Page 27 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

4 BIOMEDICAL SYSTEMS ANALYSIS

global. The microscopic modeling often leads to partial differential equations, whereas the macro-
scopic or global modeling leads to a set of ordinary differential equations. For example, the micro-
scopic approach can be used to derive the velocity profile for blood flow in an artery; the global or
macroscopic approach is needed to study the overall behavior of the circulatory system including the
flow through arteries, capillaries, and the heart. Models can also be classified into continuous time
models and models lumped in time domain. While the continuous time modeling leads to a set of
differential equations, the models lumped in time are based on the analysis of discrete events in time
and may lead to difference equations or sometimes into difference-differential equations. Random
walk models and queuing theory models are some examples of discrete time models. Nerve firing
in the central nervous system can be modeled using such discrete time event theories. Models can be
classified into deterministic and stochastic models. For example, in deterministic modeling, we
could describe the rate of change of volume of an arterial compartment to be equal to rate of flow in
minus the rate of flow out of the compartment. However, in the stochastic approach, we look at the
probability of increase in the volume of the compartment in an interval to be dependent on the prob-
ability of transition of a volume of fluid from the previous compartment and the probability of tran-
sition of a volume of fluid from the compartment to the next compartment. While the deterministic
approach gives the means or average values, the stochastic approach yields means, variances, and
covariances. The stochastic approach may be useful in describing the cellular dynamics, cell prolifer-
ations, etc. However, in this chapter, we will consider only the deterministic modeling at the macro-
scopic level.
The real world is complex, nonlinear, nonhomogeneous, often discontinuous, anisotropic, multi-
layered, multidimensional, etc. The system of interest is isolated from the rest of the world using a
boundary. The system is then conceptually reduced to that of a mathematical model using a set of
simplifying assumptions. Therefore, the model results have significant limitations and are valid only
in the regimes where the assumptions are valid.

1.1 COMPARTMENTAL MODELS

Compartment models are lumped models. The concept of a compartmental model assumes that the
system can be divided into a number of homogeneous well-mixed components called compartments.
Various characteristics of the system are determined by the movement of material from one compart-
ment to the other. Compartment models have been used to describe blood flow distribution to vari-
ous organs, population dynamics, cellular dynamics, distribution of chemical species (hormones and
metabolites) in various organs, temperature distribution, etc.
Physiological systems (e.g., cardiovascular system) are regulated by humoral mediators and can
be artificially controlled using drugs. For instance, the blood pressure depends on vascular resis-
tance. The vascular resistance in turn can be controlled by vasodilators. The principle of mass bal-
ance can be used to construct simple compartment models of drug distribution. Figure 1.1 shows a
general multicompartmental (24-compartment) model of drug distribution in the human body. The
rate of increase of mass of a drug in a compartment is equal to the rate of mass flowing into the com-
partment minus the rate of mass leaving the compartment, minus the rate of consumption of the drug
due to chemical reaction in the compartment. In the model shown in Fig. 1.1, the lungs are repre-
sented by three compartments: Compartment 3 represents the blood vessels (capillaries, etc.) of the
lung, the interstitial fluids of the lung are represented by compartment 4, and the intracellular com-
ponents of the lung are represented by compartment 5. Each other organ (e.g., kidneys) is represented
by two compartments consisting of the blood vessels (intravascular) and the tissue (extravascular
consisting of interstitial and intracellular components together). Let us consider the model equations
for a few compartments.
For compartment 3 (lung capillaries),

V dC /dt = Q C − Q C − K A (C − C ) (1.1)
3 3 2 2 3 3 3-4 3-4 3 4