Mechanical Models of Artery Walls Chapter 3: History of Artery Wall Modelling

Piotr Kalita, Robert Schaefer

History of Artery Wall Modelling

The word artery comes from Greek, it means air pipe (aer is air and terein means to keep in Greek) as in ancient Greece it was believed that arteries carry air. This was corrected by Galen in the 2nd century AD but he still did not recognize that blood circulates and he also did not connect the pulse in arteries with the beating of the heart. The circulation of blood and the role of the heart as a pump was discovered in the 18th century by William Harvey. However, the first to consider the elasticity of artery walls and its significance in the pulsatile character of blood flow was Stephen Hales (1733), who originated the Windkessel (air kettle) theory later developed by Otto Frank near the end of, the 19th century (Otto Frank is actually the originator of the name Windkessel). In this theory, the artery walls serve as a reservoir of energy, which is stored in their deformation during the systole by the stretching of the walls and then, during the diastole, is restored to the blood, which is thereby pumped to the capillaries in a constant and non- pulsatory way (as it is from the heart to the arteries). Pure Windkessel theory leads to ODEs that do not take spatial relations into account. In particular, they do not reflect the fact that due to its wavelike character, the pulse propagates along the arterial tree and is able to reflect. Further development of artery wall models followed the development of mechanics and throughout the 19th century important contributions to the field were added by Young, Poiseuille, Hagenbach, Moens and Korteweg.

Mechanical Models of Artery Walls Chapter 3: History of Artery Wall Modelling

PDE models which take into account the dependence of variables on both space and time have been in existence since the 19th century; however, their use has been boosted by the use of computers and the increase in their computational capabilities as well as the development of medical measurement techniques from the 1940s to the present day.. (Timmons [131] claims that “we can expect highly detailed models emerging within the next few years as parallel computers become economical”). Two developments important for artery wall modeling took place in the 50s and 60s: modern nonlinear mechanics, the theoretical framework most commonly used for artery models today; and the Finite Element Method (FEM), which is used to formulate relatively simple algorithms for numerical solutions of the underlying problems. From the 60s to the 80s various models based on nonlinear mechanics were used to reflect artery wall behavior. The leading researcher in this area was (and still is) Yuan Cheng Fung, who is known as the father of biomechanics.

Mechanical Models of Artery Walls Chapter 3: History of Artery Wall Modelling

Windkessel based models, being simple and fairly accurate, remain popular for global considerations, although one has to keep in mind that they are insufficient for quantitative physiological predictions. For local approaches, on the other hand, complex cardiovascular PDE models are used. PDE models range from the 1D models in which the unknown is the cross section area of the vessel (see Sect. 4 in this article) to the most detailed 3D models, by which one can find the distribution of displacement of the wall in time and space (see Sects. 7 and 8). Often PDE systems are considered with equations of hydrodynamics for flow coupled with equations of elasticity for the wall. Such approaches are the most accurate but the underlying problems are very complex: equations are highly nonlinear, difficult in theoretical considerations, and require very large computational power for numeric solution.

Mechanical Models of Artery Walls Chapter 3: History of Artery Wall Modelling

The complex 3D models are also sensitive to parameters like the geometry of the domain, initial and boundary conditions and physical parameters of the equations and all these quantities are still difficult to determine using present measurement techniques. Furthermore, the development of models is limited by the complexity of the underlying physiology. New physiological phenomena are constantly being discovered, including sophisticated control and self-regulation mechanisms, the presence of stresses and strains in unloaded arteries (residual stresses) and the structure and function of various components of the wall which determine its heterogeneous and anisotropic behavior. Models reflecting these phenomena include a large number of parameters and one must keep in mind that fitting the models to the measurement data in such cases is susceptible to mistakes and, furthermore, it is sometimes possible that a model which is actually inaccurate can be made to fit the tests in some limited measurement regime.

Because of these limitations, the simulations which are global (i.e. involve relations between various organs of the body) use only simplified cardiovascular models (like the Windkessel or its extensions). On the other hand, local specialized models are available for specialized purposes only: for instance they are used in the design of stents and grafts as well as the prediction and treatment of atherosclerosis Among the complex phenomena which have been intensively investigated and incorporated into cardiovascular models in the last ten years are:

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