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for the drug administration to a patient undergoing a certain therapy or, closer to our
topic, for an immune stimulator used in immune therapies. To these ends, the
mathematics can describe just a part of the whole system and, based on clinical data,
can help clinicians to choose among a set of possibilities spelled out on the basis of
their experience.
The present chapter aims to describe examples of the theoretical models that
scientists employ to address the second of the aforementioned problems. As a matter
of fact, in a number of situations, the molecular interactions can be included in very
simple and stylized mathematical terms in equations describing the relationships
among cells and molecules. The real value of the mathematical models is their capa-
bility to describe the complex relationships among the large number of components
of the (immune) system. By means of perturbations of the
network
of mole-
cules/cells/organs, it is possible to understand many properties of the whole system.
In other words, it is possible to answer questions like “what happens to the immune
response if we administer this drug which is able to block the action of this mole-
cule?” or “what is the best administration dose/schedule/route for interleukin-2 in
terms of a stimulation of the response?” or “what is the net effect of downregulating
the expression of such receptor on the surface of the lymphocytes?”
In the following sections we introduce some mathematical/computer models of
the immune response and in particular of the HIV infection. Then, we describe our
model (C-ImmSim) and the way we use it to uncover some unknown aspects of the
pathology and to make predictions about AIDS disease progression.
8.2 Computational Immunology
In the development of a mathematical model of the immune system of a vertebrate
animal, three levels of abstraction are usually considered: the
microscopic
level
which is the scale of subcellular activities (e.g., DNA synthesis and degradation,
gene expression, alteration mechanisms of the cell cycle, absorption of vital nutri-
ents, activation and inactivation of receptors, transduction of chemical signals within
the cell that regulate cellular activities such as duplication, motion, adhesion, or
detachment), the
mesoscopic
level (that refers to the cellular level and therefore to
the main activities of the cell population, e.g., the statistical description of the pro-
gression and activation state, cooperation/competition, aggregation properties, and
intra/extravasation processes), and the
macroscopic
level (the tissue level that refers
to the typical phenomena of continuum systems, e.g., cell migration, convection and
diffusion of nutrients and chemical factors, mechanical responses, interactions with
external tissues) (Preziosi 2003).
Besides that, the mathematical models of the immune system can be classified
according to: (1) the mechanism of regulation of immune processes (i.e., the clonal
selection theory versus. the idiotypic network); (2) the choice of the
mathematical
space
(i.e., continuous versus
.
discrete time and/or space), and (3) the presence of
stochastic (i.e., nondeterministic) components.
The implications of the first modeling choice are well known to biologists,
whereas the other two require a few more words. The difference between discrete
