This model is proposed as an exemplar application and represents an initial ap-proach towards a more “integrated” modelling of atherosclerosis. Different as-pects of atheroma formation, involving haemodynamics and the transport and biochemical interaction of species were considered, leading to a multiscale frame-work of the overall process.
The haemodynamics of the problem was solved with the help of finite volume computational fluid dynamics software, Ansys CfX v.14. In areas where WSS is extremely low (less than 5 dyn/cm2)[36], the endothelial cells become “leaky” and show higher permeability to macromolecules. The endothelial layer was modelled as a membrane, where the membrane matrix is constituted by the endothelial cells and the pores of the membrane by the intra-cellular junctions, which would allow the passage of LDL through them when “leaky”. Transport through this membrane was modelled with a shear-dependent modified version of the
Kedem-Migration of LDL from the blood stream to the arterial wall leads to an accumulation of LDL macromolecules inside the intima-media. A Convection-Diffusion-Reaction equation models the transport of LDL inside the arterial wall.
Interaction of LDL with ROS in the arterial wall leads to oxidation of LDL, mark-ing the initial stage of atherosclerosis formation[133]. Once oxidised, LDL trig-gers the activation of cytokines and signalling proteins, recruiting blood-stream monocytes. These monocytes automatically become macrophages once inside the arterial wall and their distribution follows a Diffusion-Reaction equation. The fi-nal stage of this biochemical model is the interaction of macrophages with oxLDL, leading to the formation of foam cells.
Stratifications of foam cells are the core constituents of the initial atheroma.
The growing atheroma has an impact on the overall model only when the swelling appears in the arterial wall. This swelling alters the haemodynamic patterns and redefines the behaviour of endothelial cells to LDL transport, initiating a new cycle with different transport boundaries and consequently different atheroma formation.
This integrated model allows description of the major phenomena that form the basis of atherosclerosis both separately, for a more detailed characterisation, and together to understand the dynamics of their interaction. As will be dis-cussed further in the following chapter, great effort was devoted to make this model fast and versatile to implement, allowing for various hypotheses such as different model variables (i.e. different mean blood LDL concentrations) and ar-terial geometries (from idealised geometries to patient-specific geometries) to be easily tested.
Testing of various hypotheses was fundamental, as the information available in the literature (used to validate and test models of atherosclerosis) is often heterogeneous in terms of methods, quality and outcomes and often corresponds to simplified in vitro experiments when presenting only partial aspects of the problem. A valid example to illustrate this argument is the function linking en-dothelial cell Shape Index to local values of WSS[32, 83]. In vitro experiments on human endothelium show that the endothelial cells did not change their mor-phology for more than a shape index of 0.58 in post-stenotic areas[134]. The majority of experimental data is based on experiments on non-human endothelial
cells including dog endothelial cells[32]. The SI reaction shown by these cells in a similar haemodynamic environment is 18% more severe in comparison to human cells[134]. The difference in these results might arise from different experimental conditions and inter-species differences as, canine endothelial cells and human endothelial cells might react differently to given haemodynamic stimuli[35]. The issue raised by SI data could be applied to many of the values upon which this and the majority of models in the literature are calibrated. It is beyond the purpose of this thesis to discuss the validity of experiments done on non-human speci-mens in non-physiological environments, but is worth considering them among the limitations of this type of model.
2.4 Conclusion
A mathematical model of atherosclerosis formation was presented in this chap-ter. The model is based on a continuum, conservative approach. The artery modelled was divided into three sub-domains, arterial lumen, endothelium and arterial wall. Each of these sub-domains had their own spatial and temporal scale of definition. As the finer spatial scale adopted was deemed large enough with respect to the molecular size of the agents considered, the agents were treated macroscophically and described in terms of their concentration. Mass transport across the arterial wall was coupled to the fixed lumen concentration with the help of the semi-empirical relations of Kedem and Kaltchasky, for transport across the endothelium. The change in concentration inside the arterial wall over time was modelled with the help of convection-reaction (LDL) and diffusion-reaction equations (monocytes/macrophages). The diffusion driven concentration gradient was modelled with the mathematical and observational Fick’s law. When considered, the concentration change due to convection was driven by pressure gradient across the arterial wall following Darcy’s law. The chemical reactions leading to LDL oxidation and foam cell formation were modelled using the law of mass action. Agents within the considered sub-domains were modelled with the help of these well understood and widely used formalisms, derived from exper-imental findings and in-vitro observations. Discretising the very heterogeneous
of atherosclerosis in its core processes allowed for a more detailed modelling. As atherosclerosis is ultimately the product of these processes and their interactions, a holistic modelling approach was needed to create an integrated atherosclerosis model. The subject of the next chapter is this integrated atherosclerosis model called the Atherosclerosis Remodelling Cycle.