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). After an IV dose of vancomycin, the level is 30 mg/L at 1 h and 8.7 mg/L at 17 h. Assume first-order elimination. Find the elimination rate constant k.
C1 = 30 mg/L (1 h)C2 = 8.7 mg/L (17 h)
k = h⁻¹
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A vancomycin infusion runs at 100 mg/h with no loading dose. The patient's clearance is 5 L/h and the half-life is 10 h. What is the level C after 8 h?
R = 100 mg/hCL = 5 L/ht½ = 10 ht = 8 h
C = mg/L
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:1000. You need 0.27 mg. Draw up the volume in the 1 mL syringe: what volume V is that?
1:10000.27 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 2000 mg of powder. You add 9.4 mL of diluent, and the powder adds 0.6 mL of volume. The prescription is 380 mg. Draw up the dose: what volume V do you give?
2000 mg9.4 mL + 0.6 mL380 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A bag contains 25 000 units of heparin in 250 mL. The order is 20 units/kg/h for a patient who weighs 80 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
25 000 units/250 mL20 units/kg/h80 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A bag contains dobutamine 250 mg in 250 mL. The order is 5 mcg/kg/min for a patient who weighs 82 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
250 mg/250 mL5 mcg/kg/min82 kg
R = mL/h
7.PharmacologyMedium
A patient with insulin-treated diabetes is found unconscious with very low blood glucose. There is no IV access. Which drug can a trained helper give?
AOral glucose gel poured into the mouth
BRapid-acting insulin under the skin
CIntramuscular adrenaline
DGlucagon by injection or nasal spray
8.PharmacologyHard
A patient with HIV has a CD4 count below 200 cells/mm3. Which prophylaxis is usually started?
AOseltamivir to prevent influenza
BIsoniazid to prevent hepatitis B
CCo-trimoxazole to prevent Pneumocystis pneumonia
DFluconazole to prevent urinary and skin infections
3Solutions
1.Clinical pharmacokinetics
Answer: k = 0.07737 h⁻¹
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.
Elimination rate constant
k = ln(30/8.7)/16 = 1.238/16 = 0.07737 h⁻¹
Answer
k = 0.07737 h⁻¹
2.Clinical pharmacokinetics
Answer: C = 8.513 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.
Steady-state level
Css = 100/5 = 20 mg/L
Rate constant
k = 0.693/10 = 0.06931 h⁻¹
Level after the given time
C = 20·(1 − e−0.06931·8) = 8.513 mg/L
Answer
C = 8.513 mg/L
3.Pharmacy calculations
Answer: V = 0.27 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 = 0.27/1 = 0.27 mL
Answer
V = 0.27 mL
4.Pharmacy calculations
Answer: V = 1.9 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
9.4 + 0.6 = 10 mL
C = 2000/10 = 200 mg/mL
Volume
V = 380/200 = 1.9 mL
Answer
V = 1.9 mL
5.Pediatric & critical-care dosing
Answer: R = 16 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.
Units per hour
20·80 = 1 600 units/h
Concentration
25 000/250 = 100 units/mL
Pump rate
R = 1600/100 = 16 mL/h
Answer
R = 16 mL/h
6.Pediatric & critical-care dosing
Answer: R = 24.6 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
250 mg = 250 000 mcg
250 000/250 = 1000 mcg/mL
Dose per minute and per hour
5·82 = 410 mcg/min
410·60 = 24600 mcg/h
Pump rate
R = 24600/1000 = 24.6 mL/h
Answer
R = 24.6 mL/h
7.Pharmacology
Answer: D · Glucagon by injection or nasal spray
Why this is the answer
Glucagon breaks down liver glycogen and raises blood glucose. Nothing is given by mouth to an unconscious patient because of the risk of choking and aspiration.
Answer
D · Glucagon by injection or nasal spray
8.Pharmacology
Answer: C · Co-trimoxazole to prevent Pneumocystis pneumonia
Why this is the answer
A low CD4 count greatly raises the risk of Pneumocystis jirovecii pneumonia. Co-trimoxazole prevents it and also reduces the risk of toxoplasmosis.
Answer
C · Co-trimoxazole to prevent Pneumocystis pneumonia