## 4 Proposed Methodology  
The overall proposed methodology comprises two main steps: (1) foreground cell clustering, and (2) region estimation and quantification, as shown in Figure 3. The original images of the _in vitro_ 3D cell spheroid were first converted from the \(RGB\) color space to the \(L^{*}a^{*}b^{*}\) color space. Unlike the RGB color model, the \(L^{*}a^{*}b^{*}\) color model is approximately close to human vision. This color model provides uniformity in the range of perception [19]. Extracted \(a*b*\) components from the RGB images were given as input to the TLBO clustering algorithm. The approach has two resultant phases, the teaching phase and the learner phase, as described in [12]. Initially, a K number of clusters were taken, each cluster datum or pixel datum was defined as a learner, and centroids that were selected randomly for each cluster are called teachers. After that, each learner's Euclidean distance with the centroid for all k clusters and fitness value was evaluated. Using the learner-phase steps given in [12], each learner was modified. Likewise, the centroids or the best learners and the existing solution are modified. After reaching maximal iterations \(I_{max}\), the foreground cells were separately clustered. The HCT-8 and NIH3T3 cluster cell images were then converted into a binary image using Ostu's global thresholding technique [20]. Binary images \(B_{b}andB_{r}\) were then given as input in Algorithm 1 to estimate the densest HCT-8 cell region by finding the cell count. The nearest feature voxel \(NF_{v_{i}}\) of the \(B_{b}\) and \(B_{r}\) were calculated as follows:  
\[NF_{v_{i}}=\left\{ \begin{array}{ll}0&v_{i}\in B_{v_{i}}\\ \min\big{(}\big{\|}v_{i},v_{j}\big{\|}\big{)}&(v_{i},v_{j})\in F_{v_{i}}\end{array} \right. \tag{2}\]  
where, \(\big{\|}v_{i},v_{j}\big{\|}=\big{\|}v_{i}-v_{i_{0}},v_{j}-v_{j_{0}}\big{\|}, \forall(v_{i_{0}},v_{i_{0}})\in B_{v_{i}}\) is any distance metric. The outcome of DT depends on the selection of distance metrics, which varies from application to application. However, among all other metrics, the Euclidean distance metric was the most generalized because its measurement corresponds to the way in which objects are measured in the real world, and is rotation-invariant. The metric uses the \(L_{2}\) norm and is defined as \(\big{\|}v_{i},v_{j}\big{\|}_{2}=\sqrt{v_{i}^{2}+v_{j}^{2}}\). After obtaining the nearest feature voxel matrix, maxima propagation is applied using the inverted mask matrix of \(NF_{v_{i}}\) and by limiting the propagation using some fraction (\(c_{1}=0.5\)) of the maximal distance of the nearest feature voxel matrix \(NF_{v_{i}}\). To control the extent of propagation of \(MP_{i}\), \(MP_{i}\) is dilated with the 3 \(\times\) 3 matrix of all ones. This operation transforms the finer distance image into coarser segments. After that, the different numbers of segments or levels as \(Lvl_{i}\) are found using the \"unique\" function of MATLAB. Then, background voxels were removed from the \(Lvl_{i}\) matrix. The area threshold value was calculated to impose the minimal area constraints for segments. After that, the 8-connected component (or blobs or cells) area and pixel indices were found using the \"region props()\" inbuilt function of MATLAB, which simply sums the pixels of a particular region (area) by using their assigned labels. Then, the area is thresholded by using the extracted areas and finding the count of blobs or cells encountered within that area region. This gave the count of HCT-8 and NIH3T3 cells in different regions of the _in vitro_ 3D cell spheroid. On the basis of these counts, i.e., \(B_{c}\), \(B_{p}\), \(G_{c}andG_{p}\) the densest HCT-8 cell region (\(R_{dense}\)) using Equation (1) was estimated. Figure 3: Resultant view of proposed methodology. ```
0: Binary images (\(B_{b}\) and \(B_{r}\)) of HCT-8 cluster and NIH3T3 cluster images, respectively.