Is it possible to make F1_Score differentiable and use it directly as a Loss function?
Yes there is, let's take $F_1$ score  base definition, with : 
$$
F_1 = 2 \times \frac{precision \times recall} {precision + recall} \\
F_1 = \frac{2 \times TP} {2 \times TP + FP + FN}
$$
And this is the same as the Sørensen-Dice coefficient, also known as Dice coefficient or Bray-Curtis distance. This is a statistical indicator that measures the similarity of two samples :

$$ Dice(X,Y) = \frac{2|X \cap Y|}{|X| + |Y|}$$

Concerning the implementation of this loss, we can approximate $|X \cap Y|$ as the sum of the matrix obtained using Hadamard product ($\odot$, also known as the element-wise product) between the ground truth ($y$) and the prediction ($\hat{y}$). We can then define $L_{Dice}$ as follows: 

\begin{align*}
L_{Dice} &amp;= 1 - Dice \\
L_{Dice}\left(y, \hat{y}\right) &amp;= 1 - 
        \frac{ 2\sum y \odot \hat{y}}
             {\sum y + \sum \hat{y}} 
\end{align*}

You will often find this loss in the context of segmentation problems, as well as others quite close, such as the Jaccard index (IoU).