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). Vancomycin is given as a continuous infusion to a patient whose vancomycin clearance is 75 mL/min. What infusion rate R in mg/h gives a steady-state level of 25 mg/L?
CL = 75 mL/minCss = 25 mg/L
R = mg/h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A theophylline infusion is started without a loading dose. The half-life in this patient is 6 h. How long t does it take to reach 80% of the steady-state level?
t½ = 6 h80% Css
t = h
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:1000. You need 0.34 mg. Draw up the volume in the 1 mL syringe: what volume V is that?
1:10000.34 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 500 mg of powder. You add 9.8 mL of diluent, and the powder adds 0.2 mL of volume. The prescription is 40 mg. Draw up the dose: what volume V do you give?
500 mg9.8 mL + 0.2 mL40 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion contains 250 units of regular insulin in 250 mL. The order is 10 units/h. Find the pump rate R in mL/h. Round to one decimal place.
250 units/250 mL10 units/h
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 63 kg has an order for norepinephrine at 8 mcg/min. The bag contains 8 mg in 250 mL. Find the pump rate R in mL/h. Round to one decimal place.
8 mg/250 mL8 mcg/min63 kg
R = mL/h
7.PharmacologyMedium
A patient is starting olanzapine. Which monitoring is most important over the following months?
AHearing tests and vision tests every month
BPeak expiratory flow
CWeight, blood glucose and blood lipids
DINR and platelet count
8.PharmacologyHard
A patient taking hydralazine for several months develops joint pain, fever and a rash. Anti-histone antibodies are positive. What is the most likely diagnosis?
ARheumatoid arthritis
BGout
CDrug-induced lupus
DAn acute viral infection
3Solutions
1.Clinical pharmacokinetics
Answer: R = 112.5 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 in L/h
75·60/1000 = 4.5 L/h
Infusion rate
R = 4.5·25 = 112.5 mg/h
Answer
R = 112.5 mg/h
2.Clinical pharmacokinetics
Answer: t = 13.93 h
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.
Rate constant
k = 0.693/6 = 0.1155 h⁻¹
Time to the given fraction of steady state
t = −ln(1 − 0.8)/0.1155 = 1.609/0.1155 = 13.93 h
Answer
t = 13.93 h
That is 2.322 half-lives.
3.Pharmacy calculations
Answer: V = 0.34 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.
Strength in mg/mL
1000 mg/1000 mL = 1 mg/mL
Volume
V = 0.34/1 = 0.34 mL
Answer
V = 0.34 mL
4.Pharmacy calculations
Answer: V = 0.8 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
9.8 + 0.2 = 10 mL
C = 500/10 = 50 mg/mL
Volume
V = 40/50 = 0.8 mL
Answer
V = 0.8 mL
5.Pediatric & critical-care dosing
Answer: R = 10 mL/h
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.
Concentration
250/250 = 1 units/mL
Pump rate
R = 10/1 = 10 mL/h
Answer
R = 10 mL/h
6.Pediatric & critical-care dosing
Answer: R = 15 mL/h
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
8 000/250 = 32 mcg/mL
Dose per hour
The order is per minute for the whole patient, so the weight is not needed.
8·60 = 480 mcg/h
Pump rate
R = 480/32 = 15 mL/h
Answer
R = 15 mL/h
7.Pharmacology
Answer: C · Weight, blood glucose and blood lipids
Why this is the answer
Olanzapine and several other atypical antipsychotics cause weight gain, insulin resistance and dyslipidaemia. Weight, waist size, glucose and lipids are checked regularly.
Answer
C · Weight, blood glucose and blood lipids
8.Pharmacology
Answer: C · Drug-induced lupus
Why this is the answer
Hydralazine and procainamide are classic causes of drug-induced lupus, especially in slow acetylators. Symptoms usually resolve after the drug is stopped.