## Numerical Results and Analysis
### Case Studies  
In the case studies, we are going to determine if it is possible, with the sensors, to capture the two variables of interest in a patient, to obtain a DCH that allows for monitoring the status of his/her liver. To do so, we conduct several simulations for different datasets with the information of the patients, in a case of a healthy patient taken from [15], and in others with diseased livers (cases of alcoholics (taken from [36])) and cases of other diseases (taken from [37])). Case 1:  
This is the first test case using a single sensor per variable (1 NS). We run the hyper-parameter optimization every time we launch the GA. For example, the best values for each parameter for case 1 with the dataset [15] (it is the case of non-liver sick) are shown in Table 3.  
The system with one NS was executed, and the best chromosome obtained is shown in Table 4, with the minimum best score (DCH). According to these results, the best distance from one end of the liver to the periphery, using the values of ALKPHOS and MCV for one of the patients in this dataset, is 24,225. This means that with one NS, the liver of that patient can be monitored without problem. Now, we test our technique on the other two diseased liver datasets. Table 5 shows the results obtained with our GA for one NS. Table 5 reveals two phenomena. For the first dataset, it is possible to obtain a very large DHC for the monitored patient; that is, it obtains values on the periphery of the liver to analyze its status. This indicates that a single sensor can be injected. However, the most relevant is that, in the second dataset, our GA cannot achieve a DCH, even in its periphery, to monitor this patient, which indicates that we must try with more sensors (objective of case study 2). Case 2:  
In this case study, the idea is to determine the appropriate number of sensors to be able to monitor a sick patient. In particular, we are going to use several patients from the dataset [37], which could not be monitored with one NS. On the other hand, in this case, the length of the chromosome is variable, so its length n depends on the number of NS used in that GA run. Thus, the chromosome t is represented as a vector of real numbers, with a size equal to n.  
Chromosome = [ALKPHOS1, MCV1, DCH1, ALKPHOS2, MCV2, DCH2,...,  ALKPHOSn, MCVn, DCHn] \begin{table}
\begin{tabular}{c c c} \hline \hline
**ALKPHOS** & **MCV** & **DCH (FF)** \\ \hline
76 & 85 & 24,225 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The best individual found in case 1 for the dataset [15]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cyles** & **Population Size** \\ \hline
10 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the GA for case 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dataset** & **ALKPHOS** & **MCV** & **DCH (FF)** \\ \hline
[36] & 146.00 & 99.45 & 106,092.42 \\
[37] & 150 & 120 & 168,750 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The best individual found in case 1 for datasets [36; 37]. The objective function, in this case, is given by:  
\[FF=0.05*\sum_{i=1}^{n}ALK{P}{H}{G}{S}_{i}^{2}*MC{V}_{i}) \tag{2}\]  
The hyperparameter optimization of the GA is conducted for each GA run with a different \(n\) (multisensor system) for \(n\) equal to three NSs, five NSs, ten NSs, and twenty NSs. The best results (individual) for each run are shown in Table 6.  
Table 6 indicates that at least four sensors are required to correctly monitor the liver. In addition, it indicates that having more than five sensors does not improve the accuracy of the liver monitoring results. Something interesting to determine is how to implement the NS. For example, in the case of four NSs, considering the best individual, the liver can be divided into four zones (see Figure 5): A, from 0 to 40,000; B, from 40,000 to 80,000 au; C, from 80,000 to 120,000 au; and D, from 120,000 to 156,500. Therefore, each sensor must be injected into these areas to obtain the data from the damaged area.