Drug Math for Student Nurses: Dosage Calculation and Medication Administration

Objectives of This Presentation

• Gain a general understanding of basic dimensional analysis calculations for nursing drug math.

• Verbalize basic medical terms crucial for drug math calculations.

• Identify key points to review before taking Drug Math Exams.

Medication Measurement Tools

• Liquid Medication Measurement:

  • Examples of oral syringes and cups illustrating volumes like $30 \text{ mL}$, $15 \text{ mL}$ (1 Tbsp), $10 \text{ mL}$, $5 \text{ mL}$ (1 tsp), $2 \text{ tsp}$, $1/2 \text{ tsp}$, $3/4 \text{ tsp}$, $1/4 \text{ tsp}$.

  • Specific oral syringes or cups are designated for "oral use only" to prevent inappropriate administration.

• Injectable Medication Measurement (Syringes):

  • Standard syringes are marked in milliliters (mL), typically from $1 \text{ mL}$ up to $10 \text{ mL}$ or more.

  • Subcutaneous (SC) injections are indicated, highlighting a common route for specific dosages.

  • Insulin syringes are specifically marked in "UNITS" (e.g., $20, 40, 60 \text{ units}$) for precise insulin dosing.

Reading Drug Labels Effectively

• Key Information Found on Drug Labels:

  • Drug Name: Both the generic and trade (brand) name will be available.

  • Container Size: The total volume or quantity of medication in the container.

  • Concentration (Unit of Measure): This is critical for calculations and specifies how much drug is in a given unit (e.g., $500 \text{ mg}$ per tablet, $750 \text{ mg}$ per $5 \text{ mL}$ of solution, $1 \text{ gram}$ in $3.5 \text{ mL}$).

  • Lot Number: For tracking purposes.

  • Expiration Date: Crucial for safety; expired drugs are ineffective or harmful.

  • Additional Information: May include the manufacturer, safety precautions, storage instructions, and other warnings (e.g., "Rx only", "Protect from Light", "DEA Order Form Required" for controlled substances, "Do NOT chew, crush or dissolve" for extended-release forms).

• Identifying Concentration:

  • Always locate the "strength" or "available dosage strength" of the drug.

  • Examples: "The drug is supplied as $750 \text{ mg}$ per $5 \text{ mL}$", "The available dosage strength is $1 \text{ gram}$ in $3.5 \text{ mL}$", "Pharmacy sends $325 \text{ mg}$ tablets", "The drug available is $500 \text{ mg}$ per tablet".

• Handling "Extra" Information on Labels:

  • Sometimes drug labels or questions may provide more information than needed for a specific dosage calculation, such as the total container size.

  • Example: "The nurse has a $500 \text{ mL}$ bottle of Drug A at a strength of $30 \text{ mg}$ per teaspoon."

    • This means the bottle holds $500 \text{ mL}$ of the medication.

    • Each teaspoon (which is equivalent to $5 \text{ mL}$) contains $30 \text{ mg}$ of Drug A.

    • The total amount of drug in the bottle is $(500 \text{ mL} / 5 \text{ mL/tsp}) \times 30 \text{ mg/tsp} = 100 \times 30 \text{ mg} = 3000 \text{ mg}$.

  • For dosage calculation, the key "concentration" or "unit of measure" is $30 \text{ mg}$ per teaspoon (or $30 \text{ mg}/5 \text{ mL}$), while the $500 \text{ mL}$ bottle size is often "extra" information not directly used in calculating how many mL to administer for a single dose.

• Practical Label Analysis (Example Questions):

  • Identifying generic and trade names.

  • Determining if a drug is a controlled substance (often indicated by "WARNING: May be habit forming", "DEA ORDER FORM REQUIRED", and specific logos).

  • Stating the amount of drug and its form (e.g., tablet, capsule, injection).

  • Counting the number of units (e.g., tablets, capsules) in a package.

  • Understanding storage instructions (e.g., "Store at $20^{\circ}$ to $25^{\circ}\text{C}$, Protect from light and moisture, Refrigerate").

  • Determining the concentration of injectable forms (e.g., $80 \text{ mg}$ per mL) and total volume.

Safety in Calculating Doses

• Systematic Approach: Think, Focus, and proceed step-by-step.

• Problem Comprehension:

  • Thoroughly read and understand the problem statement.

  • Be familiar with medical terminology and abbreviations.

  • Pay close attention to decimal points.

• Accurate Conversions: Essential to convert between different units accurately, especially metric units.

• Visualization: Picture the problem to aid understanding.

• Reasonableness Check: Identify an expected range for the answer; ask yourself if the calculated dose is reasonable.

• Communication: Understand the problem fully. Do not hesitate to ask for a second opinion, as a patient's life depends on your accuracy.

• Three Checks for Medication Preparation and Administration:

  1. Selection: Verify the correct drug when selecting it from storage.

  2. Preparation: Verify the correct drug and dose during preparation.

  3. Administration: Verify the correct drug, dose, patient, route, and time immediately before administering.

Metric & International System of Units (SI) Terms and Basic Conversions

• Key Terms and Abbreviations:

  • $\text{L}$ = liter

  • $\text{mL}$ = milliliter

  • $\text{cc}$ = cubic centimeter

  • $\text{tsp}$ = teaspoon

  • $\text{kg}$ = kilogram

  • $\text{g}$ = gram

  • $\text{mg}$ = milligram

  • $\text{mcg}$ = microgram

  • $\text{cm}$ = centimeter

  • $\text{mEq}$ = milliequivalent

  • $\text{gtt}$ = drop

  • $\text{PO}$ = by mouth (oral route)

• Basic Conversion Factors:

  • $1 \text{ mL} = 1 \text{ cc}$

  • $1 \text{ ounce} = 30 \text{ mL} \text{ or } 30 \text{ cc}$

  • $1 \text{ tablespoon} = 15 \text{ mL} \text{ or } 15 \text{ cc}$

  • $1 \text{ teaspoon (tsp)} = 5 \text{ mL} \text{ or } 5 \text{ cc}$

  • $3 \text{ teaspoons (tsp)} = 1 \text{ tablespoon}$

  • $1 \text{ liter (L)} = 1000 \text{ mL} \text{ or } 1000 \text{ cc}$

• Converting Pounds to Kilograms (and vice versa):

  • The conversion factor is $1 \text{ kg} = 2.2 \text{ pounds}$.

  • Pounds to Kilograms: Divide by $2.2$. The answer will be a smaller number.

    • Example: A patient weighs $73 \text{ pounds}$. $73 \text{ lbs} \div 2.2 \text{ lbs/kg} = 33.181818 \approx 33.2 \text{ kg}$ (rounded to the nearest tenth).

    • Hint: $100 \text{ lbs} \div 2.2 \text{ kg/lb} = 45.5 \text{ kg}$.

  • Kilograms to Pounds: Multiply by $2.2$. The answer will be a larger number.

    • Hint: $10 \text{ kg} \times 2.2 \text{ lbs/kg} = 22 \text{ lbs}$.

Dimensional Analysis: A Step-by-Step Approach

• Dimensional analysis is a systematic method for solving drug calculations by ensuring units cancel out correctly.

• Step 1: Identify the Desired Unit

  • Write the desired unit of measure for the final answer to the left of an equal sign.

  • This unit guides the setup of your equation.

  • Examples: mL = or tab =.

• Step 2: Start with the "Have on Hand" (Clinical Ratio)

  • Find the concentration of the drug available (what you have on hand).

  • Place this information as a ratio to the right of the equal sign.

  • Crucial Rule: The numerator of this ratio must match the unit of measure you are trying to calculate.

  • Examples of concentrations: $100 \text{ mg}/5 \text{ mL}$, $0.125 \text{ mg}/3.5 \text{ mL}$, $500 \text{ mg/tablet}$ (or $1 \text{ tablet}/500 \text{ mg}$ if calculating tablets).

  • Example Setup (calculating mL): If available is $100 \text{ mg}$ per $5 \text{ mL}$, then: $\text{mL} = \frac{5 \text{ mL}}{100 \text{ mg}}$.

• Step 3: Introduce the Ordered Dose and Conversions

  • The unit of measure in the preceding denominator must be matched in the successive numerator to facilitate cancellation.

  • Example: If the previous step ended with milligrams (mg) in the denominator and the doctor orders $50 \text{ mg}$, then: $\text{mL} = \frac{5 \text{ mL}}{100 \text{ mg}} \times \frac{50 \text{ mg}}{1}$.

  • Incorporating Metric Conversions: If your ordered dose or available strength is in a different metric unit (e.g., order in grams, available in milligrams), add the appropriate conversion factor in the middle of your equation.

    • Example (calculating tabs with a gram order and mg available): To cancel grams and introduce milligrams: $\text{tab} = \frac{1 \text{ tab}}{\text{available quantity in mg}} \times \frac{1 \text{ gram}}{1000 \text{ mg}} \times \frac{\text{ordered quantity in mg}}{1 \text{ gram}}$.

• Step 4: Cancel Units and Calculate

  • Cancel out common units that appear in both the numerator and denominator.

  • Multiply all numbers in the numerator and divide by the product of all numbers in the denominator.

  • Example (from Step 2 and 3): $\text{mL} = \frac{5 \text{ mL}}{100 \cancel{\text{ mg}}} \times \frac{50 \cancel{\text{ mg}}}{1} = \frac{5 \times 50}{100} \text{ mL} = \frac{250}{100} \text{ mL} = 2.5 \text{ mL}$.

  • Example with Reduction (Vancomycin): Order $500 \text{ mg}$, available $1000 \text{ mg}$ per $3 \text{ mL}$. $\text{mL} = \frac{3 \text{ mL}}{1000 \cancel{\text{ mg}}} \times \frac{500 \cancel{\text{ mg}}}{1} = \frac{3 \times 500}{1000} \text{ mL} = \frac{1500}{1000} \text{ mL} = 1.5 \text{ mL}$.

    • Alternatively, reduce first: $\text{mL} = \frac{3 \text{ mL}}{2 \cancel{1000 \text{ mg}}} \times \frac{1 \cancel{500 \text{ mg}}}{1} = \frac{3}{2} \text{ mL} = 1.5 \text{ mL}$.

  • Example with Metric Conversion (Tabs): Order $500 \text{ mg}$ (assume per drug available as $1 \text{ gram}$), available $1 \text{ tab}$ for $1 \text{ gram}$. Order is $500 \text{ mg}$.
    $\text{tab} = \frac{1 \text{ tab}}{1 \cancel{\text{ gram}}} \times \frac{1 \cancel{\text{ gram}}}{1000 \cancel{\text{ mg}}} \times \frac{500 \cancel{\text{ mg}}}{1} = \frac{1 \times 1 \times 500}{1 \times 1000 \times 1} \text{ tab} = \frac{500}{1000} \text{ tab} = 0.5 \text{ tab}$.

• Rounding Rules: Always apply appropriate rounding rules based on the measurement type and clinical practice (e.g., smallest calibration on a syringe, nearest tenth for weight in kg).

Body Weight-Based Dosage Calculations

• Many medications, especially in pediatrics or for potent drugs, are dosed based on a patient's weight.

• Steps:

  1. Convert weight to kilograms (kg) if the patient's weight is given in pounds (lbs) and the order is per kg.

  2. Calculate the total daily dose (mg per day).

  3. Divide into individual doses if the order specifies multiple divided doses per day.

  4. Convert the dose (mg) to the administration unit (e.g., mL, tablets) using dimensional analysis.

• Example: Fluorouracil Calculation

  • Order: Fluorouracil $12 \text{ mg}$ per kg per day (IV).

  • Patient Weight: $176 \text{ lbs}$.

  • Available: Fluorouracil $50 \text{ mg}$ per $1 \text{ mL}$. (Two vials).

  • Calculation for mL per day:
    $\text{mL/day} = \frac{1 \text{ mL}}{50 \cancel{\text{ mg}}} \times \frac{12 \cancel{\text{ mg}}}{1 \cancel{\text{ kg}}} \times \frac{1 \cancel{\text{ kg}}}{2.2 \cancel{\text{ lbs}}} \times \frac{176 \cancel{\text{ lbs}}}{1} = \frac{1 \times 12 \times 1 \times 176}{50 \times 1 \times 2.2 \times 1} \text{ mL}$
    $\text{mL/day} = \frac{2112}{110} \text{ mL} = 19.2 \text{ mL/day}$.

• Practice Problem: Valproic Acid

  • Order: Valproic acid oral solution PO $15 \text{ mg/kg/day}$ in two divided doses.

  • Available: Valproic acid oral solution $250 \text{ mg}$ per $5 \text{ mL}$.

  • Pediatric Patient Weight: $66 \text{ lbs}$.

  • Questions to answer:

    • How many kg does the child weigh?

    • How many mg per day should the child receive?

    • How many mg per dose should the child receive?

    • How many mL per dose should the child receive?

• Practice Problem: Gentamicin

  • Order: Gentamicin $2.5 \text{ mg/kg}$ q8h IM.

  • Available: Gentamicin $10 \text{ mg}$ per $1 \text{ mL}$.

  • Pediatric Patient Weight: $33 \text{ lbs}$.

  • Questions to answer:

    • How many kg does the child weigh?

    • How many mg per dose will the child receive?

    • How many mL per dose will the child receive?

Reconstituting Medications

• Purpose: Some medications are supplied in powder form because they are unstable in liquid solutions. They must be reconstituted (mixed with a liquid) before administration.

• Diluent: The liquid used for reconstitution is called a diluent. Common diluents include sterile water or $0.9\%$ saline.

• Procedure: Always follow the specific directions on the drug label for reconstitution, including the amount and type of diluent, mixing instructions (e.g., tap, shake well), and storage after mixing.

• Example: Cefadroxil Reconstitution and Dosage

  • Instructions: To prepare suspension: Tap bottle lightly. Add $30 \text{ mL}$ water in two portions. Shake well after each addition. Store in refrigerator. When mixed as directed, each $5 \text{ mL}$ will contain cefadroxil $250 \text{ mg}$.

  • Order: Cefadroxil oral solution $1 \text{ g}$ PO q12h.

  • Available (after mixing): Cefadroxil oral solution $250 \text{ mg}$ per $5 \text{ mL}$.

  • Calculation for mL to administer:
    $\text{mL/dose} = \frac{5 \text{ mL}}{250 \cancel{\text{ mg}}} \times \frac{1000 \cancel{\text{ mg}}}{1 \cancel{\text{ g}}} \times \frac{1 \cancel{\text{ g}}}{1} = \frac{5 \times 1000}{250} \text{ mL} = \frac{5000}{250} \text{ mL} = 20 \text{ mL/dose}$.

Oral Medications (PO)

• Advantages:

  • Patients can often take them without assistance.

  • Generally less expensive than parenteral medications.

  • Easy to store.

• Forms: Tablets, capsules, powders, liquids (suspensions, syrups, elixirs, tinctures).

• Considerations:

  • Absorption: Can vary significantly depending on factors like food in the stomach and gastric pH.

  • Gastric Irritation: Some oral medications can irritate the gastric mucosa. These may need to be taken with adequate fluid or food, if not contraindicated.

  • Metabolism: Can undergo destruction or partial inactivation by liver enzymes (first-pass effect).

• Specific Oral Forms and Administration:

  • Tablets: Vary in form and strength.

    • Scored Tablets: Designed to be broken in half for smaller doses.

    • Enteric Coated Tablets: Designed to dissolve in the intestines, not the stomach. Should never be crushed as this destroys the coating and can lead to gastric irritation or premature drug release.

  • Capsules: Contain either powder or timed-release beads.

    • Sustained-Release (SR) / Extended-Release (ER) Capsules: Designed to release medication over an extended period. Should never be crushed or opened unless specifically indicated, as this can lead to rapid absorption and potential toxicity.

  • Liquids: Convenient for patients who have difficulty swallowing tablets or capsules.

    • Must be measured accurately using calibrated tools (e.g., oral syringes, calibrated cups).

Administering Medications via Nasogastric (NG) Tubes

• Many oral medications can be administered through an NG tube.

• Important Guidelines:

  • Do not mix medications directly with feeding solutions as this can lead to clogged tubes or altered drug absorption.

  • Dilute Medications: Dilute individual medications in $15-30 \text{ mL}$ of water or other fluid as ordered, especially viscous liquids or powdered medications that have been crushed and mixed.

  • Administer Individually: Administer each medication separately through the tube.

  • Flush Tube: Flush the NGT with another $15-30 \text{ mL}$ of water (or as specified by policy) before and after administering medications to ensure the drug reaches the stomach and is not left in the tube, preventing adherence and ensuring full dose delivery.

Preparing Parenteral Doses

• Reasons for Parenteral Route: Used when a patient cannot swallow, has an altered level of consciousness (LOC), the drug would be inactivated by gastric juices, or for rapid and precise drug onset and effectiveness.

• Drug Forms: Medications for parenteral administration are typically supplied in:

  • Vials: Single-dose or multi-dose glass containers with a rubber stopper.

  • Ampules: Single-dose glass containers that must be broken open.

  • Preloaded Syringes: Pre-filled with a single dose of medication, ready for administration.

• Critical Equipment Considerations:

  • Syringe Size: Select the appropriate syringe size based on the volume of medication to be administered to ensure accuracy.

  • Needle Size: Choose the correct needle length and gauge for the patient, medication viscosity, and intended injection site/route.

    • Needle Length: Varies from $1/4 \text{ inch}$ to $3 \text{ inches}$.

    • Needle Gauge: Refers to the diameter of the needle. A higher gauge number indicates a smaller (thinner) needle diameter (e.g., $25 \text{ gauge}$ is thinner than $18 \text{ gauge}$).

    • Needle Components: Bevel (angled tip), Shaft (length of the needle), Hub (part that attaches to the syringe).

• Syringe Anatomy: Familiarize yourself with the plunger, barrel, and tip. Always measure the dose accurately at the correct calibration line and avoid touching sterile parts (e.g., tip, inside of barrel, plunger rod).

Resources for Further Study

• Curren, A. M. ($2010$). Dimensional Analysis for Meds. ($4^{\text{th}}$ ed). United States: Delmar Cengage Learning.

• Horntvedt, T. ($2015$) Calculating Dosages Safely: A Dimensional Analysis Approach. Philadelphia: F. A. Davis Company.

• Potter & Perry ($2021$). Fundamentals of Nursing. ($10^{\text{th}}$ ed). Elsevier.