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مدل های ریاضی رشد تومور

Mathematical models of tumor growth

Abstract

In the lecture we describe some elements of mathematical modeling of tumor growth. We present deterministic mathematical models most often used for description of tumor growth. Development of a model, based on some biological assumption, is also illustrated by one example. Described models are tested and compared by ability to describe experimental data.

1. Introduction

A mathematical model of tumor growth is a mathematical expression of the dependence of tumor size on time. In the lecture we present some results in the field of deterministic mathematical modeling of tumor growth. There are three main steps in the process of mathematical modeling: 1. Definition of model based on biological assumptions. 2. Testing the model against experimental data. 3. Acceptance of model or its rejection and change of assumptions. It is important to note that a model may be rejected due to wrong assumption(s) or inadequate number of assumptions. A particularly convenient experimental tumor paradigm is provided by the multicellular tumor spheroids (MTS) culture system [3, 27]. Spheroids provide a system for the study of the prevascular phase of tumor growth in the absence of tumor–host interactions, and for investigating the regulation of growth by three-dimensional cell–cell interactions. In MTS oxygen and nutrition come through the surface of spheroids and necrotic cells lay in the center of tumor. Figure 1 shows three layers in idealized scheme of MTS: necrotic core (I), quiescent (nonproliferating) cells (II) and proliferating cells (III). Moreover, the growth curve for tumor spheroids can conveniently be determined with uniquely dense measurements and high precision [10].

2. Mathematical models

In the case of multicellular spheroids, growth follows the sigmoid curve with the three distinct phases: the initial exponential phase, the linear phase and the plateau [13]. For this study, we selected mathematical models that reflect the sigmoid nature of growth. The models are divided into three groups: empirical models, functional models (based on cell kinetics), and structural models (developed specifically for spheroid growth).

2.1. Empirical models

These models are based on the fundamental empirical insight that growth results from the increase in size concomitant with processes that limit the size of the system. We consider two sets of empirical models developed for growth of biological systems.