Propagation characteristics of a fractal network: applications to the His-Purkinje conduction system

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Abstract
This paper considers the propagation of a pulse through a theoretical fractal network. The temporal and spectral features resulting from the decorrelation of a pulse as it transits the network including the effeci of perturbations are discussed. Application to naturally occurring networks such as the His-Purkinje system of the myocardium are considered. Both the fractal dimension (DF) and the power spectrum [P(w)are descriptive of the organizational features of the network. DF describes the geometric structure based on segment length generation and scaling factors. P(w) measures the heterogeneity of the pathways present in the network which in turn reflects the distribution of the segment length variations throughout the network. We show a direct relation between structural heterogeneity and the spectral features. Variation of perturbation size and location shows that changes in geometry occur across many size scales consistent with a fractal structure. Furthermore both location and size of the perturbation are important determinants of the extent of spectral changes produced. Simulation of the His-Purkinje conduction system demonstrates QRS complexes in good agreement with clinical observations. Additionally simulation of conduction defects in the network shows that network perturbations are reflected in the QRS and more importantly in P(w). Our results suggest that characterizing perturbations in terms of their functional wavelength (frequency) rather than their geometric dimension may be a useful method for assessing the extent of performance degradation in the system under study.

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