Page 30 - Control Theory in Biomedical Engineering

Modeling and control in physiology 17

or more these behaviors depending on the function they have evolved to
perform (Sharma, 2009).
Recently, several research works have attempted to model and to analyze
chaotic physiological systems. The glucose-insulin regulatory system is the
most popular of these dynamics (Ginoux et al., 2018; Shabestari et al., 2018;
Rajagopal et al., 2019). In (Rajagopal et al., 2019), for example, the corre-
sponding mathematical model is described by three differential equations
where the related state variables are insulin concentration, blood glucose
concentration and the population density of β-cells. Chaotic dynamic
behaviors for the late system are observed when various anomalies are
detected in the glucose-insulin regulatory system such as hypoglycemia,
hyperinsulinemia and when the body cells resist accepting insulin (type 2
diabetes). Fig. 8 describes some of these particular chaotic attractors.
Moreover, many researchers claim the importance of chaotic behaviors
in the human brain (Freeman, 1992; Schiff et al., 1994; Sarbadhikari and
Chakrabarty, 2001; Aram et al., 2017; Rostami et al., 2019) especially in
migraine headache (Bayani et al., 2018), attention deficit disorder
(Baghdadi et al., 2015) and epilepsy (Panahi et al., 2017, 2019). The math-
ematical model used to analyze epileptic seizures confirms that the normal
behavior of the human brain is chaotic. However, the abnormal epileptic
behavior of the human brain is periodic. On the other hand, the mathemat-
ical model of this system is a nonlinear neural network representing different
interconnected parts in the brain (Schiff et al., 1994).
One of the most interesting physiologic problems is cancer. Indeed, can-
cer is considered as the major cause of death worldwide (World Health
Organization, 2019). Cancer can affect different physiological systems.

2 2
1.5 1.5
z 1 z 1
0.5 0.5
0 0
2
2
1.5 1.5 2 1.5 1.5 2
1 1 1 1
(A) y 0.5 0 0 0.5 x (B) y 0.5 0 0 0.5 x
Fig. 8 Projections of the chaotic attractor related to the glucose-insulin regulatory
system onto the spaces (x, y, z): (A) Hypoglycemia, (B) Hyperinsulinemia, where x(t)is
the insulin concentration, y(t) is the blood glucose concentration and z(t) is the
population density of β-cells.