Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human Vascular System

Abstract

Aortic dissection is the most common aortic emergency requiring surgical intervention. Whether the elective endovascular repair of abdominal aortic aneurysm reduces long-term morbidity and mortality, as compared with traditional open repair, remains uncertain. The foundation of shell element based on the Reissner-Mindlin kinematics assumption is widely applicable, but this cannot model applications of shell surface stresses as needed in analysis of shell in human vascular system. The analysis is designed to assess progression of initial lesion in aortic dissection. Using general shell element analysis and tensor calculus, a higher order differential geometry-based model is proposed. Since the shell is thin, a variational formulation for initial lesion is proposed. The variational formulation for initial lesion is well posed. The weak convergence of the solution to initial lesion model is mathematically substantiated. Asymptotic analysis shows that initial lesion is membrane-dominated and bending-dominated when pure bending is inhibited and noninhibited, respectively. At least two observations are to be noted. First, the mathematical analysis of the initial lesion model is distinct from classical shell models. Second, the asymptotic analysis of the initial lesion model is based on degenerating three-dimensional continuum to bending strains in order to assess initial lesion behavior.

Interaction between Conformational Features and Hemodynamic Flows affecting Progression of Aortic Aneurysm and Dissection Disorders

“…Incidence of acute aortic syndromes recorded at emergency department in hospitals is Three-Five per 100,000 people per year. Note that this figure underestimates the actual incidence rate of acute dissection since hospital-based case accounting fails to include pre-admission deaths related to complications from aortic dissection [10][11][12][13][14]. It seems indeed the case since a prospective analysis of approximately Thirty-thousand middle-aged cohort with a mean Twenty years follow-up reported Fifteen cases per 100,000 patients per year are at risk of aortic dissection.…”

Progression of Aortic Dissection: An Application of Soliton Solutions of Nonlinear Evolution Equation in (3+1)-Dimensions

“…The directional behavior of these waves in the dynamic pathology consists of formation of unstable wave field, wherein many waves crossing each other at various angles. At a linear scale, it can be summed up as the pathological wave inclusive of long-crested and short-crested waveforms emerging from different directions [30][31][32][33]. Under abnormal anatomical and/or physiological states, these waveforms harbours destructive characteristics and contribute to the dynamic pathology with/without structural and/or functional interplay between heart and aorta.…”