Shape Modeling in CellOrganizer€¦ · Shape Modeling in CellOrganizer Robert F. Murphy Ray & Stephanie Lane Professor of Computational Biology and Professor of Biological Sciences,

ShapeModelinginCellOrganizerRobertF.Murphy

Ray & Stephanie Lane Professor of Computational Biology and Professor of Biological Sciences, Biomedical Engineering and Machine Learning

External Senior Fellow, Freiburg Institute for Advanced Studies Honorary Professor, Faculty of Biology, University of Freiburg, Germany

AnNIHBiomedicalTechnologyResearchCenter

March9,2018

Methodsformodelingcellshape•  Parametric

– Outline–  Ratio-metricrelativetonuclearshape

•  Nonparametric– Diffeomorphic– Autoencoder

Parametricshapeoutlinemodels

Kerenetal.

2008

Imagessh

owing

realsh

apes

Gene

rativ

e

shapemod

el

Shapespace

Kerenetal.2008

Limitationsofcommonoutlinemodel

Srivastavaet

al.2005

Cell shape: eigenshape ratio model

•  Conditionedonnuclearshape:–  Sampleevenlyaroundacircletorepresenttheshapebyradiusratios

–  Parameterization

–  Keep10principalcomponents(2D)–  Keep25principalcomponents(3D)

1 2/r d d=

∑ =+≈

k

i iib1 φrr d1 d2

However…•  Cellsdon’talwayssatisfyassumptionsofparametricmodels.

SegmentedPC12cell Star-polygonratiomodelrepresentation

LDDMM-LargeDeformationDiffeomorphicMetricMapping

Whatisadiffeomorphism?•  Essentiallyasmoothandinvertiblemappingfromonecoordinatespacetoanother

Adiffeomorphicmappingfromaregularrectangulargrid.

https://en.wikipedia.org/wiki/Diffeomorphism

Diffeomorphicmappingsofcontinentstoa2Dprojectionofaglobe

http://wwwx.cs.unc.edu/~mn/classes/comp875/doc/diffeomorphisms.pdf

Adiffeomorphicmappingfromoneimagetoanother.http://wwwx.cs.unc.edu/~mn/classes/comp875/doc/diffeomorphisms.pdf

Nonparametricshapeimage-basedmodels

Pengetal.2009

Real2Dnuclearshapes

http://alumni.media.mit.edu/~maov/classes/comp_photo_vision08f/

Cannotjustinterpolateimagesasiftheywerevectors

Morphingtointerpolateimages

12http://alumni.media.mit.edu/~maov/classes/comp_photo_vision08f/

ShapeBShapeAWorksofar

0.0191

Distancebetweentwoshapes

0.01650 0.0194 0.0195

Distance

Pengeta

l.2009

0.01650 0.0191 0.0194 0.0195

Distance

IterativereductionindifferencebetweendeformedshapeAandB

LDDMM-LargeDeformationDiffeomorphicMetricMapping

•  Minimalenergytransformationwithrespecttothegradientofthedeformationfieldi.e.Geodesicdistance

•  Deformationfieldisanonlinearmanifoldthatcontainstheinformationoftheimage,includinggradient,secondorderderivationetc.Theupdateisageodesicpathonthemanifold.

Shadel1974 http://wwwx.cs.unc.edu/~mn/classes/comp875/doc/diffeomorphisms.pdf

f1.png

0 0.2 0.4 0.6 0.8 1

Constructingacellshapespace•  Finddistancesofeverycelltoeveryothercell

•  Trytofinda“map”thatputseachcellthecorrectdistancefromtheothers(i.e.,putscellswithshortdistancesneareachother)

Distancematrix…BOS CHI DC DEN LA MIA NY SEA SF

BOS 0 963 429 1949 2979 1504 206 2976 3095CHI 963 0 671 996 2054 1329 802 2013 2142DC 429 671 0 1616 2631 1075 233 2684 2799DEN 1949 996 1616 0 1059 2037 1771 1307 1235LA 2979 2054 2631 1059 0 2687 2786 1131 379MIA 1504 1329 1075 2037 2687 0 1308 3273 3053NY 206 802 233 1771 2786 1308 0 2815 2934SEA 2976 2013 2684 1307 1131 3273 2815 0 808SF 3095 2142 2799 1235 379 3053 2934 808 0

http://personality-project.org/r/mds.html

…tocoordinates

http://personality-project.org/r/mds.html

Shapespacesmodeljointdistributionacrossmorphologicalfeatures

19

Shapespace

DiffeomorphicTraining

ShapestoSpace

MDS

Butthistakesalotoftime

PartialDistanceMatrixLearning

•  Mostcompleteshapespace

MDS

ShapeSpaceEmbedding•  Thisembeddingmethodrequiresallpairsofdistances•  Let’ssaywehave250cellsandittakes~30sectoregisterapair•  (30sec*2502)/2≈10days.Waytoolong…..•  canweinferembeddingwithmissingdata?•  MDSwithmissingdata

wherewi,jistheweightofobservationDi,j

PartialDistanceMatrixLearning•  LandmarkMDS

?

ApproximateMDS

DiffeomorphicSynthesis

SpacetoShapes

?

Synthesisstrategyfornewpoints

Modelingthedistributionofshapes•  Theshapespacedefinesanimplicitprobabilitydensity.

x

Nonparametricdensityestimation

p(x)=1/vin

Modelingdistributionofshapes–p(x)

Modelingthedistributionofshapes•  Theshapespacedefinesanimplicitprobabilitydensity.

x

Nonparametricdensityestimation

p(x)=1/vin

ParametricRepresentationGaussianmixturemodel2components

Modelingdistributionofshapes–p(x)n=1 n=2

n=3 n=4

ShapespacemodeledasaGaussianMixtureModel

Diffeomorphicspace•  Newfeaturespace

– Positionsinspacecorrespondtoarealimage– Featuredimensionscorrespondtodimensionswithhighesteigenvalues

– Canbetreatedexactlylikeanormalfeaturespace

HeLashapespacewithDNAintensity

Component1(R2=0.04)

Compo

nent2(R

2 =0.08)

Component3(R2=0.57)

DNAintensity

Docellandnuclearshapedependoneachother?

•  Buildshapespacesfor–  nuclearshapesonly–  cellshapesonly

•  Foreachnuclearshape,predictapositionincellshapespaceforitbyinterpolatingatthesamerelativedistancesfromthecellshapesofitsneighborsinnuclearshape(andviceversa)

•  Measurepredictionerrorasthedistanceintheshapespacebetweenthepredictedpositionandtheactualposition

Predictionofcellandnuclearshapedependency

John

sonetal.

2015

H1299shapespace,coloredbyproteinlabel

Johnsonetal.2015