This is the final part of the seven-part series on pharmaceutical calculations, this time dealing with parenteral solutions and isotonicity.

**Concentrations****Dilutions****Formulations****Calculation of Doses****Density & Displacement Values****Molecular Weights & Moles***Parenteral Solutions & Isotonicity*

##### Example 1

*A patient requires 1L of 0.9% saline solution over an 8 hour period. Assuming that 1mL is equivalent to 20 drops, how many drops per minute should be administered to this patient?*

The first thing we should note is that the question asks for ‘drops per minute’, hence we need to convert 8 hours into minutes i.e. 480 minutes.

If there are 20 drops in 1mL then this must mean that 1L contains 20,000 drops.

Hence, this infusion of 20,000 drops necessitates 480 minutes for it to complete.

So if we divide 20,000/480, we will easily find how many drops per minute should be administered. In this case, it works out to be 41.6, or 42 drops per minute.

In conclusion, the infusion of this particular saline solution should be set to 42 drops per minute.

##### Example 2

*Drug X has a recommended flow rate of 2-3mL/minute at a concentration of 5mg/mL. Drug X is only available in 500mg vials for dilution in glucose 5%. The IV administration is calibrated as 20 drops/mL. Bearing these figures in mind, answer the following two questions:*

*Using only one vial of Drug X, what volume of glucose 5% solution should be used and how many drops per minute should be delivered if the flow rate is 3mL/minute?**How long will the infusion take?*

A vial of Drug X contains 500mg therefore it should be dissolved in sufficient glucose 5% to produce **100mL of solution** to get the required concentration of 5mg/mL.

If there is 20 drops in 1mL then 3mL has a total of **60 drops**.

Therefore, 3mL is delivered in one minute so the IV infusion should be set to 60 drops per minute.

Given the total volume to be given is 100mL, as calculated above, then it will take **33.3 minutes** to deliver this 100mL (100mL/3mL).

##### Example 3

*A 56kg female patient requires an IV infusion of amphotericin at a dose of 250micrograms/kg. However, the concentration of the final solution must not exceed more than 100 micrograms/mL. A vial of amphotericin contains 50mg. Bearing this in mind, calculate the dose of amphotericin and the volume of IV solution required by the patient if the solution contains the maximum concentration. If a solution has to be delivered in 2.5 hours, what is the rate in mL/minute. If a 50mg vial is used to prepare the IV solution, what is the total volume of the solution prepared?*

The first question in this example asks us to calculate the dose of amphotericin. Given that 250 micrograms are required per every kilogram, then this necessitates that 14,000 micrograms of amphotericin is required.

The second question asks us to find the volume of IV solution required if the solution contains the maximum concentration of 100 micrograms/mL.

If there are 100 micrograms needed per every mL, then 14,000 micrograms needs 140mL to fulfil this maximum concentration.

The third question asks us to deliver a solution over 2.5 hours, or 150 minutes. This can be represented as:

\[\frac{140mL}{150\hspace{1 mm}minutes} = 0.93mL/minute\]

The final question asks us to consider a 50mg vial to prepare the IV solution and to determine the total volume of the solution prepared.

We need to convert 50mg to 50,000 micrograms because we need to be consistent in our use of units.

If we need to have a maximum concentration of 100 micrograms/mL then we would need to dissolve the 50mg vial up to 500mL of solvent to maintain this concentration. However, in terms of the infusion, we would only require 140mL for administration into Patient X.

##### Example 4

*Phenytoin has a recommended dose of 18mg/kg of body weight to be infused at a rate not exceeding 50mg/minute. Assume that Patient X is a woman weighing 56kg. Phenytoin injection is available in 5mL ampoules containing 50mg/mL. The prescriber would like an infusion volume of 100mL and a dose rate of 25mg/minute. Is the prescriber’s request even possible and, if so, what is the flow rate per minute?*

First of all, what is the total dose we need to give Patient X.

\[18mg/kg\times56kg\Rightarrow900mg\hspace{1 mm}of\hspace{1 mm}Phenytoin\]

According to the question, the injection is available as 50mg/mL. Therefore, 18mL of phenytoin injection is required for this recommended dose.

So, yes, the prescriber’s request is certainly possible. This volume, 18mL, can be made up to 100mL with saline solution and will contain the required 900mg of phenytoin.

Now that we have made the solution, we need to calculate the volume/minute that will be required to deliver 25mg/minute.

If 100mL of solution contains 900mg of phenytoin, then 2.78mL will contain 25mg of phenytoin. Hence, the flow rate should be set at 2.78mL/minute.

The calculation above should not require proportional sets. It’s simple logic. The difference between 900mg and 25mg is 36 times. Therefore, you’ll require 36 times less solution to find out how much contains 25mg. This volume, as we saw above, is 2.78mL/minute.

##### Example 5

*The required dose of furosemide by slow IV infusion is 50mg at a rate not exceeding 4mg/minute. Furosemide injection contains 10mg/mL. Calculate the volume of furosemide injection required and the infusion rate i.e. mL/minute, if the patient is to receive the correct dose.*

50mg of furosemide is required therefore we’ll require 5mL of the furosemide injection.

Thus, 5mL of furosemide injection needs to be given at a rate not exceeding 4mg/minute.

If the rate is 4mg/minute, then a 50mg dose will necessitate 12.5 minutes.

Therefore, it will take 12.5 minutes to deliver all of the injection solution at a rate of 4mg/minute.

Given that 5mL of furosemide injection is required (as above) then (5mL/12.5 minutes) the infusion device should be set at a maximum of 0.4mL/minute.

##### Example 6

*A 100mg dose of an analgesic has to be infused over a 5-hour period. The drug is available in 1mL vials each containing 50mg of the drug. The only syringe driver available has a 20mL capacity syringe, which delivers at a constant rate of 2mL/hour. Bearing this in mind, how many vials of the drug and what volume of diluent will be required to fill the 20mL syringe so that the drug is delivered at the correct concentration? The syringe driver can be set to automatically stop after 5 hours.*

If the 100mg dose needs to be delivered over a 5 hour period, then the patient requires 20mg of this drug per hour.

The rate of delivery is 2mL/hour therefore 2mL of the solution will need to contain 20mg. The syringe has a capacity of 20mL therefore we’ll need 200mg of drug.

We’re told that the drug is available in 1mL vials each containing 50mg of the drug therefore we’ll need 4 vials.

If 4 vials contain 4mL and the syringe has a capacity of 20mL, then we’ll require 16mL of diluent.

##### Example 7

*Calculate the amount of sodium chloride that should be added to the following formulation of nasal drops in order to make the final solution isotonic.*

Ephedrine Hydrochloride |
0.5g |

Water to |
100mL |

In order to answer this question, we need to know the freezing point depression of ephedrine hydrochloride and sodium chloride.

**»**A 1% w/v solution of ephedrine hydrochloride will depress the freezing point by 0.169 degrees Celsius.**»**A 1% w/v solution of sodium chloride will depress the freezing point by 0.576 degrees Celsius.

1% w/v is obviously 1g per 100mL. We’re dealing with half of that, 0.5g, so the freezing point depression will decrease by half to 0.0845.

This means that ephedrine hydrochloride will depress the freezing point to 0.0845 below 0 degrees Celsius i.e. -0.0845 degrees Celsius.

An isotonic solution freezing point, as with 0.9% saline, is -0.52 degrees Celsius.

Hence, the added Sodium Chloride will need to depress the freezing point of the ephedrine solution by a further 0.4355 degrees Celsius. A proportional set should make it easier to visualise what’s going on in this particular question.

Sodium Chloride (%w/v) | 1 | Z |

Freezing Point Depression © | 0.576 | 0.4355 |

Z works out to be 0.756. This means that 0.756% w/v of sodium chloride should be added to make sure the final solution is isotonic.