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Cardiac Muscle and Organ Mechanics Roy Kerckhoffs Dept of Bioengineering, University of California, San Diego Tutorial on heart and lungs Ohio State University, Columbus, OH, 20 sep 2006 Outline system system organ organ tissue tissue cell cell kn Ca2+ b Ca2+ R*offk k*of f kon R*on g f A*1 R0off Ca2+ R0on Ca2+ g Ca2+ A01 Ca2+ Overview • Anatomy and physiology of the heart – System and organ level – Resting cardiac tissue – Active force generation: the sarcomere • Multi-scale modeling – Anatomic models – Models of cardiac mechanics: from cell to system – Models of cardiac electromechanics Cardiac Anatomy Mitral SVC Aorta Pulmonic Pulmonary artery Tricuspid valves LA RA LV RV Septum Epicardium Endocardium Apex Base Vperi Pericardium • A sac wherein the heart sits • Limits sudden increases in volume • Increases atrio-ventricular and ventricularventricular interaction Vaw Vra Vla Vrv Vlv Vvw *Freeman & Little, AJP 1986;251:H421-H427 Physiology Conduction of System the heart right atrium sinusnode AV-node right bundle branch left atrium left bundle branch left ventricle Purkinje fibers right ventricle Coronary System 2 The Cardiac Cycle 1 3 4b 4a Systole: 1. Isovolumic contraction 2. Ejection 4a 4b Diastole: 3. Relaxation Early Isovolumic 4. Filling a) Early, rapid b) Late, diastasis 1 3 2 3 4a 16 AVC Pressure (kPa) 14 12 Aorta AVO 10 8 6 4 Left ventricle 2 MVO MVC 0 150 Volume (ml) Pressure and Volume 2 1 4b 120 90 60 30 0 100 200 300 400 Time (msec) 500 600 700 The Pressure-Volume Diagram Endsystole (ES) SV=EDV-ESV Ejection 16 AVC Ejection Fraction EF=SV/EDV AVO Stroke volume (SV) 8 Isovolumic contraction 12 Isovolumic relaxation Pressure (kPa) 20 End-diastole (ED) 4 MVO Filling MVC 0 0 50 100 Volume (ml) 150 200 The Pressure-Volume Diagram 20 ESV EW 16 P(t )d V EDV AVC AVO 8 Stroke (external) work Isovolumic contraction 12 Isovolumic relaxation Pressure (kPa) Ejection 4 MVO Filling MVC 0 0 50 100 Volume (ml) 150 200 Preload and Afterload Pressure (kPa) 20 16 control 12 preload 8 afterload 4 0 0 50 100 Volume (ml) 150 200 Time-Varying Elastance P(t) = E(t){V(t) - V0} E(200) = Emax E(160 msec) Pressure (kPa) 20 E(120 msec) 16 12 E(80 msec) 8 4 0 0 50 100 150 LV Volume (ml) 200 Starling’s Law of the Heart (The Frank-Starling Mechanism) Stroke work increased contractility (e.g. adrenergic agonist) decreased contractility (e.g. heart failure) “Preload” (EDV or EDP) Contractility (Inotropic State) increased contractility (e.g. adrenergic agonist) Pressure (kPa) 20 decreased contractility (e.g. heart failure) 16 12 8 4 0 0 50 100 Volume (ml) 150 200 Physiological Basis of Starling’s Law 20 Pressure (kPa) 16 12 8 4 0 0 50 100 Volume (ml) 150 200 Overview • Anatomy and physiology of the heart – System and organ level – Resting cardiac tissue – Active force generation: the sarcomere • Multi-scale modeling – Models of cardiac mechanics: from cell to system – Models of cardiac electromechanics Fiber and Sheet Architecture Epicardium x510 Endocardium Minimizing Stress Gradients • Residual Stress • Fiber Angles Sarcomere length (µm) • Torsion 1.94 1.90 1.86 1.82 1.78 1.74 unloaded Stress-free Epi Transmural position Endo Resting Tissue Properties • Nonlinearity • Hysteresis • Creep • Relaxation • Preconditioning Behavior • Strain Softening • Anisotropy Passive Biaxial Properties 10 Fiber stress Stress (kPa) 8 6 4 Cross-fiber stress 2 0 0.00 0.05 0.10 0.15 0.20 Equibiaxial Strain 0.25 Measurement of Myocardial strain • Radiopaque beads and biplane x-ray • video imaging of markers • ultrasound • MRI tagging Myocyte Connections • Myocytes connect to an average of 11 other cells (half end-to-end and half side-to-side) • Myocytes branch (about 12-15º) • Intercalated disks – gap junctions Overview • Anatomy and physiology of the heart – System and organ level – Resting cardiac tissue – Active force generation: the sarcomere • Multi-scale modeling – Models of cardiac mechanics: from cell to system – Models of cardiac electromechanics Cardiac Myocytes • • • • Rod-shaped Striated 80-100 m long 15-25 m diameter Striated Muscle Ultrastructure Electron micrograph of longitudinal section of freeze-substituted, relaxed rabbit psoas muscle. Sarcomere shows A band, I band, H band, M line, and Z line. Scale bar, 100 nm. From Millman BM, Physiol. Rev. 78: 359391, 1998 The Sarcomere The Sarcomere Crossbridge Cycle ExcitationContraction Coupling • Calcium-induced calcium release • Calcium current • Na+/Ca2+ exchange • Sarcolemmal Ca2+ pump • SR Ca2+ ATPdependent pump http://www.meddean.luc.edu/lumen/DeptWebs/physio/bers.html Isometric Tension in Skeletal Muscle: Sliding Filament Theory (a) Tension-length curves for frog sartorius muscle at 0ºC (b) Developed tension versus length for a single fiber of frog semitendinosus muscle Isometric Testing Sarcomere length, m 2.1 Sarcomere isometric 2.0 1.9 Muscle isometric Tension, mN 2.0 1.0 time, msec 100 200 300 400 500 600 700 Length-Dependent Activation 2.2 micrometer 1.6 micrometer Isometric peak twitch tension in cardiac muscle continues to rise at sarcomere lengths >2 m due to sarcomere-length dependent increase in myofilament calcium sensitivity Isotonic Testing Isovelocity release experiment conducting during a twitch Cardiac muscle forcevelocity relation corrected for viscous forces of passive cardiac muscle which reduce shortening velocity Ventricular Mechanics: Summary of Key Points • Ventricular geometry is 3-D and complex • Fiber angles vary smoothly across the wall • Systole consists of isovolumic contraction and ejection; diastole consists of isovolumic relaxation and filling • Area of the pressure-volume loop is ventricular stroke work which increases with filling (Starling’s Law) • Ventricles behave like time-varying elastances • The slope of the end-systolic pressure volume relation is a load-independent measure of contractility or inotropic state. Ventricular Mechanics: Summary of Key Points (cont’d) • Collagen contributes to anisotropic resting properties • Myocardial strain can be measured invasively and noninvasively • Torsion and residual stress tend to compensate for these gradients in the ventricles to maintain uniform fiber strain Overview • Anatomy and physiology of the heart – System and organ level – Resting cardiac tissue – Active force generation: the sarcomere • Multi-scale modeling – Models of cardiac mechanics: from cell to system – Models of cardiac electromechanics Integrative In-Silico Biology Functional Integration, Structural Integration • Functional integration – of interacting physiological processes • Structural integration – across scales of biological organization (c) 2004 Andrew McCulloch, UCSD Why modeling • hypothesis generation • clinical applications – diagnosis – training platforms for surgeons – predict outcomes of surgical interventions – predict outcomes of therapies Why multiscale modeling • Cardiac structure and function are heterogeneous: most pathologies are regional and nonhomogeneous • Ca2+ important ion in electrophysiology and responsible for cardiac force generation • Many interacting subsystems in basic processes: e.g. – ventricular stress coronary flow – electrical activation mechanical activation (ECC and MEF) – feedback of baroreceptors on cardiac contractility and frequency Overview • Anatomy and physiology of the heart – System and organ level – Resting cardiac tissue – Active force generation: the sarcomere • Multi-scale modeling – Models of cardiac mechanics: from cell to system – Models of cardiac electromechanics Models of cardiac mechanics cellular • Development of models of cellular cardiac mechanics have lagged behind models of cellular cardiac electrophysiology, due to – lack of available solving algorithms (and computer power) – controversies about basic mechanisms of force generation in myofilaments • 4 categories: – – – – phenomenological time-varying elastance models (algebraic) phenomenological Hill-models (ODE) A.F.Huxley type models of crossbridge formation (PDE) Landesberg type myofilament activation model (ODEs) Modeling Myofilament Force Production • Ca2+ binding to TnC causes tropomyosin to change to a permissive state • Force development occurs as actin-myosin crossbridges form • Crossbridges can ‘hold’ tropomyosin in the permissive state even after Ca2+ has dissociated Myofilament Activation/Crossbridge Cycling Kinetics kn * Roff k*off kb Ca2+ R0off Non-permissive Tropomyosin Ca2+ kon koff Ca2+ 0 Ron * Ron Permissive Tropomyosin Ca2+ g f g f Ca2+ A10 A1* Ca2+ Ca2+ bound to TnC Ca2+ not bound to TnC Permissive Tropomyosin, 1-3 crossbridges attached (force generating states) This scheme is used to find A(t), the timecourse of attached crossbridges for a given input of [Ca2](t) Myofilament Model Equations • Total force is the product of the total number of attached crossbridges, average crossbridge distortion, and crossbridge stiffness: F At xt Myofilament model: results Noff 0 N*on R*on Non 0 + Ron 0 A*1 A1 A*2 A2 0 0 A3 µtitin 0.5 0 0 0.3 [Ca] Active Force SL ηcell Simultaneous measurement of intracellular Ca2+ and shortening in single myocytes 0.6 Time (s) µgel Model validation experiments • 1 Passive mechanics of single myocyte Factive A*3 [Ca] Active Force SL Myofilament model of active force generation Relative Units Roff Relative Units 0 R*off Relative [Ca], Active Force, and SL 0 N*off 1 0.5 0 0 0.3 Time (s) 0.6 Nonlinear Elasticity of Soft Tissues • • • • • • Soft tissues are not elastic — stress depends on strain and the history of strain However, the hysteresis loop is only weakly dependent on strain rate It may be reasonable to assume that tissues in vivo are preconditioned Fung: elasticity may be suitable for soft tissues, if we use a different stress-strain relation for loading and unloading – the pseudoelasticity concept a rationale for applying elasticity theory to soft tissues Unlike in bone, linear elasticity is inappropriate for soft tissues; we need nonlinear finite elasticity Transversely Isotropic Laws for Exponential: Myocardium W C e Q 1 Exponential 2 Q 2a1E11 E 22 E33 a2 E11 2 2 2 2 2 2 2 2 a3 E 22 E33 E 23 E32 a4 E12 E 21 E13 E31 transversely isotropic, x1= "fiber" axis C=0.6kPa, a1=2.5, a2=15, a3=0, a 4=10 Isotropic + Anisotropic terms: W W I1, I 2 W 2 k1 k1 is the fiber extension ratio, e.g. (Humphrey & Yin, 1987) c 4 k112 c 2I13 W c1e c 3 e Polynomial: W I1,f C f 1 2 C 2 f 13 C 3 I1 3 C 4 I1 3 f 1 f fiber stretch C5 I1 3 2 C1 20, C 2 50, C 3 1.5, C 4 20, C 5 25 Models of cardiac mechanics tissue • Passive – strain energy functions – orthotropic (fiber – crossfiber – sheet) – heterogeneous • Active – orthotropic (fiber – crossfiber – sheet?) – heterogeneous(!) (Cordeiro et al, AJP 286, H1471-H1479, 2004) Models of cardiac mechanics organ • Solve tissue models on anatomy with e.g. finite element or finite difference method • Compute part of cardiac cycle, e.g. produce Frank-Starling curves, or • Compute full cardiac cycle with coupling to circulatory model Ventricular Geometry Truncated ellipses of revolution Prolate Spheroidal Coordinates x1 x1 = d cosh cos x2 = d sinh sin cosq q x2 b d a x3 = d sinh sin sinq Anatomic models Vetter FJ et al (1998) PBMB Stevens C et al (2003) PBMB Nielsen PMF et al (1991) AJP Grieshaber J et al (2002) LeGrice IJ et al (1997) AJP Rao J et al Smith NP et al (2000) ABME Kerckhoffs et al (2003) Models of cardiac mechanics organ: application • Diagnostic measures • Herz et al used finite element model of cardiac ischemia to generate new measures for dyskinetic cardiac tissue Herz et al. , 2005. Annals Biomed Eng 33: 912-919 Models of cardiac mechanics organ: application • Surgical training platforms • Predict outcome of surgical interventions Myosplint reduced fiber stress, but did not affect stroke volume Guccione et al. 2003. Annals of Thoracic Surgery 76,1171-1180. Circulatory models system level • 2-, 3-, 4-element windkessel Circulatory models system level windkessel = air chamber used in plunger pumps to ensure a steady flow Circulatory models system level Circulatory models system level Pressure serves as hemodynamic boundary condition Cavity pressure Cavity pressure Flow Q Qdt FE Cavity volume (c) 2004 Andrew McCulloch, UCSD Cavity volume from circulatory model Coupling PL P PR Estimate LV & RV cavity pressure Circulatory model FE model FE VLFE V FE VR Circ Cavity volumes FE Cavity volumes FE circ R V V Calculate difference R R < criterion? no yes Do not update Jacobian new P Update Jacobian next timestep old R P P circ VLcirc V circ VR 1 R p old System compliance matrix FE R V P Pi P Pi circ V P Pi VL FE PL FE VR P L FE circ VL VL PR P L FE circ VR VR PR P L circ VL P R circ VR P R FE compliance matrix Circ compliance matrix Update 1: Estimate pressure from history Update 2: Perturb LV pressure Update 3: Perturb RV pressure Updates >3: Update pressures Circulatory models system level Overview • Anatomy and physiology of the heart – System and organ level – Resting cardiac tissue – Active force generation: the sarcomere • Multi-scale modeling – Models of cardiac mechanics: from cell to system – Models of cardiac electromechanics Important components of models of cardiac electromechanics Anatomic model Hemodynamic model Electrophysiology model Mechanics model time Models of cardiac electrophysiology tissue • Couple ionic models or FitzHugh-Nagumo with – Monodomain – Bidomain Vm(x,t) Vextr(x,t) and Vintr(x,t) • Or derive wavefront from bidomain model: – Eikonal-diffusion tdep(x) Models of cardiac electromechanics cellular • Calcium • Mechano-electric feedback (MEF) – deformation – sarcomere length dependent myofilament calcium sensitivity – stretch-activated ion channels • Clinic: – Resynchronization – asymmetric hypertrophy by chronic pacing • Couple existing models of electrophysiology to models of cardiac mechanics Models of cardiac electromechanics tissue • Electromechanical Coupling: – Tight (Continuous interplay: EC and MEF) – Loose (EC only) • Due to computational demand of tightly coupled electromechanics, solve models on 2D domains Models of cardiac electromechanics tissue • Spiral waves with FitzHugh-Nagumomonodomain model and active contraction • Mechano-electric feedback through deforming tissue noncontracting contracting Nash and Panfilov. Progr Biophys Mol Biol 85, 501-522, 2004. Models of cardiac electromechanics organ • Solve tissue models on anatomy with e.g. finite element or finite difference method • Due to computational demand, either: – Phenomenological or simple ionic models – Loosely coupled (ECC only) • Tightly coupled: parallellization Models of cardiac electromechanics organ • Loosely coupled model In a computational model study1: • A physiological sequence of depolarization results in an unphysiological non-uniform distribution of shortening • An unphysiological synchronous depolarization results in a physiological homogeneous distribution of shortening End-systolic myofiber strain 0 Experiment (Mazhari et al, Circ. 104, 2001) Simulation with synchronous activation -0.1 Simulation with normal activation -0.2 0 Epi 0.25 0.5 0.75 1 Endo 1Kerckhoffs et al, ABME 31, p536-547, 2003 Models of cardiac electromechanics organ • Loosely coupled model normal dep wave synchronous myofiber strain 1Kerckhoffs et al, ABME 31, p536-547, 2003 Models of cardiac electromechanics organ From an experimental study1: The latency to onset of contraction in endocardial cells is ~20 ms longer than that of epicardial cells, which allows the impulse to traverse the LV wall and effect a coordinated contraction of the ventricular myocardium. 1Cordeiro et al, AJP 286, H1471-H1479, 2004 Models ofElectrical cardiac activation electromechanics organ: pacing in theventricular paced heart Pacing at the right ventricle activation completed in: 92 ms Pacing at the left ventricle 129 ms Kerckhoffs et al. J Eng Math. 2003;47:201-216. Models of cardiac electromechanics Contraction in the paced heart organ: ventricular pacing Pacing at the right ventricle Pacing at the left ventricle strain Maximum LV pressure increase: 137 kPa/s 240 kPa/s Kerckhoffs et al. J Eng Math. 2003;47:201-216. Models of cardiac electromechanics FE heart coupled to circulation Models of cardiac electromechanics organ • Tightly coupled model: – Simple 3-current ionic model – Model of active contraction Nickerson et al. Europace 7, 5118-5127, 2005. Models of cardiac electromechanics organ: application • Tightly coupled model • Computation time: 3 weeks on 8 parallel processors Nickerson et al. Europace 7, 5118-5127, 2005. Computational demand • With advances in biology, simulations remain large and time-consuming • Algorithm improvement • Parallel computing Computation speed at CMRG