Page 64 - Control Theory in Biomedical Engineering

Mathematical modeling of cholesterol homeostasis 51

by the lymphatic system) the parameter m diet expressed in milligrams per
minute should be equal to zero from the time the last meal was consumed
(t 0 ) to about 400–600min after the meal when it appears in the blood.
Complex and multistage cholesterol exchange processes occurring
between compartments I and II were simplified to kinetic expressions in
the form of a product of an effective kinetic constant k 12 and k 21 and an
appropriate mass. Thus, k 12 m 1 describes all the processes leading to the trans-
port of cholesterol from compartment I to II, whereas k 21 m 2 describes the
reverse process. As presented in Fig. 1 LDL and HDL are the main lipopro-
tein fractions; LDL transports cholesterol from liver to peripheral blood,
HDL transports cholesterol in opposite direction. For the stationary solu-
tions, the expected rate of cholesterol transport in both the directions is
the same; that is, k 12 m 1 is equal to k 21 m 2 . Thus, for the normalized k 21 value
of 1min 1 and according to the initial assumption that cholesterol concentra-
tions in both the compartments are the same, the parameter k 12 ¼k 21 V 2 /V 1 ,
1
which is approximately 3.6min .
Now, having determined the values of parameters k, k 21 , and k 12 , the
analysis of the stability of the model’s solutions can be continued.
The system (Eqs. 1 and 2) is locally asymptotically stable if all the eigen-
* *
values of matrix J(m 1 ,m 2 ) have negative real parts.
For this condition, we can consider the following two cases:
(a) the expression under the root is negative if
* 2
*2
*2
2
(k 12 m 1 +k 21 m 1 +k) 4k 21 k (m 1 ) is negative; and
(b) for the expression under the root to be positive,
* 2 0.5
2
* 2
* 2
((k 12 (m 1 ) +k 21 (m 1 ) +k) 4k 21 (m 1 ) ) has to be smaller than
* 2
* 2
k 12 (m 1 ) k 21 (m 1 ) k.
1
1
For k 21 ¼1min , k 12 ¼3.6min , and k in the range of
1
2
390–751mg min , case “a” never happens, while case “b” is always satis-
fied for physiological masses of cholesterol in both the compartments. Thus,
it can be concluded that Eqs. (3), (4) always lead to locally asymptotically
* *
stable points (m 1 , m 2 ).
5 Analysis of the solutions
Using the estimated values of the model parameters, the sensitivity of the
model to changes in the values of parameters can be studied. The aim of this
step is to evaluate the weight of individual parameters and to indicate the
potential ways through which a disturbed system can be optimally con-
trolled. Here, it should be noted that the results obtained are consistent with
physiological knowledge. The stationary solutions (Eqs. 3, 4) should be