For beta particles, internal conversion and Auger electrons, which deposit essentially 100% of their energy into the body, the following formula has been derived:

when using appropriate conversion factors, this equation yields:

For moderately energetic X- and gamma-rays, which deposit only part of their energy in the body, the following formula has been derived. In this case, φ typically falls in the range of about 0.1 – 0.7

when using appropriate conversion factors, this equation yields:

When the two equations are combined, the total dose equation, which represents contributions from both a beta particle and a gamma ray (non-penetrating and penetrating radiation, is:

Using a slightly different approach, we can derive a second equation that takes into account both penetrating (X, gamma) and non-penetrating (betas, electrons) radiations. The equation shown below accounts for all these factors and indicates that the total dose is a summation of both penetrating and non-penetrating contributions.

It is important to consider every radiation not only from the standpoint of self-irradiation of an organ, but also radiation transmitted to every other organ in the body. For example, we know that Tc^99m sulfur colloid localizes in the liver, spleen, and marrow. To calculate liver dose, we must be concerned with dose from the liver to the liver, from the spleen to the liver, and from the marrow to the liver.

SAMPLE CALCULATION:

Problem: calculate radiation dose to the liver from the intravenous injection of 8 mCi of Tc^99m sulfur colloid.

Assumptions: t_phys= 6 hr, t_biol= infinitely long, t_eff= 6 hr; Distribution: 90% in liver, 5% in spleen, 5% in bone marrow. While the contribution to liver dose from marrow is negligible, it has been included in the following equation for completeless.