sarah atchison dr. michael pilant - texas a&m...

SarahAtchison

Dr.MichaelPilant

MATH614.700

13May2009

TheRoleofFractalGeometryintheBiologicalSciences

Introduction

Fractalscanbeseenallthroughoutthenaturalworld.Infact,“wherevera

chaoticprocesshasshapedanenvironment,afractalstructureisleftbehind”

(KenkelandWalker1996).FromthecoastlinesofGreatBritaintotheperimeterof

acancercell,fractalsareapparentinwaysMandelbrotneverthoughtpossible.In

thepast,scientiststriedtouseclassicEuclideangeometrytoanalyzegeographical

structuresandbiologicalsystems.However,wehavesincerealizedthatamuch

morecomplextypeofgeometry,fractalgeometry,isneededtocompletethesetasks.

Euclideangeometrydoes“notyieldfiniteanswersforcertainobjects”inthenatural

world(BauerandMackenzie1995).KenkelandWalker(1996)notethatthe

“importanceoffractalscalinghasbeenrecognizedatvirtuallyeverylevelof

biologicalorganization”.Probablyoneofthegreatestobservationsoffractal

geometryinbiologyisthat,byanalyzingthefractaldimensionsoftumorsandcells,

scientistswillbeabletodetectcancergrowthinpatientsearlierandinamore

efficientmanner.Theeffectsofresearchingfractalsinbiologicalsystemsofall

levelsare,inasense,limitless.

SarahAtchisonTheRoleofFractalGeometryintheBiologicalSciences

2

SurveyofRelatedWorks

BenoitMandelbrotfirstdemonstratedtheconceptoffractaldimensionby

describingmeasuringthecoastlineofGreatBritainwitharuler.Asthelengthofthe

rulerapproacheszero,theperimeterofGreatBritainapproachesinfinity.Studies

showthatislandswithsmallerperimetershavealowermeanD,fractaldimension,

thanislandswithalargerperimeter.MandelbrotusedthedataLewisRichardson

hadfoundstudyingtheslopesofcoastlinestofindtheequationneededtocalculate

thefractaldimensionofanygivencoastline.Theequationis

N(d)=M/dD

whereN(d)isthenumberofsegmentsoflengthdneededtomeasuretheperimeter,

Misaconstant,andDisthefractaldimensionofthecoastline.

Notonlydocoastlineshavefractaldimension,sodoclouds.Whenflyingin

anairplane,“acloudtwentyfeetawaycanbeindistinguishablefrom[acloud]two

thousandfeetaway(Gleick,1987).Fractalpropertiesarealsoveryapparentin

plants.Fractaldimensionofleafedgesinplantscanbehighlyvariableinsome

species.Oaksareknowntobeanexampleofonesuchspecies.However,fractal

dimensionisthoughttobeapossibletoolfortaxonomicallyclassifyingaplant.Root

systemsalsohavefractaldimension.Rootsystemshave“beenexaminedusingthe

box‐countingmethod”andDshowstoincreaseovertime,varyingbetweenspecies

(KenkelandWalker,1996).Area‐perimeterrelationshipsarealsousedtoexamine

fractalpropertiesofmountains.Thesurfacesofrealmountainshavebeensculpted

byweatherandatmosphericchanges;however,researcherscanoftenmanufacture


3

thesameeffectsthemselvestorecreateandanalyzethesamefractaldimensionsof

mountains(Frameetal,2001).

Bacterialcolonies,liketreesandferns,growinafractalmannerunder

stressedconditions.Normally,bacteriagrowsinsoftagarwithabundantnutrient

concentration,leadingtosoftedgesandcompactness.Conversely,ifthebacteria

growsinstressfulconditionswheretheagarishardandnutrientsaremorescarce,

thecoloniestakeonafractalgrowingpatternresemblingthatofDLA(Diffusion‐

LimitedAggregation)clusters.DLAisacomputersimulationthatformsclustersby

“particlesdiffusingthroughamediumthatjostlestheparticlesastheymove”

(Frameetal.,2001).Tmorphotypecoloniesareanexampleofthisbehavior(see

Figure1).WhenTmorphotypecoloniesgrowinenvironmentswherenutrientsare

lowandagarissoft,Cmorphotypecoloniesarisethroughchiralgrowth(seeFigure

2).Consequently,whenCmorphotypesgrowunderextremelystressedconditions,

Vmorphotypesarisethroughvortexgrowth(seeFigure3).Bacteriaisnottheonly

organismthatfollowsthissamebehaviorinaharshenvironment;fungusalsogrow

fractallywhennutrientconcentrationsaredecreasing(seeFigure4).

Fractalscalingisfoundinvarioussystemsinhumananatomy.DNA

sequencesdemonstrateself‐similarityandthe“twistingsofDNAbindingproteins

havefractalproperties(KenkelandWatson,1996).Chromosomalstructure

consistsofa“concatenationof‘mini‐chromosomes’”,makingittree‐like(Kenkeland

Watson,1996).EvolutionarybiologistscanusethefractalpropertiesofDNAto

characterizeandclassifyrelationshipsbetweenorganismsthroughthemultifractal

spectra.Researchershavefoundthatcellsofalltypeshavefractalproperties.


4

Fractaldimensionhasbeenusedtomeasurecellularcomplexityandcontour

complexityofimagesofneuralcells(KenkelandWatson,1996).Thereiseven

fractalbehaviorpresentinthehumaneye.Scientistshavefoundthisbehaviorwhen

studying”homeostasisinthecornea,maintainedbyosmoticintegrityprovidedby

thecornealepithelium”(IannacconeandKhokha,1996).

Proteinandenzymeshavebeenextensivelystudiedfortheirfractal

propertiesanddimension.Enzymologistsandbiologistsstudyingmacromolecules

areactivelyresearchingfractalsinenzymesandproteins.Thefractalanalysisof

proteinshasproducedapowerlawintheformof

p~vα

wherepistheproperty,visthevariable,andαistheexponent.Variablesvandα

canberelatedtotheoreticallyorexperimentallyobtainedfractaldimensions.

Proteinsarefatfractals“sincethesurfaceoftheproteinisafractalbutatfinite

volume”(IannacconeandKhokha,1996).Liebovitchandhiscolleaguesexamined

the“kineticsofproteinionchannelsinthephospholipidbilayer”findingthatthe

fractalpropertiesofthe“timingofopeningsandclosingsofionchannels”meant

that“processesoperatingatdifferenttimescalesarerelated,notindependent”

(KenkelandWatson,1996).Proteinfractalshavebeenusedincreatingsynthetic

materials.Keratin,themainproteinfoundingoosedown,wasfoundtohavefractal

nodesandbranchesthatarethekeytodown’sabilitytotrapair(Gleick,1987).

Fractalpropertiesarefoundinthemanydichotomousbranchingsystemsof

thehumanbody,namelytherespiratory,digestive,andcirculatorysystems.Fractal

scalinginphysiologymeans“moresurfaceareaforabsorptionandtransfer”(Frame


5

etal,2001).Thelungsdemonstratemorefault‐toleranceduringgrowthduetothe

fractalscaling.Theyneedtohavetheabilitytofillupthebiggestsurfacepossible

intothesmallamountofspacetheyaregiven.Thesurfaceareaofananimal’slungs

isapproximatelyproportionaltoits“abilitytoabsorboxygen”(Gleick,1987).

Humanlungshaveasurfaceareaaboutthesizeofatenniscourt.Tissuesinthe

digestivetractshowfoldswithinfolds,givingthedigestivesystemitsfractalnature.

Theurinarycollectingsystemandthebiliaryductarealsoexamplesoffractal

structures.

Thecirculatorysystemisprobablythemostfractallycomplexsysteminthe

body.Liketherespiratorysystem,itmustfitanenormoussurfaceareaintoavery

tinyvolume.Bloodvesselsdivideandbranchoffsomanytimesthatthevesselsare

narrowenoughthebloodcellscanonlygothroughonebehindtheother.Inmost

tissues“nocellisevermorethanthreeorfourcellsawayfromavessel”,soitis

amazinghow“vesselsandbloodtakeupnomorethanaboutfivepercentofthe

body”(Gleick,1987).Theheartisfilledwithfractalnetworks.Examplesofthese

networksarethecoronaryarteriesandveins,thefibersbindingthevalvestothe

heartwall,andthecardiacmusclesthemselves.TheHis‐Purkinjesystem,”a

networkofspecialfibersinthehearthatcarrypulsesofelectriccurrenttothe

contractingmuscles”,iskeyinthefractalfrequencyspectrumofheartbeattiming

anditselfhasfractalproperties(Gleick,1987;Frameetal,2001).Theiterationof

fractalstructuresinthecirculatorysystemmakeitsostrongagainstinjurythatthe

heart“continuestofunctionevenaftertheHis‐Purkinjesystemhassuffered

considerabledamage”(Frameetal,2001;KenkelandWatson,1996).Theoretical


6

biologistshaveconcludedthatfractalscalingis“universalinmorphogenesis”

(Gleick,1987).

Manydiseasescausecellstofollowfractallaws.Histopathologistshave

turnedtofractalgeometryandmicroscopicimageprocessingtomoreeffectively

studythefractaleffectsdiseaseshaveontissueandcellgrowth.Usingthese

techniques,histopathologistscanmoreaccuratelygivediagnosesandprognoses.A

majorproblemhistopathologistshavewithdiagnosisisuniformity.Notonlywill

fractalgeometryaidinachievingartificialintelligence‐baseddiagnosis,itmightalso

provide“insightintosomediseaseprocesses”(IannacconeandKhokha,1996).Not

onlydosuchdiseasesastheherpessimplexvirushavefractalproperties,butsodo

most,ifnotall,typesofcancer.

InJanuaryof1998,itwasreleasedthatDrAndrewEinsteinandhis

colleaguesattheMountSinaiSchoolofMedicinehadbeenanalyzingthefractal

patternsofbreastcancercells.Theseresearchersanalyzedthenucleiofthesecells

andthedistributionandfractalpatternsofthechromatininside.Bymeasuring

differenceinlacunarityandbyexaminingthedifferencesinfractaldimension

betweenbenignandmalignantcells,theywereabletodeliverthecorrectdiagnosis

to39outof41patientsinablindstudy(ScheweandStein,1998).

Outlinesoftumorsrevealthelocalgrowthbehavioroftumorsto

histopathologists.Benigntumorshave“expansive”,smoothoutlineswhile

malignanttumorshave“localaggressivefeatureswithinvasionofsurrounding

tissue”(IannacconeandKhokha,1996).Theinvadingedgesaretypicallyirregular

andfragmented“withdetachmentofislandsfromthetumor”(Iannacconeand


7

Khokha,1996).Histopathologistshaveattemptedtoquantifythesepatterninto

classificationcategoriestoincreaseprognosisinoral,esophageal,andlaryngeal

carcinomas(IannacconeandKhokha,1996).Scientistsresearchedcanceroustumor

growthinmiceandfoundthatthetumorvesselsyieldedfractaldimensions1.89±

0.04whilethenormalvesselshaddimensionsof1.70±0.03.Tumorvesselshave

manysmallerbendsinsideoflargerbends.Beingableto“quantifytheirregular

structuresthatarepresentintumorshelpstoclarifywhytreatmentisso

frustratinglydifficult”(BaishandJain,2000).Thereare,however,limitationsto

applyingfractallawstocancerresearch.Whileincreasedfractaldimensionis

commoninmalignanttumorgrowth,itisnotuniversal.Normalbonemarrowcells

exhibitanincreasedfractaldimensionwhilethemalignantcellshaveadecreased

fractaldimension(BaishandJain,2000).

SummaryandConclusions

Fractalsaretheresultofchaoticprocesses.Notonlyarefractalsfoundin

mathematics,butalsointhereal,naturalworld.Fractalgeometryhasoftenbeen

dubbed“thegeometryofnature”,butitseffectsstretchmuchfurtherintobiologyas

well.Notonlydocoastlines,mountains,clouds,andplantshavefractaldimension,

thereismorecomplexfractaldimensiontobefoundinthebloodvessels,alveoliof

thelungs,andcellsinthehumanbody.Studyingfractalpropertiesofbiological

systemshasleadtomanyrevolutionarydiscoveries.Amongtheseepitomesisthe

knowledgethatfractaldimensioncanassistoncologistsindetecting,andtherefore

treating,severaltypesofcancerearlierandmoreeffectively.Ifsuchconclusionscan

befoundfromstudyingthesefractalsnow,withthecurrenttechnology,whatdoes


8

thefutureholdwithnewtechnologyandcontinuedstudyintofractalgeometryin

biologicalsciences?

MATLAB

Afterreading“CancerDetectionviaDeterminationofFractalCellDimension”

byWolfgangBauerandCharlesD.Mackenzie(1995),Idecidedtotrytoreplicate

theirfindingsfromtheirresearchofhumanlymphocytesaffectedbyhairy‐cell

leukemiausingthebox‐countingmethodandMatlab.PaulFrenchwrotetheMatlab

codeIusedandmodified.Itusesthebox‐countingmethodtofindtheHausdorff

dimensionofuploadedimages.IusedittofindtheHausdorffdimensionofan

“electronicmicroscopeimageofasectionofahumanlymphocyte,affectedwith

hairy‐cellleukemia,digitizedwith256greylevels”(BauerandMackenzie,1995).

ThearticleBauerandMackenziepublishedalreadyhadanimageoftheperimeterof

thelymphocyte,soIuploadedthatimagetopreventpossibleerrorsinconverting

theoriginal(Figure5)intothedesiredgrayscaleimage(Figure6).Ithenranthe

programandcalculatedtheHausdorffdimensiontobe1.3041(also,seefigures7‐

9).BauerandMackenzie’smethodismoreaccurateandcanbeusedonmanytypes

ofcellsororganisms,couldpossibly“deriveaquantitativemeasureforthe

raggednessofcellsorsmallbiologicalorganisms”,andalso“showeditispossibleto

distinguishbetweenhealthypersonsandhairy‐cellleukemiacancerpatients”

(BauerandMackenzie,2000).


9

Graphics

Figure1:Tmorphotypecoloniesunderincreasingstressedconditions

Nutrient=15(g/l),Agar=2.25%Nutrient=4(g/l),Agar=2.25%

Nutrient=2(g/l),Agar=1.75%Nutrient=0.01(g/l),Agar=1.75%


10

Figure2:Cmorphotypesinstressedenvironments

Figure3:Vmorphotypesinstressedenvironments

Figure4:Fungalgrowthinstressedenvironments

Figure5:Cancerpg2fig1a.jpg


11

Figure6:Cancerpg2fig1d.jpg(PerimeterofFigure1)

Figure7:DetectionoftheedgeofCancerpg2fig1d.jpg

Figure8:RawDataPlotwithBestFitLine


12

Figure9:CalculationofHausdorffDimension

AppendixA–MatlabCode

Fractaldimensionmeasurement.m(withinputfilenameschanged)%%%%%%%%%%% % this is used to find the hausdorff dimension via the box counting method % email: [email protected] % web: www.ee.ucl.ac.uk/~pfrench %%%%%%%%%%% clear clc table =[,2]; % load up original image and convert to gray-scale p = imread('Cancerpg2fig1d.jpg'); %p = rgb2gray(P); figure(1) imshow(p) % detect the edge of image 'p' using the Canny algorithm % this gives edge as 'e2' bw = im2bw(p, graythresh(p)); e = edge(double(bw)); fi = imfill(bw, 'holes'); op = imerode(fi,strel('disk',4)); e2 = edge(double(op)); figure(2)


13

imshow(e2) % once we have e2, set up a grid of blocks across the image % and scan each block too see if the edge occupies any of the blocks. % If a block is occupied then flag it and record it in boxCount -- % store both size of blocks (numBlocks) and no of occupied boxes (boxCount) % in table() Nx = size(e2,1); Ny = size(e2,2); for numBlocks = 1:25 sizeBlocks_x = floor(Nx./numBlocks); sizeBlocks_y = floor(Ny./numBlocks); flag = zeros(numBlocks,numBlocks); for i = 1:numBlocks for j = 1:numBlocks xStart = (i-1)*sizeBlocks_x + 1; xEnd = i*sizeBlocks_x; yStart = (j-1)*sizeBlocks_y + 1; yEnd = j*sizeBlocks_y; block = e2(xStart:xEnd, yStart:yEnd); flag(i,j) = any(block(:)); %mark this if ANY part of block is true end end boxCount = nnz(flag); table(numBlocks,1) = numBlocks; table(numBlocks,2) = boxCount; end % from the above table of discrete points, take a line of best fit and plot % the raw data (ro) and line of best fit (r-) x = table(:,1); % x is numBlocks y = table(:,2); % y is boxCount p = polyfit(x,y,1); BestFit = polyval(p,x);


14

figure(3) hold on grid on plot(x,y, 'ko','LineWidth',1) plot(x,BestFit, 'k-','LineWidth',2) xlabel('Number of blocks, N','FontSize',12) ylabel('Box Count, N(s)','FontSize',12) % calculate Hausdorff Dimension x2 = log(x); y2 = log(y); p2 = polyfit(x2,y2,1); BestFit2 = polyval(p2,x2); figure(4) hold on grid on plot(x2,y2, 'bo','LineWidth',1) plot(x2,BestFit2, 'b-','LineWidth',2) xlabel('Number of blocks, log N','FontSize',12) ylabel('Box Count, log N(s)' ,'FontSize',12) HausdorffDimension = p2(:,1)

AppendixB–References

Baish,J.W.,&R.K.Jain(2000).FractalsandCancer.CancerResearch.60,3683‐3688.Bauer,W.,&C.D.Mackenzie(1995).Cancerdetectionviadeterminationoffractalcelldimension.RetrievedApril20,2009,fromhttp://www.pa.msu.edu/~bauer/cancer/cancer.pdfFrame,M.,B.Mandelbrot,&N.Neger(2001).Fractalgeometrypanorama.RetrievedMay11,2009,fromFractalgeometryWebsite:http://classes.yale.edu/Fractals/Panorama/welcome.htmlFrench,P.(2007,August13).MATLABcentral‐Filedetail‐Hausdorffdimensionbytheboxcountingmethod.RetrievedMay9,2009,fromMATLABCentralWebsite:http://www.mathworks.com/matlabcentral/fileexchange/15918Gleick,J.(1987).Chaos:Makinganewscience.NewYork,NY:PenguinBooks.


15

Iannaccone,P.M.,&M.Khokha(Ed.).(1996).Fractalgeometryinbiologicalsystems:Ananalyticalapproach.BocaRaton,FL:CRCPress,Inc.Kenkel,N.C.,&D.J.Walker(1996).Fractalsinthebiologicalsciences.Coenoses,11,RetrievedApril20,2009,fromhttp://www.umanitoba.ca/faculties/science/botany/LABS/ECOLOGY/FRACTALS/fractal.htmlSchewe,P.F.,&B.Stein(1998,January,05).InsideScienceResearch.PhysicsNewsUpdate,353,RetrievedApril20,2009,fromhttp://www.aip.rg/pnu/1998/physnews.353.htm

sarah atchison dr. michael pilant - texas a&m...

Documents