Cardiovascular Engineering: An International Journal, Vol. 3, No. 2, June 2003 ( C  2003) TECHNICAL NOTE The New Apparent Compliance Concept as a Simple Lumped Model DAMIAN CRAIEM ∗,†,§ and RICARDO ARMENTANO ∗,† Apparent compliance appeared as a new concept to unify the ancient controversy that the lumped and distributed schools face to model the cardiovascular system. Perhaps its undeniable contribution, some important factors as wall vis- cosity were diminished. High frequency low-pass response was explained as a reflection phenomena, when a simple model with conceptual components can perfectly fulfill the apparent compliance attenuation behavior. Key words: apparent compliance; wall viscosity; windkessel; reflections. INTRODUCTION Recently, a new transfer function has been devel- oped to unify the ancient discrepancy between the two competing schools that model the arterial system (Quick et al., 1998). In the “transmission line school” the arterial system is modeled using distributed constants and con- cepts such as input impedance, characteristic impedance, finite pulse wave velocity (PWV), and the reflection index to name a few. The “lumped school” centers its analy- sis in the windkessel circuit where infinite PWV is as- sumed and the whole system is presented as a compliant chamber (large arteries), identified with a capacitor (C tot ), and an output load (small arteries) modeled with a re- sistor ( R p ). Both models try to characterize the arterial system and give physiological meanings to their param- eters, using a key concept: “analogy”. Simplicity is al- ways welcomed. Naturally, both have their weaknesses. ∗ Universidad Tecnol´ ogica Nacional. FRBA. Medrano 951 (C1179AAQ). Buenos Aires, Argentina. † Universidad Favaloro. Belgrano 1723 (C1093AAS). Buenos Aires, Argentina. § To whom correspondence should be addressed at Ing. Damian Craiem, Universidad Favaloro, Belgrano 1723 (C1093AAS), Ciudad de Buenos Aires, Argentina. E-mail: dcraiem@favaloro.edu.ar The “lumped school” is very simple and concentrate the main circulation components (total compliance and pe- ripheral resistance concepts) into constant elements but fails when the reflected wave effect has to be explained. The “transmission line school” shares a natural and elegant analogy with the electromagnetic wave propagation theory and has the impedance as a powerful conceptual compo- nent. Finite PWV allows reflections to be explained easily although some concepts, as the characteristic impedance, remain physiologically unclear. The apparent compliance (C app ) is presented as a so- lution to unify both models. Using the conservation of mass concept, the C app is defined as the relationship be- tween volume stored (V ) to the input pressure ( P in ) in the frequency domain: C app (ω) = V (ω)/ P in (ω). Furthermore, apparent resistance is defined as the quotient between P in and the output flow ( Q 0 ): R app (ω) = P in (ω) Q 0 (ω), where venous pressure is neglected. Both apparent components are allowed to be modeled using linear time-invariant transfer functions. Following Quick’s et al. analysis, these concepts are integrated, expressing C app in terms of R app and input impedance Z in (ω) = P in (ω)/ Q in (ω) as C app (ω) = V (ω) P in (ω) = R app − Z in j ω R app Z in (1) These transfer function relates the stored volume to the input pressure and has two remarkable properties: (1) it is based, as the windkessel, in the conservation of mass theory and 2) it includes a distributed parameter ( Z in ) al- lowing the lumped and the distributed schools to cohabit. A simple lossless distributed model with a three element windkessel (3WK) load is proposed by Quick to calcu- late the C app and analyze the arterial system (Quick et al., 1998). 81 1567-8822/03/0600-0081/0 C  2003 Plenum Publishing Corporation