A review of advances in multiscale modelings, computations, and dynamical theories of arrhythmias

Huang Xiao-Dong; He Bin-Xuan; Song Zhen; Mi Yuan-Yuan; Qu Zhi-Lin; Hu Gang

doi:10.7498/aps.73.20240977

Abstract

Biological systems are complex systems that are regulated on multiple scales, with dynamics ranging from random molecular fluctuations to spatiotemporal wave dynamics and periodic oscillations. To understand the underlying mechanisms and link the dynamics on a molecular scale to those on a tissue scale and an organ scale, the research approaches to integrating computer modeling and simulation, nonlinear dynamics, and experimental and clinical data have been widely used. In this article, we review how these approaches have been used to investigate the multiscale cardiac excitation dynamics, particularly the genesis of cardiac arrhythmias that can lead to sudden death. The specific topics covered in this review are as follows: i) mechanisms of formation of intracellular calcium sparks and waves on a subcellular scale, which can be described by the stochastic transitions between the two stable states of a bistable system and the second order phase transition, respectively; ii) mechanisms of triggered activities on a cellular scale resulting from transmembrane voltage and intracellular calcium cycling and their coupling, some of which can be well described by the bifurcation theories of the nonlinear dynamical system; iii) mechanisms for the genesis of arrhythmias on a tissue scale induced by the triggered activities, which can be regarded as dynamical instability-induced pattern formation in heterogeneous excitable media; and iv) manifestations of the excitation dynamics and transitions in the whole heart (on an organ scale) in electrocardiogram to bridge the spatiotemporal wave dynamics to clinical observations. These results indicate that nonlinear dynamics, pattern formation, and statistical physics are the fundamental components in establishing a theoretical framework for understanding cardiac arrhythmias.

Keywords:

Authors and contacts

1.
School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510641, China
2.
Peng Cheng Laboratory, Shenzhen 518000, China
3.
Deparment of Phychological and Cognitive Science, Tsinghua University, Beijing 100084, China
4.
David Geffen School of Medicine, University of California (Los Angeles), Los Angeles 90095, USA
5.
Department of Physics, Beijing Normal University, Beijing 100875, China

Corresponding author: Huang Xiao-Dong, schuangxd@scut.edu.cn

Funds: Project supported by the Basic and Applied Basic Research Foundation of Guangdong Province, China (Grant No. 2021A1515010500) and the National Natural Science Foundation of China (Grant Nos. 82172067, T2122016).

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Figure 1. Interdisciplinary subjects of experiment, modeling and simulation, and dynamical theory for life (biological) science.

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Figure 2. Multiscale behaviors of a cardiac system: (a) A record of the stochastic opening and closing of a singe RyR channel^[53]; (b) a spatiotemporal pattern of calcium release of the CRU network (horizontal direction stands for time and vertical for space)^[53], the curve in the lower column represents the total intracellular calcium release; (c) an action potential of a single cardiomyocyte, the black line represents the cell membrane potential, and the red line represents the intracellular calcium concentration; (d) spiral electrical waves in heart^[55]; (e) electrocardiogram of human being.

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图 3 心肌系统的多尺度结构(蓝色箭头指示尺度增大方向)　(a) 单个CRU的结构; (b) 几万个CRU构成的钙释放网络, 红线框中是单个CRU; (c) 单细胞的结构, 蓝灰色SR是CRU网络的整体示意, 绿色箭头指示内钙离子循环路线; (d) 多个细胞连接成的一片心肌组织; (e) 心肌细胞在三维空间排布成全心室组织

Figure 3. Hierarchical structure of a heart (the blue arrow indicates the direction of scale increase): (a) A single CRU; (b) the calcium release network composed by tens of thousands of CRUs, the red box therein shows a single CRU; (c) the scenario of a single cardiomyocyte, the grey-blue SR is the overall schematic of the CRU network, and the green arrow indicates the process of intracellular calcium cycling; (d) a piece of myocardial tissue formed by the connection of multiple cells; (e) myocardial cells are arranged in three-dimensional space into the entire ventricular tissue.

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图 4 单个CRU的钙火花动力学　(a) 钙火花的实验结果, 左图是随机钙火花, 右图是长钙火花, 一个色点(或线)对应一个释放钙的CRU ^[53]; (b) CRU钙循环的建模, $ c $表示各区域钙浓度, $ j $表示相应的流, $ {j}_{{\mathrm{D}}} $表示区域间浓度扩散流, $ {j}_{{\mathrm{R}}{\mathrm{y}}{\mathrm{R}}} $表示由RyR通道流出的钙离子流, RyR状态跃迁的随机模型示于洋红色虚线框中, $ {k}_{{\mathrm{o}}{\mathrm{c}}} $和$ {k}_{{\mathrm{c}}{\mathrm{o}}} $分别为从打开到闭合和从闭合到打开的跃迁率; (c) 钙火花的数值模拟结果, 左图为随机钙火花, 右图为长钙火花, 下方为钙火花的持续时间统计分布^[62]; (d) 细胞质内钙浓度$ {c}_{{\mathrm{i}}} $关于肌浆网钙浓度$ {c}_{{\mathrm{J}}{\mathrm{S}}{\mathrm{R}}} $的函数; (e) CRU钙动力学系统的势阱结构示意

Figure 4. Dynamics of calcium release of a single CRU: (a) The experimental results of calcium spark, the left image shows random calcium sparks, and the right image shows long lasting calcium sparks, each colored dot (or line) corresponds to a CRU that releases calcium^[53]; (b) the modeling of a CRU, $ c $ represents the calcium concentration in each compartment, j represents the associated flow, $ {j}_{{\mathrm{D}}} $ represents the concentration diffusion flow between regions, $ {j}_{{\mathrm{R}}{\mathrm{y}}{\mathrm{R}}} $ represents the calcium ion flux flowing out from the RyR channel, the random model of RyR state transition is shown in the magenta dashed box, $ {k}_{{\mathrm{o}}{\mathrm{c}}} $ and $ {k}_{{\mathrm{c}}{\mathrm{o}}} $ are the transition rates from open to closed and from closed to open, respectively; (c) the simulation results of calcium spark, the left image shows random calcium sparks, the right image shows long lasting calcium sparks, and the statistical distribution of the duration of calcium sparks is shown below^[62]; (d) the cytosolic calcium concentration $ {c}_{{\mathrm{i}}} $ as a function of junctional SR calcium concentration $ {c}_{{\mathrm{J}}{\mathrm{S}}{\mathrm{R}}}; $ (e) a schematic illustration of bistability of a CRU system.

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图 5 钙波的形成　(a) 小鼠心室肌细胞的钙波实验^[76], 自左向右4列图对应逐渐增加的外部钙浓度值, 上栏为钙波时空斑图, 中栏为内钙总量随时间变化, 下栏为CRU集团尺寸的统计; (b) CRU网络的空间结构, 每一个方盒代表一个CRU, 其结构再次示于下侧, $ {j}_{{\mathrm{D}}{\mathrm{i}}{\mathrm{f}}} $是CRU之间的钙扩散流; (c) 钙波的数值模拟结果^[76]; (d) 钙波形成的条件及相变机制示意

Figure 5. Formation of intracellular calcium wave: (a) The experimentally observed calcium waves^[76], the four columns from left to right correspond to gradually increasing external calcium concentration values, the top row shows the spatiotemporal pattern of calcium waves, the middle row shows the variation of total intracellular calcium with time, and the bottom row shows the statistics of CRU cluster size; (b) the model of the CRU network, each box represents a CRU, and its structure is shown below, $ {j}_{{\mathrm{D}}{\mathrm{i}}{\mathrm{f}}} $ is the calcium diffusion flow between CRUs; (c) the simulated results of calcium waves formation^[76]; (d) the illustration of the parametric condition and phase transition of calcium waves.

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图 6 EAD的发生机制　(a) 具有EAD的动作电位(来自兔心室肌细胞实验^[86]), 虚线表示正常动作电位; (b) EAD的霍普夫分岔机制示意^[97], 黑色线表示每个x值下快子系统的定态解, 分别用p, s, r表示, 实线代表稳定, 虚线代表不稳定, 蓝色曲线是一次动作电位在相空间中的轨道, HB处表示霍普夫分岔, HC处表示同宿轨道分岔; (c) 内钙振荡诱发EAD的实验结果^[108]; (d) 内钙振荡诱发EAD的模拟结果^[110], 竖直虚线标记钙和电位振荡的相同相位

Figure 6. Mechanism of EAD: (a) An experimental EAD-present action potential (from the rabbit ventricular cell ^[86]), the dashed line represents the normal action potential; (b) the Hopf bifurcation mechanism of EAD^[97], the black line represents the steady-state solution of the fast subsystem at each value of x, denoted by p, s, r, respectively, the solid line represents stable, while the dashed line represents unstable, the blue curve represents the orbit of an action potential in phase space, HB represents Hopf bifurcation, and HC represents homoclinic bifurcation; (c) calcium oscillation-induced EADs in experiment^[108]; (d) the simulated result of calcium induced-EADs^[110], the vertical dashed line marks the synchronized phase of calcium and voltage oscillation.

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图 7 细致细胞模型的EAD^[111]　(a) 带EAD的动作电位, 红色“*”号标示EAD诱发的DAD及其激发出的动作电位; (b) 细胞内钙总浓度(即$ {c}_{{\mathrm{i}}} $); (c) 钙波时空斑图; (d) 肌浆网内钙总浓度(即$ {c}_{{\mathrm{J}}{\mathrm{S}}{\mathrm{R}}} $).

Figure 7. EADs of the detailed spatial cell model^[111]: (a) Action potentials with EADs, the red “*” indicates the action potential triggered by DAD which is induced by the previous EADs ; (b) trace of intracellular calcium concentration ($ {c}_{{\mathrm{i}}} $); (c) spatiotemporal pattern of calcium waves; (d) trace of calcium concentration in junctional SR ($ {c}_{{\mathrm{J}}{\mathrm{S}}{\mathrm{R}}} $).

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图 8 DAD的随机动力学　(a) 自发TAP和DAD的实验^[112], 上下栏分别为动作电位和内钙浓度, 初始两次黑色为周期激发出的正常动作电位, 其后洋红色是自持续TAP, 最后蓝色是DAD; (b) 自发TAP和DAD的数值模拟结果^[117], 自上而下分别为钙波斑图、细胞质钙浓度、JSR钙浓度、动作电位; (c) TAP和DAD行为在[Ca]_i-V相空间的轨道^[117], 黑色闭合轨线为TAP的极限环, 红色为DAD, 左下方黑点是静息态, 内置小图是TAP (右边T势阱)和静息态(左边D势阱)的双吸引子结构示意

Figure 8. Stochastic dynamics of DAD: (a) An experiment of spontaneous TAP and DAD^[112], the upper and lower columns represent the action potential and intracellular calcium concentration, respectively, the initial two black traces represent the normal ones excited by the pacing, the following magenta ones represent the self sustained TAPs, and the final blue depolarization is a DAD; (b) the simulated result of TAP and DAD^[117], from top to bottom are: calcium wave pattern, cytoplasmic calcium concentration, JSR calcium concentration, and action potential; (c) the orbits of TAP and DAD in [Ca]_i-V phase plane^[117], the black closed trajectory signifies the limit cycle of TAP, while the red one denotes DAD, the black dot in the lower left corner represents the resting state, the inset schematically illustrates the dual attractor structure of TAP (T potential well on the right) and the resting state (D potential well on the left).

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图 9 早搏的机制　(a) LQTS兔心室肌组织实验的心律失常^[89], 上栏和中栏为心肌不同位置记录到的动作电位, 下栏为心电图; (b) 一维空间EAD传播导致早搏的数值模拟结果^[44], 自上而下每一条曲线对应一个细胞的动作电位, 上部洋红色代表长, 下部黑色代表短的动作电位; (c) 梯度激发早搏的数值模拟结果^[44]; (d) 非均匀一维链定态失稳导致早搏^[156], 当参数连续变化(致使梯度连续变化)时, 系统定态经历霍普夫分岔连续发展出早搏

Figure 9. The mechanism of premature ventricular complexes (PVCs): (a) A LQTS rabbit heart experiment of arrhythmia^[89], the top and middle columns display the action potentials recorded at different locations within the myocardium, while the bottom column exhibits the electrocardiogram; (b) a simulation of EAD-triggered PVC in a 1D cable^[44], each curve from top to bottom corresponds to the action potential of a cell, where the upper magenta ones represent long action potentials and the lower black ones represent short action potentials; (c) a simulation of voltage gradient-induced PVC^[44]; (d) PVC due to instability of heterogeneous tissue^[156], as the parameters undergo continuous variation, leading to a corresponding change in the gradient, the system’s steady state undergoes Hopf bifurcation, resulting in the continuous emergence of premature beats.

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图 10 早搏和基质导致心律失常的机制　(a) 早搏信号传导被阻断形成折返^[55], 4个小图分别对应4个时刻的二维空间中膜电位分布, 颜色代表V值, 红高蓝低. 自左而右分别为早搏(星号位置)、早搏传播遇到复极延长区被阻断、折返形成、螺旋波持续存在对应心律失常; (b) 基质同时导致早搏和折返的斑图演化(模拟自兔心室肌模型^[42])

Figure 10. Arrhythmia caused by PVC and substrate: (a) Formation of reentry caused by blockade of propagation of the PVC^[55], the four small graphs correspond to the two-dimensional spatial distribution of membrane potential at four successive time points, with colors representing V values, red for high and blue for low, from left to right are premature beats (from the star position), premature beat propagation blocked by delayed repolarization zone, reentry formation, and sustained spiral wave implying arrhythmia; (b) the formation of PVC and reentry caused by a unique substrate (simulated from rabbit ventricular myocyte model^[42]).

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图 11 早搏致心律失常的全心室模拟^[55]　(a) 早搏致心律失常的临床心电图; (b) 全心室模拟的数值结果

Figure 11. Whole heart simulation of PVC induced arrhythmias^[55]: (a) The clinical ECG of the arrhythmias; (b) the simulated results of the whole heart model.

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表 1 心律失常的多尺度建模和机制理论的主要进展

Table 1. Major progresses of multiscale modelings and mechanisitic theories of arrhythmias.

	临床与实验数据	建模与计算	动力学理论
微观	钙火花^[59] 长钙火花^[53] 钙波^[76]	RyR随机模型^[21] CRU网络模型^[54,78]	钙火花的势阱逃逸理论^[62,68,69] 钙波的相变理论^{[76,79,80,121]}
细胞	钙电耦合的EAD 振荡^[108,109] DAD的随机性^[112]	钙电耦合细致细胞模型^[23–25]	EAD的霍普夫分岔理论^[96,100,104] DAD的Kramers随机跃迁理论^[117]
组织	“冲动+基质”致心律失常^[41,42,89]	非均匀可激发介质^[154,156]	早搏的动力学稳定性理论^[156]; 螺旋波的形成、稳定性、调控^{[137,142,143,158]}
器官	R-on-T发展为 TdP^[87]	全心室多尺度模型^[51,160]	TdP的“R-on-T”机制理论^[55,158]

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