Methods & formulas

Every calculation in this tool is driven by the formulas below, each tied to its source. These are modeled methods for education, not clinical guidance.

• Saturable absorption — dose-dependent bioavailability (Emax/Hill)

  Definition: Per-dose bioavailable fraction F that decreases as dose rises (gabapentin behavior: ~60% at 900 mg/day → ~33% at 3600 mg/day). Emax/Hill kernel applied to the fractional form; for the linear and saturable-elimination models F is a constant parameter.

  Expression

  F(D) = Emax · D50^n / (D50^n + D^n)

  For developers

  Code reference

  bioavailability.ts#bioavailability

  Sources

  • Stewart BH, Kugler AR, Thompson PR, Bockbrader HN (1993) (opens in a new tab) A Saturable Transport Mechanism in the Intestinal Absorption of Gabapentin. Pharm Res. Identifier: PMID 8456077 Verified quote

    Clinical trials have reported dose-dependent bioavailabilities ranging from 73.8 +/- 18.3 to 35.7 +/- 18.3% when the dose was increased from 100 to 1600 mg. … The permeability of gabapentin … was consistent with carrier-mediated absorption, i.e., a 75 to 80% decrease in permeability when the drug concentration was increased from 0.01 to 50 mM.

  • Bockbrader HN, Wesche D, Miller R, Chapel S, Janiczek N, Burger P (2010) (opens in a new tab) A comparison of the pharmacokinetics and pharmacodynamics of pregabalin and gabapentin. Clin Pharmacokinet. Identifier: PMID 20818832 Verified quote

    The absolute bioavailability of gabapentin drops from 60% to 33% as the dosage increases from 900 to 3600 mg/day, while the absolute bioavailability of pregabalin remains at > or = 90% irrespective of the dosage.

  • Svensson RJ, Aarnoutse RE, Diacon AH, et al. (2017) (opens in a new tab) A Population Pharmacokinetic Model Incorporating Saturable Pharmacokinetics and Autoinduction for High Rifampicin Doses. Clin Pharmacol Ther. Identifier: PMID 28653479; PMCID PMC5888114 Verified quote

    The statistically significant relationship between dose and F was described using an Emax relationship.

• Concentration-time simulation dispatcher (oral Bateman, IV, MM ODE)

  Definition: Top-level simulate() validates inputs and dispatches by model and route: superposed Bateman for linear/saturable-absorption oral, closed-form multi-dose IV bolus/infusion, and an RK4 integration of the gut+central ODE system for saturable elimination. IV routes force F=1 (absorption bypassed). The Garrett (oral Bateman) and Saganuwan (MM elimination) citations here cover the oral and saturable-elimination paths only; the IV-route closed forms (bolus/infusion) have their own provenance in the ivBolus, ivInfusionDuring, ivInfusionPost, and cssInfusion entries.

  Expression

  linear/abs: C(t)=Σ bateman(...); iv_bolus: C(t)=Σ (D/V)·e^(−ke·(t−iτ)); MM: dC/dt=(ka·A)/V + R0/V − Vmax·C/(Km+C)

  For developers

  Code reference

  simulate.ts#simulate

  Sources

  • Saganuwan SA (2021) (opens in a new tab) Application of modified Michaelis–Menten equations for determination of enzyme inducing and inhibiting drugs. BMC Pharmacol Toxicol. Identifier: PMCID PMC8507113 Verified quote

    V0 = Vm × C / (Km + C).

  • Garrett ER (1994) (opens in a new tab) The Bateman function revisited: A critical reevaluation of the quantitative expressions to characterize concentrations in the one compartment body model as a function of time with first-order invasion and first-order elimination. J Pharmacokinet Biopharm. Identifier: DOI 10.1007/BF02353864 Verified quote

    The Bateman function, A"(e-k(e)t--e-k(a)t), quantifies the time course of a first-order invasion (rate constant ka) to, and a first-order elimination (rate constant ke) from, a one-compartment body model where A" = (gamma Dose)ka/(ka-ke)V.

  • Henthorn TK, et al. (2021) (opens in a new tab) Elimination Clearance of Dexmedetomidine: Cross-examining What the Data Say. Anesthesiology / supporting source. Identifier: consensus 3a8f19bb Quote pending

    Citation recorded, quote pending verification.

• Point concentration sampler

  Definition: Convenience wrapper that runs simulate() and returns the concentration at the simulated time-grid sample closest to t. Carries no independent math; its provenance is that of simulate() and the underlying closed forms.

  Expression

  C(t) = simulate(regimen, params) sampled at the grid point nearest t

  For developers

  Code reference

  concentration.ts#concentration

  Sources

  • Garrett ER (1994) (opens in a new tab) The Bateman function revisited. J Pharmacokinet Biopharm. Identifier: DOI 10.1007/BF02353864 Verified quote

    The Bateman function, A"(e-k(e)t--e-k(a)t), quantifies the time course of a first-order invasion (rate constant ka) to, and a first-order elimination (rate constant ke) from, a one-compartment body model where A" = (gamma Dose)ka/(ka-ke)V.

• Series metrics — Cmax, Cmin, Tmax, trapezoidal AUC, time-to-steady-state

  Definition: Cmax/Cmin/Tmax are extrema of the simulated series; AUC is the linear trapezoidal rule over the sampled window; timeToSteadyState is a numerical convergence test on per-cycle peaks (valid for linear, saturable absorption, AND Michaelis–Menten, where the 4–5·t½ rule does not hold).

  Expression

  AUC = Σ 0.5·(C_i + C_{i−1})·(t_i − t_{i−1}); timeToSteadyState = earliest cycle with |ΔpeakC|/peakC < 5%

  For developers

  Code reference

  metrics.ts#metrics

  Sources

  • Rowland M, Tozer TN (2011) (opens in a new tab) Clinical Pharmacokinetics and Pharmacodynamics: Concepts and Applications (4th ed.). Lippincott Williams & Wilkins (textbook). Identifier: ISBN 978-0781750097 Standard textbook

    The area under the plasma concentration–time curve is commonly estimated by the trapezoidal rule, in which the area between successive sampling times is approximated by that of a trapezoid.

  • Wagner JG (1978) (opens in a new tab) Time to reach steady state and prediction of steady-state concentrations for drugs obeying Michaelis-Menten kinetics. J Pharmacokinet Biopharm. Identifier: consensus 6cda5107 Quote pending

    Citation recorded, quote pending verification.

• Two-dose comparison — dose / Cmax / AUC ratios and nonlinearity factor

  Definition: Simulates two regimens differing only in dose under identical PK and reports the headline ratios. nonlinearityFactor ≈ 1 for linear PK, < 1 for saturable absorption (sub-proportional exposure), > 1 for saturable elimination (supra-proportional exposure) — the single number that operationalizes 'halving the dose ≠ halving the effect.'

  Expression

  doseRatio = D_B/D_A; aucRatio = AUC_B/AUC_A; nonlinearityFactor = aucRatio / doseRatio

  For developers

  Code reference

  compareDoses.ts#compareDoses

  Sources

  • Shargel L, Yu ABC (2016) (opens in a new tab) Applied Biopharmaceutics & Pharmacokinetics (7th ed.). McGraw-Hill (textbook). Identifier: ISBN 978-0071830935 Standard textbook

    For drugs that follow linear pharmacokinetics, the area under the curve (AUC) is directly proportional to the dose; deviation from dose proportionality indicates nonlinear (capacity-limited) pharmacokinetics.

• Linear-baseline counterfactual params

  Definition: Builds the linear-PK counterfactual used to plot 'what naive linear scaling would predict' alongside the active nonlinear model. For saturable elimination the first-order limit at C≪Km gives ke_eff = Vmax/Km (= CL/V).

  Expression

  saturableAbsorption → freeze F at bioavailability(dose); saturableElimination → ke_eff = Vmax/Km

  For developers

  Code reference

  linearize.ts#linearize

  Sources

  • Saganuwan SA (2021) (opens in a new tab) Application of modified Michaelis–Menten equations for determination of enzyme inducing and inhibiting drugs. BMC Pharmacol Toxicol. Identifier: PMCID PMC8507113 Verified quote

    V0 = Vm × C / (Km + C).

• Concentration unit conversion

  Definition: Definitional conversions among mg/L, µg/mL, g/L, and µmol/L. Molar conversions require a molecular weight (g/mol); a missing MW throws a typed MISSING_MOLECULAR_WEIGHT error.

  Expression

  mg/L ≡ µg/mL; g/L = mg/L × 1000; µmol/L = (mg/L) × 1000 / MW

  For developers

  Code reference

  units.ts#convertUnits

  Sources

  • Shargel L, Yu ABC (2016) (opens in a new tab) Applied Biopharmaceutics & Pharmacokinetics (7th ed.). McGraw-Hill (textbook). Identifier: ISBN 978-0071830935 Standard textbook

    Molar concentration (mol/L) is obtained by dividing the mass concentration by the molecular weight of the drug.

• Elimination half-life (first-order)

  Definition: First-order elimination half-life. Universal identity following from C(t) = C0·e^(−ke·t).

  Expression

  t½ = ln(2) / ke

  For developers

  Code reference

  analytical.ts#halfLife

  Sources

  • Rowland M, Tozer TN (2011) (opens in a new tab) Clinical Pharmacokinetics and Pharmacodynamics: Concepts and Applications (4th ed.). Lippincott Williams & Wilkins (textbook). Identifier: ISBN 978-0781750097 Standard textbook

    The half-life is related to the elimination rate constant by the equation t1/2 = 0.693/k.

• Clearance from volume and elimination rate

  Definition: One-compartment first-order clearance as the product of distribution volume and elimination rate constant. Universal identity.

  Expression

  CL = V · ke

  For developers

  Code reference

  analytical.ts#clearance

  Sources

  • Rowland M, Tozer TN (2011) (opens in a new tab) Clinical Pharmacokinetics and Pharmacodynamics: Concepts and Applications (4th ed.). Lippincott Williams & Wilkins (textbook). Identifier: ISBN 978-0781750097 Standard textbook

    Clearance is the product of the volume of distribution and the elimination rate constant, CL = k · V.

• Analytical Tmax for first-order oral input

  Definition: Closed-form time of peak concentration for the one-compartment first-order oral (Bateman) model; degenerate ka = ke takes the limit Tmax = 1/ke.

  Expression

  Tmax = ln(ka/ke) / (ka − ke) [→ 1/ke when ka = ke]

  For developers

  Code reference

  analytical.ts#tmaxLinear

  Sources

  • Garrett ER (1994) (opens in a new tab) The Bateman function revisited. J Pharmacokinet Biopharm. Identifier: DOI 10.1007/BF02353864 Verified quote

    Since k(e)tmax = ln m/(m-1), m can be determined from the given table relating m and k(e)tmax.

• Analytical Cmax for a single linear oral dose

  Definition: Bateman concentration evaluated at Tmax for a single oral dose; degenerate ka = ke uses (F·D·k/V)·Tmax·e^(−k·Tmax).

  Expression

  Cmax = (F·D·ka)/(V·(ka−ke)) · (e^(−ke·Tmax) − e^(−ka·Tmax))

  For developers

  Code reference

  analytical.ts#cmaxLinear

  Sources

  • Garrett ER (1994) (opens in a new tab) The Bateman function revisited. J Pharmacokinet Biopharm. Identifier: DOI 10.1007/BF02353864 Verified quote

    The Bateman function, A"(e-k(e)t--e-k(a)t), quantifies the time course of a first-order invasion (rate constant ka) to, and a first-order elimination (rate constant ke) from, a one-compartment body model where A" = (gamma Dose)ka/(ka-ke)V.

• Analytical AUC₀→∞ for a single linear oral dose

  Definition: Total exposure of a single first-order oral dose. Universal closed form; equals F·D divided by clearance.

  Expression

  AUC₀→∞ = F·D / (V·ke) = F·D / CL

  For developers

  Code reference

  analytical.ts#aucInfLinear

  Sources

  • Rowland M, Tozer TN (2011) (opens in a new tab) Clinical Pharmacokinetics and Pharmacodynamics: Concepts and Applications (4th ed.). Lippincott Williams & Wilkins (textbook). Identifier: ISBN 978-0781750097 Standard textbook

    The total area under the plasma concentration–time curve following a single dose is AUC = F·Dose/CL, where CL is clearance.

  • Garrett ER (1994) (opens in a new tab) The Bateman function revisited. J Pharmacokinet Biopharm. Identifier: DOI 10.1007/BF02353864 Verified quote

    When a drug has 100% bioavailability, regression of Dose/V/C on AUC/C in the nonabsorption phase gives ke no matter what is the ratio of m = ka/ke.

• Average steady-state concentration (multiple-dose linear)

  Definition: Mean concentration over a dosing interval at steady state for first-order multiple-dose PK; equals the single-dose AUC divided by the interval τ.

  Expression

  Css_avg = F·D / (CL·τ) = AUC_τ / τ

  For developers

  Code reference

  analytical.ts#cssAvgLinear

  Sources

  • Rowland M, Tozer TN (2011) (opens in a new tab) Clinical Pharmacokinetics and Pharmacodynamics: Concepts and Applications (4th ed.). Lippincott Williams & Wilkins (textbook). Identifier: ISBN 978-0781750097 Standard textbook

    At plateau, the average concentration is Cav = F·Dose/(CL·τ), where τ is the dosing interval.

• Closed-form steady-state concentration during a dosing interval (linear)

  Definition: Superposition steady-state Bateman concentration for an oral dose D every τ hours, 0 ≤ t ≤ τ; degenerate ka = ke uses the convergent series limit form.

  Expression

  Css(t) = (F·D·ka)/(V·(ka−ke)) · [ e^(−ke·t)/(1−e^(−ke·τ)) − e^(−ka·t)/(1−e^(−ka·τ)) ]

  For developers

  Code reference

  analytical.ts#cssLinear

  Sources

  • Wijnand HP (1988) (opens in a new tab) Equations for one- and two-compartment models with equal absorption and elimination rate constants. Pharmacokinetics (supporting source). Identifier: consensus 537ab9cb Quote pending

    Citation recorded, quote pending verification.

  • Garrett ER (1994) (opens in a new tab) The Bateman function revisited. J Pharmacokinet Biopharm. Identifier: DOI 10.1007/BF02353864 Verified quote

    The Bateman function and "feathering" fail when the rate constants are equal. The time course is then expressed by C = gamma Dtk e-kt.

• IV-bolus concentration (one-compartment, first-order)

  Definition: Single IV-bolus concentration-time profile for one-compartment first-order elimination. Also backs the multi-dose IV-bolus superposition inside simulate().

  Expression

  C(t) = (D/V) · e^(−ke·t)

  For developers

  Code reference

  analytical.ts#ivBolus

  Sources

  • Savva M (2021) (opens in a new tab) A mathematical treatment of multiple intermittent intravenous infusions in a one-compartment model. Comput Methods Programs Biomed (supporting source). Identifier: consensus 43e10516 Quote pending

    Citation recorded, quote pending verification.

• Single IV-bolus concentration (alias of ivBolus)

  Definition: Symmetry alias of ivBolus provided alongside the infusion helpers. Same closed form and provenance as ivBolus.

  Expression

  C(t) = (D/V) · e^(−ke·t)

  For developers

  Code reference

  analytical.ts#ivBolusSingle

  Sources

  • Savva M (2021) (opens in a new tab) A mathematical treatment of multiple intermittent intravenous infusions in a one-compartment model. Comput Methods Programs Biomed (supporting source). Identifier: consensus 43e10516 Quote pending

    Citation recorded, quote pending verification.

• Concentration during an ongoing IV infusion

  Definition: Rising concentration during a constant-rate (R0 mg/h) IV infusion, one-compartment first-order elimination.

  Expression

  C(t) = (R0 / (V·ke)) · (1 − e^(−ke·t)), 0 ≤ t ≤ T_inf

  For developers

  Code reference

  analytical.ts#ivInfusionDuring

  Sources

  • Savva M (2021) (opens in a new tab) A mathematical treatment of multiple intermittent intravenous infusions in a one-compartment model. Comput Methods Programs Biomed (supporting source). Identifier: consensus 744a3e72 Quote pending

    Citation recorded, quote pending verification.

• Concentration after an IV infusion stops

  Definition: Post-infusion decay tPost hours after a constant-rate infusion of duration Tinf has ended, one-compartment first-order elimination.

  Expression

  C(tPost) = (R0 / (V·ke)) · (1 − e^(−ke·Tinf)) · e^(−ke·tPost)

  For developers

  Code reference

  analytical.ts#ivInfusionPost

  Sources

  • Savva M (2021) (opens in a new tab) A mathematical treatment of multiple intermittent intravenous infusions in a one-compartment model. Comput Methods Programs Biomed (supporting source). Identifier: consensus 744a3e72 Quote pending

    Citation recorded, quote pending verification.

• Steady-state Css for continuous IV infusion (linear)

  Definition: Plateau concentration of a continuous constant-rate IV infusion under first-order elimination; the T_inf → ∞ limit of ivInfusionDuring.

  Expression

  Css = R0 / CL = R0 / (V · ke)

  For developers

  Code reference

  analytical.ts#cssInfusion

  Sources

  • Savva M (2021) (opens in a new tab) A mathematical treatment of multiple intermittent intravenous infusions in a one-compartment model. Comput Methods Programs Biomed (supporting source). Identifier: consensus 744a3e72 Quote pending

    Citation recorded, quote pending verification.

• Steady-state Css under Michaelis–Menten elimination

  Definition: Steady-state concentration for a constant dose rate R0 under capacity-limited (Michaelis–Menten) elimination. No steady state exists when R0 ≥ Vmax (toxic accumulation), so the function returns +∞ there.

  Expression

  Css = (R0 · Km) / (Vmax_mass − R0) [→ +∞ when R0 ≥ Vmax]

  For developers

  Code reference

  analytical.ts#cssMM

  Sources

  • Sumarno, Kusumastuti K, Khotib J (2023) (opens in a new tab) An analysis of the Michaelis-Menten pharmacokinetics of phenytoin in epileptic Indonesian adults. Pharmacy Education. Identifier: consensus 491dc0d9 Verified quote

    The determination of the Michaelis-Menten kinetic parameters (Vmax and Km) was calculated using the formula (equation) (Shargel & Yu, 2022): R = V max − Km.R/Css … Css: Steady level of drug in plasma … R: Dosage/day or rate of dosing … Vmax: The maximum rate of drug metabolism … Km: Michaelis-Menten constant.

  • Martin E, Tozer TN, Sheiner LB, Riegelman S (1977) (opens in a new tab) The clinical pharmacokinetics of phenytoin. J Pharmacokinet Biopharm. Identifier: consensus 10fb327d Verified quote

    For each subject the Cpss values were fitted to a rearranged Michaelis-Menten equation Cpss =KmR/(Vm-R).

• Personalised Vmax from a single steady-state TDM point

  Definition: Back-calculates an individual's maximum metabolic rate from one (dose rate R0, steady-state concentration Css1) pair given a population Km — the phenytoin dosage-individualization workflow.

  Expression

  Vmax_mass = R0 · (Km + Css1) / Css1

  For developers

  Code reference

  analytical.ts#vmaxFromTDM

  Sources

  • Nakashima E, et al. (1995) (opens in a new tab) Systematic Approach to a Dosage Regimen for Phenytoin Based on One-Point, Steady-State Plasma Concentration. Therapeutic Drug Monitoring. Identifier: consensus d2065368 Verified quote

    This system applies the Michaelis-Menten equation to the initial data pair (D1-Css1) and solves for (a) maximum metabolic rate constant (Vmax) assuming the population mean for the Michaelis constant (Km) (method 1).

• Predict new MM steady-state from personalised Vmax

  Definition: Companion to vmaxFromTDM: predicts the steady-state concentration at a new dose rate R0' using the individualized Vmax. Wraps cssMM.

  Expression

  Css2 = (R0' · Km) / (Vmax_mass − R0')

  For developers

  Code reference

  analytical.ts#predictCssMM

  Sources

  • Nakashima E, et al. (1995) (opens in a new tab) Systematic Approach to a Dosage Regimen for Phenytoin Based on One-Point, Steady-State Plasma Concentration. Therapeutic Drug Monitoring. Identifier: consensus d2065368 Verified quote

    Accurate predictions of the Css error within 5 micrograms/ml were obtained in 84% of the 25 cases, and in 30% of the 10 cases excluded.

• Power-law dose–exposure relationship

  Definition: Empirical power model for sub-proportional dose–exposure (Chen 2013 fit for gabapentin enacarbil, b ≈ 0.925 for bioavailable dose). An alternative to the Emax form when only summary data are available.

  Expression

  Exposure = a · D^b

  For developers

  Code reference

  analytical.ts#powerExposure

  Sources

  • Chen C (2013) (opens in a new tab) Meta-analyses of dose-exposure relationships for gabapentin following oral administration of gabapentin and gabapentin enacarbil. Eur J Clin Pharmacol. Identifier: consensus c44283c0 Verified quote

    For gabapentin enacarbil, a power model was most suitable, with a power of 0.925 for bioavailable dose or 0.844 for steady-state concentration.

• Steady-state accumulation ratio (linear)

  Definition: Factor by which steady-state Cmax exceeds single-dose Cmax for first-order PK dosed every τ hours. Universal identity from the geometric superposition series.

  Expression

  R = 1 / (1 − e^(−ke·τ))

  For developers

  Code reference

  analytical.ts#accumulationRatio

  Sources

  • Rowland M, Tozer TN (2011) (opens in a new tab) Clinical Pharmacokinetics and Pharmacodynamics: Concepts and Applications (4th ed.). Lippincott Williams & Wilkins (textbook). Identifier: ISBN 978-0781750097 Standard textbook

    The accumulation index, Rac = 1/(1 − e^(−k·τ)), expresses the extent of accumulation on repeated dosing relative to a single dose.

• Regimen optimizer (numerical method — NOT a literature PK formula)

  Definition: METHOD, not a pharmacokinetic law. Searches only the prescribable regimen (dose/interval/#doses) to make a candidate curve resemble a target curve while holding the drug's PK fixed (truthfulness constraint). The math cited here is the optimization method, not a drug formula.

  Expression

  argmin over (dose, interval, doses) of distance(C_target, C_candidate); 1-D via golden-section search, ≥2-D via Nelder–Mead simplex

  For developers

  Code reference

  optimize.ts#recommendRegimen

  Sources

  • Kiefer J (1953) (opens in a new tab) Sequential minimax search for a maximum. Proc Amer Math Soc (textbook method). Identifier: DOI 10.1090/S0002-9939-1953-0055639-3 Standard textbook

    Golden-section search locates the extremum of a unimodal function on an interval by successively narrowing the bracket using the golden ratio.

  • Nelder JA, Mead R (1965) (opens in a new tab) A simplex method for function minimization. The Computer Journal (textbook method). Identifier: DOI 10.1093/comjnl/7.4.308 Standard textbook

    A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n+1) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point.

• Antidepressant dose-equivalence (fluoxetine-equivalent scale)

  Definition: Approximate acute-phase switching conversion between antidepressants on the published fluoxetine-equivalent scale (Hayasaka 2015 primary-analysis table; citalopram 40 mg ≡ fluoxetine 40 mg added from Furukawa 2019). A population-average switching guide on a surrogate dose scale, NOT bioequivalence or receptor-occupancy equivalence. Furukawa 2019 also supplies the efficacy-plateau flag (above ~40 mg fluox-eq the dose-efficacy curve is flat-to-decreasing). Duloxetine is in neither source and the engine refuses to convert it.

  Expression

  targetDose = dose × (equivTarget / equivSource), where equivX = mg/day of X ≡ fluoxetine 40 mg

  For developers

  Code reference

  equivalence/antidepressants.ts#convert

  Sources

  • Hayasaka Y, Purgato M, Magni LR, Ogawa Y, Takeshima N, Cipriani A, Barbui C, Leucht S, Furukawa TA (2015) (opens in a new tab) Dose equivalents of antidepressants: Evidence-based recommendations from randomized controlled trials. J Affect Disord. Identifier: PMID 25911132; DOI 10.1016/j.jad.2015.03.021 Verified quote

    In the primary analysis, fluoxetine 40mg/day was equivalent to paroxetine dosage of 34.0mg/day, … escitalopram 18.0mg/day, … mirtazapine 50.9mg/day, … sertraline 98.5mg/day, … and venlafaxine 149.4mg/day.

  • Furukawa TA, Cipriani A, Cowen PJ, Leucht S, Egger M, Salanti G (2019) (opens in a new tab) Optimal dose of SSRIs, venlafaxine, and mirtazapine in major depression: a systematic review and dose-response meta-analysis. Lancet Psychiatry. Identifier: PMID 31178367; PMCID PMC6586944 Verified quote

    Table 1 (Current study column): "Citalopram 20" — i.e. citalopram 20 mg = 1 unit = 20 mg fluoxetine, so citalopram 40 mg ≈ fluoxetine 40 mg (fluoxetine-40-equivalent = 40.0). Efficacy plateau: "the dose-efficacy curve showed a gradual increase up to doses between 20 mg and 40 mg fluoxetine equivalents, and a flat to decreasing trend through the higher licensed doses up to 80 mg."

• Citalopram QT-prolongation maximum-dose ceiling (40 mg/day)

  Definition: Drug-specific FDA-regulated maximum-dose ceiling, NOT an equivalence ratio. A fluoxetine-equivalent conversion can mathematically land on a citalopram dose above the regulatory maximum (e.g. escitalopram 20 → citalopram 44.4 mg); this entry backs a loud safety overlay (ceilingWarning) on that branch. The engine still returns the math so the clinician sees why it is unsafe — it is a warning, not a refusal. Escitalopram has no equivalent absolute mg ceiling, so the asymmetry is intentional.

  Expression

  warn when target = citalopram AND computed dose > 40 mg/day (20 mg/day if ≥60 y, hepatic impairment, or CYP2C19 poor metaboliser)

  For developers

  Code reference

  equivalence/antidepressants.ts#convert

  Sources

  • McClelland J, Mathys M (2016) (opens in a new tab) Evaluation of QTc prolongation and dosage effect with citalopram. Ment Health Clin. Identifier: PMID 29955465; PMCID PMC6007721 Verified quote

    This warning advised against prescribing citalopram at doses >40 mg/day due to 'dose-dependent QT interval prolongation.' This safety warning also decreased the maximum dose to 20 mg in patients ≥60 years old, those with hepatic impairment, those who are CYP2C19 poor metabolizers, and/or those taking cimetidine or other CYP2C19 inhibitors.

• Antidepressant low-dose (sub-therapeutic) thresholds

  Definition: Lower-end dose thresholds below which efficacy, while superior to placebo, is inferior to higher doses (a dose-dependency confirmation). Used only as a warning flag for the three drugs Hieronymus 2016 names; no threshold is invented for unnamed drugs.

  Expression

  flag dose ≤ lower-end floor (citalopram 20 mg, paroxetine 10 mg, sertraline 50 mg)

  For developers

  Code reference

  equivalence/antidepressants.ts#lowDoseWarning

  Sources

  • Hieronymus F, Nilsson S, Eriksson E (2016) (opens in a new tab) A mega-analysis of fixed-dose trials reveals dose-dependency and a low efficacy of citalopram 10 mg, paroxetine 10 mg and sertraline 50 mg as a class for the treatment of depression. Transl Psychiatry. Identifier: PMID 27271860; PMCID PMC4931602 Verified quote

    Doses below or at the lower end of the usually recommended dose range (citalopram: 10–20 mg, paroxetine: 10 mg; sertraline: 50 mg) were superior to placebo but inferior to higher doses, hence confirming a dose-dependency to be at hand.

• Statin log-linear LDL-C dose-response

  Definition: Continuous log-linear model for the additional %LDL-C reduction per doubling of statin dose. The per-statin intercept comes from the Law 2003 anchor table; k ≈ 6 percentage points per doubling is from Oni-Orisan 2018. Drives both the source %reduction and the inverse (dose-for-%reduction) used to map an equivalent target dose.

  Expression

  %LDL-C reduction(dose) = reduction(ref) + k · log2(dose/ref), k ≈ 6 pp per doubling

  For developers

  Code reference

  equivalence/statins.ts#equivalentDose

  Sources

  • Oni-Orisan A, et al. (2018) (opens in a new tab) Characterization of Statin LDL-C Dose-response Utilizing Electronic Health Records in a Large Population-based Cohort. Circ Genom Precis Med. Identifier: PMID 30354326; PMCID PMC6214660 Verified quote

    LDL-C was lowered by an additional 6.2% (5.6% after adjustment for covariates) of the original pretreatment LDL-C for each doubling of the statin dose.

• Per-statin %LDL-C reduction anchor table (Law 2003)

  Definition: Per-statin, per-dose %LDL-C reduction anchors from the Law 2003 meta-analysis (baseline 4.8 mmol/L), the intercepts the log-linear model is anchored on. Implied potency: rosuvastatin ≈ 2× atorvastatin ≈ 4× simvastatin per mg.

  Expression

  reduction(statin, dose): e.g. rosuva 10=43, atorva 20=43, simva 40=37 (% from baseline 4.8 mmol/L)

  For developers

  Code reference

  equivalence/statins.ts#equivalentDose

  Sources

  • Law MR, Wald NJ, Rudnicka AR (2003) (opens in a new tab) Quantifying effect of statins on low density lipoprotein cholesterol, ischaemic heart disease, and stroke: systematic review and meta-analysis. BMJ. Identifier: PMID 12829554; PMCID PMC162260 Verified quote

    Reductions in LDL cholesterol (in the 164 trials) were 2.8 mmol/l (60%) with rosuvastatin 80 mg/day, 2.6 mmol/l (55%) with atorvastatin 80 mg/day, 1.8 mmol/l (40%) with atorvastatin 10 mg/day, lovastatin 40 mg/day, simvastatin 40 mg/day, or rosuvastatin 5 mg/day, all from pretreatment concentrations of 4.8 mmol/l.

• Statin individual-response band half-width (±15 pp)

  Definition: Half-width of the displayed equivalence range on the %-reduction scale, from the individual-response SD in VOYAGER (Karlson 2016: 12.8–17.9%, midpoint ≈ 15 pp). An individual-variability spread, NOT a confidence interval on the mean — it widens the displayed range to reflect real between-patient spread before inversion to a dose range.

  Expression

  displayed range = point %reduction ± 15 percentage points (SD midpoint of 12.8–17.9)

  For developers

  Code reference

  equivalence/statins.ts#equivalentDose

  Sources

  • Karlson BW, Wiklund O, Palmer MK, Nicholls SJ, Lundman P, Barter PJ (2016) (opens in a new tab) Variability of low-density lipoprotein cholesterol response with different doses of atorvastatin, rosuvastatin, and simvastatin: results from VOYAGER. Eur Heart J Cardiovasc Pharmacother. Identifier: PMID 27533947; DOI 10.1093/ehjcvp/pvw006 Verified quote

    The standard deviation of LDL-C reduction for all statins and doses ranged from 12.8 to 17.9%.

• ARB % of licensed maximum dose (Makani 2013; no mg-for-mg conversion)

  Definition: The only validated ARB (sartan) comparison axis: each ARB normalized to a fraction of its OWN licensed maximum dose, never converted mg-for-mg. The foundational ambulatory-BP meta-analysis (Makani 2013) deliberately avoided absolute-mg equivalence and stratified on 25%/50%/100% of the guideline-defined max dose, giving a shallow pooled dose-response (10.3/6.7, 11.7/7.6, 13.0/8.3 mmHg; doubling adds <2 mmHg). The per-drug licensed maxima (losartan 100, valsartan 320, telmisartan 80, olmesartan 40, candesartan 32, irbesartan 300 mg/day) come from Makani Table 1 with maxima as defined in JNC hypertension guidelines, and are the comparison denominator. This entry also backs the engine's refusal of mg-for-mg conversion (sourceForRefusal). Displayed tier B (A-grade meta-analysis ∧ guideline-defined per-drug maxima).

  Expression

  percentOfMax = dose / licensedMaxMg × 100; dose-response @ 25/50/100% max → 10.3/6.7, 11.7/7.6, 13.0/8.3 mmHg (doubling adds <2 mmHg)

  For developers

  Code reference

  equivalence/arb.ts#percentOfMax

  Sources

  • Makani H, Bangalore S, Supariwala A, Romero J, Argulian E, Messerli FH (2013) (opens in a new tab) Antihypertensive efficacy of angiotensin receptor blockers as monotherapy as evaluated by ambulatory blood pressure monitoring: a meta-analysis. Eur Heart J. Identifier: PMID 23966312; PMCID PMC5994844 Verified quote

    the dose–response curve with ARBs was shallow with decrease of 10.3/6.7 (systolic/diastolic), 11.7/7.6, and 13.0/8.3 mmHg with 25% max dose, 50% max dose, and with the max dose of ARBs, respectively. … doubling the dose merely increased the antihypertensive efficacy by <2 mmHg systolic or diastolic.

• ARB ordinal SUCRA potency rank (Zhang 2024) — badge only, never a ratio

  Definition: Optional ordinal potency annotation for the ARB comparison: a SUCRA (surface under the cumulative ranking) rank label from the Zhang 2024 network meta-analysis (olmesartan ranked highest for office systolic/diastolic BP reduction; valsartan and losartan ranked lower). It is a DISPLAY badge only — never a conversion ratio and never fed into the %-of-max math or any other number.

  Expression

  ordinal SUCRA rank: olmesartan highest; valsartan and losartan lower (NEVER a numeric ratio, NEVER fed into any calculation)

  For developers

  Code reference

  equivalence/arb.ts#percentOfMax

  Sources

  • Zhang Z, Yang H, Guo H (2024) (opens in a new tab) Comparative efficacy and safety of six angiotensin receptor blockers in hypertensive patients: a network meta-analysis. Int J Clin Pharm. Identifier: PMID 38861046 Verified quote

    Olmesartan had the highest surface under the cumulative ranking in reducing office systolic (91.4%) and diastolic blood pressure (87.2%). … Valsartan and losartan were less effective in lowering blood pressure than other drugs.

• ARB reference data — losartan dose-range + BP effect (Gradman 1995)

  Definition: Losartan dose-ranging trial (10/25/50/100/150 mg) with supine-trough change-from-baseline BP reductions including the concurrent placebo arm — REFERENCE DATA for ArbComparison.doseEffectReference, never a conversion.

  Expression

  reference data only — per-drug dose range + BP effect (no formula, no conversion)

  For developers

  Code reference

  equivalence/arb.ts#percentOfMax

  Sources

  • Gradman AH, Arcuri KE, Goldberg AI, Ikeda LS, Nelson EB, Snavely DB, Sweet CS (1995) (opens in a new tab) A randomized, placebo-controlled, double-blind, parallel study of various doses of losartan potassium compared with enalapril maleate in patients with essential hypertension. Hypertension. Identifier: PMID 7768585; DOI 10.1161/01.hyp.25.6.1345 Verified quote

    After 8 weeks of treatment, mean reductions from baseline in supine systolic/diastolic pressure 24 hours after dosing (trough) for losartan potassium 10, 25, 50, 100, and 150 mg, enalapril maleate 20 mg, and placebo were 7.6/7.9, 7.8/6.8, 13.0/10.1, 8.9/9.9, 10.5/9.7, 14.7/11.2, and 3.8/5.6 mm Hg, respectively.

• ARB reference data — valsartan dose-range + BP effect (Pool 1998 + Oparil 1996)

  Definition: Valsartan integrated dose-ranging analysis (10/20/40/80/160/320 mg) with placebo-subtracted per-dose SBP/DBP reductions, plus the Oparil 1996 optimal once-daily dose range — REFERENCE DATA for ArbComparison.doseEffectReference, never a conversion.

  Expression

  reference data only — per-drug dose range + BP effect (no formula, no conversion)

  For developers

  Code reference

  equivalence/arb.ts#percentOfMax

  Sources

  • Pool J, Oparil S, Hedner T, Glazer R, Oddou-Stock P, Hester A (1998) (opens in a new tab) Dose-responsive antihypertensive efficacy of valsartan, a new angiotensin II-receptor blocker. Clin Ther. Identifier: PMID 9916605; DOI 10.1016/s0149-2918(98)80107-0 Verified quote

    placebo-subtracted mean changes from baseline to end point for valsartan 10, 20, 40, 80, 160, and 320 mg, respectively: SDBP, -0.8, -2.8, -2.6, -3.9, -5.1, and -6.4 mm Hg; SSBP, -1.3, -5.7, -5.3, -6.8, -8.6, and -9.0 mm Hg

  • Oparil S, Dyke S, Harris F, Kief J, James D, Hester A, Fitzsimmons S (1996) (opens in a new tab) The efficacy and safety of valsartan compared with placebo in the treatment of patients with essential hypertension. Clin Ther. Identifier: PMID 8930424; DOI 10.1016/s0149-2918(96)80040-3 Verified quote

    The optimal dose range is 80 to 160 mg, given once daily.

• ARB reference data — candesartan dose-range + BP effect (Reif 1998 + Elmfeldt 1997)

  Definition: Candesartan cilexetil dose-ranging trial (2/4/8/16/32 mg) with change-from-baseline BP reductions vs the placebo arm, plus the Elmfeldt 1997 placebo-corrected sitting-DBP-by-dose support — REFERENCE DATA for ArbComparison.doseEffectReference, never a conversion.

  Expression

  reference data only — per-drug dose range + BP effect (no formula, no conversion)

  For developers

  Code reference

  equivalence/arb.ts#percentOfMax

  Sources

  • Reif M, White WB, Fagan TC, Oparil S, Flanagan TL, Edwards DT, Cushing DJ, Michelson EL (1998) (opens in a new tab) Effects of candesartan cilexetil in patients with systemic hypertension. Candesartan Cilexetil Study Investigators. Am J Cardiol. Identifier: PMID 9794352; DOI 10.1016/s0002-9149(98)00627-4 Verified quote

    Patients received either placebo or candesartan cilexetil 2, 4, 8, 16, or 32 mg once daily for 8 weeks. … Mean changes in BP were -10.7/-7.8 mm Hg and -12.6/-10.2 mm Hg in the 16- and 32-mg groups, respectively, versus -0.3/-2.6 mm Hg in the placebo group.

  • Elmfeldt D, George M, Hübner R, Olofsson B (1997) (opens in a new tab) Candesartan cilexetil, a new generation angiotensin II antagonist, provides dose dependent antihypertensive effect. J Hum Hypertens. Identifier: PMID 9331007 Verified quote

    The placebo-corrected mean reductions in sitting diastolic BP were approximately 2.5 mm Hg with 2 mg, 4.5 mm Hg with 4 mg, 6 mm Hg with 8 mg, and 8 mm Hg with 16 mg of candesartan cilexetil.

• ARB reference data — irbesartan dose-range + BP effect (Pool 1998 + Kassler-Taub 1998)

  Definition: Irbesartan dose-ranging trial (1/5/10/25/50/100/200/300 mg) with trough seated DBP reductions (systolic not reported per-dose), plus the Kassler-Taub 1998 300 mg irbesartan vs 100 mg losartan differential — REFERENCE DATA for ArbComparison.doseEffectReference, never a conversion.

  Expression

  reference data only — per-drug dose range + BP effect (no formula, no conversion)

  For developers

  Code reference

  equivalence/arb.ts#percentOfMax

  Sources

  • Pool JL, Guthrie RM, Littlejohn TW 3rd, Raskin P, Shephard AM, Weber MA, Weir MR, Wilson TW, Wright J, Kassler-Taub KB, Reeves RA (1998) (opens in a new tab) Dose-related antihypertensive effects of irbesartan in patients with mild-to-moderate hypertension. Am J Hypertens. Identifier: PMID 9607385 Verified quote

    After a placebo lead-in phase, all patients were randomized to 8 weeks of double-blind therapy with either placebo (n = 158) or irbesartan at doses of 1, 5, 10, 25, 50, 100, 200, or 300 mg (n = 731 total) orally once daily. … Reductions from baseline in trough seated diastolic blood pressure ranged from 7.5 mm Hg for 50 mg irbesartan to 11.6 mm Hg for 300 mg irbesartan.

  • Kassler-Taub K, Littlejohn T, Elliott W, Ruddy T, Adler E (1998) (opens in a new tab) Comparative efficacy of two angiotensin II receptor antagonists, irbesartan and losartan in mild-to-moderate hypertension. Am J Hypertens. Identifier: PMID 9607383; DOI 10.1016/s0895-7061(97)00491-3 Verified quote

    After 8 weeks of treatment, reductions from baseline in trough seated diastolic blood pressure (SeDBP) and trough seated systolic blood pressure (SeSBP) with 300 mg irbesartan were greater than with 100 mg losartan (P < .01 for both comparisons), by 3.0 and 5.1 mm Hg, respectively

• ARB reference data — telmisartan dose-range + BP effect (Sharpe 2001 + Smith 2000)

  Definition: Telmisartan review reporting supine-trough 'up to' BP reductions across 20-160 mg with maximum effect at 40-80 mg/day (Sharpe 2001), plus the Smith 2000 dose-ranging doses (40/80/120 mg) — REFERENCE DATA for ArbComparison.doseEffectReference, never a conversion.

  Expression

  reference data only — per-drug dose range + BP effect (no formula, no conversion)

  For developers

  Code reference

  equivalence/arb.ts#percentOfMax

  Sources

  • Sharpe M, Jarvis B, Goa KL (2001) (opens in a new tab) Telmisartan: a review of its use in hypertension. Drugs. Identifier: PMID 11558835; DOI 10.2165/00003495-200161100-00009 Verified quote

    Telmisartan 20 to 160 mg once daily produced mean reductions in supine trough systolic blood pressure and diastolic blood pressure of up to 15.5 and 10.5 mm Hg, respectively. … Maximum blood pressure reduction occurred with a dosage of 40 to 80 mg/day.

  • Smith DH, Matzek KM, Kempthorne-Rawson J (2000) (opens in a new tab) Dose response and safety of telmisartan in patients with mild to moderate hypertension. J Clin Pharmacol. Identifier: PMID 11185637 Verified quote

    After a 28-day placebo run-in period, patients were randomized to 28 days of once-daily, double-blind, double-dummy treatment with telmisartan 40, 80, or 120 mg; enalapril 20 mg; or placebo.

• ARB reference data — olmesartan dose-range + BP effect (Neutel 2002 + Zannad 2007)

  Definition: Olmesartan medoxomil dose-ranging trial (5/20/80 mg) with placebo-adjusted 24-h ambulatory SBP/DBP reductions (Neutel 2002), plus the Zannad 2007 finding that olmesartan's Emax was superior to other ARBs — supports the ceilings-differ honesty point. REFERENCE DATA for ArbComparison.doseEffectReference, never a conversion.

  Expression

  reference data only — per-drug dose range + BP effect (no formula, no conversion)

  For developers

  Code reference

  equivalence/arb.ts#percentOfMax

  Sources

  • Neutel JM, Elliott WJ, Izzo JL Jr, Chen CL, Masonson HN (2002) (opens in a new tab) Antihypertensive efficacy of olmesartan medoxomil, a new angiotensin II receptor antagonist, as assessed by ambulatory blood pressure measurements. J Clin Hypertens (Greenwich). Identifier: PMID 12368570; DOI 10.1111/j.1524-6175.2002.01051.x Verified quote

    Treatment with olmesartan medoxomil resulted in a significant placebo-adjusted reduction of mean 24-hour ambulatory diastolic blood pressure of 9.6 mm Hg, 12.2 mm Hg, and 10.6 mm Hg in the 5-, 20-, and 80-mg q.d. groups, respectively. … Corresponding reductions in mean ambulatory systolic blood pressure were 14.5 mm Hg, 16.5 mm Hg, and 15.4 mm Hg.

  • Zannad F, Fay R (2007) (opens in a new tab) Blood pressure-lowering efficacy of olmesartan relative to other angiotensin II receptor antagonists: an overview of randomized controlled studies. Fundam Clin Pharmacol. Identifier: PMID 17391291; DOI 10.1111/j.1472-8206.2007.00464.x Verified quote

    BP-lowering efficacy defined as Emax was superior with olmesartan, (DBP/SBP mmHg: -9.0/-12.4)

• Losartan is the weak-coverage ARB outlier (Makani 2013) — inferior 24-h ABPM at start and max dose

  Definition: Losartan is the ARB-class outlier: the Makani 2013 ambulatory-BP meta-analysis found losartan lowered 24-h ABP less well than the other ARBs at both starting (50 mg) and maximum dose. Backs the web losartan-outlier honesty flag (weaker per-dose + poorer 24-h once-daily coverage). Same paper as arbPercentOfMax. Not a conversion or ratio.

  Expression

  reference data only — losartan inferior 24-h ambulatory BP coverage vs other ARBs (no formula, no conversion)

  For developers

  Code reference

  equivalence/arb.ts#percentOfMax

  Sources

  • Makani H, Bangalore S, Supariwala A, Romero J, Argulian E, Messerli FH (2013) (opens in a new tab) Antihypertensive efficacy of angiotensin receptor blockers as monotherapy as evaluated by ambulatory blood pressure monitoring: a meta-analysis. Eur Heart J. Identifier: PMID 23966312; PMCID PMC5994844 Verified quote

    Losartan in the dose of 50 mg lowered ABP less well than other ARBs at 50% max dose by 2.5 mmHg systolic (P < 0.0001) and 1.8 mmHg diastolic (P = 0.0003). … Blood pressure reduction with losartan at starting dose and at max dose was consistently inferior to the other ARBs.

• ACE-inhibitor WHO DDD-anchored dose-equivalence (low-confidence, tier C)

  Definition: Low-confidence ACE-inhibitor dose-equivalence anchored on the WHO Defined Daily Dose (oral, group C09AA): captopril 50, enalapril 10, lisinopril 10, perindopril 4, ramipril 2.5 mg/day (so 1-DDD/day: ramipril 2.5 ≈ enalapril 10 ≈ lisinopril 10 ≈ perindopril 4 ≈ captopril 50). The DDD values are verified on-page; the equivalence ratio built from them is an INFERENCE, and WHO explicitly cautions that the DDD is a drug-utilization unit and does NOT reflect equipotency (§9.3). The engine is therefore pinned to tier C and carries the WHO 'not equipotency' caveat plus a duration-of-action note (ramipril/perindopril/lisinopril once-daily; enalapril/captopril often divided — Fischer & Diec 2021) as non-optional fields. The precise-PK ACE conversion (§4.1) stays a refusal and is not modelled.

  Expression

  eq_dose(Y) = dose(X) × (DDD_Y / DDD_X)

  For developers

  Code reference

  equivalence/ace.ts#convert

  Sources

  • WHO Collaborating Centre for Drug Statistics Methodology (2024) (opens in a new tab) ATC/DDD Index, group C09AA (ACE inhibitors, plain). WHO ATC/DDD Index. Identifier: ATC C09AA Verified quote

    The WHO Defined Daily Dose (DDD) values, oral, group C09AA: captopril 50 mg, enalapril 10 mg, lisinopril 10 mg, perindopril 4 mg, ramipril 2.5 mg.

  • WHO Collaborating Centre for Drug Statistics Methodology (2024) (opens in a new tab) DDD: Definition and general considerations. WHO ATC/DDD Index. Identifier: WHO ATC/DDD definition page Verified quote

    The DDD is the assumed average maintenance dose per day for a drug used for its main indication in adults. … The DDD is a unit of measurement and does not necessarily reflect the recommended or Prescribed Daily Dose. … The DDD is sometimes a 'dose' that is rarely if ever prescribed.

  • Fischer K, Diec S (2021) (opens in a new tab) Once- Versus Twice-Daily ACE Inhibitors for BP Control in Adults With Hypertension. Cureus. Identifier: PMCID PMC8451516; Cureus 13(8):e17385 Verified quote

    However, these agents' pharmacodynamics and pharmacokinetics suggest that once-daily administration may not lead to a 24-hour effect.

• Beta-blocker per-drug HFrEF target doses (no cross-agent conversion)

  Definition: Beta-blockers are pharmacologically heterogeneous (β-selectivity, α-blockade, ancillary effects — Barrese 2013) and the head-to-head COMET trial showed DIFFERENT survival at clinically matched target doses, so there is NO validated mg-for-mg conversion — convert() always refuses (NO_VALIDATED_METHOD) and no mg:mg ratio exists in the module. The only defensible numbers are each agent's own HFrEF trial target dose: bisoprolol 10 mg once daily (CIBIS-II), carvedilol 25 mg twice daily / 25–50 mg bid (COPERNICUS + Newman), and metoprolol SUCCINATE 200 mg once daily (MERIT-HF), consolidated by Newman 2024. Metoprolol succinate and tartrate are distinct keys: only succinate carries the HFrEF anchor — tartrate (COMET's 50 mg bid arm) must never inherit the succinate number.

  Expression

  targetDose(bisoprolol)=10 mg od; targetDose(carvedilol)=25 mg bid (25–50 mg bid); targetDose(metoprolol succinate)=200 mg od; cross-agent convert() ⇒ REFUSE (no mg:mg ratio)

  For developers

  Code reference

  equivalence/betaBlocker.ts#targetDose

  Sources

  • CIBIS-II Investigators and Committees (1999) (opens in a new tab) The Cardiac Insufficiency Bisoprolol Study II (CIBIS-II): a randomised trial. Lancet. Identifier: PMID 10023943 Verified quote

    bisoprolol 1.25 mg (n=1327) … the drug being progressively increased to a maximum of 10 mg per day.

  • Krum H, Roecker EB, Mohacsi P, et al. (COPERNICUS) (2003) (opens in a new tab) Effects of initiating carvedilol in patients with severe chronic heart failure: results from the COPERNICUS Study. JAMA. Identifier: PMID 12585949 Verified quote

    start dosage of at 3.125 mg twice daily with uptitration to a target dosage of 25 mg twice daily.

  • MERIT-HF Study Group (Metoprolol CR/XL Randomised Intervention Trial in Congestive Heart Failure) (1999) (opens in a new tab) Effect of metoprolol CR/XL in chronic heart failure: Metoprolol CR/XL Randomised Intervention Trial in Congestive Heart Failure (MERIT-HF). Lancet. Identifier: PMID 10376614 Verified quote

    metoprolol CR/XL 12.5 mg (NYHA III-IV) or 25.0 mg once daily (NYHA II) … The target dose was 200 mg once daily.

  • Newman E, et al. (2024) (opens in a new tab) Guideline-Directed Medical Therapy for Heart Failure with Reduced Ejection Fraction. Curr Cardiol Rep. Identifier: PMCID PMC11379751 Verified quote

    Target doses are carvedilol 25-50mg twice daily, bisoprolol 10mg daily, and metoprolol succinate 200mg daily.

  • Poole-Wilson PA, Swedberg K, Cleland JGF, et al. (COMET) (2003) (opens in a new tab) Comparison of carvedilol and metoprolol on clinical outcomes in patients with chronic heart failure in the Carvedilol Or Metoprolol European Trial (COMET): randomised controlled trial. Lancet. Identifier: PMID 12853193 Verified quote

    carvedilol (target dose 25 mg twice daily) and 1518 to metoprolol (metoprolol tartrate, target dose 50 mg twice daily) … Our results suggest that carvedilol extends survival compared with metoprolol.

  • Barrese V, Taglialatela M (2013) (opens in a new tab) New advances in beta-blocker therapy in heart failure. Front Physiol. Identifier: PMCID PMC3827547 Verified quote

    BB differ in terms of adrenergic β-receptors selectivity, adjunctive effects on α-receptors, and effects on reactive oxygen species and inflammatory cytokines production.

• Hard release-controlled class → do-not-split / do-not-crush / do-not-tube

  Definition: Enteric-coated, modified-release, and multilayer fixed-dose-combination tablets must not be split, crushed, or dispersed for a feeding tube: manipulation destroys the enteric protection / release-controlling matrix (dose dumping) or produces non-uniform fragments. A structured do-not-crush warning database is proven to cut erroneous crushing ~6-fold (van Welie 2016); dispersing for a feeding tube defeats the coating/matrix exactly as crushing does (Bifari 2023). Corroborated for the modified-release split axis by Saran 2022. (The cytotoxic/hazardous class is backed separately by splitCrushCytotoxic — its hazard is OPERATOR exposure, a different mechanism from coating/matrix dose dumping.)

  Expression

  formulation ∈ {enteric_coated, modified_release, multilayer_FDC} ⇒ split='no', crush='no', tube='no'

  For developers

  Code reference

  forms/splitCrush.ts#defaultVerdict

  Sources

  • van Welie S, Wijma L, Beerden T, van Doormaal J, Taxis K (2016) (opens in a new tab) Effect of warning symbols in combination with education on the frequency of erroneously crushing medication in nursing homes: an uncontrolled before and after study. BMJ Open. Identifier: BMJ Open 2016;6:e012286 Verified quote

    The crushing error rate decreased from 3.1% (21 wrongly crushed medicines out of 681 administrations) to 0.5% (3/636), RR=0.15 (95% CI 0.05 to 0.51) … Medications which were erroneously crushed included enteric-coated formulations (eg, omeprazole), medication with regulated release systems (eg, Persantin; dipyridamol) and toxic substances (eg, finasteride).

  • Saran AK, Pandya A, Veettil SK, Saokaew S, Kanchanasurakit S (2022) (opens in a new tab) Concerns regarding tablet splitting: a systematic review. BJGP Open. Identifier: BJGP Open 2022 (systematic review) Verified quote

    Evidence does support … the inappropriateness of splitting sustained-release preparations, given the potential for alteration of the rate of drug release for some products … sustained-release tablets, which should not be split.

  • Bifari N, Alkhalfan F, Algethami J, et al. (2023) (opens in a new tab) Unraveling medication errors in enteral tube administration: A cross-sectional study in geriatric patients receiving home health care. Saudi Pharmaceutical Journal (SPJ). Identifier: Saudi Pharm J 2023 (cross-sectional) Verified quote

    the administration of medications unsuitable for enteral feeding tubes (33.3%), predominantly due to the use of controlled release or enteric-coated formulations.

• Capsule class → do-not-split; crush='unknown' (open only if monograph allows)

  Definition: Hard and soft-gel capsules default to do-not-divide; the crush/open axis is 'unknown' (fail-closed) because the manipulation evidence for capsules is thin — Richey 2017's systematic review found NO eligible studies of oral-capsule manipulation, and results cannot be extrapolated between dosage forms or brands. A capsule is opened only when its specific monograph explicitly permits it (curated per-brand override), never by class default.

  Expression

  formulation ∈ {capsule_hard, capsule_soft_gel} ⇒ split='no', crush='unknown' (fail-closed; open only if the product monograph explicitly allows), tube='unknown'

  For developers

  Code reference

  forms/splitCrush.ts#defaultVerdict

  Sources

  • Richey RH, Hughes C, Craig JV, et al. (2017) (opens in a new tab) A systematic review of the use of dosage form manipulation to obtain required doses to inform use of manipulation in paediatric practice. International Journal of Pharmaceutics. Identifier: Int J Pharm 2017 (systematic review) Verified quote

    No eligible studies concerning manipulation of oral capsules or liquids, rectal enemas, nebuliser solutions, injections or transdermal patches were identified … The results cannot be extrapolated between dosage forms, methods of manipulation or between different brands of the same drug.

• Immediate-release class → split='halve_only' iff scored; never quartering

  Definition: Backs the SPLIT axis for immediate-release uncoated / film-coated / sugar-coated tablets. They may be split only when a score line exists, and only by HALVING — there is no verdict that licenses quartering (a half-score does not make quartering safe). Splitting accuracy is imperfect even for these classes (Helmy 2015: 15–16% of half tablets failed the USP weight/content-uniformity specification), so the split verdict is the conservative 'halve_only', never 'yes'. (The IR CRUSH axis is backed separately by splitCrushImmediateReleaseCrush/Faikoglu 2022, which is the crushability-specific source; Helmy is a splitting content-uniformity study and does not itself license crushing.)

  Expression

  formulation ∈ {IR_uncoated, IR_film_coated, IR_sugar_coated} ⇒ split='halve_only' iff scored ∈ {half, quarter} else 'no' (a half-score NEVER licenses a quartering 'yes')

  For developers

  Code reference

  forms/splitCrush.ts#defaultVerdict

  Sources

  • Helmy SA (2015) (opens in a new tab) Tablet Splitting: Is It Worthwhile? Analysis of Drug Content and Weight Uniformity for Half Tablets of 16 Commonly Used Medications in the Outpatient Setting. J Manag Care Spec Pharm. Identifier: J Manag Care Spec Pharm 2015;21(1):76-86 Verified quote

    A total of 52 of 320 half tablets (16.2%) and 48 of 320 half tablets (15.0%) fell outside of the proxy United States Pharmacopeia (USP) specification for weight and drug content … Bromazepam, carvedilol, bisoprolol, losartan, digoxin, and meloxicam half tablets failed the weight and content uniformity test … Tablets containing drugs with a wide therapeutic index and long half-life might be more suitable candidates for division.

• Immediate-release class → crush='yes' (no bioavailability change on crushing)

  Definition: Backs the CRUSH axis (and, by extension, the conservative IR tube axis) for immediate-release uncoated / film-coated / sugar-coated tablets. Crushing-specific evidence: Faikoglu 2022 reviews the clinical literature and concludes that crushing uncoated, sugar-coated, or film-coated IR tablets does not alter bioavailability or pharmacokinetics — which is the crushability claim Helmy 2015 (a tablet-SPLITTING content-uniformity study) does not itself license. Tube dispersion stays 'yes' only for uncoated IR; film/sugar-coated default to 'unknown' because the coating may impede uniform dispersion (fail conservative).

  Expression

  formulation ∈ {IR_uncoated, IR_film_coated, IR_sugar_coated} ⇒ crush='yes'; tube='yes' (IR_uncoated) else 'unknown' (coated, conservative)

  For developers

  Code reference

  forms/splitCrush.ts#defaultVerdict

  Sources

  • Faikoglu G, Saygi S, Tuncok Z, Tatar A, Tarkun BB, Demirkapu MJ, Gulec MA, Kantarcı G (2022) (opens in a new tab) The pharmacological perspective on tablet splitting or crushing. Pharmacy & Pharmacology International Journal. Identifier: Pharm Pharmacol Int J 2022;10(4) (MedCrave; identifier unconfirmed this session — Consensus record only) Verified quote

    Clinical studies report that the use of uncoated, sugar-coated, or film-coated tablets with the same dose of crushed forms and their use as a whole tablet has the same efficacy and safety data. … Physical manipulations on drugs with these coating properties do not cause any change in bioavailability and pharmacokinetic data.

• Cytotoxic / hazardous class → do-not-split / do-not-crush / do-not-tube (operator exposure)

  Definition: Cytotoxic / hazardous tablets must not be split, crushed, or dispersed for a feeding tube. The hazard mechanism is OPERATOR / CAREGIVER occupational exposure — crushing or dispersing aerosolises the drug and contaminates the work surface — NOT loss of a release-controlling coating or matrix (a cytotoxic immediate-release tablet has no such matrix). van Welie 2016 lists 'toxic substances (eg, finasteride)' among the erroneous-crush classes; Lopez 2022 (citing NIOSH 2016 + USP <800>) states crushing such tablets causes an unacceptable risk and mandates closed-system handling. This is the correct mechanism-specific provenance for the cytotoxic axis, distinct from the enteric/MR dose-dumping rationale in splitCrushHardClass.

  Expression

  formulation = cytotoxic_hazardous ⇒ split='no', crush='no', tube='no' (occupational/aerosol exposure, NOT a release-matrix mechanism)

  For developers

  Code reference

  forms/splitCrush.ts#defaultVerdict

  Sources

  • van Welie S, Wijma L, Beerden T, van Doormaal J, Taxis K (2016) (opens in a new tab) Effect of warning symbols in combination with education on the frequency of erroneously crushing medication in nursing homes: an uncontrolled before and after study. BMJ Open. Identifier: BMJ Open 2016;6:e012286 Verified quote

    Medications which were erroneously crushed included enteric-coated formulations (eg, omeprazole), medication with regulated release systems (eg, Persantin; dipyridamol) and toxic substances (eg, finasteride).

  • Lopez CV, Boix-Montañes A, Pascual-Carrasco A, et al. (2022) (opens in a new tab) Hazardous Drug Enteral Device: A Closed System Device for Crushing and Dispersing Hazardous Drug Tablets for Enteral Administration. International Journal of Pharmaceutical Compounding. Identifier: Int J Pharm Compd 2022 (identifier unconfirmed this session — Consensus record only) Verified quote

    In this list, NIOSH established that crushing tablets or making solutions from them causes an unacceptable risk at hospitals. Furthermore, United States Pharmacopeia <800> and European regulations impose the use of closed system devices and plastic pouches to contain any dust or particles generated during these operations.

• Modified-release suffix → conservative first-pass MR flag

  Definition: A conservative first-pass FLAG only, not a verdict. Pharmaceutical companies mark modified-release products with characteristic name suffixes; detecting one is a hint that the product may be modified-release and should be treated conservatively / verified against the registered інструкція. It NEVER emits a final verdict, NEVER relaxes a class default toward 'yes', and NEVER substitutes for a curated per-brand override row (silence ≠ permission). A negative detection asserts nothing.

  Expression

  name contains an EN/UA modified-release marketing suffix (SR/XR/CR/ER/XL/chrono/prolong/retard/MR/LA/SA/depot; UA ретард/тривалої дії/пролонг) at a word boundary ⇒ flag 'possible modified-release — verify' (biases toward modified_release/no/unknown; never relaxes a verdict toward 'yes')

  For developers

  Code reference

  forms/mrSuffix.ts#detectMRSuffix

  Sources

  • Lohmann K, Ferber J, Haefeli MF, Störzinger D, Schwald M, Haefeli WE, Seidling HM (2015) (opens in a new tab) Knowledge and training needs of nurses and physicians on unsuitable drugs for patients with dysphagia or feeding tubes. Journal of Clinical Nursing. Identifier: DOI 10.1111/jocn.12910; J Clin Nurs 2015;24(19-20):3016-3019 Verified quote

    in over 60% of sustained-release drugs pharmaceutical companies use specific suffixes in their drug names such as SR, chrono or prolong to highlight specific galenic formulations. Thus, these suffixes may provide a first hint that a drug is potentially inappropriate for drug modification. … Most summaries of product characteristics do not list information regarding drug modification or safety aspects.

• Cross-statin potency under-dosing caveat (VOYAGER equipotency)

  Definition: Mandatory caveat for any CROSS-statin conversion (source statin ≠ target statin). The Law-2003 equal-%LDL anchor table gives rosuva 10 ≈ atorva 20 (≈2×); the VOYAGER individual-patient-data equipotency analysis (same author group) gives a higher potency ratio (rosuva ≈ 3–3.5× atorva, ≈ 7–8× simva), so a Law-anchored cross-statin conversion can UNDER-DOSE the target. Surfaced as a caveat (not a model change): the point estimate stays the Law-anchored value; the clinician is told to verify against the patient's achieved LDL-C. Same-statin dose↔%LDL queries do not carry it.

  Expression

  VOYAGER equipotency: rosuvastatin ≈ 3–3.5× atorvastatin ≈ 7–8× simvastatin (higher than the Law-2003 equal-%LDL table)

  For developers

  Code reference

  equivalence/statins.ts#equivalentDose

  Sources

  • Karlson BW, Palmer MK, Nicholls SJ, Lundman P, Barter PJ (2016) (opens in a new tab) Doses of rosuvastatin, atorvastatin and simvastatin that induce equal reductions in LDL-C and non-HDL-C: results from the VOYAGER meta-analysis. Eur J Prev Cardiol. Identifier: DOI 10.1177/2047487315598710 Verified quote

    Rosuvastatin 10 mg reduced LDL-C by 44% … Equivalent reductions in LDL-C required atorvastatin 29 mg or simvastatin 72 mg … each rosuvastatin dose is equivalent to doses 3–3.5 times higher for atorvastatin and 7–8 times higher for simvastatin.

• Combined-use additive-hypotension caution (qualitative, non-numeric)

  Definition: Non-numeric caution emitted when two or more blood-pressure-lowering drug classes (thiazide, β-blocker, ACE inhibitor, ARB, calcium-channel blocker) are stacked. The Law-354 meta-analysis showed the BP-lowering effects of different drug categories are additive, so stacking carries an additive risk of symptomatic hypotension. The module computes NO combined dose and contains NO dose ceiling — it surfaces the additivity QUALITATIVELY and directs the clinician to titrate each drug against measured BP/HR.

  Expression

  ΔBP(A+B) ≈ ΔBP(A) + ΔBP(B) — qualitative, non-numeric additive caution; no combined dose, no ceiling

  For developers

  Code reference

  equivalence/combinedUse.ts#combinedUseCaution

  Sources

  • Law MR, Wald NJ, Morris JK, Jordan RE (2003) (opens in a new tab) Value of low dose combination treatment with blood pressure lowering drugs: analysis of 354 randomised trials. BMJ. Identifier: PMID 12829555; PMCID PMC162261 Verified quote

    The blood pressure lowering effects of different categories of drugs were additive.

• Dose-dumping illustration — verapamil crushed-vs-intact 24 h AUC anchor

  Definition: Empirical magnitude anchor for the E13 dose-dumping illustration. Kumagai 2024 measured that crushing a verapamil tablet raised the initial-24 h AUC to ≈1.7× the intact tablet's. The illustration calibrates two COMPOSED concentration-time profiles (a slow-absorption modified-release 'intact' profile and a fast-absorption 'crushed' profile, both from the shipped simulate()/metrics() engine) so their crushed:intact AUC(0–24) ratio reproduces this measured ≈1.7×. It introduces NO new pharmacokinetic formula — it is a calibrated reuse of the existing engine — and the 1.7× is the verapamil EXEMPLAR of why crushing a modified-release product dumps the dose, never a patient-specific prediction for an arbitrary product.

  Expression

  AUC_0-24(crushed) ≈ 1.7 × AUC_0-24(intact) — Kumagai 2024 verapamil anchor

  For developers

  Code reference

  forms/doseDumping.ts#doseDumpingIllustration

  Sources

  • Kumagai S, et al. (2024) (opens in a new tab) Comparative analysis of verapamil pharmacokinetics: simple suspension and crushing. J Clin Med. Identifier: Consensus record 035221cda10f5a5f9ce549d3e142d945 Verified quote

    the area under the curve for verapamil during the initial 24 h period was 1.7 and 1.3 times greater in the crushed and simple suspension groups, respectively, than in the tablet group.

• Antipsychotic dose-equivalence (olanzapine-equivalent, Classical Mean Dose)

  Definition: Approximate population-average dose equivalence between antipsychotics on the Leucht 2015 Classical-Mean-Dose olanzapine-equivalent scale (Table 1: mg/day of each drug equivalent to 1 mg/day olanzapine; a HIGHER value = a LESS potent drug). A switching guide on a surrogate dose scale, NOT bioequivalence or receptor-occupancy equivalence — the authors explicitly state a gold standard method does not exist. Clozapine's ratio is carried but accompanied by a mandatory TDM / agranulocytosis-monitoring caution (a pharmacologist may upgrade it to a hard refusal). Implemented data-only via the generic _package.ts ratio factory; this engine adds no arithmetic of its own.

  Expression

  targetDose = dose × (olanzEq_target / olanzEq_source), where olanzEq_X = mg/day of X ≡ 1 mg olanzapine

  For developers

  Code reference

  equivalence/antipsychotics.ts#convert

  Sources

  • Leucht S, Samara M, Heres S, Patel MX, Furukawa T, Cipriani A, Geddes J, Davis JM (2015) (opens in a new tab) Dose Equivalents for Second-Generation Antipsychotic Drugs: The Classical Mean Dose Method. Schizophr Bull. Identifier: PMID 25841041; PMCID PMC4601707 Verified quote

    We calculated the mean doses of each drug weighted by sample size and divided them by the weighted mean olanzapine dose to obtain olanzapine equivalents. … The doses equivalent to 1 mg/d olanzapine were: amisulpride 38.3 mg/d, aripiprazole 1.4 mg/d, asenapine 0.9 mg/d, chlorpromazine 38.9 mg/d, clozapine 30.6 mg/d, haloperidol 0.7 mg/d, quetiapine 32.3mg/d, risperidone 0.4 mg/d, sertindole 1.1 mg/d, ziprasidone 7.9 mg/d, zotepine 13.2 mg/d. … Table 1 (mean, SD): risperidone 0.38 (0.12), haloperidol 0.74 (0.22), chlorpromazine 38.88 (16.9), quetiapine 32.27 (7.4), aripiprazole 1.41 (0.3), ziprasidone 7.92 (1.56), amisulpride 38.33 (8.76), clozapine 30.62 (18.64), sertindole 1.08 (0.2), asenapine 0.89, zotepine 13.24.