Three-way and two-way analysis of in vitro breast tissue data.

<p>(a) CP factorization of the tensor with modes: <i>features</i>, <i>samples</i> and <i>time</i>. Only the 2nd component can separate the two different functional states: cancer (red-triangle sign) from normal (green-plus sign) tissue samples; (b) SVD of matrix of type: <i>features</i> by <i>samples (across all times)</i>; (c) features projected over the 2nd CP component. Cell-graph features such as <i>% of end points, number of connected components, average connected component size, average path length, average eccentricity</i> are identified as influential in the analysis.</p

Coupled factorization of in vitro bone samples represented by tensor

<p><b>and histology samples represented by matrix Y.</b> (a) Both the 1st and the 2nd column of matrix <b>B</b> extracted by a CMTF model separate cancer (blue-square sign) from normal (light blue-plus sign) samples; (b) Matrix <b>D</b> corresponding to the histology samples mode extracted using a CMTF model is useful to narrow the coupled analysis since only the 1st component can separate cancer (red-triangle sign) from healthy (green-star sign) samples; (c) features captured by the 1st CMTF component.</p

Three-way and two-way analysis of in vitro bone tissue data.

<p>(a) CP factorization of the tensor with modes: <i>features</i>, <i>samples</i> and <i>time</i>. Both the 1st and the 2nd components separate the two different functional states: cancer (red-triangle sign) from normal (green-plus sign) tissue samples; (b) SVD of matrix of type: <i>features</i> by <i>samples (across all times)</i>; (c) features projected over the 1st component of CP model. Cell-graph features such as <i>% of end points</i>, <i>number of connected components</i>, <i>giant connected component ratio, average path length, average eccentricity</i> are identified as influential in the analysis since their coefficients diverge the most from zero; (d) since the 2nd component can also distinguish between two functional states we also show the 2nd CP component in features mode. Note that the influential features are different in the 2nd component, e.g., while the <i>number of connected components</i> has a high coefficient in the 1st component, its coefficient in the 2nd component is close to 0.</p

Singular Value Decomposition and R-component CP model.

<p>Singular Value Decomposition and R-component CP model.</p

In vitro vs. histology samples of cancerous tissue (10a Brain, 10b Bone, and 10c Breast samples).

<p>The first two components of SVD analysis explain 72.4%, 65.9%, 66.5% of the variance for each tissue type, respectively. SVD yields a linear separation between in vitro and histology cancerous tissue samples. Two clusters (red and green) are very well separated, with few outliers. This defines and quantifies a structural difference between engineered tissues and the native tissues.</p

Coupled matrix and tensor factorization on in vitro breast samples represented by tensor

<p><b>and histology samples represented by matrix Y (</b><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0032227#pone-0032227-g002" target="_blank"><b>Figure 2</b></a><b>).</b> (a) The 1st column of matrix <b>B</b> corresponding to the in vitro samples mode extracted by a CMTF model can separate cancer (blue-square sign) from normal (light blue-plus sign) tissue samples; (b) Unlike for brain and bone tissues, matrix <b>D</b> corresponding to the histology samples mode extracted using a CMTF model cannot separate cancer samples (red-triangle sign) from healthy (green-star sign) samples; (c) features captured by the common component extracted by CMTF. Cell-graph features identified as influential in the coupled analysis are similar to the features in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0032227#pone-0032227-g005" target="_blank">Figure 5c</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0032227#pone-0032227-g007" target="_blank">7c</a> with some minor differences.</p

In vitro tissue samples remain structurally different from histology samples (blue) over time.

<p>We identified no time point in the development of the in vitro samples that is mathematically similar to histology data. (a) Brain cancer cultures vary considerably over time getting very close to the histology samples, demonstrating they best resemble the histology samples. In vitro bone cancer (b) and breast cancer (c) samples remain clustered over all time points and exhibit no mixing of data points with histology data.</p

Analysis of histology and in vitro data sets using coupled matrix and tensor factorization (CMTF).

<p>Time mode is slotted as 0, 1, 2, 4, 6, 10, 16, 24, 72, 120, 168 in hours. Features mode contains the cell graph features: average degree, clustering coefficient C, clustering coefficient D, clustering coefficient E, average eccentricity, diameter, radius, average eccentricity 90, diameter 90, radius 90, average path length, effective hop diameter, hop plot exponent, giant connected component ratio, # connected components, average connected component size, % isolated points, % end points, # central points, % central points, mean, std, skewness, kurtosis, # nodes, # edges. These features are defined in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0032227#pone.0032227.s001" target="_blank">Table S1</a>.</p

General methodology for quantifying collagen alignment and the structural organization of the tissue.

<p>First row: one optical slice of a 3D second harmonic generation multiphoton confocal image (Scale bar 20m). Corresponding confocal images of the cell nuclei were segmented using the Otsu Thresholding algorithm. Connected pixels were found in this segmented image and each connected component was labeled as an individual cell nuclei. Using these segmented confocal images of nuclei, we reconstructed and visualized the tissue on top of the collagen in 3D (Second row). For each nucleus, the center of mass was found and assigned as the x,y,z coordinates of that nucleus. Using the nuclei locations, cell-graphs that capture the spatial relationship between the nuclei were constructed and visualized (Third row). The collagen alignment around every edge of the graph was quantified and the Collagen Alignment Index that measures the quality of the alignment is assigned to each edge. Cell-graph edges that have an alignment greater than a given threshold (in this case 0.6) were drawn thicker to highlight areas of enhanced remodeling in 3D (Fourth row).</p

Edge-based and Node-based Voronoi partitioning.

<p>Two different Voronoi construction techniques were used in this work, done both from a cell-graph node perspective and from a cell-graph edge perspective. In <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0012783#pone-0012783-g003" target="_blank">Figure 3(a)</a>, a sample Voronoi diagram using the cell coordinates as the seed points is shown where blue (node with edges) and red (node without edges) circles are the cell nodes. Blue lines correspond to cell graph edges while dashed red lines correspond to the Voronoi compartments. In this original state of the Voronoi diagram, each cell-graph edge was shared by two separate Voronoi compartments. In <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0012783#pone-0012783-g003" target="_blank">Figure 3(b)</a>, to capture the information between cells the method was altered to set the center of the edges as the seed points. This construction ensures that each cell-graph edge is encapsulated in only one Voronoi compartment. To capture the information between cells the method was altered to set the center of the edge as the seed point. This construction ensures that each cell-graph edge is encapsulated in only one Voronoi compartment. These compartments were projected onto the corresponding SHG image to assign each pixel of type I collagen signal to a given graph node (A) or edge (B).</p