In order to do this investigation, fundamentally mathematical, coronary arteries images and their fractal dimensions, previously obtained from the arterial structural characterization study developed by Rodríguez et al. (2002) at Fundación Cardio Infantil were used. Two normal and three sick arteries evaluated with Intrinsic Mathematical Harmony concept were chosen for generalization development.
The fractal dimension was calculated using only two grids, in order to make a calculus simplification like in previous research used as reference [9
]. Two squaregrids were built, in which squares side of one of them, is double of squares side of the other one. Then each mentioned square-grids were superposed over the images, in order to do square account required for box-counting method application [2
The number of normal prototypes was calculated following the IMH established definition, where at least first decimal cipher of fractal dimension for component parts and totality must be equal, for normal arteries. The 17 arteries from the previous study [9
] which fractal dimensions are equal until the second cipher were taken as initial prototype. The fact that Box-counting fractal dimension is defined in interval(0-2) was also used. In this way, calculation of normal prototypes was done starting from the case in which three measured regions have a fractal dimension where two first significant ciphers and unit are zero, adding 0.01 consecutively to obtain successively all possible values for fractal dimension, for parts and totality simultaneously.
Based on IMH, a software in C++ language was designed, capable to simulate arterial deformation. This software allows to get all the possible arterial prototypes in occlusion process that correspond to stenosed or restenosed arteries, where each possible combination constitutes an arterial prototype, obtaining all geometric possibilities of box-counting space occupation by arterial layers and for each specific artery, including all the possibilities of experimental vascular remodeling (see figure ).
Flow chart of the functions performed by the developed software.
According to the used methodology, each arterial layer is described by a set of occupied squares in each of the grids (see figure ). The maximum size of islands occupation corresponds to the maximum number of spaces occupied by the islands in the studied prototypes. Different sets of occupied squares describe different arterial prototypes (As an example see figure ). Considering the possible sets of squares marked out by all the possible arterial contours and doing all the possible combinations, all possible arterial prototypes are obtained. The fractal dimension is calculated considering the number of squares occupied by the object evaluated, it can take real values in the range between 0 and 2, and its value changes according to the variation of the number of squares occupied by the object.
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 (more ...)
Finally, with the same software fractal dimensions were calculated and all possible sick arteries prototypes were counted, including all experimental vascular remodeling possibilities. In this work, islands represent arterial layers histologically differentiated. When external or internal elastic lamina breaks up, the minimum union way between the two extremes is taken into account for calculus. In this way, the generalization includes all lesion grades, without take into account if laminas are broken or not.
The theoretically calculated fractal dimensions were compared to those experimentally obtained [9
] to confirm correspondence with the developed generalization. Due that the software simulates arterial deformation, it leads to a unique set of possible arterial structures in a general and complete way, without requiring repetitions of experiment; avoiding statistical analysis and grate samples studies to prove the correspondence with any particular artery.
Some cases on which fractal dimension values were zero were not taken into account in results, because it does not joint to any arterial prototype, showing in this way that not all the mathematical possibilities have experimental sense.
Fractal: From the Latin fractus, it means irregularity used as substantive or irregular as an adjective.
numerical measurement to characterize irregularity degree. The fractal dimension definition used in this case is Box-Counting fractal dimension [2
Artery's Intrinsic Mathematical Harmony (IMH)
]: Similarity degree or difference between units and significant ciphers of fractal dimensions of island's parts, with artery totality.
Arterial Fractal prototype: Geometric combination of simultaneous occupation of Box-counting space by different constitutive regions, islands, and totality of arterial structure, which fractal dimensions correspond to some particular artery evaluated with IMH. (Definition done by the first author.)
Fractal object defined starting from limits of selected arterial layers [9