F. ludicello," M.W. Collins,* F.S. Henry/ J.C. Jarvis,"
A. Shortland,' R. Black," S. Salmons"
"Thermo-Fluids Engineering Research Centre, City University,
London, UK^Department of Human Anatomy and Cell Biology, University of
Liverpool, Liverpool, UK'Department of Clinical Engineering, University of Liverpool,
Liverpool, UK
Abstract
This paper is a review of numerical models of flow in large arterial vessels.The main objective of the paper is to formulate a comprehensive model ofsuch arterial flows, including the wall motion, with special attention paid tothe modelling of ventricles. In the first and main part, the fluid-dynamicalaspects of the problem will be reviewed and discussed. Blood flow modellingapplications in large vessels are reviewed, with emphasis on the modelling ofnatural and artificial ventricles (artificial hearts and cardiac assist devices).Simplified geometries of ventricles will be analyzed. An initial validationof the numerical solution for the problem of decoupled flow with a movingboundary will be presented for a simplified cylindrical ventricular model.Then rigid/compliant models will be discussed where the walls move in aprescribed sinusoidal manner. The mechanics of the wall will be the focusof the concluding section. Ventricle models reported in the literature, andsolid mechanics equations used in such models are briefly described. Theformulation of the muscle mechanics equations is reviewed.
1 Introduction
Due to the complex nature of the ventricular geometry and flow, and com-pliant nature of the wall, closed-form mathematical solutions are not avail-able and numerical solutions are the only practical way forward. Power-ful Computational Fluid Dynamics (CFD) codes developed in the last few
years allow the modelling of realistic geometries with moving walls. Thisis an important feature of physiological flows, which are mostly driven bythe periodic motion of the vessel wall. The latest computational dynamicstechniques allow a comprehensive analysis of all fluid- and haemo-dynamicparameters involved, such as shear stress at the wall or in the fluid, residencetimes and particle pathlines. Numerical simulation of blood flow througharterial systems, in particular, through prosthetic valves, bends, arterial bi-furcations, stenoses, grafts, and natural and artificial ventricles has been ofspecial interest in recent years. These have been extensively studied in thecontext of the work on blood flow modelling developed in the BiomechanicalResearch Group at City University over the last decade. Analysis of bloodflow in artificial hearts and assist devices has become recently an area ofactive interest.
Nowadays, much of the interest in blood flow modelling is focused on thefluid-structure interactions occurring at the interface between blood and ves-sel wall. In order to obtain a comprehensive solid and fluid-dynamic modelof blood vessels and the ventricle it is necessary to couple fluid-dynamicsequations with solid mechanics equations (for the wall) and (eventually) themuscle mechanics equations. In the initial numerical studies to be discussedbelow, the wall motion has been prescribed; however, the ultimate goal ofthe research is to predict wall position by solving, in a coupled manner, theequations for both fluid and solid.
In this paper blood flow modelling is discussed, concentrating on flowin the large vessels. Applications to arterial wave propagation, bifurca-tions, stenoses and grafts are reviewed in section 2. Particular attentionwill be paid to natural and artificial ventricles, that is artificial hearts andassist devices. Numerical techniques are briefly described in section 3, anda validation of the predicted flows for simplified ventricular geometries ispresented in section 4. The paper concludes with an overview of the futurework on coupled solid-fluid approaches applied to the ventricle includingconsideration of muscle behaviour.
2 Modelling of the flow in arterial vessels
The pressure increase generated by the left ventricle results in blood flowalong the aorta and throughout the circulatory system. At any given pointin the arterial tree pressure and velocity change periodically, and the flowis pulsatile. The complete cardiovascular system is far too complex to beamenable to a comprehensive analytical treatment, and most of analyseshave focused on a specific aspect of interest.
The rheological properties of blood play an important role in the physio-logy of blood circulation. Blood is non-Newtonian, its viscosity varying withshear rate. However, in large vessels, blood can reasonably be considered asa homogeneous, incompressible, Newtonian fluid, although non-Newtonian
effects can become significant in pulsatile flow [1]. Nevertheless, it has beenshown that basic features of the flow are not affected [2,3] and it may beconcluded that non-Newtonian effects can be considered as 'second-order'compared with geometry and wall distensibility effects.
An extensive account of the early historical development of ideas con-cerning the circulation is given by Fishman and Richards [4]. Womersley [5]and McDonald [6] developed an analytical linearized solution to calculatethe pulsewave propagation for an oscillatory flow of a Newtonian fluid in acircular tube. In the early analytical studies, linear and non-linear theorieswere extensively used in the mathematical analysis of wave propagation inarterial blood flow. In the last three decades, progress has been made indescribing blood flow at junctions, through stenoses, in bends and in ca-pillary blood vessels. More recently (in the last two decades), extensivetreatments of blood flow problems have been made by solving numericallythe full Navier-Stokes equations.
2.1 Arterial flow
The first derivation of the velocity of propagation of the pulsewave in bloodflow was given by Young [7]. Womersley [5] found the mathematical solutionfor an oscillatory laminar flow in a compliant tube when the pressure gradi-ent varied periodically in the form of a sine wave, and for a homogeneousNewtonian fluid. Greenfield and Fry [8] derived a pressure-flow relation thatis simpler than the more general form of the Womersley theory by linearisingthe one dimensional flow equations, the resulting equations being analogousto those for electric transmission lines. Anliker et al. [9] made a significantdevelopment of the method of characteristics applied to the Navier-Stokesand continuity equations, where the nonlinear terms of the equations areretained. The effect of the geometric complexity of the arterial system hasbeen explored recently, a principal effect of the geometric changes being thegeneration of partial reflections of the pulse wave. Comprehensive theoriesof wave propagation in the circulatory system have been developed in thepast three decades based on regarding the latter as a system of elastic tubes.
2.2 Flow at arterial bifurcations
The study of blood flow through arterial bifurcations has been driven to alarge extent by the fact that these areas of the cardiovascular system ap-pear to be preferred sites for atheroma. Modelling of flow in bifurcationsinvolves treating the complex three-dimensional geometry of the bifurca-tion, the pulsatile nature of the flow, the distensibility of the arterial walland the non-Newtonian characteristics of the blood. A review of the devel-opment of numerical models for the analysis of flow in arterial bifurcationsis given by Xu et al. [10]. Initial studies; i.e., in the 1970s, were confined totwo-dimensions. Three dimensional computer simulations are expected to
clarify the role played by haemodynamics in the formation of atheromatousplaques. Perktold and co-workers (see for example [11]), among others, havestudied three-dimensional, pulsatile flow in realistic models of the humancarotid artery bifurcation.
2.3 Flow through stenoses
Knowledge of local flow behaviour in the vicinity of an arterial stenosis hasbeen gained from both theoretical and experimental research. Several nu-merical treatments have considered symmetric constrictions in rigid axisym-metric conduits, with both steady [12], and pulsatile flow [13]. However,arterial stenoses are unlikely to be axisymmetric and a number of research-ers have considered more physiologically realistic models. Ojha et al. [14]found experimentally that flow downstream of an asymmetric stenosis in acircular tube is complex and three-dimensional. This finding was confirmedby the numerical study performed by Henry and Collins [15] of the flowthrough axisymmetric and asymmetric stenoses.
2.4 Flow through bypass graft anastomoses
Bypass grafts, used to alleviate chronic ischaemic heart disease and restoreblood supply to the distal areas of stenosed arteries, often fail due to theformation of atheromatous plaques in the anostomosis region. Numericalsimulations of flows in proximal and distal anastomosis models have aided inthe understanding of the important factors governing local flow dynamics,which may play a role in the formation of atheromatous lesions. Analysisof the predicted wall shear stress and separation sites can be used to de-termine the optimal anastomosis angle and graft to artery diameter ratiofor the particular flow situation in the presence of moderate shear levels andminimum blood damage. Most of the models reported in the literature aretwo-dimensional - see for example Kim and Chandran [16]. Recent calcula-tions on three-dimensional distal anastomoses were published by Perktoldet al. [17]. In vitro studies performed by Hughes and How [18] have shownthe highly three-dimensional nature of these flows. Henry and co-workers[19, 20] have investigated anastomosis models, and have numerically pre-dicted flow patterns which compared well with the in vitro measurementsof Hughes and How [18]. Their findings confirmed that areas in the anosto-mosis which have been found to be prone to plaque formation were shownto be areas in which high wall shear occur.
2.5 Flow in the heart, and through natural and prosthetic valves
Modelling of the flow in the heart is one of the most challenging problemsfor the present generation of blood flow modellers. Due to the complexgeometry and flow features, analytical approaches are not accurate and
numerical modelling is the only method which can be pursued. An exampleof the former is the simplified mathematical model of potential flow in apulsating bulb given by Jones [21]. More satisfactory complete analysisof the flow in the heart can be obtained using numerical methods whichcan accommodate simplified or realistic geometries of the heart and leftventricle. Fully faithful representation of these characteristics require gridstorage which is beyond current capabilities. The first comprehensive time-dependent 3-D model of flow through the left side of the heart was developedby Peskin and McQueen [22]. They have worked in the development of acoupled approach for treating cardiac wall and blood flow where the formeris modelled as a system of fibres immersed in the blood fluid (ImmersedBoundary Method).
Georgiadis et al. [23] have also recently demonstrated that computa-tional fluid dynamics techniques can be a valuable tool for simulating heartflow problems. They used an elliptical axisymmetric model for the left vent-ricle and assumed inviscid flow. Taylor et al. [24] have recently developeda realistic 3-D model to solve the LV ejection flow using a commercial CFDpackage (SCRYU). This is based on the finite volume method, and usesthe full three-dimensional Navier-Stokes equations. The time course of theventricular wall changes was assumed to follow a trigonometric function.
The most interesting fluid-dynamical events in the heart are the fillingand ejecting of blood in the ventricles associated with the motion of theheart valves. Modelling of the fluid through natural and prosthetic valveshas been the subject of many theoretical investigations [25], and more re-cently, of numerical modelling [26, 27]. McQueen and Peskin [28] usedCFD methods to conduct a parametric study of a two-dimensional modelto give design concepts of bileaflet valves (the most commonly implantedmechanical heart valves). Currently, the use of CFD models combined withexperimental verification are permitting greater understanding of the flowmechanisms downstream of the valve, and the performance of parametricdesign studies [27].
2.6 Flow in cardiac assist devices
A cardiac assist device is a blood pump which replace partially or totally thepumping function of the heart. Significant research effort is currently beingdirected towards the quite difficult task of determining the optimal haemo-dynamic conditions in which such devices should operate. High shear ratesand extended residence times causing haemolysis and platelet activationmay develop in the pump or in the cannula when inferior flow conditionsare established. Such effects may severely limit clinical applications. Ex-perimental techniques have been used extensively to test in vitro and invivo artificial ventricle models, and improve blood pump characteristics.Laser Doppler Anemometry (LDA) has become a popular tool for assess-ing haemocompatibility by measuring flow induced shear stresses, and flow
visualization techniques have been used for aiding qualitative understandingMulder et al. [29]. Jin and Clarke [30] used both techniques to carry outan experimental investigation of unsteady non-Newtonian flow behaviourwithin a sac-type ventricular device.
In investigating the requirements of an optimal design, experimentaltechniques require geometrically accurate models for testing to be carriedout in every new prototype. Numerical techniques, however, present severaladvantages, as they can provide both detailed information for conditionsnot amenable to experimental measurements, and permit geometrical para-metric studies to be carried out for a given design concept. Despite this,very few numerical studies have been reported in the literature [31-34]. Thislack of numerical optimisation study may be due to difficulties encounteredwhen modelling realistic cardiac assist devices models *. Generally, flowsinside cardiac assist devices are three-dimensional, time-dependent, and of-ten turbulent. An essential feature is the capability to accomodate thegeometries of moving boundaries. This has been introduced very recentlyin the computational fluid dynamic packages and still has not been suffi-ciently tested against experimental evidence. Another important featureis the three-dimensionality of the problem, which requires very powerfulgrid capabilities, with special attention on the refinements necessary whereboundaries are in motion. Furthermore, the grid has to undergo changesfor each time step, which in turn has to allow for the 'accurate' solution tobe established without overloading the calculation too much. Flows insidethese assist devices are often turbulent, and this is another open problemin fluid dynamics modelling.
2.7 Blood vessel modelling
For the purpose of blood flow modelling it is important also to consider thewall distensibility of blood vessels. Experimental results have shown thatmechanical properties of blood vessels are characterized by high deformab-ility, a non-linear stress-strain relationship and the existence of viscoelastictime-dependent behaviour, such as hysteresis, creep and stress relaxation.In the mathematical formulation, the arterial wall can be assumed as ahomogeneous, incompressible and non-linearly elastic material which canbe modelled as an orthotropic thin-walled cylinder undergoing finite de-formation. In the past simplified models, such as small strain analysisand incremental analysis have been applied. Finite deformation analysiscan be applied to finite non-linear elastic deformations. The problem offlow through compliant arterial models has been investigated by Henry andCollins [36], for example. Perktold and Rappitsch [37] modelled the arterysegment as a thin shell applying non-linear shell theory and incrementallylinear elastic wall behaviour.
*Even in conventional engineering, the approach has only recently become reliable forsomewhat easier complex geometry problems [35].
Computational Fluid Dynamics has been used for numerical prediction offluid flow and heat transfer since the 1960s. With the development of com-puter hardware, computational techniques have been refined to the pointwhere these are recognized as a cost-effective and convenient means of design
in modern engineering practice.The fundamental equations of fluid dynamics are represented by the
continuity equation, the momentum equations, and the energy equation.For unsteady, compressible flow, these equations are written using a vectornotation as follows:
(1)
(2)
(3)
where p is the fluid density, u = (u,v,w) is the fluid velocity vector, t istime, B is the body force, a is the stress tensor, H is the total enthalpygiven in terms of static enthalpy h as H — h + |u , p is the pressure, Tthe temperature and A the thermal conductivity. Historically, the preferrednumerical method to discretize the governing equations for fluid simulationshas been the finite difference method; however, more recently finite elementmethods have been used, particularly in the field of blood flow.
The application of CFD to the simulation of blood flow in ventriclesis demanding because it involves wall-driven unsteady flows and time de-pendent gridding. Modelling of the flow mechanics requires either transi-ent or adaptive gridding in the CFD programs, and comprehensive mod-elling of the stress-strain behaviour of the wall. The numerical modelsshould therefore make provision for: (i) arbitrary three-dimensional geo-metry, (ii) moving grid/boundary capability, (iii) (laminar) flow pulsatility,(iv) non-Newtonian blood flow characteristics, (v) wall compliance/musclebehaviour. Modern CFD codes solve the equations governing unsteady,three-dimensional flows in complex, moving geometries. Among these pro-grams, CFDS-FLOW3D [38] (now CFX-4) has the advantage of providingall the capabilities required in the modelling of blood flow; and for thisreason has been used in this work.
CFDS-FLOW3D is a general purpose code for the numerical solutionof compressible and incompressible, laminar and turbulent flows with heattransfer in three-dimensional geometries. The code uses a finite volumemethod to solve the governing equations on a general, three dimensional,non-orthogonal, body fitted grid with moving boundaries. The multi-block
grid capability allows the simulation of flows in solution domains with sig-nificantly more complex non- axisymmetric geometries. This allows flowsin curved tubes and hemispherical ends of ventricles to be modelled moreaccurately.
4 Validation of the flow predictions for simplified
ventricular geometries
Initial investigations performed by the authors in the numerical modellingof the (low in ventricles have been concerned with a progressive validationof numerical solutions generated by the code for moving boundary flowdomains. Firstly, comparisons were made with the analytical solution ofU chid a and Aoki [39] for expanding/contracting pipes. The chief feature ofthe Uchida and Aoki solution is the similarity in both x, the axial direction,and time. In this paper only two cases of expansions with Reynolds numbersequal to 0.1 and 1.67 are presented. The latter is an interesting case asa (small) reverse flow region was predicted analytically. Other rates ofexpansion and contraction were studied and these are discussed in [40].Numerical and analytical results, in terms of axial and radial velocities, aregiven in Fig. 1. They show that numerical predictions are entirely consistentwith Uchida and Aoki's solution. Of course, the solution is restricted to atype of wall motion that does not represent well that of a ventricle; i.e., it isnot periodic, but either expanding or contracting, and the Reynolds numberis rather low. The obvious limitation of the model is that a ventricle has afinite length, and hence a similarity solution in x cannot be developed. Also,the pulsatile motion of the wall means that a similarity solution in time isnot possible. Despite this, the case forms a useful and reliable validationexercise.
Figure 1: Comparisons of the distributions of axial, (a) and (c), and radialvelocities, (b) and (d); (a) and (b) expansion at Re = 0.1; (c) and (d)
expansion at Re = 1.67.
An adapted compliant ventricle model was then generated (rigid/compliantpipe) with a truncated (flat-end) apex using a sinusoidally prescribed mo-tion of the wall. A spherical-end rigid/compliant model was subsequentlyconstructed, with the flows being driven by a prescribed sinusoidal mo-tion of the wall in both radial and axial directions. The predicted flowstructures show that the number of vortex rings formed at the time of max-imum ventricular volume is a function of the frequency of the wall motion.Predicted flow patterns shown in Figs.2 and 3 for these two models arequalitatively compared with experimental flow patterns.
(a) (b)
Figure 2: Flow structure predicted at the maximum ventricular volume forthe simple rigid/compliant model, (a) Lyentride — 3 x Rm/e*,
Figure 3: Flow structure predicted at the maximum volume for thespherical-end rigid/compliant model, (a) /=1.25 Hz, (b) /=3.0 Hz.
5 Ventricle modelling including muscle effects
Numerical simulations of fluid-solid coupling in blood vessels or ventriclesis a 'forefront' topic of current research in computational methods appliedto biomechanical problems. As mentioned, the first comprehensive time-dependent 3-D model of flow through the left side of the heart was developedby Peskin and McQueen [21]. The complete computational solution of suchflow problems is complicated by the requirement to solve the complex re-lationship that exists between the wall mechanics and the fluid-dynamics.Such a solution requires that both solid- and fluid-mechanical codes arecoupled via a transient computational grid.
Muscles are not passive tissues, and therefore, in a coupled approach,not only are the fluid and solid mechanics equations coupled, but so alsoare the muscle mechanics equations. In order to have realistic ventricularmodels, solid mechanics equations have to allow for large deformation, andnon-linear elastic and visco-elastic material with orthotropic properties. Inmodelling muscle mechanics, passive and active states have to be consideredseparately. In the passive state, the muscle anysotropy cannot be neglected.In the active state, body forces have to be included, but anisotropy canbe neglected because its effect is small compared with that of the bodyforces. When the muscle is operating actively, the constitutive equationsare represented by force-length, and force-velocity relations available fromexperimental data.
Historically, the development of fluid and solid mechanics codes hasdeveloped separately, and in parallel. CFD codes commonly use a finite
volume approach, while solid mechanics codes use the finite element method.In order properly to address the compliance and muscle effects of the physiolo-gical tissue, the most comprehensive solid mechanics available today is re-quired (non-linear, anisotropic, visco-elastic, large deformation, large strain).Application of Finite Element Analysis (FEA) to structural problems hasproved to be successful in many cases, and biomechanics applications havealso been performed [41, 37].
5.1 A simple muscle mechanics model for initial ventricularstudies
The solid mechanics equations for incompressible isotropic linear-elastic ma-terial in cylindrical coordinates are represented by the following equationsof motion:
rr , rz , x , n f A\~^ = -£— + -~ — + ~(<Trr ~ <T00) + ®r (4)ot* or oz r
«9cr,r , #<7zz , 1 , n fc\"ft -- -- H -°zr + Dz (5)r
and the constitutive equations:
(Trr = 2/2- -- ft (6)
<7,, = 2/^-ft (7)r
er» = 2/i|£ - h (8)
77 and <f are the displacements in the radial, r, and axial, z, directions,respectively; Dr and DZ are the radial and the axial components of the bodyforce generated by the muscular fibre tension; p^ is the density of the muscletissue; ft = — | (o>r + aee + &zz) is the hydrostatic pressure; fj, = E/3', and Eis Young's modulus. The above equations present the same characteristicsof the Navier-Stokes equations, and therefore can be solved by using thesame equation solver [36, 42].
The classic model of muscle mechanics was given by Hill [43] for askeletal muscle. This consists of two elements in series (one passive elastic,the other contractile) which obey the relationship:
(P + a) xv = 6x (Po- P) (10)
where a and b are constants which are derived experimentally, P is themuscle force, PQ the maximum tetanic tension, and v the muscle velocity of
contraction. A simple ventricular model is represented by an infinite cyl-indrical homogeneous isotropic thin-walled ventricle. In this model, whenconsidering the coupled model, the circumferential stress balances the in-ternal blood pressure, and according to Laplace's Law: a^ = ^—, where Ris the ventricular radius, p is the blood pressure, and h is the wall thickness.If a quasi-static analysis is applied then in systole <TC is represented by globalfibre tension. In diastole, <TC is an exponential function of radius. Therefore,in both cases, a simple relation between radius and pressure, which is validfor small displacements, can be used.
5.2 More comprehensive approaches to ventricular modelling
Determining the relationship between stress and strain in the myocardiumposes the principal challenge to the study of heart mechanics due to thenon-linear time-dependent constitutive law. For several decades research-ers have been attempting to understand the mechanics of the heart. Inspite of considerable effort, no unified description of heart mechanics hasemerged. This is because of the complexity of the geometry of the heart,of its structure, and of the material behaviour of its musculature. Realisticmodelling of the complex three-dimensional geometry and fibre architec-ture, the large deformations and the non-linear elastic, anisotropic, time-dependent material properties is extremely difficult. Therefore, most modelsof the heart have been restricted to the passive left ventricle and have reliedeither on simplified geometric representations, such as cylinders or spheres,or on linear approximations to the governing equations when reproducingglobal pressure-volume relationships. However, the importance of account-ing for large deformations of the passive left ventricle when predicting stressdistributions has been demonstrated when finite element models are used[44]. This is supported by experimental measurements of myo car dial fibrestretches that may exceed 20% at physiological filling pressures.
Numerous models dealing with the mechanical activity of the heartventricles have been published in the more recent past. These models canbe divided into two main groups: (i) phenomenological models which utilizethe equations of the continuum mechanics for deformable material to modelthe mechanics of the ventricle, and (ii) structural models, that are based onthe microstructure of the heart. When studying the dynamics of the leftventricle, the choice of the geometry of the left ventricle model is ratherimportant because the relation between wall stress and left ventricle pres-sure depends on the geometry. From Back's analysis [45], the most realisticresults for the relation between left ventricular pressure and wall stress areobtained when ellipsoidal geometry is assumed. Cylindrical geometry ap-proximates these results closely, within 10%, and significantly better thanwhen spherical geometry is assumed. An extensive review of muscle mech-anics and ventricle modelling may be found in [40].
The equations of solid mechanics used by many ventricle modellers arebased on the non-linear elastic theory as treated by Green and Adkins [46].This approach has been extensively used in biomechanics by many ventriclemodellers. One of the best known approaches to the elasticity of bodiescapable of finite deformation is to postulate the form of an elastic potential,or strain energy function W [47]. Details of the form of energy functionsused in the modelling of the left ventricle as used by Guccione [48] may befound in the APPENDIX.
As regard to the application of Finite Element Analysis to biomechan-ics, recently modelling has been extended to finite deformation, and morelatterly to large strain problems, and this has made the approach verypromising in modelling biological tissues mechanics. Among the currentFEA packages for structural analysis used in the modelling of biomechanicsproblems are ABAQUS, used by Perktold and co-workers [37], and MARCused by Reuderink [41]. Very few researchers have simulated the interac-tion of fluid with cardiovascular tissue. The Immersed Boundary Methoddeveloped by Peskin [26] represents a comprehensive approach in the mod-elling of the complex structure of the heart. Reuderink [41] developed anuncoupled approach to solve the equations for the solid and for the fluidseparately. Perktold and Rappitsch [37] have developed a coupled approachusing the FEM method to solve the flow and solid mechanics problems.A number of groups are actively working on the development of coupledfluid/solids numerical codes. The Biomechanical Research group at CityUniversity are currently engaged in two projects aimed at developing nu-merical codes capable of solving the coupled problem [49]. The first is basedon the finite volume code CFDS-FLOW3D, modified to allow the modellingof the solid wall. The second project is based on the use of a FEM codecalled FEAT, developed by Nuclear Electric's Engineering Analysis Group.
6 Conclusions
In this paper, the problem of numerical modelling of blood flow inventricles has been reviewed. The final goal of this research is to generatea comprehensive, coupled, solid-fluid, ventricle model. An initial coupledmodel has been developed by our group, which uses the novel approachof solving the flow and the wall equations in an identical manner, usingthe Finite Volume Method. A few groups have considered coupling FiniteElement solutions of the flow and the solid (see for example [37]). Thisapproach is also under investigation by our group.
Acknowledgement
This project is sponsored by the British Heart Foundation (PG/94029).The work at the University of Liverpool, in the form of the Skeletal MuscleResearch Group, is also sponsored by the British Heart Foundation.
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Appendix
In diastole, a strain energy potential W can be chosen for the ventricular wall that isan exponential function of Lagrangian-Green's strain tensor components Eij referredto fibre coordinates:
Guccione [50] have chosen the following form for exponent Q to describe the specialcase of three-dimensional transverse isotropy with respect to fibre:
Material constants C, &/, &<, &/, may be found in [50]. Systolic contraction is modelledby defining the stress tensor as the sum of the passive component and an active fibre-directed component T, which is a function of external calcium concentration Ca$ andsarcomere length / [48]:
/-, b— o
where Tmax is the peak tension developed at maximum Coo and h is the Hill coeffi-cient. The calcium sensitivity ECa$Q was treated as a length-dependent relation:
where B is a constant and /o is the sarcomere length at which no active tension isdeveloped. The parameters of the active contraction model are based on experimentalmeasurements of sarcomere length and peak active tension in isolated cat trabeculae.
