Atherosclerosis Model
2.2 Multiscale Modelling
Atherosclerosis is a lifelong systemic disease. From a modelling point of view the previous sentence can be loosely translated as, “atherosclerosis is a biological process taking place over a wide spatial and temporal domain”. In this chapter, the different processes occurring in the biological domain of the model have been described. This “biological domain” is the nutshell of the model, where all pro-cesses taking place are described in both the spatial and the temporal domain (Figure 2.13). The biological domain is very heterogeneous, spanning a range of timescales from quick molecular interactions to slow organ remodelling. These features make the biological domain extremely hard to characterise in terms of time and space, as there is a very large spatial and temporal span to cover. How-ever, each of the biological processes has to be described in the most accurate way, so, difficulties arise in defining a level of characterisation and precision that are needed to satisfy each of them.
Figure 2.13: The biological domain (atherosclerosis) as a function of the time domain, represented by the life time of a human being, and the spatial domain, spanning from molecules, to tissues, to the whole organism.
One of the complexities of modelling a biological process lies in the choice of the right scale of definition. Choosing only one scale might be too coarse and lack definition in some processes, whilst a too fine scale would lead to over loading the model with data, making it very expensive or impossible to compute. This is the reason why it is important to choose the most appropriate scale for each domain, or sub-domain, using a multiscale modelling approach.
2.2.1 Spatial Scales
The three subdomains of this model have different spatial requirements. The fluid sub-domain, the arterial lumen, models the haemodynamics inside the artery. Its spatial scale needs to capture the fluid behaviour precisely, and it is represented by
the control volume in the computational fluid dynamics (CFD) mesh (Figure 2.14) (the choice of the optimal mesh for the computed domain will be discussed in the section relative to the Fluid Domain Discretisation in Chapter 4 and 5).
The surface area of endothelium and arterial wall considered (Atot) was set accordingly to the CFD mesh element area (Table 2.1, Figure 2.14). Different spatial scales were used to model the endothelium and arterial wall sub-domains in the radial direction, as this is the direction of macromolecular transport. The endothelium was modelled as a membrane, where the transport of LDL was in-dependent of its spatial discretisation. The spatial scale of the endothelium sub-domain in the radial direction was defined by the endothelium thickness[40].
In the arterial wall, a spatial scale of µmm was used. This scale allowed a precise definition of the distribution of macromolecules inside the arterial wall to be obtained that compares well with literature data[90], as will be shown in the following chapters.
Arterial wall Endothelium Arterial lumen
Lumen radial direction
A
tot
Figure 2.14: Slice of the model artery, showing: arterial lumen, endothelium and arterial wall and the considered endothelium and arterial wall surface area (Atot).
In the radial direction, the lumen is represented by a CFD mesh element, the endothelium by an element of its own thickness and the arterial wall discretised by elements on a scale of µm.
2.2.2 Temporal scale
The different processes involved in the modelling of atherosclerosis span from seconds to years. Among these different processes,biochemical interactions in the arterial wall, such as LDL oxidation chemical cascade, are the ones covering the shorter time span of action. Such processes need very short time scales in order to be properly characterised consequently, the time scale for the arterial wall, where these processes take place, was set to seconds.
The endothelium was modelled as a quasi-steady domain. Its transport prop-erties are constant in time as long as the lumen haemodynamics are unchanged.
Being the interface between the arterial wall (intima and media) and the blood flow (a domain modelled as steady state), the endothelium links information
coming from both sides, i.e., the arterial wall and the lumen. The change in LDL flux through the endothelium is dependent on its concentration gradi-ent(Equation (2.7)), which is given by a difference in the LDL concentration between the arterial lumen and the arterial wall. The concentration gradient update happens in monthly intervals, as this is the time needed for the arterial wall model to reach a steady solution for the given boundary conditions. After stability in the arterial wall concentration is reached, the change in flux due to the concentration gradient can be calculated and the endothelium LDL solute flux changed accordingly. The update in the solute flux of LDL through the endothe-lium is entirely dependent on the change in LDL concentration in the wall so long as the haemodynamics are kept constant. A change in the haemodynamics would lead to a change in the whole transport properties of the arterial endothelium, and a change in the whole endothelial-arterial wall model behaviour.
The time definition of the model will be further discussed in Chapter 3.