Mathematics Review for Anesthetists

Objectives

My overall objective is a collection of mathematical tools useful in clinical anesthesia practice.

1. Name six basic units in SI measurement.
2. Convert fractions to ratio, fraction to decimal, decimals to percent, ratios to mg/mL or mcg/mL, percent solutions to mg or mcg/mL.
3. Know common SI prefixes and be able to convert.
4. Perform temperature conversions in any scale.
5. Calculate desired rate setting for IV drips given patient weight, drug dilution, and desired dose.
6. Convert weights in pounds to kilograms.
7. Calculate FIO2 when air is being used rather than N2O.
8. State ideal weight when given actual weight and height in any units.
9. Calculate how long an E tank of oxygen will last at a given liter flow.

Math Tools Review

Further Reading

• Waugaman et al. Nurse Anesthesia 2nd ed. Chapter 4 on Metric medical mathematics by Marian Waterhouse. ISBN 0-8385-7962-0 1992 Appleton Lange.
• Kee JL, Marshall SM. Clinical Calculations. 2^nd ed. Philadelphia: WB Saunders; 1992.
• Drug Calculations for Nurses- Quiz page

Basic Approaches

Four basic techniques for solving any math problem are: Dimensional analysis, proportions, desired & available, and Lester’s Rule for drips. These are explained below.

Dimensional analysis

To convert units, multiply by an identity.

• Example 1: 760 mm Hg x 14.7 psi / 760 mm Hg = 14.7 psi
• Example 2: 29.9 in. Hg x 1013 mBar / 29.9 in. Hg = 1013 mBar
• Example 3: 454 g x 2.2 lb / 1000 g = 1 lb
• Example 4: 50 mcg/mL x 1 mg / 1000 mcg = 0.050 mg/mL

In each example, the bolded fraction is an identity (14.7 psi and 760 mm Hg are equivalent). Multiplying any quantity by an identity doesn't change the quantity, only the label (in the first example, mm Hg are changed to psi).

Proportions

Proportions are used to determine the answers to questions like "How far can I go on a half tank of gas, if a full tank will let me drive 300 miles?" or, more pertinently, "How many mL of 0.75% bupivacaine do I need to draw up to give the patient 12 mg?"

• To solve the second, you calculate that 0.75% bupivacaine contains 7.5 mg/mL (see "Percentage solutions" below on this page).
• 7.5 mg / 1 mL = 12 mg / x mL
• (Multiplying both sides by x mL gives) 7.5 mg * x = 12 mg
• (Dividing both sides by 7.5 mg gives the solution) x = 1.6 mL

Desired & Available

Working from what you have, to what you need. If you desire remifentanil 50 mg/mL, and have 1 gm available, how many mL of diluent are needed? (1 gm/50 mg = x mL/1 mL, and x = 20 mL diluent needed.)

Lester's No-Math Rule for Intravenous Infusions

Lester's No-Math Rule for intravenous infusions is The number of milligrams in 250 mL comes out in micrograms in 15 microgtts/min (Or) the number of grams in 250 mL comes out in milligrams in 15 microgtts/min).

• The basis for this is that 15 microgtts is 0.25 mL (one one-thousandth of a 250 mL bottle). And 1 mcg is 0.001 of 1 mg; just as 1 mg is 0.001 of 1 gm. Microgtts/min and mL/hr are the same.
  • Proof? 60 microgtts/min x 1 mL/60 microgtts x 60 min/1 hour = 60 mL/hour. Multiplying by the two terms which are identities doesn't change the underlying quantity, just the units. See any chemistry text under dimensional analysis.
• Example 1: What is the dose/min of lidocaine 1 gm/250 mL run at 15 mL/hr? (Answer: 1 mg/min).
• Example 2: What is the drop factor required for dopamine 3 mcg/kg/min (patient weight 70 kg; dopamine concentration 400 mg/250 mL)? (Answer: 7 microgtts/min will deliver 200 mcg/min, which is approximately 3 mcg/kg.)

Fraction to ratio

Fractions are ratios: 1/3 and 1:3 are functionally identical expressions

Fraction (or ratio) to decimal

Dividing 1 by 3 yields 0.3333

Decimals to percent

Divide numerator by denominator, then multiply by 100. 1/3 x 100 = 33.33%

Ratios in anesthesia

The ratio 1:100,000 is the one to remember. Ratios are expressed in grams/mL (it's easy to recall if you remember that 1 mL of H2O weighs 1 g), therefore, 1 gm /100,000 mL = 1000 mg /100 L.  Dividing by 100 yields 10 mg /1 L, which is the same as 10 mcg per 1 mL

Percentage Solutions

The "number of parts of drug in every hundred" is the percentage of a solution, which is conventionally expressed as number of grams/100 mL.

The shortcut is "To determine mg/mL in a % solution, you just move the decimal point one place to the right". Example: A 1% solution has 1 gm/100 mL , the same as 0.01 gm/1 mL, or 10 mg/mL (0.01 gm = 10 mg).

Pilchak's No-Math Rule

To convert pounds to kilograms:

• (lb/2) - 10% = kg
• "Take the number of pounds, subtract 50%, then subtract another 10% of what remains"

Metric System

The metric system is also known as "SI" (Systeme Internationale). The NIST Metric System Teaching Site has a wealth of information on every aspect of the metric system, very clearly presented. It's easier to learn if you start thinking of all your everyday quantities metrically- your height, your soda can, a jar of salsa- what is their volume and weight expressed in metric system units?

Remember that a gram is 1000 mg, a mg is 1000 mcg, and a mcg is 1000 ng.

Metric Prefixes

Prefix	Associated quantity	In scientific notation
mega (M)	1,000,000	106
kilo (k)	1000	103
hecto (h)	100	102
centi (c)	0.01	10-2
milli (m)	0.001	10-3
micro (mc)	0.000001	10-6
nano (n)	0.000000001	10-9

Metric pressure units (based on the Pascal) are less familiar to most than metric measures of weight.

1 atmosphere equals

100 kPa 1000 hPa 760 mm Hg 760 torr 1 Bar 1013 mBar ~1000 cm H2O 14.7 psi

Temperature Conversions

• Degrees Centigrade (C) = 5/9(F - 32)
• Degrees Farenheit (F) = (9/5 x C) + 32
• Or remember the two together with: 9C = 5F - 160
• Degrees Kelvin = C + 273

Figuring oxygen concentration when air or N2O are used

It is useful as a backup to your oxygen analyzer to know what it "should" read. You can calculate the percent composition of the oxygen as compared to the total fresh gas flow. The total volume of oxygen includes all volume from the oxygen flowmeter, 21% of the volume indicated on the air flowmeter, none of the rest of the flowmeters.

• Example: What is the percent oxygen when 2 L/min oxygen and 2 L/min air are flowing? (Answer: 60.5%, derived by 2000 mL oxygen + [0.21 x 2000] mL air = 2420 mL oxygen, which is 60.5% of the total fresh gas flow.)
• Example: What percent oxygen is inspired when flows are 2 L/min oxygen? 1 L/min oxygen? 8 L/min oxygen? (Answer: 100% in each case)
• Example: What percent oxygen is inspired when flows are 2 L/min oxygen and 2 L/min nitrous oxide? (Answer: 50%)
• Example: What percent oxygen is inspired when flows are 1 L/min oxygen and 2 L/min nitrous oxide? (Answer: 33%)

DOT-123

A clinical rule useful in (quick and dirty) estimation of respiratory parameters for infants and children:

• Dead space = 1 mL/lb,
• O2 consumption = 2 mL/lb, and
• Tidal Volume = 3 mL/lb.

Ideal Weight Estimation

1. This way is easy to remember, since we most often think of height in inches and feet:
   • Females = 105 lb + 5 lb/inch over 5 ft
   • Males = 106 lb plus 6 lb/in over 5 ft.
2. Broca's Index (Ideal weight kg = Height cm - 100) is useful if you are comfortable thinking of height in centimeters. Five feet is approximately 150 cm, six feet is approximately 180 cm.
3. Body Mass Index (BMI) is a way to relate the actual weight to the ideal.
   • BMI = Weight (kg) / Height (m)².
   • Normal is < 25, obese > 28, morbid > 35.

Calculating how long a cylinder will last

A perennial Board-type question with clinical importance. Use a proportion to solve (there are approximately 660 L in a full E tank @ 2000 psi). Don't forget to consider fresh gas flow (FGF).

• Example: Your oxygen tank pressure gauge reads 200 psi. How long can a flow of 2 L/min be maintained?
  • x L / 200 psi = 660 L / 2000 psi
  • x = 66L
  • Thus the tank will last 33 min, if the 66 L in the tank flows out at 2 L/min.

Drips by drops method for drug infusions

Works for dopamine, dobutamine, isoproterenol, epinephrine, norepinephrine, phenylephrine, theophylline, sodium nitroprusside, nitroglycerin, trimethaphan, lidocaine, procaineamide, bretyllium. Requires no calculations or tables. Donenfeld RF. Anesth Analg 1990;70:116-7.

• Method: Dilute 1 ampule of drug in one 250 mL bag of IV fluid. Infusion rates of 60, 30, and 15 mL/hr will give high, medium, and low dose rates of any of the above agents (for a 70 kg person). If a controller pump is unavailable, these rates are easy to set with microdrop tubing (1 gtt/sec, 1 gtt/2sec, 1 gtt/4sec).
• If the patient's weight differs from 70 kg, the drop rate needs adjustment accordingly, ie: Actual kg / 70kg = Adjusted rate (mL/hr).
• Delivered dose is easily calculated as follows: (mg/250 mL) x 4 = mcg/mL.
  • Proof? The number of mg/250 mL x 4 = mg/1000 mL. The number of mg/1000ml multiplied by 0.001/0.001 equals mcg/mL.)

Acceptable blood loss (ABL)

There are online calculators on the web (just in case you have internet connectivity in your OR!). I found that one by Googling "allowable blood loss".

1. EBV (mL/kg) differs with age and gender
   • Males 75 mL/kg
   • Females 65 mL/kg
   • Premature infants 90 mL/kg
   • Neonate at term 85 mL/kg
   • Children 1-2yr 75 mL/kg
2. ABL = [(Original Hct - Final Hct) / Average Hct] x EBV

Fluid maintenance

Pediatric fluid balance:

1. For the first 10 kg, give 4cc/kg/hr, then
2. For the next 10 kg (10-20 kg), give 2cc/kg/hr, then
3. Give 1cc/kg/hr for kg greater than 20 kg.

Endotracheal tube size, length

• Lip line = (age/2) + 12
• Size = (16 + age)/4
• (French - 2) / 4 = ID mm

Alveolar air

pAO2 = [(pB - pH2O) x FIO2] - paCO2
(with pH2O = 47 torr at 37 degrees C)

Arterial oxygen content

CaO2 = (1.39 x Hb x SatO2) + (paO2 x 0.003)

Expected pO2 for changes in FIO2

A rough estimator based on Henry’s Law. If you change FIO2 10%, expect paO2 to increase 50 torr (starting values FIO2 21%, paO2 100 torr approximately).

Time constant

Time constant = Capacity / Flow

• A system reaches 63% of equilibrium in one time constant; 86% of equilibrium in 2 time constants; 95% of equilibrium in 3 time constants).
• Applications are washin and washout of gases, or time to complete denitrogenation at varied fresh gas flow rates.

Remembering drug doses

It's always advisable when confronted with a family of drugs to work out their relative doses per 70 kg. That way, you have a baseline for comparison and can increase or decrease your starting dose in proportion to the patient's weight. Also, you're less likely to choose a much too big (or much too small) dose for a patient when you have memorized what the "average weight" patient would typically receive. For example, reasonable intubating doses for the muscle relaxants (assume 70 kg weight) are vecuronium 7 mg, rocuronium 42 mg, cis-atracurium 14 mg, and pancuronium 7 mg. (Of course you would modify these doses based on body habitus and past history.)

Remember "same as…" doses. For example, vecuronium and pancuronium. Or similar doses like thiopental (3-5 mg/kg) and curare (intubating dose is 0.3-0.5 mg/kg).

Work out doses for a typical person for drugs given at 1 mg/kg (70 mg for a 70 kg person), 0.5 mg/kg (35 mg), 0.1 mg.kg (7 mg), 0.05 mg/kg (3.5 mg), and 0.01 mg/kg (0.7 mg). Again, that way you have a concept of what those numbers mean for the "average" patient. You can then use this concept as a means of double-checking your answer derived from the electronic calculator, or for mental math done in haste.

Keep in mind the expected volume (number of mL) for drugs, especially narcotics and muscle relaxants. If you are drawing up and preparing to administer more than 3 to 10 mL of muscle relaxant as an intubating dose, it is likely your math is faulty (provided the patient is anywhere near normal weight).

Spinal (intrathecal) doses

1. Have somewhere to start for a reasonable level for each drug. For example tetracaine 10 mg (1 mL of 1%) for a level of T10. A reasonable lidocaine dose for the same effect is 50 to 60 mg (1 to 1.2 mL of 5%). A reasonable bupivacaine dose is 12 mg (1.6 mL of 0.75%). Know the range of doses found to be useful at your institution, in your patients.
2. Modify the dose based on height, medical history, pregnancy, or whatever other factors you like. When you add or subtract, remember the amount of drug in some usefully-small quantity: lidocaine 5% has 10 mg per 0.2 mL, tetracaine 1% has 2 mg per 0.2 mL , and bupivacaine 0.75% has 1.5 mg per 0.2 mL.
3. Alter the baricity of the drug, if needed, for the effect you want. Typically, hyperbaric spinals are given. The spinal medications may or may not come mixed in hyperbaric 7-10% dextrose. If they do not come so mixed, add an equal volume of dextrose solution to the local anesthetic.
4. Add any extras, have someone double-check your figures if you have the SLIGHTEST doubt (once given, the medication is irretrievable and has profound effects), then administer. Epinephrine (1:1000) 0.2 mL is a common additive. Preservative-free morphine 0.5-0.7 mg (0.5-0.7 mL of Duramorph 0.1%) is also common.
   • Please don't ask me how I know about the "irretrievable and profound" part.

Epidural doses

1. The rule of thumb is With a lumbar epidural placement, 10 mL of drug will generally produce a T7 to T9 level in the average size patient, 20 mL will result in a T4 level.
2. Larger volumes of drug will be required for higher blocks, more concentrated drugs for more intense motor block, and more dosage (total mass of drug) for increased sensory block (Reese CA. Spinal and epidural blocks. 2nd ed. Park Ridge, Illinois: AANA; 1996 p. 104-5).