Page 68 - Control Theory in Biomedical Engineering

54 Control theory in biomedical engineering

physiological condition of the gallbladder, and the value of the body mass
index (BMI) (Di Ciaula et al., 2012). After emptying, the gallbladder is
refilled to its maximum volume with hepatic bile. Next, if the gallbladder
is not stimulated to contract the bile accumulated within, the bile is concen-
trated by the active removal of water. This process is accompanied by influx
of the liver bile, which can be treated as a constant component of entero-
hepatic circulation. Thus, the process of cholesterol transport with bile
between the liver and the intestine can be described using constant param-
* *
eters m in and m out and variable parameters M in and M out which change over
time (associated with the gallbladder motility).
Another process that takes place over a specific time interval is the
absorption of cholesterol from food, and the amount of absorption is marked
*
as M diet . We have already shown that such processes can be expressed by the
2
function Asin (ωt), where A is the amplitude of a given process expressed in
1 ˙
mgmin (Zulpo et al., 2018). It is favorable to associate the angular pulsa-
tion ω with the length of the time interval in which a given process occurs by
dependence (Eq. 8):
π
ω ¼ (8)
t e t b
where t b and t e are the start and end times of a given process. Additionally, if
we include a shift t b relative to t 0 ¼0 (the global start time of all processes),
2
then the function Asin (ω(t t b )) takes the value 0 for t¼t b and for t¼t e
and only once the maximum value A in the interval (t e t b ). To determine
the change in the appropriate mass described by this function in the men-
tioned time interval, the integral of the expression (Eq. 9) should be
calculated:

t e 1
ð
2 t e t b
Asin ω t t b ÞÞdt ¼ A ð sin 2ω t e t b Þð ð Þ + sin 2ω t b t b ÞÞÞ (9)
ð
ð
ð
ð
2 4ω
t b
Through this shift, the solution of Eq. (9) takes a simpler form (Eq. 10):
ð
t e t e t b
2
Asin ω t t b ÞÞdt ¼ A (10)
ð
ð
2
t b
*
To estimate the variable M out , which is the mass of cholesterol leaving
compartment I together with bile and changes over time, it is necessary
to calculate the mass of bile ejected from the gallbladder. The amount of
cholesterol is proportional to the mass of bile (cholesterol makes up about