Journal of Theoretical Biology 230 (2004) 385–405

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*Correspond

E-mail addr

(L.B. Katsnelso

0022-5193/$ - se

doi:10.1016/j.jtb

Influence of viscosity on myocardium mechanical activity:a mathematical model

Leonid B. Katsnelsona,*, Larissa V. Nikitinaa, Denis Chemlab, Olga Solovyovaa,Catherine Coiraultb, Yves Lecarpentierb, Vladimir S. Markhasina

a Institute of Immunology and Physiology, Ural Division of the Russian Academy of Sciences, Bldg. 91, Pervomayskaya str., 620219,

Ekaterinburg, RussiabService de Physiologie, CHU de Bic#etre, AP-HP-Universite Paris Sud 11, Le Kremlin-Bicetre, France

Received 21 February 2003; received in revised form 30 March 2004; accepted 27 May 2004

Available online 20 July 2004

Abstract

We have previously proposed and validated a mathematical model of myocardium contraction–relaxation cycle based on current

knowledge of regulatory role of Ca2+ and cross-bridge kinetics in cardiac cell. That model did not include viscous elements. Here we

propose a modification of the model, in which two viscous elements are added, one in parallel to the contractile element, and one

more in parallel to the series elastic element. The modified model allowed us to simulate and explain some subtle experimental data

on relaxation velocity in isotonic twitches and on a mismatch between the time course of sarcomere shortening/lengthening and the

time course of active force generation in isometric twitches. Model results were compared with experimental data obtained from 28

rat LV papillary muscles contracting and relaxing against various loads. Additional model analysis suggested contribution of

viscosity to main inotropic and lusitropic characteristics of myocardium performance.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Active myocardium mechanics; Mathematical model; Rheological scheme; Muscle viscosity

1. Introduction

In an attempt to analyse the mechanical properties ofheart muscle, we have been developing a mathematicalmodel successfully simulating a number of importantfeatures of myocardium mechanical activity (it is brieflydescribed below, in Section 2.1).Together with an extensive description of active

contractile muscle properties, previous versions of themodel included equations describing passive elasticity ofthe myocardial structures. However, no passive viscosityhas so far been included in the model, as it seemeddispensable for simulating and explaining active myo-cardium performance.In the meantime, experimental data arose, proving

that viscosity may really contribute specifically to the

ing author.

esses: lbk@efif.uran.ru, leonid@cranium.uran.ru

n).

e front matter r 2004 Elsevier Ltd. All rights reserved.

i.2004.05.007

active myocardium behavior. Particularly, using laserdiffraction, we showed that sarcomere shortening lasteddistinctly longer than the increase in the muscle forceduring isometric contraction (see Fig. 1). The sameconclusion may be also drawn from some datapublished earlier (Lecarpentier et al., 1985; Stuyverset al., 2002). For instance, Fig. 2 presented in the articleby Stuyvers et al. (2002) gives such an evidence forisometric twitches recorded in a cardiac trabeculaisolated from a mouse right ventricle: sarcomere length(SL) trace corresponding to the stimulation frequencyequal 0.2Hz reveals some lateness of the peak short-ening as compared to the peak of the respective stresstrace.Besides, during isometric relaxation, the sarcomere

lengthening also lasts quite long after the muscle forcereturns to the preload level (see the same Fig. 1).Isotonic relaxation also features some peculiarities,

which presumably may result from the viscous proper-ties of the natural heart muscle. These properties are

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Fig. 1. Experimental protocol of cardiac muscle isometric force

recorded along with the simultaneous sarcomere length change

observed by means of laser diffraction. The data were obtained on a

thin trabecula from a right ventricle extracted from a Wistar rat (all the

experimental procedures have been earlier described in detail

(Lecarpentier et al., 1985; Lecarpentier and Chemla, 1990)). The

muscle was transilluminated using a 5-mW Helium–Neon laser.

During the isometric contraction sarcomeres shorten by about 7% to

the detriment of series elastic elements.

Fig. 2. Top: Classic rheological scheme of the heart muscle including

contractile element CE and two passive elastic elements PE (parallel

one) and SE (series one). The scheme has been applied in the previous

variants of our mathematical model of the active myocardium

behavior. Bottom: Modified rheological scheme of the heart muscle,

dampers included. The scheme underlies the new version of our

mathematical model of the active myocardium behavior presented in

this work. All the details of the scheme are clarified in the text.

L.B. Katsnelson et al. / Journal of Theoretical Biology 230 (2004) 385–405386

presented in the left-hand panels in Figs. 3 and 4. Thesepanels show a series of one isometric and six isotoniccontraction–relaxation cycles recorded in physiological

experiments with a rat papillary muscle (the right-handpanels demonstrate respective data of numerical experi-ments). The peculiarities are as follows.

* The peak lengthening velocity increases significantlywith decrease in the load in the real muscle (PanelsB1, C1).

* There is a very gradual (smooth) transition fromthe rounded end-systolic length plateau to theonset of isotonic relaxation in the real muscle(Panel B1).

* The time where the peak lengthening velocity occursis slightly delayed as the afterload decreases in thereal muscle (Panel C1); this fact together withincrease in lengthening velocities along with decreasein afterloads, suggests that maybe a viscous forcecauses respective delays.

These experimental observations made us modify therheological scheme of the model by including viscouselements. We tested whether our modified model couldfit both the above-mentioned experimental data andsome more features, including the roughly linearrelationship between the load and peak lengtheningvelocity (Strobeck et al., 1975; Zile et al., 1989) and thebell-shaped relationship between the load and the peakrate of force decrease after isotonic relaxation (Mattiazziet al., 1986; Parmley and Sonnenblick, 1969). The resultsof these numerical tests are presented in this paper, andthe role of viscosity in active muscle behavior isdiscussed in the light of successful simulation of theabove phenomena.Furthermore, within the framework of the model we

analysed how viscosity may contribute to the maininotropic and lusitropic characteristics of the activeheart muscle registered in isometric and isotonic modesof contraction (e.g. length–force relationship, force–velocity relationships of contraction/relaxation, load-dependence of relaxation, etc.).Finally, it is important to emphasize that the

mathematical model presented in this paper is not amodel of passive myocardium mechanical behavior ingeneral or viscous one in particular. On the contrary, themodel mainly deals with active myocardium mechanics.Therefore both the rheological scheme used in themodel (in general) and the viscous elements included (inparticular) are designed to suit this purpose andsimplified as much as possible. On the other hand,those among previously published mathematicalmodels that included viscous components, dealt mainlywith passive mechanics (Fung, 1967, 1972; Loeffler andSagawa, 1975; Chiu et al., 1982; Tsaturyan et al., 1984;Chase et al., 1998; Starc et al., 1996; Lin and Rymer,1998; Mathur et al., 2001). Thus such models did notshed a light on the contribution of viscosity to the activemyocardium performance, and we tried to fill up the gapin this work.

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Fig. 3. Natural and virtual series of myocardium contraction–relaxation protocols including one isometric cycle and six isotonic ones. Left panels

(A1–C1) represent the data of a real experiment. Right panels (A2–C2) represent the data of simulation where both types of viscous elements, VS1

and VS2 are accounted for. Top panels (A1, A2) show the time courses of the muscles’ force; medium panels (B1, B2) and bottom ones (C1, C2)

represent, respectively, the shortening–lengthening time course and the time course of the shortening–lengthening velocity.

L.B. Katsnelson et al. / Journal of Theoretical Biology 230 (2004) 385–405 387

2. Methods

2.1. Further development of the model rheological scheme

In this study, we use as a starting point the previousvariant of our mathematical model, which successfullysimulated the time courses of isometric and isotoniccontractions of the heart muscle as well as all its basicmechanical characteristics (length–force, force–velocity,end-systolic length—relaxation time, etc.) (Izakov et al.,1991; Solovyova et al., 2002), and such well-knowneffects as muscle response to quick deformations duringisometric and isotonic twitches and load-dependentrelaxation (Izakov et al., 1991; Katsnelson andMarkhasin, 1996). The model consisted of a set of

ordinary differential equations and comprised not only adescription of purely mechanical properties, but alsothat of calcium activation during contraction. It isknown that the processes of association and dissociationof calcium–troponin (CaTn) complexes along actinfilaments in the sarcomere control this activation.Special attention was therefore paid to modeling andanalysing these processes. The model included two typesof cooperativity of the contractile proteins observed inbiochemical experiments (Grabarek and Gergely, 1983;Gordon et al., 2001) as well as the so-called effect ofsarcomere lattice-spacing. The latter effect, in particular,appears as an increase in the inter-filament distance dueto sarcomere shortening (see e.g. the paper by Konhilaset al., 2002). The inclusion of both cooperativity and

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Fig. 4. Natural and virtual series of myocardium contraction–relaxation protocols including isometric one and 6 isotonic twitches. Left panels

(A1–C1) represent the data of the same real experiment as in Fig. 3 (respective panels). Right panels (A2–C2) represent the data of one more

simulation, but where the viscous elements are switched off. Top panels (A1, A2) show the force time courses, medium (B1, B2) and bottom (C1, C2)

panels do, respectively, the shortening–lengthening time course and the time course of the shortening–lengthening velocity. Comparison of medium

panels B1, B2 with each other, and especially of bottom panels C1, C2 reveals fine distinction between the results of simulation and the real muscle

relaxation features. These distinctions are set forth in the text.

L.B. Katsnelson et al. / Journal of Theoretical Biology 230 (2004) 385–405388

lattice-spacing phenomena into the model made itpossible to take into account the feedback effects ofmechanical conditions of contraction on the calciumactivation course. These feedback mechanisms manifestthemselves as a number of experimental effects includingload-dependent relaxation, and these effects weresuccessfully simulated within the framework of themodel (Izakov et al., 1991). Our model (Katsnelson andMarkhasin, 1996) has also successfully simulatedexperimentally observed effects of mechanical condi-tions of muscle contraction on the calcium transient(Allen and Kurihara, 1982; Lab et al., 1984). It was alsoshown that there must be a negative feedback effect ofcalcium concentration in the longitudinal sarcoplasmic

reticulum (LSR) on the rate of calcium uptake by theLSR (Katsnelson and Markhasin, 1996). We associatedthis effect with the experimentally found inhibition ofthe calcium pump in the reticulum (Martonosi, 1979).This allowed us to simulate, in addition to theabove-mentioned phenomena, some new ones, includingexperimentally observed disappearance of load-dependence during severe myocardium hypertrophy(Lecarpentier et al., 1987).All the above-mentioned features of the model were

described in detail and justified elsewhere (Izakov et al.,1991; Katsnelson and Markhasin, 1996). In the presentpaper, therefore, we do not specify those details of themodel, which remain unchanged in this study. However,

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the model has had to be significantly modified inrelation to its rheological scheme, to enable it tosimulate adequately and explain the mechanical phe-nomena announced in the Introduction. It seemsreasonable that we should focus on this original pieceof research in more detail in this paper.Fig. 2 (top) shows the rheological scheme of the

previous variant of the model (prior to the lastmodification). As it can be seen, the scheme consists ofcontractile element CE (that is associated with asarcomere, as usual) and two nonlinear elastic passiveelements: series (SE) and parallel (PE). There were noviscous elements within the scheme at all. However, thatscheme failed to simulate the mismatch between the timeto isometric peak force of the muscle and the time to thepeak shortening of the sarcomere. Indeed, it is clearthat the above rheological scheme determines that themaximum active isometric force is to be reached atthe very moment when elastic element SE is strained tothe maximum during the isometric twitch, i.e. with theminimum length of CE. Thus, the times to muscleisometric peak force and to sarcomere peak shorteningcoincide inevitably. This means that the rheologicalscheme must be modified to include viscosity.Fig. 2 (bottom) demonstrates the modified rheological

scheme. Firstly, two viscous elements (VS1 and VS2)appear in this scheme. VS1 is parallel to the contractileelement, and VS2 is included in the series element inparallel with nonlinear spring SE. Note that the serieselement represents to a great extent the properties of themuscle sample outer segments, which are attached to theexperimental setup’s levers. Though the contractilestructures responsible for generation of active force aredamaged in the outer segments, these segments, at leastin part, consist of the same myocardial tissue as all theinner ones. Consequently, we can assume that thepassive qualities of the outer segments are similar tothose of the inner ones. In particular, the descriptions ofVS1 and VS2 must probably be similar, whilst beingdifferent parametrically.Furthermore, some experimental observations suggest

that the coefficient of viscosity in muscles depends oninstantaneous muscle length (Noble, 1977; Loeffler andSagawa, 1975). This dependency might be explained invarious ways. For example, one of the earlier suggestedsources of muscle viscosity is extracellular fluid filtrationin the elastic pores formed by the connective tissue andmyocytes (Tsaturyan et al., 1984). It seems quite feasiblethat a change in the muscle length may affect both theshape of these pores and, consequently, the fluidpressure within them. This, in turn, might affect theintensity of fluid filtration, and result eventually in achange in the viscosity. Probably, this is not the onlypossible explanation of the mechanism hypotheticallyunderlying length-dependent viscosity. Anyway, wepostulate here for our model that the coefficients of

viscosity for both VS1 and VS2 depend on lengthchanges of the corresponding elements. Namely,

kP vis ¼ bp expðap � l1Þ and

kS vis ¼ bS expðaS � ðl2 � l1ÞÞ;

where kP vis and kS vis are the coefficients of viscosity forVS1 and VS2, respectively; l1 is the difference betweenthe instantaneous length of contractile element CE andthe length of CE corresponding to the muscle slacklength; l2 is the difference between the instantaneouslength of the element PE and the length of PE

corresponding to the muscle slack length, and thus(l2�l1) represents deformation of the element SE;aP,bP,aS,bS are the constants (parameters) of the aboveformulas describing viscosity coefficients kP vis andkS vis. The passive viscous force of damper VS1is kP vis � dl1/dt, and that of damper VS2 is kS vis � (dl2/

dt/dl1/dt).Note that in the case where ap=0 and aS=0, the

coefficients of viscosity become length-independent inour model. In other words, the variant of length-independent viscosity is also considered in the model asa particular case. Our numerical experiments, however,showed that active isometric contraction would beunrealistic in this case. Thus, the variant of lengthdependent viscosity seems preferable. Interestingly,analysis of active mechanical behavior of the heartmuscle may help to shed light on the purely passivemechanical muscle peculiarities. Respective numericalexperiments will be presented and explained in detailbelow (see Section 4).Furthermore, it has been proved experimentally that

the characteristics of muscle viscosity for stretching andreleasing processes should be different (Tsaturyan et al.,1984). Thus, we suggest:

kP vis ¼bkp expðakp � l1Þ for

dl1

dtp0;

bmp expðamp � l1Þ fordl1

dt> 0;

8><>:

kS vis ¼bkS expðakS � ðl2 � l1ÞÞ for

dl2

dt�dl1

dtp0;

bmS expðamS � ðl2 � l1ÞÞ fordl2

dt�dl1

dt> 0;

8><>:

where the arrows indicate the above constant values forthe corresponding damper’s shortening (k) and length-ening (m), respectively.Finally, an extra-series elastic element (XSE) is

inserted additionally into the scheme to avoid a specificmodel artifact. Indeed, if we consider the samerheological scheme without extra-series element XSE,then an instantaneous jump (discontinuity) of the totalmuscle force will arise during afterloaded relaxation atthe moment of the muscle length return to its initialposition, i.e. when the isotonic phase of relaxation

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changes over to its final isometric phase. The followingcircumstances will cause this jump of the force withoutXSE. The right-hand end of damper VS2 will stopimmediately, whereas the left-hand end will continue tomove at the same velocity as before. As a result, a jumpwould occur in the velocity of shortening of VS2. Thisvelocity jump, in turn, would cause an instantaneousjump in the viscous force of this damper and, conse-quently, in the total muscle force. However, realexperimental data show (see e.g. Fig. 3, Panel 1A) thatthere is no discontinuity of the muscle force at themoment of transition from afterloaded to isometricrelaxation. In fact, there is a very fast decrease in theforce immediately after this moment, but this decrease isnot instantaneous, and the curve remains continuous.Essentially, these reasons mean that series viscositycannot be concentrated at the very end of the attachedsegment of the muscle preparation, but must be out ofthis end (at least, slightly). The formula describing thedependence of the XSE force on its length is similar toones for PE and SE:

FXSE ¼ b3 ½exp ða3 � l3Þ � 1�:

We have also to stress that there are several classicvariants of rheological schemes developed for themodeling of passive myocardium mechanical properties,and the most popular ones just include an elasticelement connected in series to the cluster of all otherelements of the scheme (see e.g. Izakov et al., 2000).Moreover, this kind of the schemes corresponds muchmore accurately to the damaged ends of experimentalpreparations attached to a force transducer/servomotor.The rheological scheme used in previous variants of

our model (Izakov et al., 1991; Solovyova et al., 2002)may be deduced from the new one if we assume kP vis

=0 and kS vis =0 (i.e. bpk =bp

m =b Sk =b S

m =0) andabsolutely stiff extra-series element XSE (formally,a3=+N, but practically it is sufficient to assumea3=200 mm�1 for the physiologically reasonable rangeof other parameter values and physiologically tolerablerange of loads).The full set of the model equations corresponding

to the modified rheological scheme is presented inAppendix A.

2.2. Experimental design

2.2.1. Muscle preparation

Our routine protocol of experimental procedures(Chemla et al., 1986) was used in this work to studyleft ventricular papillary muscle mechanics in 28 Wistarrats. The experiments were carried out for this particularstudy; however the animals used in it pertained to thegeneral population described elsewhere (Suard et al.,1994). Therefore now we omit respective description.The animals were briefly anesthetized with ether, and

their hearts were quickly removed and weighed (inaccordance with the recommendations of the HelsinkiDeclaration concerning care of animals). The leftventricular papillary muscles were carefully excisedand were suspended vertically in a tissue chambercontaining the following Krebs–Ringer solution (inmM): 118 NaCl, 4.7 KCl, 1.2 Mg SO4-7H2O, 1.1 KH2

PO4, 24 NaHCO3, 2.5 CaCl2-6H2O, 4.5 glucose. Thesolutions were bubbled with a gas mixture of 95%O2–5%CO2 and maintained at pH 7.4 and 29 �C. Thepreparations were stimulated by means of two platinumelectrodes positioned parallel to the muscle and deliver-ing rectangular pulses of 5ms slightly above the thresh-old. The muscles were preloaded at the initial musclelength corresponding to the apex of the length-activeisometric tension curve, i.e., Lmax and enabling thepreparations to recover their activity during a 30-minperiod. The muscles were stimulated at 10min�1. Afterthe stabilization period, the muscles were studied. At theend of the study, the cross-sectional area (mm2) wascalculated from the length at Lmax and weight of themuscles, assuming a muscular density of 1. Suitablepapillary muscle specimens were selected on the basis ofwell-individualized cylindrical shape, whose cross-sec-tional area did not exceed 1.2mm2. Only muscles withload-dependent relaxation (Chase et al., 1998) and witha resting tension over total isometric tension ratio lessthan 0.20 entered the final analysis.

2.2.2. Electromagnetic lever system and recording

The electromagnetic lever system was previouslydescribed (Chemla et al., 1986). Briefly, this lever systemis designed for simultaneous monitoring of muscleforce and length. The load applied to the muscle isdetermined by a servo-controlled current through thecoil of an electromagnetic lever system. The equivalentmoving mass of the whole system is 155mg, and itscompliance is 0.2 mm/mN�1. The displacement of thelever is measured by means of a photoelectric transducerconsisting of an incandescent lamp, a miniature photo-diode and a preamplifier. Muscular shortening inducesa displacement of the lever, which modulates the lightemitted by the lamp. The photodiode receptor convertsthe variations in light intensity into voltage alterations.The system has a linear range of up to 2.5mm ofmuscle shortening. The force measurement amplituderanges from 0 to 140mN. During contraction, the loadon the muscle may be abruptly altered (load clamp) bystepwise change in the current. For the purposes of thisstudy, all analyses were made from digital recordingsobtained by means of an IBM PC computer. Twosignals were recorded: force and length. The recordingspeed was one analog-to-digital conversion of eachsignal every millisecond. A single recording had 512points for each signal for a total recording time of500ms.

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2.2.3. Mechanics of LV papillary muscles

Conventional mechanical parameters characterizingcontraction and relaxation phases and the couplingbetween contraction and relaxation were studied, aspreviously described (Chemla et al., 1986; Suard et al.,1994). All muscles were allowed to contract undervarious conditions of load. From zero load up tothe fully isometric twitch, 6–10 levels of load (F)were applied for each muscle (Fig. 2, left panels).The maximum rate of tension rise (+dF/dt)max and themaximum rate of tension decline (�dF/dt)max wereautomatically calculated. The (�dF/dt)max valueswere plotted as a function of relative load (F/F0, whereF0 is the maximum isometric force). Because (�dF/dt)max=0 in the fully isotonic twitch (i.e., in the twitchwith preload only), the {(�dF/dt)max vs. F/F0} relation-ships were studied for F greater than preload. Peakshortening velocity (�dL/dt)max and peak lengtheningvelocity (+dL/dt)max were also automatically calcu-lated. The (+dL/dt)max values were plotted as afunction of F/F0.

2.2.4. Sarcomere dynamics

We used laser diffraction technique to registersarcomere length (SL) during contraction/relaxation.Routine procedures that constitute this method werespecified in full elsewhere (Lecarpentier and Chemla,1990). Therefore here we do not describe the method assuch but focus only on specific procedures whichallowed us to select largely homogeneous preparationsfor this study.The laser beam was split into two equal parts. One

part was focused onto one diode producing the signal h

which represented the variation of the intensity of thefirst-order line. The second beam passed through anoptical density wedge whose transmission was related tothe lateral shift of the beam and, consequently to thegrating constant due to the diffraction angle. This beamwas collected and focused onto another diode giving thesignal g which represented the product of the variationof intensity of the first-order diffraction line and thevariation of intensity due to the beam’s lateral’s move-ment on the densitometric wedge. Thus, the ratio g/h,represented variations in sarcomere length.Furthermore, all muscle samples selected to use in this

study met the following criterion of uniformity. Whenthe laser beam illuminated a different region in themidlength of the muscle, the intensity variations (h) weremodified but not the ratio g/h, i.e., there were no spatialvariations in sarcomere shortening/lengthening. To bemore exact, the ratio g/h, which is a function of the meanlength value of about 50–100 sarcomeres illuminated bythe laser beam, turned out to be not depending onthe position within the central segment of the muscle(the noise in the SL vs. time trace was minimized bydiminishing background light. g/h was recorded with a

low-pass filter (cut-off frequency: 2 kHz)). This criterionapplied methodically during many twitches allowed usto conclude that sarcomere lengths were uniform to agreat extent along the central segments of the selectedmuscles.

3. Results

3.1. Model results Vs. experimental ones

When dealing with numerical experiments with theabove-presented mathematical model, in which values ofall the parameters are specifically set, we apply the term‘virtual muscle’ to this specified modeled sample, incontrast to biological samples. All the numericalexperiments presented in this sub-section were per-formed with one and the same set of model parametervalues to deal with one and the same virtual muscle(unless otherwise stated). It is important to note that thedata presented the particular virtual muscle are notexceptional. It is rather a typical example of absolutelyregular results of simulation within a wide range ofparameter values.

3.1.1. Relaxation in isotonic twitches. Simulation vs.

in vitro experimental data

Fig. 3 (left-hand panels) shows the results of thein vitro experiment that has already been mentioned inthe Introduction. Here we collate the data of thisexperiment with numerical data on a virtual muscle,which includes viscous elements VS1 and VS2 (the basicvalues are given in Appendix C). Fig. 4 (left-handpanels) shows the results of the same in vitro experi-ment, but in this case we collate the data of thatexperiment with numerical data on another virtualmuscle, in which both viscous elements VS1 and VS2are taken to be equal to zero. It can be seen that theconversion of the slow phase of the afterloadedrelaxation into a fast one is essentially rounder in the‘‘viscous’’ virtual muscle (Fig. 3, Panel B2) than in theother virtual one, where viscosity is eliminated (Fig. 4,Panel B2). The first variant is in good qualitativeagreement with the real experimental data (Panels B1 inboth figures). Moreover, Panel C2 in Fig. 3 shows thatthe time-to-peak lengthening velocity slightly increaseswith a decrease in the afterload corresponding to thereal experiment (Panel C1). The model without viscositytaken into account behaves conversely (the time-to-peakdecreases, as it is seen in Panel C2 of Fig. 4). It isappropriate to mention here that all previous variants ofour model corresponding to the old rheological schemegave results similar to the simulation represented inFig. 4 (right panels).Fig. 5, in turn, shows the relationships between

afterload and the peak lengthening velocity obtained

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Fig. 5. Panels I and II: Peak-lengthening velocities vs. normalized

afterloads (F/F0) plotted for the same twitches as presented in Fig. 3.

The virtual muscle (i.e. the modeled muscle) is represented in Panel I,

and the real one in Panel II: Panel III: The data (means7SE) are

analogous to the relationship shown in panel I, but obtained from

a group of 28 real papillary muscles excised from the left ventricles

of rat.

Fig. 6. Panels I and II: (�dF/dt)max vs. normalized afterloads (F/F0)

plotted for the same twitches as presented in Fig. 3. The virtual muscle

(i.e. the modeled muscle) is represented in Panel I, and the real one in

Panel II: Panel III: The data (means7SE) are analogous to the

relationship shown in Panel I, but obtained from a group of 28 real

papillary muscles excised from the left ventricles of rat.

L.B. Katsnelson et al. / Journal of Theoretical Biology 230 (2004) 385–405392

for both the numerical experiment (Panel I—the sameexperiment as in Fig. 3, A2–C2) and the real experiment(Panel II—the same experiment as in Fig. 3, A1–C1).The relationships prove to be quite concordant qualita-tively. In particular, there are quite similar plateauphases for the range of small loads in both panels.Finally, Fig. 6 presents relationships between the

normalized afterload and the maximum rate of declinein the force during the isometric phase followingisotonic relaxation ((�dFXdt)max) obtained in theabove in vitro experiments (Panel I) and in numericalones (Panel II). A (�dFXdt)max value for each afterloadis obtained during isometric relaxation (see Panels A1,A2 of Fig. 3 for the time course of these relaxationphases). It can be seen that both panels in Fig. 6demonstrate generally similar features. In particular:

(i)

the relationship is bell-shaped in both cases;

(ii)

a single maximum appears both in Panels I and II,and the afterloads giving these extremum points arehigher than 0.5F0 in both cases. However themaximum corresponds to B0.60F0 in the virtualmuscle, being B0.75F0 in the majority of the realmuscles observed. We have not found a clearexplanation of this quantitative mismatch as yet.

3.1.2. Testing the influence of viscosity on mismatch

between the time courses of isometric force development

and sarcomere shortening/lengthening

The left-hand panels in Fig. 7 show a simulation of anisometric twitch where the model accounts for bothparallel and series viscosities (VS1 and VS2) (it is thesame virtual muscle as in Fig. 3, right). The time tomuscle peak force equals 140ms (the top left panel),whereas the time to contractile element CE peakshortening equals B190ms (the left mid-panel). The

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Fig. 7. Simulation of the effect of viscosity on mismatch between the time course of isometric force development and that of sarcomere shortening/

lengthening. The role of contractile element (CE) in the virtual muscle is associated with that of sarcomeres in real cardiomyocytes. Left panels show

simulation of an isometric twitch where both series and parallel viscosity are included into the model equations. The time to muscle peak force equals

B150ms, whereas the time to CE peak shortening equals 200ms. Right panels show the simulation of an isometric twitch where the series viscosity

has been switched off, while all the other model parameters, parallel viscosity included, have the same values as in the first case.

L.B. Katsnelson et al. / Journal of Theoretical Biology 230 (2004) 385–405 393

right-hand panels show the simulation of an isometrictwitch where series viscosity VS2 is ignored (i.e. kS vis

0), while all the other model parameters, parallelviscosity VS1 included, have the same values as in theprevious case. Unlike that case, the times to peak muscleforce and to CE shortening coincide exactly.The role of contractile element CE in the virtual

muscle is associated with that of sarcomeres in realcardiomyocytes. Thus, we can conclude that seriesviscosity is very much likely to underly the effect ofmismatch between the time to muscle peak force and thetime to sarcomere peak shortening, as it was observed inreal experiments using laser diffraction (see Fig. 1).The same Fig. 1 demonstrates one more mismatch

between the time courses of isometric force development

and sarcomere lengthening: the full relaxation of theforce is followed by a slow relaxation of the sarcomerelength (lengthening up to the initial level). To explainthis mismatch with help of the model let us address toFig. 7 once more. There we see similar patterns of thefinal stage of isometric relaxation both in the left panelswhere series viscosity VS2 is involved and in the rightpanels where series viscosity VS2 is ignored. Namely,sarcomere (CE) lengthening toward the initial length(mid panels of both left and right presentations) lastsnoticeably long after the force returns to the preloadlevel (top panels). In particular, the dotted horizontallines show the level of preload in the top panels and thelevel of the initial sarcomere length in the middle ones.As it is clearly seen, when the force is getting equal to the

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preload (the right edge of all the panels, i.e. 700ms), thelength does not still reach its initial level either in rightor in left presentations. The difference between theinstant length at this moment and the initial one is7%–8% of the maximum sarcomere shortening in bothleft and right middle panels. In other words, the latenessof the final sarcomere lengthening as compared to thecomplete relaxation of the twitch force is peculiar toboth cases: with VS2 (left panels), and without VS2(right panels). Thus, this peculiarity of relaxation doesnot depend on VS2. On the other hand, it is obvious (seethe rheological scheme) that it would be impossible toobtain this effect in the model if we ignored both kindsof viscosity. This allows us to state that parallel viscosityby itself produces the discussed lateness of the finalsarcomere lengthening as compared to the completerelaxation of the force.Both these data and conclusion harmonize well with

results represented by Stuyvers et al. (1998). Theyobserved in rat cardiac trabeculae that completerelaxation of the twitch force was followed by a slowsarcomere lengthening to the initial length. They werealso inclined to believe that the effect was not due to acontracture of the damaged ends of the trabeculae.

Fig. 8. Data of numerical experiments in 3 virtual muscles: the first—with th

the third—with halved VS2. Relationships ‘Force–Velocity’ (top left panel

(bottom left panel), ‘Force—Rate of Relaxation (Decrease in Force)’ (botto

3.2. Contribution of viscosity to inotropic and lusitropic

properties of the virtual heart muscle

The above numerical experiments show that viscositymakes a contribution to the virtual heart muscleperformance. Moreover these results allow us to pointat viscosity as the most likely cause of the respectivephenomena observed in the natural muscle. At the sametime, the overall role of viscosity in the mechanicalbehavior of the active myocardium cannot be under-stood unless we assess the sensitivity of the main ino-and lusitropic characteristics to variations in viscosity.Numerical tests were done to analyse this sensitivitywithin the framework of the model. The results of thesetests are presented in Figs. 8 and 9. They are as follows.The curves used as a baseline are derived from the

same series of isometric and isotonic simulations, whichhave been already presented in Figs. 3–7. Fig. 8 showsthe main inotropic and lusitropic characteristics of thisvirtual muscle taken here as control one (the curvesmarked with circles: ) and of two other virtual muscleswhere viscosity was changed as compared to the control.The top panels of the figure show length–force andforce–velocity relationships, and the bottom ones the

e basic set of the parameter values, the second—with halved VS1, and

), ‘Force–Length’ (top right panel), ‘Force–Velocity of Lengthening’

m right panel) are obtained in series of isotonic twitches.

ARTICLE IN PRESS

Fig. 9. Data of numerical experiments on LDR effect obtained in a series of isotonic twitches in 3 virtual muscles: the first—with the basic set of the

parameter values, the second—with halved VS1, and the third—with halved VS2. LDR index is defined in the text.

Table 1

Other inotropic indices

Level of viscosity F0 (mN) time to F0 (ms) (+dF/dt]max/F0 time to (+dF/dt]max/F0

79.745 147.3786 15.2426 81.6550

95.088 147.8542 14.8072 74.5038

76.718 150.4779 14.8519 83.3340

L.B. Katsnelson et al. / Journal of Theoretical Biology 230 (2004) 385–405 395

characteristics of relaxation rates that have been alreadypresented in Panels I of Figs. 5 and 6.Such dependences were also obtained in the virtual

muscles where either parallel viscosity or series one wasreduced half as much as the control (the data aremarked in Figs. 3–7 as and , respectively).Additionally, a few inotropic indices obtained in the

same series of numerical experiments with these 3 virtualmuscles are given in Table 1.We can conclude that a change in either kind of

viscosity affects inotropic properties negligibly. The onlyexclusion is the maximum isometric force that increaseddistinctly but only in the case of change in parallelviscosity (see the second column of Table 1).On the other hand, lusitropic properties proved to be

much more sensitive to the change in the parallelviscosity. Indeed, the bottom panels of Fig. 8 demon-strate that both the peak lengthening velocities and thevalues of (�dFXdt)max increased considerably, at leastfor small and medium afterloads. At the same time,

there was no effect of a change in the series viscosity onthe lusitropic characteristics of the virtual muscle.Finally, we checked for possible effects of changes in

both types of viscosity on the load dependence ofrelaxation (LDR). Earlier we paid special attention tothe mathematical modeling of the phenomenon of LDR

and to the analysis of underlying intercellular mechan-isms (Izakov et al., 1991; Lecarpentier et al., 1987).Using the previous versions of our model we showedthat the cooperativity of contractile proteins duringCa2+ activation of thin filaments and the lattice spacingof sarcomeres produce jointly LDR of myocardium(Izakov et al., 1991; Solovyova et al., 2002). Also, wefound out that LDR in virtual heart muscles is sensitiveto the rate of sarcoplasmic reticulum Ca2+ uptake(Katsnelson and Markhasin, 1996). All the aboveintercellular mechanisms are accounted for in the newversion of our mode used in this work. The question iswhether or not viscosity affects the LDR phenomenonas well.

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Fig. 9 reveals the effect of changes in viscosity onLDR estimated for 6 afterloads in the same 3 virtualmuscles as above ( , , and ). For each afterloadthe value of LDR was calculated as a ratio between theduration of the isotonic contraction–relaxation phase(under this afterload) and the length of time where theisometric force is higher than this afterload during afully isometric twitch. This characteristic LDR iscommonly accepted (Dobrunz and Berman, 1994;Katsnelson et al., 2000). We see that though the effectof viscosity on LDR is not strong it is really quite visible.Both kinds of viscosity contributed to the effect, thoughin an opposite way. The two-fold decrease in the parallelviscosity ( ) resulted in a rise of the curve as comparedto the control ( ) for large afterloads and in a loweringfor medium and small ones. Vice versa, a two-folddecrease in the series viscosity ( ) resulted in alowering of the curve as compared to the control ( )for large afterloads and in a rise for medium andsmall ones.

4. Discussion

4.1. On physiological meaning of viscosity: its

contribution to myocardium performance

The origin of viscosity in muscles is not quite clear asyet. There are serious reasons to associate it with thefollowing intracellular structures (with either some or allof them):

* with myoplasmic viscosity;* with the slipping and friction of adjacent actin and

myosin filaments;* with viscosity of such constitutive proteins as

collagen or titin which have some visco-elasticproperties;

* with viscosity of the cytoskeletal microtubularnetwork;

* with extracellular fluid filtration in the elastic porescomposed by the connective tissue network andmyocytes.

In Section 2 we referred to some papers on theviscosity of the muscular tissue (Noble, 1977; Loefflerand Sagawa, 1975; Tsaturyan et al., 1984). Viscousproperties of single muscle cells (Mathur et al., 2001)and even of separate intracellular sub-units, e.g. of titin,the giant protein located within the sarcomere structure(Kulke et al., 2001; Linke et al., 1999), are also beingstudied. Some more works are referred to in theappendices where we justify the basic values of theviscosity parameters (Fung, 1967, 1972; Izakov et al.,2000; Mathur et al., 2001; Markhasin et al., 1994). Infact, a lot more studies have dealt with various aspects

of viscosity. For instance, viscous behavior has beenstudied for the whole isolated guinea pig left ventricle indiastole (Starc et al., 1996). Viscous properties of the catpapillary muscle have been assessed with the help ofpeculiarities of the muscle response to fast short-timestretches or shortening ramps of the muscle (Chiu et al.,1982). These deformations were applied to the muscle inboth active and passive states, and velocities ofsubsequent recoils were estimated. A variant of therheological scheme was suggested to fit those experi-mental data (Chiu et al., 1982). The viscoelastic responseof the cat heart muscle to sinusoidal perturbations of themuscle length was considered both in experiments withthe resting muscle and in isometric conditions (Loefflerand Sagawa, 1975). As a result, the authors suggested arheological scheme and a corresponding mathematicalmodel of the muscle including viscous, elastic and activebranches, although the latter was rather quasi-activebecause it did not include a contractile element. Notonly viscosity of the heart muscle but also that of theskeletal muscle in both rest and tetanus have been testedin models (Chase et al., 1998; Lin and Rymer, 1998).Despite the above and a number of other works, the

following question seems to be unanswered as yet. Howdoes myocardium viscosity particularly contributes tothe active mechanical behavior of the heart muscle? Inother words, the physiological significance of themyocardium viscous properties is still unclear. Thiswork is an attempt to clarify it in part.To this aim we added two viscous elements (i.e., VS1

parallel to the contractile element CE, and VS2 parallelto the series element SE) to the classical rheologicalscheme. VS1 generalizes the viscous properties of severalintracellular structures (e.g. cytoplasm, adjacent actinand myosin filaments slipping with a friction, collagen,and titin). VS2, in turn, is associated to a great extentwith the outer segments of a muscle sample which areattached to the experimental setup’s levers.First of all, the simulation of the heart muscle

contraction–relaxation cycle by means of this extendedmodel allowed us to analyse and explain the role ofviscosity in the below-listed effects observed experimen-tally and pertaining to the area of active myocardiumbiomechanics.

(a)

A time lag between the duration of the muscle forceincrease phase and the duration of the sarcomereshortening in fully isometric contraction (Fig. 1).

(b)

Immanent features of relaxation during isometricand isotonic twitches, which are as follows:

* sarcomere lengthening toward the initial length of thereal muscle lasts a noticeably long time after fullreturn of the muscle force to the preload level inisometric twitches;

* time-to-peak lengthening velocity is essentially load-independent during isotonic relaxation (except the

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medium range of afterloads, where this time-to-peakmay be delayed slightly, when afterload decreased).

In particular, it was shown that parallel damper VS1is responsible for the effects listed above under item (b),whereas series damper VS2 is responsible for thesingular feature of isometric contractions (above-listeditem (a)).Once established within the framework of the model,

this contribution of viscosity to myocardium perfor-mance made us try to elucidate the effect of viscosity onthe main physiological characteristics of contraction andrelaxation. To clarify this point in the model, we testedin a series of afterloaded contraction–relaxation cycleshow changes in the viscosity transformed the main ino-and lusitropic characteristics of a virtual heart muscle.In particular, we dealt with the basic values of bothkinds of viscosity in the first series, and than with a two-fold decrease either in parallel or series one. Thefollowing ino- and lusitropic characteristics were con-sidered (see Fig. 8 and Table 1):

* ‘‘force–length’’ and ‘‘force–velocity’’ of contraction;* peak isometric force and time to peak isometric force;* peak rate of isometric force development, and time

to it;* ‘‘force–velocity’’ and ‘‘force—rate of force fall’’ of

relaxation.

Furthermore, in the same series of afterloadedcontraction–relaxation cycles of the virtual muscle weverified whether viscosity might affect the phenomenonof myocardium load dependent relaxation (see Fig. 9).The following conclusions may be derived from the

series of the numerical experiments carried out.For parallel viscosity it may be concluded that:

* it does not modulate inotropic properties exceptslight influence on force amplitude F0 in isometriccontraction and on the time to [+dF/dt]max;

* it does influence markedly the lusitropic character-istics (in addition to the effect of this kind of viscosityon the relaxation features discussed earlier in the MSin regard to Figs. 4 and 8).

For series viscosity:

* despite the fact that this kind of viscosity is fullyresponsible for the experimentally observed mismatchbetween the time of the isometric peak force and thetime of sarcomere shortening (Figs. 1 and 7), itcontributes almost nothing to both inotropic andlusitropic characteristics. Practically, all of themremain unchanged in comparison with the baselines.

This result seems to be noteworthy, taking intoaccount the following reasons. The series viscosity mustbe associated to a great extent with the attached ends ofthe real experimental preparations, i.e. it is an artificial

property of such experimental samples with respect tothe intact myocardium. And the model analysis suggeststhat this experimentally imposed condition does notdistort the main myocardium characteristics obtained inthe experimental works dealing with such preparations.The above conclusion is correct with one essential

exclusion, which is the index of the LDR plotted vs.afterload (see Fig. 9). A change in the series viscositydoes turn the curve as compared to the baseline;moreover the direction of the turn is opposite to therespective effect of the parallel viscosity.As far as the interrelation between the phenomenon

of LDR and both kinds of viscosity is concerned,there seems to be one more physiologically importantconsequence of our considerations. Indeed, it isknown that LDR changes significantly in hypertrophy(Lecarpentier et al., 1987). On the other hand, it hasbeen recently shown that the viscosity of the cytoskeletalmicrotubular network increases almost four times inhypertrophied cardiocytes as compared to normal ones(Tagawa et al., 1997). Our results show that this increasemay affect LDR considerably, and this effect should beaccounted for when analysing the effect of severehypertrophy on myocardium performance.Summarizing this part of the discussion we can

conclude that not only intracellular contractile mechan-isms (i.e. calcium handling and cross-bridge kinetics) butalso viscosity, the parallel one especially, may affect therelaxation of the active myocardium. This conclusionshould be taken into account in experimental studiesdealing with either various special conditions, patholo-gical included, when an experimenter tries to explainhow and why these conditions cause experimentallyobserved changes in relaxation.

4.2. Specificity of the myocardium’s visco-elastic

elements revealing itself during active behavior

When including viscous elements into the modifiedmodel, we firmly aimed at solving particular problems ofactive myocardium mechanics. In other words, themathematical model we are dealing with in this workis a model of active myocardium mechanical behavior,not that of passive myocardium mechanical properties.In particular, it is not a model of myocardium viscosity.In accordance with our goal, passive elements play anaccessory role in the model and therefore have asimplified form as compared to the real cardiac tissue.Nevertheless, some visco-elastic properties proved to bequite peculiar in the model for the correct simulation ofthe active myocardium behavior. These peculiarities arediscussed below.The model analysis has shown that the very end

segments of real myocardium preparations must be freeof viscous features (unlike a prevailing part of thepreparations). The latter segments may occupy a very

ARTICLE IN PRESS

Fig. 10. Distorting bend of the force time-course during isometric

contraction simulated in the case of the length-independent viscous

coefficients in the model (trace (a)). The cause of this inconsistency is

discussed in the text. Trace (b) corresponds to the basic parameter

values where this defect of simulation disappears.

L.B. Katsnelson et al. / Journal of Theoretical Biology 230 (2004) 385–405398

small part of a myocardium sample tested; perhaps theyeven reside only at the points where the sample isattached to the lever of the experimental setup. Anyway,such purely elastic segments must exist (in our model itis element XSE—see Fig. 2 (bottom), because otherwiseit would be impossible to avoid a mathematical contra-diction within the rheological scheme including viscouselements. We justified this necessity in Section 2.1.Further on, we have to focus on specific features of

the included viscous elements because they are alsonecessary for correct simulation of the active muscle’sproperties. The first of them is length-dependence of theviscosity coefficients. As we mentioned in Section 2,some data obtained in the experiments with passivemyocardial tissue provide evidence in favor of length-dependent viscosity of the heart muscle (Noble, 1977;Loeffler and Sagawa, 1975). However, this is hardlysufficient to call the latter length-dependence as anestablished fact. For this reason we allow for length-independence of viscosity to be potentially achievable inthe model as a variant. Namely, if both ap and aS areequal to 0, then ordinary length independent viscosityarises in the model. However, numerical experimentscarried out with this variant elicited immediately itsprincipal inconsistency. In fact, the following dilemmaoccurs in this case:

* either parameters of linear components bp and bS

turn out to be too insufficient quantitatively toachieve even barely noticeable contribution to theactive muscle behavior;

* or, if bp and bS turn out to be sufficiently large(around 0.5mN s/mm�1 or higher), a conspicuousdistortion happens in the time course of isometricforce, as it is seen in Fig. 10.

This distortion may be explained easily taking intoaccount the fact that for ap and aS=0 the followingtime-course of the force components holds. Both theelastic force of the series element SE and the viscousforce of the damper VS2 contribute to the total force.The viscous force of VS2 increases in the model togetherwith the force of SE from the very beginning of virtualmuscle contraction until SE achieves the maximumvelocity of its lengthening. However, unlike the elasticforce, which increases further, the above viscous forcestops increasing and even decreases slightly afterachieving the maximum velocity of the element SE

lengthening. The distorting bend observed in Fig. 10(trace (a)) occurs immediately after this moment. Thelength–dependence of the viscosity coefficient (i.e. thevariant aS a0) abates this shortcoming at once(compare traces (a) and (b) in Fig. 10). As we mentionedearlier, dampers VS2 and VS2 are of the same nature.Therefore, we put ap a0 as well.The above consideration proves that ordinary dam-

pers with constant length-independent coefficients are

incorrect within the rheological scheme including anactive element (CE). It seems noteworthy that somepurely passive mechanical property of the heart muscle,which has not yet had comprehensive experimentalproof, may be confirmed when modeling the active

mechanical behavior of the muscle.Another peculiarity of the dampers involved may be

defined as orientation-dependence of the coefficients ofmuscle viscosity. This dependence means that thecharacteristics of viscosity for stretching and releasingprocesses should differ. This property was provedexperimentally (Tsaturyan et al., 1984), and the authorsof that experimental work proposed their own explana-tion of the results obtained. They developed a concept ofextracellular porous–elastic fluid filtration as themain determinant of the viscous properties of thepassive myocardium. Based on this concept theysuggested that conditions under which the fluid issqueezed in the pores (during stretching of the muscletissue) and the pores are filled (during its release) areessentially different, and this circumstance results indiverse viscous responses of the muscle to stretching orreleasing. One could accept or reject their explanation,but their experimental finding should be taken intoaccount anyway.Concluding it is necessary to repeat once more that

the model we dealt with in this work is not a model ofpassive mechanical properties of the heart muscle tissue.Hence, we cannot claim that the model can simulateadequately all the nice tests within passive myocardiummechanics. Nevertheless, the model should be able toreproduce at least some principal experimental phenom-ena within this area of heart biomechanics. That is whywe simulated two experiments with the passive myocar-dium tissue that we used for justifying our choice of

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Fig. 11. Results of numerical experiments with the same virtual muscle as in Figs. 3, 5–9, but done in passive state. Two types of passive virtual

muscle deformations are represented. Panel I shows the muscle’s passive force change owing to a series of the fast tugs. A cyclic deformation of the

same passive virtual muscle is represented in Panels II.1 and II.2. Both these numerical tests are described in detail in the text.

L.B. Katsnelson et al. / Journal of Theoretical Biology 230 (2004) 385–405 399

viscosity parameter values (the data of the respectivenumerical tests are presented in Appendix C where thechoice of all the basic parameter values is described).Both tests deal with deformation of the passive heartmuscle. The results of the tests presented in Fig. 11prove to be in good agreement with the data of realexperiments (Fung, 1967, 1972; Izakov et al., 2000;Markhasin et al., 1994).

Acknowledgements

We are grateful to the RFBR (Grant # 00-04-48323),NATO LST. CLG 975785, HHMI Grant # 55000 300,and Wellcome Trust CRIG # 074152/z/04/z for thefinancial support of our work.

Appendix A. Glossary used in the mathematical

formulations

A.1. Phase variable of the model differential equations

N—fraction of force generating attached cross-bridges;

l1—difference between the instantaneous length ofcontractile element CE and its length corresponding tothe muscle slack length;l2—difference between the instantaneous length ofparallel element PE and its length corresponding tothe muscle slack length;l3—difference between the instantaneous length of extra-series element XSE and its length corresponding to themuscle slack length;v—velocity of contractile element CE lengthening/shortening;w—velocity of parallel element PE lengthening/shortening;A—concentration of Ca–TnC complexes;B—concentration of calcium-buffer complexes;CaC—free cytosolic calcium concentration%X—vector of all the phase variables, i.e. %X ¼½N; l1; l2; l3; v;w;A;B;CaC �:

A.2. Other important output signals of the model

Fmuscle—muscle force;FCE—force of contractile element CE;FSE—force of series element SE;

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FPE—force of parallel element PE;FXSE—force of extra-series element XSE.

A.3. Dependences and parametric constants used in the

model

M(A)—dependence expressing end-to-end interactionbetween adjacent tropomyosin segments in the case ifboth of them affected by the respective CaTnCcomplexes formation;m,A1/2—constants of dependence M(A) appearing as themodel’s parameters;p(v)—steady-state dependence of an average cross-bridge’s force on the velocity of sarcomere shortening/lengthening.P�(v)—steady-state dependence of the muscle force onits shortening/lengthening velocity;G�(v)—steady-state dependence of the muscle stiffnesson its shortening/lengthening velocity;a, vmax, dh, aP, aPG—constants of dependencies G�(v)and P�(v) and p(v) appearing in these dependencies asthe model’s parameters;k+(v)—velocity-dependent rate constants of cross-bridges’ attachment;k�(v)—velocity-dependent rate constants of cross-bridges’ detachment;w,w0,m0—constants of dependencies k+(v) and k�(v)appearing in these dependencies as the model’sparameters;kP vis—length-dependent coefficient of viscosity fordamper VS1;kS vis—length-dependent coefficient of viscosity fordamper VS2;apk, bp

k, apm, bp

m, aSk, bS

k, aSm, bS

m—constants of thedependencies of either kP vis or kS vis on the sarcomerelength; all these constants are the model’s parameters;l, a1, b1, a2, b2, a3, b3—coefficients of nonlinear elasticelements SE, PE, and XSE (see formulas (F), (RIV),(RV), and (RVI) below);p(N/((l1+S0) �A))—dependence defining 1st type coop-erativity of the contractile proteins in the calciumhandling block of the model;kA, bon, boff, Btot, ac, bc, td, rCa, qCa—other constants ofthe calcium handling block appearing in this block asthe model’s parameters;

Appendix B. The model’s equations

B.1. Mechanical block of the model

This block consists of 5 ordinary differential equa-tions. Namely, it includes equations describingdNdt; dl1

dt; dl2

dt; dl3

dt; dvdt; and dw

dt(see Appendix A, where

variables N, l1, l2, l3, v, w are defined).

The equation describing the kinetics of cross-bridgeattachment–detachment and giving dN/dt looks asfollows:

dN

dt¼ *Kð %XÞ ¼ kþðvÞ � MðAÞ � n1ðl1Þ � ððl1 þ S0Þ=ð0:46þ S0ÞÞ

� ð1� NÞ � k�ðvÞ � N ðB:1Þ

where

* term M(A)=Am/(Am+A1/2m ) means end-to-end inter-

action between adjacent tropomyosin segments in thecase where both of them are affected by theformation of respective CaTnC complexes. In otherwords, the fraction of active sites on the thin filamentincreases cooperatively with Ca–TnC concentrationas a result of this interaction, i.e. this increase turnsout to be steeper then the rise of A. Here m is acoefficient of this cooperativity, and A1/2 is theconcentration of Ca–TnC for the half-maximumfraction of active sites. In the very first versions ofour model (Izakov et al., 1991) we ignored this kindof cooperativity and considered the attachment ofcross-bridges to be directly proportional to thenumber of CaTnC complexes A (instead of M(A)).However, since the third version of the model(Katsnelson and Markhasin, 1996) we have beentaking into account the end-to-end cooperativity oftropomyosin segments.

* n1(l1) is the probability of a myosin cross-bridge‘‘finding’’ a vacant site on the actin filament; in otherwords, the dependence of n1 on l1 means sarcomerelattice-spacing. In a simplified linearized form, thisdependence looks as follows: n1(l1)=0.6 � l1+0.5. Wejustified this dependence in detail elsewhere (Izakovet al., 1991).

* Loz=(l1+S0)/(0.46+S0) is a normalized linear de-pendence of the sarcomere overlap zone on thesarcomere length;

* k+(v) and k�(v) are velocity-dependent rate constantsof cross-bridges’ attachment and detachment, respec-tively. The following formulas set these dependenciesin the model:

kþðvÞ ¼ w � w0 � qðvÞ � m0 � G�ðvÞ;

k�ðvÞ ¼ w0 � qðvÞ � ð1� w � m0 � G�ðvÞÞ;

where m0 is a model parameter denoting a fraction ofcross-bridges strongly attached in steady-state isometriccondition; and w,w0 are parameters enabling one to varythe ratio between particular rates of cross-bridgeattachment and detachment in the model.We have already justified these velocity-dependencies

and established an explicit expression for q(v) elsewhere(Izakov et al., 1991). Note that w0 � q(v) is nothing elsebut a sum of k+(v) and k�(v), and this sum can beestimated by means of specific experimental data, as wehave already done (Izakov et al., 1991). We have been

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using the same expression for q(v) up till now.However, an explicit approximation for G�(v) is nowmodified as compared to the previous works. This newapproximation will be specified a few paragraphs below.The following formula gives the force developed by

the activated contractile element in the model:

FCE ¼ l � pðvÞ � N ; ðFÞ

where l is the model parameter comprising transitionfrom one sarcomere to a whole virtual muscle thatconsists of a number of myofibrils. The dependence p(v)of an average force of one cross-bridge on the sarcomereshortening/lengthening velocity v is described below.Formula (F), together with another six ones ((RI–

RVI)—see below), serves to produce a differential

dv

dt¼ jð %XÞ ¼

Isotonic mode :l � *Kð %XÞ � pðvÞ þ ap � kP vis � v2 þ a2 � b2expða2 � l2Þ � w

l � N � p0ðvÞ þ kP vis

isometric mode :l � *Kð %XÞ � pðvÞ þ ap � kP vis � v2 þ ða2 � b2 expða2 � l2Þ þ a3 � b3 expða3 � l3ÞÞ � w

l � N � p0ðvÞ þ kP vis

8>>>>>>><>>>>>>>:

ðB:5Þ

equation describing geometrical variables’ derivatives:dl1Xdt, dl2Xdt, dl3Xdt, dvXdt, and dwXdt. The firstthree parities among these six formulas follow directlyfrom the modified rheological scheme, and the secondthree express nonlinear elasticity of 3 passive elementsSE, PE, and XSE, respectively. The formulas look asfollows:

Fmuscle ¼ FXSE ; ðRIÞ

dw

dt¼

Isotonic mode :

kS vis � ðjð %XÞ � aS � ðw � vÞ2Þ � a1 � b1 expða1 � ðl2 � l1ÞÞ � ðw � vÞ � a2 � b2 expða2 � l2Þ � w

kS vis

Isometric mode :

kS vis � ðjð %XÞ � aS � ðw � vÞ2Þ � a1 � b1 expða1 � ðl2 � l1ÞÞ � ðw � vÞ � ða2 � b2 expða2 � l2Þ þ a3 � b3 expða3 � l3ÞÞ � w

kS vis

8>>>>>>><>>>>>>>:

ðB:6Þ

FXSE ¼ FPE þ FCE þ kP vis � v; ðRIIÞ

FXSE ¼ FPE þ FSE þ kS vis � ðw � vÞ; ðRIIIÞ

FSE ¼ b1 ½exp ða1 � ðl2 � l1ÞÞ � 1�; ðRIVÞ

FPE ¼ b2½exp ða2 � l2Þ � 1�; ðRVÞ

FXSE ¼ b3½exp ða3 � l3Þ � 1�: ðRVIÞ

To obtain differential equations for the isometric modewe differentiated equality (RII) with respect to time

(taking into account equalities (F), (RV), (RVI)). Toobtain equations for the isotonic mode we differentiatedthe equality (RIII) with respect to time (taking intoaccount equalities (RIV)–(RVI)). Omitting the details ofthese transformations, we provide the final shape of theequations obtained:

dl1

dt¼ v; ðB:2Þ

dl2

dt¼ w; ðB:3Þ

dl3

dt¼

0 ðisotonic modeÞ;

�w ðisometric modeÞ;

�ðB:4Þ

where

kP vis ¼bkp expðakp � l1Þ for

dl1

dtp0;

bmp expðamp � l1Þ fordl1

dt> 0;

8><>:

where the arrows indicate the respective constant valuesfor CE shortening (m) and lengthening (k).

where

kS vis ¼bkS expðakS � ðl2 � l1ÞÞ for

dl2

dt�dl1

dtp0;

bmS expðamS � ðl2 � l1ÞÞ fordl2

dt�dl1

dt> 0;

8><>:

where the arrows indicate the respective constant valuesfor SE shortening (m) and lengthening (k).Function p(v), meaning velocity-dependence of the

average cross-bridge force on the sarcomere shortening/

ARTICLE IN PRESSL.B. Katsnelson et al. / Journal of Theoretical Biology 230 (2004) 385–405402

lengthening velocity, appears in Eqs. (B.5) and (B.6). Wehave previously justified that p(v) should be expressed asfollows (Izakov et al., 1991):

pðvÞ ¼P�ðvÞG�ðvÞ

ðPÞ

where P�(v) is the dependence of the steady-statesarcomere force on the shortening/lengthening velocity,and G�(v) is the dependence of the steady-statesarcomere stiffness on the velocity. Both functions arenormalized so that P�(0)=1,G�(0)=1.The previousvariants of our model included specific formulas forP�(v) and G�(v). Those formulas supplied us withcontinuous and piecewise smooth approximations forthe above dependencies. However the new variant ofthe model (namely, Eqs. (B.5) and (B.6)), unlike theprevious ones, comprises not only p(v) but also itsderivative p0(v). Therefore, now p(v) must be fullysmooth together with both P�(v) and G�(v) participatingin formula (P). This made us pick up a new moreaccurate approximation for the latter two functions.Now they have the following formulation:

P�ðvÞ ¼

a � ð1þ v=vmaxÞa � v=vmax

for vp0;

1þ dh �d2

h � a

a � dh � ðv=vmaxÞ2=gþ ða þ 1Þ � v=vmax þ a � dh

for vX0;

8>>><>>>:

G�ðvÞ ¼

1þ 0:6 � v=vmax for � vmaxpvp0;

P�ðvÞð0:4 � a þ 1Þ=a � v=vmax þ 1

for 0pvpv1;

P�ðvÞ expð�aG � ððv � v1Þ=vmaxÞaPG Þ

0:4 � a þ 1=a � v=vmax þ 1for vXv1;

8>>>>><>>>>>:

where

g ¼a � dh � ðv1=vmaxÞ

2

3 � a � dh � ða þ 1Þ � v1=vmax

and v1, vmax, a, dh, aG, aPG are the model parameters.These new variants of P�(v) and G�(v) actually have

fully continuous derivatives.It is apposite to emphasize that the above mechanical

block is much more sophisticated than it was in all theprevious versions of our model (Izakov et al., 1991;Katsnelson and Markhasin, 1996, Solovyova et al.,2002).

B.2. Calcium-handling block

This block consists of 3 ordinary differential equa-tions. Namely, it includes equations describing dA/dt,dB/dt, dCaC/dt.

� Kinetics of Ca–TnC complexes plays a crucial role inall the versions of our model as the time-course of thisconcentration controls attachment of cross-bridges(Eq. (B.1)) and thus, the mechanical behavior of themuscle. The following equation describes dA/dt (whereA is the concentration of Ca–TnC complexes).

dA

dt¼ aon � ðAtot � AÞ � CaC � %aoff expð�kA � AÞ

� pðN=ðLoz � AÞÞ � A: ðB:7Þ

A key point of this equation is the cooperativity of Ca2+

affinity to TnC. We have previously justified two typesof this cooperativity and analysed their contribution tothe myocardium contraction/relaxation phenomena(Izakov et al., 1991; Katsnelson and Markhasin, 1996).In particular, these two types of cooperativity may bephrased in the following manner:

Type 1: The affinity increases with the averagenumber of strongly attached XBs around each Ca–TnC complex. We consider Ca–TnC off-rate to bedecreasing with increase in this number. The dependencep(N/(Loz �A)) expresses this cooperativity in above

equation [7], and its particular form may be writtendown as follows:

pðNAÞ ¼1 for NAX1;

0:02NA for NAo1;

�

where N/(Loz �A) (the average fraction of the attachedcross-bridges falling on one CaTnC complex) is denotedas NA (i.e. NA=N/(Loz �A)).

Type 2: The affinity increases with Ca–TnC concen-tration resulting from conformation of troponin–tropo-myosin complex due to Ca2+ binding to TnC.Specifically, we consider Ca–TnC off-rate to be decreas-ing with increase in A. Dependence exp(�kA �A) definesthis cooperativity in Eq. (7), where kA is the model’sparameter.

� Kinetics of calcium binding by intracellular buffer

ligands is given in as simple a form as possible in thisversion of the model. The following equation describesthis kinetics:

dB

dt¼ bon � ðBtot � BÞ � CaC � boff � B: ðB:8Þ

All the calcium-binding ligands are represented here bya unitary generalized buffer (Solovyova et al., 2002). Btot

ARTICLE IN PRESSL.B. Katsnelson et al. / Journal of Theoretical Biology 230 (2004) 385–405 403

is the total concentration of this buffer, and B denotesthe current concentration of calcium-buffer complexes.

� Kinetics of free cytosolic calcium is described indetail in another version of our model that has beenaimed especially at performing a careful analysis of therole of intracellular calcium kinetics in myocardiummechanical performance (Solovyova et al., 2002). Inaddition to equating Ca binding by TnC and otherintracellular ligands, that version of the calcium-hand-ling block also included a description (by means ofappropriate equations) of Ca influx, Ca release from theterminal cisterns (TC), Ca flow from LSR to TC, itsuptake by the LSR, and calcium-calsequestrin bindingwithin TC. In other words, that version provided amathematical description of the complete cycle ofintracellular calcium circulation. However, our focusnow is on the role of the rheological scheme in themechanical behavior of the active heart muscle. Toachieve this purpose we significantly modified themechanical block of the model, and therefore found itad rem to neglect some excessive details at this stage ofthe study. In particular, we use a reduced variant of thedescription of cytosolic calcium kinetics, which hasalso been justified earlier within the framework ofsome previous version of our model (Katsnelson andMarkhasin, 1996). In addition to Eqs. (B.7) and (B.8)this reduced block consists of only one more differentialequation that expresses a change in the time courseof free cytosolic calcium concentration CaC (i.e. inthe calcium transient) and may be written down asfollows:

dCaC

dt¼

bc � t � ð1� expð�ac � t2ÞÞ � expð�ac � t2Þ for totd

�dA=dt � dB=dt � rCa � expð�qCa � CaCÞ � CaC for tXtd :

�

ðB:9Þ

Time variable t appears in an explicit form in thelatter equation.The above Eqs (B.1)–(B.9) altogether form a full set

of differential equations, describing mechanical beha-vior of the virtual heart muscle in isotonic and isometriccontraction–relaxation cycles.

Appendix C. Basic parameter values

a1=19 mm�1,b1=2.9mN,a2=14.6 mm�1,b2=0.01mN,a3=48 mm�1,b3=0.1mN,kA=2.8,bon=2.6ms�1,boff=0.182ms�1,Btot=0.4,bc=0.03,

ac=0.005ms�2,td=33ms,rCa=0.65,qCa=50,m=3,A1/2=0.6,apk=16 mm�1ms,

bpk=15mN,

apm=16 mm�1ms,

bpm=15mN,

aSk=39 mm�1ms,

bSk=80mN,

aSm=46 mm�1ms,

bSm=60mN,

a=0.25,vmax=0.0056 mmms�1,dh=0.5,aP=4,aPG=1,w=0.705,w0=3,m0=0.9,l=960mN.

All concentration values in the above list are given infractions of [TnC] in the cardiomyocyte (where the latteris 7 10�5M). Therefore, both of these concentrationsand respective coefficients (e.g. kA) are dimensionless,and the corresponding kinetic constants are measured inms�1. Some other coefficients (e.g. a and dh) are alsodimensionless as they pertain to normalized depen-dences P�(v) and G�(v).The following considerations should be given to

explain the above choice of the parameter values. Themodel includes three main modules: a rheologicaldescription, a description of cross-bridge kinetics,and that of calcium activation of the thin filaments.Each model pertains to the respective class of physio-logical processes with its own parameters. In a realheart muscle, all the parameter may vary within someranges of values depending on both the particularityof the muscle and specific conditions of musclecontraction. This circumstance leads to a large numberof parameters in the model, which cannot be changedonce and for all with determined particular numbers,because they have to vary in the model (withinphysiologically reasonable ranges) depending on eachspecial case.Nevertheless, the choice of the parameter values was

not arbitrary; on the contrary, it was based on data ofphysiological and biochemical experiments. All the basicparameter magnitudes were chosen as average ones fromthe experimentally observed ranges. There were tworeasons for such a choice. First, values for anyparameter in real muscle samples vary in a wide range.In these conditions, averages seem to be the most

ARTICLE IN PRESSL.B. Katsnelson et al. / Journal of Theoretical Biology 230 (2004) 385–405404

representative parameter values for heart muscle on thewhole. Second, the complex character of our modelmade us use data from several experimental workswhich dealt with different experimental samples. There-fore we could not attribute a complete list of parametersto a particular muscle. Moreover, some data (e.g. timeconstants of calcium-troponin association/dissociation)were obtained in solution rather than in intact muscles.Thus, accumulating all the data in the framework of ourmathematical model we had to hang upon averagemagnitudes.The justification of the choice has been provided

earlier (Katsnelson et al., 2000) for all the parametersexcept those dealing with muscle viscosity. Thereforehere we are justifying only the latter ones.The common way to justify parameter values is to

check their validity in a series of numerical experimentsindependent of the task where the model is meant to beused. Specifically in our case, it was very reasonable tocheck the values of damper constants (ap

k, bpk, ap

m, bpm, aS

k,bSk, aS

m, bSm) by simulating the passive mechanical

properties. The passive state of the virtual muscle isdone with the help of zero-filling parameters responsiblefor Ca release and uptake in the model. All theother parameters of the model in these tests havethe same basic values presented above. Fig. 11demonstrates two numerical tests of the virtual musclein passive state.Panel I of Fig. 11 shows the response of the passive

virtual muscle to a series of fast stretches. The durationof each stretch is about 10ms. After every stretch, thelength of the passive virtual muscle increases addition-ally by 5% of its slack length. The muscle remains at thisnew length till the next stretch, which increases thelength again. As a result of each stretch, resilience arisesand then gradually falls down (relaxes) to a newstationary value of the passive force corresponding tothe new length.Two bottom panels of Fig. 11 show the results of 1%

length deformation (stretch and subsequent retrace)during 100ms of the same passive virtual muscle.Panel II.1 reveals the time course of this change inlength. The respective phase trajectory (the length–forcediagram) is presented in Panel II.2. There is a specifichysteresis loop. It can be seen that the ascending limbA1 of the loop is practically linear, whereas thedescending limb A2 is essentially concave. Line B is asegment of the steady-state passive length-force relation-ship of this virtual muscle within the same range oflengths.Both of the above tests are in good agreement

with the data of real mechanical experiments withpassive myocardium tissue (Fung, 1967, 1972;Izakov et al., 2000; Markhasin et al., 1994), whichsuggests that the choice of the damper parameter valuesis correct.

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