Simulation of the pulmonary circulation: model comparison and implementation

Pulmonary hypertension is a severe and incurable disease. It can even lead to heart failure. Thus, it is important to diagnose this type of disease as early as possible. Since blood pressure in the pulmonary circulation can only be obtained via invasive measurements, which is not an easy task, mathematical models for blood pressure and flow in this system are of great importance. Also in terms of investigating pressure and flow waveforms with respect to find significant parameters for distinguishing several disease stages of pulmonary hypertension, mathematical models are used. In this thesis three different approaches of modelling the pulmonary circulation, which are most commonly applied for simulation in literature, are investigated. The mathematical formulation of each of these modelling approaches is derived in more detail and also some examples of applications in literature are given. The first modelling theory that is considered is the so called Windkessel model. It can be described by an ordinary differential equation of first order and is often related to an electrical circuit. It is a lumped parameter model and hence does not take spatial distribution of pressure and flow into account. The second model that is examined is based on the one-dimensional linearised Navier-Stokes equations, also referred to as Womersley equations. They are derived from the Navier-Stokes equations for Newtonian fluids. An advantage of this approach is, that it also considers spatial distribution and the possibility of wave reflections. The last approach taken into account is the system of one-dimensional non-linear Navier-Stokes equations. They are commonly solved numerically by finite element schemes and by using structured trees as outlet boundary conditions. Thereby, especially the pressure and flow waveforms in the small arterioles, capillaries and small venules can be simulated by the structured trees. Furthermore, in terms of application of the Windkessel modelling method to the pulmonary circulation, an RC-tree model was implemented. The RC-tree model is a branching tree model based on 2-element Windkessel models for each vessel. For simulation 17 generations of pulmonary arterial vessels were considered. In the model the generations are connected in series while vessels within a specific generation are seen as parallel. The parallel vessels of one generation have the same resistances and compliances. This RC-tree modelling method is especially used to simulate the effects of different degrees of occlusion in different generations of the considered branching tree. Comparing the mathematical formulations with one another leads to significant differences of the models. The most remarkable distinction of the two methods based on the NavierStokes equations is that for the Womersley equations stronger assumptions are needed than for the one-dimensional non-linear Navier-Stokes equations. The main difference between the onedimensional models (one-dimensional linearised and non-linear Navier-Stokes equations) and the Windkessel modelling method is that the latter is a lumped parameter model and that