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Corresponding author. E-mail: grupoinsight2025@yahoo.es

The ventricular shape and function can be altered both in acute and chronic heart pathologies, such as acute or chronic heart failure (

The morphological measurement of the left ventricle and its respective cavities has conventionally been carried out through linear Euclidean geometrical measurements that have allowed to classify the ventricular form as normal or with mild, moderate or severe abnormality (

The forms of nature are irregular. However, they are usually evaluated with Euclidean measures with which paradoxical or inadequate results have been reached, since Euclidean geometry is designed to measure regular objects (

In medicine, fractal geometry has served as the basis for developing methodologies with which to differentiate between normal and disease states. For example, Landini and Rippin (

In the specific case of the study of the ventricular shape measured from the ventriculogram, Rodríguez et al. (

In this research, a generalization was made to obtain a complete spectrum of all the possible ventricular prototypes that can be established between the classifications of moderate to severe abnormality based on theoretical simulations generated from the results of a previous study (

Where

Systole (S): the region corresponding to the image in systole.

Diastole (D): the region corresponding to the image in diastole.

Totality (T): corresponds to the sum of the values measured in systole and diastole.

The generalization developed was based on a previous study (

The diagnostic methodology of the left ventriculogram developed by Rodríguez et al. (

Likewise, in this study it was established that the state of moderate disease is characterized by degrees of similarity with values between 1 and 900, with the presence of at least one value between 100 and 900. Severe disease is characterized by values between 1 and 9000, where at least one of these values must be between 100 and 9000. The degrees of similarity were grouped according to four sets established as follows: A = {x | 1 ≤ x ≤ 9}, B = {x | 10 ≤ x ≤ 90}, C= {x | 100 ≤ x ≤ 900} y D = {x | 1000 ≤ x ≤ 9000}. In this way, moderate disease is characterized by presenting values within sets A, B, or C, and at least one value within set C while severe disease is characterized by values between 1 and 9000, with at least one value within sets C or D.

In order to find all the possible prototypes of ventricular structure with moderate and severe disease, the maximum and minimum extremes of the degrees of similarity were taken for the ventricles with moderate and severe disease previously found (

Next, the total number of prototypes obtained for each state was quantified. These prototypes were compared with values of degrees of similarity of prototypes measured in previous studies (

The type of methodology described here did not affect any treatment or clinical decision; therefore, it did not affect the patients. This research complied simultaneously with the ethical, scientific and technical standards, and with the health research ethical guidelines of Article 11 of Resolution 008430 of 1993, of the Ministry of Health of Colombia. This methodology is classified within the category of risk-free research, since the mathematical calculations were performed based on test results of clinical practice that had been medically prescribed, protecting the integrity and anonymity of the participants (

In
total, we found 794 possible theoretical permutations of degrees of similarity that
are associated with ventricular geometric shapes with moderate abnormality, and
820 for severe abnormality. Thus, a total of 1614 prototypes were obtained for moderate
and severe disease. Tables

Fd: fractal dimension; S: systole; D: diastole; T: totality.

Fd: fractal dimension; S: systole; D: diastole; T: totality.

The measurements made to ventricles in previous studies were found within the prototypes obtained, which evidenced that the measures made in practice are included within the generalization developed.

This is the first study that establishes the totality of the prototypes of ventricular fractal structures with moderate and severe abnormality, based on a theoretical simulation generated from the results found in a previous study that allowed to establish differentiations between normality and abnormality of the left ventricular structure (

This new methodology is useful as a diagnostic aid tool, by establishing in an objective and reproducible manner all the possible geometric shapes of the left ventricle that vary between moderate and severe abnormality, regardless of the specific type of cardiac disease. The use of this generalization would facilitate the application of the diagnosis developed in both surgery-type and pharmaceutical-type interventions, because having all the possible fractal ventricular structures, one could count on all the possible evolution routes from normality to disease, useful for monitoring in the clinical practice.

Other studies have sought to establish ventricle measurements based on Euclidean measurements. For example, Kappenberger (

Among the computational methods, a methodology has been designed that traces several lines from the contour of the ventricle, which converge in a single central point that serves for the analysis of ventricular dynamics. However, it is a theoretical methodology that does not cover the differences of the thickening in the different segments of the wall. In a study related to this, a methodology was designed that more correctly reflects the movement of the ventricular cavity from an artificial line superimposed at the same distance from the ventricular edge in systole and diastole, which allows to differentiate between normality and different heart pathologies. In contrast, this methodology achieves an objective mathematical characterization of the irregularity of the ventricle, making it unnecessary an approximation to regular shapes, such as those performed in the aforementioned studies, also establishing the totality of possibilities that may be found in clinical practice, together with a diagnosis for each particular case, regardless of the epidemiological and statistical methods.

One of the limitations of the ventriculogram is that, in addition to being an invasive diagnostic test, it has the limitation that from the conventional evaluation methodologies, not always a normal ventricular diagnosis implies an absence of pathology and, in turn, not always the identification of a ventricular thickening implies an altered ventricular function (

Likewise, through mathematical methods theoretical simulations of all the particular cases of moderate and severe alterations of the ventricular form can be obtained. The statistical methods used in medicine today can only establish inferences about population groups, but cannot obtain generalizations from which particular cases can be deduced. The type of generalization developed in this study allows to establish with few particular cases all the possibilities of a phenomenon, and makes it possible to mathematically diagnose all the possible states that may occur in practice.

From this line of research other experimental and clinical applicability studies have been developed. This is the case of a methodology with which the totality of arterial prototypes was determined in an experimental restenosis model (

The approach in which this methodology has been developed is analogous to the way in which physical phenomena of chaos theory (

It is necessary to highlight that the present study corresponds to a theoretical generalization, and that it is based on mathematics and the inductive method of theoretical physics, according to which it is possible to establish generalizations with few cases studied and regardless of causal considerations such as type of pathology, risk factors, among others, since the epicenter are the underlying mathematical relationships. To continue the research process, in subsequent studies we will consider the establishment of the validation of the model according to the reference method or other aspects.

The present study optimally characterized the structure of the ventricle with moderate and severe abnormal diagnosis, based on a new methodology based on fractal geometry and the definition of degrees of similarity. Furthermore, the establishment of the total number of prototypes for the ventricular structure helps to reduce errors attributed when considering the left ventricle as a geometric object measurable from regular shapes in two and three dimensions; it can also be applied in the future to any computer system.

We appreciate the support given to this study, which is part of the results of the MED-1078 project, funded by the Research Fund of the Universidad Militar Nueva Granada. We also thank the Vice-Rector of Research and the Faculty of Medicine of the Universidad Militar Nueva Granada, for the support given to our work.

Special thanks to Doctors Jacqueline Blanco, Vice-Rector of Research, Martha Bahamón, Academic Vice-Rector, Esperanza Fajardo, Research Director of the Faculty of Medicine, and to Doctors Juan Miguel Estrada, Dean of the Faculty of Medicine, Alejandro Castro, Head of the Division of Scientific Research, and Henry Acuña, for their support of our research.

We also thank the Clínica del Country Research Center for their support of our research, Doctors Tito Tulio Roa, Director of Medical Education, Jorge Ospina, Medical Director, Alfonso Correa, Director of the Research Center, and Doctor Adriana Lizbeth Ortiz, epidemiologist, and Silvia Ortiz, Head Nurse of the Research Center.

To our children.