Internal Radiation Dosimetry
Dosimetry for injected/ingested/inhaled/absorbed radioactivity/radiopharmaceuticals.
Factors affecting internal dosimetry
- Route of exposure: inhalation, ingestion, IV injection, absorbtion
- Where did radioactivity go (e.g. thyroid, or excreted through bladder, etc)
- How long did it stay there (i.e. physical and biological half lives)
- What type of radiation at what energies?
- What type of patient (sex, age, etc.)
Absorbed dose, energy deposited per unit mass, mGy (J deposited / kg of absorber) = 100 rad (100 ergs / g).
Equivalent dose, HT, accounts for biologic damage inflicted by radiation type. $H_T = D_T × w_R$ where $D_T$ is absorbed dose in a tissue and $w_R$ is the radiation type weighting factor (photons/electrons = 1, protons = 2, alphas = 20, neutrons ≈ 10 at 1 MeV but very energy dependent). Units of Sv.
Effective dose accounts for biologic damage to a particular organ or tissue type, $H_E = \sum{D_T × w_R × w_T}$.
Table of wT
Dose rate: mGy/hr
Exposure rate (Rrhr) = air kerma rate (mGy/hr) × 0.115
K (Gy) = X (C/kg) * 33.7 = X (R) * 0.00869
Dmed = f Dair where f is factor to convert dose in air to dose in medium and depends on the mass attenuation coeffs (energy dependent).
Soft tissue f ≈ 1.1, bone f ≈ 1, increases rapidly to ~ 5 starting at E < 100 keV
Air kerma rate constant Γ in (mGy · m2) / (GBq · hr)
$\dot{X} = A \Gamma / d^2$ where exposure rate, $\dot{X}$, depends on activity $A$, gamma rate constant, $\Gamma$, and inversely with the square of distance, $1/d^2$. Applies to point source in air.
Absorbed dose rate per unit activity: Gy/(Bq · s) or mGy/(MBq · s)
MIRD
Dosimetry is generally done using the Medical Internal Radiation Dosimetry (MIRD) equation.
$D(r_t) = \sum_s{\tilde{A}(r_s) \, S(r_t \rightarrow r_s)}$
- $D(r_t)$ is dose to target organ
- $\tilde{A}(r_s)$ is time integrated activity of the source organ (e.g. mGy/MBq-s)
- $\tilde{A}(r_s) = A_o F T_{eff} / \ln{(2)}$
- $F$ is fraction of activity in source organ, depends on radionuclide and specific uptake of patient
- $A_o$ is activity administered in Bq
- $T_{eff}$ is how long activity stays in the source organ
- $T_{eff} = \frac{T_b · T_p}{T_b + T_p}$
- Effective half life depends on physical and biological half lives
- $S$ value is mean dose to target organ per integrated activity in source organ (radionuclide specific)
- $S(r_t \rightarrow r_s) = \sum_i{\Delta_i \phi_i / m_T}$
- $\Delta_i$ is mean energy per nuclear transformation for ith radiation emitted
- $M_T$ is mass of target organ
- $\phi_i$ is fraction of energy emitted by source that is absorbed by target of the ith radiation
- This is a patient size and anatomy dependent value
- $\phi_i = 1$ where target and source are same organ
- Summation over all radiations
- Specific absorbed fraction SAF = $\Phi = \phi_i/m_T$ (typically around 5e-7 to 4.5e-5 to g-1)
- Summation is over all source organs
Because of the size and anatomic dependence within the dose equation, models for man, woman and child are used as part of computational phantoms. Phantoms can range from simple (sphere) to geometric anthropomorphic, to image-based rigid 3D, to deformable moving 4D models.
Uncertainties in dose calculations are typically factor of 2 or greater.
Non-instantaneous uptake:
$\tilde{A} =A_0 T_e (T_{ue} / T_u)/\ln(2)$ where $A_0$ is administered activity to organ, $T_e$ is effective half-life, $T_{ue}$ is effective uptake half-time (combine uptake and physical half-lives) and $T_u$ is biological uptake half-time.
$\Delta_i = 1.6e-13 N_i E_i \text{(Gy · kg / Bq · sec)}$ where $N_i$ is relative frequency of ith emission and $E_i$ is average energy of the ith emission in MeV.
Dose reciprocity theorem: specific absorbed fraction, $\Phi$ is same regardless of which organ is source or target (direction of travel doesn't matter).
$S$, mean dose per cumulated activity, has been calculated for many target-source pairs for a variety of radionuclides, and tables can be used to get the values.
Limitations
Values of $\phi$ are based on simplistic models of "standard" human anatomy.
Assumes activity is uniform within an organ. Significant error from electron doses (non-penetrating).
Calculation of $\tilde{A}$ uncertain, especially if determined from animal studies. Differences within healthy subjects, and especially patients with pathophysiologic uptake/clearance/excretion.