Page 26 - Control Theory in Biomedical Engineering

14 Control theory in biomedical engineering

Table 2 Mathematical models of some endocrine systems.
Related
Gland mathematical model Reference

Pancreas Glucose-insulin Bergman et al. (1979), Mari (2002,
regulation system Kansal (2004), Boutayeb and
Chetouani (2006), Makroglou
et al. (2006), Gani et al. (2009),
Balakrishnan et al. (2011),
Ajmera et al. (2013), Palumbo
et al. (2013)
Hypothalamo- Reproductive Cl ement (2016)
pituitary- function
gonadal (HPG)
axis
Thyroid Thyroid disease Liu et al. (1994)

system is to identify and then to eliminate all foreign bodies (Gałach, 2003).
In fact, the immune response begins when tumor cells are detected as unde-
sirable cells in the body. Then, the lymphocyte cells are stimulated in order to
coordinate the counterattack. In the literature, the interaction between tumor
cells and lymphocyte cells in the immune system looks likes the interaction
between species in the predator-prey model (Bell, 1973; Ben Saad
et al., 2019). One of the most basic mathematical models describing
tumor-immune system interactions is the Bell model proposed in Bell (1973).
This model is based on the classic Lotka-Volterra model (Volterra, 1928).
Other simple models can be found in research papers (Mayer et al., 1995;
Sachs et al., 2001) and in survey papers (Adam and Bellomo, 2012; Weera-
singhe et al., 2019). Thus, the immune response can be described by the
following simple mathematical model (Sotolongo-Costa et al., 2003):
dX
8
>
< ¼ aX bXY,
dt
dY
¼ dXY fY kX + u:
>
:
dt
where X and Y are the number of tumor and lymphocyte cells, respectively,
and (a, b, d, f, k, u) are system parameters.
2.6.3 The cardiovascular system
The cardiovascular system is composed of the heart and vessels. Its main
function is to pump the blood in the body in order to supply all tissues
and organs with oxygen and other nutrients (Quarteroni et al., 2009). In fact,