Page 31 - Control Theory in Biomedical Engineering

18 Control theory in biomedical engineering

1
0.8 0.5
0.6 0
z
z 0.4
–0.5
0.2 1
0.9 0.3
0
0.8 0.25
1 0.7 0.2
0.8 1 0.6 0.15
0.6 0.8 y 0.5 0.1
0.4 0.4 0.6 x
(A) y 0.2 0 0 0.2 x (B) 0.4 0.05
Fig. 9 Projections of the attractor related to the chaotic-cancer system onto the spaces
(x, y, z): (A) chaotic attractor, (B) periodic attractor, where x(t) is the logistic growth of
cancer cells population, y(t) is the competition between tumor cells and healthy and
z(t) is the effector cells.
Thus, the complexity and diversity of cancer diseases have attracted the
interest of the scientific community in order to analyze and understand
the dynamic behaviors of cancer cells. Therefore, many mathematical
models were proposed to define tumor evolution with ordinary differential
equations. Some models exhibit chaotic attractors and regular attractors
(fixed point, limit cycle and periodic orbits) (Valle et al., 2018). One of these
models is the three-dimensional chaotic-cancer system proposed by Itik and
Banks (2010). The three state variables are the logistic growth of cancer cells
population, the competition between tumor cells and healthy cells, and the
effector cells. Both periodic and chaotic behaviors are dependent on the
strength of the immune response against cancer and describe the evolution
of tumor cells as shown in Fig. 9. A patient is healthy when the effector cells
are equal to zero, or more precisely when the chaotic-cancer system con-
verges to an equilibrium point.

3 Control in physiology
The human body contains a natural and autonomous control process that
can maintain human life. This natural control process is designed by homo-
eostasis. In this section, we present the principals and different examples of
human homeostasis.

3.1 The homeostasis principal

In physiology, control refers to the process of stabilizing a physiological var-
iable to a specified set point, either by reversing perturbations via negative
feedback closed loops or via anticipatory open loops.
In the human body, the control process is designed by homeostasis. The
term homeostasis was invented by the celebrated French physiologist Walter