A MATHEMATICAL APPROACH OF EXPLORATION TOWARDS EXTREME RISK FACTOR IN CANCER OF OPTIMAL CONDITION

A MATHEMATICAL APPROACH OF EXPLORATION TOWARDS EXTREME RISK FACTOR IN CANCER OF OPTIMAL CONDITION

Priyanka Majumder ^{* 1}, Apu Kumar Saha ¹ and Mrinmoy Majumder ²

Department of Mathematics ¹, Department of Civil Engineering ², National Institute of Technology, Agartala, Barjala, Jirania - 799046, Tripura, India.

ABSTRACT: The aim of this present investigation is to identify the most important risk factor for cancer by means of a mathematical model. The study begins with considering some types of cancer viz. breast, colorectum as well as lung and bronchus cancer as most of the cancer patients suffer from these types of cancers. Teaching-Learning-Based Optimization technique is applied to sorting out the most lethal cancer among all cancer consider in this study and it is found that lung and bronchus cancer is the most fatal. Further, we investigate the risk factor associated to lung and bronchus cancer by means of Literature, Expert and Local Hospital survey. All the risk factor has their own importance for death from cancer in medical aspects. Multi-Criteria Decision Making technique is applied to recognize the most significant risk factor among all the factors in statistical scenario. It is identified that smoking is the most concerning risk factor. The information related from the study may help to take necessary measure to control the death rate due to lung and bronchus cancer.

Breast cancer, Colorectum cancer, Lung and Bronchus cancer, Optimization, MCDM, Control

INTRODUCTION: Human body contains millions of cells; it grows divides and dies in the conventional manner. Sometimes the system goes wrong and uncontrolled no of cells grows, which leads to cancer. The cancer cells Combine and form extra mass tissue known as tumor ¹. Cancer is a common disease which spreads throughout the blood stream in the human body. Leukemia alters the blood cell and involve in its maturity and immaturity ². Some of the tumors do not spread throughout the body, but grow uncontrollably like benign tumor ³.

Normal / healthy cell controls their growth and when they become unhealthy, destroys by themselves. In Asia high prevalence of chronic viruses like hepatitis B ⁴ and C, the Epstein Barr virus and human papilloma viruses (HPV) ⁴ increases the high risk of cancer.

Mutations in p53 gene ⁵ leads to cancer as well as nutrition ⁶ Plays a vital role in mortality of cancer ⁷. Exposure to aldehydes and formaldehyde associated with high risk of lymphoma cancer ⁸. Hypoxia ⁹ is a solid tumor growth in cancer which is common and disturbs molecular pathways ¹⁰. It is not possible to find out the specific cause of cancer. Cancer cells are modulated by culture condition and extracellular microenvironment condition ¹¹. But there are many risks which increase the cancer, such as intake of tobacco, alcohol, poor diet, obesity, exposure to UV radiation, lack of physical activity ¹.

According to the literature survey, there are so many lethal cancers namely breast, colorectum, lung and bronchus ¹². Of the 7 million deaths from cancer worldwide in 2001, an estimated 2.43 million (35%) were attributable to nine potentially modifiable risk factors. Of these, 0.76 million deaths were in high-income countries and 1.67 million in low-and-middle-income nations.

Among lower and middle income regions, Europe and Central Asia had the highest proportion (39%) of deaths from cancer attributable to the risk factors studied. 1.6 million of the deaths attributable to these risk factors were in men and 0.83 million in women. Smoking, alcohol use, and low fruit and vegetable intake were the leading risk factors for death from cancer worldwide and in low and middle income countries. In high income countries, smoking, alcohol use, and overweight and obesity were the most important causes of cancer. Sexual transmission of human papilloma virus is a leading risk factor for cervical cancer in women in low and middle income countries. More than 12 million new cases of cancer occur annually worldwide. Of those 5.4 million occur in developing countries and 6.7 million in developing countries ^{13, 14}.

Aline et al., developed the Proactive Molecular Risk Classifier for Endometrial Cancer (ProMisE), a molecular classification system based on The Cancer Genome Atlas genomic subgroups, and sought to confirm both feasibility and prognostic ability in a new, large cohort of ECs ¹⁵. Evaluate the relationship between health beliefs (perceived susceptibility to breast cancer, perceived benefits of AI treatment, and perceived barriers to AI treatment) and adherence to AIs by Moriah et al., ¹⁶. Filip et al., evaluated factors associated with stage-specific cancer therapy and survival, focusing on temporal trends and sociodemographic disparities ¹⁷. To identify the variability of short- and long-term survival outcomes among closed Phase III randomized controlled trials with small sample sizes comparing SBRT (stereotactic body radiation therapy) and surgical resection in operable clinical Stage I non-small cell lung cancer (NSCLC) patients by Pamela ¹⁸.

In the present study, we heave aimed to recognize the most fatal cancer (whose death rate is maximum) and to find the most important factor (MIF) responsible for causing cancer. To carry out study, we have obtained the regression function of death rate of each cancer that have been considerable in our study of the data given in ¹⁹. Next, we have optimized the death rate due to different cancers by Teaching Learning Based Optimization (TLBO). In the second part of the study we have found out the factors. Most lethal cancer among the cancers considered in our study y literature review, expert survey and survey from a local hospital. The Analytic Hierarchy Process (AHP) has been applied to identify the most important factor responsible for that cancer.

Background: Cancer is the name given to a collection of related diseases. In all types of cancer, some of the body’s cells begin to divide without stopping and spread into surrounding tissues. Cancer can start almost anywhere in the human body, which is made up of trillions of cells. Normally, human cells grow and divide to form new cells as the body needs them. When cells grow old or become damaged, they die, and new cells take their place ²⁰. Fig. 1 showing a cancer cell. Fig. 2 showing the new cancer cases annually per 100,000 people (age-adjusted) in the world.

FIG. 1: A DIVIDING CANCER CELL (NATIONAL INSTITUTES OF HEALTH)

FIG. 2: RECENT CANCER SCENARIO IN THE WORLD

Breast Cancer: Breast cancer starts when cells in the breast begin to grow out of control. These cells usually form a tumor that can often be seen on an x-ray or felt as a lump. The tumor is malignant (cancer) if the cells can grow into (invade) surrounding tissues or spread (metastasize) to distant areas of the body. Breast cancer occurs almost entirely in women, but men can get breast cancer, too ²¹. Breast cancer is the most common malignancy affecting women worldwide ²². Breast cancer is a very important health issue in developed and developing nations of the world ²³. In several developed countries, breast cancer incidence increases faster with age before age 50 than after ²⁴. There are various type of risk factors for breast cancer, with some having greater importance than others. A review of the literature on such risk factors was thus undertaken, with these been grouped into four broad categories, namely: sociodemographic factors, reproductive factors, lifestyle factors and hormonal status. Table 1 shows all important risk factors of lung cancer. Equation 1 represents colorectal cancer risk factor ( ).

b_rf = sf + rdf + lf + hs........(1)

Where sf = b₁

rdf = b₂ + b₃

lf = b₄ + b₅ + b₆

hs = b₇ + b₈ + b₉ + b₁₀

TABLE 1: RISK FACTORS OF LUNG CANCER

Colorectum Cancer: Colorectal cancer develops in the colon or the rectum, also known as the large intestine. The colon and rectum are parts of the digestive system, also called the gastrointestinal (GI) system.

TABLE 2: RISK FACTORS OF COLORECTAL CANCER

The digestive system processes food for energy and rids the body of solid waste ²⁵. Colorectal cancer is the third and second most common cancer, respectively, in men and women worldwide ²⁶ and is a major cause of morbidity and mortality ²⁷. There are various risk factors for colorectal cancer, with some having greater importance than others. A review of the literature on such risk factors was thus undertaken, with these being grouped into four broad categories, namely: sociodemographic factors, medical factors, lifestyle factors and diet factors. Table 2 shows all important risk factors of lung cancer. Equation 2 represents colorectal cancer risk factor (cl_rf).

Lung and Bronchus: Lung cancer cells have defects in the regulatory circuits that govern normal cell proliferation and homeostasis. The transformation from a normal to malignant lung cancer phenotype is thought to arise in a multistep fashion, through a series of genetic and epigenetic alterations, ultimately evolving into invasive cancer by clonal expansion ^{28, 29}. Lung cancer has been the most common cancer in the world ³⁰. The incidence of cancer of the lung has increased precipitously since the early years of this century, and this disease is responsible for the deaths of many thousands of patients throughout the world ³¹. There are multiple risk factors for lung cancer, with some having greater importance than others. A review of the literature on such risk factors was thus undertaken, with these being grouped into two broad categories, namely: factors inherent to the individual (intrinsic factors) and factors extraneous to the individual (extrinsic or environmental factors). Table 3 shows all important risk factors of lung cancer. Equation 3 represents Lung cancer Risk Factor (l_rf).

TABLE 3: RISK FACTORS OF LUNG CANCER

Methods: In this study apply one optimization techniques (OT) viz. TLBO and one MCDM method viz. AHP. OT is apply for identifying the fatal cancer whose death rate is maximized and to find the MIF responsible for causing cancer apply MCDM.

Teaching Learning Based Optimization (TLBO): TLBO is one of the population based algorithm suggested by Rao et al., ^{32, 33} where a group of students is considered the population and different subjects offered to the learners are analogous with the various design variables of the optimization problem. The main goal of this optimization is 'in a classroom every individual attempts to learn something from teacher to improve themselves'. This algorithm is based on two period viz. teaching phase and learner phase.

Phase I: During the Teacher Phase, the teaching role is assigned to the best individual (X_teacher). The algorithm attempts to improve other individuals (X_i) by moving their positions towards the position of the X_teacher by taking into account the current mean value of the individuals (X_mean). This is constructed using the mean-values for each parameter within the problem space (dimension), and represents the qualities of all students from the current generation. Eq. (4) simulates how student improvement may be influenced by the difference between the teacher’s knowledge and the qualities of all students. For stochastic purposes, two randomly-generated parameters are applied within the equation: r ranges between 0 and 1; and TF is a teaching factor, which can be either 1 or 2, thus emphasizing the importance of student quality.

X_new = X_i+ r. (X_teacher –  (TF . X_mean))..............(4)

Phase II: During the Learner Phase, student (X_i) tries to improve his / her knowledge by peer learning from an arbitrary student Xii, where i is unequal to ii. In the case that X_ii is better than X_i, X_i is moved towards X_ii (Eq. (5)). Otherwise, it is moved away from X_ii (Eq. (6)). If student X_new performs better by following Eq. (5) or (6), he / she will be accepted into the population. The algorithm will continue its iterations until reaching the maximum number of generations.

X_new = X_i+ r. (X_ii _ X_i)...................(5)

X_new = X_i+ r. (X_i _ X_ii)...................(6)

The Analytical Hierarchy Process (AHP): The Analytical Hierarchy Process (AHP) is a decision-aiding method developed by Saaty ^{34, 35, 36, 37}. It aims at quantifying relative priorities for a given set of alternatives on a ratio scale, based on the judgment of the decision-maker, and stresses the importance of the intuitive judgments of a decision-maker as well as the consistency of the comparison of alternatives in the decision-making process ³⁴.

Saaty ^{34, 35, 36, 37} developed the following phase for applying the AHP:

Phase I: Define the problem and determine its goal.

Phase II: Structure the hierarchy from the top (the objectives from a decision - maker's viewpoint) through the intermediate levels (criteria on which sub-sequent levels depend) to the lowest level which usually contains the list of alternatives.

Phase III: Construct a set of pair-wise comparison matrices (size n × n) for each of the lower levels with one matrix for each element in the level immediately above by using the relative scale measurement shown in Table 4. The pair-wise comparisons are done in terms of which element dominates the other.

Phase IV: There are n(n-1)/2 judgments required to develop the set of matrices in step 3. Reciprocals are automatically assigned in each pair - wise comparison.

Phase V: Hierarchical synthesis is now used to weight the eigenvectors by the weights of the criteria and the sum is taken over all weighted eigenvector entries corresponding to those in the next lower level of the hierarchy.

Phase VI: Having made all the pair-wise comparisons, the consistency is determined by using the eigenvalue, λ_max to calculate the consistency index, CI as follows: CI = ( λ_max – n) / (1 - n), where n is the matrix size. Judgment consistency can be checked by taking the consistency ratio (CR) of CI with the appropriate value in Table 5. The CR is acceptable, if it does not exceed 0.10. If it is more, the judgment matrix is inconsistent. To obtain a consistent matrix, judgments should be reviewed and improved.

Phase VI: Phase III - VI are performed for all levels in the hierarchy.

TABLE 4: PAIR-WISE COMPARISON SCALE FOR AHP PREFERENCES ^{34, 35, 36, 37}

TABLE 5: AVERAGE RANDOM CONSISTENCY (RI) ^{24, 25, 26, 27}

Methodology:

FIG. 3: FIGURE DEPICTS THE BASIC METHODOLOGY OF IDENTIFYING THE MIRF OF CANCER

In the present study apply OT and MCDM method for selection of most important risk factor (MIRF) for fatal cancer. Here OT is applied for selection of fatal cancer and MCDM is applied for selection for the most impotent risk factor of fatal cancer which is selected by OT. Fig. 3 depicts the basic methodology of identifying the MIRF of fatal cancer.

Application of OT: The statistical technique multiple regression analysis (MRA) is used for developing the mathematical models by using the death rate data listed in Table 6. Here five objective functions are constructed with the help of Table 6. Using MINITAB 17 statistical software we obtain the objective function of each cancer. Each type of cancer is selected as input and average values are taken as output for this software and then obtain Regression. Here each Regression is considered as an objective function. All this data are collected from Cancer Statistics, 2016 ¹⁹. Here this data is selected because each year, the American Cancer Society estimates the numbers of new cancer cases and deaths that occurs in the United States in the current year and compiles the most recent data on cancer incidence, mortality, and survival.

TABLE 6: DEATH RATES FOR SELECTED CANCERS INDIFFERENT STATES OF USA ET 2008 TO 2012

Equation 6a, 6b and 6c represents cancer function with respect to breast, Colorectum, Lung and Bronchus respectively. Upper bound and lower bound are selected maximum and minimum value of each cancer data.

Max Z_Breast = 51.8 – 6.76 X₁+0.372 X₁²-0.006151 X₁³..(6a)

Subject to X₁ ≥ 15.1

X₁ ≤ 29

Max Z_Colorectum = –42.4+8.9X₂–0.39X₂²+0.0.0050 X₂³..(6b)

Subject to X₂ ≥ 10.2

X₂ ≤ 16.6

Max Z_{Lung and Bronchus} = 9.66+0.029 X₃+0.00745 X₃²-0.000074 X₃³  ..(6c)

Subject to X₃ ≥ 15.6

X₃ ≤ 55.2

The validity of the regression model is further tested by plotting a probability distribution shown in Fig. 4, 5 and 6. The probability values of the response parameters are scattered near to 95% line, which indicates perfect fitting of the developed regression model.

One of the aims of the present investigation is to find the fatal cancer. We select that cancer whose death rate is maximized. So in this study, we maximize the objective functions 6a, 6b and 6c also all this function 6a, 6b and 6c represents the death rate of each considering cancers.

Application of MCDM: In the present study, we have aimed 42 recognize the most important risk factor of fatal cancer (Whose death rate is maximum). The selection of that factors used AHP techniques. The steps below provide the methodology adopted to look for the weights of importance for each and every factor of the fatal cancer by the MCDM method.

Selection of Criteria: In the present study the relative score of percentage of is required for weights of importance of the criteria The following steps are required for finding the relative score alternatives:

a. Literature Survey (C₁): A survey has been carried out within the literature of related fields from where it is found that the effect of each parameters which can Induce effects of the fatal cancer. The score of literature survey is given by equation 7.

Let, f = No. of literature which prefer the alternative.

F = Total number literature studied.

S(A₁) = Score of literature survey = (f/F)%…........ (7)

Where S(C₁) denotes the percentage of C₁

b. Expert Survey (C₂): A survey has been carried out within experts in related fields are participants were asked to suggest about the fatal cancer parameters that important for analysis of fatal cancer. According to responses received from the experts a percentage was given to the factors according to Equation 8. Here Expert survey denoted by C₂.

Let, r = No. of expert survey, which prefer the criteria.

R= Total number expert survey.

S(C_{2 )}= Score of expert survey= (r/R)%………… (8)

Where S(C₂) denotes the percentage of C₂

c. Local Hospital Survey(C₃): A survey from local patient from cancer hospital has been carried out. Participants were asked to suggest about the fatal cancer parameters which can induce effect on the fatal cancer. According to the response from the local patient a percentage was given to the factors according to Equation 9. Here Local survey denoted by C₃.

Let, g = No. of local survey, which prefer the criteria.

G = Total number local survey.

S(C₃) = Score from local survey = (g/G) %……… (9)

Where S(C₂) denotes the percentage of C₂.

Selection of Alternatives: Some environmental factors will be found from a survey. We collect those factors as alternatives which are commonly found from three surveys and this survey. All these alternatives are denoted by Ai.

RESULTS AND DISCUSSION: The results for this present investigation can be sub-divided into two parts, viz., results of the OT to estimate the fatal cancer and lastly use selection of most important environmental risk of that fatal cancer use MCDM. All the three results are described in detail in the following three different Sections 5.1 and 5.2.

Result from OT: Table 7 shows the optimal value of the objective function and Fig. 7a, 7b and 7c represents the convergence graph of the objective function. Using TLBO it is found that the death rate of Lung and Bronchus cancer is maximum with respect to other two considering cancers. So it is clear that Lung and Bronchus cancer is the fatal cancer. In this study, we take population size 30 and number of iteration taken 100.

TABLE 7: TABLE SHOWING THE OPTIMAL VALUE OF CANCER

Here all of the objective function solved by MATLAB code of TLBO. Aronchick ³⁸, Gasperino ³⁹, Ruano-Ravina et al., ⁴⁰ they also say that Lung and Bronchus is now the leading cause of cancer death in men and women.

Result from MCDM: The percentage of the parameters with respect to each of the criteria is presented in Table 8. Table 9 shows the pair-wise comparison of each factor with respect to each criteria. Table 10 shows the final aggregations of AHP. According to OT result, it was found that lung and bronchus cancer is fatal. There are so many environmental factors are responsible for that cancer. According to literature, expert and local hospital survey, it was found that tobacco and smoking, second-hand tobacco smoke, diet and environmental pollutants are more responsible for lung and bronchus cancer. Here, all these factors are considered as an alternative. Here tobacco and smoking, Second-hand tobacco smoke, diet and environmental pollutants are denoted by A₁, A₂, A₃ and A₄ respectively.

TABLE 8: PERCENTAGE OF PAPERS, EXPERT AND LOCAL HOSPITAL SURVEY

According to the result of AHP it was found that Tobacco and smoking are the most important risk factor for lung and bronchus cancer because the weights of A₁ is greater than other risk factors. Vincent et al., ⁴¹, Spiro ⁴², Hecht ⁴³ also suggested that smoking is very risky for Lung and Bronchus cancer.

TABLE 9: PAIR-WISE COMPARISON OF ALTERNATIVES WITH RESPECT TO CRITERIA

TABLE 10: TABLE SHOWING EIGEN VECTOR WEIGHT FOR EACH OF THE CRITERIA AND ALTERNATIVES CONSIDERED

CONCLUSION: In this study, we first find the regression of death rate of cancer by making use of different kinds of cancers data. Investigation performed on three types of cancer viz. breast, colorectum as well as lung and bronchus cancer. The conclusions drawn from the above analysis are as follows:

a. Lung and bronchus cancer is significant over other type of cancer.

b. Smoking is the most risk factor for lung and bronchus cancer.

ACKNOWLEDGEMENT: Thank you very much to the reviewers for the kind suggestion to improve our manuscript.

CONFLICT OF INTEREST: I have no conflict of interest into the manuscript.

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    How to cite this article:

    Majumder P, Saha AK and Majumder M: A mathematical approach of exploration towards extreme risk factor in cancer of optimal condition. Int J Pharm Sci & Res 2018; 9(9): 3732-42. doi: 10.13040/IJPSR.0975-8232.9(9).3732-42.

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