Page 27 - Control Theory in Biomedical Engineering

Modeling and control in physiology 15

cardiovascular disease is one of the major causes of death worldwide (World
Health Organization, 2019). Thus, to prevent disease, it is primordial to
model cardiovascular systems. Mathematical modeling of the cardiovascular
system has a long history (Quarteroni, 2001). The first mathematical model
of the cardiovascular system was proposed in Grodins (1959). This system is
described by the following equation:

dV
RC + V ¼ CP,
dt
where V is the diastolic volume, P is the venous filling pressure, C is the
compliance of the relaxed ventricle and R is the total viscosity.
In literature, the cardiovascular system is considered a complex and crit-
ical system. Thus, many researchers have approached the modeling process
of this system via different viewpoints (Leaning et al., 1983; Shim et al.,
2004; Liang and Liu, 2005; Abdolrazaghi et al., 2010; Shi et al., 2011;
Ambrosi et al., 2012; Bessonov et al., 2016; Quarteroni et al., 2017; Bora
et al., 2019). A computational environment for human cardiovascular
system modeling and simulation is described in Larrabide et al. (2012).

2.7 Chaos in physiology

As reported by R€ossler, physiology is the mother of chaos (Rossler and
Rossler, 1994). Several researches demonstrate that chaos is a common
feature in complex physiological systems (Mackey and Glass, 1977; Glass
et al., 1988; Goldberger et al., 1990; West and Zweifel, 1992; Elbert
et al., 1994; Wagner, 1996; Glass, 2001; Aon et al., 2011; Li, 2015; Nazar-
imehr et al., 2017). A chaotic system is defined as a nonlinear system
with unpredictable dynamic behaviors and extreme sensitivity to initial
conditions (Lassoued and Boubaker, 2016; Boubaker and Jafari, 2018). In
physiology, chaos can be a sign of health or disease (Goldberger and West,
1987). In fact, some healthy functions of physiological systems are charac-
terized by chaotic dynamic behaviors, namely the human brain behaviors
(Baxt, 1994). However, other physiological systems are characterized by
ordered and regular dynamics. In this case, chaotic dynamic behaviors prove
the existence of peculiar pathologies, namely those for cardiovascular mea-
sures (blood pressure, heart rate, etc.) (Wagner, 1998). Several physiological
systems like the cardiovascular system are capable of five kinds of behavior:
(1) equilibrium (fixed point), (2) periodicity (limit cycle), (3) quasi-
periodicity (limit torus), (4) deterministic chaos (strange attractor) and (5)
random behavior (no attractor) as described in Fig. 7. Systems adopt one