A Continuum Mechanics Model of Stress Mediated Arterial Growth during Hypertension Using an Eulerian Frame
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Hypertension is a medical condition in which persistent high blood pressure causes the heart to exert more energy to circulate blood through the blood vessels and can lead to life threatening conditions including stroke, heart attack and atherosclerosis. Previous attempts to model arterial growth due to hypertension have made use of kinematic growth and mixture theory models to introduce a continuum mechanics approach to the problem. In this dissertation, we are concerned with modeling arterial growth due to hypertension using a non traditional continuum mechanics approach motivated by the belief that the arterial growth taking place during hypertension is best studied in an Eulerian frame due to its ever-changing nature where one has no a priori knowledge of a “reference” state. This study has a two-fold purpose: First, illustrate how one can formulate nonlinear elasticity in the “current configuration” and, second, apply that framework to both an isotropic constitutive relation and an anisotropic Holzapfel-Ogden constitutive relation in order to model the biologically dynamic process of stress-mediated growth that occurs during hypertension. We conclude that using an Eulerian framework allows us to solve the nonlinear elasticity problem associated with growth without needing to keep track of evolving reference configurations, with the trade-off being that the formulas are more complex. Using this framework, we test the performance of two popular competing assumptions for the increase in the rate of mass production as a function of stress; namely, a continuous growth criterion and a bang-bang method. The present model for growth during hypertension assumes that growth results from a perturbation of the arterial wall stress away from homeostasis. In particular, this means that growth only occurs at points through the thickness of the wall where the stress exceeds homeostasis. It has been conjectured that such growth occurs to drive the stress back to this homeostatic stress state. The results of this dissertation give insight that suggests it is possible for the growth process to return the wall stress back to homeostasis using both the continuous criterion and the bang-bang method for growth, although the bang-bang method does so in less time.
Johnson, Maya E (2015). A Continuum Mechanics Model of Stress Mediated Arterial Growth during Hypertension Using an Eulerian Frame. Doctoral dissertation, Texas A & M University. Available electronically from