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MRI signal intensity

  1. Dec 29, 2007 #1
    1. The problem statement, all variables and given/known data
    We will make a Spin-Echo experiment, and we know [tex]\rho_{w}, T1_{w},T2_{w}, \rho_{g},T1_{g},T2_{g}[/tex], where T1 and T2 are relaxation time,[tex]\rho[/tex] 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: [tex]SI=\rho\left(1-e^{-\frac{TR-TE}{T1}}\right)e^{-\frac{TE}{T2}}[/tex]
    also we have two equation
    [tex]SI_{w}=\rho_{w}\left(1-e^{-\frac{TR-TE}{T1_{w}}}\right)e^{-\frac{TE}{T2_{w}}}[/tex] and
    [tex]SI_{g}=\rho_{g}\left(1-e^{-\frac{TR-TE}{T1_{g}}}\right)e^{-\frac{TE}{T2_{g}}}[/tex]
    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:
    [tex]\frac{\partial}{\partial TR}f=0[/tex] and [tex]\frac{\partial}{\partial TE}f=0[/tex]
    Is this place a minimum or maximum place?
    Also my question is, divide I or subtract the two equation?
     
  2. jcsd
  3. Dec 29, 2007 #2

    marcusl

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    Gold Member

    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.
     
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