added content

content/model_mtga.html

 ---
 title: MTGA
 author: Vesa Oikonen
-updated_at: 2019-03-24
+updated_at: 2021-04-21
 created_at: 2008-04-02
 tags:
   - Logan plot
@@ -258,7 +258,8 @@ calculated from the ratio of distribution volumes for the region of interest and
 </div>
 
 <p>where linearity is achieved after the intercept (<em>Int</em>) is effectively constant 
-(Logan, <a href="https://doi.org/10.1016/S0969-8051(00)00137-2">2000</a> and
+(Logan et al., <a href="https://doi.org/10.1038/jcbfm.1990.127">1990</a>, 
+<a href="https://doi.org/10.1016/S0969-8051(00)00137-2">2000</a> and
 <a href="https://doi.org/10.1016/S0969-8051(03)00114-8">2003</a>).</p>
 
 <p><a href="https://doi.org/10.1016/j.nucmedbio.2009.06.007">Tantawy et al. (2009)</a> introduced 
@@ -284,12 +285,14 @@ formulation (as "new plot"), because it avoids the noise-induced negative biases
 <em>V<sub>T</sub></em> and <em>BP<sub>ND</sub></em> estimates of the traditional Logan plot 
 formulation.</p>
 
+
+
 <h3><a name="logan_ref" title="logan_ref"></a>Logan plot without plasma sampling</h3>
 
 <p>When <a href="./model_reference_tissue.html">reference region</a> is available, then 
 <em>V<sub>T</sub></em> ratio (distribution volume ratio, <a name="DVR"></a>DVR) can be calculated 
-directly without blood sampling by using reference region in place of the arterial plasma integral:
-</p>
+directly without blood sampling by using reference region in place of the arterial plasma integral
+(<a href="https://doi.org/10.1097/00004647-199609000-00008">Logan et al., 1996</a>):</p>
 
 <div class="eqs">
 <script type="math/tex; mode=display">
@@ -303,10 +306,23 @@ directly without blood sampling by using reference region in place of the arteri
 compartment model or <em>k<sub>2</sub>/(1+k<sub>5</sub>/k<sub>6</sub>)</em> of two-tissue compartment
 model) must be determined from studies with plasma sampling. Fortunately, in many cases the term 
 containing the population average of apparent <em>k'<sub>2</sub></em> can be omitted 
-(<a href="https://doi.org/10.1016/S0969-8051(03)00114-8">Logan, 2003</a>).</p>
+(<a href="https://doi.org/10.1097/00004647-199609000-00008">Logan et al., 1996</a>;
+<a href="https://doi.org/10.1016/S0969-8051(03)00114-8">Logan, 2003</a>). Reformulation of 
+the equation above to</p>
+
+<div class="eqs">
+<script type="math/tex; mode=display">
+  \frac{\int_0^T C_{ROI}(t)dt}{C_{ROI}(T)} = 
+  \frac{V_T}{V_T^{ref}} \times \frac{\int_0^T C_{ref}(t)dt}{C_{ROI}(T)} + 
+  \frac{V_T}{V_T^{ref} \times \bar{k_2^{,}}} \times \frac{C_{ref}(T)}{C_{ROI}(T)}  + Int^{'}
+</script>
+</div>
+
+<p>shows that when ratio <em>C<sub>ref</sub>/C<sub>ROI</sub></em> becomes reasonably constant the 
+<em>k'<sub>2</sub></em> effectively becomes part of the plot intercept.
 
-<p>The negative of the intercept in the Logan plot with reference tissue input is the <em>relative 
-residence time</em> (<em>RRT</em>), which can be used to measure the clearance of 
+<p>The negative of the intercept, -Int', in the Logan plot with reference tissue input is the 
+<em>relative residence time</em> (<em>RRT</em>), which can be used to measure the clearance of 
 PET radiopharmaceutical from region-of-interest relative to reference region 
 (<a href="https://doi.org/10.1097/00019442-200201000-00004">Shoghi-Jadid et al, 2002</a>).</p>
 
@@ -315,6 +331,24 @@ time, because Logan plot method requires both tissue and input integrals startin
 However, in some cases the analysis may be possible from late-scan data 
 (<a href="https://doi.org/10.1016/j.nucmedbio.2009.06.007">Tantawy et al., 2009</a>).</p>
 
+
+<h4><a name="alternative_logan_ref" title="alternative_logan_ref">Alternative graphical analysis 
+with reference tissue input</a></h4>
+
+<p>Based on the <a href="#alternative_logan">alternative Logan plot</a> with plasma input,
+with its relatively bold assumptions, 
+<a href="https://doi.org/10.1016/j.neuroimage.2008.09.021">Zhou et al (2009)</a> introduced 
+"new plot" in form</p>
+
+<div class="eqs">
+<script type="math/tex; mode=display">
+  \frac{\int_0^T C_{ROI}(t)dt}{C_{ref}(T)} = 
+  \frac{V_T}{V_T^{ref}} \times \frac{\int_0^T C_{ref}(t)dt}{C_{ref}(T)} + Int^{''}
+</script>
+</div>
+
+<br>
+
 <p>For each radiopharmaceutical, the reference input methods have to be validated against plasma 
 input methods. See for example 
 <a href="https://doi.org/10.1016/j.nucmedbio.2007.12.004">Anteror-Dorsey et al. (2008)</a>.</p>