relaxation tests of the papillary muscle of laboratory animals (rabbit and rat) have ... MODELING ELASTIC PROPERTIES OF PAPILLARY MUSCLE AT REST.
ISSN 00063509, Biophysics, 2011, Vol. 56, No. 3, pp. 502–509. © Pleiades Publishing, Inc., 2011. Original Russian Text © A.V. Kobelev, L.T. Smoluk, O.N. Lookin, A.A. Balakin, Yu.L. Protsenko, 2011, published in Biofizika, 2011, Vol. 56, No. 3, pp. 534–542.
COMPLEX SYSTEMS BIOPHYSICS
Modeling of SteadyState and Relaxation Elastic Properties of the Papillary Muscle at Rest A. V. Kobelevb, L. T. Smoluka, O. N. Lookina, A. A. Balakina, and Yu. L. Protsenkoa a
Institute of Immunology and Physiology, Urals Branch, Russian Academy of Sciences, Yekaterinburg, 620049 Russia b Institute of Metal Physics, Urals Branch, Russian Academy of Sciences, Yekaterinburg, 620041 Russia Received May 4, 2008; in final form, September 3, 2010
Abstract—The biomechanical modeling of a papillary muscle preparation as an adequate object for studying the properties of the myocardial tissue under uniaxial stretching has been performed. The steadystate and relaxation tests of the papillary muscle of laboratory animals (rabbit and rat) have been conducted in normal conditions and after the maceration of intracellular structures with high ionic strength solution. It has been shown that the main contribution to the viscoelastic properties in the initial range of physiological deforma tions is made by the connective tissue skeleton, whereas under large physiological deformations, by intracel lular structures. Keywords: papillary muscle, fascicule, stress–strain, stress relaxation, modeling DOI: 10.1134/S0006350911030122
INTRODUCTION The static «stress–strain» relation with a low grade in the range of small physiological values of strain and a sharp rise upon further stretching has a similar form at different levels of organization of various biological tissues, in particular in myocardial tissue [1, 2]. In consequence of the modular design of morphological structures of different scale this gives grounds for sug gesting the existence of general principles of construc tion and nature of the viscoelastic properties on the level of an elementary functional cell of the tissue, which in the case of myocardium is taken to be the fas cicule. Besides that, it is well known that a dynamic response to strain at a rate exceeding one specimen length per second is attended with relaxation of mus cle stress on all levels of tissue organization—cellular, tissue, and organ. It is also established that the viscos ity of myocardial tissue depends on the current length of the specimen [3]. This fact is of principal signifi cance for the mechanics of active muscle, inasmuch as in the course of contraction the length of a contractile element changes. In some kinds of heart pathology the viscoelastic characteristics of myocardial tissue change in a most substantial way [4, 5]. The physical «sources» of myocardial viscosity may manifest themselves in a complex of structures of a varied level of tissue organization. This is the intrinsic viscosity of biopolymers, friction of actin filaments Editor’s Note: I certify that this is a closest equivalent of the orig inal publication with all its factual statements and terminology, phrasing and style including concision and lucidity. A.G.
during their interaction with bridges and sliding rela tive to myosin filaments in the myoplasm, and also friction of connectivetissue elements in the intercel lular fluid, sliding of layers of heart wall myocardium relative to each other. Apart of that, a source of viscos ity may be served by filtration of fluid in the intercellu lar space [6]. Therewith it is necessary to take into account the linear dimensions and the sum area of the sliding surface of the structures that are subjected to strain both in cardiomyocytes and in the extracellular space. A rheological description of muscle tissue in normal conditions must include a viscous component, and it must correspond to the data of biomechanical experiments and the morphological structure of the tissue. Theoretical description of the viscoelastic properties of materials upon longitudinal strain till the present time is based on onedimensional models of Maxwell and Kelvin–Voigt in which the elastic and the viscous elements are connected in series or in parallel, and on their multielement generalizations [7]. To take into account the nonlinearity, less substantiated and less vivid approaches are applied to modeling the viscoelas tic properties of the myocardium [8, 9]. However, in the framework of onedimensional models it is impossible without special assumptions to describe such nonlinear effects as a nonelliptical shape of the hysteresis loop and the dependence of the relaxation time on the step num ber during jumpwise deformation [10]. Most recently, in connection with development of twodimensional (2D) models [11], one becomes able to understand the origin of the characteristic peculiar ities in the behavior of living soft tissues, such as the
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nonlinearity of the static stress–strain curve, the «banana» shape of the hysteresis loop, the nonexpo nentiality of the stress relaxation curves, etc. [12–14]. Such models in a certain degree appear phenomeno logical and reproduce well the experimental data, nonetheless they take into account the peculiarities of tissue morphology. The primary element of such a 2D model (a spring or damper) reflects the properties of the totally of fiber with certain spatial orientation and corresponding function conditioning the intracellular rigidity (titin, microtubules), viscosity («sliding» pro tein structures, including the actomyosin complex), and properties of the extracellular matrix (collagen, desmin, vimentin). The problem of separating the contributions of the connectivetissue complex and intracellular structures into the biomechanical properties of the muscle in pas sive state [17] has recently received attention of researchers in connection with revealing special nonlin ear properties of the giant intracellular protein connec tin (titin) [18, 19]. Titin on the molecular level makes a substantial contribution not only into the elastic but also into the viscous properties of the tissue [20]. Upon various influences on the structure of a tissue preparation there appears a possibility of isolating the contributions from intracellular and intercellular structures of various hierarchy. Such influences are diverse and can be executed on the level from fine geneengineered, such as knockout [8], whereupon in the tissue there is reduction of synthesis of definite proteins (in particular, collagen), to alteration of the structure of the network of microtubules [15] and to complete removal of intracellular structures (macera tion) with the aid of preparations destroying lipid membranes. Maceration permits removing all intrac ellular structures, leaving only the extracellular con nectivetissue carcass consisting of collagen fibers. Collagen, the main protein of the connectivetissue carcass, is known to be insoluble in salt solutions, in weak solutions of acids and alkalis. Maceration rela tively weakly influences the structure of the connec tivetissue matrix, inasmuch as collagen fibers appear stable enough to such influence, and such means is widely used in microscopic examination of connective tissue freed of intracellular structures [16]. The aim of the given work presents as biomechani cal modeling of papillary muscle upon uniaxial stretchings for elucidation of specific mechanisms of increasing stiffness of soft tissues and relaxation time with the growth of strain. We conducted static, quasis tatic and relaxation tests on papillary muscle prepara tions from laboratory animals (rabbit and rat) in the norm and in conditions when intracellular structures are removed with the aid of an alkali solution. Upon comparison with the results of modeling this has allowed making a conclusion about redistribution of the contributions of intracellular structures and con BIOPHYSICS
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nectivetissue complex into the viscoelastic properties at various levels of strain. EXPERIMENT After the isolation procedure a native preparation of myocardium for preventing its damage was placed into a physiological solution that contained (mM): NaCl 118.5; KCl 4.2; MgSO4 · 7H2O 1.2; NaHCO3 14.5; KH2PO4 1.2; CaCl2 2.5; glucose 11.1. A stable level of pH (7.35) was maintained with a phosphate carbonate buffer with bubbling carbogen (95% O2 + 5% CO2) at a temperature of 25°C. If right after fixing the preparation the active force developed by it did not exceed 0.5 mN, then the preparation was deemed damaged and further experiments with its use were not conducted. Whereas if the force developed by a prepa ration of rat myocardium constituted about 1 mN, and the force developed by a preparation of rabbit myocar dium about 4 mN, then after application of additional ligatures the preparation isometrically contracted in the course of 30–60 min (the period of «workingin» of the preparation). In the course of this period under continuous feeding with physiological solution at a temperature of 30°C, oxygen perfusion and switched on stimulation, the active force developed by the prep aration increased two–three times, while the passive tension dropped. At the expiration of the period of «workingin» of the preparation, on condition of res toration of its contractile properties, we conducted experiments in accordance with the protocol. In the capacity of testing mechanical influences we used periodic deformations of sinusoidal, rectangular, triangular «sawtooth» shapes with varying frequency (period) and different deformation rate. Quasistatic test trials consisted in setting a cyclic longitudinal strain with a period of 1–10 s of sawtooth and sinuso idal shapes with an amplitude of 200 μm at an initial strain level of 0.8–0.9 Lmax. Relaxation test trials con sisted in setting a series of stepwise stretchings and shortenings of the muscle with a step of 100 μm in the range from L0 to Lmax (L0 – the smallest length at which further stretching causes an insignificant incre ment of strain). The profile of load was chosen sinuso idal for hysteresis at a deformation rate usual for car diophysiology (from 0.1 to 10 cycles in 1 s at maximal strain 0.3), and stepped (pulse front from 10 to 100 ms) for measurements of relaxation and steady state. The strain change was set in the limits of 10% of working length with the aim of revealing nonlinearity. The saw tooth change of papillary muscle length was set with an amplitude from 0.05 to 0.3 Lmax at a rate from 0.001 to 0.1 muscle length in 1 s. The morphological structure of macerated prepa rations was subjected to histochemical analysis with the aid of section staining according to Van Gieson (for the presence of collagen fibers) and according to Weigert (for the presence of elastic fibers). At that on
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sections stained according to Van Gieson, cardiomyo cytes, which should stain with greenyellow color, were not detected. After maceration on section we dis closed only structures of endomysium, stained crim son or pink colors. Upon staining according to Weigert, elastic fibers stain to darkbrown color, but on sections they were not detected. On sections of prepa rations after soaking we disclosed structures of endomysium or of the connectivetissue carcass in the form of collagen fibers without pronounced structural changes; we observed single fibroblasts and elastic fibers, vessels of microcirculation [21]. We do not have a histochemical confirmation of a different degree of cardiomyocyte washout. A sugges tion can be made about a different degree of swelling of collagen fibers, however it will not be unequivocal because a suggestion about a different degree of cardi omyocyte washout is also possible. We suppose the lat ter variant, inasmuch as the sum volume of cardiomy ocytes in the myocardium significantly exceeds the volume of the connectivetissue carcass. After soaking at the sag length it turned out that the initial length of the preparation had substantially decreased, and therefore for comparison of the char acteristics of normal and macerated preparations we used the deformation determined according to Cauchy. Every subsequent deformation was set after relaxation of passive tension and establishment of a new steady state. First such tests were conducted in physiological solution. Then the preparations without removing them from the rods of the registering equip ment were soaked in 10% aqueous solution of NaOH for 12 h in accordance with method [16]. After soaking the preparation was again placed into normal physio logical solution and trials were repeated in accordance with the same protocol. For control of soaking quality
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Fig. 2. Hysteresis curves (unit loop) in coordinates «force– extension» of papillary muscle in normal solution before maceration (a), L0 = 3.6 mm, Lmax = 4.6 mm, Lw = 3.9 mm, ΔL = 0.22 mm, and after maceration (b), L0 = 2.8 mm, Lmax = 3.5 mm, Lw= 3.2 mm, ΔL = 0.17 mm, under sawtooth stretching with a period of 10 s at length Lw ~ 0.8 Lmax.
after all trials we prepared longitudinal and transverse histological sections of the preparations. For evaluation of the contribution of cardiomyo cytes into the elastic properties of the entire papillary muscle preparation we compared the steadystate «stress–strain» dependences of specimens in normal state and the same specimens after soaking in alkaline solution. It is shown that at relative strain values greater than 0.15 soaking leads to a sharp drop of the differential modulus of elasticity of the preparation. The «stress–strain» dependence in the soaked prepa ration becomes weakly linear (Fig. 1). In Fig. 2 we present hysteresis curves, respectively before (a) and after (b) maceration, demonstrating the substantial decrease in the viscosity parameter, to which the area of the hysteresis loop is proportional (at equal deformation rate). A record of experimental curves of the time course of force upon assigned stepped longitudinal strain with a step Δε = 0.06 is presented in Fig. 3 (gray curves) in BIOPHYSICS
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Fig. 3. An experimental record of the responses of rabbit papillary muscle force to a series of six consecutively assigned stepped jumps of deformation (gray curves, shifted left) in normal solution (a) and after maceration (b). The solid curve of stress relaxation (a) obtained in the full model, (b) after removal of the block simulating intra cellular structures.
normal state (a) and after maceration (b). Omitted are parts of experimental curves corresponding to a long pause more than 1 min as the steadystate value of force is being achieved. The characteristic peculiarities of relaxation curves consist, first, in the presence of a sharp burst («shock stiffness») at the moment coinciding with the steep front of a stepwise jump of length (deformation not less than 0.5 in the time of 1 ms). This effect represents a consequence of a «frozen state» of the preparation at such large deformation rates. Another experimentally disclosed peculiarity presents as the relaxation decay of force with an effective time on the order of 1–3 s, which is characteristic of media with a very large New tonian viscosity (Fig. 3). One more important pecu liarity of relaxation curves presents as nonlinearity both of the steadystate level of force as dependent on BIOPHYSICS
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Fig. 4. Hysteresis curves obtained in the full model (a) and after removal of the block simulating intracellular struc tures (b), in comparison with experimental data (squares and circles) in normal solution before (a) and after macer ation (b).
elongation and the dependence of the relaxation time on the current strain value (step number). The hysteresis curves (Fig. 4) obtained at signifi cant longitudinal strain (experimental points) and the model curves (see section Modeling)also demonstrate a consequence of the effect of stiffness increase (Fig. 1), manifesting itself as a nonelliptical «banana like» shape of the loop. It should be noted that the effect of stiffness increase, i.e. increase in the slope of the static «force– length» curve, and the effect of relaxation time growth are inherent in a large number of various biological soft tissues in normal state. Removal of intracellular structure led on the whole to tissue softening and almost complete disappearance of nonlinearity, and also to a substantial decrease in the relaxation time determined by the viscosity and stiffness of the tissue.
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Fig. 5. Chain of three serially connected blocks simulating the connectivetissue complex of the fascicule, with internally implanted blocks of intracellular structures.
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Fig. 6. Model of intracellular structures reflecting the viscoelastic properties of titin, microtubules, intermediate filaments, con sisting of a tenfold repeated rhombic block.
MODELING On the intermediate level of muscle tissue hierar chy (fascicule) the totality of elastic and viscous ele ments of the connectivetissue complex can be described in the framework of a parallelogram model [12] with longitudinal, oblique, and transverse ele ments. The model consists of blocks representing the matrix of the connectivetissue complex, containing oblique and longitudinal Hooke’s springs connected in parallel to Newton’s dampers (Kelvin’s element). At a nearly equal relationship of the longitudinal and transverse sizes of one model block in the initial state such a block sufficiently well describes the static «stress–strain» characteristic of papillary muscle obtained in experiment [12]. Proceeding from the real relationship of the longitudinal and transverse sizes of papillary muscle preparations, we chose a model of connectivetissue complex of three concatenated fas cicule blocks (Fig. 5). In consequence of bidimension ality such a model permits ensuring greater longitudi nal compliance of the whole construct, despite that its constituent oblique and transverse elements simulat ing the collagen fibers of connective tissue appear suf ficiently stiff. For obtaining a nonlinear longitudinal function of response, the values of elastic moduli of transverse elements, reflecting weak connections and interactions in the fiber meshwork, were chosen sev eral orders of magnitude smaller than those for oblique and longitudinal elements. For description of the intracellular structural con stituents of the cytoskeleton, such as titin, microtu bules and intermediate filaments, we used, similar to
the preceding one, a rhombic model [12, 13] with an oblique Kelvin’s element. Each of the three blocks of fascicules contains a tencomponent chain of rhombi (Fig. 6). As it is known, in contrast to the majority of one dimensional protein macromolecules (for example collagen), solitary molecules of titin possess essentially nonlinear elastic properties. Usually the viscoelastic properties of their glomerular parts (IG fragments) are described in the framework of a simple chain model consisting of serially connected rigid links [22]. The nature of the force arising in this model, nonlinearly depending on glomerule stretching, bears a statistical character. Such a model, apparently, is not quite suit able for modeling elastic properties, inasmuch as it does not contain any parameters of interaction. Here we do not pose a task of reflecting in a model the real structure of titin molecules and as a first step we use the mentioned rhombic model for phenomeno logical description of elastic and viscous properties of the entire cytoskeleton, possessing quite a compli cated spatial organization. In consequence of the effect of «toughening» under conditions of unfolding, the response of such a model to longitudinal strain appears nonlinear [23] and agrees well with the data of experiment, which is evidenced by the closeness in shape of the theoretical dependence to the experimen tal one obtained on separate titin molecules [24]. Regretfully, in consequence of the extreme simplicity of the model in its framework one cannot manage to separately take into account the contribution of titin itself and make a conclusion about the source of the observed nonlinearity of the static «force–extension» BIOPHYSICS
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curve in consequence of either the intrinsic nonlinear ity of titin or the peculiarities of the structure of the spatial network of intracellular filaments. Our aim was calculation of the steadystate «stress–strain» dependence, hysteresis curves and curves of stress relaxation upon stepped deformation. The contribution from the cytoskeleton block was taken into account or ignored depending on whether the preparation is in normal state or after maceration. For description of the process of stress relaxation, which in biological soft tissues bears an essentially nonlinear character, we used equations defining the main model suggestions. The calculation procedure consists in solving a set of differential equations that follows from conditions of quasistatic equilibrium at the point of element con nection. It is supposed that along the longitudinal axis of the model an external uniform deformation is applied in the form of a specified stepped time func tion. The resultant response to strain in the form of full force applied from the opposite ends of the construct is formed of summands pertaining to the cytoskeleton block and the connectivetissue complex block. The full longitudinal strain of the fascicule block and the cytoskeleton block is the same (see Fig. 5). Upon setting the external longitudinal deformation ε = ε(t) = (L – L0)/L0, where L0 is initial length (L0 = Ml0 = MNlc0, where M and N – number of repeats in the fascicule block and in the cytoskeleton block respectively, M = 3, N = 10), the strain of the oblique element of the cytoskeleton block εc1(t) is described by a nonlinear differential equation: k c1 ε· c1 = – ε η c1 c1 γ c – λ c3 k c3 λ c1 , – ( 1 + ε c1 ) 1 + η c1 λ c3 2 2 2 4λ c1 ( 1 + ε c1 ) – ( 1 + ε )
(2)
The set of differential equations for oblique ε1 and longitudinal ε2 strains of the fascicule block has the form: k 1 2λ 1 ( 1 + ε 1 ) ε· 1 = – 2 ε 1 + Ψ ( ε 1, ε 2 ) , λ3 μ3 η1 μ1 k 1 + ε – λ ( 1 + ε2 ) Ψ ( ε 1, ε 2 ) , ε· 2 = – 2 ε 2 + 2 η2 ( λ 3 + γ )μ 3
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The force corresponding to such strain in accor dance with the geometry of the fascicule block equals 2η 1 · ⎞ 1 + ε – λ 2 ( 1 + ε 2 ) 2 . f = ⎛ ε 1 + ε ⎝ μ1 k 2 1⎠ 2λ 1 ( 1 + ε 1 )
(4)
Here λci = λci0/λc0 and λi = li/l0 are factors con nected with the lengths of elements i of cytoskeleton and fascicule; μci = kc2/kci and μi = k2/ki – same for elasticity parameters; kci, ki, ηci, ηi – normalized stiff ness and viscosity coefficients respectively; γ = h/l0 and γc = hc/lc0 – relative dimensions of stiff transverse muffs h in models (see Figs. 5 and 6). In calculation of quasistatic hysteresis curves we supposed that initially assigned is the transverse strain εc3(t) of the cytoskeleton block in the form of a simple periodic time function: εc3(t) = εc30[1 – cos(2πt/T)]/2. (5) Taking in this case the transverse element to be vis cous, for the strain of an oblique element of the cytoskeleton block we obtain: –1
k c1 [ λ c3 ( 1 + ε c3 ( t ) ) – γ c ] ⎫ ⎧ ⎬ , ε c1 ( t ) = – ⎨ 1 + 2λ c1 [ k c3 ε c3 ( t ) + λ c3 η c3 ε c3 ( t ) ] ⎭ ⎩
(6)
while the normalized stretching force for the cytoskel eton block is: ε c1 ( t ) · [ 1 + ε c ( t ) ]. f c ( t ) = ε c ( t ) + μ c1 λ c1 [ 1 + ε c1 ( t ) ]
(7)
Inasmuch as in this case we set the full longitudinal strain of the fascicule block, the corresponding strains of all elements and the longitudinal normalized force for this block are defined by formulae (3) and (4).
(1)
while the corresponding normalized stretching force fc = fc(t) is: ε c1 η c1 · ⎞ 2 ( 1 + ε ) ⎛ f c = + ε . λ c1 ( 1 + ε ) ⎝ μ c1 μ c2 c1⎠
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DISCUSSION The geometric and elastic parameters of the model were chosen proceeding from the condition of mini mality of the deviation of curves from experimental data. In translating spring stiffness coefficients into Young’s modulus values Ei = 4ki/πdi2 we accepted the supposition that the effective diameter di of the ele ment constitutes onetenth of its length. The values of force are easy to translate into stresses, taking into account the diameters h and hc of a stiff girder to which an effort is applied in modeling the response to stretch of the fascicule block and the cytoskeleton block. The viscosity values were estimated proceeding from the area of the element side surface, determined by its length and diameter. At that the main parameters of elasticity and viscosity of model elements for calcula tion of relaxation denoted in Fig. 5 for the fascicule block and in Fig. 6 for the cytoskeleton block proved equal: E1 = 1.78, E2 = 0.38, E3 = 0.21, Ec1 = 1.48, Ec2 = 0.004, Ec3 = 0.29 (MPa), η1 = 0.011, η2 = 0.016, ηc1 = 5.5 (MPa s).
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In calculating the stress relaxation we chose a vari ant of the model taking into account the viscosity only for longitudinal and oblique elements of the fascicule block and the oblique element of the cytoskeleton block (see Figs. 5 and 6). The viscosity corresponding to all the other elastic elements of the model was taken to be negligibly small. Such a choice ensures quite good qualitative agreement with the experimentally observed effects of relaxation time growth with the step number upon stepped stretching both in normal state and after maceration (Fig. 3a,b). One should also note the qualitative coincidence of nonlinear growth of shock stiffness in the model and in the experiment. Qualitative and in some cases quantitative agreement with the results of analysis of static curves (Fig. 1) has been achieved also for the dependence of the steady state level of stress after relaxation at a certain step on the strain. The good qualitative agreement of theoretical and experimental relaxation curves (Fig. 3) and hysteresis curves (Fig. 4) allows asserting that the definitive con tribution into the static properties of papillary muscle is made by intracellular structures, among them titin. In particular, in the framework of the proposed model the nonlinearity of the static curve with the effect of increasing stiffness with growing strain and the «banana» shape of the hysteresis curves are deter mined by the block of intracellular structures. This mechanism is a consequence of the architecture of the model, in which the stiffer oblique elements of the block of intracellular structures are engaged in propor tion with a decrease of their opening angle and after compression of softer transverse elements (effect of «unfolding»). Let us note that a similar effect may take place also in the used model of the connectivetissue complex; however, at the chosen values of model parameters its attainment requires higher levels of strain exceeding the physiological limit. A conse quence of such an architecture of the model of con nectivetissue complex presents as a significantly greater compliance of such a construct under longitu dinal stretchings than it follows from the quite high values of Young’s modulus of collagen fibers as its main structural material. The values of elastic param eters for stiff elements are an order of magnitude greater than for soft ones and substantially smaller in magnitude than the elastic parameters of collagen (E ~ 1.0 GPa). Intermediate values of Young’s modulus of the longitudinal element of the fascicule block corre spond to the stiffness of elastin (E ~ 0.6 MPa). Analysis of dynamic properties, and in the first place of the curves of stress relaxation upon stepped stretching, provides additional principal possibilities for modeling of viscoelastic properties connected with manifestations of viscous friction and dissipation in a passive muscle. It is known that it is impossible to describe stress relaxation under constant strain in the framework of a onedimensional construct using a Kelvin–Voigt block with parallel connection of vis
cous and elastic elements, for this purpose they use a Maxwell block with serial connection of these ele ments [5]. It is rather difficult to find a physical analog to a serial Maxwell’s damper and its realization in bio logical structures, for example in actomyosin and titin complexes, in the structure of which there are long molecular fibers immersed into the cytoplasm. The presented twodimensional architecture of the model permits avoiding this difficulty and using the tradi tional Kelvin’s blocks for description of stress relax ation at the expense of an additional degree of freedom provided by the topology of the system. For obtaining the experimentally observed values of the relaxation time in the model it is necessary to assign very large values of viscosity parameters, which by several orders of magnitude exceed the viscosity of biological fluids. Apparently, formal consideration of the effects of viscous friction, such as stress relaxation, in the framework of Newton’s model is not quite ade quate, although it leads to correct qualitative depen dences. The question of the nature of viscous friction in biological tissues, providing significant (up to tens of seconds) relaxation times, remains, in our opinion, still open, inasmuch as, most likely, it is connected with structural changes in proteins after stretching and with the kinetics of biochemical reactions in passive muscle. CONCLUSIONS The presented model permits one without using a dependence of elastic and viscous parameters on the strain magnitude to reproduce the whole complex of nonlinear mechanical properties of a passive papillary muscle both under quasistatic strain and in the process of stress relaxation upon stepped stretching. Moreover, in the framework of one model we describe the contri butions of the connectivetissue carcass and intracel lular structures, manifesting themselves in the differ ence of viscoelastic properties before and after macer ation. The qualitative agreement of model and experimental data in this case allows making a conclu sion about the prevalent contribution of titin and other intracellular structures to the emergence of nonlinear ity of the viscoelastic properties of tissue in the range of physiological strains. The approach used for model ing the viscoelastic properties of the basic element of the myocardium—the fascicule—is consistent also with the morphological design of the tissue, allowing one to subsequently proceed to modeling of objects of a higher step in the hierarchy on the basis of the pro posed mechanism of nonlinearity. ACKNOWLEDGMENTS The work was supported by the Russian Founda tion for Basic Research (040496109). BIOPHYSICS
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