  We are currently conceiving, through (MC) simulation, a multi-channel gamma
detector array (TIARA for Time-of-flight Imaging ARrAy) for the online
monitoring of protontherapy treatments. By measuring the Time-Of-Flight (TOF)
between a beam monitor placed upstream and the Prompt-Gamma (PG) detector, our
goal is to reconstruct the PG vertex distribution to detect a possible
deviation of proton beam delivery. In this paper, two non-iterative
reconstruction strategies are proposed. The first is based on the resolution of
an analytical formula describing the PG vertex distribution in 3D. Here, it was
resolved under a one-dimensional approximation in order to measure a potential
proton range shift along the beam direction. The second is based on the
calculation of the Centre-Of-Gravity (COG) of the TIARA pixel detectors counts
and also provides 3D information on a possible beam displacement. The PG vertex
reconstruction was evaluated in two different scenarios. A coincidence time
resolution of 100 ps (rms) can be attained in single proton regime (operating a
reduction of the beam current) and using an external beam monitor to provide a
start trigger for the TOF measurement. Under these conditions, MC simulations
have shown that a millimetric proton range shift sensitivity can be achieved at
2$\sigma$ with 10$^{8}$ incident protons. This level of accuracy would allow to
act in real-time if the treatment does not conform to treatment plan. A worst
case scenario of a 1 ns (rms) TOF resolution was also considered to demonstrate
that a degraded timing information can be compensated by increasing the
acquisition statistics: in this case, a 2 mm range shift would be detectable at
2$\sigma$ with 10$^{9}$ incident protons. The COG method has shown excellent
capabilities of detecting lateral beam displacements: a 2 mm sensitivity was
found at 2$\sigma$ with 10$^{8}$ incident protons.
