Radiation Dosimetry

Radiation is known to produce a number of deleterious effects in a living system. Therefore, it is important to properly assess the benefits and risks to a patient from a given nuclear medicine procedure, ensuring that the benefits outweigh the risks, if any, by a significant margin. One factor that strongly influences the intensity or probability of radiation effects is the energy absorbed by the tissue (radiation dose). Therefore, the assessment of risk can be performed only if one knows the radiation dose that will be delivered to a patient by a particular procedure. This chapter describes the various methods for determining the radiation dose to a patient from internally administered radionuclides or radiopharmaceuticals. However, in the actual risk estimation, two other important factors, the relative biologic effectiveness of the radiation and the tissue sensitivity, are also needed; these are discussed in Chapter 15. Radiation exposure and hazard from external radioactive source are discussed in Chapter 16.

General Comments on Radiation Dose Calculations

In nuclear medicine procedures, it is almost impossible to measure the radiation dose directly using any kind of radiation detector. Instead, this has to be calculated by using a variety of physical and biologic data and mathematical equations especially developed for this purpose. It should be emphasized, however, at the outset that the accuracy of these calculations is complicated by several factors:

Two types of data are needed in the calculation of radiation dose to a patient, physical and biological. The physical data on the decay of radionuclides and interaction of radiation with biologic tissues are available fairly accurately as was explained in Chapters 2, 3, and 6. However, the biologic distribution data are quite difficult to obtain accurately. These are, in practice, extrapolated from animal data or, in some cases, limited human data and have relatively large errors compared with the physical data.
A variety of assumptions routinely made in these calculations (e.g., uniform distribution of a radionuclide in an organ, instant uptake of the radiopharmaceutical in the given organ, or single exponential biologic elimination) seldom hold true in practice.
Radiation dose calculations have usually been obtained using standard or reference anatomic models (originally referred to as “standard man”). These quite often differ appreciably in actual patients, and there are increasing efforts to incorporate such variations within the adult population, as well as children of different ages.

The calculated dose is obtained by taking an average over a large volume (>1 cm³) and therefore cannot be used to estimate radiation dose at
the cellular level. It is hypothesized that biologic effects can be better understood by knowing the radiation dose at a cellular level (microdosimetry).

These factors, when combined, make radiation dose calculations susceptible to large errors. Hence, the radiation dose calculations presented here and elsewhere represent an average dose from a given procedure that may vary by a factor of 2, or even more, in an individual case.

Definitions and Units

Obviously, to calculate the radiation dose, one should know the meaning of radiation dose and radiation dose rate and their units of measurement.

Radiation Dose, D. Radiation dose, or, more precisely, radiation absorbed dose, is a measure of the total energy absorbed from the radiation by 1 unit of mass of a substance. In the SI Unit System, absorbed dose is measured in gray (Gy). A gray equals 1 joule (J) of energy absorbed/kilogram of a substance (J/kg). Two commonly used and derived units from gray are centigray (cGy = 10^-2 Gy) and milligray (mGy = 10^-3 Gy). The old unit of radiation dose is rad, which is a short notation for the radiation absorbed dose. A rad is defined as 100 ergs (1 erg = 10^-7 J) of absorbed energy per gram of a tissue or substance. One rad equals (1/100) Gy or equals a cGy, and 1 Gy equals 100 rad.

Table 7.1 Parameters and Symbols Used in the Calculation of Radiation Dose

Parameter	Symbol	Unit
Any given radiation	i	—
Total number of radiations	n	—
Energy of radiation i	E_i	MeV
Frequency of emission of radiation i	n_i	Per decay
Equilibrium dose constant for radiation i	Δ_i	g·rad/µCi·h
Absorbed fraction	φ_i(T ← S)	—
Self-absorbed fraction	φ_i	—
Target organ	T	—
Source organ	S	—
Mass of target organ	M	g
Radiation dose rate at time t	dD/dt	rad/h
Radiation dose	D	rad
Radioactivity in source at time t	A(t)	µCi
Radioactive dosage at time 0	A₀	µCi
Fraction localized in the source	f	—
Physical half-life	T_1/2	h
Biologic half-life	T_1/2 Bio	h
Effective half-life in the source	T_1/2 (eff)	h

Radiation Dose Rate, dD/dt. Radiation dose rate, dD/dt, is defined as the amount of energy absorbed per unit time per unit of mass of tissue. Its units may be expressed in various ways, such as mGy (rad) per minute, cGy (rad) per hour, or Gy (rad) per day.

Parameters or Data Needed

In a typical situation in nuclear medicine, a known amount of radioactivity of a radiopharmaceutical is administered to a patient. A certain fraction f of the radiopharmaceutical is then localized in the organ of interest. One is interested in knowing the radiation dose delivered to this organ and sometimes to various other organs as well. Two types of data are required for these calculations: one related to the decay characteristics of the radionuclide and one to the biologic distribution and elimination of the radiopharmaceutical. Table 7.1 lists the various parameters needed for these calculations, together with the units and symbols used here.
Some parameters have been defined in Chapters 2 and 3; some are self-evident, and the others are discussed in this chapter. We have retained the old units because a large amount of the data needed in these calculations is still in old units. It is easier to convert the final equations, tables, or results in the SI units, which we have done here at the appropriate places.

Calculation of the Radiation Dose

To calculate the radiation dose, one has to determine the average amount of energy absorbed by 1 g of tissue of a target (organ of interest) from the total energy released by the decay of a given amount of radioactivity. Because x- or γ-rays are more penetrating than particulate radiation, a small x- or γ-ray emitting source, localized at any site in the body, irradiates practically every organ of the body. For example, an x- or γ-ray emitting radionuclide localized only in the liver (source S) delivers the radiation dose to all other organs (targets T) of the body in addition to the liver. Thus, if a radiopharmaceutical is localized in multiple organs (sources), the radiation dose from each source to each organ (target) has to be calculated before the final determination of the radiation dose to each organ can be made by adding all contributions. The following four steps are involved in the radiation-dose determinations:

Calculation of the rate of energy emission (erg/h) of the various types of radiation emitted by the radionuclide in the source volume.
Calculation of the rate of energy absorption from these radiations by the target volume.
Calculation of the average dose rate, dD/dt.
Calculation of the average dose D.

This method of radiation dose calculation is known as the absorbed fraction method. The first three steps require mainly physical data such as decay characteristics, organ shape and size, and so on, whereas the fourth step requires biologic distribution data.

Step 1—Rate of Energy Emission. Let us first consider a radionuclide that emits only one type of radiation (emission frequency = 1) of energy E (MeV) per decay. One microcurie (3.7 × 10⁴ decay/s) of this radionuclide will, therefore, emit energy at a rate of 3.7 × 10⁴ × E MeV/s. If we change the unit of energy from MeV to erg (1 MeV = 1.6 × 10^-6 erg) and the unit of time from second to hour (1 h = 3600 s), the rate of energy emission by this radionuclide becomes equal to 3.7 × 10⁴ × 1.6 × 10^-6 × 3600 × E erg/(h × µCi), or 213E erg/(h × µCi).

In the case of a radionuclide that emits more than one radiation, say 1, 2, 3, … n, with emission frequencies n₁, n₂, n₃, …, n_n and energies of E₁, E₂, E₃, …, E_n, respectively, the rate of energy emission for each type of radiation will be equal to 213n₁E₁ erg/(h × µCi) for radiation 1, 213n₂E₂ erg/(h × µCi) for radiation 2, and so on.

Step 2—Rate of Energy Absorption. To calculate the rate of absorption of energy by a target volume T from a radionuclidic distribution in a source volume S, we have to define a new quantity known as the absorbed fraction, φ_i(T←S). The absorbed fraction φ_i(T←S) is defined as the ratio of energy absorbed by a target volume T from a radiation i to the amount of energy released by a radionuclidic distribution in volume S in the form of radiation i. In other words

In most problems encountered in nuclear medicine, the radioactivity is distributed within the target volume T itself (i.e., T is the same as S). For example, we want to know the radiation dose to the liver when it is distributed in the liver itself. Here, the source (liver) is also the target (liver). In such cases, the absorbed fraction is known as the self-absorbed fraction and is expressed simply as φ_i.

Once φ_i(T←S) is known, the rate of energy absorption by a target volume T is simply obtained by multiplying the rate of energy emission of radiation i (from step 1 = 213n_iE_i) by the absorbed fraction φ_i(T←S), or rate of energy absorption by the target volume from radiation i = 213n_iE_i × φ_i(T←S)

Table 7.2 Self-Absorbed Fraction φ_i for Different γ-Ray Energies and Various Organs

	Energy (keV)
Organ	15	30	50	100	200	500	1000
Bladder	0.885	0.464	0.201	0.117	0.116	0.116	0.107
Stomach	0.860	0.414	0.176	0.101	0.101	0.101	0.093
Kidneys	0.787	0.298	0.112	0.066	0.068	0.073	0.067
Liver	0.898	0.543	0.278	0.165	0.158	0.157	0.144
Lungs	0.665	0.231	0.089	0.049	0.050	0.051	0.045
Pancreas	0.666	0.195	0.068	0.038	0.042	0.044	0.040
Skeleton	0.893	0.681	0.400	0.173	0.123	0.118	0.110
Spleen	0.817	0.331	0.128	0.071	0.073	0.077	0.070
Thyroid	0.592	0.149	0.048	0.028	0.031	0.032	0.029
Total body	0.933	0.774	0.548	0.370	0.338	0.340	0.321

If there are n radiations, the rate of the total energy absorption will be equal to the sum of the energies absorbed from each radiation, that is,

The above expression can be written in a concise form as

where

is the sum of all terms when i changes from 1 to n.

How does one determine φ_i(T←S)? Determination of the absorbed fraction requires the exact knowledge of the interaction of radiation with matter, discussed in the previous chapter. Computations of φ_i(T←S) for a number of source and target combinations, from these basic mechanisms of interaction of radiation with matter, are quite involved and require the use of large computers. The Journal of Nuclear Medicine published a variety of tables listing φ_i(T←S) for different x- or γ-ray energies and source and target volumes. As an illustration, Table 7.2 lists the absorbed fraction for various organs of a standard man for different x- or γ-ray energies when the radionuclidic distribution is within the same organ (i.e., T is the same as S). For other combinations, the reader is referred to the original articles.^*

General Comments on φ_i(T←S). The maximum value of φ_i(T←S) can only be 1. This occurs when all emitted energy is absorbed in the target. The minimum value is 0 and occurs when there is no absorption of energy in the target.

In the case of particulate radiations such as β particles, conversion electrons, or α particles, almost all energy emitted by a radionuclide is absorbed in the volume of distribution itself, provided the source volume is larger than 1 cm₃. Then, φ_i(T←S) = 0, unless T and S are the same, in which case φ_i = 1. The same holds true for x- or γ-radiation with energies less than 10 keV. Thus, these radiations deliver radiation doses only in the volume of distribution and not outside of it.

For x- or γ-radiations with energies higher than 10 keV, the absorbed fraction φ_i(T←S) strongly depends on the energy of the x- or γ-ray; the shape and size of the source volume; and the shape, size, and distance of the target volume. In general, φ_i first decreases with an increase in the energy of the x- or γ-ray and then eventually levels off. Notice in Table 7.2 that the absorbed fraction, φ_i drops sharply with energy up to 100 keV but then does not change significantly from 100 to 500 keV for various organs listed.

Step 3—Dose Rate, dD/dt. If one now divides the rate of energy absorption by the target with its mass M, this will give the rate of energy absorption per gram of tissue, which when divided by
100 (to convert erg/g to rad) yields the dose rate for each microcurie of activity; that is, the dose rate,

By defining Δ_i = 2.13n_iE_i, this reduces to

If the source volume contains A(t) µCi at time t, then the dose rate dD from A(t) amount of radioactivity becomes

Step 4—Average Dose, D. The radioactivity A(t) localized in an organ is generally a fraction f of the administered dosage A₀ and is being continuously eliminated, with an effective half-life of T_1/2 (eff); that is,

Therefore, the dose rate dD/dt is continuously decreasing with time and eventually becomes zero. How does one compute the total dose to the patient from the time of administration (t = 0) to the time when the dose rate has finally been reduced to zero? For this, one has to integrate the dose rate dD/dt from 0 to ∝ time, or

(dD/dt)dt. This involves integration of A(t) that, for a simple case such as equation (7.3), leads to the following expression:

In the case where the target and source volume are the same, the self-dose D is given by

Note that the factor (fA₀/M) in the above equations is the concentration of the radioactivity in the organ of localization. Therefore, it is the concentration and not the total amount of the radioactivity in an organ that is the primary determinant of the radiation dose.

From these expressions, it is evident that to minimize the radiation dose to a patient, it is desired to have smaller amounts of radioactivity (A₀), radiopharmaceuticals with shorter T_1/2 (eff), and radionuclides with smaller absorbed fractions (which means γ-ray energy higher than 100 keV and no particulate emission as discussed earlier; p. 43).

Cumulated Radioactivity

Equations (7.4) and (7.5) assume that the uptake in the organ is instantaneous and the disappearance of the radioactivity from the source can be expressed by a single exponential term, such as in equation (7.3). This does not have to be so, and often the bio-kinetics of the radionuclidic distribution is more complex. An example is shown in Figure 7.1. In this case, equation (7.3) cannot be used to describe the time behavior of the radioactivity in blood or plasma or in organs one and two. For these, the exact time activity curve, A(t), should be used to calculate the cumulated radioactivity Ã, which is defined as

A(t)dt. The unit of Ã as used here is h × µCi, which means A(t) is measured in µCi and time in hours. Because these calculations are more complicated, we shall not go into the details here.

Simplification of Radiation Dose Calculations Using “S” Factor

In an attempt to simplify the calculation of radiation dose in routine clinical situations, Snyder and colleagues combined the physical data such as radiations emitted by a radionuclide and the absorbed fractions for various source and target combinations of a standard man into one single term, which they called “S” factor. In fact, S factor is nothing but the dose rate from 1 µCi as given by equation (7.1). Therefore, S is expressed as follows:

Figure 7.1. Biokinetics of an intravenously administered radiopharmaceutical. Blood or plasma radioactivity decreases with time. A fraction of blood or plasma radioactivity is extracted by organ 1 and organ 2. Another fraction, not shown here, is excreted. For multicompartmental kinetics such as in this example, none of these curves can be fitted by a monoexponential function given by equation (7.3). In this case, cumulated activity Ã needed for radiation dose calculations is determined by either graphic methods or multiexponential curve-fitting methods.

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