Mechanism of Arrhythmogenesis Driven by Early After Depolarizations in Cardiac Tissue

Figures

• 

  Figure 1.. Phenomenological model of Ca spark recruitment in ventricular myocytes. (A) Compartment architecture. The cell is divided into 4 compartments which represent the average Ca concentration in the SR and cytosol in the vicinity of J and NJ clusters. The concentrations cb and csrb denote the average concentration in the cytosol and SR in the vicinity of J clusters, while ci and csri denote the concentrations near NJ clusters. The total SR volume near J and NJ clusters is denoted as vsrb and vsri respectively, while the total cytosolic volume is denoted as vb and vi. The currents linking these two spaces are the Ca release Jrb due to a population of sparks at J clusters, and the uptake pump current Jupb which replenishes the SR volume. Similarly, Ca release and uptake from the SR volume near NJ clusters is denoted by Jri and Jupi respectively. The currents Jdsr and Jdc are averaged diffusive currents proportional to the concentration difference between the internal cell volumes. (B) Markovian model of the LCC current. Markov states in dashed box (spark off) represent channels facing low Ca(0.1μM), while channels outside box (spark on) face a Ca spark with high local Ca(100μM). The…

• 

  Figure 2.. APD fluctuations in the absence of EADs. (A) Beat-to-beat voltage fluctuations. The cell is paced at CL=500ms to steady state (50 beats), and the voltage of the last ten beats is overlayed on the same graph. The number of J clusters is Nb=4000. (B) Distribution of APD90. Histogram is constructed by measuring the APD at 90% repolarization over 15000 beats. The distribution is normalized so that the maximum value is 1.

• 

  Figure 3.. APD variability in the presence of EADs. (A) Bifurcation diagram using a dynamic pacing protocol. The cell is paced at a CL for 200 beats and the APD is plotted for the last 50 beats. Upon completion of the 200 beats the CL is increased incrementally and the pacing protocol is repeated. In this manner the CL is swept from 440ms to 552ms. The APD is measured as the time between the AP upstroke and when the voltage crosses Vc=-40mV during repolarization. Black points denote normal conditions ACa=0.15, and the red points correspond to reduced Ca-induced inactivation (ACa=0.09). (B) The voltage time course for the last 50 beats when the cell is paced in the EAD regime (CL=520ms). (C) Distribution of APD measured from 15000 paced beats at CL=520ms. The black filled circles represent the distribution under normal conditions, while the red circles denote EAD conditions. The distributions are normalized so that the maximum value is 1.

• 

  Figure 4.. APD fluctuations in 2D tissue. (A) Steady state APD distribution for a 2D tissue of size N×N. The histogram is constructed by pacing the tissue at CL=420ms for 5000 beats and recording the APD after an initial transient of 50 beats. The tissue dimension is indicated on the inset. Distributions are normalized so that the maximum is at 1. (B) Standard deviation of the APD (σ) as a function of tissue dimension N. (C) Steady state APD distribution when the tissue is paced in the EAD regime at a CL=520ms for 5000 beats (D) Bifurcation diagram. The tissue is paced to steady state (200 beats) and the APD is plotted for the last 50 beats. The black, red, and blue points correspond to tissue sizes of N=1,5 and 10 respectively.

• 

  Figure 5.. Deterministic limit of the stochastic model. (A) Plot of ⟨APD⟩ as a function of tissue size N (black filed circles). The standard deviation σ is plotted as an error bar. The tissue is paced for 5000 beats at CL=420ms. The horizontal red line corresponds to the deterministic limit paced to steady state (50 beats). (B) Bifurcation diagram for the single cell stochastic model (blue filled circles) along with the predictions of the deterministic model (red and black filled circles). The deterministic bifurcation diagram is computed using a dynamic pacing protocol where the CL is increased gradually from CLo=475ms to CL1=550ms (black points). At each CL the cell is paced for 200 beats and the APD for the last 50 beats are plotted. Once CL1 is reached the CL is then decreased back to CLo (red filled circles). (C) The steady state voltage plotted for the last 50 beats, at CL=535ms, for the stochastic (black line) and the deterministic (red line) model.

• 

  Figure 6:. Spatial patterns and fluctuations in APD in cardiac tissue with EAD. (A) APD as a function of beat number n for a cell at the center of a 3 × 100 tissue array, paced at CL1=400ms and CL2=420ms. (B) APD as a function of the number of cells lx along the cable after 200 (black line) and 201 (red line) beats at CL2. (C) APD versus n when the tissue is paced in the bistable regime (CL2=520ms), displaying large intermittent fluctuations. (D) Spatial distribution of APD along the cable at beats 200 and 201 in the bistable regime. (E) APD vs n when the system is paced in the alternans regime (CL=530ms). (F) Spatial distribution of APD at beats 200 and 201 showing stable spatially discordant alternans.

• 

  Figure 7:. Wave Propagation in 2D Cardiac Tissue with EADs. Snapshots of planar wave propagation in a 100 × 100 cardiac tissue, paced along a 10-cell-wide strip on the left edge. The tissue is initially paced for 20 beats at CL=420ms, where the cell model does not exhibit alternans. Subsequently, the CL is switched to530ms, where EADs occur in the deterministic limit. The snapshots show the development of pronounced heterogeneities in the action potential (AP) wave back by the 40th beat, leading to wavefront break and reentrant circuit formation by the 43 rd beat. Snapshot f is 400ms after snapshot e during the 43rd beat.

• 

  Figure 8:. Analysis of Nonlinear Dynamics and EAD-Induced Bifurcations in Cardiac Tissue. (A) S1S2 restitution curve. The final APD is plotted against the previous diastolic interval (DI) after pacing the cell to steady state (50 beats) at S1=500ms. The red points are the measured APD for the S2 beat and the black line is a cubic splines fit. The inset shows voltage for the last paced beat at the indicated points on the restitution curve. The black, blue, yellow, and green plots correspond to the colored points on the restitution curve. (B) Period-doubling bifurcation diagram. The presence of EADs leads to a period-doubling bifurcation, resulting in an alternating behavior where EADs occur only on alternate beats. Blue points are computed by sweeping the period from low to high, and the red points are computed by sweeping the period from high to low. (C) A cob-web diagram demonstrating bistability of the nonlinear map where the stable fixed point (blue) coexists with an alternating behavior (red). The pacing period is fixed at T=550ms. (D) Plot of the function G2(DI)-DI, where G2(DI) is the second iteration of the S1S2 restituion map. Intersections with the dashed line indicate fixed points…

• 

  Figure 9:. Conduction block in Cardiac Tissue. (A) S1S2 restitution curve for a tissue strip 60 × 3 cells. All cells on a strip of size 10 × 3 are stimulated simulatenously using an S1S2 protocol and the APD is measured at the distal end at cell (60, 2). Black line denotes a cubic splines fit to the S1S2 restitution curve. Conduction block occurs for S2 intervals shorter than approximately DI≈25ms (green shaded region). Cobweb diagrams are drawn using a cubic spline fit to the restitution curve for initial conditions DI0=300ms (blue line) and D0=400ms (red line). Here, the red line indicates a sequence leading to conduction block. (B) Phase diagram of the system using the cubic splines fit to the S1S2 restitution curve. The y-axis represents the initial diastolic interval DI0, while the x-axis represents the pacing period T. For each initial condition, the nonlinear map was iterated 100 times. In the resulting phase diagram, blue regions indicate a stable period-1 solution, red regions indicate a stable period-2 solution, and black regions represent conduction block i.e. the system reaches DI<25ms at some point during the 100 iteration trajectory.