Medication maths: displacement volumes and values

Introduction

Many drugs are presented in a dry powder form with the intention of dissolving or mixing with an appropriate diluent prior to use. For drugs formulated in this way, it is important to consider the displacement volume of the medicine. Displacement volume is the volume occupied by the powder when the drug is dissolved or mixed with the diluent. 

The importance of the displacement volume is that some of these dry powders may take up a significant volume and this needs to be incorporated when calculating the final concentration of the mixture. Failing to consider displacement values can lead to solutions that are over-diluted and may not have the desired pharmacological effect. 

When reconstituting a dry powder with a diluent, the volume required needs to factor in the displacement volume of the drug.

For example, if the displacement volume of 2g of ceftriaxone powder is 1.37mL in water for injection, it implies that after adding 10mL of water for injection, the final volume is 11.37mL. Accordingly, the concentration would be 2g/11.37mL or 2,000mg/11.37mL, which can be simplified to 175.9mg/mL, and not 2g/10mL.

The table below shows examples of drugs presented in dry powder form that are dissolved or mixed with the diluent prior to use and the respective displacement volumes^​1–3​.

Example question

A 67-year-old man weighing 75kg is due to receive drug A for infusion at a dose of 3mg/kg/day in dextrose 5% solution for infusion. Drug A is presented as a 50mg powder vial. The directions to reconstitute each drug A vial require 12mL of sterile water to be added to 50mg of the powder. The displacement volume for 100mg of the powder is 1mL. Further dilute the reconstituted vial(s) 1 in 19 with 5% dextrose to form the infusion that is finally given to the patient. 

What volume of the final infusion (in mL) will provide the prescribed daily dose? Give your answer to the nearest 0.06mL.

Comments and explanations

To tackle the above question, you need to understand the concept of concentration and how it can be altered by displacement volume. You would also need to calculate the number of vials to be reconstituted based on the daily dose, understand and accurately follow the dilution process, and finally apply the appropriate rounding as requested. 

Here are the steps we chose, but feel free to use your own.

Step 1

Calculate the daily dose.

Daily dose = 3mg/kg 

     = 3 x 75 

     = 225mg

Step 2

Calculate the minimum number of vials needed.

Each vial contains 50mg so, 

minimum number of vials needed = 225mg ÷ 50mg

                                                     = 4.5 vials

You will need to round up to 5 vials as vials cannot be split.

Step 2

Calculate the concentration incorporating displacement volume.

Amount of drug A in a single vial = 50mg

If 100mg of drug A displaces 1mL; 50mg displaces X 

Solve for X 

X = 0.5mL 

The final volume of each vial after reconstitution is 12mL + 0.5mL = 12.5mL.

Using your knowledge of concentrations expressed as amount of drug/volume, we can express this for Drug A as 50mg/12.5mL. 

Step 3

Calculate the volume of drug if 5 vials are added.

To calculate this, you simply multiply both the volume and amount of drug by 5:

Volume of reconstituted drug A in (mL) if 5 vials are added = 12.5mL x 5

                                                                                             = 62.5mL

Amount of drug = 5 x 50mg

                           = 250mg

Therefore the concentration of the reconstituted drug = 250mg/62.5mL

Step 4

Calculate final volume of solution available for infusion.

At this stage, we have established how to reconstitute the required amount of drug A but the question states that this needs to be further diluted 1 in 19 with 5% dextrose before the infusion can be given.

After 1 in 19 dilution the final volume of solution available for infusion = 62.5mL x 19

           = 1,187.5mL

Step 5

Calculate the final volume to be infused.

The calculation so far is based on the 5 vials of drug A (5 x 50mg) used to make up the solution. This contains 250mg of the drug in total; however, the patient requires 225mg, so a final calculation is required. 

If 250mg = 1,187.5mL

225mg = X 

solve for X by cross multiplying:

X(250) = 225(1,187.5)

X = 1,068.75mL

Step 6

Round to the nearest 0.06mL

Final answer: 1,068.78mL to the nearest 0.06mL

• This article was amended on 11 October 2024 to correct an error in the example question parameters