# Toward Automatic Cardiomyocyte Clustering and Counting through Hesitant Fuzzy Sets
## 4 Experimentation and Evaluation
### Evaluating Cell Counting  
When evaluating cell counting, we use two metrics to evaluate the reliability, namely the false accept rate (FAR) and the false reject rate (FRR). FAR represents the ratio of dead cells that are mistakenly counted toward to the result; FRR represents the ratio of living cells that are not counted in the number of viable cells. Formally, FAR and FRR are defined as: \(\text{FAR}=E_{FAR}/N_{FAR}\) and \(\text{FRR}=E_{FRR}/N_{FRR}\), where \(E_{FAR}\) indicates the miscalculated number of dead cardiomyocytes, \(E_{FRR}\) indicates the number of living cardiomyocytes that were not counted in the total cardiomyocytes, and \(N_{FRR}\) and \(N_{FAR}\) indicate the total number of cardiomyocytes in both scenarios, respectively. A high FAR makes the cell-counting system inaccurate because the system accounts for too many dead cells or impurities; a high FRR causes the system to miss the living cells and the number of live cells counted is not accurate. One design goal of our proposed system is to achieve low, if not negligible, FAR and FRR rates. Although cardiomyocytes can generally survive for about seven days in the medium, they basically lose the rod-like characteristics on the third day. Therefore, in our experiment, only 50 microscope fields are collected for the first 48 h. Cardiomyocyte counts are performed in 50 different regions. For the same reason, experiments conducted at different timestamps do exhibit discrepant results. Specifically, we expect that in the initial stage of culture, the accuracy should be higher than that of the later stages because most of the cardiomyocytes in the early stage retain the rod shape and are easy to be detected by our system. In the image-processing experiments above, the cardiomyocytes are cultured for 6-8 h.  
We report the experimental results in Figure 4. The X-axis indicates the experimental serial number (ID), \(e\)(FAR) indicates the number of dead cardiomyocytes that are mistakenly counted in a specific experiment, \(e\)(FRR) indicates the number of live cardiomyocytes that should have been counted in a specific experiment, and TOTAL represents the total number of cardiomyocytes counted each time. That is, \(E_{FAR}=\sum\limits_{i=1}^{n}e_{i}(FAR)\) and \(E_{FRR}=\sum\limits_{i=1}^{n}e_{i}(FRR)\), where \(i\) corresponds to the experiment ID in Figure 4. The accuracy of the counting system is higher in the earlier stages because the myocardial cells retain the rod-like shapes. Overall, by looking at the span of 50 h, our system achieves both low error rates: FAR = 1.46%, FRR = 1.97%. However, when the cardiomyocytes are analyzed in the time range of 36-48 h, the errors rates increase to: FAR = 3.26%, FRR = 4.94%.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter Types** & **X1** & **X2** & **X3** \\ \hline A1 & [0.7,0.8,0.9] & [0.7,0.8] & [0.7,0.8,0.9] \\ A2 & [0.6,0.7] & [0.6,0.7] & [0.5,0.6] \\ A3 & [0.2,0.3] & [0.2,0.3] & [0.2,0.4] \\ A4 & [0.3,0.4] & [0.2,0.3] & [0.2,0.5] \\ A5 & [0.2,0.3] & [0.3,0.4] & [0.2] \\ A6 & [0.5] & [0.4,0.6] & [0.4] \\ A7 & [0.6,0.8] & [0.7,0.8] & [0.7,0.8,0.9] \\ A8 & [0.6,0.7] & [0.4,0.5] & [0.6] \\ A9 & [0.5,0.7] & [0.5,0.6] & [0.4,0.5] \\ A10 & [0.7,0.8] & [0.6,0.7] & [0.5,0.6] \\ \hline \end{tabular}
\end{table}
Table 2: Categorization confidence.  
Figure 4: Counting of cardiomyocyte during a 48 h period.