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

2. 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: Dynamics of Phase Singularities  Part III: Calcium Dynamics in the Myocyte

3. 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

4. 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): 197-203. [2] Gregor, T., W. Bialek, R. de Ruyter van Steveninck, et al. (2005). PNAS 102(51). [3] Nicholson, C. (2001), Rep. Prog. Phys. 64, 815-884. Need for careful characterization of diffusion constants governing various biophysical processes.

5. 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) 98:2331-2333. [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.

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

7. 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: 18000 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

8. 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

9. 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

10. 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

11. 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

12. Example of Numerical Fits to Experiments Agent ConcentrationError surface

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

14. 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

15. 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, = 2.11. (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 10 -7 cm 2 s -1  bFGF: D ~ 3 x 10 -7 cm 2 s -1 Our fitted values are in order of 10 -6 - 10 -5 cm 2 sec -1, 10 to 50 times larger !!

16. 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

17. 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, 18403 (2005).

18. 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 10 -5 cm 2 s -1, bFGF: D T = (2.4±2.9) x 10 -5 cm 2 s -1  Washout rate IGF: k = (1.4±0.8) x 10 -3 s -1, bFGF: k = (2.1±2.2) x 10 -3 s -1  Peri-epicardial boundary permeability  IGF:  = (4.6±3.2) x 10 -6 cm s -1, bFGF:  =(11.9±10.1) x 10 -6 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.

19. 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

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

21. The Heart as a Physical System

22. Motivation  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.  Goal is to use analytical and numerical tools to study the dynamics of reentrant waves in the heart on physiologically realistic domains.  And … the heart is an interesting arena for applying the ideas of pattern formation. W.F. Witkowksi, et al., Nature 392, 78 (1998) Patch size: 5 cm x 5 cm Time spacing: 5 msec

23. Big Picture What are the mechanisms underlying the transition from ventricular tachychardia to fibrillation? How can we control it? TachychardiaFibrillation Paradigm: Breakdown of a single spiral (scroll) wave into disordered state, resulting from various mechanisms of spiral wave instability (Courtesty of Sasha Panfilov, University of Utrecht)

24. Focus of Our 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. * Jianfeng Lv: Analytical and computational studies of the bidomain model of cardiac tissue as a conducting medium

25. Motivated by … “Numerical experiments”: Winfree, A. T. in Progress in Biophysics and Molecular Biology (1997)… Panfilov, A. V. and Keener, J. P. Physica D (1995): Scroll wave breakup due to rotating anisotropy Fenton, F. and Karma, A. Chaos (1998): Rotating anisotropy leads to “twistons”, eventually destabilizing scroll filament Analytical work: In isotropic excitable media Keener, J. P. Physica D (1988) … Biktashev, V. N. and Holden, A. V. Physica D (1994) … In anisotropic excitable media Setayeshgar, S. and Bernoff, A. J. PRL (2002)

26. 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

27. 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, revealing the nested ventricular fiber surfaces, from C. E. Thomas, Am. J. Anatomy (1957). From Textbook of Medical Physiology, Guyton and Hall. 3d conduction pathway with uniaxial anisotropy: Enhanced conduction along fiber directions. c par = 0.5 m/sec c perp = 0.17 m/sec

28. Peskin Asymptotic Analysis of the Fiber Architecture of the LV: 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.

29. Peskin Asymptotic Model: Results 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

30. 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

31. Electrophysiology: Governing Equations  Transmembrane potential propagation  Transmembrane current, I m, described by simplified FitzHugh-Nagumo type dynamics [1] v: gate variable Parameters: a=0.1,  1 =0.07,  2 =0.3, k=8,  =0.01, C m =1 [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

32. 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.

33. Conductivity Tensor Local CoordinateLab Coordinate Transformation matrix R

34. Parallelization  The communication can be minimized when parallelized along azimuthal direction.  Computational results show the model has a very good scalability. CPUsSpeed up 21.42 ± 0.10 43.58 ± 0.16 87.61 ±0.46 1614.95 ±0.46 3228.04 ± 0.85

35. 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.

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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

48. 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

49. 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

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

51. 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 [1] [1] Hunter, P. J., A. J. Pullan, et al. (2003), Annual Review of Biomedical Engineering 5(1): 147-177.

52. 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. [1] [1] 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

53. Conclusions so far…  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

54. Work in Progress  Computational: Investigate role of geometry and fiber architecture on scroll wave stability (Preliminary results indicate filament instability is suppressed in minimally realistic model versus rectangular slab!)  Analytical: Extend perturbation analysis of scroll waves in the presence of rotating anisotropy [1] to include filament motion [1] Setayeshgar, S. and Bernoff, A. J. PRL (2002).

55. Rotating anisotropy

56. Coordinate System

57. Governing Equations

58. Perturbation Analysis

59. Scroll Twist Solutions Scroll Twist,  z Rotating anisotropy generated scroll twist, either at the boundaries or in the bulk. Twist

60. Significance? In isotropic excitable media (  = 1), for twist > twist critical, straight filament undergoes buckling (“sproing”) instability [1] Henzi, Lugosi and Winfree, Can. J. Phys. (1990). What happens in the presence of rotating anisotropy (  > 1)??

61. Filament Motion

62. Filament motion (cont’d)

63. Filament Tension Destabilizing or restabilizing role of rotating anisotropy!!

64. Part III: Calcium Dynamics in the Myocyte Xianfeng Song, Department of Physics, Indiana University Sima Setayeshgar, Department of Physics, Indiana University

65. Part III: Outline  Importance and background on calcium signaling in myocytes  Future directions

66. Overview of Calcium Signaling From Berridge, M. J., M. D. Bootman, et al. (1998). "Calcium - a life and death signal." Nature 395(6703): 645-648.  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.

67. 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

68. Ventricular Myocyte  Some facts about myocytes  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. From Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer Ventricular Myocyte Structure Calcium induced Calcium release

69. Future Directions  What is the role of receptor clustering on calcium signaling?  What is the role of the buffer Calsequestrin in facilitating calcium release?

70. 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. Analogy with chemotaxis receptor clustering in E.coli shown to be important in [1]  Signal amplification  Noise reduction 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: 533-553.) [1] Skoge, M. L., R. G. Endres, et al. (2006), Biophys. J. 90(12): 4317.

71. The Role of Calsequestrin  Calsequestrin is the buffer inside SR, most of which are located close to RyRs.  Calsequestrin plays an important role during CICR.  Role of Calsequestrin polymerization/depolymerization on its diffusive uptake of Ca 2+ as a store? (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 et al, PNAS 103(8) 2982-7 (2005)) Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer

72. Thanks!!

73. 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

74. 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): 1871-1884.)  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

75. 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

76. 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