Page 17 - Control Theory in Biomedical Engineering

6 Control theory in biomedical engineering

variables. For more details about this approach, see the survey paper by
Brown (1980), Compartmental Modeling and Tracer Kinetics (Anderson,
2013), Compartmental Analysis in Biology and Medicine ( Jacquez, 1972), and
chapter 7 in Introduction to Biomedical Engineering (Enderle and Bronzino,
2012). In this framework, several applications in different areas of physiology
were represented by the compartmental approach, such as in diabetes
dynamics (Chiarella and Shannon, 1986), the respiratory system
(Similowski and Bates, 1991), tumor resistance to chemotherapy
(Alvarez-Arenas et al., 2019), metabolic systems (Cobelli and Foster,
1998; Staub et al., 2003), pharmacokinetics (Garcia-Sevilla et al., 2012b),
and so on. A software tool allowing implementation of linear compartmental
models is also available and is described in Garcia-Sevilla et al. (2012a).
Compartmental modeling is very attractive to users because it formalizes
physical intuition in a simple and reasonable way. According to this method,
the governing law is conservation of mass. Compartmental models are
lumped parameter models, in that the events in the system are described
by a finite number of changing variables (Cobelli and Foster, 1998). Each
compartment characterizes both the physical-chemical proprieties and its
environment and the corresponding mathematical model is a collection
of ordinary differential equations (ODEs). Each equation defines the time
rate of change of amount material in a particular compartment. Thus, the
basic equations of compartment model with n compartments are defined
as (Brown, 1980):
n
dx i X
¼ f i0 + f ij f ji f 0i ;x i 0ðÞ ¼ x 0i ;i ¼ 1,2,…,n:
dt
i¼1
j6¼i
where x i is the amount of material in compartment i, x i0 is its corresponding
initial value and f ji is the mass flow rate of compartment j from compartment i.
Fig. 2 illustrates the compartment structure. The index 0 denotes the environ-
ment of the physiological system.

2.2.2 Equivalent modeling approach
The main goal of the equivalent approach is to describe physiological
systems by using equivalent physical systems such as electrical or mechanical
systems. The equivalent models are used in physiology to simplify their
dynamic analysis. Each variable in a physical system has its corresponding
variable in an analog physical system. In the literature, some basic