pp. 1–14 (2017) Mechanical model of the left ventricle of the heart approximated by axisymmetric geometry F. A. Syomin∗ and A. K. Tsaturyan∗ Abstract — An axisymmetric model is suggested to simulate mechanical performance of the left vent- ricle of the heart. Cardiac muscle is treated as incompressible anisotropic material with active tension directed along muscle fibres. This tension depends on kinetic variables that characterize interaction of contractile proteins and regulation of muscle contraction by calcium ions. For numerical simulation of heartbeats the finite element method was implemented. The model reproduces well changes in vent- ricle geometry between systole and diastole, ejection fraction, pulse wave of ventricular and arterial pressure typical for normal human heart. The model also reproduces well the dependence of the stroke volume on end-diastolic and arterial pressures (the Frank–Starling law of the heart and Anrep effect). The results demonstrate that our model of cardiac muscle can be successfully applied to multiscale 3D simulation of the heart. Keywords: Heart, cardiac muscle, muscle contraction, mathematical model, finite elements. MSC 2010: 74L15, 92C10, 92C30, 74S05 Computer modelling of the heart mechanics is a fast developing field of computa- tional physiology. During the last two decades a number of electromechanical mod- els of the whole heart or its left ventricle has been developed [26]. Such multiscale models usually combine several models that describe electrical and chemical pro- cesses at the level of a single cell with mechanical properties of cardiac muscle tissue. Generally, a model of the heart consists of a model of ionic currents in the cardiomyocytes, model of myocardial mechanics, model of blood circulation (hae- modynamics model), and a geometrical approximation of the heart. In spite of the presence of 3D models with patient-specific geometry of hearts and very detailed description of ionic currents, those models do not provide accurate description of cardiac muscle mechanics. The problem is that fine detailed models of actin-myosin interaction that underlies development of active stress in cardiac muscle are spe- cified by systems of partial differential equations, and thus they are too complicated for numerical simulation. On the other hand, prevailed simple models that are set by the systems of a few ODEs do not reproduce some major properties of cardiac muscle. To overcome the problem stated above, we have developed a mechanical model of myocardium specified by a system of ODEs. The model is based on a kinetic ∗Institute of mechanics MSU, Mitchurinsky prosp. 1, Moscow 119192, Russia. E-mail: [email protected], [email protected] The work was supported by the RFBR (grants 15-04-02174 and 16-04-00693). 2 F. A. Syomin and A. K. Tsaturyan model of muscle contraction and its regulation [21, 22]. The model reproduces a series of different uniaxial experiments performed on striated muscles. Some of them are steady-state shortening and lengthening at full activation, steady-state force-calcium dependencies, responses of force or length to step-like changes in muscle length or load, isometric and isotonic contractions at full activation and with consideration of regulation processes. Later we have applied this model to the sim- ulation of contraction of the left ventricle approximated by a thick-walled cylin- der [23], describing blood circulation by a simple compartmental model (Windkes- sel model). This approbation of the model has shown satisfactory results. We were able to reproduce the time course of the major haemodynamical and geometrical values during a heartbeat of an average ’healthy’ heart and under conditions that are typical for hypertrophic and dilated cardiomyopathies. After improving our circula- tion model, we have also investigated numerically the dependence of ventricular per- formance (ejection fraction of the ventricle) on the ventricle preload (end-diastolic pressure) and afterload (peripheral resistance or arterial pressure) [24]. The simula- tion reproduces the Frank–Starling law of the heart, pressure-volume loops, and the Anrep effect. At the next step of 3D modelling of the heart with patient-specific geometry we approximated the ventricle shape by a thick-walled body of revolution more similar to real heart than a cylinder. Here we describe a finite element model of pumping function of the ventricle based on our model of cardiac muscle. 1. Statement of the problem 1.1. Material and geometry We approximated the ventricle by a thick-walled body of revolution with the shape close to semi-ellipsoid. To set up the geometry we used curvilinear coordinate sys- tem (g, y) introduced in [17]. Here g corresponds to the position of a point in the wall of the ventricle: g = 0 for inner points of the ventricle (subendocardium), and g = 1 for outer points of the ventricle (subepicardium). Coordinate y corresponds to the position of the point between base (y = 0) and apex (y = p=2) of the ventricle. These coordinates can be expressed in terms of cylindrical coordinates as follows r =(r + g (r − r ))(e cosy + (1 − e)(1 − siny)) in out in (1.1) z =(hin + g (hout − hin))(1 − siny) + (1 − g)(hout − hin) where rin and rout are the inner and outer radii at y = 0, hin and hout are the lengths of the ventricle axis between the ventricular base and the inner and outer axial point, respectively. Parameter e sets the curvature of the ventricle. The shape is conic for e = 0, and when e = 1 the shape is a body of revolution with y being inclination angle of a point. We have chosen e > 1 in order to approximate the region of tapering from the widest part of the ventricle to its region of fibrous valve. The myocardium was considered to be hyperelastic incompressible transversely- isotropic medium with active stress caused by mechanochemical processes. The Mechanical model of the left ventricle of the heart 3 constitutive equation in its general form looks like Fact + Ftit T = Tis − pE + · B: (1.2) Ls=Ls0 Here Ls0 is a length of unstrained sarcomere, Ls is a length of deformed sarcomere, T is Cauchy stress tensor, Tis is an isotropic part of passive stress tensor, p is a Lagrange multiplier, or pressure, E is a unit tensor, B is a tensor of fibres orienta- tion, which is equal to dyadic product of deformed unit vectors aligned with muscle fibres. The scalar Fact is active tension of cardiac muscle, and Ftit is a component of passive tension caused by non-linearly elastic titin fibres. Both tensions are caused by the forces aligned with fibres and applied at the cross-section of a fibre. Passive stress of hyperelastic material was expressed using strain energy function W, which depends on the first and second invariants I1 and I2 of the right Cauchy–Green de- formation tensor G: i j ¶W Tis = ¶ei j 1 Q(I1;I2) (1.3) W = cise 2 2 Q = cis · 0:25(I1 − 3) − 0:5(I2 − 2I1 + 3) : 1 2 Here ei j are the components of the Gauchy–Green strain tensor, cis and cis are the material parameters. The strain energy function was based on that suggested in [5]. The general forms of expressions for the first and the second invariants are as fol- lows: i 2 2 I1 (G) = E ••G = ∑Gi; I2 (G) = I1 (G) − I1 G =2 i where ‘••’ is a notation for double tensor contraction. Titin force Ftit was specified by so-called worm-like chain model [13], which is often used for description of stresses in long molecular chains, and was set by equation ! 6k Tr 0:25 0:5 · (L − L ) F = B m · − 0:25 + s s0 : (1.4) tit L 2 L p 1 − 0:5 · (Ls − Ls0) Lc c Here Lc is a contour length of titin, Lp is a persistent length of titin, rm is a number of myosin filaments per unit of cross-section area of unstrained muscle, kB is a Boltzmann constant, and T is an absolute temperature. Active force Fact is defined by a kinetic model of cardiac muscle [21, 22] based on the following concept. Muscle contraction is caused by relative sliding of two sets of protein filaments in an elementary contractile unit of muscle—sarcomere. These filaments are thick myosin filament and thin actin filament. Myosin heads protruding from the backbone of thick filaments are molecular motors, which pro- duce force during their interaction with actin molecules. During contraction these 4 F. A. Syomin and A. K. Tsaturyan myosin heads go through Lymn–Taylor cycle. According to the cycle a myosin head can be in detached from actin state (state 0), or it can attach to actin and form a cross-bridge. A cross-bridge can attach actin weakly (state 1), or strongly (state 2). In the latter case, it generates active force and/or causes displacement. Our model of contraction is based on the Lymn–Taylor cycle and considers muscle activation as well. In the absence of calcium ions, Ca2+, a complex of regulatory proteins covers myosin binding sites on actin and causes muscle relaxation. The binding of Ca2+ to regulatory protein troponin-C and formation of CaTnC (calcium-troponin C) complex suspends the inhibition and opens actin filaments for myosin binding. The formation and dissociation of the CaTnC complexes depends on a number of factors. The affinity of the complexes to Ca2+ ions increases with an increase in sar- comere length (length-dependent activation), a number of already formed CaTnC complexes, and a number of strongly bound cross-bridges [4]. These cooperative effects are also taken into consideration in our model. Main variables of our con- traction model are: 1. n is a probability of a myosin head to be attached to thin filament.