## 5 Results and Discussion  
Before the ANN can predict the cell stiffness, it must be trained with known data. I used the DMP model to simulate cell populations with different levels of stiffness and, at the same time, train the ANN. I ran three hundred simulations: half of these were used for training and half for validation. Since these data allow for effective training of the ANN, how the accuracy of the ANN changes with the number of simulations is not specifically investigated in this study. Under real conditions, both the stiffness \(k\) and the diameter \(d\) of the population are not constant but vary within a certain range. To account for this, at the beginning of each simulation, both the \(k\) and \(d\) are randomly selected (uniform distribution) within \(\pm 10\%\) of their given values. Another random variable is the initial position of the cell in the channel. In Figure 6, the cell is initially placed at the  
Figure 8: Deep multiphysics model of the separation device. The number of neurons/layers is not representative of the actual Artificial Neural Network (ANN) used but is only for illustrative purposes. centre of the channel height, but, in reality, it can be located at any height compatible with its diameter. The initial position of the cell, therefore, is also randomly allocated. In order to move along the channel, the cell must bend the cilia. When the cilia bend, they generate stress at their bases. The value of the total force acting at the base of each of the 11 cilia is recorded every 0.2 s for 15 timeframes. After the cell passes through the ciliated section, therefore, we have \(15\times 11\) stress values that can be stored in a matrix \(F\). If the force \(f\) is normalized according to  
\[f^{*}=\frac{f-f_{\min}}{f_{\max}-f_{\min}}, \tag{5}\]  
the normalized matrix \(F^{*}\) can be seen as a greyscale image (Figure 9). Each cell produces a slightly different image according to its stiffness, size and initial position. This image represents a sort of fingerprint of the cell, and the ANN is trained to distinguish between soft and rigid fingerprints. In all the simulations, I assume the two populations have the same average diameter \(d=30\)\(\upmu\)m (\(\pm 3\)\(\upmu\)m) and their only difference consists of their stiffness (and initial position). I consider three different cases where the stiffness of the two populations becomes gradually closer (Table 1). The same ANN, with one hidden layer, is used in all three cases. The input layer has 165 nodes (the size of the matrix \(F^{*}\)), the hidden layer has 3 nodes and the output layer has 1 node. All the layers are fully connected, and the logistic function is used as activation for all the nodes. The output of the ANN is the normalized stiffness  
\[k^{*}=\frac{k-k_{\min}}{k_{\max}-k_{\min}}, \tag{6}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{\(k\) **Soft Population [J-10\({}^{15}\)]**} & \multicolumn{3}{c}{\(k\) **Rigid Population [J-10\({}^{15}\)]**} \\ \hline
**Case** & **Min.** & **Ave.** & **Max** & **Min.** & **Ave.** & **Max** & **Err \%** \\ \hline
**1** & 1.8 & 2 & 2.2 & 18 & 20 & 22 & 0 \\ \hline
**2** & 3.6 & 4 & 4.4 & 9 & 10 & 11 & 0 \\ \hline
**3** & 4.05 & 4.5 & 4.95 & 4.5 & 5 & 5.5 & 11 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Minimal, average and maximal stiffness of the two populations for the three cases considered. Err is the percentage of soft cells erroneously classified as rigid by the ANN, and vice versa, in the validation set. Figure 9: The ’stress’ fingerprint of a cell with stiffness \(k=4.5\times 10^{-15}\) J.  
where \(k\) is the stiffness of the cell, and \(k_{\rm min}\) and \(k_{\rm max}\) are, respectively, the minimal and maximal stiffness of the set. The reader should not confuse the max. and min. stiffnesses in Table 1 and in Equation (6): the former are the maximal and minimal values _of the population_, the latter _of the data set_, which include instances of both populations.