Quite a few unfortunately. Being relatively heavy and positively charged, alpha particles in alpha radiation tend to have a very short mean free path, and quickly lose kinetic energy within a short distance of their source. This results in several MeV being deposited in a relatively small volume of material. This increases the chance of cellular damage in cases of internal contamination. In general, external alpha radiation is not harmful since alpha particles are effectively shielded by a few centimeters of air, a piece of paper, or the thin layer of dead skin cells. Even touching an alpha source is usually not harmful, though many alpha sources also are accompanied by beta-emitting radio daughters, and alpha emission is also accompanied by gamma photon emission. If substances emitting alpha particles are ingested, inhaled, injected or introduced through the skin, then it could result in a measurable dose.
The Relative Biological Effectiveness (RBE) is a measure of the fact that alpha radiation is more effective at causing certain biological effects, notably either cancer or cell-death, compared to photon or beta radiation, for equivalent radiation exposure. This is generally attributable to the high Linear Energy Transfer (LET), which is about one ionization of a chemical bond for every Angstrom of travel by the alpha particle. The RBE has been set at the value of 20 for alpha radiation by various government regulations. The RBE is set at 10 for neutron irradiation, and at 1 for beta and ionizing photon radiation.
However, another component of alpha radiation is the recoil of the parent nucleus, due to the conservation of momentum requiring the parent nucleus to recoil, much like the 'kick' of a rifle butt when a bullet goes in the opposite direction. This gives a significant amount of energy to the recoil nucleus, which also causes ionizaton damage. The total energy of the recoil nucleus is readily calculable, and is roughly the weight of the alpha (4 amu) divided by the weight of the parent (typically about 200 amu) times the total energy of the alpha. By some estimates, this might account for most of the internal radiation damage, as the recoil nuclei are typically heavy metals which preferentially collect on the chromosomes. In some studies, this has resulted in a RBE approaching 1,000 instead of the value used in governmental regulations.