Cycle-averaged dynamics of a periodically driven, closed-loop circulation model☆
Introduction
Over the past 30 years, computational models of cardiovascular function have become abundant in both basic research and teaching, with increasingly sophisticated models becoming available at any biological size and time scale. At the system level, time-varying ventricular elastance models have proven to be useful representations of the right and left heart (see e.g., Sunagawa & Sagawa, 1982). When coupled to appropriate models of the peripheral systemic and pulmonary circulations, such models allow for simulation of realistic pulsatile, quasi-periodic pressure and flow waveforms. The dynamics of these models, however, are quite complex, representing cardiovascular physiology at a variety of time scales that include very fast dynamics (such as cardiac contraction) and slower dynamics (such as peripheral blood flow). Frequently one is not interested in an instantaneous value of a particular variable, or in the details of a specific waveform, but rather in the response of the variable's short-term average to perturbations in its parameters. This response typically occurs over time scales that are large compared to the dynamics of cardiac contraction.
In these cases, a cycle-averaged model, which tracks cycle-to-cycle (i.e. inter-cycle) dynamics rather than intra-cycle dynamics, seems desirable for several reasons. First, by ignoring the fine intra-cycle structure of each waveform, one can expect to reduce computational cost significantly. Second, one can anticipate that analysis of the dynamics of interest can be simplified if the model structure is reduced sufficiently. Third, it is typically the time-averages and not instantaneous values of key state variables that are regulated through feedback control. This situation is typical of domains (from biology to power systems) in which the component processes and their interactions are fairly well mapped out. In such cases, the natural models for computation and simulation of the global behavior of the system comprise complex interconnections of models for the component processes. However, the transients produced by such complex non-linear models may look similar to the transients produced by low-order linear models.
The goal of this paper, which builds on Chang (2002), is to study a periodically driven, closed-loop, lumped-parameter recirculation model and to derive a cycle-averaged version of it by applying circuit-averaging techniques from the power electronics literature (Verghese, 1996). The process of cycle-averaging preserves the state–space description of the model. Furthermore, the resulting model structure turns out to be linear and time-invariant (LTI), which allows for further insight into and simplification of the model structure.
While the following discussion is inevitably framed from the perspective of physiological systems, we have tried to keep the domain-specific terminology to a minimum in order to make this work accessible to the broader engineering audience.
Section snippets
Pulsatile model
We implemented a simplified version of a previously published lumped-parameter, closed-loop pulsatile model (Heldt, Shim, Kamm, & Mark, 2002). While the simplified model only represents the left ventricle and the systemic circulation, it is still rich enough to capture the essential time-varying dynamics of the pulsatile model and to serve as a sufficient testbed for our development of an averaging methodology.
As shown in Fig. 1, the model is in circuit form, and consists of three segments,
Developing a cycle-averaged model
In developing a cycle-averaged version of the simplified cardiovascular model, we make use of the definition of the symmetric running (or local) time-average of a waveform over a period T:If is periodic with period T, then will evidently be constant, but departures from this periodicity result in time-varying .
An important consequence of Eq. (3) is that the derivative of the time-averaged waveform equals the time-average of its derivative, i.e.
Comparison of simulations
To evaluate the performance of the cycle-averaged model we will compare its simulation results and simulation time to that of the pulsatile model.
Model reduction
As noted in Section 3.3, the cycle-averaged model has one very fast time constant . As mentioned before, this time constant is much smaller than our averaging interval and is therefore irrelevant to the cycle-averaged model. Using ideas from singular perturbation theory (see, e.g., Caliskan, Verghese, & Stanković, 1999), we can accordingly partition the cycle-averaged state-space model as follows:where and correspond to rapidly and slowly varying
Conclusions
In this paper, we have applied circuit-averaging techniques to a simplified lumped-parameter model of the cardiovascular system. We have shown that the resultant model structure is linear and time invariant, which allows for further insight into the model structure as demonstrated by our analysis of eigenvalues and eigenvectors. The realization of a fast time-constant also leads to the development of a reduced-order model using singular perturbation techniques. The cycle-averaged models allow
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Cited by (0)
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This publication was partially supported by the National Aeronautics and Space Administration through the NASA Cooperative Agreement NR 9-58 with the National Space Biomedical Research Institute and by Grant number 1 RO1 EB001659 from the National Institute of Biomedical Imaging and Bioengineering (NIBIB). Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the NIBIB or National Institutes of Health.