European Journal of Cancer, Vol. 35, No. 6, pp. 886±891, 1999# 1999 Elsevier Science Ltd. All rights reservedPrinted in Great Britain0959-8049/99/$ - see front matterPergamonPII: S0959-8049(99)00067-2Point of ViewDoes Breast Cancer Exist in a State of Chaos?M. Baum,1 M.A.J. Chaplain,2 A.R.A. Anderson,2 M. Douek1 and J.S. Vaidya11Department of Surgery, Royal Free and University College Medical School, University College London, 67±73Riding House Street, London W1P 7LD; and 2Department of Mathematics, University of Dundee, Dundee, U.K.INTRODUCTIONÐMODELS OF DISEASEThroughout the history of medicine mankind has demonstrated extraordinary feats of imagination in elaboratinghypothetical models to explain the nature of disease. Thesemodels then suggest therapeutic strategies to influence thenatural history of disease processes. Until the early 17th Century in Western medicine and to this day in classical Chinesemedicine these models were largely metaphysical [1]. However, Ayurveda, the Indian system of medicine, is based on adynamic balance between three principles: combustion/energy-production/anabolism (Pitta), transport/communication (Vaata), and excretion/catabolism (Kapha). Disease wassupposed to be a manifestation of the disturbance of thisbalance and the doctor's duty was to maintain and restorethis balance [2]. In the last century or so models of diseasehave progressed from the mechanical to the biological andmore recently to the mathematical. The change of allegiancefrom one model to another has been likened to a Kuhnianrevolution or paradigm shift [3]. The two most famous in thehistory of medicine were the overthrowing of the Gallenicdoctrine by William Harvey's description of the circulation ofthe blood and the replacement of the miasma theory ofinfection by the bacterial theory of infection which ultimatelyled to the success of anti-microbial therapy.MODELS OF BREAST CANCERBreast cancer, an enigmatic disease with an unpredictablenatural history has been a fertile soil for the development ofhypothetical models each with their therapeutic consequence.Until the discovery of the cellular nature of cancer, the disease was managed according to Gallenic principles, the disease being visualised as an excess of melancholia (black bile)that coagulated within the breast [4]. Treatments aimed atridding the body of this excess of black bile involved venesection, purgation, cupping, leaching, enemas and bizarrediets (many `alternative' treatments of breast cancer to thisday are in fact a form of neo-gallenism).In the mid 19th Century the humoral theory of breastcancer was overturned by a mechanistic model which descriCorrespondence to M. Baum, e-mail: m.baum@ucl.ac.ukReceived 28 Oct. 1998; revised 24 Feb. 1999; accepted 24 Feb. 1999.bed the disease as a phenomenon arising locally within thebreast and then spreading centrifugally along lymphatics tobe arrested in the ®rst echelon of lymph nodes which acted asa barrier to onward spread by their innate ®ltering capacity. Asecond echelon of lymph nodes existed like the casementwalls of a medieval town protecting the citadel at its centre.The therapeutic consequence of such a belief was the development of the Halsted radical mastectomy, almost exactly100 years ago [5].THE CONTEMPORARY MODEL OF BREASTCANCERLargely due to the seminal work of Professor BernardFisher in the late 1960s, the mechanistic model of the diseasewas overturned to be replaced by a biological model whichlooked upon the outcome of the disease as pre-determined bythe extent of micrometastases disseminated via the microvasculature of the tumour very early on in its natural history,instead of looking at breast cancer as a chronological eventi.e. `early' and `late' we have been taught to look upon breastcancer as a biological challenge with disease being eitherfavourable or unfavourable [6]. The important therapeuticconsequence of this belief system has been the developmentof adjuvant systemic therapy using cytotoxic drugs or endocrine manipulation. This approach has undoubtedly demonstrated the ®rst major impact on breast cancer mortality sincethe disease was recognised. Unfortunately, the treatmentshave not lived up to their promise and, although in relativeterms, mortality can be reduced by 25%, in absolute termsthis translates into a gain of between 5 and 10% over 10±15years [7, 8]. The pace has now slowed down and ®ne-tuningof the therapeutic regimens may only oVer us incrementalpoints of progress. At the same time, frustrated by this lack ofprogress, many medical oncologists have been advocating the`reductio ab adsurdum' of high-dose chemotherapy and stemcell rescue. Hopes of any important improvement in resultswith this approach have been dashed with the recent publication of two randomised-controlled trials [9, 10].FLAWS IN THE CONTEMPORARY MODELThe triumphalism of the last 20 years of endeavour hasnow been muted as we begin to recognise the limitations of886Does Cancer Exist in a State of Chaos?adjuvant systemic therapy but along the way we seem tohave forgotten one salient feature of the contemporaryparadigm. Hidden within the biological model is a linearmathematical model. It has been assumed that the putativemicrometastases, that ultimately express themselves as clinical disease, are autonomous and growing according to themathematical formulae of deterministic or Gompertziankinetics [11]. Furthermore a second order hypothesis hassuggested that in order to achieve maximum cell kill, cytotoxic drugs have to be given at a maximum tolerated dosein cycles to achieve log cell kill until the residual tumourburden can be `mopped up' by the patient's natural hostresistance mechanisms [12]. It is now long forgotten thatthis mathematical model was based on the behaviour ofanimal experimental systems like the hamster lymphomadescribed by Skipper and colleagues in the 1960s [13].What this model fails to explain are the various inconsistencies listed in Table 1. In particular we wish to drawattention to the extraordinary phenomenon of the biphasicnature of hazard rates for relapse and death. Instead ofthese hazard rates being constant with time, they demonstrate twin peaks, the ®rst at approximately 3 years and asecond ¯atter peak between approximately 7 and 9 years[14, 15]. The other extraordinary and chilling fact is that,although the disease-free interval of women who have beenoperated upon for breast cancer may be extremely variable,ranging from a few weeks to 30 years, once metastasesbecome symptomatic, death is inevitable usually within 2±3years. Randomised trials which have tested the bene®t ofregular screening for metastatic disease after treatment ofprimary breast cancer have failed to show any bene®t interms of mortality [16, 17] suggesting that even asymptomatic metastatic disease is not curable at present. Thisextraordinary phenomenon of dormancy and awakening ofmetastatic disease has challenged researchers for a longtime. Recently, there have been suggestions that the mostpromising model to explain the dormancy of metastaticdisease is the one based on the process of angiogenesis[18].In this paper, we wish to describe an alternative biologicalmodel of metastases in breast cancer that suggests that theseare complex organisms existing in a state of dynamic equilibrium close to a chaotic boundary. Furthermore, the mathematics to describe the natural history of these organismsinvokes non-linear dynamics or the chaos theory. Central to887the understanding of this model has been the pioneering workof Judah Folkman on tumour angiogenesis. [19, 20].ANGIOGENESIS AND CANCERSolid tumours cannot grow beyond 106 cells or approximately 1±2 mm in diameter in the absence of a blood supply[21]. The initial pre-vascular phase of growth is followed by avascular phase in which tumour-induced angiogenesis is therate limiting step for further growth and provides malignantcells direct access to the circulation [15, 22]. The poor prognostic indication of extensive angiogenesis quanti®ed bymicrovessel density in histological sections is well recognisedin a wide variety of cancers including breast cancer [23, 24].The high vascularity of breast tumours may be imaged bycontrast-enhanced magnetic resonance imaging (MRI) sinceenhancement relies on tumour vascularity and vascular permeability, as demonstrated by histopathologic correlationalstudies [25] (Figure 1).The relationship between angiogenesis and tumour cellshas been summarised by Folkman's endothelial cell, tumourcell compartment theory. In the tumour cell compartment,cells may stimulate endothelial cell proliferation by the production of tumour angiogenic factors such as beta FGF andVEGF [26]. Alternatively, endothelial cells may themselvesstimulate the growth of tumour cells by producing factorssuch as platelet derived growth factor (PDGF), heparin likegrowth factor and interleukin 6 (IL6) [27]. It has also beenproposed that the primary tumour secretes anti-angiogenicfactors and, therefore, removal of the primary tumour mighttrigger the outgrowth of occult foci into clinically apparentsecondary disease [28]. In addition to the importance of themicro-vasculature, we have also to visualise these microscopicfoci as existing in a `soup' of cytokines and endocrine polypeptides and steroids, with cells interacting with each otherand with the surrounding stroma. with competing signalsdirecting the cancer cells towards proliferation or apoptosis[18]. It is then possible to visualise sub-clinical metastasis as acomplex structure with its future determined by the balanceof angiogenic and anti-angiogenic factors as well as factorsthat stimulate or inhibit epithelial proliferation and stimulateor inhibit apoptosis. Such complexity cannot be modelled bylinear dynamics, or even a full understanding of the completecatalogue of genetic mutations at the cellular level, becausethe critical events of multiple cell to cell interaction requiresthorough understanding of epigenetic phenomena. A modelTable 1. Inconsistencies and paradoxes in the conventional mechanistic model of breast cancerBreast cancercharacteristicsNot linearly related toTumour sizeNumber of lymph nodesinvolved, i.e.One can have a large tumour with no lymph nodes involved and an occult primarywith axillary lymph nodes involved.Incidence of metastasis at Whether the tumour is T1, T2 or even early T3, the incidence of metastases atpresentation, i.e.presentation does not go beyond 5%, but within 2±3 years of diagnosis/therapy,more of the larger tumours present with metastases.Hazards of recurrenceTime from diagnosis, i.e.The hazards of recurrence and death have a sharp peak at 3 years; the stage atdiagnosis only in¯uences the amplitude of the peak but not the timing.Timing of surgery in relation Survival, i.e.to menstrual periodThe ®nding that women operated in the follicular phase of their menstrual cyclehave a worse prognosis than those operated on in the luteal phase.Site of metastasisTumours can metastasise to either bone or liver bypassing the intervening lung.Vascular or lymphaticaccess from the primarytumour, i.e.888M. Baum et al.somewhat similar to ours has been proposed by Demicheliand colleagues in 1997 [29] and this paper is the ®rst attemptto apply new mathematics of complexity to make predictionsabout the factors in¯uencing angiogenesis that might one dayprovide a therapeutic window.ejc99). This type of approach may enable a more accuratemeans of monitoring response to treatment. Integration ofdata derived from various imaging modalities (Figure 3) withhistopathological tumour assessment in such a non-linearmodel, might better approximate the natural history of cancer.A NEW MATHEMATICAL MODEL FOR BREASTCANCER MICROMETASTASESOver the last few years several mathematical models havebeen developed to describe some of the important features oftumour induced angiogenesis. In particular Anderson andChaplain have described a model using only three importantvariables involved in tumour angiogenesis [30]. We havecalled these endothelial cells (EC), tumour angiogenic factors(TAF), and matrix angiogenic factors (MAF). The relativelysimple formulae using non-linear dynamics are shown in theappendix and like other `chaotic systems' produce beautifulfractal-like images which can be shown to be exquisitely sensitive to initial conditions (e.g. diVerent concentrations ordiVerent gradients of the three biological variables). The 3dimensional (3-D) mathematical simulation of vasculargrowth towards a tumour (Figure 2), shows a striking similarity with the vasculature of a breast tumour as visualisedusing CT (computerised tomography) specimen angiographyand 3-D reconstruction (Figure 3). Further illustrationsincluding animations of diVerent simulations can be seen onthe internet (http://www.mcs.dundee.ac.uk:8080/$sanderso/HOW DOES THE NEW MODEL EXPLAIN THEINCONSISTENCIES OF THE OLD MODEL?To explain the late appearance of metastatic disease over alatent period of more than 10 years, we no longer have toinvoke a dormant cell theory. It has to be remembered thatthe cancer cells are not foreign organisms but closer-to-selfthan non-self [31], and there is no surprise that these complex organisms can exist in a state of dynamic equilibriumuntil some chance, late event perturbs the status quo. Inaddition, we can perhaps explain the phenomenon where thebreast cancer presents with axillary metastases and the primary is undetectable. The original cancer might have outgrown its blood supply but not before it had a chance toFigure 2. Computer simulation of angiogenesis: computersimulation of the evolution of a capillary network in responseto a layer of tumour cells (or alternatively, a large solid circular tumour) over a 15 day period. The simulation shows themigration, branching and anastomosis of the capillary sproutsas they make their way from the parent vessel through thebreast tissue.Figure 1. T1 weighted pre- (a) and post- (b) contrastenhanced breast MRI: a breast tumour (large arrow) withnipple involvement (small arrow) were clearly visible aftercontrast enhancement (b). Digital subtraction of image (a)from image (b) and subsequent colour coding (c), demonstrated a very good correlation between areas of highenhancement intensity (red) and areas rich in blood vessels(brown staining) as seen with factor VIII-immunohistochem-Figure 3. Three-dimensional CT-reconstruction of breastcancer vascularity. The lateral thoracic artery in a fresh mastectomy specimen was identi®ed and cannulated. A radioopaque mixture containing barium sulphate was injected intothe specimen and a CT scan performed. This 3-dimensionalreconstruction of the resulting 106 slices (a) shows the tumour(large orange mass) surrounded by an extensive vascular network (yellow); the small orange mass is the nipple, shown fororientation. The growth pattern of vessels from the lateralthoracic artery (left to right) agree with mathematical predictions of vascular growth, as seen when the tumour and nippleare removed from the image (b).Does Cancer Exist in a State of Chaos?disseminate cells along the lymphatics, and in the lymphnodes the conditions might exist for the successful establishment of a microvasculature.The twin peaks can be explained as follows. The removal ofthe primary tumour disinhibits angiogenesis and encouragesepithelial proliferation with the surgical trauma encouragingthe secretion of insulin-like growth factor, so in other wordsthe act of surgery kick-starts metastasis. This phenomenon isseen at its most dramatic in women developing breast cancerunder the age of 35 years [32]. It could even be argued that ifby chance the operation was performed during the follicularphase at a point of maximum epithelial stimulus from oestradiol and minimum apoptotic stimulus through low progesterone levels, then the prognosis might be even worse[33, 34]. Furthermore, it has been demonstrated that tumourexpression of genes that govern proliferation and metastaticpotential, change during the two phases of menstrual cycle[35]. In other words the system is exquisitely sensitive toinitial conditions. The second lower peak could be explainedas a stochastic summation of adverse events in a woman's lifesuch as inter-current infections, second operations or evenbereavement.THERAPEUTIC CONSEQUENCES OF THE NEWMODELThe speci®c role of the mathematical model in all of this isto provide objective and quantitative data. The computersimulation that is presented in Figure 2 was realised throughapplying quantitative estimates of experimentally measurableparameter values such as cell random motility, cell migrationrates in response to TAF and MAF gradients etc. The vascular network that appears is, therefore, not only morphologically similar in a qualitative manner (i.e. it looks like a realcapillary network) but also, and more importantly, it is morphologically similar in a quantitative manner, i.e. it has thesame growth rate as a real network and reaches the tumour inthe same time as a real network. Furthermore, the modelprovides additional data regarding capillary branching patterns, vessel length, numbers of new vessels created, the areaand volume of the vessels, etc. Further re®nements of themathematical model can include details such as the rate of¯ow of drugs through its network, interactions with tumourcells and the rate of supply of drugs to the tumour cells.Therefore, it would be possible to incorporate the generallyavailable clinicopathological and radiological data speci®c toa particular patient into this model which would then providesimulations and predictions that are not directly derivable bystandard linear mathematics. Simulating treatments in themodel would help to choose the best guess of several possiblenew treatments.For example, the therapeutic intervention would be antiangiogenic and the timing of the intervention would be preoperative so that at the time of surgery the system was primedto protect against the sudden ¯ooding with angiogenic signals. It might indeed be the case that some of the successattributed to adjuvant tamoxifen is as a result of its antiangiogenic potential rather than its anti-oestrogenic function[36]. Assuming we can then protect the subject from the ®rstpeak of metastatic outgrowth, then we will have to monitorher with extreme vigilance. By the time the metastases areclinically apparent it is perhaps too late, so monitoring thepatient with tumour markers and re-introducing an antiangiogenic strategy at the ®rst rise in tumour markers might889prove successful. Better still using PCR techniques to detectvariations in the anti-angiogenic milieu may provide an evengreater lead time than the use of conventional tumourmarkers such as CEA and CA15. In the meantime, we cancontinue to add additional layers of complexity to the simulations of our mathematical model to help develop alternativestrategies for biological interventions to maintain the statusquo.Such modeling could potentially precede clinical trials, andaccelerate progress of cancer therapeutics. Unlike the hamster lymphoma models of the past, the new model feeds oncomplexity and gets closer and closer to simulating nature inall its awesome beauty.1. Porter R. The Greatest Bene®t to Mankind. London, HarperCollins, 1997, pp. 147±162.2. Vaidya JS. The concept of scienti®c holism. Natl Med J Ind1996, 9, 299±300.3. Strohman RC. The coming Kuhnian revolution in biology. Nature Biotechnology 1997, 15, 194±200.4. Porter R. The Greatest Bene®t to Mankind. London, HarperCollins, 1997, pp. 73±82.5. Halsted WS. The results of operation for the cure of cancer ofthe breast performed at the Johns Hopkins Hospital from June1889 to January 1894. Ann Surg 1894, 20, 497±555.6. Fisher B. 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Intensive diagnostic follow-up after treatment ofprimary breast cancer. A randomized trial. National ResearchCouncil Project on Breast Cancer follow-up (see comments).JAMA 1994, 271, 1593±1597.18. Baum M, Benson JR. Current and future roles of adjuvantendocrine therapy in the management of early carcinoma of thebreast. In Senn H.-J., Gelber RD, Goldhirsch A, ThuÈrlimann B,eds. Recent Results in Cancer ResearchÐAdjuvant Therapy of BreastCancer V. Berlin, Heidelberg, Springer Verlag, 1996, 215±226.19. Folkman J. Tumor angiogenesis: therapeutic implications. NEngl J Med 1971, 285(21), 1182±1186.890M. Baum et al.20. Folkman J. Angiogenesis: initiation and control. Ann N Y AcadSci 1982, 401, 212±227.21. Folkman J, Watson K, Ingber D, Hanahan D. Induction ofangiogenesis during the transition from hyperplasia to neoplasia.Nature 1989, 339(6219), 58±61.22. Folkman J. What is the evidence that tumors are angiogenesisdependent? J Nat Cancer Inst 1990, 82, 4±6.23. 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Demicheli R, Retsky MW, Swartzendruber DE, Bonadonna G.Proposal for a new model of breast cancer metastatic development. Ann Oncol 1997, 8, 1075±1080.30. Anderson ARA, Chaplain MAJ. Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull Math Biol1998, 60, 857±899.31. Schipper H, Turley EA, Baum M. A new biological frameworkfor cancer research. Lancet 1996, 348, 1149±1151.32. De La Rochefordiere A, Asselain B, Campana F, et al. Age asprognostic factor in premenopausal breast carcinoma. Lancet1993, 341, 1039±1043.33. Hrushesky WJ, Bluming AZ, Gruber SA, Sothern RB. Menstrualin¯uence on surgical cure of breast cancer. Lancet 1989, 2, 949±952.34. Badwe RA, Gregory WM, Chaudary MA, et al. Timing of surgeryduring menstrual cycle and survival of premenopausal womenwith operable breast cancer. Lancet 1991, 337(8752), 1261±1264.35. Saad Z, Bramwell VHC, Wilson SM, O'Malley FP, Jeacock J,Chambers AF. Expression of genes that contribute to proliferative and metastatic ability in breast cancer resected duringvarious menstrual phases. Lancet 1998, 351, 1170±1173.36. Haran E, Maretzek A, Goldberg I, Horowitz A, Degani H.Tamoxifen enhances cell death in implanted MCF7 breastcancer by inhibiting endothelial growth. Cancer Res 1994, 54,5511±5514.37. Hynes RO. Fibronectins. New York, Springer-Verlag, 1990.38. Chaplain MAJ, Anderson ARA. Mathematical modelling, simulation and prediction of tumour-induced angiogenesis. InvasionMetastasis 1997, 16, 222±234.39. Stokes CL, Rupnick MA, Williams SK, LawVenburger DA.Chemotaxis of human microvessel endothelial cells in responseto acidic ®broblast growth factor. Lab Invest 1990, 63, 657±668.AcknowledgementsÐProfessor Michael Baum gratefully acknowledges the Cancer Research Campaign's support for a Senior Fellowship. Dr Chaplain and Dr Anderson gratefully acknowledge supportfrom grants 94/MMI09008 (BBSRC) and GR/K92641 (EPSRC).APPENDIX: THE MATHEMATICAL MODELIn this appendix we present the mathematical model which wedeveloped to describe the evolution of a capillary network in responseto a nearby tumour. This model is an extension of the work ofAnderson and Chaplain [30]. The model focuses on three particularly important variables involved in tumour angiogenesis; namely,endothelial cells (EC), tumour angiogenic factors (TAF); e.g. VEGF,a-FGF, bFGF, angiogenin), and matrix angiogenic factors (MAF;e.g. ®bronectin, laminin). Each of these variables has a crucial role toplay in orchestrating the sequence of events which constitutes angiogenesis.In order to achieve the vascularisation of the tumour, the EC haveto make their way through the stroma and extracellular matrix whichconsists of various components including interstitial tissue, collagen®bre, ®bronectin and laminin. Various MAFs (e.g. ®bronectin, laminin) are known to enhance EC adhesion to collagen and are alsoproduced by EC themselves [37]. EC are known to migrate inresponse to diVerential adhesive gradients of MAF created in thesurrounding tissue. This type of motion is known as haptotaxis. TheEC also detect and respond to changes in the TAF concentration viacell-surface receptors and move up concentration gradients in thedirection of increasing concentration. This type of motion is knownas chemotaxis. In addition to these two types of directed motion, wealso assume that there is some random motion of the EC.The mathematical model which we derive considers the development of a vascular bed in response to a solid tumour in the breast(e.g. carcinoma). We consider the domain to be a small cube ofbreast tissue, the length of each side being 2 mm. Therefore, everypoint in the tissue has an (x, y, z) Cartesian co-ordinate triple givingits location in the breast tissue space. The concentration of TAF atany point in the breast tissue can thus be described as a function of itslocation in space and in time and we denote this symbolically by c (x,y, z, t). In a similar manner we write the endothelial cell densitywithin the breast tissue as n (x, y, z, t) and the concentration of MAFwithin the breast tissue as m (x, y, z, t). This mathematical description takes account of the fact that these concentrations and cell densities are dynamic, i.e. varying in space and changing with time. Wecan calculate the rates of change of these variables by using partialdiVerential equations. The mathematical model, therefore, consists ofthree coupled non-linear partial diVerential equations describing theevolution in space and time of EC density (random motion, chemotaxis and haptotaxis), TAF concentration (uptake by EC) and MAFconcentration (production and uptake by EC) and is given by:where D is the random motility coeYcient, 1 is the chemotacticresponse parameter and & is the haptotactic response parameter ofthe EC, respectively. is a measure of the TAF uptake rate by EC, and are measures of the MAF production and uptake rate by EC,respectively. Taking each equation in turn, the terms and equationsymbols have the following meanings:Equation 1: The ®rst term in the equation represents the rate ofchange of EC density with respect to time; the second term in theequation represents the random motility of the EC, i.e. movement ofEC in no particular direction; the third term represents the directedmigration of EC in response to gradients of TAF, i.e. chemotaxis, thestrength of this response measured by the parameter 1; the ®nal termin the equation represents the directed migration of EC to gradientsof MAF within the matrix, i.e. haptotaxis, the strength of thisresponse measured by the parameter &.Equation 2: the ®rst term of this equation represents the rate ofchange of TAF concentration with respect to time; the second termrepresents the amount of TAF that is being uptaken by the EC, withthe rate of uptake being measured by the parameter .Equation 3: the ®rst term represents the rate of change of MAFconcentration with respect to time; the second term represents theamount of MAF produced (synthesised) by the EC, the rate ofsynthesis being measured by the parameter ; the ®nal term represents the amount of MAF being uptaken by the EC, with the rate ofuptake being measured by the parameter .Parameter estimates were available from experimental observations for the random motility coeYcient D of the EC (10 À 9 cm2.s À 1.), and the chemotaxis coeYcient 1 (2600 cm2. s À 1. M À 1.)[38, 39].Does Cancer Exist in a State of Chaos?We solve the above system of mathematical equations using anovel adaptation of a standard numerical analysis technique used inobtaining the numerical solution of partial diVerential equationsÐtheso-called ®nite diVerence method. In essence we ®rstly discretise theabove system of partial diVerential equations to obtain a new systemof discrete equations, and then add an element of stochasticity (randomness). This is accomplished by solving the system of discreteequations via computer simulation on a spatially discrete cube atregular intervals of time. In eVect, this means that the2 mmÂ2 mmÂ2 mm breast tissue domain is divided into a meshwork of smaller cubes with sides of 10 microns. At this small lengthscale the system of partial diVerential equations is used to generate aset of probabilities for a single cell to move from one location in themeshwork to an adjacent location or to remain stationary. Theseprobabilities re¯ect the relative in¯uence of the local TAF and MAFconcentrations on the endothelial cells. For example, if there exists a891local concentration gradient around the cell from left to right (i.e.lower concentration to the left, higher concentration to the right)then the cell will be more likely to move to the right at the next timestep. Given the local in¯uences of the TAF and MAF concentrations,a cell then moves up, down, right or left, or remains stationary. Thisprocess of generating the cell motion is repeated at each time step ofthe simulation. We assume that the cell we are tracking is located atthe tip of a sprout and hence by following this cell as it moves inresponse to its surroundings, we in eVect simulate the growth of thewhole sprout since the remaining cells simply follow the path of theleading EC. In addition to the rules for cell movement we alsoincorporate rules for sprout branching, anastomosis and EC proliferation. A detailed description of the mathematical techniques andprocesses used can be found in [30, 38] and at the web-site: http://www.mcs.dundee.ac.uk:8080/$sanderso/angiocont.htm.