Pick the relation: C = C₀·e−kt, LD = Vd·C/(S·F), maintenance rate = CL·Css/(S·F), τ = ln(Cpeak/Ctrough)/k.
For kidney function, estimate CrCl with Cockcroft–Gault first, then scale the dose or the interval.
Check the result: levels fall by half every half-life, and steady state takes about five half-lives.
Pharmacy calculations
Percentage strength
% w/v = g per 100 mL
Amount in a solution
g = % × mL / 100
Ratio strength
1:X = 1 g in X mL; % = 100 / X
Reconstitution
final volume = diluent + powder volume
Concentration
mg/mL = drug in vial / final volume
Alligation
parts of H = T − L, parts of L = H − T
Water or base
0%
Pediatric & critical-care dosing
Daily dose
mg/day = mg/kg/day × weight (kg)
Doses per day
number = 24 h / dosing interval (h)
Pounds to kilograms
kg = lb / 2.2
Infusion concentration
units/mL = units in bag / bag volume
Pump rate
mL/h = dose per hour / concentration
Titrated drip
mL/h = mcg/kg/min × kg × 60 / (mcg/mL)
Dose from pump rate
mcg/kg/min = mL/h × (mcg/mL) / (60 × kg)
Units
1 mg = 1000 mcg
2Practice problems
1.Clinical pharmacokineticsMedium
Practice exercise (made-up values, not clinical guidance). A 91 kg patient is to receive an aminophylline infusion (salt factor S = 0.8: 80% of aminophylline is theophylline). The theophylline clearance is 0.04 L/kg/h. What aminophylline infusion rate R in mg/h gives a steady-state theophylline level of 10 mg/L?
91 kgCL = 0.04 L/kg/hCss = 10 mg/LS = 0.8
R = mg/h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 52 kg patient takes oral aminophylline (S = 0.8, F = 1) every 8 h. The theophylline clearance is 0.05 L/kg/h. Each dose is 450 mg of aminophylline. Find the average steady-state level Css,avg.
A solution has a ratio strength of 1:1000. You need 0.43 mg. Draw up the volume in the 1 mL syringe: what volume V is that?
1:10000.43 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 250 mg of powder. You add 4.4 mL of diluent, and the powder adds 0.6 mL of volume. The prescription is 205 mg. Draw up the dose: what volume V do you give?
250 mg4.4 mL + 0.6 mL205 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A practice protocol calls for a heparin bolus of 60 units/kg for a patient who weighs 85 kg. The vial contains 5 000 units/mL. What volume V do you draw up? Round to one decimal place.
60 units/kg85 kg5 000 units/mL
V = mL
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A dobutamine infusion of 500 mg in 250 mL runs at 23 mL/h. The patient weighs 72 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
500 mg/250 mL23 mL/h72 kg
D = mcg/kg/min
7.PharmacologyMedium
Bethanechol may be used for non-obstructive urinary retention. How does it work?
AIt increases urine production in the kidney
BIt blocks muscarinic receptors and relaxes the bladder wall muscle
CIt shrinks the prostate gland
DIt stimulates muscarinic receptors and contracts the bladder muscle
8.PharmacologyHard
A patient with epilepsy needs pain relief. Why is tramadol used with caution?
AIt speeds up the metabolism of all antiepileptic drugs
BIt always causes respiratory arrest in epilepsy
CIt lowers the seizure threshold
DIt causes a sharp rise in blood glucose
3Solutions
1.Clinical pharmacokinetics
Answer: R = 45.5 mg/h
Clearance and volume: at steady state the dosing rate equals the elimination rate, so the maintenance rate = CL × Css. A loading dose is LD = Vd × C/(S × F), with S the salt factor (fraction of active drug in the salt) and F the bioavailability (1 for IV). From two levels, k = ln(C₁/C₂)/Δt; for an IV bolus Vd = dose/C₀ and CL = k × Vd. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Clearance
CL = 0.04·91 = 3.64 L/h
Infusion rate
R = CL·Css/S = 3.64·10/0.8 = 45.5 mg/h
Answer
R = 45.5 mg/h
2.Clinical pharmacokinetics
Answer: Css,avg = 17.31 mg/L
Designing a regimen: with rapid IV doses the level falls from peak to trough during the interval, so τ = ln(Cpeak/Ctrough)/k and the dose that restores the peak is Vd × (Cpeak − Ctrough). The average steady-state level is Css,avg = S·F·dose/(CL·τ). An infusion rises as C = (R/CL)(1 − e−kt). For a drug cleared by the kidneys, the dose is scaled with the creatinine clearance. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Strength to volume: turn the strength into mg per mL first. A w/v percentage is grams per 100 mL, so x% = 10x mg/mL. A ratio strength 1:n is 1 g in n mL. After reconstitution the concentration is the powder's dose over the final volume (diluent + powder volume). Then volume = dose/concentration.
Strength in mg/mL
1000 mg/1000 mL = 1 mg/mL
Volume
V = 0.43/1 = 0.43 mL
Answer
V = 0.43 mL
4.Pharmacy calculations
Answer: V = 4.1 mL
Strength to volume: turn the strength into mg per mL first. A w/v percentage is grams per 100 mL, so x% = 10x mg/mL. A ratio strength 1:n is 1 g in n mL. After reconstitution the concentration is the powder's dose over the final volume (diluent + powder volume). Then volume = dose/concentration.
Final volume and concentration
4.4 + 0.6 = 5 mL
C = 250/5 = 50 mg/mL
Volume
V = 205/50 = 4.1 mL
Answer
V = 4.1 mL
5.Pediatric & critical-care dosing
Answer: V = 1 mL
Heparin and insulin infusions: concentration = units in the bag/bag volume (units/mL). Pump rate (mL/h) = units per hour/concentration, and units per hour = mL/h × concentration. For a weight-based order, units/h = units/kg/h × weight first. A bolus volume = units needed/vial strength. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Units needed
60·85 = 5 100 units
Volume
V = 5100/5000 = 1.02 ≈ 1 mL
Answer
V = 1 mL
6.Pediatric & critical-care dosing
Answer: D = 10.6 mcg/kg/min
Titrated drips: first find the concentration in mcg/mL (mg × 1000/mL). For a weight-based order, mcg/min = mcg/kg/min × weight. Then mL/h = mcg/min × 60/concentration. Working backwards, mcg/kg/min = mL/h × concentration/(60 × weight). The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Concentration
500 000/250 = 2000 mcg/mL
Micrograms per minute
23·2000 = 46000 mcg/h
46000/60 = 766.7 mcg/min
Per kilogram
D = 766.7/72 = 10.65 ≈ 10.6 mcg/kg/min
Answer
D = 10.6 mcg/kg/min
7.Pharmacology
Answer: D · It stimulates muscarinic receptors and contracts the bladder muscle
Why this is the answer
Bethanechol mimics acetylcholine at muscarinic receptors, contracting the detrusor muscle. It must not be used if there is mechanical obstruction to outflow.
Answer
D · It stimulates muscarinic receptors and contracts the bladder muscle
8.Pharmacology
Answer: C · It lowers the seizure threshold
Why this is the answer
Tramadol can cause seizures, especially at high doses or with other drugs that lower the seizure threshold. Its serotonergic action also raises the risk of serotonin syndrome.