Pharmacy Math: Mastering Dosing Calculations and Prescriptions

Module 10 Outcomes Outcome 1: Interpret prescription orders Outcome 2: Solve oral and parenteral dosage problems including pediatric and weight-based dosing Outcome 3: Create dimensional analysis formulas to solve for necessary units

10.1 Pharmacy Math Introduction This module will cover the basic mathematics involved in most medication dosage calculations. Medication dosage calculations are often necessary for a variety of reasons. Examples include when a medication is dosed based on a patient’s weight, when the dose needed for a patient is not commercially available, and when the rate at which a medication is to be administered intravenously needs to be calculated. These are a few common examples of why pharmacy math is so important. Before discussing the calculation portion of this module, some helpful information regarding prescription orders will be introduced. It is critical that one be able to interpret the prescription, or order (as it is referred to in the hospital setting), to know what the provider is ordering for the patient and to ensure the dose administered is the intended dose. Table 1.1 below contains a list of commonly used abbreviations in prescriptions. Table 1.1 Abbreviations Commonly Used in Prescriptions Abbreviation Interpretation Abbreviation Interpretation Route Frequency IM Intramuscular ac Before meals IV Intravenous pc After meals IVPB Intravenous piggyback prn As needed SC Subcutaneous hs Bedtime SL Sublingual stat Immediately, at once PO By mouth qd Once a day PR Rectally qod Every other day OD Right eye bid Twice a day OS Left eye tid Three times a day

OU Both eyes qid Four times a day AD Right ear qh Every hour AS Left ear q4h Every 4 hours AU Both ears q6h Every 6 hours General General a before mEq Milliequivalent c with npo Nothing by mouth cap capsule qs A sufficient quantity et and Sig Write on label dc discontinue Tbs, tbsp Tablespoon disp dispense tsp teaspoon gtt drop Ud As directed IU International Units Rx Prescription Misc. Drugs and Conditions IV Products ASA Aspirin DW Distilled Water APAP Acetaminophen D5LR Dextrose 5% in Lactated Ringers Solution BP Blood Pressure D5NS Dextrose 5% in normal saline BS Blood Sugar D10W Dextrose 10% in water CA Cancer NS Normal Saline

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CHF Congestive Heart Failure HA Headache HC Hydrocortisone NTG Nitroglycerin SOB Shortness of breath Note: The abbreviation tsp is usually lowercase so as not to be confused with Tbs. Table 1.2 below outlines some common prescription sigs, or directions for use of the medicine, that are seen in the clinical setting. This will help illustrate how the abbreviations above are used together to make a complete prescription order that can be interpreted by the healthcare professional to make sure the patient receives the appropriate dose. Table 1.2 Common Prescription Sigs Directions as written by prescriber “Sig” Interpretation 1 tab po q 4-6h prn Take 1 tablet by mouth every 4-6 hours as needed 1 cap po bid Take 1 capsule by mouth twice daily 2 gtts OU HS Place 2 drops in both eyes at bedtime Prn sob Use as needed for shortness of breath Figures 1.1 and 1.2 provide examples of prescriptions. Pay attention to the information included in each prescription. Figure 1.1 identifies the different components of the order that must be present. These include the following: patient name, date of birth, name of the medication, strength of the medication, amount of medication, route and frequency, how much total medication should be dispensed (this is not necessary in a hospital setting),

refills given (also not necessary in a hospital setting), and prescriber signature and date. Figure 1.1 Example outpatient prescription for a liquid medication highlighting the necessary components of a prescription. The Sig indicates for the patient to take 2.5 ml by mouth every 4 hours. Figure 1.2 Example prescription ordering 30 tablets of Lipitor 20mg. The Sig indicates for the patient to take 1 tablet by mouth every day.

Calculations One very common type of pharmacy calculation is the conversion between different units of measure. The most common examples are listed below, along with either their conversion factor or the formula needed to convert between units. Basic Conversion Factors and Formulas 1. Pounds to Kilograms (Kg) 2.2 lb = 1 Kg It is recommended to round to the nearest tenth when converting between pounds to Kg. 2. Tablespoon and teaspoon to m 1 Tbsp = 15 ml 1 tsp = 5 ml 3. Celsius to Fahrenheit and vice versa To convert temperatures in degrees Celsius to Fahrenheit, multiply by 1.8 (or 9/5) and add 32.  Example: (30°C x 1.8) + 32 = 86°F To convert temperatures in degrees Fahrenheit to Celsius, subtract 32 and multiply by .5556 (or 5/9).  Example: (50°F - 32) x .5556 = 10°C Unit Conversions Units given X conversion factor = Units desired Example 1. How many Kg is 100 pounds? 2. How many tablespoons is 45 ml?

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3. If someone has a temperature of 101.4 °F, what is their temperature in Celsius? 10.2 Metric Conversion The metric system, also known as the Système Internationale (SI), is used around the world. It is built upon multiples of ten. This makes converting between units simpler without even needing a calculator. You can easily convert between metric units simply by moving the decimal point. The most common units that you will need to convert between when dealing with medications include milligrams, grams, and micrograms. Table 1.3 is a summary chart of the common metric or System International (SI) unit prefixes. Table 1.3 Common Metric Unit Prefixes Prefix Symbol Meaning Order of Magnitude giga- G 1000000000 10 9 mega- M 1000000 10 6 kilo- K 1000 10 3 hector- H 100 10 2 deka- Da 10 10 1 Base unit (i.e., meter, gram, liter) 1 10 0 deci- d 0.1 10 -1 centi- c 0.01 10 -2 milli- m 0.001 10 -3

micro- µ 0.000001 10 -6 nano- n 0.000000001 10 -9 How to Convert between Different Metric Units How many meters are in 5.5 Km? First, the starting order of magnitude of the unit must be determined. In this case, Kilo is 10 3 , meaning the decimal will need to be moved 3 places. In this case, the decimal will be moved 3 places to the right since this is a conversion from a larger unit of measure to a smaller one. The first example involved going from a larger unit (Km) to a smaller unit (meter). The process is the same just in reverse when starting with a smaller unit like centimeter and converting it to a larger unit like hectometer. Centi is 10 -2 , and hecto is 10 2 , meaning the decimal will need to be moved 4 places. How many hectometers are in 770 cm? It helps to put the decimal point in, even when starting with a whole number. The decimal will be moved 4 places to the left since the conversion involves moving from a smaller unit of measure to a larger one. 770.0 cm = 0.077 hm Medication Calculation Application for Metric Conversion Dopamine 5 mcg/kg per minute is ordered. How many mg will the nurse administer to a patient weighing 100 Kg? Solution: Step 1: Determine the dose based on the patient’s weight: Step 2: Convert the 500 mcg to mg.

Micro is 10 -6 , and milli is 10 -3 , meaning the decimal needs to be moved 3 places to the left. 500.0 mcg = 0.5 mg of dopamine 10.3 Determining the Amount of Medication to Administer Pharmacy calculations are more than just basic unit conversions. The amount of medication being administered may need to be calculated because it is not always straightforward depending on the amount ordered and the strength or formulation that is on hand. The Basic Formula below will help to set up the equation and correctly solve for the amount to administer. Basic Formulas to Us to Calculate How Much Medication to Administer Basic Formula: Example The dose ordered by the physician is 7.5 mg. The medication is only available in 5-mg tablets, which are scored. How many tablets should a patient take? Desired dose = 7.5 mg Vehicle = 1 tablet On hand = 5 mg Ratio Method This equation can also be set up as a simple ratio and then cross multiplied to determine the unknown. When using this method, the 7.5 would

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be multiplied by 1 and then divided by 5. The same answer of 1.5 tablets is calculated. Dimensional Analysis Dimensional analysis is a process that considers the units to figure out how to solve the problem. In general, when setting up and solving medication dose calculations, you want to start with what you know and solve for what you need. The equation can usually be set up using factor labeling, which is also known as dimensional analysis. Essentially, you will start with what you have and use the information given to you to solve for the unknown. GIVEN CONVERSION FACTOR UNKNOWN VALUE Unit A Unit B = _____________ Unit B Unit A To calculate the unknown value, the given can be multiplied by the conversion factor, as shown below: Example Convert 25 pounds into Kilograms. GIVEN CONVERSION FACTOR UNKNOWN VALUE 25 pounds 1 Kg = 11.4 Kg 2.2 pounds

To calculate the unknown value, the given is multiplied by the conversion factor, as shown below: The above example involves the most basic dimensional analysis calculation, in that it was a single unit conversion. In fact, this was what was shown in the beginning of the unit when basic unit conversions were presented. However, the dimensional analysis method can be used to calculate dosages using a series of ratios and factors, organized in the form of fractions that are then multiplied. To start the problem, identify what is given, what needs to be determined, or the unknown value, and find the correct conversion factor between the two units. In general, the unit of measure being calculated (this is what you want to end up with) needs to be known, the amount of medication ordered, and the formulation or strength available to you. Arrange the equation so that the unit of measurement in the denominator of one factor is placed in the numerator of the following factor and so on. Unwanted units are then canceled. Below is an example of how this can work: You are given mg/min to start with and you need to convert to liters per hour. The drug is available in mg/ml. The equation started with mg/min, but the desired units were liters per hour. Converting the mg to ml was straightforward because the drug was available in mg/ml. The resulting unit was ml/min. Units of time and measure are easy to convert by setting up the ratio using how many minutes are in an hour and how many milliliters are in a liter. This can be done with any unit that is a constant. Other examples might be pounds to Kg using a drop factor (gtt/ml). The equation needs to simply be set up so the unwanted units can be crossed out. The units being crossed out must be diagonal from each other (one above the fraction line and one below). These can be completed one calculation at a time (stepwise) or as one complete calculation. Both options will be presented below. It is recommended to start with the stepwise approach until you become more comfortable with the process. Once you are comfortable with the process, it may be easier to move on to the complete calculation approach (see below). Example

Medication is ordered 10 mcg/kg/min. The medication is available in 25 mg/50 ml of D5W. The patient weighs 125 pounds. Calculate the dose in ml/hour. Stepwise : Step 1. Convert pounds to Kg. Step 2. Determine the dose in mcg/min. Step 3. Metric Conversion: In order to determine how many ml are needed, the conversion of mcg to mg needs to be performed first because the medication is available in mg/ml. Step 4. Determine how many ml of the drug are needed to equal that dose. Step 5. Convert minutes to hours. Complete Calculation : The same calculation can be done in one long dimensional analysis equation that involves setting up a series of factor conversions. It is the same process as the stepwise process outlined above, and the answer is the same. Close attention must be paid to the units and work to

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cancel out in order to arrive at the desired units. You start with the weight of the patient and then use the amount of medication ordered and the strength the medication is available in, as well as a metric conversion, and the constant of how many minutes are in an hour to reach the desired units of ml/hr. This next formula is helpful when determining the rate at which an IV should be set. IV Flow Rate Formula Example You have 50 mg of a medication in 50 ml of D5W that needs to run over 30 minutes. The tubing has a drop factor of 15. How many drops per minute does the IV need to be set at? Step 1. Step 2.

10.4 Terminology Related to Parenteral Dosing Parenteral dosage calculations often involve something called a drop factor. Drop factor refers to the number of drops it takes to make up 1 ml of fluid. This is dependent on the size of the tubing being used. The larger the tube, the smaller the drop factor. Some other terms to be familiar with include D5W and Lactated Ringers. D5W is an i ntravenous sugar solution, also known as dextrose solution, that is a mixture of dextrose and water. It is used to treat low blood sugar or water loss without electrolyte loss. Lactated Ringers is a mixture of sodium chloride, sodium lactate, potassium chloride, and calcium chloride in water. It is used for replacing fluids and electrolytes in those who have low blood volume or low blood pressure. This mixture is often referred to as “fluids”. Calculating Drip Rate You have 2,000 ml of NS to be infused over 12 hours. The drop factor for the tubing is 20. Calculate the drip rate in drops per minute. Stepwise: Step 1 . Take the known value of 2000 ml over 12 hours and the drop factor (gtts/min). This step converts from ml/hour to drops/hour. Step 2 . You now have drops per hour but the question asked for drops per minute. You need to convert the hours to minutes. This is easy enough to do by using the constant ratio of 60 minutes in 1 hour.

Complete Calculation: Once you become comfortable with the stepwise process, it may be easier to just set up one complete calculation. This allows all unit conversions to be calculated in a single equation. This also avoids rounding after each individual calculation. As can be seen, the same answer is arrived at as that calculated using the stepwise approach above. Setting a Pump You have 1000 ml of D5W that you need to infuse over 8 hours. How many ml/hour would you set the controller for? In order to solve this calculation, the total volume in ml needs to simply be divided by the total infusion time of 8 hours. This will provide the infusion rate in ml/hr. Weight-Based Dosing Weight-based dosing can be used for oral or parenteral dosing. Typically, the dose is given in terms of Kilograms, such as 10 mg/kg. This makes it necessary to convert a patient’s weight from pounds to Kilograms. Additionally, in some cases, a range that the dose may fall in, such as 80-90 mg/kg, may be given. Depending on the formulation available (what strength the medication comes in), the exact dose may need to be selected when making the calculation. The following should be ensured: 1. The dose falls within the recommended range. 2. The dose is a practical dose for the patient or nurse to measure. This is especially true with oral medication that will be measured using either a medicine cup or possibly an oral syringe. A rounded number should be chosen. For example, if the dose comes out to 2.436 ml, it is probably best to round that up to 2.5 ml, as long as 2.5 ml still falls within the recommended dosing range. If it does not, then the value may need to be rounded down. The dosage form and the measuring device must both be taken into account. Oral liquid medications can be measured in measuring cups, dosing spoons, or oral syringes, as seen in Figure 1.3 below. The accuracy of

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measuring the dose depends on the marking on the measuring device. For example, the dosing cup would be hard to be accurate beyond 0.5- ml doses. A dose of 7.5 ml would be possible to estimate halfway between the 5-ml mark and the 10-ml mark. However, a 3.75-ml dose would be hard to measure accurately with a cup. An oral syringe may be a better option for that type of dose. Figure 1.3 shows a variety of liquid medication dosing devices. Figure 1.3 A variety of liquid medication dosing devices. As can be seen in the image below, depending on the size of the oral syringe and the markings, it may be possible to accurately measure different doses. Figure 1.4 depicts three different sizes of oral dosing syringes. Depending on the dose being ordered, one must be able to select the appropriate syringe to accurately measure the correct dose. Figure 1.4 Three different sizes of oral dosing syringes.

Example The order is written for Amoxicillin 25-30 mg/kg Divided q8 hours. The patient is 50 pounds. Amoxicillin suspension is available in 200 mg/5 ml. Stepwise: Step 1 . Convert the pounds to Kg. Step 2. Determine the dose in mg. Step 3. Choose the dose that is a multiple of 200 mg and then determine the total daily dose in ml. Step 4. Divide the dose by 3. Weight-Based Pediatric Dosing Examples Example 1: The doctor orders Acetaminophen 500 mg q6 hours for your pediatric patient. Safe dose range is 10-16 mg/kg/dose. The patient weighs 25 kg. Calculate the safe dose range. Is this dose safe?

The safe dose range is between a 250-mg and 400-mg dose. A dose of 500 mg is too high for this patient’s weight. Example 2: Ceftriaxone 1100-mg IV once daily for 7 days is ordered for a pediatric patient. The recommended dose is 50 mg/kg/day. The child weighs 22 kg. Is this dose safe? This is a safe dose for the weight of the patient. Final Tips Regarding Medication Safety and Dosing Calculations Always remember that the key to medication calculations is to pay close attention to details. A decimal place off just by one place can be the difference between life or death. Additionally, the units being used must be accurate. There is a big difference between mcg and mg. Be sure you are working with the correct units. Double-check your calculations. Remember, correct dosing calculations are vital when caring for patients, and it is a critical piece of medication safety.

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Module 10 - Dosing Calculations

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