Formulas and calculations | Order entry and processing

Conversions

Knowing the conversions is important when using formulas and calculations while compounding in pharmacy. Following are the important conversions to know

$1$ kilogram = $1, 000$ grams

$1$ kilogram = $2.2$ pounds

$1$ liter = $1, 000$ milliliters

$1$ gram = $1, 000$ milligrams

$1$ milligram = $1, 000$ micrograms

$1$ grain = $64.8$ milligrams

$1$ ounce = $28.35$ grams

$1$ ounce = $30 ml$

$1$ teaspoon = $5 ml$

$1$ tablespoon = $15$ milliliters

$1$ pound = $454$ grams

Roman numerals

Following are the Roman numerals and their corresponding numbers.

I $1$ - One

V $5$ - Five

X $10$ - Ten

L $50$ - Fifty

C $100$ - One hundred

D $500$ - Five hundred

M $1, 000$ - One thousand

SS $\frac{1}{2}$ - Half

Calculations

Concentration

Concentration is the amount of active ingredient per total substance weight. Liquids are expressed as weight/volume ( $w / v$ ), with the weight being the amount of drug and the volume representing a specific volume of drug and vehicle. For example, if the concentration of a suspension is $10 mg /5 ml$ , that means there is $10$ mg of the active drug in $5 ml$ of the suspension. Solid topical medication concentrations are usually expressed as weight/weight ( $w / w$ ), with the numerator representing the weight (mass) of the drug present in the denominator, which is the total weight of the drug plus vehicle. For example, nystatin cream is available in a concentration of $100, 000 units / g$ . This means that each gram of cream has $100, 000$ units of nystatin. If the concentration of a liquid medication is stated as ( $v / v$ ), e.g., $10 ml /100 ml$ , then it has $10 ml$ of the active drug in $100 ml$ of the total liquid.

Specific gravity refers to the ratio of the weight of a substance to the weight of an equal volume of water at the same time. If the specific gravity is known, the volume or weight of the desired quantity can be determined.

Specific gravity $= \frac{number of grams of a substance}{number of milliliters of a substance}$

Number of tablets $= \frac{desired dose}{stock strength}$

The above formula can be used when the prescription strength is not in stock, e.g., if the prescription states to dispense $1000 mg$ of drug A, but you only have $250 mg$ available.

Number of tablets of drug A $= \frac{1000 mg}{250 mg} = 4$ tablets.

Hence, you can dispense four tablets of $250 mg$ each to replace one tablet of $1000 mg$ .

Amount of solution to be given $= (\frac{desired dose}{stock strength}) \times stock volume$

The above formula can be used when calculating the volume of a liquid medication to be dispensed, assuming the strength or concentration of the stock solution is known. For example, if the stock solution has a strength of $100 mg /10 ml$ and you need to dispense $500 mg$ , how much volume of the stock solution will you dispense?

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Using the formula above, the amount of solution to be given $= (\frac{500 mg}{100 mg}) \times 10 ml = 5 \times 10 = 50 ml$ of stock solution needs to be dispensed.

Ratios

Ratios are a way to express the strength of a solution or liquid preparation. In ratio strength, the first number is a $1$ , and a colon follows it and then another number, e.g., $1 : 100$ . The units are always grams or milliliters, depending upon whether you are dealing with a $w / w$ or $w / v$ preparation. Thus, a $1 : 100$ ratio strength means a solution with $1$ g in $100$ ml or a solid preparation, say $1$ g of drug in $100$ g of ointment. For example, a $1 : 1, 000$ ratio for epinephrine represents $1$ gram of epinephrine in $1, 000$ ml of solution, so the amount per unit of volume is $1 mg / ml$ . A $1 : 10, 000$ ratio for epinephrine represents $1$ gram of epinephrine in $10, 000$ ml of solution, so the amount per unit volume is $0.1 mg / ml$ .

Percent strength: Percent strength represents the number of grams contained in $100 mL$ of product.

Percent weight in volume (w/v): The number of grams in $100 mL$ of solution is expressed as $% w / v$ . Powdered substances suspended in a liquid vehicle would be calculated as $% w / v$ . For example, $1%$ $% w / v$ solution will have 1 gram of the powder in $100 mL$ of the solution.

Practice problem: A prescription calls for $150 mL$ of $2%$ $% w / v$ gentamicin solution. How much gentamicin powder is needed to make the desired solution? Let’s assume the unknown quantity is “G”.

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$\frac{2}{100} = \frac{G}{150}$

$G = \frac{2 \times 150}{100} = \frac{300}{100} = 3$

Hence, 3 grams of gentamicin powder are needed to make $150 mL$ of a $2%$ $% w / v$ solution. Dissolve 3 grams in $150 mL$ to get the desired strength.

Percent volume in volume (v/v): The number of milliliters in $100 mL$ of solution and is expressed as $% v / v$ . A liquid component in a liquid preparation would be calculated on a $% v / v$ basis.

Practice problem: How much alcohol is in $100 mL$ of a $10%$ $% v / v$ solution?

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$10%$ solution of alcohol will have $10 mL$ alcohol in $100 mL$ of the solution. Hence, the answer is $10 mL$ .

Percent weight in weight (w/w): The number of grams in $100$ grams of total dosage form and is expressed as $% w / w$ . Powdered substances mixed with a solid or semisolid would be calculated as $% w / w$ , e.g., ointments.

Practice problem: How much betamethasone is in $150 mg$ of a $0.1%$ $% w / w$ betamethasone cream? Assuming the unknown quantity of betamethasone is “B”,

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$\frac{1}{1000} = \frac{B}{150}$

$B = \frac{1 \times 150}{1000} = 0.15$ gram

The steps are the same as calculating $% w / v$ , but in this example, strength is $0.1%$ $% w / w$ ; converting to a fraction will become $\frac{1}{1000}$ .

If a solution requires a dilution, the active drug in the solution will remain constant, but the volume will increase. When two solutions have equal osmotic pressure and salt concentration, they are said to be isotonic. Normal saline has a concentration of $0.90%$ $% w / v$ of NaCl in sterile water and, therefore, is an isotonic crystalloid.

Calculating drug dosage based on body surface area: In some conditions, drug dosage is calculated according to body surface area (BSA).

$BSA (m^{2}) = \frac{1}{6} (W H)^{0.5}$ , where $W$ is body weight in kg, $H$ is body height in meters

Pediatric dosing: Clark’s and Young’s rules are used to calculate drug dosages in the pediatric age group ( from birth to about 18 years of age).

According to Clark’s rule, the pediatric dose is obtained by dividing the patient’s weight in pounds by the average standard weight of $150$ pounds multiplied by the adult dose of a drug.

$Pediatric dose = [\frac{weight of child (lbs)}{150}] \times adult dose$

Young’s rule uses the patient’s age to calculate the pediatric dosage.

$Pediatric dose = adult dose \times [\frac{age}{age + 12}]$

Fried’s rule also uses age to calculate the pediatric dosage.

$Pediatric dose = (\frac{age in months}{150}) \times adult dose$

Pediatric doses are often stated as per body weight, e.g., $X mg / kg / day$ .

Practice problem: If the prescription calls for Amoxicillin $30 mg / kg / day$ for ten days, then how much amoxicillin will you dispense? Assume body weight is 22 pounds.

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First, convert pounds to kg. $1 kg = 2.2$ pounds.

$\frac{22}{2.2} = 10 kg$

Applying it to dosing, $30 \times 10 = 300 mg / day$ .

Total dosage $= 300 \times 10 = 3000$ mg

Hence, you will have to dispense a total of $3000$ mg.

Typically, this is further mixed to form a solution of a particular concentration, e.g., $400 mg /5 mL$ . To get the volume to dispense, you must use the formula

$Amount of Solution to be given = (\frac{Desired Dose}{Stock Strength}) \times Stock Volume$ .

Estimating intravenous (IV) flow rates: Intravenous infusions administer fluids directly into the veins. The rate of infusion can be calculated in milliliters or drops.

Drops per minute are abbreviated as gtts/min. The drop factor is the number of drops in $1 mL$ . The width of the tube used for administering the infusion determines the size of the drop. Macrodrip tubing administers a larger drop and may be used for $10 gtts / mL$ , $15 gtts / mL$ , or $20 gtts / mL$ . Macrodrips are used when rapidly infusing large amounts of fluids. Microdrip tubing administers $60 gtts / mL$ . Microdrips are typically used in children. All IV packages clearly label the gtts/mL for that set.

The following formulae can be used to set the rate of the IV infusion:

Total IV volume/time (hour or minute) $= mL per hour or minute$

$[\frac{Total IV volume}{time (minute)}] \times drop factor = drops per minute$

Practice problem: The physician has ordered D5W $1200$ milliliters in $12$ hours using a $15$ -drop per milliliter infusion rate. Calculate the number of drops per minute.

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We have to convert $12$ hours into minutes as the infusion rate is in drops per minute.

$12 hours = 12 \times 60 = 720 minutes$

Using the formula:
$Drops per minute = (\frac{1200 mL}{720 min}) \times 15 = 1.6667 \times 15 = 25$

International units (IU): International units are used to denote doses for hormones like insulin, vaccines, vitamins, blood products, etc. They measure the drug’s effect or biological activity, not its weight or mass.

Milliequivalent (mEq): The milliequivalent (mEq) is the unit of measure often used for electrolytes. It indicates the chemical activity, or combining power, of an element relative to the activity of $1 mg$ of hydrogen.

$m Eq / l = \frac{( mg / l ) \times valence}{molecular weight}$

Alligation: The method of mixing two liquids or solids of different concentrations to produce a mixture of the desired concentrations. The end product has a different concentration than the original. We need to mix a specific proportion of each ingredient to get the desired concentration, which is given by the alligation ratio.

$Alligation ratio = \frac{H}{L}$

Considering “m” as the desired concentration, ‘l’ is the lower concentration, and “h” is the higher concentration.

$H = m - l$

$L = h - m$

Once the alligation ratio is known, e.g., $2 : 3$ , the proportion is two parts of higher concentration to three parts of lower concentration. This is followed by calculating the alligation volume, which is simply converting the parts into volume. For example, if we are compounding $500 ml$ of the final solution, then a ratio of $2 : 3$ corresponds to $200 ml$ of solution of higher concentration and $300 ml$ of solution of lower concentration.

For example, an order comes in for $10$ grams of $2%$ hydrocortisone cream. You only have $1%$ and $3%$ hydrocortisone available. Using the alligation method,

$m = 2$

$h = 3$

$l = 1$

$H = 2 - 1 = 1$

$L = 3 - 2 = 1$

Hence, alligation ratio = $\frac{1}{1} = 1$ .

You need to mix equal parts of $1%$ and $3%$ to get $2%$ concentration. As we need $10$ grams, mix $5$ grams of $1%$ hydrocortisone with $5$ grams of $3%$ hydrocortisone to get $10$ grams of $2%$ hydrocortisone.

In another example, suppose you get an order for $100 ml$ of a solution with a $50 mg/ml$ concentration. You have $30 mg/ml$ and $100 mg/ml$ concentrations available. How will you use the alligation method to calculate the volume of each concentration required?

$m = 50$

$h = 100$

$l = 30$

$H = 50 - 30 = 20$

$L = 100 - 50 = 50$

Alligation ratio = $\frac{H}{L} = \frac{20}{50} = \frac{2}{5}$

We need two parts of higher concentration $100 mg/ml$ and five parts of lower concentration $30 mg/ml$ . There are many ways to get the volume. You can use the following equation -

$2 x + 5 x = 100$

$7 x = 100$

$x = \frac{100}{7} = 14.286$

$2 x = 2 \times 14.286 = 28.572 ml$

$5 x = 5 \times 14.286 = 71.43 ml$

Hence, mix $28.572 ml$ of $100 mg/ml$ and $71.43 ml$ of $30 mg/ml$ to get $100 ml$ of $50 mg/ml$ .

Dilution: Dilution is a method of preparing a solution of desired concentration by diluting a concentrated stock solution. Water is typically used to dilute the solution. Dilution can be done by using the following formula:

$M_{1} V_{1} = M_{2} V_{2}$

where $M_{1}$ and $M_{2}$ are equal to the molarity of the solutions, measured as mol/L or M, and $V_{1}$ and $V_{2}$ are equal to the volume of the solutions. Sometimes, concentration is given in $v / v$ or $w / v$ concentrations. In that case substitute $M_{1}$ and $M_{2}$ by the respective concentrations.

How much water should you add to $150 ml$ of a $0.4%$ $v / v$ solution to reduce its strength to a $0.02%$ $v / v$ solution?

Using the formula above, $0.4 \times 150 = 0.02 \times V_{2}$

$V_{2} = \frac{0.4 \times 150}{0.02} = 3000 ml$

Since we already have $150 ml$ of the solution, subtract that from $3000 ml$ .

$3000 - 150 = 2850 ml$ of water needs to be added to $150 ml$ of $0.4%$ $v / v$ solution to get a $0.02%$ $v / v$ solution. Make sure to keep the same unit system of measurement while doing calculations. For example, if using volume in “ml,” keep it the same on both sides of the equation by converting “L” to “ml.”

Aliquot: The dilution method used when a minimal quantity of the drug is required. It is used when the minimum measurable or weighable quantity (MMQ or MWQ) of the equipment available is much higher than the quantity of drug needed. For example, if MWQ is $100 mg$ and the amount of drug required is $25 mg$ .

In the above example, measure out the MWQ of $100 mg$ and dilute it to a desirable dilution, e.g., you can dilute with $20 ml$ of water to make a $100 mg /20 ml = 5 mg/ml$ solution. Measure $5 ml$ of the $5 mg/ml$ solution to get $25 mg$ of the drug.