Reentry via high-frequency pacing in a mathematical model for human-ventricular cardiac tissue with a localized fibrotic region

Figures

• 

  Figure 1. Fibroblast clumps. (a) Pseudocolour plot of a domain with a fibroblast clump; inside the circular clump fibroblasts are distributed randomly such that the percentage of sites with fibroblasts is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}$$\end{document}pf = 30%. In our model, a site can be occupied either by a myocyte or by a fibroblast, but not both; blue and yellowish-green colours indicate normal myocytes and fibroblasts, respectively. (b) A fibroblast clump, as in (a), where the excitability of the myocytes, inside and around the clump, is modified. The red colour indicates myocytes with modified excitability (see text).

• 

  Figure 2. Wave distortions and spiral waves, around a fibroblast clump, caused by high-frequency pacing (PP stimulation protocol). Pseudocolour plots of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm{V}}}_{{\rm{m}}}$$\end{document}Vm showing the formation of wave distortions, because of high-frequency pacing and with the protocol PP (see text), at a fibroblast clump; in this figure we use the illustrative parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}\mathrm{=30}$$\end{document}pf=30% and radius R = 1 cm. Note that WD occurs first at the periphery of the fibroblast clump at time t = 15.18 s; eventually this WD evolves into spiral waves at t = 17.38 s.

• 

  Figure 3. Action potentials. Figure showing the action potentials of an isolated myocyte (blue curve) and a myocyte coupled to a fibroblast (red curve). We see that the APD of the red-curve is slightly larger than the APD of the blue curve, which indicates that coupling of a myocyte to a fibroblast slightly increases the APD of the myocyte.

• 

  Figure 4. Plots of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}τ, the mean time for the formation of wave distortions (WDs), versus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}$$\end{document}pf. The variation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}τ with the percentage \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}$$\end{document}pf of fibroblasts, in the clump, for clump radii R = 0.6 cm (blue…

• 

  Figure 5. Early occurrence of WD on the reduction of excitability inside and around a fibroblast clump. Pseudocolour plot of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{m}$$\end{document}Vm showing the early occurrence of WD at t \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\simeq $$\end{document}≃ 1.8 s for the illustrative case of a fibroblast clump with R = 1 cm and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}$$\end{document}pf = 30%. Here, we reduce the excitability inside the clump and in an annular region around it (marked by red colour) within a circular region with radius of 1.4 cm (Fig. 1(b)), by reducing the conductance…

• 

  Figure 6. Increase of the wave velocity around the fibroblast clump with enhanced excitability in and around the clump. Pseudocolour plot of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{m}$$\end{document}Vm showing the increase in the wave velocity in the vicinity of the fibroblast clump with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}$$\end{document}pf = 30%, R = 1 cm when the excitability of the medium is increased (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G}_{Na}$$\end{document}GNa \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}…

• 

  Figure 7. Reentry via the high-frequency TP protocol. Pseudocolour plot of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{m}$$\end{document}Vm showing the initiation of reentry, because of our high-frequency pacing (TP protocol), in a medium with a fibroblast clump with R = 2.4 cm and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}$$\end{document}pf = 33% (top row) and 43% (bottom row).

• 

  Figure 8. Stability diagram, showing regions in which reentry occurs in the R-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}$$\end{document}pf plane when we stimulate a medium with a clump of fibroblasts by using the high-frequency TP stimulation protocol. We find no reentry in the outermost, dark-blue region of the R-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}$$\end{document}pf plane. The different colours indicate the number of times we obtain reentry in the 10 simulations (for each pair of values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}$$\end{document}pf and R, we perform 10 simulations for 10 different…

• 

  Figure 9. Region in the R-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}$$\end{document}pf plane in which reentry occurs via our low-frequency TP stimulation protocol. A stability diagram showing regions, in the R-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}$$\end{document}pf plane, where reentry occurs (red) and where it does not (blue), with our TP-pacing protocol for PCL = 200 ms. The colours also indicate the total number of reentries we observe in our simulations for 10 different realizations of the random distribution of fibroblasts.

• 

  Figure 10. Reentry via our low-frequency TP stimulation protocol. Pseudocolour plot of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{m}$$\end{document}Vm showing the formation of reentry, via our low-frequency TP pacing protocol, in a medium with a fibroblast clump of size R = 3 cm and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}$$\end{document}pf = 41%. After the passage of the last (20th) wave, the irregular wave pattern (t = 4.3 s) left inside the clump triggers excitations that leads to reentry (see text).

• 

  Figure 11. TP-pacing induced reentry in our 2D and 3D simulation domains with fibroblast clumps of R = 2.4 cm. Figure showing the range of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}$$\end{document}pf (horizontal axis) in which reentry occurs, via our TP-pacing protocol with PCL = 152 ms, for 2D (blue curves) and 3D (black curves) simulation domains with fibroblast clumps of R = 2.4 cm (in the 3D case the clump is cylindrical). The vertical axis shows the total number of reentry episodes that we observe in our simulations for 10 different realizations of the fibroblast distribution.

• 

  Figure 12. TP-pacing induced reentry in our 3D simulations with a cylindrical fibroblast clump. Pseudocolour plot of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{m}$$\end{document}Vm showing the formation of reentry, via our TP stimulation protocol with PCL = 152 ms, in a 3D domain with a cylindrical fibroblast clump, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{f}$$\end{document}pf = 65%, and R = 2.4 cm.