## Cellular Automata Model of Hypersensitivity

The model to be employed here for studying the role of drug schedules in the eruption of hypersensitivity, is based on the clonal selection theory of the Nobel Price laureate F.M. Burnet (1959) developed on the tracks first highlighted by P. Ehrlich at the beginning of the twentieth century. In our model, a cubic millimetre of blood serum of a vertebrate is mapped onto a two-dimensional l x l hexagonal lattice (six neighbours), with periodic boundary conditions. Physical proximity is modelled...

## Modelling at the Cellular Scale

This section provides a review of models suitable to describe the immune competition at the microscopic scale. Modelling the behaviour of the immune system at the cellular scale is tackled by many authors using a variety of different techniques and with a variety of different aims. The first aim of modelling consists in reproducing the immune response (primary and secondary response). However many other aspects like the auto-immune disease, selection and hyper-mutation of antibodies during an...

## An Overview of Discrete Models

While continuous models have been formulated in the framework of both im-munological theories (see 5 and Perelson 21 ), discrete models are mostly based on Jerne's theory 16 . Celada-Seiden's 19 model, which may include both theories, rests its foundation on the clone selection theory. The main task of the immune system is to perform a pattern recognition between cell receptors and antigens. The binding mechanism, mostly unknown in details, is based on different physical effects (short range...

## Mathematical Models of the Immune System

The immune system has some unique features, which render it appealing for mathematical modelling It is a highly distributed system, which carries out a complex recognition and classification task It evolves and matures using combinatorial, evolutionary and adaptation mechanisms It is able to remember Immune system models can generally be classified into continuous models, describing the immune process by sets of differential equations, and discrete models, describing the immune process as a...

## Automata Model of the Immune System

Filippo Castiglione, Vera Sleitser, and Zvia Agur Institute for Medical Biomathematics (IMBM), Bene Ataroth (Israel) 12.2.11mmunoglobulins and the Isotype Switch 12.2.2 Cytokines Production and the Role of Th1 Th2 Shift 12.2.3 Mathematical Models of the Immune System 12.3 A Cellular Automata Model of Hypersensitivity 12.3.1 Choosing Parameters 12.4 Model Validation and Simulation Results 12.4.1 Healthy Subjects Primary and Secondary Immune Response to a Generic Antigen 12.4.2 Allergic Subjects...

## Adhesive Interactions between Cancer Cells and Endothelium

Despite the striking similarities between the process of leukocyte diapedesis and tumour cell extravasation, there are differences between leukocytes and circulating tumour cells. Leukocytes are very motile small cells whereas tumour cells are larger, with a far less ability to migrate. However, these bigger cells can be arrested easily by size constraints in the microcirculation. In addition, these cells can form multicellular aggregate, by interacting with themselves or with leukocytes and...

## Experimental Techniques to Quantify Interstitial Transport

Most of the experimental techniques used to quantify interstitial transport parameters have limited capability for noninvasivemeasurements of diffusion in a small sample. Results obtained with invasive methods 59 must be interpreted with caution since these techniques may alter the fluid balance and damage the structure of the tissue by causing oedema that may strongly affect the diffusion characteristics of the tissue. Intravital microscopy coupled with quantitative fluorescence has been shown...

## N N 1101 N 1 N

The following question naturally arises what is the probability W(m, N) that the particle arrives at the point to (to being an integer), after suffering TO (N > to ) displacements Suppose for instance that to > 0. Then W(m, N) is the probability of taking (JV+TO) steps to the right (indeed, (N + m) must be an even number), out of a total of TO steps. It then turns out that W(m, N) (probability corresponding to an arbitrary sequence of paths) x (number of paths leading to place to ) Formula...

## V0 Hx

Actually, the argument in 49 is carried out for more general initial values than that in Equation (13.39), but consideration of this case is enough for the discussion that follows. A direct check shows that the function vo(x,t) + T df - F(vo( ,i ))d , -rrr k r + F(vo(x,t)) , -oo < x < oo , t> 0. at ox1 A sequence of functions (x, i) with i> 1 can be now constructed by means of the rule vi+1 (x,t) V0(x,t) + df - F(vi(t,r ))d . + F(vi(x,t)) , -oo < x < oo , f> 0, The argument then...

## References

1 Auckland, K. and Nicolaysen, G., Interstitial fluid volume local regulatory mechanisms, Physiol. Rev. 61, 556-643,1981. 2 Baxter, L.T. and Jain, R.K., Transport of fluid and macromolecules in tumors. I. Role of interstitial pressure and convection, Microvasc. Res. 37, 77-104, 1989. 3 Baxter, L.T. and Jain, R.K., Transport of fluid and macromolecules in tumors. II. Role of heterogeneous perfusion and lymphatics, Microvasc. Res. 40, 246263, 1990. 4 Baxter, L.T. and Jain, R.K., Transport of...

## The Celada Seiden Model

One of the most prominent attempts to reproduce, with the quest for biological fidelity, is the Immune Simulator automaton, also known as the Celada-Seiden model developed in 19 and 31 . The Immune Simulator belongs to the class of immunological cellular automata, but its degree of sophistication sets it apart from simpler CA in the Ising-like class 24 . The Celada-Seiden model explicitly implements the cellular and humoral im mune response in one comprehensive set of rules which apply to a...

## Discussion and Conclusions

This work presents a mathematical model for tumour invasion using a novel blend of continuum, deterministic modelling, and discrete, stochastic modelling in one and two space dimensions. The continuum model consists of a system of nonlinear partial differential equations and examines how tumour cells respond to ECM gradients via haptotaxis, created both by the tumour cells through MDE degradation of the matrix and those already in existence within the matrix. The results from the one...

## Multicellular Spheroid as a Mixture

Recently a mathematical description of avascular tumour as a multiphase system, namely a saturated porous material, has been proposed 3 , 7 , 18 . This approach starts from the observation that multicell spheroids are basically made of two constituents a solid skeleton constituted by an ensemble of sticky cells and by an organic liquid filling the extra-cellular space, which is used by the cells to grow. The introduction of such a mechanical framework allows one to deal with stresses and with...

## Introduction

This chapter deals with the modelling and with some mathematical problems related to the interaction and competition between the immune system and cancer cells. It is well understood that the above competition can play a crucial role in the cancer self-organised defence developed by the immune system, but also in connection with therapeutic actions. The above competition may possibly end up with the elimination of the host, while in some cases the opposite behaviour is observed. Medical...

## Overview

Most chemotherapeutic agents have proven to induce hypersensitivity. All four types of allergic reactions have been reported in literature, but type I, or IgE-mediated (see below) is the most common one 1 . In the clinical practice these complications are usually overcome by means of either suitable premedication with antiallergic agents, or by postponing drug administration. Nevertheless, the risk of a severe anaphylactic reaction is a major concern, severity strongly depending on the drug...

## Modelling by Generalised Boltzmann Models

The competition between immune cells and aggressive hosts has been modelled also by methods which are typical of nonequilibrium statistical mechanics. The above approach was first proposed by Bellomo and Forni 37 and subsequently developed by various authors as documented in the review papers by Bellomo and De Angelis 38 . The pertinent bibliography is reported in the above cited papers. The various models proposed in the literature may differ for technical aspects, but all refer to the...

## Spatially Uniform Models of Avascular Tumour Growth

In this section we present a number of mathematical models that have been used to describe the growth dynamics of solid tumours when spatial effects are neglected. The models are amongst the earliest that were used to describe solid tumour growth and are formulated as systems of differential equations. As we show, models of this type may be used to describe how the numbers of proliferating, quiescent, and dead cells contained within a tumour change over time. Equally, differential equation...

## The Tissue Specific Angiogenic Inducers

Depending on the phenotypic features and the growth rate of the different tissue compartments, the endothelium of the various vascular beds is diverse and distinct 48 , 49 . This happens despite the fact that the expression of VEGF and Ang (see above) is almost ubiquitous. The morphology and architecture of ECs also differs among different capillary beds. For example, fenestrae are associated with highly permeable vessels, such as those serving the endocrine tissues. The contribution of tissue...

## Role of Solute Dimension and Charge

Diffusion of small molecular weight molecules, such as oxygen and small molecular weight drugs, within interstitial space is scarcely hindered since the molecular size is very small compared to the matrix pore size 87 . Transport of small molecular weight molecules may therefore be envisaged as occurring in the free fluid phase within the extracellular matrix with a negligible steric obstruction resulting from the solid matrix network. A wide range of novel cancer therapies seek to utilise...

## Cancer Modelling and Simulation

A CRC Press Company Boca Raton London New York Washington, D.C. Pictured on the cover left, vasculature surrounding a tumour center (top), in vitro vasculogenesis center (bottom), trajectories of some endothelial cells involved in vasculogenesis on a background of the simulated concentration field of chemoattractant produced by the endothelial cells right (top), results of the vasculogenesis simulation right (bottom), comparison of the experimental trajectory of an endothelial cell with the...

## Inducers of Angiogenesis The Example of the VEGF Family

Direct angiogenic inducers cause ECs to migrate, proliferate, and differentiate into nascent blood vessels, which require the involvement of other molecules (e.g., Angs) to acquire whole function and stability. Before 1989, the prototypic members of the fibroblast growth factor family, were the leading candidates as positive regulators of angiogenesis they are potent angiogenic inducers both in vitro and in vivo and widely distributed in tissues and organs. However, cloning of genes encoding...

## List of Abbreviations

Fluorescence Recovery After Photobleaching Hypoxia-Inducible transcription Factor 1 Intercellular Adhesion Molecule-1 (or CD54) Lymphocyte Function-Associated Antigen-1 Mixed Lineage Leukemia trithorax protein Nucleosome Remodeling and Deacetylation Platelet Endothelial Cell Adhesion Molecules-1 Reflexion Interference Contrast Microscopy Rho-associated, coiled-coil-forming protein kinase Vascular Endothelial Growth Factor receptor-1

## Cell Cell Interactions and Signalling

Cells communicate with each other through different means, in order to coordinate growth, differentiation, and metabolism. Soluble signalling molecules (growth factors, cytokines, etc.) released by cells and targeting receptors on target cells allow 1 Extracellular matrix refers to the environment filling the spaces between cells. The extracellular matrix is a complex three dimensional network of proteins and carbohydrates secreted and remodelled by the cell. It helps bind the cells together in...

## Applying the Models From Theory to the Clinic

One can utilise angiogenesis mathematical modelling to serve several purposes. As we saw above, empirical data may reflect very intriguing phenomena, the analysis of which is enabled using such tools. The better understanding of such phenomena will lead to novel ideas for research and therapy. In addition, this work gives new options for the evaluation of novel antiangiogenic therapies. This will be demonstrated in the results section below. Clearly, if one wishes to apply such models for...

## Key Steps of Cancer Metastasis

The leading cause of death among cancer patients is the occurrence of metastases, which are secondary tumours arising at a distant site from the primary tumour. Tumours of comparable size and histology can have widely divergent metastatic potential, depending on their genotype and their local environmental influences, such as angiogenesis, stroma-tumour interactions, and production of cytokines by the local tissue. Metastasis is a cascade of linked sequential steps involving multiple...

## Automata Based Models

We now consider in more detail some automata-based models. Since an exhaustive review is beyond the aim of this section we will present only a few of the most representative models. Specifically, the Kaufman, Urbain, and Thomas model (KUT) is one of the first applications of discrete automata to investigate the logic of the normal immune response and introduced in 24 . These authors were interested in the simplest way to describe the logic of interactions among a number of different cell types...

## Cytokines Production and the Role of Th1Th2 Shift

T helper lymphocytes are mainly classified according to the types of cytokines they secrete 26 . Two distinct kinds of T helper lymphocytes can be distinguished, namely Th1 and Th2 lymphocytes. Th1 lymphocytes participate in cell-mediated immunity. They secrete interleukin-2 (IL-2), IFN-7, and TNF to enhance inflammation and antiviral responses, and are essential for controlling such intracellular pathogens as listeria and Mycobacterium tuberculosis (the bacillus that causes tuberculosis). In...

## Healthy Subjects Primary and Secondary Immune Response to a Generic Antigen

There are several ways in which normal immune response to a generic antigen can be simulated using our model. For instance one can inhibit the production of IL-4 in the model, thus knocking-out IL-4 activity 60 . Another possibility is to force Th cells to be of class 1 only (i.e., 1 ). Here we mimic a healthy subject by using the first method, that is, we set things so that no IL-4 can be released by Th2 cells. The consequence is a bias towards the Th1 response, i.e., a normal immune response....

## In Vitro Flow Studies of Circulating Cell Endothelium Adhesion

The rolling of leukocytes on activated endothelium is a critical step in the inflammatory cascade and has received considerable attention in the literature 32 . Leukocytes rolling occurs via the following steps 1. a receptor-ligand bond forms, exerting an adhesive stress which slows cell velocity 2. the slower motion of the cell promotes additional bonding 3. bonds dissociate at the back edge of contact, causing the cell to tumble forward in the direction of flow. These receptor-ligand bonds...

## Cell Basement Membrane Interactions during Tumour Progression

More than 80 of cancers originate from epithelial tissue and one of the earliest modifications in the neoplastic epithelium is the alteration of tissue polarity. Tissue polarisation is a characteristic of epithelial differentiation and is accompanied with the compartmentalisation of proteins along the polarisation axis (Figure 2.1). This phenomenon induces an asymmetrical intracellular organisation in which different subcellular areas are morphologically or functionally distinct. Tissue...

## Simple Applications

In order to show how the general theory illustrated above can be applied, in this section we solve some simple problems, namely the homogeneous growth inside a rigid cylinder and the inhomogeneous growth of a sphere under no applied external loads. The following procedure is adopted the motion is assumed to have some symmetry, and the function is assumed to have a certain simple form. Then we seek a deformation of the material that satisfies the equilibrium equation and appropriate boundary...

## Cancer Cell Migration

Finally, combining the different approaches above, it may be concluded that migration speed is somehow related to the level of affinity between cell receptors and ligands. Also it is the short-term adhesion which seems to control this migration. On the other hand, it seems like cancer cell migration is different from model cells which are usually studied (fibroblasts, keratocytes mainly). In particular 47 , it has been shown that tumour cells develop migrating cell clusters, therefore single...

## One Dimensional Spatial Models of Avascular Tumour Growth

The earliest spatially-structured models of avascular tumour growth are due to Burton and Greenspan 3,16 . At this time, biologists were focusing on the effect that changes in the composition of the medium surrounding the tumours had on their growth 13,26 . They recorded the radii of the (approximately radially-symmetric) tumours over time, supplementing their measurements, where possible, with information about the oxygen distribution within the tumours and the proportion of the tumours that...

## Immunoglobulins and the Isotype Switch

During the primary response of a normal individual, B cells produce antibodies of the IgM type. Several hours after the onset of IgM production, stimulated by the presence of interferon (IFN-7), IgG-producing B cells swing into action. Eventually, blood serum concentration of IgG antibodies increases above that of IgM, but as long as the antigen is present in the body, both IgM and IgG antibodies continue to be produced. Upon complete antigen removal, B cell stimulation is shut off and the...