Improved Diffusion Rate with Triple Well Potential for Stem Cell

2015iMedPub Journalshttp://www.imedpub.com Vol. 4 No. 2:12

Journal of Biomedical SciencesISSN 2254-609X

Research Article

DOI: 10.4172/2254-609X.100012

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R. Nathiya1 and G. Sivaradje2

1 Research Scholar, Pondicherry Engineering College, Kalapet, Puducherry, India

2 ElectronicsandCommunicationEngineering, Pondicherry Engineering College, Kalapet, Puducherry, India

Corresponding author: R. Nathiya

Research Scholar, Pondicherry Engineering College, Kalapet, Puducherry -605014, India

nathiya.reshmi@gmail.com

Citation: Nathiya R, Sivaradje G. Improved DiffusionRatewithTripleWellPotentialforStemCellImageSegmentation.JBiomedicalSci. 2016, 4:2.

Improved Diffusion Rate with Triple Well Potential for Stem Cell Image Segmentation

IntroductionStemcellsareuniquecellsofthebodyinthattheyareunspecializedandhavetheabilitytodevelopintoseveraldifferenttypesofcells.Beforeinjectingstemcellstothepatients,firstwehavetoanalysewhetheritishealthyornot.Theyaredifferentfromspecializedcells,suchasheartorbloodcellsandcanreplicatethemselves.Theanalysisof stemcell includes segmentation, featureextraction,and pattern recognition. The preliminary process to find thegrowth rateof stemcell is segmentation.Themainaimof thispaperistoobtaintheimprovedsegmentationmethodtoacquirethepreliminaryprocess inordertoexaminethegrowthrateofstemcellsinfuture.Segmentationisoneoftheemphasisemethodinanimageprocessing.Itdealswithsectionalisationofanimageintomultiplepartstorecognizetheobjects likecolour,texture,grey value, etc. andalso image segmentation is theprocessofassigningalabeltoeverypixelinanimagesuchthatpixelswiththesamelabelsharecertaincharacteristics.Thesuccessofimageanalysisdependsonreliabilityofsegmentation,butanaccuratepartitioningofanimageisgenerallyaverychallengingproblem.Inimageprocessing,therearemanytypesofsegmentationlikethresholding, colour-based segmentation, transform method,

texturebasedsegmentation,partialdifferentialequation-basedmethod, etc. One of the partial differential equation-basedtechniques used here is called level setmethod. The level setmethod(LSM)isbasedonpartialdifferentialequations(PDE),i.e.activeevaluationof thedifferencesamongneighbouringpixelstofindobjectboundaries.Idyllically,theLSMwillconvergeattheboundaryoftheobjectwherethedifferencesarethehighest.LSMisusedasa tool for numerical analysis of surfaces and shapes. TheLSMmethodismainlyusedfortheimagesegmentationanditalsofinditsapplicationinimagescience.

History of Level Set MethodsIncontemporaryyears,thelevelsetmethodsfinditsapplicationinmanyfieldslikecomputervision,imageprocessing,computergraphics, fluid dynamics computational geometry andoptimization. The active contour based methods are used todealaboutvariousimagesegmentationproblems.Therearetwocategories of active contour such as parametric active contourmodel [1] and geometric active contourmodel [2]. Parametricactive contour models are as parametric curves described byLangrangian framework and geometric active contour modelsuse curve evolution theory and denoted as level set function(LSF). The level setmethodwas first introduced byOsher and

AbstractSegmentationof stemcells is a challengingprocess in imageprocessingwherelevelsetmethodisusedextensively.Thislevelsetcriterionwithre-initialisationprocessreducesitswiderrangeofapplication.Thusre-initialisationiseradicatedbyamethodcalleddistanceregularisationinwhichthesigneddistancepropertyismaintainednearthezerolevelsetviaforwardandbackward(FAB)diffusioneffect.Apotential function is habituated to sustain thedesired shapeof the level setfunction.Theinitialpotentialfunctionusedindistanceregularisationissingle-wellpotentialtechniquethatcausesexcessbackwarddiffusionandoscillationinLSF,soitissucceededbydouble-wellpotential.Butthisdoublewelltechniquelossesthecapabilitytoconservetheregularityatsomepoints.Thisproblemiseliminatedbyanewnovelmethodcalledimproveddiffusionratewithtriple-wellpotential.Theproposeddiffusionratecanmaintainthesigneddistancepropertyofthelevelsetfunction(LSF)andovercomethesideeffectofdouble-wellpotentialwithreducednumberofiterations.

Keywords: Levelsetmethod(LSM);Imagesegmentation;Re-Initialization;Forwardandbackwarddiffusion(FAB);Singlewellpotential;Doublewellpotential;Triplewellpotential

Received: June29,2015; Accepted: August04,2015; Published: August06,2015

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Sethian [3] in1987.Theconceptoflevelsetistosymbolizetheactive contour as zero level set which is a higher dimensionalfunctioncalledlevelsetfunction(LSF)anddevisethemotionofcontourastheevolutionoflevelsetfunction.Levelsetmethodsareabletorepresentthecontoursofmultifacetedtopologyandcanhandlethetopologicalvariationslikesplittingandmerginginefficientwaywhichcannotbedonebyparametricactivecontourmodel[1,4,5].Thechangesintheshapeandtopologyofcontourcanbeexpressedascurveandlevelsetevolution.Theevolutionofcurve[6]isexpressedas

( )C tâN

t=

∂∂

(1)

Where â is the speed function that controls the speed ofthe contour and

N is the inward normal vector of contourC(t). This C(t) can be described as zero level set function ,

( ) ( ) ( )C t x, y x, y, 0 | t = = , here 0φ > and 0φ < represent the external (positive value) and internal (negative value) of thecurve. The level setmethod is defined by the signed distancefunction(SDF)as

(x, y, t) d((x, y),C(t))φ = ± (2)

Whered((x,y),C(t))indicatesthedistanceofapointontheplanetothecurve. Ifthepoint locates insidethecurvethenthesign

ofthedistance isnegativeorelse it ispositive.φ (x,y,t)satisfies

1∇∅ = withthedefinitionofSDF.Thevariationrateofφ (x,y,t)isuniformanditisusefulfornumericalcalculation[7].Theinward

normalvectorinthecurveevolutionisgivenas∇φ

= −∇φ

N where

∇ is the gradient operator. By using this, curve evolution in equation(1)isexpressedintermsofpartialdifferentialequationas

∂φφ

∂= ∇â

t (3)

Theaboveequation ismentionedas level setequation.Duringthe evolution of contour due to the variation in velocity fieldof the imageplane thereoccur irregularitiesof LSF,asa resultthe stability of the level set evolution is ruined and producenumericalerrors.Togetridofthisproblemanumericalremedycalled re-initialisation [8,9] is used to restore the regularity ofLSFandmaintain stable level setevolution.The conceptof re-initialization is to stop the evolution instantly and remodellingthe disgraced LSF as a signed distance function [9,10]. The re-initialisationmethod ismainly used to elucidate the evolutionequation:

sign( )(1 )∂φ − ∇

∂=

ø øt

(4)

Where ( )φ istheLSFtobereinitialisedandsign ( )φ is the sign function.The solution for thisequationwill be signeddistancefunction. The re-initializationmethod [10,11] ismostly used inmanylevelsetmethodsandothertechniquewithfastmarchingalgorithm[8].Eventhoughthere-initializationisabletomaintaintheregularityofLSF,itmayerroneouslymovethezerolevelsettotheunwantedposition[8,9,12].

Invariationlevelsetmethod,thereisnoneedforre-initialisationinsteaditusesapenaltytermwhichwillcorrectthedeviationoftheLSFfromasigneddistancefunction[13].Thispenaltytermwill avoid the use of re-initialisation and also it is easy to getnumericalscheme.HoweverthiscausesanadversesideeffectontheLSFinsomepointsandaffectsthenumericalaccuracy.

To overcome the side effects, a new variation level set calleddistance regularization level set evolution (DRLSE) [14] usesdistance regularization termandanexternal energy term. Thismethodwillavoidthere-initialisationbyusingthepenaltytermaspotentialfunctionparticularlydouble-wellpotentialwhichwillmove gradientmagnitudeof the LSF. The regularity of the LSFismaintainedbytheforwardandbackward(FAB)diffusion[15].AnothermethodofDRLSEisproposedwhichusesanimproveddiffusion rate [16] to overcome the side effects in doublewellpotential. InSRLSE[17]thedistanceregularizationisappliedtotheimagesegmentation.DRLSEmethodisupgradedbyprovidingan improveddiffusionratewithtriplewellpotential technique.ThiswillavoidthesideeffectsinDRLSEmethodandincreasestheaccuracyandmaintainsthesigneddistanceproperty[18].

System MethodSystemdesigndescribesblockdiagramforDRLSEwithtriplewellpotential. The time-lapse series of stem cell image is hired asinputimageandthedistanceregularisationmethodisenforcedto segment the stem cell image. Here the triple-well potentialis instigated to improve the segmentation process and thesegmentedoutputiscomparedwiththepreliminarysingle-wellanddouble-wellpotential.Thethreepotentialsareexplainedinthedowncomingsections(Figure 1).

Input Image

Single-well potential

Double-well Potential

Triple-well Potential

segmented output with LSF

Figure 1 BlockDiagramofproposedwork.

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Input imageThetime-lapseimagedataofstem cells is chosen as an input image. Many experiments have been done for time lapse series [18]imagesegmentation;trackingandanalysisofthemorphologicalchanges of cells are the challenging tasks in imageprocessing.Stemcellsareverypowerfulcellsfoundinbothhumansandnon-humananimals.Stemcellsarecapableofdividingandrenewingthemselvesfor longperiods.Theyareunspecializedcellswhichcangiverisetospecializedcelltypes(Figure 2).

Single well potentialSegmentationistheprocessofpartitioningthegivenimagesuchthattherequiredareaofobjectcanbeobtained.Therearemanymethods for the segmentation process and here it deals withdistanceregularizationlevelsetevolution[15].

In level setmethods [3], there is aproblemof re-initialization.Toovercome thisproblemapenalty term [14] is introduced inthedistanceregularizationprocess.Thelevelsetenergyfunction( )φå onadomainΩ,isexpressed as

( ) ( ) ( )φ = φ + φp extå ì å (5)

Where, ( )φp is the level set regularization term ì is a constant and ( )φextå is the data term that depends on the image data. The level set regularization term is given by

( ) ( )φ ∇φ∫p p dxΩ

(6)

Wherep:[0,∞]isapotentialfunctionwhichactsasthepenaltyterm and it [14] is given as

( ) ( )21

1 12

= ∆ −

p p s s (7)

Whichhasoneminimumpoints=1i.e.singlewellpotential.Thelevelsetregularisationtermwiththispotentialisexpressedas

( ) ( )21 12

φ ∇φ −∫ dxΩ

(8)

Thisisconsideredasthepenaltyterminourprimarywork [14,17].Thispotentialfunctionmaintainsthesigneddistancepropertyin

thewholedomainbutithassomesideeffectsinLSFφ at some positions.

Double well potential

To avoid the side effects in the single well potential, Li [15]proposedadifferentvariationmethodinwhichanewpotentialfunction ( )2p s isproposedwhichisdefinedas

( ) ( )( )

( )2

2

1 1 cos 2 , 12 2

1 1 ,(s)

12

− ≤ − ≥

=ðs if s

ðp

s if s (9)

Therearetwominimumpointsinthispotentialfunction ( )2p s andtheyares=0ands=1.Thefirstderivativeforthispotentialisderived as

( )( )2

1 sin 2 , 12

1,(

1s)

≤ − ≥

=' ðs if sp ð

s if s (10)

Thisdoublewellpotentialisabletomaintainthesigneddistanceproperty 1∇φ = butwhen 0.5∇φ < ,itapproachesto0insteadof1whichwillaffecttheregularityoftheLSF.

Though the double-well potential has many advantages overpreliminary work, it has certain drawbacks due to which thistechnique lags good results. If the initial LSF signed distancefunction ∇φ is smaller than 0.5 on all LSF domain, the DRLSE with ( )2pd s fails tosegmenttheobjectaccuratelybecausethedistanceregularizationwith ( )2pd s losestheabilitytoregularizethe LSF for 0.5∇φ < andmakes ∇φ to approach 0 instead of 1whichleadstoundesirablesideeffectstotheevolution.

Triple-well potential for distance regularizationTheenergyfunction ( )φ isproposedasapenaltytermininitialworkinanefforttomaintainthesigneddistancepropertyintheentire domain. The consequent level set evolution for energyminimizationhasanundesirablesideeffectontheLSFφ in some circumstances.Toavoidthissideeffect,anewpotentialfunctionin thedistanceregularizationtermp is introduced.Thisnewpotential function is aimed to maintain the signed distanceproperty 1∇φ = only in an area of the zero level set, whilekeepingtheLSFconstantwith 0∇φ = atlocationsfarawayfromthezerolevelset.

TomaintainsuchaprofileoftheLSF,thepotentialfunctionmusthaveminimumpointsats=2,s=1ands=0.Suchapotential isatriple-wellpotentialasithasthreeminimumpoints(wells).Usingthis triple-well potential p=p3 not only avoids the side effectthatoccurs in thecaseofp=p2,butalsooffers some temptingtheoreticalandarithmeticalpropertiesofthelevelsetevolution.

This potential functionmaintains the signed distance functionshapearoundtheSDB(SignedDistanceBand)withflatoutsidetheband.Theconstantc0willcontrolthewidthoftheSDB,becausewhenthefunction φ becomesasigneddistancefunctioninthe

Figure 2 Stem Cell Image.

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SDB, itsvaluesvaryfrom-C0 to 1∇φ =acrossthebandatarateof1∇φ = . This implies that thewidthof theSDB isapproximately

2co.Thepotentialfunctionisderivedas

( ) ( )( )

( )23

1 1 cos 2 12 2

1 1 12

0.5

(s)

s 1,

=

− ≤

− ≥

<

−

,

ðs , if sð

p s if s

if s

Thederivativeoftheaboveequationisgivenas

( ) ( )( )'3

1 sin 2 12

1(s)1, 0.5

≤ ≥ <

=

s - 1,

ðs , if sð

p if s if s

The above equation is used to find the regularization effect intheLSF.Herethethreeminimumpointss=0,s=1,s=0.5areusedwhich is calledas triplewellpotential.Thecondition is chosenass-1ifs<0.5sincethegradientvalues-1willbe1sothattheedgecanbesegmentedmoreclearly.Byusingthispotential,LSFismaintainedbelow0.5valuesof∇φ .Thusthesigneddistancepropertyismaintainedinthedesiredlocations.

Result and AnalysisThe level set function of the desired region is obtained byimplementationofdistanceregularisationeffectasfollows:

Step1:Parametersetting

Step2:SmoothimagebyGaussianconvolution

Step3:InitializeLSFasbinarystepfunction

Step4:Obtaintheinitialzerolevelcontour

Step 5: Choose the potentialfunction(single,doubleortriple)

Step6:Startlevelsetevolution

Step7:Obtainthefinalzerolevelcontour

Step8:Display thefinal zero levelcontour inmeshtogetfinallevelsetfunction.

Theparameterslikeµ,α,λareselectedfortheimplementation.ThentheimageissmoothenbyusingtheGaussianconvolutionandthenappliedtothesegmentationprocess.Theinitiallevelsetfunctionisinrectangularform.Thepotentialfunctionischosenaseithersingle-well ( ) ( )2

1 0.5 1= −p s * s ,whichisgoodforregion-basedmodelordouble-well(eqn.8,whichisgoodforbothedgeand region basedmodels) or triple-well (eqn.10,which is alsogoodforbothedgeandregionbasedmodels).Theaccuracyofthe triple-well potential is betterwhen compared to theothertwo functions. This can be proved by using the precision and recall values.

Single-well potential analysisIn the singlewellpotential,thepotentialfunctionisusedtochoosethesingleminimumpoint.Thebinarystepfunctionisusedastheinitial functiontostimulate thedistanceregularizationeffect.In the interiorof theregions 0 0φ = c and 0 0φ = −c thevalueof

0∇φ = forthebinaryinitialfunction.So,thesinglewellpotentialhasbackwarddiffusion intheFABdiffusionwith largediffusionrate. The strong backward diffusion significantly increases ∇φandcauseoscillationinφ ,whichfinallyappearsasperiodiccrestsand dales in the convergedLSF.ThebelowFigure 3 (a,b) showsthe segmentation and their corresponding LSF it is clear thatthereismorepeaksandvalleysduetotheoscillations.

Figure 3a SegmentedImageofStemCellUsingSingle-WellPotential.

Figure 3b LevelsetfunctionofSegmentedImage.

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Double-well potential analysisThediffusionrateisincreasedbyusingthedoublewellpotentialin which the diffusion rate is maintained by using both theforward ( )1∇φ < and backward ( )1∇φ < diffusion so that thesigned distance property is maintained and the oscillationsoccurred in the diffusion rate with 1=p p is eliminated. Thesegmented output of double-well is more accurate than thesingle-wellpotential.ThiscanbeclearlyseenfromtheFigure 4 (a, b)convergedLSFgivenbelow.

Triple-well potential analysisIn double-well potential, the signed distance function below0.5willdecreasethevaluetozerobutitshouldbeonesothatthe signed distance property is not achieved. This problem isovercomebyaddinganotherpotentialcalledtriple-wellpotentialwhich will increase the diffusion rate by choosing anotherminimum point.

The accuracy is also increased and the number of iterations isreducedby increasing thetimegap. The segmentedoutput is soclearthanthepreliminaryworkthiscanbeseenbytheFigure 5(a, b).

Theachievementofsigneddistancefunctioninthesigneddistancebandbyusingtriple-wellpotentialisclearlyobserved.The Figure 6(a) showsthatthesigneddistancefunctionisnotattainedeveninthezerolevelsetwhichwillleadtotheboundedness.Byusingthe ( )2pd s , Figure 6(b)thesigneddistancefunctionisachievedabovethe0.5valuebutfailed inalldomainsandzero levelsetis also missing. Figure 6(c) representsthedistanceregularisationresult of ( )3pd s . The final LSF shows that the signed distanceproperty is obtained in all domains by using three minimumpoints (wells) and also it possess the signed distance functionshape in the signed distance band and flat outside the bandwhichiscontrolledbytheconstant 0c .

Table 1 illustrates that the precision, recall and PSNR value ofTriplewellpotentialisgraduallyimprovedwhencomparedtothesinglewellandDoublewell.Becausethismethodhastheabilitytomaintainthesigneddistancepropertyevenwhensigneddistancefunction is less than0.5.And itdisables thehurdlespresent inthe previous technique. Duetoitsfastprocessingschemetheremaybechancesofoversegmentation.Oversegmentationcanbeavoidedinfuturebycontrollingthedirectionofevolutioninlevelsetmethodbyintroducingballoonforces.

Figure 4a SegmentedImageofStemCellUsingDouble-WellPotential.

Figure 4b LSFofSegmentedImageofStemCellUsingDouble-WellPotential.

Figure 5a SegmentedImageOfStemCellUsingTriple-WellPotential.

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Monitoring cell confluence is valuable for optimization of cellculture parameters to provide efficient approach for detectionofnecrosis. Successfuloutcomeof clinical studiesareessentialandnextstepistoimplementinnovativetechnologicalsolutionto deliver affordable clinical results to artificially reconstructhumantissuewhichhasbeenpreviouslydamaged.Thecellsareextractedfromthepatientsandtheyaregrownundercontrolledin vitro condition in lab so that they canmultiply and lead togrowwithsomeadditionofproteinsorgrowthhormonestotheculturemediumandthen implant to thepatients.Using imageprocessing on stem cell the future healthy parameters of the

stem cell is analysed and viewed through results without anypracticalneedsbutinclinicalapproachitisquitedifficultandalsoitneedsmoretimefortheprocess.Computerizedanalysisgivespriorinformationforclinicalapproachtopredictthehealthinessofstemcellbyitsgrowthrate.

ConclusionThustheoscillationsindistanceregularizationeffectusingsingle-wellpotential and the sideeffectsofdouble-wellpotential areovercome by using the triple-well potential. The segmentationof thetime lapseseries imageofstemcell isdoneproperlybyusingthetriple-wellpotentialdistanceregularizationeffect.The

Figure 5b LSFSegmentedImageOfStemCellUsingTriple-WellPotential.

Figure 6 Effectofdistanceregularisation,(a)crosssectionofLSFusingsingle-wellpotential,(b)crosssectionofLSFusingdouble-wellpotential,(c)crosssectionofLSFusingtriple-wellpotential.

Stem cell image Single well potential Double well potential Triple well potential

Hours Precision Recall PSNR Precision Recall PSNR Precision Recall PSNR0 76.62 93.71 32.14 80.42 94.71 38.3 81.98 96.25 45.15

24 81.62 9.27 34.18 82.42 94.27 40.15 83.98 95.81 46.4048 83.62 92.84 35.15 84.42 93.84 40.94 85.98 95.38 46.9172 85.62 9.41 35.73 86.42 93.41 41.40 87.98 94.95 47.1996 87.62 91.97 36.12 88.42 92.97 41.70 89.98 94.51 47.37

120 89.62 91.54 36.40 90.42 92.54 41.91 91.98 94.08 47.49144 91.62 91.11 3.61 92.42 92.11 42.07 93.98 93.65 47.58168 9.62 90.68 36.78 94.42 91.68 42.19 95.98 93.22 47.45

Table 1.Performanceanalysisofsinglewell,doublewell,triplewellpotentialsegmentation.

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signeddistancefunctionisachievedinthezerocrossingandthesigneddistancepropertyisachievedinthesigneddistanceband.Theexperimental analysis clears that theaccuracy is increasedbythismethodandthenumberof iterationsalsoreduced.Thelevelsetfunctionisregularandthesigneddistanceproperty is

alsoachievedusingthetriple-wellpotential.Hencetheimprovedsegmentationmethodachieveshighprecision, recall andPSNRvalues.Thisproposedtechniquepayswayfortheexplorationofgrowthrateandhealthinessofstemcells.

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