Radiation and Medical Imaging

Lessons

Senior 6 · Physics

Radiation and Medical Imaging

5 lessons

Lesson 2 of 5

Radiation Dose, Effective Half-Life and Safety

25 min

A badge that keeps the score

Every radiographer, nuclear-medicine technologist and radiotherapy physicist wears a small badge clipped to the chest. It does not beep or flash. Once a month it is posted to a laboratory, which heats or scans the material inside and sends back a single number in millisieverts. Over a working life those numbers are added up, and the law says the total must stay below fixed limits. To understand what the badge measures, you need three dose quantities, each answering a different question: how much energy was absorbed, how harmful that kind of radiation is, and how much the exposed organs matter to the whole person.

Absorbed dose: energy per kilogram

When ionising radiation passes through tissue, the secondary electrons and ions of Lesson 1 eventually turn the energy into chemical change and a little heat. The basic quantity is the energy deposited per unit mass:

$D = \frac{E}{m}$

Definition

Absorbed dose $D$ is the energy of ionising radiation absorbed per unit mass of material. Its SI unit is the gray (Gy): $1\,\mathrm{Gy} = 1\,\mathrm{J\,kg^{-1}}$ . Example: a tumour of mass 0.050 kg that absorbs 0.10 J has received $D = \frac{0.10}{0.050} = 2.0\,\mathrm{Gy}$ .

Worked example 1: a dangerous dose is a tiny amount of heat

The World Health Organization gives the threshold for acute radiation syndrome as about 1 Sv; for gamma rays this is an absorbed dose of about 1 Gy. A person of mass 60 kg receives 1.0 Gy to the whole body. (a) How much energy is absorbed? (b) Treating the body as water ( $c = 4200\,\mathrm{J\,kg^{-1}\,K^{-1}}$ ), by how much does its temperature rise?

Solution

(a) $E = Dm = 1.0 \times 60 = 60\,\mathrm{J}$ .

(b) $\Delta T = \frac{E}{mc} = \frac{60}{60 \times 4200} = 2.4 \times 10^{-4}\,\mathrm{K}$ .

Interpretation: 60 J is the energy of a cup of tea cooling by a fraction of a degree. The harm comes not from heat but from where the energy goes: into breaking bonds in DNA and other key molecules, as in Lesson 1.

Equivalent dose: not all grays are equal

The same absorbed dose does more biological damage when the radiation has a high LET, because the ionisation is packed so closely that DNA is broken in several places at once. The absorbed dose is therefore multiplied by a radiation weighting factor $w_{R}$ :

$H = w_R\,D$

Radiation	$w_{R}$
photons (X-rays, gamma rays) of all energies	1
electrons (β⁻, β⁺) and muons	1
protons	2
alpha particles, fission fragments, heavy ions	20
neutrons	2.5 to 20, depending on energy

Radiation weighting factors recommended by the International Commission on Radiological Protection (ICRP Publication 103, 2007).

Definition

Equivalent dose $H$ in an organ or tissue is the absorbed dose multiplied by the radiation weighting factor, summed over the types of radiation: $H = \sum w_R D_R$ . Its unit is the sievert (Sv); $w_{R}$ has no unit, so $1\,\mathrm{Sv} = 1\,\mathrm{J\,kg^{-1}}$ too. Example: 1 mGy of alpha radiation in the lung gives $H = 20 \times 1 = 20\,\mathrm{mSv}$ .

Effective dose: the risk to the whole person

Medical exposures are rarely uniform: a dental radiograph irradiates the jaw, a thyroid scan the neck. Organs also differ in how readily radiation induces cancer in them. Each equivalent dose is therefore multiplied by a tissue weighting factor $w_{T}$ , and the results are added:

$E = \sum_T w_T H_T, \qquad \sum_T w_T = 1$

Tissue or organ	$w_{T}$ each
red bone marrow, colon, lung, stomach, breast, remaining tissues	0.12
gonads	0.08
bladder, oesophagus, liver, thyroid	0.04
bone surface, brain, salivary glands, skin	0.01

Tissue weighting factors from ICRP Publication 103 (2007). Check: $6 (0.12) + 0.08 + 4 (0.04) + 4 (0.01) = 1.00$ .

Because the factors add up to 1, a uniform whole-body equivalent dose of 2 mSv is an effective dose of 2 mSv. The effective dose is the number a personal dosimeter estimates and the number used to compare examinations.

Definition

Effective dose $E$ is the sum, over all irradiated organs, of each organ's equivalent dose multiplied by its tissue weighting factor. Unit: sievert (Sv). It is a protection quantity, designed for comparing exposures and setting limits for people in general, not a precise measure of one individual's risk.

Worked example 2: two partial-body exposures

(a) Radon gas is the main source of natural radiation (WHO). Over a year, the alpha particles from radon and its decay products give a person's lungs an absorbed dose of 0.50 mGy. Find the equivalent dose to the lungs and the effective dose. (b) A thyroid scan gives the thyroid an absorbed dose of 20 mGy from gamma rays, and other organs a negligible dose. Find the effective dose.

Solution

(a) $H_{\text{lung}} = w_R D = 20 \times 0.50 = 10\,\mathrm{mSv}$ . $E = w_T H = 0.12 \times 10 = 1.2\,\mathrm{mSv}$ .

(b) $H_{\text{thyroid}} = 1 \times 20 = 20\,\mathrm{mSv}$ . $E = 0.04 \times 20 = 0.80\,\mathrm{mSv}$ .

Comment: the thyroid in (b) received a far larger absorbed dose (20 mGy against 0.50 mGy), yet its effective dose is smaller, because gamma rays have $w_{R} = 1$ and the thyroid is a smaller fraction of the whole-body risk.

Two kinds of harm

Figure 2.1: The two kinds of biological effect. In (b) the straight line below about 100 mSv is an assumption used for protection, not a measurement.

Deterministic effects (tissue reactions) happen only when so many cells in an organ are killed that it stops working properly. There is a threshold dose; above it, the effect is certain and its severity grows with dose. Examples: skin burns, hair loss, cataracts, and acute radiation syndrome, whose threshold the WHO gives as about 1 Sv.
Stochastic effects come from a cell that survives with damaged DNA and may, years later, grow into a cancer; damage to germ cells could in principle be passed on. Their probability, not their severity, rises with dose. The WHO notes a significant increase in cancer risk above about 100 mSv. Below that the risk is too small to measure directly, so for protection it is assumed to fall in proportion to the dose, with no threshold: the linear no-threshold (LNT) model. On this basis the ICRP uses a nominal risk of about 5% of fatal and non-fatal cancers per sievert, averaged over a whole population.

Source of exposure	Typical effective dose
natural background, world average (UNSCEAR)	2.4 mSv per year
dental radiograph	0.005 mSv
chest radiograph	0.1 mSv
CT of the brain	1.6 mSv
CT of the abdomen and pelvis	7.7 mSv
whole-body PET/CT	22.7 mSv

Medical values for adults from RadiologyInfo.org (American College of Radiology and RSNA); individual doses vary with the machine and the patient. Ultrasound and MRI give no ionising dose at all.

Half-lives inside the body

A radionuclide given to a patient (Lesson 5) leaves the body in two ways at once: its nuclei decay, with the physical half-life $T_{P}$ of Unit 5, and the body excretes the substance (in urine, sweat and breath), with a biological half-life $T_{B}$ . Both are random processes with constant probabilities per unit time, so their rates add:

$\frac{\mathrm{d}N}{\mathrm{d}t} = -\lambda_P N - \lambda_B N = -(\lambda_P + \lambda_B)N \quad\Longrightarrow\quad \lambda_E = \lambda_P + \lambda_B$

Since $\lambda = \frac{\ln 2}{T}$ , dividing by $\ln 2$ gives the effective half-life:

$\frac{1}{T_E} = \frac{1}{T_P} + \frac{1}{T_B} \qquad\text{or}\qquad T_E = \frac{T_P T_B}{T_P + T_B}$

Condition: biological removal must itself be exponential (a constant fraction excreted per unit time), which is a good first model. $T_{E}$ is always shorter than both $T_{P}$ and $T_{B}$ ; if one is much shorter than the other, $T_{E}$ is close to the shorter one.

Figure 2.2: Physical decay and biological removal together empty the body faster than either alone (biological half-life illustrative).

Worked example 3: technetium-99m in a patient

Technetium-99m has a physical half-life of 6.01 h. In a certain compound its biological half-life is 24 h. (a) Find the effective half-life. (b) What fraction of the injected activity remains in the patient after 24 h? (c) Compare with physical decay alone.

Solution

(a) $T_E = \frac{6.01 \times 24}{6.01 + 24} = \frac{144.2}{30.01} = 4.81\,\mathrm{h}$ .

(b) $\frac{A}{A_0} = \left(\frac{1}{2}\right)^{\frac{t}{T_E}} = \left(\frac{1}{2}\right)^{\frac{24}{4.806}} = 2^{-4.99} = 0.031$ , about 3.1%.

(c) Physical decay alone: $2^{-\frac{24}{6.01}} = 2^{-3.99} = 0.063$ , about 6.3%. Excretion halves what remains after a day.

The system of protection

The international Basic Safety Standards (IAEA General Safety Requirements Part 3, 2014, co-sponsored by the WHO) apply three principles to every use of ionising radiation:

Justification: an exposure must do more good than harm. A CT scan to find internal bleeding after a road accident is justified; a CT scan "just to check" is not.
Optimisation (ALARA): every justified dose is kept as low as reasonably achievable, economic and social factors being taken into account. In imaging this means the lowest tube current and time that still give a usable picture, collimating the beam to the region of interest, child-sized settings for children, avoiding repeat exposures, and choosing ultrasound or MRI where they answer the clinical question. The time, distance and shielding rules of Unit 5 are how ALARA is achieved for staff.
Dose limits: for workers and the public, the total from all planned sources must stay below fixed limits. Patients have no dose limit, because the dose is given for their own benefit; justification and optimisation protect them instead.

Group (IAEA Basic Safety Standards)	Effective dose limit
radiation workers	20 mSv per year averaged over 5 years, and not more than 50 mSv in any single year
members of the public	1 mSv per year

Workers' doses are measured with personal dosimeters. In a thermoluminescent dosimeter (TLD), radiation lifts electrons into traps in a crystal such as lithium fluoride; when the crystal is later heated, the electrons fall back and emit light whose total amount is proportional to the dose. Electronic personal dosimeters use a small semiconductor detector, show the dose at once and sound an alarm at a preset dose rate.

Worked example 4: weighing a risk against a benefit

A CT scan of the abdomen and pelvis gives an adult an effective dose of about 7.7 mSv. (a) How many years of average natural background (2.4 mSv per year) is this? (b) Using the LNT model and a nominal risk of 5% per sievert, estimate the extra lifetime chance of cancer. (c) Evaluate whether the scan should be done for a patient with suspected appendicitis.

Solution

(a) $\frac{7.7}{2.4} = 3.2$ years.

(b) $7.7 \times 10^{-3}\,\mathrm{Sv} \times 0.05\,\mathrm{Sv^{-1}} = 3.9 \times 10^{-4}$ , roughly 1 in 2600.

(c) The estimated risk is small and is itself uncertain: it rests on the LNT assumption and is a population average, higher for children and lower for older adults. A missed, burst appendix is an immediate threat to life. The scan is justified if it will change the treatment, but ALARA still applies: in a child or a pregnant woman, ultrasound, which gives no ionising dose, would usually be tried first.

Warning

Common mistakes. (1) Treating gray and sievert as the same: they have the same base units, but the sievert includes the weighting factors, so 1 Gy of alpha radiation is 20 Sv. (2) Adding half-lives: the effective half-life is shorter than both, so it can never be $T_{P} + T_{B}$ . (3) Saying "below 100 mSv there is no risk": the risk is too small to measure, and protection assumes it is not zero. (4) Using the 5% per sievert figure to predict one person's fate: it is a population average for planning.

Tip

Climb the ladder of dose with three questions. Gray: how much energy per kilogram? Sievert (equivalent): how damaging is this radiation? ( $\times w_R$ ) Sievert (effective): how much do these organs matter to the whole person? ( $\times w_T$ , then add.) For half-lives, remember that rates add, so reciprocals add.

Radiation units: Absorbed, Equivalent & Effective dose, by Dr. Paulien Moyaert. Watch how the gray becomes a sievert through the two weighting factors, then check her factors against the tables in this lesson.

✦

Practice

Define the gray and the sievert, and explain why radiation protection needs both.
A worker of mass 75 kg absorbs 15 mJ of gamma-ray energy, spread evenly through the body. Find the absorbed dose and the effective dose.
In a year, a person's lungs absorb 0.20 mGy from alpha particles and 0.30 mGy from gamma rays. Find the equivalent dose to the lungs and its contribution to the effective dose.
Fluorine-18 has a physical half-life of 110 min. If its biological half-life in a patient is 120 min, find its effective half-life.
A patient is treated with iodine-131 (physical half-life 8.0 days). The measured effective half-life in the thyroid is 6.0 days. Find the biological half-life.
Classify each as deterministic or stochastic, giving a reason: a cataract after an accident; leukaemia 10 years after exposure; a skin burn from a radiotherapy beam; a mutation passed to a child.
A radiographer's annual effective doses over five years are 30, 25, 15, 18 and 10 mSv. Did she stay within the IAEA limits? Explain.
Keza says: "A chest X-ray gives only 0.1 mSv, so the ALARA principle does not apply to it." Evaluate her statement.

Show answersHide answers

Gray: absorbed energy per kilogram ( $\mathrm{J\,kg^{-1}}$ ). Sievert: absorbed dose weighted for the type of radiation (and, for effective dose, the organs exposed). Equal grays of different radiations, or to different organs, carry different risks, so the gray alone cannot compare harm.
$D = \frac{0.015}{75} = 2.0 \times 10^{-4}\,\mathrm{Gy} = 0.20\,\mathrm{mGy}$ ; gamma rays have $w_{R} = 1$ and the exposure is uniform, so $E = 0.20\,\mathrm{mSv}$ .
$H = 20(0.20) + 1(0.30) = 4.3\,\mathrm{mSv}$ ; contribution $= 0.12 \times 4.3 = 0.52\,\mathrm{mSv}$ .
$T_E = \frac{110 \times 120}{230} = 57\,\mathrm{min}$ .
$\frac{1}{T_B} = \frac{1}{6.0} - \frac{1}{8.0} = \frac{1}{24}$ , so $T_{B} = 24$ days.
Cataract: deterministic (threshold; cell killing in the lens). Leukaemia: stochastic (a chance effect of surviving damaged cells, long delay). Skin burn: deterministic. Inherited mutation: stochastic.
Total $= 98\,\mathrm{mSv}$ , an average of 19.6 mSv per year, within 20 mSv per year averaged over 5 years; no single year exceeds 50 mSv. She stayed within the limits, but only just, so her work practice (time, distance, shielding) should be reviewed under ALARA.
Wrong. ALARA applies to every justified exposure, however small; the dose should still be the lowest that gives a useful image (collimation, correct settings, no repeats), especially because the LNT model assumes no safe threshold. ALARA does not mean refusing the X-ray: when it is justified, the small risk is outweighed by the benefit.

Key points

$D = \frac{E}{m}$ (gray); $H = w_{R} D$ (sievert; $w_{R} = 1$ for photons and electrons, 20 for alpha); $E = \sum w_T H_T$ (sievert).
Deterministic effects have a threshold and grow in severity; stochastic effects grow in probability, assumed linear with no threshold (about 5% per sievert).
$\frac{1}{T_E} = \frac{1}{T_P} + \frac{1}{T_B}$ : the effective half-life is shorter than both.
Protection: justification, optimisation (ALARA), dose limits (workers 20 mSv per year averaged over 5 years; public 1 mSv per year); personal dosimeters record staff doses.

Sources and credits

32.2: Biological Effects of Ionizing Radiation, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Facts quoted from Ionizing radiation and health effects, World Health Organization fact sheet, 27 July 2023.
Weighting factors and nominal risk from The 2007 Recommendations of the International Commission on Radiological Protection, ICRP Publication 103; dose limits from Radiation Protection and Safety of Radiation Sources: International Basic Safety Standards, IAEA Safety Standards Series No. GSR Part 3, 2014. World-average background from UNSCEAR, 2008 Report.
Typical medical doses quoted from Radiation Dose, RadiologyInfo.org (American College of Radiology and Radiological Society of North America).
Photo: Radiation and the International Atomic Energy Agency - Personal Dosimeters by IAEA Imagebank, Wikimedia Commons, licensed under CC BY 2.0.
Photo: Mikael Häggström at radiology in 2019 by Mikael Häggström, Wikimedia Commons, CC0.
Video: Radiation units: Absorbed, Equivalent & Effective dose, Dr. Paulien Moyaert, embedded from YouTube.

The two diagrams, the worked examples and the practice exercises are written by Ganzaa.

1A badge that keeps the score

2Absorbed dose: energy per kilogram

3Equivalent dose: not all grays are equal

4Effective dose: the risk to the whole person

5Two kinds of harm

6Half-lives inside the body

7The system of protection

A badge that keeps the score

Absorbed dose: energy per kilogram

Equivalent dose: not all grays are equal

Effective dose: the risk to the whole person

Two kinds of harm

Half-lives inside the body

The system of protection