Health-e-Child - IST-2004-027749 - Deliverable D.11.4

Brain Tumours

Tumour Growth Modelling in the Brain

1. Literature review and general overview

1.1 Introduction

Tumour growth modelling is the study of complex dynamics of cancer progression using mathematical descriptions. Internal dynamics of cancerous cells, their interactions with each other and with healthy tissue, nutrition and oxygen transport from the extracellular matrix (ECM) and from the vascular network, chemicals secreted by tumour cells, type of the underlying tissue and many other factors are modelled by mathematical abstractions. These abstractions rely on biological and clinical observations coming from different sources at different scales. Example observations include in-vitro experiments, in-vivo experiments on animal subjects, biopsy and autopsies results, medical images of patients like computed tomography (CT) scans or magnetic resonance images (MRIs). These experimental results and images are keys to develop models describing the tumour growth process accurately.

Mathematical growth models offer important tools for both clinical and research communities in oncology. By providing a common mathematical ground to combine experimental findings made in many diverse fields of cancer research they allow us to interpret such experimental results and understand the underlying mechanisms of tumour growth. Virtual experiments and simulations give the opportunity to observe effects of different treatments on cancerous cells, and could lead to improve these treatments or suggest new ones. Personalised models, adapted to patient specific cases, could be used in therapy planning by suggesting irradiation regions adapted to growth dynamics or optimal temporal distribution of chemotherapy.

1.2 Mathematical Growth Models

Tumour growth modelling has received considerable attention in the literature and there has been a vast amount of work during the last 15 years. A complete review of all the work done on this subject is outside the scope of this page. Here, we synthesise the main approaches towards this problem and review some of the latest work, in view of giving the reader an overview of the current trends.

Tumour growth is an extremely complicated process consisting of different interacting sub-processes at different scales. Proposed models concentrate on one or several of these sub-processes. Research conducted on tumour growth modelling can be coarsely classified into two large groups: microscopic models and macroscopic models. The main difference between these two classes is the scale of observations they are trying to explain and formulate. Microscopic models concentrate on observations in the microscopic scale, like in-vitro and in-vivo experiments. As a result they formulate the growth phenomena at this scale. Macroscopic models on the other hand, concentrate on observations at the macroscopic scale like the ones provided by medical images. They formulate the average behaviour of tumour cells and their interactions with underlying tissue structures, which are visible at this scale of observation (grey matter, white matter, bones...). These models try to describe the behaviour of the tumour as a whole, consisting of millions of cells.

Further classification within these groups can be made based on the stage of the tumour growth being analysed (avascular growth, angiogenesis and vascular growth) or the effect of the growth on the brain (diffusion/invasion of tumour cells and mass-effect). In this review we use the classification based on the stage for the microscopic models and the one based on the effect for the macroscopic models for better comprehension.

Classification of tumour growth models: First level classification is based on the scale of the model and scale of the observation model uses. Second level classification is based on the stage of the tumour growth for microscopic models and effect of the tumour for the macroscopic models.

1.2.1 Microscopic Models

Microscopic growth models aim to describe the tumour growth process at the cellular level based on the experimental observations at this level. They take into account physical and chemical interactions between cancer cells, the extra-cellular matrix and the healthy cells. Mechanical phenomenon like pressure, cohesion and adhesion forces are often included in these descriptions. As for chemical interactions, microscopic models formulate phenomena like diffusion of nutrition and oxygen, secretion of different factors by tumour cells and their effects on the surrounding. Mathematical systems obtained for these models are usually very detailed because they try to take into account all the observed factors. As a consequence, the number of parameters for such models are very high. From the technical point of view, formulations used in microscopic models enjoy a large variety of mathematical methods. Most commonly used methods are partial differential equation (PDE) systems, cellular automata and statistical models.

1.2.2 Macroscopic Models

Observations at the macroscopic scale consists of medical images like computed tomography scans (CT), magnetic resonance images (MRI) and MR diffusion tensor images (MR-DTI). Since the resolution of these observations is limited, typically around 1mm x 1mm x 1mm in the best case, observable factors are limited. Mathematical models at the macro-scale try to formulate the tumour growth using observations coming from this scale. For this reason, these models include fewer factors and are mathematically simpler than the microscopic models of Section 1.2.1. On the other hand, while microscopic models simulate the tumour growth in theoretical settings (infinite boundaries, known location of different structures,...), macroscopic models use real settings personalised for each patient , e.g. real boundaries of the brain, grey-white matter segmentation...

Based on the effect of the tumour on the brain, macroscopic models can be classified in two different classes: mechanical models, which concentrate on the mass-effect of the tumour on the brain tissue, and diffusive models, which concentrates on the invasion of surrounding tissue by tumour cells. From the mathematical point of view, all macroscopic models use continuum formulations based on a continuous local density or proportion of tumour cells. As a result, formulations contain several ordinary and/or partial differential equations to describe the growth process.

1.3 References