Fig 1.
Components of the multi-scale computational framework.
A–Schematic of single cell Ca2+ handling and ion current models. (i)– 3D, microscopic Ca2+ handling model, illustrating the 3D grid of calcium release units (CRUs; upper panel) and the compartments and Ca2+ fluxes within a single CRU (lower panel). Labelled are the dyadic cleft space (DS), sub-space (SS), bulk cytosolic space (CYTO), network and junctional SR spaces (NSR, JSR), and a T-tubule (TT); fluxes through the LTCCs (JCaL), RyRs (Jrel), NCX (JNaCa) and SERCA (Jup) are illustrated according to the key; double-headed black arrows indicate transfer between compartments; double-headed red arrows indicate diffusion between neighbouring CRUs. (ii)– 0D, non-spatial cell model, illustrating the same fluxes as in (i) but without inter-CRU diffusion. Global ion currents are illustrated along the membrane (which apply to both models). B–Whole-cell voltage (i) and calcium transient (ii) of the different ion-current models used in the present study, showing the hybrid-minimal model (left; minimal), O’Hara et al., human ventricular model [26] (middle; ORd) and Colman et al., human atrial model [27] (right; Col 2013). C–Tissue models, showing schematic of a 2D sheet model (i), and the 3D anatomical reconstructions: (ii)—human ventricular wedge [28]; (iii) whole canine ventricle [29]; and (iv) whole human atria [27,30–32].
Fig 2.
Illustration of Ca2+ clamp protocol.
A–Ca2+ clamp protocol illustrated for 9 steps of SR-Ca2+, showing traces for: (i) proportion open RyR; (ii) intracellular- (purple) and SR- (blue) Ca2+ concentration. B–Snapshots of the spatio-temporal Ca2+ dynamics at different SR-Ca2+ concentrations, showing: (i) non-propagating sparks; (ii) slow Ca2+ wave; (iii) multiple and rapid Ca2+ waves; the time range for the snapshots is shown in the square brackets. The data shown are clipped to the first of the two seconds associated with each clamp step in order to clearly visualise the waveforms.
Fig 3.
Derivation of the Spontaneous Release Functions.
A–Traces of open RyR (NRyR_O/NRyR) associated with 250 simulations of SCRE at different SR-Ca2+ concentrations (low = 1125 mM, purple; and high = 1200 mM, blue; 100 traces for each condition shown) in the 3D cell model. Each trace represents an individual simulation, and two simulations are highlighted for each of the two SR-Ca2+ values (note the two blue traces overlap, but are separate simulations); A(inset)–spontaneous Ca2+ transients which correspond to the four highlighted RyR traces. B–Histogram describing the initiation time for SCRE and C–the corresponding cumulative frequency plots, associated with 250 simulations of each condition. Inset–example of fitting the cumulative frequency with two sigmoidal functions (F1(ti) and F2(ti), orange and green), separated at a specific point (ti_Sep, F(ti)|ti = ti_Sep, red triangular marker). D–Examples of two types of waveform, corresponding to those highlighted in A (upper panel) and the SRF which approximate them (lower panel). Labelled are the parameters which fully describe the waveforms: the initiation time (ti), peak time (tp), final time (tf), initiation time of spike during plateau (tispike), peak open RyR (NRyR_Opeak), plateau open RyR (NRyR_Oplateau). E–Histogram illustrating the distribution of RyR waveform duration for the two conditions, with the median duration (MD) and width of distribution either side of the median (DW1, DW2) labelled for Distribution 1 (upper panel); relationship between the distribution widths and median (lower panel, points–data; lines–fit by Eqs (18 and 19)). F–Correlation between peak of open RyR and the duration, shown for the two conditions featured in the Figure (purple, blue) and all other simulations (yellow). Low amplitude SCRE occurring near the threshold SR-Ca2+ are shown in orange. Fit by Eq (12) is shown by the red line.
Fig 4.
SR-Ca2+ dependence of SRF distribution parameters.
A–The Dynamic Fit SRF parameters: summary data (points) and the fit from the relevant functions (lines) for the three Ca2+ handling conditions (purple–control; blue–CRU-CRU coupling enhancement; orange–SERCA upregulation and NCX downregulation model) against SR-Ca2+ for: i) probability of whole-cell SCRE; ii) initiation time corresponding to the separation point, ti; iii) the normalised cumulative frequency at this point, CFti,Sep = F(ti)|ti = ti,Sep; iv) the k parameter for F1(ti) and (v) for F2(ti); (vi) the median duration, MD. B–The General Dynamic SRF parameters: i) Illustration of the curve for probability of SCRE and its relation to the user defined parameters (CaSRthresdhold, CaSRP_range) and derived parameters (CaSRmin). ii) Illustration of the function form describing ti,Sep and median duration (MD) (purple line) and its relation to CaSRmin, CaSRmax, ti,Sepmin, ti,Sepmax, MDmin, MDmax, and how the range of the distributions varies with SR-Ca2+ (shaded regions) at two different non-linearity factors (Hwidth = 0.75, blue; = 2.5, orange).
Fig 5.
Schematic of the algorithms used to integrate the SRF with the non-spatial cell models for the three implementations. Note that the Δ[Ca2+]SR clause is only calculated if the SRF parameters have been set but SCRE has not yet been initiated, and RyR recovery is only calculated if the model has undergone an AP.
Fig 6.
Validation of the SRF under dynamic pacing conditions.
Results of 250 simulations for 20 different pacing conditions in which notable SCRE occurred. A– 100 examples (with one highlighted) of SCRE occurring in the 3D model (purple, i-iii) and 0D model (blue, iv-v) for two different conditions which resulted in SR-Ca2+ close to threshold (a, corresponding to Vent EPI, BCL = 400 ms, RCRU-CRU) and above it (b, corresponding to Vent ENDO, BCL = 400 ms, ISO + RSERCA/NCX). The linescans in (iii) correspond to the highlighted trace in (i-ii). Final paced beat and subsequent quiescent period is shown. B–Histograms of SCRE initiation time (left of each panel) and incidence of DADs and TA (bars, right of each panel) for the 20 different conditions (panel titles correspond to cell model, pre-pacing BCL and pro-SCRE conditions); the x-axis label for the histogram plots refers to the total range over which the plot is shown, rather than absolute values. Col. RA refers to the simplified Colman et al. 2013 [27] human atrial model; ORd refers to the simplified O’Hara et al. [26] human ventricular cell model; all other labels refer to the cell-type used in the hybrid minimal model presented in this study.
Fig 7.
The emergence of SCRE at the tissue scale in different models.
The BCL range over which activity corresponding to single cell DADs (purple) and TA (blue) and tissue focal excitation (orange) is shown for each of the different cell models/regions (A-F) under different conditions (x-axis labels–remodelling, and control and remodelling + ISO). The BCL range for which at least one SCRE/TA/focal-excitation occurs is indicated by the extent of the lines. Note that no significant SCRE was observed for control conditions for any model, and so this condition has not been included in the figure.
Fig 8.
The role of electrotonic coupling in overcoming source-sink mismatch.
A–Activation maps for intracellular Ca2+ (i) and membrane potential (ii) associated with a spontaneous focal excitation. Note that the time of the initiation of focal excitation (t = 0 ms) corresponds to the halfway point of the colour map. B–Temporal snapshots of Ca2+ (i) and Vm (ii) associated with the onset of focal excitation. C–Vm (i) and Ca2+ (ii) traces from individual cells from two regions with the tissue (labelled in Aii) illustrating the independent (Ca2+) and coupled (Vm) cellular behaviour.
Fig 9.
The effect of SCRE heterogeneity on the SR-Ca2+-TA relationship.
A–Dependence of single cell DADs (purple), single cell TA (blue), and ectopic focal activity in 2D tissue (green; square markers) on the SR-Ca2+ concentration in control (i) and reduced IK1 conditions (ii); homogeneous SCRE dynamics. B–Dependence of ectopic focal activity in tissue on SR-Ca2+ in the heterogeneous SRF conditions with small and large variability (blue and orange) and with small and large variability with higher SR-Ca2+ threshold cells reassigned to baseline (blue and red; triangular markers) compared to the homogeneous condition (green); single cell DADs in the baseline condition are shown for reference (purple).
Fig 10.
Mechanisms of SCRE mediated conduction block.
A–Demonstration of DAD-mediated conduction block in 2D (upper panels) and 3D (lower panels). In both cases, two stimuli were applied to one side of the tissue (left edge of the 2D sheet; ENDO surface in 3D) at a coupling interval of 500 ms, with SCRE induced DADs interrupting the second applied stimulus. Spatial snapshots cover the time just before and during this second stimulus. The locations of the cells from which the AP traces are taken are indicated by the triangular markers in the 2D sheets; for the 3D case, the traces correspond to a region which did (blue) and did not (purple) exhibit conduction block. Solid white lines represent sites of conduction block. B–Demonstration of spontaneous focal excitation leading to different behaviour in electrically homogeneous or heterogeneous tissue. The purple trace corresponds to the homogeneous condition, in which the focal excitation propagates uniformly; the blue trace corresponds to the heterogeneous condition in which focal excitation propagates non-uniformly following conduction block. The triangular marker indicates the site from which the AP traces were extracted, and the region of reduced IK1 is highlighted by the dashed-white rectangle. The stimulus is applied once to the left edge (ENDO region) of the tissue at t = 0 ms; the second excitation is spontaneously induced.
Fig 11.
Coupling between re-entry and SCRE.
A–SR-Ca2+ concentration (i) and Vm (ii) associated with sustained re-entry followed by self-termination (at around 13 s), simulations without SCRE (purple) and with the General Dynamic SRF model with two different thresholds (orange, 1.125 mM; blue, 1.0 mM). B–Temporal snapshots of voltage in the 2D sheet associated with the traces shown in A, showing self-termination (i) and the emergence of delayed (ii, corresponding to the orange traces in A) and rapid (iii, corresponding to the blue trace in A) focal excitations. C–Examples of non-localised focal excitations emerging in the 2D sheet (i) and 3D whole atria models (ii). Baseline General Dynamic SRF parameters, corresponding to panel A(orange)/Bii: CaSRthreshold = 1.25 mM; CaSRmax = 1.525 mM; CaSRP_range = 0.05 mM; ti,Sepmin = 300 ms; ti,Sepmax = 870 ms; ti,widthmin = 200 ms; ti,widthmax = 1000 ms; MDmin = 150ms; MDmax = 600 ms; λwidthmin = 70 ms; λwidthmax = 300 ms; Hwidth = 2.5. Parameter differences for panel A(blue)/Biii: CaSRthreshold = 1.00 mM; CaSRmax = 1.2 mM; ti,Sepmin = 30 ms; MDmin = 50ms; λwidthmin = 20 ms. For panel Ci: MDmin = 160ms; λwidthmin = 75 ms; λwidthmax = 300 ms. For panel Cii: CaSRthreshold = 0.9 mM; CaSRmax = 1.3 mM; ti,Sepmin = 30 ms; ti,widthmin = 20 ms; MDmin = 80ms; MDmax = 800 ms; λwidthmin = 20 ms.
Fig 12.
Functional localisation of re-entry and focal activity.
A–Temporal snapshots (i-viii) of voltage (a), intracellular Ca2+ (b), and SR-Ca2+ (c) associated with the final, self-terminating re-entrant cycle and first focal excitation. Arrows in (a) indicate the conduction of the re-entrant and then focal excitations. Highlighted region (circle) illustrates the island of large SR-Ca2+ associated with the unexcited scroll wave core (i-iv) and its correlation with the focus of ectopic activation (vi-vii). B–Examples of recovery time maps (a) and focal activation maps (b) for 5 independent simulations (i-v; selected from simulations covering the full range of coupled AP models and tissue parameters which led to transient re-entry which sustained sufficiently to load the SR-Ca2+) associated with the self-termination of re-entry followed by ectopic excitation. The contour surrounding the region of longest recovery time (corresponding to the unexcited core illustrated in A) is highlighted in red in the recovery time map and green in the activation maps. (vi)—summary of the correlation between distance between from centre of the focal source to the closest edge of the region of longest recovery and the time of the focal excitation, t(focal), relative to the latest activation of the non-focal excitation. C–Mechanism switching between re-entrant and focal excitation, showing the AP from a randomly selected cell (a) and temporal snapshots associated with the transition from re-entry to focal activity (b) and focal activity to re-entry (c). Snapshots corresponds to the temporal range illustrated by the grey and white bars with solid (re-entry to focal) and dashed (focal to re-entry) borders. Parameters which led to the mechanism switching simulation: CaSRthreshold = 1.0 mM; CaSRmax = 1.2 mM; CaSRP_range = 0.05 mM; ti,Sepmin = 30 ms; ti,Sepmax = 870 ms; ti,widthmin = 20 ms; ti,widthmax = 200 ms; MDmin = 50ms; MDmax = 800 ms; λwidthmin = 20 ms; λwidthmax = 300 ms; Hwidth = 2.5.
Fig 13.
Comparison of focal excitations following re-entry vs regular pacing.
Ai–Focal excitation time, t(focal), at different SR-Ca2+ thresholds for release under re-entry (purple, circle markers) and matched regular pacing (blue, triangular markers) conditions. t(focal) is calculated relative to the latest activation time of the final paced or re-entrant excitation. (ii)–Temporal snapshots of spatial SR-Ca2+ concentration in the 2D sheets, comparing the heterogeneity at equivalent time points. Bi–Categorisation of the outcomes of focal excitation under the two conditions into either non-focal (blue), symmetric focal (orange) or asymmetric focal (red). (ii)–Examples of symmetrical vs asymmetrical conduction patterns emerging following each type of excitation. The times labelled in Bii are relative to the onset of focal excitation.