The arterial Windkessel

Nico Westerhof 0 1 2 3 Jan-Willem Lankhaar 0 1 2 3 Berend E. Westerhof 0 1 2 3 0 J.-W. Lankhaar Departments of Physics and Medical Technology and Pulmonary Diseases, Institute for Cardiovascular Research , ICaR-VU, VU University Medical Center , Amsterdam, The Netherlands 1 N. Westerhof (&) Departments of Physiology and Pulmonary Diseases, Institute for Cardiovascular Research , ICaR-VU, VU University Medical Center , van der Boechorststraat 7, 1081 BT Amsterdam, The Netherlands 2 J.-W. Lankhaar is supported by a grant from the Netherlands Heart Foundation , the Hague, the Netherlands (NHS2003B274) 3 B. E. Westerhof BMEYE, Amsterdam, The Netherlands Frank's Windkessel model described the hemodynamics of the arterial system in terms of resistance and compliance. It explained aortic pressure decay in diastole, but fell short in systole. Therefore characteristic impedance was introduced as a third element of the Windkessel model. Characteristic impedance links the lumped Windkessel to transmission phenomena (e.g., wave travel). Windkessels are used as hydraulic load for isolated hearts and in studies of the entire circulation. Furthermore, they are used to estimate total arterial compliance from pressure and flow; several of these methods are reviewed. Windkessels describe the general features of the input impedance, with physiologically interpretable parameters. Since it is a lumped model it is not suitable for the assessment of spatially distributed phenomena and aspects of wave travel, but it is a simple and fairly accurate approximation of ventricular afterload. 1 Introduction Models are a simplification of reality which help to understand function. The arterial system has been modeled in many ways: lumped models [18, 73], tube models [8, 41, 80] and anatomically based distributed models [42, 64, 71]. In this paper we will discuss the lumped or Windkessel models. Lumped models of the venous system [67] will not be discussed. Hales (1735) was the first to measure blood pressure and noticed that pressure in the arterial system is not constant, but varies over the heart beat. He already suggested that the variations in pressure are related to the elasticity of the large arteries. Weber (Weber EH (1827) as cited by Wetterer and Kenner [80]), was probably the first who proposed comparison of the volume elasticity of the large arteries with the Windkessel present in fire engines (Fig. 1). It was Frank [18] who quantitatively formulated and popularized the so-called two-element Windkessel model consisting of a resistance and a compliance element. Poiseuilles law states that resistance is inversely proportional to blood vessel radius to the fourth power. The resistance to flow in the arterial system is therefore mainly found in the resistance vessels: the smallest arteries and the arterioles. When all individual resistances in the microcirculation are properly added, the resistance of the entire systemic vascular bed is obtained and we call this (total) peripheral resistance. The peripheral resistance, R, can simply be calculated as: Fig. 1 The concept of the Windkessel. The air reservoir is the actual Windkessel, and the large arteries act as the Windkessel. The combination of compliance, together with aortic valves and peripheral resistance, results in a rather constant peripheral flow Pven;mean =CO Pao;mean=CO with Pao,mean and Pven,mean mean aortic and venous pressure and CO cardiac output. The compliant element is mainly determined by the elasticity of the large, or conduit, arteries. It can be obtained by addition of the compliances of all vessels and is therefore called total arterial compliance. The value of total arterial compliance, C, is the ratio of a volume change, DV, and the resulting pressure change DP: C DV =DP However, it is very difficult to perform an experiment were a volume is injected into the arterial system without any volume losses through the periphery. Therefore several methods to derive total arterial compliance were developed, and those based on the Windkessel are discussed in detail below. Actually the compliance of the large arteries acts as the Windkessel, but over time it became customary to call these lumped arterial models, made up of resistance and compliance, Windkessel models. The strict separation of conduit (compliant) arteries and small arteries and arterioles (resistance vessels) is not possible, because large, compliant, arteries have small resistive properties as well and resistive vessels have, some, compliance. When accounting for R and C only, we deal with the Frank or two-element Windkessel model. The two-element Windkessel predicts that in diastole, when the aortic valve is closed, pressure will decay exponentially with a characteristic decay time RC (see below). Franks goal was to derive cardiac output. With the characteristic decay time RC, derived from the aortic pressure in diastole and an independent estimate of total arterial compliance the peripheral resistance could be calculated. Mean flow (i.e. cardiac output) is then simply mean aortic pressure divided by peripheral resistance. Frank estimated total arterial compliance from pulse wave velocity in the aorta. This example shows that Windkessel models and wave transmission of pressure in the aorta give complementary information. The Windkessel is a so-called lumped model. In other words this lumped model describes the whole arterial system, in terms of a pressure-flow relation at its entrance, by two parameters that have a physiological meaning. One cannot study phenomena that take place inside the arterial tree such as wave travel and reflections of waves, etc. It is interesting to note that in the past hypertension research focused mainly on peripheral resistance, while the contribution of total arterial compliance to blood pressure was often neglected. (The groups of Safar [58] and Westerhof [46] were exceptions in this respect, Fig. 2). In 1997, however, it was shown that pulse pressure is a major predictor of cardiovascular morbidity and mortality [3, 36]. This observation made researchers realize that arterial compliance is also of great importance, especially in old age (systolic) hypertension. The two-element Windkessel model tells us that the load on the heart consists of peripheral resistance and total arterial compliance and that both are important. 2 Improvement of Franks Windkessel: the three-element Windkessel Frank had only (aortic) pressure to base the two-element Windkessel on. The diastolic pressure, Pdia(t), in the proximal aorta with closed valves can be described by an exponential decay and the two-element Windkessel indeed predicts such a decay (Fig. 3): Pdiat Pese t=RC with Pes = end-systolic aortic pressure. In the 1930s and 1940s a number of researchers tried to improve the two-element Windkessel by adding resistance Fig. 2 A sudden decrease in arterial compliance, but constant peripheral resistance results in an increase sy (...truncated)