# Dosimetry: Calculating Radiation Dose for Medical Applications ## Mastering Dosimetry: Pivotal Science for Radiological Protection, Nuclear Medicine, and Radiotherapy Applications

Dosimetry, the science of measuring and calculating the radiation dose absorbed by matter, plays a pivotal role in radiological protection, nuclear medicine, and radiotherapy. This field is a cornerstone for accurate radiation dose estimation, minimising potential health hazards, particularly in medical treatment and occupational settings. The mathematics of dosimetry enables accurate and consistent dose calculation by integrating various aspects, such as the physical properties of radiation, interaction with matter, and biological effects.

Radiation can be ionising or non-ionising, depending on its energy level. Ionising radiation, such as alpha, beta, gamma, and X-rays, possesses enough energy to ionise atoms and molecules in the matter it interacts with, causing biological damage. Non-ionising radiation, such as ultraviolet, visible light, and radio waves, typically does not cause ionisation but can still cause harm at high intensities. Dosimetry primarily focuses on ionising radiation due to its potential risks and in various applications, including medical diagnosis and treatment, industrial processes, and research.

The interaction of ionising radiation with matter leads to energy deposition, which can cause localised damage to biological tissues. To quantify this damage, dosimetry employs several dose units, such as the Gray (Gy) for absorbed dose, Sievert (Sv) for equivalent and effective dose, and Becquerel (Bq) for activity. These units provide a standardised means of comparing radiation exposure and evaluating risks across different scenarios.

Dosimetry calculations often rely on mathematical models and simulations to account for the complex behaviour of radiation within matter. Monte Carlo simulations, for example, use random sampling techniques to model radiation transport and interaction, providing a statistically accurate representation of dose distribution in a given medium. In addition, these models can be customised to consider various factors, such as radiation type, energy spectrum, and the geometrical arrangement of the source, medium, and detectors.

In medical applications, dosimetry is critical in determining the optimal radiation dose for diagnostic and therapeutic procedures. For example, in diagnostic imaging, such as computed tomography (CT) scans and X-ray examinations, dosimetry helps balance image quality and patient dose to ensure accurate diagnosis with minimal risk. In radiotherapy, dosimetry helps plan the radiation dose distribution to deliver the maximum dose to the tumour while sparing healthy tissues, improving the likelihood of successful treatment.

Occupational dosimetry monitors and controls radiation exposure in workplaces such as nuclear power plants, research facilities, and hospitals. Tracking radiation dose using personal dosimeters ensures compliance with regulatory limits, minimising health risks and promoting a safety culture.

## Basic Concepts and Units in Dosimetry

Radiation can be classified into two main categories: ionising and non-ionising. Ionising radiation carries enough energy to remove tightly bound electrons from atoms and molecules, forming charged particles or ions. Non-ionising radiation, on the other hand, does not possess sufficient energy to cause ionisation but can still produce thermal and photochemical effects.

Ionising radiation is further divided into directly ionising and indirectly ionising radiation. Directly ionising radiation consists of charged particles, such as alpha and beta particles, which can ionise atoms and molecules in their path due to their inherent electric charge. Alpha particles are heavy, positively charged particles emitted from the nucleus of heavy elements during radioactive decay. They have low penetration power and can be stopped by a sheet of paper or a few centimetres of air. Beta particles are lighter, negatively charged particles (electrons) or positively charged particles (positrons) also emitted during radioactive decay. Their penetration power is higher than alpha particles but can be stopped by thin plastic, glass, or aluminium, directly ionising radiation, including photons, gamma and X-rays, and neutrons.

Gamma rays are high-energy electromagnetic radiation emitted during nuclear reactions or radioactive decay. They are highly penetrating and require dense materials, like lead or concrete, for effective shielding. X-rays are similar to gamma rays in nature but are produced through different mechanisms, such as the interaction between high-energy electrons and matter. Neutrons are uncharged particles that can cause ionisation indirectly by interacting with atomic nuclei, causing the emission of charged particles. Neutrons can be highly penetrating, and their shielding requires materials rich in hydrogen, like water or polyethene.

These various types of radiation possess different energies and interaction properties with matter. Understanding their characteristics is essential for effective radiation protection, dosimetry, and the development of radiation-based technologies in medicine, industry, and research. Knowing the energy levels and properties of each radiation type makes it possible to design appropriate shielding, monitoring, and control strategies to minimise potential health risks and optimise the beneficial applications of radiation in numerous fields.

## Dosimetric Quantities

In dosimetry, two key quantities are essential for understanding and assessing the impact of ionising radiation on matter and living organisms: the absorbed dose (D) and the equivalent dose (H). These quantities provide a standardised way to quantify the potential biological effects of different types of radiation, allowing for informed decision-making in medical, industrial, and research settings.

The absorbed dose (D) represents the mean energy imparted by ionising radiation per unit mass of matter. It is typically expressed in Gray (Gy) units, where 1 Gy equals the absorption of 1 joule of radiation energy per kilogram of matter. The absorbed dose indicates the amount of energy deposited in a material or tissue, which can cause ionisation and potential biological damage in the case of living organisms. However, the absorbed dose does not account for the varying effectiveness of different radiation types in causing harm.

The equivalent dose (H) considers the biological effectiveness of the radiation by accounting for the type and energy of the radiation. The absorbed dose is multiplied by a radiation weighting factor (wR) to calculate the equivalent dose, reflecting the specific radiation type’s relative biological effectiveness (RBE). The unit for equivalent dose is the sievert (Sv), and 1 Sv equals 1 Gy multiplied by the radiation weighting factor.

Equivalent dose (H) = Absorbed dose (D) × Radiation weighting factor (wR)

The radiation weighting factors account for the differences in the ionising potential and the density of ionisation events caused by various types of radiation. For example, alpha particles have a higher radiation weighting factor than beta particles, gamma rays, or X-rays because of their greater ability to cause biological damage.

Creating a comprehensive table of dosimetry values for the human body is challenging due to the factors involved, such as the type of radiation, energy, and individual tissue sensitivities. However, I can provide a simplified table illustrating the organ and tissue weighting factors (wT) used for calculating the effective dose (E) from the equivalent dose (H) in radiation protection:

Organ/TissueWeighting Factor (wT)
Red Bone Marrow0.12
Colon0.12
Lung0.12
Stomach0.12
Breast0.05
Liver0.05
Oesophagus0.05
Thyroid0.05
Skin0.01
Bone Surface0.01
The Remainder (Other Organs/Tissues)0.12

Please note that these values are based on the International Commission on Radiological Protection (ICRP) Publication 103 recommendations and are subject to change with updated recommendations.

The effective dose (E) can be calculated by summing the product of the equivalent dose (H) and the weighting factor (wT) for each organ or tissue:

E = Σ [wT * H(T)]

This calculation allows for the comparison of radiation exposure across different organs, tissues, and radiation types, accounting for the varying sensitivities of different organs to ionising radiation.

## Interaction Of Radiation With Matter

Attenuation is a critical concept in understanding the behaviour of radiation as it interacts with and passes through matter. As radiation traverses a medium, it loses intensity due to attenuation, including absorption (energy transfer to the medium) and scattering (deflection of radiation in different directions). Understanding attenuation is vital for designing effective shielding, radiation protection, and dosimetry strategies in various applications, including medicine, industry, and research.

The intensity of the radiation (I) after passing through a given thickness (x) of material can be determined using the Beer-Lambert law:

I(x) = I0 * e-μx

Where I0 is the initial intensity, μ is the linear attenuation coefficient, and x is the thickness of the material. The linear attenuation coefficient (μ) is a property of the material and the type and energy of the radiation. It quantifies the likelihood of interaction between the radiation and the material, with higher values indicating greater attenuation.

The linear attenuation coefficient (μ) varies depending on the type and energy of radiation and the composition of the specific tissue in the human body. Here is a simplified list of approximate linear attenuation coefficients for some common tissues and materials in the human body at a photon energy of 100 keV:

Tissue / MaterialLinear Attenuation Coefficient (μ) (cm⁻¹)
Air0.0002
Water0.17
Fat0.19
Muscle0,20
Lung Tissue0.23
Blood0.21
Bone0.35 – 0.6

Note that these values are approximate and specific to the given photon energy of 100 keV. The linear attenuation coefficients for other energies or types of radiation, such as alpha or beta particles, will be different. Additionally, the values may vary for specific tissues or materials, even within the same general category.

For a more detailed and accurate list of linear attenuation coefficients, refer to the National Institute of Standards and Technology (NIST) XCOM database, which provides values for various materials and energies.

Attenuation is a significant factor when designing radiation shielding, as it determines the thickness and material properties required to reduce radiation exposure to acceptable levels. Different materials have varying attenuation characteristics for specific types of radiation. For instance, dense materials like lead or concrete effectively attenuate gamma rays, while materials rich in hydrogen, such as water or polyethene, are suitable for shielding against neutrons.

Understanding attenuation is essential for optimising image quality and radiation dose in medical imaging and radiotherapy. By knowing how radiation interacts with different tissues and materials, medical professionals can adjust imaging parameters and plan therapeutic interventions to minimise the radiation dose to patients while maximising diagnostic or therapeutic outcomes.

## Linear Energy Transfer (LET)

Linear energy transfer (LET) is critical in radiation physics and dosimetry. It quantifies the energy a charged particle deposits per unit length of its path through a medium. It is typically expressed in units of keV/µm (kilo-electron volts per micrometre) or MeV/cm (mega-electron volts per centimetre). The LET of radiation significantly impacts its biological effects and interaction properties with matter.

High LET radiation, such as alpha particles, is characterised by a dense ionisation pattern along the particle’s track, causing more ionisations and excitations per unit length than low LET radiation, such as gamma rays or X-rays. Due to the high ionisation density, high LET radiation has a greater probability of causing severe and clustered DNA damage in living cells, which is less likely to be accurately repaired by cellular repair mechanisms. Consequently, high LET radiation is generally considered more biologically effective than low LET radiation of the same absorbed dose.

Low LET radiation, such as gamma rays and X-rays, has a sparse ionisation pattern and deposits its energy over a longer distance in the medium. This type of radiation typically causes less severe and more isolated DNA damage, which is more likely to be repaired by cellular mechanisms. However, low LET radiation can still cause significant biological effects, especially at high doses or dose rates.

Below is a table summarising the properties of alpha, beta, and gamma radiation, including their LET values and other relevant information:

SymbolParticle/PhotonChargeMass (amu)LET (keV/μm)Penetrating PowerShielding Material
AlphaαHelium nucleus+2450-200LowPaper, clothing
BetaβElectron-1~00.1-10ModeratePlastic, glass
• Alpha (α) radiation consists of helium nuclei (2 protons and 2 neutrons) and has a high LET value, which means it causes many ionisations and excitations per unit length. Due to its high LET, alpha radiation has low penetrating power and can be stopped by paper or clothing.
• Beta (β) radiation consists of electrons, which are much lighter and have a negative charge. Beta particles have a lower LET value than alpha particles, indicating fewer ionisations and excitations per unit length. Beta radiation has moderate penetrating power and can penetrate materials like plastic or glass but can be shielded by denser materials.
• Gamma (γ) radiation is a type of electromagnetic radiation consisting of photons with no charge or mass. Gamma rays have the lowest LET values, implying minimal ionisations and excitations per unit length. However, they have high penetrating power and can pass through many materials, requiring lead or concrete to provide effective shielding.

It is important to note that these values are approximate and may change on the specific radionuclide and energy of the radiation.

Understanding the LET of radiation is essential for various applications, including radiation protection, medical imaging, and radiotherapy. Knowing the LET can guide the choice of appropriate shielding materials and designs to minimise exposure to radiation protection. In medical imaging, the knowledge of LET can help balance image quality and patient dose by optimising imaging parameters. In radiotherapy, considering the LET can help design treatment plans that maximise the radiation dose to tumours while minimising the dose to healthy tissues.

## Mathematical Models in Dosimetry

The point-source model is the simplest dosimetric model used to estimate the radiation dose from a source, assuming that the radiation source is an infinitesimally small point. This assumption simplifies the mathematical calculations and provides a good approximation when the source dimensions are much smaller than the distance to the point of interest or when the dose is calculated far from the source.

In this model, the dose at any point in space can be calculated using the inverse square law:

D(r) = K * (1/r2)

D(r) is the dose at a distance r from the source, and K is a constant representing the source strength. The inverse square law demonstrates that the dose decreases rapidly as the distance from the source increases. This relationship is valid for isotropic sources that emit radiation uniformly in all directions.

The point-source model is widely used in basic dosimetry calculations, particularly for initial estimates and simple geometries. However, it has limitations when applied to more complex situations or when the source dimensions cannot be ignored. In such cases, more advanced models like line-source or volumetric-source models may be required, which account for the spatial distribution of the radiation source.

Despite its simplicity, the point-source model provides valuable insight into the fundamental behaviour of radiation in space and serves as a basis for more complex dosimetric models. By understanding the relationship between dose and distance from the source, radiation protection professionals can design effective shielding, develop safe work practices, and estimate potential radiation exposures. In medical applications, the point-source model can be used as a starting point for treatment planning in radiotherapy, allowing clinicians to estimate dose distributions around tumours and minimise the dose to healthy tissues.

## Line-Source Model

The line-source model is an extension of the point-source model, addressing the limitations of the point-source model by considering a one-dimensional linear radiation source instead of an infinitesimally small point. This model is useful when the radiation source has a significant length, or the source geometry is more complex than a single point.

To calculate the dose at a point in space, the line-source model integrates the contribution of the infinitesimal point sources along the line:

The line-source model equation you provided calculates the dose at a point in space by integrating the contribution of the infinitesimal point sources along the line:

D(r, θ) = ∫ K(x) * 1 / [(r2 + x2) – 2rx*cos(θ)]3/2 dx

Where:

• D(r, θ) is the dose at a distance r from the source and an angle θ from the line,
• K(x) is the source strength per unit length (e.g., radiation intensity),
• r is the radial distance from the centre of the line source,
• x is the position along the line,
• θ is the angle between the point of interest and the line source.

The line-source model is particularly useful in situations where the geometry of the radiation source is long and thin, such as in brachytherapy or when analyzing radiation from transmission lines. It is important to note that the line-source model is still a simplification and might not fully account for all factors in real-world situations.

The line-source model is more versatile than the point-source model, as it can account for the spatial distribution of the radiation source and provide a more accurate estimation of the dose distribution around the source. This model is particularly relevant when the radiation source is linear or elongated, such as radioactive wires used in brachytherapy or linear accelerator beams used in external beam radiotherapy.

Although it increased complexity compared to the point-source model, the line-source model still relies on simplifying assumptions, such as the one-dimensional nature of the source. In cases where the source geometry is more complex, other models, such as the volumetric-source model, may be required for accurate dose estimation.

## Volume-Source Model

The volume-source model is a more advanced dosimetric model that accounts for a three-dimensional radiation source, such as a solid radioactive material or an irradiated volume within a patient. This model offers a higher level of accuracy in dose estimation, particularly when the radiation source has a complex geometry or when the dose distribution needs to be determined within a three-dimensional volume.

The volume-source model integrates the contribution of infinitesimal point sources within the volume to calculate the dose at a point in space. For example, the dose at a point (r, θ, φ) in a spherical coordinate system can be expressed as:

D(r, θ, φ) = ∫∫∫ K(r’, θ’, φ’) * 1/[(r2 + r’2 ) – 2rr’cos(θ)sin(θ’)cos(φ-φ’)]3/2 * r’2 sin(θ’) dr’ dθ’ dφ’

Where D(r, θ, φ) is the dose at a distance r from the source, K(r’, θ’, φ’) is the source strength per unit volume at a point (r’, θ’, φ’) within the source, and r’, θ’, and φ’ are the spherical coordinates of the infinitesimal point sources within the volume.

The volume-source model is especially relevant in medical applications, such as brachytherapy and external beam radiotherapy, where the radiation source or target volume has a complex three-dimensional shape. By accurately modelling the spatial distribution of the source, this model allows for more precise treatment planning and dose optimisation.

In radiation protection, the volume-source model can help design effective shielding and containment for radioactive materials and estimate potential exposure doses in various scenarios.

Calculation of Equivalent Dose

Estimating the equivalent dose (H) is crucial in radiation protection and dosimetry, as it takes into account not only the absorbed dose (D) but also the relative biological effectiveness (RBE) of the radiation, which varies depending on the type and energy of the radiation. The equivalent dose is calculated by multiplying the absorbed dose by a radiation weighting factor (wR):

H = wR * D

Here is a table with some example values of absorbed dose (D), radiation weighting factor (wR), and the resulting equivalent dose (H). Please note that these values are for illustrative purposes only and may not represent real-life situations.

X-rays1001100
Gamma Rays50150
Electrons2001200
Neutrons305 (for 1 MeV neutrons)150
Alpha Particles1020200

Remember that the radiation weighting factor (wR) depends on the radiation’s type and energy, and the values provided in the table are examples based on general recommendations by the International Commission on Radiological Protection (ICRP). In practice, the absorbed dose (D) and equivalent dose (H) values will vary depending on the specific radiation source, energy, and exposure conditions.

Radiation weighting factors (wR) account for the differences in biological damage caused by different types of radiation. These factors depend on the radiation type and energy. They represent the ratio of the biological effectiveness of a specific type of radiation to that of a reference radiation (usually X-rays or gamma rays).

For example, wR values for photons (X-rays and gamma rays) and electrons are 1, indicating that these low-LET radiations have similar biological effects to the reference radiation. On the other hand, alpha particles have a wR value of 20, reflecting their higher LET and greater capacity to cause biological damage.

For neutrons, wR values vary with energy, ranging from 2 for low-energy neutrons to 20 for high-energy neutrons. This variation reflects the complex energy-dependent interactions of neutrons with matter and their wide range of biological effectiveness.

In radiation protection, an equivalent dose is used to compare the potential biological effects of exposure to different radiation types and establish dose limits for occupational and public exposure. An equivalent dose is essential for treatment planning and evaluating the risks associated with radiation-based diagnostics and therapies in medical applications.

In radiation therapy, dosimetric calculations are essential for developing an effective treatment plan that delivers an optimal radiation dose to the tumour while minimising the dose to surrounding healthy tissues. Accurate dosimetry is crucial for maximising the likelihood of tumour control and reducing the risk of complications in normal tissues. The mathematical models used in radiation therapy planning include the following:

The dose-volume histogram (DVH): A DVH is a graphical representation of the dose distribution within a target volume (e.g., tumour) and critical structures (e.g., organs at risk). It displays the volume of the target or structure receiving a specific dose or higher. DVHs allow clinicians to evaluate the treatment plan’s effectiveness in terms of tumour coverage and dose homogeneity, as well as the potential toxicity to nearby healthy tissues. By comparing different treatment plans using DVHs, clinicians can choose the optimal plan to maximise tumour control while minimising the risk of side effects.

The linear-quadratic (LQ) model: The LQ model is used to describe the cell survival probability (S) after exposure to radiation as a function of dose (D) and two radiosensitivity parameters, α and β:

S = exp(-αD – βD2)

The LQ model is widely used in radiation therapy to estimate the tumour control probability (TCP) and normal tissue complication probability (NTCP). TCP represents the likelihood of eradicating all tumour cells and achieving local control, while NTCP quantifies the risk of complications in healthy tissues. In addition, the LQ model guides the optimisation of fractionation schemes, which involves dividing the total radiation dose into smaller doses (fractions) delivered over several sessions, and dose escalation strategies to improve treatment outcomes.

Dosimetry calculations are essential for estimating the radiation dose received by an individual or a material. Here’s an example of a simple dosimetry calculation for an external radiation source

Assumptions:

1. The source is a point source of gamma radiation.
2. The individual is at a fixed distance from the source.
3. The source emits radiation isotropically.
4. The radiation dose rate is constant during the exposure time.
5. No shielding is present between the source and the individual.

Parameters:

1. The activity of the source (A) in Becquerels (Bq)
2. Distance from the source to the individual (d) in meters (m)
3. Exposure time (t) in hours (h)
4. Dose rate constant (Γ) in microsieverts per hour per becquerel (µSv/h/Bq)

Step 1: Determine the dose rate at the distance d. The dose rate (Drate) can be calculated using the inverse square law and the dose rate constant:

Drate = (Γ × A) / (4 × π × d²)

Step 2: Calculate the total dose. To calculate the total dose (Dtotal) received by the individual during the exposure time, multiply the dose rate by the exposure time:

Dtotal = Drate * t

Example: Suppose a source with an activity of 1,000 Bq emits gamma radiation. The individual is located 2 meters away from the source and is exposed for 1 hour. The dose rate constant for the particular gamma radiation is 0.2 µSv/h/Bq.

Step 1: Calculate the dose rate. Drate = (0.2 µSv/h/Bq × 1,000 Bq) / (4 × π × 2² m²) Drate ≈ 7.96 µSv/h

Step 2: Calculate the total dose. Dtotal = 7.96 µSv/h × 1 h Dtotal ≈ 7.96 µSv

In this example, the individual would receive a total dose of approximately 7.96 µSv from the gamma radiation source.

### Conclusion

The mathematics of dosimetry is fundamental in understanding and quantifying the interaction of radiation with matter. Accurate dosimetry is essential in various applications, such as radiation protection, nuclear medicine, and radiotherapy, to minimise potential health risks and ensure effective treatment. Mathematical models, including point-source, line-source, and volume-source models, facilitate the calculation of the absorbed dose and equivalent dose for different radiation types and source geometries.

In radiation therapy, dosimetric calculations are vital for treatment planning, ensuring that an optimal radiation dose is delivered to the tumour while minimising the dose to surrounding healthy tissues. Mathematical models, such as the dose-volume histogram (DVH) and the linear-quadratic (LQ) model, are employed to evaluate treatment efficacy and potential toxicity. DVHs offer a graphical representation of dose distribution within the target volume and critical structures, allowing clinicians to assess the treatment plan’s effectiveness and compare alternative plans. The LQ model helps estimate the tumour control probability (TCP) and normal tissue complication probability (NTCP), guiding the optimisation of fractionation schemes and dose escalation strategies.

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