Radionuclides used in PET scanning are typically isotopes with short half-lives

Question

Radionuclides used in PET scanning are typically isotopes with short half-lives such as carbon-11 (~20 min), nitrogen-13 (~10 min), oxygen-15 (~2 min), and fluorine-18 (~110 min). You are given 3 mystery materials, which contain different amounts of these four isotopes. You have a dumb counter, which can only measure total counts but cannot discriminate. You measure every minute for 2 hours (121 measurements). Your goal is to determine the amount of each isotope in the 3 mystery containers.

Q.Set up the least squares problem that you are going to solve. Compute the governing matrix.

Note: you don't have to write out every single element of this 121 x 4 matrix, just write down a formula/representation that provides a formula for each element. (Each column corresponds to a different decay constant.)

" I have three noise-free data sets and three noisy data sets. They are representing counts-1, counts-2 and counts-3 for each noisy and noise-free. Each one of them has 121 elements"

Explanation / Answer

f = @(B,t) B(1).*exp(K(1).*t) + ... + B(4).*exp(K(4).*t); f = @(B,t) B(1).*exp(B(2).*t) + ... + B(7).*exp(B(8).*t);

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