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Pharmaceutical Calculations: Complete Guide
For SPLE Preparation (Saudi Pharmacy Licensure Exam)
PART 1: PHARMACOKINETICS (PK) DOSING
1.1 Core PK Parameters
| Parameter | Symbol | Formula | Units |
|---|
| Volume of Distribution | Vd | Vd = Dose / C₀ | L or L/kg |
| Clearance | CL | CL = Rate of elimination / C | L/hr |
| Half-life | t½ | t½ = 0.693 × Vd / CL | hours |
| Bioavailability | F | F = AUC(oral) / AUC(IV) × 100% | % |
| Elimination rate constant | Ke | Ke = 0.693 / t½ | hr⁻¹ |
1.2 Loading Dose
Used when you need to quickly reach therapeutic concentration.
Formula:
LD = (Vd × Target Cp) / F
- Vd = Volume of distribution (L/kg × patient weight)
- Target Cp = desired plasma concentration
- F = bioavailability (= 1.0 for IV)
Example:
Digoxin: Vd = 7 L/kg, target Cp = 1.5 mcg/L, patient weight = 70 kg, oral (F = 0.7)
LD = (7 × 70 × 1.5) / 0.7 = 735 / 0.7 = 1,050 mcg = ~1 mg oral
1.3 Maintenance Dose
Formula:
MD = CL × Target Cp × Dosing interval / F
Or equivalently:
MD = (Target Cp × CL × τ) / F
Example:
Theophylline: Target Cp = 10 mg/L, CL = 2.8 L/hr, τ = 12 hr, oral (F = 0.96)
MD = (10 × 2.8 × 12) / 0.96 = 336 / 0.96 = 350 mg every 12 hours
1.4 Creatinine Clearance (CrCl) - Renal Dose Adjustment
Cockcroft-Gault Formula:
CrCl (mL/min) = [(140 - age) × weight(kg)] / [72 × serum creatinine (mg/dL)]
× 0.85 if female
Example:
65-year-old female, 60 kg, SCr = 1.2 mg/dL
CrCl = [(140-65) × 60] / [72 × 1.2] × 0.85
= [75 × 60] / 86.4 × 0.85
= 4500 / 86.4 × 0.85
= 52.1 × 0.85 = 44.3 mL/min
Clinical use: Dose adjust renally-cleared drugs (aminoglycosides, vancomycin, metformin, etc.)
1.5 Vancomycin AUC-Based Dosing (High-Yield SPLE Topic)
Target: AUC/MIC ratio of 400-600 mg·h/L
Ke = (ln C1 - ln C2) / (t2 - t1)
Vd = Dose / [AUC × Ke] (simplified)
t½ = 0.693 / Ke
Example:
Peak (C1) = 30 mg/L at t1 = 1 hr post-infusion
Trough (C2) = 8 mg/L at t2 = 12 hr
Ke = (ln 30 - ln 8) / (12 - 1) = (3.40 - 2.08) / 11 = 0.12 hr⁻¹
t½ = 0.693 / 0.12 = 5.8 hours
1.6 Aminoglycoside Dosing (Gentamicin / Tobramycin)
Conventional dosing:
- Peak target: 6-10 mcg/mL (gentamicin)
- Trough target: < 2 mcg/mL
Extended-interval (once-daily) dosing:
- Dose: 5-7 mg/kg IV q24h (based on IBW or AdjBW)
- Use Hartford nomogram to check timing
Ideal Body Weight (IBW):
Male: IBW = 50 + 2.3 × (height in inches - 60)
Female: IBW = 45.5 + 2.3 × (height in inches - 60)
Adjusted Body Weight (for obese patients):
AdjBW = IBW + 0.4 × (Actual BW - IBW)
PART 2: IV INFUSION CALCULATIONS
2.1 IV Flow Rate (mL/hr)
Rate (mL/hr) = Volume (mL) / Time (hr)
Example:
1000 mL NS over 8 hours
Rate = 1000 / 8 = 125 mL/hr
2.2 Drip Rate (drops/min)
Drip rate = [Volume (mL) × Drop factor (gtt/mL)] / Time (min)
Common drop factors:
- Macrodrip: 10, 15, or 20 gtt/mL
- Microdrip: 60 gtt/mL
Example:
500 mL D5W over 4 hours, drop factor = 20 gtt/mL
Time = 4 × 60 = 240 min
Drip rate = (500 × 20) / 240 = 10,000 / 240 = 41.7 ≈ 42 gtt/min
2.3 Drug Infusion Rate (mcg/kg/min or mg/hr)
Concentration = Amount of drug (mg) / Volume of solution (mL)
Rate (mL/hr) = [Dose (mcg/kg/min) × Weight (kg) × 60] / Concentration (mcg/mL)
Example:
Dopamine: ordered at 5 mcg/kg/min for 70 kg patient
Available: 400 mg dopamine in 250 mL NS
Concentration = 400 mg / 250 mL = 1.6 mg/mL = 1600 mcg/mL
Rate = (5 × 70 × 60) / 1600 = 21,000 / 1600 = 13.1 mL/hr
2.4 Drug Amount Being Infused Per Minute/Hour
Dose rate (mg/hr) = Concentration (mg/mL) × Rate (mL/hr)
Example:
Heparin 25,000 units in 500 mL NS running at 18 mL/hr
Concentration = 25,000 / 500 = 50 units/mL
Dose = 50 × 18 = 900 units/hr
2.5 Time to Complete an Infusion
Time (hr) = Volume (mL) / Rate (mL/hr)
PART 3: DILUTIONS
3.1 Simple Dilution (C1V1 = C2V2)
C1 × V1 = C2 × V2
- C1 = initial (stock) concentration
- V1 = volume of stock needed
- C2 = final desired concentration
- V2 = final total volume
Example:
Prepare 200 mL of 2% solution from a 10% stock
V1 = (C2 × V2) / C1 = (2 × 200) / 10 = 40 mL of stock
Add 40 mL of 10% stock + 160 mL diluent = 200 mL of 2%
3.2 Alligation Method (Mixing Two Strengths)
Used when mixing two concentrations to get a desired middle concentration.
Steps:
- Place the higher concentration at top-left, lower at bottom-left, desired in center
- Subtract diagonally (always subtract from the desired concentration)
- Result gives the ratio of parts needed
Example:
Mix 70% alcohol with 30% alcohol to make 50% alcohol
- Parts of 70%: 50 - 30 = 20 parts
- Parts of 30%: 70 - 50 = 20 parts
- Ratio = 20:20 = 1:1 (equal volumes of each)
To make 1000 mL: mix 500 mL of 70% + 500 mL of 30%
3.3 Percent Concentrations
| Type | Formula | Example |
|---|
| % w/v (weight/volume) | g per 100 mL | 5% dextrose = 5 g/100 mL |
| % w/w (weight/weight) | g per 100 g | 2% ointment = 2 g/100 g |
| % v/v (volume/volume) | mL per 100 mL | 70% isopropyl = 70 mL/100 mL |
Example:
How many mg of NaCl in 500 mL of 0.9% NaCl?
0.9% w/v = 0.9 g/100 mL = 9 mg/mL
Amount = 9 × 500 = 4,500 mg = 4.5 g
3.4 Parts Per Million (PPM)
1 ppm = 1 mg/L = 1 mcg/mL
Example:
Fluoride water at 1 ppm = 1 mg fluoride per liter of water
PART 4: ADULT DRUG DOSING
4.1 Basic Dose Calculation
Dose to give = (Desired dose / Available concentration) × Volume
Example:
Order: Amoxicillin 500 mg. Available: 250 mg/5 mL
Volume = (500 / 250) × 5 = 10 mL
4.2 Weight-Based Dosing (mg/kg)
Total dose = Dose (mg/kg) × Patient weight (kg)
Example:
Metformin: not weight-based - fixed 500 mg twice daily
Enoxaparin DVT treatment: 1 mg/kg SC q12h
80 kg patient: 1 × 80 = 80 mg SC every 12 hours
4.3 Body Surface Area (BSA) Dosing (mainly chemotherapy)
Mosteller Formula:
BSA (m²) = √[(Height(cm) × Weight(kg)) / 3600]
DuBois Formula:
BSA = 0.007184 × Height(cm)^0.725 × Weight(kg)^0.425
Example (Mosteller):
Patient: 170 cm, 70 kg
BSA = √[(170 × 70) / 3600] = √[11900 / 3600] = √3.306 = 1.82 m²
Cyclophosphamide 750 mg/m²:
Dose = 750 × 1.82 = 1,365 mg
4.4 Renal & Hepatic Dose Adjustments
Renal:
- Calculate CrCl (Cockcroft-Gault, see Part 1.4)
- Adjust dose or interval per drug monograph
| CrCl | Renal Function |
|---|
| > 90 mL/min | Normal |
| 60-89 | Mildly reduced |
| 30-59 | Moderately reduced |
| 15-29 | Severely reduced |
| < 15 | Kidney failure |
Hepatic (Child-Pugh Score):
Used for drugs with high hepatic extraction (e.g., morphine, propranolol, lidocaine)
| Child-Pugh Class | Severity | Action |
|---|
| A (5-6 pts) | Mild | No dose change usually |
| B (7-9 pts) | Moderate | Reduce dose 25-50% |
| C (10-15 pts) | Severe | Avoid or use very cautiously |
PART 5: PEDIATRIC DOSING
5.1 Weight-Based Dosing (Most Common Method)
Dose = mg/kg/dose × weight (kg)
Always check:
- Maximum single dose (do not exceed adult dose)
- Daily maximum dose
- Age range (neonate vs infant vs child vs adolescent)
Example:
Paracetamol (acetaminophen): 15 mg/kg/dose q4-6h (max 75 mg/kg/day)
Child weighing 20 kg:
Single dose = 15 × 20 = 300 mg (available as 250 mg/5 mL)
Volume = (300/250) × 5 = 6 mL
Daily max = 75 × 20 = 1,500 mg/day
5.2 Pediatric Age Groups
| Group | Age Range |
|---|
| Neonate (newborn) | 0-28 days |
| Infant | 1-12 months |
| Toddler | 1-3 years |
| Child | 3-12 years |
| Adolescent | 12-18 years |
| Premature neonate | < 37 weeks gestational age |
Neonates have reduced renal/hepatic function, altered Vd, and immature enzyme systems - always check neonatal-specific dosing.
5.3 Pediatric Fluid Maintenance - Holliday-Segar Method
- First 10 kg: 100 mL/kg/day
- Next 10 kg: 50 mL/kg/day
- Each kg above 20 kg: 20 mL/kg/day
Example:
Child, 25 kg:
First 10 kg = 10 × 100 = 1,000 mL
Next 10 kg = 10 × 50 = 500 mL
Last 5 kg = 5 × 20 = 100 mL
Total = 1,600 mL/day = 66.7 mL/hr
Shortcut for 4-2-1 rule (mL/hr):
4 mL/kg/hr for first 10 kg + 2 mL/kg/hr for next 10 kg + 1 mL/kg/hr for each kg above 20
5.4 Young's Rule (Age-based, less commonly used)
Child dose = [Age(yr) / (Age + 12)] × Adult dose
Example:
Adult dose of drug = 500 mg, child is 6 years old
Child dose = [6 / (6+12)] × 500 = (6/18) × 500 = 0.33 × 500 = 167 mg
5.5 Clark's Rule (Weight-based, less commonly used)
Child dose = [Weight(lb) / 150] × Adult dose
Note: Weight in pounds (1 kg = 2.2 lb)
5.6 Fried's Rule (For infants < 1 year)
Infant dose = [Age(months) / 150] × Adult dose
5.7 BSA-Based Pediatric Dosing
Child dose = (Child BSA / 1.73 m²) × Adult dose
Where 1.73 m² is average adult BSA.
Example:
Child BSA = 0.8 m², adult dose = 100 mg
Child dose = (0.8 / 1.73) × 100 = 46.2 mg
5.8 Neonatal Dosing Considerations
| Factor | Effect |
|---|
| Reduced albumin | Higher free drug levels (e.g., phenytoin, warfarin) |
| Immature CYP450 | Slower drug metabolism |
| Higher % body water | Higher Vd for water-soluble drugs |
| Less body fat | Lower Vd for lipid-soluble drugs |
| Low GFR | Accumulation of renally cleared drugs |
| Immature BBB | Higher CNS drug penetration risk |
Example - Gentamicin in neonates:
- Preterm (< 30 wks): 3.5 mg/kg q36h
- Term neonate (≥ 37 wks): 4 mg/kg q24h
Compared to adult: 5-7 mg/kg q24h
PART 6: ELECTROLYTE & OSMOLARITY CALCULATIONS
6.1 Milliequivalents (mEq)
mEq = (mg × valence) / molecular weight
Example:
mEq of calcium in 1 g CaCl₂ (MW = 111, valence = 2)
mEq = (1000 × 2) / 111 = 18 mEq
6.2 Osmolarity
Osmolarity (mOsm/L) = 2[Na⁺] + [Glucose/18] + [BUN/2.8]
Normal serum osmolarity: 280-295 mOsm/L
Example:
Na = 140, Glucose = 90 mg/dL, BUN = 14 mg/dL
Osm = 2(140) + (90/18) + (14/2.8) = 280 + 5 + 5 = 290 mOsm/L ✓
6.3 Osmol Gap
Osmol Gap = Measured Osm - Calculated Osm
Normal: < 10 mOsm/L
Elevated suggests: ethanol, methanol, ethylene glycol toxicity
PART 7: QUICK REFERENCE - COMMON FORMULAS SUMMARY
| Calculation | Formula |
|---|
| Loading dose | LD = (Vd × Cp target) / F |
| Maintenance dose | MD = (CL × Cp × τ) / F |
| Half-life | t½ = 0.693 × Vd / CL |
| Creatinine clearance | CrCl = [(140-age) × wt] / (72 × SCr) × 0.85 ♀ |
| IV flow rate | Rate = Volume / Time |
| Drip rate | gtt/min = (Vol × drop factor) / time(min) |
| Drug infusion | Rate = (mcg/kg/min × wt × 60) / conc(mcg/mL) |
| Dilution | C1V1 = C2V2 |
| BSA (Mosteller) | √[(Ht × Wt) / 3600] |
| Peds fluid (Holliday-Segar) | 100/50/20 mL/kg/day rule |
| Young's rule | [Age/(Age+12)] × Adult dose |
| Clark's rule | [Wt(lb)/150] × Adult dose |
| Fried's rule | [Age(mo)/150] × Adult dose |
| mEq | (mg × valence) / MW |
| Osmolarity | 2[Na] + Glu/18 + BUN/2.8 |
EXAM TIPS FOR SPLE
- Always identify units first - convert everything to consistent units before calculating (mg vs mcg, mL vs L, hr vs min)
- Check if patient is obese - use IBW or AdjBW for aminoglycosides and vancomycin, not actual body weight
- Renally cleared drugs - always calculate CrCl if SCr is given
- Pediatric: never exceed adult dose - even if the mg/kg calculation gives a higher number
- Alligation - draw the grid out visually, never subtract wrong direction
- Infusion rate questions - if mcg/kg/min is given, convert to mL/hr using the preparation concentration
- % w/v - always 1% = 10 mg/mL (this shortcut is tested frequently)
1% = 10 mg/mL = 10,000 mcg/mL
This guide covers all major calculation categories tested in the SPLE (Saudi Pharmacist Licensure Exam). Practice each formula type with multiple examples - the exam heavily favors applied case-based calculation questions.