Development of a new method
of dynamic modeling of tumor
growth when exposing the laser
hyperthermia
Elena M. Revzina, Irina L. Maksimova
Saratov State University., Russia
Today the effort to combat malignant tumors
is not only one of the most urgent medical
problems, but it is also a matter influencing
many aspects of social life.
The aim of the project: Laser action
optimization in case of tumors hyperthermia.
Building a mathematical model of malignant
tumor growth which allows for the impact of a
certain therapy type will make it possible to
predict the tumor’s behavior after treatment
termination and therefore to determine an
appropriate treatment course for the patient.
Multiple laboratory and clinical experiments
show that immune system plays one of the most
important roles in tumor growth inhibition.
In real life immune response scheme is very complicated.
Building such a model by simulation method using differential equations is difficult as the
equations’ set comes out lengthy and difficult to analyze.
Therefore for implementing the project we use cellular automata modeling method. Cellular
automata modeling is one of the practical methods used in many natural-science researches.
Unlike the modeling method with differential equations, cellular automata modeling allows to
include significantly more types of immunocompetent cells for examination and to make the
model more suitable for a certain situation.
The deviations of immunity state indexes in dogs with spontaneous malignant epithelial tumors were
statistically analyzed. The researchwas conducted in the First Veterinary Clinic of Saratov.The aim of the second R&D stage was algorithm development for the cellular model of immune response.The following immunity state indexes were taken into consideration in the modeling process: Ig A ,Ig G, Ig M¸
Т- lymphocytes, B-lymphocytes, IL-2.Tumor and immune system interaction model which allows for the dynamics of Ig A ,Ig G, Ig M¸ Т-
lymphocytes,B-lymphocytes, IL-2 was defined. Intercellular interactionscheme was used for this purpose.Initially the level of antitumor factors and tumor cells population magnitude (population “capacity”) are set. A
limit number of tumor cells (N1) is introduced.
(Axes X – time, axes Y – tumor cells number (N))
Tumor cells growth can be described by nonlinear logarithmic function.
It is assumed that starting from N1 the probability of tumor cells loss increases dramatically. The speed of cancer cells annihilation can be described by nonlinear function. If N>N1, then
1
1
N
NNkf
The following assumptions are made in this model. First, it is assumed that intracellular biological
cascades and extracellular signals transmission can be measured in units and defined by severalmathematical equations.
Second, it is assumed that natural mechanisms of selective cancer cells’ loss in living
organisms exist; otherwise the percentage of cancer incidences would have been significantly
higher. Cancer cells can be eliminated by immune response.
Third, we link the two seemingly contradictory biological facts about cancer and apoptosis: a
classical cancer sign is that cancer develops when there is no apoptosis and the natural
mechanism of transformed cells annihilation is prevented, and apoptosis induction of tumor cell
due to irradiation and most chemotherapy agents. It was further suggested that these skills
require communication with the other cells in the system at viability status (dead or alive) and
coordination with external commands for apoptosis.
The model: a dynamic model which simulates the tissue composed from the cells, fluctuating
in 2-dimensional space.
The numerical implementation of the algorithm will be carried out through cellular
automata method in CAME&L environment.
For a start, what are "cellular automata"? They are simple models, which are used for studying
complex systems behavior in different fields of science. For example, cellular automata found
applications in physics, mathematics, computer sciences, biology, chemistry, meteorology, social
sciences and many others.
These automata are discrete dynamic systems, which functioning can be completely described with
the terms of local interactions. In fact differential equations describe continuous dynamic systems.
So, everything that can be defined with differential equations can be modeled with cellular
automata. Actually they represent discrete analogue of the "field" concept.
Moreover these automata form the common paradigm of parallel computations as Turing machines
do for the consecutive computations.
It should be clear that cellular automata are rather useful for problems solving. But these systems
have specific parallel architecture, so if it is wanted to get some benefits from using them they are
to be implemented on specialized hardware or software platform.
Software CAME&L consists of three main parts:
Environment - application with rich and friendly user interface for solving problems with the help of
cellular automata. Basic abstraction, in terms of which environment is functioning, is concept
"experiment" - computational task. Here "experiment" is a synonym of common term "document".
Environment contains tools for computations control, studying and analysis, cluster arrangement,
workstations management and many others
•Cellular Automata Development Library
(CADLib) - C++ class library, which is designed
to present an easy-to-use and rich set of
instruments for creating so named
"components", elementary unities of solutions in
CAME&L. It is provided for reusing and
enlarging by researchers for their specific
problems. Library also contains useful functions,
macrodefinitions and constants, which make
development of components as easy as
possible.
•
•Standard components - basic set of "bricks",
from which solutions and experiments can be
built.