Iodine-131 is used to treat hypo-thyroidism since it is preferentially absorbed by the thyroid and typically involves a total radiation dosage of 10,000,000 millirem. Iodine-131 has a half-life of eight days.
1. Set up the appropriate integral to represent the radiation dosage delivered by the absorbed iodine in eight days.
2. Use the fact that eight days is the half-life of the isotope to find the initial radiation intensity in millrems/hour.
3. To the nearest 10 millirems, how much of the total radiation is delivered in six weeks?
I got this:
D = int dt Io e^-.0866 t from t=0 to t = 8
but what do I do after that to get B and C?
Now I did this yesterday :)
However I did it in days and for part 2 you need it in hours.
To do it in hours, get 8 days in hours
8*24 = 192 hours half life
so
I = Io e^-kt now t in hours
.5 = e^-192 t
ln .5 = -.693 = -192 k
k = .00361
so
I = Io e^-.00361 t
Now do your integral from t = 0 to t = 8 days
D = int I dt = int Io e^-.00361 t from t = 0 to t = 192
or
D = (Io/.00361)(1/2)= 138 Io
so
10^7 millirem = 10^4 rem = 138 Io
so
Io = 10,000/138 = 72.5 rem/hr = 72.5*10^3 millirem/hr
now in 6 weeks
6 weeks = 7*24*6 = 1008 hr
D = (Io /.00361)(1-e^-(.00631*1008))
D = (72.5*10^3/.00361)(1 essentially)
D = 20083*10^3 millirems = 20*10^6 millirems essentially or about twice the total dose we got in one half life logically enough
A patient is given a 5 mCi dose of 131I. Estimate the activity of her thyroid after 4 days. (Assume that all remaining 131I has been absorbed by her thyroid. Radioactive half-life of 131I is 8 days and it has a biological half-life of 138 days
To solve the problem and find B and C, we need to follow these steps:
1. Set up the appropriate integral to represent the radiation dosage delivered by the absorbed iodine in eight days:
The given integral is correct:
D = ∫ Io e^(-0.0866t) dt, where t varies from 0 to 8.
2. Use the fact that eight days is the half-life of the isotope to find the initial radiation intensity in millirems/hour:
Since iodine-131 has a half-life of eight days, we can use the concept of exponential decay to find the initial radiation intensity (Io).
The formula for the half-life is:
Io * e^(-0.0866 * 8) = Io / 2
Simplifying this equation, we get:
e^(-0.0866 * 8) = 1 / 2
Solving for e^(-0.0866 * 8), we find:
e^(-0.0866 * 8) ≈ 0.5
Therefore, the initial radiation intensity (Io) is approximately half of the total radiation dosage over eight days: Io = 10,000,000 / 2 = 5,000,000 millirem.
3. To find the radiation delivered in six weeks (42 days) to the nearest 10 millirems:
We can use the integral equation above and evaluate it for the interval of t from 0 to 42. Let's denote the result for this integral as B.
B = ∫5,000,000 * e^(-0.0866t) dt from t = 0 to 42
To approximate the value of B to the nearest 10 millirems, we need to solve this integral either numerically or using suitable software.
Alternatively, if you have access to a scientific calculator, you can break down the integral into smaller intervals and manually calculate them. For example, you could evaluate the integral from t = 0 to t = 8, then from t = 8 to t = 16, and so on until t = 42, and sum up the results.
Once you find the value of B, round it to the nearest 10 millirems.
I hope this helps! Let me know if you have any further questions.