In order to estimate the health risk associated with a low dose radiation, the fundamental process of the radiation effects in a living cell must be understood. It is desired that an electron bunch or photon pulse precisely knock a cell nucleus and DNA. The required electron energy and electronic charge of the bunch are several tens keV to 1 MeV and 0.1 fC to 1 fC, respectively. The smaller beam size than micron is better for the precise observation. Since the laser-driven dielectric electron accelerator seems to suite for the compact micro-beam source, a phase-modulation-masked-type laser-driven dielectric accelerator was studied. Although the preliminary analysis made a conclusion that a grating period and an electron speed must satisfy the matching condition of LG/λ = v/c, a deformation of a wavefront in a pillar of the grating relaxed the matching condition and enabled the slow electron to be accelerated. The simulation results by using the free FDTD code, Meep, showed that the low energy electron of 20 keV felt the acceleration field strength of 20 MV/m and gradually felt higher field as the speed was increased. Finally the ultra relativistic electron felt the field strength of 600 MV/m. The Meep code also showed that a length of the accelerator to get energy of 1 MeV was 3.8 mm, the required laser power and energy were 11 GW and 350 mJ, respectively. Restrictions on the laser was eased by adopting sequential laser pulses. If the accelerator is illuminated by sequential N pulses, the pulse power, pulse width and the pulse energy are reduced to 1/N, 1/N and 1/N2, respectively. The required laser power per pulse is estimated to be 2.2 GW when ten pairs of sequential laser pulse is irradiated.