Pharmacology Basis

The Degeneracy Problem

In SI, drug amounts are measured in mass (kg or mg). But:

Drug	Dose	Equivalent Effect
Morphine	10 mg	Baseline
Fentanyl	0.1 mg	Same as 10 mg morphine
Codeine	100 mg	Same as 10 mg morphine

All are "milligrams" in SI. Treating them as equivalent is fatal.

Similar problems exist with: - International Units (IU) — different for each substance - Bioavailability — IV vs oral doses - Receptor binding — agonist vs antagonist potency

---

The Extended Pharmacology Basis

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

PHARMA = Basis("Pharma", [
    BasisComponent("mass", "M"),              # 0 — physical mass
    BasisComponent("potency", "P"),           # 1 — biological activity
    BasisComponent("bioavailability", "F"),   # 2 — fraction reaching systemic circulation
])

---

Dimensional Vectors

Quantity	Vector	Interpretation
Drug mass	M¹P⁰F⁰	Raw substance amount
Potency factor	M⁰P¹F⁰	Activity per unit mass
Bioactive dose	M¹P¹F⁰	Mass × potency
Systemically available dose	M¹P¹F¹	Mass × potency × bioavailability

---

Implementation

# Dimensions
drug_mass = Dimension(
    vector=Vector(PHARMA, (1, 0, 0)),
    name="drug_mass"
)

potency_factor = Dimension(
    vector=Vector(PHARMA, (0, 1, 0)),
    name="potency_factor"
)

bioactive_dose = Dimension(
    vector=Vector(PHARMA, (1, 1, 0)),
    name="bioactive_dose"
)

systemic_dose = Dimension(
    vector=Vector(PHARMA, (1, 1, 1)),
    name="systemic_dose"
)

# Units with potency baked in
morphine_mg = Unit(
    name="morphine_mg",
    shorthand="mg(morphine)",
    dimension=bioactive_dose
)  # potency = 1 (reference)

fentanyl_mg = Unit(
    name="fentanyl_mg",
    shorthand="mg(fentanyl)",
    dimension=bioactive_dose
)  # potency = 100

codeine_mg = Unit(
    name="codeine_mg",
    shorthand="mg(codeine)",
    dimension=bioactive_dose
)  # potency = 0.1

---

Morphine Milligram Equivalents (MME)

The CDC uses MME for opioid risk assessment:

# Conversion factors to MME
OPIOID_MME = {
    "morphine": 1.0,
    "fentanyl": 100.0,      # 0.1 mg fentanyl = 10 MME
    "oxycodone": 1.5,
    "hydrocodone": 1.0,
    "codeine": 0.15,
    "methadone": 4.0,       # variable, dose-dependent
    "buprenorphine": 30.0,
}

# With dimensional tracking:
fentanyl_dose = fentanyl_mg(0.1)
mme = fentanyl_dose.to(morphine_equivalent)  # → 10 MME

# CDC risk thresholds
# ≥50 MME/day: increased overdose risk
# ≥90 MME/day: avoid or justify

---

International Units (IU)

IU is a measure of biological activity, not mass. Each substance has its own definition:

Substance	1 IU =	Notes
Insulin	0.0347 mg (human)	Based on blood glucose effect
Vitamin D	0.025 μg	Based on antirachitic activity
Vitamin A	0.3 μg retinol	Based on growth assay
Heparin	~0.002 mg	Based on anticoagulation
Penicillin	0.6 μg	Historical bioassay

Problem: 1 IU of insulin ≠ 1 IU of vitamin D, but SI treats both as dimensionless counts.

Solution: Model IU as substance-specific potency:

# Each IU type is a distinct unit with P¹ dimension
insulin_iu = Unit(name="insulin_IU", dimension=potency_factor)
vitamin_d_iu = Unit(name="vitamin_d_IU", dimension=potency_factor)

# These cannot be added
insulin_iu(100) + vitamin_d_iu(1000)
# raises: incompatible (different substances, even though both are "IU")

For full safety, extend the basis with a substance tag:

PHARMA_TYPED = Basis("Pharma-Typed", [
    BasisComponent("mass", "M"),
    BasisComponent("potency", "P"),
    BasisComponent("bioavailability", "F"),
    BasisComponent("substance_class", "S"),  # insulin=1, vitamin_d=2, etc.
])

---

Bioavailability

The fraction of drug reaching systemic circulation:

Route	Typical F
IV	1.0 (by definition)
IM	0.8-1.0
Oral	0.1-0.9 (highly variable)
Transdermal	0.1-0.5
Sublingual	0.3-0.8

# Oral morphine vs IV morphine
oral_morphine = Unit(
    name="oral_morphine_mg",
    dimension=systemic_dose  # M¹P¹F¹
)
# F ≈ 0.3 for oral morphine

iv_morphine = Unit(
    name="iv_morphine_mg",
    dimension=systemic_dose
)
# F = 1.0 for IV

# 30 mg oral ≈ 10 mg IV (same systemic exposure)

---

Dimensional Algebra

Mass → Bioactive dose:

bioactive = mass × potency
M¹P¹F⁰ = (M¹P⁰F⁰) × (P¹)  ✓

Bioactive → Systemic:

systemic = bioactive × bioavailability
M¹P¹F¹ = (M¹P¹F⁰) × (F¹)  ✓

---

Safety Guarantees

# These are now dimension mismatches:

morphine_mg(10) + fentanyl_mg(0.1)
# Must convert to common equivalents first

oral_morphine(30) + iv_morphine(10)
# Must account for bioavailability

insulin_iu(100) + vitamin_d_iu(1000)
# Cannot add different IU types

---

Clinical Decision Support

With dimensional tracking, you can build safer systems:

def check_mme_risk(prescriptions: list[Number]) -> str:
    """Check total daily MME against CDC thresholds."""
    total_mme = sum(
        dose.to(morphine_equivalent)
        for dose in prescriptions
    )

    if total_mme.quantity >= 90:
        return "HIGH RISK: ≥90 MME/day"
    elif total_mme.quantity >= 50:
        return "ELEVATED RISK: ≥50 MME/day"
    else:
        return "Standard risk"

The dimension system ensures all inputs are converted to a common basis before summation — no accidental fentanyl + morphine addition.

---

Corticosteroid Equivalents

Similar pattern for steroid conversions:

Drug	Equivalent Dose	Relative Potency
Prednisone	5 mg	1.0 (reference)
Prednisolone	5 mg	1.0
Methylprednisolone	4 mg	1.25
Dexamethasone	0.75 mg	6.7
Hydrocortisone	20 mg	0.25

The same extended basis applies — mass alone is insufficient.