Drug Calculations & Pharmacokinetics

Drug calculation errors kill people. Studies consistently show that medication errors are one of the leading causes of preventable patient harm in hospitals worldwide. A tenfold overdose of morphine, an incorrect dilution of a potassium infusion, or a miscalculated paediatric dose can be fatal. Every doctor, nurse, and pharmacist needs to be competent in basic drug calculations. The mathematics is not complicated — it is mostly multiplication, division, and ratio — but it must be applied correctly, every time, under pressure. The key areas are: 1. Concentration calculations — what does 5 mg/mL mean, and how much do you draw up? 2. Dose calculations — if the dose is 0.1 mg/kg and the patient weighs 70 kg, what is the total dose? 3. Infusion rate calculations — if the drug runs at 10 mL/hr and the concentration is 2 mg/mL, how many mg/hr is the patient receiving? 4. Unit conversions — confidently converting between micrograms, milligrams, grams, millilitres, and litres

The metric system in medicine: Drugs are measured in mass (grams, milligrams, micrograms) and volume (litres, millilitres). Key conversions to memorise: - 1 gram (g) = 1,000 milligrams (mg) - 1 milligram (mg) = 1,000 micrograms (mcg or μg) - 1 litre (L) = 1,000 millilitres (mL) Never confuse mg and mcg — this is one of the most common fatal drug errors. A patient prescribed 125 micrograms of digoxin who receives 125 milligrams would receive a dose 1,000 times too high. Concentration: Drug concentration is expressed as mass per volume: - mg/mL — milligrams per millilitre (most common) - % — grams per 100 mL. So 1% lidocaine = 1 g per 100 mL = 10 mg/mL - 1 in X — e.g. adrenaline 1:1000 = 1 g per 1000 mL = 1 mg/mL; adrenaline 1:10,000 = 0.1 mg/mL The core calculation: Volume to give (mL) = Dose required (mg) ÷ Concentration (mg/mL) Example: A patient needs 75 mg of pethidine. The ampoule contains 100 mg in 2 mL. Concentration = 100 ÷ 2 = 50 mg/mL Volume = 75 ÷ 50 = 1.5 mL Always ask yourself: does this look reasonable? A volume of 0.01 mL or 50 mL for a single injection should make you stop and recheck.

Weight-based dosing: Many drugs — especially in children and in critical care — are dosed per kilogram of body weight (mg/kg or mcg/kg/min). Total dose (mg) = Dose per kg (mg/kg) × Patient weight (kg) Example: Amoxicillin 25 mg/kg for a child weighing 20 kg. Total dose = 25 × 20 = 500 mg Paediatric drug calculations are particularly high-risk because children cannot report symptoms as clearly as adults, doses are small (making errors more proportionally significant), and weights vary enormously. Infusion rate calculations: Many drugs in intensive care are given as continuous infusions rather than bolus doses. Rate (mL/hr) = Dose required (mg/hr) ÷ Concentration (mg/mL) Example: A patient needs dopamine at 5 mcg/kg/min. They weigh 70 kg. The infusion is made up as 200 mg in 50 mL (= 4 mg/mL = 4,000 mcg/mL). Step 1: Convert dose to mcg/hr 5 mcg/kg/min × 70 kg = 350 mcg/min × 60 = 21,000 mcg/hr = 21 mg/hr Step 2: Calculate rate Rate = 21 mg/hr ÷ 4 mg/mL = 5.25 mL/hr This type of calculation is performed multiple times per shift in any ICU. Getting it wrong can be immediately life-threatening.

Pharmacokinetics describes the journey of a drug through the body: how it is absorbed, distributed, metabolised, and excreted — summarised as ADME. A — Absorption How the drug gets from where it is given into the bloodstream. Factors affecting absorption: - Route — intravenous (IV) gives 100% bioavailability instantly; oral drugs must survive the gut and first-pass metabolism - Bioavailability — the fraction of the administered dose that reaches the systemic circulation. Oral morphine has ~30% bioavailability because the liver breaks much of it down before it reaches the circulation (first-pass effect) - pH — aspirin (a weak acid) is absorbed better in the acidic stomach; basic drugs absorb better in the alkaline small intestine D — Distribution Once in the blood, drugs distribute throughout the body. The volume of distribution (Vd) describes how widely a drug spreads — a large Vd means the drug leaves the blood and enters tissues extensively. Highly lipid-soluble drugs (like diazepam) distribute widely into fat — this is why benzodiazepines have long durations of action and why obese patients need higher loading doses. Protein binding: many drugs bind to plasma proteins (mainly albumin). Only the unbound (free) drug is active. In patients with low albumin (liver disease, malnutrition), more drug is free — leading to unexpectedly strong effects at normal doses. M — Metabolism Most drugs are metabolised (chemically altered) by the liver, mainly by enzymes in the cytochrome P450 (CYP) system. Metabolism usually converts active drugs to inactive metabolites that can be excreted. Some drugs (pro-drugs) are inactive until metabolised — codeine is converted to morphine in the liver. CYP enzyme interactions are a major source of drug interactions: - Inducers (e.g. rifampicin, some anticonvulsants) speed up metabolism → reduce drug levels → treatment failure - Inhibitors (e.g. fluconazole, grapefruit juice) slow metabolism → increase drug levels → toxicity E — Excretion Most drugs are excreted by the kidneys in urine. Kidney failure reduces excretion — drugs accumulate and doses must be reduced. Some drugs are excreted in bile and faeces.

Half-life (t½) The half-life of a drug is the time it takes for the plasma concentration to fall by 50%. After one half-life, 50% remains. After two, 25%. After five half-lives, less than 3% remains — the drug is considered effectively eliminated. Half-life determines: - How often to dose — a drug with a short half-life (e.g. adenosine: 10 seconds) must be given as a rapid bolus; a drug with a long half-life (e.g. amiodarone: 40–55 days) can be given once daily and takes weeks to reach steady state - How long toxicity lasts — paracetamol overdose requires treatment within hours; warfarin overdose effects persist for days Steady state: When a drug is given repeatedly, it accumulates until the rate of administration equals the rate of elimination. Steady state is reached after approximately five half-lives. This is clinically important — the true therapeutic effect of a new drug or dose change cannot be assessed until steady state is reached. The therapeutic window: The range of plasma drug concentrations that is effective without being toxic. A wide therapeutic window (penicillin) means dosing is forgiving. A narrow therapeutic window means small changes in dose can cause either treatment failure or toxicity. Drugs with narrow therapeutic windows requiring monitoring: - Digoxin — heart failure drug; toxic at levels just above therapeutic - Warfarin — anticoagulant; INR monitoring required - Lithium — bipolar treatment; requires regular blood level checks - Gentamicin — antibiotic; toxic to kidneys and hearing at high levels - Phenytoin — anticonvulsant; non-linear kinetics make it particularly tricky Toxicity: Drug toxicity can be dose-dependent (more drug = more toxicity, e.g. paracetamol hepatotoxicity) or idiosyncratic (unpredictable allergic or immune reactions, e.g. penicillin anaphylaxis). Paracetamol overdose is the most common cause of acute liver failure in the UK. At therapeutic doses, paracetamol is safely metabolised. In overdose, a toxic metabolite (NAPQI) accumulates and destroys liver cells. Treatment with N-acetylcysteine (NAC) replenishes glutathione and neutralises NAPQI — but must be given within hours.