Physica A: Statistical Mechanics and its Applications
Cellular automata models of tumour natural shrinkage
Highlights
► We present two three dimensional cellular automata models for tumour growth. ► We focus on the tumour’s natural shrinkage. ► We offer several computational adaptations instead of traditional chain shifting. ► Ignoring shrinkage results in wrong tumour volume dynamics and growth pattern.
Introduction
Cancer is the second leading cause of death (after cardiovascular diseases) in developed countries [1]. Cancer represents a wide spectrum of diseases, which can be commonly characterised by uncontrolled cell division, caused by a series of DNA mutations. Normally, the balance between proliferation and programmed death (apoptosis) is strictly controlled to save the integrity and the structure of tissue. This harmony is broken as soon as at least one cell will start to divide rapidly and unsupervised [2].
An effective cancer treatment is an objective for many scientists from different fields. Not only biologists and clinicians should participate in the fight against the disease. The role of mathematical modelling is gaining importance. Modern biological approaches are often unable to separate the underlying mechanisms of tumour growth. This is where statistics and mathematical modelling make their contribution [3].
A large range of models for tumour growth in different tissues have been proposed [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. A key problem is validation. The fact is that as soon as a tumour has been detected treatment starts. That is why it is impossible to get a detailed description of the dynamics of untreated tumour growth in time. Nevertheless people are still creating new models of fair validity, usually neglecting relevant subphenomena of the complex process of tumour growth. For example, most models do not take into account shrinkage due to necrosis of the malignant cells [16].
Usually shrinkage is associated with tumour treatment, but in fact any neoplasm, where cells are subject to necrosis demonstrates shrinkage. After a cell dies its membrane disintegrates, then all the fluid content flows out and diffuses away [17]. It is usually assumed that the mortal remains occupy nearly one third of volume of the living cell [8], [9], [10], [11]. As a result, accounting for this phenomenon should influence volumetrical dynamics of tumours. At the same time it is important to note that cells, which underwent apoptosis, do not leave remains at all, because all occurrences of the “programmed cells death” are known by the immune system and such remains are immediately phagocytized.
In this paper we will show the relevance of shrinkage and its influence on the growth regime by presenting a three-dimensional cellular automata based model of tumour growth, with special attention to shrinkage simulation. Only early stages of avascular growth will be considered. Earlier cellular automata which modelled a single biological cell by a single automaton cell have not payed special attention to shrinkage. Some involve it implicitly [18, for example], but a distinct rate of shrinkage was not determined.
Cellular automata [19], [20], [21], [22], [23] were chosen as a basic simulations vehicle, because they have proven their suitability in the field of biomodelling [4].
The scientific community has produced significant results and software [24], [25] based on cellular automata. The model was implemented with the help of the software environment [25], [26]. Results of the simulation were validated against in vitro growth of the LoVo cells spheroid [27].
Section snippets
General considerations
Traditionally, tumour growth models are based on the assumption that the process starts from a single malignant cell [28].
In our version of the model one nutrient metabolism is considered. While neglecting angiogenesis, the healthy tissue around the tumourous clot is supposed to be perfectly vascularised. Nutrient diffuses from the healthy area to the clot of malignant cells. As a result, normal cells act as nutrition sources for the cancerous tissue. We model the tumour during the early stage,
Algorithm 1. Basic
The algorithm is implemented with a three dimensional cellular automaton with Cartesian metrics and Moore’s neighbourhood [34]. The timestep is assumed to be equal to one hour. This value was chosen as an appropriate compromise to represent the various phases in the cell cycle, and to allow for some cell to cell variability in the duration of the cell cycles.
Each automaton cell represents a single biological cell, with size 10 μm×10 μm×10 μm. Each cell can be in one of the following biological
Algorithm 2 optimised
In this approach we look for a position for a newborn cell not near the mother cell, but on the surface of the tumour. This substitutes the outward chain shifting by placing the daughter cell directly at the outer end of the chain. Despite the fact that this does not reproduce the biology of the process, it can be used when such local cellular phenomena as adhesion and influence of pressure are neglected. Taking Hayflick’s phenomenon [36] into account also needs to abandon the usage of such an
Validation based on Gompertz law
One approach to validation of in silico tumour growth [37], [38], [39], [40] consists in the volumetrical comparison with in vivo or in vitro data. The kinetic behavior of a tumour’s growth process can be reproduced by the Gompertz law [27], [37], [41], [42], [43] which is widely used for these purposes.
The law can be written in the following form: where is the tumour’s volume in time; —the initial volume; —the initial instantaneous tumour’s growth rate;
Conclusions
This paper described two models of avascular tumour growth, including the phenomenon of tumour shrinkage. Despite its simplicity, the results demonstrate a compliance with biological data in terms of the parameters of Gompertzian growth. In any case, validation studies of cellular automata models need a much finer definition of the underlying parameters. Detailed experimental studies are required to assess these parameters, in order to really validate the model. Nevertheless it was illustrated
Acknowledgement
This research was partly sponsored by a grant from the ‘Leading Scientist Program’ of the Government of the Russian Federation, under contract 11.G34.31.0019.
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