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Heart: From macroscopic to microscopic Xianfeng Song Advisor: Sima Setayeshgar January 9, 2007 Outline Transport Through the Myocardium of Pharmocokinetic Agents Placed in the Pericardial Sac: Insights From Physical Modeling Electrical Wave Propagation in a Minimally Realistic Fiber Architecture Model of the Left Ventricle Calcium dynamics: exploring the stochastic effect 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 Pericardial Delivery: Motivation 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 Recent experimental feasibility of pericardial access 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. 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. Vperi (human) =10ml – 50ml Diffusion in biological processes Diffusion plays a crucial role in brain function Nicholson, C. (2001), “Diffusion and related transport mechanisms in brain tissue”, Rep. Prog. Phys. 64, 815-884 Diffusion is important during early Drosophila embryonic pattern formation Gregor, T., W. Bialek, et al. (2005). "Diffusion and scaling during early embryonic pattern formation." Proceedings of the National Academy of Sciences 102(51): Protein diffusion in single cells M. B. Elowitz et al. “Proteins diffusion in single cells”, J. Bacteriology Pericardial Delivery: 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 (125I-IGF, MW: 7734 Da) Basic Fibroblast Growth Factor (125I-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 Experimental Procedure At t = T (1h or 24h), sac fluid is distilled: CP(T) Tissue strips are submerged in liquid nitrogen to fix concentration. Cylindrical transmyocardial specimens are sectioned into slices: CiT(x,T) x denotes i CT(x,T) = i CiT(x,T) x: depth in tissue Mathematical Modeling Goals Determine key physical processes, and extract governing parameters Assess the efficacy of drug 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: DT Substrate washout in myocardium (through the intramural vascular and lymphatic capillaries): k Idealized Spherical Geometry Pericardial sac: R2 – R3 Myocardium: R1 – R2 Chamber: 0 – R1 R1 = 2.5cm R2 = 3.5cm Vperi= 10ml - 40ml Governing Equations and Boundary Conditions Governing equation in myocardium: diffusion + washout CT: concentration of agent in tissue DT: 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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2005, Los Angeles Numerical Fits to Experiments Conce Drug Concentration 1 Molecule per ml = 1.3 x10-11 picograms per ml Error surface Fit Results Numerical values for DT, k, consistent for IGF, bFGF to within experimental errors Time Course from Simulation Parameters: DT=7×10-6cm2s-1 k=5×10-4s-1 =3.2×10-6cm2s2 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 cm2s-1 bFGF: D ~ 3 x 10-7 cm2s-1 Our fitted values are in order of 10-6 - 10-5 cm2sec-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 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2005, Los Angeles 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" results in larger diffusion constants. Conclusion 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: DT = (9±3) x 10-6 cm2s-1, bFGF: DT = (6±3) x 10-6 cm2s-1 Washout rate IGF: k = (8±3) x 10-4 s-1, bFGF: k = (9±3) x 10-4 s-1 Peri-epicardial boundary permeability IGF: = (2.7±0.8) x 10-6 cm s-1, bFGF: = (6.0±1.6) 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. Electrical Wave Propagation in a Minimally Realistic Fiber Architecture Model of the Left Ventricle Xianfeng Song, Department of Physics, Indiana University Sima Setayeshgar, Department of Physics, Indiana University Outline Motivation Model Construction Numerical Results Conclusions and Future Work 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 selfsustained 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 slab J.P. Keener, et al., in Cardiac Electrophysiology, eds. D. P. Zipes et al. (1995) Anatomical canine ventricular model Courtesy of A. V. Panfilov, in Physics Today, Part 1, August 1996 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, 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). Peskin asymptotic model: first principles derivation of toroidal fiber surfaces and fiber trajectories as approximate geodesics Fiber angle profile through LV thickness: Comparison of Peskin asymptotic model and dissection results, from C. S. Peskin, Comm. in Pure and Appl. Math. (1989). Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Model Construction Nested cone geometry and fiber surfaces Fiber paths Geodesics on fiber surfaces Circumferential at midwall 2 L f ( , 1 d , )d d f d f d ' z subject to: 0 0 Fiber trajectory: L 0 1 1 1 a 2 sec 1 Fiber trajectories on nested pair of conical surfaces: inner surface outer surface Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Governing Equations Transmembrane potential propagation Cm u ( Du ) I m t Cm: capacitance per unit ar D: diffusion tensor u: transmembrane potentia Im: transmembrane current Transmembrane current, Im, described by simplified FitzHugh-Nagumo type dynamics* I m ku(u a)(u 1) uv v v 1 v ku(u a 1 t 2 u v: gate variable Parameters: a=0.1, 1=0.07, 2=0.3, k=8, =0.01, Cm=1 * R. R. Aliev and A. V. Panfilov, Chaos Solitons Fractals 7, 293 (1996) Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore 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. Diffusion Tensor Transformation matrix R Local Coordinate Dlocal D// 0 0 0 D p1 0 0 0 D p 2 Lab Coordinate Dlab R 1 Dlocal R Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Parallelization The communication can be minimized when parallelized along azimuthal direction. Computational results show the model has a very good scalability. CPUs Speed up 2 1.42 ± 0.10 4 3.58 ± 0.16 8 7.61 ±0.46 16 14.95 ±0.46 32 28.04 ± 0.85 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Phase Singularities 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. Color denotes the transmembrane potential. Movie shows the spread of excitation for 0 < t < 30, characterized by a single filament. Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding Algorithm “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 Find all tips Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding Algorithm “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 Random choose a tip Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding Algorithm “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 Search for the closest tip Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding Algorithm “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 Make connection Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding Algorithm “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 Continue doing search Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding Algorithm “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 Continue Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding Algorithm “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 Continue Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding Algorithm “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 Continue Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding Algorithm “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 The closest tip is too far Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding Algorithm “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 Reverse the search direction Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding Algorithm “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 Continue Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding Algorithm “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 Complete the filament Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding Algorithm “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 Start a new filament Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding Algorithm “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 Repeat until all tips are consumed Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Filament-finding result t=2 FHN Model: t = 999 Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Numerical Convergence 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. Filament Number and Filament Length versus Heart size 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. Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Scaling of Ventricular Turbulence Log(total filament length) and Log(filament number) The average filament length, normalized by Both filament length versus Log(heart size) average heart thickness, versus heart size 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) Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore 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 Biodomain description of tissue Electromechanical coupling Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2006, Baltimore Calcium Dynamics: Exploring the stochastic effect Xianfeng Song, Department of Physics, Indiana University Sima Setayeshgar, Department of Physics, Indiana University Outline Brief introduction to calcium dynamics in myocyte Motivation: why stochastic Future work Overview of Calcium Signals • Calcium sparks and waves Ca sparks in an isolated mouse ventricular myocyte. Spiral Ca2+ wave in the Xenopus oocyte. 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. • Calcium serves as an important signaling messenger. – – – Extracellular sensing Ca2+ signaling during embryogenesis The regulation of cardiac contractility by Ca2+ Mechanically stimulated intercellular wave in airway epithelial cells Fundamental elements of Ca2+ signaling machinery •Calcium Stores: external stores and internal stores: Endoplasmic Reticulum (ER), Sarcoplasmic Reticulum (SR), Mitochondria •Calcium pumps: Ca2+ is moved to these stores by a Ca2+/Na+ exchanger, plasma membrane Ca2+ pumps and SERCA pumps. •Calcium channels: Ca2+ can enter the cytoplasm via receptor-operated channels (ROC), store-operated channels (SOC), voltage-operated channels (VOC), ryanodine receptors (RyR) and inositol trisphosphate receptors (IP3R). Ventricular Myocyte The typical cardiac myocyte is a cylindrical cell approximately 100 um in length by 10um 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 primary function of SR is to store Ca for release upon cellular excitation. The junctional cleft is a very narrow space between the SL and the SR membrane. The SR release channel, or ryanodine receptor (RyR) is found almost entirely within the part of the SR membrane which communicates with the juntional cleft. Ventricular Myocyte and Excitation-Contraction coupling • Ca-Induced Ca Release (CICR) • • • • From the resting state (channel closed), Ca may bind rapidly to a relatively low affinity site (1), therby activating the RyR. Ca may then bind more slowly to a second higher affinity site (2) moving the release channel to an inavitive state. As cytoplasmic [Ca] decreases, Ca would be expected to dissocaiate from the lower affinity activating site first and then more slowly from the inactivating site to return the channel to the resting state. Excitation-Contraction Coupling (ECC) A small amount of Ca is initiated by depolarization of the membrane, thus induce CICR, initiate contraction. Motivation: why stochastic Binding kinetics is by itself a stochastic process. Receptor number is small, i.e., in Xenopus oocytes, IPR are arranged in clusters with a density of about 1 per 30μm2 with each cluster containing about 25 IPRs. Diffusive noise is large. The noise is limited by Schematic representation of a cluster ofmreceptors of size b, distributed uniformly on a ring of size a. l is the effective size of receptors or receptor array. (W. Bialek, and S. Setayeshgar, PNAS 102, 10040(2005)) The global Calcium wave are comprised by local release events, called puffs. From single localized Calcium response to a global calcium wave Future work