Study of the impact of cardiovascular exercises and their functional responses on diabetic peripheral neuropathy using neutrosophic statistics. (2024)

1 Introduction

Today there are approximately 382 million people who have diabetes mellitus (DM) worldwide and the projection towards 2030 is not encouraging at all, since the WHO considers that this disease will become in the leading cause of death worldwide, which indicates a complex health panorama shortly, furthermore that at least 10% of diabetic patients present diabetic peripheral neuropathy, a figure that reaches up to 50% in patients who have had the disease for at least 10 years, and at least 75% develops a very high risk of amputations.

In Ecuador the prevalence is very high since it is estimated that at ages between 20-79 years old, it reaches up to 8.5% of the population; It is even shortly proposed that diabetes and associated neuropathies would be the second cause of deaths in general.

Diabetic peripheral neuropathy is a very common complication of type II diabetes mellitus. It is usually characterized by significant deficits in tactile sensitivity, the sense of vibration, and proprioception of the lower extremities. Performing cardiovascular exercises is one of the most effective and beneficial strategies to reduce the symptoms of diabetic peripheral neuropathy.

So, the main objective of this research is to develop cardiovascular exercise strategies that allow improving functional responses in patients with diabetic peripheral neuropathy; to verify its effectiveness. The procedure begins with a 6-minute walk test called 6MWT, to answer the question posed in this work: whether the development of strategies based on cardiovascular exercises can improve functional responses in patients with diabetic peripheral neuropathy.

For the study, we decided to use neutrosophic models since the clinical problems and treatment of diseases such as diabetes and its complications can only be studied if the uncertainty and indeterminacy of both the diagnosis and the treatment are taken into account. In this type of disease converge a series of biological, as well as sociological and psychological factors. The patient must be educated to live with his (or her) illness and avoid complications. In addition, health parameters do not correspond to a single value, but to a range of values.

Due to the aforementioned, Neutrosophic Statistics is the tool that we proposed to apply in the study of the strategies to follow in the improvement of patients who suffer from diabetic peripheral neuropathy, since Neutrosophic Statistics is the generalization of classical statistics to situations where data or parameters exist in the form of intervals, also where the size of the population cannot be precisely defined [1-3]. In our case, due to the nature of the problem that we set out to study, where an exact normal value of heart rate or laboratory test results cannot be determined, it is, therefore, necessary to use values in the form of interval or neutrosophic numbers. In this way, a greater number of measurement situations for each individual are taken into account, beyond the specific situation in which the study is carried out, which increases the reliability of the experiment. In addition, we obtained an equation to determine the number of meters that the patient can travel concerning their state of dyspnea. For this end, we use the neutrosophic least square method ([4]). In this way, the patient can determine his (her) physical condition for walking by measuring the state of his dyspnea. This can be extended to other variables.

This paper is divided into a preliminary section, where we present the main concepts of neutrosophic numbers and Neutrosophic Statistics. Section 3 contains the results of the study carried out. The last section is to give the conclusions.

2 Preliminaries

This section contains the fundamental concepts about neutrosophic numbers and neutrosophic statistics.

Neutrosophic statistics refers to a set of data, such that the data or a part of it is indeterminate to some degree, and to the methods used to analyze these data ([1]).

In classical statistics all data are determined, this is the distinction between neutrosophic statistics and classical statistics. In many cases, when the indeterminacy is zero, the neutrosophic statistics coincide with the classical statistics. The neutrosophic measurement can be used to measure indeterminate data. Neutrosophic statistical methods will allow us to interpret and organize neutrosophic data (data that may have some indeterminacies) to reveal underlying patterns. Many approaches can be used in neutrosophic statistics.

In neutrosophic probability, indeterminacy is different from randomness. While classical statistics refers only to randomness, neutrosophic statistics refers to both randomness and especially indeterminacy.

Neutrosophic descriptive statistics is composed of all techniques for summarizing and describing the characteristics of neutrosophic numerical data. Since neutrosophic numerical data contain indeterminacies, neutrosophic line graphs and neutrosophic histograms are represented in 3D spaces, rather than 2D spaces as in classical statistics. The third dimension, in addition to the Cartesian system XOY, is that of indeterminacy (I). From unclear graphic data, we can extract neutrosophic (unclear) information.

Neutrosophic inferential statistics consist of methods that allow the generalization of neutrosophic sampling to a population from which the sample was selected.

Neutrosophic data are data that contain some indeterminacy. In a similar way to classical statistics, it can be classified as:

- Discrete neutrosophic data, if the values are isolated points; for example: 7 + [i.sub.1], where [i.sub.1] [member of] [0,1], 2, 38 + [i.sub.1], where [i.sub.2] [member of] [10,12];

- and Continuous neutrosophic data, if the values form one or more intervals, for example [0.05, 0.1] or [0.9, 1.0] (i.e., not sure which one).

Other classification:

- Quantitative (numerical) neutrosophic data;

For example: a number in the interval [3, 8] (we do not know exactly), or; 50, 53, 58, or 61 (we do not know exactly);

- and Qualitative (categorical) neutrosophic data; for example: blue or red (we do not know exactly), white, black or green or yellow (we do not know exactly). Additionally, we can have:

- Neutrosophic data univariate, i.e. neutrosophic data consisting of observations on a single neutrosophic attribute;

- and Multivariate neutrosophic data, that is neutrosophic data consisting of observations on two or more attributes. In particular cases, we mention the bivariate neutrosophic data and the trivariate neutrosophic data.

A neutrosophic statistical number N has the following form:

N = a + bI, where a is the determinate (known) part of N, and bI is the indeterminate (unknown) part of N ([1]).

The arithmetic operations between these numbers are summarized below ([5-8]):

Given [N.sub.1] = [a.sub.1] + [b.sub.1]I and [N.sub.2] = [a.sub.2] + [b.sub.2]I two neutrosophic numbers, some operations between them are defined as follows:

* [N.sub.1]+[N.sub.2]=[a.sub.1] + [a.sub.2] + ([b.sub.1] + [b.sub.2])I (Addition);

* [N.sub.1] - [N.sub.2] = [a.sub.1] - [a.sub.2] + {[b.sub.1] - [b.sub.2])I (Difference),

* [N.sub.1] x [N.sub.2] = [a.sub.1][a.sub.2] + {[a.sub.1][b.sub.2] +[b.sub.1][a.sub.2]+ [b.sub.1][b.sub.2] I (Product),

* [Please download the PDF to view the mathematical expression] (Division)

For example, a = 4 + I, where I [member of] [0, 0.5], is equivalent to a [member of] [4,4.5], so for sure a [greater than or equal to] 4 (meaning that the determinate part of a is 4), while the indeterminate part i [member of] [0,0.5] means the possibility that the number a is a little greater than 4. For example, if we have the following neutrosophic data: 3 + [I.sub.1] with [I.sub.1] [member of] (0,0.1); 5 + [I.sub.2] with [I.sub.2] [member of] [4,6]; 5 + [I.sub.3], with [I.sub.3] [member of] [0,1]; 10 + [I.sub.4], with [I.sub.4] [member of] [1.1,1.5); 9 + [I.sub.1].

A neutrosophic sample is a selected subset of a population, a subset that contains some indeterminacy: either concerning several of its individuals (who may not belong to the population we study or may only partially belong to it) or concerning the subset as a whole.

While classical samples provide precise information, neutrosophic samples provide vague or incomplete information. By abuse of language, it can be said that any sample is a neutrosophic sample since its determination can be considered equal to zero.

The results of the neutrosophic survey are survey results that contain some indeterminacy. A neutrosophic population is a population that is not well determined at the level of membership (i.e., it is not certain whether some individuals belong or do not belong to the population). For example, as in the neutrosophic set, a generic x(t, i, f) [member of] M element i% the belonging of x to M is indeterminate (unknown, unclear, neutral: neither in the population nor outside).

A simple random neutrosophic sample of size N from a classical or neutrosophic population is a sample of N individuals such that at least one of them has some indeterminacy.

A neutrosophic normal distribution of a continuous variable [[sigma].sup.2], for example, [mu], or [sigma], or both can be set with two or more elements. The most common distributions are when [mu], [sigma], or both are intervals.

The formula for the neutrosophic frequency function is the same, except replaced [[mu].sub.N] by [mu] and [[sigma].sub.N] by [sigma]:

[Please download the PDF to view the mathematical expression], where [X.sub.N] means [N.sub.N](*,*) that instead of one bell-shaped curve for X, we can have two or more bell-shaped curves that have common and uncommon regions between them and are above the x-axis. Each of them is symmetrical concerning the vertical line passing through the mean (x = [mu]).

Let us illustrate this with a neutrosophic example for the normal distribution, let us consider a normal distribution with [mu] = 0 and [sigma] = [1,2]. Therefore, the standard deviation is indeterminate.

"Within one standard deviation of the mean" is translated in this example by [mu] [+ or -] [sigma] = 10 [+ or -] [2,3] = [10-3, 10 + 3] = [7,13], or approximately 68% of the values are in x[member of] [7,13].

"Within two times the standard deviations of the mean" translates to [mu][+ or -]2[sigma] = 10 [+ or -] 2 * [2,3] = 10 [+ or -] [4,6] = [10 - 6,10 + 6] = [4,16], or approximately 95.4% of the values are in x [member of] [4,16]. We could also calculate the last interval as: [7,13] [+ or -] [sigma] = [7,13] [+ or -][2,3] = [7-3,13 + 3] = [4,16].

Similar to classical statistics, a neutrosophic null hypothesis, denoted by [NH.sub.0], is the statement that is initially assumed to be true. The alternative neutrosophic hypothesis, denoted by [NH.sub.a], is the other hypothesis.

When carrying out a test of [NH.sub.0] versus [NH.sub.a] there are two possible conclusions: to reject [NH.sub.0] (if the sample evidence strongly suggests that [NH.sub.0] is false) or do not reject [NH.sub.0] (if the sample does not support evidence against [NH.sub.0]).

Examples:

[NH.sub.0] : [mu][member of] [90,100] [NH.sub.a] : [mu]< 90,

[NH.sub.a]: [mu] > 100, [NH.sub.a]: [mu] [??] [90,100], where [mu] represents the classical average Intelligence Quotient of all children born since January 1, 2001.

For reading applications of Neutrosophic Statistics, see [9-14].

3 Results

The study was carried out in the province of Tungurahua, Canton of Ambato-Cevallos, Ecuador. With a group of patients suffering from diabetic peripheral neuropathy. Table 1 contains a summary of the distribution in terms of gender, age, and obesity, among other characteristics of the subjects studied.

For a patient to be included in the study, the following inclusion criteria were used:

* Patients from Atahualpa and Cevallos who present diabetic peripheral neuropathy, to whom fractional exercises will be applied, during the period from April to September 2022.

* Patients with ages ranging from 30 to 80 years old who present diabetic peripheral neuropathy.

* Patients who freely express their participation in the study by signing the informed consent.

The exclusion criteria used were the following:

* Patients who present pathologies other than diabetic peripheral neuropathy.

* Patients with recent surgeries.

* Patients who, due to associated comorbidities, cannot comply with the protocol in its entirety.

* Patients with severe cardiovascular disease.

To determine the sample, probabilistic sampling was selected, a method that is characterized by seeking with great dedication to obtain qualitatively representative samples, through the inclusion of apparently typical groups, that is, they meet characteristics of interest to the researcher with patients who are located in Atahualpa and Cevallos, who present diabetic peripheral neuropathy, to whom intense cardiovascular exercises 70% to 80% will be applied, during the period October 2022-January 2023, after evaluation and from whom the data and information required to the development of the study.

In the study, an initial and final evaluation was developed with the implementation of an accessible cardiovascular exercise protocol for patients with diabetic peripheral neuropathy, 15-minute cardiovascular exercises were selected. A form was used to record the information on the 6-minute walk test data for each patient who applied the test and completed it in its entirety.

The 6-minute walk test is a variety of the Cooper test, which aims to measure the maximum distance that a person can walk for 6 minutes. The speed at which the patient walks will determine the distance in meters, that is, a test that evaluates, in an integrated manner, the response of the respiratory, cardiovascular, metabolic, skeletal muscle, and neurosensory systems that the individual develops during exercise.

The 6-minute test is a valid and reliable method to evaluate functional capacity in a population with cardiovascular problems in phase II/III. This test is very valuable for smaller healthcare facilities that wish to document functional improvements but do not have access to conventional treadmill testing.

Table 2 shows a summary of the other diseases associated with the patients under study.

To perform numerical calculations we use neutrosophic numbers to represent the collected data. For each aspect to be measured, the symbolic value I represents the normal range of what is measured, and in numerical calculations it is replaced by the equivalent range. For example, for oxygen saturation the normal range is between 95-100% resting, which is why we take I = [95,100], this guarantees that if a patient has a resting saturation equal to 96 before the study and after the study he (she) has 98%, then the difference is 0. In this case, a reference is obtained from what is normal to what is not normal. Although this means loss of precision, in reality, it is quite the opposite, since at the time of the study the patient may have a certain saturation that may vary at another time, for example, upon awakening and when the interval I is considered instead of a specific value is taking into account, then more situations are studied than the only one in which the experiment is carried out, and in this way the dynamic behavior of these indicators is more accurately reflected.

Before performing the statistical t Test, we checked that the data satisfied the normality condition; in this case, it was a non-parametric test of normality with the help of the Kolmogorov-Smirnov test adapted to data in the form of intervals. Although the test resulted in the data not being distributed normally, as the sample is large enough I = 34 [greater than or equal to] 30, it is well known that the t-test is robust enough to give reliable results in the case of large samples, greater than 30 [15,16].

Table 3 shows the results obtained for heart rate, it reflects the average of the initial evaluation and the final evaluation, after having the training with the exercises.

The following table (Table 4) shows the results regarding the saturation levels during the initial and final evaluation process regarding the execution of the exercises.

Table 5 shows the difference in the degree of fatigue between the execution of exercises between the initial evaluation and the final evaluation.

Table 6 below shows the results regarding dyspnea between the initial and final evaluation regarding the execution of the exercise.

Table 7 summarizes the results regarding blood pressure.

Table 8 shows the final results regarding the distance traveled in meters and the volume of oxygen used. In the results of the table, we do not consider neutrosophic numbers, since the measured parameters have no limits of what is a normal parameter.

Finally, in the following, we consider the neutrosophic method least squares that appear in ([4]). This is an extension of the well-known statistical method. The objective is to link the variable dyspnea as a dependent variable with the variable meters traveled as an independent variable. To do this, we carry out a statistical approximation using a linear function. The Equations are the following represented by intervals.

First, we wish to obtain the coefficients of the following linear equation:

[Please download the PDF to view the mathematical expression] (1)

Where [??] is the approximation in interval form (or its equivalent in neutrosophic number) of the dependent variable, [a.sub.N], and [b.sub.N] are the coefficients in numbers within intervals of the linear equation, while [x.sub.N] is the data of the independent variable given in the form of intervals/neutrosophic numbers.

The approximation of the first coefficient is obtained from Equation 2.

[Please download the PDF to view the mathematical expression] (2)

Where [Please download the PDF to view the mathematical expression] and for the approximation of [b.sub.N] Equation 3 is used.

[Please download the PDF to view the mathematical expression] (3)

[Please download the PDF to view the mathematical expression] and [n.sub.N] is the number of elements in the sample.

In this way, we obtained the equation [I.sub.dist] = 338.601 - 77.0[I.sub.day].

Conclusion

In this article, we carry out a study of the effectiveness of applying the 6-minute walk test (6MWT) in patients suffering from diabetic peripheral neuropathy in Ecuador. The study was carried out with 34 patients who suffer from this complication due to diabetes. Measurements were made of the results of different medical indicators applied before and after the training of the patients with the 6MWT, they are namely, "heart rate", "oxygen saturation", "fatigue", "dyspnea", "blood pressure", "distance traveled" and "condition physical cardiorespiratory (VO2max)". Within the study, we realized that the data collected are crisp and respond to the patient's state at a precise moment of measurement, although these parameters change over time and moment, it is for this reason that we converted the data from crisp to neutrosophic numbers and we apply neutrosophic statistics methods. Additionally, we found a statistical relationship between dyspnea and the number of meters traveled by patients with diabetic peripheral neuropathy, based on the neutrosophic least square method.

Specifically, about the results of the method, we conclude the following:

* It is concluded that from the state of health and physical condition of diabetic patients, the vast majority suffer from high blood pressure that affects their health condition and is a critical factor in diabetic patients if it is not adequately controlled.

* At the end of the application of the accessible cardiovascular exercise protocol for patients with diabetic peripheral neuropathy, which was designed with information from the initial diagnosis of the patient's clinical histories that allowed us to know the patient's pathologies, it was determined that the test of the 6 minutes is effective to evaluate the maximum travel distance and VO2 level in the effort between distance and walking, to apply 15-minute cardiovascular exercises working at an intense intensity of 70% to 80% that were carried out in 8 weeks.

* At the end of the intervention, the results in older adults with diabetic peripheral neuropathy after having trained in cardiovascular exercises established that the development of the intervention favors systolic blood pressure resting and during the 6-minute exercise because it adapts the need and physical activity favors the health of patients when it is developed in a planned manner and based on their needs, regular exercise can contribute to diabetic patients by improving their quality of life.

Acknowledgment

To the Directorate of Research and Development (DIDE) and the Technical University of Ambato to which the Project entitled: "Multidisciplinary health intervention strategy in patients with diabetic peripheral neuropathy to promote their quality of life. Ambato Canton", approved in Resolution No. UTA-CONIN-2022-0169-R.

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Lisbeth Reales Chacon (1), Angela Campos Moposita (2), Diana Marquina Amon (3) Victor Garcia Camacho (4) Josselyn Bonilla Ayala (5) Javier Caiza Lema (6) Yadira Aguilar Zapata (7) Paul Cantuna Vallejo (8)

(1) Universidad Tecnica de Ambato, Campus Ingahurco y Avenida Colombia, Ambato, 180101, Ecuador, Email: [emailprotected]

(2) Universidad Tecnica de Ambato, Campus Ingahurco y Avenida Colombia, Ambato, 180101, Ecuador, Email: [emailprotected]

(3) Universidad Tecnica de Ambato, Campus Ingahurco y Avenida Colombia, Ambato, 180101, Ecuador, Email: [emailprotected]

(4) Universidad Tecnica de Ambato, Campus Ingahurco y Avenida Colombia, Ambato, 180101, Ecuador, Email: [emailprotected]

(5) Universidad Tecnica de Ambato, Campus Ingahurco y Avenida Colombia, Ambato, 180101, Ecuador, Email [emailprotected]

(6) Universidad Tecnica de Ambato, Campus Ingahurco y Avenida Colombia, Ambato, 180101, Ecuador, Email: [emailprotected]

(7) Universidad Tecnica de Ambato, Campus Ingahurco y Avenida Colombia, Ambato, 180101, Ecuador, Email ya.aguilar @uta.edu.ec

(8) Universidad Tecnica de Ambato, Campus Ingahurco y Avenida Colombia, Ambato, 180101, Ecuador, Email [emailprotected]

Received: 16 Dec,2023Accepted:15 Feb,2024

Table 1. Distribution of the patients under study according to their age, gender, and body mass index. % of the Total number of boardAge Elderly 70.6% Adults 29.4%Sex Male 32.4% Female 67.6%Index of Low weight 2.9%massbodily Normal 32.4% Overweight 26.5 Obesity 29.4

Table 2. Percentage of person pathological history with diseases not directly related to diabetes and its complications.BackgroundPathologicalPersonalDisease Frequency PercentageNone 6 35.3Hypertension 2 11.8Hypothyroidism 6 35.3Respiratory 3 17.6

Table 3. Results regarding average heart rate. Average initial Average final evaluation evaluationFrequency cardiac resting -5.1 + I -4.6 + IFrequency cardiac 3min -0.34 + I 2.41 + IFrequency cardiac 6 min -0.12+ I 1.41 + I Difference average pFrequency cardiac resting 0.76 0.683Frequency cardiac 3min 2.75 0.561Frequency cardiac 6 min 1.53 0.644

Table 4. Levels of average saturation. Average initial Average final Evaluation evaluationSaturation resting -1.81 + I -1.53 + ISaturation 3 min -2.59 + I -1.24 + ISaturation 6 min -0.76 +Yo 0.41 + Yo Difference average pSaturation resting 0.28 0.988Saturation 3 min 1.35 0.921Saturation 6 min 1.17 0.371

Table 5. Fatigue average. Average initial Average final Difference p evaluation evaluation AverageFatigue resting 0.02+ I -0.1 + I -0.12 0.253Fatigue 3 min 2.78 + I 2.25 + I -0.53 0.253Fatigue 6 min 5.66 + I 4.78 + I -0.88 0.121

Table 6. Dyspnea average. Average Average Difference p initial final average evaluation evaluationDyspnea resting 0.02 + I -0.1 + I -0.12 0.09Dyspnea 3 min 0.31 + I -0.1 + I -0.41 0.14Dyspnea 6 min 0.84 + I 0.66 + I -0.18 0.34

Table 7. Arterial Pressure average. Average initial Average final Difference p evaluation evaluation average Pressure arterial 7.35 + I 10.12+ I 2.77 0.002systolic resting Pressure arterial -0.24 + I 0.12 + I 2.36 0.013diastolic resting Pressure arterial 16.18 + I 16.41 + I 0.23 0.010systolic 6 min Pressure arterial 2.88 + I 2.76 + I -0.12 0.501diastolic 6 min 

Table 8. Distance traveled and average VO2. Average initial Average final Difference p evaluation evaluation averageMeters traveled 325.71 336.18 10.47 0.00VO2 max 22.89 23.55 0.66 0.00