Cardiovascular Modeling: Physiological Concepts and Simulation

Abstract

Modeling concepts are utilized whenever relationships are described between two or more quantities. One of the most elementary cardiovascular models is the relationship between mean arterial blood pressure and mean arterial blood flow in a vascular bed, described as vascular resistance. For simulation purposes, the electrical analogy is conventionally used where voltage (V) is analogous to pressure (P) and current (I) is analogous to flow (Q), such that the equivalent ohmic resistance (R) is obtained from the relationship P = Q.R. This analogy to Ohm’s Law (V=I.R) implies a linear relationship, such that R can be determined for any value of P and Q. However, the physiological constituents of R, that is, vascular geometry and blood viscosity, are also functions of P and Q. Hence the relationship between P and Q is inherently non-linear, with the degree of nonlinearity depending on the anatomical location along the arterial tree and so the value of velocity of blood flow. This concept applying to steady values of P and Q is extended to time varying signals, P(t) and Q(t), where steady state oscillations are described in the frequency (ω) domain such that the relationship between oscillatory pressure (P(ω)) and flow (Q(ω)) is vascular impedance (Z(ω)), and where R is the zero-frequency value of Z(ω).

The non-linearities are also inherent in the impedance model, since the elastic properties of the arterial wall makes constitutive parameters such as wave speed and vessel diameter pressure dependent. However, in the range of pressure excursions during the cardiac cycle, the effects of non-linearity are relatively small, hence allowing closed form expressions of impedance based on vascular and blood properties, such that the input impedance spectrum can describe the complete hemodynamics of the vascular bed. When applied to the ascending aorta or the pulmonary artery, it describes the dynamic load on the left and right ventricles respectively. When the impedance model is applied to distributed structures such as branching vascular trees, it can be used to investigate underlying concepts related to optimal functions determined by allometric relationships of cardiovascular parameters (eg heart rate) and body size. Simulation of wave propagation in arterial models can be used determine factors that contribute to the change in pulse waveform throughout the arterial tree and inclusion of non-linear properties, such as pressure-dependent elasticity, can simulate changes in arterial hemodynamics due to gravitational effects on arteries, as occurs with changes from supine to upright posture.