No abstract

Aortic dissection (AD) is the most common catastrophic disease reported at cardiovascular emergency in hospitals. Herein, a tear in the tunica intima results into separation of layers of aortic wall leading to rupture and torrential bleed. Hypoxia and oxidative stress are associated with AD. The release of hypoxia inducible factor (HIF)-1[Formula: see text] from the initial flap lesion in the tunica intima is the basis for aneurysmal prone factors. We framed a boundary value problem (BVP) to evaluate homeostatic saturation for oxygen dynamics using steady-state analysis. We prove uniqueness and existence of the solution of the BVP for gas exchange at capillary–tissue interface as a normal physiological function. Failure of homeostatic mechanism establishes hypoxia, a new quasi-steady-state in AD. We model permeation of two-layer fluid comprised of blood and HIF-1[Formula: see text] through tunica media as a generalized [Formula: see text]-dimensional nonlinear evolution equation and solve it using Lie group of transformations method. We note that the two-layer fluid permeates the tunica media as solitary wave including solitons such as bright soliton, dark soliton, peregrine soliton, topological soliton, kink soliton, breather soliton and multi-soliton complex. Also, we introduce the main result and discuss the implications of soliton solution, using graphic interpretation, to describe the early stage of progression of AD.

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.

Hemodynamic flows are subjected to quasi-periodic velocity modulations. Currently, there is no methodology to predict the rate of progression of aortic aneurysm and dissection disorders. In this paper, the association between conformational features of the vessel wall and intra-luminal hemodynamic flows is explored to understand shear forces affecting the structural integrity of the vessel wall. The infinitesimal component of the vessel wall with cystic medial necrosis in tunica media is considered as nonlinear ballooning in the aorta with aortic aneurysm and dissection disorders. Assuming the hemodynamic system as a finite dissipative system, we frame the equation of the vessel wall of the aorta as well as lesion related to aortic aneurysm and dissection disorders. We introduce the main results for the generalized (3+1)-dimensional nonlinear evolution equation, using the Lie group of transformations method. Furthermore, we discuss the implications of traveling wave solutions and the shape, height, and width of the aneurysm to describe the impact of ballooning on hemodynamic flow in the aorta with aortic aneurysm and dissection disorders. We find that cumulative accretion of potential energy contributes to the creation of bright soliton in non-ballooned regions. The wave speed is minimum at the center of the ballooned region and maximum at the center of the non-ballooned region of the aorta. Interactions of topological defects and multi-breathers cause thrombus formation and/or turbulent flow. Interaction between breathers and standing waves is associated with thrombus formation.

Aortic dissection is a serious pathology involving the vessel wall of the aorta with significant societal impact. To understand aortic dissection we explain the role of the dynamic pathology in the absence or presence of structural and/or functional abnormalities. We frame a differential equation to evaluate the impact of mean blood pressure on the aortic wall and prove the existence and uniqueness of its solution for homeostatic recoil and relaxation for infinitesimal aortic tissue. We model and analyze generalized (3+1)-dimensional nonlinear partial differential equation for aortic wave dynamics. We use the Lie group of transformations on this nonlinear evolution equation to obtain invariant solutions, traveling wave solutions including solitons. We find that abnormalities in the dynamic pathology of aortic dissection act as triggers for the progression of disease in early-stage through the formation of soliton-like pulses and their interaction. We address the role of unstable wavefields in waveform dynamics when waves are unidirectional. Moreover, the notion of dynamic pathology within the domain of vascular geometry may explain the evolution of aneurysms in cerebral arteries and cardiomyopathies even in the absence of anatomical and physiological abnormalities.