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6 Magnetization Transfer Imaging/3 Magnetization Transfer Saturation/01-Introduction.md

@@ -22,7 +22,7 @@ This content of this section is still a work-in-progress and has not been proofr
 
 Magnetization Transfer Saturation (MTsat) is a semi-quantitative MRI technique that offers unique insights into tissue microstructure. Built upon the spoiled gradient-recalled echo (SPGR) sequence, the MTsat protocol acquires images with and without an MT-preparation off-resonance pulse to acquire different contrast that varies with macromolecular density and _T_{sub}`1`.
 
-The foundation of MTsat lies in a 2008 model by Helms and colleagues (Helms et al. 2008), which treats the off-resonance pulse as a second excitation pulse, allowing us to model the effects of MT analytically without the need of the complex Bloch-McConnel equations. Following some reasonable approximations and the acquisition of three distinct MRI images, this model allows for analytical computation of a parameter that models the % reduction in free-pool longitudinal magnetization due to a single off-resonance pulse, MTsat. 
+The foundation of MTsat lies in a 2008 model by Helms and colleagues [@Helms2008-wf], which treats the off-resonance pulse as a second excitation pulse, allowing us to model the effects of MT analytically without the need of the complex Bloch-McConnel equations. Following some reasonable approximations and the acquisition of three distinct MRI images, this model allows for analytical computation of a parameter that models the % reduction in free-pool longitudinal magnetization due to a single off-resonance pulse, MTsat. 
 
 This introduction provides a glimpse into the theoretical basis of MTsat, its practical applications, and sensitivity to variables like tissue _T_{sub}`1` and _B_{sub}`1`. By exploring the unique properties and potential of MTsat, we hope to give readers a better understanding of the advantages and limitations of this MRI technique in both research and clinical practice, as well as give a deeper conceptual understanding of what the MTsat value means.
 
@@ -32,7 +32,7 @@ This introduction provides a glimpse into the theoretical basis of MTsat, its pr
 Simplified pulse sequence diagram of an MTR imaging sequence. An off-resonance and high powered MT-preparation pulse is followed by a spoiler gradient to destroy any transverse magnetization prior the application of the imaging sequence, in this case a spoiled gradient recalled echo (SPGR).
 ```
 
-In the initial MTsat paper (Helms et al. 2008, 2010), the main innovation stems from a new model of the MT-weighted SPGR sequence shown in [](#mtsatFig1). There, (Helms et al. 2008) proposed to interpret the effects of the MT-preparation pulse as a second excitation RF pulse of an unknown flip angle. That is to say, they modeled the reduction of the longitudinal magnetization of the free pool due to the MT pulse to be the same reduction caused by the flip angle rotation of a second instantaneous excitation RF pulse. [](#mtsatFig2) presents the Helms model, where to be consistent with the convention presented in mathematical derivations in (Helms et al. 2008, 2010), the order of the pulses are switched such that the readout excitation pulse comes first ({math}`\alpha_{1}`), and the excitation pulse modelling the effects of the MT pulse comes second ({math}`\alpha_{2}`). Note that, after a steady-state is reached, this order would not not impact the signal value during image readout. 
+In the initial MTsat paper [@Helms2008-wf;@Helms2010-kv], the main innovation stems from a new model of the MT-weighted SPGR sequence shown in [](#mtsatFig1). There, [@Helms2008-wf] proposed to interpret the effects of the MT-preparation pulse as a second excitation RF pulse of an unknown flip angle. That is to say, they modeled the reduction of the longitudinal magnetization of the free pool due to the MT pulse to be the same reduction caused by the flip angle rotation of a second instantaneous excitation RF pulse. [](#mtsatFig2) presents the Helms model, where to be consistent with the convention presented in mathematical derivations in [@Helms2008-wf;@Helms2010-kv], the order of the pulses are switched such that the readout excitation pulse comes first ({math}`\alpha_{1}`), and the excitation pulse modelling the effects of the MT pulse comes second ({math}`\alpha_{2}`). Note that, after a steady-state is reached, this order would not not impact the signal value during image readout. 
 
 ```{figure} img/mtsat_model_sequence.png
 :label: mtsatFig2

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

@@ -18,7 +18,7 @@ numbering:
 This content of this section is still a work-in-progress and has not been proofread and/or reviewed.
 :::
 
-MTsat, like MTR and many flavours of quantitative MT, is based on spoiled gradient recalled echo (SPGR) images (Haase et al. 1986; Sekihara 1987; Hargreaves 2012) preceded by an off-resonance RF pulse to provide magnetization transfer contrast (Wolff and Balaban 1989; Henkelman et al. 1993; J. G. Sled and Pike 2000; John G. Sled 2018). [](#mtsatFig1) presents a simplified diagram of this MT-prepared SPGR pulse sequence (imaging gradients are not shown). A standard SPGR sequence (low flip angle [~5-10°], short TR [~10-30ms], and a strong spoiler gradient) are preceded by a long (~10 ms) off-resonance (~1-5 kHz) pulse with a strong peak amplitude (the total pulse has an equivalent on-resonance flip angle of 200°-700°). A smooth shape (e.g. Gaussian or Fermi) is typically used for the off-resonance pulse in order to have a single off-resonance frequency (from Fourier analysis). A strong spoiler gradient is also added between the off-resonance MT-preparation pulse and the on-resonance excitation pulse in order to destroy residual transverse magnetization that may have been created by the off-resonance pulse. Images acquired without MT saturation are acquired using the same timing as this sequence, but with the off-resonance RF pulse either completely off or using a very large off-resonance frequency (e.g. ~30+ kHz).
+MTsat, like MTR and many flavours of quantitative MT, is based on spoiled gradient recalled echo (SPGR) images [@Haase1986-kt;@Sekihara1987-bs;@Hargreaves2012-kj] preceded by an off-resonance RF pulse to provide magnetization transfer contrast [@Wolff1989-ag;@Henkelman1993-lt;@Sled2000-pc;@Sled2018-zr]. [](#mtsatFig1) presents a simplified diagram of this MT-prepared SPGR pulse sequence (imaging gradients are not shown). A standard SPGR sequence (low flip angle [~5-10°], short TR [~10-30ms], and a strong spoiler gradient) are preceded by a long (~10 ms) off-resonance (~1-5 kHz) pulse with a strong peak amplitude (the total pulse has an equivalent on-resonance flip angle of 200°-700°). A smooth shape (e.g. Gaussian or Fermi) is typically used for the off-resonance pulse in order to have a single off-resonance frequency (from Fourier analysis). A strong spoiler gradient is also added between the off-resonance MT-preparation pulse and the on-resonance excitation pulse in order to destroy residual transverse magnetization that may have been created by the off-resonance pulse. Images acquired without MT saturation are acquired using the same timing as this sequence, but with the off-resonance RF pulse either completely off or using a very large off-resonance frequency (e.g. ~30+ kHz).
 
 ```{math}
 :label: mtsatEq1
@@ -28,7 +28,7 @@ S\left( \alpha,\text{TR} \right)=A\text{sin}\left( \alpha \right)\frac{1-\text{e
 \end{equation}
 ```
 
-where A is some proportionality constant (e.g. gyromagnetic ratio, density, coil sensitivity, etc), ɑ is the excitation flip angle, _R_{sub}`1` = 1/_T_{sub}`1` (assuming a monoexponential longitudinal relaxation curve), and TR is the repetition time. Similarly, an analytical equation for the steady-state signal of a dual-excitation SPGR experiment ([](#mtsatFig2)) can be derived, and (Helms et al. 2008) demonstrated it to be:
+where A is some proportionality constant (e.g. gyromagnetic ratio, density, coil sensitivity, etc), ɑ is the excitation flip angle, _R_{sub}`1` = 1/_T_{sub}`1` (assuming a monoexponential longitudinal relaxation curve), and TR is the repetition time. Similarly, an analytical equation for the steady-state signal of a dual-excitation SPGR experiment ([](#mtsatFig2)) can be derived, and [@Helms2008-wf] demonstrated it to be:
 
 ```{math}
 :label: mtsatEq2
@@ -40,7 +40,7 @@ S\left( \alpha_{1},\text{TR}_{1},\alpha_{2},\text{TR}_{2} \right)=A\text{sin}\le
 
 where {math}`\alpha_{1}` is the imaging excitation flip angle, {math}`\alpha_{2}` is the excitation flip angle representing the MT saturation pulse, TR{sub}`1` is the time between {math}`\alpha_{1}` to {math}`\alpha_{2}`, TR{sub}`2` is the time between {math}`\alpha_{2}` and the following {math}`\alpha_{1}`, and TR = TR{sub}`1` + TR{sub}`2`. 
 
- [](#mtsatEq2) has three unknowns: _A_, _R_{sub}`1`, and {math}`\alpha_{2}`. Of these three, {math}`\alpha_{2}` is expected to be the most sensitive to macromolecular density via the MT effect, and as such is the parameter that we’d like to calculate or fit using this dual-excitation SPGR model for the MT-prepared SPGR pulse sequence. Although there would be some ways to acquire additional measurements (three unknowns, so at a minimum three measurements are needed) and apply a nonlinear fit to  [](#mtsatEq2) to extract {math}`\alpha_{2}`, this method has a long numerical processing time. To shorten the calculation of the parameter maps, (Helms et al. 2008, 2010) proposed some reasonable assumptions that can be made to simplify  [](#mtsatEq2). The first proposed assumption is that _R_{sub}`1`*TR << 1, which is true when using typical MT-weighted SPGR protocol parameters (TR ~ 0.01-0.05 s) and in the brain at clinical field strengths (_T_{sub}`1` ~ 1 s, thus _R_{sub}`1` ~ 1 s{sup}`-1`). The same approximation applies to TR{sub}`1` and TR{sub}`2`, which are shorter than TR. This leads to the removal of all exponential functions in  [](#mtsatEq2), as via the Taylor expansion of the exponential function, exp(x) ~ 1 + x when abs(x) << 1, and the removal of another term via R{sub}`1`TR{sub}`1` * R{sub}`1`TR{sub}`2` ~ 0 when R{sub}`1`TR{sub}`1` and R{sub}`1`TR{sub}`2` are both << 1. The simplifications result in
+ [](#mtsatEq2) has three unknowns: _A_, _R_{sub}`1`, and {math}`\alpha_{2}`. Of these three, {math}`\alpha_{2}` is expected to be the most sensitive to macromolecular density via the MT effect, and as such is the parameter that we’d like to calculate or fit using this dual-excitation SPGR model for the MT-prepared SPGR pulse sequence. Although there would be some ways to acquire additional measurements (three unknowns, so at a minimum three measurements are needed) and apply a nonlinear fit to  [](#mtsatEq2) to extract {math}`\alpha_{2}`, this method has a long numerical processing time. To shorten the calculation of the parameter maps, [@Helms2008-wf;@Helms2010-kv] (Helms et al. 2008, 2010) proposed some reasonable assumptions that can be made to simplify  [](#mtsatEq2). The first proposed assumption is that _R_{sub}`1`TR << 1, which is true when using typical MT-weighted SPGR protocol parameters (TR ~ 0.01-0.05 s) and in the brain at clinical field strengths (_T_{sub}`1` ~ 1 s, thus _R_{sub}`1` ~ 1 s{sup}`-1`). The same approximation applies to TR{sub}`1` and TR{sub}`2`, which are shorter than TR. This leads to the removal of all exponential functions in  [](#mtsatEq2), as via the Taylor expansion of the exponential function, exp(x) ~ 1 + x when abs(x) << 1, and the removal of another term via R{sub}`1`TR{sub}`1`R{sub}`1`TR{sub}`2` ~ 0 when R{sub}`1`TR{sub}`1` and R{sub}`1`TR{sub}`2` are both << 1. The simplifications result in
 
 ```{math}
 :label: mtsatEq3
@@ -50,7 +50,7 @@ S\left( \alpha_{1},\text{TR}_{1},\alpha_{2},\text{TR}_{2} \right)=A\text{sin}\le
 \end{equation}
 ```
 
-The second approximation is that {math}`\alpha_{2}` is small (less than 30 degrees), which is to say that the MT saturation is relatively small. This is expected to be true for the tissue properties of the brain (mostly, myelin), but care must be taken with the planned MT pulse parameters as the MT saturation increases with smaller offset frequency and high peak pulse amplitude. Later, we’ll calculate if this is a reasonable assumption for the calculated {math}`\alpha_{2}`. This assumption is integrated into [](#mtsatEq2) via the Taylor series expansion of the {math}`\text{cos} \left( \alpha_{2} \right)`, where {math}`\text{cos} \left( x \right) \approx 1-x^{2}/2`for small x (this relationship is true for x < 30 degrees or 0.5 radians). Introducing this approximation in [3] and with the additional simplifications {math}`\alpha_{2}^{2}` * R{sub}`1`TR ~ 0 (from the assumptions above), this results in
+The second approximation is that {math}`\alpha_{2}` is small (less than 30 degrees), which is to say that the MT saturation is relatively small. This is expected to be true for the tissue properties of the brain (mostly, myelin), but care must be taken with the planned MT pulse parameters as the MT saturation increases with smaller offset frequency and high peak pulse amplitude. Later, we’ll calculate if this is a reasonable assumption for the calculated {math}`\alpha_{2}`. This assumption is integrated into [](#mtsatEq2) via the Taylor series expansion of the {math}`\text{cos} \left( \alpha_{2} \right)`, where {math}`\text{cos} \left( x \right) \approx 1-x^{2}/2`for small x (this relationship is true for x < 30 degrees or 0.5 radians). Introducing this approximation in [3] and with the additional simplifications {math}`\alpha_{2}^{2}`R{sub}`1`TR ~ 0 (from the assumptions above), this results in
 
 ```{math}
 :label: mtsatEq4
@@ -80,15 +80,15 @@ S\left( \alpha_{1},\alpha_{2},\text{TR} \right)=A \alpha _{1}\frac{R_{1}\text{TR
 \end{equation}
 ```
 
-[](#mtsatFig3) demonstrates how {math}`\delta`, which represents MTsat as was defined in (Helms et al. 2008), is the fractional reduction in longitudinal magnetization after the MT pulse in the MTsat model illustrated in [](#mtsatFig2) relative to the Mz prior to the pulse. Conventionally, MTsat ({math}`\delta`) is reported in percentage, so {math}`\text{MTsat} = \delta \cdot 100` .
+[](#mtsatFig3) demonstrates how {math}`\delta`, which represents MTsat as was defined in [@Helms2008-wf], is the fractional reduction in longitudinal magnetization after the MT pulse in the MTsat model illustrated in [](#mtsatFig2) relative to the Mz prior to the pulse. Conventionally, MTsat ({math}`\delta`) is reported in percentage, so {math}`\text{MTsat} = \delta \cdot 100` .
 
 ```{figure} img/mtsat_trig.png
 :label: mtsatFig3
 :enumerator: 6.16
 Demonstration through trigonometry of how following a small flip angle {math}`\alpha_{2}` (eg MT saturation), the value {math}`\delta \equiv \alpha_{2}^{2}/2` represents the fraction of the reduction in longitudinal magnetization due to the pulse (bigDelta) relative to the value prior to the pulse (Mz{sub}`before`).
 ```
 
-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 _T_{sub}`1` at 1.5T (field strength that Brown used) of 0.55 s in healthy WM, so R{sub}`1` = 1.8. First off, [](#mtsatFig3) with no MT pulse (thus {math}`\delta` = 0) should converge close to the well-known SPGR equation [1]. Inputting the values in each equations, we get 0.0816A for [1], and 0.0815A, 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 {math}`\delta` using the MTR equation to bring everything together. Doing so is shown in [Appendix A](#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% ({math}`\delta` = 0.0492), which is close to some reported MTsat values in the literature (Karakuzu et al. 2022). From there, and by definition of {math}`\delta`, the modeled {math}`\alpha_{2}` in [](#mtsatFig2) for this example is 18 degrees, confirming that earlier assumption that {math}`\alpha_{2}` < 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 [@Brown2013-eg] of the MTR section, {math}`\alpha_{1}`=15 deg and TR = 0.03 s, so assuming a _T_{sub}`1` at 1.5T (field strength that Brown used) of 0.55 s in healthy WM, so R{sub}`1` = 1.8. First off, [](#mtsatFig3) with no MT pulse (thus {math}`\delta` = 0) should converge close to the well-known SPGR equation [1]. Inputting the values in each equations, we get 0.0816A for [1], and 0.0815A, 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 {math}`\delta` using the MTR equation to bring everything together. Doing so is shown in [Appendix A](#mtsatAppendixA), from there and using our simulations in the MTR post with [@Brown2013-eg] for healthier WM (MTR = 46%), we get an MTsat value of 4.92% ({math}`\delta` = 0.0492), which is close to some reported MTsat values in the literature (Karakuzu et al. 2022). From there, and by definition of {math}`\delta`, the modeled {math}`\alpha_{2}` in [](#mtsatFig2) for this example is 18 degrees, confirming that earlier assumption that {math}`\alpha_{2}` < 30 degrees for that approximation.
 
 In that example, we used a known _T_{sub}`1` 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 _T_{sub}`1` 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 {math}`\alpha_{1}`/TR than the MTon (MTw) and MToff (PDw) measurements used for MTR, that MTsat can be calculated analytically, and as a bonus a _T_{sub}`1` map is also calculated in the process. (This makes sense, as the VFA _T_{sub}`1` mapping sequence is often just two SPGR measurements with different {math}`\alpha` values). Thus, using this three measurement protocol (MTw/PDw/T1w, which we’ll call the MTsat protocol), MTsat and _T_{sub}`1` (1/R{sub}`1`) can be calculated analytically pixelwise using the following set of equations (derived from [](#mtsatEq5)):
 
@@ -122,11 +122,11 @@ Remember, like MTR, MTsat is calculated from the equations above following the a
 <table>
    <tr>
       <th colspan="2" align="center"></th>
-      <th colspan="1" align="center">Helms 2008</th>
-      <th colspan="1" align="center">Weiskopf 2013</th>
-      <th colspan="1" align="center">Campbell 2018</th>
-      <th colspan="2" align="center">Karakuzu 2022</th>
-      <th colspan="1" align="center">York 2022</th>
+      <th colspan="1" align="center">[@Helms2008-wf]</th>
+      <th colspan="1" align="center">[@Weiskopf2013-lp]</th>
+      <th colspan="1" align="center">[@Campbell2018-hi]</th>
+      <th colspan="2" align="center">[@Karakuzu2022-af]</th>
+      <th colspan="1" align="center">[@York2022-fl]</th>
 
    </tr>
    <tr>