The Quantitative Relationship among the Number of the Pacing Cells Required, the Dimension, and the Diffusion Coefficient.

The purpose of the paper is to derive a formula to describe the quantitative relationship among the number of the pacing cells required (NPR), the dimension , and the diffusion coefficient (electrical coupling or gap junction ). The relationship between NPR and has been investigated in different dimensions, respectively. That is, for each fixed , there is a formula to describe the relationship between NPR and ; and three formulas are required for the three dimensions. However, there is not a universal expression to describe the relationship among NPR, , and together. In the manuscript, surveying and investigating the basic law among the existed data, we speculate the preliminary formula of the relationship among the NPR, , and ; and then, employing the cftool in MATLAB, the explicit formulas are derived for different cases. In addition, the goodness of fit ( ) is computed to evaluate the fitting of the formulas. Moreover, the 1D and 2D ventricular tissue models containing biological pacemakers are developed to derive more data to validate the formula. The results suggest that the relationship among the NPR, , and the () could be described by a universal formula, where the NPR scales with the (the dimension) power of the product of the square root of () and a constant which is dependent on the strength of the pacing cells and so on.