Homogenization in Cardiac Electrophysiology and Blow-Up in Bacterial Chemotaxis

Transcription

1 Homogenization in Cardiac Electrophysiology and Blow-Up in Bacterial Chemotaxis by Paul Earl Hand A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics New York University May 2009 Professor Charles Peskin Professor Nader Masmoudi

2 c Paul Earl Hand All Rights Reserved, 2009

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4 Acknowledgements I am indebted to many people who made this dissertation possible. First and foremost, I would like to thank my mother, father, and the rest of my family for the support and sacrifices they made to allow me to pursue my interests in mathematics. I would also like to thank my advisors, Charlie Peskin and Nader Masmoudi, for giving me the freedom and guidance to pursue these interests as they developed. I gratefully acknowledge Boyce Griffith, Yoichiro Mori, Glenn Fishmann, and Greg Morley for many discussions about mathematical and experimental cardiac electrophysiology. I am thankful for several years of funding by the United States Department of Defense through a National Defense Science and Engineering Graduate Fellowship. Finally, I would like to thank many of my friends for their help and distraction, including Ben Olsen, Saverio Spagnolie, Will Findley, Mike Damron, Murphy Stein, Giulio Trigila, Alex Rubinsteyn, Al Momin, Ross Tulloch, Dan Goldberg, Tom Alberts, Jeff Ryan, Kela Lushi, Shilpa Khatri, Jens Jørgensen, and Thomas Fai. iv

5 Abstract In the first part of this dissertation, we investigate three different issues involving homogenization in cardiac electrophysiology. We present a modification for how heart tissue is typically modeled in order to derive values for intracellular and extracellular conductivities needed for bidomain simulations. In our model, cardiac myocytes are rectangular prisms and gap junctions appear in a distributed manner as flux boundary conditions for Laplace s equation. In other models, gap junctions tend to be explicit geometrical entities. Using directly measurable microproperties such as cellular dimensions and endto-end and side-to-side gap junction coupling strengths, we inexpensively obtain effective conductivities close to those given by simulations with a detailed cytoarchitecture. This model provides a convenient framework for studying the effect on conductivities of aligned vs. brick-like arrangements of cells and the effect of different distributions of gap junctions between the sides and ends of myocytes. We further illustrate this framework by investigating the effect on conductivity of non-uniform distributions of gap junctions within the ends of cells. We show that uniform distributions are local maximizers of conductivity through analytical perturbation arguments. We also derive a homogenized description of an ephaptic communication mechanism along a single strand of cells. We perform numerical simulations of the full model and its homogenization. We observe that the two descriptions agree when gap junctional coupling is at physiologically normal levels. When gap junctional coupling is low, the homogenized description does not capture the behavior that the ephaptic mechanism can speed up action potential propagation. In the second part of this dissertation, we investigate finite-time blow-up and stability of the Keller-Segel model for bacterial chemotaxis. We use a second moment calculation to establish finite-time blow-up for the Keller-Segel system on a disk with Dirichlet boundary conditions and a supercritical mass. We numerically investigate the evolution and stability of the Keller-Segel system in order to provide a conjecture about the generality of boundary blow-up for supercritical mass under the Jäger-Luckhaus boundary conditions. Finally, we use the free energy of solutions to Keller-Segel equations to derive a functional inequality that may be helpful for analyzing the stability of steady states. v

6 Contents Acknowledgements iv Abstract v List of Figures xiii List of Tables xv I Homogenization in Cardiac Electrophysiology 1 1 Introduction Cellular Biophysics The Hodgkin-Huxley Ionic Model The Luo-Rudy Ionic Model The Cable Equation The Bidomain Equations Homogenization of Partial Differential Equations Outline of Part I Homogenization of Cardiac Models that Describe Gap Junctions Through Boundary Conditions Introduction Bidomain Equations and Effective Conductivity Full Cellular Model in an Aligned Arrangement Nondimensionlization Statement of the Effective Conductivity Problem Homogenization in an Aligned Arrangement Analytical Solution to the Corrector Problem Resulting Effective Conductivities Equivalent Resistor Network Full Cellular Model and Homogenization in a Brick-like Arrangement Resulting Effective Conductivities Discussion Effective Conductivity Values vi

7 2.6.2 Comparison of PDE and Resistor Network Methods Application to Electromechanical Simulations Conclusion Extracellular Conductivities Parameters and Variables Gap Junction Distributions for Optimal Effective Conductivity Introduction Model and Statement of the Gap Junction Distribution Problem Nondimensionlization and Homogenization Statement of the Gap Junction Distribution Problem Local optimality of a uniform gap junctional distribution Discussion Homogenization of a Model for Ephaptic Cardiac Communication Introduction The Full Ephaptic Model Nondimensionalization of the Full System The Homogenized Ephaptic Model Derivation of Homogenized System Numerical Simulation of the Full and Homogenized Models Initial Value Problems for Numerical Simulation Numerical Results for the Full System Numerical Results for the Homogenized System Discussion Simulation Parameters Numerical Schemes Full System Homogenized System II Blow-Up in Bacterial Chemotaxis 73 5 Introduction Derivation of Keller-Segel Equations Finite-Time Blow-Up in the Whole Plane Free Energy for Keller-Segel Systems Outline of Part II Finite-Time Blow-Up of Keller-Segel on a Disk with Dirichlet Boundary Conditions Introduction vii

8 6.2 Finite-Time Blow-Up under Dirichlet Boundary Conditions for Supercritical Mass Numerically Motivated Conjecture on Boundary Blow-Up with Jäger-Luckhaus Boundary Conditions Introduction and Blow-Up Conjecture Outline of Chapter Numerical Simulation of Keller-Segel Evolution Initial Value Problem Numerical Scheme Results Numerical Stability of Keller-Segel Steady States Discretization Method Results for Numerical Stability Analysis Discussion Convergence Study for Evolution Simulation Spatial Convergence Study Temporal Convergence Study A Free Energy Stability Criterion Introduction Free Energy Inequality Motivation and Derivation Application and Discussion Stability of Uniform Profiles Differences with Linear Stability Does the Free Energy Inequality Aid Analysis? Bibliography 102 viii

9 List of Figures 1.1 The electrical circuit model of an isopotential cell (shaded). Any current injected inside would either charge the plasma membrane as a capacitor or flow across the membrane through ion channels A cable of electrically active membrane filled with conducting fluid. A flux balance calculation allows for a derivation of the cable equation A depiction of the complicated geometrical structure of the intracellular and extracellular space of cardiac tissue. The thick, dark vertical stripes separating cells are known as intercalated disks. Reprinted with permission from Guyton and Hall (Fig. 9-2, p. 108) In this section, we homogenize Laplace s equation over the periodic domain Y ε (a). This domain is composed of translated and scaled versions of the unit cell Y (b) Instead of modeling gap junctions through complex cellular geometry (a), we model them through flux boundary conditions on simple geometry (b). The multiple resistors shown in (b) represent a continuous boundary condition, see equations (2.2) - (2.7) The aligned cellular architecture: (a) cells of dimension l w c w c are arranged in three space with period l w p w p ; (b) an x 1, x 2 cross-section In order to compute effective conductivity, we impose a potential difference across a large number of cells and calculate the resulting current density (a). In this figure, extracellular space has been omitted for clarity. The resulting electric potential is depicted in (b) The unit cell in the aligned geometry is composed of an intracellular region Y i and an extracellular region Y e The brick-like cellular architecture. Cells of dimension l w c w c are arranged with a period of l w p w p, except that adjacent fibers are offset by half a cell length (a). The arrangement can be viewed as the periodic extension of the region inside the dashed prism. An x 1, x 2 cross-section is shown in (b) ix

10 2.6 Electric potential in several cells in a brick-like arrangement with a longitudinally applied potential gradient using cellular parameters from Table 2.4. The second and fourth cells from the left are offset by half a cell length because they are in a neighboring fiber to the first and third cells. The units of the vertical axis are arbitrary. As the cellular domain is three dimensional, the plot shows only the x 3 -averaged potential within a cell Effective intracellular transverse and longitudinal conductivities as a function of the fraction of gap junctions expressed on the cell sides, keeping the total conductance constant. As gap junctions are moved from the ends to the sides of a cell, intracellular longitudinal conductivity decreases (a). With an aligned arrangement it decreases to zero, but with the brick arrangement there is a nonzero intracellular longitudinal conductivity even when all gap junctions are on the sides of the cells. Meanwhile, in the parameter regime given by cellular measurements, intracellular transverse conductivity increases linearly with the fraction of gap junctions on the cell sides (b). The side-to-side and end-to-end conductance measurements given in Table 2.4 correspond to a fraction 0.68 of gap junctions on the sides of cells. A uniform density of gap junctions between the sides and ends of cells with dimensions given in Table 2.4 corresponds to a fraction 0.92 of gap junctions on the sides of cells In this cross-section of tissue, the longitudinal direction points out of the page. The dark shaded regions indicate cells, and the remaining regions indicate extracellular space. We treat the light shaded regions as insulators for the calculation of transverse conductivity in the x 2 direction A simplified, two-dimensional depiction of the three-dimensional, two cell model of [8]. Each cell is represented as a cubic lattice. Gap junctions are represented by lattice points connecting the two cells. The gap junctions can be arranged as a single placque of adjacent vertices (a) or can be scattered randomly (b) The cellular model of [29]. Cells in a row are modeled as rectangular regions. Gap junctions are represented as holes in the boundaries between them, making intracellular space contiguous. Keeping the total length of the gap junctions fixed, few large holes correspond to gap junctions aggregated into placques (a). Many small holes represent a scattered gap junctional distribution (b) x

11 3.3 Our model of cells and gap junctions. Cells in a sequence are modeled as squares whose interiors are not physically connected. Gap junctions are represented in a continuous manner as resistive connections between neighboring cells. The multiple resistors shown represent a flux boundary condition between the cells. A non-uniform distribution of resistance (a) corresponds to a gap junction distribution with placques. A uniform distribution of resistance (b) corresponds to a scattered distribution of gap junctions The tissue level cellular architecture. Cells are modeled as squares adjoined without extracellular space. The thick horizonal lines represent membranes that prevent vertical current flow. Hence, the behavior along each row of cells is identical. An individual cell need only be identified by its horizontal cell number, i A cartoon of the ephaptic mechanism between two adjacent cells (shaded). Initially, the Na + channels of both the pre-junctional (left) and post-junctional (right) cells are closed (a). When an action potential reaches the left cell, its Na + channels open, allowing a flux of Na + current in from extracellular space via the clefts (b). As per Ohm s law, the potential inside the cleft decreases, resulting in the depolarization of the post-junctional membrande. If this effect is sufficiently strong, this membrance may reach threshold, causing the Na + channels to open and allowing current to flow inward (c) The geometry and circuit diagram for our model of an ephaptic mechanism. We describe cells as active cables coupled through direct resistive connections and active membranes involving shared cleft potentials. We model extracellular space as grounded. We ignore the effects of any changes in ion concentration A schematic of the domains for the full (a) and homogonized (b) models. The full model is posed over discrete cells of length ε with an equipotential cleft between adjacent cells. The intracellular potential within the i-th cell is φ i (t, x), and the cleft potential to the right of the i-th cell is φ i c (t). Note that the cleft potential is defined only over a discrete set of points. The homogenized model is posed over the entire length L tissue of tissue, as it does not resolve individual cells. The intracellular potential φ 0 (t, x) and the cleft potential φ c,0 (t, x) are defined over the entire domain xi

12 4.4 Conduction velocity under the full ephaptic model as a function of the nondimensional cleft-to-ground resistance for gap junction expression ranging from 1% to 100% of normal. For these simulations, ε = Na + channels are distributed such that (a) all are located at the intercalated disks, (b) half are located at the intercalated disks, or (c) channel density is uniform Conduction velocity under the homogenized ephaptic model as a function of the nondimensional cleft-to-ground resistance for gap junction expression ranging from 1% to 100% of normal. Na + channels are distributed such that (a) all are located at the intercalated disks, (b) half are located at the intercalated disks, or (c) channel density is uniform An overlay of the conduction speeds computed under the full and homogenized systems. At normal gap junction expression levels (κ = 1) the systems agree well, but that agreement disappears as the gap junctional coupling is reduced An overlay of the computed solutions to the full and homogenized systems under β = 10 3, κ = 1, f Na = 1 at two different times. The top panels show intracellular potentials. Note that individual cells can be resolved in the upstroke of the full simulations. The bottom panels show cleft potential. For clarity, the intracellular potentials of the full simulations are plotted for only 5 of the 20 interior nodes An overlay of the computed solutions to the full and homogenized systems under β = 10 3, κ = 0.01, f Na = 1 at two different times. The top panels show intracellular potentials. Note that individual cells can be resolved in the upstroke of the full simulations. The bottom panels show cleft potential. For clarity, the intracellular potentials of the full simulations are plotted for only 5 of the 20 interior nodes The spatial discretization of each biological cell in the full model. Each biological cell is broken into n computational nodes and two ghost nodes, represented by hollow dots. Note that there are no computational nodes for the clefts as the cleft potential can be deduced from the potential at the interior and ghost nodes. Also note that the rightward ghost node of a cell is distinct from the leftward ghost node of its neighbor. The variable φ i k denotes the potential at the k-th node of the i-th cell, where k = 0 and k = n+1 correspond to the ghost nodes xii

13 7.1 Evolution of the Keller-Segel system with (JL) boundary conditions over the singly periodic square, T [0, 1] with mass M = 4.5π initially distributed in a noisy uniform shape. Note that between times t = 0.5 and t = 2.5, the solution has the form of a growing cosinusoidal disturbance from uniform. After that, the solution approaches the steady state of the 1d simulation with the same mass. In this simulation t = 0.005, x = y = 1/40. For clarity, the plot only shows a sampling of the simulated grid points Evolution of the Keller-Segel system with (JL) boundary conditions over the singly periodic squre, T [0, 1] with mass M = 4.6π initially distributed in a noisy uniform shape. The solution follows the same description as the simulation shown in Figure 7.1, but it stays near the 1d steady state only until t 50. At this time, a disturbance at the boundary grows until the mass concentrates there by finite time. In this simulation t = 0.005, x = y = 1/40. For clarity, the plot only shows a sampling of the simulated grid points Plot of the smallest eigenvalue of the discretization of the Keller- Segel equations over T [0, 1] with (JL) boundary conditions, linearized about the one-dimensional non-uniform steady state with mass M. A negative eigenvalue indicates a linear instability. The 1d steady states change from being linearly stable to linearly unstable as mass increased beyond M 4.57π xiii

14 List of Tables 1.1 Differential Equations, conductivities, reversal potentials, and resting values used in simulations in Chapter 4, based on the Luo Rudy 1991 dynamic [34]. All ordinary differential equations carry units ms 1. From [34], the resting value of X was unclear. For it, we selected a small, non-zero initial value for our evolution simulations in Chapter Rate constants and values for gating variables Intracellular, longitudinal conductivities σ i,l (ms/cm) obtained from the aligned and brick-like arrangements for various values of g GJ,end and g GJ,side. The numbers before the commas are for the aligned arrangement. The numbers after the commas are for the brick-like arrangement Conductivity values obtained by fitting macroscopic wavespeed data to solutions of bidomain equations. Values are obtained for ventricular tissue from various animals, such as dogs, cows, and sheep Conductivity values obtained directly from microscopic measurements via homogenization in the present work The physical parameters, derived parameters, and variables that enter our cellular model. Measured parameters from [56, 18] correspond to mouse ventricular myocytes The physical and derived parameters that enter our full and homogenized models Computational parameters that enter our numerical simulations of the full and homogenized models Spatial convergence rates for full system with ε = 0.01 computed at nondimensional time Temporal convergence rates for full system computed at nondimensional time Spatial convergence rates for homogenized system computed at nondimensional time xiv

15 4.6 Temporal convergence rates for homogenized system computed at nondimensional time Spatial convergence rates for Keller-Segel simulation computed at time Temporal convergence rates computed at time xv

16 Part I Homogenization in Cardiac Electrophysiology 1

17 Chapter 1 Introduction 1.1 Cellular Biophysics Many cells require electricity to function. Some, such as epithelial cells, control the concentrations of ions like Na + and K +, primarily for the purpose of controlling cell volume [29, 24]. Others, such as neurons and cardiac muscle cells, are called excitable and control ionic flow for the purpose of electrical communication. If the electric potential of excitable cells is slightly disturbed due to a current injection, it relaxes to its resting value. If, instead, the potential is disturbed by a large enough current, it undergoes a large spike, called an action potential, before returning to rest. For a more thorough introduction, see [29, 16]. These action potentials can propagate down nerve axons and through cardiac muscle, resulting in communication between distant cells. In the electric circuit model of cells, the membrane is described by a capacitor Figure 1.1: The electrical circuit model of an isopotential cell (shaded). Any current injected inside would either charge the plasma membrane as a capacitor or flow across the membrane through ion channels. 2

18 in parallel with non-ohmic resistors. The phospholipid bilayer acts as a capacitor because it is an insulator which can separate charge. Membrane ion channels are the resistors as they permit current flow in response to electric potentials. Figure 1.1 shows the resulting model circuit for an electrically active isopotential cell. With it, we can see that current injected into the intracellular region either charges the cell as a capacitor or flows through the ion channels. In the absense of such an injection, the potential evolves according to the differential equation C dφ dt + I ion(φ, w) = 0, (1.1) where φ is the transmembrane potential, C is the membrane capacitance per unit area, and I ion is the outward ionic current per unit area at specific values of φ and any relevant gating variables w. The specific ions and currents relevant for physiology varies between cell types and animal species. Mathematically, these differences alter the I ion function. The most famous ionic model is the Hodgkin-Huxley model [23] of the squid giant axon. A similar model for mammalian ventricular muscle is the Luo-Rudy dynamic [34, 35, 14]. There are many other realistic models, but scientists also study nonphysiological ones, such as the Fitzhugh-Nagumo model [39, 19, 43] or McKean s piecewise linear model [38], because of their mathematical simplicity. We detail the Hodgkin-Huxley and Luo-Rudy models now The Hodgkin-Huxley Ionic Model The Hodgkin-Huxley model is composed of three ionic currents: a fast-activating, slow-inactivating Na + current; a slow-rectifying K + current; and a leak current. The resistances underlying the Na + and K + currents are governed by gating variables which evolve in a voltage dependent way. Precisely, the Hodgkin-Huxley ionic model is given by w = (n, m, h), I ion (φ, w) = g Na m 3 h (φ E Na ) + g K n 4 (φ E K ) + g l (φ E l ), ds dt = α s(φ)(1 s) β s (φ)s for s = n, m, or h, where g Na, g K, g l are the maximal conductances for the respective Na +, K +, and leak currents; n, m, h are gating variables between 0 and 1; E Na, E K, E l are the reversal potentials for the Na +, K +, and leak currents; and α s (φ), β s (φ) are the experimentally obtained voltage dependent rate constants governing the opening and closing of gates within the relevant ion channels. The precise form of α s (φ) and β s (φ) can be found in [29]. In the above equations, w is a three component 3

19 vector, and s stands for any single one of those components. The reversal potentials depend on the ionic concentrations inside and outside the cell through the Nernst equation. Technically, any current through the channels alters this concentration, but such changes are typically small enough that the concentrations, and hence the reversal potentials, are constants in time The Luo-Rudy Ionic Model The Luo-Rudy ionic model, also known as the Luo-Rudy dynamic (LRd) is an ionic model of mammalian cardiac ventricular cells. The 1991 version of the model [34] consists of six currents, all with separate dynamics. More recent versions are even more detailed [35, 15, 14]. Precisely, the LRd ionic model is given by w = (m, h, j, d, f, X, [Ca] i ), I ion (φ, w) = I Na + I si + I K + I K1 + I Kp + I b, I Na = g Na m 3 h j (φ E Na ), I si = g si d f (φ E si ([Ca] i )), I K = g K X X i (φ E K ), I K1 = g K1 K1 (φ E K1 ), I Kp = g Kp Kp (φ E Kp ), I b = g b (φ E b ), dw = LRD(φ, w), dt K1 = α K1 (φ)/ (α K1 (φ) + β K1 (φ)), X i = X i (φ), Kp = Kp(φ), where I Na is the fast inward sodium current, I si is the slow inward current, I K is the time-dependent potassium current, I K1 is the time-independent potassium current, I Kp is the plateau potassium current, I b is the background current; the time dependent gating variables are m, h, j, d, f, and X; and the g s and E s are the maximal conductances and reversal potentials for their corresponding variables. The K1 variable relaxes quickly, so it is replaced with its steady state value. Further, the X i and Kp variables have no time dependence and are only a function of potential. The differential equations for w given by LRD(φ, w), the values for unspecified constants, and the resting values used in our simulations are given in Table 1.1. Table 1.2 presents the voltage dependent rate constants and gating variables. 4

20 Differential Equations ds dt = α s(φ) (α s (φ) + β s (φ))s for s = m, h, j, d, f, X, d[ca] i = 10 4 I si (10 4 [Ca] i ). dt Conductivities and Reversal Potentials g Na = 23 ms/cm 2, E Na = 54.4 mv, g si = 0.09 ms/cm 2, E si = ln([ca] i ) mv, g K = [K] 0 /5.4 ms/cm 2, E K = 77 mv, g K1 = [K] 0 /5.4 ms/cm 2, E K1 = 10 3RT F ln [K] 0 mv, [K] i g Kp = ms/cm 2, E Kp = E K1, g b = ms/cm 2, E b = mv. Other Constants [K] 0 = 5.4 mm, R = J / mol K, T = K. [K] i = 145 mm, F = C / mol, Resting Values φ 0 = mv, m 0 = , h 0 = , j 0 = , d 0 = , f 0 = , X 0 = , [Ca] i,0 = Table 1.1: Differential Equations, conductivities, reversal potentials, and resting values used in simulations in Chapter 4, based on the Luo Rudy 1991 dynamic [34]. All ordinary differential equations carry units ms 1. From [34], the resting value of X was unclear. For it, we selected a small, non-zero initial value for our evolution simulations in Chapter 4. 5

21 Rate Constants and Values for Gating Variables { 0 φ < 40 mv, α h = exp[(80 + φ)/ 6.8] φ 40 mv, { 1/(0.13{1 + exp[(v )/ 11.1]}) φ < 40 mv, β h = 3.56 exp(0.079φ) exp(0.35φ) φ 40 mv, 0 φ < 40 mv, α j = [ exp(0.2444φ) exp( φ)] (V )/{1 + exp[0.311 (φ )]} φ 40 mv, { 0.3 exp( φ)/{1 + exp[ 0.1(φ + 32)]} φ < 40 mv, β j = exp( φ)/{1 + exp[ (φ )]} φ 40 mv, α m = 0.32(φ )/{1 exp[ 0.1(φ )]}, β m = 0.08 exp( φ/11), α d = exp[ 0.01(φ 5)]/{1 + exp[ 0.072(φ 5)]}, β d = 0.07 exp[ 0.017(φ + 44)]/{1 + exp[0.05(φ + 44)]}, α f = exp[ 0.008(φ + 28)]/{1 + exp[0.15(φ + 28)]}, β f = exp[ 0.02(φ + 30)]/{1 + exp[ 0.2(φ + 30)]}, α X = exp[0.083(φ + 50)]/{1 + exp[0.057(φ + 50)]}, β X = exp[ 0.06(φ + 20)]/{1 + exp[ 0.04(φ + 20)]}, α K1 = 1.02/{1 + exp[ (φ E K )]}, β K1 = { exp[ (φ E K )] + exp[ (φ E K )]} / {1 + exp[ (φ E K )]}, { 1 φ 100 mv, X i = {exp[0.04(φ + 77)] 1}/{(V + 77) exp[0.04(φ + 35)]} φ > 100 mv, Kp = 1/{1 + exp[(7.488 φ)/5.98]}. Table 1.2: Rate constants and values for gating variables. 6

22 r x 1 x 2 Figure 1.2: A cable of electrically active membrane filled with conducting fluid. A flux balance calculation allows for a derivation of the cable equation. 1.2 The Cable Equation As mentioned in Section 1.1, action potentials can propagate down nerve axons or cardiac muscle. In order to study such propagation mathematically, we must modify the differential equation (1.1) to incorporate spatial effets. The resulting partial differential equation (PDE) is called the cable equation, which we now derive. Consider a long cylinder of radius r of electrically active membrane, as in Figure 1.2, filled with conducting cytosolic fluid. Assuming there are no appreciable variations of the potential within a cross-section of the cable, φ varies only with time and the x coordinate. As per Ohm s law, φ j(t, x) = σ c (t, x) (1.2) x where j is the current density, and σ c is the cytoplasmic conductivity. The net current into the region between x = x 1 and x = x 2 either charges the membrane as a capacitor or passes through the ion channels. Thus πr 2 ( σ c φ x (t, x 1) + σ c φ x (t, x 2) x 1 ) = x 1 x2 x 1 2πr ( C φ t (t, x) + I ion(φ, w) ) dx. (1.3) Writing the left hand side as an integral, we find x2 πr 2 2 φ x2 ( σ c (t, x)dx = 2πr C φ ) x2 t (t, x) + I ion(φ, w) dx. (1.4) Since this equation holds for all x 1, x 2, we obtain the cable equation σ c r 2 2 φ x 2 = C φ t + I ion(φ, w). (1.5) 7

23 Figure 1.3: A depiction of the complicated geometrical structure of the intracellular and extracellular space of cardiac tissue. The thick, dark vertical stripes separating cells are known as intercalated disks. Reprinted with permission from Guyton and Hall (Fig. 9-2, p. 108). 1.3 The Bidomain Equations Heart muscle is an irregular, three dimensional arrangement of cells with intricate structure. The cells are typically around 100 µ in length and about 20 µ in width [29], and are connected to each other at their ends. As can be seen in Figure 1.3, the cells form fibers which can branch. Adjacent cells are connected through the intercalated disks, shown by the thick dark lines in the figure. The cells are also surrounded by a conducting, irregularly shaped extracellular space. Action potential propagation in cardiac tissue results from ion flow in and out of cells in this convoluted geometry. In order to study cardiac action potential propagation mathematically, we model the tissue in an averaged way. The resulting partial differential equations are called the bidomain equations. Their averaged nature avoids the cumbersome task of detailing the complexity of cellular geometry and arrangements at the micron level. To derive the bidomain equations, we let Ω i denote the intracellular space of a tissue, Ω e denote the extracellular space, and Ω = Ω i Ω e. Let φ i (x) be the intracellular potential and φ e (x) be the extracellular potential. Technically, φ i is defined only over Ω i and φ e only over Ω e. Viewing Ω as being a combination of intracellular and extracellular space for all x, we let φ i be defined over all Ω. We interpret its value at a point x as the intracellular potential in a region near x. Similarly, we consider φ e to be defined over all Ω. 8

24 We assume that the intracellular and extracellular domains give rise to an anisotropic, ohmic current-voltage relationship, j i = σ i φ i, (1.6) j e = σ e φ e, (1.7) where j i and j e are the intracellular and extracellular current densities, σ i and σ e are the intracellular and extracellular conductivity tensors, which could, in principle, vary with spatial position. Barring outside current injections, there can be no source or sink of current in the combination of extracellular and intracellular space. Any apparent sink in the intracellular current must then be a source in the extracellular current. Thus ( σ i φ i σ e φ e ) = 0. (1.8) The current flowing from intracellular to extracellular space acts either to charge the local membrane as a capacitor or to flow through the ion channels. Thus ( (σ i φ i ) = β C (φ ) i φ e ) + I ion (φ i φ e, w), (1.9) t where β is the membrane surface area per unit volume of tissue. By combining (1.8) and (1.9), we obtain the bidomain equations ( (σ i φ i ) = β C (φ ) i φ e ) + I ion (φ i φ e, w), (1.10) t ( (σ e φ e ) = β C (φ ) i φ e ) + I ion (φ i φ e, w). (1.11) t These equations are the commonly accepted macroscopic description of of cardiac tissue under normal and pathological conditions [22]. 1.4 Homogenization of Partial Differential Equations Homogenization is a two-scale asymptotic technique used to describe an averaged description of a partial differential equation with periodic structure. Examples of such periodic structure include highly oscillatory conductivity and periodic geometry. Such an averaged description allows us to determine the effective conductivities of a periodic medium. We now formally demonstrate the homogenization of a material with periodic geometry for the purpose of determining its 9

25 (a) (b) Y ε Y Figure 1.4: In this section, we homogenize Laplace s equation over the periodic domain Y ε (a). This domain is composed of translated and scaled versions of the unit cell Y (b). effective conductivities. See [13] for a similar demonstration with periodic conductivity. Let Y T 3 have a smooth boundary and a connected periodic extension. Let Y ε = i,j,k ε (Y + (i, j, k)), which is depicted in Figure 1.4. Let Ω = [0, 1] 3 and Ω ε = Ω Y ε. Consider Laplace s equation in Ω ε, We make the two-scale homogenization ansatz σ φ ε (x) = f(x) in Ω ε, (1.12) ν φ ε = 0 on Ω ε \ Ω, (1.13) φ ε (x) = 0 on Ω. (1.14) φ ε (x) = φ 0 (x) + εφ 1 (x; x/ε) + ε 2 φ 2 (x; x/ε) +, (1.15) where φ 1 and φ 2 are periodic in the second variable y = x/ε. Plugging the ansatz (1.15) into the boundary value problem (1.12) (1.14) and extracting the leading order terms gives The next order terms in ε are σ y φ 1 (x; y) = 0 in Y, (1.16) y φ 1 ν = x φ 0 ν on Y. (1.17) y (σ y φ 2 + σ x φ 1 ) = x (σ x φ 0 + σ y φ 1 ) in Y, (1.18) y φ 2 ν = x φ 1 ν on Y. (1.19) 10

26 The boundary value problem for φ 1 can be written in terms of the corrector functions w i by φ 1 (x; y) = xi (x)w i (y), (1.20) where w i (y) solves Laplace s equation over the periodic domain Y. Specifically, y w i (y) = 0 in Y, (1.21) ν w i (y) = e k ν on Y. (1.22) As the macroscopic φ 0 does not satisfy the boundary conditions on the fast spatial scale of the periodic domain, the O(ε) term in the ansatz functions to correct the solutions normal derivative on the microscale domain boundary. Finally, a PDE for φ 0 can be obtained by applying the solvability condition for the φ 2 equation. Integrating (1.18) over Ω gives y (σ y φ 2 + σ x φ 1 ) + x (σ x φ 0 ) + x (σ y φ 1 )dx = f(x) Y. (1.23) Ω where Y is the volume of Ω. The first two terms cancel by applying the divergence theorem and the boundary condition (1.19). The equation, which inherits the boundary condition on Ω, then becomes xi (σ ( δ ik + 1 Y Y yk w i (y)dy ) ) xk φ 0 (x) = f in Ω, (1.24) φ 0 (x) = 0 on Ω, (1.25) from which we can read off the conductivity tensor. Alternatively, effective conductivity could be determined by placing a potential difference along one direction of the macroscopic domain Ω and calculating the drawn current. This approach is used in Chapter Outline of Part I In Chapter 2, we present a modification to the conventional framework for describing cardiac myocytes in calculations of effective macroscopic conductivity. Instead of modeling gap junctions as discrete geometrical entities, we model their effect through Neumann boundary conditions on simple cellular geometry. We then derive effective conductivity values based on measured cellular parameters in the cases of aligned and brick-like cellular arrangements. We compare our conductivity values to those obtained in the literature either (1) by fitting macroscopic 11

27 wavespeed data to solutions of the bidomain equations or (2) by inference based on microscopic measurements. We also discuss the applicability of our framework to electromechanical simulations. In Chapter 3, we use this modified framework in order to determine which distribution of gap junctions along the ends of cells provides the most macroscopic conductivity. In agreement with random walk and PDE models describing gap junctions through geometry, we establish that a uniform distribution is a local maximizer of conductivity when the total number of gap junctions is held fixed. In Chapter 4, we present a model of ephaptic cardiac communication through extracellular clefts which are resistively connected to extracellular space. We nondimensionalize and homogenize the differential equations arising from the biophysics. We investigate the full and homogenized models numerically and compare the computed wavespeeds and waveforms over physiologically relevant parameter regimes. We observe that the two models agree when gap junctional coupling is at physiologically normal levels but disagree when gap junction levels are low. 12

28 Chapter 2 Homogenization of Cardiac Models that Describe Gap Junctions Through Boundary Conditions 2.1 Introduction Computer simulations have the potential to increase our understanding of normal and pathological cardiac function, and to improve the effectiveness of clinical therapies. Although the biophysics at the cellular level is well understood, whole heart simulations that resolve every cell are computationally infeasible at present. Instead, a more tractable approach is to perform macroscopic simulations based on the bidomain equations [22], which govern locally averaged potentials inside and outside cells. These equations require physical values of the effective conductivity of intracellular and extracellular regions in both the longitudinal (fiber) and the transverse (cross-fiber) directions. Ideally, such values should be directly obtained from measurable cellular properties, such as geometry and gap junctional conductivity. Otherwise, the values are free parameters which must be chosen by matching simulation results to experiments. There are relatively few studies which attempt to derive the macroscopic parameters required by the bidomain equations directly from measurable microscale quantities. One approach to obtain these values is homogenization of partial differential equations (PDEs) as in Neu and Krassowska [44]. In this approach, the intracellular region of tissue is modeled as a collection of periodically arranged cells connected through physical openings that correspond to gap junctions. They can compute effective conductivitities by solving Laplace s equation on a periodic domain. Although these authors present a detailed derivation involving homoge- 13

29 nization, they resort to a resistor network model in order to find analytical formulae for the effective bidomain conductivities. A shared inconvenience of their PDE and resistor network models is that they intertwine transverse and longitudinal gap junctional connections, making it difficult to assign proper values based on separate experimental measurements of side-to-side and end-to-end coupling strengths [56]. An alternative approach to computing passive conductivities is given by Stinstra et al. [51], who create a detailed tissue model designed to account for realistically complex cell shapes with random variability. Sinstra et al. then solve Laplace s equation over a domain containing several cells to obtain the effective conductivities. Although their simulations involve a high level of detail in the cellular microstructure, their calculation for intracellular transverse conductivity yields values an order of magnitude less than those in the experimental literature. They propose two possible explanations of this discrepancy: the total gap junctional conductivity may be larger than measured, and the gap junctions may be more preferentially located on the cell sides than is measured. In the present chapter, we explore the feasibility of both explanations. In this chapter, we follow a homogenization approach similar to that of Neu and Krassowska [44] to derive the effective conductivites of cardiac tissue. Unlike this earlier homogenization work, however, we employ our microscale PDE model to obtain the macroscopic bidomain parameters. In particular, we do not resort to a resistor network to obtain the macroscopic parameters from the measured microscale quantities. In our approach, we separate the structure and placement of cells from the gap junctional connections between them. Specifically, we idealize cells as rectangular prisms, inside of which the electric potential satisifies Laplace s equation. Instead of modeling gap junctions as discrete geometrical entities akin to Figure 2.1a, we include their effect as boundary conditions on each cell membrane as in Figure 2.1b. This approach assumes no more detail than is provided by direct measurements, such as those in [56]. These modeling decisions make a mathematically natural framework within which to study the effects on conductivity of the arrangement of cells and of the distribution of gap junctions on cell membranes. In some cases, we are able to obtain analytical formulae for the effective conductivitites. In cases where we cannot do so, we need only to solve Laplace s equation on one cell of fixed geometry. The remainder of this chapter proceeds as follows: In Section 2.2, we motivate the calculation of effective conductivities by considering their role in the bidomain equations. Section 2.3 sets up the electrostatic Laplace s equation in the aligned cellular arrangement. In Section 2.4, we perform the homogenization in the aligned arrangement and calculate the numerical values of the effective conductivities. Similarly, in Section 2.5, we formulate the electrostatic problem in a brick-like cellular arrangement, perform the homogenization, and calculate the corresponding 14

30 a b Figure 2.1: Instead of modeling gap junctions through complex cellular geometry (a), we model them through flux boundary conditions on simple geometry (b). The multiple resistors shown in (b) represent a continuous boundary condition, see equations (2.2) - (2.7). conductivities. In Section 2.6, we compare our conductivity values to those in the literature and discuss the relevance of our computations to electromechanical simulations. In Section 2.7 we present our conclusions. Finally, in a brief Appendix, we present additional details regarding the derivation of the extracellular effective conductivities. 2.2 Bidomain Equations and Effective Conductivity The bidomain equations provide the most realistic macroscopic description of the electrical activity of cardiac tissue under normal and pathological conditions [22]. They govern the intracellular and extracellular electric potential in an averaged sense and are given by (σ i φ 0 i ) = β( C t (φ 0 i φ 0 e ) + I ion(φ 0 i φ 0 e, ω)), (σ e φ 0 e ) = β( C t (φ 0 i φ 0 e ) + I ion(φ 0 i φ 0 e, ω)), where σ i and σ e are the macroscopic conductivity tensors of the intracellular and extracellular spaces, β is the membrane surface area per unit volume of tissue, C is the membrane capacitance per unit area, ω stands for relevant gating variables, and I ion is the ionic current per unit area of the membrane. Here, φ 0 i (t, x) and φ 0 e (t, x) are each defined for all x, and are to be interpreted as the locally averaged intracellular or extracellular potential near the point x. Note that the intracellular and extracellular potential are each separately averaged, not averaged with each other. 15

31 a b φ i,j+1,k i l φ i+1,j+1,k i w c x x 3 2 x 1 x 2 x 1 φ i,j,k i φ i+1,j,k i w p Figure 2.2: The aligned cellular architecture: (a) cells of dimension l w c w c are arranged in three space with period l w p w p ; (b) an x 1, x 2 cross-section. We write the superscript 0 because φ 0 i is the leading order term of the asymptotic expansion (2.15) for intracellular potential. Similarly, φ 0 e is the leading order term of a corresponding expansion for extracellular potential. If we locally align the coordinates with the myocardial fibers so that the x 1 direction coincides with the fiber direction and assume that the x 2 and x 3 directions are locally indistinguishable, then the conductivity tensors are diagonal and involve only the longitudinal and transverse conductivities. That is, σ i,l 0 0 σ e,l 0 0 σ i = 0 σ i,t 0 and σ e = 0 σ e,t σ i,t 0 0 σ e,t A goal of the present work is to determine σ i,l, σ i,t, σ e,l, σ e,t directly from measured microscopic quantities. 2.3 Full Cellular Model in an Aligned Arrangement We model myocytes as l w c w c prisms arranged periodically in three-space with period l w p w p, see Figure 2.2. The length of the cells l is assumed to equal the longitudinal period, but the width and height of the cells w c are smaller than the transverse periods w p in order to provide an extracellular volume fraction α = 1 (w c /w p ) 2. In this model, the intracellular space is not physically contiguous. Instead, resistive connections allow current to flow directly between the interiors of adjacent cells. Note that we do not attempt to account for the branching of cells in the present work. Additionally, we ignore fiber rotation because our analysis is purely local. To identify the bidomain parameters σ i and σ e, we consider the degenerate case of a steady state without transmembrane ionic current. In this situation, 16

32 the equations for intracellular and extracellular potential decouple. In the following, we consider only the intracellular potential; the similar calculations for the extracellular potential are described in Appendix 2.8. The electric potential in the intracellular space satisfies Laplace s equation, φ i,j,k i ( x) = 0 in Ω i,j,k i, (2.1) where Ω i,j,k i = [i l, (i + 1) l] [j w p, j w p + w c ] [k w p, k w p + w c ] is the region occupied by the (i, j, k)-th cell, and φ i,j,k i is the intracellular potential of the (i, j, k)-th cell, indicated in Figure 2.2b. For ease of notation, we identify the domain of the function φ i,j,k i with [0, l] [0, w c ] [0, w c ]. We denote position by x = (x 1, x 2, x 3 ). We model the gap junctions between cells in a continuous manner through boundary conditions on equation (2.1). The current density between two neighboring cells is proportional to the potential difference between the positions on opposite sides of the gap junctions: σ c x1 φ i,j,k i (l, x 2, x 3 ) = g GJ,end w 2 c σ c x2 φ i,j,k i (x 1, w c, x 3 ) = g GJ,side ( φ i,j,k i l w c σ c x3 φ i,j,k i (x 1, x 2, w c ) = g GJ,side ( φ i,j,k i (l, x 2, x 3 ) φ i+1,j,k i (0, x 2, x 3 ) ), (2.2) (x 1, w c, x 3 ) φ i,j+1,k i (x 1, 0, x 3 ) ), (2.3) (x 1, x 2, w c ) φ i,j,k+1 i (x 1, x 2, 0) ), (2.4) ( φ i,j,k i l w c σ c x1 φ i,j,k i (l, x 2, x 3 ) = σ c x1 φ i+1,j,k i (0, x 2, x 3 ), (2.5) σ c x2 φ i,j,k i (x 1, w c, x 3 ) = σ c x2 φ i,j+1,k i (x 1, 0, x 3 ), (2.6) σ c x3 φ i,j,k i (x 1, x 2, w c ) = σ c x3 φ i,j,k+1 i (x 1, x 2, 0), (2.7) where σ c is the cytoplasmic conductivity (ms/cm), g GJ,end is the total conductance (ms) of all gap junctions on one end of a cell, and g GJ,side is the total conductance (ms) of all gap junctions on one side of a cell. Equations (2.2) (2.4) balance cytosolic and gap junctional current, whereas equations (2.5) (2.7) equate the current leaving each cell with the current entering its appropriate neighbor. See Table 2.4 for the physical parameters and variables introduced for this model. In the remainder of the present section, we nondimensionalize equations (2.1) (2.7) and state the effective conductivity problem in this aligned cellular arrangement. Note that the model could easily be generalized to allow for more complex geometries, such as the jutting cells analyzed in [25]. It could also allow gap junctional density to be varying within the ends or sides of cells. As an illustration, we explore the effects of a brick-like cellular arrangement in Section