Radiation Dosimetry Basis

The Degeneracy Problem

In SI, all of these have dimension L²T⁻² (equivalent to J/kg):

Quantity	Unit	What It Measures
Absorbed dose	Gray (Gy)	Energy deposited per mass
Equivalent dose	Sievert (Sv)	Biologically weighted dose (radiation type)
Effective dose	Sievert (Sv)	Whole-body risk-weighted dose
RBE-weighted dose	Gy(RBE)	Therapeutically weighted dose
Kerma	Gray (Gy)	Kinetic energy released in matter
Specific energy	J/kg	Generic energy per mass

All dimensionally identical. All clinically distinct.

Confusing Gray with Sievert, or equivalent dose with effective dose, is a medical error.

---

The Extended Dose Basis

from ucon.basis import Basis, BasisComponent, Vector
from ucon.core import Dimension, Unit
from fractions import Fraction

DOSE = Basis("Dose", [
    BasisComponent("energy", "E"),              # 0
    BasisComponent("mass", "M"),                # 1
    BasisComponent("radiation_weighting", "R"), # 2 — w_R (regulatory)
    BasisComponent("rbe", "B"),                 # 3 — RBE (therapeutic)
    BasisComponent("tissue_weighting", "T"),    # 4 — w_T (organ sensitivity)
])

---

Dimensional Vectors

Quantity	Vector	Interpretation
Absorbed dose (Gy)	E¹M⁻¹R⁰B⁰T⁰	Raw energy deposition
Equivalent dose (Sv)	E¹M⁻¹R¹B⁰T⁰	w_R weighted (organ level)
Effective dose (Sv)	E¹M⁻¹R¹B⁰T¹	w_R × w_T weighted (whole body)
RBE-weighted dose Gy(RBE)	E¹M⁻¹R⁰B¹T⁰	Therapeutic RBE weighted

---

Implementation

# Dimensions
absorbed_dose = Dimension(
    vector=Vector(DOSE, (1, -1, 0, 0, 0)),
    name="absorbed_dose",
    symbol="D"
)

equivalent_dose = Dimension(
    vector=Vector(DOSE, (1, -1, 1, 0, 0)),
    name="equivalent_dose",
    symbol="H"
)

effective_dose = Dimension(
    vector=Vector(DOSE, (1, -1, 1, 0, 1)),
    name="effective_dose",
    symbol="E_eff"
)

rbe_weighted_dose = Dimension(
    vector=Vector(DOSE, (1, -1, 0, 1, 0)),
    name="rbe_weighted_dose",
    symbol="D_RBE"
)

# Weighting factor dimensions
radiation_weighting = Dimension(
    vector=Vector(DOSE, (0, 0, 1, 0, 0)),
    name="radiation_weighting"
)  # R¹ — w_R values: gamma=1, alpha=20

tissue_weighting = Dimension(
    vector=Vector(DOSE, (0, 0, 0, 0, 1)),
    name="tissue_weighting"
)  # T¹ — w_T values: lung=0.12, gonads=0.08, etc.

rbe_factor = Dimension(
    vector=Vector(DOSE, (0, 0, 0, 1, 0)),
    name="rbe_factor"
)  # B¹ — RBE values: proton≈1.1, carbon≈2-3

# Units
gray = Unit(name="gray", shorthand="Gy", dimension=absorbed_dose)
sievert_eq = Unit(name="sievert_equivalent", shorthand="Sv", dimension=equivalent_dose)
sievert_eff = Unit(name="sievert_effective", shorthand="Sv_eff", dimension=effective_dose)
gray_rbe = Unit(name="gray_rbe", shorthand="Gy(RBE)", dimension=rbe_weighted_dose)

---

Dimensional Algebra

Absorbed → Equivalent:

H = D × w_R
E¹M⁻¹R¹B⁰T⁰ = (E¹M⁻¹R⁰B⁰T⁰) × (R¹)  ✓

Equivalent → Effective:

E_eff = H × w_T
E¹M⁻¹R¹B⁰T¹ = (E¹M⁻¹R¹B⁰T⁰) × (T¹)  ✓

Absorbed → RBE-weighted:

D_RBE = D × RBE
E¹M⁻¹R⁰B¹T⁰ = (E¹M⁻¹R⁰B⁰T⁰) × (B¹)  ✓

---

Safety Guarantees

# These are now compile-time errors (dimension mismatch):

gray(2.0) + sievert_eq(1.0)
# raises: incompatible dimensions (R⁰ vs R¹)

sievert_eq(1.0) + sievert_eff(0.5)
# raises: incompatible dimensions (T⁰ vs T¹)

gray(2.0) + gray_rbe(2.2)
# raises: incompatible dimensions (B⁰ vs B¹)

---

Weighting Factor Values

Radiation Weighting Factors (w_R)

Radiation Type	w_R
Photons (all energies)	1
Electrons, muons	1
Protons	2
Alpha particles	20
Neutrons	5-20 (energy dependent)

Tissue Weighting Factors (w_T)

Tissue	w_T
Bone marrow, colon, lung, stomach, breast	0.12
Gonads	0.08
Bladder, liver, esophagus, thyroid	0.04
Skin, bone surface, brain, salivary glands	0.01
Remainder	0.12 (distributed)

RBE Values (Therapeutic)

Particle	RBE (typical)
Photons	1.0 (reference)
Protons	1.1 (clinical convention)
Carbon ions	2-3 (varies with LET)

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Clinical Context

Radiation Protection (ICRP)

Uses w_R and w_T for regulatory dose limits: - Occupational limit: 20 mSv/year (effective dose) - Public limit: 1 mSv/year (effective dose)

Radiation Therapy

Uses RBE for treatment planning: - Proton therapy: Gy(RBE) = Gy × 1.1 - Carbon ion therapy: Gy(RBE) = Gy × RBE(LET)

The distinction matters: A patient receiving 60 Gy(RBE) proton therapy has a different physical dose than 60 Gy photon therapy, and both differ from regulatory effective dose calculations.

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Projection to SI

If you need to collapse back to SI (L²T⁻²), use a ConstantBoundBasisTransform with bindings that record which weighting factors were absorbed:

DOSE_TO_SI = ConstantBoundBasisTransform(
    source=DOSE,
    target=SI,
    matrix=(...),  # Maps E¹M⁻¹ → L²T⁻²
    bindings=(
        ConstantBinding(
            source_component=DOSE["radiation_weighting"],
            target_expression=Vector(SI, (0, 0, 0, ...)),  # dimensionless
            constant_symbol="w_R",
            exponent=Fraction(1),
        ),
        # ... similar for tissue_weighting, rbe
    ),
)

The bindings enable round-trip conversion while tracking which weighting was applied.