JOURNAL OF LA Prediction of Growth Factor Dependent Cleft Formation …dhulena/compsoc_journal.pdf · 2015-07-08 · JOURNAL OF LATEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 1

JOURNAL OF LATEX CLASS FILES, VOL. 13, NO. 9, SEPTEMBER 2014 1

Prediction of Growth Factor Dependent CleftFormation During Branching Morphogenesis

Using A Dynamic Graph-Based Growth ModelNimit Dhulekar1, Shayoni Ray2, Daniel Yuan3, Abhirami Baskaran1, Basak Oztan4, Melinda Larsen5, and

Bulent Yener1

1Department of Computer Science, Rensselaer Polytechnic Institute, Troy, New York, USA2ISIS Pharmaceuticals Inc, Carlsbad, CA, USA

3Epic Systems Corporation, Verona, Wisconsin, USA4American Science and Engineering Inc., Billerica, Massachusetts, USA

5Department of Biological Sciences, University at Albany, State University of New York, Albany, NewYork, USA

Abstract—This study considers the problem of describing and predicting cleft formation during the early stages of branchingmorphogenesis in mouse submandibular salivary glands (SMG) under the influence of varied concentrations of epidermal growthfactors (EGF). Given a time-lapse video of a growing SMG, first we build a descriptive model that captures the underlying biologicalprocess and quantifies the ground truth. Tissue-scale (global) and morphological features related to regions of interest (local features)are used to characterize the biological ground truth. Second, we devise a predictive growth model that simulates EGF-modulatedbranching morphogenesis using a dynamic graph algorithm, which is driven by biological parameters such as EGF concentration,mitosis rate, and cleft progression rate. Given the initial configuration of the SMG, the evolution of the dynamic graph predicts the cleftformation, while maintaining the local structural characteristics of the SMG. We determined that higher EGF concentrations cause theformation of higher number of buds and comparatively shallow cleft depths. Third, we compared the prediction accuracy of our modelto the Glazier-Graner-Hogeweg (GGH) model, an on-lattice Monte-Carlo simulation model, under a specific energy function parameterset that allows new rounds of de novo cleft formation. The results demonstrate that the dynamic graph model yields comparablesimulations of gland growth to that of the GGH model with a significantly lower computational complexity. Fourth, we enhanced thismodel to predict the SMG morphology for an EGF concentration without the assistance of a ground truth time-lapse biological videodata; this is a substantial benefit of our model over other similar models that are guided and terminated by information regarding thefinal SMG morphology. Hence, our model is suitable for testing the impact of different biological parameters involved with the processof branching morphogenesis in silico, while reducing the requirement of in vivo experiments.

Index Terms—Statistical learning, unsupervised learning, predictive models, network theory (graphs), image segmentation, systemsbiology, Monte Carlo methods

F

1 INTRODUCTION

Branching morphogenesis is a developmentally con-served process occurring in many organs, including thelungs, pancreas, kidneys, salivary and mammary glands [1],[2]. Branching morphogenesis is a temporally regulatedhighly dynamic, multiscale process involving mRNA mod-ifications, protein signaling pathways and reciprocal in-teractions between epithelial and mesenchymal cell types;leading to tissue level structural changes affecting organo-genesis [2]. Although the branching structures in developingorgans have been studied in detail, we are still far fromcomprehending the integrated process.

Since the early developmental processes in branchingmorphogenesis in several branched organs are conserved,we used mouse embryonic submandibular salivary gland(SMG) in our investigations [3]. The ability to produce salivais important in maintaining oral health, and continuedefforts are being targeted to identify methods to restorefunctionality or design artificial salivary glands. Compu-

tational modeling of the developing organ can not onlyadd to the basic knowledge of developmental mechanismsbut can also facilitate organ engineering efforts. EmbryonicSMG organ explants have long been used as a biologicalmodel system to study pattern formation during the processof branching morphogenesis [3], [4]. The embryonic SMGexplants undergo branching morphogenesis when grown onfilters at the air/media interface in serum-free medium in away that reproduces the branching pattern that occurs invivo [5].

The SMG initiates as a thickening of the primitive oralcavity epithelium on embryonic day 11 (E11). At E12 theprotrusion of the primitive oral epithelium into the sur-rounding condensed mesenchyme forms a single cellular-ized epithelial primary bud on an epithelial stalk, as shownin Fig. 1. By E12.5 clefts, or invaginations of the basementmembrane begin to form on the epithelial surface of thisinitial bud, as seen in Fig. 1(a). These clefts are stabilized


before they begin to progress deeper into the gland andseparate the initial bud into multiple secondary buds, asshown in Fig. 1(b). Epithelial proliferation occurs duringcleft progression aiding in tissue growth [5]. Clefts even-tually stop progressing further into the tissue and beginto widen at their base during cleft termination, as seen inthe left-most cleft in Fig. 1(c), and ultimately transition intonewly forming ducts. The gland undergoes multiple roundsof cleft and bud formation, and duct elongation throughoutdevelopment and, as a result, a progressively complex andhighly arborized structure is formed. Detectable epithelialcellular differentiation starts by E15; thereafter creatingfunctional ductal structures to transport saliva and secretoryacinar units capable of saliva secretion.

Branching of the salivary gland epithelial tissue isknown to be dependent upon growth factors and exogenousbasement membrane [6]–[8]. Epidermal growth factor (EGF)is one such growth factor, which is known to be involved inthe morphogenesis and fetal development of several organs,including the lungs [9], kidney [10], mammary gland [11],and pancreas [12]. The role of several growth factors in SMGbranching morphogenesis, including EGF, was previouslyinvestigated using mesenchyme-free epithelial rudimentscultured in a basement membrane extract in the presenceof exogenously added growth factors [13]–[15]. Addition ofEGF induced cleft formation and development of a highlylobed structure with little ductal elongation. The EpidermalGrowth Factor Receptors (EGFR) family displays receptortyrosine kinase activity and ligand binding induces severaldownstream signaling cascades that modulate EGFR activ-ity affecting global growth patterns in a tissue [16]–[19].EGF is known to activate several developmental processesincluding growth, survival, migration, and cell-fate deter-mination [20]; however, the exact mechanism by which EGFregulates cleft formation has not yet been investigated.

Conventional cellular and molecular biological tech-niques are limited in their ability to explain complex biolog-ical phenomenon, and thus computational approaches havebeen introduced as a means to model branching morpho-genesis [21]. Computational modeling of morphogenesisdates back to the mid 20th century with important mathe-matical models that advanced our understanding of funda-mental properties of clusters of cells [22], [23]. These theorieswere followed by biochemical and mechanochemical mod-els [24] that led to the use of continuum mechanics whichconsidered a tissue to be composed of cells and extracellularmatrix (ECM) and described the stress forces between thecells and the ECM [25]–[27]. Such models have also beenused for modeling of epithelial morphogenesis in 3D breastculture acini [28], and lung [29]–[31] and kidney branchingmorphogenesis [32]. Each of these models was tailored tothe particular biological process in question to account forthe structurally different final branching patterns in theseorgans, even though mechanistic pathways are conservedacross several branching organs.

The continuum mechanics models laid the foundationfor utilizing computational approaches to model complexbiological processes; however, they often made oversim-plified assumptions regarding small deformations at thetissue-scale. Also, solutions for continuum mechanical prob-lems have higher computational complexity, and require

a b c

Early Stage

Intermediate Stage

Advanced Stage

d e f

g h i

j k l

m n o

Fig. 1. Three stages in cleft formation during early branching morpho-genesis between embryonic days E12 and E13. Selected frames fromthe time-lapse confocal microscopy image sets demonstrate progres-sively deeper clefts and bud outgrowth. In (a), multiple nascent cleftsare visible on a single large bud, which deepen in (b), and begin to formbuds. By image (c), some clefts have terminated, and the gland has atleast partially separated into distinct buds. Scale, 100 µm. Figures (d)-(o) show the progression of individual clefts in two data sets, with theupper rows ((d)-(f) and (j)-(l) Scale, 100 µm) displaying the entire glandand a white rectangle around the particular cleft, and the lower rows((g)-(i) and (m)-(o) Scale, 50 µm) displaying the zoomed in version ofthis rectangle.

knowledge of the strength of bonds between cell typesor the knowledge of various force fields, which are notknown. Although recent studies have considered the tissueto behave as a viscous liquid under the assumption that theepithelium and mesenchyme are immiscible Stokes fluids;these models also fell short in reproducing actual salivarygland cleft shapes [33]–[35]. To overcome the shortcomingsin these earlier modeling techniques and to better replicatethe complicated dynamics governing biological processes,stochastic models were constructed based on Monte-Carlo(MC) methods. MC-based approaches can provide approx-imate solutions to complex, sometimes intractable mathe-matical problems when a large percentage of the possible


configurations of the system have high energies and thushave a low probability of being attained [36]. The Glazier-Graner-Hogeweg model, an on-lattice MC model, was usedto determine cellular parameters regulating cleft progres-sion during branching morphogenesis in the epithelial tis-sue of an early embryonic SMG [37]. The disadvantages ofon-lattice MC-based approaches include a time-consumingsampling step to reach desired solution, potential negativeeffects of lattice discretization, and the use of variance-reduction techniques [38].

Over the past 20 years, graph theoretical models [39]have become significantly important in analyzing large-scale networks with complex interactions between mul-tiple participating entities. Biological networks have alsocommonly benefitted from the advent of network anal-ysis tools and techniques [40] that have been used tomodel protein-protein interactions [41]–[45], metabolic net-works [46]–[49], genetic and transcriptional regulatory net-works [50], [51], disease progression [52], [53], and neuronalconnectivity [54]. We previously developed a graph theo-retical model called cell-graphs to study the structure ofcellular networks [55]–[58]. A cell-graph is an unweightedand undirected graph where the topological organization ofthe cells within tissues is characterized by graph theoreticfeatures. The graph vertices (nodes) represent the cellularnuclei within the tissue and graph edges (links) capture cell-to-cell interactions. Cell-graphs enable quantification of thespatial uniformity, connectedness, and compactness at mul-tiple scales. Conventionally, graph models have been usedto depict structural properties of tissues at fixed time-pointsenabling characterization and quantification of the spatialevolution of tissue shape and integrity, without explicitlyaddressing the temporal component.

1.0.1 Our ContributionsIn this study, we utilize a novel approach towards quan-tifying the spatio-temporal evolution of tissue shape andgrowth pattern using a graph-based growth model. Weutilized time-lapse confocal images of SMG grown for 12hours under the influence of varying concentrations of EGFand constant concentration of Fibroblast Growth Factor 7(FGF7). FGF7 is critical to the processes of mitosis and celldifferentiation. The primary contributions of this study areas follows:

• Descriptive model: We extracted morphometric param-eters from multiple time-lapse confocal images ofmesenchyme-free epithelial rudiments. A dynamicgraph-based growth model was constructed thatsimulated the first round of branching morphogen-esis with elongation of existing clefts, tissue prolifer-ation and de novo cleft formation. From this graphmodel, novel local cleft and bud-based features suchas median cleft depth and median bud perimeter per-centage, and global morphological features such asarea and perimeter of the growing epithelial tissue,were analyzed for each concentration of EGF. Weshow that the dynamic graph-based growth modelsuccessfully characterized the developmental stagesof the SMG growth pattern and simulated temporalchanges of structural properties of the tissue.

• Predictive growth model: Given only the initial config-uration of an SMG organ explant, the dynamic graphmodel predicts the time-evolving development of theSMG between E12 and E13 as a function of initialgland morphology, EGF concentration, and mitosisand cleft progression rates. Varying concentrationsof EGF was used to adjust rates of epithelial prolifer-ation in the model [59]–[61] that accounted for globaltissue growth.

• Interpolation-driven prediction modeling: We enhancedthe dynamic graph-based growth model to predictthe time-evolving morphology and expected config-uration of an SMG organ explant without (i) time-lapse data obtained from biological experiments,and (ii) apriori knowledge to determine the haltingconfiguration, or cleft termination. Our approachis based on building a linear regression model ofcleft deepening dependent on the perimeter of theadjoining buds and the EGF concentration; this de-pendency allows us to better modulate the growth ofthe gland. This approach is novel since it automati-cally calculates the target configuration to determinethe halting condition. We present simulation resultsthat demonstrate significant qualitative agreementbetween the target configuration predicted by ourlinear regression model and the biological insight.

We also show that while the dynamic graph model isnondeterministic, multiple executions of the model success-fully produce the same number of clefts that vary in theirindividual depths and locations across the epithelial periph-ery. This observed variance in cleft depths and locationsin the the dynamic graph model closely mimics the highdegree of variability in cleft depths and place of initiationfound in the ex vivo grown samples. This realistic modelingby the dynamic graph model is because it is driven by EGFconcentration, and mitosis and cleft progression rates, anddoes not enforce assumptions regarding bonds between celltypes; also leads to reduced computational complexity.

1.0.2 Organization of the paperThe rest of the paper is organized into four primary sec-tions. Following this introduction (Section 1), Section 2describes the biological data sets used for our experiments,preliminary image processing techniques applied on thesebiological data sets, algorithms for detection and develop-ment of clefts, and extraction of features including globalmorphological features, and local region of interest features.Section 3 describes the descriptive model based on dynamicgraphs, the feature-based clustering of the biological datasets, comparison of the predictive growth model with aMonte-Carlo based on-lattice model, and the augmentedprediction model based on dynamic graphs. Section 4 sum-marizes the findings of the study and alludes to potentialfuture directions.

2 MATERIALS AND METHODS

2.1 Data Acquisition: Ex Vivo submandibular salivarygland epithelial organ culturesTimed-pregnant female mice (strain CD-1, Charles RiverLaboratories) at E12, with day of plug discovery desig-


nated as E0, were used to obtain SMG rudiments follow-ing protocols approved by the National Institute of Dentaland Craniofacial Research IACUUC committee, as reportedpreviously [13]. E12 SMGs that contain a single primaryepithelial bud were microdissected, the mesenchyme wasremoved and the epithelial rudiments were cultured inpresence of 100ng/ml FGF7, as described previously [13].For three of the glands, the media was also supplementedwith 20 ng/mL epidermal growth factor (EGF), while forthe other three 1 ng/mL EGF (R&D Systems) was used.Images were collected as described in the next section A.1.Henceforth, we will refer to the image sets as EGF-20a,EGF-20b and EGF-20c (20 ng/mL EGF), and EGF-1a, EGF-1b and EGF-1c (1 ng/mL EGF). Description of the otherdata acquisition techniques can be found in the Appendixsections A.1 and A.2.

2.2 Quantification of ground truth: Image Processingand Segmentation

The first step in characterizing the SMG morphology con-sists of segmenting the SMG regions in the FN (via Im-ageJ) and GFP (manual segmentation due to noisy images)time-lapse data sets. The FN images were segmented viaOtsu’s technique [62] by calculating an optimal thresh-old to separate the tissue (foreground) from the Matrigelmedium (background). Biologists visually inspected thismanual segmentation to ensure that it correctly capturedthe morphology. To obtain nuclear information regardingcell distribution and cell morphologies, we referred to theex vivo data set (in Section 2.1).

2.3 Quantification of ground truth: Detection of cleftregions

The first important step in characterizing the SMG morphol-ogy is the detection of clefts as they form and deepen. TheSMG is comprised of alternating buds and clefts, whereclefts are narrow valley-shaped indentations that form inthe basement membrane. Figures 1(d)-(o) illustrate the pro-gression stages of typical clefts from shallow nascent cleftsto narrow and deep progressive clefts in the EGF-1a andEGF-20a data sets. We characterize the cleft region usingcleft center defined as the deepest point of the cleft, withthe walls of the cleft extending on either side of the surfacenormal at the cleft center, and the corresponding left andright extrema points that determine the extent of the cleft;the buds are considered to be starting beyond the pointsmarked as cleft extrema. The cleft center and cleft extremaare illustrated in Fig 2(a). Automated detection of these keypoints is carried out as follows:

1) We identify local extrema of the gradient alongthe SMG boundary by detecting angular variationsgreater than 35◦ at regular intervals of 14 successive(x,y) coordinate points measured via Euclidean dis-tance on a rectangular Cartesian grid. This intervalconstituted by 14 successive boundary points wasfound to be optimal based on the successful identi-fication of inflection points along the boundary. An-gular thresholds lower than 35◦ identified multipleoutliers. The extrema thus identified correspond to

potential cleft centers or peaks of boundary irregu-larities.

2) These peaks are eliminated using the signedarea of the triangle formed by the candidatepoint, t, and two of its immediate 8-connectedneighbors, t-1 and t+1, along the boundary or-dered in clock-wise direction. This is obtained as xt−1 yt−1 1

xt yt 1xt+1 yt+1 1

,where (xt, yt), (xt−1, yt−1),

and (xt+1, yt+1) represent the horizontal and ver-tical coordinates of the candidate point and itsprevious and next neighbors along the boundary,respectively. This expression is positive for cleftsand negative for peaks.

3) After the peaks are eliminated, we identify the cleftextrema points using the mean-squared error (MSE)between the best-fit line and SMG boundary pointson either side of the potential cleft centers. As pointsfrom the curved buds are included in the best-fitline, a higher MSE is obtained in comparison tothe steeper cleft walls. The algorithm progressesfrom the cleft center incrementally adding pointson either side of the cleft center to the cleft region.When the MSE exceeds a threshold the boundarypoint is labeled as a cleft extrema. We set a dy-namic threshold for the MSE that is computed asa function of depth of the cleft from the closestconvex hull vertices obtained after fitting a convexhull around the SMG. For every cleft center detectedby the algorithm, we identify the vertices lying onthe convex hull to its immediate left and right.The depth is then calculated as the perpendiculardistance from the cleft center to the mid-point of theline segment joining these closest vertices identifiedon the convex hull. We implement cleft tracking aspart of the algorithm to track the progress of thecleft.

4) As a final filtering step to eliminate boundary ir-regularities or nascent clefts, we exploit the cleftdepth and spanning angle as illustrated for a samplecleft in Fig. 2(a). Cleft depth is described as theshortest Euclidean distance from the cleft center tothe line segment joining the two-extrema points,and the spanning angle is the angle formed by thetwo line segments joining the extrema points to thecleft center. Indentations that had a depth of lessthan 9µm and spanning angle greater than 150◦

were not considered since our analysis of time-lapsedata indicated that such regions were boundary ir-regularities that might not form a stable cleft. Thesethresholds were decided based on discussions withbiologists and measurements from empirical data.Figures 2(b)-(e) show original images from four datasets with detected clefts highlighted in green andtheir cleft centers marked in red (or maroon as inFig. 2(c)).


Fig. 2. Characterization of clefts and sample results of the cleft detectionalgorithm. The figure shows cleft extrema (marked in green) and cleftcenter (marked in maroon) points that characterize the cleft in (a) (Scale,50 µm). Spanning angle and cleft depth are calculated from these pointsas illustrated. In (b)-(e) (Scale, 100 µm), results of applying the cleftdetection algorithm to four different data sets is shown. The cleft regionsare highlighted in the DIC microscopy ((b) and (d)), GFP-labeled (c),and FN-labeled (e) images. The cleft centers are highlighted in red (ormaroon in (c)) and the cleft regions are marked in green.

2.4 Quantification of ground truth: Extraction of globalSMG morphological featuresThe morphology of the SMG undergoes quantifiable trans-formations as a consequence of creating the ramified struc-ture. We capture these transformations by extracting sevenmorphological features, namely area, perimeter, eccentricity,elliptical variance, convexity, solidity, and box-count dimen-sion. We label this data matrix of morphological features asD. When referring to a feature matrix in subsequent text,we allude to the data matrix D ∈ RM×7, consisting of thevalues of the seven morphological features over M time-steps. Appendix Table A1 lists the definitions of the variousmorphological features.2.5 Quantification of ground truth: Extraction of novellocal cleft and bud featuresEarly branching morphogenesis is characterized primarilyby bud outgrowth and cleft deepening. Early clefts tendto get deeper and narrower with time, with the end re-sult being that the initial buds are split into multiple sec-ondary buds. We ascertained novel local cleft and bud-based features to analyze the effects of cleft deepeningon early branching morphogenesis. Using QR factorizationwith column pivoting, we sorted these features in accor-dance with their ability to capture the variance within thedata [63]. This factorization is performed when the fea-ture matrix, A, is not of full rank. QR factorization withcolumn pivoting is given as A = QRPT , where Q is anorthogonal matrix, R is an upper triangular matrix, andP is a permutation matrix chosen such that the diagonalelements of R are non-increasing |r11| ≥ |r22| . . . ≥ |rnn|.The selection of features (columns) from A is based onfinding the feature with the maximum Euclidean norm, andsuccessively finding the features maximally orthogonal to

the subspace spanned by the previously such determinedfeatures. The sequence of selection of features is stored inP. Other algorithms including singular value decomposition(SVD) may also be used for feature selection (please referhttp://featureselection.asu.edu/ for other feature selectiontechniques). The lower computational cost of QR factoriza-tion as compared to SVD was the reason we chose it as thefeature selection algorithm.2.6 Modeling cleft progression as a function of EGFconcentration and adjacent bud perimetersWe observed that although EGF stimulates branching,higher EGF concentrations produced quantitatively shal-lower clefts as compared to lower EGF concentrations. Wethus determined that cleft depth is a function of the EGFconcentration levels. We also found that cleft depth is a func-tion of the perimeter of the adjacent buds. Larger adjacentbuds allow the cleft to progress much deeper into the tissue,and higher EGF concentration levels create more buds butshallower clefts. Appendix Table A2 lists correlation coeffi-cients, represented by ρ, between cleft depth and adjacentbud perimeters for three of the data sets. In all we identified524 cleft segments, where a cleft segment is defined onlyfor the sequence of images where its adjacent buds do notsplit. We collected information regarding cleft depth andadjacent bud perimeters from all the data sets to formulatecleft progression as a linear regression model with equationsof the form c = BA + E, where c ∈ R524×1 is a vectorof depths attained by the cleft before one of its adjacentbud splits creating a new cleft, B ∈ R524×3 is the matrixof adjacent bud perimeters (with a column of 1s for theintercept), A ∈ R3×1 is the vector of equation coefficients,and E ∈ R524×1 is the vector of errors. For our simulations,to determine the depth a cleft can achieve, we first calculatedistances to every cleft in the database by comparing theadjacent bud perimeters in Euclidean space. We then applytwo levels of weights to these Euclidean distances, oneweight for the EGF concentration, and the other weight foreach cleft in the database. For the EGF concentration, wecalculate the absolute difference of the EGF concentrationfrom 1 ng/mL and 20 ng/mL (our two sample EGF con-centrations), calculate the inverse of these differences andnormalize the inverse differences by dividing by their sum.We call these weights W1. We repeat this procedure for thedistances of the bud perimeters adjacent to the cleft underinvestigation to the corresponding bud perimeters for eachcleft in the database: calculate the inverse of the distancesand normalize by dividing the inverse distances by theirsum. We call these weights W2. We split the equationcoefficients (B) into two groups for the two sample EGFconcentrations, and weight the coefficients correspondingto each cleft in either group by the appropriate weight inW2. We then take the product of these new coefficientswith W1 and sum the cleft depth values estimated by thetwo EGF concentrations. This gives us the final interpolatedcleft depth. The simulated cleft is then assigned this cleftdepth, which is updated when there is further splitting of itsadjacent buds, or it exceeds the ascribed depth. We decreasethe rate of growth of the cleft for higher EGF concentrations,and vice versa for lower EGF concentrations. This allows usto create shallow clefts for higher EGF concentrations, anddeeper clefts for lower EGF concentrations.


SMG

Growth

Expected Configuration

Compare against

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SMG

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and Segmentationand Segmentation

Stop

Deepening Update

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Dynamic CleftCleft

New Vertices

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Cleft Regions

Detection of

Ground TruthCharacterization of

Dhulekar et. al. 2012

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Model Parameters

No

Yes

ConfigurationPredicted

ConfigurationPredicted

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Dynamic Graph

Fig. 3. Overview of dynamic graph-based growth model. We start withacquisition of the biological data that gives us the SMG images. Wethen quantify the biological data by identifying clefts and computingglobal tissue-scale morphological features and local features. We runour dynamic graph-based growth model with the model parameterscomputed from the biological data as shown in the figure. Terminationis based on reaching the optimal iteration that minimizes the distance tothe target configuration.

2.7 Predictive dynamic graph growth model: Construc-tion of a biologically data driven dynamic graph-basedgrowth model of epithelial branchingThis section details one of the most important contributionsof this study – the development of a dynamic graph modelto describe the first round of cleft formation and progressionin SMG branching morphogenesis. The dynamic graph-based growth model advances our prior work on a graph-theoretical model called cell-graphs, which was used forhistopathological image analysis [64], tissue modeling [55],and characterization of the response of E13 salivary glandsto 24 hours of inhibition of the Rho kinase (ROCK) signalingpathway [56], [58]. A cell-graph G = (V,E) consists of a setof vertices V representing cell nuclei, and a set of edges Erepresenting cell to cell interactions. An edge is present inthe graph where the distance between two cell nuclei is lessthan a predetermined threshold. The model takes the initialgland morphology, nuclei locations, and EGF concentrationas input. The initial image in each time-lapse image set wasused for gland morphology. The EGF concentration levelsdetermine the mitosis and cleft deepening rates. We assumecells to be circular in shape, and cell size is approximatedby the diameter. We start with a uniform grid-graph whereV is the intersection of the grid lines, and the grid linesthemselves constitute E. The grid is continuously distortedon every iteration of the algorithm.

The outline of the dynamic graph-based growth modelis given in Fig 3. The steps involved in the algorithm arelisted below:

1) Detection of cleft regions: The first step of the modelinvolves identifying clefts. Please refer to Section 2.3for details regarding the methodology of detectingclefts.

2) Creation of new vertices: Each iteration of thegrowth algorithm divides the cells into two popu-lations based on the distance from the gland bound-ary, namely internal (I) and periphery (P ). SubsetsI ′ ⊂ I and P ′ ⊂ P are chosen to undergo aproliferation attempt. Cells in P ′ that successfullyundergo mitosis create new cells (or vertices) V ′

that are added to V. For I ′, we compute the shortestdistance to the boundary of the gland (not includingthe cleft region) and find the cell in P closest tothat boundary point; new cells V thus created areadded to V . New edges, E′ and E for peripheryand internal cells respectively, are also constructedbased on the distances from the new cells to existingcells in G.∀I ′ ⊂ I : mitosis(I ′) = V then V ∈ V, E ∈ E

∀P ′ ⊂ P : mitosis(P ′) = V ′ then V ′ ∈ V,E′ ∈ E

We measured average mitosis rates for EGF-1 andEGF-20 concentration levels as 4 cells/minute and 6cells/minute, respectively. Additional assumptionsthat build upon the Eden model [23] were imposedto model mitosis. These assumptions included cellswith identical topology and growth permitted onlyat the gland boundary where a hypothesized “nutri-ent medium” provided by the mesenchyme is acces-sible. In our dynamic graph-based growth model,this similarity is enforced via the local structuralproperties of cell-graphs that maintain consistencyin the topology of the SMG throughout the develop-ment stages. When first created, potential daughtervertices are placed outside the initial gland bound-ary in a region within 20◦ of the surface normal fromthe parent vertex at a minimum distance of one celldiameter, but less than the specified maximum edgelength. Five possible candidate daughter verticessatisfying these spatial and angular constraints arechosen, and the daughter vertex with the closestlocal cell-graph features to the parent vertex is se-lected as the optimal daughter vertex. These localstructural features (refer to Appendix C) assess thespatial uniformity (clustering coefficient), connect-edness (degree, closeness centrality, betweennesscentrality), and compactness (edge length statis-tics) of the cell-graph. We distribute the extensiondistances to the neighbors of the parent vertex tomodel bud outgrowth in a local region and pre-vent spikes in the gland boundary. Figures 4(a-c) show a sample illustration of the configurationwith creation of new vertices at different magnifica-tion levels. Supplementary Movie V1 shows mitosisevents occurring over 3 hours (movie can be down-loaded here: http://dsrc.rpi.edu/cellgraph/SMGmodeling/Supplementary Video V1.mp4). As de-scribed above, mitosis events are only allowed tooccur on the SMG boundary; this was done pri-marily to reduce computational overload involvingbookkeeping tasks.

3) Cleft deepening: To create dynamic clefts and todeepen existing clefts, we delete edges to verticesthat would now lie in the cleft region (C).

∀E(Vi, Vj) : Vi ∈ C, Vj ∈ C thenE(Vi, Vj) /∈ G,Vi /∈ V, Vj /∈ V

This edge deletion effectively isolates these verticesfrom the rest of the graph. All edges that have ver-tices lying on opposite sides of a cleft, i.e. edges goacross the cleft, are also deleted. Figure 5 illustrates


the cleft deepening algorithm. Once a cleft deepens,the cleft centers (marked in red) are removed fromthe cell-graph – the edges from these vertices toall other vertices are deleted. The deleted verticesare marked in brown in the second panel. Cleftprogression is based on the linear regression modeldescribed previously in. The cleft deepening rateis modulated by the EGF concentration. For lowerEGF concentrations, we use a higher cleft deepen-ing rate, thereby producing deeper clefts, and weuse a lower cleft deepening rate for higher EGFconcentrations to produce shallower clefts. For asample cleft we list the observed and estimated cleftdepth values in Table 1, based on the growth coef-ficients computed earlier. We can see that the linearregression model estimates the cleft depth betterfor deeper clefts than for shallow clefts. We foundthat the coefficient of determination,R2, was 0.9873,confirming our hypothesis that the cleft depths canbe computed as a function of the adjacent budperimeters. Restricting mitosis in the clefts as wellas progressively increasing cleft depth causes thecleft to narrow and to deepen, both characteristicsof progressive cleft formation.

TABLE 1Observed Cleft Depth Values as Calculated by the Cleft Detection

Algorithm vs. the Cleft Depth Values Predicted by the LinearRegression Model. Cleft Depth is reported in µm.

Observed Cleft Depth Estimated Cleft Depthfrom Linear Regression Model

17.475 20.29615.878 21.51421.808 20.33322.097 21.93522.443 21.60322.550 21.59722.903 20.93122.505 20.91922.654 21.62622.229 22.73222.094 22.218

4) Dynamic cleft creation: We first determined thesmallest possible perimeter at which a bud splits,and the percent increase in the perimeter of budsbefore they split. These values were also found to bedependent on the EGF concentration. Since higherEGF concentrations stimulate branching morpho-genesis and create more buds, we use progressivelysmaller increases in bud perimeter under increasedEGF concentration levels. The position of the splitis determined probabilistically. We select coordinatepoints along the boundary that lie in the vicinityof the center of the bud, and we randomly choose apoint from this list as the cleft center for the dynamiccleft. We isolate the vertices that lie in the newlycreated cleft region from the rest of the graph bydeleting edges incident to these vertices. All edgesthat go across the cleft are also deleted.

5) Maintaining boundary smoothness: We use the spa-tial orientation of vertices to create a smoother glandboundary. The smoothness algorithm is based on

the hypothesis that if daughter vertices are alignedsimilarly to the parent vertices, then smoothnesswill be maintained when the daughter vertices areintegrated into the boundary. This is accomplishedby minimizing the quantity |φ′i−φi|, where φi is theangle ∠pi−1pipi+1, and φ′i is the angle ∠pi−1p′ipi+1,as shown in Fig. 4(d). The previous and next ver-tices pi−1 and pi+1, respectively, are fixed, and theposition of the daughter vertex p′i is varied along theline segment pip′i. This process is repeated from thesecond till the (n − 1)th daughter vertex, keepingthe first and nth daughter vertices fixed.

6) Updating the gland boundary: An interpolating cu-bic spline curve is used to insert the daughter ver-tices into the existing gland boundary. If the distancebetween the current and next daughter vertices isgreater than a predetermined threshold, we connectthe current daughter vertex to the +3 neighbor ofthe parent vertex along the SMG boundary.

7) Compare against expected target configuration: Thesimulation is run for 100 iterations, after which theoptimal terminating iteration is selected via post-processing by determining the iteration number thatminimizes the weighted Euclidean distance to thetarget configuration. From our analysis, we foundthat about 6 hours of experimental data translatesto running 100 iterations of the model. We computethe morphological feature vectors for all the itera-tions, and project this data matrix into the reducedspace (xj) by post-multiplying by the right singularvectors of the particular ground truth data set. Wethen compare this projected feature matrix to theprojected feature vector corresponding to the targetconfiguration of the corresponding ground truthdata set (yj) using a weighted Euclidean distancegiven as:

dxy =

√√√√ N∑j=1

wj(xj − yj)2 (1)

Weights are computed as the square of the posi-tive singular values of the projected ground truthdata set divided by the sum of the squares of allpositive singular values. We look for the globalminima in the distance dxy values and select thisoptimal iteration as the terminal configuration forour simulation. Table 2 lists biological processes andproperties, and the corresponding mechanisms tohandle them in the dynamic graph-based growthmodel.

3 RESULTS

We present three different types of results in this section.First, we validate the accuracy of our representation of theground truth. Next, we compare the prediction accuracyof the dynamic graph-based growth model with a Monte-Carlo-based simulation model. Finally, we present resultsfor the interpolation-driven prediction modeling.

3.1 Time-Series Analysis of SMG structural featuresBased on the features described in the Sections 2.4 and 2.5,we present in this section the growth trends of the respective


b c

p!i

Parent Node

Daughter Node

BoundaryPotential Updated

Current Boundary

!!i

!i

pi

pi!1

pi+1

Fig. 4. Creation of new vertices and maintaining boundary smoothness.The configuration of the cell-graph (initially a grid-graph where verticesare found at the intersection of the grid lines, and the grid lines are theedges of the graph) after a single iteration of creation of new vertices(step 2 of the dynamic graph-based growth algorithm) is shown in (a).The black rectangle in (a) represents the closer snapshot of the sub-graph viewed in (b). A further black rectangle in (b) represents a smallersub-graph shown in (c). The spatial positions of parent vertices in thecurrent boundary are used to identify the optimal location for daughtervertices, as shown in (d). The uniformity in the grid-graph is distortedwith every iteration of the dynamic graph-based growth algorithm.

Arrows indicate direction of growth

a b

Fig. 5. Illustration of the algorithm driving cleft deepening and dynamiccleft creation. The two panels show the initial (in a) and final (in b)conditions of the SMG before and after cleft deepening. In (a), verticesthat are designated as cleft centers are marked in red. All other verticesare marked in green. After cleft deepening, all cleft centers lie in thecleft region (these original cleft centers are marked in brown) and arereplaced by new vertices. The edges from these original cleft centers toother vertices are deleted. This effectively isolates these vertices fromthe graph. Supplementary Movie V1 shows mitosis events occurringover 3 hours.

features for two of the ground truth data sets. Figure A2displays the trends for the global and local SMG morpho-logical features for the EGF-1a and EGF-20b data sets. Thereare 29 images in the EGF-1a data set stretching over aperiod of about 3 hours. There are 47 images in the EGF-20b data set stretching over a period of about 8 hours. Areaand perimeter (Appendix Figures A2(a) and (b)) display anincreasing trend over time as the SMG matures. Eccentricity(Appendix Fig. A2(c)) is a measure of the circularity of theellipse fitted to the SMG that has the same second-momentsas the SMG, and quantifies the elongation of the SMG.

TABLE 2Biological Processes and Properties, and their Corresponding

Interpretations in the Dynamic Graph-Based Growth Model, and theState-of-art Monte-Carlo-Based-Simulation Model Used for

Comparison.

Biology Dynamic Graph Model GGH ModelGland Structure Graph Geometry Effective EnergyMitosis Rate New Vertex Creation Rate MitosisCell-cell Adhesion Maximum Link Length Contact EnergyCell Volume Cell Diameter, or Mini-

mum Link LengthCell Area

Cell Surface Area Not Included Cell PerimeterCleft Deepening Edge Deletion Manual Specifica-

tion

Eccentricity increases as the SMG becomes more elongatedwith growth. The drop in eccentricity values for the EGF-20bdata set can be attributed to the drop in perimeter values forthe same range. Elliptical variance (Appendix Fig. A2(d))displays an increasing trend as the clefts deepen since theerror of fitting an ellipse to the SMG with deeper cleftswould be higher than a SMG with shallow clefts. Convexity(Appendix Fig. A2(e)) displays a decreasing trend over thetime-lapse images. As the clefts progress, the perimeter ofthe SMG increases, but there is only a minor increase in theperimeter of the convex hull. The perimeter of the convexhull is dependent on bud outgrowth, and this is a muchslower process as compared to cleft progression [5]. Solidity(Appendix Fig. A2(f)) decreases over time since deepen-ing of the clefts reduces the rate of growth of the SMGas compared to its convex hull. The box-count dimension(Appendix Fig. A2(g)) is a measure of a shape’s space-filling capacity. Cleft deepening is expected to increase thebox-count dimension. Although the rates of change of thefeatures differ for the EGF-1 and EGF-20 data sets, allthe global morphological features follow similar trends, i.e.either the features increase for both data sets or the featuresdecrease for both data sets.

From the feature analysis, we identified median cleftdepth and median bud perimeter percentage as the mostimportant local features. We considered median cleft depthsince some clefts progress towards termination faster thanother nascent clefts. Median bud perimeter percentage is themedian of the percentage of the SMG perimeter belongingto individual buds. Appendix Fig. A2(h) shows the trendin median cleft depth. An increasing trend is seen for thisfeature since the clefts deepen with time. Sudden dips inthe trends indicate formation of nascent clefts. The deeperclefts that are characteristic of EGF-1 data sets are the reasona higher slope is observed in the graph for EGF-1a ascompared to the slope for the EGF-20b graph. Median budperimeter percentage drops over time as more buds arecreated that are smaller in perimeter than the original buds.

3.2 Validation of the morphological featuresTo verify whether the set of global morphological and localfeatures described in the previous section was sufficient tocapture the tissue-scale and local changes in the SMG, weattempted to cluster the EGF-1 and EGF-20 biological datasets (ground truth). The hypothesis being that the two EGFconcentrations give rise to different morphological changes,


and thus would cluster separately. As an illustration, con-sider the area of the SMG. It is known that the mitosisrate is dependent on EGF concentration [60]. A higherEGF concentration increases mitosis rate as compared to alower mitosis rate for lower EGF concentration. This in turnimplies that the area values for the target configuration ofthe EGF-1 data sets will differ from the area values for thetarget configuration of the EGF-20 data sets.

We ran QR factorization with column pivoting to deter-mine the importance of each of the nine morphological fea-tures, seven global and two local region of interest features(see Section 2.5). We ran the factorization for each of the sixdata sets and found that no feature was consistently rankedwith the least important score indicating that we needed toconsider all nine features for our analysis. We ran k-meansclustering [65] in the full nine-dimensional space, with kequals to 2, to separate the sets into two classes as shownin Fig. 6 with the three EGF-1 data sets listed first, followedby the three EGF-20 data sets. All the data sets are orderedchronologically within themselves from the first frame in theset to the last frame in that set. The EGF-1 data sets are eithercompletely contained in cluster C2, or transition fairly earlyfrom cluster C1 to cluster C2 as they develop deeper cleftsand larger buds. The EGF-20 data sets are either completelycontained in cluster C1 or transition comparatively latefrom cluster C1 to cluster C2 exhibiting behavior similarto advanced EGF-1 data sets. The majority presence of theEGF-1 data sets in cluster C2 supports our claim that thiscluster signifies larger but fewer buds and deeper clefts,whereas the majority presence of EGF-20 data sets in clusterC1 verifies that this cluster is characteristic of smaller butmore buds and shallower clefts. Thus, we verify that ourfeatures were able to represent the morphological changesthat occur in the SMG during branching morphogenesis. Aninteresting observation that was revealed via this analysiswas that although increased EGF concentration stimulatesbranching morphogenesis by creating more buds, the cleftsare shallower than when compared to lower EGF concentra-tions. Table 3 shows multiple clustering measures includingrecall and precision, F-score, and entropy to validate thatthe clustering is able to sufficiently distinguish the two datasets [66].

TABLE 3Purity Measures for Evaluating the Effectiveness of Clustering

Cluster Numbers Purity MeasuresRecall /Purity

Precision F-score Entropy

Cluster C1 0.87 0.57 0.69 0.98Cluster C2 0.69 0.92 0.79 0.41

3.3 Performance Evaluation of the Predictive GrowthModel on the basis of the Accuracy of Predicted SMGMorphologyTo understand the efficacy of the predictive growth model,we compare it with a Monte-Carlo-based simulation modelthat works on the principle of energy minimization of thecombined effective energy function (constructed as a Hamil-tonian expression).

3.3.1 A Brief Overview of the GGH modelThe Glazier-Graner-Hogeweg (GGH) model [67], [68] isbuilt upon the energy minimization-based Ising model [69],

1 29 1 19 1 15 1 57 1 21 1 47C1

C2

Clu

ste

r In

de

x

Frame Indices

EGF−1a EGF−1b EGF−1c EGF−20a EGF−20b EGF−20c

Fig. 6. k-means clustering of EGF-1 and EGF-20 data sets based onmorphological features. The markers represent the frames that belongto each data set. The EGF-1 data sets are listed first, followed by theEGF-20 data sets. All the data sets are ordered chronologically from thefirst image in the set to the last image in that set, where the frame indicesfor the initial and final image are indicated in the abscissa. We utilizethe k-means clustering algorithm with k = 2 clusters. It is observed thatcluster C1 is characteristic of smaller buds and shallows clefts, whereascluster C2 is characteristic of larger buds and deeper clefts. The majorityof EGF-1 data sets are present in cluster C2 indicating that this clusterrepresents larger but fewer buds with deep clefts. The majority of EGF-20 data sets are present in cluster C1 indicating that this cluster repre-sents smaller but multiple buds with shallow clefts. Although, increasedEGF stimulates branching morphogenesis by creating more buds, cleftsare shallow in comparison to lower EGF concentrations.

using imposed fluctuations via a Monte Carlo (MC) ap-proach. Previously, we performed an in-depth investiga-tion where we constructed a local GGH-based model ofepithelial cleft formation and analyzed the ranges of thecellular parameters [37]. In this study, we utilized similarvalues for each of the parameters in the simulation ofcleft formation occurring on a tissue scale. Please refer toAppendix B for further details about implementation of theGGH model. The authors would also like to mention thatthey discussed the initial development of the specific GGHmodel parameter set with members of the Glazier Lab atIndiana University.

We compared the ability of the dynamic graph-basedgrowth model to simulate EGF-stimulated cleft formationto that of the GGH model. To make a fair comparisonbetween the dynamic graph-based growth model and theGGH model, we included the dynamic cleft creation mod-ule, described in the Section 2.7 to our implementation ofthe GGH model. This ability of the GGH model to generatede novo clefts is an improvement on our earlier work [37],[70].

3.3.2 Evaluation of the predictive growth model

The morphological feature-based comparison of the dy-namic graph-based growth model with a quantitative anal-ysis of the biological data set (ground truth) and the GGHmodel under the specific parameter set described above(and in Appendix B and [37]) for organ explants grown inthe presence of low or high levels of EGF (EGF-1a and EGF-20b datasets, respectively) is shown in Fig. 7 and AppendixFig. A3. For the sake of brevity and page restrictions, onlythe area feature for both EGF concentrations is shown in thefigure. Please refer to Appendix Figs. A3 and A4 for furtherfeature comparisons. Although, both models are constructedby very different modeling techniques, one is a graph-basedmodel whereas the other minimizes a Hamiltonian formu-lation, our comparison is solely based on the final shape ofthe epithelial tissue produced by them. No comparison is


a

(a)

b

(b)

Fig. 7. Comparison of SMG morphological features between ground truth (green), dynamic graph-based growth model (red), and the GGH simulation(blue) for EGF-1a and EGF-20b data sets. The features are area (a) and (b), perimeter (c) and (d), eccentricity (e) and (f), and median cleft depth(g) and (h). “Start” refers to cleft initiation, “Intermediate” refers to mid cleft progression stage, and “End” refers to beginning of cleft termination.Area and perimeter display increasing trends and similar trends are also seen for the dynamic graph model as well as the GGH simulation. GGHshows the opposite trend for eccentricity as it tries to create circular shapes. Increasing median cleft depth trends are shown by both dynamic graphmodel and the GGH simulation. For more detailed explanation of the feature trends, please refer to Section 3.3.

done based on the outputs of the models in their originalform. The ground truth trends are displayed in green, thedynamic graph-based-growth-model’s trends are displayedin red, and the GGH model’s trends are displayed in blue.As observed in the ground truth, an increase in area andperimeter is seen in both models. The dynamic graph-basedgrowth model is able to replicate the increase in area andperimeter more effectively than GGH, and usually remainsfaithful to the increasing trend. Eccentricity increases for theground truth as the gland becomes more elongated. This is atrend that the dynamic graph model is also able to replicate,whereas GGH fails to reproduce the appropriate trend. TheGGH models inability to properly reproduce the eccentricitytrend could be attributed to the fact that it tries to acquirea circular structure because of the anisotropic growth of themodel. Both models are able to model the trends in mediancleft depth that are observed in the ground truth, althoughGGH performs better in a few cases since it can exactlyspecify the number of clefts, location of the clefts, and thedepth of each cleft. The sudden drops in the value of themedian cleft depth in the ground truth as well as the modelssignify creation of new clefts.

Although the range of values of elliptical variance isfairly small (10−2), both models show an increasing trendfor this feature. Convexity drops as the rate of increase ofperimeter of the SMG increases at a faster rate than theperimeter of its convex hull. In keeping with the groundtruth, both models show the appropriate decreasing trend.Solidity decreases with deepening of the clefts. While thedynamic graph-based growth model reproduces the ap-propriate trend, since the GGH model tends to grow thegland more circular, solidity is also affected by this trait ofthe GGH model. Also, the clefts generated by GGH are inconstant flux, appearing and disappearing from one MCSstep to the next, and this may also be causing the soliditypatterns to be modeled incorrectly. In general, the dynamic

a c b

d e f

Fig. 8. Target configuration of the ground truth data sets, and the finalconfigurations reached by the simulations for the dynamic graph-basedgrowth model and the GGH simulation. The target ground truth configu-ration is shown in (a) and (d), the dynamic graph model’s configurationsare shown in (b) and (e), and the GGH model’s configurations are shownin (c) and (f). Dynamic graph-based growth model creates more cleftsthan the GGH model and this is one of the primary reasons that it hasbetter quantitative agreement with the ground truth in regards to themorphological features.

graph model has a higher box-count dimension than theGGH model. This could be because it tends to create moreclefts than the GGH model, thereby creating more area ofconcavity. These plots illustrate that the dynamic graph-based growth model is able to predict the growth of theclefts during early branching morphogenesis. Figure 8 dis-plays target ground truth configuration, and sample ter-minal configurations for both models for the EGF-1a andEGF-20b data sets. As can be noticed from the terminalconfigurations, the dynamic graph-based growth model isable to produce de novo clefts and the final configuration iscomparable to that produced by the GGH model.


3.3.3 Computational Complexity ComparisonWe analyzed the time complexity of both approaches toillustrate the savings in time afforded by the dynamic graphmodel. The dynamic graph model has worst case timecomplexity of O(n2), whereas the GGH model has worstcase time complexity of O(n3), omitting the constants thatcome from convergence criteria (or number of iterations).This is unsurprising given that the GGH model calculatesan energy function between each pair of cells, and considersincreased cellular-level detail thereby further increasing itscomplexity.

As an illustration, we compared the simulation times forboth models. We ran 50 experiments on a 2.4 GHz Intel Core2 Duo processor with 4GB RAM. The dynamic graph modelwas on average 10.24 times faster than the GGH model,taking an average of 4.83 min ± 1.41 sec to complete theexperiment, whereas GGH took 49.47 min ± 34.21 sec.

3.4 Interpolation-driven prediction modeling: Predic-tion of growth factor dependent branching morphogen-esisFor the majority of comparisons to biological data, both thetime-lapse data set and the expected target configurationsare available in advance. Our objective was to enhance ourdynamic graph growth model such that when presentedwith an initial SMG boundary image and EGF concentra-tion, the model could predict gland morphologies withoutthe aid of a time-lapse data set, and thus no informationregarding the target configuration.

We start by artificially creating ground truth morpholog-ical feature vectors from the initial SMG boundary imageand EGF concentration, using as a first attempt, lineargrowth equations under the assumption that all featuresare uncorrelated. We computed two sets of average linearregression models for all features for EGF-1 and EGF-20concentrations, given in Appendix Table A3. The lineargrowth rates follow the expected behavior as explained inSection 3.1. For any intermediate EGF concentration, weinterpolate between these two sets of models, and applya normalized inverse distance function for the EGF concen-tration as a weighting factor (similar to the function definedin 2.3). With these individual feature growth equations,we determine the values the features would attain after acertain time interval. For our experiments, we consider timeintervals of 3, 4.5, and 6 hours. Since the GGH model re-quires apriori knowledge of expected target configurations(i.e. final cleft depth) it was not possible to make suchpredictions with this model.

Using the formula mentioned earlier in Section 2.2, wecalculated average mitosis rates (MR) for EGF-1 and EGF-20concentrations as 4 cells/minute and 6 cells/minute, respec-tively. Mitosis rates for intermediate EGF concentrationswere predicted by interpolating between these two mitosisrates, weighted by a normalized inverse distance function.We also interpolated cleft deepening rates and maximumcleft depth using a linear interpolation scheme (describedin Section 2.6), and ran our algorithm (Section 2.7) basedon these interpolated mitosis and cleft deepening rates,and bud-splitting statistics for about 100 iterations, approxi-mately the number of iterations required to simulate 6 hoursof growth. For all iterations of the algorithm, we computed

the morphological feature vector (yj) and compared it tothe expected target feature vector, or target configuration,as determined by the ground truth generated by the lineargrowth model (xj). The comparison is done as following:

dxy =

√√√√ N∑j=1

(yj − xjxj

)2

The comparison was performed in the original featurespace, since the rank of the matrix of the linearly generatedground truth is 1. We look for the iteration number thatminimizes the distance dxy , and use this iteration numberas the terminal configuration for each of the three time-intervals mentioned above.

This interpolation-driven dynamic graph growth modelwas used to predict gland morphology at specific timepoints for different EGF concentrations. Figure 9 showsthe same starting image grown under EGF-1, EGF-10,and EGF-20 concentrations with predicted outcomes us-ing the dynamic graph-based growth model for 3, 4.5,and 6 hours. Table 4 lists the area, perimeter, numberof clefts, and median cleft depth for the three configu-rations. We can observe that higher EGF concentrationsstimulate branching morphogenesis by creating more buds.As time progresses, the lower EGF concentrations havedeeper clefts. This can be observed at 6 hours where EGF-1 has a much higher median cleft depth, whereas EGF-20has relatively shallower clefts that have hardly progressedbeyond 3 hours. Supplementary Movie V2 shows a sim-ulation time-lapse movie produced by the dynamic graphmodel of all three EGF concentrations together (movie canbe downloaded from http://dsrc.rpi.edu/cellgraph/SMGmodeling/Supplementary Video V2.mp4).

TABLE 4Comparison of Final Configurations of the Three EGF Concentrations

as Predicted by the Dynamic Graph-Based Growth Model.

Area (µm2) Perimeter(µm)

Numberof clefts

Mediancleft depth(µm)

EGF-1 74132.43 1057.03 5 36.42EGF-10 77473.87 1041.53 6 23.11EGF-20 80715.32 1067.93 6 17.09

4 DISCUSSION

The objective of this study was to quantify and predict thecore processes involved in the initial stages of branchingmorphogenesis to initiate the process of branching morpho-genesis: cleft initiation, stabilization, and progression, underdifferent concentrations of EGF. These cleft stages occursimultaneously with bud outgrowth and subsequently leadto cleft termination and duct formation. For this purpose, weextracted morphometric parameters from time-lapse mousesubmandibular salivary gland (SMG) images. We developeda biological data driven descriptive model utilizing dynamicgraphs. This dynamic graph model describes and predictsearly branching morphogenesis in SMG. Given an initialSMG boundary image and EGF concentration level, thedynamic graph model is able to predict the growth of theSMG between embryonic days E12 and E13. The modelprobabilistically adds daughter cells and integrates thesecells into the SMG by appropriately expanding its boundary.


EGF-1

EGF-10

EGF-20

a b c

d e f

g h i

3 hours 4.5 hours 6 hours

Fig. 9. Dynamic graph-based prediction model results. Same startingimage grown under different EGF concentrations for 3, 4.5, and 6 hours.The configuration of the salivary gland after 3 hours under EGF-1,EGF-10, and EGF-20 concentrations are shown in (a), (d), and (g),respectively. The configuration of the salivary gland after 4.5 hoursunder under the same concentrations are shown in (b), (e), and (h),respectively. The configuration of the salivary gland after 6 hours underthe same concentrations are shown in (c), (f), and (i), respectively. Thenumber of buds increases from EGF-1 to EGF-20, with clefting occurringmore frequently in higher EGF concentrations. Buds are larger and cleftsare deeper in lower EGF concentrations. Please recall that we do notcompare the prediction model to a Monte-Carlo-based simulation (MCS)model. For further information about the prediction model, please referto Section 3.4

The model also creates new edges between the daughtercells and cells existing in the graph representing the initialgland. This augmented cell-graph (with the daughter cells)maintains the local structural properties of the original cell-graph. Cleft deepening and creation of dynamic clefts arecrucial components of the model allowing it to producemore realistic branched structures and deeper clefts, andare based on the rules captured from the time-lapse datasets. The process of de novo cleft creation is modeled by firstidentifying the regions of initiation of the cleft, as well asthe increase required in the perimeter of the buds. Giventhis information, we then use a probabilistic model to createde novo clefts.

Our results indicate that the dynamic graph model cancorrectly capture and represent the tissue-level morpho-logical changes during cleft formation in the developmen-tal stages of the SMG branching morphogenesis. We alsoshowed that cleft progression is linearly dependent on theperimeters of the adjacent buds and is modulated by theEGF concentration. As was expected, higher EGF stimu-lated branching morphogenesis by producing more clefts,whereas lower EGF concentrations produced fewer clefts.Our analysis revealed an interesting observation regardingthe depth of the clefts. Higher EGF concentrations pro-duced shallower clefts, whereas lower EGF concentrationsproduced deeper clefts, for reasons that remain unclear.Future studies will be required to understand the cellularprocesses activated by EGF during cleft formation. Sincethis dynamic graph modeling approach can model cleft

formation in response to modulation of EGF signaling, thisapproach could be employed to evaluate the contribution ofother signaling pathways to cleft formation. Similar model-ing approaches could be employed towards understandingother developmental processes in which large changes inshape occur.

Interestingly, the morphology of the epithelial budstends to be slightly more circular than the buds producedby the dynamic graph-based growth model. It is not pos-sible to capture this difference in the circularity of budswith the methods used here and would require a morecomplicated model. Currently, this level of detail in shapemodeling is beyond the consideration of the dynamic graph-based growth model, and would be a potential directionof enhancement for the model. We compared our resultsagainst a well-known on-lattice Monte-Carlo-based simu-lation model, the Glazier-Graner-Hogeweg (GGH) model,under a specific parameter set consisting of energy functionsthat have biologically relevant equivalents, and demon-strated that our results are in a similar quantitative agree-ment with the biological data as those of the GGH model,but converge significantly faster to the target configuration.The authors would like to point out that the GGH modelhandles cellular-level changes at a higher resolution thanthe dynamic graph-based growth model. We also presenteda method to introduce de novo clefts in the SMG using theGGH model thus adding to the dynamic nature of the GGHmodel.

We enhanced the dynamic graph-based growth modelto predict the growth of the SMG at any specified timebetween embryonic days E12 and E13 without requiring atime-lapse data set. This is one of the primary benefits of theinterpolation-driven prediction modeling approach – it doesnot require apriori knowledge of the target configuration –the initial configuration and growth rules determined fromthe biological data are sufficient for the algorithm to predictthe gland morphology at a future time point. The predictivenature of the model reduces its dependence on in vivoexperiments, allowing the biologists to view a simulationof the experiment prior to performing it. Most other com-putational biological techniques, including the GGH model,require information regarding the final configuration of theground truth data in advance. Thus, it was not possible tocompare our predictive model to other simulative models.We examined the growth trends in the biological data fromthe viewpoint of morphological features and discoveredthat individual linear growth models are able to predictthe evolution of each feature. This allowed us to identifythe expected configuration of the morphological features atdifferent time points.

While analyzing the results of the dynamic-graphgrowth model, we observed that it has certain shortcomings.Since individual tissue-scale and cellular-scale models havetheir deficiencies with regards to modeling different as-pects of cleft formation in branching morphogenesis, futureefforts should be aimed at creating a multi-scale hybridmodel that can achieve better tissue-scale modeling, and beable to more realistically capture cellular-level events suchas mitosis and cellular reorganization in cleft regions. Thedynamic graph model is a generic model and as such canbe used in conjunction with other models to create this


hybrid model. Another future direction for the dynamicgraph-based growth model involves including dynamic cellmovement information for accurate construction of cellgraphs with a better estimation of the spatial distributionof cells [13].

ACKNOWLEDGMENTS

The authors would like to thank Dr. James A. Glazier and his researchgroup members Dr. Abbas Shirinifard and Dr. Srividhya Jeyaramanat Indiana University, Bloomington, IN for discussions, suggestions,and help with the use of GGH model for SMG branching morpho-genesis. The authors would also like to thank former research groupmember, Lauren Bange, for her efforts on the initial understandingand implementation of the GGH model. We also thank Dr. KennethYamada for use of the Zeiss 510 Meta confocal microscope for time-lapse imaging. This work was supported by a grant from the NIH toMelinda Larsen and Bulent Yener (R01 DE019244) and by NIH C06RR015464 to University at Albany, SUNY.

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Nimit Dhulekar is an Application Support Engi-neer at MathWorks, Inc. He received his Ph. D. inComputer Science from Rensselaer PolytechnicInstitute (RPI) in 2015. He received his B. E.degree in Computer Engineering from MumbaiUniversity in 2006, and his M. S. degree in Com-puter Science from Dartmouth College in 2010.His research interests include analysis of time-series data for various complex systems, imageinformatics, computer vision, and computationalmodeling.Shayoni Ray is a Post-Doctoral Research Sci-entist in the Antisense Drug Discovery, Cardio-vascular Group at ISIS Pharmaceuticals, Inc.She received her Ph. D. in Cellular/Molecularand Computational Biology from SUNY Albanyin 2014. She received her B. S. degree in Chem-istry from Calcutta University in 2004, and her M.S. degree in Biotechnology and Bioinformaticsfrom Jadavpur University in 2007. Her researchinterests include elucidating cellular and molec-ular mechanisms during organogenesis, tumor

biology and skin fibrosis, and computational modeling of complex bio-logical processes.Daniel Yuan received his B. S. degree in Chemistry from MassachusettsInstitute of Technology in 2008 and his M. S. degree in Cellular andMolecular Biology from University of Wisconsin - Madison in 2011. Hestudied Computer Science at RPI where he contributed to a computa-tional model for mouse salivary gland development.Abhirami Baskaran received her B. E. degree in Computer Sciencefrom People’s Education Society Institute of Technology in India in 2011,and her MS degree in Computer Science from RPI in 2013.

Basak Oztan is a Senior Image Processing Sci-entist with American Science and EngineeringInc., Billerica, MA. He received his B. S. degreein Electrical and Electronics Engineering fromMiddle East Technical University, Ankara, Turkey,in 2003 and his M. S. and Ph. D. degrees inElectrical and Computer Engineering from theUniversity of Rochester, in 2004 and 2010, re-spectively. He was a Post-Doctoral ResearchAssociate with the Department of Computer Sci-ence, RPI between 2010 and 2013. His research

interests include image informatics, computer vision, and machine learn-ing.

Melinda Larsen is as Associate Professor inthe Department of Biological Sciences at theUniversity at Albany, SUNY. Dr. Larsen obtainedher B. S. degree from the University of Utah in1993, and her Ph. D. from Baylor College ofMedicine in 1999. She completed postdoctoraltraining at the NIH/NIDCR from 1999-2006. Hercore research interests are in understanding themolecular mechanisms driving salivary gland or-gan formation and in deciphering the contribu-tion of the extracellular environment to epithelial

cell function for application in engineering artificial organs.

Bulent Yener is a Professor in the Departmentof Computer Science with joint appointmentswith the Department of Electrical, Computer andSystems Engineering, and with the Departmentof Mathematical Sciences at RPI in Troy, NewYork. He received his M. Sc. and Ph. D. degreesin Computer Science, both from Columbia Uni-versity, in 1987 and 1994, respectively. He is thefounding director of the Data Science ResearchCenter at RPI. Dr. Yener is a Fellow of IEEE, anda senior member of ACM. His research focus is

on various aspects of data science spanning from modeling of complexsystems to data security.

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