Advisor: Sima Setayeshgar

Advisor: Sima Setayeshgar
Physics of the Heart: From the macroscopic to the microscopic Xianfeng Song Advisor: Sima Setayeshgar April 17, 2007 Good afternoon. Thanks everyone for coming to my candidacy seminar. My name is Xianfeng Song, and my advisor is Sima Setayeshgar. The topic of my talk is physics of Heart: from macroscopic to microscopic.

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 There are three parts in my talk. The first part is about “Transport through the myocardium of pharmocokinetic Agents placed in the pericardial sac: insights from physical modeling. This part of my work is directly motivated by experiments and makes use of experimental data. The second part is about the electrical wave propagation in a minimally realistic fiber architecture model of the left ventricle. The third part is about calcium dynamics: exploring the stochastic effect. I will briefly introduce the calcium dynamics and myocyte and the future works. The unifying theme is understanding biophysical mechanisms in the heart, from the macroscopic to the cellular scales.

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 This is the first part of my talk. It was done in collaboration with Keith March etc from IUPUI medical school. The focus of this work is by modelling the transport through the myocardium of drugs placed in the pericardial sac to infer the information of effective diffusion constant to evaluate the efficacy of this drug delivery method

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. First, I will briefly introduce the background and motivation of this work. As we all know, diffusion is a very important physical processes in varied biological systems, from the cell level to organ level. Here are the points made by the literature. The rate of protein diffusion in bacterial cytoplasm may constrain a variety of cellular functions and limit the rates of many biochemical reactions in vivo. (response times and reaction rates in E. coli often depend on the movement of proteins from one location to another in the cell. These proteins amy have regulatory or signaling functions, or they may act as enzymes or substrates for cellular reactions. In bacterial, diffusion may be the primary means of intracellular movement. ) In drosophila embryo, diffusion sets a relationship between length and time scales. Diffusion also play a crucial role in brain function. Diffusion control the delivering of glucose and oxygen from the vascular system to brain cells and also moves informational substances between cells. Need for careful characterization of diffusion constants governing various biophysical processes. [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,

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] This is the background of this work. The pericardial sac is a self-contained fluid-filled layer surrounding the heart, whose purpose is to lubricate the heart in its motion in the chest cavity. As such, it can be used as a drug reservoir for delivery of various therapeutic agents, such as anti-arrhythmic and angiogenic drugs, to the heart muscle as well as coronary vasculature. Indeed, catheter-based placement of agents into the pericardial space has recently become feasible and safe. So pericardial delivery become feasible and promising. Vperi (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 In this part, I will discuss our work on mathematical modeling and analysis of experimental data on pericardial delivery of test agents to the myocardium. The goal is to assess the efficacy of considering the pericardial space as a novel route for drug administration and extract the effective diffusion constant. The outline of this part is as follows: First, I will briefly describe the experiments performed by Keith March’s group at the IUPUI medical school. Next, I will present our mathematical model for this process and results from its comparison with experimental data. Finally, I will give our conclusions from the modeling and data analysis.

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: Da) Initial concentration delivered to the pericardial sac at t=0 200 or 2000 mg in 10 ml of injectate Harvesting at t=1h or 24h after delivery The experiments were performed on juvenile farms pigs using the radiotracer method to determine the concentration of radio-iodinated test agents in the tissue from rate of radiactive decay. These agents, IGF and bFGF, are relevant therapeutic growth factors. Different initial amounts (200 and 2000 micrograms in an injectate volume of 10 ml) were delivered to the pericardial space of an anesthetisized animal at t=0. At t=1 hour or t=24 hours, the heart was harvested and measurements were made as follows:

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 Samples were taken from the pericardial sac fluid, giving C_P(T). Tissue strips were excised and fixed in liquid nitrogen. Cylindrical transmyocardial specimens were sectioned into slices as shown, giving C_T(x,T), where x is the thickness through the tissue. We focus on the data obtained from the left ventricle only, and average C_T^i(x,t) obtained at different (total of 9) spatial locations to obtain a single concentration profile C_T(x, T). Hence, the experimental data provides only two snapshots of the dynamics, at T = 1 hr and T = 24 hours. Each snapshot corresponds to a single sacrificed animal. For each agent and each initial concentration, several experiments were carried out at each time snapshot. The data used here corresponds to a total of 17 animals. CT(x,T) = Si CiT(x,T) x: depth in tissue

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: DT Substrate washout in myocardium (through the intramural vascular and lymphatic capillaries): k Our goals are to establish a minimal physical model for drug penetration in the myocardium using this mode of delivery and to extract numerical values for the governing parameters by comparison with experimental data. The key processes in our model are: Substrate transport across boundary layer between pericardial sac and myocardium, described by the parameter alpha which is the permeability of the peri/epicardium boundary Substrate diffusion in the myocardium, described by the effective diffusion constant D_T Substrate washout through the vascular and lymphatic capillaries, described by the rate k

Idealized Spherical Geometry
Pericardial sac: R2 – R3 Myocardium: R1 – R2 Chamber: 0 – R1 R1 = 2.5cm R2 = 3.5cm Vperi= 10ml - 40ml We adopt an idealized spherical geometry, with dimensions consistent with the porcine heart, where the inner radius of the heart wall is 2.5 cm, the outer radius is 3.5 cm and the volume of the pericardial space is allowed to vary between 10 and 40 ml.

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 The governing equation in the myocardium includes diffusion and washout. The pericardial sac is assumed to be well-mixed and self- contained. The boundary conditions at the peri/epicardial boundary are given by 1) conservation of drug number (number of agent molecules leaving the pericardial sac equal the number entering the tissue) and 2) the fact that the drug current at this boundary is proportional to the difference in concentration in the tissue and pericardial space, with the constant of proportionality given by the permeability of this boundary layer. The concentration at the inner radius of the heart wall is taken to be zero.

Example of Numerical Fits to Experiments
We numerically fitted simulations of the model equations to the experimental data, extracting best-fit values for D_T, k, and alpha. On the left, is an example of the data showing the concentration of IGF at 24hr through the thickness of the tissue and the resulting fit for an initial delivery amount of 2000 micrograms. Each slice corresponds to 0.4 mm; we have included only 10 slices in the fits since the concentration below this point was at the background. Each data point corresponds to the concentration at that depth averaged over pigs (4 in this case) and 9 LV locations. The error bars are the error on the mean concentration obtained in this way. On the right is the Chi-square surface as a function of D and k (for example) clearly showing a minimum. Agent Concentration Error surface

Fit Results Numerical values for DT, k,  consistent for IGF, bFGF
Our fit results, obtained separately for each experimental group, show consistency in the values of D_T, k and alpha for IGF and bFGF to within the same order. Given the large variation of the experimental data, this result is reasonable. Numerical values for DT, k,  consistent for IGF, bFGF

Time Course from Simulation
For example, using these experimentally determined parameters we can study the time-course of agent concentration in the myocardium shown here for IGF. Here shows two movies with reasonable parameter values for IGF. The time-course of concentration in the tissue increase everywhere during the first 1 hour, but decrease afterwards. The turning point of the trend is roughly at 1 hour. Different parameter values gives different turning point. But the whole trend would be like this: the concentration in the tissue reaching a maximum at some time after delivery and then decaying and the concentration in pericardial sac will always be decaying . This makes the fitting process non-trivial. Note that for different parameter values, such as with a smaller washout rate (?), the concentration ??? *** Xianfeng: here I would like us to mention that the time course of concentration in the tissue can be nontrivial, with the concentration reaching a maximum at some time after delivery and then decaying. ********** Such computational experiments can help guide details of pericardial delivery in terms of initial drug amount in one-time or continuous, pumped administration (which we have not discussed here). Parameters: DT = 7×10-6cm2s-1 k = 5×10-4s-1 a = 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, l= 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 To verify our fit results, we consider the diffusion coefficients for IGF and bFGF obtained from the Stokes-Einstein relation, using values for the viscosity of the extracellular space assumed to mostly water and the molecular radii of these agents, and also accounting for tortuosity of the myocardium. We find these values to be on the order 10^-7 cm^2/sec, 10 to 50 times smaller than our results from fits to the experimental data. We note that unlike earlier experiments on measurement of diffusion constants in excised tissue samples, our experiments were carried out in the live, beating heart. This raises two possible mechanisms that may account for the enhanced effective diffusion constants. Our fitted values are in order of 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 One possible mechanism is that the agent 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.

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. A second possible mechanism is the following: The extracellular volume consists to first approximation of water which is incompressible. Hence, expansion and contraction of the fiber bundles leading to changes in tissue pore size can generate extracellular fluid flow, resulting in effective stirring. The reference here by Gregor etc, also shows the enhanced diffusion constant for dextran molecules in early drosophila embryo which is consistent with a random “stirring of the cytoplasm. [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: DT = (1.7±1.5) x 10-5 cm2s-1, bFGF: DT = (2.4±2.9) x 10-5 cm2s-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: a = (4.6±3.2) x 10-6 cm s-1, bFGF: a =(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. In conclusion to the first part, our model for pericardial delivery accounting for effective diffusion and washout in the myocardium appears to be consistent with experiments, despite its simplicity. We have extracted the order for numerical values of the key physical parameters for IGF and bFGF. An important finding is that the values of the effective diffusion constants for IGF and bFGF are larger than predicted for diffusion of agents of this size in myocardium. Our work demonstrates the feasibility of using models of the kind we have developed here for computational studies of the amount and time course of pericardial drug delivery.

Part II: Electrical Wave Propagation in a Minimally Realistic Fiber Architecture Model of the Left Ventricle: Dynamics of Phase Singularies Here comes my second part: “Electrical wave propagation in a minimally realistic fibre architecture model of the left ventricle”. 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 outline of this part is as the follows: First, I will briefly discuss the motivation and goals for constructing such a model. Next, I will talk about how we construct it. Later, I will present some numerical results and their comparison with fully realistic models. Finally, I will give our conclusions and prospects for future work.

The Heart as a Physical System
Image appeared on cover of Physics Today approximately 10 years ago, showing among other physics isn’t heartless. It shows computational results on the propagation of a spiral wave of electrical excitation in an anatomically realistic model of the canine ventricles. The colors denote regions in the tissue where the transmembrane potential (which signal contraction) is high. The range in colors denotes time. If this were a movie, one would see a spiral rotating about a fixed core as a function of time. It is important to note that a spiral wave of electrical activity corresponds to unhealthy function of the heart (where the healthy functioning corresponds to plane wave propagation). Any number of mechanisms that give rise to this single spiral can lead to its further breakup intomultiple spirals, ultimately resulting in a spatiotemporally disordered wave state that is fatal within minutes of onset.

W.F. Witkowksi, et al., Nature 392, 78 (1998)
Motivation W.F. Witkowksi, et al., Nature 392, 78 (1998) 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. Ventricular fibrillation (VF) is the main cause of sudden cardiac death in industrialized nations, accounting for 1 out of 10 deaths. There is strong experimental evidence that self-sustained waves of electrical wave activity in cardiac tissue (shown on the right) are related to fatal arrhythmias. Mechanisms that generate and sustain VF are poorly understood. One conjectured mechanism is that VF arises from the breakdown of a single spiral (scroll) wave into a disordered state, resulting from various mechanisms of spiral wave instability. Patch size: 5 cm x 5 cm Time spacing: 5 msec

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

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. The main thrust of our work is to distinguish the role in the generation of electrical wave instabilities of the “passive” properties of cardiac tissue as a conduction medium, such as -- geometrical factors (aspect ratio and curvature) -- rotating anisotropy (rotation of the mean fiber direction through the 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

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)

From Idealized to Fully Realistic Geometrical Modeling
Rectangular slab Anatomical canine ventricular model 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 Much progress has been made in our understanding of the heart, from modeling it as homogeneous and isotropic excitable medium. However, the heart is a highly complicated, heterogeneous physical system, with strong anisotropy and nontrivial geometry. It has been well established that geometry and fiber architecture strongly influence patterns of electrical wave activity, where for example: -- numerical studies indicate that rotating anisotropy of cardiac tissue leads to twist-induced scroll wave instability -- clinical studies indicate that enlarged hearts are more susceptible to heart attacks Numerical work has ranged from consideration of the heart as an idealized rectangular slab with rotating anisotropy, to anatomically realistic whole heart models. In the latter case, reliable, numerically resolved simulations of detailed models of electrophysiology and mechanics continue to be at the limit of computational capability. To this end, we have constructed a minimally realistic fiber architecture model of the left ventricle, which is the powerhouse of the heart. Such a model offers a good compromise between computational resources, complexity of electrophysiological models and electromechanical coupling, and applicability to basic research. 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. 3d conduction pathway with uniaxial anisotropy: Enhanced conduction along fiber directions. Dissection results (sketched in one of the early, classic papers on the left) have shown that the ventricles consist of nested fiber surfaces, with muscle fibers tracing out approximate geodesics on these surfaces. The first picture shows the anterior view of the fibers on hog ventricles, which reveals the nested ventricular fiber surfaces, with the fiber on each fiber surfaces are approximate geodesics. The second pictures shows only the inner and outer fiber surfaces around left ventricle which shows the outer fiber goes this direction and inner fibers are going this direction. From Textbook of Medical Physiology, Guyton and Hall. cpar = 0.5 m/sec cperp = 0.17 m/sec Anterior view of the fibers on hog ventricles, revealing the nested ventricular fiber surfaces, from C. E. Thomas, Am. J. Anatomy (1957). Fibers on a nested pair of surfaces in the LV, from C. E. Thomas, Am. J. Anatomy (1957).

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. Motivated by these results, Peskin has derived an asymptotic fiber architecture model of the left ventricle from mechanical first principles. These principles include: The fiber structure has axial symmetry and cross section area look like the picture, which is quite right for left ventricle around which has a large cylindrical part. The fiber structure of the left ventricle is in near equilibrium with the pressure gradient in the wall, which gives this equilibrium equation. The state of stress in the ventricular wall is the sum of a hydrostatic pressure and a fiber stress which give this equation. The cross-sectional area of a fiber tube does not vary along its length, gives the equation. The thickness of the fiber structure is considerably smaller than its other dimensions, which is right for human heart.

Peskin Asymptotic Model: Results
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 Cross-section of the predicted middle surface (red line) and fiber surfaces (solid lines) in the r, z-plane. Given by the principles above, Peskin derived an asymptotic model, which gives the following conclusions: The fiber run on a nested family of toroidal surfaces which are centerd on a degenerate torus which is a circular fiber in the equatorial plane of the ventricle, like the first picture shows. The fiber are approximate geodesics on fiber surfaces which is consistent with experimental results. The fiber-angle distribution through the thickness of the wall follows an inverse-sine relationship. The second picture here shows the comparison between the derived result and the experimental results. Fiber angle profile through LV thickness: Comparison of Peskin asymptotic model and dissection results

Fiber trajectories on nested pair of conical surfaces:
Model Construction Nested cone geometry and fiber surfaces Fiber paths Geodesics on fiber surfaces Circumferential at midwall Fiber trajectories on nested pair of conical surfaces: Based on Peskin’s derived geometry but seeking a geometry that is computationally more tractable, we have constructed a nested cone architecture. The cones are taken to be of circular cross-section (shown on the upper right): the heavy lines denote the endo- and epicardium, and the dotted line represents the midwall. The inner and outer cones of each nested pair are given by the slopes describing the endo- and epicardium, respectively. The geometry we used here is the same as the initial condition of the model computed by immersed boundary condition by Mcqueen and Peksin. The fiber paths are obtained from the Euler-Lagrange equations minimizing the fiber length, subject to the condition that the fibers be circumferential at the midwall. Fiber trajectories on the inner and outer sheets of a nest cone pair are shown on the lower right. subject to: Fiber trajectory: inner surface outer surface

Electrophysiology: Governing Equations
Transmembrane potential propagation Transmembrane current, Im, described by simplified FitzHugh-Nagumo type dynamics [1] Cm: capacitance per unit area of membrane D: conductivity tensor u: transmembrane potential Im: transmembrane current The governing equation describing transmembrane potential propagation, treating cardiac tissue in the monodomain approximation, is a conventional parabolic partial differential equation. Here Cm is the capacitance per unit area of membrane, D is the conductivity tensor, u is the transmembrane potential. Im is the transmembrane current. It is described by electrophysiological models of varying complexity. Here, we use a simplified description of the FHN type, where v denotes the collective role of the ion channels. v: gate variable Parameters: a=0.1, m1=0.07, m2=0.3, k=8, e=0.01, Cm=1 [1] R. R. Aliev and A. V. Panfilov, Chaos Solitons Fractals 7, 293 (1996)

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. We discretize the computational domain model in spherical coordinates, as shown. Working on this coordinate system, with the boundaries of the computational domain described by two nested cones, is equivalent to computing in a rectangular box with periodical boundary condition on azimuthally direction. We use the second order center finite difference scheme to treat spatial derivatives and explicit Euler to treat time derivative.

Conductivity Tensor Local Coordinate Lab Coordinate
Transformation matrix R Local Coordinate Lab Coordinate The fiber field is used to construct the conductivity tensor. The local principal axes of conductivity at each point are given by the fiber direction (fast diffusion, D_parallel), the direction perpendicular to the fiber and in the fiber plane (slow in-plane diffusion, D_perp1), and the direction perpendicular to the previous two directions (slow out-of-plane diffusion, D_perp2). The transformation matrix R from the local fiber frame to the lab frame, obtained from the fiber trajectories, is used to calculate the conductivity tensor in the lab frame.

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 Our numerical implementation is easily parallelized. The optimal choice is to divide the computational domain along the azimuthal direction as shown. The communication between adjacent computational nodes only occurs on the boundary and the boundary condition on each node would be same. The plot shows the speed up ratio for different CPU numbers. The green curve shows the ideal speed up, demonstrating that our results are very close to the ideal ones, indicating our model has a good scalability.

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. This movie is an example of our simulation results. The color denotes the transmembrane potential (red: high, blue: low). Tips and filaments are phase singularities that act as organizing centers for spiral (2D) and scroll (3D) dynamics, respectively, about which the wave rotates. They offer a way to quantify and simplify the full spatiotemporal dynamics. To describe the resulting filament dynamics quantitively, we have developed a robust algorithm for constructing filaments.

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 First, we use a standard tip finding algorithm (as the intersection between excitation and recovery isoclines) to find all tips on all conical surfaces, which are approximately the fiber surfaces. Then we randomly choose a point, search for its nearest neighbor. If the nearest tip is within a certain threshold, as shown in this picture, we connect these two tips. Then we continue to search for the closest tip again, until we reach this situation: The nearest tip is too far away. At this point, we reverse the search direction, until we no longer have a close tip. This completes the filament. Then we start a new filament and repeat the same procedure until we find all filaments. Because we don’t want to connect the tips not on the adjacent surfaces, in our algorithm, the distance between two tips is calculated as follows: if two tips are not on the same fiber surface or on adjacent surfaces, we simply consider the distance to be infinity. Then we randomly choose a tip as the current tip and find the nearest tip. If the nearest tip is within a certain threshold distance, we consider these two tips are in the same filament and using the new tip as current tip to repeat the same loop. If the nearest tip is not within a certain threshold, we reverse the search direction and repeat the same loop here. We only reverse the search direction once before considering the current filament is complete. After that, we repeat the whole process until we consume all tips. Here, we redefined the distance between two tips in the algorithm, makes the distance calculated between two tip on non-adjacent surfaces to be very big, which inhibit the connections between two tips on non-adjacent surfaces. 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 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 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 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 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 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 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 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 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 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 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 Repeat until all tips are consumed

Filament-finding result
FHN Model: t = 2 Here we show the filament finding results. The top left picture shows the simulation at time=2. The top right picture shows that it is characterized by a single filament, corresponding to the scroll wave. The bottom two pictures are the results at time=999, when the system is in the fully-developed turbulent or fibrillation state, characterized by many filaments. Animation: heart size: (wave length approximately 40) r1=87.5 r2=210 D//=1, Dp1=Dp2=0.5 Mesh points: 168*42*672 (dr=0.7) time:30 (parfile.42.1) Simulation: 7 hours for one CPU and 15 minutes for 32 CPUs t = 999

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. 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): It is imperative to demonstrate numerical convergence of simulation results, to demonstrate that fundamental, dynamical instabilities are not numerical artifacts. We use three different mesh sizes to simulate the model and compare the total filament number and filament length for each case. As we can see from this plot, for filament lengths, the result for three different mesh sizes are all consistent to within the error. For filament number, the mesh sizes dr=0.7 and dr=0.5 are consistent to within error bars (and agree with dr=1.1 to within two standard deviations). The computation time for dr=0.7 for one wave period in normal heart size is less than 1 hours of cpu time. Filament Number and Filament Length versus Heart size

Scaling of Ventricular Turbulence
Here we explore the relationship between filaments and heart size: On the left, the blue curve shows the logarithm of filament length versus the logarithm of heart size and the red curve shows the logarithm of filament number versus the logarithm of the heart size, clearly demonstrating a linear relationship. This is consistent with simulations on the fully realistic anatomical canine ventricular model using the same electro-physiological model. The right figure shows the average filament length normalized by average heart thickness vs the heart size, demonstrating that the average filament length normalized by heart thickness tends to be a constant. This is also consistent with the result from the fully realistic model. Filament number: y=2.787x-9.992 a_err=1.48 b_err=0.30 Filament length: y=3.739x-11.07 a_err=1.35 b_err=0.27 Panflov: Filament number a=2.21 +/ Filament length a=2.98 +/ !! The xlabel is wrong!! Log(total filament length) and Log(filament number) versus Log(heart size) The average filament length, normalized by average heart thickness, versus heart size 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)

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 We have constructed a minimally realistic 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 agreement of filament dynamics with those 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 as future work.

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).

Rotating anisotropy

Coordinate System

Governing Equations

Perturbation Analysis

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

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

Filament Motion

Filament motion (cont’d)

Filament Tension Destabilizing or restabilizing role of rotating anisotropy!!

Part III: Calcium Dynamics in the Myocyte
Here is my third part of my talk: “Calcium dynamcis” Xianfeng Song, Department of Physics, Indiana University Sima Setayeshgar, Department of Physics, Indiana University

Part III: Outline Importance and background on calcium signaling in myocytes Future directions Here comes my thrid part: about calcium dynamics. This is the direction we are heading in and just recently started to explore this area. I will first briefly introduce the importance and the background of calcium dynamics, especially in myocyte. First I will briefly introduce the background of calcium dynamics in myocyte. Then the motivation of this work: why go stochastic Finally the future work

Overview of Calcium Signaling
From Berridge, M. J., M. D. Bootman, et al. (1998). "Calcium - a life and death signal." Nature 395(6703): Calcium signal ranges from very small event like puff to very large event. Elementary events (red) result from the entry of external Ca2+ across the plasma membrane or release from internal stores in the endolasmic or sarcoplasmic reticulum (ER/SR). Global Ca2+signals are produced by coordinating the activity of elementary events to produce a Ca2+ 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 Ca2+ Signaling Machinery
Here shows the fundamental elements of Ca2+ signaling pathways. Calcium stores are important elements in this pathway. As the name indicate, it store calcium. The calcium store can be external stores which are the extracelluar space. Inside the cell, calcium stores have Endoplasmic Reticulum (ER), sarcoplasmic reticulum (SR, mostly in excitable cells), and mitochondria. In the calcium stores, calcium is heavily buffered in all cells, with at least 99% of the available Ca2+ bound to large Ca2+ binding proteins. These binding proteins are called calcium buffers, for example, the calsequestrin are calcium buffers inside SR. Calcium pumps also play an important role: Ca2+ is moved to calcium stores by all kinds of pumps, which includes Ca2+/Na+ exchanger, plasm membrane Ca pumps and SERCA pumps. (Sacroplasmic/endoplasmic reticulum calcium ATPase, SERCA, outer plasma cell membrane ATPases, PCMAs) Unlike calcium pumps, calcium channels are passive channels and most of them don’t consume energy. Ca2+ enter the cytoplasm via varies channels, among which, Ryanodine receptors(RyR) and inositol trisphosphate receptores (IP3R) are most important ones. DHPR:dihydropyridine receptorĀ꾈 PLC: phospholipase C Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer 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 Ca2+ bound to large Ca2+-binding proteins., such as Calmodulin, Calsequestrin. Calcium pumps: Ca2+ is moved to Calcium stores by varies pumps. Calcium channels: Ca2+ can enter the cytoplasm from calcium stores via varies channels, i.e. ryanodine receptors (RyR) and inositol trisphosphate receptors (IP3R).

Ventricular Myocyte Some facts about myocytes
The typical cardiac myocyte is a cylindrical cell approximately 100 mm in length by 10mm 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 Here the first picture shows the structure of a ventricular myocyte and the second picture is the schematic version. Typically the cardiac myocyte is a cylindrical cell approximately 100 um in length by 10 um in diameter and is surrounded by a cell membrane known as the sarcolemma (SL). There are three physical compartments: the cytoplasm, the sarcoplasmic reticulum (SR) and the mitochondria which is not shown in picture. 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 which serves an important role during Excitation contraction coupling. The SR release channel or Ryr is found almost entirely within the part of the SR membrane which communicates with the junctional cleft. The SR release channel, or ryanodine receptor (RyR) is found almost entirely within the part of the SR membrane which communicates with the junctional cleft. Calcium induced ca release is the key process for excitation contraction coupling. The picture shows how this process happens. First, in the resting state, the RyR channels are closed. And calcium concentration inside the cytoplasm is at the normal. When small amount of calcium flow into the junctional cleft which induced by depolarization of the membrane, these small amount of calcium bind rapidly to a relatively low affinity site (1), thereby activating the RyR channel and induce the calcium release. The calcium concentraion inside cytoplasm thus become higher than normal level and then induce the contraction of the myocyte. Calcium may then bind more slowly to a second higher affinity site (2) moving the release channel to an inactivate state, thus close channel. And the calcium in cytoplasm slowly decreases because the calcium pumps pump these calcium into calcium stores. Finally, the channel and the calcium return back to the resting state. Because CICR, excitation and contraction of myocyte are coupling together. [text] 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 inacvitive state. As cytoplasmic [Ca] decreases, Ca would be expected to dissociate from the lower affinity activating site first and then more slowly from the inactivating site to return the channel to the resting state. Calcium induced Calcium release

Future Directions What is the role of receptor clustering on calcium signaling? What is the role of the buffer Calsequestrin in facilitating calcium release? From above, here comes our possible future directions: The first thing we want to do is to understand the role of receptor clustering in calcium signaling and try to understand what is the design principle underlying the receptor clustering. There are two possible method we could adopt: one is by using the method given in the paper: physical limits to biochemical signaling, by sima and william bialek to calculate the diffusive noise in this system to determine whether the clustering serves as noise reduction. Another possible is to use Monto-carlo method to simulate this system to explore the efficiency by clustering the receptors together. The another future direction is to try to fully understand the role of calsequestrin during CICR: can calsequestrin really facilitate the calcium release in the physics point of view? How best can it facilitate this release? The third possible direction is to explore the role of cooperativity of calcium binding to calmodulin as noise reduction in calcium signaling path Data: from Falcke, M., L. Tsimring, et al. (2000). "Stochastic spreading of intracellular Ca^{2+} release." Physical Review E 62(2): 2636. In which is modeling the intracellular calcium dynamics arising via diffusion and via the IP3 receptor, uses: the receptor cluster radius R=0.225 um and the distance between cluster is d=2.5um, each clusters contains 40 channels Methodology: by using the methods described by paper: physical limits to biochemical signaling, by sima and william bialek to calculate the diffusive noise in the system and thus determine whether the clustering serves as noise reduction.

Receptor Clustering RyR and IP3R 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 Experiments show that for both RyR and Ip3R, the channels are spatially organized in clusters, with the distance between clusters are approximately two order magnitude larger than the distances between channels within one cluster. The distance between clusters are on the order of several microns and the distances between channels within one cluster are on the order of serveral hundreds of nms. The picture shows a high resolution image on a single Ca puff which was done on xenopus oocyte. The single puff demonstrates an Ip3R cluster lies in the middle of the puff. The density of the clusters are estimated to be 1 per 30um^2 for Ip3R in xenopus oocyte. The clustering of chemotaxis receptors in bacteria have been proven to do noise reduction and signal amplification. Notes: why there is only one puff in the picture: whthin the limit of our resolution, it appeared that the distribution of cytosolic Ca was consistent with Ca ions diffusing from a point source of release, rather than being released over an appreciable area or from multiple closely spaced sites. Data: the distance between clusters: several microns The radius of clusters: microns The distance between channels: order of 10nm High resolution image showing a Ca2+ puff evoked by photoreleased InsP3 which demonstrate an IP3R cluster (From Yao, Y. etc, Journal of Physiology 482: ) [1] Skoge, M. L., R. G. Endres, et al. (2006), Biophys. J. 90(12): 4317.

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 Ca2+ as a store? Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer Calsequestrin is the mail buffer inside SR, most of which are located close to RyRs. Calsequestrin has a linear structure and can interact with RyR through triadin and/or junctin thus control the activity of RyR. Experiment has shown that calsequestrins play an important role during calcium induced calcium release. The paper by Launikonis and Rios demonstrate that the calsequestrins forms a proximate Ca stores. One proposal about the proximate ca stores is shown in the pictures: (1) when the channel first opens, the calcium adsorbed to linear CSQ polymers feeds repidly release. This induce the fast calcium release. After the polymers are depleted Ca, the polymers then disassemble. After the depletion becomes deeper as calcium replenishes the calsequestrins, the polymers reassembles and goes back to RyR. (A) The channel opens, Ca2+ adsorbed to linear CSQ polymers feeds rapid release. (B) The polymers are depleted Ca2+ thus disassemble. (C) Depletion becomes deeper as Ca2+ replenishes the proximate store and the CSQ polymers reassembles (from Launikonis et al, PNAS 103(8) (2005))

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Advisor: Sima Setayeshgar

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