Chapter 03: Ratios, Proportions, and Dosage Calculation
03
CHAPTER
Ratios, Proportions, and Dosage Calculation
This chapter will focus on these topics:
- Basic and complex proportion problems
- Dose conversions (mg/kg, units, mEq, mOsm)
- IV flow rates and infusion calculations
Chapter 3 focuses on dosage calculation and conversions. This skills in this chapter apply directly to day-to-day activities a pharmacist will encounter in both a community and clinical setting.
Basic and complex proportion problems
We have discussed proportion and how to set one up, but we have not discussed in full. A proportion is a mathematical equation stating that 2 ratios are equal. In pharmaceutics, proportions compare known relationships between quantities to solve an unknown value. As you’ve practiced, they are expressed as:
\(\frac{a}{b} = \frac{c}{d}\)
The letters a – d are quantities. a and b are assumed to have an equivalent relationship as c and d.
Some scenarios where proportions are commonly used in pharmaceutical practice is conversions of certain medications classes like glucocorticoid, iron, and opioids. Proportions are also used in Dilution calculations, concentration doses (e.g oral/IV medications), and dose adjustments.
Let’s start with some simple proportions that are not like previous questions:
Example 1:
If 450 capsules contain 1575 mg of an active ingredient, how many milligrams of the active ingredient will 72 capsules contain?
The question is similar to previous questions but has a different concept as it shows the relationship between capsules and doses. For this problem you set up a proportion like so:
\(\frac{450 \ capsules}{1575 \ mg} = \frac{72 \ capsules}{x \ mg}\)
Even though capsules are not technically a unit, it is good practice to put it out beside like a unit. You can also do this with tablet as well. When solving, the answer should be 252 mg. This problem shows the established relationship between individual capsules and mg. You can even find how many mg is in 1 capsule by dividing milligrams by capsules.
Example 2:
If phenobarbital elixir contains 22.7 mg of phenobarbital per 10 ml, how many grams of phenobarbital would be used in preparing a pint of the elixir? Round to the nearest thousandth?
Now we need to use prior knowledge from household conversions to solve this. 1 pint equals 473 milliliters. Using this knowledge, we can now set up the proportion like so:
\(\frac{22.7\ mg}{10\ mL} = \frac{x\ mg}{473\ mL}\)
The answer is 1.074 g. This question highlights the importance of household conversions, which are frequently encountered in the pharmaceutical world and commonly tested on the NAPLEX.
Example 3:
WL is a 70-year-old male patient hospitalized with decompensated heart failure and fever. Cultures are positive far aspergillosis. WL weighs 213.4 lbs. and will receive 0.35 mg/kg per day of amphotericin B (reconstituted and diluted to 0.1 mg/mL) by IV infusion. How many milliliters of amphotericin solution is required to deliver the daily dose? Round to the nearest mL.
First, we need to convert 213.4 lbs. to kg in which you divide 213.4 by 2.2 to get 97 kg. You then want to isolate the mg from kg dose. We would then multiply 0.35 by the weight in kg of the patient which is 97 kg to get 33.95 mg. 33.95 is the mg per dose the patient needs to receive of amphotericin. Now we need to find the milliliters equivalent to the dilution of 0.1 mg/mL. one way of solving is by setting up a proportion.
\(\frac{0.1\ mg}{1\ mL} = \frac{33.95\ mg}{x\ mL}\)
By setting up a proportion, you should get an answer of 339.5 mL which is 340 mL when rounded to the nearest mL.
There will be times when you might switch a patient on a different medication from the same drug class. For instance, a patient might be on methylprednisolone 15 mg and then be switched on Dexamethasone 3 mg for the same treatment. Why is that? Different medications have different potency. Potency refers to the amount of drug needed to reach a certain effect. A more potent drug requires a less amount while less potent drug requires a higher amount to reach a therapeutic window. In this case, Dexamethasone is more potent than methylprednisolone.
As pharmacists or pharmacy students, we are required to understand the conversions between certain medications. Classes like opioids, glucocorticoid steroids, and iron and some examples in which we are required to know for future jobs and to pass the NAPLEX. Here is a table of the steroid conversions followed by some example problems.
| Medication | Equivalent Oral Dose |
|---|---|
| Cortisone | 25 mg |
| Hydrocortisone | 20 mg |
| Prednisone | 5 mg |
| Prednisolone | 5 mg |
| Methylprednisolone | 4 mg |
| Triamcinolone | 4 mg |
| Betamethasone | 0.75 mg |
| Dexamethasone | 0.75 mg |
Example 1:
A patient takes 20 mg of prednisone daily for an autoimmune condition. What would be an equivalent daily dose of dexamethasone in milligrams?
Solving conversions for glucocorticoids is easy when utilizing a proportion. Thus can be structured in the following way.
\(\frac{pred \ 5 \ mg}{dex \ 0.75 \ mg} = \frac{pred \ 20 \ mg}{dex \ x \ mg}\)
You do not have to put the name of the medication beside the dose, but it is easier to keep track of what medications you are comparing. Solving this follows the same principles as any other proportion problem. The answer should be 3 mg of dexamethasone is equivalent to 20 mg of prednisone.
Example 2:
A hospital receives a patient transfer, The patient is taking methylprednisolone 12 mg. The hospital does not carry methylprednisolone in the formulary. The doctor wants to switch the patient to prednisone. What would the equivalent dose of prednisone be in milligrams?
We would need to set a proportion of the equivalency dosage of methyltrienolone to prednisone.
\(\frac{met\ 4\ mg}{pred\ 5\ mg} = \frac{met\ 12\ mg}{pred\ x\ mg}\)
Now, solving it is just like the previous example. You should get an answer that comes out to 15 mg of prednisone.
Now when switching opioids, the principle of solving them is the same as glucocorticoids, but with a small change. Opioids are converted back to MME which is morphine milligram equivalent.
MME is the standardized way to show the potency of opioids that is used to assess opioid exposure, ensure safe opioid conversions, and to help guide risk stratification. Table 1 will show IV and PO conversions.
Opioid Conversion | |||
| Medication | IV/IM | PO | MME |
| Morphine | 10 mg | 30 mg | 30 MME |
| Hydrocodone | _ | 30 mg | 30 MME |
| Oxycodone | _ | 20 mg | 30 MME |
| Oxymorphone | 1 mg | 10 | 30 MME |
| Hydromorphone | 1.5 mg | 7.5 mg | 30 MME |
| Codeine | 130 mg | 200 mg | 30 MME |
| Fentanyl | 0.1 mg | _ | 30 MME |
| Meperidine | 75 mg | 300 mg | 30 MME |
Example 1:
A patient is taking 20 mg of oxycodone PO every 6 hours for chronic pain. You want to switch them to oral morphine. What is the equivalent total daily dose of morphine?
First, we need to calculate the daily dose of oxycodone. The patient receives 20 mg of oxycodone every 6 hours which means the patient receives 4 doses or oxycodone a day.
\(\frac{4 \ \cancel{dose}}{day} \times \frac{20 \ mg}{\cancel{dose}} = \frac{80 \ mg}{day}\)
By multiplying the number of doses by the mg per dose, the answer gives us the daily dose of oxycodone. To find the equivalent daily dose of oxycodone to morphine, we can set up a proportion using the conversion factors from the provided table.
\(\frac{oxycodone\ 20\ mg}{morphine\ 30\ mg} &= \frac{oxycodone\ 80\ mg}{morphine\ x\ mg}\)
The equivalent dose of morphine to 80 mg of oxycodone/day is 120 mg/day of morphine.
Example 2:
A patient receives 10 mg of Hydromorphone PO for pain. The pharmacist wants to know how many MME the patient is receiving when discharged to figure out if they need to recommend Narcan. What would be the MME of a Hydromorphone?
For this problem a proportion can be used to solve the problem. The proportion set up the MME of the hydromorphone dose.
\(\frac{hydromorphone\ 7.5\ mg}{30\ MME} = \frac{hydromorphone\ 10\ mg}{x\ MME}\)
By solving the proportion, you should get 40 MME. You could also solve the first part by taking 30 MME and divide it by 7.5 mg of hydromorphone to get the conversion factor of hydromorphone which is 4. You can then multiply that by 10 mg dose the patient is receiving to get the same answer of 40 MME.
\(\frac{30\ MME}{7.5\ mg} = 4 \times 10\ mg = 40\ MME\)
Conversion factor is not required information to know as a student can easily find them by using the table. However, Memorizing conversion factors in the future can make finding MME easier.
Example 3:
A hospitalized patient is receiving 4 mg of hydromorphone IV/day. They are being discharged on oral oxycodone. What is the equivalent total daily oral oxycodone dose?
We can either find the MME daily dose of Hydromorphone or we can find the PO dose of Hydromorphone. To find the daily MME of Hydromorphone, a proportion of Hydromorphone IV mg to MME can be set up.
\(\frac{1.5\,mg\,IV}{30\,MME} = \frac{4\,mg\,IV}{x\,MME}\)
By setting up a proportion, we should get an answer of 80 MME. We can then convert this to equivalent oxycodone PO dose. We can set up another proportion to find the mg needed.
\(\frac{20\ mg\ PO}{30\ MME} = \frac{x\ mg\ PO}{80\ MME}\)
The mg equivalent is 53.33 mg PO of oxycodone. The other way is by setting a proportion comparing the PO mg to IV mg of hydromorphone.
\(\frac{1.5\ mg\ IV}{7.5\ mg\ PO} = \frac{4\ mg\ IV}{x\ mg\ PO}\)
In this proportion you get a PO dose of 20 mg of hydromorphone. We can then set up a proportion comparing PO hydromorphone to PO oxycodone by using the 30 MME standard.
\(\frac{hydromorphone\ 7.5\ mg}{oxycodone\ 20\ mg} &= \frac{hdyromorphone\ 20\ mg}{oxycodone\ x\ mg}\)
This produces the same answer of 53.33 mg of PO oxycodone daily.
The last conversion we will talk about in this topic is the iron conversions. Iron conversions are a little different from both opioids and glucocorticoid conversions. Iron conversions are based off elemental iron which is the actual amount of iron the patient absorbs and receives contrary to the dose of medication. For example, if a patient receives 325 mg of ferrous sulfate, the patient is only going to absorb 65 mg of elemental iron and not the whole 325 mg. iron conversions are required to know for everyday application and for the NAPLEX because dosing is based on elemental iron, different iron medication with different salts have a difference in potency with elemental iron, and it an prevent a patient from being overdosed or underdosed. The following table outlines commonly used iron formulations along with their corresponding elemental iron content.
|
Oral Dosing Iron Conversion Table |
|||
|
Medication |
Dose (mg) |
Elemental Iron (mg) |
Approximate Elemental Iron % |
|
Ferrous
Sulfate |
325 mg |
65 mg |
20% |
|
Ferrous
Sulfate dried (ER tablet) |
160
mg |
50
mg |
30% |
|
Ferrous
Gluconate |
324 mg |
38 mg |
12% |
|
Ferrous
Fumarate |
324
mg |
106
mg |
33% |
|
Carbonyl |
90 mg |
90 mg |
100% |
|
Polysaccharide
Iron Complex |
150
mg |
150
mg |
100% |
|
Ferric
Maltol |
30 mg |
30 mg |
100% |
Ferrous Sulfate is going to be the most common iron supplement used on this list. Let’s go through some examples.
Example 1:
A provide prescribes 100 mg of elemental iron daily. You recommend putting the patient on ferrous sulfate. What is the equivalent dose of ferrous sulfate to achieve 100mg of elemental iron?
In this problem we would set up a proportion that shows the comparison of elemental iron in mg to ferrous sulfate dose.
\(\frac{elemental \ 65 \ mg}{ferrous \ sulfate \ 325 \ mg} = \frac{elemental \ 100 \ mg}{ferrous \ sulfate \ x \ mg}\)
The answer should be 500 mg of ferrous sulfate.
Example 2:
A patient is currently taking ferrous fumarate 106 mg once daily. Due to shortages, the pharmacist wants to switch them to ferrous gluconate. How many mg of ferrous gluconate would be a similar dose to ferrous fumarate achieving the same elemental iron. Round to the nearest milligram.
In this example, its good practice to convert the ferrous fumarate dose to elemental iron then find the equivalent dose in ferrous gluconate. Setting up 2 proportions would be ideal for solving this problem. The first is comparing ferrous fumarate dose to elemental iron
\(\frac{fumarate\ 324\ mg}{elemental\ 106\ mg} = \frac{fumarate\ 106\ mg}{elemental\ x\ mg}\)
Setting up the proportion produces 35 mg 34.679012 mg of elemental iron which is the daily amount of iron the patient is receiving. The next proportion compares the elemental iron to ferrous gluconate.
\(\frac{gluconate\ 324\ mg}{elemental\ 38\ mg} = \frac{gluconate\ x\ mg}{elemental\ 32.679012}\)
The proportion should yield you an answer of around ~ 290 mg of ferrous gluconate. Depending on how many digits you kept after the decimal point will determine how accurate the number will be.
To conclude this topic, proportions are a very useful application to a multitude of different problems from opioid conversions to finding how many milliliters is needed for a dose of medication.
Dose conversions (mg/kg, units, mEq, mOsm)
Dose conversions are a main part of pharmacy practice, allowing accurate medication dosing over a wide range of drug formulations. This section will focus on the commonly used units such as milligrams per kilogram (mg/kg), units, milliequivalents (mEq), and milliosmoles (mOsm), all of which required to know to ensure appropriate therapeutic outcomes.
mg/kg dosing is a very common dosing regimen for medications like acetaminophen, antibiotics, and chemotherapy agents will dose according to weight. In previous chapters, we have used some examples of mg/kg dosing. These next examples are more tailored to this topic.
Example 1:
A 5-year-old male child weighs 39.6 lbs. and is prescribed acetaminophen 15 mg/kg every 6 hours as needed for fever. What would the dosing you would recommend of acetaminophen to the prescriber in milligrams? Round to the nearest milligram
To solve this example, we would need to first convert the patient’s weight of lbs. to kg by dividing 39.6 lbs. by 2.2 and get an answer of 18 kg. We would then use dimensional analysis to solve solve for milligrams by also canceling out kg.
\(\frac{15\ mg}{1\ \cancel{kg}} \times \frac{18\ \cancel{kg}}{1} = 270\ mg\)
By multiplying 15 mg/kg by 18 kg, the dimensional analysis yields an answer of 270 mg of acetaminophen. You recommend 270 mg of acetaminophen every 6 hours for fever to the prescriber.
Example 2:
A male patient weighing 231 lbs. comes into the hospital with an infection. The doctor prescribes gentamicin at 5 mg/kg IV once daily. The hospital protocol states that the maximum dose should not exceed 500 mg. would dose should be administered to the patient?
The first part of the problem is finding the dose of the gentamicin from the prescribers dosing instructions. The patients weight needs to be converted into kg by dividing 231 lbs by 2.2. the patient weight in kg is 105 kg. the next part is calculating the dose of 5 mg/kg by 105 kg. dimensional analysis can be used to calculate the dose of gentamicin.
\(\frac{5\ mg}{1\ k\cancel{g}} = \frac{105\ \cancel{kg}}{1} = 525\ mg\_
The dose calculated should be 525 mg, but that is not the answer to the question. Since the protocol wants to cap the dosing to 500 mg and we have a dose of 525 mg, the dose administered to the patient should be 500 mg.
Units area unique dosing measurement used for medications whose potency is based on biological activity, rather than mass (mg or mEq). The value of a unit is determined through standardized biological assays and can var between medications. Medications that commonly use units are insulin, heparin, and erythropoietin. Each medication has a different dosing for units. Here are some examples.
Example 1: Insulin Conversion
A 70 kg patient with DM type 1 is starting a new insulin therapy while hospitalized. The provider wants to initiate 0.5 units/kg/day of total insulin as follows:
- 50% basal glargine
- 50% bolus aspart, divided equally before 3 meals
Additionally, the patient will receive 1 unit of insulin aspart for every 10 grams of carbohydrates consumed each meal. How many units of insulin should be administered:
- Asbasal insulin glargine
- Asbolus insulin aspart before each meal
- Forcarbohydrate coverage at lunch if the patient eats 60 grams of carbohydrates
Round to the nearest unit.
To solve the first part, we need to figure out how many units daily the patient should receive. we can multiply 70 kg by 0.5 units/kg/day to get 35 units of insulin a day. If the insulin are split 50/50 then we can divide the units by 2 to get the basal glargine needed which would be 17.5 units which we would round to 18 units.
The second part used the other 17.5 units of insulin. We know the 17.5 units will be split up into 3 doses for each meal the patients have a day. By dividing 17.5 by 3 we get 5.8 which is rounded to 6 units before each meal.
The last part asks how many additional units of insulin aspart the patient needs for every 10 g of carbohydrates the patient consumes at lunch time. the patient has 60 grams of carbohydrates and divided by 10 g/unit; we are left with 6 units that we would add to his lunch administration on top of the 6 units administered before carbohydrate coverage.
Example 2: Heparin Dosing
A female patient weighing 154 kg is prescribed a heparin bolus 80 units/kg IV injection. How many units of heparin should be administered to the patient?
Solving this is similar to the mg/kg problems as you want to eliminate the kg in the dosing. To do that we need to convert the patient’s weight into kg which is 70 kg. We then multiply 70 kg by 80 units/kg to eliminate kg. We should be left with an answer of 5600 units.
mEq
mOsm
Understanding dosing units such as mg/kg or mEq is critical to ensure a effective medication management for a plethora of medications may have different dosing strategies involving different units. Understanding different units gives students and pharmacists a better clinical knowledge of dosing which can help informed clinical decision making and help verify appropriate doses.
IV Flow Rates and Infusion Calculations
IV flow rates and infusion calculations determine the rate of which IV fluids or medications are administered. IV flow rates specifically determine the speed of infusion. IV flow rates are typically measured by mL/hr or gtt/min. Flow rates can also be measured by dosage per time such as mg/hr. Infusion calculations is the broader calculation term used as steps involved with determining the flow rate. Infusion calculation is going to encompass the whole infusion plan. flow rates and calculation are important parts in inpatient and outpatient treatments. IV flow rates can help determine the maintenance fluids, continuous medications such as continuous a heparin drip to help anticoagulation, and titrated medication drips such as norepinephrine which we monitor the patient’s hemodynamic status like MAP to determine if we titrate the drug up or down. Learning and practicing IV flow rates and calculations will aid pharmacists and students recommend an effective delivery of a safe and effective rate of fluids and medications. Let’s look at some examples.
Example 1:
A prescriber orders 1500 mL of 0.45% NaCl to be infused over 18.5 hr. what would be the flow rate in mL/hr? round to the nearest whole number.
For this problem, we only must divide 1500 mL by 18.5 hr to get a flow rate of 81.08 mL/hr which can be rounded to 81 mL/hr.
Example 2:
A pharmacist recommends administering a 1 L bag of D5W over 8 hours with a macrodrip tubing of a drop factor of 15 gtt/mL. what is that rate in gtt/min? Round to the nearest whole number.
In this problem, its better use dimensional analysis to better construct a calculation strategy. It would be 15 gtt/mL first fraction as that is what is given. The second fraction can either be hours mL or hours but choosing to cancel out mL first would be a better practice. Then hours would go next and remember to change L to mL and hours to mins to get the correct units. In all the dimensional analysis should look something similar to this:
\(\frac{15\ gtt}{\cancel{ml}} \times \frac{1\ \cancel{L}}{1} \times \frac{1000\ \cancel{ml}}{1\ \cancel{L}} \times \frac{1}{8\ \cancel{hr}} \times \frac{1\ \cancel{hr}}{60\ mins} = \frac{31.25\ gtt}{min}\)
You do not have to put the 1 L and cancel out if a student wants to just use 1000 mL, but it is good practice to leave in there to understand where the 1000 mL came from if a backtrack is needed.
The answer to the problem should be 31 gtt/min as the question asks to round to the nearest whole number.
Example 3:
A male patient weighing 165 lbs. is ordered dopamine at a dose of 5 mcg/kg/min. the solution is prepared as 400 mg of dopamine that is added to a 250 mL bag of D5W. what is the infusion rate in mL/hr? Round to the nearest tenth
We first start by converting the patient’s weight in kg which is 75 kg. Now we can set up a dimensional analysis to answer the question. Preferably start with the dose of dopamine then next multiply the dose by the weight of the patient to cancel out kilograms. Next convert the dose to hr instead of mins. Then lastly convert the dose to mg or convert prepared dopamine from mg to mcg. Make sure to either add the conversion in or already to convert it before putting it as a proportion over the 250 mL as the prepared dopamine is in the 250 mL bag of fluids. The dimensional analysis stated should be constructed like so:
\(\frac{5 \ \cancel{mcg}}{\cancel{kg} * \cancel{min}} \times \frac{75 \ \cancel{kg}}{1} \times \frac{60 \ \cancel{mins}}{1 \ hr} \times \frac{250 \ mL}{400 \ \cancel{mg}} \times \frac{1 \ \cancel{mg}}{1000 \ \cancel{mcg}} = \frac{14.06 \ mL}{hr}\)
The answer to the question is 14.1 mL/hr because we round to the nearest tenth. You can go about calculating the infusion rate, but solving through dimensional analysis is an efficient way.
Example 4:
You want to administer fentanyl at 100 mcg/hr, but the pharmacy can only provide a bag labeled 250 mcg/mL. What would the flow rate be in mL/hr? round to the nearest tenth is needed.
In this scenario be can simply either put a dimensional analysis of 100 mcg/hr to 1 mL/250 mcg or you can simply just divide. It is harder to visualize why you can divide 100 by 250 without writing it out to cancel the necessary units.
\(\frac{100 \ \cancel{mcg}}{hr} \times \frac{1 \ mL}{250 \ \cancel{mcg}} = 0.4 \frac{mL}{hr}\)
The point of chapter of 3 is to reinforce the mathematical and clinical reasoning skills that a student must possess. By exploring more into proportions, dose conversions and, IV flow rates, a student can become familiar with different application problems and units associated with dosing regimens of different medications. These topics are more tailored to outpatient and inpatient practice that a student will encounter on APPEs and on the NAPLEX.
Part 1 Conclusion
Part 1 of the pharmacy Calculation guidebook for PharmD students to lay the groundwork and for more advanced calculations in future parts. These chapters provide a student with the mathematical framework to solve simple application problems that a pharmacist can encounter daily in clinics, hospitals, and even community practice. The overarching theme for part 1 is make sure a student is accurate and in calculation while also becoming well versed in different application of knowledge like Iv flow rates, or unit conversions that happen daily. Chapter one is a universal problem-solving method that supports accuracy and precision when a calculation is being conducted. Chapter 2 introduces conversions and from multiple measurement systems student si required to know systems interchangeably to understand prescriptions. Learning different measurement systems strengthens a student knowledge when constructing the mathematic concepts such as proportions and dimensional analysis. Chapter 3 bridges topics from chapters 1 and 2 to create an efficient foundation for applied problems like IV flow rates and dosing calculations. Overall, Part 1 ensures a pharmacist can minimize calculation errors and help with clinical decision making that happens throughout a pharmacists career.
