Arrhythmia Mechanisms and Spontaneous Calcium Release: I - Multi-scale Modelling Approaches

Motivation Spontaneous sub-cellular calcium release events (SCRE), controlled by microscopic stochastic fluctuations of the proteins responsible for intracellular calcium release, are conjectured to promote the initiation and perpetuation of rapid arrhythmia associated with conditions such as heart failure and atrial fibrillation: SCRE may underlie the emergence of spontaneous excitation in single cells, resulting in arrhythmic triggers in tissue. However, translation of single-cell data to the tissue scale is non-trivial due to complex substrate considerations. Computational modelling provides a viable approach to dissect these multi-scale mechanisms, yet there remains a significant challenge in accurately and efficiently modelling this probabilistic behaviour in large-scale tissue models. The aim of this study was to develop an approach to overcome this challenge. Methods The dynamics of SCRE under multiple conditions (pacing rate, beta-stimulation, disease remodelling) in a computational model of stochastic, spatio-temporal calcium handling were analysed in order to develop Spontaneous Release Functions, which capture the variability and properties of SCRE matched to the full cell model. These functions were then integrated with tissue models, comprising idealised 2D sheets as well as full reconstructions of ventricular and atrial anatomy. Results The Spontaneous Release Functions accurately reproduced the dynamics of SCRE and its dependence on environment variables under multiple different conditions observed in the full single-cell model. Differences between cellular models and conditions where enhanced at the tissue scale, where the emergence of a focal excitation is largely an all-or-nothing response. Generalisation of the approaches was demonstrated through integration with an independent cell model, and parameterisation to an experimental dataset. Conclusions A novel approach has been developed to dynamically model SCRE at the tissue scale, in-line with behaviour observed in detailed single-cell models. Such an approach allows evaluation of the potential importance of SCRE in arrhythmia in both general mechanistic and disease-specific investigation.


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Methods: The dynamics of SCRE under multiple conditions (pacing rate, beta-stimulation, disease 18 remodelling) in a computational model of stochastic, spatio-temporal calcium handling were 19 analysed in order to develop Spontaneous Release Functions, which capture the variability and 20 properties of SCRE matched to the full cell model. These functions were then integrated with 21 tissue models, comprising idealised 2D sheets as well as full reconstructions of ventricular and 22 atrial anatomy.

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Results: The Spontaneous Release Functions accurately reproduced the dynamics of SCRE and its 24 dependence on environment variables under multiple different conditions observed in the full 25 single-cell model. Differences between cellular models and conditions where enhanced at the 26 tissue scale, where the emergence of a focal excitation is largely an all-or-nothing response. 27 Generalisation of the approaches was demonstrated through integration with an independent cell 28 model, and parameterisation to an experimental dataset.
1 Introduction 2 Cardiovascular disease is one of the major healthcare problems faced by the developed world, 3 with increasing prevalence associated with aging populations [1][2][3]. Improved understanding of 4 the mechanisms underlying cardiac arrhythmias, a major component of cardiovascular diseases' 5 impact on morbidity and mortality, is therefore vital to the effort to improve both lifespan and 6 quality of life therein. Rapid arrhythmias, such as tachycardia and fibrillation, are associated with 7 highly non-linear electrical excitation patterns and are challenging to manage [4,5], in part 8 because the complex multi-scale mechanisms have yet to be full elucidated. 9 Malfunction of the intracellular calcium (Ca 2+ ) handling system has been implicated in the 10 development of rapid arrhythmias, linking sub-cellular spontaneous Ca 2+ release events (SCRE, 11 described below) to pro-arrhythmic triggers in single cell [6][7][8][9][10]. However, translation of these 12 cellular data to assess the mechanisms and importance of SCRE in tissue-scale arrhythmia 13 remains a significant challenge [11]. Computational modelling provides a viable approach for 14 detailed multi-scale evaluation of cardiac arrhythmia mechanisms. Never-the-less, simulating 15 SCRE in tissue-scale models is non-trivial due to the complex cellular structures and non-linear, 16 spatio-temporal dynamics underlying the associated phenomena. These are described in detail 17 below to facilitate understanding of the challenges involved.

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The intracellular Ca 2+  cell membrane potential as a delayed-after-depolarisation (DAD) or full triggered AP (TA; Figure   10 1Aii), which in tissue may present as a focal excitation. Simulating this behaviour therefore 11 requires computational models which explicitly account for the underlying stochastic spatio-12 temporal dynamics (Figure 1Bi-  illustrating the same fluxes as in (i) but without inter-CRU diffusion. Global ion currents are 31 illustrated along the membrane (which apply to both models Where Φ x is a general reaction term comprising the relevant channel fluxes, β x is the 21 instantaneous buffering term [48], v x is the volume of the compartment in the subscript, τ x is the 22 time constant of inter-compartmental diffusion, ∇ 2 is the spatial Laplacian operator in 3D and D 23 is the diffusion coefficient. Full reaction terms are given in the Supplementary Material S1 Text Where g RyR is the maximal flux factor for Ca 2+ release, v ds is the volume of the dyadic cleft space, 19 The N RyR_O waveforms can be grouped into two primary types: spike-like associated with short, 20 large-amplitude release, and plateau-like associated with long, small-amplitude release ( Figure   21 4D). For the spike-like morphology, the waveform can be well approximated with the simple 22 function: where t i is the initiation time of the SCRE, t f is the end time (duration, λ, thus = t f -t i ), t p is the time 29 of the peak of the waveform and N RyR_O peak is the peak of open proportion RyR ( Figure 4D). The 30 function for the plateau-like waveform (corresponding to durations longer than 300 ms) is 31 derived from the same parameters: Where N RyR_O plateau is the amplitude of the plateau ( Figure 4D). This equation assumes the same 3 form for the spike occurring within the plateau, with its upstroke time being 50 ms and its decay 4 time 35ms; t i spike ( Figure 4D) therefore corresponds to t p -50 (and its half maximal activation time The waveform is therefore completely described by four-five parameters: (1) initiation time, t i ; 7 (2) duration (λ = t f -t i ); (3) peak time, t p ; and (4-5) amplitude (N RyR_O peak ; N RyR_O plateau ). In order to 8 maintain physiological waveforms and randomly sample the parameter values from appropriate 9 distributions, the nature of stochastic variation of these four parameters is discussed below.
The distribution for t i is therefore determined by four parameters: the initiation time Where the widths (DW 1 , DW 2 , in ms) are a function of the MD (Figure 4E), given by:   28 concentration ( Figure 5). Relation to this single variable was chosen for practicality and simplicity 1 of the resulting equations, which is in particularly valuable for reproducing variable Ca 2+ handling 2 system states.
Duration, λ; inverse function of equation (17) functions: first, a random number between 0 and 1 is generated and compared to the probability 5 of release, P(SCR); if rand < P(SCR), then the input parameters are used to determine the inverse 6 functions (equations 28-30) and 4 more random numbers determine the actual SRF waveform 7 parameters; if rand > P(SCR), then no SRF parameters are set. Note that the t i_sep must be set 8 relative to excitation time. The model will set SRF parameters based on these single distributions 9 with every cellular excitation (note that the parameters may give an SCRE timing later than the 10 next stimulated excitation, in which case it will be reset on the next excitation).

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For the Dynamic Fit model, the calculation is performed multiple times, dynamically determined 12 during the simulation: After an initial stimulated AP, when the RyR availability has recovered 13 above a set threshold, the SR-Ca 2+ is input to first define the probability of release, P(SCR), from 14 equation (22)     titles correspond to cell model, pre-pacing BCL and pro-SCRE conditions); the x-axis label for the 37 histogram plots refers to the total range over which the plot is shown, rather than absolute values.

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Generalisation of the approaches 39 To demonstrate generalisation of the developed approaches, the SRF were first integrated with 40 an independent cell model and its native Ca 2+ handling system ( Figure 12A). The human atrial AP 41 model of Grandi et al., 2011 [51] was selected due to its detailed Ca 2+ handling system containing 1 a physiologically representative model of CICR and RyR model suitable for integration with SCRE 2 waveforms. A General Dynamic implementation was parameterized to the SR-Ca 2+ observed in 3 that model in order to reproduce, for example, rate dependent susceptibility to SCRE ( Figure   4 12A). This demonstrates the potential suitability for direct integration with available 5 contemporary, non-spatial AP models, without the requirement to replace the native intracellular 6 Ca 2+ handling system. and thus the precise mechanisms and potential importance of these events manifesting as tissue-25 scale arrhythmia have yet to be fully described.

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In this study, a multi-scale computational approach was developed to simulate the dynamics of Functions (SRF; Figure 4) were used to reproduce the morphology of SCRE in the 0D cell and 31 tissue models (Figures 9-11, 13-14). Briefly, the waveform is fully described by its initiation time 32 and duration, and randomly sampling these parameters from physiological distributions ( Figure   33 7) ensured accurate and congruent stochastic variation in SCRE dynamics.

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It is intended that these approaches will be further developed and incorporated with