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In the present study, our work is focused on a prey predator model with a stage structure for the prey. The objective of the study is to find the behavior of the model using parameter values in the presence of Pentagonal fuzzy numbers. The interaction between the species is done by using functional responses, such as the Holling type I reaction for immature prey and the Crowley Martin functional response for mature prey. Prey population categorized as immature and mature prey. The idea of the problem is to construct a fuzzy theoretical method which helps us to create a model to solve this uncertainty problem by using fuzzy parameters and initial values. The mathematical formulation is developed by using a prey predator model in a fuzzy environment. The existence of equilibrium points is carried out. By using the concept of alpha cut the parameters which are used in the prey – predator model is treated as pentagonal fuzzy numbers. The parameters which they have used in the mathematical formulation can be taken as a crisp value by applying the defuzzification method. Here a Robust ranking technique method is utilized. The stability of each equilibrium point is studied by computing the Jacobian matrix and finding the eigenvalues evaluated at each equilibrium point. By utilizing the prey stage structure stability analysis is also discovered. For the dynamical system, numerical simulations are provided using a MATLAB program and also, they show the behavior of the system and determine if it is stable or unstable.

Abstract Background Case–control studies show that mammographic density is a better risk factor when defined at higher than conventional pixel-brightness thresholds. We asked if this applied to interval and/or screen-detected cancers. Method We conducted a nested case–control study within the prospective Melbourne Collaborative Cohort Study including 168 women with interval and 422 with screen-detected breast cancers, and 498 and 1197 matched controls, respectively. We measured absolute and percent mammographic density using the Cumulus software at the conventional threshold (Cumulus) and two increasingly higher thresholds (Altocumulus and Cirrocumulus, respectively). Measures were transformed and adjusted for age and body mass index (BMI). Using conditional logistic regression and adjusting for BMI by age at mammogram, we estimated risk discrimination by the odds ratio per adjusted standard deviation (OPERA), calculated the area under the receiver operating characteristic curve (AUC) and compared nested models using the likelihood ratio criterion and models with the same number of parameters using the difference in Bayesian information criterion (ΔBIC). Results For interval cancer, there was very strong evidence that the association was best predicted by Cumulus as a percentage (OPERA = 2.33 (95% confidence interval (CI) 1.85–2.92); all ΔBIC > 14), and the association with BMI was independent of age at mammogram. After adjusting for percent Cumulus, no other measure was associated with risk (all P > 0.1). For screen-detected cancer, however, the associations were strongest for the absolute and percent Cirrocumulus measures (all ΔBIC > 6), and after adjusting for Cirrocumulus, no other measure was associated with risk (all P > 0.07). Conclusion The amount of brighter areas is the best mammogram-based measure of screen-detected breast cancer risk, while the percentage of the breast covered by white or bright areas is the best mammogram-based measure of interval breast cancer risk, irrespective of BMI. Therefore, there are different features of mammographic images that give clinically important information about different outcomes.

Abstract Background In a previous paper, we had assumed that the risk of screen-detected breast cancer mostly reflects inherent risk, and the risk of whether a breast cancer is interval versus screen-detected mostly reflects risk of masking. We found that inherent risk was predicted by body mass index (BMI) and dense area (DA) or percent dense area (PDA), but not by non-dense area (NDA). Masking, however, was best predicted by PDA but not BMI. In this study, we aimed to investigate if these associations vary by tumor characteristics and mode of detection. Methods We conducted a case-control study nested within the Melbourne Collaborative Cohort Study of 244 screen-detected cases matched to 700 controls and 148 interval cases matched to 446 controls. DA, NDA and PDA were measured using the Cumulus software. Tumor characteristics included size, grade, lymph node involvement, and ER, PR, and HER2 status. Conditional and unconditional logistic regression were applied as appropriate to estimate the Odds per Adjusted Standard Deviation (OPERA) adjusted for age and BMI, allowing the association with BMI to be a function of age at diagnosis. Results For screen-detected cancer, both DA and PDA were associated to an increased risk of tumors of large size (OPERA ~ 1.6) and positive lymph node involvement (OPERA ~ 1.8); no association was observed for BMI and NDA. For risk of interval versus screen-detected breast cancer, the association with risk for any of the three mammographic measures did not vary by tumor characteristics; an association was observed for BMI for positive lymph nodes (OPERA ~ 0.6). No associations were observed for tumor grade and ER, PR and HER2 status of tumor. Conclusions Both DA and PDA were predictors of inherent risk of larger breast tumors and positive nodal status, whereas for each of the three mammographic density measures the association with risk of masking did not vary by tumor characteristics. This might raise the hypothesis that the risk of breast tumours with poorer prognosis, such as larger and node positive tumours, is intrinsically associated with increased mammographic density and not through delay of diagnosis due to masking.