1. Radioactivity Law of Decay, Half Life and Statistics Weak 3
Dr. Hanan Y. Abbas-

2. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
Radioactivity – Definition and Units
Radioactivity (or the activity of a given sample of a radionuclide)is defined as number of disintegrations per unit of time (decay rate).
(1)
Where dNt is a number of radioatoms, dt is a small time interval and Rt is the radioactivity.
Equation (1) shows that a number of radioatoms, dNt, decays in a small time interval dt.
The minus sign denotes that the radioatoms are decreasing in number.
Or Radioactivity is the number of nuclei disintegrating per unit time at time (t)
Dr. Hanan Y. Abbas-

3. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
The Units for Radioactivity are the Curie (Ci), named in the honor of Madame Curie, and Becquerel (Bq). 1Curie (Ci)= 3.7x 1010 dis/sec A curie is a large unit and defined as the radioactivity of a sample which is decaying at a rate of 3.7x 1010 dis/sec. mCi= 10-3xCi=3.7x107 dis/sec μCi=10-6xCi=3.7x104 dis/sec nCi=10-9xCi=3.7x10 dis/sec pCi=10-12xCi=3.7x10-2 dis/sec
Curie is a large unit, the smaller units derived from the curie are more commonly used.
Dr. Hanan Y. Abbas-

4. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
1Bq=1 dis/sec. A becquerel is a small unit and defined as the radioactivity of a sample which is decaying at a rate of one disintegration per second. Relationshop between (Bq) and (Ci) 1Bq=27.03x10-12Ci= pCi. KBq=103Bq=27.03 nCi MBq=106 Bq=27.03 μCi GBq=109 Bq=27.03 mCi
Dr. Hanan Y. Abbas-

5. Dr. Hanan Y. Abbas- hanan_abbas1@windowslive.com
Law of Decay
Rt=λ . Nt (2)
Where λ is known as the decay constant, Nt is the number of the radioatoms present at a time t.
decay constant λ is defined as the probability of decay per unit time for a single radioatom.
λ is a characteristic of a given radionuclide and different for different radionuclides.
From Eq. (2) for the different radionuclides (that is different λ), the number of radioatoms (Nt ) that must be present to produce the same amount of radioactivity (Rt)varies.
For example , the no of radioatoms present in a sample of 99mTc, which has a radioactivity of 1 mCi, is not the same as the no of radioatoms present in a sample of 131I, which has a radioactivity of 1 mCi.
Dr. Hanan Y. Abbas-

6. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
Calculation of the mass of a radioactive sample
Substituting value of Nt (from eq. (2), for 1-mCi sample, Nt =Rt/ λ =(3.7x107)/λ
Unit: gm/mCi
Therefore mass of 1 mCi can be calculated.
Dr. Hanan Y. Abbas-

7. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
Example (1): What is the mass of a 1-mCi sample of 99mTc? The decay constant is 3.2x10-5/sec. What is the specific activity of 99mTc sample? If the technetium radioactivity is present as pertechnetate e rather than as a simple element (99mTc), find the specific radioactivity of the sample per unit mass of the pertechnetate exist in the sample? (molecular weight=163).
Dr. Hanan Y. Abbas-

8. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
Specific Activity
The specific activity of a radioactive sample is defined as the radioactivity per unit mass of a radionuclide
Unit: mCi/gm (or) dis/min/mg.
The specific activity is simply the inverse of M
The specific activity= 1/M=λ/(Ax6x 10-17)
It is assumed that all the atoms of technetium were radioactive and there were no stable or longer lived technetium atoms present in the sample.
Sample which contains only the radionuclide of interest and no other isotope or longer lived radioisotope of that radionuclide is known as ‘ carrier free’ sample.T
From above the specific activity applied to a carrier free sample
Dr. Hanan Y. Abbas-

9. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
When a radionuclide is not present in its elemental form, but in a molecule form, the concept of specific activity can be generalized to the mass of the molecule rather than the element.
If the sample is carrier free, (i.e., sample does not contain 99mTc or molecules)
The specific activity of the sample per unit mass of molecule= 1/M=λ/(Ax6x 10-17)
Where (A) is the molecular weight in this case
Dr. Hanan Y. Abbas-

10. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
The Exponential Law of Decay
If we combine equation (1) and (2), then (–dNt /dt)=λNt
This is a differential eq. whose solution is given by the following expression:
Nt=Noe-(λ.t) (3)
No Is the initial number of radioatoms (at t=0), Nt is the remaining number at time (t). Eq. (3) is known as the exponential law of decay.T
By substituting eq. (3) on the right hand side of eq. (2) we get:
Rt=λ . Nt (2)
Rt= λ. Noe-(λ.t)
Or Rt = Roe-(λ.t) (4)
Where Ro (Ro = λ. No)is the radioactivity at time Zero.
Dr. Hanan Y. Abbas-

11. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
Half-Life
Have life is defined as the time interval for a given number of nuclei (or their radioactivity) to decay to one-half of the original value.
From definition of the half-life, when Nt= No/2, then t= T1/2
Substitute these values in eq. (3) we obtain
(No/2)=Noe-(λ.T1/2)
½=e-(λ.T1/2)
Then
λ.T1/2 = (5) unit of T1/2 =second
From this eq. if one knows the half-life of the radionuclide, one can calculate the decay constant or vice versa.
Dr. Hanan Y. Abbas-

12. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
Example (2): Half-life of 131I=8 days. Determine the decay constant for 131I.
Dr. Hanan Y. Abbas-

13. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
Problems in Radioactive Decay Eqs. (1) to (5) are the basic eqs. Necessary for solving routine problems in nuclear medicine. However, in special cases, where multiple of half-life are involved, eq. (3) can be written in a simple form by substituting the value of λ from eq. (5) Nt=Noe-(λ.t) λ.T1/2 =0.693 Nt=Noe-(0.693t)/ T1/2 =Noe(-0.693)t/ T1/2 =No(1/2)t/ T1/2 For example, if one is interested in finding the number of radioatoms remaining after an interval t=3x the half-time of a radionuclide, the above expression become Nt=No/8
Dr. Hanan Y. Abbas-

14. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
Rt =Ro(1/2)t/ T1/2 (6b) Where t is time interval. Example (3): A radioactive sample of 99mTc contains 10-mCi radioactivity at 9:00A.M What will be the radioactivity of this sample at 3:00P.M. on the same day? (The half-life of 99mTc is 6 hr).
Dr. Hanan Y. Abbas-

15. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
Average Life (Tav)
Occasionally we encounter the use of another term known as average life of a radionuclide. It is related to the half-life of the radionuclide or the decay constant λ as follows:
Tav=1.44 x T1/2=1/ λ
Since in radioactive decay not all radioatoms decay at the same time, each individual radioatom exists for a different period of time. Average life is therefore the mean of all these intervals for which individual radioatoms remain in existence
Dr. Hanan Y. Abbas-

16. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
Biological Half-Life
The disappearance with time of a biochemical or drug in a biological system, thyroid, liver, lung, bone, blood, and plasma, can be described by an exponential law similar to that for radionuclidic decay.
Mt = Moe-(λBio.t) (7)
Mothe original amount of biochemical or drug , while Mt the amount of the biochemical or drug present in the biological system at time t. λBio is the probability of its disappearance in the biological system under consideration
λBio.T1/2 (Bio)= (8a)
The biological half-life is defined as the interval during which a given amount of a biochemical or drug in biological system is reduced to one-half its original value.
Mt = Mo(A1e-λ1Bio.t+A2e -λ2Bio.t+-----)(8b)
Where A1, A2,---- and λ1Bio, λ2Bio… are constants and are determined experimentally for a given drug in a given biological system.
Dr. Hanan Y. Abbas-

17. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
Effective Half-Life
In nuclear medicine one is interested in the disappearance with time of radioactive substance or drugs from biological systems as a results of both physical decay and biological elimination (metabolism, diffusion or excretion).
λ+ λBio= λeff (9)
Where λeff is the effective disappearance probability for a radioactive substance or drug.
We can convert the effective disappearance constant to its respective half-life
(10)
Dr. Hanan Y. Abbas-

18. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
From this relationship eq. (10), one can determine the effective half-life of a radiopharmaceutical if the physical half-life of the radionuclide and the biological half-life of the drug (or chemical) are known.
However, T1/2(eff) will always be less than or equal to the shorter of the two T1/2 and T1/2 (Bio)
Dr. Hanan Y. Abbas-

19. Dr. Hanan Y. Abbas- dhmd70@hotmail.com
Example (4): The biological half-life of iodine in human thyroid is about 64 days, and the physical half-life of 131I is 8 days. Determine the effective half-life of 131I in the thyroid.
Dr. Hanan Y. Abbas-