2.4 Dosage Calculations

Apothecary System

One of the older systems of measurement is the apothecary system, and it has been used by apothecaries or pharmacists for the last couple of centuries. This system is very difficult to use, and the Joint Commission, the FDA, and the Institute of Safe Medication Practice (ISMP) have recommended that it be discontinued. It is now seldom used, and the metric system was adopted. The nurse will rarely see an apothecary measurement, but it is important to understand how to correctly adapt an order using the apothecary system to the metric system. This can be accomplished through collaboration with the pharmacist or by using a table of weights and measures. The units of measure in the apothecary system are grains, drams, scruples, and the minim; the values are sometimes expressed as Roman numerals (e.g., X, IV) instead of Arabic numbers (e.g., 1, 2, 3, 4). The nurse should be aware of this; however, it won’t be a focus of this chapter.

Household System

The household system (also known as the customary system) includes units of measure such as tablespoons, teaspoons, pounds, and ounces. Unfortunately, there has been much confusion in using this system, which leads to dosing errors. Thus, it is no longer used. The ISMP has issued recommendations to avoid household measurements. For conversion equivalents, see Appendix C: Drug Conversion Tables.

Metric System

Three types of metric measures are commonly used: length, volume, and weight. This section will cover only volume and weight. It is important to understand that the units can be smaller or larger in relation to their metric measure. The basic units are multiplied or divided by multiples of 10, which increases the ease of use when converting from one unit of measure to another.

Volume measures a liquid and is commonly used in dosage calculation when measuring liquid drugs, in the reconstitution of drugs, or with intravenous therapy. The metric system defines volume in units of liters. Weight measures mass and is commonly used in dosage calculation when a particular drug is administered based on the client’s weight. The metric system defines weight in units of grams. A few examples are listed in Table 2.10, followed by their abbreviations and their equivalent number of grams. (See also Appendix A: International System of Units.)

Kilo indicates a larger unit of measurement and equals 1000. Milli indicates a smaller unit of measurement and equals 1/1000. Micro is also a smaller measurement and equals 1/1,000,000. Conversion between metric system units routinely occurs when you are preparing to administer drugs to clients.

Note that the abbreviation for “micro” is sometimes shown as the Greek mu (µ, as in µg or µL); however, that practice is considered to be error prone by the Institute for Safe Medication Practices and has been supplanted by “mc,” as in “mcg” for microgram or “mcL” for microliter.

Basic Formula Method

The basic formula method, also known as Desire Over Have, is the simplest of the drug calculation methods.

$\frac{D}{H}\times \mathit{Q}=\mathit{X}$

$\begin{array}{rcl}\mathit{D} & =& \mathrm{Desired\; dose\; (the\; dose\; ordered)}\\ \mathit{H} & =& \mathrm{Amount} \mathrm{on} \mathrm{hand} \mathrm{or} \mathrm{available}\\ \mathit{Q} & =& \mathrm{Quantity\; or\; volume\; of\; the\; drug\; form\; (tablet,\; capsule,\; liquid)}\\ \mathit{X} & =& \mathrm{Amount\; calculated\; to\; be\; administered\; to\; the\; client}\end{array}$

To use this method:

1. Write the desired dose or the dose ordered.
2. Write the drug dose on hand (available).
3. Divide the desired dose by the dose on hand and multiply by the quantity or volume of the drug form to get the amount calculated to be administered to the client.

Note: It is extremely important that the units of measure are the same when doing these calculations. If the desired dose is in mg, the on-hand amount must also be in mg. The amount to be administered will be in the same units as the quantity or volume of the drug form.

Example:

Desired dose: Lisinopril 10 mg orally BID (twice a day)
On hand: Lisinopril 2.5 mg per 1 tablet
How many tablets should the client receive per dose?

$\begin{array}{rcl}\mathit{D} & =& \mathrm{Desired} \mathrm{dose} \left(\mathrm{order}\right) = 10 \mathrm{mg}\\ \mathit{H} & =& \mathrm{Drug} \mathrm{on} \mathrm{hand} (\mathrm{what} \mathrm{is} \mathrm{available}) = 2.5 \mathrm{mg} \mathrm{per} 1 \mathrm{tablet} \\ \mathit{Q} & =& 1 \mathrm{tablet} \end{array}$

$\frac{10 \mathrm{mg}}{2.5 \mathrm{mg}}\times 1 \mathrm{tablet}=4 \mathrm{tablets}$

$$\begin{gathered} 10 \mathrm{mg} \mathrm{divided} \mathrm{by} 2.5 \mathrm{mg} = 4 \\ 4 \times 1 \mathrm{tablet} = 4 \mathrm{tablets} \end{gathered}$$

The amount to be administered to the client is 4 tablets.

Ratio and Proportion Method

The ratio and proportion method uses a linear equation to solve the dosage calculation problem.

$\begin{array}{rcl}\mathit{K} & =& \mathrm{Known\; dose\; (available)}\\ \mathit{M} & =& \mathrm{Known\; unit\; of\; measure}\\ \mathit{D} & =& \mathrm{Desired/ordered\; dose}\\ \mathit{X} & =& \mathrm{Desired\; amount\; to\; be\; administered}\end{array}$ $\mathit{K} \mathrm{(known\; dose)} :\mathit{M} \mathrm{(known\; unit\; of\; measure)}= \mathit{D} \mathrm{(desired\; or\; ordered\; dose)} : \mathit{X} (\mathrm{desired} \mathrm{amount})$

To use this method:

1. Write the known dose (dosage strength from drug label). (See K in the above formula.)
2. Write the known unit of measurement (also found on the drug label). (See M in the above formula.)
3. Write the desired ordered dose (usually found in the physician’s orders). (See D in the above formula.)
4. Write X in the above formula as the placeholder for the desired amount to be administered. You will solve for X.
5. Check the units of measurement to make sure they are the same (e.g., mg : mg or tablets : tablets). Note that if they are not the same, you must convert them to the same unit of measurement prior to solving for X.
6. Solve for X (the desired amount to be administered to the client) by multiplying the means (M and D in the middle of the formula above) and the extremes (K and X on the outside of the formula). (Work only with the numbers and not the units of measurement.) Clear the X by dividing both sides of the equation by the number in front of the X, which solves X.

Another method of setting up a ratio and proportion equation is as follows:

$\frac{K (\mathrm{Known} \mathrm{dose})}{M (\mathrm{Known} \mathrm{unit} \mathrm{of} \mathrm{measure})}=\frac{D (\mathrm{Desired/ordered} \mathrm{dose})}{X (\mathrm{Desired} \mathrm{amount} \mathrm{to} \mathrm{be} \mathrm{administered})}$

To solve this type of equation, cross multiply: $\left(KX\right)=\left(MD\right)$. (Work only with the numbers and not the units of measurement.). Clear the X by dividing both sides of the equation by the number in front of the X, which solves for X.

Example:

Ordered: Lisinopril 10 mg orally BID
Available: Lisinopril 2.5 mg
How many tablets should the client receive per dose?

Figure 2.17 These formulas show two different ways to solve the example problem using the ratio and proportion method. (attribution: Copyright Rice University, OpenStax, under CC BY 4.0 license)

Dimensional Analysis Method

Dimensional analysis is one of several methods for determining the correct dose of medication. Dimensional analysis uses a series of equivalent measurements to change one unit of measurement to another to solve a problem. This method uses equivalent measurements that are set up as a series of “fractions,” called conversion factors (or dimensions), that are used to cancel unnecessary units of measurements, leaving only the desired answer. There are times when there will be a number on top but not on the bottom. (These are not true fractions.)

To use this method:

1. Determine what you want to find (or solve for). Consider this a road map to your destination—where you want to go. Place the units of measure or quantity you want to find on the right side of the equation following the equal sign.
2. Typically, you will then start on the far left with the provider’s medication order. Once you know what you wish to find (the provider’s order), place that number and unit at the top left portion of the equation.
3. Use the information provided to slowly move forward, canceling units of measure when possible and circling units of measure you are looking for (using the road map as a guide).

In the first example below:

1. X number of tablets are placed on the right side of the equation following the equal sign because you are solving for how many tablets should be given.
2. 10 mg is placed in the upper left side in the numerator position.
3. Because you are solving for the correct number of tablets, tablet must be on the top left of the equation and not canceled out. And because mg is not on the right side of the equation, mg should be canceled out. That helps determine the placement of the next part of the equation.
4. One tablet must be on top so that the mg in 2.5 mg can be canceled out.
   $OA \times DS = DA$
   $\mathit{OA} = \mathrm{Ordered} \mathrm{amount} (\mathrm{or} \mathrm{desired} \mathrm{dose})$
   $\mathit{DS} = \mathrm{Dosage} \mathrm{strength}$
   $\mathit{DA} = \mathrm{Desired} \mathrm{amount}$

Example 1:
Ordered: Lisinopril 10 mg PO BID
Available: Lisinopril 2.5 mg

How many tablets should the client receive per dose?

$\begin{array}{rcl}OA \times DS & =& DA \end{array}$

1. First, determine what goes on the right side of the equation. What units of measure are you solving for? In this example, it is tablets.
2. Then, identify the information on the top left—in this example, 10 mg have been ordered. Note that there is no number or letter under the 10 mg; leave it blank.
3. Cross out the units of measure—mg (numerator) and mg (denominator).
4. Now solve the math: $10 \mathrm{mg} \times 1 = 10$. Divide by 2.5 and solve for X (see Figure 2.18).

Figure 2.18 This equation demonstrates dimensional analysis for solving the number of tablets that the client should be given. (attribution: Copyright Rice University, OpenStax, under CC BY 4.0 license)

Example 2:
Ordered: Give digoxin 0.5 mg IV push × 1 dose now
Available: Digoxin 0.25 mg/1 mL

How many mL should be given?

1. First, determine what goes on the right side of the equation. What units of measure are you solving for? In this example, it is mL.
2. The next step is to identify the information on the top left of the equation, which is the ordered amount of 0.5 mg. Note that there is nothing under 0.5 mg. Leave it blank.
3. Cross out the units of measure—mg (numerator) and mg (denominator).
4. Now solve the math: $0.5 \mathrm{mg} \times 1 = 0.5$. Divide by 0.25.
5. The answer is 2 mL (see Figure 2.19).

Figure 2.19 This example of dimensional analysis is for an IV push medication. (attribution: Copyright Rice University, OpenStax, under CC BY 4.0 license)

Dimensional analysis is particularly helpful for more complicated equations, such as the following one. Note how the information can unfold as you add units of measure and cancel out as needed.

Example 3:
Ordered: Initiate dopamine IV infusion at 5 mcg/kg/minute
Available: Dopamine hydrochloride 800 mg in 500 mL normal saline
Client’s weight: 176 lb

How many mL/hour should the medication infuse?

1. First, determine what goes on the right side of the equation. What units of measure are you solving for? In this example, it is mL/hour.
2. Now identify the information on the top left of the equation. In this example, it is 5 mcg. The order reads 5 mcg/kg/min. Therefore, 5 mcg goes on the top left, and kg/min is on the bottom left.
3. To solve the equation, the client’s weight of 176 lb will also need to be converted within the equation. The following shows how the equation for solving for this infusion rate should be laid out. (It does not have to be laid out in this specific order, but all units of measurement should be included; for example, 1 kg/2.2 lb could switch places with 500 mL/800 mg).
   $\frac{5 \mathrm{mcg}}{\mathrm{kg/min}}\times \frac{1 \mathrm{kg}}{2.2 \mathrm{lb}}\times \frac{176 \mathrm{lb}}{}\times \frac{60 \min }{1 \mathrm{hour}} \times \frac{1 \mathrm{mg}}{1000 \mathrm{mcg}} \times \frac{500 \mathrm{mL}}{800 \mathrm{mg}}=\mathit{X} \mathrm{mL/hour}$
4. Because you are solving for mL/hour, circle mL on top and hour on the bottom, ensuring that those units of measure will not be crossed out.
5. To solve the problem, you need to cross out duplicate units of measure. For this equation you would cross out (a) mcg, (b) kg, (c) min, (d) lb, and (e) mg.
6. This leaves the desired measurement as mL/hour (see Figure 2.20).

Figure 2.20 This shows the equation with the units of measure crossed out and the desired units of measure circled. (attribution: Copyright Rice University, OpenStax, under CC BY 4.0 license)

7. Then multiply the numerators (top numbers) and multiply the denominators (bottom numbers) (see Figure 2.21).

Figure 2.21 This stage of dimensional analysis shows the multiplication of numerators and denominators. (attribution: Copyright Rice University, OpenStax, under CC BY 4.0 license)

8. You are now ready to solve for X by dividing the numerator of 26,400,000 by the denominator of 1,760,000. $\begin{array}{rcl}\frac{26,400,000}{1,760,000}& = & X \mathrm{mL/hour}\\ X& = & 15 \mathrm{mL/hour} \end{array}$

Body Weight Method

The body weight method allows dosage calculation for individualization. This method is commonly seen with medications with a narrow therapeutic index, drugs being adjusted due to the individual’s ability to metabolize the drug, and pediatric clients.

$\mathrm{Drug} \mathrm{dose} \times \mathrm{Body} \mathrm{weight} = \mathrm{Client} \mathrm{dose}$

To use this method:

1. Convert the client’s weight from pounds to kilograms $\left(2.2 \mathrm{lb}=1 \mathrm{kg}\right)$.
2. Determine the drug dose for body weight by multiplying them together.
3. Use your preferred dosage calculation method (formula, ratio proportion, or dimensional analysis) to solve for the dose of the drug to be administered to the client.

Example:
Ordered: Amoxicillin 10 mg per kg orally QID (four times a day)
Available: Amoxicillin 125 mg/5 mL
Client’s weight: 110 lb

How many mL should the client receive per dose?

$\begin{array}{rcl}\frac{110 \mathrm{lb}}{2.2 \mathrm{kg}}& = & 50 \mathrm{kg}\\ 10 \mathrm{mg} (\mathrm{drug} \mathrm{dose}) \times 50 \mathrm{kg} & =& 500 \mathrm{mg} \mathrm{per} \mathrm{dose} \end{array}$

Use the label shown in Figure 2.22 to calculate the number of mL per dose.

Figure 2.22 Use this label to calculate the desired dose using dimensional analysis. (credit: “AMOXICILLIN powder, for suspension, for suspension”by National Library of Medicine/DAILYMED, Public Domain. Drug Company Logo All Rights Reserved.)

Using dimensional analysis, it would look like this:

$\frac{500 \mathrm{mg}}{1}\times \frac{5 \mathrm{mL}}{125 \mathrm{mg}}=20 \mathrm{mL}$

Body Surface Area Method

The body surface area (BSA) method allows dosage calculation for individualization. This method is commonly seen with medications that have a narrow therapeutic index, drugs that are being adjusted due to the individual’s ability to metabolize the drug, chemotherapeutic agents, pediatric clients, and clients with burns. It is an approximation of the total skin area of an individual measured in meters squared (m²). BSA is calculated using the square root calculation as determined by the client’s height and weight.

To use this method:

1. Convert pounds to kilograms.
2. Calculate BSA. The formula is: $\sqrt{\frac{[\mathrm{height} \left(\mathrm{cm}\right) \times \mathrm{weight} \left(\mathrm{kg}\right)]}{3600}}$
3. Calculate the dose in mg.
4. Calculate the dose in mL.

Example:

Calculate the dose of ceftriaxone in mL for a client who weighs 37 lb and is 97 cm tall. The dosing required is 2 mg/m², and the drug comes in 1 mg/mL.

1. $37 \mathrm{lb} ÷ 2.2 \mathrm{kg} = 16.8 \mathrm{kg}$
2. $\sqrt{16.8 \mathrm{kg}\times 97 \mathrm{cm} ÷ 3600} = 0.67 {\mathrm{m}}^2$
3. $2 \mathrm{mg}/{\mathrm{m}}^2 \times 0.67 {\mathrm{m}}^2 = 1.34 \mathrm{mg}$
4. $1.34 \mathrm{mg} ÷ 1 \mathrm{mg}/\mathrm{mL} = 1.34 \mathrm{mL}$

The dose to be administered is 1.34 mL.

To calculate the dose using dimensional analysis, some of the same steps are used.

1. $37 \mathrm{lb} ÷ 2.2 \mathrm{kg} = 16.8 \mathrm{kg}$
2. $\sqrt{16.8 \mathrm{kg}\times 97 \mathrm{cm} ÷ 3600} = 0.67 {\mathrm{m}}^2$
3. $0.67{\mathrm{m}}^2 \times 2 \mathrm{mg}/{\mathrm{m}}^2 = 1.34 \mathrm{mL}$

Remember that body surface area is measured in meters squared (m²). Begin the dosage calculation by calculating the body surface area of the client, then multiply the size of the client by the ordered dose to obtain the dose needed for the client.
$\mathrm{Client} \mathrm{size} \times \mathrm{Order} = \mathrm{Dose}$

Rounding Rules

Nurses must know when and how to round medication doses. Caplet and tablet doses can be rounded to the nearest half-tablet if they are scored. Some are scored in fourths and can be rounded accordingly. Caplets and tablets that are controlled release, extended release, sustained release, or enteric-coated should not be split or crushed. Capsules are rounded to the nearest whole number because they cannot be divided. Liquid drugs are rounded to the nearest tenth. If the nurse is calculating a liquid medication for drops, then it is to be rounded to the nearest whole number because it is not possible to administer a partial drop. These rounding rules guidelines assist the nurse with rounding medication doses. The National Library of Medicine provides another resource for rounding as well as performing calculations.

Next, rounding is determined by how many places from the decimal point are appropriate. If a number to the right of the decimal point needs to be rounded and it is 4 or less, then it is rounded down. For example, if the nurse calculated that they were to give 1.243 and they were rounding to the nearest tenth, the number would be rounded to 1.2. If a number to the right of the decimal point is 5 or greater, then it should be rounded up. For example, if the nurse calculated that they were to give 1.251 and they were rounding to the nearest tenth, the number would be rounded to 1.3.