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

I suggest you start by considering this a sum of exponentials. You have the isotope half-lives, so you can easily compute the decay constants [ K(n) here ] from them, for example as an anonymous function (with the K(n) vector previously defined in your code so the function can access it from the workspace): f = @(B,t) B(1).*exp(K(1).*t) + ... + B(4).*exp(K(4).*t); This can be considered a linear problem. If you also have to estimate the decay constants, then this becomes a nonlinear problem: f = @(B,t) B(1).*exp(B(2).*t) + ... + B(7).*exp(B(8).*t); In either situation, with a 121-element data vector, estimating four or eight parameters should not be difficult. You can then compare the decay constants with those you already calculated from the half-lives to see what initial concentrations correspond to what isotopes. The term ‘governing matrix’ is new to me. I can't even find it on a search. (The best I can guess is that it's the Jacobian, or the variance-covariance matrix of the parameters that can be derived from the Jacobian and the residuals.) ‘CAS’ is also new to me. It doesn't seem to be a sufficiently unique acronym that a search suggests anything specific.

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