Try adding link to github discussion

2 T1 Mapping/2-4 Exercises/01-Exercises.md

@@ -11,7 +11,7 @@ numbering:
     template: Fig. %s
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-**[Problem 1 ⭑](gh-discussion:4)**
+**Problem 1 ⭑**<gh-discussion:4>
 :  **a.** Using the Plotly figure in the inversion recovery section that displays the signal curves for white matter, gray matter, and CSF, determine the inversion times that null the signal from each of these tissues.
 :  **b.** In practice, you may not have this type of interactive figure at the scanner. Using the three values nulling inversion time values above, can you find an easy way to approximate the nulling time for any arbitrary T1 value?
 :  **c.** Assuming that the images acquired at the MRI scanner displays magnitude-only images, at approximately which inversion time will the white matter and grey matter have the lowest contrast?

6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/02-Theory.md

@@ -88,4 +88,6 @@ Figure 3 demonstrates how delta, which represents MTsat as was defined in (Helms
 Demonstration through trigonometry of how following a small flip angle alpha2 (eg MT saturation), the value delta = alpha2^2/2 represents the fraction of the reduction in longitudinal magnetization due to the pulse (bigDelta) relative to the value prior to the pulse (Mzbefore).
 ```
 
-Before jumping into how to measure MTsat, let's demonstrate some expected properties and values using known values from a simpler MTR experiment. From the MTR protocol in (Brown, Narayanan, and Arnold 2013) of the MTR blog post, 1=15 deg and TR = 0.03 s, so assuming a T1 at 1.5T (field strength that Brown used) of 0.55 s in healthy WM, so R1 = 1.8. First off, Eq. 5 with no MT pulse (thus delta = 0) should converge close to the well-known SPGR equation [1]. Inputting the values in each equations, we get 0.0816*A for [1], and 0.0815*A, thus they are in close agreement. Next, we can get an estimated value of MTsat, using a known MTR value, the calculated S0 value (which we just did), and then solving [5] for delta using the MTR equation to bring everything together. Doing so is shown in [Appendix 6A](#mtsatAppendixA), from there and using our simulations in the MTR post with Brown2013 for healthier WM (MTR = 46%), we get an MTsat value of 4.92% (delta = 0.0492), which is close to some reported MTsat values in the literature (Karakuzu et al. 2022). From there, and by definition of delta, the modeled alpha2 in Figure 2 for this example is 18 degrees, confirming that earlier assumption that alpha2 < 30 degrees for that approximation.
+Before jumping into how to measure MTsat, let's demonstrate some expected properties and values using known values from a simpler MTR experiment. From the MTR protocol in (Brown, Narayanan, and Arnold 2013) of the MTR blog post, 1=15 deg and TR = 0.03 s, so assuming a T1 at 1.5T (field strength that Brown used) of 0.55 s in healthy WM, so R1 = 1.8. First off, Eq. 5 with no MT pulse (thus delta = 0) should converge close to the well-known SPGR equation [1]. Inputting the values in each equations, we get 0.0816*A for [1], and 0.0815*A, thus they are in close agreement. Next, we can get an estimated value of MTsat, using a known MTR value, the calculated S0 value (which we just did), and then solving [5] for delta using the MTR equation to bring everything together. Doing so is shown in [Appendix 6A](#mtsatAppendixA), from there and using our simulations in the MTR post with Brown2013 for healthier WM (MTR = 46%), we get an MTsat value of 4.92% (delta = 0.0492), which is close to some reported MTsat values in the literature (Karakuzu et al. 2022). From there, and by definition of delta, the modeled alpha2 in Figure 2 for this example is 18 degrees, confirming that earlier assumption that alpha2 < 30 degrees for that approximation.
+
+In that example, we used a known T1 value to extract MTsat using a two-measurement MTR experiment, but in practice this value is not known and varies per-pixel across tissues. Although we could use an additionally measured T1 map to do this, this can be time consuming depending on the method used. (Helms et al. 2008, 2010) thus demonstrated that with one additional T1w measurement that uses no MT preparation pulse but has different alpha1/TR than the MTon (MTw) and MToff (PDw) measurements used for MTR, that MTsat can be calculated analytically, and as a bonus a T1 map is also calculated in the process. (This makes sense, as the VFA T1 mapping sequence is often just two SPGR measurements with different alpha values). Thus, using this three measurement protocol (MTw/PDw/T1w, which we’ll call the MTsat protocol), MTsat and T1 (1/R1) can be calculated analytically pixelwise using the following set of equations (derived from Eq. 5):