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.