2013 Pascal Problem 18

In the diagram, PQRS is a square with side length 2. Each of P, Q, R, and S is the centre of a circle with radius 1. What is the area of the shaded region?

\textbf{(A)}\ 16-\pi^2\quad \textbf{(B)}\ 16-4\pi\quad \textbf{(C)}\ 4-4\pi\quad \textbf{(D)}\ 4-4\pi^2\quad \textbf{(E)}\ 4-\pi

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Pascal contests here.