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In this paper, stochastic 1D and 2D models based on the two-pool calcium-induced calcium release (CICR) process are presented. An anisotropic diffusion model is presented that approximates the calcium concentration in the cytosol as well as the sarcoplasmic reticulum (SR). The relative source strengths in 1D and 2D are adjusted to simulate the behaviour of a point source array in the real 3D problem. The numerical conversion presented involves the convolution between the Green's and Heaviside functions. The models are solved using the Crank Nicolson (CN) and alternating direction implicit (ADI) for the 1D and 2D cases, respectively. These methods are known to be stable. Results are presented for the comparison between the two models.

BackgroundSince the discovery of Ca2+ sparks and their stochastic behaviour in cardiac myocytes, models have focused on the inclusion of stochasticity in their studies. While most models pay much attention to the stochastic modelling of cytosolic Ca2+ concentration the coupling of Ca2+ sparks and blinks in a stochastic model has not been explored fully. The cell morphology in in silico studies in the past is assumed to be Cartesian, spherical or cylindrical. The application on curvilinear grids can easily address certain restrictions posed by such grid set up and provide more realistic cell morphology. In this paper, we present a stochastic reaction–diffusion model that couples Ca2+ sparks and blinks in realistic shapes of cells in curvilinear domains.Methodology: Transformation of the model was performed to the curvilinear coordinate system. The set of equations is used to produce Ca2+ waves initiated from sparks and blinks. A non-buffered and non-dyed version as well as a buffered and dyed version of these equations were studied in light of observing the dynamics on the two different systems. For comparison, results for both the Cartesian and curvilinear grids are provided.Results and conclusions: A successful demonstration of the application of curvilinear grids serving as basis for future developments.

Several computational studies have been undertaken to explore the Ca2+-induced Ca2+ release (CICR) events in cardiac myocytes and along with experimental studies it has given us invaluable insight into the mechanism of CICR from spark/blink initiation to termination and regulation, and their interplay under normal and pathological conditions. The computational modelling of this mechanism has mainly been investigated using coupled differential equations (DEs). However, there is a lack of computational investigation into (1) how the different formulation of coupled DEs capture the Ca2+ movement in the cytosol and sarcoplasmic reticulum (SR), (2) the buffer and dye inclusion in both compartments, and (3) the effect of buffer and dye properties on the calcium behaviour. This work is set out to explore (1) the effect of different coupled formulation of DEs on spark/blink occurrence, (2) the inclusion of improved sarcoplasmic buffering properties, and (3) the effects of cytosolic and sarcoplasmic dye and buffer properties on Ca2+ movement. The simulation results show large discrepancies between different formulations of the governing equations. Additionally, extension of the model to include sarcoplasmic buffering properties show normalised fluorescent dye profiles to be in good agreement with experimental and amongst its one- and two-dimensional representations.