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Math for Radiographers

Robert H. Posteraro, MD, MBI, FACR

Observation

•Many radiography students have difficulty solving calculation problems … especially “word” problems

What’s *not* the issue

Knowing the formulas

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What are the issues?

•Setting up the problem•Evaluating the “answer” … using critical thinking skills

Why is math important in radiography?

• It’s fundamental to understanding radiation physics

• It’s used to convert traditional units to SI units

• It’s used in calculations of spatial resolution (object size vs lp/mm)

• It’s used to calculate file size to determine memory requirements for storing digital images

• It’s used to calculate exposure

• It’s used to calculate shielding requirements

• It’s used to calculate nuclear medicine dosages

• It’s on the Registry exam

What you need to know• Basic algebra

• Positive and negative numbers• Absolute value• Identities (0 is the additive identity; 1 is the multiplicative identity)

• Inverses (additive inverse; multiplicative inverse)• Operations (+, - , x, /)• Associative, commutative, and distributive properties

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More of what you need to know

•Maintaining equality in an equation•Cross multiplying•Isolating variables•Simplifying expressions

I’ll assume that you’re familiar with those

Still more of what you need to know

•Powers and roots•Logarithms (base 10 and natural logs)•Order of operations – “Please excuse my dear aunt Sally”

•Working with parentheses and brackets and why they’re needed at times

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Powers and roots• A number raised to a power (the exponent) means multiply that number times itself however many times is indicated by the exponent:

34 = 3 x 3 x 3 x 3 = 8128 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256

• N.B. – a power (exponent) can be indicated by a superscript, or by using the caret symbol ‘^’ (e.g., 34 is the same as 3^4 = 81)

•There are some powers that have special names:

Squared (the power is ‘2’)

9 squared = 92 = 9 x 9 = 81

Cubed (the power is ‘3’)6 cubed = 63 = 6 x 6 x 6 = 216

Why “squared”?• Well, if you have a square and multiply the length of a side

times itself, you get the area of the square

b

bb x b = b2 = the area of the square

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Why “cubed”?• If you have a cube and multiply the length of a side

times itself, and times itself again, you’ll get the volume of the cube

aa

a

a x a x a = a3 = the volume of the cube

Where do we see powers in radiography?

•Photomultiplier tube gain

PM gain = gn where ‘g’ is the gain of one dynode and ‘n’ is the number of dynodes in the PM tube

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Where else do we see powers in radiography?

• Single-target, single-hit model of cell survivability

S = N / N0 = e–D/D37

Where,S = surviving fractionN = number of cells that survived dose DN0 = initial number of cellsD = the dose of radiation applied to the cellsD37 = a constant related to the cell radiosensitivitye = Euler’s number, the constant, 2.71828…, the

base of the natural logarithms

• Multi-target, single-hit model of cell survivability

• S = N/N0 = 1 - (1 – e-D/D0)n

Where,S = surviving fractionN = number of cells that survived dose DN0 = initial number of cellsD = the dose administeredD0 = the dose that would result in an average of one hit per

targetn = the “extrapolation number”e = Euler’s number, the constant, 2.71828…

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Roots• Taking a root is the reverse of raising a number to a power.

Given a number (say ‘Y’), and a power, “taking a root” answers the question, “What is the number (X) which, when raised to that power, will give me the number Y?”

• Example:What is the 4th root of 625? That is, what number do I

have to raise to the 4th power to give me 625?Ans: 5

54 = 5 x 5 x 5 x 5 = 625

Where do we see roots in radiography?

• The formula for noise in a CT systemNoise () = sqrt {[ (xi – xav)2 ] / (n – 1)}

Where, = Greek uppercase sigma, the symbol meaning “sum” (add ‘em all up)xi = the CT value of each pixelxav = the average of a number (> 100) of CT pixel valuesn = the number of CT pixel values that were averaged

Powers and roots, interesting facts

•You can have powers and roots that are not whole numbers!

(How can you multiply something times itself a partial number of times???? But you can!)

32.4 = 13.9666 …

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Powers and roots, interesting facts

• You can have negative powers and roots!(How can you multiply something times itself a negative number of times??? But you can!) Remember the formula for the single-target, single-hit model of cell survivability? S = N / N0 = e–D/D37)

E.g., 2-3 = 0.125

Powers and roots, interesting facts

•A number raised to a negative power is the same as 1 divided by the number raised to that positive power

5-2 is the same as1 / 52 = 1 / 25 = 0.04

FWIW, the opposite is also true!

•A number raised to a positive power is the same as 1 divided by the number raised to that negative power

34 is the same as1 / 3-4 = 81

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***It’s important to become familiar with the calculator that you use and know how to enter the operations for raising a number to a power, for taking the root of a number, for calculating using ‘e’, and for calculating logarithms

During an exam is not the time to figure that out

Logarithms (logs)

What is a logarithm?

“A logarithm is when you enter a number in your calculator and you press the button that says ‘LOG’ it gives you a different number.”

--- Actual answer by a student in a college biochemistry class!

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Logarithms (logs)• A logarithm is the power to which a fixed number (called the

‘base’) must be raised to produce a given number

• Any number can be the ‘base’. If no base is indicated, the word “LOG” implies that the base is ’10’. If the indication for the logarithm is ‘ln’, that’s “natural logarithm” and the base is the number 2.71828 …, also known as ‘e’, which stands for “Euler’s number” (remember that from the single-target, single-hit and multi-target, single-hit cell survivability formulas?)

Logarithms (logs)

•Example:•What is the log of 10,000,000? That is, what power do I have to raise ’10’ to in order to get 10,000,000?

•10? = 10,000,000•On your calculator: 10,000,000 > LOG = 7•107 = 10,000,000

• What is the log of 97? That is, what power do I have to raise ’10’ to to get 97?

• On your calculator: 97 > LOG = 1.9867717 …

101.9867717 … = 97

• This should be understandable. Since 102 = 100, and 97 is just a little bit less than100, the power that you raise 10 to in order to get 97 will be a number that’s just a little less than ‘2’

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Where do we see logs in radiography?

• The formula for optical density (OD)

OD = log (Io / It)

Where,Io = light intensity incident to the processed filmIt = light intensity transmitted through that

processed film

Working through equations

Order of Operations

•“Please excuse my dear aunt Sally.”

•Parentheses (brackets, and braces … they work from the inside out if they’re “stacked”)

•Exponents (and roots)•Multiplication and Division•Addition and Subtraction

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2 + 5 x 6 / 3 – 7 = ???? What do you do first??

There is no indication of what operation we do first, so we follow the order of operations and do multiplication and division first, then addition and subtraction, and we go from left to right.

2 + 5 x 6 / 3 – 7 = ?= 2 + 30 / 3 – 7 =?= 2 + 10 – 7 = 5

Multiplication and division have equal precedence, so we could have done the multiplication & division in the reverse order without changing anything

2 + 5 x 6 / 3 – 7 = ?; do the division first= 2 + 5 x 2 – 7 = ?; do the multiplication= 2 + 10 – 7 = 5

But suppose we intended for something else? Then we need parentheses, brackets, braces

• Suppose we meant for the ‘2’ and ‘5’ to be added, first, and that sum multiplied by ‘6’ and all that divided by ‘3’, before subtracting the ‘7’?

• 2 + 5 x 6 / 3 – 7 = ????= {[(2 + 5) x 6] / 3} – 7 = ?= [(7 x 6) / 3] – 7 =?= (42 / 3) – 7 =?= 14 – 7 = 7, which is different from the ‘5’ we got when we

followed the Order of Operations

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***N.B. – some calculators automatically follow the order of operations, some do not! Get familiar with your calculator so that you know how it will perform a series of operations

To be on the safe side, use parentheses when performing a series of operations or, if your calculator doesn’t have parentheses buttons, write down your intermediate answers as you’re doing your calculations

Problem set

Inverse square law

•The intensity of radiation is 4.5 R at 1.5 meters from the source. What is the intensity at 5 meters?

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•I1 / I2 = (d2)2 / (d1)2

•Or is it (d1)2 / (d2)2 ???

Inverse square law

If you can’t remember how the formula is set up, try to reason it out

•The name of the formula is the Inverse Square Law

•The word “Inverse” should tell you that the elements are flipped (inverted) on opposite sides of the equals sign

•Therefore, if I1 is in the numerator on one side of the equals sign, then (d2)2 should be in the numerator on the opposite side of the equals sign

•So, it has to be: I1 / I2 = (d2)2 / (d1)2

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•Also, you know that as the distance from the source increases, the intensity of the radiation decreases (as one value goes up, the other goes down), which tells you that your fractions have to be opposite (flipped) with respect to one another

Or, work it out and see if your answer makes sense

• A) • I1 / I2 = (d2)2 / (d1)2

• I1 / 4.5 R = (1.5 m)2 / (5 m)2

• I1 = 4.5 R x (1.5 m)2 / (5 m)2

• I1 = 4.5 R x 2.25 m2 / 25 m2

• I1 = 4.5 R x 0.09• I1 = 0.405 R

• B) • I1 / I2 = (d1)2 / (d2)2

• I1 = 4.5 R x (5 m)2 / (1.5 m)2

• I1 = 4.5 R x 25 m2 / 2.25 m2

• I1 = 4.5 R x 11.11… • I1 = 50 R

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•You know that at the longer distance the intensity should decrease, therefore, answer ‘A’ makes sense and answer ‘B’ does not

Oxygen Enhancement Ratio

•It takes 6 R of x-ray radiation to kill 90% of tumor cells in a cell culture under anoxic conditions. If it takes 2.5 R of x-ray radiation to kill the same percentage of tumor cells under oxygenated conditions, what is the OER with respect to this cell line?

Oxygen Enhancement Ratio

•OER = dose anoxic to produce a given effect / dose aerobic to produce the same effect

•Or is it dose aerobic / dose anoxic ???

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If you can’t remember how the formula is set up, reason it out

• We know that oxygen makes tissues more sensitive to radiation, so you don’t need as much radiation, when oxygen is present, to get the effect. Therefore, the radiation dose under aerobic (oxygenated) conditions should be *lower* (a smaller number) than the dose under anoxic conditions (a larger number).

• The only way you can get a fraction to be greater than ‘1’ (enhancement) is if the larger number is in the numerator and the smaller number is in the denominator!

• Therefore, the anoxic dose needs to be in the numerator and the aerobic dose in the denominator

• So, it has to be: OER = dose anoxic / dose aerobic

Or, work it out and see if your answer makes sense

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• A)

• OER = dose anoxic to produce a given effect / dose aerobic to produce the same effect

• OER = 6 R / 2.5 R

• OER = 2.4

• B)

• OER = dose aerobic to produce a given effect / dose anoxic to produce the same effect

• OER = 2.5 R / 6 R

• OER = 0.4167

•You know that the OER should be larger than ‘1’ (that’s what we mean by “enhancement”), so answer ‘A’ makes sense and answer ‘B’ does not

Word problems!!

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http://www.caracaschronicles.com/2016/03/17/maduro-fiddles-rearranges-deckchairs/

You want me to do a word problem??!!!

Rule #1 – Take a deep breath!

• You know this material• So, you know how to solve this problem

• All you have to do is:• 1) Get the information and data organized• 2) Choose the appropriate formula(s)• 3) Do the calculations

• Which of these steps is the most difficult (or the most commonly avoided) one?

Number 1 – organizing the information and data!

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Skipping that first step is like starting out on a trip to an unfamiliar destination

without looking at a map or turning on your GPS … it’s like baking a cake for the

first time without looking at the recipe

Let’s work one out

•A chest unit is directed at the image receptor on the wall, six feet away. The wall has 3 HVL of shielding, and two meters from the opposite side of the wall is the office secretary’s chair. If the chest unit generates an average of 0.05 R, per image, at the image receptor, and the office has an average of 6 requests for PA and lateral chest studies per day, what is the weekly radiation exposure, in rads, at the location of the secretary’s chair? Assume a 5 day work week. Assume negligible wall thickness.

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Lots of information and data!!! We need to organize it all!

Your organizational template

•Given: the data that is given

•To find: the question(s) being asked

•Solution: the formula(s) and calculations

Given:

Chest unit is 6 ft from the image receptor (on the wall)Chest unit generates an average of 0.05 R per image at the image receptor3 HVL of shielding in the wallNegligible wall thickness

Secretary’s chair is 2 m from the wall6 PA and lateral studies done per day (average)

5 day work week

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To Find:

What is the weekly radiation exposure, in rads, at the secretary’s chair?

Solution:

1) Draw and label a diagram that represents the problem

6 ft2 m

3 HVL

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Solution:

2) Label the drawing with data and unknowns

6 ft2 m

3 HVL

0.05 R / image6 PA & Lat images / day = 12 images / day5 day week5 days / wk x 12 images / day = 60 images / wk

0.05 R / image x # images / wk = R / wkbefore shielding

? R / wk after shielding

? R / wk after 2 m

Solution:3) Break the problem down into pieces that you will work on individually

• A) How much radiation is being delivered to the image receptor (wall) over the course of one week?

• B) What is the radiation dose over one week after the radiation passes through the 3 HVL in the wall?

• C) What is the radiation dose, in rads, at the secretary’s chair 2 m away from the wall?

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Part A

• How much radiation is being delivered to the image receptor (wall) over the course of one week?

• 6 requests per day for PA and lateral CXR – that’s 2images per study

• 2 images/study x 6 studies/day = 12 images per day• 12 images/day x 5 day work week = 60 images/week• 60 images/week x 0.05 R/image = 3 R/wk at the wall

We’ve solved the first part of the problem

Part B

• What is the radiation dose over one week after the radiation passes through the 3 HVL in the wall?

• We’ve already calculated the weekly exposure (3 R/wk) at the wall, so now we just need to factor in the 3 HVL

• 3 R/wk x ½ x ½ x ½ = 0.375 R/wk on the opposite side of the wall

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We’ve solved the second part of the problem!

Part C• What is the radiation dose, in rads, at the secretary’s chair 2 m

away?

• Use the inverse square law formula … I1 / I2 = d22 / d1

2

• I1 is the intensity at the secretary’s chair = ????

• I2 is the intensity after passing through the wall = 0.375 R / wk• d2 is the distance from the chest unit to the wall = 6 ft• d1 is the distance from the chest unit to the secretary’s chair =

6 ft + 2 m *** Different units!! We need to convert feet to meters or meters to feet

Part C, continued

1 ft = 0.3048 m

d2 (the distance from the chest unit to the wall)

= 6 ft = 6 ft x 0.3048 m / ft = 1.8288 md1 (the distance from the chest unit to the secretary’s chair

= 1.8288 m + 2 m = 3.8288 m

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Part C, conclusion• Substitute into the formula, isolate, and solve for the unknown• I1 / I2 = d2

2 / d12

• I1 / 0.375 R / wk = (1.8288 m)2 / (3.8288 m)2

• I1 = 0.375 R / wk x [(1.8288 m)2 / (3.8288 m)2]

• I1 = 0.375 R / wk x (3.345 m2 / 14.66 m2)

• I1 = 0.0086 R / wk

• But our answer had to be in rads, not R, but 1 R is approximately equal to 1 rad, so we don’t need to make a conversion of units

• I1 = 0.0086 rads / wk at the secretary’s chair

Review your answer!!!•Does your answer make sense?•We’d expect that the radiation exposure farther from the wall would be less than the radiation exposure at the wall. This is the case.

•We’d also expect the radiation at the chair to be less than the radiation at the chest unit because of the shielding in the wall. This is also the case.

•Our answer makes sense.

•If your answer doesn’t make sense, you did something wrong. Go back and check … Is my diagram a correct representation of the problem? Are my formulas correct? Did I substitute the numbers into the formulas correctly? Did I inadvertently invert a fraction? Did I do the calculations correctly (e.g., enter the correct numbers into my calculator)?

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Regardless of what your calculator says, if your answer doesn’t make sense, redo the problem!

Just because a number is in the calculator display doesn’t mean it’s a correct answer. It has to make sense.

Units!

•Make sure all your units are the same type (English or metric)

•Convert units when you have to … it’s best to do this FIRST, when you enter your “Given:” data. Then you won’t have to worry about it in the middle of your calculations

Keep your units with you!!•**Most important!**•Keep your units with you during your calculations and treat the units as if they were numbers!

•Do not just write down the numbers, perform your calculations without units, and then plug in the units at the end.

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Keep your units with you!!

•Keeping the units throughout the calculations and treating them like numbers will tell you whether your answer is reasonable or not.

•E.g., if your answer is supposed to be in Ci, and you end up with an answer in Ci / msec, you know that you’ve done something wrong and can go back and correct it

The importance of units!

The Mars Climate Orbiter (1999)

Our $327.6 million “oops”!

The Mars Climate OrbiterOne team on the project worked in English units (feet, pounds of force); another team on the project worked in metric units (meters, newtons of force)

When the data and formulas were combined to calculate thrust, no one noticed that different units were being used. The thrust that was supposed to put the spacecraft into a nice orbit around Mars ended up sending it deep into the Martian atmosphere and then back out into space.--- http://mars.nasa.gov/msp98/news/mco990930.html

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Take home points …

1) Know your material

•Know your subject•Know your formulas

2) Organize•Use the template:

•Given: Organize and write down your data **with units**

•To Find: Write down what it is you’re looking for **with units**

•Solution: Draw and fully label a diagram

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3) If it’s complex, break it down

•Break complex problems down into their constituent parts and work each part out individually

4) Perform the calculations

•Perform your calculations **keeping units with your calculations** and treating units like numbers

5) Review your answer(s)

•Review your answer to see if it makes sense! If it does, fine; if it doesn’t, go back and find and correct the error

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6) You’re smart; calculators are stupid

•Do *not* blindly accept the display on your calculator as being correct

Questions?

Thank you!

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robert.posteraro@ttuhsc.edu