Atherosclerosis remodelling initialisation cycle

As already mentioned in the previous chapter, the Navier-Stokes equations, de-scribing the haemodynamics in the arterial lumen for the given geometry and boundary conditions, are solved in Ansys CfX v.14. Once the solution for a given geometry has been computed, the variables of interest are extracted from the CFD model and saved in an Excel file. For a surface of n nodes, the data would

∗http://www.mathworks.it/products/matlab/

†http://office.microsoft.com/en-gb/excel

be exported in matrix form with each string representing a different variable:

where the first column is the node number, from second to the fourth column x, y and z coordinates of the node respectively, from the fifth to seventh column the projection of the normal vector in the x, y and z direction and the last column the WSS value at that nodal location.

Each variable stored in Equation (3.1) is individually processed in Matlab.

The WSS nodal values are extracted from this matrix and selected following a specified criteria. The nodal number (N_1:n) is used as an identifying index for each of the extracted WSS values. If the WSS value at the nodal location is greater or equal to the Poiseuille WSS value for the given arterial branch as calculated in Equation (2.4), the node is discarded, and no arterial wall displacement (∆h, Equation (2.42)) is assigned to that node.

If the criterion is satisfied and the extracted WSS value is less than the calcu-lated WSS Poiseuille value (Equation (2.4)), the node having that WSS value will be selected. This WSS value is used as the lumen side condition (IV in Figure 3.1) for the endothelium and the arterial wall model, implemented in Matlab.

Following the WSS selecting criteria, a new matrix of selected surface nodes are formed:

where the first column is the node number, representing also the node index in the parent matrix (Equation (3.1)). The second column is the node positioning

index in the current matrix, the third to fifth column are the x, y and z coor-dinates of the node respectively, the sixth to eight column the projection of the normal vector in the x, y and z direction and the last column the WSS value on that surface node. Matrix (3.2) is a sub-matrix of (3.1), as it is the product of the element selection performed on the Matrix (3.1). WSS based node selective criteria, identifies the atherosclerosis-prone areas. Narrowing the area of interest allows for reduced data handling.

Once the lumen side boundary conditions have been set, the endothelium-arterial wall model can be implemented in Matlab. One of the assumptions of the model is that the transport of species occurs in the radial direction (direction normal to the lumen surface) only, hence, there is no exchange of information between neighbouring nodes (longitudinal direction) (Figure 2.11). The Matlab model of the endothelium and the arterial wall can be solved for each node sep-arately. For increased computational efficiency, the selected surface nodes of the CFD model with their section of endothelium and arterial wall model are clus-tered in packages and solved in parallel. For parallelisation purposes a remote cluster (UCL Computer Science Cluster, total of 2500 cores, typical node with dual socket E5-2620, 48GB of memory and 128GB SSD (solid-state drive)) was used. The Matlab implementation would have to handle a much larger number of nodes at once without parallelisation. It follows that this parallelisation allows for a further decrease in the computational power required for this model.

The result of the endothelium and arterial wall model is an arterial wall growth value for each one of the CFD surface nodes considered. This growth value represents the thickening of the arterial wall and will result in a deformation of the arterial surface to accommodate the growth in IMT (∆h), as calculated from Equation (2.42). The displacement (∆h) value will be imposed on each of the surface nodes of the arterial lumen model. An increase in the arterial wall thickness would result in a displacement of the arterial lumen surface nodes towards the centre, as atherosclerosis formation leads to a decrease in the arterial lumen diameter.

Using the indexing number (N1:n) the displacement (∆h) corresponding to each one of the surface nodes for both the pro-atherosclerotic areas and for areas

for the surface nodes. The following matrix will be created:

Where N_1:n is the indexing number, the second to fourth entry are the node coordinates, the fifth to seventh entry are the nodes normal vector projections, and the last entry is the node displacement.

The information stored in Matrix (3.3) is used to calculate the nodal displace-ment to be imposed on the surface nodes of the CFD lumen geometry, using the point translation technique (Figure 3.4). For a single node, the new position coordinates X_new, Y_new and Z_new are given by:

X_new = X ± ˆnx· ∆h (3.4)

Y_new = Y ± ˆny· ∆h (3.5)

Z_new= Z± ˆnz· ∆h (3.6)

Where X, Y , Z are the nodal coordinates of the initial geometry and ˆn_x· ∆h, ˆ

n_y· ∆h, ˆnz· ∆h, are the displacement vectors with magnitude ∆h and direction

±ˆnx, ±ˆny, ±ˆnz. Applied to the Ansys CfX v14 model geometry surface, the new coordinates of the nodes are given by:



Figure 3.4 shows the mesh deformation technique used for modelling lumen-imposed deformation caused by the arterial wall growth. A node (Figure 3.4a) is translated to a position at a distance ∆h in the direction normal to the node (Figure 3.4b). Considering the node as part of the surface mesh (Figure 3.4c), the mesh deformation coming from the node translation can be visualised in Figure 3.4d.

(a) (b)

(c) (d)

Figure 3.4: a. Node of the surface mesh represented in its x, y, z spacing with the projection of the normal vectors in the x, y and z directions. b. Translation of the node along a vector with magnitude ∆h in the direction normal to its surface.

c. Node located in a portion of the surface mesh grid. d. Displacement of that node downwards (towards the arterial lumen) and remodelling of the grid around it.

The node translation calculated can be imposed onto the surface nodes of the Ansys CfX v.14 model.

In order to instruct Ansys to perform the correct mesh deformation the

fol-lowing matrix needs to be utilised:

Where the first three entries are the nodal coordinates and the fifth to seventh column are the new x, y and z coordinates that the nodes will have after the translation has been performed.

As mentioned at the beginning of this section, the end of an atherosclerosis remodelling cycle is marked by the arterial wall deformation. In this case, the deformation marks the end of the first cycle, or initialisation cycle. To characterise the plaque development further in time, after the initialisation cycle has finished, another cycle will start. This is represented in Figure 3.3 by the green framed part of the workflow.

Once the atherosclerosis remodelling cycle has ended, the endothelium and arterial wall models are temporary interrupted as the wall growth threshold cri-terion (∆h^∗) has been reached (Figure 3.1). Among the surface nodes, if at least one had a wall displacement value that satisfied the criteria:

∆h ∼= ∆h^∗ (3.9)

where ∆h^∗ is 1% of the artery radius. The developed growth threshold crite-rion has been satisfied and the CFD geometry is remodelled accordingly.