v=vmax þ 1; �vmax6 v6 v1;

ð0:6v1=vmax þ 1Þ v0�vv0�v1

h i0:1; v16 v6 v0;

(ðA:15Þ

PðvÞ ¼a ð1þv=vmaxÞa�v=vmax

; �vmax6 v6 0;ðs1 v

vmaxþ 1ÞGðvÞ; 06 v6 v1;

d; v16 v6 v0:

8<: ðA:16Þ

Coefficient s1 in Eq. (A.16) is determined providing continuity of P ðvÞ.

qnðvÞ ¼0:02� 0:26 v

vmax

�; �vmax6 v6 0;

0:02� 0:005 v0:99v0

�; 06 v6 0:99v0;

0:015

1þ5000 ðv�0:99v0Þ½ 10 ; 0:99v06 v6 v0:

8>>><>>>:

ðA:17Þ

O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711 1707

A.2. Ca2þ handling block of the model (see Fig. 1(B) for the scheme)

Ca2þ handling block of the model equations contains ordinary differential equations for following variables:

CaC ðlMÞ – cytosolic [Ca2þ ];CaTC ðlMÞ – [ Ca2þ] within terminal cisterns (TC) of SR;CaLR ðlMÞ – ½Ca2þ within longitudinal reticulum (LR) of SR;

A (lM) – concentration of Ca–TnC;B1 ðlMÞ – concentration of Ca2þ bound with ‘‘fast’’ Ca2þ binding ligands;B2 ðlMÞ – concentration of Ca2þ bound with ‘‘slow’’ Ca2þ binding ligands;C (lM) – concentration of Ca–calsequestrin complexes.

These equations are as follows:

dCaC

dt¼ Fin � Fout þ kTC=C FSRrel � FSRpump �

dAdt

� dB1dt

� dB2dt

; ðA:18Þ

dCaTC

dt¼ �FSRrel þ kLR=TC FSRflow �

dCdt

; ðA:19Þ

dCaLR

dt¼ kC=LR FSRpump � FSRflow; ðA:20Þ

dAdt

¼ aon ðAtot � AÞCaC � aaoff expð�kA AÞpðNAÞA; ðA:21Þ

dB1dt

¼ b1on ðB1tot � B1ÞCaC � b1off B1; ðA:22Þ

dB2dt

¼ b2on ðB2tot � B2ÞCaC � b2off B2; ðA:23Þ

dCdt

¼ con ðCtot � CÞCaTC � coff C; ðA:24Þ

Fin ¼JðtÞ

zF 0:5VC; ðA:25Þ

Fout ¼ koutCaC

CaC þ Kout; ðA:26Þ

FSRrel ¼ krelQðtÞ ðCaTC � CaCÞ; ðA:27Þ

FSRpump ¼ kpumpCaC

Kpump þ CaCexpð�kinh CaLRÞ; ðA:28Þ

FSRflow ¼ kflow CaLRð � CaTCÞ; ðA:29Þ

kTC=C ¼ VTCVC

; kC=LR ¼ VCVLR

; kLR=TC ¼ VLRVTC

; ðA:30Þ

NA ¼ NðA=AtotÞLoz

: ðA:31Þ

Functions used in the above equations are listed below:

pðNAÞ ¼1; 06NA < 0:25;0:022NA�0:5; 0:256NA < 0:75;0:02; 0:756NA 6 1;

8<: ðA:32Þ

JðtÞ ¼ kin0:8� 0:008 ðt � 10Þ2; 06 t < 10;0:5 expf10�t

10g þ 0:3 expf10�t

55g; 106 t6 30;

8<: ðA:33Þ

QðtÞ ¼ t=t1; 06 t < t1;t2�tt2�t1

; t16 t6 t2:

�ðA:34Þ

1708 O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711

Differential equations (A.9)–(A.11), (A.18)–(A.24) present a complete system for the phase variables of the model.

Hill curve for Ca2þ–force relationship

F ¼ FpðCa2þÞnH

ðCa2þÞnH þ ðCa2þ1=2ÞnH

; ðA:35Þ

where F is a force, Fp is the force maximum plateau, nH is the Hill coefficient, and Ca1=2 is the Ca2þ concentration for

the half-maximum force.

Hill coefficient for Ca2þ–force relationship

nH ¼�4� F 0ðpCa1=2Þ

Fp � lnð10Þ; ðA:36Þ

where pCa ¼ lgðCa2þÞ; F 0ðpCa1=2Þ is derivative of function F ðpCaÞ calculated at the point pCa1=2.

Appendix B. Additional equations used for combining two virtual muscles in a duplex

DL – duplex shortening;DL1;2 – muscle shortening;F – duplex force;

F1;2 – muscle force.Connection in parallel (see Fig. 8(A) for the scheme)

DL ¼ DL1 ¼ DL2; ðB:1Þ

F ¼ F1 þ F2: ðB:2Þ

Connection in series (see Fig. 8(B) for the scheme)

DL ¼ DL1 þ DL2; ðB:3Þ

F ¼ F1 ¼ F2: ðB:4Þ

Eqs. (B.1)–(B.4) were rewritten using phase variables ðl1Þi; ðl2Þi; i ¼ 1; 2;, where i denotes a virtual muscle number.From these relations implicit equations for dðl2Þi=dt are obtained which substitute corresponding equation (A.10) foreach duplex’s muscle. Other equations A.9,A.11, (A.18)–(A.24) are duplicated for each virtual muscle in the duplex

system.

Basic values of the model parameters

a1 14:6 lm�1 B2tot 100 lMb1 0.56 mN b2on 0:001 lM�1 ms�1

a2 14.6 lm�1 b2off 0.003 ms�1

b2 0.0012 mN Ctot 40,000 lMk 20 mN lm�1 con 0.05 lM�1 ms�1

KA 35 lM coff 50 ms�1

l 2.5 krel 2.13 ms�1

vmax 0.0055 lm ms�1 kpump 1.5 lM ms�1

v0 0:5vmax lm ms�1 Kpump 1 lMv1 0:1vmax lm ms�1 kinh 400 lM�1

A 0.25 kflow 0.0039 ms�1

D 0.6 kin 5 C ms�1

Atot 70 lM kout 0 lM ms�1

aon 0.02285 lM�1 ms�1 Kout 0 lMaaoff 0.6 ms�1 VC 20; 000� 10�15 lkA 0.4 lM�1 VTC 120� 10�15 lB1tot 80 lM VLR 1200� 10�15 lb1on 0.1 lM�1 ms�1 z 2

B1off 0.182 ms�1 F 9:648� 1010 C lM�1

O. Solovyova et al. / Chaos, Solitons and Fractals 13 (2002) 1685–1711 1709

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