MRI Signal Intensity: Optimizing TR & TE

[schulzy]

Homework Statement

We will make a Spin-Echo experiment, and we know $$\rho_{w}, T1_{w},T2_{w}, \rho_{g},T1_{g},T2_{g}$$, where T1 and T2 are relaxation time,$$\rho$$ is a proton density, and g mean gray matter, w mean white matter.
We search TR and TE, TR is a repetition time, TE is an echo time. We need this two time define, and be the different of signal intensity biggest. Signal intensity: $$SI=\rho\left(1-e^{-\frac{TR-TE}{T1}}\right)e^{-\frac{TE}{T2}}$$
also we have two equation
$$SI_{w}=\rho_{w}\left(1-e^{-\frac{TR-TE}{T1_{w}}}\right)e^{-\frac{TE}{T2_{w}}}$$ and
$$SI_{g}=\rho_{g}\left(1-e^{-\frac{TR-TE}{T1_{g}}}\right)e^{-\frac{TE}{T2_{g}}}$$
and I don't know, what should I do, I divide or subtract the two equation. And after this operation we make a partial derivation:
$$\frac{\partial}{\partial TR}f=0$$ and $$\frac{\partial}{\partial TE}f=0$$
Is this place a minimum or maximum place?
Also my question is, divide I or subtract the two equation?

 

[marcusl]

Since you presumably want to differentiate grey and white matter, you want the biggest ratio of the two signals--so you divide the two equations.

The classic way to tell whether you have found a minimum or maximum is to then evaluate the 2nd derivative at the value of extremum that you found from the 1st. If the 2nd derivative is positive it's a minimum, negative = max, 0 = inflection point. Your text, it probably has additional info on extrema and 2nd derivatives.