The U-70 Synchrotron from the article produced 76 GeV protons. Electron-volts (eV) are a unit of kinetic energy. That works out to 0.012 microjoules per proton.
This was a pulsed machine, with a repetition rate of 0.11 Hz (9 seconds between pulses). In one pulse there were 1.7×10^13 protons. That adds up to about 200 kJ per pulse: About the same kinetic energy as a Toyota Corolla (1200kg) travelling at 65km/h.
As for exactly how the interaction would happen:
The particles would have bounced around the atoms in Mr Bugorski's head. They would lose energy I think mostly via interactions with atomic electrons. As they decelerated, they would also throw off Bremsstrahlung photons which would create a pencil-beam of x-rays. At that energy there would also be plenty of nuclear interactions which would tend to produce neutrons and prompt gamma rays.
Not all of the kinetic energy in the pulse would have been absorbed by Mr Bugorski's head. The energy deposited into a medium by a proton beam is described by the Bragg Peak. If you look it up, protons of ~200MeV would be almost entirely stopped by a human head (~20cm). But at almost 400 times that energy, I think a large fraction of the kinetic energy would just come out the other side in the form of neutrons, gammas, x-rays, and slightly slowed protons.
Every type of particle, and band of wavelengths, has its own damage mechanism in the body, making the choice of a unit of measure complicated.
I'm not a nuclear physicist, but remember just a bit from my radiation safety training to work in an accelerator lab for a brief time period. Today I work with optics, and there are formulas in the regulations for laser and UV safety. They are quite definitely scientifically based. For instance, the UV wavelength with the highest weighting factor for hazard determination is the absorbance peak of DNA.
Hope that doesn't answer your question (since I'm not an expert anyway), but sheds some light on the complexity of the problem.
Where I went to grad school, they had one of the earliest accelerator physics programs. I remember a story, that the old timers would adjust the beam by closing their eyes and letting the particles cause scintillation inside their eyeballs. That was when beams were vastly less energetic, and in any event, I don't think they did this for very long without realizing the danger. In addition to the beam itself, an accelerator can give off a lot of X rays due to secondary radiation processes. The entire accelerator lab was behind a labyrinth of brick walls, and had an interlock system to make sure nobody was in the tunnel when the machine got started up.
The most important factors are energy per particle, number of particles, width of beam, and is it matter or antimatter. The most general case is it’s going to bore right through you, so cancer and heating is a lesser concern than the gaping hole.
A microwave beam is a different story, but again the specifics make a huge difference. A microwave oven isn’t going to do much though heating the eyeballs can cause issues. Cancer isn’t a serious concern, but boost the power enough and you can get serious and eventually fatal burns.
If someone had a microwave beam of a certain wattage aimed at their head, it is possible to calculate the total absorbed energy and estimated the heating. Is heating the main mechanism for damage with a particle beam too, or is it the ionizing effects? If it's the latter, is it more a radiation exposure problem? If so, how would be frame this in term of Siverets (sp?) or whatever the relevant unit is, and compare it to background and to other medical procedures that use radiation?