Doses and weighting¶

The gray and the sievert are both J/kg, and each measures more than one quantity. The SI gives the gray to absorbed dose and to kerma (SI Brochure, 9^th ed., Table 4). The sievert carries three quantities that are defined differently and used for different things:

• Dose equivalent \(H = Q \cdot D\) at a point, with the quality factor \(Q\) a function of linear energy transfer (CIPM 2002, Recommendation 2). This is the SI's quantity. The operational quantities a monitor or a dosimeter reads, \(H^*(10)\) and \(H_{\mathrm{p}}(10)\), are dose equivalents at a stated depth.
• Equivalent dose \(H_{\mathrm{T}} = \sum w_{\mathrm{R}} \cdot D_{\mathrm{T,R}}\), the mean absorbed dose in a tissue weighted by a radiation weighting factor \(w_{\mathrm{R}}\) (ICRP Publication 103, Eq. 4.2).
• Effective dose \(E = \sum w_{\mathrm{T}} \cdot H_{\mathrm{T}}\), the tissue-weighted sum over the body of a Reference Person (ICRP Publication 103). Dose limits are written in it.

A number in Sv that doesn't say which of these it is reads three ways. So exact_units gives each its own quantity (Quantities), all of them kinds:

graph TD
  D["AbsorbedDose<br/>Gray's quantity"] --> K["Kerma"]
  H["DoseEquivalent, H = Q·D<br/>Sievert's quantity"] --> A["AmbientDoseEquivalent10<br/>H*(10)"]
  H --> P10["PersonalDoseEquivalent10<br/>Hp(10)"]
  H --> P007["PersonalDoseEquivalent0_07<br/>Hp(0.07)"]
  H --> HT["EquivalentDose<br/>H_T"]
  H --> E["EffectiveDose<br/>E"]

Every box is a kind: none converts or adds to another. The ones under DoseEquivalent are kinds under it so that a sievert can measure them, as Of<EffectiveDose, Sievert> (units::EffectiveDose). A bare Sievert is the SI's dose equivalent \(H\).

use exact_units::{Fixed, prefix::Milli, units::{EffectiveDose, EquivalentDose}};

// an organ's equivalent dose is not an effective dose
let h = Fixed::<EquivalentDose, Milli, u32>::new(20);
let _ = h.convert::<EffectiveDose, Milli, u32>();

Each dose has a rate in the same shape: AbsorbedDoseRate (GrayPerSecond), DoseEquivalentRate (SievertPerHour) with AmbientDoseEquivalentRate10 and the personal rates under it, KermaRate, EquivalentDoseRate, EffectiveDoseRate. A survey meter's µSv/h is Fixed<Of<AmbientDoseEquivalentRate10, SievertPerHour>, Micro, u32>. Gy/s does not convert to Sv/h.

From a gray to a sievert: a weighting¶

Neither convert nor as_quantity takes a gray to a sievert. The step that does is a weighting, and the weight comes from a named, versioned set in exact_units::radiation, as a measured constant comes from a CODATA edition (Constants):

use exact_units::{Fixed, prefix::Micro, radiation::{Icrp103, Icrp60}};
use exact_units::units::{EquivalentDose, Gray};

let d = Fixed::<Gray, Micro, u32>::new(2000);
let h: Fixed<EquivalentDose, Micro, u32> = d.equivalent_dose(Icrp103::PROTONS_AND_CHARGED_PIONS).unwrap();
assert_eq!(h.to_string(), "4000 µSv");
// the same protons by the 1991 recommendations
let h: Fixed<EquivalentDose, Micro, u32> = d.equivalent_dose(Icrp60::PROTONS_ABOVE_2_MEV).unwrap();
assert_eq!(h.to_string(), "10000 µSv");

The sets carry the tables as printed:

A weight a set doesn't give is a compile error: ICRP 60 has no PROTONS_AND_CHARGED_PIONS. The edition is named at the call, not carried in the type: an equivalent dose by ICRP 60 and one by ICRP 103 are the same quantity, as dose records add them in practice. Code that must keep editions apart declares a kind of its own under EquivalentDose with quantity!.

The step is exact or None. An integer weight that overflows the storage is None, and so is a fractional one whose product isn't a whole number of the storage's step; equivalent_dose_rounded names a rounding instead. ICRP 103's neutron weight is \(e^{-\ln^2(E)/6}\) and has no exact value, so it is a float edge: Icrp103::neutrons_f64 (feature libm) gives a WeightF64, and equivalent_dose_f64 multiplies into an f64.

A kerma takes no weighting until it is named an absorbed dose, the named step as_quantity::<AbsorbedDose>() that charged-particle equilibrium justifies:

use exact_units::{Fixed, prefix::Micro, radiation::Icrp103, units::{EquivalentDose, Kerma}};

let k = Fixed::<Kerma, Micro, u32>::new(2000);
let _: Option<Fixed<EquivalentDose, Micro, u32>> = k.equivalent_dose(Icrp103::PHOTONS);

The dose equivalent: H = Q·D¶

The SI's dose equivalent weights by the quality factor \(Q(L)\) of the charged particles' linear energy transfer \(L\) in water, defined by ICRP 60 and kept by ICRP 103 (Eq. B.4.2): 1 below 10 keV/µm, \(0.32 \cdot L - 2.2\) up to 100 keV/µm, \(300/\sqrt{L}\) above. Icrp60::q_of_let takes an \(L\) in KiloElectronVoltPerMicrometer (a LinearEnergyTransfer, not a force) and gives \(Q\) exactly where it is a fraction: always up to 100 keV/µm, above where \(\sqrt{L}\) is rational. Elsewhere it is None, and q_of_let_f64 is the float edge.

use exact_units::{Fixed, prefix::{Micro, One}, radiation::Icrp60, quantities::AmbientDoseEquivalent10};
use exact_units::units::{Gray, KiloElectronVoltPerMicrometer, Of, Sievert};

let q = Icrp60::q_of_let(Fixed::<KiloElectronVoltPerMicrometer, One, u32>::new(50)).unwrap();
assert_eq!(q.ratio(), (69, 5)); // 13.8
let h: Fixed<Sievert, Micro, u32> = Fixed::<Gray, Micro, u32>::new(500).dose_equivalent(q).unwrap();
// what the calibration says this point is, named
let ambient: Fixed<Of<AmbientDoseEquivalent10, Sievert>, Micro, u32> = h.as_quantity();
assert_eq!(ambient.get(), 6900);

A \(Q\) from a measured spectrum is a Weight of the caller's, Weight::new(num, den, of, source, year), with its source said.

Effective dose: E = Σ w_T·H_T¶

The effective dose weights the equivalent doses of the organs and tissues of a Reference Person by the tissue weighting factors \(w_{\mathrm{T}}\) of ICRP 103's Table 3 (the French edition's Tableau 4.2, again as Tableau B.2). Icrp103::TISSUES carries them, and Icrp103Tissues holds one value per tissue, so no tissue can be left out of the sum:

use exact_units::{Fixed, prefix::Micro, radiation::{Icrp103, Icrp103Tissues}};
use exact_units::units::{EffectiveDose, EquivalentDose};

let mut h = Icrp103Tissues::<Fixed<EquivalentDose, Micro, u32>>::default();
h.thyroid = Fixed::new(50_000); // 50 mSv to the thyroid
h.lung = Fixed::new(1000);
let e: Fixed<EffectiveDose, Micro, u32> = Icrp103::effective_dose(&h).unwrap();
assert_eq!(e.to_string(), "2120 µSv");

effective_dose takes the sex-averaged \(H_{\mathrm{T}}\). effective_dose_by_sex(&male, &female) is Eq. 4.5 itself, \(E = \sum w_{\mathrm{T}} \cdot (H_{\mathrm{T}}^{\mathrm{M}} + H_{\mathrm{T}}^{\mathrm{F}})/2\), with nothing rounded between the two. The remainder field is the arithmetic mean of the 13 remainder tissues of each sex (Eq. 4.6), as ICRP's dose coefficient tables give it; ICRP 60's splitting rule for the remainder is gone, so \(E\) is a plain sum.

The \(w_{\mathrm{T}}\) are hundredths, so a sum is often no whole number of the storage's step. The step is exact or None, as the weightings are, and effective_dose_rounded rounds once, at the end. A float storage is exact or None too: \(0.04 \cdot 0.5\,\unit{mSv}\) is no f64. Equivalent dose rates give an effective dose rate the same way. An absorbed dose is refused: it takes its \(w_{\mathrm{R}}\) first.

use exact_units::{Fixed, prefix::Micro, radiation::{Icrp103, Icrp103Tissues}, units::{EffectiveDose, Gray}};

let d = Icrp103Tissues::<Fixed<Gray, Micro, u32>>::default();
let _: Option<Fixed<EffectiveDose, Micro, u32>> = Icrp103::effective_dose(&d);

A rate and its dose¶

A dose rate times a time is a plain product, Derived, and as_quantity could name it any quantity of that dimension. The named steps keep the pair: rate.dose_over(t) gives the dose of the rate's own kind, and dose.rate_over(t) the mean rate of the dose's kind. \(\dot{H}^*(10)\) over a time is an \(H^*(10)\), never an \(H_{\mathrm{p}}(10)\) or an effective dose. The pairs are each dose rate of the tree above with its dose: AbsorbedDoseRate and AbsorbedDose, KermaRate and Kerma, and so on for the dose equivalent, the three operational quantities, the equivalent dose and the effective dose.

use exact_units::{Fixed, prefix::{Micro, Nano, One}};
use exact_units::quantities::{AmbientDoseEquivalent10, AmbientDoseEquivalentRate10};
use exact_units::units::{Minute, Of, Sievert, SievertPerHour};

// a survey meter reads 3 µSv/h; 20 minutes there
let rate = Fixed::<Of<AmbientDoseEquivalentRate10, SievertPerHour>, Micro, u32>::new(3);
let t = Fixed::<Minute, One, u32>::new(20);
let h: Fixed<Of<AmbientDoseEquivalent10, Sievert>, Nano, u32> = rate.dose_over(t).unwrap();
assert_eq!(h.to_string(), "1000 nSv");
// and back
let back: Fixed<Of<AmbientDoseEquivalentRate10, SievertPerHour>, Micro, u32> = h.rate_over(t).unwrap();
assert_eq!(back, rate);

The result's unit, scale and storage are the caller's, usually given by the binding; the factor between the units (Sv/h times minutes into Sv) is worked out at compile time. The step is exact or None (a result that isn't a whole number of its step, a zero time, a value past the storage); dose_over_rounded and rate_over_rounded round once.

use exact_units::{Fixed, prefix::{Micro, One}};
use exact_units::quantities::{AmbientDoseEquivalentRate10, PersonalDoseEquivalent10};
use exact_units::units::{Of, Second, Sievert, SievertPerHour};

// an area monitor's rate gives no personal dose equivalent
let rate = Fixed::<Of<AmbientDoseEquivalentRate10, SievertPerHour>, Micro, u32>::new(3);
let _: Option<Fixed<Of<PersonalDoseEquivalent10, Sievert>, Micro, u32>> = rate.dose_over(Fixed::<Second, One, u32>::new(1));

What the crate doesn't do¶

• Rate arithmetic keeps no kind. rate * t and dose / t are plain products; only dose_over and rate_over keep the pair. A quantity that declares its rate, so the operators keep it too, is an idea (UNIT-16).
• The remainder's mean is the caller's. Eq. 4.6's mean of 13 tissues per sex goes into the remainder field as a value; there is no step from the 13 tissues.
• ICRU Report 95's operational quantities (2020: \(H^*\), \(H_{\mathrm{p}}\) from conversion coefficients to \(E\), no depth) are not here, nor \(H'(0.07, \Omega)\), \(H'(3, \Omega)\) and \(H_{\mathrm{p}}(3)\) of ICRU 51.
• Older and national quantities are the user's to declare: ICRP 26's effective dose equivalent, and the US NRC's TEDE, deep-dose and shallow-dose equivalents (10 CFR 20), which use ICRP 26/30 weights.