A Novel Dynamic GraphBased Computational Model forPredicting Salivary Gland Branching Morphogenesis
Nimit Dhulekar, Lauren Bange, Abhirami Baskaran,Daniel Yuan, Basak Oztan, B¨ulent Yener
CS Dept., Rensselaer Polytechnic Institute, Troy, NY, USAdhulena@cs.rpi.edu, bangel@alum.rpi.edu, baskaa@rpi.edu, yuand@rpi.edu, oztanb@cs.rpi.edu, yener@cs.rpi.edu
Shayoni Ray, Melinda Larsen
Biological Sciences Department University at Albany, SUNY Albany, NY, USAsray@albany.edu, mlarsen@albany.edu
Abstract
—In this paper, we introduce a biologically motivateddynamic graphbased growth model to describe and predict the stagesof cleft formation during the process of branching morphogenesis inthe submandibular mouse gland (SMG) from
3
hrs after embryonicday
E
12
to
8
hrs after embryonic day
E
12
, which can be consideredas
E
12
.
5
. Branching morphogenesis is the process by which manymammalian exocrine and endocrine glands undergo signiﬁcant morphological transformations, from a primary bud to an adult organ.Although many studies have investigated the cellular and molecularmechanisms driving branching morphogenesis, it is not clear howthe shape changes that are inherent to establishing organ structureare produced. Using morphological features extracted from sequentialimages of SMG organ cultures we were able to develop a dynamicgraphbased predictive model that is able to mimic the process of cleftformation and predict the ﬁnal state. In addition, we compare ourmodel to a stateoftheart GlazierGranerHogeweg (GGH) simulativetool, and demonstrate that the dynamic graphbased predictive modelhas comparable accuracy in modeling growth of clefts across SMGdevelopmental stages, as well as faster convergence to the target SMGmorphology.
Keywords
morphogenesis, cell behavior, epithelial, mathematicalmodel, cellgraph, dynamic graph growth
I. I
NTRODUCTION
Branching morphogenesis is the process of development of many mammalian exocrine and endocrine glands such as the lungs,kidney, pancreas, mammary, and salivary glands [1]. The organization of a branched morphology maximizes the total area of contactbetween the metabolic exchange surfaces and the surroundingenvironment while minimizing the total volume of the organ. Thisallows for an efﬁcient exchange of gases, nutrients, metabolites,and wastes.During branching morphogenesis of the mouse submandibulargland (SMG), the morphology of the gland undergoes signiﬁcanttransformation [2]. For purpose of illustration, Fig. 1 shows threestages of growth of a submandibular mouse gland (SMG) startingfrom a primary bud in Fig. 1a, where small clefts, or indentations,have initiated on the surface of the primary bud. In Fig. 1b clefts,or indentations, have appeared in the surface of the bud and havestarted to undergo cleft progression to separate the primary budinto multiple smaller buds. Cleft progression includes a cellularcontraction as well as cellular proliferation that result in budoutgrowth [3]. Clefts ultimately cease progressing and begin towiden at their base, as seen in the leftmost bud in Fig. 1c thatshows the SMG at a much advanced stage. This widening isfollowed by outgrowth of the cells at the base of the clefts to formducts. Once the new buds are fully formed, cleft formation beginson the new buds, and occurs reiteratively throughout developmentto create the ramiﬁed structure of the adult organ.
(a)
E
12
(b)
E
12 + 5
hrs (c)
E
12 + 14
hrs
Figure 1: Three stages of branching morphogenesis in SMGdemonstrating progressively deeper clefts and bud outgrowth.In (a), multiple nascent clefts are visible on a single large bud,which deepen in (b) and begin to form buds. By image (c), someclefts have terminated, and the gland has been separated intodistinct buds.Branching morphogenesis requires an understanding of themolecular mechanisms regulating epitheliummesenchyme interactions during development of the glands. The process is highlydynamic, involving interactions between multiple participating cellsand molecules. These interactions cannot be completely understoodusing conventional cellular and molecular biology methods alone,which has increased interest in this area of systems biology [4],[5]. Initial systems approaches to understanding morphogenesiswere based on the Turing morphogenesis model [6] and focusedon pattern formation and its applications to morphogenesis. TheEden model [7] was a competing method that grows the tissue byselecting a periphery cell with at least one empty nearest neighbor,and then with equal probability choosing and occupying one of itsempty nearest neighbors. Since then, many advanced mathematicalmodels based on differentialalgebraic equations, ﬂux balanceanalysis, and stochastic algorithms, have been utilized to develop abetter understanding of morphogenesis [8]–[10]. Simulation toolshave also made it possible to analyze various cellular behaviorsthat are difﬁcult to examine
in vivo
[11].In this paper, we introduce a dynamic graphbased growthtechnique to predict branching morphogenesis of the SMG asa function of initial gland morphology, cell proliferation rate,cleft progression rate, and cellcell adhesion. The SMG is widelyused to study branching morphogenesis [12], and shares commonmechanisms with the other types of salivary glands [13] and otherbranching organs [2]. In the current study, we focus on developinga predictive and descriptive model for the process of cleft formationin the ﬁrst round of branching morphogenesis, which initiates
©
in the SMG at
E
12
, starting from the initiation of clefts andproceeding through cleft progression. Our model terminates prior tothe apparent widening of clefts that occurs during ductal formationand does not address the second or subsequent rounds of branchingthat occur as the gland continues to develop. Note that branchingmorphogenesis progresses beyond this stage (embryonic day
E
13
);however, we are limiting our focus in this paper to the earliest stageof morphogenesis and excluding the processes of duct formationand hollowing of the structure [14].Graphbased models are commonly used to model highly complex events at various scales with multiple participants. In particular, biological networks have beneﬁtted from the use of graphtheoretical concepts that have been used to model proteinproteininteractions [15], [16], disease progression [17], and neuronalconnectivity [18]. Our proposed model builds upon our earlier work on histopathological image analysis [19], tissue modeling [21],and characterization of branching morphogenesis at embryonic day
E
13
[22], at cellular and tissue scales using a graphtheoreticalmethod called
cellgraphs
. Cellgraphs are unweighted and undirected graphs where the cell nuclei are represented by graph nodes,and interactions between cells are represented by edges. Cellgraphs enable us to assess the spatial uniformity, connectedness,and compactness at multiple scales. When a cell undergoes mitosis,the daughter cell inherits not only the genetic characteristics of the parent cell but also the local topological characteristics of its neighborhood [23]. In our proposed model, this similarity isenforced via the local structural properties of cellgraphs thatmaintain consistency in the topology of the SMG throughout thedevelopment stages. The SMG also maintains a smooth boundaryduring the growth stages, and we utilize a function of the locationsof the nodes in the cellgraph to encode this smoothness into ourmodel.To evaluate the efﬁcacy of our model, we extract morphological features that characterize developmental stages of the SMGbranching morphogenesis. Our results closely mimic the observeddevelopmental stages. In addition, we compare to a stateoftheartsimulative model based on how well the dynamic graph modelrepresents the process of cleft formation observable in SMGdevelopmental stages, and the computational time complexity.The rest of this paper is organized as follows: in Section II,we present the materials and methods; in particular, the proposedSMG branching morphogenesis dynamic growth model is describedin detail in Section IIC. We present our experimental results anda discussion in Section III and ﬁnally, in Section IV, we presentconclusions.II. M
ATERIALS AND
M
ETHODS
In this section, we ﬁrst introduce the timelapse data that formsour ground truth in Section IIA. Next, we present features forthe characterization of the SMG morphology in Section IIB, andﬁnally, our dynamic graphbased SMG branching morphogenesisprediction model is presented in detail in Section IIC.
A. Acquisition of Data
Our ground truth is a timelapse image set of an embryonic day
E
12
SMG that was treated with dispase, and had the mesenchymephysically removed by microdissection, as described in [24].The epithelial rudiment was grown in Matrigel (1:1 dilution inDMEM/F12 growth media) supplemented with
100
U/ml penicillin,
100
mg/ml streptomycin,
50
mg/ml transferrin,
150
mg/ml ascorbic acid,
20
ng/ml epidermal growth factor (EGF), and
100
ng/mlﬁbroblast growth factor (FGF). The gland was imaged using timelapse microscopy at 200X magniﬁcation using a Zeiss 510 Metaconfocal microscope.
87
images were captured as
7
µ
m sectionsat
10
minute intervals using the
488
nm laser to capture a nearDIC image. Images were captured at a
512
×
512
pixel resolutionusing a scan speed of
8
in line averaging mode. These imageswere visually inspected to identify the morphological changes inprogressing clefts that occurs in the period from
3
hrs to
8
hrs after
E
12
.To obtain nuclear information regarding cell distribution, cellproliferation rates, and cell morphologies, we used separate
ex vivo
data set. Intact E12 SMGs were cultured for
2
hours,
8
hours,
12
hours, and
24
hours and pulsed for
2
hours with ClickiT Edu.Following this process SMGs were ﬁxed and permeabilized in
4%
PFA. Labeled DNA was detected using ClickiT EdU Alexa Flour647 kit (Invitrogen). Following EdU detection, glands were blockedand immunostained with antibody recognizing Ecadherin (1:100)and Cy5conjugated Donkey F(ab)2 secondary antibody (1:100,Jackson ImmunoResearch Lab) to detect the epithelial area. Totalnuclei were detected using SYBR Green I (Invitrogen) counterstaining. Immunostained glands were imaged using a laser scanningconfocal microscope (Zeiss 510 Meta) at 20x (Plan APO/0.75NA) using identical settings for all samples. Multiple imagesoverlapping each other by approximately
10%
were acquired atthe center of each explant (depth direction) such that the entireexplant was imaged.
B. Characterization of the SMG morphology
In the following subsections we describe the initial imageprocessing steps to segment the SMG regions from the timelapse image set (Section IIB1), identify clefts in the SMG (Section IIB2), and characterize the SMG using morphological features(Section IIB3).
1) Preliminary Image Processing:
In order to characterize theSMG morphology, the SMG regions in the timelapse data set needto be segmented. Due to the signiﬁcantly low contrast and highnoise of these images, it was difﬁcult to automatically segmentthe SMG regions with high accuracy. Therefore, we manuallysegmented the SMG regions using ImageJ [25]. Since SMG growthis characterized by bud outgrowth and cleft deepening, the stemregions were also eliminated during the manual segmentation.
2) Initial Cleft Detection:
A critical step in modeling SMGbranching morphogenesis is the detection and characterization of clefts as they form and deepen. The SMG boundary is comprisedof alternating bud and cleft regions, with clefts as narrow valleyshaped formations that separate growing buds. For the purposeof illustration Fig. 2 shows the progression stages of a typicalcleft. Narrowing and deepening of the cleft can be observed fromFig. 2(a) to (c) as branching morphogenesis progresses.We characterize the cleft regions using their center and twoextrema points as illustrated in Fig. 3a. The cleft center is thedeepest point of the cleft, with the walls extending on eitherside of the surface normal at the cleft center. The cleft extremapoints determine the extent of the cleft; the bud region starts
(a) Early stage(b) Intermediate stage(c) Advanced Stage
Figure 2: Stages of cleft formation during branching morphogenesisillustrating (a) early, (b) intermediate and (c) advanced stages.Images in the top row are enlarged segments of the images shown inthe bottom row wherein other clefts at different progressive stagescan also be seen.beyond these points. Automated detection of these key pointsis carried out as follows: ﬁrst we compute the angles at eachpoint with its eighth neighboring points on either side alongthe SMG boundary. These extrema correspond to not only thepotential cleft centers but also the peaks of boundary irregularities.Next, the peak points are eliminated using the signed area of the triangle formed by the cleft center, and two of its immediateneighbors along the boundary ordered in clockwise direction. Thisis obtained as
x
c
−
1
y
c
−
1
1
x
c
y
c
1
x
c
+1
y
c
+1
1
, where
(
x
c
,y
c
)
,
(
x
c
−
1
,y
c
−
1
)
,and
(
x
c
+1
,y
c
+1
)
represent the horizontal and vertical coordinatesof the candidate point and its previous and next neighbors alongthe boundary, respectively. This expression is positive for clefts andnegative for peaks. After the peaks are eliminated, we identify thecleft extrema points using the meansquared error (MSE) betweenthe bestﬁt line and SMG boundary points on each side of thepotential cleft centers. For each side, the algorithm progresses fromthe cleft center including a neighboring boundary node, and ﬁttinga line between the center and the set of points in consideration.When the MSE exceeds a threshold the node is labeled as a cleftextrema. We set a dynamic threshold for the MSE, varying between
10
and
40
µm
which is computed as a function of cleft depthobtained from ﬁtting a convex hull around the SMG. A convexhull is the smallest convex polygon that fully contains the SMGwithout progressing into concavities such as the cleft regions. Thecleft depth is calculated as the shortest distance from the cleft centerto the line joining points on either side of it that lie on the convexhull.In order to eliminate boundary irregularities or nascent clefts, weexploit the cleft depth and spanning angle as illustrated for a samplecleft in Fig. 3a. Cleft depth is taken as the Euclidean distance fromthe cleft center to the line segment joining the two extrema pointsand the spanning angle is formed by the two line segments joiningthe extrema points to the center. We considered clefts that have a
Spanning AngleCleft ExtremasCleft CenterCleft Depth
(a) Cleft Properties
Nascent cleftDetected clefts
(b) Cleft detection
Figure 3: Characterization of clefts. Subﬁgure (a) shows extremaand cleft center points that characterize the cleft. Spanning angleand cleft depth are calculated from these points as illustrated.Subﬁgure (b) shows an intermediate stage SMG where the detectedand nascent clefts are marked in green and yellow, respectively.depth less than
22
µm
as boundary irregularities or nascent cleftsand eliminated them from our model. In our analysis we observethat the spanning angles of the detected clefts were between
8
◦
and
125
◦
. Figure 3b shows a ground truth image with detectedclefts highlighted in green and nascent clefts in yellow. Note thatthe detected clefts have varying depths and spanning angles thatare representative of the cleft developmental stages illustrated inFig. 2.The timelapse data from a stereotypical gland was partitionedinto three developmental stages of SMG morphogenesis based onthe number of clefts detected in each image by our algorithm.
Earlystage
glands were deﬁned as those that have three clefts detectedthat progress from
3
hrs after
E
12
to
4
hrs
40
mins after
E
12
.From that point
intermediate stage
glands that have four detectedclefts progress until
6
hrs after
E
12
and ﬁnally
advanced stage
glands that have ﬁve detected clefts progress until
8
hrs after
E
12
.While this stereotypical gland was assumed to be the “ground truth”data for this study, minor variability can be detected in independentsamples in terms of how many and what quality of clefts are presentat any given time of development.
3) SMG Morphological Features:
Visual inspection of thetimelapse data reveals signiﬁcant differences in morphology. Toquantitatively capture these changes, we extract six morphologicalfeatures, area, perimeter, eccentricity, solidity, and boxcount dimension, described next. Figure 4 shows plots of these features, aswell as a linear regression to indicate general trends.Area and perimeter are counts of pixels within the SMG regionand on the boundary of the SMG region, respectively. Both increasewith time as the SMG grows, as seen in Figs. 4a and 4b. Isoperimetric quotient and eccentricity are measurements that quantify theelongation of the SMG. The isoperimetric quotient is the ratio of the area of the SMG region to the square of its perimeter, andwhich decreases as the SMG becomes more elliptical, as shown inFigure 4c. Eccentricity is a measure of the circularity of the ellipseﬁtted to the SMG that has the same secondmoments as the SMG.It is deﬁned as the ratio of the distance between the foci of theellipse and its major axis length. Eccentricity increases as the SMGbecomes more elongated throughout the growth stages, visible inFig. 4d. Solidity is computed as the ratio of areas of the SMGregion to its convex hull. Because deepening clefts cause a relative
33.544.555.566.577.580.280.290.30.310.320.33
Slope = 0.006mm
2
/hrRMSE = 0.0010mm
2
Slope = 0.0074mm
2
/hrRMSE = 0.0008mm
2
Slope = 0.0092mm
2
/hrRMSE = 0.0012mm
2
Hours after E12 Developmental Stage
A r e a ( m m
2
)
EarlyIntermediateAdvanced
(a) SMG Area
33.544.555.566.577.582.22.42.62.833.23.4
Slope = 0.340mm/hrRMSE = 0.0127mmSlope = 0.1784mm/hrRMSE = 0.0134mmSlope = 0.2716mm/hrRMSE = 0.0173mm
Hours after E12 Developmental Stage
P e r i m e t e r ( m m )
EarlyIntermediateAdvanced
(b) SMG Perimeter
33.544.555.566.577.580.30.350.40.450.50.550.60.650.7
Slope = −0.144/hrRMSE = 0.0077Slope = −0.0429/hrRMSE = 0.0043Slope = −0.0493/hrRMSE = 0.0033
Hours after E12 Developmental Stage
I s o p e r i m e t r i c Q u o t i e n t
EarlyIntermediateAdvanced
(c) SMG Isoperimetric Quotient
33.544.555.566.577.580.620.640.660.680.70.72
RMSE = 0.0051Slope = 0.022/hrRMSE = 0.0030Slope = 0.0198/hrRMSE = 0.0031Slope = 0.0109/hr
Hours after E12 Developmental Stage
E c c e n t r i c i t y
EarlyIntermediateAdvanced
(d) SMG Eccentricity
33.544.555.566.577.580.890.90.910.920.930.940.950.96
Slope = −0.013/hrRMSE = 0.0022Slope = −0.0160/hrRMSE = 0.0053Slope = −0.0092/hrRMSE = 0.0034
Hours after E12 Developmental Stage
S o l i d i t y
EarlyIntermediateAdvanced
(e) SMG Solidity
33.544.555.566.577.581.031.041.051.061.071.081.091.11.111.12
RMSE = 0.0020Slope = 0.026/hrRMSE = 0.0012Slope = 0.0065/hrSlope = 0.0152/hrRMSE = 0.0018
Hours after E12 Developmental Stage
B o x − C o u n t D i m e n s i o n
EarlyIntermediateAdvanced
(f) SMG Boxcount Dimension
Figure 4: Features characterizing the morphology of SMG developmental stages. Area, perimeter, isoperimetric quotient, eccentricity,solidity, and boxcount dimension of the three stages are shown in (a)–(f), respectively. The trends in each feature are indicated with thebestﬁt lines (dashed lines), the ﬁtting rootmeansquare errors, and bestﬁt line slopes within the plots.decrease in the SMG region’s area but do not affect the convex hull,solidity decreases over time, conﬁrmed in Fig. 4e. The boxcountdimension is a measure of a shape’s spaceﬁlling capacity. Themethod overlays the SMG region with boxes of increasing size,recording the number of boxes required to cover the boundary. Weuse the slope of the bestﬁt line in
log

log
space with least MSEto the vector containing boxcount dimension values for squareboxes of size
2
n
×
2
n
where
n
varies from
0
to
9
. Cleft deepeningis expected to increase the boxcount dimension, and is indeedobserved in Fig 4f.
C. Biologicallydriven Dynamic GraphBased Growth Model for Prediction of Branching Morphogenesis
The dynamic graph growth technique is an extension of staticcellgraphs to capture and model timevarying data. A static cellgraph
G
= (
V,E
)
consists of a set of nodes
V
representing cellnuclei, and a set of edges
E
representing the interactions betweencells. An edge is inserted into
E
when the pairwise Euclideandistance between two nuclei in set
V
is less than a predeterminedthreshold.In addition to the initial gland morphology and cell locations, ourmodel incorporates cellular proliferation and cleft progression rates.We assume cells to be circular in shape, and size is approximatedby the diameter. Here, the initial image in each developmental stagewas used for gland morphology; information from the
ex vivo
dataset was used to determine the cell proliferation rate and diameter.The cleft progression rate was determined from the timelapse databy the cleft detection algorithm described in Section IIB2. Sincethe timelapse data does not provide nuclear information, and the
exvivo
data consists of different tissue samples, starting cell locationswere approximated by a uniform grid overlaid on the SMG. Thisapproximation was based on cell density measurements made onthe
ex vivo
data set. At each growth step, cells are divided intotwo populations based on the distance from the gland boundary,namely internal and periphery. A subset of both cell populationsare chosen to undergo a proliferation attempt. For the internal cellsthat are selected for proliferation, we compute the shortest distanceto the boundary of the gland (not including the cleft region) andﬁnd the periphery cell closest to that boundary point.To model cell proliferation we impose additional assumptionsthat build upon the Eden model, which considers all cells to beidentical, and permits growth only at the gland boundary wherethe mesenchymal nutrient medium is accessible [7]. Creation of new cells in the cleft region is disallowed to prevent the cleftfrom closing, but the SMG is allowed to grow around the cleft.This inhibition of cell proliferation in the cleft region is equivalentto the replacement of the epithelial cellcell junctions with thecellextracellular matrix (ECM) junctions where ﬁbronectin (FN)translocates along SMG epithelial cells into nascent clefts andkeeps them in their current state [24]. To model increase in cleftdepth, we use the cleft progression rate obtained from the timelapse data to determine the distance that the cleft needs to beextended at every iteration of the algorithm. The cleft progression
rate changes over time, and thus we model the length by whichto extend the cleft depth as a function of time. We move the cleftcenter by this distance along the surface normal in the oppositedirection to the cleft extrema. We use a cubic spline interpolationbetween the cleft center and its
−
2
and
+2
neighbors alongthe SMG boundary to form the extended cleft. Restricting cellproliferation in the cleft region as well as increasing cleft depthcauses the cleft to narrow and deepen, both characteristics of progressive cleft formation. Table I lists biological processes andproperties, and the corresponding mechanisms to handle them inour model. We run separate simulations for the three stages of SMG
BiologyDynamic Graph Model GGH Model
Gland Structure Graph Geometry Effective EnergyMitosis New Node Creation MitosisCellcell AdhesionGraph Links (Edges) Contact EnergyLink Length FPPCell Volume Min. Link Length Cell AreaCell Surface Area Not Included Cell Perimeter
Table I: Biological processes and properties, and their corresponding interpretations in our model, and the stateofart simulativemodel used for comparison in Section III.development (early, intermediate, and advanced) starting with theinitial image of each set. The steps involved in each iteration of our dynamic growth algorithm are as follows:1)
Creation of new nodes:
The cell proliferation rate is calculated from the
ex vivo
data set as a percentage of the totalcell population. Periphery nodes are probabilistically chosento undergo mitosis, creating new daughter nodes. Thesedaughter nodes are placed outside the initial gland boundaryin a region within
20
◦
of the surface normal at a minimumdistance of one cell diameter, but less than the speciﬁedmaximum edge length. Five possible candidate daughternodes satisfying these spatial and angular constraints arechosen, and the daughter node with the shortest Euclideandistance to the parent node is selected as the optimal daughternode. The local structural features assess the spatial uniformity (clustering coefﬁcients C,D,E), connectedness (degree,closeness, betweenness), and compactness (nearest neighbordistance, mean edge length) of the cellgraph. Please refer [22] for further details about local structural features. Tomodel bud outgrowth in a local region and prevent spikes inthe gland boundary, we distribute the extension distances tothe neighbors of the parent node.2)
Maintaining boundary smoothness:
After all daughter nodeshave been created, the spatial orientation of nodes is usedto create a smoother gland boundary. The smoothness algorithm is based on the intuition that if daughter nodes arealigned similarly to the parent nodes, then smoothness willbe maintained when the daughter nodes are included into theboundary. This is accomplished by minimizing the quantity

φ
i
−
φ
i

, where
φ
i
is the angle
∠
p
i
−
1
p
i
p
i
+1
, and
φ
i
isthe angle
∠
p
i
−
1
p
i
p
i
+1
, as shown in Fig. 5. The previousand next nodes
p
i
−
1
and
p
i
+1
, respectively, are ﬁxed, andthe position of the daughter node
p
i
is varied along theline segment
p
i
p
i
. This process is repeated from the secondtill the
(
n
−
1)
th daughter node, keeping the ﬁrst and
n
thdaughter nodes ﬁxed.3)
Updating the gland boundary:
Use a interpolating cubicspline curve to join the daughter nodes to the gland boundary.If the distance between the current and next daughter nodesis greater than a threshold, we connect the current daughternode to the
+3
neighbor of the parent node along the SMGboundary.4)
Terminating the algorithm:
The simulation is run for
20
iterations, after which the terminal step is calculated. Wecompute the morphological feature vector and compare itto the feature vectors of the ﬁrst and last images in theparticular set (
3
,
4
, or
5
clefts) of the ground truth usinga weighted Euclidean distance. The weights for the featuresare determined from the ground truth using singular valuedecomposition [26]. The iteration that has the minimumdistance to the last image is considered as the terminal stage.
p
′
i
Parent NodeDaughter NodeBoundaryPotential UpdatedCurrent Boundary
φ
′
i
φ
i
p
i
p
i
−
1
p
i
+1
Figure 5: Smoothing of SMG boundary in dynamic graph model.Spatial positions of parent nodes in the current boundary are usedto identify optimal location for daughter nodes.III. R
ESULTS AND
D
ISCUSSION
We follow the methodology explained in the previous sectionand generate SMG branching morphogenesis states using theproposed dynamic graphbased growth model starting from theimage of the
E
12
gland grown for
3
hrs where initiated clefts aredetected. We were able to mimic the developmental stages of theground truth data based on the number of clefts as identiﬁed by ourcleft detection algorithm (Section IIB2). Figure 6 (a)–(c) showsearly, intermediate, and advanced stages of growth generated by ourmodel. The ﬁgure shows a gradual increase in the area and perimeter, and deepening of the clefts over time. We also compare ourresults against a wellknown simulative model, the GlazierGranerHogeweg (GGH) model. Note that in simulative models the targetconﬁgurations are speciﬁed along with the initial conﬁgurations,and therefore there is a high likelihood of reproducing the targetconﬁgurations. In contrast to a simulative model, for a predictivemodel only the initial conﬁguration is speciﬁed. In this section, weﬁrst describe the GGH model brieﬂy, and then present comparativeresults between the models and against the ground truth data.
A. GlazierGranerHogeweg (GGH) model
To measure the efﬁcacy of the proposed dynamic graphbasedmodel, we compare it against the GGH model, also known as theCellular Potts model. This simulative model has been successfullyused in a variety of biological modeling applications such as tumorgrowth and organ development [27], [28]. The GGH model treatscells as a cluster of sites on a ﬁxed lattice. Interactions betweenthese cells and their adjacent lattice sites impose an energy penalty,