Multiscale modelling of aortic dissections

Brown, Andrew (2025) Multiscale modelling of aortic dissections. PhD thesis, University of Glasgow.

Abstract

The human aorta is the main and largest artery in the human body, transporting oxygenated blood, water and vital nutrients to all parts of our bodies. An aortic dissection is an acute life threatening cardiovascular disease. It occurs when a small tear in the artery wall is pressurised by the blood flow, thus causing the tear to propagate. This process can quickly lead to death due to a restricted supply of blood flow, damage to the aortic valve or a full rupture of the artery wall. Understanding the fundamental mechanics of this process is crucial for the prevention and treatment of the disease. The current understanding of this process comes from a number of phenomenological studies which do not fully account for the role of the aortic microstructure. Our arteries are complicated three-layered structures, with each layer of the artery playing a different crucial mechanical role. The mechanical properties of each layer are due to their differing microstructures. These microstructures vary at the cellular level, with some layers being made up on laminated rows of smooth muscle cells and other layers being made up fibrous bundles of tissue. It follows that understanding the tearing process from a microscopic perspective is key to creating a mathematical model of aortic dissections.

The goal of this thesis is to develop a new multiscale model of material failure which fully explains the role of an arbitrary microstructure at a macroscopic level. To do this we first begin by introducing the damage phase field method. Using this method, we create a damage model comprising of two partial differential equations: an equation of motion and a damage evolution equation. Our model also has several physical constraints, including the irreversibility of damage and an energetic criterion for damage evolution, such constraints ground the model in the physical world. By applying this model over an arbitrary microscopic domain we can approximate the aortic microstructure as a composite material made of many different constituents. This provides us with an extremely large set of partial differential equations which one cannot practically work with. However, by upscaling this microscopic model we create a simpler macroscopic model comprising of an equation of motion and a damage evolution equation. This is done by employing asymptotic homogenisation and an assumption of local periodicity at the micro scale. With careful analysis we can fully upscale all the microscopic modelling constraints to the macro scale. Our macroscopic model accounts for the microscopic material properties and geometry by encoding them in a series of effective macroscopic coefficients. These coefficients themselves are calculated by solving a series of local cell problems relating the microscale to the macroscale.

For completeness, we introduce numerical methods for solving our damage phase model. We apply these numerical methods to both our microscopic and macroscopic models of material failure. By solving our models in one-dimension we perform a parametric analysis of both models. This analysis allows us to demonstrate a good agreement between the microscopic and macroscopic models, as one would expect. We also demonstrate that the microscopic model converges to the macroscopic model as the ratio of the microscopic and macroscopic length scales tends towards zero. Next, we use the finite element method to create a numerical method for solving our damage model in higher dimensions. To illustrate the utility of our model we consider a set of toy simulations. We achieve this by simulating a trouser test, where a rectangular configuration is clamped at one end and at the other end we pull the material apart. This action essentially creates a tear through the material resulting in a final configuration that looks like a pair of trousers. By performing a trousers test we are able to convey how our multiscale model can be used to study the effects of small changes at the micro scale. We find that these small changes can have a large effect on the outcome of the tearing process at the macroscale.

It follows, then that our developed multiscale model of the damage phase field method will be a valuable tool for further study into how changes in a materials microstructure can affect material failure. Throughout the thesis we remark upon how this model can be applied to the phenomena of aortic dissections. However, before this application can be performed it is commented upon how we require more data about the aortic microstructure to be collated. With this information we can calculate the appropriate ranges of effective coefficients describing the aorta. Allowing us to accurately study a wide variety of physical situations leading to aortic dissections.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Funder's Name: Engineering and Physical Sciences Research Council (EPSRC)
• Supervisor's Name: Hill, Professor Nicholas, Roper, Dr. Steven and Penta, Dr. Raimondo
• Date of Award: 2025
• Depositing User: Theses Team
• Unique ID: glathesis:2025-85239
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 06 Jun 2025 17:33
• Last Modified: 06 Jun 2025 17:35
• Thesis DOI: 10.5525/gla.thesis.85239
• URI: https://theses.gla.ac.uk/id/eprint/85239