Unit 10: Radiation Dosimetry, Radiobiology & Radioprotection (ALARA)

Radiation dosimetry is the quantitative science of measuring and calculating the energy deposited by ionizing radiation in matter and biological tissue.

1. Exposure ($X$)

The historical quantity defining the ionizing capacity of X-ray and gamma-ray photons in dry air under conditions of electronic equilibrium:

where $dQ$ is the total electrical charge of ions of one sign produced in air when all secondary electrons liberated by photons in dry air of mass $dm$ are completely stopped.

• SI Unit: Coulomb per kilogram ($\text{C/kg}$).
• Traditional Unit: The Roentgen (R):

Because producing one ion pair in dry air requires $W_{\text{air}} \approx 33.97\text{ eV} = 33.97\text{ J/C}$, an exposure of $1\text{ R}$ corresponds to an energy absorption of:

2. Absorbed Dose ($D$)

The fundamental physical quantity applicable to all types of ionizing radiation (photons, electrons, neutrons, heavy ions) in any absorbing material:

where $d\bar{\epsilon}$ is the mean energy imparted by ionizing radiation to matter of mass $dm$.

• SI Unit: The Gray (Gy):

• Traditional Unit: The rad (radiation absorbed dose):

3. Kerma ($K$, Kinetic Energy Released per unit MAss)

For uncharged radiation (photons and neutrons), Kerma quantifies the kinetic energy transferred to initial secondary charged particles:

Under conditions of Charged Particle Equilibrium (CPE) where radiative losses are negligible:

4. The Bragg-Gray Cavity Principle

To measure absorbed dose inside a solid medium (e.g., patient tissue or water phantom), an ionization gas cavity is introduced. According to the Bragg-Gray theorem, if the cavity is sufficiently small that it does not perturb the fluence of secondary electrons crossing it:

where $\bar{s}_{\text{med, gas}} = (S/\rho)_{\text{med}} / (S/\rho)_{\text{gas}}$ is the ratio of mass stopping powers of the medium to the cavity gas. This allows ionization current measured in a gas chamber to be converted directly into absorbed dose in patient tissue.

Absorbed dose ($D$, in Grays) quantifies physical energy deposition, but does not describe biological damage. A dose of $1\text{ Gy}$ delivered by dense alpha particles produces vastly greater biological lethality than $1\text{ Gy}$ delivered by dispersed gamma photons.

1. Equivalent Dose ($H_T$)

To quantify biological risk across different radiation modalities, the International Commission on Radiological Protection (ICRP) defines the Equivalent Dose $H_T$ in an organ or tissue $T$:

where $D_{T, R}$ is the absorbed dose delivered by radiation type $R$, and $w_R$ is the dimensionless Radiation Weighting Factor:

| Radiation Type and Energy Spectrum | Radiation Weighting Factor ($w_R$) | Biological Justification | | :--- | :--- | :--- | | Photons (X-rays, $\gamma$-rays, all energies) | $1$ | Reference low-LET standard | | Electrons, positrons, muons (all energies) | $1$ | Sparsely ionizing track structure | | Protons and charged pions | $2$ | Moderate linear energy transfer | | Alpha particles, fission fragments, heavy ions | $20$ | High LET ($\sim 100\text{ keV}/\mu\text{m}$), dense double-strand breaks | | Neutrons: Thermal ($< 1\text{ keV}$) | $2.5$ | Indirect proton recoil / capture | | Neutrons: Epithermal & Fast ($0.1 - 2\text{ MeV}$) | $20$ (Peak at $1\text{ MeV}$) | Maximum recoil proton stopping power | | Neutrons: High Energy ($> 20\text{ MeV}$) | $5 - 10$ | Nuclear spallation reactions |

• SI Unit: The Sievert (Sv):

• Traditional Unit: The rem (Roentgen equivalent man):

2. Effective Dose ($E$)

Different human organs and tissues exhibit vastly different sensitivities to radiation-induced cancer and genetic damage. The Effective Dose $E$ quantifies the overall stochastic health risk to the entire individual:

where $w_T$ is the Tissue Weighting Factor representing the relative contribution of organ $T$ to total stochastic risk:

``` TISSUE / ORGAN (ICRP Publication 103) w_T FRACTION ───────────────────────────────────────────────────────────────────────────── Red Bone Marrow, Colon, Lung, Stomach, Breast, Remainder 0.12 ea 72.0% Gonads (Testes / Ovaries - Genetic Hereditary Detriment) 0.08 8.0% Urinary Bladder, Esophagus, Liver, Thyroid 0.04 ea 16.0% Bone Surface, Brain, Salivary Glands, Skin 0.01 ea 4.0% ───────────────────────────────────────────────────────────────────────────── TOTAL SUM (Whole Body Normalized): 1.00 100.0% ```

Effective dose allows partial-body medical exposures (e.g., a chest CT scan delivering $7\text{ mSv}$) to be compared directly with whole-body natural background radiation ($\approx 2.4 - 3.0\text{ mSv/year}$).

Living biological cells consist of approximately $70 - 85\%$ liquid water. When ionizing radiation traverses biological tissue, energy deposition damages critical cellular targets via two distinct pathways:

``` IONIZING RADIATION │ ┌───────────────┴───────────────┐ ▼ (~35%) ▼ (~65%) DIRECT ACTION INDIRECT ACTION Direct ionization of Radiolysis of cellular water nuclear DNA macromolecule (Free radical generation: •OH) │ │ │ ▼ └───────────────────────► DNA LESIONS (Single & Double Strand Breaks) ```

The Radiolysis of Water

Within $10^{-16}\text{ to }10^{-12}\text{ seconds}$ of radiation passage, water molecules undergo ionization and electronic excitation:

1. Hydrated Electron Formation:

The ejected fast electron thermalizes and becomes trapped in the dielectric dipole cage of surrounding water molecules within $\sim 1\text{ picosecond}$:

2. Hydroxyl Radical Generation:

The radical cation $\text{H}_2\text{O}^{+\bullet}$ reacts with neighboring water via ultrafast proton transfer:

The hydroxyl radical ($\cdot\text{OH}$) is an extraordinarily reactive, neutral oxidizer ($E^\circ = +2.80\text{ V}$) responsible for over $65\%$ of all indirect DNA damage in biological cells.

3. Hydrogen Radicals and Radiolytic Products:

DNA Lesion Spectrum and Double-Strand Breaks (DSBs)

A typical mammalian cell nucleus ($V \approx 500\,\mu\text{m}^3$) contains $6 \times 10^9$ base pairs of double-helical genomic DNA. A uniform absorbed dose of $1\text{ Gy}$ of low-LET X-rays produces:

• $\sim 100,000$ water ionization events.
• $\sim 1,000 - 2,000$ base damages (oxidized guanines, e.g., 8-oxo-dG).
• $\sim 1,000$ Single-Strand Breaks (SSBs).
• $\sim 40$ Double-Strand Breaks (DSBs).

While SSBs are repaired with high fidelity by DNA ligases using the intact complementary strand as a template, Double-Strand Breaks (where both opposing phosphodiester backbones are severed within $10 - 20$ base pairs) represent the critical lethal lesion. Misrepair of DSBs via Non-Homologous End Joining (NHEJ) causes dicentric chromosomal aberrations, ring chromosomes, translocations, and apoptotic cell death.

The biological consequence of a given absorbed dose depends fundamentally on the spatial track structure of ionization events, characterized by Linear Energy Transfer (LET).

Linear Energy Transfer (LET)

LET quantifies the average energy transferred locally to the absorbing medium per unit path length ($\text{keV}/\mu\text{m}$):

• Low-LET Radiation ($< 10\text{ keV}/\mu\text{m}$): Cobalt-60 gamma rays ($0.2\text{ keV}/\mu\text{m}$), $250\text{ kVp}$ X-rays ($2\text{ keV}/\mu\text{m}$), fast electrons. Ionization events are spaced hundreds of nanometers apart, producing isolated, easily repairable lesions.
• High-LET Radiation ($> 20\text{ keV}/\mu\text{m}$): Alpha particles ($100\text{ keV}/\mu\text{m}$), heavy recoil ions ($>1000\text{ keV}/\mu\text{m}$). Ionization events form dense, continuous columns that deposit dozens of ion pairs across a single $2\text{ nm}$ DNA diameter, producing complex, unrepairable clustered damage.

Relative Biological Effectiveness (RBE)

RBE is defined as the ratio of absorbed dose of reference $250\text{ kVp}$ X-rays ($D_{\text{ref}}$) to the dose of test radiation ($D_{\text{test}}$) required to achieve an identical biological endpoint (e.g., $10\%$ cell survival):

``` RBE (Relative Biological Effectiveness) ▲ │ PEAK RBE (~ 100 keV/μm) │ /\ │ / \ │ / \ Overkill Effect │ Low-LET / \ (Wasted Dose) │ ───────┐ / \_____ │ └─────────/ └───────────────────┴──────────┴────► LET (keV / μm) 0.1 10 100 1000 ```

• As LET increases from $1$ to $100\text{ keV}/\mu\text{m}$, RBE climbs steeply to a maximum peak at $\approx 100\text{ keV}/\mu\text{m}$. At this optimal density, the average spacing between ionizing events ($\sim 2\text{ nm}$) corresponds precisely to the diameter of the DNA double helix!
• Beyond $100\text{ keV}/\mu\text{m}$, RBE drops due to the overkill effect: more energy is deposited in the cell nucleus than is required for cell sterilization, wasting excess dose.

The Linear-Quadratic (LQ) Cell Survival Model

Cell survival curves plot surviving fraction $S(D)$ versus absorbed dose $D$. The universally accepted biophysical model is the Linear-Quadratic Model:

where:

• $\alpha$ is the linear coefficient ($\text{Gy}^{-1}$), representing lethal single-hit "unrepairable" track damage (intra-track DSB).
• $\beta$ is the quadratic coefficient ($\text{Gy}^{-2}$), representing two independent radiation tracks interacting to produce a lethal lesion (inter-track repairable damage).
• The ratio $\alpha/\beta$ (units of $\text{Gy}$) defines the dose at which the linear and quadratic cell-killing contributions are equal ($\alpha D = \beta D^2 \implies D = \alpha/\beta$).
• Early-responding tissues / Tumors: High $\alpha/\beta \approx 10\text{ Gy}$ (steep linear survival, minimal fractionation sparing).
• Late-responding normal tissues: Low $\alpha/\beta \approx 2 - 3\text{ Gy}$ (broad curve shoulder, massive sparing with fractionated radiation therapy).

The biological consequences of ionizing radiation are categorized into two fundamentally distinct classes:

1. Deterministic Effects (Tissue Reactions)

Arise from radiation-induced killing of large populations of functional tissue cells:

• Threshold Dose: A clear, finite dose threshold $D_{\text{th}}$ exists below which no clinical effect is observable. Above the threshold, the severity of the injury increases monotonically with dose.
• Pathology: Acute cell depletion, vascular damage, and fibrotic tissue death.
• Representative Thresholds:
• Temporary Sterility (Testes): $0.15\text{ Gy}$ ($150\text{ mGy}$).
• Depression of Hematopoiesis (Bone Marrow): $0.50\text{ Gy}$.
• Skin Erythema (Reddening): $3 - 5\text{ Gy}$; Dry Desquamation: $10\text{ Gy}$; Necrosis: $>15\text{ Gy}$.
• Radiation Cataractogenesis (Lens of Eye): $0.5\text{ Gy}$ (ICRP 118).
• Acute Radiation Syndrome (ARS, Whole Body):
• Hematopoietic Syndrome: $1 - 6\text{ Gy}$ ($50\%$ lethal dose without medical care: $\text{LD}_{50/60} \approx 3.5 - 4.0\text{ Gy}$).
• Gastrointestinal (GI) Syndrome: $6 - 20\text{ Gy}$ (destruction of crypt stem cells; lethal within $1 - 2\text{ weeks}$).
• Cerebrovascular Syndrome: $>20 - 50\text{ Gy}$ (cardiovascular collapse and brain edema; lethal within $24 - 48\text{ hours}$).

2. Stochastic Effects (Cancer and Heritable Mutations)

Arise from non-lethal, mutagenic alteration of a single somatic stem cell that survives with an oncogenic mutation:

• No Threshold: Governed by probability rather than severity. Even an infinitesimal dose has a non-zero probability of inducing a malignant transformation.
• Severity is Independent of Dose: A cancer induced by a $10\text{ mSv}$ dose is clinically identical to one induced by a $1000\text{ mSv}$ dose; only the probability of occurrence scales with dose.
• Latent Period: Solid cancers manifest after long latency periods ($10 - 40\text{ years}$); leukemias manifest after $2 - 10\text{ years}$.

``` STOCHASTIC EFFECTS DETERMINISTIC EFFECTS Probability of Cancer Severity of Tissue Injury ▲ ▲ │ / │ / │ / │ / │ / │ / │ LNT Model / │ / │ / │ / │ / │ / │ / │ / │_________/ │______________/_ └─────────┴────────────────► Dose D └──────────────┴─────► Dose D 0 0 D_th (Threshold) ```

The Linear No-Threshold (LNT) Model

For radiation protection regulation, the ICRP, NCRP, and IAEA adopt the Linear No-Threshold (LNT) hypothesis: The excess lifetime risk of fatal stochastic cancer is assumed to be strictly proportional to effective dose, extrapolating linearly from high-dose epidemiological data (Hiroshima and Nagasaki atomic bomb survivors, Life Span Study LSS) down to zero dose.

• ICRP Detriment Coefficient:

For an occupational worker receiving an annual effective dose of $20\text{ mSv}$, the lifetime excess cancer mortality risk is:

The operational foundation of external radiation protection rests on the three classical pillars: Time, Distance, and Shielding.

1. Time Optimization

The total accumulated absorbed dose $D$ is directly proportional to exposure duration $t$:

Minimizing residence time in a radiation field reduces dose proportionally. Practicing complex manipulation protocols using non-radioactive mockups ("dry runs") before handling high-activity sources minimizes hands-on handling time.

2. Distance Optimization (The Inverse Square Law)

For an isotropic point source emitting radiation, photon flux spreads over the spherical surface area $4\pi d^2$. The radiation intensity and dose rate $\dot{D}$ decrease inversely with the square of the distance $d$:

Doubling distance ($2d$) reduces dose rate by a factor of $4$ ($25\%$); increasing distance tenfold ($10d$) reduces dose rate by a factor of $100$ ($1\%$)!

• Practical Application: Never touch unshielded gamma or beta sources with bare hands! Using a $30\text{ cm}$ remote handling tongs instead of direct fingertip contact ($1\text{ cm}$) reduces the dose rate by a factor of:

Gamma Constant ($\Gamma$) for Point Sources

The exposure rate $\dot{X}$ at distance $d$ from a point gamma emitter of activity $A$ is parameterized by the Specific Gamma-Ray Constant $\Gamma$:

For air kerma rate constant $\Gamma_\delta$ ($\mu\text{Gy}\cdot\text{m}^2/\text{GBq}\cdot\text{h}$):

• $^{60}\text{Co}$: $\Gamma \approx 308\,\mu\text{Gy}\cdot\text{m}^2 / (\text{GBq}\cdot\text{h}) = 1.32\text{ R}\cdot\text{m}^2 / (\text{Ci}\cdot\text{h})$
• $^{137}\text{Cs}$: $\Gamma \approx 78\,\mu\text{Gy}\cdot\text{m}^2 / (\text{GBq}\cdot\text{h}) = 0.33\text{ R}\cdot\text{m}^2 / (\text{Ci}\cdot\text{h})$
• $^{192}\text{Ir}$: $\Gamma \approx 115\,\mu\text{Gy}\cdot\text{m}^2 / (\text{GBq}\cdot\text{h}) = 0.48\text{ R}\cdot\text{m}^2 / (\text{Ci}\cdot\text{h})$
• $^{99m}\text{Tc}$: $\Gamma \approx 17\,\mu\text{Gy}\cdot\text{m}^2 / (\text{GBq}\cdot\text{h}) = 0.076\text{ R}\cdot\text{m}^2 / (\text{Ci}\cdot\text{h})$

3. Shielding Optimization

When distance and time limits are reached, physical barriers attenuate radiation exponentially:

Combining distance and shielding:

Under the guidance of the International Commission on Radiological Protection (ICRP Publication 103), global radiation safety is governed by three fundamental ethical and operational principles:

1. Justification: No practice involving exposure to radiation should be adopted unless it produces a net positive societal or individual benefit sufficient to offset the radiation detriment.

2. Optimization (The ALARA Principle): All exposures must be maintained As Low As Reasonably Achievable (ALARA), economic and societal factors being taken into account.

3. Dose Limitation: Total individual doses must not exceed statutory regulatory limits to prevent deterministic effects and limit stochastic risk.

Statutory Dose Limits (ICRP & IAEA Basic Safety Standards)

| Exposed Population Group | Effective Dose Limit (Whole Body) | Equivalent Dose: Lens of Eye | Equivalent Dose: Skin & Extremities | | :--- | :--- | :--- | :--- | | Occupational Radiation Workers | $20\text{ mSv/year}$ (averaged over 5 yr; max $50\text{ mSv}$ in any single yr) | $20\text{ mSv/year}$ (reduced from $150$) | $500\text{ mSv/year}$ ($50\text{ rem}$) | | Pregnant Radiation Workers | $1\text{ mSv}$ to fetus post-declaration | — | — | | General Public | $1.0\text{ mSv/year}$ ($0.1\text{ rem/year}$) | $15\text{ mSv/year}$ | $50\text{ mSv/year}$ |

Notice that medical patients undergoing diagnostic or therapeutic procedures are strictly exempt from dose limits; their exposure is governed exclusively by clinical justification and protocol optimization.

Internal Dosimetry & Biokinetic Compartmental Models

When radionuclides are inhaled, ingested, or absorbed into wounds, they distribute throughout biological compartments, delivering continuous internal radiation until eliminated by physical radioactive decay ($\lambda_p$) and biological clearance ($\lambda_b$). The effective elimination constant $\lambda_{\text{eff}}$ is:

The effective half-life $T_{\text{eff}}$ is always strictly shorter than both the physical half-life $T_p$ and biological clearance half-life $T_b$.

• Example: Cesium-137 ($^{137}\text{Cs}$): Physical $T_p = 30.17\text{ years}$, but biological clearance half-life in human tissue is $T_b \approx 70 - 100\text{ days}$. The internal effective half-life is $T_{\text{eff}} \approx 70 - 100\text{ days}$, preventing multi-decade internal retention.
• Example: Strontium-90 ($^{90}\text{Sr}$): Bone-seeker incorporated into hydroxyapatite mineral matrix ($T_b \approx 50\text{ years}$), giving an effective half-life $T_{\text{eff}} \approx 18\text{ years}$ and delivering high cumulative bone marrow dose.

The cumulative internal dose delivered over 50 years ($70\text{ years}$ for children) is quantified as the Committed Effective Dose $E(50)$:

where $e(50)$ is the radionuclide-specific dose coefficient ($\text{Sv/Bq}$).

Operational radiation safety transforms fundamental physical principles into engineering practices designed to keep occupational and public doses As Low As Reasonably Achievable (ALARA).

1. ICRP Human Respiratory Tract Model (HRTM, ICRP Publication 66 & 130)

Inhaled radioactive aerosols are deposited in respiratory compartments according to aerodynamic diameter ($AMAD$, Activity Median Aerodynamic Diameter, typically $1 - 5\,\mu\text{m}$):

• Extrathoracic airways ($ET_1, ET_2$): Nose, pharynx, larynx.
• Bronchial ($BB$) and bronchiolar ($bb$) tree: Ciliated mucociliary escalator transports particles upward to the esophagus within hours.
• Alveolar-interstitial region ($AI$): Gas-exchange region lacking cilia; clearance is rate-limited by chemical solubility in alveolar macrophages.

Particles are classified by their chemical absorption rate into blood:

• Type F (Fast): $100\%$ absorbed into systemic blood within minutes to hours (e.g., soluble nitrates, fluorides, pertechnetates).
• Type M (Moderate): Half-times of days to weeks (e.g., oxides of uranium, carbonates, cobalt).
• Type S (Slow): Highly insoluble refractory particulates (e.g., high-fired $\text{UO}_2, \text{PuO}_2$) retained in lung tissue and pulmonary lymph nodes for decades ($T_{\text{clearance}} > 7,000\text{ days}$).

2. In Vivo and In Vitro Bioassay Monitoring

To monitor occupational internal contamination:

• In Vivo Whole-Body Counting (WBC): The worker sits in a heavily shielded low-background room (pre-WWII steel plate shielding) surrounded by large HPGe or $\text{NaI(Tl)}$ detectors to quantify high-energy gamma emitters ($^{60}\text{Co}, ^{137}\text{Cs}$). Specialized low-energy Germanium (LEGe) lung counters detect the faint $59.5\text{ keV}$ photons of $^{241}\text{Am}$ to assay insoluble plutonium contamination.
• In Vitro Bioassay: Radiochemical separation and alpha spectrometry of 24-hour urine and fecal samples to detect sub-picocurie levels of alpha-emitting actinides ($^{239}\text{Pu}, ^{238}\text{U}, ^{232}\text{Th}$).

3. ALARA Engineering Controls

In nuclear engineering facility design:

1. Zoned Ventilation: Air flows strictly from zones of lowest contamination potential to zones of highest contamination potential (Zone 1: Offices $\to$ Zone 2: Hallways $\to$ Zone 3: Radiation Laboratories $\to$ Zone 4: Hot Cells). Hot cells operate under permanent negative gauge pressure ($-250\text{ Pa}$) to prevent aerosol leakage.

2. HEPA Filtration: Exhaust air passes through redundant series of Nuclear-Grade High-Efficiency Particulate Air (HEPA) filters ($99.97\%$ capture efficiency for $0.3\,\mu\text{m}$ particles) and charcoal beds to trap radioiodine vapors.

ICRP International System of Radiological Protection: Dose Limits & Detriment Metrics

``` ICRP DOSE LIMIT ARCHITECTURE (ICRP 103) │ ┌───────────────────────┴───────────────────────┐ ▼ ▼ OCCUPATIONAL WORKERS GENERAL PUBLIC • Effective Dose: 20 mSv/yr • Effective Dose: 1.0 mSv/yr (Averaged over 5 yr; max 50 mSv in any 1 yr) (Higher in special circumstances) • Equivalent Dose Lens: 20 mSv/yr • Equivalent Dose Lens: 15 mSv/yr • Equivalent Dose Skin: 500 mSv/yr • Equivalent Dose Skin: 50 mSv/yr • Equivalent Dose Hands/Feet: 500 mSv/yr • Pregnant Public: Unrestricted ```

Major Radiological Accidents and Lessons Learned

1. Chernobyl (1986, INES Level 7):

Prompt criticality power excursion driven by positive void coefficient ($+4.5\,\beta$) during low-power turbine test with control rods withdrawn, followed by xenon pit poisoning. Release of $1.8 \times 10^{18}\text{ Bq}$ of $^{131}\text{I}$ and $8.5 \times 10^{16}\text{ Bq}$ of $^{137}\text{Cs}$. Resulted in 28 acute radiation deaths and over 6,000 thyroid cancers among children who drank milk contaminated with $^{131}\text{I}$.

2. Fukushima Daiichi (2011, INES Level 7):

Loss of all off-site power and on-site emergency diesel generators following a $14\text{-meter}$ tsunami. Loss of decay heat cooling caused core meltdowns in Units 1, 2, and 3. Zircaloy-water reaction ($Zr + 2\text{H}_2\text{O} \to \text{ZrO}_2 + 2\text{H}_2$) produced explosive hydrogen gas, blowing off reactor building roofs. Prompt evacuation and food bans prevented any acute radiation sickness or fatal deterministic exposures.

3. Goiânia (1987, INES Level 5):

Scrap metal scavengers dismantled an abandoned teletherapy machine, breaching a sealed cesium-137 chloride source ($50.9\text{ TBq} \approx 1375\text{ Ci}$). Attracted by its glowing blue luminescence in the dark, individuals spread the soluble powder across family members and neighborhoods. 4 fatalities from acute bone marrow syndrome; over 112,000 individuals screened.