Fig. 10.1
Overview of the operation of a model of heart valve physiology, based on a Computational Fluid Dynamics (CFD) work flow developed by EurValve (Horizon 2020, Project Number 689617)
The model is a set of mathematical equations. These equations are relevant to the question the model has been designed to answer. They describe the governing dynamics (e.g. stress–strain behaviour of the wall, shear–shear-rate behaviour of the blood). In the real-world things exist in four dimensions; x, y, z and t. The model may also exist in four dimensions but lower dimensional models might be able to represent the system with sufficient accuracy, and might even produce additional insight that is obscured in a more complex model. There is some inconsistency in the description of the dimensionality of models. Generally 0D, 1D, 2D and 3D indicate the number of spatial dimensions in the model. For 0D models variation in time is implicit, for the higher dimensions, transient analyses are sometimes described for example as 3D + t and sometimes as 4D. There are even references, particularly in the medical imaging community, to higher dimensions such as ‘7D’. This refers to the three-dimensional field of the three components of velocity, changing in time.
Additional constraints relevant to the problem being solved are needed which are applied to the model in order that a solution can be obtained from the equations which make up the model. It is useful to separate these additional constraints into ‘boundary conditions’ and ‘input data’. For example, in a 3D flow model one might be interested in the variation in wall shear stress for different input flow waveforms. In this case the boundary conditions would be the 3D geometry and the pressure which would be set the same for all simulations. The input data would be the flow-time waveform which would vary from one simulation to the next.
The output data is the data which arises from the simulation. For the example above this would be a 3D time-varying dataset of blood velocity from which wall shear stress could be calculated.
10.2 Zero Dimensional Models
Zero dimensional (0D) models are a reduced order abstraction of reality. In these models it is assumed that there is no spatial variation of a quantity within any individual compartment of the model. The system within any compartment is described by a series of ordinary differential equations, with only time derivatives. Zero-D models can be powerful tools to describe system-level interactions between components. Characteristics such as ventricular pressure-volume loops, systolic-diastolic pressure ratios, temporal pressure gradients, cardiac output, ejection fractions, ventricular work, etc., can all be computed using 0D models. The most commonly used 0D model of a chamber of the heart is the variable elastance model. In this model, a relationship is described between elastance, which is specified as a function of time, chamber volume and chamber pressure. The elastance terms describe the active contraction of the heart. The 0D vascular components lump all of the inertia, resistance and compliance of a portion of the vasculature into simple electrical analogue representations. A very simple model of the left side of the heart and the systemic circulation, including a snapshot of the model representation from the CellML repository (www.cellml.org), is presented in Fig. 10.2. Despite its simplicity, a typical implementation of this model actually has 23 input parameters. It is a real challenge to select appropriate parameters and parameter ranges for the study of the mechanics of the system, and an even greater challenge to personalise them to represent an individual.
Fig. 10.2
Simple model of the left side of the heart and the systemic circulation (‘Westkessel’), including a snapshot of the model representation from the CellML repository (www.cellml.org)
One important class of applications of 0D models includes the representation of changes under prospective interventions. As the human system is not passive, but is actively regulated by both central and local control systems, it is necessary to model control mechanisms if the model is to accurately represent the effects of interventions (pharmacological or surgical), or changes of physiological state (e.g. exercise). Such control models can describe regulation, neural signals to control pressure, volume and flow fluctuations, physiological and interventional changes, as well as haemorrhage. The CellML repository contains a wealth of curated models that are freely available for download. For further reading on all the above aspects of 0D models of the cardiovascular system, please see Shi et al. (2011).
10.3 One Dimensional Models
One dimensional (1D) models are useful to capture gross features of the circulation and are particularly useful downstream of the heart. They are described by partial differential equations that relate pressure and the axial component of velocity and their spatial and temporal derivatives. The 1D equations can be derived directly from the full 3D Navier-Stokes equations (see for example Landau and Lifshitz 1959), represented in polar co-ordinates, by making the assumption of axisymmetry. The characteristics of these equations are studied in detail by Canic and Kim (2003), who show for example that, if the radius of the vessel is small relative to a characteristic length, the pressure is constant over the radius of the vessel. This is an important practical result for flow in arteries. A radial distribution of the axial velocity is assumed, the form of which depends on the flow regime that is to be represented. One-D models are particularly useful to capture wave transmission in the cardiovascular system without the computational expense of a 3D model. Wave transmission arises from the elasticity of the vessels and their capacity to store fluid as they are pressurised.
A 1D representation of the circulation can be important in the use of system models that characterise, for example, cardiac load under a range of physiological conditions, or the impact of disease on the elevation of cardiac and vascular pressures. An example of the application of state-of-the-art branching tree models to the physiology of the pulmonary circulation is presented by Qureshi et al. (2014). In some applications, particularly in the context of coronary and pulmonary physiology, 1D models are used to describe and to characterise the relationship between forward and backward travelling waves in the circulation. Analysis might be conducted in the frequency domain or in the time domain: for the latter the concept of wave intensity analysis, reviewed by Hughes et al. (2008) is a useful device. The healthy system maximises the efficiency of blood transmission and minimises the load on the heart by matching the impedance of different branching vessels. In disease, this impedance matching can be diminished, with measureable differences in power of the forward and backward travelling waves. For a comprehensive review of 1D models including theoretical considerations, see van de Vosse and Stergiopulos (2011).
10.4 Three-Dimensional Models
Three-dimensional (3D) models aim to incorporate 3D geometry. A 3D model requires geometry, which can either be created using computer aided design (CAD) software, or can be obtained from medical images (described in Chap. 11). For the heart and the larger arteries and veins, blood is usually assumed to be a fluid continuum, ignoring its microscopic make-up, and incompressible. Under these conditions there are:
Two fundamental variables of interest, pressure (p) and velocity (v), and, since velocity is a vector with three direction components (u, v, w), there are four degrees of freedom (p, u, v, w) that together describe the physical state at each point within the fluid domain.
Two governing equations accurately describe the variation of the fundamental variables (p, u, v, w) throughout the domain, given material properties and boundary conditions. The first is based on the conservation of mass and the second is based on the conservation of momentum. Since the velocity has three component directions, there are three momentum equations. The equation of conservation of mass is often called the continuity equation and the equations of conservation of momentum are called the Navier–Stokes equations.
10.4.1 Rigid Wall Models
A rigid wall model is one in which the vessel walls do not move during the cardiac cycle. This is opposed to the reality in vivo where the vessel wall in an artery typically changes in diameter by 5–10 % during the cardiac cycle. Rigid wall analysis is suited to applications where the purpose of the model is to evaluate flow characteristics that are not strongly influenced by shape changes. It is a necessary, but not sufficient, condition that the fractional volume change of the domain of interest over the cardiac cycle is small. Examples in which the rigid wall assumption might be adequate include the modelling of flow separation and of vortical structures, wall shear stress distributions in the region of bifurcations, aneurysms, stenoses or anastomoses. In particular, wall shear stress in arteries is relatively insensitive to the motion of the vessel wall, and therefore, it is often adequate to assume a rigid wall model.
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