Mathematics, Developmental Biology and Tumour Growth

CONTEMPORARY MATHEMATICS 492

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American Mathematical Society

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Mathematics, Developmental Biology and Tumour Growth

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CONTEMPORARY MATHEMATICS

492

Mathematics, Developmental Biology and Tumour Growth

UIMP–RSME Lluis A. Santaló Summer School September 11-15, 2006 Universidad Internacional Menéndez Pelayo, Santander, Spain

Fernando Giráldez Miguel A. Herrero Editors

American Mathematical Society Real Sociedad Matemática Española

American Mathematical Society Providence, Rhode Island

Editorial Board of Contemporary Mathematics Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss

Editorial Committee of the Real Sociedad Matemática Española Guillermo P. Curbera, Director Luis Al´ıas Linares Alberto Elduque Palomo Emilio Carrizosa Priego Pablo Pedregal Tercero Bernardo Cascales Salinas Rosa Mar´ıa Miró-Roig Javier Duoandikoetxea Zuazo Juan Soler Vizca´ıno

2000 Mathematics Subject Classiﬁcation. Primary 34K10, 34K25, 35B40, 35F25, 92C50.

Library of Congress Cataloging-in-Publication Data UIMP-RSME Santaló Summer School (2006 : Universidad Internacional Menéndez Pelayo) Mathematics, developmental biology, and tumour growth : UIMP-RSME Santaló Summer School, September 11–15, 2006, Universidad Internacional Menéndez Pelayo, Santander, Spain / Fernando Giráldez, Miguel A. Herrero, editors. p. cm. — (Contemporary mathematics ; v. 492) Includes bibliographical references and index. ISBN 978-0218-4663-6 (alk. paper) 1. Carcinogenesis—Mathematical models—Congresses. 2. Developmental biology—Mathe- matical models—Congresses. I. Giráldez, Fernando, 1952– II. Herrero, M. A. (Miguel Angel) III. Universidad Internacional Menéndez Pelayo. IV. Title. RC2.5.U46 2009 571.9′780151—dc22 2009009818

Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientiﬁc purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the ﬁrst page of each article.) ⃝c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ⃝∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009

Contents

Preface vii Developmental Biology and Mathematics: The Rules of an Embryo Berta Alsina, Adrian´ L. Garc´ıa de Lomana, Jordi Villa-Freixa` and Fernando Giraldez´ 1 From Lineage to Shape: Modeling Dorsal-Ventral Speciﬁcation in the Developing Mouse Limb Carlos G. Arques and Miguel Torres 13 Notch-Mathics Rita Fior and Domingos Henrique 27 Modelling Tumour-Induced Angiogenesis: A Review of Individual-Based Models and Multiscale Approaches Tomas´ Alarcon´ 45 Tumour Radiotherapy and Its Mathematical Modelling Antonio Cappuccio, Miguel A. Herrero and Luis Nunez˜ 77 Multiphase and Individual Cell-Based Models of Tumour Growth J. Galle and L. Preziosi 103

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Preface

This volume contains a number of selected survey papers on the topics presented and discussed at the “Lluis A. Santaló Summer School”, that was devoted to Mathematics of Development and Cancer. The School was held in Santander (Spain) on September 11–15, 2006, as part of the activities of the Universidad Internacional MenéndezPelayo (UIMP), in collaboration with the Real Sociedad MatemáticaEspañola(RSME). Lecturers came from different scientific fields, including Biology, Mathematics, Medicine and Physics. They were selected in an attempt to present an outline of ongoing research in selected areas of mathematics and biology, in a manner that could be widely accessible to an audience consisting mainly of advanced undergraduates and graduate students on Mathematics. These were thus given a front seat at the research currently done by a number of groups worldwide working in Biology and Mathematics. The course was centred on De- velopmental Biology and Tumour Growth. These are topics where Mathematics is increasingly being used as a new and powerful technique to gain new insights. Mathematics is in its turn receiving a significant scientific pay-off in the form of new and challenging mathematical problems to be added to their own ones. For instance, Developmental Biology is an area of basic research in Biology and Medicine that has fascinated mankind since the earliest recorded scientific thought. Developmental Biology, which has experienced a great impulse during last years, deals with the basic problem of understanding the unfolding of utterly complex living structures “from egg to embryo”. In this manner, it has been seen to raise a number of challenging quantitative problems that immediately appeal at the imagination of mathematicians. This is the case of some central problems in Biology related to pattern formation, where the issue of how space and time evolution is first coded in the genome, and then set in action during development, has generated a great interest in the use of mathematical tools. The first part of this volume contains three views on different aspects of De- velopmental Biology as seen with the eyes of biologists interested in incorporating Mathematics to their technical tools. More precisely, Alsina et al. summarise the state of the art in Developmental Biology by describing some of the basic questions in the field, and by commenting on some examples of formal approaches to the specific problem of patterning in development. Arques and Torres describe the problem of how three dimensional patterns are established in the limbs of vertebrates, and show how an interesting model on mesenchymal compartments formation can be derived from a clever clonal analysis of cell lineages. Finally, Henrique and Fior discuss on one of the key signalling pathways in development, the Notch pathway, a subject on which a good deal of work is being currently done both by biologists and mathematicians.

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viii PREFACE

On the other hand, Cancer Research is a major priority in health care and a challenge to the scientific community as a whole. Cancer is a complex biological process that raises issues in Genetics, Cell Differentiation, Environmental Sciences, Immunology, Pharmacology, Physics,... to mention but a few active fields of scientific endeavour. Interdisciplinary research is thus a growing need of mathematicians working along physicians and biologists in order to develop models and new therapeutic approaches. In this volume, the reader will find contributions by Alarcón,Cappuccio, Her- rero and Núñez, and Galle and Preziosi , all of them dealing with the mathematical modelling and subsequent analysis of problems related to Tumour Growth. More precisely, Alarcón provides an updated account of current views on angiogenesis, the formation of new blood vessels from a preexisting vasculature, which is known to play a key role in many types of tumour progression. Cappuccio et al. give an overview of modelling issues arising in radiotherapy, a commonly used technique to treat a number of malignant (and also benign) tumours. On their turn, Galle and Preziosi present a comprehensive picture of tumour growth insisting on individual cell-aspects and their relation to a multiphase -fluid flow picture of that process. The book is addressed to PhD students and advanced undergraduate students in sciences willing to start their research in the exciting, and as yet unchartered, interface among Biology and Mathematics. There is a widely perceived interest in finding common spaces for discussion between mathematical and biological sciences, and this volume is expected to provide a contribution to that goal. The aim was at presenting an updated view on some basic problems of Biology, and to illustrate how Mathematics may help to improve knowledge on some of them. It goes without saying that the choice of topics has no claim to be exhaustive, and that the authors are fully aware that many interesting subjects have been left out of the selection herein made. The editors wish to thank the RSME for giving them the opportunity to plan the School from which this work has unfolded. Our thanks also go to the UIMP, which provided excellent organization, and outstanding facilities, to organise the series of lectures which are at the origin of this book.

Fernando Gir´aldez and Miguel A. Herrero

Contemporary Mathematics Volume 492, 2009

Developmental Biology and Mathematics: the rules of an embryo

Berta Alsina, Adrián L. Garc´ıa de Lomana, Jordi Villà-Freixa, and Fernando Giráldez

Introduction The use of mathematical concepts and theories to understand our world has a long journey. Although the interrelations between Mathematics and Physics have been the most prevalent, some biological phenomena such as genetic evolution and epidemiology embraced mathematical modelling long ago, using it to get a better understanding of such typically non-linear underlying mechanisms. Developmental Biology deals with how specialized cells emerge in a spatial and temporal pattern during embryogenesis to properly organize an adult organism. Is the development of an embryo just genes and proteins with no inherent formulations? The intersection of Mathematics and Developmental Biology was pioneered in the 1950s but it is just recently that this new area is developing rapidly. In this chapter we will review some concepts and biological processes of Developmental Biology to then pinpoint the emerging areas of Applied Mathematics and computer simulations in Developmental Biology. Developmental Biology has to do with the instructions that are required to build up an organism. Those are coded in the genome of each animal species. The unfolding of this genetic information is done from the fertilized egg to the embryo, in a sequence of gene expression patterns and cellular interactions that is invariant for any given animal species, and that generates the diﬀerent stages of the embryo along its development. The state of this area of biology is the result of more than one hundred years of observation and experimental work. During the last thirty years it has plunged into the molecular description. But it is still at the level of the description of the elements and their gross interactions where Developmental Biology resides. Most of the literature on Developmental Biology is devoted to identify genes that are relevant to development, and to characterise their basic eﬀects on cells and on other developmental genes. More recently, genome-wide techniques permitted this work to be done on more massive basis, but the approach is, still, to identify the actors and their basic interactions. The understanding of

The work was supported by grants BFU 2005-03045 Ministerio de Educaci´on y Ciencia, CTQ20-05/BQU Ministerio de Ciencia e Innovaci´ on and the EC funded QosCosGrid project (IST-033883)and VPH NOE (ICT-223920).

⃝c ⃝XXXXc 2009 American Mathematical Society 1

2 ALSINA ET AL. the transit ”from the egg to the embryo” requires, no doubt, the identification of the basic elements, genes and cellular states, but also and most importantly of deciphering the laws of transformation between states. On the mathematical side, the framework provided by the already established field of systems biology often distinguishes between top-down and bottom-up approaches. A top-down approach aims at building explanations for complex problems by analysing high information containing models. On the contrary, a bottom-up approach is synonymous of a synthesis of (sometimes) massive but poorly informative data into a given conceptual framework. Historically, the use of Mathematics in Developmental Biology has been based with the first schema, trying to design abstract and general principles governing the different observed biological events. Lately, though, the boom in the generation of molecular biology data has lead to the design of new synthetic models of the cell behaviour based on this bottom-up schema. Both approaches have pros and cons, and only joint efforts will explain the complexity of developmental processes. Speaking of Developmental Biology is to speak of genetics, cell biology, cell signalling, organ formation, regeneration and tissue engineering, stem-cells, neuro- biology and so on. The particularity of Developmental Biology is that the focus of study is not limited to any dimensional scale (proteins or the cellular level) but needs to integrate studies of all biological levels for a higher view of the problem. This particularity has made Developmental Biology one of the best disciplines for the use of synthetic biology (Ismagilov and Maharbiz, 2007), integrating elementary modules for the construction of higher order models, and to analyze this complexity with the tools that non-linear mathematics provides. Developmental Biology is, thus, agglutinative and it occupies a central position in Biology (Slack, 2007). It goes from genes to cells and to the structure of organisms in a way in which all three levels of complexity are integrated. Such integrative discipline is desperately in need of formal treatments not only to allow quantitative computations of multiple interactions, simulations and model- based predictions, which can be dealt with the emerging Applied Mathematics field of multi-scale simulations, but also to get an insight into the internal logic of the process: into the formal structure of the language of the embryo. Developmental Biology is surely one area of biology that will benefit enormously of formal approaches and Mathematics in the years to come. The deciphering of the genome and the introduction of highly sophisticated techniques of molecular biology and in vivo imaging has presented biologists with an opportunity to study genetic processes on a genomic scale, and to obtain (semi-) quantitative understanding, not just of individual molecular mechanisms but also of their dynamical interactions and regulation at the systems level. In Developmental Biology, ensemble properties emerge at each level of organization from interactions of heterogeneous biological units (cells, tissues and organs) coordinated in the space and time dimensions. Biologists recognize that appropriate mathematics can help interpret any kind of data but scepticism has prevailed until today due to the lack of proper quantitative measurements and integrative vision. Conversely, Mathematics will benefit from its involvement with biology, just as has already benefit from its involvement with physics. Finally, a note of caution should be taken on the misuse of modelling. On the one hand, models are often built with excessively detailed compartments that

THE RULES OF AN EMBRYO 3 are sometimes irrelevant for the proposed question while other important aspects are not considered even in an approximate way. On the other, very well known reference models may have limited applicability in the daily biological research, even if they proved to be very helpful in the identiﬁcation of the basic bricks for complex biological behaviour in a top-down fashion. Thus, building models is in a way the art of balancing the complexity of the question with the completeness of the answer. And, of course, a model cannot, by deﬁnition, be used to explain phenomena outside its underlying assumptions.

The basic problems of Developmental Biology:time space, construction from building blocks The development of an organism is like the building of an opera house. First, the major architectural plan has to be set up, and progressively subdivide and give functionality to each of the contained parts. Thus, embryogenesis is a sequential process in which the coordination of time and space must be exquisitely regulated to achieve that, for instance, the limb forms in the right position relative to the head and this occurs not before establishing the major axial structure. The essence of Developmental Biology is the progressive increase of complexity. In informational terms the increase in developmental complexity is measured by the generation of new population of cells, each of which reads out a defined genetic subprogram, and each of which arises in a particular spatial domain of the embryo. The difference between building an embryo and building an opera house relies in the fact that in the embryo, the arcs, vaults and walls once erected must ”communicate” to the ceiling in order to place the windows in the right position (Davidson, 2006). Information must constantly flow between all the components to arrange them correctly. Thus, in Developmental Biology the information of the construction of the organism is intrinsically encoded and progressively readjusted. Much of the genomic regulatory program for development is devoted to the progressive organization of spatial interactions between cells of different character (Davidson, 2006). Cell specification or the acquisition of a given cell’s identity requires communication between cells, what is termed intercellular signalling. Each cell defines what is going to be in the adult when it incorporates the receiving inputs and activates a particular genetic program. The organism is built by switching on different genetic programs in different locations and times. Until terminal differentiation, this genetic program is transient and multiple genes are constantly being switched on and off, and, in a similar way as in the Morse language, the combination of on and off signals generates the final order. This leads to the realisation that stochastic and discrete events may play relevant roles in both cell specification and intercellular signalling events. Indeed, as we will outline below, biology can be seen as a gradient from fully deterministic/continuous processes, including eventually some white (or Gaussian) noise, to fully stochastic/discrete events, mainly depending on the number of elements (molecules, cells) interacting in each process.

The fate of cells is coded by the combinatorial expression of speciﬁc gene ensembles One of the most breathtaking biology processes is the development of a complex creature. In a matter of just a day (a ﬂy maggot), a few weeks (a mouse) or several months (ourselves), an egg grows into millions, billions, or, in the case of

4 ALSINA ET AL. humans, ten trillion cells formed into organs, tissues and parts of the body. So, Developmental Biology’s main question is to understand how do cells arising from division of a single cell become different from each other. Despite the huge amount of mathematically possible combinations of gene expression states in a given cell (for a simplified situation of 2 states on/off -see below- of the 30,000 genes in the human genome, such number could be of the order of 230000, which is augmented to an, obviously non-compatible with life, astronom- ical number by the grading of gene expression (Soneji et al., 2007), there are in the order of just one hundred different cell types in the human body. “The major paradigm of developmental genetics is differential gene expression from the same nuclear repertoire” (Gilbert, 2003). Developmental Biology studies how specialised cells develop in the embryo, that is, how the process of differentiation occurs. Dif- ferentiation implies changes in cellular biochemistry and function of cells. But those changes are preceded by a set of instructions that specify the commitment of the cell to a certain fate. Cells in the embryo progressively loose their initial extended potential and transit through successive states in which they restrict their fate by the selective expression of a set of genes and the suppression of others. There are conditions, therefore, that anticipate differentiation and that dictate the fate of the cell. For instance, a domain of the embryo is specified to become the neuroectoderm that contains the progenitors of the cells of the nervous system. Those cells and their progeny are then specified to become either motor, sensory neurones, or glia, and then further to develop into specific neuronal subtypes1. The ability of the embryonic cells to change their fates to compensate for the missing parts is called regulation. Regulative development is seen in most vertebrate embryos, and its example is the development of identical twins. In the formation of such twins, the cleavage-stage cells of a single embryo divide into two groups, and each group of cells produces a fully developed individual (Gilbert, 2003). In 1961 Jacob and Monod showed that the levels of a gene product in E.coli were under regulation as a result of a feedback on the transcription of DNA into RNA2 (Jacob and Monod, 1961). This brought the concept of gene expression at

1When the state of commitment is spontaneously expressed in a neutral environment, but still reversible or modifiable, it is called specification, and the cell is said to be specified for a certain fate. When the process of commitment is irreversibly fixed, then it is called determination (Slack, 20). The process of commitment is said to be autonomous when the fate of the cell is fixed with independence of the interactions with neighbouring cells, i.e. based only on the information inherited after each cell division. It is referred to as mosaic development and it depends on cytoplasmic determinants (proteins or messenger RNAs)that are placed in different regions of the egg cytoplasm and are distributed to the different cells as the embryo divides (Gilbert, 2003). Commitment is said to be conditional when it requires the interaction of one cell with the neighbours. In this type of specification, each cell originally has the ability to become many different cell types, but the interactions with other cells restrict the fate of one or both of the participants. 2Two crucial discoveries in the early sixties provided the basis for the understanding of the molecular basis of cell specification and cell fate acquisition. Briggs and King published a paper in 1952 showing that normal hatched tadpoles can be obtained by transplanting the nucleus of a blastula cell to the enucleated eggs of Rana pipiens. But John Gurdon and collaborators, described in a series of papers the developmental capacity of nuclei transplanted from the endoderm lineage using donors from blastulae up to the intestinal epithelium of feeding larvae of Xenopus (Gurdon, 1962). The nucleus of a somatic cell contains all the necessary information to develop a new organism, and its corollary: the genome of a cell can be reprogrammed so to reinitiate the expression of embryonic genes. ”Although complete nuclear reprogramming takes place in only a small percentage of nuclear transfers from differentiated cells, it is remarkable that it takes

THE RULES OF AN EMBRYO 5 the core of genetic regulation and provided a way of understanding how cells with an identical genome may have different genetic properties. Indeed, different cell types make different sets of proteins out of identical genomes. As shown above, every cell of a human being has roughly twenty to thirty thousand genes in each nucleus, but each cell uses only a fraction of these genes, and different cell types use different subsets of these genes.

Gene regulation is at the core of development:Differential gene expression How a tissue becomes different from the rest relies on the activity of a specific set of genes, whether they are transcribed (on state) or not (off state). Tran- scription of a gene is initiated at a specific base pair in the template genomic DNA and is controlled, in addition to the transcription machinery (RNA polymerases), by trans-acting proteins, transcription factors that bind to specific sites of the genomic DNA, the cis-acting regulatory DNA sequences (Fig. 1). These proteins are either repressors and/or activators that control the transcriptional state of genes. The cis- acting regulatory elements, also named enhancers, are genomic sequences that can be further away from the coding unit and to which large number of transcription factors bind. As a result of this arrangement, transcription from a single promoter may be regulated by binding of multiple transcription factors to alternative control elements, permitting complex control of gene expression (Davidson, 2006). Fig. 1 schematically shows three genes (A, B, C in yellow, red and green boxes) with their upstream genomic regulatory regions. In these regions transcription factors such as X, Y, A, B bind to specific consensus sequences that will enhance or repress the transcription of that gene. The combination of this consensus sequences and the occupancy state of the site by transcription factors will determine the mode of the transcription, an on or an off state as well as the relative levels of product generated. During development, sequential developmental stages are acquired with the activation of new genes. If A activates B and then B represses C, in the cells where X and Y were present, B but not C will be turned on. In the embryo, gene interactions finally results in areas of differential gene expression, a region with C plus a region with B but not C. Ultimately, regulative interactions between genes create a pattern (see Fig. 1, bottom right). During the last 70 years mutation analysis in several invertebrate and vertebrate organisms (Drosophila,yeast,C.elegans,mouse)have described some of the molecular interactions between genes, building simple genetic networks. Recently, Developmental Biology has also incorporated high-throughput techniques that allow massive analysis of the transcriptome (detection of all mR- NAs in a given cell or tissue) or proteome (detection at the protein level). Large regulatory networks are now being constructed when comparing several sets of experimental data. This revolution in molecular biology has incorporated two new components into biology: first, a huge amount of data and second, the possibility of obtaining quantitative measurements. On the other hand, it is well established now that the output of the biological response depends not only on the presence of place at all” Gurdon and Byrne (2004). These findings provided an answer to the long-standing question of whether the process of development and cell differentiation requires a loss or stable change in the genetic constitution of cells (Gurdon and Byrne, 2004). In 1997, Wilmut announced that a sheep had been cloned from a somatic cell nucleus from an adult female sheep (Wilmut et al., 1997).

6 ALSINA ET AL.

Figure 1. A. Schematic drawing of the genomic organization of genes A, B and C. Large rectangles represent the coding unit (part of the DNA that will finally generate the protein) and small squares at left of each coding unit represent the binding sites for the transcription factors (free shaped). A combinatorial binding of distinct transcription factors will determine the final outcome of the response (either activation or repression of gene transcription). The bent arrows indicate active transcription while truncated arrows indicate repressed transcription. B. Linear representation of sequential genetic interactions between X, Y, A, B and Z. C. Sequence of gene activation over time in the embryo and final outcome of genetic network giving rise to a pattern of gene expression. the inputs but also on their levels. Developmental Biology therefore must change its language from qualitative to quantitative terms, something that was completely neglected and left in a back-side drawer until very recently (Tomlin and Axelrod, 2007; Lewis, 2008).

Coding for space in the embryo:Regional specification is at the heart of development (or at least of many developmental biologists) As with zebras, butterflies and giraffes, we are attracted to organism patterns. As mentioned, these patterns are originated during their embryonic life. Devel- opmental biologists have focused on the genes that originate spatial growth and determine the fates of individual cells. Regional specification is the process by which spatial patterns are set in the embryo. During embryonic development there are successive ”decisions made by the embryo” that result in the allocation of spatial domains to specific fates. For instance, in vertebrates, during the first divisions of the zygote some cells are set aside and destined to become the embryo, while others are to become the nourishing structures. Or later in development embryonic

THE RULES OF AN EMBRYO 7 cells that occupy certain domains are segregated into three different germ layers, ectoderm, mesoderm and endoderm- that give rise to specific tissues and structures. Even later, some tissues become different parts of the brain, arms, or legs, and so on. Therefore pattern formation, regional specification, focuses on how genetic information is translated into space, rather than how it results in cellular differentiation. Positional information is the intrinsic information carried by a cell or a group of cells that are equipotent, by which they acquire a specific developmental fate because of the position they occupy relative to the three axis of the embryo (Wolpert, 1996). There are several mechanisms by which spatial patterns can arise in the embryo. Pioneered by British mathematician Alan Turing (1952), several researchers have formulated that body patterns can arise when chemicals diffusing through a developing organism interact to trigger a response in some places and inhibit it in others (Fig. 2 upper part). To make a pattern from scratch, a developing organism needs to strike a balance between two processes: one that stimulates growth/gene activation, and one that inhibits it. Such reaction-diffusion theories, initially developed in Gierer and Meinhardt (1972), have been used to model biological phenomena from the regeneration of the tentacles of the freshwater hydra to the arrangement of leaves and flowers on plants; from embryonic development to the spotting of seashells. Although some researchers believe that these models can be applied to any problem, others also demand proofs that the hypothesized activators and inhibitors, which are known as Turing morphogens, actually exist. For example, the proteins Nodal and Lefty possess all the properties of the activator and inhibitor molecules. In vertebrate embryos, these proteins control the generation of three distinct layers of cells-the endoderm, the ectoderm, and the mesoderm-that will generate different sets of organs and it was already known that Nodal spurs the growth of mesoderm and endoderm and that Lefty counters the effects of Nodal (reviewed in Cho, 2004). The initial concept of morphogen was expanded to the definition of a morphogen gradient in which the emission of a substance from one part of the embryo provides a signal that is interpreted by the neighbouring cells depending on the distance from the source (Fig. 2, middle part). Lewis Wolpert illustrated this type of positional information using what he called “the French flag problem” (revisited in Wolpert, 1996): “What mechanism would ensure that a line of totipotent cells, no matter how long, would always have a French flag pattern -a third blue, a third white and a third red?” Let us have a row of equivalent -equipotent- cells, each of which is capable of differentiating into a red, white, or blue cell, and a signal -a morphogen- whose source is on the left-hand edge of the row, and its sink is at the right hand edge. This generates a concentration gradient that is highest at one end of the “flag tissue” and lowest at the other. All cells are taken to be equivalent -equipotent- and to respond to the signal with a threshold, so that high concentrations result in one fate (blue), intermediate concentrations result in another fate (white) and low concentrations in once another fate (red). The colour is in this case the cell parameter (positional value) that is related to the position of the cell (Kerszberg and Wolpert, 2007). An example of such type of gradient morphogen activity is the one mediated by the gradient of Retinoic Acid along the brain and spinal cord in such a way that high doses of Retinoic Acid are required in the spinal cord and

8 ALSINA ET AL.

Figure 2. A. Reaction-diffusion model for pattern formation. a) typical stripped pattern found in the animal kingdom. b) Gene network proposed by Turing; the activator activates the inhibitor in parallel to itself (positive feed-back) while the inhibitor represses the activator (negative feed-back mechanism). c) classical 3D representation of the outcome of the network with spaced peaks and valleys. B: Positional information mediated by a morphogen gradient. d) gradient of the morphogen (concentration in the ordinates) throughout the embryonic space in the abscissa. e) cells that measure high concentration of morphogen activate blue genes, in intermediate levels white genes are activated and finally cells receiving low morphogen signals activate red genes generating the french flag pattern. C. Pattern formation mediated by lateral inhibition. f) distribution of hair-cells in the mouse cochlea detected by labelling a specific hair-cell protein (green fluorescent signal) g) hair-cells in green are orderly spaced by supporting cells in red. h) Green cells (stochastically or non-stochastically) obtain high levels of Delta ligand. Delta then activates the Notch receptor in adjacent cell membranes. Notch pathway inhibits the production of Delta in the same cell and the adoption of primary fate. The Notch activated cell becomes red. Inhibition of Delta results in lesser activation of Notch in green cells, thus adopting primary fate.

THE RULES OF AN EMBRYO 9 posterior hindbrain, while low doses develop into anterior hindbrain (Maden and Holder, 1992). Jaeger et al. (2008) note, however, that the underlying biochemical mechanisms are likely to be diverse and change rapidly over time, involving a range of regulatory feedbacks on multiple levels, and that experimental design may have been biased by classical positional information ideas. Thus, systems biology emerges as the alternative to unravel complex phenomena involving physics and biochemistry at multiple scales and simultaneous reaction channels (Salazar-Ciudad, 2008), beyond reductionist interpretations. In fact, as more is known on the biological action of morphogens new considerations are incorporated into the models of spatial specification such as i) the dynamics of the gradient as the developing field grows, ii) the exposure time of cells to the morphogen molecule, iii) the stability of the source and sink and, iii) feed-back and feed-forward interactions between the gradient and the specified tissue (Ashe and Briscoe, 2006). In the lateral inhibition model the same principle is applied with an inhibitory feed-back loop (Fig. 2 lower part). This mechanism was described to explain the cellular mosaic patterns found in Drosophila notum, hair-cells in the cochlea or feather buds in birds. The major difference relies on the fact that the pattern is generated by local cell to cell interactions, the activating molecule is anchored to the cell membrane thus allowing only interactions with the neighbouring cells. The activation of the adjacent cells causes turning off the activator and inhibition of the primary fate. The result of this lateral mechanism is that one cell inhibits all the neighbouring cells to become as the activating cell. The Notch-Delta signalling pathway is one of the main pathways involved in this type of response, the singling out of one cell from an equivalence group of cells. It has been shown that this mechanism is repetitively used during development when binary choices must occur, as to say “you become a neurone or you do not.” Following the top-down/bottom-up description above, there exist a plethora of mathematical methods that can be applied to unravel complex problems in De- velopmental Biology. From the top down approach, methods using different types of simulation protocols have been used to formalize gradients of concentrations of different species through space and time, both intra- and intercellular or to describe the characteristics of feedback loops as modular elements of more complex behaviours. Among them, Mathematics provides methods to describe continuous to discrete events processes. Such mathematical frameworks include stochasticity in the form of white (Gaussian-like) noise in otherwise continuous variables modelling or in the form of discrete events firing in fully stochastic simulations. From the other site, bottom-up approaches rely in statistical and probabilistic inference of models from raw data obtained from high throughput experiments. Multiscale simulations are arising as the natural way to merge different methodologies to explain the complexity of biological systems (Newman et al., 2008). In the following section we present with some detail one of the best characterized examples of interpreting patterning by the use of mathematical models, although many others can be found in the references we have introduced during the chapter.

10 ALSINA ET AL.

Segmentation in Drosophila, formulating dynamical gene regulatory networks Drosophila anterior-posterior patterning refers to the segmentation process: the formation of serially repeated units (segments). The order and identity of these segments is controlled by the segmentation genes, which constitute a hierarchical system of interacting genes. Initially, the maternal genes distribute in a gradient and lead to the activation of gap genes. Combined action of maternal and gap genes activate pair-rule genes and finally the latter regulate the activity of segment polarity genes. This regulatory network of genes has been extensively used for mathematical modelling, mainly due to the wealth of data on the temporal sequence of activation, their cross-talk and lately to the quantitative data on their expression levels (Reinitz et al., 1998; von Dassow et al., 2000; Jaeger et al., 2004; Sanchez et al., 2008). The most frequent approach to model developmental genetic networks relies on setting up differential equations stating how the concentration of each regulatory product (variable) varies (the time derivative of the variable) depending on the concentration of its regulators. In general, these equations are coupled and non-linear due to synergistic effects (reviewed in Thieffry and Sanchez, 2003). However, different mathematical methodologies are applied, for example in Von Dassow et al., (2000) mainly simulations and random/directed parameter space exploration were used, in Reinitz et al. (1998) reverse engineering and parameter fitting was used and finally in Jaeger et al. (2004) recent work data-driven models based on quantification of protein levels were employed. On the other hand, advances in mathematical models are walking hand by hand to sophisticated imaging and quantitation techniques of biological processes. Is in this last work in which was shown that the “French flag model” based on morphogen concentrations could not explain some of the patterns observed, and asymmetric feedback repression of posterior genes to the anterior define the final boundaries. Drosophila embryogenesis provides one of the best models for studying gene regulatory networks for several reasons: first, the number of genes implicated in the segmentation process is small, their cross-interactions are well studied, it is possible to visualize the expression of mRNA and proteins but most importantly, the Drosophila embryo is a syncytium (cells are not totally segregated) and in the models continuous intranuclear dynamics of protein can be considered. This is of course an oversimplification, as future realistic mathematical models will need to consider in detail the structure of the nucleus and its dynamics (Kosak and Groudine, 2004). Gregor et al. ( 2007) have also quantified the molecules of bicoid found in Drosophila embryo and finally modeled that about five molecules of bicoid arerequiredtobindtothetargetDNAtotrigger the patterning effects. But, models become difficult when communication between cells must be considered and when different processes and scales in the system such as transcriptional regulation, signal transduction and tissue-level-patterning should be coupled. Quantitative data is unavailable yet for the vertebrate embryo but is in the way as the technology for non-invasive imaging in medicine has proven to successfully measure physiological parameters within tissues. Interestingly, a common view of robustness of the segment polarity network is arising from such mathematical models, often by showing that a wide interval of model parameter values are possible to explain it (von Dassow et al., 2000) or in other cases showing how positive feedback loops are essential to allow for the variation in the parameter space (Ingolia, 2004).

THE RULES OF AN EMBRYO 11

Other types of mathematical models are not based on ordinary differential equations but on discrete-state models such as Boolean and Bayesian models. Those are used to represent relationships, influences and interactions of cellular components as graphs. These models are more abstract because they ignore kinetic properties that describe how the system changes over time, but they can be used to determine dependencies and causal relationships between variables in the model (Soneji et al., 2007). As seen by new mathematical models of limb growth, lateral inhibition process during neurogenesis, left-right asymmetry or dynamics of axon growth navigation, more and more collaborations between modellers and biologists are springing up (Shimmi et al., 2005) (Newman et al., 2008). Finally, it seems that biologists are ready to step into the world of formalism and numerical data and mathematicians enter into the molecular world of life sciences. In the upcoming years new developments are expected to arise, in particular in the field of multiscale simulations, which will benefit from the knowledge of both top-down and bottom-up visions of complex biological systems. Acknowledgements. We are grateful to Cristina Pujades for comments on the manuscript.

References

[1] Briggs, R. and King, T.J. (1952) Transplantation of Living Nuclei From Blastula Cells into Enucleated Frogs’ Eggs. Proc Natl Acad Sci U S A. 38:455-63. [2] Carroll, S.B., Grenier, J.K., Weatherbee, S. D. (2001). From DNA to Diversity. Molecular Genetics and the Evolution of Animal Design. Firs Edition. Blackwell Science Inc. USA. [3] Carroll, S.B. (2005). Endless Forms Most Beautiful. The new science of evo devo. W.W Norton & Company, Inc., New York, USA.. [4] Cho, A.(2004)Life’s Patterns: No need to spell it out? Science 303: 7823. [5] Davidson, E.H. (2006). The regulatory Genome. Gene Regulatory Networks in Development and Evolution. Elsevier Academic Press. Canada. [6] Giepmans, B.N.G., Adams, S.R., Ellisman, M.H. and Tsien, R.Y. (2006). The fluorescent toolbox for assessing protein location and function. Science 312: 217-224. [7] Gierer, A. and Meinhardt, H. (1972). A theory of biological pattern formation. Kybernetik 12: 30-39. [8] Gilbert, S. F. (2003). Developmental Biology. Seventh Edition. Sinauer Associates, Inc USA. [9] Gurdon, J.B. (1962)Adult frogs derived from the nuclei of single somatic cells. Dev Biol. 4:2563. [10] Gurdon, J.B. and Byrne, J.A. (2004)The first half-century of nuclear transplantation. Proc Natl Acad Sci U S A. 100048-52. [11] Jacob, F. and Monod, J. (1961). Genetic Regulatory Mechanisms in the Synthesis of Proteins. Journal of Molecular Biology 3: 318-356. [12] Jaeger, J., Surkova, S., Blagov, M., Janssens, H., Kosman, D., Kozlov, K.N., Myasnikova, E., Vanario-Alonso, C.E., Samsonova, M., Sharp, D. H., Reinitz, J. (2004). Dynamic control of positional information in the early Drosohpila embryo. Nature 430: 3-371. [13] Kerszberg, M. and Wolpert, L.. (20)Specify ing positional information in the embryo: looking beyond morphogens. Cell. 130:205-9. [14] May, R. M. (2004). Uses and abuses of mathematics in biology. Science 303: 7903. [15] Reinitz, J., Kosman, D., Vanario Alonso, C.E, Sharp, D.H. (1998). Stripe forming architecture of the gap gene system. Dev. Genet. 23: 11-. [16] Slack, J.M.W. (20). From egg to embryo: regional specification in early development. 2nd Edition. Series: Developmental and Cell Biology Series (26). [17] Thieffry, D. and Sanchez, L.. (2003). Dynamical modelling of pattern formation during embryonic development. Current Opinion in Genetics and Development. 13: 326-330. [18] Tomlin, C.J. and Axelrod, J.D. (20). Biology by numbers: mathematical modelling in Developmental Biology. Nature Reviews Genetics 8: 331-340.

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[19] Turing, A.M. (1952).The chemical basis of morphogenesis. Philos. Trans. Soc. London B, 237: 372. [20] Von Dassow, G., Meir, E., Munro, E.M, Odell, G.M. (2000). The segment polarity network is a robust developmental module. Nature 406: 188-192. [21] Wilmut, I., Schnieke, A. E., Mcwhir, J., Kind, A. J., Campebell, K.H.S. (1997). Viable oﬀspring derived from fetal and adlult mammalian cells. Nature 385: 81013. [22] Wolpert, L., The French ﬂag problem: a contribution to the discussion on pattern development and regulation, Towards a Theoretical Biology, v.2 (Ed. C. Waddington, Pub.: Edin- burgh University Press, 19), 125-133. [23] Wolpert, L. (1996). One hundred years of positional information. Trends Genet. 9: 359-364.

Developmental Biology Group; Departament de Ciencies` Experimentals i de la Salut, Universitat Pompeu Fabra (UPF), Parc de Recerca Biomedica` de Barcelona, (PRBB) c/Doctor Aiguader, 88, 08003 Barcelona, Spain E-mail address: [email protected]

Computational Biochemistry and Biophysics laboratory; Departament de Ciencies` Experimentals i de la Salut, Universitat Pompeu Fabra (UPF), Parc de Recerca Biomedica` de Barcelona, (PRBB) c/Doctor Aiguader, 88, 08003 Barcelona, Spain E-mail address: [email protected]

Contemporary Mathematics Volume 492, 2009

From lineage to shape: modeling dorsal-ventral speciﬁcation in the developing mouse limb

Carlos G. Arques and Miguel Torres

Abstract. A hallmark of the development of multicellular organisms is the appearance of discrete domains of expression of developmentally important genes. The correct development of the organism requires that these domains adopt a precise location and shape. These domain maps become progressively more refined and complex as development advances, and a question of major and long-standing interest for developmental biologists is how information from domains already present at one stage is used to generate correctly positioned and shaped new domains. Along the way, work in this area has produced some important concepts such as that of the morphogen—a diffusible molecule that forms a concentration gradient across a developmental field, working through a threshold mechanism to activate different genes at specified distances from its source, effectively regionalizing the developmental field. An example of a developmental gene expression domain whose emergence has lacked a satisfactory explanation is that of Lmx1b in the dorsal limb bud. This domain occupies the dorsal half of the limb bud mesenchyme, and its ventral limit matches reasonably well with the central plane of the organ. The only molecule identified as being responsible for the induction of Lmx1b expression in the limb bud is Wna, which diffuses from the dorsal ectoderm and is known to be a short range signal. However, it has been suggested that such a signal cannot, on its own, account for the three-dimensional shape of the 3D Lmx1b domain. In recent years light has begun to be shed on this question with the discovery of dorsal-ventral compartmentalization of the limb bud mensenchyme, and evidence has been presented showing that the Lmx1b expression domain is coincident with the dorsal compartment. To further expand our understanding of how dorsal-ventral patterning of the vertebrate limb takes place, we have devised a model that we call inverted cistern. In this model, dorsal-ventral compartmentalization is central to the question of how a short-range dorsal ectodermal signal is able to generate a solid domain comprising the dorsal half of the limb bud mesenchyme. We have computer-simulated this model, and the results satisfactorily match what is known for the wild-type organ and, interestingly, also for the anti-intuitive En1 mutant.

This work was supported by grants from the Spanish Ministry of Science and Innovation, (BFU2006-10978/BMC), the Human Frontiers Science Program (RGP00/2004) and the Euro- pean Commission (LSHG-CT-2003-5032259). The CNIC is supported by the Spanish Ministry of Science and Innovation and the Pro-CNIC Foundation.

⃝c ⃝XXXXc 2009 American Mathematical Society 13

14 CARLOS G. ARQUES AND MIGUEL TORRES

1. Introduction

1.1. Developmental biology, complex systems, and computer-assisted modeling Developmental biology is defined—plainly speaking—as the study of the mechanisms and processes through which an adult multicellular animal, plant, or fungus forms from a single cell—the fertilized egg, or zygote. In a developing multicellular organism a great many cells (typically thousands or millions) simultaneously interact and divide, differentiate into diverse cell types, and organize as tissues and organs, resulting in an adult organism that shows certain fixed and reproducible features. This characteristic, of being constituted by many parts that interact to produce a totality that is not reducible to the interactions between these parts, identifies the developing multicellular organism as a complex system. For the study of complex systems, the researcher’s intuition is of limited utility, and computer-assisted modeling is especially valuable—whether the system under study is biological or otherwise. Here, we have designed a computer simulation that attempts to deepen our understanding of the phenomenon—recently described in our lab—of dorsal-ventral compartmentalization of mesenchyme in the developing mouse limb.

1.2. Limb development The vertebrate limb is a model system traditionally used in developmental biology. Being an exposed, non-essential, organ, it is amenable to several surgical techniques impossible in other organs. Additionally, the limb nonetheless develops a complex pattern, some of whose underlying molecular mechanisms are the same as those used in the development of other structures. After decades of research, the main molecular mechanisms that generate pattern along all three limb axes (proximal-distal, anterior-posterior, dorsal-ventral) are known[1]–[3]. Nonetheless, many important details have yet to be elucidated.

Dorsal-ventral patterning Vertebrate limbs have a complex dorsal-ventral pattern that is especially patent in the morphology of muscle and tendon. The fundamental molecular mechanisms underlying the generation of this pattern are known [4]–[18]. The dorsal ectoderm of the limb bud produces the diﬀusible factor Wnt7a. In the ventral ectoderm, expression of this factor is impeded by the presence there of the transcription factor En1. Wnt7a diﬀuses from dorsal ectoderm to the mesenchyme under it, where it induces the expression of the transcription factor Lmx1b. The Lmx1b-positive domain extends to occupy the dorsal half of limb bud mesenchyme, and the presence of Lmx1b determines dorsal identity in mesenchymal cells (Figure 1). Thus mutants lacking Lmx1b have striking doubly-ventral limbs, and mutants in which Lmx1b is ectopically expressed in ventral regions show dorsalized limbs.

Clonal analysis Clonal analysis has historically been a very productive technique in developmental biology. It consists of placing an indelible mark on a cell at a certain stage during development. This mark must be such that it is inherited by all the progeny of the cell. In this way we can, at some later point, examine the shape, distribution,

FROM LINEAGE TO SHAPE 15

Figure 1. Molecules known to be involved in dorsal-ventral limb bud patterning, shown here in drawings of two diﬀerent sectional views of a limb bud.

and contribution to diﬀerent organs and tissues of the clonal descendants of the originally marked cell [19]–[28]. This technique, although very fruitful in the study of insect development, has nonetheless had very limited application in the study of mouse embryo development— mainly because of technical diﬃculties—and this is especially true for limb development.

A new system for clonal analysis in the mouse Part of the work of our laboratory has been [29] to establish a clonal analysis tool of sufficient power and versatility for use in the mouse embryo (Figures 2 and 3). This tool is based on the combined use of a ubiquitously expressed inducible recombinase and a ubiquitous reporter for that recombinase. This allows us, at the desired point during embryo development, to induce the genetic modification in a cell. From that point on, the cell will express the reporter gene 𝛽-galactosidase. This genetic modification is inherited by all progeny, which can be later visualized in the embryo as blue-stained cells.

1.3. The developing limb is compartmentalized along the dorsal-ventral axis The use of this technique in the study of the limb has led us to a striking discovery— mesenchymal cells of the dorsal and ventral halves of the developing limb are of distinct lineages that remain physically separated (Figure 3). It is the cells of dorsal lineage that express the transcription factor Lmx1b [29]. This phenomenon, in which the ancestry of each cell determines its allocation to speciﬁc cellular territories that remain coherent and unmixed, is known in the developmental biology tradition as compartmentalization, and each of those territories is called a compartment [30]. The organization of the developing embryo into compartments was ﬁrst described in the fruitfully, and since then in many other organisms, including vertebrates. Compartmentalization is considered to be a fundamental process during development [31], providing the means through which development becomes hier- archized and modularized. Compartments have been suggested to be autonomous developmental units, possessing their own mechanisms of pattern generation and size control [32].

16 CARLOS G. ARQUES AND MIGUEL TORRES

Figure 2. Flow diagram of the clonal analysis system established in our laboratory.

The division of a main model organ—the vertebrate limb—into two compartments is by itself significant; in addition to this, these compartments have certain atypical characteristics that make them even more interesting: 1. They are the first compartments—in any organism—that have been described in a non-epithelial tissue. Unlike all compartments described up until now—which were in epithelial tissues and thus bi-dimensional—these new compartments exist in a tissue, mesenchyme, that fills volumes. They are thus the first 3D compartments described, and their frontiers are not lines, but surfaces. 2. This is the first compartment division described in vertebrates whose fron- tier does not match or anticipate an anatomical border. The frontiers of all compartments described in vertebrates until now have anticipated or matched a later anatomical border. This is not the case for many of the compartments described in insects and neither is it the case for these new vertebrate compartments. 3. The plane separating the two compartments does not appear to function as a signaling center. It is a typical feature of compartments—especially in insects—that their borders are signaling centers, producing diffusible molecules that

FROM LINEAGE TO SHAPE 17

Figure 3. Developing limbs showing ventral (A) and dorsal (B) somatic clones generated using the clonal analysis system referred to in the article. The dorsal-ventral compartmentalization of the developing limb is clear (From Arques et al. 2007).

provide positional information for cells in neighboring compartments and contribute to their patterning. At the time of writing, no molecule is known that is speciﬁcally produced in the plane that separates the dorsal and ventral domains of the limb mesenchyme.

New compartments, new questions These features of the newly discovered dorsal and ventral compartments of the limb mesenchyme raise new questions: 1. What are the mechanisms of the formation and maintenance of these 3D compartments? Are they the same or diﬀerent from those that regulate bi-dimensional compartments? 2. If these borders do not match or anticipate anatomical borders, and neither form signaling centers, what is the function of these new compartments? Why has evolution selected and conserved the molecular machinery needed for their formation and maintenance? In an attempt to address these questions we have modeled the essential features of the system in a computer simulation.

18 CARLOS G. ARQUES AND MIGUEL TORRES

2. Results

2.1. A simplified EM model To model dorsal-ventral patterning in the vertebrate limb (including compartments) we used a simplified version of extended cellular Potts model [33] (ECPM) imple- mented in Netlogo programming environment. ECPM type models have been successfully used to model very diverse biological processes, such as the aggregation, migration and fructification of the slime mold Dictyostelium discoideum [34], generation of capillary networks [35], and the formation of skeletal precursors in the chicken limb [36]. The Netlogo programming environment has been used previously for the qualitative simulation of complex systems [37], including biological systems, and is suitable for rapid model prototyping when qualitative information is needed. However, its use forced some restrictions on model size and complexity. In an ECPM model every cell is represented by a set of pixels (or voxels if the model is 3D). A global energy function is defined for the whole system that depends on factors such as cell shape and the extent of contact between different types of cells. This system is allowed to evolve according to the Metropolis algorithm [38]– [40], effectively minimizing the energy function. The system states compared in the Metropolis algorithm are the current state and another state in which one cell is extended to a random neighboring pixel. In our simplified ECPM system, each cell is represented by one pixel, and the Metropolis algorithm works by comparing the current state to a candidate state in which two random neighboring cells have swapped places.

2.2. The inverted cistern model In this model—which we have named inverted cistern—a section of the developing limb is assumed to be composed of ectoderm and mesenchyme. For simplicity, sections are circular, with ectoderm on the exterior circumference (two cells deep) and mesenchyme making up the rest. Ectoderm is divided into dorsal and ventral, and only dorsal cells are the source of diffusible factor Wnt7a. This factor diffuses (according to Netlogo function diffuse) and is degraded in every cell, so that in a short time it forms a stable gradient (Figure 4). The total energy of the system (Hamiltonian) is considered to be simply the sum of contact energies between every cell and its closest eight neighbors. The model is then defined by the following features: 1. Activation of Lmx1b expression depends on the presence of Wnt7a. So at the beginning of the simulation all cells are ventral type (no Lmx1b expressed) but can change their state if exposed to Wnt7a. 2. The Wnt7a signal is short range. A threshold is defined such that when a cell is exposed to a Wnt7a concentration above this, Lmx1b expression is activated. This threshold is set sufficiently high to restrict activation to mesenchyme regions very close to the dorsal ectoderm. 3. Lmx1b activation is all-or-nothing. 4. Once a cell has activated Lmx1b expression (it has become dorsal), it will never deactivate it. Therefore, if a cell later migrates away from the high Wnt7a region it will keep its Lmx1b activated state and its dorsal identity.

FROM LINEAGE TO SHAPE 19

Figure 4. A) Representation of a section of a limb bud used in the simulation. The Wnt7a gradient originating in the dorsal ectoderm is shown (shades of grey). In this initial state the whole mesenchyme is of ventral identity. B), C), and D) Progression of the simulation from the initial state. Cells that have been exposed to above-threshold concentrations of Wnt7a and have converted to dorsal identity are shown in dark grey. E) Resultant pattern that reasonably matches that of a real limb bud. F) En1 mutant simulation. All the ectoderm has been made to express Wnt7a. In the resultant pattern a central negative domain is found that, as in the real case, includes a region that is positive in the wildtype.

5. Ventral and dorsal cells must be able to separate into two coherent domains that do not mix. This has been achieved by assigning specific affinity energies to homotypic and heterotypic cell contacts. Inverted cistern reproduces limb dorsal-ventral pattern To keep the model as simple as possible, the only contact energies we permitted to differ from zero were those between dorsal and ventral mesenchymal cells (Edv). We tried to manually adjust the Edv value for the model to reach and maintain the desired biological-like pattern. This proved to be possible and the empirical best

20 CARLOS G. ARQUES AND MIGUEL TORRES

Figure 5. Evolution of the system during the ﬁrst 2⋅105 Monte- Carlo steps. The 𝑦 axis shows the percentage of the vertical dimension (radius for En1 KO) of the simulated limb occupied by dorsal cells. Edv (arbitrary units) represents the (repulsive) contact energy between two neighboring dorsal and a ventral cells. All other contact energies are set to 0. For the En1 KO (dotted line) the Edv value is 4.0. Each line shows the average of 4 independent runs.

value for Edv was found to be 4.0 (arbitrary units). This value had a repulsive effect, and, as said, all other contact energies were kept at 0. The outcome of the simulations was that, for the selected affinity values, the system reaches a stable configuration that is remarkably coincident with that observed experimentally (Figure 4). The generation of this configuration can be explained qualitatively by the occurrence of the following phenomena (Figure 4): 1. At first, the mesenchyme cells closest to the dorsal ectoderm (and therefore exposed to an above-threshold Wnt7a concentration) change their identity to dorsal. 2. These same cells, as a result of differences in homo and hererotypic affinities, aggregate, freeing space in high Wnt7a regions that become occupied by new ventral cells, which will thus change their type to dorsal. 3. Eventually enough cells will be converted to dorsal identity to occupy the entire dorsal half of mesenchyme. This configuration is highly stable, because no new ventral cells can access the high Wnt7a regions and change type. Of course, due to the stochastic nature of the system, given enough time all mesenchymal cells will become dorsal. How long this takes depends on parameter values (Figure 5) and can be extremely long. For our purposes, the system behaves ”correctly” when its evolution shows two clearly distinct phases: one in which the dorsal compartment grows rapidly until it occupies about half the limb height, and a second one of very slow dorsal growth.

FROM LINEAGE TO SHAPE 21

As shown (Figure 5), this behaviour is essentially resistant to variations in the value of Edv of up to 25%, affecting only the height of dorsal compartment at what the second, stable phase is reached. This height varies between approximately 40% and 60% of the total section height. Reductions of greater than 25% in Edv value disrupt the stability of the second phase, allowing relatively fast filling of the limb with dorsal cells. Increases of greater than 25% simply lock the system in local minima, impeding its progression toward the desired configuration.

Inverted cistern reproduces the anti-intuitive pattern of En1 KO In En1 loss-of-function (KO) mutant embryos, the Wnt7a ectodermal domain expands to occupy the whole limb ectoderm. However, contrary to what might be expected these mutants do not express Lmx1b throughout the mesenchyme. Instead Lmx1b is expressed in a subectodermal domain, which surrounds a circular Lmx1b- negative mesenchymal domain in the center [41].This result was used to show that the Wnt7a signal is short range. Interestingly, the central Lmx1b-negative domain includes a region that is positive for Lmx1b in wild-type embryos. The inverted cistern model satisfactorily reproduces the pattern observed in this mutant. When all simulated ectoderm is made to express Wnt7a, a subectodermal domain of Lmx1b-expressing cells is created that is stable. This process stops when this domain is deep enough to prevent new ventral-type cells (which now occupy a central position) from reaching the high Wnt7a region and converting to dorsal-type. So ﬁnally a central circular domain of ventral-type (Lmx1b-negative) cells is formed that is surrounded by a domain of dorsal-type (Lmx1b-positive) cells under the ectoderm (Figure 4). As in the real mutant limbs, the central domain includes a region that is Lmx1b-positive in the wild-type model. As can be seen (Figure 5) the obtained pattern for En1 KO is not as stable as that for the wild-type limb. Nonetheless, after 2⋅105 Monte-Carlo steps, cells of ventral identity still occupy a circular central domain with a radius approximately 30% that of the whole section.

Dorsal-ventral compartmentalization is required for correct Lmx1b domain shape In our model, which does not include cell division, compartmentalization is represented by the irreversibility of the ventral-to-dorsal transition and the values of the intercellular cell affinities that make ventral and dorsal cells form coherent non- mixable domains. Trivially, if any of these fails, the shape of the simulated Lmx1b domain will no longer resemble the real one. So, if irreversibility of ventral to dorsal transition is removed (so that cells revert to a ventral identity when they exit the high Wnt7a zone), then the shape of the Lmx1b domain is merely a shallow subectodermal dorsal domain that does not fill the dorsal half of the mesenchyme. In a similar way, if differences in homotypic and heterotypic affinities are removed, then no coherent dorsal domain forms and new ventral cells can always access high Wnt7a areas, so that after a while the whole mesenchyme acquires a dorsal identity.

22 CARLOS G. ARQUES AND MIGUEL TORRES

3. Discussion

3.1. Dorsal-ventral compartmentalization could explain the shape of Lmx1b domain Limb dorsal-ventral patterning can be explained on the basis of the known facts As we have tried to show, this model, based solely on the known facts, is able to reproduce the dorsal-ventral pattern observed in the developing vertebrate limb. So, an important part of its value is to show that what is already known is probably sufficient for a qualitative explanation of this pattern formation. Furthermore, its ability to naturally reproduce an anti-intuitive mutant phenotype such as the En1 KO suggests that at least some essential aspect of the working of the model is faithful to reality. A novel mechanism for compartment maintenance From a more general theoretical viewpoint, the model introduces the possibility that compartment formation and maintenance could be, at least in some cases, based on the stabilization of the final pattern by compartment dynamics itself. In this scenario, lineage restriction is achieved not because cells are intrinsically unable to change from one type to another, but because the spatial configuration of compartments itself impedes cells from coming into the range of the inductor molecules they require to make that change. From semi-circumference to semicircle via compartments From our point of view, the most interesting contribution this model makes is its suggestion that the reason for the existence of this atypical dorsal-ventral compartmentalization in the developing limb is that it is needed to achieve the right shape of the Lmx1b expression domain (and therefore of the dorsal and ventral domains themselves). Compartments have traditionally been assigned functions such as the formation of signaling centers at their borders, modulation of patterning processes, or size control. The output of the inverted cistern model suggests that at least sometimes acompartment’sraison d’être might be something more basic: simply to achieve the right shape. The model thus proposes that dorsal-ventral compartmentalization in the developing limb mesenchyme is needed to ensure that dorsal and ventral halves of the limb form real halves, and not some other shape. In fact, this might be its sole function. So, faced with the problem of how to convert the shape of a 2D dorsal ectodermal domain (represented as a semi-circumference in the model) into a 3D half-limb domain (represented as a semicircle) with minimal means (one short-range diffusible factor, Wnt7a), vertebrate limb evolution could have come up with compartmentalization as a practical solution. 3.2. Testability of the model An important aspect of every biological model is that it should be possible to test the extent to which reality does and does not conform to it. The inverted cistern model makes the following testable predictions: 1. If, in a clonal analysis experiment in which clones are induced after compartment set up, a clone is composed of a mixture of cells expressing

FROM LINEAGE TO SHAPE 23

and not expressing Lmx1b (thus violating dorsoventral restriction), the majority should be ventral. This is because the proposed mechanism of compartment maintenance could allow a small probability of a ventral cell reaching the high Wnt7a region and converting to dorsal, but the opposite conversion should not be possible. 2. This kind of violation (with changes in the state of expression of Lmx1b) should happen mostly in the subectodermal region. This is because this is the only place where a ventral cell can change its Lmx1b expression state. 3. Any violating clone that is mostly dorsal should express Lmx1b in all cells, even those in the ventral region. One can imagine that a dorsal cell might, with low probability, be displaced transiently to a ventral region, but the model would predict that there would be no change in Lmx1b expression state of these cells. 4. If at some point adhesion molecules controlled by Lmx1b are discovered, their loss-of-function mutant should present dorsalized limbs, and mutants in which those molecules are ectopically expressed in ventral cells should also present dorsalized limbs. This is because, as we have seen, if dorsal and ventral cells can freely mix, according to the model every mesenchymal cell should end up being dorsal. With the exclusion of the last prediction, which depends upon a hypothetical discovery, these predictions should be to some extent testable on the basis of current knowledge; and we expect this will be done in the future. Additionally, for the cistern model to work in the real limb, cell mixing should be relatively high, at least in the dorsal region. As has been shown [29], this is in fact the case not only for the dorsal region, but for the whole limb bud. 3.3. Limitations The model has some limitations that must be taken into account. As presented here, the model does not consider three-dimensionality or growth of the real organ. It is our opinion that these two factors are not of essential importance, but also that they cannot be discarded apriori. So, an obvious and desirable extension to the model would be to include these two features. On the other hand, the eventual acquisition of dorsal identity by the whole section does not render the model theoretically implausible. As stated above, the time need for this can be very long (with Edv=4.0, the dorsal compartment doesn’t reach 60% after 2⋅106 MCS). Moreover, in the real organ the pattern only needs to be stable enough to maintain itself in the context of organ development. Acknowledgements. We thank Cristina Claver´ıa, Cristina Pujades, and Fer- nando Gir´aldez for valuable advice and suggestions during the early stages of this work.

References

[1] Niswander, Pattern formation: old models out on a limb, Nature Reviews, Genetics 4, (2003) 735–741. [2] Tickle, Patterning systems-from one end of the limbto the other , Developmental Cell 4, (2003)6103–6116.

24 CARLOS G. ARQUES AND MIGUEL TORRES

[3] Robert and Lallemand, Anteroposterior patterning in the limband digit specification: contribution of mouse genetics, Developmental Dynamics 235, (2006). [4] Patou, Analyse de la morphogenese due pled des Oiseaux a la’aide de malange cellulaires interspecifiques, I. Etude morphologie. J. Embryol. Exp. Morph. 29, (1973)175–196. [5] MacCabe, Errick and Saunders, Ectodermal control of the dorsoventral axis in the leg bud of the chick embryo, Developmental Biology 39, (1974)473–481. [6] Geduspan and MacCabe, Transfer of dorsoventral information from mesoderm to ectoderm at the onset of limbdevelopment , The Anatomical Record 224, (1989)79–87. [7] Michaud, Lapointe and Le Douarin, The dorsoventral polarity of the presumptive limbis determined by signals produced by the somites and by the lateral somatopleure, Development 124, (1997)66–72. [8] Parr and McMahon, Dorsalizing signal Wnt7a required for normal polarity of D-V and A-P axes of mouse limb,Nature374, (1995)5252–5260. [9] Wurst, Auerbach and Joyner, Multiple developmental defects in Engrailed-1 mutant mice: an early mid-hindbrain deletion and patterning defects in forelimbs and sternum, Development 120, (1994)2065–25. [10] Loomis, Kimmel, Tong, Michaud and Joyner, Analysis of the genetic pathway leading to formation of ectopic apical ectodermal ridges in mouse Engrailed-1 mutant limbs, Development 125,(1998)3521–3532. [11] Ahn, Mishima, Hanks, Behringer and Crenshaw, BMPR-1A signalling is required for the formation of the apical ectodermal ridge and dorsal-ventral patterning of the limb, Development 128, (2001)–30. [12] Pizzete and Niswander, Early steps in limbpatterning and chrondogenesis , Novartis Foun- dation symposium 232, (2001)23–36; discussion 36–46. [13] Riddle, Ensini, Nelson, Tsuchida, Jessell and Tabin, Induction of the LIM homeobox gene Lmx1 by WNT7a establishes dorsoventral pattern in the vertebrate limb,Cell83, (1995) 631–640. [14] Vogel, Rodr´ıguez, Warnken and Izpisua Belmonte, Dorsal cell fate specified by chick Lmx1 during vertebrate limb development,Nature78, (1995)221–230. [15] Chen, Lun, Ovchinnikov, Kokubo, Oberg, Pepicelli, Gan, Lee and Johnson, Limband kidney defects in Lmx1bmutant mice suggest an i nvolvement of LMX1B in human nail-patella syndrome, Nature Genetics 19, (1998)209–213. [16] Schweizer, Johnson and Brand-Saberi, Characterization of migration behaviour of myogenic precursor cells in the limbbudwith respect to Lmx1bexpression , Anatomy and Embryology 208, (2004)78–84. [17] Gonzalez, Swales, Bejsovec, Skaer and Mart´ınez Arias, Secretion and movement of wingless protein in the epidermis of the Drosophila embryo, Mechanisms of Development 35, (1991) 43–54. [18] Nusse and Varmus, Wnt genes,Cell69, (1992)13–17. [19] Blair, Genetic mosaic techniques for studying Drosophila development, Development 130, (2003)22–24, –30. [20] Garcia-Bellido, Ripoll and Morata, Developmental compartmentalisation of the wing disk of Drosophila, Nature: New Biology (1973) 245, 251–253. [21] Patterson, The production of mutations in somatic cells of Drosophila melanogaster by means of x-rays, Journal of Experimental Zoology 53, (1929)3–372. [22] Harrison and Perrimon, Simple and efficient generation of marked clones in Drosophila, Current Biology 3, (1993)16–20. [23] Liu, Jenkins and Copeland, Efficient Cre-losP-induced mitotic recombination in mouse embryonic stem cells, Nature Genetics 30,(2002)66–72. [24] Zong, Espinosa, Su, Muzumdar and Luo, Mosaic analysis with double markers in mice,Cell 121, (2005)66–72. [25] Nicolas, Mathis, Bonnerot and Saurin, Evidence in the mouse for self-renewing stem cells in the formation of a segmented longitudinal structure, the myotome, Development 122, (1996) 981–984. [26] Mathis, Bonnerot, Puelles and Nicolas, Retrospective clonal analysis of the cerebellum using genetic laacZ/lacZ mouse mosaics, Development 124, (1997)49–4104. [] Bonnerot and Nicolas, Application of LacZ gene fusions to postimplantation development, Methods in Enzymology 225, (1993)434–451.

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[] Petit, Legue and Nicolas, Methods in clonal analysis and applications, Reproduction, Nutri- tion, Development 45, (2005)11–17 [29] Arques, Doohan, Sharpe and Torres, Cell tracing reveals a dorsoventral lineage restriction plane in the mouse limbbudmesenchyme , Development 134, (20)3713–3722. [30] Garcia-Bellido, Genetic control of wing disc development in Drosophila, Ciba Foundation Symposium, (1975)66–72. [31] Lawrence and Struhl, Morphogens, compartments, and pattern: lessons from drosophil , Cell 85, (1996)66–72. [32] Martin and Morata, Compartments and the control of growth in the Drosophila wing imaginal disc, Development 133, (2006)66–72 [33] Graner and Glazier, Simulation of biological cell sorting using a two-dimensional extended Potts model, Physical Review Letters 69, (1992)2013–2016. [34] Mar´ee and Hogeweg, How amoeboids self-organize into a fruiting body: multicellular coordination in Dictyostelium discoideum, PNAS 98, (2001)3879–83. [35] Merks, Brodsky, Goligorksy, Newman and Glazier, Cell elongation is key to in silico repli- cation of in vitro vasculogenesis and subsequent remodeling, Developmental Biology 289, (2006)44–54. [36] George, Hentschel, Glimm, Glazier and Newman, Dynamical mechanisms for skeletal pattern formation in the vertebrate limb, Proceedings of the Royal Society London B 271, (2004) 1713–1722. [37] Chitnis and Itoh, Exploring alternative models of rostral-caudal patterning in the zebraﬁsh neurectoderm with computer simulations, Current Opinion in Genetics & Development 14, (2004)415–421. [38] Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller, Equations of State Calculations by Fast Computing Machines, Journal of Chemical Physics 21, (1953)17–1092. [39] Hastings, Monte Carlo Sampling Methods Using Markov Chains and Their Applications, Biometrika 57, (1970)97–109. [40] Siddhartha and Greenberg, Understanding the Metropolisastings Algorithm , American Statistician 49, (1995)3–335. [41] Cygan, Johnson and McMahon, Novel regulatory interactions revealed by studies of murine limbpattern in Wnt7a and En1 mutatns, Development 124, (1997)5021–5032.

Fundacion´ Centro Nacional de Investigaciones Cardiovasculares Carlos III, Mel- chor Fernandez Almagro 3, 28029 Madrid, Spain E-mail address: [email protected]

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Contemporary Mathematics Volume 492, 2009

Notch-Mathics

Rita Fior and Domingos Henrique

Abstract. In this review, we will first present a brief overview of the Notch pathway and its components and then focus on its function during neural development. By analogy with the proposed model for the somitogenesis clock [1], we propose that neurogenesis might also be controlled by two interacting circuits that could generate oscillations of gene expression. These two circuits are very similar but have completely different outcomes: one seems to tune cells for the same behaviour- making a somite, whereas the other makes two cells acquire different fates. As Julian Lewis pointed out [1], intuition is not enough: we need mathematics - and that is the challenge that we propose here.

1. The Notch signaling pathway The building of a multicellular organism relies on the ability to form organized cellular patterns - and this is only possible if cells ‘talk’ to each other and inﬂuence one another’s fate and behaviour. One of the most important cell-cell communication mechanisms is mediated by the Notch signalling pathway. This pathway is implicated in probably all developmental programs in vertebrate embryos, not just neural development; it is implicated in vascular and kidney development, body segmentation, intestine, skin and hematopoetic development and many other developmental processes (reviewed in [2], [3]). The ﬁrst reports of Notch mutants in Drosophila, by Donald Poulson [4], [5], described that absence of Notch activity in the embryo results in hypertrophy of the neural tissue at the expense of the epidermis, a phenotype that was named “neurogenic” and later shown to be characteristic of other Drosophila mutants [6]. At the core of the Notch signalling pathway is the transmembrane Notch receptor in one cell, interacting with a transmembrane ligand in a neighbouring cell. Notch is a large type-I transmembrane receptor that accumulates at the plasma membrane as a dimer of two polypeptides, covalently bound, the Notch Extracellular Domain (NECD) and a membrane-bound intracellular domain (NTM). These two polypeptides are formed from the original Notch protein in the trans-Golgi as the result of proteolytic activity by a Furin protease that constitutively cleaves Notch molecules at the S1 site (Fig. 1). The known Notch ligands belong to Delta-Serrate-Lag2 (DSL) family and are also type-I transmembrane receptors [7]. Upon ligand-receptor interaction, the Notch receptor undergoes successive proteolytic cleavages that lead to the release of the Notch intracellular domain (NICD).

⃝c ⃝XXXXc 2009 American Mathematical Society 27

28 RITA FIOR AND DOMINGOS HENRIQUE

Figure 1. A partial view of Notch in action, mediating communication between two adjacent cells. The mechanisms that lead to Notch activity in the receiving cell are depicted, including the cleavage of Notch at the cell membrane and the assem- bly of a tripartite nuclear complex with CSL and MAM. NECD (Notch extracellular domain), NTM (NICD + transmembrane domain), NICD (Notch intracellular domain). Delta at the surface of the signal-sending cell binds the Notch extracellular domain. Upon ligand-receptor interaction, S2 proteolytical cleavage occurs releasing NEXT, NTM is then further processed at the S3 and sites, releasing NICD. NICD then translocates into the nucleus where it associates with CSL and MAM, displacing the co-repressor (CoR) and thereby triggering a switch from repression to activation. The best studied targets of the Notch pathway are the hes family genes. Adapted from [8].

Cleavage at the S2 site is triggered by ligand binding to NECD and is carried out by the ADAM/TACE/Kuzbanian family of metaloproteases (reviewed in [8]). This S2 cleavage generates an activated membrane-bound form of Notch, NEXT (Notch Extracellular Truncation). Subsequently, NEXT is further processed at two more cleavages sites - S3 and , releasing the NICD into the cytoplasm and a small peptide (N𝛽) to the extracellular space. The S3 and cleavages sites are located within the transmembrane domain and are catalyzed by the -secretase activity of the Presenilin-Nicastrin-Aph1-Pen2 protein complex (reviewed in [1], [8]–[11]).

NOTCH-MATHICS 29

The NICD fragment translocates to the nucleus and is the active form of the receptor, binding to the CSL transcription factor (mammalian CBF-1, Drosophila Supressor of Hairless and C.elegans Lag-1) and to the Mastermind (MAM and C. elegans Lag-3) co-activator, forming a tripartite complex that activates transcription of target genes (reviewed in [12]). In the absence of NICD, the CSL transcription factor assembles a repressor complex at the cis-regulatory regions of the CSL/NICD target genes, which are therefore transcriptionally inactive [13]–[17]. When NICD translocates to the nucleus and binds to CSL and MAM, the co-repressor complexes are displaced, repression is relieved and transcription is activated, thereby switching from CSL-mediated repression to NICD/CSL/MAM-mediated activation (Fig. 1). The best characterized NICD targets are genes of the hes (hairy and Enhancer of split)andhrt (hairy-related) families, encoding bHLH (basic Helix-Loop-Helix) transcriptional repressors (reviewed in [18]), which function as effectors to implement the cell fate decisions mediated by Notch signalling. Although the Notch pathway is probably involved in the development of all tissues, regulating cell fate specification, patterning, proliferation, cell death and cell morphology, all these different outputs tend to fall in two types of operational logic: lateral inhibition- that mostly mediates binary cell fate decisions, and lateral induction - that in most of the cases is involved in boundary formation separating different cellular fields (reviewed in [12], [19]). In this review, we will focus on lateral inhibition, which is the operation that occurs during neurogenesis and that is better understood. We will also discuss the role of Notch signalling during somitogenesis, where Notch seems to be synchronizing cells for the same behaviour, perhaps reflecting yet another type of operational logic.

2. Lateral Inhibition:binary cell fate decisions The Notch-mediated lateral inhibition mechanism regulates interactions between cells with equivalent or similar potential, ensuring that they acquire one of two alternative fates. In this process, Notch signalling has no instructive role on the decision, functioning mainly to guarantee that the interacting cells follow alternative fates and, simultaneously, ensuring that both fates are adopted in the end. These binary cell-fate decisions have been shown to involve cells with similar or different developmental potential, either within a field of cells or between sibling cells. The classical example of a binary cell fate decision mediated by Lateral In- hibition is the Drosophila neural-epidermal choice. This decision has been well studied during the singling out of neural precursors (neuroblasts) of the embryonic Central Nervous System (CNS) and during formation of the adult Peripheral Nervous System (PNS) sensory bristles (reviewed in [20], [21]. The areas of the Drosophila embryonic ectoderm and imaginal discs from which neural precursors arise are known as proneural clusters, whose cells have a neural potential due to the expression of proneural genes of the Achaete-Scute Complex (AS-C) [22]. Absence of achaete (ac)andscute (sc) in the imaginal discs leads to the loss of sensory bristles, whereas ectopic expression of these genes results in ectopic differentiation of bristles [23], [24]. In the neuroectoderm, proneural genes are both necessary and sufficient to initiate and drive the neural differentiation program. Nevertheless, only one cell in these proneural clusters fulfils its neural potential and is chosen to

30 RITA FIOR AND DOMINGOS HENRIQUE be the neural precursor - the neuroblast. However, if this cell is eliminated by laser ablation, a neighbouring cell will take up the job [25]. This implies two things: first, that all the cells in the cluster have the potential to be a neuroblast, meaning that cells within the proneural cluster are equipotential or at least have similar developmental potential; and second, that neuroblasts prevent their neighbours from adopting the same fate. This last phenomenon is known as lateral inhibition and is mediated by Notch signalling. Absence of Notch activity in the Drosophila embryo results in the so called “neurogenic” phenotype, where all cells in the ventral neuroectoderm develop as neuroblasts and no ventral epidermis is formed [6], [26]. By contrast, constitutive Notch signalling has the opposite phenotype - suppresses neuronal differentiation [27]. However, Notch is not necessary for the acquisition of the epidermal fate, it is only required to inhibit the neural fate, since double mutant cells for Notch and ac-sc differentiate into epidermis [28], [29]. Thus, Notch does not play an instructive role to induce the epidermal fate. Instead, it inhibits the neuronal fate and allows cells to adopt the alternative epidermal fate. The seminal work by Heitzler and Simpson [30] showed that Notch mutant cells develop cell autonomously as bristle precursors while, at the same time, neighbouring wild-type cells reliably adopt the epidermal cell fate and never the neural fate (Fig. 2D). Moreover, inhibition of the neural fate by Notch activity depends on the ligand Delta, as double mutant Notch/Delta cells are no longer able to inhibit their neighbours [28]. In addition, Delta mutant cells, when adjacent to wild-type cells, are able to differentiate as epidermis, implying that Delta is not required for the reception of the inhibitory signal. In parallel, wild-type cells when adjacent to Delta mutant cells, become neural precursors in the majority of cases, implying that wild type cells are not receiving the inhibitory signal. Furthermore, dosage experiments showed that wild-type cells will always adopt the epidermal fate if they are adjacent to cells expressing lower levels of Notch than themselves, but will become neural precursors if they are next to cells that are expressing a higher level of Notch (Fig. 2G). These experiments show that lateral inhibition is a competitive process, where cells compare the relative amounts of Notch activity with their neighbours, prior to their fate choices. Furthermore, these experiments also suggested the existence of a feedback mechanism where cells that have less Notch activity, relatively to their neighbours, acquire higher signal-sending ability. It was suggested that stochastic fluctuations in the expression of the neurogenic or proneural genes can provide a small difference in Notch activity between neighbouring cells that would then be amplified by the negative feedback on Delta expression [30]. Therefore, a cell that activates Notch will presumably have progressively less signal-sending capacity, biasing directionality of the signal and the singling out of the adjacent signalling cell. This was confirmed later by the discovery that Notch activity leads to the upregulation of genes encoding the E(spl) transcriptional repressors, which inhibit transcription of proneural genes [13], [31], [32]. Proneural proteins are bHLH transcriptional activators that positively regulate expression of the Delta gene [33], thus establishing a self-amplifying feedback loop. The mosaic analysis performed by Heitzler and Simpson provided a basis for this model: cells mutant for the E(spl) complex cell-autonomously differentiate as bristles even when adjacent to wild-type cells. However they have the ability of influencing their wild- type neighbouring cells- E(spl) mutant cells prevent wild-type neighbouring cells from adopting a neural fate [34]. These authors also showed that this capacity of

NOTCH-MATHICS 31

Figure 2. Drosophila neural / epidermal cell fate decision. A. Absence of Notch signalling results in the “neurogenic” phenotype- Notch mutant (N-) cells develop as neural precursors (red) and additional bristles (green) are formed at the expense of epidermis. B. Constitutive Notch signalling (NICD) suppresses neuronal differentiation and all cells adopt the epidermal fate (light pink). C. Double mutant cells for Notch/proneurals (N-/pro-N-) differentiate into epidermis, implying that Notch is not necessary for the epidermal fate, is only necessary to suppress proneural activity and therefore the neural fate. D. Notch mutant cells develop cell autonomously as bristle precursors but at the same time force neighbouring wild-type (wt) cells to adopt the epidermal cell fate and never the neural fate. E. Wt cells adjacent to Notch/Delta double mutant (N-/Dl-) cells are able to acquire the neural fate, implying that N-/Dl- cells are no longer able to inhibit their neighbours. F. Delta mutant cells (Dl-) when adjacent to wt cells, are able to differentiate as epidermis, implying that Delta is not required for the reception of the inhibitory signal. In parallel, wt cells when adjacent to Dl mutant cells, in the majority of cases, become neural precursors, implying that the wt cells are not receiving the inhibitory signal. G. Dosage experiments showed that wt cells will always become neural precursors if they are next to cells that are expressing a higher dosage of the N receptor. Diagram of some experiments performed by Heitzler and Simpson (1991).

32 RITA FIOR AND DOMINGOS HENRIQUE

Notch or E(spl) mutant cells to inﬂuence the neighbour’s fate is dependent on the proneural proteins, which positively regulate the inhibitory signal - Delta. These experiments, together with others [35] (reviewed in [19]), have provided a solid model for the process of lateral inhibition, where Delta-Notch signalling between cells with similar developmental potential is resolved over time into unidirectional signalling, with one cell becoming the signalling cell while inhibiting its neighbours from adopting the same fate.

3. Notch during vertebrate neurogenesis:maintaining neural progenitors Notch signalling most probably regulates production of all neurons and glia, both in the vertebrate CNS as well in the PNS. Moreover, Notch signalling is not only used during different steps of the embryonic development of the nervous system, but is also active in adulthood [36]. Here, we will focus our attention to the role of Notch signalling in maintaining neural progenitors throughout vertebrate neural development. In vertebrate embryos, neurogenesis occurs over a long period of time (around 3 weeks in mouse and 8 months in humans) starting at different regions at different times. Also within each region, neurons do not differentiate simultaneously. Thus, mechanisms must exist to regulate this prolonged and controlled production of neurons, ensuring that a pool of progenitors is maintained until all types of neurons and glia are generated. This long-lasting process of differentiation allows progenitor cells to be exposed to different changing cues that instruct progenitors to generate different types of neurons and glia. This implies that neural progenitors have to be maintained throughout the course of development, and one of the most important mechanisms to ensure the maintenance of progenitors is mediated by Notch signalling. During neurogenesis, neural progenitors have to make a choice: either remain as progenitors or embark on neuronal differentiation. If they choose to remain as progenitors, they retain their dividing capacity and stay within the Ventricular zone (Vz) of the neuroepithelium. In contrast, if they decide to differentiate as neurons, they withdraw from the cell cycle and migrate out of the Vz to the Mantle Layer (ML), where they differentiate and accumulate (Fig. 3A). This crucial choice is dictated by the balance between two different sets of bHLH transcription factors: positive factors- the proneural proteins which drive progenitors to neuronal differentiation, and negative factors- the HES proteins which, by inhibiting the positive factors, repress neuronal differentiation and maintain the progenitors in an uncom- mitted state. However, this choice is not a selfish and deaf choice: progenitors choose their fates taking into account what their neighbours have chosen. This cell ‘talking’ is mediated by Notch, in a remarkably similar manner to the choice between the epidermal and neural decision in the Drosophila neuroectoderm. Over the past decade, several lines of evidence, from experiments in Xeno- pus, chick, zebrafish and mouse, in different parts of the CNS (retina, spinal cord, cortex) support a conserved and crucial role for Notch in maintaining the neural progenitor population. The Notch1 [37], [38]andhes genes [39], [40] are expressed in the Vz, where the progenitors are located, whereas Delta1 and proneural genes are preferentially expressed in cells that have undergone or are undergoing their last division-mentioned as nascent neurons [41], [42] (Fig. 3A) and are in their

NOTCH-MATHICS 33

Figure 3. Neurogenesis in the vertebrate spinal cord. A. Picture of four day chick spinal cord illustrating how the neural tube is regionalized in two zones: the ventricular zone (Vz-in red), where progenitors are located and the Notch receptor and targets (hes genes) are expressed; and the mantle zone (Mz-in green) where neurons accumulate. B. Nascent neurons express high levels of proneural proteins that activate Delta expression. Delta then activates the Notch receptor in neighbouring cells, which then translocates into the nucleus where it triggers transcription of the downstream target genes-the hes genes. HES proteins then act as eﬀectors of the Notch pathway by repressing transcription and activity of the proneural genes, thereby inhibiting neural progenitors from diﬀerentiating as neurons.

way towards the mantle layer. These nascent neurons, which have high levels of proneural proteins, trigger the process of lateral inhibition by activating expression of the Notch ligand Delta1 [43] which then activates Notch in neighbouring progenitor cells that express Notch receptors (Figs. 3B, 4). Activation of the pathway in progenitor cells leads to upregulation of downstream target/effectors genes-the hes genes, which will then suppress proneural activity in progenitors, thereby preventing these cells from differentiating prematurely into neurons and from expressing the ligand Delta1. In this way, cells that express the ligand Delta1 undergo neuronal differentiation, becoming neurons, but simultaneously ensure that the neighbouring cells do not make the same choice. Thus, whereas in the Drosophila neuroectoderm lateral inhibition controls the decision between becoming a neural precursor (neuroblast) or an epidermal cell, during vertebrate neurogenesis it controls the decision between becoming a neuron versus remaining as a neural progenitor. In vertebrates, lateral inhibition works like a homeostat or “quorum sensing”, providing a feedback mechanism to control the production of neurons: excessive production of neurons will result in an excess of inhibitory signal, thus preventing further production of neurons, whereas a low production of neurons will result in a reduction of inhibitory signal, relieving inhibition of neuronal differentiation. In

34 RITA FIOR AND DOMINGOS HENRIQUE this way, any perturbations on the balance of production of neurons will be self- correcting, and will tend back to the equilibrium point [44].

4. Proneural-Notch-HES signalling syntagm From this overview of Notch signalling and neural development, a major feature emerges: the existence of a conserved interconnected circuitry between the proneural genes and the genes of the hes family. Actually, one can say that neurogenesis is based on the antagonistic relationship between these two different sets of bHLH proteins: proneural proteins playing a positive role in promoting the commitment to a neural fate and HES proteins repressing this cell fate decision. Therefore this conserved circuitry could be considered as a “syntagm” or a developmental cassette, a concept developed by A. Garcia-Bellido to describe “a group of genes that interact to perform a discrete developmental operation” [45], [46]. Here, we propose that the proneural genes, together with the Notch/Delta/hes pathway, constitute the “syntagm” that regulates neurogenesis, controlling the balanced production of neurons and progenitors from flies to vertebrates. In this syntagm, the proneural genes activate a signalling process (lateral inhibition) that results in the inhibition of its own activity, creating a non-cell-autonomous negative feedback loop (in trans). However, proneural proteins are also able to directly activate expression of the hes genes in cis [13], [43], [47]–[50]. In Drosophila, it was demonstrated that AS-C proteins bind in vitro the E(spl)m8 and m7 promoters and that activation of E(spl)m8 in vivo requires an intact E box [47], [48]. In vertebrates, mash1 mutant embryos show a loss in hes5 expression and it was shown that the MASH1 protein is able to bind the hes5 promoter [51]. Another example comes from Xenopus, where the Esr10 expression is dependent on the direct binding of NGN1 to the promoter of esr10 [16]. In addition, proneural transcription factors have the capacity to positively regulate their own expression and also activate other members of the same family, leading to cascades of proneural activity (reviewed in [52], [53]. This auto-regulative ability may provide a crucial mechanism to reinforce and up-regulate proneural expression in the selected progenitors that will undergo neuronal differentiation and, ultimately, maintain expression even after the induc- tive signal is OFF. This can be achieved through a direct mechanism, by binding to E boxes on their own promoters or by activating indirect positive auto-regulatory loops through transcriptional activation of genes that increase their own proneural activity [52]. We propose that these interactions between proneural and hes genes during neurogenesis are part of two conserved and interconnected circuits (Fig. 5): ∙ An inter-cellular circuitry mediated by Notch signalling, where proneural proteins activate hes expression in a non-cell autonomous manner, thereby contributing to inhibit neighbouring cells from embarking on neuronal differentiation. ∙ An intra-cellular circuitry, where proneural proteins cell-autonomously activate hes expression. In addition, one should also consider the existence of auto and cross-regulation between each family of bHLH factors within the same cell: auto and cross-regulation between proneural genes generating a positive feedback and the auto and cross- regulation between hes genes generating a negative feedback.

NOTCH-MATHICS 35

Figure 4. Lateral Inhibition controls the balance between neuronal progenitors and neurons. A. Nascent neurons inhibit neighbouring progenitor cells from adopting the neural fate. B. When all cells deliver LI, by ectopic expression of Delta1, all cells inhibit one another from embarking on neuronal diﬀerentia- tion therefore, no neurons are generated. C. When all Delta-Notch signalling is impaired, no LI occurs and all progenitors diﬀerentiate as neurons and no progenitors remain.

The logic of this circuitry with counteracting activities (proneural proteins activating expression of their own repressors cell-autonomously) is still not known. However, by making an analogy with an electrical circuit, Meir and colleagues [54] proposed that the proneural/hes intra-cellular loop could be working as a homeostat, reducing proneural activity and thereby working as a buﬀer to reduce the sensitivity to developmental noise (as stochastic changes in transcription or translation rates). This design would prevent the network from switching individual cells ON or OFF by noise, leading to a new state-a neither ON-nor-OFF steady state, which would delay the ON or OFF switch until some extrinsic cue forces the system to choose one of the states.

5. Neurogenesis as a reiterative and dynamic process During neurogenesis, progenitor cells have to decide again and again whether they are to remain as progenitors or diﬀerentiate. Since this decision is based on Notch-mediated lateral inhibition, this implies that the Notch signaling cascade is activated transiently in neural progenitors, in a reiterative manner, producing

36 RITA FIOR AND DOMINGOS HENRIQUE

Figure 5. The Neurogenesis syntagm. The controlled production of neurons relies on the balance between the activity of proneural bHLH activators and bHLH HES repressors, which is mediated by Delta/Notch signalling. pulses of Notch activity and, therefore, of hes expression. This also means that, after each pulse of Notch activity, the signaling cascade has to be downregulated in order to allow progenitors to become competent again to respond to environmental cues and start a new cell-fate decision process. One way to achieve this down-regulation involves the rapid degradation of the nuclear form of the Notch receptor (NICD) triggered by the co-activator MAM [55]. However, the activity of the downstream effectors, namely the hes genes, has also to be restrained.We propose that the hes auto and cross-circuitry of negative feedback regulation functions to restrict the duration of Notch signalling in neural progenitors, ensuring that the pathway is shut down after each event of Notch activation. Progenitors could then go back to a state where they are again competent to respond to environmental cues and decide their fate. Vertebrate neurogenesis can thus be viewed as a reiterative process in which progenitors activate the Notch signalling cascade transiently, time and again, until they finally commit to differentiation. Thus, in a simple scenario, when Notch is activated in a neural progenitor, in response to a Delta signal from a neighbouring cell, transcription of the hes genes will follow. As Notch effectors, their activity will be essential to implement the decision to stay as a neural progenitor, by repressing the proneural genes. Later on, negative auto and cross regulation of hes genes would lead to a downregulation of their own expression finally closing a cycle of Notch activity allowing the cell to again embark on a new process of cell fate decision. This would involve a choice between continuing as a neural progenitor (which requires a new cycle of Notch activity), or commit to neuronal differentiation (which involves a definitive release from Notch signalling). Therefore, it is conceivable that neural progenitors may go through cyclic bursts of Notch activity, until they finally commit to differentiation or instead switch to another fate, like glial progenitor, and we

NOTCH-MATHICS 37 propose that the hes circuitry of negative feedback regulation might play a central role in this mechanism by shutting down the pathway after each Notch activation event. This is consistent with previous findings in Drosophila, where it has been shown that neurectodermal cells which expressed E(spl) genes as result of Notch activation can subsequently re-enter the neural pathway [56]. Thus, although Notch-E(spl) activity does inhibit neuroblast segregation, inhibited cells are competent to respond to a subsequent signal and become neuroblasts. Similar findings in the mouse embryo are also consistent with the above hypothesis: as shown by Mizutani and Saito [57], neural progenitors which have been temporarily subjected to Notch activation at an early stage might generate neurons at later stages, skipping the early neural fate. These experiments show that Notch activity does not cause a permanent block on the progenitors’ competence to differentiate, which are still able to respond to the appropriate cues.

6. Neurogenesis could be controlled by two interacting loops During vertebrate somitogenesis, Notch signalling reveals a cyclic activation pattern that coincides with the period of somite formation (reviewed in [58]). Sev- eral genes from the Notch pathway and from other signalling pathways [59]have been shown to transcriptionally oscillate in cells of the presomitic mesoderm, as part of a molecular machinery that regulates the periodicity of somite formation. This means that each cell in the PSM (Presomitic mesoderm) transcribes and degrades the message and protein encoded by these genes, not only in a periodic manner but also in synchrony with its neighbours. It has been proposed [1]that the generation of such cyclic gene oscillations relies on two interacting loops: an intracellular negative feedback-loop established by HES transcription factors on the promoter of their own genes, and an inter-cellular loop involving Notch activation by Delta in adjacent cells. The ﬁrst loop should drive a cell-autonomous oscilla- tion of hes gene expression based on a time-delayed feedback mechanism, while the second should account for the rhythmic activation of Notch to maintain synchrony between adjacent cells. The periodic Notch activation that enables synchrony is driven by cyclic DeltaC expression in Zebraﬁsh [60], or by the cyclic expression of Lunatic Fringe in the mouse (a regulator of the Delta-Notch interaction) [61]that makes that group of cells signal and receive Notch activity at the same time. We propose a model where the neurogenesis ‘syntagm’ involves two similar interacting loops, which may also generate an oscillatory behaviour of hes gene expression (Fig. 6): i) An inter-cellular loop mediating lateral inhibition (Fig. 6A) where high levels of proneural proteins up-regulate Delta expression in the nascent neuron (cell A), which will activate Notch in a neighbouring progenitor (cell B). Notch activation leads to a burst of hes transcription in cell B and HES repressors will then inhibit proneural gene transcription and activity, thus impairing neuronal commitment in cell B. Next, negative auto-and cross-regulation of HES repressors on their own genes would downregulate hes expression in the neural progenitor. Then, if this progenitor contacts with a Delta-expressing cell, it would activate Notch again and up-regulate hes expression once more.

38 RITA FIOR AND DOMINGOS HENRIQUE

Figure 6. Neurogenesis could be controlled by two interacting loops. A. An inter-cellular loop based on Delta-Notch signalling would activate HES expression to high levels, inhibiting neurogenesis, thus instructing the cell to remain as progenitors. B. An intracellular loop based on HES negative feedback with a time delay could possibly generate low amplitude oscillations of hes5 expression.

This intercellular loop would therefore result in an oscillatory behaviour of hes expression with high amplitude. However, in contrast with what occurs in the PSM, cells in the asynchronous neuroepithelium do not signal at the same time, so these oscillations would have a variable period, dependent on the frequency in which this cell contacts with a Delta-expressing cell. ii) An intra-cellular loop established by two negative feedback loops. A ﬁrst loop would involve the activity of proneural proteins activating expression of their own repressors - the hes genes, generating intermediate levels of proneural expression. A second loop would result from the negative feedback of HES transcription factors on the promoter of their coding genes. Mathematical modelling showed that feedback inhibition with transcriptional delay may account for oscillatory gene expression [1], [62]. Thus, it is possible to envision that this intracellular loop (negative auto-regulation of hes genes) could generate cell-autonomous oscillations of hes5 expression, with low amplitude [since these oscillations would occur in the absence of Notch activity]. These hes5 oscillations would have a ﬁxed period,

NOTCH-MATHICS 39

Figure 7. Lateral Inhibition generates progenitors with diﬀerent levels of hes expression. At t3 there will be progenitors at several states of hes expression, some with high levels (activated at t3-black), others with intermediate levels (that were activated by Notch at t2-dark grey) and others already with low levels of hes (that were activated by Notch at t1-light grey).

or not, depending on the time-delays of proteins and mRNA synthesis. To- gether, the intermediate levels of proneural proteins and the fluctuations of hes5 expression could characterize the “neither-ON-nor-OFF” steady state, in which progenitors are ready and competent to receive information to differentiate or not. The competent state would thus lie within this ‘noisy’ low amplitude range of hes expression. It is noteworthy, however, that the negative feedback on proneural expression and their positive autoregulation (Fig. 6B) could possibly generate an oscillating behaviour of proneural expression. In this case, the “neither-ON-nor-OFF” could possibly have oscillating antiphasic expression of hes and proneural genes. Never- theless, in order to simplify the rational, we will consider that the “neither-ON-nor- OFF” state is characterized by intermediate levels of proneural expression together with oscillating hes expression (Fig. 6B). As already mentioned, in contrast with the PSM, cells in the asynchronous neuroepithelium do not signal at the same time, so the high amplitude oscillations that result from the inter-cellular loop would have a variable period, dependent on the encounter with a Delta-expressing cell. Interestingly, this asynchronous activation of Notch in the neuroepithelium could also generate a “side-effect” of lateral inhibition - leading to the appearance of a heterogeneous population of progenitors with different levels of hes expression (Fig. 7). For instance, in time1 (Fig. 7.t1), when the first nascent neuron expresses Delta, Notch will be activated in the neighbouring cells, causing a burst of HES expression (Fig. 7.t2) which, after a time delay, will start decreasing, due to negative auto and cross-regulation, and mRNA and protein decay (Fig. 7.t3). If in t2 a second Delta-expressing nascent neuron is “born”, it will activate Notch in the neighbouring cells, leading again to a burst of HES expression in t3. Finally, if in t3 a third Delta expressing neuron is “born”, activating Notch in the neighbouring cells, this would generate several states of hes expression within the neuroepithelium.

40 RITA FIOR AND DOMINGOS HENRIQUE

Thus, in contrast to what occurs during somitogenesis where Notch seems to ‘tune’ cells to the same behaviour [63], the ‘de-phased’ and asynchronous Notch activation in the neuroepithelium leads to a heterogeneous progenitor population, with variable and varying levels of hes expression (Fig. 7).

7. Feedback-loops, fluctuations of hes5 expression and physiological relevance Small differences in protein abundance (“cellular noise”) may confer advantages or disadvantages to development [64]. A positive example comes from the nervous system, where stochastic activation of an odorant receptor (OR) gene followed by negative feedback may generate the diversity of olfactory neurons (ON), each expressing only one type of the 1500 OR genes [65]. Since ONs expressing a particular OR gene project their axons to a specific set of glomeri in the olfactory bulb (OB), the odorant stimuli is converted into a topographic map of activated glomeri on the OB. In addition, it has been proposed that intrinsic noise can generate fluctuations in the relative levels of two alleles of the same gene, which may potentially result in cells expressing no allele, one or both alleles. If the two alleles are functionally different, the population of cells may acquire heterogeneity, which may contribute, for instance, to the phenomenon of hybrid vigour [64]. In this way, a pool of genetically identical cells may exhibit significant diversity even when they have identical histories of environmental exposure. Similarly, cells in an equivalent group may have similar potential but not identical. According to this view, it is possible that the postulated ‘side efect’ of lateral inhibition together with the oscillatory ‘noisy’ expression of hes genes (neither- ON-nor-OFF steady state), could have a positive impact on neurogenesis. Since HES repressors inhibit proneural expression and function, it is possible to envision that the different levels of hes expression within progenitors could provide different states of receptiveness to differentiating signals. This implies that even cells within the same equivalence group would respond differently to these signals, thereby increasing the heterogeneity of the progenitor population. For instance, if a progenitor cell receives a differentiating cue (e.g. RA, in the spinal cord) when it has just experienced a Notch activation event, it will not respond since it has very high levels of HES proteins to counteract the proneural proteins. However, if a progenitor cell has downregulated Notch activity and lies within the “neither-ON-nor-OFF” competent state, it can respond to the differentiation cues in two ways: if it is at the higher peak of HES expression (Fig. 8A), it could be less responsive to these cues. In contrast, if the progenitor cell receives a differentiation signal when the levels of HES repressors are low (Fig. 8.B), this cell would be more prompt to upregulate the proneural genes and therefore embark on differentiation. In this way, the possible oscillatory ‘noisy’ hes expression could result into different states of receptiveness to differentiating cues. These different states of competence might provide yet another mechanism, besides lateral inhibition, to delay neuronal differentiation, providing time to allow changes in the competence state of neural progenitors and in the extracellular cues, thus permitting the formation of the correct numbers and types of neural cells.

NOTCH-MATHICS 41

Figure 8. Different levels of HES expression may provide different states of receptiveness to the differentiating signals. A. High relative levels of HES repressors may delay the response to differentiating cues. B. High relative levels of proneural proteins may render cells to respond more rapidly to the differentiating cues.

8. Final remarks It is well established that the oscillatory expression of several Notch targets plays a crucial role in the molecular mechanism that times the periodic and sequential formation of segments in vertebrates (reviewed in [58]). The existence of negative auto-regulatory mechanisms, as well as cross-regulatory interactions between hes genes in the developing neural tube [66], raises the attractive possibility that Notch signalling might also have an oscillatory behaviour during neurogenesis. The challenge is to experimentally conﬁrm this oscillatory behaviour in the neuroepithelium and understand its functional relevance. And, through mathematical modelling, compare the circuitries that control somitogenesis and neurogenesis and understand how such two similar circuitries can generate completely diﬀerent outcomes.

References

[1] Lewis, J., Autoinhibition with transcriptional delay: a simple mechanism for the zebraﬁsh somitogenesis oscillator. Curr Biol, (2003). 13 (16), 1398–14. [2] Artavanis-Tsakonas, S., M.D. Rand, and R.J. Lake, Notch signaling: cell fate control and signal integration in development. Science, (1999). 284 (5415), 770–776.

42 RITA FIOR AND DOMINGOS HENRIQUE

[3] Lai, E.C., Notch signaling: control of cell communication and cell fate. Development, (2004). 131 (5), 965–973. [4] Poulson, D.F., Chromosomal Control of Embryogenesis in Drosophila. The America Natu- ralist, (1945). 79 83), 340–363. [5] Poulson, D.F., Chromosomal Deficiencies and the Embryonic Development of Drosophila Melanogaster. Proc Natl Acad Sci U S A, (1937). 23 (3), 133–137. [6] Lehmann, R., et al., On the phenotype and development of mutants of early neurogenesis in Drosophila melanogaster. Roux’s Arch. Dev. Biol., (1983). 192, 62–74. [7] Fleming, R.J., Structural conservation of Notch receptors and ligands. Semin Cell Dev Biol, (1998). 9 (6), 599–6. [8] Schweisguth, F., Notch signaling activity. Curr. Biol, (2004). 14 (3), R129–138. [9] Mumm, J.S. and R. Kopan, Notch signaling: from the outside in. Dev Biol, (2000). 228 (2), 151–165. [10] Ehebauer, M., P. Hayward, and A. Martinez-Arias, Notch signaling pathway.SciSTKE, (2006). 2006 (364), c. [11] Ehebauer, M., P. Hayward, and A.M. Arias, Notch, a universal arbiter of cell fate decisions. Science, (2006). 314 (5804), 1414-1415. [12] Bray, S.J., Notch signalling: a simple pathway becomes complex. Nat Rev Mol Cell Biol, (2006). 7 (9), 8–9. [13] Bailey, A.M. and J.W. Posakony, Suppressor of hairless directly activates transcription of enhancer of split complex genes in response to Notch receptor activity. Genes Dev, (1995). 9 (21), 2609–2622. [14] Nellesen, D.T., E.C. Lai, and J.W. Posakony, Discrete enhancer elements mediate selective responsiveness of enhancer of split complex genes to common transcriptional activators.Dev Biol, (1999). 213 (1), 33–53. [15] Cave, J.W., et al., A DNA transcription code for cell-specific gene activation by notch signaling. Curr Biol, (2005). 15 (2), 94–104. [16] Lamar, E. and C. Kintner, The Notch targets Esr1 and Esr10 are differentially regulated in Xenopus neural precursors. Development, (2005). 132 (16), 3619–3630. [17] Ong, C.T., et al., Target selectivity of vertebrate notch proteins. Collaboration between discrete domains and CSL-binding site architecture determines activation probability.JBiol Chem, (2006). 281 ), 5106–5119. [18] Davis, R.L. and D.L. Turner, Vertebrate hairy and Enhancer of split related proteins: transcriptional repressors regulating cellular differentiation and embryonic patterning. Oncogene, (2001). 20 (58), 8342–57. [19] Greenwald, I., LIN-12/Notch signaling: lessons from worms and flies. Genes Dev, (1998). 12 (12), 1751–1762. [20] Ghysen, A., et al., Cell interactions and gene interactions in peripheral neurogenesis. Genes Dev., (1993). 7 (5), 723–733. [21] Simpson, P., Notch signalling in development: on equivalence groups and asymmetric developmental potential. Curr Opin Genet Dev, (1997). 7 (4), 537–542. [22] Modolell, J. and S. Campuzano, The achaete-scute complex as an integrating device.IntJ Dev Biol, (1998). 42 (3), 5–2. [23] Garc´ıa-Bellido, A., Genetic Analysis of the Achaete-Scute System of Drosophila melanogaster. Genetics, (1979). 91 (3), 491–520. [24] Ghysen, A. and C. Dambly-Chaudiere, From DNA to form: the achaete-scute complex. Genes Dev, (1988). 2 (5), 495–501. [25] Doe, C.Q., J.Y. Kuwada, and C.S. Goodman, From epithelium to neuroblasts to neurons: the role of cell interactions and cell lineage during insect neurogenesis. Philos Trans R Soc Lond B Biol Sci, (1985). 312 (1153), –81. [26] Campos-Ortega, J.A., Cellular interactions during early neurogenesis of Drosophila melanogaster. Trends Neurosci, (1988). 11 (9), 400–405. [] Struhl, G., K. Fitzgerald, and I. Greenwald, Intrinsic activity of the Lin-12 and Notch intracellular domains in vivo. Cell, (1993). 74 (2), 331–345. [28] Heitzler, P. and P. Simpson, Altered epidermal growth factor-like sequences provide evidence for a role of Notch as a receptor in cell fate decisions. Development, (1993). 117 (3), 1113– 1123.

NOTCH-MATHICS 43

[29] Simpson, P. and C. Carteret, A study of shaggy reveals spatial domains of expression of achaete- scute alleles on the thorax of Drosophila. Development, (1989). 106 (1), 57–66. [30] Heitzler, P. and P. Simpson, The choice of cell fate in the epidermis of Drosophila. Cell, (1991). 64 (6), 13–1092. [31] Lecourtois, M. and F. Schweisguth, Role of suppressor of hairless in the delta-activated Notch signaling pathway. Perspect Dev Neurobiol, (1997). 4 (4), 305–311. [32] Lecourtois, M. and F. Schweisguth, The neurogenic suppressor of hairless DNA-binding protein mediates the transcriptional activation of the enhancer of split complex genes triggered by Notch signaling. Genes Dev, (1995). 9 (21), 2598–26. [33] Kunisch, M., M. Haenlin, and J.A. Campos-Ortega, Lateral inhibition mediated by the Drosophila neurogenic gene delta is enhanced by proneural proteins.ProcNatlAcadSci U S A, (1994). 91 (21), 10139–1043. [34] Heitzler, P., et al., Genes of the Enhancer of split and achaete-scute complexes are required for a regulatory loop between Notch and Delta during lateral signalling in Drosophila. Devel- opment, (1996). 122 (1), 161–171. [35] Wilkinson, H.A., K. Fitzgerald, and I. Greenwald, Reciprocal changes in expression of the receptor lin-12 and its ligand lag-2 prior to commitment in a C. elegans cell fate decision. Cell, (1994). 79 ), 1187–1198. [36] Louvi, A. and S. Artavanis-Tsakonas, Notch signalling in vertebrate neural development.Nat Rev Neurosci, (2006). 7 (2), 93–102. [37] Weinmaster, G., V.J. Roberts, and G. Lemke, A homolog of Drosophila Notch expressed during mammalian development. Development, (1991). 113 (1), 199–205. [38] Myat, A., et al., A chick homologue of Serrate and its relationship with Notch and Delta homologues during central neurogenesis. Dev Biol, (1996). 174 (2), 233–247. [39] Kageyama, R., et al., Regulation of mammalian neural development by helix-loop-helix transcription factors. Crit Rev Neurobiol, (1995). 9 (2–3), 177–188. [40] Sasai, Y., et al., Two mammalian helix-loop-helix factors structurally related to Drosophila hairy and Enhancer of split. Genes Dev, (1992). 6 (12B), 2620–234. [41] Chitnis, A., et al., Primary neurogenesis in Xenopus embryos regulated by a homologue of the Drosophila neurogenic gene Delta [see comments]. Nature, (1995). 375 (6534), 761–766. [42] Henrique, D., et al., Expression of a Delta homologue in prospective neurons in the chick. Nature, (1995). 375 (6534), 787–790. [43] Castro, B., et al., Lateral inhibition in proneural clusters: cis-regulatory logic and default repression by Suppressor of Hairless. Development, (2005). 132 (15), 3333–3344. [44] Henrique, D., et al., Maintenance of neuroepithelial progenitor cells by Delta-Notch signalling in the embryonic chick retina. Curr Biol, (1997). 7 (9), 661–0. [45] Garc´ıa-Bellido, A., The Bithorax syntagma. Advances in Genetics, Development and Evolu- tion in Drosophila, ed. S. Lakovaara. (1981, New York: Plenum. 135–148. [46] Huang, F., Syntagms in development and evolution. International Journal of Developmental Biology, (1998). 42 (3), 487–494. [47] Kramatschek, B. and J.A. Campos-Ortega, Neuroectodermal transcription of the Drosophila neurogenic genes E(spl) and H-m5 is regulated by proneural genes . Development, (1994). 120 (4), 815–826. [48] Singson, A., et al., Direct downstream targets of proneural activators in the imaginal disc include genes involved in lateral inhibitory signaling. Genes Dev, (1994). 8 (17), 2058–21. [49] Oellers, N., M. Dehio, and E. Knust, bH proteins encoded by the Enhancer of split complex of Drosophila negatively interfere with transcriptional activation mediated by proneural genes. Mol Gen Genet, (1994). 244 (5), 465–473. [50] Cooper, M.T., et al., Spatially Restricted Factors Cooperate with Notch in the Regulation of Enhancer of split Genes. Dev Biol, (2000. 221 (2), 390–403. [51] Gazit, R., V. Krizhanovsky, and N. Ben-Arie, Math1 controls cerebellar granule cell diﬀer- entiation by regulating multiple components of the Notch signaling pathway. Development, (2004). 131 (4), 903–913. [52] Gibert, J.M. and P. Simpson, Evolution of cis-regulation of the proneural genes.IntJDev Biol, (2003). 47 –8), 643–651. [53] Bertrand, N., D.S. Castro, and F. Guillemot, Proneural genes and the speciﬁcation of neural cell types. Nat Rev Neurosci, (2002). 3 ), 517–530.

44 RITA FIOR AND DOMINGOS HENRIQUE

[54] Meir, E., et al., Robustness, flexibility, and the role of lateral inhibition in the neurogenic network. Curr Biol, (2002). 12 (10), 778–786. [55] Fryer, C.J., J.B. White, and K.A. Jones, Mastermind recruits CycC:CDK8 to phosphorylate the Notch ICD and coordinate activation with turnover. Mol Cell, (2004). 16 (4), 509–520. [56] Jennings, B., et al., The Notch signalling pathway is required for Enhancer of split bH protein expression during neurogenesis in the Drosophila embryo. Development, (1994). 120 (12), 3537–3548. [57] Mizutani, K. and T. Saito, Progenitors resume generating neurons after temporary inhibition of neurogenesis by Notch activation in the mammalian cerebral cortex. Development, (2005). 132 (6), 1295–1304. [58] Giudicelli, F. and J. Lewis, The vertebrate segmentation clock. Curr Opin Genet Dev, (2004). 14 (4), 4–414. [59] Dequeant, M.L., et al., A complex oscillating network of signaling genes underlies the mouse segmentation clock. Science, (2006). 314 (5805), 1595–1598. [60] Julich, D., et al., beamter/deltaC and the role of Notch ligands in the zebrafish somite segmentation, hindbrain neurogenesis and hypochord differentiation. Dev Biol, (2005). [61] Dale, J.K., et al., Periodic notch inhibition by lunatic fringe underlies the chick segmentation clock. Nature, (2003). 421 (6920), 5–8. [62] Monk, N.A., Oscillatory expression of Hes1, p53, and NF-kappaB driven by transcriptional time delays. Curr Biol, (2003). 13 (16), 1409–1413. [63] Jiang, Y.J., et al., Notch signalling and the synchronization of the somite segmentation clock. Nature, (2000). 408 (11), 475–479. [64] Raser, J.M. and E.K. O’Shea, Noise in gene expression: origins, consequences, and control. Science, (2005). 309 (5743), 2010–2013. [65] Serizawa, S., K. Miyamichi, and H. Sakano, One neuron-one receptor rule in the mouse olfactory system. Trends Genet, (2004). 20 (12), 648–563. [66] Fior, R. and D. Henrique, A novel hes5/hes6 circuitry of negative regulation controls Notch activity during neurogenesis. Dev Biol, (2005). 281 (2), 318–333.

Instituto de Medicina Molecular, Universidade de Lisboa, Av. Prof. Egas Moniz, 164928 Lisboa, Portugal E-mail address: [email protected]

Contemporary Mathematics Volume 492, 2009

Modelling tumour-induced angiogenesis: A review of individual-based models and multiscale approaches

Tom´as Alarc´on

Abstract. A non-technical review of the main developments of the last 10 years in mathematical modelling of tumour-induced angiogenesis is presented. This field has seen much progress in recent times, therefore, for the sake of concreteness and brevity, we will focus on hybrid models in which the progression of the vascular network, described in terms of individual-based models, is coupled to blood flow and/or external factors due to the dynamics of its sorrounding tissue. The interest of such models is twofold: whilst they constitute the state-of-the-art as far as (potential) predictive power and clinical applicability is concerned, their analysis and numerical implementation also offer remarkable mathematical challenges.

1. Introduction Mathematical Biology1 has experienced an spectacular growth in the last decade. Such growth has been driven by necessity. In the postgenomic era, Biology has been flooded by massive amounts of data. This has brought about a number of the so- called “OMICS” disciplines (eg genomics, proteomics, metabolomics, etc.) in an attempt to organise this flow of information into different layers and make sense of it. Fundamental to such attempts is the use of mathematical methods which constitute the foundations of Systems Biology and Bioinformatics. This flow of information has also uncovered the fact that many biological processes, including many human diseases, are far more complex than it had been expected previously. This has lead to the realisation that the traditional attempts amongst biologists to formulate theories based on verbal models and lineal rea- soning are essentially obsolete [GM]. A possible remedy to this situation is to

1991 Mathematics Subject Classiﬁcation. Primary; Secondary . Key words and phrases. mathematical modelling, multiscale modelling, angiogenesis, cancer, developmental biology. The author would like to thank his collaborators Helen Byrne, Philip Maini and Markus Owen for much enjoyable work done together. He also gratefully acknowledges ﬁnancial support from the EPSRC. 1In the present chapter, the term “Mathematical Biology” will be used in a broad sense, en- compassing any attempt to tackle a biological problem by means of a quantitative approach. This will include the mathematical foundations of disciplines such as Systems Biology and Bioinformatics.

⃝c 2009 Americanc 0000 Mathematical (copyright Societyholder) 45

46 TOMAS´ ALARCON´ formulate mathematical models of such complex systems, which, in turn, might help to formulate hypothesis as to how the particular system under examination works, which should then be tested against and compared to experimental results. A paradigmatic example of this situation is cancer. The term “cancer” is a heading under which a large number of different disorders is grouped. All these diseases share a common feature, namely, the control mechanisms maintaining home- ostasis in normal tissue are bypassed and uncontrolled growth ensues [W]. This deregulation of normal control mechanisms affects and is influenced by processes occurring at all levels of biological organisation, which, physically, correspond to different time and length scales: from alterations in the patterns of gene regulation and expression to aberrant behaviour of whole tissues, such as abnormal elastic properties or pathological remodelling of the vasculature. This fact implies that to understand cancer evolution and design efficient treatments, information from all these different levels must be integrated into a unified conceptual framework. A natural approach to achieve such description is the use of multiscale models, in particular hybrid models2 [ABb, ABc]. Multiscale models in their hybrid in- carnation consist of a hierarchy of models, each of them dealing with the processes relevant at a different scale. Each of these processes may be described using a different mathematical description, but the common denominator of all these models is the use of an individual-based model to deal with the “cellular phase” of the system. For example the multiscale model of vascular tumour growth developed in [ABc] uses ordinary differential equations (ODEs) to describe intracellular processes (cell cycle progression, apoptosis, etc.), a cellular automaton to describe cell-to-cell interactions and partial differential equations (PDEs) to describe diffusion of nutrients and signalling cues. A number of models based on this concept have been developed to analyse different aspects of tumour growth [A, DD, PG, SG]. The present review concentrates in a particular but rather crucial step in the development of solid tumours: angiogenesis. Angiogenesis, the process whereby new vessels are generated from the existing vasculature, is thought to be one of the landmarks in tumour development. According to the usual picture of tumour development, which goes back to Folkman’s seminal work [F], a tumour starts off as a small lession whose cells obtain oxygen and other vital substances by diffusion accross its surface. Its size during this initial stage, the so-called avascular phase, is therefore limited by the diffusion rate of nutrients, typically to 1-2 mm in diameter [RC]. Thus, during this phase, tumours remain small and localised lessions and, consequently, deemed harmless. However at some point, cancer cells, under the stress of the lack of oxygen and other nutrients, start secreting a number of cytokines known as angiogenic factors [FK], most importantly, the so-called vascular endothelium growth factor (VEGF). These cytokines diffuse through the tissue until they finally reach the vasculature of the host organism, which triggers activation of the endothelial cells3 (ECs). Active ECs migrate chemotactically towards the source of angiogenic factor (i.e. the tumour) and start dividing. This marks the onset of the angiogenic process which culminates with the formation of

2This term is often used in a loose and rather imprecise way. For example, the angiogenesis model of Anderson and Chaplain [AC] to be described in Sec. 3 is sometimes presented as a hybrid model. This model is an individual-based model based on an approach which combines continuous and discrete approaches in a particular way. It therefore would not fall within the category of hybrid models as we deﬁne them here. 3Endothelial cells are the cells that line the vessels of the vascular system

MODELLING TUMOUR-INDUCED ANGIOGENESIS 47 a new vascular network which, in turn, provides the tumour with virtually endless resources and unbound growth ensues. In addition to this, malignant cells are shed by the tumour into the circulation. These cells eventually extravasate, and start colonies in remote parts of the host organism called metastasis, which is, in fact, the primary cause of mortality in cancer patients [CG]. Angiogenesis, therefore, would appear to mark the transition from a tumour being a loclaised, harmless lession to it becoming a systemic, potentially fatal disease. It was suggested by Folkman [F] that anti-angiogenic therapy, whereby the tumour would be deprived of the vascular network supplying it with oxygen and other vital metabolites, should starve the tumour, shrinking it to a harmless size. Such hypothesis has been extensively confirmed by in vitro experiments and mice models. However, in spite of the success in laboratory experiments and animal models, anti-angiogenic therapies have fared very poorly when in clinical trials on human patients with extremely modest results (see [Jb] and references therein for a full account). The reasons for this failure are far from clear, specially concerning the explanation of why anti-angiogenic drugs are efficient in mice but not in humans. In view of this state of affairs, new avenues are beginning to be explored in relation with combination of anti-angiogenic and conventional therapies (i.e. cytotoxic drugs and radiotherapy). It has been observed that when chemo- or radio-therapy are appropriately combined with antio-angiogenic drugs, the later notably increases the efficiency of the former [Jb]: a few days after anti-angiogenic therapy is administered, a period of time, the so-called window of opportunity, opens such that if conventional therapy is administered, its efficiency is greatly increased within this time window. The mechanisms for this phenomenon are, again, largely unknown, although there is strong evidence pointing towards vessel normalisation to be reponsible for it. Tumour vasculture has very different properties from its normal counterpart. In fact, whereas “normal” vasculature has a very definite anatomical structure and a very regular spatial distribution to ensure no portion of tissue receive insufficient supply of nutrients, tumour vasculature lacks most of these properties: it is immature, structurally unstable and spatially disorganised. Anti-angiogenic drugs seem to partly remediate these anomalies to produce a more normal-looking vasculature. Such normalisation contributes to create a more uniform supply of oxygen and drugs, which yields an increase in their efficiency [Jb]. This discussion illustrates that we are looking at a complex problem which needs to be examined from an integrative perspective to achieve a proper understanding. Angiogenesis is, therefore, a very complex sequence of well-orchestrated steps involving sprouting, chemotactic cell migration, proliferation, sprout fusion and maturation, leading to onset of blood flow, and complex interactions with the surrounding tissue [R, Ja, Jb]. It thus offers a natural ground for the application of the hybrid modelling methods described in previous paragraphs. The present chapter is devoted to review advances in angiogenesis modelling using the hybrid approach. For the interested reader, other aspects of cancer modelling have been dealt with at length in other topical or general reviews. A general review of several topics in mathematical approaches to tumour growth may be found in the book edited by Preziosi [P]. More recent developments have been reviewed in [BA]. For an excellent review of angiogenesis modelling, mostly dealing with earlier work but with some overlap with the present chapter, see reference [MW]. Roose et

48 TOMAS´ ALARCON´ al present a review of avascular tumour modelling in [RC]. The use of cellular automata in Biology and, in particular, in tumour growth, has been reviewed in [CG] and [MD], respectively. It must be remarked that the aim of this paper is to illustrate the different approaches to modelling tumour-induced angiogenesis by means of a collection of selected examples, rather than producing an exhaustive review of the literature on the subject. This chapter is organised as follows. Section 2 presents a brief biological summary of angiogenesis. Section 3 is devoted to review individual-based models of network formation. Section 4 deals with models of blood flow and vascular adaptation, leading to a discussion of how the complexities involved in blood flow through the microvasculature produce a heterogenous microenvironment and imposes barriers to drug delivery. In Section 5 , we introduce the issue of cooption-induced vessel dematuration and collapse and the models attempting to study its effects of tumour growth and its therapy. Section 6 presents models which include a dynamic coupling between angiogenesis and vascular adaptation. Finally, we present our conclusions in Section 7.

2. Brief introduction to the biology of angiogenesis Angiogenesis is part of the homeostatic mechanisms in place to maintain tissue well-oxygenated in response to acute increase in metabolic demands, and as such is part of many physiological, normal processes [C]. In fact, the vascular system is the first organ to be formed during development, a process called vasculogenesis, which is different from, but related to, angiogenesis that has been the object of a number of mathematical models [ABe, Me, MG, MB, SP]. Hypoxia (i.e. oxygen starvation) is a major factor in triggering angiogenesis. When solid tumours grow beyond some point, the existent vasculature is unable to supply it with the oxygen and other substances it needs to keep on growing, extensive regions within the tumour become hypoxic. In response to such stress, cancer cells secrete and release a number of signalling molecules, the so-called tumour- angiogenesis factors (TAFs), in particular a cytokine called VEGF which has potent angiogenic effects [BB, PR]4. TAFs secreted by tumour cells under metabolic stress (e.g. hypoxia) are released into the tissue where they diffuse to eventually reach the pre-existent vasculature. On receiving the corresponding stimulus, ECs lining these vessels are activated and a complex series of events ensues. One of the earliest of these events seems to consist of ECs activating their proteolytic machinery, i.e. the pathways which regulate the ability of the cell for protein degradation, being activated [PBa, PBb], which appears to be a fundamental step in helping the angiogenenic process. ECs need to degrade the basement membrane in order to pass into the surrounding tissue. Once the ECs have entered the sorrounding tissue, they are able detect gradients of TAF and respond chemotactically, migrating towards the source of angiogenic factor (i.e. the tumour), thus forming cords of ECs [C]. In addition to the chemotactic response, EC migration has been recently shown to be also dependent

4The mechanism just described is by no means exclusive of cancer cells, as normal cells respond essentially in the same way to hypoxic stress. However, it often seems to be the case that tumours take advantage of normal physiological process. For example, it has been observed that in many solid tumours the hypoxia-induced factor (HIF), which is part of the normal response system to hypoxia, is upregulated to the beneﬁt of the tumour [PR].

MODELLING TUMOUR-INDUCED ANGIOGENESIS 49 upon the activation of the proteolyitc machinery, which degrades the extracellular matrix of the tissue, thus facilitating EC movement [PBb]5. During their migration towards the tumour, some of these cords meet and fuse with each other. This process of cord fusion is called anastomosis and is the first step towards a functional vascular bed. The aforementioned process generates a proto-vascular 6 network, which does not aleviate the physiological stress on the tumour, as it is not circulated by blood and, therefore, is not supplying the tumour with any nutrients. Before becoming a functional network, a few steps must still be accomplished. These steps involve capillary strand and tube formation, which, eventually, leads to the establishment of blood flow. Stabilisation of this nascent vasculature is a very complex process with a number of different signalling pathways involved. It appears, however, that, in both vasculogenesis and physiological angiogenesis, such stablisation depends upon recruitment of mural cells (i.e. cells belonging to the wall of blood vessel) and generation of extracellular matrices, which trigger survival signals within the ECs and contribute towards preventing vessel collapse [C, Ja]. Another system that is critical for vessel maturation in embryonic vasculogenesis is the Tie receptors, Tie1 and Tie2. The corresponding ligands are the an- giopoietins, Ang1 and Ang2. These two ligands are secreted by the mural cells and the ECs, respectively. Ang1 stabilises new vessels and makes them leak-resistant, possibly by facilitating the interaction between ECs and mural cells although the mechanisms by which Ang1 acts are largely unknown7. The effects of Ang2 appear to depend upon the actual environment: in the presence of VEGF, Ang2 promotes sprouting and, thus, angiogenesis. When VEGF is absent, Ang2 acts as an antagonist of Ang1, destabilising the vessel and promoting collapse. Many of the mechanisms involved in physiological angionesis, as it occurs, for example, in wound healing, are similar to those occurring in embryonic vasculogenesis. However, the different environment, as produced, for example, by low pH and abnormal hydro- static pressure or shear stress, may remarkably influence the formation, maturation and remodelling of angiogenic vessels. 2.1. Abnormal vessel maturation in tumour angiogenesis. 8 Most of the process described above is common to physiological (normal) and pathological angiogenesis, specially those concerning its earliest stages. However, as far as vessel maturation is concerned, normal and tumour angiogenesis differ greatly. In fact, abnormal vasculature is one of the hallmarks of solid tumours [Ja]. Tumour vasculature presents abnormalities with respect to its normal counterpart at all levels of organisation and function. Whereas the normal vasculature has a well structured, hierarchical organisation, tumour vessels are organised in a chaotic fashion. This has a number a consequences specially regarding transport of

5Tissue degradation by activation proteolysis appears to play a major role in other stages of the development of solid tumours, such as invasion. See [CLa, CLb] for detailed reviews and models. 6The term proto-vessel to refer to these uncirculated, immature cords of ECs was (unoﬃcially) coined by Markus Owen. 7For example, in absence of mural cells, upregulation of Ang1 is enough to produce normal vessels [Ja]. 8Here we give a brief summary of this important issue. The interested reader is referred to the reviews [YD, Ja, Jb] and references therein for a more detailed account.

50 TOMAS´ ALARCON´ oxygen and other nutrients and metabolites to the tissue. In normal tissue no cell is farther apart from any vessel by a distance superior to the oxygen diﬀusion length, which imposes a uniform distribution of the size of the spaces between vessels. In cancerous tissue, such distribution of sizes does not follow such a regular pattern, which leads to extensive hypoxic areas within the tissue. ECs in tumour vessels lack common endothelial markers and exhibit disreg- ulation of expression of a number of genes, in particular those corresponding to adhesion molecules. As a consequence, tumour vasculature also lacks the normal wall structure whereby ECs do not line perfectly with wide junctions in some regions. Mural-cell coverage of the vessels seems to be impaired. All these factors contribute towards tumour vessels being immature, leaky and prone to collapse. To make the situation even worse, as has been mentioned in the Introduction, solid tumours appear to have the ability of, as it grows and engulfs the native, originally normal vasculature, reversing the maturation process. As the tumour invades the normal tissue and co-opts the normal vasculature in place, vessels, sensing inappropriate co-option, start expressing autocrine Ang2. According to the picture drawn in previous paragraphs, this increase in the levels of Ang2 triggers apoptotic death in the vessels, thus producing vascular regression. Such regression leads to the development of hypoxic regions within the tumour and, consequently, to the secretion of VEGF which, together with high Ang2 levels, triggers the onset of angiogenesis. In addition, due to the fact that the levels of Ang2 stay high, the resulting vessels fail to mature normally, leading to a defective vasculature as the one described in the previous paragraphs.

3. Models of network formation based on random tip motion Broadly speaking, the models we deal with in this section correspond to the earliest stages of the angiogenic processes, i.e. to the generation of a network of uncirculated proto-vessels as described in Section 2. In other words, these models generate the skeleton of the network, without dealing with issues such as blood flow or transport of oxygen to the tissue. These models share a common structure, schematically represented in Fig. 1. The basic ingredient of such models is an individual-based model of the random movement of a point, reffered to as tip, on a lattice. This tip represents the ECs at the leading edge of a growing vessel. Its random walk on the lattice is biased according to several stimuli (which, in biological terms, correspond to VEGF concentration, ECM concentration, etc) spatially distributed over the lattice. The trail left behind by the tip (defined as the locus of lattice sites visited by the tip) is identified with the vessel. This type of model has its precendents in the stochastic model of EC response to chemotactic cues by Stokes & Lauffenburger [SL] and the models of reinforced random walks [OS]. 3.1. The continuous-discrete approach of Anderson & Chaplain. The first example of this sort of model we discuss is presented in [AC] in which the so-called continuous-discrete approach is introduced. The core of this model, as we have already mentioned, is the biased random walk of the tip of the vessel. In this model, the movement of the tip is modulated by two factors: the gradient of TAF and the gradient of adhesivity. In every individual-based model (IBM) some rules for the time evolution of the system must be defined. These rules dictate how the state at time t +∆t depends

MODELLING TUMOUR-INDUCED ANGIOGENESIS 51

Tumour (high VEGF concentration)

∆∆ Pk,l =f( V, f,i,j)

P i,j+1

P P i−1,j i+1,j

Parent vessel (low VEGF concentration)

Figure 1. This figure is an schematic representation of a typical model of network formation by tip random migration. In the plot the tumour is assumed to at the top and a vessel within the organism native vasculature is at the bottom. The grid represents the space between both. The figure depicts the random movement of the tip (in yellow in the figure), which is biased by the VEGF gradient (chemotaxis) and by the adhesivity gradient (haptotaxis). The vessel is then assumed to be the set of lattice sites visited by the tip in its random motion (black spots). on the state of the system at time t. The way in which these rules for the dynamical evolution of the system are defined in a somewhat arbitrary way. The novelty of the model presented by Anderson & Chaplain [AC] rests on the way in which these rules are defined9. The starting point of the method presented in [AC] is a continuous, PDE model for EC motility. This model incorporates diffusion, chemotactic movement in response to a signalling cue, and haptotaxis (i.e. migration up adhesivity gradients which, in the present case is assumed to be proportional to the gradient of extracellular matrix). The model is given by: ∂n χ = D∇2 −∇· n∇c −∇·(ρn∇f) ∂t 1+αc ∂f = βn − γnf ∂t ∂c (3.1) = −ηnc ∂t

9There exist some precedents for the model proposed by Anderson & Chaplain [AC]. For example, there is a Monte Carlo method for solving the diﬀusion equation based on discretising the Laplace operator using an Euler scheme, interpreting the corresponding coeﬃcients as transition probabilities and generating an ensemble of random walk trajectories and averaging over that ensemble to produce the solution [LB].

52 TOMAS´ ALARCON´

In these equations, n is the density of ECs, c is the concentration of TAF (chemotactic signal) and f is the concentration of extracellular matrix proteins10 the second term on the right hand side of the first equation corresponds to the chemotactic response to the signalling cue, c, whereas the thrid term corresponds to haptotactic response to gradients of extracellular matrix, f. Furthermore, this model assumes that the extracellular matrix is both degraded and diposited by the ECs. TAF is uptaken by the ECs. The next step in Anderson & Chaplain is discretising the PDE in Eq. (3.1) using Euler finite difference scheme, which involves discretising the domain where the model Eq. (3.1) is solved as a square lattice of a given mesh size h. They further discretise the equation in the time domain by discrete increments k:

q+1 q q q q q (3.2) nl,m = nl,mP0 + nl+1,mP1 + nl−1,mP2 + nl,m+1P3 + nl,m−1P4, where the coefficients Pi depend on the values of c and f on the lattice site l, m and in the corresponding first neighbour in a way such that Pi grows with the difference between them. Now, similarly to what is done in the Monte Carlo solution of the diffusion equation, Eq. (3.2) is reintrepreted as a Master Equation and the coefficients Pi as the transition probabilities for the tip of the vessel to move from site l, m to the corresponding first neighbour, i.e. P0 is the probability per unit time of not moving from l, m, P1 the probability of moving one lattice site in the x direction, and so on and so forth. In order to generate a network, these rules for tip movement need to be complemented with rules for branching and anastomosis. These processes are not part of the original continuous model as given by Eqs. (3.2) and they have to be added ad hoc. As far as the process of branching is concerned, three rules are taken into account: (1) The age of the current sprout is greater than some threshold branching age Ψ, i.e., new sprouts must mature for a length of time at least equal to Ψ before being able to branch. (2) There is sufficient space locally for a new sprout to form, i.e., branching into a space occupied by another sprout is not possible. (3) The endothelial-cell density is greater than a threshold level nb, where −1 nb ∝ cl,m. These rules incorporate a number of biologically feasible assumptions, namely, a nascent sprout must reach a minimum size before a new sprout branches off, and branching becomes more frequent as the vessel approaches the tumour. Anatomosis, i.e. the fusion of two sprouts, is supposed to occur whenever two tips encounter at the same lattice site. Once such fusion has occurred, it is assumed that only one of the two sprouts carry on growing. In addition an initial profile of c and f is given, which then evoves according to the discretise version of Eqs. (3.1). The initial distribution of TAF, c, is such that its concentration is higher in the tumour side and decreases towards the parent vessel side (see Fig. 1).

10Anderson & Chaplain consider a particular component of the extra cellular matrix: ﬁbronectin.

MODELLING TUMOUR-INDUCED ANGIOGENESIS 53

3.1.1. Results. The first model results obtained by Anderson & Chaplain con- cern the effect of haptotaxis on the structure of the resulting network. They observe that when haptotaxis is absent (ρ = 0), the sprouts grow towards the tumour in paralel with no anastomosis and minimum branching. In the presence of haptotaxis (ρ = 0) the results are drammatically different: lateral movement of the sprouts ensues and, from very early on, anastomosis occurs. Branching is very much increased with respect to the previous case, specially in the close neighbourhood of the tumour. The rate at which, on average, the network approaches the tumour is smaller when haptotaxis is included than when migration is only due to chemotaxis. Furthermore, for ρ = 0 the leading edge of the network slows down as the tumour is approached, which prevents the completion of angiogenesis, as the network stops advancing before reaching the tumour. A remedy to such situation, which allows angiogenesis to be completed, is the introduction of EC proliferation. It is assumed that EC proliferation occur at regular intervals after an initial transient has lapsed during which there is no proliferation (as observed experimentally). Anderson & Chaplain [AC] assume that this has the effect on their model of asynchronously (i.e. at different times for each sprout) increasing the length of each sprout by one lattice space.

3.2. Reinforced random walk models. Closely related to the model described in Section 3.1, there is a class of individual-based of angiogenesis models based on the concept of the reinforced random walks [OS]. Originally introduced in a biological context as model for biological dispersal [OD], these models assume that the random walker diposits or consume some substance whose concentration regulates the transition probabilities. In the case of the model discussed in the previous section, these substances would correspond to ﬁbronectin and TAF, re- spectively11. The starting point of this type of models is the stochastic Master Equation. For example, in one dimension [SW]:

∂Pn + − + − (3.3) = τ (W )Pn− + τ (W )Pn − (τ (W )+τ (W ))Pn. ∂t n−1 1 n+1 +1 n n Here we assume a one dimensional grid, where the position of the EC cell is given by x = nh, h being the lattice space. Pn(t) is the probability of ﬁnding an EC ± at position x = nh at time t. τn (W ) is the probability per unit time of, being at x = nh, moving to (n +1)h and (n − 1)h, respectively. These transition rates are modulated by the control substances, denoted by W . By making the assumption that the decision of when to jump is taken independently of the decision of where to jump [OS], the diﬀusion limit:

D = lim λh h → 0,λ→∞ can be taken, yielding a Fokker-Planck-like equation [SW]:

11The reader is reminded that the model by Anderson & Chaplain [AC] is not a reinforced random walk in the strict sense.

54 TOMAS´ ALARCON´

∂P ∂ ∂ P (3.4) = D P ln ∂t ∂x ∂x τ(W ) where P is now to be interpreted as the density of ECs at position x at time t.In these models, the connection between the microscopic properties (i.e. the behaviour of individual ECs) and the macroscopic behaviour (i.e. the evolution of the density of ECs), as given by Eq. (3.4), is via the transition rate τ(W ) and this is the object that this sort of approach aims to model. Usually, these transition rates are modelled so as to reproduce a given macroscopic behaviour. For example, if the view that ECs undergo simple diﬀusion, chemotaxis and haptotaxis (i.e. the corresponding transport coeﬃcients are constant) is adopted, a possible way of choosing τ(W ) is:

τ W τ c τ f ( )= 1() 2( ) χ0 τ1(c)=exp c D ρ (3.5) τ (f)=exp 0 f 2 D where c is the concentration of TAF and f, the concentration of fibronectin. χ0 and ρ0 are the chemotactic and haptotactic coefficients, respectively. It is possible to incorporate more complex assumptions. For example, if we assume that ECs move in response to simple chemo- and hapto-taxis, but instead of simple diffusion, they are supposed to diffuse through a porous medium, the corresponding transition rate reads [SW]: P m χ ρ (3.6) τ(P, c, f)=P exp − + 0 c + 0 f ,m>0. m D D In the particular case of the model presented by Sleeman & Wallis [SW], Eqs. (3.5) is the model of choice. Sleeman & Wallis present individual-based simulations of a random walk in 2 and 3 dimensions with the transition rates determined by Eqs. (3.5). In general, angiogenesis models based on the reinforced random walk concept do not incorporate cell proliferation (note that Eq. (3.4) describes the evolution of a conserved density). Therefore, they do not lead to successful angiogenesis (see discussion in Section 3.1), however they are expected to produce useful insights on the migration of ECs under tactic stimuli. This is the case with the model presented in [SW]. Otherwise, their results are very similar to those discussed by Anderson & Chaplain in [AC]. Where haptotaxis is not present, sprouts move parallel to each other, thus preventing anastomosis and the formation of a connected network. Consideration of haptotaxis remediates this situation. 3.2.1. Off-lattice models of angiogenesis. An interesting generalisation of the reinforced random walk model of angiogenesis (in its proto-network formation stage) was presented by Planck & Sleeman [PSa]. Their model, which could be dubbed an off-lattice reinforced random walk, tries to unify the approach of Stokes & Lauf- fenburger [SL] with individual-based models such as those discussed in Sections 3.1 and 3.2. The model by Stokes & Lauffenburger [SL] is a model for the movement of ECs in two dimensions under chemotactic stimulus. The model is formulated in

MODELLING TUMOUR-INDUCED ANGIOGENESIS 55 terms of a (system of two) stochastic differential equation(s) for the two components of the velocity of an EC. Both the drift and the (white) noise terms depend on the gradient of TAF. The model developed in [PSa] is too a model for the random variation of EC velocity in two dimensions. Their basic assumption is that the modulus of the velocity and its direction vary independently from each other. It is further assumed that the modulus of the velocity is a constant, which we denote by s. This reduces the problem to study the biased random walk of a particle on the surface of the unit sphere. A model, i.e. the equivalent of the function τ(W ), is proposed for the probability per unit rate of the velocity vector to turn clockwise and counter-clockwise. The model they propose incorporate chemo- and haptotaxis by introducing two bias directions, parallel to the direction of the gradient of TAF and adhesivity, respectively, in their model transition probability12. Similarly to what is done in the two models previously described they supplement the EC migration model with rules for branching and anastomosis. They further assume that, at high TAF concentrations, ECs become desensitised so that they are not able to detect gradients of TAF, i.e. there is no bias direction in the transition probability for the orientation of the velocity vector. Planck & Sleeman simulations show how each of the factors included influences the model and then proceed to make a comparison with the results obtained in lattice-based simulations [PSa]. In particular, they compare to the models presented in [AC, SW], which we have discussed in Sections 3.1 and 3.2. To summarise, the results of the off-lattice simulations exhibit qualitative agreement with their lattice-based counterparts, specially those regarding the role of haptotaxis in producing a functional network in which the different vessels fuse to form arcades through which, at a later stage, blood may flow. There are, however, some significant differences in the behaviour of their off-lattice model, namely, the effects of increasing the concentration of fibronectin has more severe effects on the lattice- based models than on their off-lattice counterpart. It has been shown in [AC, SW] that fibronectin delays the chemotactic movement of EC. Planck & Sleeman show that if the effect of haptotaxis is increased in the lattice-based simulations ECs are not able to separate from the parent vessel and migrate towards the tumour. By contrast, in their off-lattice model, ECs have an intial momentum that, under equivalent circumstances, may allow the ECs to overcome the brake imposed by extracellular matrix.

4. Models of blood flow and vascular adaptation So far, we have discussed some examples of individual-based models which, by tracking the movement of a tip EC, produce a model network of proto-vessels.The main outcome of these models is the clarification of how hapto- and chemo-taxis combined to produce a network with the appropriate structure to sustain blood flow, although the models described in Section 3 do not explicitely model the problems inherent to blood flow through complex structures such as their resulting networks. However, the properties of blood flow through such networks are a major issue in the overall dynamics of tumour growth and in the elaboration of efficient therapeutic strategies.

12Planck & Sleeman [PSa] adopt and generalise a model of a biased circular random walk previously developed in the context of gravitotactic swimming organisms. See [PSa] for references.

56 TOMAS´ ALARCON´

There are several intrinsic difficulties in the modelling of blood flow through vascular networks. The first one is the complex nature of blood itself. Blood is far from being a simple, Newtonian fluid. It is a complex suspension of cells and particles with a wide range of characteristic sizes. As a consequence, it exhibits a strong non-Newtonian behaviour. Luckily, it is possible to make a number of approximations that render the problem of computing the blood flow through a complex vascular network workable. The most popular of these approaches consists of assuming that the flow through each vessel can be described by a Poiseuille flow but with an effective viscosity, which may depend on a number of factors, including properties of both vessels and blood [PSb].

5 H=0.6

4 rel µ H=0.45 3

H=0.3 2

H=0.15

1 10 100 1000 R (µm)

Figure 2. Plot of the relative viscosity, µrel, as a function of the radius of the vessel for various values of the haematocrit. This plot corresponds to a ﬁt from experimental data obtained by Pries et al. [PSb].

This approach has been followed by Pries et al. [PSb] who found that the main contributors to this effective viscosity are the radius of the vessel, R,andthe red blood cells contained within a vessel, i.e. haematocrit13, H, thereby implicitely considering blood as a two-phase fluid: plasma (liquid phase) and haematocrit (erythrocites, i.e. solid phase). Blood flow through a vessel within a microvascular network is thus described by:

πR4∆P (4.1) Q˙ = 8µ0µrel(R, H)L where Q˙ is the ﬂow rate, ∆P is pressure drop between the two ends of the vessel, L is the length of the vessel, µ0 is the viscosity of plasma, and µrel(R, H) is the so- called relative viscosity, which is deﬁned as the ration between blood viscosity and plasma viscosity. The dependence of µrel on R and H is shown in Fig. 2, where we

13The haematocrit is actually deﬁned as the proportion of the total volume of a vessel which is occupied by red blood cells.

MODELLING TUMOUR-INDUCED ANGIOGENESIS 57 have plot the analytical fit obtained by Pries et al. from experimental data [PSb]. We can see that the overall blood viscosity increases with increasing haematocrit, as blood becomes thicker. We can observe, that due to Fahreaus-Lindquist effect [Fu], viscosity decreases with decreasing vessel radius, until a minimum is reached whereby blood becomes more viscous as radius decreases further. A further problem when trying to model blood flow through microvascular networks is the interaction between the structure of the network, the influence of the surrounding tissue and blood flow itself. This is a long standing problem in the study of the physiology of the vascular system which relates to fundamental issues of how its function (i.e. providing oxygen and other essential substances to every cell within the organism) and its structure are related [L]. Investigations on these issues go back to D’Arcy Thomson’s seminal book [T]andtheworkof Murray [Mu]. He hypothesised a design principle for the vascular system whereby it should be organised so as to minimise the energetic cost necessary to run it. Murray further assumed that the main sources of energy dissipation are blood flow and blood’s metabolic rate:

˙ 2 8µLQ 2 (4.2) D = + αbπR L, πR4 where αb is the metabolic rate per unit volume of blood, and, therefore, the second term on the right hand side of this equation represents the energetic cost of the volume of blood contained in a vessel of radius R and length L. The first term corresponds to the energy dissipated by Poiseuille flow. Note that Murray assumes that the viscosity of blood is constant. By minimising D with respect to the radius we obtain Q˙ ∼ R3. This dependency of flow rate on the third power of the radius of a vessel in an optimal network is known as Murray’s law. Further analysis by Zamir [Z] lead to the realisation that Murray’s law yields constant wall shear stress (WSS) throughout the vascular system, which he could check to be a valid prediction against experimental data for the bigger arteries. Some 20 years after Zamir’s work, more data had become available and the predictions of Murray’s law had to be reconsidered. Pries et al. [PSc] collected a more complete set of data than the one Zamir was able to obtain, including data for arteries, arterioles, veins, venules and capillaries. Strikingly, when the WSS is plotted against the corresponding physiological pressure the different type of vessels are subjected to, their whole data set collapsed into a sigmoidal-like curve: for bigger vessels the WSS is constant (pressure-independent), consistently with Zamir’s analysis, but as we move down the hierarchy of the vascular system, both the WSS and the pressure decrease in such a way that the relation between them fits the aforementioned sigmoidal relationship (see Fig. 3 for an schematic representation). In view of these results, Pries et al. [PSc] formulated a design principle for vascular beds whereby vascular networks should be organised in such a way that the WSS-pressure relationship adopts the sigmoidal form they have found experimentally. More recent work on this issue include design principles based on geometrical constraints, such as the network being space-filling [GL] or being generated with an stochastic growth process whose fractal properties match those observed experimentally [GB]. Further research has been done on design principles based

58 TOMAS´ ALARCON´

100 WSS (dyn/cm2)

40 20 100

Pressure (mmHg)

Figure 3. Schematic representation of the WSS-pressure relationship, based on the data collected by Pries et al. [PSc]. The blue region in the plot correspond to the regions in which data from arterioles, venules and capillaries collapse. The red region corresponds to the region corresponding to arteries and veins. on minimisation of the energy cost necessary to run the vascular system, including a model able to predict the allometric scaling between metabolic rate and body mass [WB], and a generalisation of Murray’s design principle which, by taking into account the dependency of the blood viscosity on vessel radius and haematocrit shown in Fig. 2, is able to reproduce the WSS-pressure relationships found both experimentally and theoretically [ABd]. The design principles described in previous paragraphs represent a static picture and do not take into account another fundamental property of the vascular system: adaptability. The vascular system is not simply a collection of passive pipes, it is under tight regulation to adapt to the changing necessities of the diﬀer- ent tissues within the body. In order to achieve that, a number of mechanisms have been developed to act upon the vasculature both acutely, eg transient structural adaptation by means of controlling vessel radius, and cronically, e.g. remodelling involving vessel pruning and angiogenesis. Mechanisms of structural adaptation, i.e. changes in vessel radius in response to a number of stimuli, has been proposed and studied at length by Pries et al. [PSd, PSe]. The main assumption is that vessel radius change in time to adapt to signals both from the blood ﬂow itself, so the vascular system will have a general tendency to comply to the structure imposed by a given design principle14 and from the tissue, eg if some portion of the tissue undergoes hypoxia, it will produce and release signalling cues which will tend to increase the vessel radius so the supply of oxygen is locally increased. These two stimuli are referred to as hydrodynamic stimulus and metabolic stimulus respectively. There is a further stimulus, the so-called conducted stimulus, which ensures that the vascular system remains functional upon local adaptation. This stimulus consists of signals sent both up- and down-stream. Last,

14In the case of [PSd, PSe], this design principle consists of the vasculature to adapt itself so that WSS a pressure exhibit a sigmoidal dependency [PSc].

MODELLING TUMOUR-INDUCED ANGIOGENESIS 59 and to ensure that the system is stable, a global shrinking tendency is assumed, whereby all vessels shrink at a given rate in the absence of signals that prevent them to do so.

4.1. Simulations on complex networks with simple rheology. The picture emerging from this summary is that modelling blood flow through the mi- crocirculation is a daunting task, as blood is not only a complex, non-Newtonian fluid but the network of vessels it flows through is itself a dynamical system which responds to both blood flow and external signals. In view of this, the first attempts on modelling blood flow through vascular networks in cancer were performed on static networks. In particular, the work by McDougall et al. [MAa] takes as starting point a network generated by the model of Anderson & Chaplain [AC]. The outcome of the simulation of the network formation process is a set of nodes, corresponding to the points where two vessels have fuse to form an arcade, and a connectivity matrix which contains the information of how the nodes are connected. This simulated network is then projected onto a square lattice and the vessels are constructed by selecting at random their lengths (in units of the lattice spacing) and joining the corresponding nodes along the edges of the lattice (see Fig. 4 for an schematic representation). Radii of each vessel are also drawn randomly from a uniform distribution.

Figure 4. Schematic representation of the projection of the vascular network on a square lattice. The vessel on the top is assumed to be the parent vessel. A pressure drop is applied between the ends of the parent vessel. This is the only pressure drop assumed. The tumour is assumed to be on the far bottom side of this cartoon.

The basic blood flow simulation technique used in [MAa], which is shared by most of the works to be discussed later on, is to assume that blood flow through a vessel can be described by Poiseuillle’s law. This law (see Eq. 4.1) has the same form as Ohm’s law in electrical current. As blood flow and electric current are both under the same type of conservation constrains, it is to possible to map the blood flow problem onto an electric current problem and use Kirchoff’s laws to solve it.

60 TOMAS´ ALARCON´

McDougall et al. [MAa] make some rather strong simplifying assumptions: they consider blood as a Newtonian fluid with constant viscosity and they consider that the vessel radii do not change. In spite of these assumptions, and as a consequence of the complex structure of the vascular network, they obtain some rather interesting results concerning transport of drugs to the tumour. To analyse this issue, McDougall et al. first solve for the nodal pressures and elemental flow rates and then inject a substance (i.e. drug) into the upstream end of the parent vessel. At each time step, the amount of drug entering each node is calculate and, by assuming perfect mixing within each node, the new drug concentrations for each outlet vessel are calculated (for the technicalities involved in this calculation the reader is refered to [MAa]). The results from simulations of drug transport with continuous infusion reported in [MAa] are as expected: due to its continuous infusion, the drug eventually saturates the capillary netwrok and significant amounts of drug reach the tumour. In fact, after an initial transient where no drug has reached the tumour (corresponding to the transit time for the leading edge of the drug distribution within the network to reach the tumour) the amount of drug being delivered grows linearly with time. As expected, increasing the blood viscosity and decreasing the average vessel radius has the same qualitative effect, namely, reducing the amount of drug being delivered. Simulations with drug being administered as a bolus injection (i.e. drug being delivered into the inlet of the parent vessel only for a given period of time) appear to produce more interesting results. As it is injected, drug initially propagates through the network very much in the same way as in the continuous infusion case. However, after the bolus injection, fresh clean blood comes in and start diluting the drug within the vasculature. This, combined with high tansit times due to the complex structure of the network, may have important consequences on drug efficiency, specially if, for example, the drug needs to accumulate within the tissue beyond some threshold before becoming effective [MAa].

4.2. Investigations on simple networks with complex rheology. The results of the model by McDougall et al. [MAa] are a reflection of the complex network structure, due to the simplifying assumptions made on blood rheology (constant viscosity) and structural adaptation (vessel radii held static throughout the simulations). The models by Alarcón et al. [ABa, ABb, ABc, AO] take the opposite approach: whilst the topology of the network considered is far simpler than the one considered in [MAa], the model by Alarcón et al. incorporates the complexities involved in blood rheology and vascular structural adaptation. There is another substantial difference between the models discussed in this section and those formulated by McDougall et al. [MAa, SM, MAb]. Whilst the later ones focus exclusively on the formation of the vascular network and the properties of flux and drug transport, the former ones couple the evolution of the network and blood flow, and the dynamics of the tissue. The basic model as proposed in [ABa] builds up on the empirical description of the relative viscosity as discussed earlier on this Section [PSb] and on the structural adaptation mechanism proposed by Pries et al. [PSd], with the difference that we are not considering the conducted stimuli. The main aim of this model is to study the effect the complexities inherent to blood flow have on the transport of oxygen

MODELLING TUMOUR-INDUCED ANGIOGENESIS 61

a) b) P P H L

(µ,ν) 3

3 (µ,ν)

Figure 5. Representation of the hexagonal vascular lattice used in [ABa, ABb, ABc]. to the tissue. This forces the introduction of a factor not taken into account in the model discussed in Section 4.1: the haematocrit. By doing so we can calculate the amount of red blood cells within each vessel of the network shown in Fig. 5. This network is then projected onto an square lattice (very much in the same way as it is done in [MAa] and discussed in Section 4.1). This information is then used as a spatially extended source of oxygen, which, in turn, by solving the corresponding diffusion equation allows us to calculate the concentration of oxygen on each lattice site (for the technical details involved in this, the reader is referred to [ABa]). Simultaneously, on this lattice, there is a cellular automaton describing the population dynamics of and the competition between normal and cancer cells. This dynamics depends on the local oxygen levels: where the concentration of oxygen is high, cells are likely to proliferate, whereas in those regions where oxygen concentration falls below some threshold, cells are unable to survive. In this way we can analyse how blood flow and vascular dynamics influence the growth of populations of cells.

v Q H 1 1 1 P 1

v Q H P RBC 0 0 0 0

P 2 v >v => P

Figure 6. Schematic representation of the mechanism for phase separation, i.e. uneven distribution of haematocrit, at bifurcations.

Due to the inclusion of the haematocrit in the model, some consideration to the two-phase nature of blood and to phase separation eﬀects must be given. In the model presented in [ABa] such issues take the form of a number of rules on how the haematocrit is splitted at bifurcations (see Fig. 6 for an schematic representation). These rules are rather simpliﬁed and constitute a rough approximation of what is

62 TOMAS´ ALARCON´ actually happening but they still seem to provide a reasonable description. The rule for haematocrit splitting is divided in two parts, based on the ratio of the velocities of the two daughter vessels. If this ratio is smaller than a given threshold, the haematocrit is splitted so that the ratio between the haematocrits corresponding to each of the daughter vessels equals the corresponding ratio of velocities. If, rather, the velocity ratio is larger than the threshold, due to the so-called plasma skimming effect, all the haematocrit of the parent vessels passes onto the faster branch, whereas the slowest one gets none. All these ingredients contribute towards a very complex behaviour: blood flow, which determines the distribution of haematocrit and therefore tissue behaviour, depends, via the viscosity and the structural adaptation mechanism, on the distribution of haematocrit. This coupling is treated in the usual iterative way until some stationarity condition is satisfied. As a consequence of this coupling the distributions of both flow and haematocrit are very inhomogeneous. This, in turn, induces an inhomogeneous oxygen distribution which, in turn, strongly influence the growth properties of the cell populations. This implies that the view usually held whereby wherevere there are endothelial cells there is blood and, therefore, the tissue is going to be well oxygenated, is unlikely to be accurate [ABa]. These results are likely to have some bearing on the way anti-angiogenic therapy is conceived. This issue was partially addressed in [ABb]. Here, simulations of the model presented in [ABa] are carried out for different vascular densities. It is observed that eliminating vessels does not necessarily leads to smaller tumours. In fact, if the fraction of vessels being eliminated is moderate, a larger tumour burden is sustained by a less dense vasculature with respect to the original (more dense) vasculature. This is caused by the fact that the less dense vasculature produces a more homogeneous distribution of haematocrit and, therefore, as well of oxygen over the tissue, which leads to more cells being able to survive than under the more dense vasclature. Of course, if the fraction of removed vessels is very big, large pockets of hypoxia appear, resulting in smaller tumours than in the “control” case.

4.3. Models incorporating complex vasculature and complex rheology. These main ideas of the models proposed in [MAa, ABa] have been further developed into models which aim to couple the formation and later remodelling of the network with blood flow and structural adaptation. An example of a first step towards such an integrative model is presented by Stephanou et al. in [SM]. This model represents an extension of the model originally formulated in [MAa]intwo different directions. One one hand, Stephanou et al. couple the process of network formation to blood flow introducing shear stress-modulated branching probabilities. The model for network formation considered in [MAa] is essentially the one proposed in [AC]. In [SM], this model is modified in two ways: the effects of matrix-degrading enzymes secreted by active ECs is explicitely considered, and the branching probabilities are considered to depend upon both the local concentration of TAF and the wall shear stress exerted by blood flow upon a particular vessel. As the network formation process progresses, arcades able to sustain blood flow are formed. Whenever one of these is formed, the simulation solves for the new distribution of nodal pressures and elemental flows, which allows to calculate the WSS acting upon the circulated vessels. This information is, in turn, used to update the branching

MODELLING TUMOUR-INDUCED ANGIOGENESIS 63 probabilities. Generally speaking, these probabilities are monotonically increasing functions of both TAF concentration and WSS [SM]. A comparison between the networks produced by WSS-TAF modulated branching and TAF-only induced branching yields interesting results. From the point of view of the structure the resulting network, the former mechanism appears to produce a more dense network close to the surface of the tumour than the later. Fur- thermore, the model incoporating WSS-TAF modulated branching leads to a global redistribution of the WSS, as the vasculature evolves continuously. Such WSS redistribution leads to reinforcement of vessel connectivity in other parts of the network and, consequently, modification of the blood flow [SM]. A closer, more quantitative analysis reveals important structural differences between networks generated by WSS-TAF or TAF-only branching. Whereas density increases as the surface of the tumour is approached in both cases, an analysis of the degree distribution of the networks (the amount of nodes connected to a given number of nodes) reveals that, close to the surface of the tumour, the network generated by TAF-only regulated branching is dominated by nodes with two vessels. On the contrary, the network corresponding to the WSS-TAF modulated branching the network is much more interconnected as the proportion of nodes with three and four vessels is similar to that of nodes with two vessels [SM]. A second difference between the approach by Stephanou et al. [SM] and its counterpart as formulated in [MAa] concerns the consideration of mechanisms for vascular structural adaptation similar to those accounted for in [ABa, ABb]. The networks described in previous paragraphs generated by a WSS-TAF modulated branching probability are homogeneous in the sense that the vessel radii are held constant throughout the simulation. The implementation of these mechanisms are similar to the model described in [ABa]: after the network has been prescribed, as generated by the mechanisms described in [SM] and described in the previous paragraphs, the radii of the vessels is modified accordingly to the structural adaptation mechanisms discused in 4.2. Once these two issues have been introduced in the model and the corresponding networks have been generated, Stephanou et al. [SM] analyse the transport of drugs to the tumour. As in [MAa] two different regimes are considered: continuous infusion and bollus injection. In the former case, it is observed that the amount of drug that reaches the tumour is significantly higher when structural adaptation is considered. The transit time is reduced. This is due to the fact that structural adaptation increases the average radius, and to the active remodelling of the network which reduces the bypassing effect, typical of very dense networks. In the bolus injection case, the amount of drug that reaches the tumour is greater for the adapted vasculature.

5. Models of vasculature degradation and vessel normalisation The models discussed in Sections 3 and 4 assume that the evolution of the network is independent from the dynamics of the tumour. For example, in the models discussed in Sections 3, 4.1 and 4.3 assume that the formation of the network occurs under an static distribution of TAF without any feed-back between network and tissue dynamics.

64 TOMAS´ ALARCON´

In this section, we discuss some modelling approaches to how the dynamics of the tissue alters the dynamics of the corresponding vascular network. As mentioned in Section 2, an outstanding example of this is the destabilasing eﬀect that engulfment by the growing tumour has on the vasculature. The analysis of these phenomena forces the introduction of a model that explicitely accounts for tumour growth, including the secretion and releasing of angiogenic factors in the absence of oxygen.

5.1. Flow-correlated percolation model of vascular remodelling. Bartha & Rieger [BRa, BRb] have studied the process of vascular remodelling upon engulfment by a growing tumour. Their starting point is a vascular network in two dimensions of a given density with a small tumour in the centre. The vessels are, for simplicity, arranged in a regular square mesh with lattice space a. This vascular mesh is, in turn, embeded in a lattice of smaller lattice space, which sustains the growth .of the tumour. The main aim of the model by Bartha & Rieger is to study how vessel collapse and degradation upon cooption by the growing tumour remodels the vascular network and acts on the dynamics and structure of the tissue. The dynamics of the model consists of a succession of Monte Carlo steps, each of them, in turn, consisting of a number of random steps which determine the dynamics of the diﬀerent elements (vessels and tumour cells) which make up the system. The dynamics of the tissue is driven by the local concentration of oxygen and the availability of space. At each time (Monte Carlo) step cells divide with a probability ∆τ/Tc,where∆τ is the time step and Tc the average proliferation time for cancer cells, provided there is at least one free space in the neighbourhood of the cell attempting division and there is enough oxygen. Otherwise the cell does not divide. Tumour cell death is also controlled by the local concentration of oxygen: if a cell has spent longer than some threshold time span under a threshold oxygen concentration, it is killed with probability 1/2. Angiogenesis is accounted for in Bartha & Rieger’s model in a much less detailed way than the models discussed in previous sections. Bartha & Rieger [BRa] introduce whole vessels between two existing ones depending on several factors. TAF concentration must exceed a threshold value. Furthermore, no cancer cell can be sitting in the path to be occupied by the new vessel. Last, the distance between the ends of the new vessel must not exceed a maximum number of lattice spaces. If these three conditions are satisﬁed, a new vessel is introduced between to circulated vessels with probability ∆τ/Te, where Te is the estimated proliferation time of ECs. Vessels are also allowed to increase their radii provided the local concentration of TAF exceeds a given threshold. Then the radius of the vessel is increased by a given amount with probabily ∆τ/Te, provided the radius does not exceed a maximum value. Vessel collapse occurs upon engulfment by the growing tumour. As summarised in Section 2, vessel co-option triggers a process of vessel dematuration which render the vessel more susceptible to collapse. Bartha & Rieger model this process by letting a vessel to be removed from the network with probability ∆τ/Tcollapse if the vessel is sorrounded by cancer cells and the wall shear stress is below a certain threshold. Vessels are also assumed to regress if they have spent longer than a threshold time span under low oxygen concentration15.

15Recall that vessels may be uncirculated and, consequently, carrying no oxygen.

MODELLING TUMOUR-INDUCED ANGIOGENESIS 65

The distributions of TAF and oxygen in the model by Bartha & Rieger [BRa] are introduced in a phenomenoligical way. Cancer cells are assumed to be sources of TAF (regardless the concentration of oxygen), the concentration of TAF being defined by: (5.1) cT ( r)= f(| r − r |) r where the summation is over the locus of lattice sites, r, occupied by tumour cells. The function f(x)=(RT − x)/N forx ≤ RT and f(x) = 0 otherwise. N is a normalisation function chosen so that r f(| r|) = 1. The oxygen concentration is defined in a similar way: (5.2) cO( r)= g(| r − r |) r where now the summation is over the locus of all lattice sites occupied by circulated vessels. The function g(x)=(RO − x)/Nfor x ≤ RO and g(x) = 0 otherwise. N is a normalisation function chosen so that r g(| r|) = 1. The constants RT and RO represent the diffusion lengths of TAF and oxygen, respectively16. Due to the presence of vessel collapse and regression not all the vessels are circulated: only those which are connected by an uninterrupted path to both the inlet and outlet of the network are. Hence, after each Monte Carlo step, such vessels need to be identified17. Once, this has been done the nodal pressures and elemental flows and WSS are computed using Kirchoff’s laws. Bartha & Rieger further assume that blood is Newtonian and consider constant viscosity. 5.1.1. Results. Bartha & Rieger [BRa] obtain a number of interesting results. The first one concerns the structure of the network and the tumour. Their model is able to reproduce experimental results in melanoma whereby the density of vessels in the interior of the tumour is reduced, as a consequence of vessel remodelling upon vessel co-option, with the remaining vessels having large radii. They also observe that the density of vessels greatly increases towards the edge of the tumour. As far as the tumour morphology is concerned, they observe, due to the extensive remodelling and pruning of the network, equally extensive necrotic regions in the centre of the tumour, again in agreement to observations in melanoma. A most interesting conclusion of Bartha & Rieger investigations [BRa, BRb] is the fact that the resulting structure of the vascular network in the interior of the tumour is a direct consequence of the fact that the pruning of the network upon engulfment by the tumour is coupled to the blood flow. If vessels were removed in a purely random fashion, due to fundamental theorems in percolation theory [CM], the interior of the tumour would be either completely deprived of or totally full of vessels. Only for a very particular value of the removal probability (corresponding to the critical value of the percolation transition) it would be possible to obtain an intermediate vascular density. The fact that the removal probability depends upon the WSS is of basic importance to obtain a more realistic morphology: pruning the

16The functions f and g are linear approximations to the proper (exponential) propagator of the diﬀusion equation. In this sense, Bartha & Rieger [BRa] are ignoring that oxygen is being consumed by the cells. 17Technically speaking, this is achieved by computing the bipartite component of the associated network.

66 TOMAS´ ALARCON´ vessels with lower WSS deviates the ﬂow through a increasingly smaller number of bigger vessels where the WSS exceeds the removal threshold. A further interesting observation is the fact that, due to the spatial and temporal inhomogeneities of the vascular dynamics, the microvascular density is not a particularly informative index as to what is going to be the future evolution of the tumour. The initial vascular density, instead, seems to produce more interesting predictions [BRa].

5.2. A multiscale model of the combination of anti-TAF and chemotherapy. As it has been mentioned in the Introduction, in spite of soundness of the anti-angiogenic approach to treating solid tumours and its early success both in vitro and in animal experiments, its failure in clinical trials has triggered the investigation as to why anti-angiogenic therapy has not fared as well as expected and possible ways to salvage the considerable amount of effort and resources invested in it. One of these approaches consists of trying to exploit the phenomenon of vessel normalisation to try and combine anti-angiogenic and conventional chemo- or radiotherapy [Jb]. In Section 2, we have mentioned that, due, at least in part, to the high concentration of angiogeneic factors tumour vasculature is not allowed to mature properly as its normal counterpart does. As a consequence, it is now believed that anti-angiogenic drugs acts as stabilisers of the abnormal vasculature generated by tumour-induced angiogenesis: under the action of such drugs, the vascular network is prunned, vessels have a more normal-looking structure, as blood flow does. Oxygen transport to the tissue is thus similarly normalised and, therefore, becomes more evenly distributed (see the review by Jain [Jb] and references therein). As hypoxia is a well-known factor promoting resistance to conventional therapy18, combination of anti-angiogenic therapy, which appears to remediate hypoxia within the tissue, and conventional therapy should produce improved therapeutic outcome. Such results are partially hinted by the simulation results presented in [ABb], where moderate pruning of the vessel network leads to a more homogeneous oxygen distribution to the benefit of the tumour, which is then able to grow bigger than under a more dense vasculature, although a more in-depth investigation is needed. To this end, Alarcón et al. [AO] present a generalisation of the multiscale model formulated in [ABc]. The model formulated in [ABc] is a multiscale model of vascular tumour growth, which couples within an integrated framework models for cell behaviour (i.e. models of how extracellular oxygen modulates cell-cycle progression, apoptosis and VEGF secretion), models for the competition between normal and cancer cells (i.e. a cellular automaton model which accounts for cell-to-cell interactions and space limitations), and a model for blood flow, structural adaptation (coupled to dynamics of the tissue via VEGF) and oxygen transport19. The original model has subsequently been subject to several improvements in issues concerning

18Radio- and chemo-therapy target dividing cells. Hypoxia delays progression through the cell-cycle, thus reducing the duplication rate of cancer cells, hence the role of hypoxia in resistance to therapy. For a recent review of the role of hypoxia on resistance to radiation therapy the reader is referred to [BH] 19The last two ingredients of this model, i.e. the models concerning the cellular phase and vascular dynamics are based on the model presented in [ABa] and discussed in Section 4.2.

MODELLING TUMOUR-INDUCED ANGIOGENESIS 67 the inclusion of drug transport and delivery to the tissue [BA] and cell movement [BOa]. In [AO], this modelling framework is further developed to account for different states of vessel maturation in terms of whether they are engulfed by the tumour and the local concentration of VEGF, which are the two factors that according to the discussion in Section 2 determine the maturation status of a vessel. According to these factors we classify the vessels in two types: NORMAL and CO-OPTED. These two types of vessels adapt according to different adaptation mechanisms. The normal vessels behave according to the general principles for structural adaptation as per the model of Pries et al. [PSe]20 and discussed in Section 4. The behaviour of the CO-OPTED vessels depend, in turn, on the local levels of VEGF. If the level of VEGF exceeds some threshold value vessels adapt according to an angiogenic mechanism [ABc], whereby vessel adaptation is affected by the hydrodynamic, metabolic and shrinking tendency stimuli as discussed in Section 4 with the intensity of the metabolic stimulus depending on the concentration of VEGF. If, on the contrary, the local concentration of VEGF falls below its threshold the vessel adapt according to the collapsing mechanism: vessels collapse at rate η, with collapse being opposed by the hydrodynamics stimulus21 [AO]. Thus, the algorithm to determine vessel behaviour is as follows: (1) All vessels are initially labelled NORMAL. (2) The numbers of cancer and normal cells within one lattice site distance from the vessel are counted. If the former exceed the latter the vessel is termed CO-OPTED, otherwise it remains NORMAL. Vessel co-option is reversible, i.e. if this condition ceases to hold, a co-opted vessel is re-labelled NORMAL. (3) Normal vessels undergo structural adaptation according to the normal structural adaptation mechanism as proposed in [AO] (4) Co-opted vessels undergo structural adaptation according to the coopted adaptation mechanism if the local concentration of VEGF is below its threshold value. Otherwise, they adapt according to the angiogenic adaptation mechanism. It must be noted that angiogenesis is not explicitely included in this model: the angiogenic adaptation mechanism refers only to the increase in the intensity of the adaptation to metabolic demands on the part of the tissue when as the concentration of VEGF becomes bigger. Anti-angiogenic therapy in this model is assumed to take the form of an antibody against the VEGF receptor (VEGFR): this drug binds to the receptor but does not elicit any cellular response. In the model being discussed here [AO], this is taken into account within intesity of the metabolic stimulus, reducing the “effective” concentration of VEGF. Cytotoxic drug is introduced in the same way as in [BOb]: it is assumed that the cytotoxic drug acts only on cells that are proliferating. The cells in a quiescent state due to lack of oxygen are immune to it.

20The reader is warned that the conducted stimuli used in [AO] is somewhat diﬀerent from the one proposed in [PSe]. 21Note that this is pretty similar to the ﬂow-correlated vessel removal considered in [BRa, BRb]

68 TOMAS´ ALARCON´

5.2.1. Results. This model reproduces the main experimental results in spite of the fact that angiogenesis is not explicitely included, only modulation of the adaptation mechanism by the concentration of VEGF. The model studies the behaviour under two bolus injections of anti-angiogenic and cytotoxic drugs, respectively. The application of the anti-VEGFR antibody leads to a more homogenous distribution of oxygen, as reflected by the reduction of the population of cells under hypoxic stress22. Such reduction in the size of the hypoxic pockets within the tissue starts shortly after the application of the bolus injection, and after a given time span during which the size hypoxic population stays low, the size of hypoxic pockets recovers to a size comparable to its size previous to the anti-VEGFR injection. In agreement with the clinical observations the number of cancer cells killed by the injection of anti-VEGFR antibody is very small [AO]. The reduction of the size of the quiescent population after the bolus injection of anti-VEGFR antibody implies that the efficiency of the cytotoxic drug is increased if applied within this period of time. This behaviour is identified with the existence of the so-called window of opportunity. In fact, if the cytotoxic drug is administered either simultaneously with the anti-VEGFR antibody or after the window of opportunity has been closed, the combination does not yield superior results than the application of the cytotoxic drug on its own [AO].

6. Models with a dynamic coupling between angiogenesis and vascular adaptation The last two models we consider in this review consist of two different attempts to model the dynamical coupling between blood flow, vascular adaptation and angiogenesis [MAb, OA]. 6.1. A model for dynamic adaptive tumour-induced angiogenesis. The model presented by McDougall et al. in [MAb] constitutes a further step forward with respect to [SM], as it incorporates dynamic coupling between vascular adaptation and network formation, rather than adapting an already generated network. This raises a series of technical issues, mostly concerning how to achieve an efficient scheme for time stepping. This issue is, in fact, ubiquitous in the simulation of multiscale systems which, in a natural way, exhibit manifold time scales. Given that, in the present case, the time scale for EC migration and network formation (of the order of days) is much bigger than the time scale for network perfusion (of the order of minutes). An ideal scenario would involve two simulation time steps: one for the migration/network formation process, and another, much finer one for the blood/flow remodelling. This ideal scenario would then simulate network formation using the former time step, stop the simulation whenever a new arcade is formed, switch to the shorter time step, remodel the network accordingly, and then switch back to the longer time scale [MAb]. Unfortunately, in the case of the model of McDougall et al. [MAb], this becomes prohibitingly time consuming, as more and more anastomoses form as the vasculature approaches the tumour. Given these restrictions, McDougall et al. decided to flow and adapt the network at regular intervals of intermediate duration (shorter than the naturally corresponding to EC

22In the model proposed in [AO], such population is identiﬁed with the quiescent subpopulation.

MODELLING TUMOUR-INDUCED ANGIOGENESIS 69 migration but longer than the characteristic perfusion times) until it reaches an steady state. The main elements of the model, apart from the dynamical coupling between angiogenesis and vascular adaptation, most notably the inclusion of WSS-dependent branching, were already present in [SM] and discussed in Section 4.1. However, the extensive simulations carried out in [MAb] concerning the sensitivity of the system to various biochemical and physical factors produce a number of interesting results which we proceed to summarise. 6.1.1. Results. One of the most relevant results of the model presented in [MAb] concerns the role played by the WSS-dependent branching. In general, this type of flow dependent branching enforces dilated anastomoses to appear earlier (i.e. in a more proximal position) than VEGF-only-driven branching. In addition, this process is positively reinforced, yielding to more branching and to the accum- mulation of more, bigger proximal vessels. All these factors contribute towards posing important barriers to the delivery of both nutrient and drug to the tissue. In fact, it is shown that increasing the sensitivity of the branching process to the levels of WSS leads to an increase of the rate of branching in proximal regions, leading to the creation of a capillary shunt, which would seriously hinder delivery of drug and nutrient to the tumour. Making branching more robust to WSS yields, on the contrary, a more dendritic pattern of vessels which would lead to a more fluid pattern of drug and nutrient delivery. According to results presented in [MAb], vascular adaptation would also have an important role in drug delivery and nutrient supply. Transport in adapted vasculatures appear to be mostly dominated by a small23 number of vessels which sustains most of the flow. It so happens that most of the so-called brush border, i.e. a region in the proximity of the tumour characterised by exhibiting a vascular density much bigger than that observed in the proximal regions, are narrow and poorly perfused. This implies that most of the injected drug finds its way back into the circulation without actually reaching the tumour. McDougall et al. [MAb] also study the influence of some biochemical parameters on drug and nutrient delivery, specially the effect of varying the haptotactic sensitivity of EC. Simulation results indicate that a reduction of the haptotactic sensitivity leads to a vascular morphology with less branching in the proximal regions and a more direct, less lateral pattern of EC migration, which leads to a more directed pattern in both vascular morphology and flow. Such effect yields, in turn, to an increased amount of tracer drug being delivered to the tumour. As pointed out by McDougall et al., this is a nice example of how two effects which, at first, do not seem to have much mutual relation happen to be interconnected, and how mathematical modelling can help to reveal such unevident couplings.

6.2. A multiscale approach to adaptive angiogenesis. Whereas the model formulated by McDougall et al. [MAb] relyis on a predetermined gradient of TAF for network formation without considering any coupling between dynamics within the tissue and the evolution of the network, the model considered in this Section, due to Owen et al. [OA], constitutes the ﬁrst attempt to a genuinely multiscale model of angiogenesis, where the formation of the vascular network and tumour

23That is, small in relation to the total number of vessels within the generated capillary bed.

70 TOMAS´ ALARCON´ growth are coupled via oxygen transport to the tissue and secretion of TAF on the part of tumour. The backbone of this model consists of the multiscale model formulated in [ABc] and the model of cell migration considered in [BOa]. The multiscale model has been briefly summarised in Sections 4.2 and 5.2 and, therefore, will not be dicussed any further. The model presented in [OA] is not under a network of fix topology, but under an arbitrary network which evolves in time according to two main factors, namely, angiogenesis due to signalling cues released by regions within the tumour under hypoxic stress, and remodelling (pruning) of the network due to blood flow-related factors, mostly WSS. In addition to survival signals from mural cells, ECs receive vital signals from the blood flow. It is believed that WSS plays a main role as a transductor of these signals [RY]. Thus, the model by Owen et al. assumes that if the WSS within a vessel is below some threshold value, the vessel is removed from the network. This removal is not instantaneous: WSS must be below its threshold for a given time span before the vessel is actually removed. Such an element, which is not taken into account in the model by McDougall et al. [MAb], is reminiscent of the WSS-dependent removal probability mechanism proposed by Bartha & Rieger [BRa]. In Owen et al. [OA], EC migration is modelled as a biased random walk (see the models discussed in Section 3), although with some differences with respect previous treatments [AC, SW, PSa]. EC migration is controlled by several factors. Owen et al. assume, as it is standard, that EC migration is biased towards sites where the concentration of TAF is higher. Two other factors are taken here into account that have not been considered before in previous modelling efforts. The first one of them is that a carrying capacity is introduced whereby if a site is fully occupied the cell cannot move into that site24. A second factor is an inertia parameter which measures the propensity of cells to stay put. Although the model by Owen et al. [OA] does not account explicitely by adhesivity effects, this parameter could be argued to play a similar role. Branching is assumed to be controlled by the concentration of TAF only, with the branching probability being a monotonically increasing function of TAF levels [OA]. When a sprout branches off, it is uncirculated until it fuses with either another uncirculated sprout or a circulated, mature vessel. Anastomosis has to happen within a given threshold time span for the sprout to survive and become a mature vessel, otherwise the sprout dies off. Once a new arcade has been formed, flow is recalculated for the new network and the new capillary bed is adapted using the adaptation mechanisms discussed previously25. 6.2.1. Results. Preliminary simulations are carried out with no angiogenesis and a pressure drop applied between one single inlet and one single outlet vessels to assess the effect of WSS-dependent pruning of the vascular network. These simulations show that this mechanism actually yields to the generation of a backbone of dilated vessels which sustain all of the blood flow [OA]. It is shown that pruning

24Note that in the model by Owen et al. [OA] ECs migrate into space that is already occupied by cells and therefore such excluded volume eﬀects mus be taken into account. 25Owen et al. [OA] can proceed in this way without running into the time stepping issues discussed in Section 6.1 because the typical size of the networks considered in [OA] is considerably smaller than tose considered in [MAb].

MODELLING TUMOUR-INDUCED ANGIOGENESIS 71 due to low WSS levels is increased as the pressure drop between inlet and outlet of the network is reduced. Simulations performed under no WSS-dependent remodelling show that the final vascular density is influenced by its initial condition, a similar result to that obtained by Bartha & Rieger [BRa], as well as in the time evolution in the approach towards a steady state: better vascularised tissues reach the steady state earlier than the ones with poorer initial vasculatures. It also seems that the steady state vascular density tends to be larger for tissued with poor initial vascularisation [OA]. Owen et al. [OA] also analyse the relative importance of the different factors involved in the model. In particular, it is shown that under low chemotactic sensitivity to TAF, new sprouts are unable to form anastomosis thus yielding to irregular development of the angiogenic network, with extensive regions of the tissue being poorly vascularised. Larger chemotactic sensitivity allow sprouts to form anastomosis and, therefore, become circulated, mature vessels. These vessels criti- cally depend for survival of stealing enough flow from the vessels originally in place. This fact, in turn, forces the vascular density to fluctuate around an average which depends on the pressure drop, since this controls the amount of WSS-dependent pruning: whenever that, due to angiogenesis, too many vessels have been generated they become poorly perfused and thereby removed. This, in turn, increases the (local) concentration of VEGF, which triggers angiogenesis.

7. Conclusions and discussion The aim of this review is presenting an overview of the modelling approaches to tumour-induced angiogenesis restricting ourselves to individual-based and multiscale models. This review has been organised so that the models discussed at each stage account for the different processes involved in angiogenesis: formation of a backbone of immature vessels, maturation and establishment of blood flow, structural adaptation, and remodelling. All these processes, in practice, occur simultaneously. However, due to the complexity of the process, modelling approaches have proceeded in order of increasing complexity. The state-of-the-art models of angiogenesis [MAb, OA] are starting to approach the modelling of the dynamically coupled process described above. The way in which the review is organised intends to reproduce the development of the different modelling approaches, from the simpler, although still challenging, situations described in Section 3 to the more complex situations dealt with by the hybrid [MAb] and multiscale [OA] models discussed in Section 6. In this process, the difficulties inherent to each of the levels of complexities have been made explicit and how the approximations made to render the problem tractable at one stage have been tackled in the next step using new approaches and techniques. In spite of the sophistication of models such as those described in Section 6, there is still a long way to run in terms of turning these models into efficient tools for helping experimentalists to formulate new hypothesis to be tested later on in the laboratory and guiding clinicians in designing effective therapeutical protocols. This issue is a critical one as far as angiogenesis is concerned. Now that it has become evident that anti-angiogenic therapy performs poorly in humans, experimentalists and clinicians are turning to strategies consisting of combining traditional chemo- or radio-therapy with anti-angiogenic agents. For reasons explained above, coming up with efficient schemes will involve understanding such complex issues as the relation

72 TOMAS´ ALARCON´ between vascular network structure and drug and nutrient delivery. Mathematical models, specially those of the type dealt with in this review, will be more than likely to play a role in such efforts. However, as the models become more complex to account for more biologically realistic scenarios, their mathematical treatment and numerical implementation become daunting tasks. An example of such difficulties is the time stepping issues discussed in Section 6.1. New numerical and analytical techniques in combination with state-of-the-art computational techniques will have to be used. A particularly promising avenue seems to be along the lines of the hybrid methods proposed by Quarteroni and coworkers to deal with multiscale problems in the circulatory system [FN, QV]. These techniques concentrate the computational effort in spatially- resolved models of the region of interest with lumped models of the rest of the circulatory system. This approach could be generalised to deal with the complex problems we are dealing with. Simpler models could be used in combination with more complex models focusing on a particular region. For example, in tumour growth there is strong evidence that most of the proliferation within the tumour occurs at the border [BA]. This implies that if we are interested in assessing the effects of a cytotoxic drug, we could focus the computational effort in an accurate, multiscale model of the border and model the rest of the tissue in a less detailed way, provided proper boundary conditions are established.

References [ABa] T. Alarcón, H.M. Byrne, P.K. Maini, A cellular automaton model for tumour growth in inhomogeneous environment, J. theor. Biol. 225 (2003), 257-274. [ABb] T. Alarcón,H.M. Byrne, P.K. Maini, Towards whole-organ modelling of tumour growth, Progr. Biophys. Mol. Biol. 85 (2004), 451-472. [ABc] T. Alarcón, H.M. Byrne, P.K. Maini, A multiple scale model of tumour growth, Multiscale Model. Simul. 3 (2005), 440-475. [ABd] T. Alarcón,H.M. Byrne, P.K. Maini, A design principle for vascular beds: the role of complex blood rheology, Microvasc. Res. 69 (2005), 156-172. [AO] T. Alarcón,M.R. Owen, H.M. Byrne, P.K. Maini, Multiscale modelling of tumour growth and therapy: the influence of vessel normalisation on chemotherapy, Computational and Mathematical Methods in Medicine. 7 (2006), 85-119. [AK] M.S. Alber, M.A. Kiskowski, J.A. Glazier, Y. Jiang, On Cellular Automaton Approaches to Modeling Biological Cells.InMathematical Systems Theory in Biology, Communication, and Finance, J. Rosenthal, and D. S. Gilliam, eds. IMA 142, Springer-Verlag, New York (2002). [A] A.R.A. Anderson, A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion, Math. Med. Biol. 22, (2005), 163-186. [AC] A.R.A. Anderson, M.A.J. Chaplain, Continuous and discrete mathematical models of tumour-induced angiogenesis, Bull. Math. Biol. 60 (1998), 857-900. [ABe] D. Ambrosi, F. Bussolino, L. Preziosi, A review of vasculogenesis models, J. Theor. Med. 6 (2005), 1-19. [AG] B.P. Ayati, G.F. Webb, A.R.A. Anderson, Computational methods and results for structured multiscale models of tumour invasion, Multiscale Model. Simul. 5 (2006), 1-20. [BRa] K. Bartha, H. Rieger, Vascular network remodelling via vessel cooption, regression and growth in tumours, J. theor. Biol. 241 (2006), 903-918. [BRb] K. Bartha, H. Rieger, Flow correlated percolation during vascular remodelling in tumours, Phys. Rev. Lett. 96 (2006), 058104. [BB] G. Bergers, L.E. Benjamin, Tumorigenesis and the angiogenic switch, Nat. Rev. Cancer 3 (2003), 401-410. [BOa] R. Betteridge, M.R. Owen, H.M. Byrne, T. Alarcón, J. Murphy, P.K. Maini, The impact of cell crowding and active cell movement on vascular tumour growth, Networks and Heterogeneous Media. 1 (2006), 515-535.

MODELLING TUMOUR-INDUCED ANGIOGENESIS 73

[BA] H.M. Byrne, T. Alarcón, M.R. Owen, S.D. Webb, P.K. Maini, Modelling aspects of cancer dynamics: a review, Phil. Trans. Roy. Soc. A, 364 (2006), 1563-1678. [BOb] H.M. Byrne, M.R. Owen, T. Alarcón, J. Murphy, P.K. Maini, Modelling the response of vascular tumours to chemotherapy: A multiscale approach, Math. Mod. Meth. Appl. Sci. 16 (2006), 1219-1241. [BH] R.G. Bristow, R.P. Hill. Hypoxia, DNA repair and genetic instability. Nat. Rev. Cancer. 8, 180-192. [BA] A. Brú, S. Albertos, J.L. Subiza, J.L. Garc´ıa-Asenjo, I. Brú. The universal dynamics of tumour growth. Biophys. J. 83 (2003), 2948-2961. [C] P. Carmeliet, Angiogenesis in health and disease, Nat. Med. 9 (2003), 653-660. [CG] A.F. Chambers, A.C. Groom, I.C. MacDonald, Dissemination and growth of cancer cells in metastatic sites, Nat. Rev. Cancer, 2 (2002), 563-572. [CLa] M.A.J. Chaplain, G. Lolas. Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system. Math. Models Meth. Appl. Sci. 15 (2005), 1685-1734. [CLb] M.A.J. Chaplain, G. Lolas. Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity. Networks and Heterogeneous Media. 1 (2006), 399-439. [CM] K. Christensen, N.R. Moloney. Complexity and criticality, Imperial College Press, London, UK. (2005). [DD] S. Dormann, A. Deutsch, Modeling of self-organized avascular tumor growth with a hybrid cellular automaton, In Silico Biol. 2 (2002), 0035. [F] J. Folkman, Tumour angiogenesis: therapeutic implications, New Eng. J. Med. 285 (1971), 1182-1186. [FK] J. Folkman, M. Klagsburn, Angiogenic factors, Science, 235 (1987), 442-447. [FN] L.Formaggia, F.Nobile, A. Qarteroni, A.Veneziani. Multiscale modelling of the circulatory system: a preliminary analysis. Comput. Visual Sci. 2 (1999), 75-83. [Fu] Y.C. Fung. Biomechanics. Springer, New York, USA, (1993). [GL] V.V. Gafyichuk, I.A. Lubashevsky. On the Principles of the Vascular Network Branching. J. theor. Biol. 212 (2001), 1-9. [GB] Y. Gazit, D.A. Berk, M. Leunig, L.T. Baxter, R.K. Jain. Scale invariant behaviour and vascular network formation in normal tumour tissue. Phys. Rev. Lett. 75 (1995), 2428-2431. [GM] R.A. Gatenby, P.K. Maini, Mathematical oncology: cancer summed up, Nature, 421 (2003), 321. [Ja] R.K. Jain, Molecular regulation of vessel maturation, Nat. Med. 9 (2003), 685-693. [Jb] R.K. Jain, Normalization of Tumor Vasculature: An Emerging Concept in Antiangiogenic Therapy, Science. 307 (2005), 58-62. [L] M. LaBarbera. Principles of design of fluid transport systems in Zoology. Science. 249 (2000), 992-1000. [LB] D.P. Landau, K. Binder, A guide to Monte Carlo simulations in statistical physics,Cam- bridge University Press, Cambridge, UK, (2000) [MW] N.V. Mantzaris, S. Webb, H.G. Othmer, Mathematical modelling of tumour-induced angiogenesis, J. Math. Biol. 49 (2004), 111-187. [MAa] S.R. McDougall, A.R.A. Anderson, M.A.J. Chaplain, J.A. Sherratt, Mathematical modelling of through vascular networks: Implications for tumour-induced angiogenesis, Bull. Math. Biol. 64 (2002), 673-702. [MAb] S.R. McDougall, A.R.A. Anderson, M.A.J. Chaplain, Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: Clinical implications and therapeutic strategies,J. theor. Biol. 241 (2006), 564-589. [Me] H. Meinhardt, Morphogenesis of lines and nets. Differentiation 6 (1976), 117-123. [MG] R.M.H. Merks, J.A. Glazier, Dynamic mechanisms of blood vessel growth, Nonlinearity 19 (2006), C1-C10. [MB] R.M.H. Merks, S.V. Brodsky, M.S. Goligorsky, S.A. Newman, J.A. Glazier, Cell elongation is keyto in silicoreplication of in vitro vasculogenesis and subsequent remodelling, Dev. Biol. 289 (2006), 44-54. [MD] J. Moreira, A. Deutsch, Cellular automaton models of tumour development: a critical review, Adv. Complex Sys. 5 (2002), 1-21. [Mu] G.D. Murray. The physiological of minimum work I: the vascular system and the cost of blood flow. Proc. Natl. Acad. Sci. USA. 12 (1926), 207-214.

74 TOMAS´ ALARCON´

[OD] H.G. Othmer, S.R. Dunbar, W. Alt. Models of dispersal in biological systems. J. Math. Biol. 26 (1988), 263-298. [OS] H.G. Othmer, A. Stevens. Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math. 57 (1997), 1044-1087. [OA] M.R. Owen, T. Alarcón,H.M. Byrne, P.K. Maini, Angiogenesis and vascular remodelling in normal and cancerous tissues, J. Math. Biol. Submitted (2007). [PG] A.A. Patel, E.T. Gawlinski, S.K. Lemieux, R.A. Gatenby, A Cellular Automaton Model of Early Tumor Growth and Invasion: The Effects of Native Tissue Vascularity and Increased Anaerobic Tumor Metabolism, J. theor. Biol. 213 (2001), 315-331. [PSa] M.J. Plank, B.D. Sleeman, Lattice and non-lattice models of tumour angiogenesis, Bull. Math. Biol. 66 (2004), 1785-1819. [PBa] G.W. Prager, J.M. Breuss, S. Steurer, J. Mihaly, B.R. Binder. Vascular endothelial growth factor (VEGF) induces rapid prourokinase (pro-uPA) activation on the surface of endothelial cells. Blood. 103 (2004), 955-962. [PBb] G.W. Prager, J.M. Breuss, S. Steurer, D. Olcaydu, J. Mihaly, P.M. Brunner, H. Stockinger, B.R. Binder. Vascular endothelial growth factor receptor-2-induced initial endothelial cell migration depends on the presence of the urokinase receptor. Circ. Res. 94 (2004), 1562-1570. [P] L. Preziosi (Ed.), Cancer modelling and simulation, Mathematical & Computational Biology Series, vol. 3, Chapman and Hall/CRC (2003). [PSb] A.R. Pries, T.W. Secomb, T. Gessner, M.B. Sperandio, J.F. Gross, P. Gaehtgens. Resistance to blood flowin microvessels in vivo. Circ. Res. 75 (1994), 904-915. [PSc] A.R. Pries, T.W. Secomb, P. Gaehtgens. Design principles of vascular beds. Circ. Res. 757 (1995), 1017-1023. [PSd] A.R. Pries, T.W. Secomb, P. Gaehtgens. Structural adaptation and stability of microvascular networks: theory and simulations. Am. J. Physiol. 275 (1998), H349-H360. [PSe] A.R. Pries, B. Reglin, T.W. Secomb. Structural adaptation of microvascular networks: functional response to adaptive responses. Am. J. Physiol. 281 (2001), H1015-H1025. [PR] C.W. Pugh, P.J. Ratcliffe, Regulation of angiogenesis by hypoxia: role of the HIF system. Nat. Med. 9 (2003), 677-684. [QV] A. Qarteroni, A.Veneziani. Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations. Miltiscale Model. Simul. 1 (2003), 173-195. [RY] N. Resnick, H. Yahav, A. Shay-Salit, M. Shushy, S. Schubert, L.C.M. Zilbermann, E. Wofowitz. Fluid shear stress an the vascular endothelium: for better and foe worse. Progr. Biophys. Mol. Biol. 81 (2003), 177-199. [R] W. Risau, Mechanisms of angiogenesis, Nature, 386 (1997), 671-674. [RC] T. Roose, S.J. Chapman, P.K. Maini, Mathematical models of avascular growth, SIAM Rev. 49 (2007), 179-208. [SW] B.D. Sleeman, I.P. Wallis, Tumour induced angiogenesis as a reinforced random walk: Mod- elling capillary network formation without endothelial cell proliferation. Math. Comp. Modell. 36 (2002), 339-358. [SG] K. Smallbone, R.A. Gatenby, R.J. Gillies, P.K. Maini, D.J. Gavaghan, Metabolic changes during carcinogenesis: Potential impact on invasiveness, J. theor. Biol. 244 (2007), 703-713. [SM] A. Stéphanou, S.R. McDougall, A.R.A. Anderson, M.A.J. Chaplain, Mathematical modelling of the influence of blood rheological properties upon adaptive tumour-induced angiogenesis, Math. Comp. Model. 44 (2006), 96-123. [SL] C.L. Stokes, D.A. Lauffenburger. Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J. theor. Biol. 152 (1991), 377-403. [SP] A. Szabo, E.D. Perryn, A. Czirok. Network formation of tissue cells via Preferential attrac- tion to elongated tructures. Phys. Rev. Lett. 98 (2007), 038102. [T] D’A. Thomson. On growth and form. Cambridge University Press, Cambridge, UK, (1917). [W] R.A. Weinberg, The biology of cancer, Garland Sciences, (2006). [WB] G.B. West, J.H. Brown, B.J. Enquist. A General Model for the Origin of Allometric Scaling Laws in Biology. Science. 276 (1997), 122 - 126. [YD] G.D. Yancopoulos, S. David, N.W. Gale, J.S. Rudge, S.J. Wiegand, J. Holash. Vascular- specific growth factors and blood vessel formation. Nature. 407 (2000), 242-248. [Z] M. Zamir. Shear forces and blood vessel radii in the cardiovascular system. J. Gen. Physiol. 69 (1977), 449-461.

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Institute for Mathematical Sciences, Imperial College, 53 Princes Gate, London SW7 2PG, United Kingdom E-mail address: [email protected]

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Contemporary Mathematics Volume 492, 2009

Tumour radiotherapy and its mathematical modelling

Antonio Cappuccio, Miguel A. Herrero, and Luis Nu˜nez

Abstract. Radiotherapy consists in the delivery of ionizing radiation with curative goals. It represents a major modality for treating solid malignancies in any anatomical site. It requires extremely precise dosimetry and delivery to control the lesion while sparing the surrounding normal tissue. This is currently achieved by means of a combination of techniques. Among these, mathematical modelling may contribute to design improved treatment plans. In the present work, we provide an overview of the basic features of radiotherapy. Thereafter, we describe concrete examples of how mathematical modelling has so far been employed to simulate radiotherapy and to theoretically explore alternative strategies. Open problems and possible research directions are also discussed.

1. Introduction The first reported cure of cancer by radiation treatment appeared in the literature in 1899, just a few years after the discovery of X-rays [1]. During the following decades, radiation therapy was administered with little knowledge of its effects. The equipment used for delivery was primitive and there was no method of calculating the dose produced. These initial attempts involved a massive exposure to radiation, with poor curative effects and high damage to normal tissue. From the 1960s on, radiotherapy has been steadily increasing its field of application. Nowadays, it represents a major treatment modality for various benign and malignant tumours. For instance, it is estimated that about 50% of cancer patients in the United States receive radiotherapy in any of its forms [2]. Such extensive use is closely related to a parallel progress in various scientific disciplines, that has led to dramatic improvements in what concerns success rate, prevention of recurrence and efficactoxicity ratios. Radiotherapy is a fully interdisciplinary technique which provides major challenges in medicine, biology, physics, and engineering. In this framework, mathematics has gained an established role. A rich repertoire of mathematical methods,

Key words and phrases. Radiotherapy; Tumour processes; Radiobiology; Mathematical models. This workhas been partially supported by the European Contract MRTN-CT-2004-503361, as well as by Universidad Complutense Acci´on Especial AE 10/07-15449, Junta de Andalucia Project E-1268 and MICINN Research Grant MTM2008-01867.

⃝c ⃝XXXXc 2009 American Mathematical Society 77

78 CAPPUCCIO ET AL. including filtering, harmonic analysis and inverse problems, are essential for target definition during radiotherapy planning [3]. Intensity Modulated Radiation Therapy (IT), a widespread mode of high-precision radiotherapy that utilizes computer-controlled X-ray accelerators, makes extensive use of optimization algorithms to calculate the radiation distribution that best adapts to the tumour shape, with minimal damage to the surrounding tissue [2]. Mathematical methods have also been employed to develop a quantitative radiobiology, that is, a detailed description of the processes following the interaction between radiation and the living matter. Since the introduction of the Linear- Quadratic (LQ) model in the sixties, much research has been devoted to predict the biological response to radiation in terms of Tumour Control Probability (TCP) and Normal Tissue Complication Probability (NTCP). As we shall see, some of the existing TCP and NTCP models have been statistically validated and assist radiologists and physicists during treatment planning. It may be foreseen that mathematical modelling of the biological effects of radiation will play an increasing role in the radiotherapy to come. The efforts made so far have shown that mathematical methods and computer simulations make it possible to investigate how radiotherapy affects the fundamental characteristics of 𝑖𝑛 𝑣𝑖𝑣𝑜 tumours, such as DNA damage and repair dynamics, reoxygenation, repopulation and vascularization. This may favor the transition from the standard protocols currently in use, designed for group characteristics, to fully-personalized treatments. In the present work, we aim at providing a basic overview of tumour radiotherapy and its basic terminology, in an attempt to bridge the gap between modelling and clinical practice. Further, we illustrate concrete examples of what modelling has achieved so far. To this end, we organize the work as follows. In Section 2, we recall the main concepts of radiobiology. In Section 3, we describe the basic devices of radiotherapy, namely medical imaging and radiation emitters. In Section 4, we report on some available treatment modalities and illustrate how they are selected at the stage of planning. In Section 5, we summarize which malignancies are treated by radiotherapy and point out some frequent causes of failure. We then review some of the existing radiobiological models. In Section 6, we focus on the static models, that is, models that neglect the changes in time induced by radiotherapy at the gene, cell and tissue level. This class encompasses the linear quadratic model, as well as the TCP and NTCP models. Next, in Section 7 we report four examples of how mathematical modelling, ranging from ordinary differential equations to multiscale descriptions, has been employed to simulate and optimize radiotherapy. Finally, Section 8 is devoted to some concluding remarks and open problems in radiotherapy modelling.

2. Radiobiology and the five R’s of radiotherapy Radiobiology studies the effects of radiation on the living matter. One of its central issues is understanding which factors affect cell radiosensitivity, that is, the overall cell response to a specific radiation dose generally measured in Grays (1 Gray=1 Gy =1 JoulKg). Among other effects, the interaction between radiation and cell produces (i) molecular excitation and (ii) ejection of fast electrons from the atoms of the cell, a process referred to as ionization. While molecular excitation is dissipated as heat, ionization creates highly unstable compounds called free radicals,

TUMOUR RADIOTHERAPY AND ITS MATHEMATICAL MODELLING 79 bearing an unpaired electron in the outer orbital shell. In a remarkably short life of 10−5 seconds, free radicals initiate a cascade of reversible and irreversible changes involving the cell nucleus. Nuclear damage may comprise various DNA lesions, including single and double DNA strand breaks [4]. A central aspect of radiobiology is that cells have the ability to repair DNA lesions. Repair may be error-free, but misrepair is also possible. In this case, lethal errors induce cell apoptosis in a time ranging from hours to days, while non- lethal errors are incorporated and passed on to daughter cells, possibly leading to genomic instability and subsequent carcinogenesis. While repair of DNA damage is common to all cells, the extent and efficacy at which it takes place is variable. 𝐼𝑛 𝑣𝑖𝑡𝑟𝑜 assessments of cell survival after radiation exposure corroborate that most tumour lines show reduced capacity to repair DNA damage if compared to normal cells [1]. This finding provides an important rationale behind the idea of using radiation to treat tumours, i.e., tumour radiotherapy. Unfortunately, the 𝑖𝑛 𝑣𝑖𝑣𝑜 situation is by far more complicated. 𝐼𝑛 𝑣𝑖𝑣𝑜 tumour radiosensitivity is a result of multiple interdependent factors, that include DNA repair as a crucial but not exclusive aspect. On top of repair capabilities, each cell line has an intrinsic radiosensitivity, due to its structural properties. Ra- diosensitivity shows a dependence on the cell-cycle phase. Proliferating cells are generally more radiosensitive than resting cells, as measured by several parameters of radiation damage [4]. Cell response to radiation is also modulated by the distribution of oxygen. Hypoxia, which is defined as a low oxygen concentration, is known to increase cell radioresistance. This is due to reduced fixation of DNA- damaging free radicals, but also to modification of signal transduction pathways of various genes and proteins [5]. As oxygen is supplied by blood vessels, 𝑖𝑛 𝑣𝑖𝑣𝑜 radioresistance is accordingly affected by the underlying vasculature. It is worth mentioning that tumour vasculature often forms an intricate, highly unpredictable network [6]. Finally, both normal and tumour cells exposed to radiation respond with increased proliferation. Taken together, (intrinsic) radiosensitivity, repair, redistribution over the cell- cycle, reoxygenation and repopulation are named the five R’s of radiobiology [7]. Recently, the role of other factors has been stressed. For instance, an abnormal cell radiosensitivity at low doses has been observed. Such process, referred to as hypersensitivity, may have clinical implications [8]. Cell radiosensitivity has also been shown to depend on the rate of radiation delivery, the so called dose- rate effect [9, 10]. In spite of this, the R’s are still acknowledged as the most influential processes in tumour radiotherapy. These factors should be considered as highly interdependent, a fact particularly relevant when tumour radiotherapy is delivered in several sessions. After the first session, a fraction of tumour cells is eliminated and the oxygen consumption decreases. Moreover, radiation is known to directly reshape the blood vessel geometry, increasing blood perfusion. Due to lower consumption and higher blood perfusion, oxygen availability grows [11, 12]. As a consequence, quiescent cells may enter the proliferating phase, thereby modifying the cell-cycle distribution and the overall radiosensitivity to subsequent irradiations. Mathematical investigation of these interactions is challenging and implies potential benefits in terms of improved therapeutic strategies. Before reviewing the attempts made in this direction, we now introduce the main technological and clinical aspects of radiotherapy.

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3. The tools of radiotherapy Tumour radiotherapy uses ionizing radiations to kill pathological cells. Given that the tumour may be located anywhere in the body and is always surrounded by normal tissue, the success of radiotherapy relies on both precise target deﬁnition and accurate radiation delivery. In the clinical practice, these goals are obtained, on the one hand, by using various medical imaging procedures and, on the other hand, by accurate radiation emitters. In what follows, we describe how these tools contribute to treatment preparation and execution.

3.1. Medical imaging. Medical imaging plays a pivotal role in radiotherapy for disease detection, target definition, treatment decision making, and follow-up. The goal for medical imaging in radiotherapy is to characterize a pathological area and then select an appropriate course of therapy. Imaging information used in radiotherapy can be classified as anatomical or biological. Anatomical images provides geometric information on the shape and location of the lesion via computer tomography (CT), magnetic resonance () and ultrasound (US) techniques. The vast majority of studies is currently performed on the basis of anatomical information. However, an emerging idea is that anatomical information should be complemented by data acquisition of the tumour biological activities, such as its metabolic and proliferative state. As seen in Section 2, these features affect tumour radioresistance and their elucidation may lead to enhanced therapeutic strategies [13]. Nowadays, biological information on the tumour target can be captured mainly by two imaging techniques: positron emission tomography and magnetic resonance spectroscopy. 3.2. Radiation emitters. Ionizing radiation for clinical operations is commonly produced by a linear accelerator (LINAC). A LINAC can produce X-rays or electron beams in a wide range of energies (Fig. 1). The beams penetrate from the exterior into the body tissue. The in-depth energy deposition in the tissue depends on the nature and energy of the beams. These characteristics, along with the selection of the beam incidence, enables the operator to focus the emitted radiation to the tumour target. The biological impact of radiation increases with the ionizing density, which is defined as the frequency of ionizing events produced during penetration. X-ray and electrons induce similar ionizing densities. Accordingly, their biological effects are similar. Heavy particles (protons or accelerated ions) produce higher ionizing density. The correspondingly higher biological effect may be suitable to treat tumours of high radioresistance. Heavy particle accelerators require complex and expensive installations (cyclotrons) and are owned only by a few developed countries. So far, techniques based in combinations of X-rays or electron beams remain the most common ones. However, systematic cost-benefit studies are in progress, to address the question as to what extent heavy particle accelerators would improve therapeutic success [14]. Clinical accelerators are designed to guarantee precise beam outputs in terms of dose-rate, energy, uniformity, symmetry and stability, in an attempt to ensure the reproducibility of treatments. A key accelerator feature is the collimation technology. Modern collimation technologies are based on multileaf systems. These last are made up of individual ’leaves’ of a high atomic-numbered material (usually tungsten), that can move independently in and out of the path of a particle beam in order to block it. The multileaf systems can accurately adjust the radiation profile

TUMOUR RADIOTHERAPY AND ITS MATHEMATICAL MODELLING 81

Figure 1. Schematic representation of a medical LINAC. The ﬁg- ure shows three main components of linear accelerators: the gantry, the beam collimator and the couch (with SIEMENS permission). to provide the best adaptation of the dose distribution to the tumour volume (Fig. 2). Such property is generally referred to as conformal shaping. Further tools of radiotherapy are the brachytherapy units. These are based on electromechanical systems that allow to insert a small (about 5 mm in length and 1mm in diameter) radioactive source (usually containing Iridium 192 and Yodium 125) in the interior of a tumour by means of a catheter.

4. Planning radiotherapy Radiation can be delivered according to diﬀerent doses and protocols. The treatment modalities are designed at the stage of planning on the basis of the speciﬁc lesion (size and location), and other criteria of medical and statistical nature. In this section, we outline some basic features of the planning stage.

4.1. Protocols. Radiation treatments can be delivered according to a variety of established protocols [15]. A protocol is speciﬁed, among other parameters, by the total dose, the number of daily and weekly sessions, and the dose administered at any of them. For instance, conventional practice typically prescribes a total dose of 50-60 Gy to be delivered by means of a LINAC once a day, ﬁve days a week, with 2 Gy per session. Such a conventional scheme can be changed for reasons of medical and also of logistic nature. More therapeutic sessions with lower doses per fraction can be selected (hyper-fractionation) or, conversely, less sessions with increased doses may be chosen (hypo-fractionation).

82 CAPPUCCIO ET AL.

Figure 2. A multileaf system. It consists of various leaves whose conﬁguration allows conformal shaping of a LINAC beam to match the borders of the target tumour (with VARIAN permission).

A limit case of hypo-fractionation is Radiosurgery. In this case, the dose is administered externally in one single session. Radiosurgery is used to treat brain lesions of small volume located in areas which are difficult to access via surgical procedures. It requires devices able to provide high dosimetric and geometric accuracy, less than 1 mm in space and a few percent in dose, respectively. A further limit case of hypo-fractionation is the single fraction targeted intraoperative radiotherapy (Targit) [16]. It is surgically administrated to patients under general anesthesia, when the lesion is still accessible because of a previous incision. This allows to insert in the tumour bed a specific applicator that delivers electron beams. The prescribed dose is 5 and 20 Gy at 1 cm and 0.2 cm respectively, from the tumour bed, and is released in about 25 minutes [17]. In addition to evident logistic advantages, it has been speculated that Targit may have clinical advantages in terms of local tumour control. Its efficacy is currently being tested in randomised trials. Finally, brachytherapy can be delivered according to a variety of protocols, depending on whether it is used alone or to boost a previous external irradiation. A general distinction can be made between high-dose rate and low dose-rate brachytherapy. In the first case, a given dose is delivered in a short burst, lasting only a few minutes. Patients may receive up to 12 separate treatments over one or more weeks. In the low-dose rate brachytherapy procedure, the patient is treated with radiation delivered at a continuous rate over several hours or days. A detailed description of radiation dosing in brachytherapy can be found in [18].

4.2. Dosimetry. The deﬁnitions and criteria for prescribing and reporting the doses in radiation therapy are studied and recommended by international orga- nizations [19, 20]. A critical issue is the deﬁnition of the target to irradiate. This

TUMOUR RADIOTHERAPY AND ITS MATHEMATICAL MODELLING 83 may be difficult to identify, in view of uncertainties on edge detection and radiation delivery. Several categories of volumes have been established, that account for the margins necessary to minimize the effect of the uncertainties in the treatment outcome. The Gross Target Volume (GTV) is defined as the gross demonstrable extent and location of the malignant growth. A margin is added around the GTV to take into account potential ”subclinical” invasion. GTV and this margin define the Clinical Target Volume (CTV). To ensure that all parts of CTV receive the prescribed dose, additional safety margins are considered. An Internal Margin (IM) is added for the variations in position anor shape and size of CTV. This defines the Internal Target Volume (ITV). A set-up margin (SM) is added to take into account all the variationuncertainties in patient-beam positioning. Finally, the union of CTV, IM, SM defines the Planning Target Volume (PTV) (Fig 3). Unfortunately, no standard criteria exist to define the contours of the various target volumes on the basis of image analysis. This leads to ambiguities in the definition of the region of interest (Fig 4).

,0 60 *7 &7 ,7

Figure 3. Target volumes in external beam radiotherapy. GTV and CTV are purely oncological concepts. PTV is related to technical or setup issues. These regions are built by adding margins in such a way that the CTV is obtained from the GTV and the PTV from the CTV [20]. See text for further details.

In order to analyze the coverage of the dose distribution, other dosimetric parameters have been introduced by the Radiotherapy Oncology Group (http://www.rtog.or). This is the case, for instance, of the Homogeneity In- dex (HI). The HI is commonly defined as the ratio between the maximum and the minimum dose in the PTV. The design of dose distributions with low HI (1.5-2.5) has been considered an optimal prescription. Yet, a properly optimized inhomogeneous dose may enhance treatment outcome, if biological information on the target radiosensitivity is available [21]. Other parameters used during treatment planning link a specific treatment strategy with a prediction of its radiobiological effects based on mathematical models and statistical validation. Among these, the most important are the tumour control probability (TCP) and normal tissue complication probability (NTCP). The first is defined as the probability of local tumour control, i.e., the probability that

84 CAPPUCCIO ET AL.

Figure 4. Example of disagreement when delineating the gross volume of a brain tumour (top-left side). The closed contours correspond to the definition of the gross tumour volume by different radiation oncologists. Such variability is due to the lack of standard or automatic procedures to establish the region of interest. all tumour cells are lethally damaged after therapy. NTCP is defined as the probability that a given treatment induce severe side effects. We shall describe TCP and NTCP models in more details later in Section 6. 4.3. The dose-volume histograms. A plot of a cumulative dose-volume frequency distribution, known as a dose-volume histogram (DVH), graphically sum- marizes the simulated radiation distribution which would result from a proposed plan. DVHs display the fraction of tumour and normal tissues that receives a given dose. DVHs are tools for comparing alternative treatments [22]. However, they do not provide positional information in the volume under consideration. For this reason, a new concept (the z-dependent DVH) was introduced as a supplement to the DVH to provide the spatial variation, the size and magnitude of the different irradiated regions within the target [23].

5. Lights and shadows of tumour radiotherapy We devote this section to review the clinical use of tumour radiotherapy and to emphasize some of its limitations. 5.1. Malignancies treated with radiation. The following list is not exhaustive and its aim is simply to indicate how important radiotherapy is in the modern antitumour arsenal. Radiotherapy slightly improves the poor prognosis of brain tumours, such as glioblastoma and astrocytoma. In head and neck tumours, radiotherapy can be used alone or with surgery, either pre-operatively or more often post-operatively, treating both the primitive tumour and the satellite node territories. Apart from a few rare exceptions, bronchial tumours cannot be cured by

TUMOUR RADIOTHERAPY AND ITS MATHEMATICAL MODELLING 85

Figure 5. A dose-volume histogram corresponding to one session of tumour radiotherapy. It provides a clear visualization of the percentage of various tissues (represented by diﬀerent curves) that receives a particular dose. radiotherapy alone. It is generally used for inoperable tumours, to obtain a long palliative period. Randomised trials demonstrate the importance of postoperative radiotherapy for breast and chest tumours. In pancreas tumours, results are poor even when associated with chemotherapy: it is very diﬃcult to deliver a correct dose to this deeply-situated organ which is surrounded by many critical structures. It has been observed that pre-surgical irradiation of the pelvis improves the local control of rectal tumours.

5.2. Recurrence and complications: why? Preventing tumour recurrence is a difficult goal. In principle, survival of a single tumour clone may result in a future outbreak. Further, even removal of all tumour clones may not suffice to prevent recurrence. This could also arise from morphologically normal cells around the primary tumour that have an increased predisposition to genetic changes. For instance, one common genetic mutation in the carcinogenesis cascade is the so- called loss of heterozygosity (LOH) in tumour suppressor genes (TSGs). LOH occurring in TSGs in tissues adjacent to the primary tumour was shown to increase the probability of recurrence 4-5-fold in breast cancer [24]. Such evidence on local recurrence may have therapeutic implications, some of which will be discussed in Section 7.2. The achievement of local tumour control, difficult in itself, is antagonized by the need of sparing patients from severe side effects. Radiotherapy is in itself painless and many low-dose palliative treatments (for example, radiotherapy to bone metastases) cause minimal complications. Conversely, treatments to higher doses are known to cause diverse side effects from weeks to years after therapy. The severity of these complications is related to the irradiated volume, the functional

86 CAPPUCCIO ET AL. importance of the organs involved and the dose administered. All these facts give rise to an extensive bibliography in radiotherapy journals. Many side effects can be anticipated. Damage to epithelial surfaces (skin, oral, pharyngeal and bowel mucosa, urothelium) is frequently observed. The rate of recovery depends on the turnover rate of the epithelial cells, but it is usually quick. The lining of the mouth, throat, esophagus, and bowel may be harmed by radiation. If the head and neck area is treated, temporary soreness and ulceration may occur in the mouth and throat. The esophagus can also become sore, inducing diarrhoea and nausea. As part of the general inflammation that occurs, swelling of soft tissues may cause problems during radiotherapy, particularly in brain tumours. The gonads (ovaries and testicles) are highly sensitive to radiation. Hair loss may be most pronounced in patients receiving brain radiotherapy. Dry mouth and dry eyes can become irritating long-term problems. Finally, radiation itself is a potential cause of cancer. However, secondary malignancies are seen in a minority of patients and it is generally acknowledged that this risk is outweighed by the benefits conferred by irradiating the primary cancer. Since the goal of radiation therapy is tumour control with minimal harm to patients, failure corresponds to recurrence, induction of severe side effects, or both. Some frequent causes of failure are: inadequate dosage, inaccurate target definition, target missing due to patients’ motion, lack of knowledge about tumour vasculature, and treatment interruptions. Although the relative importance of these factors remains difficult to assess, technological progress is reducing the corresponding negative impact. Further contributions could hopefully come from mathematical models which would be able to accurately represent the underlying tumoral processes.

6. Radiobiological static models As seen in Section 4, the parameters considered at the stage of planning generally contain little biological information on the target. As a matter of fact, radiologists have considered models as TCP and NTCP to be described below only as rough indications. However, speciﬁc computer softwares based on these radiobiological models have recently been developed [25, 26]. The role of biological descriptors in optimization of treatment planning is growing. In this section, we review some of the most classical static radiobiological models.

6.1. The Linear-Quadratic model. The Linear-Quadratic (LQ) model has represented the main tool of radiobiology during the last four decades. LQ is used for analyzing cell survival, for comparing radiotherapy strategies and for studying other radiobiological endpoints. Although its theoretical foundations are still con- troversial, the LQ model can be derived as the asymptotic limit of a dynamic model describing the kinetics of radiation damage production, repair and misrepairs [27], as we shall see in Section 7.1. The LQ model quantiﬁes the 𝑖𝑛 𝑣𝑖𝑡𝑟𝑜 survival fraction of irradiated cells under the assumption that (i) the probability that a single hit produces two double-strand breaks grows linearly with the dosage, and (ii) the probability that two hits induce a double strand break grows quadratically. If 𝐹 denotes the surviving fraction and 𝐷 the dosage, the LQ model states that

𝐹 = 𝑒−𝛼𝐷−𝛽𝐷2 .

TUMOUR RADIOTHERAPY AND ITS MATHEMATICAL MODELLING 87

TissuOrgan 𝛼/𝛽 Brain parenchyma 2.1-2.9 Gy Optic nerve 0.5-1 Gy Cranial nerves 3-5 Gy Brain stem 2 Slowly growing Tumours 2-5 Gy Rapidly growing Tumours 10 Gy Metastases 10 Gy Table 1. Standard 𝛼/𝛽 ratios for diﬀerent normal and tumour cell lines.

The positive constants 𝛼 and 𝛽 depend on the specific cell-line. Remarkably, a dependence of these parameters upon the specific irradiation device has been shown [28]. Graphically, the ratio 𝛼/𝛽 determines how the survival curves bend with increasing dose (Fig. 5). From a biological point of view, the 𝛼/𝛽 ratio is related to cell repair ability. A tissue with a low alphbeta ratio has a higher capacity for self-repair than one with a high alphbeta ratio. In vitro assessments show that for most normal tissues 𝛼𝑁 /𝛽𝑁 ≃ 1 − 3 𝐺𝑦, while most tumours have 𝛼𝑇 /𝛽𝑇 ≃ 10 𝐺𝑦 (see Table 1). This difference has important therapeutic significance. Indeed,

Figure 6. 𝐼𝑛 𝑣𝑖𝑡𝑟𝑜 cell survival curves of diﬀerent human tumour lines ﬁtted to the LQ model [29]. LNCaP and DU145 correspond to prostate cancer; T47D-B8, MCF-7, MCF-7-Bus, MCF-7-BB to breast cancers; RT112 to bladder and testicular cancers; SCC4451 to squamous head carcinoma; HeLa to epithelial cancer. Reprinted by permission from MacMillan Publishers Ltd (R. A. El-Awady, E. Dikomey and J. Dahm-Daphi, Br. J. Cancer, 89; 593-601), copyright (2003).

88 CAPPUCCIO ET AL. it implies that delivering a given dose in fractions have a sparing eﬀect in normal tissue.

2 9

1.8 8

1.6 7 1.4 6 1.2 5 1 g(T) h(D) 4 0.8 3 0.6 2 0.4

0.2 1

0 0 0 2 4 6 8 10 0 1 2 3 4 5 T (hours) D (Gy)

Figure 7. Sketch of the functions 𝑔(𝑇 )andℎ(𝐷).

The LQ model has been reﬁned in various ways to incorporate the relevant features of tumour radiotherapy, some of which are described below. : Repopulation. To account for the accelerated tumour repopulation observed in conventional therapy, it is often assumed that, after a given delay 𝑇𝑘 since the onset of therapy, tumour cells start to grow exponen- tially. Thus, the LQ model can be modiﬁed as in [30]

2 𝐹 = 𝑒−𝛼𝐷−𝛽𝐷 +𝛾(𝑇 −𝑇𝑘),

where 𝑇 is the duration of treatment, and 𝛾 and 𝑇𝑘 are tumour-specific constants. For instance, head and neck cancers show 𝛾 values of 0.94-0.99 Gday and 𝑇𝑘 in the range 17-31 days in [31]. : Dose-rate effect. Beyond a dependence on the dose, the biological effect of radiation is also affected by the rate at which the dose is delivered. Indeed, lethal damage depends on double-strand breaks, which in turn depend on the frequency of the ionizing events. The dose-rate effect can be introduced by incorporating into the LQ model a term modulating the 𝛽 coefficient [10] 𝐹 = 𝑒−𝛼𝐷−𝑔(𝑇 )𝛽𝐷2 . Here 𝑇 is the duration of radiation delivery and the function 𝑔(𝑇 )(Fig. 7) is chosen as ( ( )) 2 1 − 𝑒−𝜇𝑇 𝑔(𝑇 )= 1 − . 𝜇𝑇 𝜇𝑇 The constant 𝜇 expresses the sub-lethal damage repair (log 2/𝜇 =0.5 − 3 h). : Hypersensitivity. The LQ model has been shown to underestimate response in the dose range up to 1 Gy, at which cells may show an abnormally high

TUMOUR RADIOTHERAPY AND ITS MATHEMATICAL MODELLING 89

radiosensitivity called hypersensitivity. Hypersensitivity was included in the standard LQ model in the Induced Repair (IR) model [8] as follows 𝐹 = 𝑒−𝛼ℎ(𝐷)𝐷−𝛽𝐷2 ,

with ( ) −𝐷/𝑑𝑐 ℎ(𝐷)= 1+(𝛼𝑠/𝛼 − 1)𝑒 .

This model is based on the threshold dose 𝑑𝑐. Prior to this threshold dose (generally up to 𝑑𝑐 ∼ 0.5 Gy), the tissue is hypersensitive, but when the dose has exceeded the threshold repair mechanisms are induced in the tissue, and cell sensitivity is reduced (Fig. 7). Note that, if 𝐷>>𝑑𝑐,the IR model reduces to the standard LQ model. Closely related to the LQ model is the concept of Biologically Effective Dose, originally proposed in [32]. Roughly speaking, BED accounts for the fact that the same physical dose has different biological effects, in dependence of the particular fractionation modality [33]. The BED corresponding to 𝑛 fractions of dose 𝑑 for a given 𝛼/𝛽 ratio is 𝑑 (1) 𝐵𝐸𝐷 = 𝑛𝑑(1 + ). 𝛼/𝛽 By definition, two schemes delivering the same BED are called iso-effective. Eq.(1) can be used to calculate the alternative iso-effective fractionation modalities. Any scheme with dose per fraction 𝑑1 and 𝑛1 number of fractions is iso-effective to eq.(1) if 𝑑 𝑑 𝐵𝐸𝐷 = 𝑛𝑑(1 + )=𝑛 𝑑 (1 + 1 ). 𝛼/𝛽 1 1 𝛼/𝛽 The calculation of alternative iso-effective regimens is particularly relevant to the problem of compensating missed treatment days in radiotherapy [34].

6.2. TCPmodels. The TCP is deﬁned as the probability that no tumour cell survives after treatment. This quantity can be directly derived from the LQ model. For simplicity, let us disregard the eﬀect of tumour repopulation, which will be discussed at the end of this section. If 𝐹0 is the initial number of tumour cells and treatment consists of 𝑛 deliveries of dose 𝑑, the fraction 𝐹 of surviving tumour cells is ( ( )) 𝛽 (2) 𝐹 = 𝐹 𝑒𝑥𝑝 −𝛼𝐷 1+ 𝐷 , 0 𝛼 where 𝐷 = 𝑛𝑑 and 𝛼, 𝛽 are the tumour radiosensitivity parameters. Tumour killing is a binary event. According to the Poisson statistics, the probability of having 𝑦 surviving tumor cells, being 𝐹 the expected value, is given by 𝑒−𝐹 𝐹 𝑦 𝑃 (𝐹, 𝑦)= . 𝑦! The TCP then corresponds to (3) 𝑇𝐶𝑃 = 𝑃 (𝐹, 0) = 𝑒−𝐹 . Substituting the expression (2) in (3), one obtains

−𝛼𝐷−𝛽𝐷2 (4) 𝑇𝐶𝑃 = 𝑒−𝐹0𝑒 .

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The family of curves described in (4) fails to adequately fit to the experimental TCP data. This is due to the fact that such model neglects heterogeneity of the dose distribution and inter-patient diversity of the radiosensitivity parameters. To incorporate such features, it has been proposed that (1) due to heterogeneity in dose distribution, 𝑛 initial tumour sub-populations 𝐹0,𝑖 receive doses 𝐷𝑖 (2) the parameter 𝛼 is normally distributed among patients with parameters (¯𝛼,𝛼 𝜎 ) These assumptions lead to the more realistic expression ∫ ∞ 1 −𝛼𝐷𝑖(1+𝑑𝑖𝛼/𝛽) 2 2 𝑛 −𝜌𝑑𝑉𝑖𝑒 −(𝛼−𝛼¯) /2𝜎𝛼 (5) 𝑇𝐶𝑃 = √ Π𝑖=1𝑒 𝑒 𝑑𝛼, 2𝜋𝜎𝛼 0 𝑛 where Π𝑖=1 denotes the product of the 𝑛 factors at its right. Equation (5) is known as the Webb-Nahum model. Its parameters have been recently assessed by fitting the model to clinical data [35, 36]. TCP models can also be extended to include the effect of tumour repopulation. The simplest way to do so is to start from the version of the LQ-model which accounts for repopulation, and to repeat the steps in eq. (1), (2) and (3). A more sophisticated way to include repopulation in inhomogeneously irradiated tumours can be found in [37]. A further TCP model, which is a function of the temporal protocol of dose delivery, can be found in [38]. 6.3. NTCPmodels. The concept of Normal Tissue Complication Probabil- ity has been formulated in several ways, both empirical and theoretical. An example of empirical NTCP is the Lyman-Kutcher-Burman model [39]. It states that NTCP canbewrittenintheform ∫ 𝑡 1 2 (6) 𝑁𝑇𝐶𝑃 = √ 𝑒−𝑥 /2𝑑𝑥, 2𝜋 −∞ where 𝐷 − 𝑇𝐷50(𝑣) −𝑛 𝑡 = ,𝑇𝐷50(𝑣)=𝑇𝐷50(𝑙)𝑣 . 𝑚 ⋅ 𝑇𝐷50(𝑣) Here 𝑣 is the fraction of the organ uniformly irradiated; 𝑇𝐷50(𝑙) is the dose that gives a 50% probability of damage when uniformly administered to the whole organ; 𝑚 and 𝑛 are organ-specific parameters. The numerical values of 𝑇𝐷50(𝑙),𝑣,𝑚,𝑛 have been estimated by fitting the analytic expression in eq. (6) to clinical tissue tolerance data [40]. The relative seriality model is an alternative way of calculating NTCP. It is based on binomial statistics and takes into account the organ structure introducing the concept of seriality [41]. This concept focuses on the distinction between parallel and serial structure organs. For parallel organs, a damage to a subunit does not endanger the overall functionality. Conversely, correct functioning of serial organs can be compromised by harming one subunit. Typically, the structure of human organs is something in between a completely parallel and a completely serial one. The relative seriality model is [ ] 𝑘 𝑠 1/𝑠 𝑁𝑇𝐶𝑃 = 1 − Π𝑗=1(1 − 𝑁𝑇𝐶𝑃(𝐷𝑗 ) , where −𝑒𝛾(1−𝐷𝑗 /𝐷50) 𝑁𝑇𝐶𝑃(𝐷𝑗 )=2 .

TUMOUR RADIOTHERAPY AND ITS MATHEMATICAL MODELLING 91

As before, the term 𝐷50 is the dose that gives a 50% probability of damage if uniformly administered to the whole organ; 𝑠 is the seriality constant: 𝑠 =0 corresponds to a serial organ and 𝑠 = 1 to a parallel organ. The critical volume model is based on the assumptions that organs are composed of functional subunits (FSUs) and that organ function is compromised when a certain critical fraction (𝜇𝑐𝑟) of these FSUs is damaged. For a uniformly irradiated organ with N FSUs, and a reserve capacity of L FSUs (i.e., 𝜇𝑐𝑟 = 𝐿/𝑁), the NTCP can be mathematically expressed as

∑𝑁 𝑁! (7) 𝑁𝑇𝐶𝑃 = 𝑝𝑀 (𝐷)(1 − 𝑝𝑀 (𝐷))𝑁−𝑀 𝑀!(𝑁 − 𝑀)! 𝑆𝐹𝑈 𝑆𝐹𝑈 𝑀=𝐿 where 𝑝𝐹𝑆𝑈(𝐷) is the probability of damage to a FSU after receiving a dose 𝐷. Since the number of FSUs is always quite large, the cumulative binomial distribution can be approximated by a cumulative normal distribution ⎛ √ ⎞ 𝑁(𝑝𝑀 (𝐷) − 𝜇 𝑟) (8) 𝑁𝑇𝐶𝑃 =Φ⎝√ 𝑆𝐹𝑈 𝑐 ⎠ , 𝑀 𝑀 𝑝𝑆𝐹𝑈(𝐷)(1 − 𝑝𝑆𝐹𝑈(𝐷)) where the function Φ denotes the standard Gaussian function. If the organ is heterogeneously irradiated, the previous expression becomes ⎛ ⎞ √ ∑ 𝑀 ⎜ 𝑁( 𝜈𝑖𝑝 (𝐷𝑖) − 𝜇𝑐𝑟) ⎟ ⎜ 𝑆𝐹𝑈 ⎟ ⎜ 𝑖 ⎟ (9) 𝑁𝑇𝐶𝑃 =Φ⎜√∑ ⎟ ⎝ 𝑀 𝑀 ⎠ 𝑝𝑆𝐹𝑈(𝐷𝑖)(1 − 𝑝𝑆𝐹𝑈(𝐷𝑖)) 𝑖

In the previous formula, it is stated that the total damage to the organ can be ∑treated as the sum of damage produced in the functional subunits. The sum 𝑀 𝑝𝑆𝐹𝑈(𝐷𝑖) can be identiﬁed as the mean relative damaged volume,𝜇 ¯𝑑.The 𝑖 parameters 𝛼 and 𝑁0 describe the cellular radiosensitivity and the number of cells in the 𝐹𝑆𝑈, respectively. It is assumed that the 𝐹𝑆𝑈 is only irreparably damaged when all cells are killed. Considerable attention has been devoted to validating and comparing NTCP models [42, 43].

7. Radiobiological dynamic models As seen in the previous section, TCP and NTCP models can be actually used for treatment planning. However, therapeutic use of these models is limited by their static nature, as they do not provide a dynamic description of how radiotherapy aﬀects normal and tumour tissue at the subcellular, cellular and tissue scale. A deeper knowledge of these features may lead to predict the overall response to alternative therapeutic scenarios. To this end, several models have been introduced that make use of the established tools of applied mathematics. As illustrative examples, we review four models of increasing sophistication that address diﬀerent problems of tumour radiotherapy.

92 CAPPUCCIO ET AL.

7.1. The kinetics of radiation damage, repair and misrepair. The kinetics of damage production, repair and misrepair following irradiation was in- vestigated by means of a two compartment ODE model in [44]. Let 𝑁 denote the fraction of viable cells and 𝑈 the mean number of DNA double-strand breaks (DSB) per cell. If radiation is assumed to induce lethal damage either by direct action or by binary misrepair of DSBs, we can write 1 (10) 𝑁˙ = −[𝛼𝐷˙ + 𝑘𝑈2]𝑁, 2 (11) 𝑈˙ = 𝛿𝐷˙ − 𝜔𝑈 − 2𝑘𝑈2. Here 𝐷˙ denotes the dose rate, 𝛼 is the direct lethal action of radiation due to non-repairable lesions, 𝛿 is the production of (potentially) non-repairable DSBs, 𝜔 is a constant of DSB repair and 𝑘 is the rate of DSB misrepair. In addition to the correct repair process happening at rate 𝜔, DSBs may undergo misrepair due to the encounter of two DNA fragments belonging to different chromosomes. For each binary misrepair, occurring at rate 𝑘𝑈2, two DSBs are removed. On average, one-half of these turns out to be lethal because of the formation of a diocentric chromosome plus an acentric fragment. Consider an impulsive dose 𝐷 delivered at 𝑡 = 0. In this case, the equations (10)-(11) reduce to 1 (12) 𝑁˙ = − 𝑘𝑈2𝑁, 2 (13) 𝑈˙ = −𝜔𝑈 − 2𝑘𝑈2, with initial conditions (14) 𝑁(0+)=𝑁(0−)𝑒−𝛼𝐷, (15) 𝑈(0+)=𝑈(0−)+𝛿𝐷, 𝑈(0−)=0. It can be verified that the solution of the previous system satisfies ( ) 𝑁(𝑡) 𝜔 2𝑘𝛿 2𝑘𝛿𝐷 (𝜔 +2𝑘𝛿𝐷)(1 − 𝑒−𝜔𝑡) log = log[1 + (1 − 𝑒−𝜔𝑡)] − 𝑁(0+) 8𝑘 𝜔 𝜔 𝜔 +2𝑘𝛿𝐷(1 − 𝑒−𝜔𝑡) For 𝑡 →∞, the previous equation implies the equality ( ) 𝜔 2𝑘𝛿𝐷 2𝑘𝛿𝐷 (16) log 𝐹 = −𝛼𝐷 + log[1 + ] − . 8𝑘 𝜔 𝜔 2𝑘𝛿𝐷 Finally, for 𝜔 << 1, the relation in (16) reduces to the LQ model log 𝐹 = −𝛼𝐷 − 𝛽𝐷2, with 𝑘𝛿2 𝛽 = . 4𝜔 Thus, the linear quadratic model can be obtained as an asymptotic limit from the system (12)-(13). If a given dose 𝐷 is split in two half doses delivered with a time interval 𝑇 , the following relation can be derived [45]

2 −𝛼𝐷−𝛽(1−𝑒𝜔𝑇 ) 𝐷 (17) 𝐹 = 𝑒 2 . Equation (17) will be useful in Section 7.3 to compare single dose and split-dose delivery.

TUMOUR RADIOTHERAPY AND ITS MATHEMATICAL MODELLING 93

7.2. A tumour invasion model and Targit. Partial differential equations describing tumour growth and invasion can be applied to simulate various therapeutic interventions, including radiotherapy. As an example, we next illustrate an invasion model that has been employed to theoretically compare the outcome of postoperative conventional radiotherapy and Targit (see Section 4.1) in breast cancer [46]. The model has three variables: tumour cells 𝑛, extracellular matrix (ECM) 𝑓 and matrix-degrading enzymes 𝑚 [47]. In a few words, tumour cells proliferate while moving by diffusion and haptotaxis, a motion oriented to areas of lower ECM density. Moreover, the tumour produces enzymes that diffuse, degrade the surrounding matrix and decay. These assumptions lead to the following system of equations

𝑝𝑟𝑜𝑙𝑖𝑓𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 ℎ𝑎𝑝𝑡𝑜𝑡𝑎𝑥𝑖𝑠 ∂𝑛 (18) = 𝜆𝑛(1 − 𝑓 − 𝑛)+ 𝑑 ∇2𝑛 − 𝛾∇⋅(𝑛∇𝑓), ∂𝑡 𝑛 𝑑𝑒𝑔𝑟𝑎𝑑𝑎𝑡𝑖𝑜𝑛 ∂𝑓 (19) = −𝜂𝑚𝑓 , ∂𝑡 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑑𝑒𝑐𝑎𝑦 ∂𝑚 (20) = 𝑑 ∇2𝑚 + 𝛼𝑛(1 − 𝑚) − 𝛽𝑚 . ∂𝑡 𝑚

Tumour growth is assumed to be radially symmetric. Therefore, the system is considered to hold in a one-dimensional domain. On the basis of the model in (18)-(20), simulation of postoperative radiotherapy was performed in three phases: (i) a solid tumour invading the host breast tissue until a critical size is reached, (ii) surgery and radiation therapy, (iii) development (or not) of local recurrence. At phase (ii), surgery was simulated by removing all areas of high tumour cell density, defined by the concentration dominance of tumour cells over tissue cells. In these areas, each model component is set to zero after surgery. The effect of radiation therapy was incorporated by applying the LQ model to normal and tumour cell populations. When simulating fractional radiotherapy, the invasion model (18)-(20) is applied between two subsequent sessions, and it is assumed that the dose delivered is uniform throughout the domain. Targit, in contrast, is characterized by a rapid dose falloff over distance. In both conventional treatment and Targit, radiation delivery is considered instantaneous. At phase (iii), the outcome of treatments are compared. The simulation results display that both external radiotherapy and Targit following breast conserving surgery eliminate tumour cells that may have escaped into the surrounding healthy tissue. Under the hypothesis that local recurrence arises out of tumour cells left behind after surgery, the model predicts that Targit would have at least a similar curative effect as the conventional method. The model allows to compare conventional therapy and Targit also on the alternative assumption that local recurrence arises irrespective of whether or not stray tumour cells were found within a certain margin of the primary lump pre- treatment zone. In this scenario, it is plausible that genetic mutations in healthy cells close to the primary tumour increase their susceptibility to further mutations, giving rise to a new tumour (see Section 5.2). This concept was modelled by

94 CAPPUCCIO ET AL. hypothesizing that, after surgery, a proper margin around the primary tumour contains mutated cells that are potential sources of tumour recurrence. Simulations suggest that conventional radiotherapy would not only spare the healthy cells, but also the cells with crucial mutations adjacent to the breast cancer. Only massive genetic damage, as could be caused by localized high-dose irradiation with Targit, would eradicate the mutated cells. Therefore, Targit may be able to eliminate not only the residual tumour cells after surgery, but also the potentially malignant cells in the immediate neighborhood of the tumour. Taken together, the above mentioned results suggest that Targit may imply therapeutic as well as logistic advantages over conventional fractional therapy.

7.3. Tumour cords and the role of oxygenation. The tumour cord model is a suitable mathematical framework to simulate reoxygenation and split-dose response to radiotherapy [45]. Tumour cords are cylindrical structures of tumour cells forming around the blood vessels. The existence of such cell arrangements is demonstrated by several lines of evidence [48]. Though the complexity of tumour vascular networks is extremely high, the tumour cord model is based on an idealized geometry of regularly spaced and parallel vessels [49]. It is assumed that the tumour tissue is partitioned into circular cylinders of radius 𝐵 around central blood vessels. The vessels move with the tumour. If the distance between the vessels exceeds a threshold, necrotic regions appear. Each cord, that can be studied independently, contains the volume fractions occupied by viable proliferating cells(𝜈𝑃 ), quiescent † cells (𝜈𝑄), lethally damaged cells (𝜈 ) and dead cells (𝜈𝑁 ). The model assumptions are: (i) The cord has cylindrical symmetry and all the variables have only a radial component. (ii) The velocity field, denoted by 𝑢(𝑟, 𝑡) is common to both live and dead cells. (iii) The cell-cycle is regulated by the oxygen concentration 𝜎(𝑟, 𝑡). (iv) cells die instantly when 𝜎 is below a threshold 𝜎𝑁 . (v) only impulsive irradiation is considered and the radiosensitivity parameters 𝛼 and 𝛽 are functions of 𝜎.(vi) Lethally damaged cells die at rate 𝜇. (vii) Dead cells are degraded at rate 𝜇𝑁 and drained away by the flow of extracellular fluid. Finally, (viii) the total volume ∗ † fraction of cells 𝜈 = 𝜈𝑃 + 𝜈𝑄 + 𝜈 + 𝜈𝑁 is constant. The mass balance assumption gives the following conservation equations ∂𝜈 1 ∂ (21) 𝑃 + (𝑟𝑢𝜈 )=𝜒𝜈 + 𝛾(𝜎)𝜈 − 𝜆(𝜎)𝜈 − 𝑚 (𝑟, 𝑡)𝜈 , ∂𝑡 𝑟 ∂𝑟 𝑃 𝑃 𝑄 𝑃 𝑃 𝑃 ∂𝜈 1 ∂ (22) 𝑄 + (𝑟𝑢𝜈 )=−𝛾(𝜎)𝜈 + 𝜆(𝜎)𝜈 − 𝑚 (𝑟, 𝑡)𝜈 , ∂𝑡 𝑟 ∂𝑟 𝑄 𝑄 𝑃 𝑄 𝑄 ∂𝜈† 1 ∂ (23) + (𝑟𝑢𝜈†)=𝑚 (𝑟, 𝑡)𝜈 + 𝑚 (𝑟, 𝑡)𝜈 − 𝜇𝜈†, ∂𝑡 𝑟 ∂𝑟 𝑃 𝑃 𝑄 𝑄 ∂𝜈 1 ∂ (24) 𝑁 + (𝑟𝑢𝜈 )=𝜇𝜈† − 𝜇 𝜈 . ∂𝑡 𝑟 ∂𝑟 𝑁 𝑁 𝑁 It is worth mentioning that, from a historical point of view, the system in (21)- (24) is linked with various previous works [50, 51, 52, 53]. The functions 𝜆 and 𝛾 correspond to the direct and inverse transition rates between the 𝑃 and 𝑄 compartments and are assumed to be a decreasing and an increasing function of the oxygen concentration, respectively. Adding up the equations (21)-(24) one obtains a relation for the velocity field 1 ∂ 𝜈∗ (𝑟𝑢)=𝜒𝜈 − 𝜇 𝜈 ,𝑢(𝑟 ,𝑡)=0, 𝑟 ∂𝑟 𝑃 𝑁 𝑁 0

TUMOUR RADIOTHERAPY AND ITS MATHEMATICAL MODELLING 95 where 𝜈∗ is constant by assumption (viii). The dynamics of repair and misrepair, incorporated in the functions 𝑚𝑃 and 𝑚𝑄, is chosen as in (10)-(11). The oxygen is thought to diffuse in a quasi-static regimen, in line with experimental findings [48]. Its dynamics is thus governed by the equation 1 ∂ ∂𝜎 (𝑟 )=𝑓(𝜎)(𝜈 + 𝜈 + 𝜈†), 𝑟 ∂𝑟 ∂𝑟 𝑃 𝑄 where 𝑓(𝜎) is the ratio between the per cell consumption rate and the diffusion coefficient. At the inner boundary 𝑟 = 𝑟0, a constant source term is prescribed

𝜎(𝑟0,𝑡)=𝜎𝑏. To assign the outer boundary condition for 𝑟 = 𝐵(𝑡), the cases of absence and presence of necrosis should be distinguished. When necrosis is absent, the symmetry of vascularization implies a homogeneous boundary condition ∂𝜎 (25) =0, ∂𝑟 𝑟=𝐵(𝑡) and, from the assumption that vessels move solidly with the tissue 𝐵˙ = 𝑢(𝐵(𝑡),𝑡).

In the presence of necrosis, the cornecrosis interface 𝑟 = 𝜌𝑁 (𝑡) becomes a free boundary of the domain. This boundary can be determined by noting that the necrotic material cannot be converted to living cells and that assumption (iv) for- bids living cells to exist for 𝜎<𝜎𝑁 . Thus the following inequalities must be satisﬁed 𝑢(𝜌𝑁 (𝑡),𝑡) − 𝜌˙𝑁 (𝑡) ≥ 0,𝜎(𝜌𝑁 (𝑡),𝑡) ≥ 𝜎𝑁 , together with the no-ﬂux condition ∂𝜎 =0. ∂𝑟 𝑟=𝜌𝑁 (𝑡)

If the cells cross the interface 𝜌𝑁 (𝑡), the cord radius is deﬁned by

𝜎(𝜌𝑁 (𝑡),𝑡)=𝜎𝑁 , and the interface is a non-material free boundary. Otherwise, the cord boundary becomes a material free boundary whose motion is given by

𝜌˙𝑁 = 𝑢(𝜌𝑁 ,𝑡). To represent the eﬀects of therapy, radiation delivery is considered instantaneous. The surviving fractions after irradiation are computed through the LQ model. The radiosensitivity parameters are allowed to be both cell-cycle and oxygen-dependent 𝑃 𝑄 (26) 𝛼𝑃 (𝜎)=𝛼𝑀 𝜓𝛼(𝜎),𝛼𝑄(𝜎)=𝛼𝑀 𝜓𝛼(𝜎), 𝑃 𝑄 (27) 𝛽𝑃 (𝜎)=𝛽𝑀 𝜓𝛽(𝜎),𝛽𝑄(𝜎)=𝛽𝑀 𝜓𝛽(𝜎), where 𝜓𝛼 and 𝜓𝛽 are functions chosen according to previous assessments [54]. Simulations of the cord model for physiological parameter values were used to study how radiotherapy aﬀects tumour radiosensitivity. After the delivery of a single dose 𝐷, the model reproduced the decrease in cell number and the subsequent reoxygenation of the cord. According to the model, the shrinking of the cord size following irradiation additionally contributes to reoxygenation due to the boundary

96 CAPPUCCIO ET AL. condition in (25). To investigate how reoxygenation affects subsequent irradiations, the delivery of two doses of 𝐷/2 was simulated for different inter-dosing intervals 𝑇 . The response to a single dose 𝐷 was compared with the response to two 𝐷/2- doses by computing in each case the total number of proliferating and quiescent cells defined by

∫ 𝐵(𝑡) ∫ 𝐵(𝑡) 𝑃𝑖(𝑡)= 𝑟𝜈𝑃 (𝑟, 𝑡)𝑑𝑟, 𝑄𝑖(𝑡)= 𝑟𝜈𝑄(𝑟, 𝑡)𝑑𝑟, 𝑟0 𝑟0 where the subscripts refer to single-dose (i=1) and split-dose (i=2) response, respectively. The survival ratio 𝑆𝑅, defined by 𝑚𝑖𝑛[𝑃 (𝑡)+𝑄 (𝑡)] 𝑆𝑅 = 2 2 , 𝑚𝑖𝑛[𝑃1(𝑡)+𝑄1(𝑡)] was then evaluated of the SR in correspondence to different fractionation intervals 𝑇 . The tumour cord model predicted that reoxygenation generally makes the split-dose response more effective, though values of SR smaller than one were also achieved in the case of cords surrounded by necrosis. The cord model also stressed the role of the inter-vessel distance as a factor having an impact into the therapeutic outcome.

7.4. A multiscale model. Multiscale modelling is an established approach to study biological systems [55, 56]. It first proceeds by means of a separate modelling of processes taking place at different spatial scales. The models are then connected to generate a global description. Multiscale modelling has been applied, for instance, to study the gene-dependent cell cycle regulation of tumour radiosensitivity in colorectal carcinoma [57]. More precisely, the model derived in that work focuses on the following scales : Gene Level. Five key regulatory genes are assumed to be responsible for the evolution of colorectal cancer and the response to radiation therapy: APC, K-RAS, TGF, SMAD and p53 are considered. If not mutated, p53 is activated when DNA is damaged and leads to apoptosis; SMAD is activated during hypoxia and inhibits proliferation; APC is activated during overpopulation and inhibits proliferation; RAS promotes proliferation when there is no hypoxia. The genes are considered boolean variables 0 or 1, corresponding to an activated and inactivated state, respectively. The state of each gene is updated according to a logical function representing the gene-gene interactions. Outputs of the genetic code are then linked to the cell scale. : Cell level. Cell cycle is depicted by a discrete mathematical model including the phases S, G1, G2, M [58]. Cells move in a 2D-domain according to the following assumptions: if oxygen concentration is below a threshold and the gene SMAD is not mutated, cells are forced to a resting state through SMAD activation; if cell density increases above a threshold and APC is not mutated, cells are forced to quiescence through APC activation; if neither hypoxia nor overpopulation signals are sensed, cells proceed with the cell-cycle, generating new cells. : Tissue level. A continuous fluid dynamics model is used to represent the tissue behavior. The flow of tumour cells in the extracellular matrix is

TUMOUR RADIOTHERAPY AND ITS MATHEMATICAL MODELLING 97

assumed to follow Darcy’s law 𝑣 = −𝑘∇𝑝,

where 𝑝 is the pressure ﬁeld. The evolution of cell densities 𝑛𝜙(𝑥, 𝑦)at position (𝑥, 𝑦) in the cycle phase 𝜙 is derived by applying the mass balance equation ∂𝑛 𝜙 + ∇⋅(𝑣𝑛 )=𝑃 , ∂𝑡 𝜙 𝜙 where 𝑃𝜙 is the cell density proliferation of cells in phase 𝜙. The oxygen concentration 𝐶 is assumed to diﬀuse across the computation domain Ω and is consumed by the cells ∂𝐶 ∑ (28) −∇⋅(𝐷∇𝐶)=− 𝑎 𝑛 𝑜𝑛 Ω /Ω . ∂𝑡 𝜙 𝜙 𝑏𝑣 𝜙

Here Ω𝑏𝑣 represents the regions of the domain occupied by blood vessels. Equation (28) is complemented by the boundary conditions

𝐶 = 𝐶𝑚𝑎𝑥 𝑜𝑛 Ω𝑏𝑣,𝐶∂Ω =0. The first condition means that the oxygen concentration is constantly equal to 𝐶𝑚𝑎𝑥 on the blood vessels, while the last condition is the standard Dirichlet boundary condition. Preliminary simulations with no therapeutic intervention show that the multiscale model qualitatively reproduces the experimental gene-dependent growth regulation of a cell colony. Normal cells grow until activation of APC due to overpopulation. APC-mutated cells continue proliferation until regulation by hypoxia through SMAD activation. At this point APC and SMAD-mutated cells switch to permanent, uncontrolled growth. Therapy is then modelled by assuming that radiation is uniformly distributed in the tumour region and that the number of DNA strand breaks is proportional to the dose. The response to a conventional fractional protocol is then explored. Altogether, simulation results allow to appreciate the importance of considering regulating factors (hypoxia and overpopulation), as well as tumour geometry and tissue dynamics, in improving radiotherapeutic efficacy. The model predicts that the conventional fractional scheme may fail to be optimal, as heuristically designed regimens based on the cell-cycle regulation show higher efficiency.

8. Concluding remarks Radiotherapy represents a source of deep mathematical problems, whose solution may enhance treatment efficacy and result in great social benefits. It is acknowledged that the current efficactoxicity ratios may be improved by a progressive shift from the traditional plans based on mere physical dose-optimization to biologically-optimized plans [21]. This transition will be favored by further progress in biological imaging technologies, but also by the availability of quantitative radiobiological models. We have provided a short introduction to radiotherapy, and reported on significant examples of how mathematics can be applied to formulate radiobiological models. In particular, we have illustrated how some of the well recognized features of 𝑖𝑛 𝑣𝑖𝑣𝑜 tumours, namely the R’s of tumour radiobiology, can be integrated in models of increasing complexity.

98 CAPPUCCIO ET AL.

It is worth remarking that, in a modelling perspective, radiation delivery is generally assumed instantaneous. However, real treatments do have a duration of a few minutes. During this time, the target volume may be subjected to non- negligible motion, particularly due to patients’ respiration. Target motion may represent a frequent cause of target missing and damage to critical structures in a number of malignancies, including lung, breast and colorectal cancer. The problem of synchronizing radiation delivery and patient’s motion, further complicated by uncertainties in the target spatial location and dose delivery, has received so far little attention in the mathematical literature. Models may prove fundamental to explore synergistic combinations of radiotherapy with other therapeutic interventions, such as radiation sensitizers, anti- angiogenic factors, chemotherapy and immunotherapy. Pending on proper validation procedures, models can also be used for optimization of dose and fractionation schemes. Optimization of radiotherapy protocols is a particularly interesting subject. As more and more biological aspects are being incorporated into the models, the question arises of how to design improved therapeutic strategies via optimization methods. For relatively simple models, established optimization techniques can be applied [59, 60]. However, multiscale and hybrid models pose major problems of global optimization which require further theoretical and computational eﬀorts. Outstanding theoretical questions are related, for instance, to show the actual existence of the optimal protocols for a given model. Computational methods are mandatory to develop robust algorithms able to perform the search for such optimal solutions in reasonable times. Further modelling eﬀorts, in alliance with modern technological capabilities, have the potential to contribute towards the ultimate goal of radiotherapy: the design of personalized plans able to control the malignancies with minor complications.

References [1] F. Mould. A Century of X-rays and Radioactivity in Medicine. Institute of Physics Publishing, London, 1993. [2] D.M. Shepard, M.C. Ferris, G.H. Olivera, and T. Mackie. Optimizing the delivery of radiation therapy to cancer patients. SIAM rev, 41(4):721–744, 1999. [3] F. Natterer and F. W¨ubbeling. Mathematical methods in image reconstruction. SIAM, Philadelphia, 2001. [4] F.D. Malkinson. Some principles of radiobiology: A selective review. J Invest Dermatol, 76:32–36, 1981. [5] J.G. Eriksen and M. R. Horsman. Tumour hypoxia -a characteristic feature with a complex molecular background. Radiother Oncol, 81(2):119–121, 2006. [6] R.K. Jain. Normalization of tumor vasculature: An emerging concept in antiangiogenic therapy. Science, 307(5706):58–62, 2005. [7] C.S. Wong and R.P. Hill. Experimental radiotherapy. McGraw-Hill, New York, 1998. [8] B. Marples and M. C. Joiner. The response of chinese hamster v79 cells to low radiation doses: Evidence of enhanced sensitivity of the whole cell population. Radiat Res, 133(1):41–51, 1993.

REFERENCES 99

[9] H.P. Leenhouts and K.H. Chadwidr. The influence of dose rate on the dose- effect relationship. J Radiol Prot, 10(2):95–102, 1990. [10] R.G. Dale. Dose-rate effects in targeted radiotherapy. Phys Med Biol, 41:1871– 1884, 1996. [11] Crockart N., Jordan B.F., C. Baudelet, R. Ansiaux, P. Sonveaux, V. Grégoire, N. Beghein, J. DeWever, C. Bouzin, O. Feron, and B. Gallez. Early reoxygenation in tumors after irradiation: determining factors and consequences for radiotherapy regimens using daily multiple fractions. Int J Radiat Oncol Biol Phys, 63(3):901–910, 2005. [12] P. Sonveaux, C. Dessy, A. Brouet, B.F. Jordan, V. Grégoire, B. Gallez, J.L. Ballingard, and O. Feron. Modulation of the tumor vasculature functionality by ionizing radiation accounts for tumor radiosensitization and promotes gene delivery. FASEB J, 16(14):1979–1981, 2002. [13] M.M. Miften, S.K. Das, M. Su, and L.B. Marks. Incorporation of functional imaging data in the evaluation of dose distributions using the generalized concept of equivalent uniform dose. Phys Med Biol, 49(9):1711–1721, 2004. [14] M. Lodge, M. Pijls-Johannesma, L. Stirk, A.J. Munro, D. De Ruysscher, and T. Jefferson. A systematic literature review of the clinical and cost-effectiveness of hadron therapy in cancer. Radiother Oncol, 83(2):110–122, 2007. [15] E. Hall and A.J. Giaccia. Radiobiology for the Radiologist (sixth edition). Lippincott Wilkins & Williams, Philadelphia, 2006. [16] J.S Vaidya, M. Baum, J.S. Tobias, D.P. D’Souza, S.V. Naidu, S. Morgan, K.J. Harte, A.P. Sliski, and E. Thomson. Targeted intra-operative radiotherapy (targit): An innovative method of treatment for early breast cancer. Ann Oncol, 12(8):1075–1080, 2001. [17] J.S. Vaidya, M. Baum, J.S. Tobias, S. Morgan, and D. D’Souza. The novel technique of delivering targeted intraoperative radiotherapy (targit) for early breast cancer. EJSO, 28:447–454, 2002. [18] B. Thomadsen, M.J. Rivard, and W.M. Butler. Brachytherapy Physics (second edition). Medical Physics Publishing, Madison, 2005. [19] International Commission on Radiation and Measurements. Prescribing, recording, and reporting photon beam therapy. ICRU Report, 50, 1993. [20] International Commission on Radiation and Measurements. Prescribing, recording, and reporting photon beam therapy (supplement to icru report 50). ICRU Report, 62, 1999. [21] A. Brahme, J. Nilsson, and D. Belkic. Biologically optimized radiation therapy. Acta Oncol, 40(6):725–734, 2001. [22] R.E. Drzymala, R. Mohan, L. Brewster, J. Chu, M. Goitein, W. Harms, and M. Urie. Dose-volume histograms. Int J Radiat Oncol Biol Phys, 21(1):71–8, 1991. [23] C.W. Cheng and I.J. Das. Treatment plan evaluation using dose-volume histogram (dvh) and spatial dose-volume histogram (zdvh). Int J Radiat Oncol Biol Phys, 43(5):1143–50, 1992. [24] Z. Li, Moore, D.H., Z.H. Meng, B.M. Ljung, J.W. Gray, and S.H. Dairkee. Increased risk of local recurrence is associated with allelic loss in normal lobules of breast cancer patients. Cancer Res, 62:1000–1003, 2002. [25] B. Sanchez-Nieto and A.E. Nahum. Bioplan: software for the biological evaluation of radiotherapy treatment plans. Med Dosim, 25(2):71–8, 2000.

100 REFERENCES

[26] I. El Naqa, G. Suneja, P.E. Lindsay, A.J. Hope, J. Alaly, M. Vicic, J.D. Bradley, A. Apte, and J.O. Deasy. Dose response explorer: an integrated open- source tool for exploring and modelling radiotherapy dose-volume outcome relationships. Phys Med Biol, 51(22):5719–5735, 2006. [27] R.K. Sachs and D.J. Brenner. The mechanistic basis of the linear-quadratic formalism. Med Phys, 75:2071–2073, 1998. [28] C.K. Wang and X. Zhang. A nanodosimetry-based linear-quadratic model of cell survival for mixed-let radiations. Phys Med Biol, 51(23):6087–6098, 2006. [29] R.A. El-Awady, E. Dikomey, and J. Dahm-Daphi. Radiosensitivity of human tumour cells is correlated with the induction but not with the repair of dna double-strand breaks. Br J Cancer, 89:593–601, 2003. [30] R. Muller-Runkel and S. Vijayakumar. Equivalent total doses for different fractionation schemes, based on the linear quadratic model. Radiology, 179(2):573– 577, 1991. [31] R.G. Dale, J.H. Hendry, B. Jones, A.G. Robertson, C. Deehan, and J.A. Sin- clair. Practical methods for compensating for missed treatment days in radiotherapy, with particular reference to head and neck schedules. Clin Oncol, 14:382–393, 2002. [32] G.W. Barendsen. Dose fractionation, dose rate and iso-effect relationships for normal tissue responses. Radiat Oncol Biol Phys, 8(11):1981–1987, 1982. [33] B. Jones, R.G. Dale, C. Deehan, K.I. Hopkins, and D.A.L. Morgan. The role of biologically effective dose (bed) in clinical oncology. Clin Oncol, 13:71–78, 2001. [34]J.H.Hendry,S.M.Bentzen.G.Dale,J.F.Fowler,T.E.Wheldon,B.Jones, A.J. Munro, N.J. Slevin, and A.G. Robertson. A modelled comparison of the effects of using different ways to compensate for missed treatment days in radiotherapy. Clin Oncol, 8:297–307, 1996. [35] P.J. Keall and S. Webb. Optimum parameters in a model for tumour control probability, including interpatient heterogeneity: evaluation of the log-normal distribution. Phys Med Biol, 52:291–302, 2007. [36] P. Stavrev, M. Weldon, B. Warkentin, N. Stavreva, and B.G. Fallone. Radi- ation damage, repopulation and cell recovery analysis of 𝑖𝑛 𝑣𝑖𝑡𝑟𝑜 tumour cell megacolony culture data using a non-poissonian cell repopulation tcp model. Phys Med Biol, 50:3053–3061, 2005. [37] A. T. Wolfgang and J.F. Fowler. On the inclusion of proliferation in tumour control probability calculations for inhomogeneously irradiated tumours. Phys Med Biol, 48:261–268, 2003. [38] N Zaider and G.N. Minerbo. Tumor control probability: a formulation appli- cable to any temporal protocol of dose delivery. Phys Med Biol, 45:279–293, 2000. [39] J.T. Lyman. Complication probabilities as assessed from dose-volume histograms. Radiat Res Suppl, 104:13–19, 1985. [40] C. Burman, G.J. Kutcher, B. Emami, and M. Goiten. Fitting of normal tissue tolerance data to an analytic function. Int J Radiat Oncol Biol Phys, 21:123– 135, 1991. [41] P. Kallman, A. Agren, and A. Brahme. Tumour and normal tissue responses to fractionated non-uniform dose delivery. Int J Radiat Biol, 62(2):249–262, 1992.

REFERENCES 101

[42] I. Tsougos and P. Mavroidis. Evaluation of dose-response models and parameters predicting radiation induced pneumonitis using clinical data from breast cancer radiotherapy. Phys Med Biol, 50:3535–3554, 2005. [43] M.A. Alexander, W.A. Brooks, and S.W. Blake. Normal tissue complication probability modelling of tissue fibrosis following breast radiotherapy. Phys Med Biol, 52:1831–1843, 2007. [44] R.K. Sachs, P. Hahnfeld, and D.J. Brenner. The link between low let dose-response relations and the underlying kinetics of damage produc- tiorepaimisrepairs. Int J Radiat Biol, 72(4):351–374, 1997. [45] A. Bertuzzi, A. Fasano, A. Gandolfi, and C. Sinisgalli. Reoxygenation and split- dose response to radiation in tumours with krogh-like vasculature. Preprint, 2007. [46] H. Enderling, A.A. Andersona, M.A.J. Chaplain, A.J. Munrob, and J.S. Vaidyab. Mathematical modelling of radiotherapy strategies for early breast cancer. JTheorBiol, 241(1):158–171, 2006. [47] A.A. Anderson, Chaplain M.A.J., E.L. Newman, R.J.C. Steele, and A.M. Thompson. Mathematical modelling of tumour invasion and metastasis. J Theor Med, 2(2):129–154, 2000. [48] I.F. Tannock. The relation between cell proliferation and the vascular system in a transplanted mouse mammary tumor. Br J Cancer, 22(2):258–273, 1968. [49] A. Krogh. The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary to supply the tissue. JPhysiol, 52:409–415, 1919. [50] J.P Ward and J. King. Mathematical modelling of avascular-tumour growth. MathAppl Med Biol , 14:39–69, 1997. [51] H.M. Byrne and M.A.J. Chaplain. Free boundary value problems associated with the growth and development of multicellular spheroids. European J Appl Math, 8:639–658, 1997. [52] C.P. Please, G.J. Pettet, and D.L.S. McElwain. A new approach to modelling the formation of necrotic regions in tumours. Appl MathLett, 11:89–94, 1998. [53] C.P. Please, G.J. Pettet, and D.L.S. McElwain. Avascular tumour dynamics and necrosis. MathModels Methods Appl Sci , 9(4):569–579, 1999. [54] B.G. Wouters and J.M. Brown. Cells at intermediate oxygen levels can be more important than the hypoxic fraction in determining tumor response to fractionated therapy. Radiat Res, 147(5):541–550, 1997. [55] A. Bellouquid and M. Delitalia. Mathematical methods and tools of kinetic theory towards modelling of complex biological systems. MathModel Meth Appl Sci, 15(11):1639–1666, 2005. [56] N. Bellomo and G. Forni. Looking for new paradigms towards a biological mathematical theory of complex multicellular systems. MathModel MethAppl Sci, 16(7):1001–1029, 2006. [57] B. Ribba, T. Colin, and S. Schnell. A multiscale mathematical model of cancer, and its use in analyzing irradiation therapies. Theor Biol Med Model, 3(7), 2006. [58] P. Nurse, Y. Mausi, and L. Hartwell. Understanding the cell cycle. Nature Med, 4(10):1103–1106, 1998. [59] A. Cappuccio, M.A. Herrero, and L. Nuñez. Biological optimization of tumour radiosurgery. Med Phys, 36(1):98–104, 2009.

102 REFERENCES

[60] G.P Chen, E. Ahunbay, C. Schultz, and X. Lia, Allen Lia. Development of an inverse optimization package to plan nonuniform dose distributions based on spatially inhomogeneous radiosensitivity extracted from biological images. Med Phys, 34(4):1198–1205, 2007.

Departamento de Matematica´ Aplicada, Facultad de Ciencias Matematicas,´ Univer- sidad Complutense, Plaza de las Ciencias 3, 28040 Madrid, Spain E-mail address: [email protected]

Departamento de Matematica´ Aplicada, Facultad de Ciencias Matematicas,´ Univer- sidad Complutense, Plaza de las Ciencias 3, 28040 Madrid, Spain E-mail address: Miguel [email protected] ¯ Servicio de Radiof´ısica, Hospital Puerta de Hierro, Manuel de Falla 1, 28222, Ma- jadahonda, Madrid E-mail address: [email protected]

Contemporary Mathematics Volume 492, 2009

Multiphase and Individual Cell-Based Models of Tumour Growth

J. Galle and L. Preziosi

Abstract. Focusing as an example on the description of the phenomenon of contact inhibition of growth, the aim of this chapter is to compare from the qualitative and whenever possible the quantitative point of view the results obtained using two very diﬀerent modelling approaches, an individual based model and a continuum multiphase model.

1. Introduction In typical experiments of cell culture it can be observed that after an initial exponential growth the rate of proliferation decreases as a consequence of the fact that cells come in contact. Cell density then saturates and it is said that the colony has grown to conﬂuence. This phenomenon is often called contact inhibition of growth (Deleu et al. 1998, Dietrich et al. 1997, Kato et al. 1997, Nelson and Chen 2003, Polyak et al. 1994, St. Croix et al. 1998) and is governed by some transmembrane proteins called cadherins that act as mechanotrasducers. The formation of the intercellular cadherin junctions inhibits proliferation, while on the contrary, their disruption triggers the production of growth factors which contribute to induce proliferation. In pathological conditions the loss of contact responsiveness can lead to deregulated growth, a phenomenon which is commonly associated with the formation of hyperplasia (Uglow et al. 2000, Gottardi et al. 2001) and malignant transfor- mations (Tzukita et al. 1993, Becker et al. 1994, Oda et al. 1994, Risinger et al. 1994, Christofori and Semb 1999, Cavallaro et al. 2002). Actually, in the last paper the misperception of the presence of neighbouring cells is considered such a fundamental milestone in the development of tumours to be named “cadherin switch” in analogy with the “angiogenic switch” leading to the vascularization of tumours. The other main mechanism considered in this paper is extracellular matrix remodelling. The extracellular matrix (ECM) is composed by many constituents produced by a variety of stromal cells, mainly ﬁbroblasts. The ECM is continuously renewed and remodelled through both the production of matrix metalloproteinases (MMP) and the synthesis of new ECM. In stationary physiological conditions the remodelling of ECM is much slower than when new tissue has to be produced. For

104 J. GALLE AND L. PREZIOSI instance, in the human lung the physiological turnover of total ECM is 10-15% per day (Johnson 2001), which means that a complete turnover takes nearly a week. On the other hand, at the initial stages of growth, in vitro endogenous proteins such as fibronectin and von Willebrand factor are released by endothelial cells and complex matrix is organised within a few hours after seeding (Chiquet et al. 1996, Dejana et al. 1990). Our understanding is that upon reaching confluency the cells slow down the production of ECM components. On the other hand, when cells become malignant, they activate a proteolytic programme to cleave the extracellular matrix and penetrate more easily. The composition of ECM changes with tumour progression. This process was considered such important in determining cell growth, differentiation and movement that Liotta and Kohn (2001) mention the possibility of finding a “stromal therapy as a new strategy” against cancer. Actually, Zhang et al. (2003) showed that the content of collagen fibers increased with prostate cancer grade ranging from nearly 7% to 26% of the area analysed. Moreover, it was found that many tumours are characterised by the formation of fibrotic tissue (Brewster et al. 1990, Johnson 2001, Pujuguet et al. 1996) or by degradation of ECM due to excessive production of MMP-13 (Yang 2004). Johnson (2001) argued that the formation of fibrosis can be due to several probably con- curring reasons: increased de novo synthesis of ECM proteins, decreased activity of its degrading enzyme (MMP), and upregulation of the tissue-specific inhibitors of metalloproteinases (TIMP), that is, the molecules that make matrix degrading enzymes (MDEs) ineffective. In this respect, the mathematical model proposed by Chaplain et al. (2006) showed that all possibilities are valid, a possible discriminant being the amount of homeostatic MDEs present in the tissue.

2. The Multiphase Model To take into account the different constituents present in a tumour and the mechanical structure of biological tissues the use of the theory of mixture was proposed by Byrne and Preziosi (2004), Byrne et al. (2003), and Ambrosi and Preziosi (2002). Many other applications of multiphase models have been proposed e.g. by Breward et al. (2002, 2003), Frank and King (2003), Graziano and Preziosi (2007), some of them also including the effect of the mechanical interactions with the surrounding tissues. In constructing the multiphase model presented here we distinguish two subpopulations, one more sensitive to contact inhibition (the normal or host cells) and the other less sensitive to it (the tumour cells). Cells grow in a porous material formed by the extracellular matrix (ECM) and wet by the extracellular liquid. At first we will assume for sake of simplicity that the ECM network is rigid and fixed. Later on, we will drop the latter assumption allowing for ECM remodelling, but keeping its rigidity.

2.1. Cell populations. Referring to Ambrosi and Preziosi (2002) and Preziosi and Tosin (2007) for more details and to Astanin and Preziosi (2007) for a ped- agogical chapter on how to develop multiphase models of tumour growth, let us denote by φi ∈ [0, 1] the volume ratios occupied by the i-th cell clone and by m the

MULTIPHASE AND INDIVIDUAL CELL-BASED MODELS OF TUMOUR GROWTH105 volume ratio occupied by the ECM. Saturation implies that φi + m ≡ ψ ≤ 1 , i with the remaining portion of space 1 − ψ occupied by the liquid. For these quantities one can write the following mass balance equations ∂φi (1) + ∇·(φivi)=Γiφi − δiφi ,i=1,...,n, ∂t where following Chaplain et al. (2006) the velocity of the subpopulation i is given by

(2) vi = −Ki∇Σi(ψ) .

The coeﬃcient Ki is not only related to the pure permeability of the porous medium but also to the ability of cells to move through the ECM network, which involves two main aspects: space occupation and activation of adhesion mechanisms. The functions Σi measure the response to compression that depends on the amount of cells present in the tissue. As far as the r.h.s. is concerned, the cell apoptotic rate may depend on the compression δi = δi(ψ), and the cell duplication rate is given by

(3) Γi = γiHε(ψ − ψi) , where Hε is a monotonic mollifier of the step function with the properties that it is a continuous function with Hε(φ)=1ifφ ≤ 0 and Hε(φ)=0ifφ ≥ ε. Equation (3) represents a simple switch mechanism depending on the compression level. Referring to the phenomenological discussion presented in the previous section, it states that the cells belonging to the i-th population replicate if there is sufficient space to do this, that is if they sense a sustainable level of compression (ψ ≤ ψi), or equivalently stress (Σi(ψ) ≤ σi with σi =Σi(ψi)). Actually, it has been observed that growing tissues are usually not in a stress- free configuration (see Skalak et al. 1996). A residual stress is in fact present in many cases and may be related to differential growth (Ambrosi and Guana 2007, Ambrosi and Mollica 2003, Humphrey and Rajagopal 2002). Actually, it seems that cells prefer to feel a moderate amount of stress (see Rouslahti (1997) and references therein). In the model presented here a local balance in the growth term is obtained for −1 δi (4) ψ = ψi + Hε . γi

Our assumption is that the threshold value ψi is nearly equal to the stress-free value ψ0 for normal cells and slightly larger for abnormal ones. Actually, growth might even be independent of ψ, meaning that the tumour cells are completely insensitive to mechanical cues and continue replicating independently of the compression level. Even in the case ψn = ψ0, Eq.(4) implies that the stationary value is larger than ψ0. Of course, the growth term depends on other quantities, e.g. the amount of nutrient and growth factors. In this chapter, however, we want to focus on contact inhibition as a test case and therefore assume that all nutrients are abundantly supplied and neglect the eﬀect of growth factors. In doing this we tacitly assume that all the constituents necessary for the cell to grow and undergo mitosis can be found in the extracellular liquid.

106 J. GALLE AND L. PREZIOSI

In the following examples we will consider only two populations identified by the index i = t, n for tumour and normal cells, and the case in which they only differ for the stress perception, i.e., ψn <ψt, or for the motility characteristics, i.e., different Ki. All the other coefficients are the same, in particular, γn = γt and δn = δt meaning that the reproduction and apoptotic rates are the same. For sake of clarity the model that will be used in the following simulation is summarised here ⎧ ⎪ ∂φn ⎨⎪ = ∇·(φnKn∇Σ(ψ)) + [γHε(ψ − ψn) − δ] φn , ∂t (5) ⎪ ⎩⎪ ∂φt = ∇·(φtKt∇Σ(ψ)) + [γHε(ψ − ψt) − δ] φt . ∂t

2.2. ECM proteases and remodelling. The extracellular matrix is an intricate network of fibrous material made of many macromolecules, including fibronectin, laminin and collagen, which can be degraded by matrix degrading enzymes, usually shortened as MDEs (Matrisian 1992, Parson et al. 1997). Active MDEs are produced (or activated) by the cells, diffuse throughout the tissue and undergo some form of decay, either passive or active. The equation governing the evolution of MDE concentration might therefore be written as

∂c 2 c (6) = κ∇ c + Pn + Pt − , ∂t τ where the functions Pn and Pt model the production of active MDEs by normal and tumour cells, respectively. In order to introduce metalloproteinasis as an eﬀect of the transition of tumor cells to the mesenchymal state in Eq. (6) we allow Pt to depend on the cell speed, which in turn is related to the stress gradient through (2). From the simulations shown in Figure 3 to follow it is evident the stronger gradients and higher velocities occur next to the tumour boundary, so that the action of the extra-production of MMP by the cells at the boundary of the tumour is to cleave the ECM located close to the boundary and increase cell motility. In particular, we will take

(7) Pn = πnφn ,Pt =(πt + πv|v|)φt .

Upon contact, MDEs degrade the extracellular matrix produced by the cells in a stress-dependent way. Hence the degradation process can be modelled by the following simple equation

∂m (8) = µn(Σ)φn + µt(Σ)φt − νcm, ∂t where ν is a positive constant and µn and µa depend on whether the cells are in a confluent situation or not, or on the fact that the tissue is subject to strain. In the simulations to follow we will however take them as constant. In (8) we distinguish the production of ECM by normal and abnormal cells, because the production rates can be different. Actually, in Preziosi and Graziano (2007) it was shown how fibrosis can be caused by the excessive production of ECM by the tumour cells.

MULTIPHASE AND INDIVIDUAL CELL-BASED MODELS OF TUMOUR GROWTH107

Hence the following system of equations will be added to (5) in the simulation that model ECM proteases and remodelling ⎧ ⎪ ∂m ⎨⎪ = µnφn + µtφt − νcm, ∂t (9) ⎪ ⎩ ∂c 2 = κ∇ c + πnφn +(πt + πv|v|)φt . ∂t

3. The Individual Cell-Based Approach The impact of multiple cellular alterations, as failures of the cellular growth control mechanisms, on tumor growth and invasion can be alternatively studied applying individual based computational approaches, so called individual cell-based or agent-based models. In these models individual cells are able to move, to grow and to divide and can form (adhesive) contacts to neighbor cells and/or matrix components. A number of different individual based models of cell populations have been studied so far. They comprise: • cellular automaton models where each cell is represented by a single lattice site (Anders 2005, Dormann and Deutsch 2002, Paulus et al. 1993, Loeffler et al. 1987), • cellular automaton models where each cell is represented by many lattice sites (Bauer et al. 2007, Jiang et al. 2005, Hogeweg 2000, and Graner and Glazier 1993) and • off-lattice models where cells are either modeled as deformable particles (spheres: Galle et al. 2005, Drasdo et al. 1995; ellipsoids: Dallon and Othmer 2004, Palsson and Othmer 2000), or as Voronoi polygons (Schaller and Müller-Hermann 2005, Brodland and Veldhuis 2002, Meineke et al. 2001, Honda et al. 2000). In the following we focus on off-lattice models. In off-lattice models the dynamics of the cells is described either by equations of motion for each individual cell or by Monte-Carlo approaches. Both approaches were shown to yield comparable results (Drasdo et al. 2007, Drasdo and Hoehme 2005). Testing the robustness of simulation results against modifications of the assumed mechanisms, it was demonstrated, that neither details of the assumptions regarding cell shape and cell division mechanisms nor assumptions about the precise shape of the interaction force between the cells significantly affect the growth behavior on long time scales (Drasdo et al. 2007, Galle et al. 2006a). As a specific feature individual cell-based models allow for integrating subcellular regulation processes. Following this strategy environmental signaling was linked to intracellular decision processes controlling cell growth and apoptosis as well as to cell migration and proteolysis (Hoehme et al. 2007, Galle et al 2006b, 2005). These studies demonstrate that if these models are parameterized by characteristic, experimentally accessible cell and substrate properties and involve growth control mechanisms, these models are capable of explaining the complex spatial growth and pattern formation processes of cell populations. In the following the model described by Galle and co-workers (Galle et al. 2005) will be considered as a representative example of 3D individual cell-based models. The model was originally introduced analyzing growth control mechanisms of transformed epithelial cell populations expanding in standard Petri dishes and

108 J. GALLE AND L. PREZIOSI

Figure 1. Contact formation. Cells which form contacts to other cells or the substrate deform. Consequently, the actual cell volume VA changes. VA may adopt the intrinsic volume VT again by subsequent changes of their radius R (from: Galle et al. 2005). was further developed linking the growth properties under these conditions to the invasion dynamics of colorectal carcinoma in vivo (Galle et al. 2006b). Including an explicit control mechanism of the cell volume, the model enables straightforward simulations of regulation dynamics related to dense cellular packing. Thus, it is well suited comparing IBM behavior with that of the PDE systems introduced above. 3.1. The basic model for tumour growth and invasion. An isolated cell is represented by an elastic sphere of radius R and the substrate of the Petri dish by an impenetrable sphere with inﬁnite radius. If a cell gets into contact with the substrate or with other cells, the cell adheres. The adhesive cell-substrate and cell-cell interaction energy is approximated by

(10) WA = kAC , where k denotes the average adhesion energy per unit contact area and AC the actual contact area for contact to the substrate (k = s) or another cell (k = c), respectively. As a result of the contact the shape of the cell changes by ﬂattening at the contact area. Consequently, the volume of the cell changes as well. The energy related with a volume change of the cell is approximated by the energy of a uniform compression (or inﬂation) of a homogeneous elastic solid with bulk modulus K

K 2 (11) WK = (VT − VA) 2VT Here, VA is the actual volume of the cell and VT the target volume, i.e. the volume the cell would adopt if it were isolated.

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Moreover, the deformation energy for the contact is calculated from the Hertz model 5/2 i,j 2xk RiRj (12) WD = i,j 5D Ri + Rj

Here, Ri, Rj are the cell radii and the terms xk (k = c, s) are deﬁned by the distances of the cells (Fig. 1). The constant Di,j is related to the Young modulus and the Poisson ratio of the interaction partners (Landau and Lifschitz 1986). The total interaction energy between two cells i and j is deﬁned by K D A (13) Wi,j = Wi,j + Wi,j + Wi,j.

Wi,j depends on the distance between the cells and the radius of both cells. Thus, the cell-cell contacts can equilibrate via cell displacements or by changes of the cell radius R. The total interaction energy between a cell and the substratum Wi,s is defined in an analogous way. The generalized forces on cell i which drive the dynamics are given by det ∂Wi,j ∂Wi,s det ∂Wi,j ∂Wi,s (14) Fi = ni,j + ni,s and Gi = + . j ∂ri,j ∂ri,s j ∂Ri ∂Ri ri,j = |ri,j| = |ri − rj| where ri and rj are the position vectors of cell i and j, respectively. In the same way ri,s is the distance between cell i and the substrate s. ni,j = ri,j/|ri,j| and ni,s = ri,s/|ri,s|. The dynamics of each individual cell i is modeled by Langevin equations in a friction dominated regime. Thus, in the absence of an external stimulus the cells perform a random movement. The displacement and radius change of cell i are modeled in separate equations as fr det st (15) Fi ≡ Ci,j (wi − wj)+Ci,swi + cM wi = Fi + Fi j for the force balance determining the translational cell movement with velocity wi = dri/dt and fr det (16) Gi ≡ Bi,j (ui + uj)+(Bi,s + bV ) ui = Gi j for the variation of the radius with velocity ui = dRi/dt. fr Here, Fi is the sum over all friction forces during translational movement, fr Gi denote the friction forces during changes of the radius of cell i. Ci,j is a (3 × 3) friction coefficient matrix, Bi,j a scalar friction coefficient for the friction between cells i and j. Ci,s and Bi,s denote the corresponding quantities for the friction between cell i and the substrate s. cm and bv are coefficients for cell medium friction and for resistance to cell volume changes, respectively. The right hand sides of both equations denote the sums over all generalized forces causing det det the displacement and the radius change, respectively. In particular, Fi (Gi ) st summarize the deterministic forces, Fi denotes the stochastic force with zero mean and delta-correlated autocorrelation function that models the random component of cell movement. A two phase cell cycle is assumed in the model to consider cell proliferation. During Phase I (interphase), a cell doubles its target volume VT by stochastic increments. This growth process results in an approximately Γ-distributed, i.e.

110 J. GALLE AND L. PREZIOSI variable, growth time τ of the cells. During Phase II (mitotic phase), a cell divides into two daughter cells of equal target volume VT = V0. The orientation of cell division occurs perpendicular to the direction of the maximum force exerted on a cell. Further details can be found in (Galle et al. 2005).

4. Application to Colony Growth Dynamics In this section we will apply the two modelling approaches brieﬂy introduced in Section 2 and 3 to describe the phenomenon of contact inhibition. Cellular growth control mechanisms were introduced into the IBM model to study the impact of failures of these mechanisms on population growth, tumor expansion and invasion. In particular, a cell-cell contact dependent form of growth inhibition was considered assuming: – that if the actual volume VA of a cell is smaller than a threshold value Vinh

4.1. Contact inhibition affects the growth of tumour cell colonies in vitro. Generally, two phases can be distinguished during colony growth in vitro: an initial phase where the number of cells grows exponential allowing to derive the cell doubling times and a subsequent phase where the colony diameter grows linearly, i.e. with a constant spreading velocity. Eventually, if the colony is close to confluence its growth decreases till it stops. Analyzing the growth of tumor cell monolayers, our experimental collaborators in Galle et al. (2006b) found for Widr cells (cells of a colorectal adenocarcinoma cell line) two morphological identical subpopulations (SP), SPfast and SPslow.Their doubling times are the same, but SPfast spreads faster than SPslow. Assuming a smaller value of the threshold volume Vinh for the subpopulation SPfast compared to SPslow, results in a quantitative description of the observed colony growth even regarding the distribution of proliferation events (Fig. 2). Assuming that cells of SPfast are characterised by a larger Young modulus or smaller cell-substrate friction, i.e. are more motile, one would get comparable results; as long as the cell layer remains stable. Thus, differences in the sensitivity to contact inhibition as well as differences in cell properties affecting the relaxation of cell compression can explain the dynamics of the different subclones. In Figure 3 we use the multiphase PDE model (5) to describe a situation similar to the experiment shown in Fig. 2B. Two populations of tumor cells are virtually seeded in the points (0.4, 0.4) and (0.4, 0.6), i.e., at a distance of 200µm. Here,

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Figure 2. Experimental and simulated colony growth.A) Vertical section through a growing model population. The grey value of the cells is a marker of the cell volume. Black cells (ar- row heads) are in the act of starting division, white cells (arrows) underwent contact inhibition of growth. B) Top view on two cul- tured Widr populations after 14 days stained for proliferation by Brdu. They grow from cells of two different subpopulations SPfast and SPslow. C) Computer simulation results related to B). Scale Bars: 200 µm. D) Cell number of Widr cell colonies versus time t. The thick line indicates exponential growth according to a cell doubling time of 25h. E) Diameter of the colonies shown in D). Lines are computer simulation results. Reprinted from Am. J. Pathol. 2006 169: 1802–1811 with permission from the American Society for Investigative Pathology. the only difference between the two populations is that we assume the left one to have a ten times higher motility coefficient. However, we mention that comparable results would be achieved if cells of the colony on left were assumed to be stiffer or characterised by a higher stress perception. At the very beginning the assumption does not cause any difference in the growth of the two colonies. However, from a plot of the growth rates (Figure 3B) it is evident that while all cells duplicate on the SPfast colony, in the SPslow colony the cell in the center duplicate less. This is due to the fact that the less motile colony

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Figure 3. Simulated colony growth by the multiphase PDE model. The left column (A,C,E) reports the volume ratio at t =1, 6, 11 days. The right column (B,D,F) the growth rates. The border of the tumour is represented by the white line. does not release the inner stress so fast. The cells in the center are too compressed to activate their duplication programme. The radius of the colony on the right is a bit smaller than that on the left. However, if one integrates the densities (better saying the volume ratio) of the two clones to get the mass of the colonies (better saying the volume occupied by the two colonies), they are identical (see Fig. 4A). At this stage the growth of both colonies is nearly exponential (see Fig. 4B). Going on with time the diﬀerence between the two clones becomes more and more evident with cell proliferation segregated to the proliferating rim. This is similar to what observed in many papers on tumour growth as an eﬀect of nutrient availability inside the tumour. However, we remind that in the models presented

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fast 0.55 fast slow slow

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0 5 10 15 0 5 10 15 t [days] t [days] AB

Figure 4. Simulated colony growth by the multiphase PDE model. A) Total mass of the two colonies versus time. B) Temporal evolution of the effective diameter, i.e., of the diameter of the circle having the same areas as the colonies. nutrient distribution is not taken into account and it is not an issue in the experiment as the colonies grow on a Petri dish. The proliferating rim is simply due to contact inhibition and by the same reason duplication is mainly restricted to cells close to the boundary in the IBM model. As the populations grow, the mass of the population on the right departs from the one on the left (see Fig. 4A), like in the IBM model (Fig. 2D). After nearly 6 days the two colonies touch and growth is inhibited on the contact line (see Fig. 3C,D). However, the cells on the outer border keep duplicating and those on the left are always more active, so at the end the colony on the right is almost engulfed by the one on the left (Fig. 3E,F). This last configuration is close to the one presented in the experiment shown in Figure 2B. From Figures 3D, and 3F, it can be appreciated how the contact between the two clones progressively inhibits also the proliferation in the population on the left. The evolution of the cell mass (Fig. 4A) and of the cell effective diameter (Fig. 4B), defined as the diameter of the circle having the same area as the one occupied by the clone, should be compared with the equivalent Figures 2E,F. A qualitative comparison gives an amazing similarity of the results of these two very different models. Thus, in both modelling approaches differences in cell properties that affect contact inhibition of growth can explain the dynamics of different subclones of growing tumor cell populations.

4.2. Contact Inhibition Affects Population Morphology and Age. A detailed analysis of the effect of contact inhibition of growth on the expansion of tumor cell colonies in vitro showed that the mechanisms has also significant impact on the spatial growth patterns (Galle et al. 2005). In stable cell monolayers the sensitivity of the cells to contact inhibition of growth affects the width of their

114 J. GALLE AND L. PREZIOSI proliferating zone. Under these growth conditions, decreasing the sensitivity of the cells results in a broader proliferation zone and an increased spreading velocity as mentioned in the example above. However, decreasing the sensitivity below a critical value, results in a decreased spreading velocity of the population due to monolayer destabilization. Cells in the interior of the monolayer are pushed out of the monolayer forcing a continuous renewing of the population. Accordingly, proliferation events are found throughout the growing colony. Switching off the anchorage dependent control mechanisms the model allows studying the formation of 3d multicellular aggregates, as tumor spheroids. In principle, the 3d growth dynamics simulated by IBMs parallel that observed in monolayers. So, in tumor spheroids contact inhibition of growth was shown to control the width of the proliferating rim and consequently lead to a constant growth velocity of the spheroid diameter (Drasdo and Hoehme 2005). Changes of the width of the proliferating zone are also obtained using the multiphase PDE model. Looking at the profile of the growth terms, one can see that it is characterized by a plateau near the boundary and then decreases very fast going to the core of the colonies. This makes it difficult to define a proliferating thickness, but it can be qualitatively stated that the thickness of the plateau region linearly depends on the sensitivity to contact inhibition, here measured through the quantity D = ψt − ψn, while the thickness of the decay is almost independent of ψt − ψn.

5. Application to Clonal Competition and Proteolytic Invasion In this section we will focus on the eﬀects of contact inhibition of growth on tumor expansion due to clonal competition and proteolytic invasion. In all simulations tumor cells were characterized by a decreased sensitivity to contact inhibition tumor normal of growth (Vinh

ψt ψ − 1 F ≈ n , ψt − ψt ψ0 ψn which however is nearly proportional to D. For both models all other parameters appearing in the model related to tumor and normal cells are identical facilitating comparative analysis. The parameter sets described in Galle et al. (2005) and Chaplain et al. (2006) were applied.

5.1. Clonal competition by diﬀerent sensitivity to contact inhibition. To prepare a scenario of clonal competition within the IBM we started with a population of normal cells with full contact inhibition ﬁlling a closed box. The cells induce a persistent ‘within-tissue’ pressure corresponding in the multiphase PDE model to the volume ratio −1 δn (18) φn = ψn + Hε . γn

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0.9

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Figure 5. Clonal competition by the multiphase PDE model. A) Phases of growth in the space independent situation. B) Tissue invasion for t =5, 10, 20, 40, 60, 80 days. The plots on the top refer to φt + φn,thosebelowtoφn only. The compression of the normal tissue due to the expansion of the tumour is put in evidence by the peaks near the tumour boundary. Note the travelling wave characteristics. (C) Velocity of propagation as a function of D = ψt − ψn for δ/γ =0.1, 0.2, 0.4 (from lower to upper curve).

This pressure keeps the normal cells quiescent due to contact inhibition of growth. Subsequently, an initial tumor clone is generated selecting a few cells at the center of the box and decreasing their sensitivity to contact inhibition of growth. This growth advantage would enable them to expand under the initial conditions but only until the pressure is high enough to stop proliferation of the tumor cells as well (Galle et al. 2006b, Chaplain et al. 2006).

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The PDE model simulation starts with an extremely small initial value for −5 the tumour cells φt =10 , seeded in a virtual environment made of 50% of cells and 20% of ECM. In Figure 5A we plot for a homogeneous situation the volume ratio of the cell populations and the overall volume ratio ψ. The evolution can be divided into three phases. The ﬁrst one (say up to 10 days) is characterized by a simple partial replacement of normal cells by tumour cells, keeping the total volume ratio unchanged and constantly equal to ψn. Therefore in this phase there is no compression. After that, in the second phase, the amount of tumour cells undergoing mitosis is larger than the amount of normal cells undergoing apoptosis, so that the total volume ratio ψ starts exceeding the stress-free value ψn, till it reaches the value for which also tumor cells start feeling the pressure and undergoing contact inhibition themselves. However, in this third stage there are still normal cells that progressively die. Eventually, the tissue is entirely ﬁlled by tumour cells, surviving in an overcompressed state and subject to contact inhibition. Figure 5B focuses on the invasion process. The space dependent integration puts in evidence in particular how the invasion velocity becomes constant. Through a travelling wave analysis that can be performed for small D = ψt − ψn, Chaplain et al. (2006) evaluated analytically that the invasion velocity can be approximated by dΣ δ (19) v ≈ 2K (ψn)δ 1 − (ψt − ψn) . dψ γ

In particular, the invasion velocity goes like the square root of the sensitivity parameter. Coherently with the multiphase PDE model which is characterized by an apoptotic rate δ, to induce ongoing competition between the two different cell clones, we considered in the IBM model a defined overall rate of spontaneous apoptosis wapop (additional to the substrate contact dependent one). Assuming the same rate wapop for both clones, the clone with the low sensitivity to contact inhibition of growth expands continuously and eventually completely replaces the other one as in the PDE model. Figure 6A shows the total cell number as well as the cell numbers of the individual clones; Figure 6B the tumor radius, which grows linearly with time coherently with the constancy of the invasion velocity. Again a close relation to the dynamics observed in the PDE model is demonstrated for monolayers stable adhering to the underlying substrate (compare Figs. 5A and 6A). Moreover, by variation of the sensitivity parameter F we found that the less sensitive the tumor cells to contact inhibition of growth are the faster their clones spread. We calculated the initial expansion velocity of the tumor clones Vexpa = tumor tumor dR /dt considering clones with a radius R smaller√ than L/4. As shown in Fig. 7A we found Vexpa to be roughly proportional to F as observed in the PDE system (see Fig. 5C). However, IBM specific effects and phenomena at the cellular scale are more visible in this case. Except for very small apoptosis rates wapop, we observed only a weak dependence of the expansion velocity on this parameter. This effect is related to a limited relaxation of the tissue, i.e. it is related to the reduced motility of the individual cells due to the presence of strong cellular friction forces. As demonstrated in Fig. 7B decreasing the friction coefficient for the cell-cell and cells-substrate friction µ

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Figure 6. Tumor growth dynamics by the IBM model:A) Total number of cells (solid lines), number of tumor cells (dashed lines), and number of normal/stromal cells (dotted lines) and B) tumor radius Rtumor for single simulations of clonal competition (thin lines, F =9,wapop =0.2/d) and proteolytic invasion (thick lines, F =9,wdegr =0.4/d). While during clonal competition a quasi linear increase of the tumor size was observed already for the initial tumor (a), during the proteolytic invasion such behavior was found only after some non-linear expansion (a-b). C)-F) Top views on clones (tumor cells: dark grey, dividing tumor cells: black, box length 1 mm). C)-D) Clonal competition at time t = 0 C) and after 6 days D). E)-F) Proteolytic invasion at time t = 0 E) and after 12 days F).

broadens the range where the dynamics depends substantially on the apoptosis rate and increases the expansion velocity.

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The observation obtained above using the IBM model is compatible with (19) that shows that the invasion velocity goes like δ(γ − δ) with other parameters kept fixed. Thus, regarding clonal competition in the IBM model, contact inhibition of growth acts in concert with the motility of the cells to determine the expansion dynamics. 5.2. Modeling proteolytic tumor cell invasion. The clonal competition model described above can be utilized to simulate tumor invasion. For that purpose the normal cells of the model are considered as “stroma cells” which represent both cell and matrix components of the tumor stroma (Galle et al. 2006b). In the following we focus on 2d invasion dynamics facilitating direct comparison with the clonal competition situation. For the invasion simulations using the IBM model we assumed the spontaneous apoptosis rate wapop to vanish. Thus, the initial conditions refer to complete contact inhibition of growth in the system (Fig. 2). To ensure invasion the tumor cells are assumed to degrade the surrounding stroma cells with a defined rate wprot per cell- cell contact, i.e., we assume the tumor cells to be proteolytic active. Changes of the proteolytic activity of tumor cells are modeled changing this rate wprot. From the physiological point of view such an effect can be also related to the fact that the anaerobic metabolism used by tumour cells generates a low pH-environment in which normal cells have more difficulties in surviving (Gatenby and Gawlinski 1996). Considering this kind of proteolysis we found in the IBM simulations a two- phase invasion dynamics (Fig. 6B). An initial phase, where the diameter of the tumor increases non-linearly, is followed by a phase which is characterized by a tumor constant invasion velocity of the tumor clone vinva = dR /dt. Analyzing vinva for Rtumor 0 is required for invasion, for large values of F the invasion velocity becomes independent of F (Fig. 7C). Moreover, as shown in Fig. 7D, we found vinva to be directly proportional to the degradation rate wprot. Within the considered parameter ranges these results were found to depend not on the tissue relaxation properties. Thus, regarding tumor invasion in the IBM model, contact inhibition of growth represent a prerequisite of the process but does not determine the invasion velocity. For sake of completeness, we briefly report some results obtained by integrating the model (5, 9) in the case in which strong ECM cleavage due to mesenchymal transition and overproduction of ECM by tumour cells are co- present. As shown in Figure 8 the proteolytic mechanism (obtained setting πv = 0 produces an overex- pression of MDEs near the tumour border, which cleaves locally the ECM. However, the cells in the center of the tumour produce new ECM at a higher rate than physiological (µt >µn). Eventually the ECM occupies more space and cells less space than in the physiological case. In fact, in the fibrotic tissue the ECM to cell ratio raises from 0.4 up to 1.

6. Final Remarks on Hybrid Modeling In this chapter we have shown how IBM and PDE models can be used to study the same biological problem obtaining similar results from the qualitative and to some extent quantitative point of view. Of course each method has its fors and

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Figure 7. Tumor growth dynamics by the IBM model.A) In the clonal competition model the expansion velocity vexpa depends on the sensitivity√ to contact inhibition of growth F and is roughly proportional to F (solid line). B) For large apoptosis rates wapop vexpa becomes independent of wapop depending on the friction considered. C) In the proteolytic invasion model the invasion velocity Vinva is independent of F for large F .D)Vinva is proportional to the degradation rate wprot. Note that the reduced expansion velocity for large F and small wapop encircled in A) refers to monolayer destabilization. µ0 denotes the reference friction constant equal to 1011 Ns/m3 (Galle et al. 2005). againsts, starting from the fact that PDE models are more proper to describe the behavior of tissues from the macroscopic point of view will IBM models are able to look more closely at what happens at the cellular level. The main difficulty in relating the two models stays, in our opinion, in the fact that sometimes it is not easy to relate parameters measured at the tissue level with those measured at the cellular level. For instance, it is difficult to transfer the concept of constitutive model from continuum mechanics to cell-based models. At the same time, it is difficult to transfer measurements made on single cells to the tissue level, e.g., strength of adhesion bonds, or receptor activation. On the background we are aware of the fact that the challenge nowadays seems to be to describe biological process not using a single-scale view but in a multi-scale

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Figure 8. Proteolitic invasion by the multiphase PDE model. (A) From below to above, host cells, tumor cells and their sum. (B) The top curve refers to ψ, the others to MDE concentration and ECM volume ratio, with degradation at the tumour boundary and fibrosis formation in its central region. landscape. In this respect, IBM are more flexible and suited to insert in each cell sub-cellular mechanisms, and to exploit the most recent discoveries in the fields of biochemistry, genomics and proteomics. On the other hand, unfortunately it is well known that IBM might become computationally expensive, when using a large number of cells. It would then be computationally prohibitive to increase the complexity of the behaviour of single cells. On the other hand, it is often not important to keep this level of details throughout the tissue. For this reason it would be important to link IBM and PDE models, building hybrid models that combine and exploit the advantages brought by the different frameworks. Some steps in this direction are presented in Drasdo (2005) and Kim et al. (2007) and we believe that in general such an attitude and attention will produce the key to unlock the function of complex tissues based on their genetic and cellular composition.

References [1] L. Deleu, F. Fuks, D. Spitkovsky, R. Hörlein,S. Faisst, and J. Rommelaere (1998). Op- posite transcriptional effects of cyclic AMP-responsive elements in confluent or p27kip- overexpressing cells versus serum-starved or growing cells, Molec. Cell. Biol. 18: 409–419. [2] C. Dietrich, K. Wallenfrang, F. Oesch, and R. Wieser (1997). Differences in the mechanisms of growth control in contact-inhibited and serum-deprived human fibroblasts, Oncogene 15: 2743–2747. [3] A. Kato, H. Takahashi, Y. Takahashi, and H. Matsushime (1997). Inactivation of the cyclin D-dependent kinase in the rat fibroblast cell line, 3Y1, induced by contact inhibition, J. Biol. Chem. 272: 8065–8070. [4] C.M. Nelson and C.S. Chen (2003). VE-cadherin simultaneously stimulates ad inhibits cell proliferation by altering cytoskeletal structure and tension, J. Cell Science 116: 3571–3581. [5] K. Polyak, J. Kato, M.J. Solomon, C.J. Sherr, J. Massague, J.M. Roberts, and A. Koff (1994). p27Kip1, a cyclin-Cdk inhibitor, links transforming growth factor-β and contact inhibition to cell cycle arrest, Genes & Develop. 8: 9–22.

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[6] B. St. Croix, C. Sheehan, J.W. Rak, V.A. Florenes, J.M. Slingerland, and R.S. Kerbel (1998). E-cadherin-dependent growth suppression is mediated by the cyclin-dependent kinase inhibitor p27(Kip1), J. Cell Biol. 142: 557–571. [7] K.F. Becker, M.J. Atkinson, U. Reich, I. Becker, H. Nekarda, J.R. Siewert, and H. Hofler (1994). E-cadherin gene mutation provide clues to diffuse gastric carcinoma, Cancer Res. 54: 3845–3852. [8] T. Oda, Y. Kanai, T. Okama, K. Yoshiura, Y. Shimoyama, W. Birchmeier, T. Sugimura, and S. Hirohashi (1994) E-cadherin gene mutation in human gastric carcinoma cell lines, Proc. Natl. Acad. Sci. USA 91: 1858–1862. [9] S. Tzukita, M. Itoh, A. Nagafuchi, S. Yonemura, and S. Tsukita (1993). Submembrane junc- tional plaque proteins include potential tumor suppressor molecules, J. Cell Biol. 123:1049– 1053. [10] C.J. Gottardi, E. Wong, and B.M. Gumbiner (2001). E-cadherin suppresses cellular transformation by inhibiting β-catenin signalling in an adhesion-independent manner, J. Cell. Biol. 153: 1049–1060. [11] J.I. Risinger, A. Berchuck, M.F. Kohler, and J. Boyd (1994). Mutation of E-cadherin gene in human gynecological cancers, Nature Genetics 7: 98–102. [12] E.B. Uglow, G.D. Angelini, and S.J. George (2000). Cadherin expression is altered during intimal thickening in humal sapphenous vein, J. Submicrosc. Cytol. Pathol. 32: C113–C119. [13] G. Christofori and H. Semb (1999). The role of cell-adhesion molecule E-cadherin as a tumors- suppressor gene, Trends Biochem. Sci. 24: 73–76. [14] U. Cavallaro, B. Schaffhauser, and G. Christofori (2002). Cadherin and the tumour progression: Is it all in a switch?, Cancer Lett. 176: 123–128. [15] P.R.A. Johnson (2001). Role of human airway smooth muscle in altered extracellular matrix production in asthma, Clin. Exp. Pharm. Physiol. 28: 233–236. [16] M. Chiquet, M. Matthisson, M. Koch, M. Tannheimer, and R. Chiquet-Ehrismann (1996). Regulation of extracellular matrix synthesis by mechanical stress, Biochem. Cell Biol. 74: 737–744. [17] E. Dejana, M.G. Lampugnani, M. Giorgi, M. Gaboli, and P.C. Marchisio (1990). Fibrinogen induces endothelial cell adhesion and spreading via the release of endogenous matrix proteins and the recruitment of more than one integrin receptor, Blood 75: 1509–1517. [18] L.A. Liotta and E.C. Kohn (2001). The microenvironment of the tumor-host interface, Nature 411: 375–379. [19] C.E. Brewster, P.H. Howarth, R. Djukanovic, J. Wilson, S.T. Holgate, and W.R. Roche (1990). Myofibroblasts and subepithelial fibrosis in bronchial asthma, Am. J. Respir. Cell Mol. Biol. 3: 507–511. [20] P. Pujuguet, A. Hammann, M. Moutet, J. L. Samuel, F. Martin, and M. Martin (1996). Expression of fibronectin EDA+ and EDB+ isoforms by human and experimental colorectal cancer, Am. J. Patho. 148: 579–592. [21] C.-M. Yang, C.-S. Chien, C.-C. Yao, L.-D. Hsiao, Y.-C. Huang, and C.B. Wu (2004). Me- chanical strain induces collagenases-3 (MMP-13) expression in MC3T3-E1 osteoblastic cells, J. Biol. Chemistry 279: 22158–22165. [22] Y. Zhang, S. Nojima, H. Nakayama, J. Yulan, and H. Enza (2003). Characteristics of normal stromal components and their correlation with cancer occurrence in human prostate, Oncol. Rep. 10: 207–211. [23] M. Chaplain, L. Graziano, and L. Preziosi (2006). Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development, Math. Med. Biol. 23: 197–229. [24] H.M. Byrne and L. Preziosi (2004). Modeling solid tumor growth using the theory of mixtures, Math. Med. Biol. 20 : 341–366. [25] H.M. Byrne, J.R. King, D.L.S. McElwain, and L. Preziosi, (2003). A two-phase model of solid tumor growth, Appl. Math. Letters 16: 567–573. [26] D. Ambrosi and L. Preziosi (2002). On the closure of mass balance models for tumor growth, Math. Mod. Meth. Appl. Sci. 12: 737–754. [27] C.J.W. Breward, H.M. Byrne, and C.E. Lewis (2002). The role of cell-cell interactions in a two-phase model for avascular tumor growth, J. Math. Biol. 45: 125–152. [28] C.J.W. Breward, H.M. Byrne, and C.E. Lewis (2003). A multiphase model describing vascular tumor growth, Bull. Math. Biol. 65: 609–640.

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[29] S.J. Franks and J.R. King (2003). Interactions between a uniformly proliferating tumor and its surrounding. Uniform material properties, Math. Med. Biol. 20: 47–89. [30] L. Graziano and L. Preziosi (2007). Mechanics in tumour growth, in: Modeling of Biological Materials, F. Mollica, L. Preziosi, and K.R. Rajagopal, Eds., Birkhäuser, 267–328. [31] L. Preziosi and A. Tosin (2007). Multiphase modeling of tumor growth and extracellular matrix interaction: Mathematical tools and applications, J. Math. Biol. in press. [32] S. Astanin and L. Preziosi (2007). Multiphase models of tumour growth, in: Selected Topics on Cancer Modelling: Genesis - Evolution - Immune Competition - Therapy, Nicola Bellomo, Mark Chaplain, and Elena De Angelis, Eds., Birkhauser (2007). [33] R. Skalak, S. Zargaryan, R.K. Jain, P.A. Netti, and A. Hoger, (1996). Compatibility and genesis of residual stress by volumetric growth, J. Math. Biol. 34: 889–914. [34] D. Ambrosi and F. Guana (2007). Stress modulated growth, Quart. J. Mech. Appl. Math. 12, 319–342. [35] D. Ambrosi and F. Mollica (2004). The role of stress in the growth of a multicell spheroid, J. Math. Biol. 48: 477-499. [36] J.D. Humphrey and K.R. Rajagopal (2002). A constrained mixture model for growth and remodeling of soft tissues, Math. Mod. Meth. Appl. Sci. 12: 407–430. [37] E. Rouslahti (1997). Stretching is good for a cell, Science 276: 1345–1346. [38] L.M. Matrisian (1992). The matrix–degrading metalloproteinases, Bioessays 14: 455–463. [39] S.L. Parson, S.A. Watson, P.D. Brown, H.M. Collins, and R.J.C. Steele (1997). Matrix metalloproteinases, Brit. J. Surg. 84: 160–166. [40] A.R. Anderson (2005). A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion, Math. Med. Biol. 22: 163–186. [41] S. Dormann and A. Deutsch (2002). Modelling of self-organized avascular tumor growth with a hybrid cellular automaton, in Silico Biol. 2: 393–406. [42] U. Paulus, M. Loeffler, J. Zeidler, G. Owen, and C.S. Potten (1993). The differentiation and lineage development of globet cells in the murine small intestinal crypt: experimental and modelling studies. J. Cell Sci. 106: 473–484. [43] M. Loeffler, C.S. Potten, and H.E. Wichmann (1987). Epidermal cell proliferation, Virchows Arch. B 53: 286–300 [44] A.L. Bauer, T.L. Jackson, and Y. Jiang (2007). A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis, Biophys. J. 92: 3105–3121. [45] Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell, J.P. Freyer (2005). A multiscale model for avascular tumor growth. Biophys. J. 89: 3884-3894. [46] P. Hogeweg (2000). Evolving mechanisms of morphogenesis: on the interplay between differential adhesion and cell differentiation, J. Theor. Biol. 203: 317–333. [47] F. Graner and J.A. Glazier (1993). Simulation of biological cell sorting using a two- dimensional extended Potts model. Phys. Rev. Lett. 69: 2013–2016. [48] J. Galle, M. Loeffler, and D. Drasdo (2005). Modelling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro, Biophys. J. 88: 62–75. [49] D. Drasdo, R. Kree, and J.S. McCaskill (1995). Monte-Carlo approach to tissue-cell populations, Phys. Rev. E 52: 6635–6657. [50] J. Dallon and H.G. Othmer (2004). How cellular movement determines the collective force generated by the Dictyostelium discoideum slug. J. Theor. Biol. 231: 203–222. [51] E. Palsson and H.G. Othmer (2000). A model for individual and collective cell movement in Dictyostelium discoideum. Proc. Nat. Acad. of Sciences. 97: 10448–10453. [52] G. Schaller and M. Meyer-Hermann (2005). Multicellular tumor spheroid in an off-lattice Voronoi-Delaunay cell model. Phys. Rev. E 71, 051910. [53] G.W. Brodland and J.H. Veldhuis (2002). Computer simulations of mitosis and interdepen- dencies between orientation, cell shape and epithelia reshaping, J. Biomech 35: 673–681. [54] F.A. Meineke, C.S. Potten, and M. Loeffler (2001). Cell migration and organisation in the intestinal crypt using a lattice-free model, Cell. Prolif. 34: 253–266. [55] H. Honda, M. Tanemura, and A. Yoshida (2000). Differentiation of wing epidermal scale cells in a butterfly under the lateral inhibition model - appearance of large cells in a polygonal pattern, Acta Biotheo. 48: 121–136.

MULTIPHASE AND INDIVIDUAL CELL-BASED MODELS OF TUMOUR GROWTH123

[56] D. Drasdo, S. Hoehme, M. Block (2007). On the role of physics in the growth and pattern formation of multi-cellular systems: What can we learn from individual-cell based models? J. Stat. Phys. 128, 287–345. [57] D. Drasdo and S. Höhme (2005). A single-cell-based model of tumor growth in vitro: monolayers and spheroids, Phys. Biol. 2, 133–147. [58] J. Galle, G. Aust, G. Schaller, T. Beyer, D. Drasdo (2006a). Individual cell-based models of the spatial-temporal organization of multicellular systems – Achievements and limitations, Cytometry A. 69: 704–710. [59] J. Galle, D. Sittig, I. Hanisch, M. Wobus, E. Wandel, M. Loeffler, and G. Aust (2006b). Individual cell-based models of tumor-environment interactions: Multiple effects of CD97 on tumor invasion. Am. J. Pathol. 169: 1802–1811. [60] L.D. Landau and E.M. Lifschitz (1986). Theory of Elasticity, 3rd rev., Oxford, Pergamon Press. [61] J. Moreira and A. Deutsch (2002). Cellular automaton models of tumor development: A critical review, Adv. Compl. Sys. 5: 247–267. [62] G. Schaller and M. Meyer-Hermann (2006). Continuum versus discrete model: a comparison for multicellular tumour spheroids, Philos. Transact. A: Math. Phys. Eng. Sci. 15;364(1843): 1443–1464. [63] D. Drasdo and S. Höhme (2003). Individual-based approaches to birth and death in avascular tumors, Math. Comp. Mod. 37: 1163–1175. [64] I. Roeder, L.M. Kamminga, K. Braesel, B. Dontje, D. De Haan, and M. Loeffler (2005). Competitive clonal hematopoiesis in mouse chimeras explained by a stochastic model of stem cell organization, Blood 105:609–616. [65] M. Loeffler and I. Roeder (2002). Tissue stem cells: Definition, plasticity, heterogeneity, self organization and models - A conceptual approach, Cells Tissues Organs 171: 8–26. [66] M. Loeffler and C.S. Potten (1997). Stem cells and cellular pedigrees - A conceptual introduction, in: Stem Cells, C.S. Potten (ed.), Academic Press, 1–27. [67] R.A. Gatenby and E.T. Gawlinski (1996). A reaction–diffusion model of cancer invasion, Cancer Research 56: 5745–5753. [68] D. Drasdo (2005). Coarse graining in simulated cell populations, Adv. Complex Syst. 2&3, 319–363. [69] Y. Kim, M. Stolarska, and H. Othmer (2007). A hybrid model for tumor spheroid growth in vitro I: Theoretical development and early results , Math. Models Methods in Appl. Sci.,to appear.

Interdisciplinary Center for Bioinformatics, University Leipzig, Germany E-mail address: [email protected] Dipartimento di Matematica, Politecnico di Torino, Italy E-mail address: [email protected]

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Titles in This Series

492 Fernando Giráldez and Miguel A. Herrero, Editors, Mathematics, developmental biology and tumour growth, 2009 491 Carolyn S. Gordon, Juan Tirao, Jorge A. Vargas, and Joseph A. Wolf, Editors, New developments in Lie theory and geometry, 2009 490 Donald Babbitt, Vyjayanthi Chari, and Rita Fioresi, Editors, Symmetry in mathematics and physics, 2009 489 David Ginzburg, Erez Lapid, and David Soudry, Editors, Automorphic Forms and 𝐿-functions II. Local aspects, 2009 488 David Ginzburg, Erez Lapid, and David Soudry, Editors, Automorphic forms and 𝐿-functions I. Global aspects, 2009 487 Gilles Lachaud, Christophe Ritzenthaler, and Michael A. Tsfasman, Editors, Arithmetic, geometry, cryptography and coding theory, 2009 486 Frédéric Mynard and Elliott Pearl, Editors, Beyond topology, 2009 485 Idris Assani, Editor, Ergodic theory, 2009 484 Motoko Kotani, Hisashi Naito, and Tatsuya Tate, Editors, Spectral analysis in geometry and number theory, 2009 483 Vyacheslav Futorny, Victor Kac, Iryna Kashuba, and Efim Zelmanov, Editors, Algebras, representations and applications, 2009 482 Kazem Mahdavi and Deborah Koslover, Editors, Advances in quantum computation, 2009 481 Aydın Aytuna, Reinhold Meise, Tosun Terzio˘glu, and Dietmar Vogt, Editors, Functional analysis and complex analysis, 2009 480 Nguyen Viet Dung, Franco Guerriero, Lakhdar Hammoudi, and Pramod Kanwar, Editors, Rings, modules and representations, 2008 479 Timothy Y. Chow and Daniel C. Isaksen, Editors, Communicating mathematics, 2008 478 Zongzhu Lin and Jianpan Wang, Editors, Representation theory, 2008 477 Ignacio Luengo, Editor, Recent Trends in Cryptography, 2008 476 Carlos Villegas-Blas, Editor, Fourth summer school in analysis and mathematical physic Topics in spectral theory and quantum mechanics, 2008 475 Jean-Paul Brasselet, José Luis Cisneros-Molina, David Massey, JoséSeade, and Bernard Teissier, Editors, Singularities I Geometric and topological aspects, 2008 474 Jean-Paul Brasselet, José Luis Cisneros-Molina, David Massey, JoséSeade, and Bernard Teissier, Editors, Singularities Algebraic and analytic aspects, 2008 473 Alberto Farina and Jean-Claude Saut, Editors, Stationary and time dependent Gross-Pitaevskii equations, 2008 472 James Arthur, Wilfried Schmid, and Peter E. Trapa, Editors, Representation Theory of Real Reductive Lie Groups, 2008 471 Diego Dominici and Robert S. Maier, Editors, Special functions and orthogonal polynomials, 2008 470 Luise-Charlotte Kappe, Arturo Magidin, and Robert Fitzgerald Morse, Editors, Computational group theory and the theory of groups, 2008 469 Keith Burns, Dmitry Dolgopyat, and Yakov Pesin, Editors, Geometric and probabilistic structures in dynamics, 2008 468 Bruce Gilligan and Guy J. Roos, Editors, Symmetries in complex analysis, 2008 467 Alfred G. Noël, Donald R. King, Gaston M. N’Guérékata, and Edray H. Goins, Editors, Council for African American researchers in the mathematical science Volume V, 2008 466 Boo Cheong Khoo, Zhilin Li, and Ping Lin, Editors, Moving interface problems and applications in fluid dynamics, 2008

TIES IN THIS SERIES

465 Valery Aleev, Arnaud Beauville, C. Herbert Clemens, and Elham Izadi, Editors, Curves and Abelian varieties, 2008 464 Gestur Olafsson,´ Eric L. Grinberg, David Larson, Palle E. T. Jorgensen, Peter R. Massopust, Eric Todd Quinto, and Boris Rubin, Editors, Radon transforms, geometry, and wavelets, 2008 463 Kristin E. Lauter and Kenneth A. Ribet, Editors, Computational arithmetic geometry, 2008 462 Giuseppe Dito, Hugo Garc´ıa-Compeán, Ernesto Lupercio, and Francisco J. Turrubiates, Editors, Non-commutative geometry in mathematics and physics, 2008 461 Gary L. Mullen, Daniel Panario, and Igor Shparlinski, Editors, Finite fields and applications, 2008 460 Megumi Harada, Yael Karshon, Mikiya Masuda, and Taras Panov, Editors, Toric topology, 2008 459 Marcelo J. Saia and José Seade, Editors, Real and complex singularities, 2008 458 Jinho Baik, Thomas Kriecherbauer, Luen-Chau Li, Kenneth D. T-R McLaughlin, and Carlos Tomei, Editors, Integrable systems and random matrices, 2008 457 Tewodros Amdeberhan and Victor H. Moll, Editors, Tapas in experimental mathematics, 2008 456 S. K. Jain and S. Parvathi, Editors, Noncommutative rings, group rings, diagram algebras and their applications, 2008 455 Mark Agranovsky, Daoud Bshouty, Lavi Karp, Simeon Reich, David Shoikhet, and Lawrence Zalcman, Editors, Complex analysis and dynamical systems III, 2008 454 Rita A. Hibschweiler and Thomas H. MacGregor, Editors, Banach spaces of analytic functions, 2008 453 Jacob E. Goodman, János Pach, and Richard Pollack, Editors, Surveys on Discrete and Computational Geometry–Twenty Years Later, 2008 452 Matthias Beck, Christian Haase, Bruce Reznick, Michèle Vergne, Volkmar Welker, and Ruriko Yoshida, Editors, Integer points in polyhedra, 2008 451 David R. Larson, Peter Massopust, Zuhair Nashed, Minh Chuong Nguyen, Manos Papadakis, and Ahmed Zayed, Editors, Frames and operator theory in analysis and signal processing, 2008 450 Giuseppe Dito, Jiang-Hua Lu, Yoshiaki Maeda, and Alan Weinstein, Editors, Poisson geometry in mathematics and physics, 2008 449 Robert S. Doran, Calvin C. Moore, and Robert J. Zimmer, Editors, Group representations, ergodic theory, and mathematical physic A tribute to George W. Mackey, 2007 448 Alberto Corso, Juan Migliore, and Claudia Polini, Editors, Algebra, geometry and their interactions, 2007 447 Fran¸cois Germinet and Peter Hislop, Editors, Adventures in mathematical physics, 2007 446 Henri Berestycki, Michiel Bertsch, Fel E. Browder, Louis Nirenberg, Lambertus A. Peletier, and Laurent Véron, Editors, Perspectives in Nonlinear Partial Differential Equations, 2007 445 Laura De Carli and Mario Milman, Editors, Interpolation Theory and Applications, 2007

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Developmental biology and tumour growth are two important areas of current research where mathematics increasingly provides powerful new techniques and insights. The unfolding complexity of living structures from egg to embryo gives rise to a number of difficult quantitative problems that are ripe for mathematical models and analysis. Understanding this early development process involves the study of pattern formation, which mathematicians view through the lens of dynamical systems. This book addresses several issues in developmental biology, including Notch signalling pathway integration and mesenchymal compartment formation. Tumour growth is one of the primary challenges of cancer research. Its study requires interdisciplinary approaches involving the close collaboration of mathematicians, biologists and physicians. The summer school addressed angiogenesis, modelling issues arising in radiotherapy, and tumour growth viewed from the individual cell and the relation to a multiphase-fluid flow picture of that process. This book is suitable for researchers, graduate students, and advanced undergraduates interested in mathematical methods of developmental biology or tumour growth.

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