Theoretical generalization of normal and sick coronary arteries with fractal dimensions and the arterial intrinsic mathematical harmony

Abstract

Background: Fractal geometry is employ to characterize the irregular objects and had been used in experimental and clinic applications. Starting from a previous work, here we made a theoretical research based on a geometric generalization of the experimental results, to develop a theoretical generalization of the stenotic and restenotic process, based on fractal geometry and Intrinsic Mathematical Harmony.

Methods: Starting from all the possibilities of space occupation in box-counting space, all arterial prototypes differentiating normality and disease were obtained with a computational simulation. Measures from 2 normal and 3 re-stenosed arteries were used as spatial limits of the generalization.

Results: A new methodology in animal experimentation was developed, based on fractal geometric generalization. With this methodology, it was founded that the occupation space possibilities in the stenotic process are finite and that 69,249 arterial prototypes are obtained as a total.

Conclusions: The Intrinsic Mathematical Harmony reveals a supra-molecular geometric self-organization, where the finite and discrete fractal dimensions of arterial layers evaluate objectively the arterial stenosis and restenosis process.

Figure 1

Flow chart of the functions performed by the developed software.

Figure 2

Island 1 with the two superposed Box-Counting grids. The green area on the right image is an example of the theoretical remodeling of this island, obtained with the developed software.

Figure 3

Example of two arterial prototypes theoretically obtained using the developed software. The green area corresponds to the remodeling simulation of the island 1, the blue zone corresponds to the remodeling simulation of the island 2. The left image corresponds to a lower level of occlusion, with respect to the right one.