Mechanical Properties Of The Heart Muscle

Mechanical properties of the heartmuscleINF 5610 – p.1/45

OutlineCrossbridge theory. How does a muscle contract?A mathematical model for heart muscle contraction.Coupling to electrophysiology(Notes on passive mechanics and full-scale heartmechanics models)INF 5610 – p.2/45

What will not be covered?Non-linear solid mechanicsConstitutive laws for passive properties of heart tissueINF 5610 – p.3/45

Possible (advanced) readingCell contraction: Hunter PJ, McCulloch AD, ter KeursHE. Modelling the mechanical properties of cardiacmuscle. Prog Biophys Mol Biol.1998;69(2-3):289-331.Basic continuum mechanics: George E. Mase,Continuum mechanicsNon-linear mechanics: Gerhard Holzapfel, Non-linearsolid mechanics, a continuum approach for engineeringINF 5610 – p.4/45

Muscle cellsSmooth muscleStriated muscleCardiac muscleSkeletal muscleMost mathematical models have been developed for skeletalmuscle.INF 5610 – p.5/45

Striated muscle cellsSkeletal muscle cells andcardiac muscle cells havesimilar, but not identical,contractile mechanisms.A muscle cell (cardiac orskeletal) contains smallerunits called myofibrils,which in turn are made upof sarcomeres.The sarcomere containsoverlapping thin and thickfilaments, which are responsible for the force development in the musclecells.INF 5610 – p.6/45

Thick filaments are made up of the protein myosin. Themyosin molecules have heads which form cross-bridgesthat interact with the thin filaments to generate force.Thin filaments contain the three proteins actin,tropomyosin and troponin.The actin forms a double helix around a backboneformed by tropomyosin.INF 5610 – p.7/45

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In the base configuration, tropomyosin blocks thecross-bridge binding sites on the actin.Troponin contains binding sites for calcium, and bindingof calcium causes the tropomyosin to move, exposingthe actin binding sites for the cross-bridges to attach.INF 5610 – p.9/45

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After calcium has bound to the troponin to expose thebinding sites, the force development in the muscle happensin four stages:1. An energized cross-bridge binds to actin.2. The cross-bridge moves to its energetically preferredposition, pulling the thin filament.3. ATP binds to the myosin, causing the cross-bridge todetach.4. Hydrolysis of ATP energizes the cross-bridge.During muscle contraction, each cross-bridge goes throughthis cycle repeatedly.INF 5610 – p.11/45

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Cardiac muscleThe ability of a muscle to produce tension depends onthe overlap between thick and thin filaments.Skeletal muscle; always close to optimal overlapNot the case for cardiac muscle; force dependent onlengthINF 5610 – p.13/45

Cross bridge binding and detachment depends ontension. The rate of detachment is higher at lowertensionExperiments show that attachment and detachment ofcross-bridges depends not only on the current state ofthe muscle, but also on the history of length changes.INF 5610 – p.14/45

Important quantitiesIsometric tension (T0 ): the tension generated by amuscle contracting at a fixed length. The maximumisometric tension (for a maximally activated muscle) isapproximately constant for skeletal muscle, but forcardiac muscle it is dependent on length.Tension (T ): Actively developed tension. Normally afunction of isometric tension and the rate of shortening:T T0 f (V ),where V is the rate of shortening and f (V ) is someforce-velocity relation.Fibre extension ratio (λ): Current sarcomere lengthdivided by the slack length.INF 5610 – p.15/45

Force-velocity relationsThe classical equation of Hill (1938) describes therelation between velocity and tension in a muscle thatcontracts against a constant load (isotonic contraction).(T a)V b(T0 T )T0 is the isometric tension and V is the velocity. a and bare parameters which are fitted to experimental data.Recall that T0 is constant for skeletal muscle cells,dependent on length in cardiac cellsINF 5610 – p.16/45

Velocity as function of force:T0 TV bT aForce as function of velocity:bT0 aVT b VINF 5610 – p.17/45

Inserting T 0 in the Hill equation givesbT0V0 ,awhich is the maximum contraction velocity of the muscle.The maximum velocity V0 is sometimes regarded as aparameter in the model, and used to eliminate b.VT /T0 1 V0T aINF 5610 – p.18/45

A typical xis; force (g/cm2 )y -axis; velocity (cm/s)INF 5610 – p.19/45

To summarize, the force development in muscle fibersdepends on the rate of cross-bridges binding and detachingto the the actin sites. This in turn depends onSarcomere lengthShortening velocity(History of length changes.)The proportion of actin sites available, which dependson the amount of calcium bound to Troponin C (which inturn depends on the intracellular calcium concentrationand tension).INF 5610 – p.20/45

A model for the contracting muscleA detailed mathematical model for the actively contractingmuscle fiber should include the following:The intracellular calcium concentration, [Ca2 ]i .The concentration of calcium bound to Troponin C,[Ca2 ]b . This depends on [Ca2 ]i and the tension T .The proportion of actin sites available for cross-bridgebinding. Depends on [Ca2 ]b .The length-tension dependence.Force-velocity relation.INF 5610 – p.21/45

An example model: HMTThe Hunter-McCulloch-terKeurs (HMT) model waspublished in 1998Includes all features presented on the previous slidesSystem of ODEs coupled with algebraic relationsOriginal paper contains detailed description ofexperiments and parameter fittingINF 5610 – p.22/45

Ca2 bindingWe regard [Ca2 i ] as an input parameter (obtained fromcell electrophysiology models)Calcium binding is described with an ODEd[Ca2 ]bdt T2 2 2 [Ca2 ρ0 [Ca ]i ([Ca ]bmax [Ca ]b ) ρ1 1 γT0Attachment rate increases with increased [Ca2 ]i anddecreases with increasing [Ca2 ]bDetachment rate decreases with increasing tension T ,and increases with increasing [Ca2 ]bINF 5610 – p.23/45

Binding site kineticsThe process from calcium binding to exposure ofbinding sites is not instant, but subject to a time delayA parameter z [0, 1] represents the proportion of actinsites available for cross-bridge binding.Dynamics described by an ODE 2 n dz[Ca ]b(1 z) z α0dtC50INF 5610 – p.24/45

Length dependenceIsometric tension T0 depends on length (λ) and numberof available binding sites (z )The tension is given by an algebraic relationT0 Tref (1 β0 (λ 1))z,where z is given by the previous equation.INF 5610 – p.25/45

Force-velocity relationActive tension development depends on isometrictension and rate of shorteningForce-velocity relation given by a Hill function(T a)V b(T0 T )INF 5610 – p.26/45

(More advanced T-V relation)Experimental data shows that the binding anddetachment of cross-bridges depends not only on thepresent state of the muscle fiber, but also on the historyof length changesThe Hill function only includes the current velocity, so itis not able to describe this behaviorThe HMT model uses a standard Hill function, but withvelocity V replaced by a so-called fading memorymodel, which contains information on the history oflength changesFor simplicity we here assume a classical Hill-typerelationINF 5610 – p.27/45

Active tension from Hill model1 aVT T0,1 Va is a parameter describing the steepness of theforce-velocity curve (fitted to experimental data)INF 5610 – p.28/45

HMT model summaryTension T is computed from two ODEs and two algebraicrelations :d[Ca2 ]b f1 ([Ca2 ]i , [Ca2 ]b , Tactive , T0 )dtdz f2 (z, λ, [Ca2 ]b )dtT0 f3 (λ, z)Tactive f4 (T0 , λ, t)(1)(2)(3)(4)INF 5610 – p.29/45

Coupling to electrophysiologyCoupling of the HMT model to an electrophysiologymodel is straight-forward.To increase the realism of the coupled model the cellmodel should include stretch-activated channels. Thisallows a two-way coupling between theelectrophysiology and the mechanics of the muscle,excitation-contraction coupling and mechano-electricfeedback.INF 5610 – p.30/45

Summary (1)The force-development in muscles is caused by thebinding of cross-bridges to actin sites on the thinfilaments.The cross-bridge binding depends on the intracellularcalcium concentration, providing the link betweenelectrical activation and contraction(excitation-contraction coupling).Accurate models should include stretch-activatedchannels in the ionic current models (mechano-electricfeedback).Heart muscle is more complicated to model thanskeletal muscle, because the force development islength-dependent.INF 5610 – p.31/45

Summary (2)The model for cross-bridge binding and forcedevelopment is expressed as a system of ordinarydifferential equations and algebraic expressionsThe models can easily be coupled to ODE systems forcell electrophysiology, because of the dependence onintracellular calciumINF 5610 – p.32/45

Modeling the complete muscle (1)The HMT model only gives the force development in asingle muscle fibre.The deformation of the muscle is the result of activeforce developed in the cells, and passive forcesdeveloped by the elastic properties of the tissue.Modeling the deformation of the muscle requiresadvanced continuum mechanicsDetailed description beyond the scope of this course,simple overview provided for completenessINF 5610 – p.33/45

Modeling the complete muscle (2)The key variables in solid mechanics problems arestresses and strainsStress force per area, strain relative deformationStress tensor: σ11 σ12 σ13 σij σ21 σ22 σ23 σ31 σ32 σ33Strain tensor:ε11 ε12 ε13 ǫij ε21 ε22 ε23 ε31 ε32 ε33INF 5610 – p.34/45

Modeling the complete muscle (3)The equilibrium equation relevant for the heart reads ·σ 0(The divergence of the stress tensor is zero)Vector equation 3 scalar equations, symmetric stresstensor 6 scalar unknownsEquation is valid for any material, need to becomplemented with information on material behaviorMaterial described by constitutive laws, typically astress-strain relationINF 5610 – p.35/45

Simple stress-strain relationSay we pull a rod with length L and cross-sectional areaA using a force F . This results in a length increase L.The following relation is valid for small deformation inmany construction materials: LF EALThe quantity L/L is called the strain, F/A is thestress, and E is a parameter characterizing the stiffnessof the material (Young’s modulus).This relation is called a stress-strain relation. This linearrelation is known as Hooke’s law.Stress-strain relations in the heart are much more compli-INF 5610 – p.36/45cated, but the principle is exactly the same.

A linear elastic materialHooke’s lawNormally applicable only for small 30.40.50.60.70.80.91INF 5610 – p.37/45

Non-linear (hyper)elastic materialsFor materials undergoing large elastic deformations, thestress-strain relation is normally 0.80.91For the heart, the tissue is also anisotropic, withdifferent material characteristics in different directionsINF 5610 – p.38/45

Coupling active and passive stressesTo model both the active contraction and the passivematerial properties of the heart, we introduce a stress thatconsists of two parts.T σp σa .Passive stress σp is computed from a stress-strainrelation.Active stress σa is computed from a muscle model likethe HMT model.The sum of the two stresses is inserted into theequibrium equation, which is then solved to determinethe deformationsINF 5610 – p.39/45

Complete modelThe complete electrical and mechanical activity of oneheart beat consists of the follwoing components:Cell model describing electrical activation.Cell model describing contraction (for instance HMT).Receives calcium concentration from el-phys model andgives tension as output.Elasticity equation describing the passive materialproperties. Takes the tension from the HMT model asinput, returns the deformation of the muscle.Equation describing the propagation of the electricalsignal through the tissue (bidomain model).INF 5610 – p.40/45