It has been noted in previous chapters that there are governing equations which describe the behaviour between strain and stress in a solid and between strain rate and shear rate in a fluid. There are a very few geometries for which exact solutions of these equations exist. For a fluid, the governing equations are the ‘Navier–Stokes equations’. Exact solutions of the Navier–Stokes equations may be found for simple geometries such as motion of an infinite plate and steady or pulsatile flow in a cylinder. For more complex geometries, computational mechanics provides a framework to allow the governing equations in a fluid and a solid to be solved. The mathematics of computational mechanics is complex and outside the scope of this book, but is available in standard texts and review articles for interested readers (Zienkiewicz 2004; Zienkiewicz et al. 2005; Chapra and Canale 2014). The basic steps involved in computational mechanics are detailed below:

Digital computation. The governing equations are applicable to a continuous media (i.e. all x, y, z, t). Computational mechanics operates using a digital or discrete model in which equations are solved at many specific values of x, y, z, t. This digital model is then suitable for calculation using a computer. The set of points is referred to as the ‘mesh’ or ‘grid’. The smallest unit of the mesh is called an ‘element’ which consists of a number of nodes. Each node of the mesh has a number of values related to the variables in the governing equations.

Estimation of flow field data using computational mechanics is referred to as ‘computational fluid dynamics’ or CFD. Commonly the finite difference and finite volume discretization methods are used in CFD software packages. Estimation of stresses in solids usually involves finite element discretization, so that solid modelling is usually referred to as ‘finite element analysis’ or FEA. A number of commercial packages are available for flow and solid modelling including CFX and Fluent (ANSYS, Canonsburg, PA, USA) and Abaqus (Dassault Systemes, Simulia, Rhode Island, Providence, USA). For use in blood flow, several groups have developed their own CFD packages (for example, Minev and Ethier 1998; Ethier et al. 1999; Sherwin and Karniadakis 1995; Witherden et al. 2014).

The starting point of the PSM processing chain involves the acquisition of medical imaging data. Medical imaging systems were described in Chap. 9, where it was noted that a number of imaging modalities provide high-resolution 3D data. The medical imaging dataset provides 3D data on the tissues of interest in the patient. In the case of arteries this will be the arterial wall, any thrombus (e.g. present in clinically relevant aneurysms) and the vessel lumen. The ideal imaging modality for PSM should have the following features: high resolution, low noise, artefact-free, high contrast between tissues. A brief summary of the use of different imaging modalities in PSM is described below:

Magnetic resonance imaging (MRI). Image contrast is high which enables different soft tissues to be distinguished. The good soft tissue contrast also enables some visualisation of the aortic wall, which is often not possible on CT. For use in atherosclerosis, MRI has been used to acquire 2D and 3D data in atherosclerotic plaque, where its excellent soft tissue discrimination enables visualisation of the different plaque components. However, MRI has a number of limitations for PSM. Acquisition times for 3D data can be long. For data acquired from the thorax and upper abdomen, the patient must hold their breath to limit the displacement of the chest. Image registration tools are required to register the images together and remove this motion. MRI has much larger slice intervals compared to CT, typically around 5–6 mm for abdominal imaging protocols. So although, MRI has excellent in-plane (x, y) pixel resolution, <1 mm, the resulting geometries may lack detail in z-direction and be unsuitable for PSM.

Ultrasound. In principle, 3D ultrasound data may be acquired using externally applied transducers. However, in practice, there are problems associated with registration, low resolution, loss of data due to calcifications and bowel gas (Hammer et al. 2009). These problems are mostly resolved by the use of intravascular ultrasound where the transducer is much higher frequency (resulting in much improved spatial resolution of 50–100 μm), and where the transducer images the tissues from inside the vessel. IVUS has been used to provide 3D geometries for PSM since the very early days of PSM (Chandran et al. 1996; Krams et al. 1997). The position and orientation of the IVUS scan-plane needs to be known so that the IVUS data can be positioned within a 3D geometry. Krams et al. (1997) used an angiography system to obtain this information; the overall system was known as ANGUS (ANGiography and UltraSound).

Optical coherence tomography (OCT). This is also an invasive technique which can be used for imaging arteries and has very good spatial resolution of 10–20 μm. It was noted in Chap. 9 that the critical thickness of the cap in atherosclerotic plaque is around 70 μm, far below the resolution of CT, MRI or transcutaneous ultrasound, and the only technique capable of providing accurate measurements of cap thickness in vivo is OCT.

Segmentation is the process whereby the surfaces of the organ of interest are identified. Segmentation may also involve defining the boundaries between different regions in the organ. Typically for fluid modelling the inner lumen of the vessel is required. For solid modelling ideally both inner and outer lumen of the vessel wall should be identified. For abdominal aortic aneurysms the region of thrombus is required. For atherosclerosis, the regions need to be identified corresponding to different plaque constituents. Segmentation concerns the detection of edges in an image and is commonly used in 3D imaging for visualisation of structures. For example, in CT imaging in the Radiology Department, the soft tissues can be ‘peeled back’ to reveal underlying organs, skeleton etc. Segmentation can be manual, automated or semi-automated.

Threshold methods. These are the simplest automated segmentation methods and involve looking for differences in intensity values between adjacent voxels. Figure 11.2 shows a threshold-based method in operation for identification of the inner lumen of an abdominal aortic aneurysm in a CT dataset. The algorithm starts in the middle of the lumen and works outwards radially. When the intensity value exceeds a threshold value, the edge has been found. These simple methods work best when image noise is low. Some commercial software such as Mimics (Materialise, Belgium) allows manual definition of the organ boundaries which are then smoothed to produce a 3D surface mesh suitable for modelling.

Deformable models. These are often referred to as ‘active contours’ (2D), ‘active surfaces’ (3D) or ‘snakes’ (Xu et al. 2000). These provide a widely used and powerful segmentation technique. The snake is a connected set of points which can deform dependent on the local image content. Typically an initial contour is seeded and grows outwards until it reaches the organ boundary. Local image measures which have high values where there is an edge are used to constrain the snake. The snake will often be ‘trained’ from learning datasets to exhibit some sort of geometrical behaviour, e.g. when segmenting the left ventricle the snake knows roughly what shape a typical left ventricle is and this is used as a constraint in the process. The result from this approach is a robust 3D reconstruction with good reproducibility.

Prior to meshing, the surfaces obtained from segmentation must be prepared. For flow modelling, the blood is only in contact with the inner surface of the vessel. In this case, a single layer surface will suffice. For solid modelling in a vessel both an inner and outer surface are required. Due to imaging constraints, especially in CT, it is often not possible to identify the outer surface of the vessel. In this case, it is commonly assumed that the vessel has a particular wall thickness. In the case of abdominal aortic aneurysm, it is common to assume that the wall has a constant thickness of 1.9 mm (Raghavan et al. 2000), or that the wall thickness varies between 1.5 and 1.13 mm at thrombus-free and covered sites respectively (Gasser et al. 2010). The surfaces of resulting geometries must be smoothed to remove artefacts of the reconstruction algorithm and to ensure that surfaces do not cause undesirable issues during the meshing or modelling stages.

Meshing is the process where the 3D segmented geometry is divided into many elements (Figs. 11.3 and 11.4). Mesh generation is one of the critical parts of the patient specific modelling process. Increased solution accuracy is produced with a larger number of elements, but at the expense of increased processing time. The total number of elements employed for a given model is a balance between solution accuracy and processing time. While the mesh is generated using a specialist computer programme, the operator has considerable input in defining element types and element sizes. For example, when there are large velocity gradients or stress gradients then the mesh density needs to be higher. The process of mesh optimisation is therefore essential and involves adjusting the local mesh following examination of the solution. In practice, several iterations between mesh and solution may be required.

Different types of element may be used. Common elements are tetrahedron (shaped like a pyramid) or hexahedron (shaped like a brick). Meshes may involve both types of elements. Hexahedral elements are generally more suited to structures with straight edges, hence in PSM the use of tetrahedral elements is more common. For CFD, some codes such as STAR-CCM+ (CD-adapco Group) use polyhedral elements which offer improved computation time over tetrahedral elements, while maintaining the flexibility of tetrahedral elements to mesh complex geometries.

The next stage is computational modelling in which the solution is produced after several iterations. A number of different modelling regimes can be adopted. Computational fluid dynamics can be performed using a rigid-walled approach, which is simple and sufficient for providing output data on basic haemodynamics. A moving-wall method can be adopted with input of moving-wall geometry data. Solid modelling is used for estimation of tissue stress. Solid modelling alone is suitable for estimation of tissue stress in abdominal aortic aneurysms and for 2D studies in atherosclerotic plaque. Combined solid-fluid modelling is called fluid structure interaction or FSI. This is needed when the pressure distribution is not uniform within the 3D geometry and is essential for 3D studies of tissue stress in stenosed arteries.