## Gentle Drug Math

Quite an assignment I have: “Write about drug math conversationally. Keep it light. Don’t make it sound like a lecture.”

Light? Conversational? What was I thinking when I suggested that? How informally can I present such a dry, technical topic that many paramedics find scarier than breech births?

If you’re nervous about milligrams and kilograms, you’re not alone. Four studies published since 2000 portray paramedics’ drug math proficiency as frighteningly low.^{1,2} During one experiment in a simulated high-stress environment, “advanced paramedics” scored only 61% on a series of calculations.^{1} That’s the good news; less experienced medics averaged not quite 40%.

It gets worse. Only three of 47 paramedics from two U.S. sites were able to solve more than seven of 10 drug problems correctly.^{2} The mean was three of 10. In North Carolina, 109 medics averaged 51% on a 10-question math test—with calculators and unlimited time.^{1}

Researchers Eastwood, Boyle and Williams speculated that “pressure” was a “major factor” in the above results. However, none of those studies involved actual patients. Practical exams are stressful, but more than real calls? I don’t think so.

I blame technology and our school systems for two generations of practitioners with mediocre math skills. I think we started to devalue manual number crunching in the early 1970s, as first calculators, then computers, made mental arithmetic—a major component of my primary education—seem as outdated as iceboxes. Most paramedics know that even hi-tech devices don’t tell us how to frame drug math problems.

My job is to help you make sense of that process by beginning with a review of terminology, adding arithmetic, then focusing on the simplest ways I know to handle drug math in the field. What I won’t be doing is preaching gimmicks that work only for very small subsets of real-world problems. For example, I saw a trick for dopamine drips that reads something like, “Take your waist size, divide it by the number of minutes to defrost a strip steak, add 37 if the patient has red hair, etc.” What good are such shortcuts if you have to remember a different one for each med? I’d rather help you polish skills that you can use for a variety of calculations.

Memo to the engineers and physicists among you (including you, Dad): I’m going to bend the definitions of some terms and compromise the precision of certain calculations to achieve a balance between simplicity and accuracy that I believe is acceptable when treating anyone except manikins. If it bothers you to see me call the kilogram a unit of weight instead of mass, please keep two things in mind: (1) I’m assuming all patients are treated on Earth, even if they weren’t born here, and (2) I still suffer post-traumatic stress from early-morning physics classes. If you send me e-mails quoting Newton or Einstein about anything more complicated than growing tomatoes, you might push me over the edge.

Let’s begin with some basics:

*Weight*** **is the amount of drug we’re giving. Forget about pounds and ounces; almost all doses are in metric units—grams, milligrams and micrograms. Kilograms are used, too—for patient weights. A kilogram equals a thousand grams, a gram is a thousand milligrams, and a milligram has a thousand micrograms—field-friendly multiples of 10 that make our “English” system of grains, ounces, pounds and tons seem primitive. You disagree? Okay, how many grains in an ounce? I’ll give you a hint: It’s not a round number.

The only conversion factor you might need between English and metric weights for drug math is 2.2 pounds per kilogram. Usually we start with the weight of our patient in pounds, then divide by 2.2 to get kilos. Since we’re often guessing people’s weights, and weight-related drug doses are estimates, also, we could drop the decimal point and just divide by two (except during exams!), which is the same as half the pounds. My approximate weight in kilograms is 150 lbs / 2, or 75. (If I were cooking for myself, it would be 120 lbs / 2.)

Conservative, prudent estimating is a fundamental part of medicine. Doctors do it, drug companies do it, and we do it. If you doubt that, try to find field doses that aren’t round numbers. Not many, right? Consider epinephrine. Is it a coincidence that we prescribe boluses of 1 mg—not 0.94 mg or 1.12 mg—for a human body in cardiac arrest? No, it’s an estimate—a nice, round estimate. You won’t convince me that changing almost any drug dose by 10% in either direction makes a difference. Go ahead, try.

When I was teaching, some of my students had trouble deciding whether to multiply or divide when converting from one unit of measure to another. Here’s a simple rule: When you’re going from a smaller unit to a bigger one, divide; otherwise multiply. Since we need approximately 2 pounds to make one kilogram, a pound must be the smaller of the two; therefore, we’re going to divide by 2 when converting pounds to kilos, and multiply by 2 for kilos to pounds. We can apply that logic to grams and milligrams, too: Multiply by the conversion factor, 1000, to go from grams (bigger) to milligrams (smaller), and divide by 1000 to change milligrams to grams. Same with milligrams and micrograms. Remember, multiplying by 1000 moves the decimal point three places to the right, and dividing by 1000 moves it three places left. When you’re redirecting decimal points and you run out of digits, add zeroes; for example, 2 * 1000 = 2000 and 2 / 1000 = .002.

Another way to handle conversions is with ratios—comparisons between numbers, often expressed as fractions. Let’s try calculating my weight in kilograms using that approach:

- Express the conversion factor as a ratio: 2 pounds per 1 kilogram, or 2 lbs / 1 kg.
- We need another ratio in the same format that expresses my known weight in pounds versus my calculated weight in kilograms: 150 lbs / X kg, where X is a placeholder for the still-unknown number of kilos that equals 150 pounds.
- We can combine those two ratios by saying 2 pounds is to 1 kilogram as 150 pounds is to X kilograms, or 2 lbs / 1 kg = 150 lbs / X kg. Notice that pounds and kilos occupy the same relative positions on both sides of the equation.
- It’s easier to do our calculation if X is on the top instead of the bottom. No problem; we can turn that fraction upside-down, as long as we do the same to the ratio on the other side of the equal sign: 1 kg / 2 lbs = X kg / 150 lbs.
- Now we want to isolate X by multiplying both sides of the equation by 150 pounds. On the left we get 150 * 1 / 2, which is 75 (omitting all units of measure to keep it simple). On the right, the 150s cancel out, leaving only X. We could show the result as 75 = X. More commonly, though, we put X on the left: X = 75 kg. If we’d used 2.2 instead of 2 as the conversion factor, the answer would have been 68 kg—a difference that’s critical only in the classroom.

When we solved for X, we did algebra. Pretty simple, right? The only tool you need for almost any drug math problem is algebra just like that. Read on for more examples.

** Volume** is the amount of space occupied by a drug. Once again, metric units—liters and milliliters (the same as cubic centimeters)—are used much more commonly in medicine than English measures such as cups, pints, quarts and gallons. One thousand milliliters (ml) (or cubic centimeters (cc)) equals a liter (L), which is a little larger than a quart.

We use units of volume to specify syringes, IV bags and drops. If I asked you to get a 3-cc syringe, or a liter bag of normal saline (NS), you’d know exactly what I meant. Not so obvious is the volume of drops; they drip from administration sets classified by the number of drops per cc. For example, a 60-drop set produces drops so small that 60 of them fit in one cubic centimeter (about the size of an aspirin).

Would the drops dripping from a 60-drop administration set be bigger or smaller than those from a 10-drop set? Only 10 of those drops are needed to fill 1 cc; therefore, each of those drops must be six times bigger (60 / 10) than the ones dripping from a 60-drop set. It takes more of a smaller item to equal fewer of a bigger one. That’s an important concept—well, maybe not as important as stretchers fitting into elevators, but one that we can use to quickly adjust IV drips for different administration sets. For example, an IV running at 100 drops per minute (gtts/min) through a 60-drop set should be slowed to 25 gtts/min (100 * 15 / 60) to maintain the same flow through a 15-drop set. Bigger drops means a slower drip. You’ll find much more about drip rates a few pages from here.

Let’s try some algebra to convert a half-liter into cubic centimeters:

- Start with the conversion factor: 1000 cubic centimeters per 1 liter, or 1000 cc / 1 L.
- Using the same format, X cc / 0.5 L (0.5 is one half expressed as a decimal).
- Our equation is 1000 cc / 1 L = X cc / 0.5 L.
- We don’t have to flip fractions to solve for X this time, because X is already on top. Just multiply both sides by 0.5 L to isolate X: 0.5 * 1000 / 1 = 500 = X, or X = 500 cc.

** Concentration **is weight per volume. Let me explain:

Sometimes it’s not enough to state how many milligrams or milliliters of a drug we’re administering; we also need to know how much space the desired dose of a drug is occupying. Consider epinephrine; it wouldn’t be acceptable to inject 3 cc (0.3 mg) of a 1-mg-per-10-cc bolus of epi—the kind we use in cardiac arrests—into the deltoid muscle of an asthmatic or anaphylactic patient. We’d be pushing too much fluid into a limited space. The correct procedure is to draw up 0.3 cc, instead of 3 cc, of 1:1000 epi—much more concentrated than the 1:10,000 epi mentioned above. The amount of epinephrine—0.3 mg—would be the same in both cases, but the concentration differs by a factor of 10.

1:1000 and 1:10,000 are just ways of expressing ratios. They look different than the fractions we used above, but 1:1000 is the same as 1 g per 1000 cc, or 1 g / 1000 cc. Since 1 g = 1000 mg, that’s the same as 1000 mg / 1000 cc, or 1 mg / 1 cc. That’s exactly how 1:1000 epi is usually packaged: 1 milligram in that tiny, 1-cc glass ampule that you have to break without embedding either end in your fingertips.

We can show concentration as a percentage, too. D50 (50% dextrose), for example, is 25 grams of dextrose in a 50-cubic-centimeter solution. 25 / 50 = 0.5 = 50%. (To convert decimals to percentages, move decimal points two places to the right.)

How would we express 1:1000 epi as a percentage? Easy: 1 / 1000 = 0.001 (decimal point moved three places left) = 0.1%. What about 1:10,000 epi? 1 / 10,000 = 0.0001 = 0.01%.

Once you’re comfortable with weight and volume, it’s easy to calculate IVP drug doses. You just need to know the weight of the dose and the volume of the vial. To administer 60 mg of lidocaine from a 5-cc, 100-mg vial, for example, start with your ratios: 5 cc / 100 mg = X cc / 60 mg. To solve for X, multiply both sides of the equation by 60: 60 * 5 / 100 = 300 / 100 = 3 = X. X = 3 cc of lidocaine.

** Time** is used in some drug calculations. There’s not much to say about time, except that there’s never enough of it. I think we’re all comfortable with 60 seconds in a minute and 60 minutes in an hour. Just remember that fractions of hours and minutes can be expressed as decimals, too; for example, 1 / 2 hr = 0.5 hr and 15 sec = 0.25 min.

A ** rate** is something that gets measured per unit of time. Miles per hour, beats per minute and alimony per month are rates. So is drops per minute, the unit we’re usually seeking when we calculate drip rates—the practical math problems that seem to worry medics the most.

I’ve thought a lot about how to simplify IV drips for you. I haven’t found a better way than the technique taught by my ex-instructor, ex-boss and fellow medic, Eric Niegelberg. We called his method “Eric’s math” when he presented it during my original paramedic class. Eric’s math (EM) is what we call a “black box”—a process or device that works without requiring the user to know how. Black boxes are even more useful when they’re flexible enough to handle variables—changes in conditions or assumptions. Common IV-drip variables are the size of the bag, the amount of drug you add to it and the administration set you run it through. EM handles all of them.

Let’s start with a classic example: a lidocaine drip. Your goal is to administer 2 mg / min. Start by selecting your bag and administration set, just as you would on an actual call. Suppose you chose a 500-cc bag of NS and a 60-drop set. We’ll also assume you’ve added a whole 2-gram vial of lidocaine to that bag. Now what?

EM begins with a few easy questions:

1. Is the dose weight-related? No. If it were, we’d have to insert some simple arithmetic here (see the dopamine example below).

2. How many of those 500-cc IV bags would equal 1 liter (1000 cc)? The answer is 2 (500 cc * 2 = 1000 cc)

3. What do we get if we multiply the amount of the drug (2 g) we’ve added to the bag by that same factor, 2? That’s easy: 2 * 2 = 4.

Next we build a fraction. On top is the amount of drug we want to administer per minute: 2. On the bottom is the answer to Question #3: 4. The fraction is 2 / 4.

Multiply that fraction by the size of the administration set (60): (2 / 4) * 60. The answer is 30 gtts/min.

The black box that drives EM substitutes the answers to questions #2 and #3 for a more complicated concentration calculation, and ensures the units of measure are manipulated to yield drops per minute. You don’t have to worry about those details; however, for the intellectual aristocrats among you, here’s how that works:

The 4 on the bottom of the fraction is the concentration of lidocaine in milligrams per cc. How did we get there? We started with 2 grams, or 2000 milligrams of lidocaine in 500 cc of normal saline. If we divide 2000 by 500, we get 4 mg/cc.

The dose of lidocaine we’re administering to the patient is 2 milligrams per minute—the top part of the fraction. With all the units of measure included, that fraction now reads: (2 mg/min) / (4 mg/cc). The milligrams cancel. When we add the size of the administration set to the equation, the cubic centimeters also cancel:

((2 mg/min) / (4 mg/cc)) * 60 gtts/cc = 30 gtts/min

Like I said, don’t worry about the units.

Let’s try a weight-related IV drip problem: 5 micrograms (mcg) per kilogram per minute of dopamine administered to a 160-pound patient from a 400-mg vial. Assume we’re using a 250-cc bag of NS and a 60-drop set:

This time the answer to EM Question #1 is yes. First we need to convert the patient’s weight into kilograms—160 / 2 = 80—then multiply those kilos by the dose: 80 kg * 5 mcg/kg/min = 400 mcg/min. That’s the top of the EM fraction.

EM Question #2 asks how many 250-cc bags are equivalent to 1 liter. A liter is the same as 1000 cc, so 4 * 250 cc = 1000 cc = 1 L. The answer to Question #3 is 4 * 400 = 1600—the bottom of the EM fraction. Our drip-rate calculation looks like this:

(400 / 1600) * 60 = 15 gtts/min

If we’d used the more precise 2.2-pounds-per-kilo conversion factor for the patient’s weight, the answer would have been 13.6 gtts/min—hardly enough of a difference to justify the extra arithmetic. Decreasing the dopamine dose from 5 to 4 mcg/kg/min, still well within the lower range for that drug, would have caused an even bigger change in the drip rate—to 12 gtts/min. I don’t think you’d find a doctor who’d argue that a 1 mcg/kg/min difference in that med is significant.

Our review of drug math has taken us from decimals to drips. We built on everyday measures—weight, volume, concentration, time and rate—and reintroduced the concept of ratios to develop a comfort level with simple algebra. We ended with Eric’s math, a black-box approach to IV drips that minimizes the complexity of those calculations. The techniques we presented are not the only ways to determine dosing, but they are among the easiest. Let me know how they work for you.

##### References

##### 1. Eastwood K, Boyle M, Williams B. Paramedics’ ability to perform drug calculations. *Western Journal of Emergency Medicine*. 10(4):240-243, Nov 2009.

##### 2. Henderson C, Clouatre A, Mayfield R, et al. Missing the mark: an assessment of paramedics’ ability to correctly perform drug calculations. *Journal of Emergency Medical Services*. 36(3):44, Mar 2011.

Mike Rubin, BS, NREMT-P is a paramedic in Nashville, TN, and a member of EMS World’s editorial advisory board. Contact him at mgr22@prodigy.net.

**Women Just as Good (or Bad) at Math as Men**

The common stereotype, “men are better at math” doesn’t stand up to scrutiny, at least according to one new study.

A recent University of Missouri review casts doubt on the accuracy of a popular theory that attempted to explain why there are more men than women in top levels of mathematic fields. The researchers found that numerous studies had major methodological flaws, utilized improper statistical techniques and many studies had no scientific evidence of this stereotype.

This theory, called stereotype threat, says that due to the stereotype that women are worse than men in math skills, females develop a poor self-image in this area, which leads to mathematics underachievement.

In the study, David Geary, curators professor of psychological sciences in the MU College of Arts and Science, and Giljsbert Stoet, from the University of Leeds in the United Kingdom, examined 20 influential replications of the original stereotype theory study. The researchers found that many subsequent studies had serious scientific flaws, including a lack of a male control group and improperly applied statistical techniques.

“We were surprised the researchers did not subject males to the same experimental manipulations as female participants,” Geary said. “It is reasonable to think that men also would not do well if told ‘men normally do worse on this test’ right before they take the test. When we adjusted the findings based on this and other statistical factors, we found little to no significant stereotype theory effect.”

Which means nature doesn’t distinguish between who is better at math, men or women. If you’re bad at math it’s probably just because you’re bad at math.