Clinical specialty kernels
Textbook clinical formulas for seven specialties -- cardiac hemodynamics, dental/oral-health indices, hormone physiology, kidney function, eye optics, orthodontic biomechanics, and lung function -- as
Seven modules, one per medical specialty, each a direct implementation of the named-equation formulas a clinician already uses on paper: Cockcroft-Gault and CKD-EPI for kidney function, the alveolar gas equation for lung function, HOMA-IR for insulin resistance, Snellen-to-decimal conversion for visual acuity, DMFT for dental caries burden, and Euler beam theory for orthodontic wire force. Every function takes and returns plain Float64 (dental counts also take/return Int) — there are no patient records, no structs, and no clinical-decision output; each call is one named formula evaluated on the numbers you already measured; interpreting the result against a reference range is left to the caller.
from algenta import MojoRuntime
runtime = MojoRuntime(mode="local")
# CKD-EPI eGFR: 60-year-old woman, serum creatinine 1.0 mg/dL
runtime.execute("nephrology", "gfr_ckd_epi", [1.0, 60.0, 1.0])["result"]
# 61.18505161746657 -- mL/min/1.73m^2, stage G2 (mildly reduced)
# the alveolar gas equation on room air at sea level
runtime.execute("pulmonology", "alveolar_oxygen", [0.21, 40.0, 760.0])["result"]
# 99.73 -- PAO2 in mmHg, the textbook ~100 mmHg room-air valueThe modules
cardiology
Hemodynamics: cardiac output (rate * volume and Fick's principle), stroke volume, ejection fraction, MAP, pulse pressure, systemic vascular resistance, rate-pressure product, LV stroke work, and coronary perfusion pressure.
dentistry
General dental/oral-health metrics: the DMFT caries index and its population prevalence, plaque and gingival indices, clinical attachment loss, alveolar bone loss, local-anesthetic dosing, restoration-longevity and treatment-cost estimates.
endocrinology
Hormone physiology: HOMA-IR insulin resistance, cortisol's diurnal cosine rhythm, free-hormone fractions (thyroid, testosterone), closed-loop feedback gain, receptor Bmax from Scatchard analysis, and IGF-1-driven growth rate.
nephrology
Kidney function: eGFR (CKD-EPI) and creatinine clearance (Cockcroft-Gault), fractional excretion of sodium, urine anion gap, TTKG, renal plasma flow, filtration fraction, osmolar and free-water clearance, and urine protein/creatinine ratio.
ophthalmology
Eye optics: Snellen-to-decimal visual acuity, spherical refractive error, the lensmaker's equation, Goldmann IOP with corneal-thickness correction, keratometry-to-corneal-radius, retinal image size, depth of focus, CFF, contrast sensitivity, and dark-adaptation kinetics.
orthodontics
Tooth-movement biomechanics: force/optimal-force movement rate, moment-to-force ratio, anchorage load per tooth, Euler-beam arch-wire force and bracket torque, Bolton and Pont indices, Little's-irregularity arch perimeter, cephalometric growth discrepancy, and viscoelastic retainer-force decay.
pulmonology
Lung function: predicted (ARDSnet) tidal volume, minute and alveolar ventilation, FEV1/FVC, %-predicted PEF, the alveolar gas equation, A-a gradient, respiratory quotient, static compliance, and airway resistance.
Two mismatches with a naive reading of the specialty names
cardiology is entirely hemodynamics — there is no ECG or QT-interval arithmetic here at all. Reading mojo/engine/cardiology.mojo's ten functions end to end: cardiac output (two ways — rate * stroke volume, and the Fick principle from oxygen consumption), stroke volume, ejection fraction, mean arterial pressure, pulse pressure, systemic vascular resistance, rate-pressure product, left ventricular stroke work, and coronary perfusion pressure. Every input and output is a pressure, volume, flow, or rate — QT correction (Bazett/Fridericia), heart-rate-corrected intervals, or any other electrophysiology formula simply is not part of this module's scope, contrary to what its name alone might suggest to someone expecting ECG arithmetic. If a corrected-QT formula exists anywhere in this codebase it is not here.
dentistry and orthodontics are real siblings, but they share zero functions. Both files' docstrings frame themselves as dental: dentistry.mojo is "Dental Science Library," and orthodontics.mojo is "Dental and orthodontic biomechanics calculations." Reading both source files in full confirms the split is clean rather than accidental duplication: dentistry is general oral-health epidemiology and restorative dentistry — DMFT caries scoring, plaque/gingival indices, periodontal attachment loss, fluoride dosing, anesthetic dosing, restoration longevity, treatment cost — none of it involves an applied force or a wire. orthodontics is exclusively the physics of moving teeth — PDL force response, moment-to-force ratios, anchorage load, Euler-beam wire deflection force, bracket torque, Bolton/Pont arch-size indices, Little's irregularity index, cephalometric growth prediction, and viscoelastic retainer relaxation — none of it involves an index score, a caries count, or a drug dose. There is no name collision and no formula computed twice under two names; the hint's framing of orthodontics as "a specialization of dentistry" holds at the level of medical taxonomy, but at the code level these are two disjoint, non-overlapping libraries that happen to sit in adjacent files.
A docstring that omits a real unit-conversion factor
ophthalmology.depth_of_focus's docstring states the formula as DoF ~= 2 * VA_arcmin / (3438 * pupil_mm) diopters, but the code itself computes 2.0 * visual_acuity_arcmin / (3438.0 * pupil_diameter_mm / 1000.0) — which is exactly 1000x larger than what the docstring's own algebraic expression describes. Deriving the formula independently from thin-lens defocus geometry confirms the code is the one that is dimensionally correct, not the docstring: a defocus of delta diopters spreads a blur circle of diameter c = f' * a * delta on the retina (f' the eye's ~17 mm posterior focal length, a the pupil diameter), and setting that blur circle's angular subtense equal to the eye's minimum angle of resolution and solving for delta reproduces the code's 2000 * VA_arcmin / (3438 * pupil_mm) term-for-term — the 1000 comes from converting the 17 mm focal length into millimeters twice over in the derivation, a step the docstring's shorthand silently drops. Numerically, a 3 mm pupil at 20/20 acuity (VA_arcmin = 1) gives depth_of_focus(3.0, 1.0) = 0.1939 diopters under the code — physiologically the right order of magnitude for a photopic pupil — while the docstring's literal formula, evaluated the same way, would give 0.000194 diopters, three orders of magnitude too small to be a real depth-of-focus value. The function is correct; its docstring's written-out formula is missing the conversion factor that makes it correct, the same category of docstring/code mismatch flagged in earlier kernel-family write-ups in this corpus.
Two independent formulas for the same measurement, deliberately kept simple
nephrology carries both gfr_ckd_epi and creatinine_clearance_cockroft rather than picking one. The CKD-EPI implementation is the original 2009 Levey et al. coefficients (kappa, alpha = -0.411 male / -0.329 female, the 0.993^age term, *1.018 for female) with the race multiplier simply omitted rather than replaced — it is not the separately re-calibrated 2021 race-free refit (which uses different constants, 142 / -1.200 / 0.9938 / *1.012). Cockcroft-Gault is the plain 1976 formula, (140-age) * weight / (72 * Scr), with *0.85 for female and no upper weight cap or lean-body-weight substitution. Both are simplifications of formulas that, in full clinical use, carry adjustments this module does not implement; the docstrings do not claim otherwise.
Guards and refusal behavior
Every division in all seven modules is guarded against its degenerate input, and every guard returns a plain 0.0 sentinel rather than propagating inf or nan:
cardiology.ejection_fraction, .systemic_vascular_resistance, .fick_cardiac_output
edv_ml <= 0, co_l_per_min <= 0, cao2 - cvo2 <= 0
0.0
cardiology.stroke_volume
edv_ml - esv_ml < 0
0.0 (never negative)
dentistry.dmft_prevalence, .plaque_index_score, .gingival_index, .bone_loss_percentage, .fluoride_concentration_ppm, .restoration_longevity_estimate
denominator <= 0
0.0
endocrinology.hormone_clearance_rate, .hormone_pulsatility
auc <= 0, observation_hours <= 0
0.0
endocrinology.receptor_binding_bmax
free_conc <= 0
0.0
endocrinology.feedback_gain
1 + G_ol * f <= 0
0.0
endocrinology.thyroid_free_fraction
1 + Ka * [TBG] <= 0
1.0 (all hormone free)
endocrinology.testosterone_free
1 + Ka_SHBG * SHBG + Ka_Alb * Alb <= 0
total_t (all hormone free)
nephrology.creatinine_clearance_cockroft, .fractional_excretion_sodium, .transtubular_potassium_gradient, .renal_plasma_flow, .filtration_fraction, .osmolar_clearance, .protein_creatinine_ratio (7 of the module's 10 functions)
the relevant plasma/urine denominator <= 0
0.0
ophthalmology.visual_acuity_decimal, .corneal_radius_from_keratometry, .retinal_image_size, .depth_of_focus, .contrast_sensitivity
denominator <= 0
0.0
ophthalmology.refractive_error_spherical
far_point_m == 0
0.0
ophthalmology.critical_flicker_frequency
luminance_cd_m2 <= 0
k (the intercept alone)
orthodontics.tooth_movement_rate, .moment_to_force_ratio, .anchorage_loss_estimate, .arch_wire_force, .levelling_torque, .bolton_ratio
the relevant force/span/tooth-count denominator <= 0
0.0
pulmonology.alveolar_ventilation
tidal_volume_ml - dead_space_ml <= 0
0.0 (dead space exceeding tidal volume is refused, not made negative)
pulmonology.fev1_fvc_ratio, .peak_expiratory_flow, .respiratory_quotient, .compliance_static, .airway_resistance
denominator <= 0
0.0
endocrinology.thyroid_free_fraction and .testosterone_free are the asymmetric cases in the table: every other guard answers "no signal" with 0.0, but a non-positive binding denominator in either of these two means no carrier protein is binding the hormone at all, so the physically correct answer is that all of it is free — thyroid_free_fraction returns 1.0 (100% free fraction) and testosterone_free returns the input total_t unchanged (all of it free), neither one the 0.0 every division-guard elsewhere in this family falls back to.
nephrology.gfr_ckd_epi is the one function in this family whose division has no guard at all. Every other nephrology function with a denominator or power base that can vanish returns an early 0.0; gfr_ckd_epi does not — its formula raises min(Scr/kappa, 1) to a negative power (alpha = -0.411 male, -0.329 female), and a creatinine_mg_dl of 0.0 makes that base exactly 0.0, i.e. 0.0 raised to a negative exponent, a divide-by-zero one level up inside pow rather than in a plain division the source could check with an if. Confirming this independently in Python — the same IEEE-754 arithmetic Mojo's pow is built on — 0.0 ** -0.411 raises a domain/zero-division error rather than silently returning a finite number (math.pow(0.0, -0.411) raises ValueError: math domain error; some C libm implementations instead return +inf for the identical input). Either behavior is a crash or an inf, not the 0.0 sentinel every sibling function in this module falls back to — and creatinine_clearance_cockroft, defined immediately below gfr_ckd_epi in the same file, guards its own identical zero-creatinine input explicitly. That makes the missing guard here a genuine gap in this specific function, not a stylistic inconsistency across the module.
Worked examples, hand-verified against textbook values
None of these seven modules has a pinned numeric test (see Verification below), so every number below was computed independently from the real formulas in mojo/engine/*.mojo and checked, where a standard clinical reference value exists, against that value.
Cardiology — a full hemodynamic profile from EDV/ESV and blood pressure. EDV 120 mL, ESV 50 mL, heart rate 72 bpm, BP 120/80 mmHg, LVEDP 10 mmHg:
ejection_fraction recomputes (EDV-ESV)/EDV from its own two arguments rather than taking a precomputed stroke volume, so it agrees with a separately-called stroke_volume only because both start from the same EDV/ESV pair — the two functions do not call each other.
Nephrology — Cockcroft-Gault vs. CKD-EPI on the same patient. A 60-year-old woman, 70 kg, serum creatinine 1.0 mg/dL:
The two formulas estimate different quantities (creatinine clearance overestimates true GFR because some creatinine is also secreted, not just filtered) and use different equations entirely, so their ~5 mL/min gap on this patient is not evidence either one is wrong — it is exactly the kind of discrepancy both formulas are known to produce against each other, and against measured GFR, in real patients. A prerenal-vs-intrinsic AKI check on the same patient with urine studies (Na 40 mEq/L, creatinine 60 mg/dL against plasma Na 140 mEq/L, creatinine 1.2 mg/dL):
Pulmonology — the alveolar gas equation and A-a gradient at sea level. FiO2 = 0.21 (room air), PaCO2 = 40 mmHg, barometric pressure = 760 mmHg, measured arterial PaO2 = 95 mmHg:
Endocrinology — HOMA-IR at the insulin-resistance boundary. Fasting glucose 5.5 mmol/L, fasting insulin 60 pmol/L:
Ophthalmology — pupil size and depth of focus. At 20/20 acuity (visual_acuity_arcmin = 1.0), a 3 mm photopic pupil and a 6 mm scotopic pupil:
Halving the pupil diameter doubles the depth of focus, matching the formula's direct inverse proportionality — a large pupil admits a narrower range of in-focus object distances, which is why pinhole optics (an extremely small effective pupil) produces the deepest focus of all.
Orthodontics — arch-wire force from Euler beam theory. A 0.016 in (0.4064 mm) round NiTi wire (E = 41 GPa), 10 mm interbracket span, deflected 1 mm:
Dentistry — DMFT and treatment cost, lifted directly from the real pinned test. These two are not hand-computed; they are read straight from tests/mojo/test_wave91.mojo's test_dentistry:
Verification
Six of these seven modules (cardiology, endocrinology, nephrology, ophthalmology, orthodontics, pulmonology) have only the auto-generated smoke test found at mojo/tests/libs/test_<module>.mojo: each calls every function once with literal 0.5 arguments and asserts only that the result is finite (not isnan and not isinf), matching the pattern documented for the mixture-of-experts family — no specific numeric output is pinned by these files.
dentistry is the exception: tests/mojo/test_wave91.mojo's test_dentistry function pins 24 real assertions against exact expected values (dmft_index(3, 2, 5) == 10, treatment_cost_estimate(3, 200.0, 1.5) == 900.0, and 22 more), which is where the two dentistry numbers above come from verbatim. Every other worked example in this page was hand-traced against the formula as written in the corresponding mojo/engine/*.mojo file and cross-checked, where a standard textbook or clinical-reference value exists (CKD-EPI, Cockcroft-Gault, the alveolar gas equation, HOMA-IR), against that value.
Next: see Runtime library catalog for every module and function, or Engine health & verification for confirming the runtime executing these formulas is live.
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