Radioiodine treatment Roughly 12,000 Americans are diagnosed with thyroid cancer

Question

Radioiodine treatment Roughly 12,000 Americans are diagnosed with thyroid cancer every year, which accounts for 1% of all cancer cases. It occurs in women three times as frequently as in men. Fortunately, thyroid cancer can be treated successfully in many cases with radioactive iodine, or I-131. This unstable form of iodine has a half-life of 8 days and is given in small doses measured in millicuries.

a. Suppose a patient is given an initial dose of 100 millicuries. Find the function that gives the amount of I-131 in the body after t0t0 t greater than or equal to 0 days.

b. How long does it take the amount of I-131 to reach 10% of the initial dose?

c. Finding the initial dose to give a particular patient is a critical calculation. How does the time to reach 10% of the initial dose change if the initial dose is increased by 5%?

Explanation / Answer

Inititally dose of 100 mullicuries is given
Half life is 8 days which means the volume present in the body decreases by half every 8 days;
Going by eqution of Half life :
Nt = N0 e-kt where N0 is the intial dosage, Nt is the amount left in the body after 't' days
'k' is a constant given by ln 2/half life
Nt = 100 e-t * ln 2/ k

Here half life = 8 days
t1/2 = 8 days so ln 2/k = 8 and k = ln 2/ 8= 0.0866
k=0.0866

Now the equation is
Nt = 100 e-t * 0.0866 = 100 e-0.0866t  
where t>= 0 days and units of Nt is millicuries

b) to reach 10% of initial odse, Nt = 100/10 = 10
10 = 100 * e-0.0866t
0.0866t = ln (100/10)
t = 2.3025/ 0.0866 = 26.5877
Thus, it'll take 26.5877 days to make it 10% of initial dose

c) if the initail dose was 5% higher then 10% of it will be 10% of 105 = 10.5
substituting the values in the equation we get

10.5 = 105 * e-0.0866t
0.0866t = ln (105/10.5)
t = 2.3025/ 0.0866 = 26.5877
Thus, it'll take 26.5877 days to make it 10% of initial dose even if initial dose is increased by 5%

Note: here 10% of initial dose means 10% of the new intial dose which is 10% of 105 which comes out to be 10.5 millicuries

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