FORMULASNMT 1002 :GROUP iv Guillermo Chea irisbel Martinez Jose pachon Maria del pozo Stephen carmona FORMULASNMT 1002


objectives :objectives To explain step-by-step the simple methods used to solve math problems in nuclear medicine. Review some math basics and how to apply them into the formulas.


Chapter 2 :statistics Chapter 2


How to calculate the standard deviation :How to calculate the standard deviation SD = √∑(n-ñ)2 √ N-1 Where: ∑ = is the summation symbol which indicates that the values immediately following the symbol are to be summed. n = individual values ñ = mean value N = total number of values


Example :Example Calculate the standard deviation for the following series: 175,180,178,169,184 Calculate the mean ñ= 175+180+178+169+184=177.2 5 Calculate (n-ñ)2 for each value (175-177.2)2=-2.22=4.84 (180-177.2)2=2.82=7.84 (178-177.2)2=0.82=0.64 (169-177.2)2=6.82=46.24


2nd part :2nd part Calculate ∑(n-ñ) : 4.84+7.84+0.64+67.2+46.24=126.8 Calculate the SD: SD =√126.8 =√31.7=5.6 5-1


How to calculate the standard deviation from a single value at a desired confidence interval :How to calculate the standard deviation from a single value at a desired confidence interval This number represents the range of acceptable values at that level of confidence.   CI= interval   For the 68% confidence interval ( +1 SD): CI68% =n +√n   Where n= the single   For the 95% confidence interval ( + 2SD): CI95% =n + 2√n   For the 99% confidence interval ( + 3SD): CI99% =n + 3√n


example :example What range of values is included in one standard deviation of 1000 counts? Two standard deviations? Three standard deviations?   1SD or CI68%: 1000 ct + √1000 = 1000 ct + 32 or 968 to 1032 counts   2SD or CI95% 1000 ct + 2√1000 = 1000 + 63 or 937 1063 counts   3SD or CI99% 1000 ct + 3√1000 = 1000 + 95 or 905 to 1095 counts


Chapter 3 formulas :Radiation safety Chapter 3 formulas


How to convert counts per minute to disintegrations per minute using the well-counter efficiency :How to convert counts per minute to disintegrations per minute using the well-counter efficiency Disintegrations per minute better represent the actual amount of radioactivity present on a survey swab than do counts per minute (cpm). To convert cpm to dpm, divide the cpm by the well counter efficiency. FORMULA: Dpm = gross cpm – background cpm Efficiency expressed as decimal


example :example A wipe test swab produces a gross count of 1225 cpm in a well counter with a 56% efficiency. The background is 395 cpm. Calculate the dpm produced by the swab. 1225 cpm- 395 cpm = 1482 dpm 0.56


total radiation dose formula :total radiation dose formula Total dose = (dose rate) (time) There is a direct linear relationship between radiation dose and time. The radiation dose increases in a linear fashion as the time of exposure increases.


example :example If a technologist is exposed to a source with a dose rate of 3.0 mRem /hr for a period of 2hrs, what is the total radiation dose? (3.0 mRem)(2 hr)= 6.0 mRem


Inverse Square Law formula :Inverse Square Law formula (I1)(D1)2 = (I2)(D2)2 Where: I1 = Intensity at original distance (D1) I2 = Intensity at new distance (D2) Close to the source, you interact with many emissions. As you move further from the source, the number of interactions drops off rapidly, by the square of the distance.


Inverse square law :Inverse square law


example :example You are standing 1 foot from a radioactive source and receiving a radiation exposure dose of 20 mRem/hr. If you move to a position 2 feet from the source, what will be your rate of exposure? (20mRem/hr)(1ft)2 = (x) (2ft)2 X= (20mRem/hr)(1ft)2 = 20 = 5 mRem/hr (2ft)2 4


How to calculate the change in exposure rate due to shielding :How to calculate the change in exposure rate due to shielding Formula I = I0e-(0.693)(x/HVL) WHERE: I = Exposure rate (intensity) being calculated I0= Original exposure rate (intensity) e = 2.718… This may be represented on a specific calculator as either an ex key or Ln x. ex and Ln x are the inverse of one another. 0.693 is the natural log of 2. It represents the halving of values with the addition of each HVL x = thickness of shielding material HVL = half value layer for given shielding material.


steps :steps First, calculate the number of half value layers that have been added Second, use the chart to find the fraction of the original exposure rate. Multiply the original exposure rate by the fraction.


example :example The half-value layer of lead for Tc99m is 0.3mm. If a radiopharmaceutical dose is producing 11 R/hr, what will be the exposure rate if the dose is placed in a lead syringe shield that has a thickness of 2.5mm? I = 11R/hr e-(0.693)(2.5mm/0.3mm) = 11e-5.775 = (11)(0.0031) = 0.034 R/hr or 34 mR /hr


Chapter 4 :instrumentation Chapter 4


%error or %difference formula :%error or %difference formula %error or %difference = |expected reading – actual reading| X 100% Expected reading The absolute value of the difference between expected and actual readings is used, if a negative number is obtained in the numerator, the negative sign is simply ignored. Because the absolute value is used, reversing the elements of the numerator will still yield the same % error or % difference. Example: what is the % error in accuracy if the expected dose calibrator reading is 1.35mCi and the actual reading is 1.26mCi? % error = |1.35mCi – 1.26mCi | x 100% = 6.67% 1.35mCi This percent error is within is NRC limits, so the dose calibrator can be used.


Correction factor formula :Correction factor formula Correction factor = expected activity Actual activity If the actual value is less than the expected value, then the correction factor must be greater than 1. in other words, the actual value must be made larger and this can only be done by multiplying it by a number greater than 1. the actual value must be made smaller to achieve accuracy. Example: the expected reading for a geometry test on a 5ml syringe was 2.95mCi. The actual reading was 2.63mCi. Calculate the correction factor. 2.95mCi = 1.12 2.63mCi


How to apply correction factor :How to apply correction factor When a dose calibrated, it is then multiplied by the appropriate correction factor to obtain the accurate activity reading. True activity = (actual activity reading)(correction factor) Example : if a correction factor of 1.18 is required for a 5 cc syringe, what will be the true activity of a dose with a reading of 12.6mCi? (12.6mCi)(1.18) = 14.9mCi


Chi-square Formula (x2) :Chi-square Formula (x2) The chi-square test allows you to determine how close a series of counts would come to a true Poisson distribution. It provides a measure of the precision or the constancy of an instrument’s performance X2 = ∑ (n - ñ)2 ñ Where: ∑ = is the summation symbol which indicates that the values immediately following the symbol are to be summed. n = individual values ñ = mean value


Chi-square Table :Chi-square Table For the purpose of testing a well-counter or an uptake probe, the x2 is acceptable if it falls between P values of 0.9 to 0.1.


Example :Example Calculate the x2 for the following data that was obtained from a sample counted in a well-counter. Determine if the variation in counts is acceptable. n n-ñ (n-ñ)2 1 14231 45 2025 2 14103 -83 6889 3 14267 81 6561 4 14391 205 42025 5 14088 -98 9604 6 14219 33 1089 7 14112 -74 5476 8 14097 -89 7921 9 14183 -3 9 10 14166 -20 400 ∑n = 141,857 ∑(n – ñ)2 = 81, 999 ñ = 141857 = 14186 x2 = ∑(n – ñ)2 = 81999 = 5.780 10 ñ 14186


How to calculate the chi-square value :How to calculate the chi-square value First, calculate the mean (ñ) Second, calculate (n-ñ) for each value. Third, square the number (n – ñ)2 Fourth, calculate the sum of the (n – ñ)2 values. And finally solve chi-square equation.


Well counter efficiency :Well counter efficiency % efficiency = counts per minute x 100% (Disintegrations per unit time)(Mean number per disintegration) Where: counts per minute = cpm or cps produced by source during preset time disintegrations per unit time = dpm or dps produced by the activity (uCi) used for the test mean number per disintegration = abundance of emissions at selected energy setting in decimal form. The units of time in the numerator and denominator must match. If you use cpm in the numerator, then dpm must be used in the denominator. 1uCi = 3.7 x 104 dps. To convert dps to dpm: (3.7 x 104 dps) = 2.22 x 106 dpm or 2,220,000 dpm/uCi


example :example The current activity of Cs137 stick source is 0.2 uCi. Cs137 has an 85% abundance at 662KeV. A 1 minute acquisition produces 175,000 counts. What is the well counter efficiency? First, convert 85% to its decimal form: .85 then 175,000cpm x100% (0.2uCi)(2,220,000dpm/uCi)(0.85)


Chapter 5 :radiopharmacy Chapter 5


DECAY CALCULATION :DECAY CALCULATION A radionuclide is constantly decaying at a specific rate while the activity is constantly decreasing. The amount of decrease per unit time is dependent upon the half life of the radionuclide. At = Ao e -0.693 x (t/t ½) Where: At = activity at specified time Ao = original acivity e = euler’s number t = elapsed time t1/2 = half life


EXAMPLE: :EXAMPLE: A vial of DTPA is reconstituted with 200 mCI of Tc99m. What will be its activity three hours later? The half life of Tc99m is 6.01 hours At = 200 mCi x e - 0.693 x (3hr/6hr) = 200 mCi x e-0.693x0.4992 = 200 mCi x e-0.3459 A = 200 mCi x 0.7076 = 142 mCi


PRE-CALIBRATION CALCULATIONS :PRE-CALIBRATION CALCULATIONS When preparing unit doses and bulk kits, you need to determine the amount of activity to be used based on the activity needed at future time (time t). Pre- calibration factors allow you to determine the activity needed at time 0. Ao = At x pre-calibration factor Where: At = activity at time (calibration time) Ao = activity at time 0 (preparation time)


Example :Example A 22 mCi Tc99m HDP unit dose is needed at 10:00 am. The dose is being prepared at 6:00am. How many mCi must be placed in the syringe? Calculate elapsed time = 4hours Pre calibration factor = e 0.693(4hr/6hr) = 1.587 Ao = 22mCi x 1.587 = 35 mCi


DOSE VOLUME CALCULATIONS :DOSE VOLUME CALCULATIONS In order to calculate the volume of solution needed to provide the desire quantity of radioactivity, the specific concentration of the solution must be known.


example :example A vial of Tc99m sodium pertechnetate contains 375 mCI in 8.2 ml. If a 20 mCi dose is needed, how many ml must be withdrawn from the vial?


CALCULATION OF TOTAL ACTIVITY NEEDED TO PROVIDE SPEIFIC NUMBER OF KITS OR DOSE :CALCULATION OF TOTAL ACTIVITY NEEDED TO PROVIDE SPEIFIC NUMBER OF KITS OR DOSE The total activity needed to reconstitute a radiopharmaceutical kit can be determined when you know the number of doses to be administered and the approximate times the doses will be required. Total activity required = Where ∑ = indicates that the values immediately following the symbol are to be summed. Total activity required =


Example :Example An MDP kit is being reconstituted to provide one 20 mCI dose at 9:00am, at 10:00 am, at 11:00 am, and at 12 noon. What is the minimum activity that must be placed in the vial when it is being prepared at 7:00am? Calibration time elapsed time dose x decay factor = activity needed at 7:00am 9:00 am 2hr 20mCI x 1.259 =25 mCi 10:00am 3hr 20mCI x 1.414 = 28mCI 11:00am 4hr 20mCI x 1.587 = 32mCi 12:00pm 5hr 20mCI x 1.779 = 36mCi Total minimum activity needed at 7:00am = 121 mCi


How to calculate a dose based on activity per weight :How to calculate a dose based on activity per weight Patient dose= (dose/kg)(patient weight in kg) The doses of some radiopharmaceuticals and many of the interventional drugs used in nuclear medicine are calculated according to the patient’s weight.


Example :Example A 35 kg pediatric patient is scheduled for a Meckels scan. The protocol requires a dose of 75 uCi /kg. What dose should be administered? (75uCi/Kg)(35Kg) = 2625 uCi or 2.6mCi


conclusion :conclusion It is very important to understand the mathematics that we are going to experience in the practice of clinical nuclear medicine technology. Demonstration and practice are essential aspects of learning.


references :references Wells, Patricia: Practical Mathematics in Nuclear Medicine Technology. New Jersey: Society of Nuclear Medicine, Inc. 1999. Christian, Paul: Nuclear Medicine and PET/CT. St. Louis, Missouri: Elsevier Inc. 2007