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 67 kg patient is to be loaded intravenously with phenytoin sodium to reach a phenytoin level of 15 mg/L. The volume of distribution is 0.7 L/kg and the salt factor is S = 0.92. Find the loading dose LD of phenytoin sodium in mg.
67 kgVd = 0.7 L/kgC = 15 mg/LS = 0.92
LD = mg
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 9 h. How long t does it take to reach 75% of the steady-state level?
t½ = 9 h75% Css
t = h
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:10 000. You need 0.38 mg. Draw up the volume in the 10 mL syringe: what volume V is that?
1:10 0000.38 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 600 mL of 10% hydrogen peroxide from a 30% stock and purified water. How many mL V of water do you need? Round to the nearest whole mL.
30%0%10%600 mL
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion contains 100 units of regular insulin in 250 mL. The order is 4 units/h. Find the pump rate R in mL/h. Round to one decimal place.
100 units/250 mL4 units/h
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 74 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
250 mg/250 mL5 mcg/kg/min74 kg
R = mL/h
7.PharmacologyMedium
A patient takes isoniazid. Which advice is most important?
ATake it with antacids to protect the stomach from irritation
BStop the drug when the cough improves
CAvoid alcohol and report jaundice, dark urine or poor appetite
DExpect the urine to turn orange
8.PharmacologyHard
A nurse gave a patient ten times the prescribed dose of hydromorphone. The patient is currently alert. What is the most appropriate sequence of actions?
AMonitor closely, have naloxone ready, inform the prescriber and report
BGive another dose later to balance it out
CChange the chart so that it shows the correct dose, and continue to monitor
DSay nothing because the patient is alert
3Solutions
1.Clinical pharmacokinetics
Answer: LD = 764.7 mg
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.
Volume of distribution
Vd = 0.7·67 = 46.9 L
Amount of active drug needed
Vd·C = 46.9·15 = 703.5 mg
Loading dose
LD = 703.5/(0.92·1) = 764.7 mg
Answer
LD = 764.7 mg
In practice the dose is rounded to a convenient amount.
2.Clinical pharmacokinetics
Answer: t = 18 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 = 0.693/9 = 0.07702 h⁻¹
Time to the given fraction of steady state
t = −ln(1 − 0.75)/0.07702 = 1.386/0.07702 = 18 h
Answer
t = 18 h
That is 2 half-lives.
3.Pharmacy calculations
Answer: V = 3.8 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.
Strength in mg/mL
1000 mg/10 000 mL = 0.1 mg/mL
Volume
V = 0.38/0.1 = 3.8 mL
Answer
V = 3.8 mL
4.Pharmacy calculations
Answer: V = 400 mL
Alligation: to mix a higher strength H and a lower strength L into a target strength T, take (T − L) parts of H and (H − T) parts of L. The parts are crossed: each stock gets the distance from the target to the other stock. Water or a plain base counts as 0%. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Alligation parts
30%: 10 − 0 = 10 parts
0%: 30 − 10 = 20 parts
total: 30 parts
Water
V = 600·20/30 = 400 mL
Answer
V = 400 mL
Check: 200 mL of 30% contains 60 g of drug, which is 10% of 600 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
100/250 = 0.4 units/mL
Pump rate
R = 4/0.4 = 10 mL/h
Answer
R = 10 mL/h
6.Pediatric & critical-care dosing
Answer: R = 22.2 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
5·74 = 370 mcg/min
370·60 = 22200 mcg/h
Pump rate
R = 22200/1000 = 22.2 mL/h
Answer
R = 22.2 mL/h
7.Pharmacology
Answer: C · Avoid alcohol and report jaundice, dark urine or poor appetite
Why this is the answer
Isoniazid can cause hepatitis, and alcohol increases the risk. Signs of liver injury are reported at once, and liver tests may be checked.
Answer
C · Avoid alcohol and report jaundice, dark urine or poor appetite
8.Pharmacology
Answer: A · Monitor closely, have naloxone ready, inform the prescriber and report
Why this is the answer
The patient may become sedated and develop respiratory depression later, so close monitoring and a ready antidote are needed. The prescriber is informed, the patient is told honestly, and the error is reported to improve the system.
Answer
A · Monitor closely, have naloxone ready, inform the prescriber and report