MD. Insight Group Director - Investigation Center, Clínica del Country. Professor and Director of the research area: "Mathematical and Physical Theories Applied to Medicine". Universidad Militar Nueva Granada, Cr 11 No. 101-80, Bogotá, Colombia

Insight Group researcher - Investigation Center, Clínica del Country. Bogotá, Colombia

Psychologist. Insight Group researcher- Investigation Center, Clínica del Country. Bogotá, Colombia

Systems Engineering Student. Universidad Nacional de Colombia. Insight Group researcher - Investigation Center, Clínica del Country. Bogotá, Colombia

Medicine student. Universidad Militar Nueva Granada, Bogotá, Colombia

Degree in Physics student. Universidad Pedagógica Nacional. Insight Group researcher- Investigation Center, Clínica del Country. Bogotá, Colombia

Medicine student. Universidad Militar Nueva Granada, Bogotá, Colombia. Special Internship: "Mathematical and Physical Theories Applied to Medicine

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.

Background

The fractal geometry, developed by Benoit Mandelbrot, allows irregular objects characterization, through fractal dimensions

This problem was analyzed

The Intrinsic Mathematical Harmony concept was adapted and used for the differentiation between ventricles with ejection fraction less than 40% compared to normal ones. Rodríguez et al. ^{0.9 }and 2^{10}, while those with a ventricular ejection fraction less than 40% are between 2^{10 }and 2^{500 }at least in one of the comparisons made, successfully distinguishing normal of severe cases with mathematical, objective and reproducible measures.

The purpose of this investigation is to develop a generalization of arterial fractal geometric structure evaluation, taking as fundament the Intrinsic Mathematical Harmony concept, through of a software and in this way obtain the finite set of possible normal and sick arteries, designed as prototypes.

Methods

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

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

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

**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

Island 1 with the two superposed Box-Counting grids

**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.

Example of two arterial prototypes theoretically obtained using the developed software

**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.

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.

Mathematical Analysis

The theoretically calculated fractal dimensions were compared to those experimentally obtained

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.

Definitions

**Fractal: **From the Latin fractus, it means irregularity used as substantive or irregular as an adjective.

**Fractal Dimension: **numerical measurement to characterize irregularity degree. The fractal dimension definition used in this case is Box-Counting fractal dimension

**Artery's Intrinsic Mathematical Harmony (IMH) **

**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.)

**Island: **Fractal object defined starting from limits of selected arterial layers

Results

At the execution of the software and the fractal dimensions calculus for the three defined regions from sick arteries, it was found that fractal dimension of island 1 can take values between 0.0443 and 1.5670; Island 2, between 0.7520 and 1.3168 and Total Island, between 0.0395 and 1.6147. The interval of fractal dimensions from Total Island is bigger than the other islands. The interval of values for fractal dimensions of island 1 was bigger than the island 2.

The arterial prototypes that correspond to all possible sick arteries are 69 049, no matter if those are associated to stenosis or restenosis, while 200 prototypes of healthy arteries were obtained, starting from the difference between healthy and unhealthy arteries based on IMH concept. So, considering every possible normal and sick arteries prototypes, there are 69 249 in total. This result shows how can a normal artery evolves into a sick one, without significance of the cutting place. Some data obtained in simulation is showed in tables

Fractal dimensions of three normal prototypes obtained from the normal arteries simulation based on IMH.

**Island 1**

**Island 2**

**Total island**

1

0.8715

0.8726

0.8737

2

0.8434*

0.8479*

0.8930*

3

1.0565*

1.0524*

1.0544*

The numbers with an asterisk are fractal dimensions present in empirical arteries from the previous study

Fractal dimensions of ten sick prototypes obtained from the simulation.

**Island 1**

**Island 2**

**Total island**

1

0.9368

0.9643

1.0604

2

1.0641

1.0494

1.0780

3

0.9328

1.0931

1.0199

4

0.7629

0.8883

0.7608

5

0.8826

0.9845

0.7776

6

1.0230

1.1868

1.0824

7

0.9765

0.8771

0.9349

8

1.2016

1.1015

1.2854

9

0.9510

1.0435

0.8382

10

0.9259*

1.143*

1.2094*

The numbers with an asterisk are fractal dimensions present in empirical arteries from the previous study

The experimental measures already obtained

Extreme values of the fractal dimensions of theoretical and experimental sick arteries.

**Theoretical**

**superior value**

**Experimental**

**superior value**

**Experimental**

**inferior value**

**Theoretical**

**inferior value**

Island 1

1.5670

1.0919

0.8073

0.0443

Island 2

1.3166

1.0809

0.8821

0.7520

Total Island

1.6147

1.3599

0.9068

0.0395

The experimental values are a previous result

Discussion

This is the first work in which a mathematical generalization is constructed starting from fractal dimensions calculus and IMH concept, based on a new theoretical-practical investigation methodology in animal experimentation area. With this methodology, every possible fractal prototypes of normal and sick arteries were calculated, finding a finite quantity. The numeric limits used can change without affecting the geometric characterization. This methodology does not require descriptive classifications of stenosis or restenosis grades.

The Standard methodology to evaluate arterial occlusion proposed by some investigators

In a previous work

Multiple researches in medicine based on physics and mathematics theories have been developed, based on a-causality of theoretical physics and geometry. Among these some characterizations that allow an objective mathematical differentiation for binding and not binding peptides to red blood cell receptor, based on sets theory and with probability and entropy theories too

In this methodology only with the determination of harmonic relation between parts and whole object is possible to know the occupied space by an irregular object. This shows the simplicity of complexity and that it is possible to create a methodology able to find the subjacent order into irregularity. This kind of methodology can be used in human body studies and in experimental models with animals, being able to obtain results with small samples, regardless statistical and epidemiological studies

Limitations

The developed generalization shows all the possibilities of occupation of a defined fractal space, but it doesn't establish specific artery thicknesses or longitudes.

Conclusions

A new methodology of scientific investigation in animal experimentation was developed, based on IMH in fractal space of Box-counting, that allows a geometric and numeric objective generalization, capable of simulate variability and complexity of coronary stenosis and restenosis, without take into account experimental classifications.

From this new methodology perspective, arterial stenosis and restenosis are revealed as a fractal supra-molecular geometric auto-organization phenomenon.

Based on the developed methodology, a finite quantity of 200 normal and 69049 sick arterial prototypes are obtained, calculating 69 249 in total.

This methodology allows getting objective and precise mathematical results with simple experiments, without a great sample, avoiding the unnecessary death of animals, and additionally allowing an optimum use of financial sources and time.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

JOR conceived the study, and participated in its design and coordination, SEP drafted the document, participated in data analysis, CC drafted the document, participated in the design of the study and data analysis, PAB developed software and mathematical - fractal calculations and drafted the document, GEP participated in data recollection and its systematization, SV participated in data recollection and its systematization, YS drafted the document and participated in mathematical calculations, DM drafted the document and participated in mathematical calculations.

All authors read and approved the final manuscript.

Acknowledgements

To Universidad Militar Nueva Granada, especially to Dr. José Ricardo Cure Hakim sub-dean of researches; Dr. Henry Acuña, chief of the Scientific Research Area; Dra. Sandra Moreno, director of the Medicine Research Center; Dr. Juan Miguel Estrada, dean of Medicine Faculty; Dra. Clara Benavides, director of the Medicine Program, and Dr. Germán Forero, for their support to the research developed at the university. We thank the support of the Research Fund of Universidad Militar Nueva Granada, that financed this work.

To Fundación Cardio-Infantil, especially to Dr. Darío Echeverri, for his support to our investigations.

To Stella Huerfano PhD, mathematician and teacher at Universidad Nacional de Colombia, for always support our research group.

Pre-publication history

The pre-publication history for this paper can be accessed here: