Math and Measurement

Order of Operations

• Order of operations should be considered first
• Four principal arithmetic operations include addition, subtraction, multiplication, and division
• When an equation involves multiplication and/or division as well as addition and/or subtraction, multiplication and division operations must be executed first
• For example, x = 12 + 3 · 10 → x = 12 + 30 → x = 42
• If an equation includes symbols of enclosure (parentheses and division bars), the arithmetic operations inside the symbol of enclosure must be executed first
• As an example 12+3.(4+2) / 3.5 = 12+3.(6) / 3.5 = 30 / 3.5

Algebra

• When solving an equation that involves addition and/or subtraction as well as multiplication and/or division, the order of operations reverses
• First, clear the addition and/or subtraction operations, and then the multiplication and/or division
• Charles's Law (temp's effect on gas volume) is an example; V1/T1 = V2/T2

Scientific Notation

• Scientific notation consists of a number multiplied by a power of 10
• If the decimal point is moved to the left, the exponent is positive, and if the decimal point is moved to the right, the exponent is negative
• Large numbers have positive exponents when expressed in scientific notation, while small numbers (less than 1) have negative exponents
• In proper scientific notation, numbers have only one digit in front of the decimal place
• For example, 0.0034 x 10^25 is not in proper form, which would be expressed as 3.4 x 10^22

Measurements and Significant Figures

• Significant figures are digits in a measured value that have a physical meaning and can be reproducibly determined
• Nonzero digits are always significant
• Captive zeros are always significant
• Leading zeros are never significant
• Trailing zeros are significant only when the number contains a decimal point (540.0 vs. 540)
• Exact numbers and definitions are considered to have an infinite number of significant digits
• One operational way to identify significant zeros is to convert the number into scientific notation
• If the decimal point does not cross a zero when converting a number into scientific notation, the zero is always significant
• Ambiguity can be removed by expressing the number in scientific notation
• 100 is expressed as 1 sig. fig. as (1x10^2), 2 sig. fig. as (1.0x10^2), and 3 sig. fig. as (1.00x10^2)

Significant Figures in Calculations

• The result of a calculation cannot be more reliable than any of the measurements on which the calculation is based
• The statistical way in which the inherent uncertainties in measurement are propagated differ between addition and subtraction operations vs. multiplication and division operations
• When adding or subtracting, keep the smaller number of decimal places
• When multiplying or dividing, keep the smaller number of significant figures

Accuracy and Precision

• There are two ways to determine how good a measurement is
• Accuracy is the agreement between experimental data and the "true" value
• Precision is the agreement between replicate measurements
• Accuracy can be improved by making replicate measurements and taking the average
• Accuracy is the percent error is measured according to the formula: %error = (measured value – “true” value) / "true" value) * 100%
• Precision is improved by careful lab technique and/or using instruments capable of yielding greater precision
• Repeating a measurement may not improve precision, but it allows determination if the measurement is reproducible
• Precision can be quantified by standard deviation; the smaller the ratio of the standard deviation to the average value, the better the precision

SI (International System)/Metric System

• The metric system consists of a base unit and (sometimes) a prefix multiplier
• Base units describe the quantity measured, like length, mass, and time
• Meter (m) measures length
• Kilogram (kg) measures mass (pounds are a is weight unit; kilograms are a mass unit, which is not a clinical distinction)
• Kelvin (K) measures temperature but is not measured in degrees
• Mole (mol) measures amount of material

Absolute Zero and Kelvin Scale

• The Kelvin temperature scale begins at absolute zero
• Temperatures cannot be lower than 0 K (zero kelvin, not "degrees kelvin")
• 0 K is equivalent to -273.15°C
• K = °C + 273.15 is the conversion between degrees celcius and kelvins
• Converting between kelvins and °F requires first converting °F to °C; °F = 1.8°C + 32
• Then °C = (98.6°F – 32)/1.8 = 37.0°C
• Conversion from celsius to kelvins is 37.0°C + 273.15 = 310.2 K

Conversion Factors

• A conversion factor is a statement of equality between two measurements of the same object or property
• A conversion factor times is multiplied by the beginning quantity so that the units of the beginning quantity are mathematically replaced by the units of the desired quanitity
• When using a conversion factor between metric quantities, one side of the equation gets the prefix multiplier letter, and the other side gets the numerical equivalent of the multiplier

Important Points about Conversion Factors

• This process works only if you are meticulous in including units with all numerical data
• A final answer that has the correct units is probably correct, and an answer with the wrong units is almost certainly incorrect

Density

• Density is the ratio of the mass of an object to its volume
• Density will always have two units, a mass unit divided by a volume unit
• Density is a conversion factor between mass and volume
• Density can be measured, for example, in units of kg/m3, g/ml, g/L
• When object density is equal to fluid density, objects are neither floating nor sinking
• Objects that are less dense than fluid density float
• Objects that are denser than fluid density sink

Density and Specific Gravity

• Specific gravity is the ratio between an object's density and the density of water
• The formula for this is specific gravity = sg = density of object / density of water
• Any sample with a specific gravity greater than 1 is denser than water (blood)
• Any sample with a specific gravity less than 1 is less dense than water (ice)
• Specific gravity equals density only when the units of measure are grams per milliliter, because the density of water is 1 g/ml

Clinical Mathematics for SRNAs

• Convert fractions to ratio, fraction to decimal, decimals to percent, ratios to mg/mL or mcg/mL, percent solutions to mg or mcg/ml
• Perform temperature conversions in any scale
• Calculate desired rate setting for IV drips given patient weight, drug dilution, and desired dose
• Convert weights in pounds to kilograms
• Calculate FiO2 when air is being used rather than N2O
• State the ideal weight when given actual weight and height in any units
• Calculate how long an E tank of oxygen will last at a given liter flow
• Know fluid and blood administration, and how much blood loss is acceptable
• Understand tube sizing and tidal volume

Dimensional Analysis

• Convert units by multiplying by an identity (these are sometimes called conversion factors)
• Examples:
  • 760 mm Hg × ( 14.7 psi / 760 mm Hg) = 14.7 psi
  • 454 gm x (2.2 lb./1000 gm) = 1 lb.
  • 50 mcg/ml × (1 mg/1000 mcg) = 0.05 mg/ml

Proportions and Desired/Available

• When calculating doses of drugs, setting up proportions can be helpful
• For example, how many ml of 0.75% bupivacaine are required to give the patient 12 mg when the available solution is 1 ml/ 7.5 mg?
• 1 ml/ 7.5 mg = X ml/12 mg → (12 mg) 1 ml/7.5 mg = X ml/12mg →12/7.5 ml = X → 1.6 ml = X
• As another example, if a 1 ml/10 mg sample of ephedrine is desired, and 50 mg are available, how many ml of diluent are needed?
• Setting up the equation, 1 ml/10 mg = X ml/50 mg → (50 mg) 1 ml/10 mg = X ml/50mg → 5 ml = x

Examples of Drug Dilutions

• Epi 1:1,000 contains 1 mg/ml, which is the same as 1,000 mcg/ml
• Taking 1 ml and adding 9 ml diluent yields a syringe with 1,000 mcg/10 ml (100 mcg/1 ml)
• Taking 1 ml of this mixture (containing 100 mcg/ml) and adding 9 ml diluent to it yields a syringe containing 100 mcg/10 ml, with the "double-diluted” new mixture containing 10 mcg/ml
• For example, using epirdural preparation using Bupivacaine 0.25% (2.5 mg/ml), fentanyl 50 mcg/ml, and sterile diluent as available ingredients, a 60 ml syringe containing 0.0625% (often called 1/16%) bupivacaine with fentanyl 5 mcg/ml is desired

Example Epidural Preparation

• A 60 ml syringe with 0.625 mg/ml (0.0625%) means the syringe will contain a total mass of bupivacaine of 37.5 mg (= 60 ml * 0.625 mg/ml)
• A total mass of fentanyl is desired to be 5 mcg/ml * 60 ml = 300 mcg
• A syringe requires 37.5 mg/60 ml of bupivacaine so 15 ml (37.5/2.5 = 15) of bupivacaine 0.25% (2.5 mg/ml) is drawn up
• A syringe requires 300 mcg/60 ml of fentanyl so 6 ml of fentanyl (50 mcg/ml) is drawn up
• Up to this point, 15 ml (bupivacaine) + 6 ml (fentanyl) = 21 ml has been used

Epinephrine

• Epinephrine concentration is measured differently
• A 1 mg ampule of 1:1000 epinephrine means the solution contains 1 mg of epinephrine per ml
• Epinephrine is commercially available in a 1 ml ampule containing 1 mg (i.e., 1:1000 or 1000 mcg per ml), and a 10 ml ampule containing 1 mg (i.e., 1:10,000 or 100 mcg per ml)
• 1:10,000 contains 1 gram/10,000 = 1000 mg/10,000 or 1 mg/10 ml or 1000 mcg/10 ml = 100 mcg/ml
• Avoid injecting espinephrine in locations lacking collateral vessels like fingers, nose, toes, and ears

Ratios and Percentages

• A solution containing epinephrine 1:100,000 has 10 micrograms of epinephrine per ml (1 million mcg (or 1 gram)/100,000 ml)
• A solution marked 1:200,000 has 5 micrograms epinephrine per ml
• 1 gram = 1000 mg = 1,000,000 mcg
• 1:100,000 solution contains 10 micrograms per ml
• 1:200,000 solution contains 5 micrograms per ml
• 1:250,000 solution contains 4 micrograms per ml
• 1:400,000 solution contains 2.5 micrograms per ml

How to Prepare Epinephrine

• Preparing 25 ml of 2% lidocaine with 1:250,000 epinephrine requires using an ampule of 1:1000 epinephrine (i.e., 1 mg/ml), adding 10 ml of saline to give 1:10,000 (i.e., 100 micrograms per ml), and adding 1 ml of this epinephrine solution to 24 ml of 2% plain lidocaine, for a final concentration of 100 micrograms per 25 ml = 4 micrograms per ml
• Preparing a 1:200,000 solution of epinephrine in 20 ml of 1% lidocaine requires taking 0.1 ml of epinephrine from a 1:1000 ampule and adding it to 19.9 ml of 1% plain lidocaine (100 mcg/20 ml = 5 mcg/ml)

Percentages

• To determine mg/ml in a % solution, move the decimal point one place to the right.
• For example, A 1% solution has 1 gm/100 ml (1000/100 = 10 mg/ml)

Weight Math

• A quick estimator of converting convert pounds to kg is (pounds/2) - 10% = Weight in kilograms
• Exact conversion from pounds to kilograms: Weight in pounds x kg/2.2 lbs
• Estimating weight in kilograms of pediatrics patients: (Age in years x 2) + 9 = Weight in kilograms
• Ideal Body Weight can also be estimated using simple rules
  • The ideal body weight of females in pounds is 100 lb. + (5 lb. per inch over 5 ft)
  • The ideal body weight of males in pounds is 105 lb. + (6 lb. per inch over 5 ft)