1. Lecture 4: Mathematics of Decay and Units of Radioactivity
Unit I: the Physics of Nuclear Medicine
From “Researchers Find Cancer in Ancient Egyptian Mummy.” The Cosmos News. Retrieved 27 Aug 2012 from

2. Lecture 2 Objectives Define Decay Constant.
Use the General form of the radioactive decay equation to calculate precalibration and post calibration quantities of radioactivity.
Define Decay Factor and Precalibration Factor.
List the radioactive units and define curie and becquerel.
Write the equations for average half-life and effective half-life and calculate effective and biological half-lives.
Recognize and discuss the use of the Bateman Equation.

3. Mathematics of Decay Decay Constant (λ):
Average proportion of atoms present that will decay over a selected period of time.
ex: 0.33/hour, means that 1/3 of atoms will decay in an hour
Paul Christian & Kristen M. Waterstram-Rich, Nuclear Medicine and Pet/CT: Technology and Techniques, 6th Ed. (St. Louis: Mosby 2004), p 50.

4. Mathematics of Decay What is e????
Using calculus, we can derive the following:
We can figure out the number of parent atoms remaining after the passage of a given amount of time (t).
What is e????
Paul Christian & Kristen M. Waterstram-Rich, Nuclear Medicine and Pet/CT: Technology and Techniques, 6th Ed. (St. Louis: Mosby 2004), p 50.

5. Mathematics of Decay
Change of parent atoms is directly related to decay and therefore atomic disintegration
Disintegration is directly related to radioactivity (or Activity), Therefore…
At is the remaining activity after time t
A0 is the activity at time = 0
e is Euler’s number (base of natural log)
-λ is the decay constant
t is the time that has elapsed
Paul Christian & Kristen M. Waterstram-Rich, Nuclear Medicine and Pet/CT: Technology and Techniques, 6th Ed. (St. Louis: Mosby 2004), p 51.

6. Mathematics of Decay
Easier to use half-life (T1/2) than decay constant (λ).
We know that we’ll have half of the activity after one half life (T1/2)
The natural log (ln) of the natural log base (e) to a given power is equal to that exponent.
Using substitution…
The natural log (ln) of 2 = (try it in your calculator)

7. The Radionuclide Decay Equation
At is the Radioactivity after the passage of time t
A0 is the Radioactivity at time = 0 (or Original activity)
t is time elapsed
T1/2 is physical half-life of the radionuclide

8. Mathematics of Decay Decay Factor
The “Decay Factor” (DF) can be multiplied against an original amount of activity to determine the amount of activity present after a period of time.
Paul Christian & Kristen M. Waterstram-Rich, Nuclear Medicine and Pet/CT: Technology and Techniques, 6th Ed. (St. Louis: Mosby 2004), p 51.

9. Mathematics of Decay What are these???? A list of DFs
for Tc-99m (based on 6.02 hour half-life) for 1-hour increments of time.
What are these????
DFs are useful shortcuts for approximating activity for commonly used radionuclides.
Paul Christian & Kristen M. Waterstram-Rich, Nuclear Medicine and Pet/CT: Technology and Techniques, 6th Ed. (St. Louis: Mosby 2004), p 51.

10. The Radionuclide Precalibration Equation
Sometimes we know the activity after time has elapsed but need to find out what it was originally…
To do this we can simply use algebra to solve for A0 instead of At:
Yet if we simply remove the minus sign from the exponent, we can also use the decay equation, but trade places between A0 and At:

11. Mathematics of Decay Precalibration Factor
The “Precalibration Factor” (PCF) can be multiplied against an amount of activity after a period of time (t) has elapsed to determine the amount of activity originally present.
Paul Christian & Kristen M. Waterstram-Rich, Nuclear Medicine and Pet/CT: Technology and Techniques, 6th Ed. (St. Louis: Mosby 2004), p 51.

12. Testing these out… Tc-99m has a half-life of 6 hours.
You have a 30 mCi unit dose of Tc-99m MDP for a bone scan calibrated for 8 AM. Your patient calls to say he’s running late and won’t be in until 11:00 AM. Will you have enough dose to fall in the mCi acceptable range?
A0 = 30 mCi
T1/2 = 6 hours
t = 11:00 – 8:00 = 3 hours
At = ???

13. Testing these out…
Your patient arrives at 11:00 AM, but when you assay your dose it reads 18 mCi. To complain to the vendor, you need to precalibrate your assayed reading back to 8 AM:
A0 = ???
T1/2 = 6 hours
t = 11:00 – 8:00 = 3 hours
At = 18 mCi
Or…

14. Radioactivity Units
Anytime a nuclide changes form it “disintegrates” to take on the new form.
Radioactivity is measured according to the number of these disintegrations per unit time.
Paul Christian & Kristen M. Waterstram-Rich, Nuclear Medicine and Pet/CT: Technology and Techniques, 6th Ed. (St. Louis: Mosby 2004), p 52.

15. Radioactivity Units Becquerel (Bq) 1 mCi = 37 MBq SI unit
Based on a radioactive sample that decays at 1 disintegration per second (dps)
Because NM doses are much larger (mCi—3.7 X 107 dps), we usually convert to mega Becquerels (Mbq—a million [106] Becquerels)
1 mCi = 37 MBq
Example:

16. Radioactivity Units

17. Other Decay Calculations
Average decay time:
The average time it takes for a parent radionuclide atom to transform to its daughter
Paul Christian & Kristen M. Waterstram-Rich, Nuclear Medicine and Pet/CT: Technology and Techniques, 6th Ed. (St. Louis: Mosby 2004), p 52.

18. Other Decay Calculations
Effective Half-life
Paul Christian & Kristen M. Waterstram-Rich, Nuclear Medicine and Pet/CT: Technology and Techniques, 6th Ed. (St. Louis: Mosby 2004), p 52.

19. Other Decay Calculations
Effective half-life
Besides decaying the radionuclide is also being metabolized by biological process, so effective half-life tells us about the activity remaining in the body after time.
The decay constants for biological process and decay are cumulative.
Paul Christian & Kristen M. Waterstram-Rich, Nuclear Medicine and Pet/CT: Technology and Techniques, 6th Ed. (St. Louis: Mosby 2004), p 52.

20. Other Decay Calculations
λ = 0.693/T1/2 for both bio and radioactive (or physical) half-lives
Dividing each side of the decay constant equation by 0.693, we get…
Paul Christian & Kristen M. Waterstram-Rich, Nuclear Medicine and Pet/CT: Technology and Techniques, 6th Ed. (St. Louis: Mosby 2004), p 52.

21. Other Decay Calculations
That equation can also be expressed as…
Paul Christian & Kristen M. Waterstram-Rich, Nuclear Medicine and Pet/CT: Technology and Techniques, 6th Ed. (St. Louis: Mosby 2004), p 52.

22. Other Decay Calculations
Effective Half-life Example (from book)
Tc-99m MAA in the lungs
Physical half-life of Tc-99m is 6 hrs
Biological half-life of MAA is 3 hrs OR
(The answer is 2 hrs)

23. Parent-daughter Radionuclide Relationships
The Bateman Equation
A1,0 = Parent Activity at t=0
Remember…
The Decay Constant =
A2,0 = Daughter Activity at t=0
A2,t = Parent Activity after time t
λ1 = Parent Decay Constant
λ2 = Daughter Decay Constant

24. Next time: Interaction of Radiation with Matter