CALCULATION OF DRUG DOSAGES: DOSE CALCULATIONS, DIMENSIONAL ANALYSIS
In mathematics and science, dimensional analysis is a tool to understand the properties of physical quantities independent of the units used to measure them. Every physical quantity is some combination of mass, length, time, electric charge, and temperature, (denoted M, L, T, Q, and Θ (theta), respectively). For example, speed, which may be measured in meters per second (m/s) or miles per hour (mi/h), has the dimension L/T, or alternatively LT -1.
Dimensional analysis is routinely used to check the plausibility of derived equations and computations. It is also used to form reasonable hypotheses about complex physical situations that can be tested by experiment or by more developed theories of the phenomena, and to categorize types of physical quantities and units based on their relations to or dependence on other units, or their dimensions if any.
The basic principle of dimensional analysis was known to Isaac Newton (1686) who referred to it as the “Great Principle of Similitude“.[1] The 19th-century French mathematician Joseph Fourier made important contributions based on the idea that physical laws like F = ma should be independent of the units employed to measure the physical variables. This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually formalized in the Buckingham π theorem. This theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n − m dimensionless parameters, where m is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.
A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization
, which begins with dimensional analysis, and involves scaling quantities bycharacteristic
units of a system or natural units of nature. This gives insight into the fundamental properties of the system, as illustrated in the examples below.
When you’re doing applied math numbers have units of measure, or “dimensions,” attached to them. There are lots of formulas out there, but here’s the big idea: when you plug values into a formula and pay close attention to what happens to the units as the formula is simplified, you’ll see that all the units cancel out except those units that end up in your answer. This always happens if the formula is correct and you plug in the appropriate factors.
So what someone figured out is that you don’t need formulas at all. For every problem you can just take the factors associated with it and arrange them so all the units you don’t want cancel out. You’re then left with only the units you do want (the ones in your answer). This process is fairly trivial, and with only slight attention to detail, you always get the right answer, bing-bang-boom, every time.
The technique has been taught to students of applied science for longer than You have been able to determine and for the sole reason that students using it make fewer mistakes. You pay attention to the units of measure and if they’re not canceling out right, you know that you’re doing something wrong and that your answer is guaranteed to be wrong.
As nurses doing calculations, error is not an option. Passing med-math class may require getting only 80% of test problems right, but coming up with the right answer only four out of five times isn’t good enough when real patients are at risk. While mistakes can still be made using any technique, dimensional analysis does the best job of minimizing them. The only fault lies in the name. Perhaps the Math-Weenie-No-Brainer technique would be more appropriate. At any rate, give dimensional analysis a try. At the end of a 12-hour shift, when you’re tired, things are crazy, and you have to do a med-math calculation, you’ll be glad you did.
The following outline can be used as a guide for doing dimensional analysis (DA). Some familiarity with DA is assumed. See the 25 Med-math Problems Solved for an introduction to DA. While not all steps listed below will be needed to solve all problems, I have found that any problem that can be solved using DA will yield its answer if the following steps are followed. I would not suggest memorizing the sequence of steps, but rather understanding and practicing them. Understanding is more durable than memory.
1. Determine what you want to know. Read the problem and identify what you’re being asked to figure out, e.g. “how many milligrams are in a liter of solution.”
a.Rephrase if necessary using “per.” Example: You want to know “milligrams per liter.”
b. Translate into “math terms” using appropriate abbreviations to end up with “mg/L” as your answer unit (AU). Write this down, e.g. “AU= mg/L”
2. Determine what you already know.
a. What are you given by the problem, if anything? Example: “In one minute, you counted 45 drops.”
• Rephrase if necessary. Think: “Drip rate is 45 drops per minute.”
• Translate into math terms using abbreviations, e.g. “45 gtt/min”
If a given is in the form mg/kg/day, rewrite as mg/kg x day.
If a percentage is given, e.g. 25%, rewrite as 25/100 with appropriate labels.
b. Determine conversion factors that may be needed and write them in a form you can use, such as 60 min/1 hour.” You will need enough to form a “bridge” to your answer unit(s).• Factors known from memory: You may know that 1 kg = 2.2 lb, so write down “1 kg/2.2 lb” and/or “2.2 lb/1 kg” as conversion factors you may need.• Factors from a conversion table: If the table says “to convert from lb to kg multiply by 2.2,” then write down “2.2 lb/1 kg”
3. Setup the problem using only what you need to know.
a. Pick a starting factor.• If possible, pick from what you know a factor having one of the units that’s also in your answer unit and that’s in the right place.• Or pick a factor that is given, such as what the physician ordered.
• Note that the starting factor will always have at least one unit not in the desired answer unit(s) that will need to be changed by canceling it out.
b. Pick from what you know a conversion factor that cancels out a unit in the starting factor that you don’t want.
c. Keep picking from what you know factors that cancel out units you don’t want until you end up with only the units (answer units) you do want.
d. If you can’t get to what you want, try picking a different starting factor, or checking for a needed conversion factor.
e. If an intermediate result must be rounded to a whole number, such as drops/dose which can only be administered in whole drops, setup as a separate sub-problem, solve, then use the rounded off answer as a new starting factor.
4. Solve: Make sure all the units other than the answer units cancel out, then do the math.
a. Simplify the numbers by cancellation. If the same number is on the top and bottom, cancel them out.
b. Multiply all the top numbers together, then divide into that number all the bottom numbers.
c. Double check to make sure you didn’t press a wrong calculator key by dividing the first top number by the first bottom number, alternating until finished, then comparing the answer to the first one.Miskeying is a significant source of error, so always double check.
d. Round off the calculated answer.
• Be realistic. If you round off 74.733333 to 74.73 mL that implies that all measurements were of an extreme accuracy and that the answer is known to fall between 74.725 and 74.735, or 74.73 + 0.005 mL. A more realistic answer would probably be 74.7 mL or 75 mL.
See example 6.
• If you round to a whole number that implies a greater accuracy than is appropriate, write your answer to indicate a range, such as 75 + 5 mL.
e. Add labels (the answer unit) to the appropriately rounded number to get your answer. Compare units in answer to answer units recorded from first step.
5. Take a few seconds and ask yourself if the answer you came up with makes sense. If it doesn’t, start over.
There are, as is ofteoted, more than one technique for doing med-math problems. If the one you are using works for you, then don’t read any further. If, however, med-math is still a bit of a struggle, consider using the technique preferred by chemists, physicists, and engineers for decades called, somewhat intimidatingly, “dimensional analysis” hereafter referred to as “DA.”
Advantages include:
• One technique, not several
• Works with all problems
• No formulas to know, look up, or apply
• Problems are not solved piecemeal, but in one step
• You get to the right answer quicker—less error prone
• All calculations done at one time; no rounding errors
• You focus only on units of measure, not numbers, so math phobics can rejoice
• Stepwise approach makes solving almost all problems a virtual no-brainer
To illustrate I’ll do the sample problems we were given using DA. I’ll do the first one the long way, with explanations, then the rest as I would normally set them up.
Example 1:
The patient weighs 73 kg. The MD orders dopamine at 3 mcg/kg/min. The dopamine is mixed as 400 mg in 250 mL of solution. What is the infusion rate in mL/hr?
First you focus on what units of measure you want in the answer. In this problem we are kindly given “mL/hr.” We are also given that there are 400 mg dopamine in 250 mL (or 400 mg/250 mL), but also that in 250 mL there are 400 mg dopamine (or 250 mL/ 400 mg). It is important to realize that factors can be turned over or inverted as needed.
The other important bit to realize in order to do DA is that 3 mcg/kg/min can also be written as 3 mcg/(kg x min). This may seem a little weird, but if asked to divide 1/4 by 2 you have 1/4/2. But dividing is the same as inverting and multiplying, so inverting 2 to get 1/2 and multiplying you have 1/4 x 1/2, or 1/(4 x 2), or 1/8. Another example is acceleration, which is measured in ft/sec/sec. This can be written as ft/(sec x sec) or, more familiarly, as ft/sec2.
Since you want “mL” on top in your answer you won’t go wrong starting with 250 mL/400 mg as a logical starting factor.
250 mL
400 mg
You are now ready to play a game called “plug in other factors to cancel out the units you don’t want until you end up with the units you do want.” Here goes:
The horizontal bar means “divide,” as usual, and the vertical bars mean “multiply.” If the units cancel out properly, then your set up is correct and you can be quite sure the answer will be correct if you just manage to punch the right keys on your calculator. The most twisted of med-math problems devised by the most fiendish minds can be solved, bing-bang-boom, in this manner.
If this introduction to DA is too brief, visit the following nursing math website for more:http://www.alysion.org/dimensional/analysis.htm
Example 2:
The patient is receiving nitroprusside at 23 mL/hr. The bag has 50 mg of nitroprusside in 250 mL of solution. The patient weighs 67 kg. What dosage of nitroprusside in mcg/kg/min is the patient receiving?
Example 3:
Your patient receives an order for procainamide at 3 mg/min. She weighs 58 kg. The pharmacy has mixed 2 g of procainamide in 500 mL of solution. What is the infusion rate in mL/hr?
Note that weight is not used, extra details are often included in problems.
Example 4:
The patient weighs 117 pounds. Dopamine is running at 30 mL/hr. There is 400 mg of dopamine in 500 mL of solution. How much dopamine is the patient receiving in mcg/kg/min?
Since you want kg, a unit of weight (mass) on the bottom, starting with pounds on the bottom makes sense.
Example 5:
You need to start a continuous drip of amiodarone at 1 mg per minute (by pump). The standard IV mixture is 450 mg in 250 mL.
Pumps like to be programmed in mL/hr, so mL/hr are your answer units.
Example 6:
Milrinone Lactate (Primacor) has been ordered for a patient at 0.4 mcg/kg/min. The patient weighs 100 kg. If the pharmacy mixes 20 mg of Milrinone in 100 mL of total solution, what would be the rate of the infusion?
First, think what units will be in the answer. Since you’re using a pump, it’s mL/hr.
Example 7:
A patient has a confirmed pulmonary embolus and the physician has ordered a heparin drip. The initial rate is ordered at 1000 units per hour. Premixed heparin infusions are 25,000 units of heparin in 250 mL of D5W. What is the rate of the infusion?
You want mL/hr again, so start with mL on top.
Example 8:
An IV solution containing 2 grams of Lidocaine in 500 mL of D5W is infusing at 15 mL per hour. What should the ordered dose be in milligrams per minute?
• Don’t panic. Break THE PROBLEM down into small ones you CAN solve.
• Figure out what answer unit(s) you want to end up with. This is usually easy.
• Write down, in math terms, everything you know that relates to the problem. You may need to read the problem several times, rephrasing parts of it, so you can translate everything into math terms. You may need to look up a few conversion factors, but that’s inconvenient, not difficult.
• You now need to pick a starting factor. If possible pick one that already has one of the units you want in the right place. Otherwise start with something you are given that is not a conversion factor.
• Plug in conversion factors that allow you to cancel out any units you don’t want until you are left with only the units you do want (your answer units).
• If you can’t solve the problem, pick a different starting factor and start over.
• Do the math and solve it. Now double-check your calculations.
• Ask yourself if the answer seems right or reasonable. If not, recheck everything.
The steps for doing dimensional analysis are:
1. Determine the starting factor and answer unit.
2. Formulate a conversion equation.
3. Solve the conversion equation.
Determining the answer unit or units is crucial; they are not always obvious and can be challenging to determine. For some problems, reading the problem correctly is the only challenge. Students need to be able to translate sometimes convoluted English descriptions of a problem into clear, properly labeled factors they can later use to solve the problem. This skill is not emphasized in the textbook. If the answer unit is always given in the examples used, then this is because the examples have been contrived to be more simple and consistent than actual problems tend to be.
In some real-world problems no starting factor is given, or several possible starting factors are given with no way to decide, initially, which to use. It is preferable, in such cases, to determine everything you know that might be relevant to solving the problem, then decide, after the answer unit is determined, which of the factors you know would make an appropriate starting factor.
All examples used throughout the text use only numbers having a single unit attached for starting factors. Apparently “1 hour” is an acceptable starting unit, but “250 mL/hour” is not. This is not correct as starting factors are often in the form of “something per something.” Indeed, some problems cannot be solved if they have a single unit starting factor (see example 3 in Appendix A).
While many conversion factors are approximations, and fraction of a percent errors are unimportant in medication math, 10 percent errors are a bit worrisome. Equating 1 grain with 60 milligrams when the actual equivalency is closer to 64.8 mg, is questionable, as is equating liters and quarts, or 1 mL to 15 minims (actually 1 mL = 16.23 minims). It is possible to solve a problem and come up with answers that differ by as much as 10% depending on which approximate conversion factors you decide to use. If + 5% errors are acceptable, then, as an aside, any answer to a test question that is within 5% of the correct answer should be counted as correct. It is oddly inconsistent to insist on carrying out calculations to two decimals, rounding to the nearest tenth, when far greater errors can be introduced by using loose approximations.
When, in chapter 6, a problem involving amount/body weigh/day comes up, the solution is presented in an unorthodox way. The problem (p. 49) gives 25 mg/kg/24 hr. When doing dimensional analysis it is essential that all the units given should be used and accounted for. Ignoring a given unit, then pulling it out of thin air at the end is poor technique, yet this is what the textbook does. The solution is given as:
The problem is that the correct answer units should be how many mL should be administered per day, or “mL/day.” Omitting the “per day” part doesn’t alter the fact that that is what you want to know—not per hour, not per dose, but per day. There is actually a simple rule that applies here. For example, when acceleration is measured in feet per second per second, it is not written as ft/sec/sec, but as ft/sec 2 because ft/sec/sec is equal to ft/sec x sec. So if you’re given mg/kg/day, the preferred way to deal with such a “triple decker” is to rewrite it as mg/kg x day. In this form it can be used, all undesired units cancel, and you end up with the desired answer with the right units attached:
If the problem called for “mL/dose” given 4 doses per day, then the solution is straightforward:
If “day” were omitted, however, this problem would become more difficult to solve. The textbook method is to calculate “mL,” then divide by 4 to get “mL/dose.” Students must remember to perform this final “critically important” step which would not exist if better technique were used. As the text acknowledges, “it is easy to forget to divide the total daily dose into the prescribed number of doses, thus greatly increasing the risk of administering an overdosage (sic).”
Problems of this sort are common, and it is unfortunate that the authors neglect to show students how to logically deal with them. The risk of confusing some students by introducing a new rule can hardly be worth the risk of error introduced by teaching a flawed technique.
In Chapter 10, page 184, an example is shown, as a model for students to follow, to determine how many mcg/min must be administered to a 215 lb patient at 3 mcg/kg/min:
In this example, at least, minutes are not omitted then added at the end, and the technique is not even erroneous, but merely confusing to many students and visually awkward. A student might try to logically extend this technique to determine mL/hr:
The student who notices that the answer doesn’t make sense might wonder what went wrong. Would they realize that when “mcg” was cancelled that “3 1/min” was left requiring the use of 60 min/1 hr instead of 1 hr/60 min? Trying to explain how to work around the poor technique employed by this example only digs a deeper hole. The better response to student confusion would be to have them put a big X mark over this section of the textbook and show them a sensible way to set it up:
Another case of flawed technique arises in Chapter 10. Students are given problems that require converting from mL/hr to gtt/min, and are shown conversion equations like the following:
The problem, again, is that the correct answer unit is “gtt/min” and not “gtt” as it appears. The correct answer is just pulled out of nowhere and declared to be “33 gtt/min.” The initial starting factor of “1 min” is spurious. It is not a given, and it means absolutely nothing to say that you know “1 min” or “1 hour” or “1 cabbage.” If such meaningless starting factors are simply omitted from such examples, the problems are perfectly setup to yield the correct answers with the correct answer units. It seems that the pseudo-starting factor is used to avoid having a starting factor with more than one unit attached. As mentioned, however, there is no such requirement when doing dimension analysis. In the above example “90 mL/1 hr” would make a logical and perfectly good starting factor.
Students should be told to just ignore the nonsensical “1 min” and “1 hour” starting factors. If you were to introduce “1 hour” as a starting factor in example 3 in Appendix A, you would be committing mathematical suicide as the problem would be rendered unsolvable once “hour” is cancelled out.
Here’s an actual example from chapter 10:
Calculation of IV Flow Rate When Total Infusion Time is Specified
Order: 1000 mL of D5W (5% Dextrose in water) IV to infuse over a period of 5 hr
Drop Factor: 10 gtt/mL
Starting Factor Answer Unit
1 min gtt (drops)
Equivalents: 1000 mL = 5 hr, 10 gtt = 1 mL, 60 min = 1 hr
Conversion Equation:
1 min x 1 hr x 1000 mL x 10 gtt = 33.3 = 33 gtt
60 min 5 hr 1 mL
Flow Rate: 33 gtt/min
For review, let’s go over this problem.
1. There are two errors relating to the starting factor. One is procedural—there is no logical way to pick a starting factor as the first step. The other is that “1 min” is a meaningless factor. I can meaningfully say that I know there are 10 drops per mL, but it means nothing to say that I know “1 min” in the context of this problem.
2. The answer unit is wrong. I want to know a rate of flow in drops per some unit of time. Just “gtt” doesn’t cut it.
3. Factors are expressed as equalities. It should read “something per something” and not “something equals something” which leads to absurd statements like “25 mg = 1 kg”
4. By introducing a spurious starting factor the setup is in error, as is the resultant answer. The number is correct, but the answer unit is not.
5. The final statement, that the flow rate is 33 gtt/min, is the only part of the example that is correct, but it is logically disconnected from everything that precedes it.
So, let’s see, the text manages to state an incorrect answer unit, then introduces a spurious starting factor, which makes the setup wrong, which yields 33 gtt for an answer, which is also wrong. But through some sort of mental slight-of-mind, they finally come up with the correct answer, which they simply declare to be 33 gtt/min.
Is there a better way to do this problem? First ask, what do I want to know? The flow rate in gtt/min, which is my answer unit, not just gtt (drops). What do I know? I’m given that there are 10 gtt/mL and that the infusion rate is 1000 mL/5 hr. Since I want gtt on top and 10 gtt/mL has gtt in the right place, 10 gtt/mL makes a perfectly good starting factor—I just need to get from mL to min. My set up then:
10 gtt x 1000 mL x 1 hr = 33 gtt
1 mL 5 hr 60 min min
Just omitting the “1 min” from the textbook’s setup would also work.
As to what the authors might be thinking, the only clue to their reasoning was given in the following paragraph that preceded this example:
“In calculating the flow rate for drops per minute , one minute becomes the labeled value that must be converted to an equivalent value: number of drops. One minute , therefore, is the starting factor and drops is the answer unit and these, as in all dimensional analysis conversions, form an equivalent relationship.”
On page 9 is the following table:
Table 1-2 Conversion Equation
This table reveals how the authors think about dimensional analysis. They see the starting factor as something given; there can be only one starting factor; it has only one unit associated with it, and it forms a special “equivalent relationship” with the answer unit, which, being equivalent, must also have only a single unit associated with it. In between are conversion factors that are fundamentally different from the starting factor.
All of these assumptions are incorrect as generalizations about dimensional analysis. The only equivalent relationship is between what is on the left side of the equal sign and what is on the right side. One could speak of an equivalent relationship between the “numerator” and “denominator” of a conversion factor (2.2 lb/1 kg means 2.2 lb = 1 kg), but otherwise there is no necessary “equivalent relationship” implied.
There is a particular type of DA problem, the simple conversion problem, that does involve going from one unit of measure to another equivalent measure (such as converting from feet to meters). In this subtype of problem you have only one logical starting factor, which can be said to be equivalent to your answer (10 inches x 2.54 cm/1 inch = 25.4 cm), but such problems should not be taken as a model for all DA problems, which appears to be what has happened.
By the Commutative Law of Multiplication, it doesn’t matter what order the factors on the left side are multiplied in. Therefore any factor could be first, and thus be the starting factor, although usually only one or two factors qualify to be thought of as logical starting factors. Both starting factors and answer units are often in the form of something per something. You could start with miles/hour and end up with seconds in your answer, for example, without any equivalence between starting factor and answer unit.
It appears that such fundamental misunderstandings underlie the errors in the textbook. Problems that do not conform to their notions are tortured into compliance by introducing spurious starting factors and using obviously incorrect answer units. I don’t think it is going too far to suggest that the poor technique exhibited by the textbook makes it difficult for students to master med-math. Indeed, those who do must do so in spite of the textbook and not because of it.
Recommended Corrections to:
Clinical Calculations: A unified approach (4th ed.)
A Google search shows that only this textbook and a few nursing sites associate “label factor” with dimensional analysis (DA). Likewise “unit conversion” is not a synonym for DA. The only synonym commonly used is “factor-label method.” While this point is nit-picky, I would expect the authors to use the same terminology as everyone else by the 4th edition.
At the bottom, “Step I: Determining the Starting Factor and Answer Unit,” should read, “Step I: Determining the Answer Unit.” Determining the starting factor should come after Step II, since the starting factor is not always given, there can be more than one possible starting factor, and the best starting factor to use may be one of the factors determined in Step II. Picking a starting factor from what you are given or know is the first step of Step III—setting up/solving the conversion equation.
In the box is the statement: “When the conversion equation is solved, it will be seen that the starting factor and the labeled answer have formed an equivalent relationship.” The belief that this is true leads to serious error and confusion in Chapter 10. If true, the collorilary would be that if the starting factor has one unit of measure associated with it, then the answer unit can have only one unit of measure associated with it and vice versa .
Emphasize that several of the equivalents in the table are fairly rough approximations. Give the actual equivalents—some students will want to know. Also, if the value of an equivalent can be 5% off, then, to be consistent, any test answer that is within + 5% of the correct value should be counted correct. In some (unlikely) cases answers could be as much as 10% off when several approximate equivalents are used to compound the error.
In the example at the bottom of the page you are given 25 mg/kg/24-hr (or day). The third unit given should not be dropped. There is a way to deal with problems of this type (25 mg/kg/day = 25 mg/kg-day) that can be consistently applied to all problems of this type. Triple unit factors are common and the difficulty they pose should be dealt with head on. All the various ad hoc attempts to get around these problems result in endless trouble in the long run. In this example the answer unit is given as “mL,” whereas the correct answer unit is “mL/day.” The problem should be setup as:
Whatever initial difficulty this technique may present for students not already familiar with it, it is still the technique of choice and will save a lot of grief later on. Some of the techniques contrived to deal with these problems work on some problems, but not others. The technique used above has the virtue of working with all problems involving triple unit factors.
In the two examples on this page the Answer Unit is incorrectly given as “cap” whereas “cap/dose” is what is really desired. In the first example, you are given 50 mg/kg/day and 4 doses/day, but not knowing what to do with “mg/kg/day” the problem is broken into two problems. The “day” is initially ignored, then brought back in the second part of the problem, thus paving the way for confusion and error. The logically consistent one-step setup would be:
For the second example the setup should be:
In the box at the bottom on the page are several warnings (“critically important,” “easy to forget”) that do not apply when the problems are done in a single step.
Avoid the two-step technique, and ignore the two examples at the bottom of the page. Work out as above.
Cross out the second paragraph: “In calculating the flow rate for drops per minute , one minute becomes the labeled value that must be converted to an equivalent value: number of drops. One minute , therefore, is the starting factor and drops is the answer unit and these, as in all dimensional analysis conversions, form an equivalent relationship.”
Ignore examples. Omit the spurious “1 min” Starting Factors. Note that Answer Units are also wrong (should be “gtt/min,” not just “gtt“). All that needs to be done is to cross out the “1 min” at the beginning of each example and add “/min” to “gtt” (to get the correct answer unit).
Another ad hoc variation in technique is introduced without comment in step 1 of the first example. Students will get into trouble if they try to extend this example to other problems. Also, what if the desired answer units were “mcg/hr?” Would students have trouble canceling out “min” with “min” apparently on top? Putting “mcg/min” on top invites confusion. A better setup for step 1 would be:
For step 2:
For steps 3 and 4, just omit the “1 min.” and “1 gtt”
Cross out the meaningless Starting Factors in examples 1, 3, 4, 5, and 6. In example 2, change “mcg/min” over “kg” to “mcg” over “kg x min.”
In Example a., the setup is in error due to a failure to fully label units. The 10 mL is “10 mL water.” You have to ask, “10 mL of what?” Your answer unit is “mL Chloromycetin sol” and not just “mL.” You can’t use “mL water” and end up with “mL Chlor. sol.” When you add 10 mL water to reconstitute you will end up with somewhat more than 10 mL Chlor. solution. Since you want “mL Chlor. sol” in your answer, pick a factor that has “mL Chlor. sol” in it and in the right place. You are given “100 mg/mL” which should be more completely written as “100 mg Chlor./mL Chlor. sol” and “10 mL/g” should be “10 mL water/1 g Chlor.” which is quite an unnecessary bit of information for solving this problem, though the text incorrectly uses it (and by luck gets away with it). The correct setup should be:
Omit spurious Starting Factors from example.
The first example asks, “How many mL should the child receive per dose?” The answer unit, therefore, should be “mL/dose” and not “mL.” You are given 15 mcg/kg/dose, so solve as shown above for examples on pages 49 and 50—likewise with the second example on page 221.
Page 225: Again, example gives 50,000 U/kg/day and 4 doses/day, so a one-step setup would be:
That’s about it. The other 96% of the text is okay.
Textbook Guide to Dimensional Analysis
(as compiled from various pages throughout the textbook)
Determine the starting factor* and answer unit.
Initially, it is essential to determine exactly what information is sought: the known quantity called the starting factor , and the desired unit to which the starting factor will be converted, the answer unit.
When the conversion equation is solved, it will be seen that the starting factor and the labeled answer have formed an equivalent relationship.
In calculating the flow rate for drops per minute (or mL per hour) one minute (or one hour) becomes the labeled value that must be converted to an equivalent value: number of drops (or mL). One minute , therefore, is the starting factor and drops is the answer unit and these, as in all dimensional analysis conversions, form an equivalent relationship.
Formulate a conversion equation consisting of a sequence of labeled factors, in which successive units can be cancelled until the desired answer unit is reached.
If a given is in the form mg/kg/day, ignore the third unit, do the conversion, then remember to factor the omitted unit back in. If in the form mcg/kg/min, change to mcg/min over kg if mcg/min is the answer unit.
If a percentage is given, e.g. 25%, rewrite as 25/100 with appropriate labels.
Determine conversion factors that may be needed. You will need enough to form a “bridge” to your answer unit(s).
Use only conversion factors that have a 1:1 relationship
It is desirable that conversion factors be arranged in a sequence so that identical units are placed diagonally.
In setting up the conversion factors, it is helpful to write the denominator first, as this contains the unit of the preceding numerator and facilitates cancellation of successive units.
Solve the conversion equation by use of cancellation and simple arithmetic.
Cancel units first
Reduce numbers to lowest terms.
Multiply/divide to solve the equation.
Reduce answer to lowest terms, convert to decimal, and/or round off.
Take a few seconds and ask yourself if the answer you came up with makes sense. If it doesn’t, start over.
* The text in red represents weak or erroneous technique. Errors of omission are not indicated.
Conclusions
This may be a case of a book being the worst textbook on dimensional analysis available—with the exception of all the others. I’ve heard that it is much better than its predecessor. Several medication math textbook titles are currently available, but not having reviewed them, I can’t assume any do a better job, but I think other titles should be looked into.
There are errors of omission where students are not given a complete enough understanding of dimensional analysis to do all problems that could crop up. There are errors of commission where students are taught flawed or even erroneous technique. Throughout the textbook, overly simplified examples are used that fail to show the range of problems that students may encounter. A wider range of problems, however, would have illustrated the shortcomings of the techniques as taught, and may have been omitted for that reason.
Overall, however, I would say that this book is quite useable provided its shortcomings and flaws are amended. A better rounded, more robust presentation of dimensional analysis is definitely needed. Students should not only do well solving test problems, but come away feeling confident in their ability to handle any problems that may come their way in the future.
There are, as is ofteoted, more than one technique for doing med-math problems. If the one you are using works for you, then don’t read any further. If, however, med-math is still a bit of a struggle, consider using the technique preferred by chemists, physicists, and engineers for decades called, somewhat intimidatingly, “dimensional analysis” hereafter referred to as “DA.”
Advantages include:
• One technique, not several
• Works with all problems
• No formulas to know, look up, or apply
• Problems are not solved piecemeal, but in one step
• You get to the right answer quicker—less error prone
• All calculations done at one time; no rounding errors
• You focus only on units of measure, not numbers, so math phobics can rejoice
• Stepwise approach makes solving almost all problems a virtual no-brainer
To illustrate I’ll do the sample problems we were given using DA. I’ll do the first one the long way, with explanations, then the rest as I would normally set them up.
Example 1:
The patient weighs 73 kg. The MD orders dopamine at 3 mcg/kg/min. The dopamine is mixed as 400 mg in 250 mL of solution. What is the infusion rate in mL/hr?
First you focus on what units of measure you want in the answer. In this problem we are kindly given “mL/hr.” We are also given that there are 400 mg dopamine in 250 mL (or 400 mg/250 mL), but also that in 250 mL there are 400 mg dopamine (or 250 mL/ 400 mg). It is important to realize that factors can be turned over or inverted as needed.
The other important bit to realize in order to do DA is that 3 mcg/kg/min can also be written as 3 mcg/(kg x min). This may seem a little weird, but if asked to divide 1/4 by 2 you have 1/4/2. But dividing is the same as inverting and multiplying, so inverting 2 to get 1/2 and multiplying you have 1/4 x 1/2, or 1/(4 x 2), or 1/8. Another example is acceleration, which is measured in ft/sec/sec. This can be written as ft/(sec x sec) or, more familiarly, as ft/sec2.
Since you want “mL” on top in your answer you won’t go wrong starting with 250 mL/400 mg as a logical starting factor.
250 mL
400 mg
You are now ready to play a game called “plug in other factors to cancel out the units you don’t want until you end up with the units you do want.” Here goes:
The horizontal bar means “divide,” as usual, and the vertical bars mean “multiply.” If the units cancel out properly, then your set up is correct and you can be quite sure the answer will be correct if you just manage to punch the right keys on your calculator. The most twisted of med-math problems devised by the most fiendish minds can be solved, bing-bang-boom, in this manner.
If this introduction to DA is too brief, visit the following nursing math website for more:
http://www.alysion.org/dimensional/analysis.htm
Example 2:
The patient is receiving nitroprusside at 23 mL/hr. The bag has 50 mg of nitroprusside in 250 mL of solution. The patient weighs 67 kg. What dosage of nitroprusside in mcg/kg/min is the patient receiving?
Example 3:
Your patient receives an order for procainamide at 3 mg/min. She weighs 58 kg. The pharmacy has mixed 2 g of procainamide in 500 mL of solution. What is the infusion rate in mL/hr?
Note that weight is not used, extra details are often included in problems.
Example 4:
The patient weighs 117 pounds. Dopamine is running at 30 mL/hr. There is 400 mg of dopamine in 500 mL of solution. How much dopamine is the patient receiving in mcg/kg/min?
Since you want kg, a unit of weight (mass) on the bottom, starting with pounds on the bottom makes sense.
Example 5:
You need to start a continuous drip of amiodarone at 1 mg per minute (by pump). The standard IV mixture is 450 mg in 250 mL.
Pumps like to be programmed in mL/hr, so mL/hr are your answer units.
Example 6:
Milrinone Lactate (Primacor) has been ordered for a patient at 0.4 mcg/kg/min. The patient weighs 100 kg. If the pharmacy mixes 20 mg of Milrinone in 100 mL of total solution, what would be the rate of the infusion?
First, think what units will be in the answer. Since you’re using a pump, it’s mL/hr.
Example 7:
A patient has a confirmed pulmonary embolus and the physician has ordered a heparin drip. The initial rate is ordered at 1000 units per hour. Premixed heparin infusions are 25,000 units of heparin in 250 mL of D5W. What is the rate of the infusion?
You want mL/hr again, so start with mL on top.
Example 8:
An IV solution containing 2 grams of Lidocaine in 500 mL of D5W is infusing at 15 mL per hour. What should the ordered dose be in milligrams per minute?
CALCULATING IV DRUG DOSES
Principles
1. For drugs already in solution, check the amount of drug in each ml and the total amount of drug in the container.
2.Make sure you are clear about the dose units used.Most commonly prescribed are milligrams (mg) or micrograms.
3. Beware of drugs such as insulin and heparin, for which doses are prescribed in international units (which is sometimes, but should never be abbreviated to i.u. which can be misread as 10).
4. Check the dose on the prescription and that it is expressed in the same units as on the medicine label.
5. If the prescription and the medicines label use different units of strength, refer to the conversion table and calculation examples on page 4 and 5.
6. Once you are sure that the units are the same, divide the required dose by the amount of the drug in the ampoule and multiply by the volume of solution in the vial or ampoule.
7. The answer is the volume needed for each dose.
CALCULATING DRIP RATES FOR GRAVITY FLOW INFUSIONS
PRINCIPLES
1. Without a flow control device such as a pump, infusion rates depend entirely on gravity. Rate of flow is measured by counting drops per minute.
2. Administration sets deliver controlled amounts of fluid at a predetermined fixed rate, measured in drops per minute.
3. It is also important to check the number of drops per ml delivered by the administration set (which is printed on the outer packaging). This may vary between sets, between manufacturers and between different infusion fluids or drug solutions.
4. A (drug) solution administration set will usually deliver 20 drops per ml of clear infusion fluid such as 0.9% sodium chloride injection.
5. A blood administration set will deliver 15 drops per ml of blood.
6. A burette administration set will usually deliver 60 drops per ml of infusion fluid or drug solution.
CALCULATING INFUSION RATES FOR INFUSION DEVICES
1. All infusions require rate control. This can be achieved using a roller clamp (gravity flow), an infusion pump, a syringe driver, a syringe pump or a disposable device.
2. When using any sort of rate control device, check at least the following parameters at regular intervals in accordance with local policy:
3. • Volume given
4. • Volume remaining
5. • Administration rate
6. • Condition of the patient including the administration site.
7. Before and after transfer of care between units or teams, make sure you repeat the above checks.
8. You should always check the manufacturer’s instructions or refer to local policy to ensure you use the correct administration set for the device and that the device is programmed correctly.
9. An administration device should only be used by practitioners who have been trained and are competent in the use of the particular device.
10. The rate may be prescribed in terms of:
11. Volume: For example ml per hour or ml per min OR amount of drug:
For example mg per min or international units per hour.
CALCULATING RATES FOR SYRINGE DRIVERS
A syringe driver pushes the plunger of a syringe forward at an accurately controlled rate.
• For most syringe pumps the rate is set according to the volume of solution injected per hour i.e. in ml per hour.
• For some syringe drivers the rate is set according to the distance travelled by the plunger in mm per hour or mm per 24 hour.
If the rate is to be set in mm, the volume to be adminstered by a syringe driver depends on the diameter of the syringe barrel as well as on the rate setting. Different makes of syringe may have different barrel sizes. It is essential that the brand of syringe to be used is specified and the stroke length is measured.
Serious errors have occurred when settings in mm per hour and ml per hour have been confused.
1. Prepare prescribed infusion.
2. Prime the extension set with fluid.
3. If using a syringe driver, measure the stroke length (the distance the plunger has to travel) in mm.
4. Check carefully the units of time in which the syringe driver operates:
Is the rate set in mm per hour or mm per day (24 hours)?
See diagram for example.
CALCULATING IV DRUG DOSES FOR CHILDREN AND NEONATES
1. Remember that many injections are made for adults. For children’s doses, you may need as little as one tenth (1/10) or even one hundredth (1/100) of the contents of one ampoule or vial.
2.When calculating infusions, consider the child’s total daily fluid allowance.More concentrated individual infusions may be required. Discuss with the pharmacist or prescriber.
3. Ensure the prescribed infusion fluid/diluent is appropriate for the child e.g. the sodium content of an infusion contributes to the child’s total daily sodium requirement.
4. Displacement Values.
For many injections presented as powders for reconstitution, the powder adds to the volume of the final solution after the diluent has been added. This ‘displacement value’ must be taken into account when the dose needed is less than the full contents of the vial or ampoule.
The displacement value can be found on the package insert. It may vary with brands, so it is crucial to check the package insert for the product you are actually using.
5. For the above reasons, the calculations involved in preparing and administering infusions for children are often particularly complex. It is most important that these calculations are independently checked
DRUGS CALCULATION.
There are several ways to determine how much of a medication you are supposed to administer to a patient. No matter what method you choose to use, if performed properly, they should all come up with the same answer. Following are three methods for determining the appropriate dose based on information that you have available to you.
Method 1
The first method is based on the following formula:
An Example: Medical control orders you to administer 5 mg of morphine sulfate IV to your 84-year-old female patient who has signs and symptoms of a hip fracture. The morphine in your formulary contains 10 mg in 1 mL. How many milliliters of morphine sulfate do you need to administer to this patient in order to deliver 5 mg?
You have the following information:
Order: 5 mg morphine sulfate IV
On hand: 10 mg/1 mL
Fill in the formula:
Cancel any common values (volumes or concentrations) that exist on the top and on the bottom, and multiply across the top.
You need to administer 0.5 mL of morphine sulfate to your patient.
Method 2
This second method involves ratio and proportion. The symbol for proportion is and the symbol for ratio is using the same problem as in method 1, start with the known ratio on the left side of the proportion:
Place the unknown ratio on the right side of the proportion in the same sequence as the ratio on the left side of the proportion. This ratio is usually the physician order or the dosage that you are permitted to administer based on standing orders:
First, multiply the extremes ( the far outside values: 10 mg and X mL) and place the result on the left side of the equation. Second, multiply the means ( the numbers on either side of the proportion symbol: 1 mL and 5 mg) and place this value on the right side of the expression:
Multiply:
Divide both sides by the number in front of the X
You need to administer 0.5 mL of morphine sulfate to your patient.
Method 3
The third method is referred to as the cross multiplication method. This method sets the problem up using fractions. The fi rst fraction is the concentration, and the second fraction is the physician’s order over the volume of medication being administered.
Cross multiply the fractions by multiplying numerators by the denominator on the opposite side. Express the results as an algebraic equation the same as used in the proportion method.
You need to administer 0.5 mL of morphine sulfate to your patient.
Many nurses are weak with drug calculations of all sorts. This article will help to review the major concepts related to drug calculations, help walk you through a few exercises, and provide a few exercises you can perform on your own to check your skills. There are many reference books available to review basic math skills, if you find that you have difficulty with even the basic conversion exercises.
TOP CALCULATION TIPS
Drugs are formulated into medicines in such a way that most adult doses are easily calculated and predictable, e.g.1 or 2 tablets, 1 or 2 capsules, 1 vial or ampoule of an injectable medicine, 1 suppository.
Before doing a calculation, it is sensible to estimate the dose you are likely to require so that you know whether your calculated answer seems reasonable i.e. roughly what you expected.
To check doses use a reliable reference source, such as the BNF or BNF for Children.
For recommended administration methods, see local drug policies or national guides such as The IV Guide.
Dose volumes of oral liquid medicines are typically 5-20mls for adults and 5mls or less for children.
Crushing tablets should be avoided wherever possible. Some tablets, such as ‘modified release’ products should never be crushed.Always ask your pharmacists’ advice before crushing tablets. If itmust be done, a pestle and mortar or tablet crusher should be used and the tablet ground to as fine a powder as possible.
Always check childrens’ and babies’ weights carefully.Make sure they are weighed in kg and that their weight is recorded in kg.
If a calculation using weight or surface area gives an answer equivalent to or greater than the normal adult dose, reconfirm that it is what is really required.
If you are in any doubt about a calculation, stop, and contact the ward pharmacist, an on-call pharmacist or the prescriber.
Principles
The way the strength of a drug in a solution is described will affect the way a dose calculation is carried out
Doses may be expressed in a number of different ways:
1.Mass (weight) per volume of solution e.g. mg in 10ml, mMol/L.
2. Units of activity per volume of solution e.g. units per ml.
3. Percentage This is the weight of the drug in grams that is contained in each 100ml of the solution. Common examples are 0.9% sodium chloride; 5% glucose
If you know the number of grams in 1000ml, divide by 10 to convert to % strength.
If you know the % strength, multiply by 10 to give the number of grams of drug in 1000ml.
If you know the % strength, divide by 100 to calculate the amount of drug in 1ml.
4. Ratios Strengths of some drugs such as adrenaline (epinephrine) are commonly expressed in ratios
CALCULATING ORAL DOSES IN TABLETS
Principles
1. Check the strength of (amount of drug in) each tablet or capsule.
2.Make sure you are clear about the dose units used, most commonly prescribed are milligrams or micrograms.
3. Check the dose on the prescription and that it is expressed in the same units as on the medicine label.
4. If the prescription and the medicines label use different units of strength, refer to the conversion table and calculation examples on page 4 and 5.
5. Once you are sure that the units are the same, divide the required dose by the strength of the tablet or capsule.
6. The answer is the number of tablets/capsules needed for each dose.
Extra safety tip
If your first calculation gives a dose of more than two tablets, double check the calculation and confirm thatthe dose doesn’t exceed the manufacturer’s recommended maximum. If it does, or if you are still unsure that the dose is correct, check with the prescriber or pharmacist.
CALCULATING ORAL DOSES FOR CHILDREN AND NEONATES
Principles
1. Always use the smallest oral syringe that will hold the volume you need to measure.
2. If the dose prescribed means that less than a whole tablet or capsule is required, check with the pharmacy that it is appropriate to break a tablet or split a capsule before doing so.
3. If it is essential, dissolve or disperse the powder/crushed tablet in an accurately measured amount of water (e.g. 5ml). Stir and draw up the required volume immediately
4. If the result cannot be accurately measured e.g. 0.33ml, it is generally acceptable to round the dose up or down. However, the actual dose given must be within 10% of the calculated dose. If this cannot be achieved, discuss with the prescriber and pharmacist.
mcg/kg/min
If a certain dose is ordered:
_______ mcg/kg/min X _______ kg ÷ ______ mcg/ML = ______ mL/h
Dosage Pt weight concentration pump setting
If infusion pump is already set and running:
______ mcg/Ml X ______ mL/h ÷ 60 ÷ ______ kg = ______ mcg/kg/min
Concentration pump setting Pt weight dosage
mcg/min
If a certain dose is ordered:
_______ mcg/min X 60 ÷ ______ mcg/mL = ______ mL/h
Dosage concentration pump setting
If infusion pump is already set and running:
______ mcg/mL X ______ mL/h ÷ 60 = ______ mcg/min
Concentration pump setting dosage
mg/min
If a certain dose is ordered:
_______ mg/min X 60 ÷ ______ mg/mL = ______ mL/h
Dosage concentration pump setting
If infusion pump is already set and running:
______ mg/mL X ______ mL/h ÷ 60 = ______ mg/min
Concentration pump setting dosage
mg/h
If a certain dose is ordered:
_______ mg/h ÷ ______ mg/mL = ______ mL/h
Dosage concentration pump setting
If infusion pump is already set and running:
______ mg/mL X ______ mL/h = ______ mg/h
Concentration pump setting dosage
Most drugs in children are dosed according to body weight (mg/kg) or body surface area (BSA) (mg/m2). Care must be taken to properly convert body weight from pounds to kilograms (1 kg= 2.2 lb) before calculating doses based on body weight. Doses are often expressed as mg/kg/day or mg/kg/dose, therefore orders written “mg/kg/d” which is confusing, require further clarification from the prescriber.
Chemotherapeutic drugs are commonly dosed according to body surface area which requires an extra verification step (BSA calculation) prior to dosing. Medications are available in multiple concentrations, therefore orders written in “mL” rather than “mg” are not acceptable and require further clarification.
Dosing also varies by indication, therefore diagnostic information is helpful when calculating doses. The following examples are typically encountered when dosing medication in children.
· purpose: body surface area is used in the medical field when calculating drug doses, though is relevant to sports when looking at responses to the heat and cold.
· equipment required: scales for measuring weight, stadiometer for measuring height, calculator for working out the formula.
· procedure: determine height and weight using standard procedures. Use the relevant formula (whether you used kg/cm or lbs/in). A calculator is available to convert cm and inches and convert kg and lbs.
· formula: This is the formula by Mosteller (1987):
If using cm and kilograms:
BSA (m²) = ( [Height(cm) x Weight(kg) ]/ 3600 )^½
e.g. BSA = SQRT( (cm*kg)/3600 )
If using inches and pounds:
BSA (m²) = ( [Height(in) x Weight(lbs) ]/ 3131 )^½
Example 1.
Calculate the dose of amoxicillin suspension in mLs for otitis media for a 1-yr-old child weighing 22 lb. The dose required is 40 mg/kg/day divided BID and the suspension comes in a concentration of 400 mg/5 mL.
Example 2.
Calculate the dose of ceftriaxone in mLs for meningitis for a 5-yr-old weighing 18 kg. The dose required is 100 mg/kg/day given IV once daily and the drug comes pre-diluted in a concentration of 40 mg/mL.
Example 3.
Calculate the dose of vincristine in mLs for a 4-yr-old with leukemia weighing 37 lb and is 97 cm tall. The dose required in 2 mg/m2 and the drug comes in 1 mg/mL concentration.
HOW TO CALCULATE CONTINUOUS INFUSIONS
mg/min (For example – Lidocaine, Pronestyl)
mcg/min (For example – Nitroglycerin)
mcg/kg/min (For example – Dopamine, Dobutamine, Nipride, etc.)
To calculate cc/hr (gtts/min)
TO CALCULATE MCG/KG/MIN
Example:
Nipride 100 mg/250 cc D5W was ordered to decrease your patient’s blood pressure.
The patient’s weight is 143 lbs, and the IV pump is set at 25 cc/hr. How many mcg/kg/min of Nipride is the patient receiving?
How to calculate mcg/kg/min if you know the rate of the infusion
For example:
26.6 is the dosage concentration for Dopamine in mcg/cc/min based on having 400 mg in 250 cc of IV fluid. You need this to calculate this dosage concentration first for all drug calculations. Once you do this step, you can do anything!
NOW DO THE REST!
If you have a 75 kg patient for example…
How to calculate drips in cc per hour when you know the mcg/kg/min that is ordered or desired
For example:
400 mg Dopamine in 250 cc D5W = 26.6 mcg/cc/min
ALWAYS WORK THE EQUATION BACKWARDS AGAIN TO DOUBLE CHECK YOUR MATH!
For example:
For example:
400mg of Dopamine in 250 cc D5W = 1600 mcg/cc 60 min/hr = 26.6 mcg/cc/min
26.6 is the dosage concentration for Dopamine in mcg/cc/min based on having 400 mg in 250 cc of IV fluid. You need this to calculate this dosage concentration first for all drug calculations. Once you do this step, you can do anything!
NOW DO THE REST!!
If you have a 75 kg patient for example
The concentration of a chemical solution refers to the amount ofsolute
that is dissolved in a solvent. We normally think of a solute as a solid that is added to a solvent (e.g., adding table salt to water), but the solute could just as easily exist in another phase. For example, if we add a small amount of ethanol to water, then the ethanol is the solute and the water is the solvent. If we add a smaller amount of water to a larger amount of ethanol, then the water could be the solute!
Units of Concentration
Once you have identified the solute and solvent in a solution, you are ready to determine its concentration. Concentration may be expressed several different ways, usingpercent composition by mass, volume percent,mole fraction, molarity,molality, or normality.
1. Percent Composition by Mass (%)
This is the mass of the solute divided by the mass of the solution (mass of solute plus mass of solvent), multiplied by 100.
Example:
Determine the percent composition by mass of a 100 g salt solution which contains 20 g salt.
1. Solution:
20 g NaCl / 100 g solution x 100 = 20% NaCl solution
2. Volume Percent (% v/v)
Volume percent or volume/volume percent most often is used when preparing solutions of liquids. Volume percent is defined as:
v/v % = [(volume of solute)/(volume of solution)] x 100%
Note that volume percent is relative to volume of solution, not volume of solvent. For example, wine is about 12% v/v ethanol. This means there are 12 ml ethanol for every 100 ml of wine. It is important to realize liqud and gas volumes are not necessarily additive. If you mix 12 ml of ethanol and 100 ml of wine, you will get less than 112 ml of solution.
As another example. 70% v/v rubbing alcohol may be prepared by taking 700 ml of isopropyl alcohol and adding sufficient water to obtain 1000 ml of solution (which will not be 300 ml).
3. Mole Fraction (X)
This is the number of moles of a compound divided by the total number of moles of all chemical species in the solution. Keep in mind, the sum of all mole fractions in a solution always equals 1.
Example:
What are the mole fractions of the components of the solution formed when 92 g glycerol is mixed with 90 g water? (molecular weight water = 18; molecular weight of glycerol = 92)
Solution:
90 g water = 90 g x 1 mol / 18 g = 5 mol water
92 g glycerol = 92 g x 1 mol / 92 g = 1 mol glycerol
total mol = 5 + 1 = 6 mol
xwater = 5 mol / 6 mol = 0.833
x glycerol = 1 mol / 6 mol = 0.167
It’s a good idea to check your math by making sure the mole fractions add up to 1:
xwater + xglycerol = .833 + 0.167 = 1.000
4. Molarity (M)
Molarity is probably the most commonly used unit of concentration. It is the number of moles of solute per liter of solution (not necessarily the same as the volume of solvent!).
Example:
What is the molarity of a solution made when water is added to 11 g CaCl2 to make 100 mL of solution?
Solution:
11 g CaCl2 / (110 g CaCl2 / mol CaCl2) = 0.10 mol CaCl2
100 mL x 1 L / 1000 mL = 0.10 L
molarity = 0.10 mol / 0.10 L
molarity = 1.0 M
5. Molality (m)
Molality is the number of moles of solute per kilogram of solvent. Because the density of water at 25°C is about 1 kilogram per liter, molality is approximately equal to molarity for dilute aqueous solutions at this temperature. This is a useful approximation, but remember that it is only an approximation and doesn’t apply when the solution is at a different temperature, isn’t dilute, or uses a solvent other than water.
Example:
What is the molality of a solution of 10 g NaOH in 500 g water?
Solution:
10 g NaOH / (40 g NaOH / 1 mol NaOH) = 0.25 mol NaOH
500 g water x 1 kg / 1000 g = 0.50 kg water
molality = 0.25 mol / 0.50 kg
molality = 0.05 M / kg
molality = 0.50 m
6. Normality (N)
Normality is equal to the gram equivalent weight of a solute per liter of solution. A gram equivalent weight or equivalent is a measure of the reactive capcity of a given molecule. Normality is the only concentration unit that is reaction dependent.
Example:
1 M sulfuric acid (H2SO4) is 2 N for acid-base reactions because each mole of sulfuric acid provides 2 moles of H+ ions. On the other hand, 1 M sulfuric acid is 1 N for sulfate precipitation, since 1 mole of sulfuric acid provides 1 mole of sulfate ions.
Dilutions
You dilute a solution whenever you add solvent to a solution. Adding solvent results in a solution of lower concentration. You can calculate the concentration of a solution following a dilution by applying this equation:
MiVi = MfVf
where M is molarity, V is volume, and the subscripts i and f refer to the initial and final values.
Example:
How many millilieters of 5.5 M NaOH are needed to prepare 300 mL of 1.2 M NaOH?
Solution:
5.5 M x V1 = 1.2 M x 0.3 L
V1 = 1.2 M x 0.3 L / 5.5 M
V1 = 0.065 L
V1 = 65 mL
So, to prepare the 1.2 M NaOH solution, you pour 65 mL of 5.5 M NaOH into your container and add water to get 300 mL final volume.
INFORMATION FOR REFRESHING YOUR KNOWELAGE
The health professional must be meticulous in converting those measurements into the proper amount of liquid or solid medication that a patient requires. This takes converting from or within one or more of the three systems of measurements that health professionals use.
The Apothecary system is one of the oldest system of pharmacologic measurements. It’s expressed in romaumerals and special symbols. A unit of liquid measure is a minim and a unit for weight is a grain. You may even see some romaumeric symbols written in medication orders. Last, the household system is less accurate measurement that is based upon drops, teaspoons, tablespoons, cups and glasses.
Metric Conversions
Apothecary Conversions
Other symbols that the Apothecary system uses are: ss = ½, i = one, and ounces=
Household Conversions
Medications are prescribed in a specific amount or weight per volume. For example a single tablet has 100mg of medication; the volume of that tablet is 1. A medication that comes in 80 mg per 2mL has a volume of 2. Some liquid medications are prescribed as volume alone because they are only available in one strength. Common formulas for calculating medications are ratio and proportion, “desired over have”, and dimensional analysis. If you can remember these basic conversions, you should be able to pass these quarterly tests and use this knowledge throughout your nursing career.
The metric system of measurement is the most widely used system of measurement in the world. It is the preferred system for administering medications, because it is based on a series of 10 measures or multiples of 10. It is a simple and accurate form of measurement between health care professionals.
Metric Weight Measures
1 kilogram (kg, Kg) = 1000 grams or 1000 g
1 gram = 1000 milligrams or 1000 mg
1 milligram (mg) = 1000 micrograms or 1000 mcg
1 microgram (mcg) = 0.001 milligrams or 0.001 mg
1 milligram = 0.001 gram or 0.001 g
1 microgram (mcg) = 0.000001 gram or 0.000001 g
Metric Volume Measures
1 milliliter (ml) = 0.001 liter or 0.001 L
1 liter = 1000 milliliters or 1000 ml
1kiloliter = 1000 liters or 1000 L
Metric Length Measures
1 millimeter (mm) = 0.001 meter
1 centimeter (cm) = 0.01 meter or 0.01 m
1 decimeter (dm) = 0.1 meter or 0.1 m
1 kilometer (km) = 1000 meters or 1000 m
1 meter (m) = 100 centimeters or 100 cm
1 meter (m) = 1000 millimeters or 1000 mm
1 centimeter (cm) = 10 millimeters or 10 mm
The apothecaries system of measurement is the oldest system of drug measurement. In fact, it was the first system used to measure medication amounts. It is infrequently used as a drug measurement. There are a few medications that are still measured in grains (gr). To ensure administration of the correct dose of medication to a patient, it is important to know the conversion of grains to milligrams and how to convert from one system of measurement to another.
The household system of measurement is based on the apothecary system of measures. Household measures are used to measure liquid medications. Parents understand one teaspoon of liquid medication more clearly than ordering 15 milliliters. The important thing to realize with household measures is that these measures are not as exact as the metric system of measurement. Also the comparison of metric measures to household measures are not equal. These measures are called equivalent measures because the measurement is close enough. A liter is very close in equal measurement to a quart. It is not an exact measure. It is an equivalent.
Summary
Many nurses have difficulty with drug calculations. Mostly because they don’t enjoy or understand math. Practicing drug calculations will help nurses develop stronger and more confident math skills. Many drugs require some type of calculation prior to administration. The drug calculations range in complexity from requiring a simple conversion calculation to a more complex calculation for drugs administered by mcg/kg/min. Regardless of the drug to be administered, careful and accurate calculations are important to help prevent medication errors. Many nurses become overwhelmed when performing the drug calculations, when they require multiple steps or involve life-threatening drugs. The main principle is to remain focused on what you are doing and try to not let outside distractions cause you to make a error in calculations. It is always a good idea to have another nurse double check your calculations. Sometimes nurses have difficulty calculating dosages on drugs that are potentially life threatening. This is often because they become focused on the actual drug and the possible consequences of an error in calculation. The best way to prevent this is to remember that the drug calculations are performed the same way regardless of what the drug is. For example, whether the infusion is a big bag of vitamins or a life threatening vasoactive cardiac drug, the calculation is done exactly the same way.
Many facilities use monitors to calculate the infusion rates, by plugging the numbers in the computer or monitor with a keypad and getting the exact infusion titration chart specifically for that patient. If you use this method for beginning your infusions and titrating the infusion rates, be very careful that you have entered the correct data to obtain the chart. Many errors take place because erroneous data is first entered and not identified. The nurses then titrate the drugs or administer the drugs based on an incorrect chart. A method to help prevent errors with this type of system is to have another nurse double check the data and the chart, or to do a hand calculation for comparison. The use of computers for drug calculations also causes nurses to get “rusty” in their abilities to perform drug calculations. It is suggested that the nurse perform the hand calculations from time to time, to maintain her/his math skills.
