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Control of spiral wave and spatiotemporal chaos in two-layer aging cardiac tissues

Li Qian-Yun Bai Jing Tang Guo-Ning

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Control of spiral wave and spatiotemporal chaos in two-layer aging cardiac tissues

Li Qian-Yun, Bai Jing, Tang Guo-Ning
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  • Cardiac arrhythmias can be caused by the occurrence of electrical spiral waves and spatiotemporal chaos in the cardiac tissues, as well as by the changes of cardiac tissues resulting from the electrical coupling of cardiomyocytes to fibroblasts (M-F coupling). How to control the arrhythmia induced by spiral wave or spatiotemporal chaos is the problem which has attracted much attention of scientists. In this paper, a two-layer composite medium is constructed by using cardiomyocytes and fibroblasts. Luo-Rudy phase I cardiac model and passive model of fibroblast are used to study the effects of the M-F coupling on the formation of spiral wave and the control of spiral wave and spatiotemporal chaos in a two-layer composite medium. A control scheme that the spiral waves and spatiotemporal chaos are controlled by increasing the coupling strength between cells is proposed. The numerical results show that the M-F coupling has an important influence on the dynamics of spiral wave. With the increase of the density of fibroblasts, the M-F coupling may result in spiral wave meandering and spiral wave breaking into spatiotemporal chaos, and even induce the transition from spatiotemporal chaos (or spiral wave) to no wave. The eliminating spiral wave and spatiotemporal chaos in the composite medium by increasing the coupling strength between cells is only effective in most of cases, depending on the role played by fibroblasts. When fibroblasts act as current sinks for the cardiomyocyte, the spiral wave and spatiotemporal chaos are effectively eliminated only in most of cases by increasing the coupling strength between cells, and the controlled area is small. When fibroblasts act as a current source for the cardiomyocyte, increasing the coupling strength between cells to a value higher than a critical value can effectively terminate spiral wave and spatiotemporal chaos, and the controllable area is greatly increased compared with the former. Increasing the coupling strength between cardiomyocytes is a key factor in controlling the spiral waves and spatiotemporal chaos.
      Corresponding author: Tang Guo-Ning, tangguoning@sohu.com
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11565005, 11947413)
    [1]

    Johnson R D, Camelliti P 2018 Int. J. Mol. Sci. 19 866Google Scholar

    [2]

    Camelliti P, Borg T K, Kohl P 2005 Cardiovasc. Res. 65 40Google Scholar

    [3]

    Greisas A, Zlochiver S 2016 Cardiovasc. Eng. Techn. 7 290Google Scholar

    [4]

    Sachse F B, Moreno A P, Abildskov J A 2008 Ann. Biomed. Eng. 36 41Google Scholar

    [5]

    Peters N S, Wit A L 1998 Circulation 97 1746Google Scholar

    [6]

    Jacquemet V, Henriquez C S 2008 Am. J. Physiol. Heart Circ. Physiol. 294 H2040Google Scholar

    [7]

    Xie Y F, Garfinkel A, Weiss J N, Qu Z L 2009 Am. J. Physiol. Heart Circ. Physiol. 297 H775Google Scholar

    [8]

    Nguyen T P, Xie Y F, Garfinkel A, Qu Z L, Weiss J N 2012 Cardiovasc. Res. 93 242Google Scholar

    [9]

    Brown T R, Krogh-Madsen T, Christini D J 2015 BioMed. Res. Int. 2015 465714Google Scholar

    [10]

    Chilton L, Giles W R, Smith G L 2007 J. Physi. 583 225Google Scholar

    [11]

    Louault C, Benamer N, Faivre J F, Potreau D, Bescond J 2008 BBA-Biomembranes 1778 2097Google Scholar

    [12]

    Vasquez C, Mohandas P, Louie K L, Benamer N, Bapat A C, Morley G E 2010 Circulation Research 107 1011Google Scholar

    [13]

    Baudino T A, McFadden A, Fix C, Hastings J, Price R, Borg T K 2008 Microsc. Microanal. 14 117Google Scholar

    [14]

    Shajahan T K, Nayak A R, Pandit R 2009 Plos One 4 e4738Google Scholar

    [15]

    Nayak A R, Shajahan T K, Panfilov A V, Pandit R 2013 Plos One 8 e72950Google Scholar

    [16]

    Kofron C M, Kim T Y, King M E, Xie A, Feng F, Park E, Qu Z, Choi B R, Mende U 2017 Am. J. Physiol. Heart Circ. Physiol. 313 H810Google Scholar

    [17]

    Kohl P, Camelliti P, Burton F L, Smith G L 2005 J. Electrocardiol 38 45Google Scholar

    [18]

    Luo C H, Rudy Y 1991 Circ. Res. 68 1501Google Scholar

  • 图 1  不同参数下在$\rho \text{-} n$参数平面上的相图(■代表稳定螺旋波; ●代表漫游单螺旋波; ◆代表不稳定螺旋波; ▼代表时空混沌态; ▲代表介质无波) (a) $ {G}_{\rm{f}}=1\;{\rm{ns}}$, ${G}_{\rm{gap}}= $$ 3\;{\rm{ns}}$; (b) $ {G}_{\rm{f}}=4\;{\rm{ns}}$, $ {G}_{\rm{gap}}=1\;{\rm{ns}}$

    Figure 1.  Phase diagrams on the $\rho \text{-} n$ parameter plane with different parameters: (a) $ {G}_{\rm{f}}=1\;{\rm{ns}}$, $ {G}_{\rm{gap}}=3\;{\rm{ns}}$; (b) ${G}_{\rm{f}}= $$ 4\;{\rm{ns}}$, $ {G}_{\rm{gap}}=1\;{\rm{ns}}$. Black square, stable spiral wave; black circle, meandering of single spiral wave; black rhombus, unstable spiral wave; black down-triangle, spatiotemporal chaos; black up-triangle, no wave can propagate in the medium.

    图 2  不同参数下其中一层心肌层的螺旋波斑图 (a) ${G}_{\rm{f}}= $$ 4\;{\rm{ns}}$, $ {G}_{\rm{gap}}=1\;{\rm{ns}}$, $\rho = 20{\text{%}}$, $n = {\rm{1}}$; (b) $ {G}_{\rm{f}}=4\;{\rm{ns}}$, $ {G}_{\rm{gap}}=1\;{\rm{ns}}$, $\rho = 30{\text{%}}$, $n = {\rm{3}}$; (c) $ {G}_{\rm{f}}=4\;{\rm{ns}}$, $ {G}_{\rm{gap}}=1\;{\rm{ns}}$, $\rho = 40{\text{%}}$, $n = {\rm{2}}$; (d) $ {G}_{\rm{f}}= $$ 1\;{\rm{ns}}$, $ {G}_{\rm{gap}}=3\;{\rm{ns}}$, $\rho = {\rm{4}}0{\text{%}}$, $n = $4

    Figure 2.  Pattern of spiral wave in one layer cardiac tissue for different parameters: (a) $ {G}_{\rm{f}}=4\;{\rm{ns}}$, $ {G}_{\rm{gap}}=1\;{\rm{ns}}$, $\rho = 20{\text{%}}$, $n = {\rm{1}}$; (b) $ {G}_{\rm{f}}=4\;{\rm{ns}}$, $ {G}_{\rm{gap}}=1\;{\rm{ns}}$, $\rho = 30{\text{%}}$, $n = {\rm{3}}$; (c) ${G}_{\rm{f}}= 4\;{\rm{ns}}$, $ {G}_{\rm{gap}}=1\;{\rm{ns}}$, $\rho = 40{\text{%}}$, $n = {\rm{2}}$; (d) ${G}_{\rm{f}}= $$ 1 \;{\rm{ns}}$, $ {G}_{\rm{gap}}=3\;{\rm{ns}}$, $\rho = {\rm{4}}0{\text{%}}$, $n = {\rm{4}}$.

    图 3  不同参数下成纤维细胞静息电位随时间变化 (a) ${G_{\rm{f}}} = 1\;{\rm{ns}}$, ${G_{{\rm{gap}}}} = 3\;{\rm{ns}}$, $n = 2$, $\rho = 30{\text{%}}$; (b) ${G_{\rm{f}}} = 1\;{\rm{ns}}$, ${G_{{\rm{gap}}}} = 3\;{\rm{ns}}$, $n = 5$, $\rho = 30{\text{%}}$; (c) ${G_{\rm{f}}} = 4\;{\rm{ns}}$, ${G_{{\rm{gap}}}} = 1\;{\rm{ns}}$, $n = 2$, $\rho = 30{\text{%}}$; (d) ${G_{\rm{f}}} = 4\;{\rm{ns}}$, ${G_{{\rm{gap}}}} = 1\;{\rm{ns}}$, $n = 5$, $\rho = 30{\text{%}}$

    Figure 3.  Evolution of the resting potential of a fibroblast for different control parameters: (a) ${G_{\rm{f}}} = 1\;{\rm{ns}}$, ${G_{{\rm{gap}}}} = 3\;{\rm{ns}}$, $n = 2$, $\rho = 30{\text{%}}$; (b) ${G_{\rm{f}}} = 1\;{\rm{ns}}$, ${G_{{\rm{gap}}}} = 3\;{\rm{ns}}$, $n = 5$, $\rho = 30{\text{%}}$; (c) ${G_{\rm{f}}} = 4\;{\rm{ns}}$, ${G_{{\rm{gap}}}} = 1\;{\rm{ns}}$, $n = 2$, $\rho = 30{\text{%}}$; (d) ${G_{\rm{f}}} = 4\;{\rm{ns}}$, ${G_{{\rm{gap}}}} = 1\;{\rm{ns}}$, $n = 5$, $\rho = 30{\text{%}}$.

    图 4  图1(a)对应的各初态下在$D \text{-} {G_{{\rm{gap}}}}$参数平面上的相图, 其中, 第1行到第4行各图参数分别对应$\rho = $ 10%, 20%, 30%, 40%; 第1列到第5列各图参数分别对应$n = 1, \;2, \;3, \;4, \;5$; ■代表不可控, Δ代表慢可控, □代表快可控

    Figure 4.  Phase diagram in the $D \text{-} {G_{{\rm{gap}}}}$ parameter plane for different initial states showed in Fig. 1(a). The parameter $\rho $ of the panels from the first row to the fourth row equals to 10%, 20%, 30%, 40%, respectively. The parameter $n$ of the panels from the first column to the fifth column equals to 1, 2, 3, 4, 5, respectively. Black and hollow squares represent the uncontrolled point and fast control point, respectively. Hollow up-triangle represents the slow control point.

    图 5  图1(b) 对应的各初态下在$D \text{-} {G_{{\rm{gap}}}}$参数平面上的相图, 其中, 第1行到第4行各图参数分别对应$\rho =$ 10%, 20%, 30%, 40%; 第1列到第5列各图参数分别对应$n = 1, \;2, \;3, \;4, \;5$; ■代表不可控, Δ代表慢可控, □代表快可控

    Figure 5.  Phase diagram in the $D \text{-} {G_{{\rm{gap}}}}$ parameter plane for different initial states showed in Fig. 1(b). The parameter $\rho $ of the panels from the first row to the fourth row equals to 10%, 20%, 30%, 40%, respectively. The parameter $n$ of the panels from the first column row to the fifth column equals to 1, 2, 3, 4, 5, respectively. Black and hollow squares represent the uncontrolled point and fast control point, respectively. Hollow up-triangle represents the slow control point.

    图 6  在不同的初态斑图和控制参数下平均膜电位差随时间的变化 (a)初态为图2(b), $D = 0{\rm{.004}}\;{\rm{c}}{{\rm{m}}^2}/{\rm{ms}}$, ${G_{{\rm{gap}}}} = {\rm{10\; ns}}$; (b)初态为图2(a), $D = 0{\rm{.003}}\;{\rm{c}}{{\rm{m}}^2}/{\rm{ms}}$, ${G_{{\rm{gap}}}} = $$ {\rm{10\; ns}}$; (c)初态为图2(c), $D = 0{\rm{.004}}\;{\rm{c}}{{\rm{m}}^2}/{\rm{ms}}$, ${G_{{\rm{gap}}}} = {\rm{12\; ns}}$

    Figure 6.  Evolution of the average membrane potential difference for different control parameters and initial states: (a) The initial state is shown in Fig. 2(b), $D = 0{\rm{.004}}\;{\rm{c}}{{\rm{m}}^2}/{\rm{ms}}$, ${G_{{\rm{gap}}}} = {\rm{10\; ns}}$; (b) the initial state is shown in Fig. 2(a), $D = 0{\rm{.003}}\;{\rm{c}}{{\rm{m}}^2}/{\rm{ms}}$, ${G_{{\rm{gap}}}} = {\rm{10\; ns}}$; (c) the initial state is shown in Fig. 2(c), $D = 0{\rm{.004}}\;{\rm{c}}{{\rm{m}}^2}/{\rm{ms}}$, ${G_{{\rm{gap}}}} = {\rm{12\; ns}}$.

    图 7  图6(a)参数下第一层复合介质中$j = 150$这一行(a)与$i = 150$这一列(b)各细胞膜电位随时间的变化斑图

    Figure 7.  Spatiotemporal pattern of the membrane potential of the grid points in the 150th row (a) and the 150th column (b) in the first layer composite medium for the parameters corresponding to Fig. 6(a).

    图 8  图6(b)参数下不同时刻的膜电位斑图 (a) $t = $$ {\rm{10\; ms}}$; (b) $t = {\rm{40\; ms}}$; (c) $t = {\rm{8}}0\;{\rm{ms}}$; (d) $t = {\rm{12}}0\;{\rm{ms}}$; (e) $t = $$ {\rm{18}}0\;{\rm{ms}}$; (f) $t = {\rm{22}}0\;{\rm{ms}}$; (g) $t = {\rm{32}}0\;{\rm{ms}}$; (h) $t = {\rm{38}}0\;{\rm{ms}}$; (i) $t = {\rm{44}}0\;{\rm{ms}}$

    Figure 8.  Pattern of membrane potential at different time moments for the parameters corresponding to Fig. 6(b): (a) $t = $$ {\rm{10\; ms}}$; (b) $t = {\rm{40\; ms}}$; (c) $t = {\rm{8}}0\;{\rm{ms}}$; (d) $t = {\rm{12}}0\;{\rm{ms}}$; (e) $t = $$ {\rm{18}}0\;{\rm{ms}}$; (f) $t = {\rm{22}}0\;{\rm{ms}}$; (g) $t = {\rm{32}}0\;{\rm{ms}}$; (h) $t = {\rm{38}}0\;{\rm{ms}}$; (i) $t = {\rm{44}}0\;{\rm{ms}}$.

    图 9  图6(c)参数不同时刻的膜电位斑图 (a) $t = $$ {\rm{1}}0\;{\rm{ms}}$; (b) $t = {\rm{30\; ms}}$; (c) $t = {\rm{8}}0\;{\rm{ms}}$; (d) $t = {\rm{11}}0\;{\rm{ms}}$; (e) $t = $$ {\rm{14}}0\;{\rm{ms}}$; (f) $t = {\rm{17}}0\;{\rm{ms}}$; (g) $t = {\rm{26}}0\;{\rm{ms}}$; (h) $t = {\rm{36}}0\;{\rm{ms}}$; (i) $t = {\rm{48}}0\;{\rm{ms}}$

    Figure 9.  Pattern of membrane potential at different time moments for the parameters corresponding to Fig. 6(c): (a) $t = $$ {\rm{1}}0\;{\rm{ms}}$; (b) $t = {\rm{30\; ms}}$; (c) $t = {\rm{8}}0\;{\rm{ms}}$; (d) $t = {\rm{11}}0\;{\rm{ms}}$; (e) $t = $$ {\rm{14}}0\;{\rm{ms}}$; (f) $t = {\rm{17}}0\;{\rm{ms}}$; (g) $t = {\rm{26}}0\;{\rm{ms}}$; (h) $t = {\rm{36}}0\;{\rm{ms}}$; (i) $t = {\rm{48}}0\;{\rm{ms}}$.

  • [1]

    Johnson R D, Camelliti P 2018 Int. J. Mol. Sci. 19 866Google Scholar

    [2]

    Camelliti P, Borg T K, Kohl P 2005 Cardiovasc. Res. 65 40Google Scholar

    [3]

    Greisas A, Zlochiver S 2016 Cardiovasc. Eng. Techn. 7 290Google Scholar

    [4]

    Sachse F B, Moreno A P, Abildskov J A 2008 Ann. Biomed. Eng. 36 41Google Scholar

    [5]

    Peters N S, Wit A L 1998 Circulation 97 1746Google Scholar

    [6]

    Jacquemet V, Henriquez C S 2008 Am. J. Physiol. Heart Circ. Physiol. 294 H2040Google Scholar

    [7]

    Xie Y F, Garfinkel A, Weiss J N, Qu Z L 2009 Am. J. Physiol. Heart Circ. Physiol. 297 H775Google Scholar

    [8]

    Nguyen T P, Xie Y F, Garfinkel A, Qu Z L, Weiss J N 2012 Cardiovasc. Res. 93 242Google Scholar

    [9]

    Brown T R, Krogh-Madsen T, Christini D J 2015 BioMed. Res. Int. 2015 465714Google Scholar

    [10]

    Chilton L, Giles W R, Smith G L 2007 J. Physi. 583 225Google Scholar

    [11]

    Louault C, Benamer N, Faivre J F, Potreau D, Bescond J 2008 BBA-Biomembranes 1778 2097Google Scholar

    [12]

    Vasquez C, Mohandas P, Louie K L, Benamer N, Bapat A C, Morley G E 2010 Circulation Research 107 1011Google Scholar

    [13]

    Baudino T A, McFadden A, Fix C, Hastings J, Price R, Borg T K 2008 Microsc. Microanal. 14 117Google Scholar

    [14]

    Shajahan T K, Nayak A R, Pandit R 2009 Plos One 4 e4738Google Scholar

    [15]

    Nayak A R, Shajahan T K, Panfilov A V, Pandit R 2013 Plos One 8 e72950Google Scholar

    [16]

    Kofron C M, Kim T Y, King M E, Xie A, Feng F, Park E, Qu Z, Choi B R, Mende U 2017 Am. J. Physiol. Heart Circ. Physiol. 313 H810Google Scholar

    [17]

    Kohl P, Camelliti P, Burton F L, Smith G L 2005 J. Electrocardiol 38 45Google Scholar

    [18]

    Luo C H, Rudy Y 1991 Circ. Res. 68 1501Google Scholar

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Publishing process
  • Received Date:  09 August 2020
  • Accepted Date:  26 November 2020
  • Available Online:  16 April 2021
  • Published Online:  05 May 2021

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