Math for Radiographers. Observation. What s *not* the issue

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1 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 1

2 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 2

3 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 3

4 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: 3 4 = 3 x 3 x 3 x 3 = = 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., 3 4 is the same as 3^4 = 81) There are some powers that have special names: Squared (the power is 2 ) 9 squared = 9 2 = 9 x 9 = 81 Cubed (the power is 3 ) 6 cubed = 6 3 = 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 b bx b = b 2 = the area of the square 4

5 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 a a a x a x a = a 3 = the volume of the cube a Where do we see powers in radiography? Photomultiplier tube gain PM gain = g n where g is the gain of one dynode and n is the number of dynodes in the PM tube 5

6 Where else do we see powers in radiography? Single-target, single-hit model of cell survivability S = N / N 0 = e D/D37 Where, S = surviving fraction N = number of cells that survived dose D N 0 = initial number of cells D = the dose of radiation applied to the cells D 37 = a constant related to the cell radiosensitivity e = Euler s number, the constant, , the base of the natural logarithms Multi-target, single-hit model of cell survivability S = N/N 0 = 1 - (1 e -D/D0 ) n Where, S = surviving fraction N = number of cells that survived dose D N 0 = initial number of cells D = the dose administered D 0 = the dose that would result in an average of one hit per target n = the extrapolation number e = Euler s number, the constant,

7 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 4 th root of 625? That is, what number do I have to raise to the 4 th power to give me 625? Ans: = 5 x 5 x 5 x 5 = 625 Where do we see roots in radiography? The formula for noise in a CT system Noise ( ) = sqrt {[ (x i x av ) 2 ] / (n 1)} Where, = Greek uppercase sigma, the symbol meaning sum (add em all up) x i = the CT value of each pixel x av = the average of a number (> 100) of CT pixel values n = 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!) =

8 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 / N 0 = e D/D37 ) E.g., 2-3 = 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 as 1 / 5 2 = 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 3 4 is the same as 1 / 3-4 = 81 8

9 ***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! 9

10 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 , 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 = = 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 = = 97 This should be understandable. Since 10 2 = 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 10

11 Where do we see logs in radiography? The formula for optical density (OD) OD = log (I o / I t ) Where, I o = light intensity incident to the processed film I t = 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 11

12 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 x 6 / 3 7 =? = / 3 7 =? = = 5 Multiplication and division have equal precedence, so we could have done the multiplication & division in the reverse order without changing anything x 6 / 3 7 =?; do the division first = x 2 7 =?; do the multiplication = = 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? 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 12

13 ***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? 13

14 Inverse square law I 1 / I 2 = (d 2 ) 2 / (d 1 ) 2 Or is it (d 1 ) 2 / (d 2 ) 2??? 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 I 1 is in the numerator on one side of the equals sign, then (d 2 ) 2 should be in the numerator on the opposite side of the equals sign So, it has to be: I 1 / I 2 = (d 2 ) 2 / (d 1 ) 2 14

15 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) I 1 / I 2 = (d 2 ) 2 / (d 1 ) 2 I 1 / 4.5 R = (1.5 m) 2 / (5 m) 2 I 1 = 4.5 R x (1.5 m) 2 / (5 m) 2 I 1 = 4.5 R x 2.25 m 2 / 25 m 2 I 1 = 4.5 R x 0.09 I 1 = R B) I 1 / I 2 = (d 1 ) 2 / (d 2 ) 2 I 1 = 4.5 R x (5 m) 2 / (1.5 m) 2 I 1 = 4.5 R x 25 m 2 / 2.25 m 2 I 1 = 4.5 R x I 1 = 50 R 15

16 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??? 16

17 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 17

18 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 = 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!! 18

19 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! 19

20 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. 20

21 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 receptor 3 HVL of shielding in the wall Negligible wall thickness Secretary s chair is 2 m from the wall 6 PA and lateral studies done per day (average) 5 day work week 21

22 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 3 HVL 6 ft 2 m 22

23 Solution: 2) Label the drawing with data and unknowns 0.05 R / image x # images / wk = R / wk before shielding 3 HVL? R / wk after shielding 6 ft 2 m 0.05 R / image 6 PA & Lat images / day = 12 images / day 5 day week 5 days / wk x 12 images / day = 60 images / wk? 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? 23

24 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 2 images 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 = R/wk on the opposite side of the wall 24

25 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 I 1 / I 2 = d 22 / d 1 2 I 1 is the intensity at the secretary s chair =???? I 2 is the intensity after passing through the wall = R / wk d 2 is the distance from the chest unit to the wall = 6 ft d 1 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 = m d 2 (the distance from the chest unit to the wall) = 6 ft = 6 ft x m / ft = m d 1 (the distance from the chest unit to the secretary s chair = m + 2 m = m 25

26 Part C, conclusion Substitute into the formula, isolate, and solve for the unknown I 1 / I 2 = d 22 / d 1 2 I 1 / R / wk = ( m) 2 / ( m) 2 I 1 = R / wk x [( m) 2 / ( m) 2 ] I 1 = R / wk x (3.345 m 2 / m 2 ) I 1 = 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 I 1 = 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)? 26

27 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. 27

28 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 Orbiter One 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

29 Take home points 1) Know your material Know your subject Know your formulas Use the template: 2) Organize 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 29

30 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 30

31 6) You re smart; calculators are stupid Do *not* blindly accept the display on your calculator as being correct Questions? Thank you! 31

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