1. Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.
TREATMENT PLANNING
Mathematics Review
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

2. Algebraic Equations with One Unknown
When an unknown in multiplied by some quantity, divide both sides by that quantity to isolate the unknown.
When a quantity is added to an unknown, subtract it from both sides of the equation to isolate the unknown. When quantity is subtracted, add it to both sides.
When equation appears in fractional form, cross-multiply both sides by that quantity.
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

3. Ratios and Proportions
Ratio – comparison of two numbers, values, or terms
Often written as x/y or x:y
Ratios remain unchanged if both terms undergo operations by the same number
Proportion – two ratios that are equal or an equation relating to two ratios
5/7 = n:49
(49x5) = 7n
245 = 7n
n = 35
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4. Inverse Proportionality
Airplane example
Used in radiation therapy to determine
Depth and percentage dose
Beam energy and penumbra width
Inverse square law
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5. Direct Proportionality
Airplane example
Used in radiation therapy to determine
Field size and percentage depth dose
Beam energy and tissue air ratio/tissue maximum ratio
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6. Trigonometric Ratios and Right Angle Triangle
Right triangle – three-sided polygon in which one corner measures 90 degrees
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7. Trigonometric Ratios and Right Angle Triangle
Sine = opposite/hypotenuse = AC/BC
Cosine = adjacent/hypotenuse = AB/BC
Tangent = opposite/adjacent = AC/AB
Used in radiation therapy to
Match divergences of two abutting treatment fields
Measure angle or thickness of chest wall
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8. Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.
Linear Interpolation
Linear interpolation – process of calculating unknown values from known values
Assumptions
Two particular values are known
Rate of change between known values is constant
An unknown data point must be found
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

9. Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.
Exponents
Exponent – shorthand notation that represents the multiplication of a number by itself a given number of times
43 = 4 x 4 x 4 = 64
3 = exponent or “power”
4 = base
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

10. Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.
Exponents
Rules for exponents
X0 = 1
Xa x Xb = Xa+b
(Xa)b = Xab
(xy)a = xaya
(x/y)a = xa/ya
x-a = 1/(xa) and 1/(x-a) = xa
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

11. Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.
Exponents
Scientific notation – special use of exponents that uses base 10 notation. Used to represent very large or very small numbers.
n.nnn x 10P
n.nnn = first four numerical values of the specific number
For example: = x 103
Or x 102 but not widely accepted
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

12. Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.
Significant Figures
Significant figure – number of figures in a measurement that are known with some degree of reliability
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13. Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.
Significant Figures
Rules
All nonzero digits are significant
1.234 g has 4 significant figures
Zeroes between nonzero digits are significant
1002 kg has 4 significant figures
Zeroes to the left of the first nonzero digits are not significant
0.001° C has only 1 significant figure
Zeroes to the right of a decimal point in a number are significant
0.200 g has 3 significant figures
When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant
5040 cGy may be 3 or 4 significant figures
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

14. Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.
Significant Figures
When combining measurements with different degrees of accuracy and precision, accuracy of the final answer can be no greater than the least accurate measurement. This principle can be translated into the following rules:
When measurements are added or subtracted, the answer can contain no more decimal places than the least accurate measurement.
When measurements are multiplied or divided, the answer can contain no more significant figures than the least accurate measurement.
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

15. Natural Logarithms and Exponential Function
Logarithm – operates as the reverse of exponential notation
Logb (N) = x
43 = “four raised to the third power” in exponential notation = 64
Logarithm base 4 of 64 = 3
Log4(64) = 3
b = base, N = product, x = power
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

16. Natural Logarithms and Exponential Function
Natural logarithms – logarithms based on Euler’s constant
Euler’s constant – special number discovered by Euler equal to …
Exponential function = ex = N
Natural logarithm base e = ln(N) = x
ln(ex) = x
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

17. Natural Logarithms and Exponential Function
Natural logs and exponential functions are inverses. Properties beneficial in radiation therapy:
If the power (x) is greater than 0, then the value of ex is greater than 1.
If the power is less than 0, then the value of ex is a number greater than 0 and less than 1.
If the power is exactly 0, then the value of ex is exactly equal to 1.
ex > 1 if x > 0
0 < ex < 1 if x < 0
ex = 1 if x = 0
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

18. Basic Units of Measurement
Metric or International System of Units (SI)
World standard for scientific and technical work
Four fundamental units
Second (time)
Meter (distance)
Kilogram (mass)
Ampere (electrical current)
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

19. Basic Units of Measurement
Roentgen (r) – unit of radiation exposure that represents a measure of the amount of ionization created by radiation in the air
Coulomb/kilogram (C/kg)
1 roentgen = 2.58 x 10-4 C/kg
Gray (Gy) – amount of radiation × energy absorbed by a medium
Joule/kilogram (J/kg)
Outdated unit replaced by the gray is the rad
1 rad = 0.01 gray
1 rad = one centigray (cGy)
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

20. Basic Units of Measurement
Joule (J) – 1 kilogram-meter2 per second2
1 kgm2/s2
Electron volt (eV)
1 eV = × 10–19 J
Kiloelectron volt (keV = 103 eV) and megaelectron volt (MeV = 10 eV)
Most common energy units used in the radiation therapy clinic
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

21. Measurements and Experimental Uncertainty
Source-to-axis distance (SAD)
Source-to-skin distance (SSD)
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

22. Measurements and Experimental Uncertainty
Systematic errors
Inherent within the measuring device
Always affects measurement in same way
Vision inaccuracies
Imperfect techniques
Unacceptable instrument calibrations
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

23. Measurements and Experimental Uncertainty
Random errors
Unavoidable variations attributed to chance
To correct, one may take several measurements and average the results
May be reduced by improving the measuring device/technique
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

24. Measurements and Experimental Uncertainty
Blunders
Human error in calculations or from improper use of measuring device
Can be avoided by comparing measurements with previous measurements that are known to be accurate
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

25. Measurements and Experimental Uncertainty
Accuracy and precision
Accuracy – how close a measurement comes to the true value
Precision – how reproducible/consistent a particular measurement is
A measurement can be precise but not accurate
Therapists must be concerned with both
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

26. Measurements and Experimental Uncertainty
Percent relative error – percentage of error in a measurement relative to the accepted value
% Relative certainty = [(Experimental value – Accepted value)/Accepted value] x 100
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27. Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.
Dimensional Analysis
Dimensional analysis – process that involves careful assessment of units of measurement used in calculating a specific quantity
Rules of Thumb
Any quantity divided by 1 = quantity itself
Any quantity divided by itself = 1
Division = multiplying numerator by inverse of denominator
Copyright © 2010 by Mosby, Inc., an affiliate of Elsevier Inc.

28. Tx Planning Math Review
Similar Triangle
Inverse Square
Radioactivity
Equivalent Square
Gap Formula
Mayneord’s F-factor
Hinge Angle
Mag Factor
….and something new !!

29. If a 14 x 3cm field gives a 21 x 4.5 cm image, what is the magnification factor?

30. 1.5

31. If the ring placed on the patient measures 3cm, and the image of the ring measures 7cm, what is the mag factor?

32. 2.3

33. If the SID is 100cm and the SAD is 94cm, what is the mag factor?

34. 1.06

35. A film is taken to determine the depth of a kidney
A film is taken to determine the depth of a kidney. A 4cm ring is placed on the patient’s skin. It measures 9cm on the film. If the kidney depth below the skin on the film measures 5cm, what is the kidney depth in the patient?

36. 2.2

37. At 100cm SSD the field size is 20cm, what is the field size at 120cm SSD?

38. 24

39. A 3cm block is placed at 45cm SSD, what is the block size at isocenter when using 100cm SSD?

40. 5.3

41. The magnification factor is 1
The magnification factor is 1.8, the physical measurement of the field on the patient is 10 x 13cm, what should the field measure on the film?

42. 18 x 23.4cm

43. At 60cm SAD the field size is 12 x 15, what will the field size be at 100cm SAD?

44. 20 x 25cm

45. A patient receives an isocentric parallel opposed field with the isocenter placed at midplane. If the patient seperation is 30cm and the treatment unit is a 6MV accelerator, what is the SSD?

46. 85cm SSD

47. A patient is having isocentric fields planned
A patient is having isocentric fields planned. The simulator field size display shows 10 x 16cm at 100cm SAD. What will the field size measure on the patients skin at 89cm SSD?

48. 8.9 x 14.2cm

49.   A 3.5cm block on a blocking tray 45cm from the source will block what width at 80cm?

50. 6.2cm

51. A treatment is prescribed to be treated isocentrically on the 6MV Linac at 100 SAD. Calipers have measured the patient separation to be 21cm. What SSD will the therapist(s) set to the patients skin for a POP treatment?

52. 89.5cm SSD

53. A treatment is prescribed at 80cm SAD
A treatment is prescribed at 80cm SAD. If the field size indicator is set for 14cm, what would the field size be at the surface of the patient if the SSD was 75cm?

54. 13.1cm

55.   A Co60 unit is set for 10 x 10cm field at 80cm SSD what is the field size at 60cm SSD?

56. 7.5 x 7.5cm

57. The dose rate at 100cm is 50cGy. What is the dose rate at 120cm?

58. 34.7cGy

59. What will be the activity after 2days if the original activity is 6 curies of a sample of Iridium? A= Aoe -λt

60. 5.05

61. The %DD for a 10MV beam at a depth of 5cm and 80cm SSD is 91. 8%
The %DD for a 10MV beam at a depth of 5cm and 80cm SSD is 91.8%. Calculate the %DD for the same Tx field at 100cm SSD.

62. 92.7

63. What is the wedge angle for a 150 hinge angle?

64. 15