Pharmaceutical calculations
Introduction
Most pharmaceutical calculations are simple arithmetic. Calculating quantities in the dispensing process requires careful, methodical working which will minimize the risk of errors. Always try to relate the calculation to practice, visualize what you are doing and double-check everything.
How to minimize errors
As in all dispensing procedures, an organized, methodical approach is essential:
Write out the calculation clearly – it is all too easy to end up reading from the wrong line
If you are transferring data from a reference source, double-check what you have written down is correct
Do not take short cuts – you are more likely to make a mistake
Try not to be totally dependent on your calculator – have an approximate idea of what the answer should be and then if you happen to hit the wrong button on the calculator you are more likely to be aware that an error has been made
Finally, always double-check your calculation. There is frequently more than one way of doing a calculation, so if you get the same answer by two different methods the chances are that your answer will be correct. Alternatively, try working it in reverse and see if you get the starting numbers
Expressions of concentration
The metric system is the International System of Units (SI Units) for weight, volume and length. The basic unit for weight is the kilogram (kg), the basic unit for volume is the litre (L) and the basic unit of length is the metre (m). The prefix ‘milli’ indicates one-thousandth (10−3) and ‘micro’ one-millionth (10−6).
In some countries, the avoirdupois (or imperial) system (pounds and ounces) is still used in commerce and daily life. The imperial system of volume (pints and gallons) is still a common system for commerce and household measurement. Pharmacists need to know about these systems in order to avoid serious errors in interpretation of prescriptions. It is important to be able to change between the systems. Some conversion factors for the metric and avoirdupois systems are shown in Box 19.1 (see Examples 19.1–19.3).
Expressions of strength
Ratio is the relative magnitude of two like quantities. Thus:
If 1 g of sucrose is in 10 g of solution, the ratio is 1:10. Therefore, 10 g of sucrose is in 100 g of solution. This can be expressed as a percentage, so it is equivalent to a 10% w/w (weight in weight) solution.
Ratio strength is the expression of a concentration by means of a ratio, e.g. 1:10. Percentage strength is a ratio of parts per hundred, e.g. 10% (see Examples 19.4–19.6).
Percentage weight in weight (w/w)
Percentage weight in weight (w/w) is the number of grams of an ingredient in 100 grams (solid or liquid) (Example 19.7).
Percentage weight in volume (w/v)
Percentage weight in volume (w/v) is the number of grams of an ingredient in 100 mL of liquid (Example 19.8).
Percentage volume in volume (v/v)
Percentage volume in volume (v/v) indicates the number of millilitres (mL) of an ingredient in 100 mL of liquid (Example 19.9).
Moles and molarity
Concentrations can also be expressed in moles or millimoles (see Ch. 41). When a mixture contains the molecular weight of a drug in grams in 1 litre of solution, the concentration is defined as a 1 molar solution (1 mol). It has a molarity of 1. Thus, for example, the molecular weight of potassium hydroxide (KOH) is the sum of the atomic weights of its elements, i.e. KOH = 39 + 16 + 1 = 56. Therefore a 1 molar solution (1 mol) of KOH contains 56 g of KOH in 1 litre of solution.
A 1 millimole (mmol) solution of KOH contains one-thousandth of a mole in 1 litre = 56 mg (see Examples 19.14–19.16).
Calculating quantities from a master formula
In extemporaneous dispensing, a list of the ingredients is provided on the prescription or is obtained from a reference source. It may be that this ‘formula’ is for the quantity requested, but more often, the quantities provided by the master formula have to be scaled up or down, depending on the quantity of the product required. This can be achieved using proportion or by deriving a ‘multiplying factor’. The latter is the ratio of the required quantity divided by the formula quantity. The following examples illustrate this process (Examples 19.17 and 19.18).
Example 19.17
Calculate the quantities to prepare the following prescription:
50 g Compound Benzoic Acid Ointment BPC.
The master formula is for 100 g, the prescription is for 50 g, therefore the multiplying factor is 50/100, i.e. each quantity in the master formula is multiplied by 50/100 = 0.5 to give the scaled quantity.
Double-check: the quantities for the master formula add up to 100 g and the scaled quantities add up to 50 g.
In most formulae where a combination of weights and volumes is required, the formula will indicate that the preparation is to be made up to the required weight or volume with the designated vehicle. However, occasionally, as can be seen in the next example, a combination of stated weights and volumes is used and it is not possible to indicate what the exact final weight or volume of the preparation will be. In these instances an excess quantity is normally calculated for and the required amount measured (Example 19.19)
Calculations involving parts
In the following example the quantities are expressed as parts of the whole. The number of parts is added up and the quantity of each ingredient calculated by proportion or multiplying factor, to provide the correct amounts (Example 19.20).
There are some situations when extra care is necessary in reading the prescription (see Example 19.21).
Calculations involving percentages
There are conventions which apply when dealing with formulae which include percentages:
A solid in a formula where the final quantity is stated as a weight is calculated as weight in weight (w/w)
A solid in a formula where the final quantity is stated as a volume is calculated as weight in volume (w/v)
A liquid in a formula where the final quantity is stated as a volume is calculated as volume in volume (v/v)
A liquid in a formula where the final quantity is stated as a weight is calculated as weight in weight (w/w) (see Example 19.22).
In the following example, a liquid ingredient, the coal tar solution, is stated as a percentage and a weight in grams of final product is requested. The convention of % w/w is applied (Example 19.23).
When dealing with preparations where ingredients are expressed as a percentage concentration, it is important to check that the standard conventions apply because there are some situations where they do not apply. Two examples are given below:
1. Syrup BP is a liquid – a solution of sucrose and water. If the normal convention applied it would be w/v, i.e. a certain weight of sucrose in a final volume of syrup. However, in the BP formula the concentration of sucrose is quoted as w/w. Therefore Syrup BP is:
This means that when preparing Syrup BP the appropriate weight of sucrose is weighed out and water is added to the required weight, not volume.
2. A gas in a solution is always calculated as w/w, unless specified otherwise. Formaldehyde Solution BP is a solution of 34–38% w/w formaldehyde in water.
Changing concentrations
Sometimes it is necessary to increase or decrease the concentration of a medicine by the addition of more drug or a diluent. On other occasions, instructions have to be provided to prepare a dilution for use. These problems can be solved by the dilution equation:
where C1 and V1 are the initial concentration and initial volume respectively; and C2 and V2 are the final concentration and final quantity of the mixture, respectively.
When three terms of the equation are known, the fourth term can be made the subject of the formula, and solved (Examples 19.24–19.27).
Alligation
Alligation is a method for solving the number of parts of two or more components of known concentration to be mixed when the final desired concentration is known. When the relative amounts of components must be calculated for making a mixture of a desired concentration, the problem is most easily solved by alligation (Examples 19.28 and 19.29).
Example 19.28
Calculate the amounts of a 2% w/w metronidazole cream and of metronidazole powder required to produce 150 g of 6% w/w metronidazole cream (to 2 decimal places).
In alligation, the two starting material concentrations are placed above each other on the left hand side of the calculation. The target concentration is placed in the centre. The arithmetic difference between the starting material and the target is calculated and the answer recorded on the right hand end of the diagonal. The proportions of the two starting materials are then given by reading horizontally across the diagram.
As shown above, the difference between the concentration of the pure drug powder (100%, recorded top left) and the desired concentration (6%) is 94 (recorded bottom right). This is equivalent to the number of parts of 2% cream required (read horizontally across the bottom). Similarly, the difference between the concentration of 2% cream (recorded bottom left) and the desired concentration (6%) is 4 (recorded top right). This is equivalent to the number of parts of 100% drug (metronidazole powder) needed for the mixture (read horizontally across the top).
Example 19.29
Promazine oral syrup is available as 25 mg/5 mL and 50 mg/5 mL. Calculate the quantities used to prepare 150 mL of 40 mg/5 mL of the oral syrup.
Convert all the concentrations to percentages.
Therefore, 25 mg/5 mL is equivalent to 0.025 g in 5 mL = 0.500 g in 100 mL = 0.5% w/v
Similarly, 50 mg/5 mL = 1% w/v; and 40 mg/5 mL = 0.8% w/v.
Calculations where quantity of ingredients is too small to weigh or measure accurately
When preparing medicines by extemporaneous dispensing, the quantity of active ingredient required may be too small to weigh or measure with the equipment available. In these situations, a measurable quantity has to be diluted with an inert diluent. The process is called ‘trituration’ (see Ch. 38).
Small quantities in powders
The method for preparing divided powders is described in Chapter 38 (see Example 19.30).
Small quantities in liquids
If the quantity of a solid to be incorporated into a solution is too small to weigh, again dilutions are used. In this case, a solution is prepared, so the solubility of the substance needs to be considered. Normally a 1 in 10 or 1 in 100 dilution is used (Example 19.31)
Solubilities
When preparing pharmaceutical products, the solubility of any solid ingredients should be checked. This will give useful information on how the product should be prepared. The objective of this section is to clarify the terminology used when solubilities are stated.
The solubility of a drug can be found in reference sources such as the drug monograph in Martindale. The method of stating solubilities is as follows:
This means that 1 g of sodium chloride requires 2.8 mL of water, 250 mL of alcohol or 10 mL of glycerol to dissolve it. An example of how knowledge of a substance’s solubility can help in extemporaneous dispensing can be found in Chapter 30.
Calculations involving doses
A simple calculation which pharmacists sometimes have to make while dispensing, is to calculate the number of tablets or capsules or volume of a liquid medicine to be dispensed (Examples 19.32 and 19.33).
Calculating doses
An overdose of a drug, if given to a patient, can have very serious consequences and may be fatal. It is the responsibility of everyone involved in supplying or administering drugs to ensure that the accuracy and suitability of the dose are checked.
The standard way to check whether a drug dose is appropriate is to consult a recognized reference book. One of the commonest used for this purpose is the BNF. When first using any reference source it is important to be aware of the terminology used, to avoid misinterpreting the entries, especially where doses are quoted as ‘×milligrams daily, in divided doses’. An explanation of the terminology will usually be found in the introduction to the book (Example 19.34).
Calculations of children’s doses
Children require different doses from those of adults. Ideally these should be arrived at as a result of extensive clinical studies, although this is often not possible. When this is the case, an estimate of the dose has to be made. This is best carried out using body weight (see next section), but where this is not available, there are three formulas which relate the child’s dose to the adult dose.
Calculation of doses by weight and surface area
For some drugs the amount of drug has to be calculated accurately for the particular patient. This is normally carried out using either body weight or body surface area. When body weight is being used, the dose will be expressed as mg/kg. In countries which still use pounds, it will be necessary to convert the patient’s weight in pounds into kilograms by dividing by 2.2. The total dose required is then obtained by multiplying the weight of the patient by the dose per kilogram.
Body surface area is a more accurate method for calculating doses and is used where extreme accuracy is required. This is necessary where there is a very narrow range of plasma concentration between the desired therapeutic effect and severe toxicity, such as with the drugs used to treat cancer. The body surface area can be calculated from body weight and height using the equation given below, but it is more usual to use a nomogram for its determination. The actual nomogram is published in many reference sources.
Reconstitution and infusion
Some drugs are not chemically stable in solution and so are supplied as dry powders for reconstitution just before use. Many of these are antibiotics, but there is also a range of chemotherapeutic agents used in cancer treatment. The antibiotics may be for oral use or for injection. An oral antibiotic for reconstitution comes as a powder in a bottle with sufficient space to add the water. Those for injection are sterile powders and are dissolved in sterile water aseptically (see Ch. 40). There are a number of calculations which may be required around the reconstitution processes (Example 19.35).
Sometimes, the doctor may request a more or less concentrated syrup to be produced which requires altering the amount of water added from that indicated by the manufacturer (Example 19.36).
However, this type of oral mixture is likely to have other ingredients – thickeners, colours, flavours, etc. – which will occupy some of the final volume. So this may not be correct and we need to be able to calculate exactly how much water to add (Examples 19.37 and 19.38).
Drugs for injection solutions do not normally contain ingredients other than the drug (or they make an insignificant contribution to the final volume). However, they are usually packed as a quantity of drug with the final volume left to be calculated by the pharmacist (Example 19.39).
Calculation of infusion rates
Drugs may be given to patients intravenously by adding them to an intravenous (IV) infusion (see Ch. 41). Calculations involve working out how much drug solution should be added, working out how fast, in terms of mL/min, the infusion should be administered, and calculating what this means in terms of ‘drops per minute’ through the giving set. When an infusion pump is used, this can be set to deliver a specified number of mL/min (Example 19.40).
When administering intravenous infusions, the rate of administration is first calculated in terms of millilitres per minute (Example 19.41).
Most infusions are administered using a giving set with a dropping device on the tube Partial clamping of the tube can be used to adjust the rate of dropping. Depending on the drop size – that is the number of drops per millilitre – it is then possible to convert a rate of millilitres per minute into drops per minute which the nurse can adjust (Example 19.42).
A variation on this is when the doctor wishes a drug solution to be added to the infusion (Example 19.43).
Key Points
Always work methodically and write down calculations clearly
Check calculations, using a different method where possible
Estimate the answer before you start
Try to visualize the quantities you are using in the calculation
Look carefully to see if a formula gives the quantities of all ingredients or uses ‘to’ for the vehicle
Always check the units being used and be careful not to mix them during a calculation
Be careful to read the wording; small changes in terminology can alter the calculation
Triturates with solids and liquids normally use a 1 in 10 dilution per step
Be very careful in checking doses, particularly with ‘in divided doses’ and ‘mg/kg’ statements in the reference books
On completion of a calculation, ask yourself whether the answer is ‘reasonable’ given the numbers you are using