Physics of the Heart: From the macroscopic to the microscopic Xianfeng Song Advisor: Sima Setayeshgar January 9, 2007

Outline  Part I: Transport Through the Myocardium of Pharmocokinetic Agents Placed in the Pericardial Sac: Insights From Physical Modeling  Part II: Electrical Wave Propagation in a Minimally Realistic Fiber Architecture Model of the Left Ventricle  Part III: Calcium Dynamics in the Myocyte

Part I: Transport Through the Myocardium of Pharmocokinetic Agents Placed in the Pericardial Sac: Insights From Physical Modeling Xianfeng Song, Department of Physics, Indiana University Keith L. March, IUPUI Medical School Sima Setayeshgar, Department of Physics, Indiana University

Motivation: Diffusion in Biological Processes Diffusion is the dominant transport mechanism in biology, operative on many scales: Intracellular [1]  The rate of protein diffusion in the cytoplasm constrains a variety of cellular functions and limit the rates and accuracy of biochemical signaling in vivo. Multicellular [2]  Diffusion plays an important role during the early embryonic pattern formation in establishing and constraining accuracy of morphogen prepatterns. Tissue-level [3]  Diffusion controls delivery of glucose and oxygen from the vascular system to tissue cells and also governs movement of signaling molecules between cells. [1] Elowitz, M. B., M. G. Surette, et al. (1999). J. Bact. 181(1): [2] Gregor, T., W. Bialek, R. de Ruyter van Steveninck, et al. (2005). PNAS 102(51). [3] Nicholson, C. (2001), Rep. Prog. Phys. 64, Need for careful characterization of diffusion constants governing various biophysical processes.

Background: Pericardial Delivery  The pericardial sac is a fluid-filled self-contained space surrounding the heart. As such, it can be potentially used therapeutically as a “drug reservoir.”  Delivery of anti-arrhythmic, gene therapeutic agents to  Coronary vasculature  Myocardium via diffusion.  Recent experimental feasibility of pericardial access [1], [2] V peri (human) =10ml – 50ml [1] Verrier VL, et al., “Transatrial access to the normal pericardial space: a novel approach for diagnostic sampling, pericardiocentesis and therapeutic interventions,” Circulation (1998) 98: [2] Stoll HP, et al., “Pharmacokinetic and consistency of pericardial delivery directed to coronary arteries: direct comparison with endoluminal delivery,” Clin Cardiol (1999) 22(Suppl-I): I-10-I-16.

Part 1: Outline  Experiments  Mathematical modeling  Comparison with data  Conclusions

Experiments  Experimental subjects: juvenile farm pigs  Radiotracer method to determine the spatial concentration profile from gamma radiation rate, using radio-iodinated test agents  Insulin-like Growth Factor ( 125 I-IGF, MW: 7734 Da)  Basic Fibroblast Growth Factor ( 125 I-bFGF, MW: Da)  Initial concentration delivered to the pericardial sac at t=0  200 or 2000  g in 10 ml of injectate  Harvesting at t=1h or 24h after delivery

Experimental Procedure  At t = T (1h or 24h), sac fluid is distilled: C P (T)  Tissue strips are submerged in liquid nitrogen to fix concentration.  Cylindrical transmyocardial specimens are sectioned into slices: C i T (x,T) x denotes C T (x,T) =  i C i T (x,T) x: depth in tissue i

Mathematical Modeling  Goals  Determine key physical processes, and extract governing parameters  Assess the efficacy of agent penetration in the myocardium using this mode of delivery  Key physical processes  Substrate transport across boundary layer between pericardial sac and myocardium:   Substrate diffusion in myocardium: D T  Substrate washout in myocardium (through the intramural vascular and lymphatic capillaries): k

Idealized Spherical Geometry Pericardial sac: R 2 – R 3 Myocardium: R 1 – R 2 Chamber: 0 – R 1 R 1 = 2.5cm R 2 = 3.5cm V peri = 10ml - 40ml

Governing Equations and Boundary Conditions  Governing equation in myocardium: diffusion + washout C T : concentration of agent in tissue D T : effective diffusion constant in tissue k: washout rate  Pericardial sac as a drug reservoir (well-mixed and no washout): drug number conservation  Boundary condition: drug current at peri/epicardial boundary

Example of Numerical Fits to Experiments Agent ConcentrationError surface

Fit Results Numerical values for D T, k,  consistent for IGF, bFGF

Time Course from Simulation Parameters: D T = 7×10 -6 cm 2 s -1 k = 5×10 -4 s -1  = 3.2×10 -6 cm 2 s 2

Effective Diffusion, D *, in Tortuous Media  Stokes-Einstein relation D: diffusion constant R: hydrodynamic radius : viscosity T: temperature  Diffusion in tortuous medium D * : effective diffusion constant D: diffusion constant in fluid : tortuosity For myocardium, = (from M. Suenson, D.R. Richmond, J.B. Bassingthwaighte, “Diffusion of sucrose, sodium, and water in ventricular myocardium, American Joural of Physiology,” 227(5), 1974 )  Numerical estimates for diffusion constants  IGF : D ~ 4 x cm 2 s -1  bFGF: D ~ 3 x cm 2 s -1 Our fitted values are in order of cm 2 sec -1, 10 to 50 times larger !!

Transport via Intramural Vasculature Drug permeates into vasculature from extracellular space at high concentration and permeates out of the vasculature into the extracellular space at low concentration, thereby increasing the effective diffusion constant in the tissue. Epi Endo

Diffusion in Active Viscoelastic Media Heart tissue is a porous medium consisting of extracellular space and muscle fibers. The extracellular space consists of an incompressible fluid (mostly water) and collagen. Expansion and contraction of the fiber bundles and sheets leads to changes in pore size at the tissue level and therefore mixing of the extracellular volume. This effective "stirring" [1] results in larger diffusion constants. [1] T. Gregor, W. Bialek, R. R. de Ruyter, van Steveninck, et al., PNAS 102, (2005).

Part I: Conclusions  Model accounting for effective diffusion and washout is consistent with experiments despite its simplicity.  Quantitative determination of numerical values for physical parameters  Effective diffusion constant IGF: D T = (1.7±1.5) x cm 2 s -1, bFGF: D T = (2.4±2.9) x cm 2 s -1  Washout rate IGF: k = (1.4±0.8) x s -1, bFGF: k = (2.1±2.2) x s -1  Peri-epicardial boundary permeability  IGF:  = (4.6±3.2) x cm s -1, bFGF:  =(11.9±10.1) x cm s -1  Enhanced effective diffusion, allowing for improved transport  Feasibility of computational studies of amount and time course of pericardial drug delivery to cardiac tissue, using experimentally derived values for physical parameters.

Part II: Electrical Wave Propagation in a Minimally Realistic Fiber Architecture Model of the Left Ventricle: Dynamics of Phase Singularies Xianfeng Song, Department of Physics, Indiana University Sima Setayeshgar, Department of Physics, Indiana University

Part II: Outline  Motivation  Model Construction  Numerical Results  Conclusions and Future Work

The Heart as a Physical System  Ventricular fibrillation (VF) is the main cause of sudden cardiac death in industrialized nations, accounting for 1 out of 10 deaths.  Strong experimental evidence suggests that self- sustained waves of electrical wave activity in cardiac tissue are related to fatal arrhythmias.  Mechanisms that generate and sustain VF are poorly understood.  Conjectured mechanism: Breakdown of a single spiral (scroll) wave into a disordered state, resulting from various mechanisms of spiral wave instability. W.F. Witkowksi, et al., Nature 392, 78 (1998) Patch size: 5 cm x 5 cm Time spacing: 5 msec

Focus of this work Distinguish the role in the generation of electrical wave instabilities of the “passive” properties of cardiac tissue as a conducting medium  geometrical factors (aspect ratio and curvature)  rotating anisotropy (rotation of mean fiber direction through heart wall)  bidomain description (intra- and extra-cellular spaces treated separately) from its “active” properties, determined by cardiac cell electrophysiology.

From idealized to fully realistic geometrical modeling Rectangular slabAnatomical canine ventricular model Minimally realistic model of LV for studying electrical wave propagation in three dimensional anisotropic myocardium that adequately addresses the role of geometry and fiber architecture and is:  Simpler and computationally more tractable than fully realistic models  Easily parallelizable and with good scalability  More feasible for incorporating realistic electrophysiology, electromechanical coupling, J.P. Keener, et al., in Cardiac Electrophysiology, eds. D. P. Zipes et al. (1995) Courtesy of A. V. Panfilov, in Physics Today, Part 1, August 1996 bidomain description

LV Fiber Architecture Early dissection results revealed nested ventricular fiber surfaces, with fibers given approximately by geodesics on these surfaces. Fibers on a nested pair of surfaces in the LV, from C. E. Thomas, Am. J. Anatomy (1957). Anterior view of the fibers on Hog Ventricles, which reveals the nested ventricular fiber surfaces,from C. E. Thomas, Am. J. Anatomy (1957).

Peskin asymptotic model: Fundamental principles and Assumptions  The fiber structure has axial symmetry  The fiber structure of the left ventricle is in near- equilibrium with the pressure gradient in the wall  The state of stress in the ventricular wall is the sum of a hydrostatic pressure and a fiber stress  The cross-sectional area of a fiber tube does not vary along its length  The thickness of the fiber structure is considerably smaller than its other dimensions. Mcqueen & Peskin Mathematical model on heart Crossection area of Peskin’s asymptotic model

Peskin Asymptotic model: Conclusions Fiber angle profile through LV thickness: Comparison of Peskin asymptotic model and dissection results Cross-section of the predicted middle surface (red line) and fiber surfaces (solid lines) in the r, z-plane.  The fibers run on a nested family of toroidal surfaces which are centered on a degenerate torus which is a circular fiber in the equatorial plane of the ventricle  The fiber are approximate geodesics on fiber surfaces, and the fiber tension is approximately constant on each surface  The fiber-angle distribution through the thickness of the wall follows an inverse- sine relationship

Model Construction  Nested cone geometry and fiber surfaces  Fiber paths  Geodesics on fiber surfaces  Circumferential at midwall subject to: Fiber trajectory: Fiber trajectories on nested pair of conical surfaces: inner surfaceouter surface

Governing Equations  Transmembrane potential propagation  Transmembrane current, I m, described by simplified FitzHugh-Nagumo type dynamics* v: gate variable Parameters: a=0.1,  1 =0.07,  2 =0.3, k=8,  =0.01, C m =1 * R. R. Aliev and A. V. Panfilov, Chaos Solitons Fractals 7, 293 (1996) C m : capacitance per unit area of membrane D: conductivity tensor u: transmembrane potential I m : transmembrane current

Numerical Implementation  Working in spherical coordinates, with the boundaries of the computational domain described by two nested cones, is equivalent to computing in a box.  Standard centered finite difference scheme is used to treat the spatial derivatives, along with first-order explicit Euler time-stepping.

Conductivity Tensor Local CoordinateLab Coordinate Transformation matrix R

Parallelization  The communication can be minimized when parallelized along azimuthal direction.  Computational results show the model has a very good scalability. CPUsSpeed up ± ± ± ± ± 0.85

Phase Singularities Color denotes the transmembrane potential. Movie shows the spread of excitation for 0 < t < 30, characterized by a single filament. Tips and filaments are phase singularities that act as organizing centers for spiral (2D) and scroll (3D) dynamics, respectively, offering a way to quantify and simplify the full spatiotemporal dynamics.

Keener’s Theory of Filament Dynamics  Consider the system described by a "reaction-diffusion" system of the form  Assume the system has the two-dimensional solution (i.e. independent of one spatial dimensions, say z) in the form of a stationary rotating vortex  By using singular perturbation theory, one can get the evolution equation for filament:  is the vortex rotation phase R: is a point on the filament N: the normal T: the tangent B: the binormal unit vector k: the curvature  : tortion of the filament J. Keener, P., Physica D 31, 269 (1988).

Tension of organizing filaments of scroll wave V. N. Biktashev etc simplify evolution equation to By analyzing the total length of filament, one can get: The coefficient b 2 plays a very special role in the dynamics of the filament. Its sign uniquely determines whether the length of the filament increases or decreases irrespective of the form of the filament and the values of the other parameters of the medium. Due to this property, the coefficient b 2 can be reasonably called the filament tension. V. N. Biktashev, A. V. Holden, and H. Zhang, Philosophical Transactions: Physical Sciences and engineering 347, 611 (1994)

Filament-finding Algorithm Find all tips “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Random choose a tip “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Search for the closest tip “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Make connection “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Continue doing search “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Continue “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Continue “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Continue “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm The closest tip is too far “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Reverse the search direction “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Continue “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Complete the filament “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Start a new filament “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Repeat until all tips are consumed “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding result FHN Model: t = 2 t = 999

Numerical Convergence Filament Number and Filament Length versus Heart size  The results for filament length agree to within error bars for three different mesh sizes.  The results for filament number agree to within error bars for dr=0.7 and dr=0.5. The result for dr=1.1 is slightly off, which could be due to the filament finding algorithm.  The computation time for dr=0.7 for one wave period in a normal heart size is less than 1 hour of CPU time using FHN-like electrophysiological model. Fully realistic model requires several days per heart cycle on a high-performance machine (Hunter, P. J., A. J. Pullan, et al. (2003). "MODELING TOTAL HEART FUNCTION." Annual Review of Biomedical Engineering 5(1): )

Scaling of Ventricular Turbulence Both filament length These results are in agreement with those obtained with the fully realistic canine anatomical model, using the same electrophysiology. A. V. Panfilov, Phys. Rev. E 59, R6251 (1999) Log(total filament length) and Log(filament number) versus Log(heart size) The average filament length, normalized by average heart thickness, versus heart size

Work in progress  The role of fiber architecture Whether the anisotropy in our model induce instability or suppress instability of wave propagation  The role of geometry What is the role of cone geometry comparing with the simple rectangular slab given the same physiological model.

Rotating anisotropy

Coordinate System

Governing Equations

Perturbation Analysis

Scroll Twist Solutions

Filament Motion

Filament motion (cont’d)

Filament Tension

Conclusions and Future Work  We have constructed and implemented a minimally realistic fiber architecture model of the left ventricle for studying electrical wave propagation in the three dimensional myocardium.  Our model adequately addresses the geometry and fiber architecture of the LV, as indicated by the agreement of filament dynamics with that from fully realistic geometrical models.  Our model is computationally more tractable, allowing reliable numerical studies. It is easily parallelizable and has good scalability.  As such, it is more feasible for incorporating  Realistic electrophysiology  Bidomain description of tissue  Electromechanical coupling

Part III: Calcium Dynamics in myocyte Xianfeng Song, Department of Physics, Indiana University Sima Setayeshgar, Department of Physics, Indiana University

Part III: Outline  The importance and background of calcium signaling in myocyte  Future work

Overview of Calcium Signaling Berridge, M. J., M. D. Bootman, et al. (1998). "Calcium - a life and death signal." Nature 395(6703):  Elementary events (red) result from the entry of external Ca 2+ across the plasma membrane or release from internal stores in the endolasmic or sarcoplasmic reticulum (ER/SR).  Global Ca 2+ signals are produced by coordinating the activity of elementary events to produce a Ca 2+ wave that spreads throughout the cell.  The activity of neighboring cells within a tissue can be coordinated by an intercellular wave that spreads from one cell o the next.

Fundamental elements of Ca 2+ signaling machinery  Calcium stores: External and internal stores, i.e. Endoplasmic Reticulum (ER), Sarcoplasmic Reticulum (SR), Mitochondria  Calcium buffers: Calcium is heavily buffered in all cells, with at least 99% of the available Ca 2+ bound to large Ca 2+ -binding proteins., such as Calmodulin, Calsequestrin.  Calcium pumps: Ca 2+ is moved to Calcium stores by varies pumps.  Calcium channels: Ca 2+ can enter the cytoplasm from calcium stores via varies channels, i.e. ryanodine receptors (RyR) and inositol trisphosphate receptors (IP 3 R). Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer

Receptor clustering  RyR and IP 3 R channels are spatially organized in clusters, with the distance between clusters are approximate two order magnitude larger than the distances between channels within one cluster.  Chemotaxis receptors in E.coli are organized in clusters, proven to make:  Noise reduction  Signal amplification High resolution image showing a Ca 2+ puff evoked by photoreleased InsP 3 which demonstrate an IP 3 R cluster (From Yao, Y. etc, Journal of Physiology 482: ) Skoge, M. L., R. G. Endres, et al. (2006), Biophys. J. 90(12):

Ventricular Myocyte  Some facts about myocyte  The typical cardiac myocyte is a cylindrical cell approximately 100  m in length by 10  m in diameter  Three physical compartments: the cytoplasm, the sarcoplasmic reticulum (SR) and the mitochondria.  The junctional cleft is a very narrow space between the SL and the SR membrane.  Calcium Induced Cacium Release (CICR) A small amount of Ca2+ goes into the junctional cleft thus induce large scale of Ca2+ release from calcium stores (mainly SR).  Excitation-Contraction Coupling (ECC) The depolarization of the membrane initial a small amount of Ca2+, thus induce CICR and initiate contraction. Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer Ventricular Myocyte Structure Calcium induced Calcium release

The role of Calsequestrin  Calsequestrin is the buffer inside SR, most of which are located close to RyRs.  Calsequestrins play an important role during CICR. One proposal about Calsequestrin: (A) The channel opens, Ca 2+ adsorbed to linear CSQ polymers feeds rapid release. (B) The polymers are depleted Ca 2+ thus disassemble. (C) Depletion becomes deeper as Ca 2+ replenishes the proximate store and the CSQ polymers reassembles. (from Launikonis etc, PNAS 103(8) Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer

Calmodulin  A kind of calcium-binding proteins which contains four calcium binding sites. The binding on both terminal domains are in the cooperative manner.  Calmodulin undergoes a conformational change upon binding to calcium, which enables it to bind to specific proteins for a specific response. The Protein Data Bank. Nucleic Acids Research, 28 pp (2000). Shifman et al. PNAS, 103 (38): 13968

Future directions  Why receptors are clustering together? What is the design principle underlying the receptor clustering?  To fully understand the role of Calsequestrin: how best can Calsequestrin facilitate the calcium release?  The role of cooperativity of Calcium binding to Calmodulin as noise reduction in calcium signaling pathway?

Thanks!!

Overview of Calcium Signals  Calcium serves as an important signaling messenger.  Extracellular sensing  The regulation of cardiac contractility by Ca 2+  Ca 2+ signaling during embryogenesis  Calcium sparks and waves Spiral Ca 2+ wave in the Xenopus oocytes. The image size is 420x420 um. The spiral has a wavelength of about 150 um and a period of about 8 seconds. Part B is simulation. Ca sparks in an isolated mouse ventricular myocyte. Mechanically stimulated intercellular wave in airway epithelial cells Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer

Motivation: Why stochastic  The global Calcium wave is comprised by local release events, called puffs.  Binding kinetics is by itself a stochastic process.  Receptor number is small, i.e., Calcium sparks are thought to consist of Ca 2+ release from between 6 and 20 RyRs. (Rice, J. J., M. S. Jafri, et al. (1999). "Modeling Gain and Gradedness of Ca2+ Release in the Functional Unit of the Cardiac Diadic Space." Biophys. J. 77(4): )  Diffusive noise is large. The noise is limited by l is the effective size of receptors or receptor array. (W. Bialek, and S. Setayeshgar, PNAS 102, 10040(2005)) From single localized Calcium response to a global calcium wave Schematic representation of a cluster of m receptors of size b, distributed uniformly on a ring of size a. W. Bialek, and S. Setayeshgar, PNAS 102,10040(2005) Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer

Ventricular Myocyte  The typical cardiac myocyte is a cylindrical cell approximately 100  m in length by 10  m in diameter and is surrounded by a cell membrane known as the sarcolemma (SL)  Three physical compartments: the cytoplasm, the sarcoplasmic reticulum (SR) and the mitochondria.  The junctional cleft is a very narrow space between the SL and the SR membrane. Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer Ventricular Myocyte Structure

Calcium signaling in Ventricular Myocyte  Ca-Induced Ca Release (CICR) A small amount of Ca 2+ goes into the junctional cleft thus induce large scale of Ca 2+ release from calcium stores (mainly SR).  Excitation-Contraction Coupling (ECC) The depolarization of the membrane initial a small amount of Ca 2+, thus induce CICR and initiate contraction. Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer