Dimensional Analysis Nursing Practice

Start with the unit needed, mL, and arrange the factors so that mcg and mg cancel. mL = 100 mcg × (1 mg ÷ 1,000 mcg) × (1 mL ÷ 0.4 mg) mcg cancels with mcg and mg cancels with mg, leaving mL, so the set-up is right. Multiply the tops: 100 × 1 × 1 = 100. Multiply the bottoms: 1,000 × 0.4 = 400. Divide: 100 ÷ 400 = 0.25 mL. A volume under 1 mL is given to the hundredth, with a leading zero. Leaving out the mcg-to-mg factor gives 250 mL, an impossible injection volume. Small doses such as this one reverse the respiratory depression without taking away pain control. Takeaway: check that every unit except the one needed cancels before doing any arithmetic.

Sources: Open RN Nursing Skills 2e, 5.6 Using Dimensional Analysis; StatPearls: Dose Calculation Dimensional Analysis Factor-Label Method; DailyMed: Naloxone hydrochloride injection label (0.4 mg/mL; 0.1 to 0.2 mg steps after surgery)

Start with the unit needed, minutes, and build a chain in which g and mg cancel. minutes = 1 g × (1,000 mg ÷ 1 g) × (1 min ÷ 10 mg) g cancels with g and mg cancels with mg, leaving minutes. Multiply the tops: 1 × 1,000 × 1 = 1,000. Multiply the bottoms: 1 × 10 = 10. Divide: 1,000 ÷ 10 = 100 minutes. The rate is written upside down, as 1 min ÷ 10 mg, so that mg cancels. The label takes the longer of the two times, and 100 minutes is longer than 60 minutes, so 100 minutes is the shortest time allowed. Answering 60 minutes would run the drug at about 16.7 mg/min. A faster infusion raises the risk of flushing of the face, neck and upper body and of low blood pressure. Takeaway: write a rate as a fraction and flip it when that makes the unwanted unit cancel; when a rule sets two limits, use the one that takes longer.

Sources: Open RN Nursing Skills 2e, 5.6 Using Dimensional Analysis; StatPearls: Dose Calculation Dimensional Analysis Factor-Label Method; DailyMed: Vancomycin hydrochloride for injection label, 500 mg and 1 g vials (1 g every 12 hours; no more than 10 mg/min or over at least 60 minutes, whichever is longer); DailyMed: Vancomycin injection label, premixed bags (infusion reactions with a fast infusion: flushing of the face, neck and upper body, low blood pressure)

The equivalence of 1 mg per 100 units works as a conversion factor. mL = 3,500 units × (1 mg ÷ 100 units) × (1 mL ÷ 10 mg) Units cancel with units and mg cancels with mg, leaving mL. Multiply the tops: 3,500 × 1 × 1 = 3,500. Multiply the bottoms: 100 × 10 = 1,000. Divide: 3,500 ÷ 1,000 = 3.5 mL. This volume holds 35 mg of protamine, which is within the limit of 50 mg in any 10-minute period. Stopping after the first factor gives 35, which is the dose in mg, not the volume in mL. Takeaway: any stated equivalence, such as 1 mg per 100 units, can be written as a fraction and placed in the chain.

Sources: Open RN Nursing Skills 2e, 5.6 Using Dimensional Analysis; StatPearls: Dose Calculation Dimensional Analysis Factor-Label Method; DailyMed: Protamine sulfate injection label (1 mg neutralizes about 100 units of heparin; 10 mg/mL; no more than 50 mg in 10 minutes)

Convert the weight first and round it: 161 lb ÷ 2.2 = 73.18 kg, so 73.2 kg. Then build the chain from the unit needed, mL: mL = 73.2 kg × (150 mg ÷ 1 kg) × (1 g ÷ 1,000 mg) × (30 mL ÷ 6 g) kg, mg and g cancel, leaving mL. Multiply the tops: 73.2 × 150 × 1 × 30 = 329,400. Multiply the bottoms: 1 × 1,000 × 6 = 6,000. Divide: 329,400 ÷ 6,000 = 54.9 mL. The dose is 10,980 mg, so more than one vial is needed. Using 161 as if it were kg gives 120.8 mL, more than double the dose. Takeaway: round the weight in kg once, then let the chain carry the dose per kg, the mg-to-g conversion and the vial strength.

Sources: Open RN Nursing Skills 2e, 5.6 Using Dimensional Analysis; Open RN Nursing Skills (ACC adaptation), 5.11 Weight-Based Calculations; DailyMed: Acetylcysteine injection label (three-bag regimen: 150 mg/kg loading dose; 6 g in 30 mL vials)

Step 1: find the dose now due. The infusion started at 1 milliunit/min and was raised by 1 milliunit/min at 0830 and at 0900. The 0930 increase makes it 4 milliunits/min. Step 2: build the chain so that everything except mL/hr cancels. mL/hr = (4 milliunits ÷ 1 min) × (60 min ÷ 1 hr) × (1 unit ÷ 1,000 milliunits) × (1,000 mL ÷ 10 units) Multiply the tops: 4 × 60 × 1 × 1,000 = 240,000. Multiply the bottoms: 1 × 1 × 1,000 × 10 = 10,000. Divide: 240,000 ÷ 10,000 = 24 mL/hr. The contraction timing and the fetal heart rate guide the decision to increase but are not part of the arithmetic. Takeaway: read the current dose from the chart, then convert it with a chain that includes the bag strength and 60 min/hr.

Sources: Open RN Nursing Skills 2e, 5.6 Using Dimensional Analysis; StatPearls: Dose Calculation Dimensional Analysis Factor-Label Method; DailyMed: Oxytocin injection label (10 units in 1,000 mL; start 0.5 to 1 milliunit/min, raise by 1 to 2 milliunits/min every 30 to 60 minutes); StatPearls: Fetal Heart Rate Tracing, Assessment and Clinical Interpretation (baseline 110 to 160/min; tachysystole)