OSIPI · Aarshpatel12 · Jan 26, 2026
diff --git a/docs/perfusionModels.md b/docs/perfusionModels.md
@@ -119,7 +119,7 @@ We provide the differential equations and impulse response functions using the c
 | M.IC1.005 <button class="md-button md-button--hyperlink">COPY LINK</button> | <a id="ETM"></a> Extended Tofts Model | Modified Tofts Model, Extended Generalized Kinetic Model, Modified Kety model | ETM | The extended Tofts model describes bi-directional exchange of indicator between vascular to extravascular extracellular spaces. The capillary and EES are modeled as well-mixed compartments. It is equivalent to the 2CXM in the highly perfused limit. Dispersion of indicator within the capillary bed is assumed negligible: </br> $C_{c,p} = C_{a,p}$ </br> The forward model is given by the following differential equation: </br> $v_{e}\frac{dC_{e}(t)}{dt} = PSC_{c,p} - PSC_{e}(t)$ </br> The impulse response function is given by: </br> $I(t) = v_{p}\delta(t) + K^{trans}e^{{-\frac{K^{trans}}{v_{e}}t}}$ </br> with </br> [[$C_{c,p}$ (Q.IC1.001.c,p)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [[$C_e$ (Q.IC1.001.e)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}],</br>  [[I (Q.IC1.005)](quantities.md#IRF){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [$\delta$ (M.DM1.009)](quantities.md#delta){:target="_blank"}, </br> [PS (Q.PH1.004)](quantities.md#PS){:target="_blank"}, </br> [$v_{e}$ (Q.PH1.001.e)](quantities.md#v){:target="_blank"}, </br> [$K^{trans}$ (Q.PH1.008)](quantities.md#Ktrans){:target="_blank"} | (Tofts 1997) |
 | M.IC1.006 <button class="md-button md-button--hyperlink">COPY LINK</button> | <a id="Patlak"></a> Patlak Model | -- | PM | The Patlak model allows uni-directional exchange of indicator from vascular to extravascular extracellular spaces. Indicator exchange from the EES to the intravascular space is considered negligible during the timeframe of the imaging experiment. The capillary and EES are modeled as well-mixed compartments. It is equivalent to the two compartment uptake model in the highly perfused limit. Dispersion of indicator within the capillary bed is assumed negligible: </br> $C_{c,p} = C_{a,p}$ </br> The forward model is given by the following differential equation: </br> $v_{e}\frac{dC_{e}(t)}{dt} = PSC_{c,p}$ </br> The impulse response function is given by: </br> $I(t) = v_{p}\delta(t) + PS$ </br> with </br> [[$C_{c,p}$ (Q.IC1.001.c,p)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [[$C_e$ (Q.IC1.001.e)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}],</br>  [[I (Q.IC1.005)](quantities.md#IRF){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [$\delta$ (Q.PH1.009)](quantities.md#delta){:target="_blank"}, </br> [PS (Q.PH1.004)](quantities.md#PS){:target="_blank"}, </br> [$v_{p}$ (Q.PH1.001.p)](quantities.md#v){:target="_blank"} | (Patlak et al. 1983) |
 | M.IC1.007 <button class="md-button md-button--hyperlink">COPY LINK</button> | <a id="2CUM"></a> Two compartment uptake model | -- | 2CUM | The 2CU model allows uni-directional exchange of indicator from vascular to extravascular extracellular spaces. Indicator exchange from the EES to the intravascular space is considered negligible during the timeframe of the imaging experiment. The capillary and EES are modeled as well-mixed compartments. The forward model is given by the following differential equations:  </br> $v_{p}\frac{dC_{c,p}(t)}{dt} = F_{p}C_{a,p} - F_{p}C_{c,p} - PSC_{a,p}$ </br> </br> $v_{e}\frac{dC_{e}(t)}{dt} = PSC_{a,p}$ </br> The impulse response function is given by: </br> $I(t) = F_{p}e^{-({\frac{F_{p} + PS}{v_{p}}})t} + E(1 - e^{-({\frac{F_{p} + PS}{v_{p}}})t})$ </br> with </br> [[$C_{a,p}$ (Q.IC1.001.a,p)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [[$C_{c,p}$ (Q.IC1.001.c,p)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [[$C_{e}$ (Q.IC1.001.e)](quantities.md#C), [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [[I (Q.IC1.005)](quantities.md#IRF){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [$F_p$ (Q.PH1.002)](quantities.md#Fp){:target="_blank"}, </br> [PS (Q.PH1.004)](quantities.md#PS){:target="_blank"}, </br> [E (Q.PH1.005)](quantities.md#E){:target="_blank"}, </br> [$v_{e}$ (Q.PH1.001.e)](quantities.md#v){:target="_blank"}, </br> [$v_{p}$ (Q.PH1.001.p)](quantities.md#v){:target="_blank"} | (Pradel et al. 2003), (Sourbron 2009) |
-| M.IC1.008 <button class="md-button md-button--hyperlink">COPY LINK</button> | <a id="PFUM"></a> Plug flow uptake model | -- | PFUM | The plug flow uptake model allows uni-directional exchange of indicator from vascular to extravascular extracellular spaces. Indicator exchange from the EES to the intravascular space is considered negligible during the timeframe of the imaging experiment. The capillary space is modeled as a plug flow system and the EES as a well-mixed compartment. The forward model is given by the following differential equations: </br> $v_{p}\frac{\partial C_{c,p}(x_{ax}, t)}{\partial t} = -F_{p}L_{ax}\frac{\partial C_{a,p}(x_{ax}, t)}{\partial x_{ax}} - PSC_{a,p}(x_{ax}, t)$ </br> </br> $v_{e}\frac{dC_{e}(t)}{dt} = PS \int_{0}^{L_{ax}} C_{c,p} (x_{ax},t) dx$ </br> The impulse response function is ... TO ADD IRF <br/> with </br> [[$C_{c,p}$ (Q.IC1.001.c,p)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [[$C_{e}$ (Q.IC1.001.e)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [[I (Q.IC1.005)](quantities.md#IRF){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [$F_p$ (Q.PH1.002)](quantities.md#Fp){:target="_blank"}, </br> [PS (Q.PH1.004)](quantities.md#PS){:target="_blank"}, </br> [$v_{e}$ (Q.PH1.001.e)](quantities.md#v){:target="_blank"}, </br> [$v_{p}$ (Q.PH1.001.p)](quantities.md#v){:target="_blank"}, <br/> [$L_{ax}$ (Q.GE1.007)](quantities.md#Lax){:target="_blank"}, <br/> [$x_{ax}$ (Q.GE1.008)](quantities.md#xax){:target="_blank"} | (St. Lawrence and Frank 2000) |
+| M.IC1.008 <button class="md-button md-button--hyperlink">COPY LINK</button> | <a id="PFUM"></a> Plug flow uptake model | -- | PFUM | The plug flow uptake model allows uni-directional exchange of indicator from vascular to extravascular extracellular spaces. Indicator exchange from the EES to the intravascular space is considered negligible during the timeframe of the imaging experiment. The capillary space is modeled as a plug flow system and the EES as a well-mixed compartment. The forward model is given by the following differential equations: <br /> $v_{p}\frac{\partial C_{c,p}(x_{ax}, t)}{\partial t} = -F_{p}L_{ax}\frac{\partial C_{a,p}(x_{ax}, t)}{\partial x_{ax}} - PSC_{a,p}(x_{ax}, t)$ <br /> <br /> $v_{e}\frac{dC_{e}(t)}{dt} = PS \int_{0}^{L_{ax}} C_{c,p} (x_{ax},t) dx$ <br /> The impulse response function is given by: <br /> $I(t) = F_p \left[ u(t) - e^{-\frac{PS}{F_p}} u\left(t - \frac{v_p}{F_p}\right) \right]$ <br /> with <br /> [[$C_{c,p}$ (Q.IC1.001.c,p)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], <br /> [[$C_{e}$ (Q.IC1.001.e)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], <br /> [[I (Q.IC1.005)](quantities.md#IRF){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], <br /> [$F_p$ (Q.PH1.002)](quantities.md#Fp){:target="_blank"}, <br /> [PS (Q.PH1.004)](quantities.md#PS){:target="_blank"}, <br /> [$v_{e}$ (Q.PH1.001.e)](quantities.md#v){:target="_blank"}, <br /> [$v_{p}$ (Q.PH1.001.p)](quantities.md#v){:target="_blank"}, <br /> [$L_{ax}$ (Q.GE1.007)](quantities.md#Lax){:target="_blank"}, <br /> [$x_{ax}$ (Q.GE1.008)](quantities.md#xax){:target="_blank"} | [St. Lawrence and Frank 2000](https://doi.org/10.1002/1522-2594(200012)12:6%3C975::AID-MRE23%3E3.0.CO;2-J){:target="_blank"} |
 | M.IC1.009 <button class="md-button md-button--hyperlink">COPY LINK</button> | <a id="2CXM"></a> Two compartment exchange model | -- | 2CXM | The 2CX model allows bi-directional exchange of indicator between vascular and extravascular extracellular compartments. Indicator is assumed to be well mixed within each compartment. The forward model is given by the following differential equations: </br> $v_{p}\frac{dC_{c,p}}{dt} = F_{p}C_{a,p}(t) - F_{p}C_{c,p}(t) - PSC_{c,p}(t) + PSC_{e}(t)$ </br> </br> $v_{e}\frac{dC_{e}}{dt} = PSC_{c,p}(t) - PSC_{e}(t)$ </br> The impulse response function is given by </br> $I(t) = F_{p}e^{-K_{+}t} + E_{-}(e^{-K_{+}t} - e^{-K_{-}t})$ </br> </br> $K_{\pm} = \frac{1}{2}\Biggl(\frac{F_{p} + PS}{v_{p}} + \frac{PS}{v_{p}} \pm \sqrt{(\frac{F_{p} + PS}{v_{p}} + \frac{PS}{v_{e}})^{2} - 4\frac{F_{p}PS}{v_{p}v_{e}}}\Biggl)$ </br> </br> $E_{-} = \frac{K_{+} + \frac{F_{p}}{v_{p}}}{K_{+} + K_{-}}$ </br> with </br> [[$C_{a,p}$ (Q.IC1.001.a,p)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [[$C_{c,p}$ (Q.IC1.001.c,p)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [[$C_{e}$ (Q.IC1.001.e)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [[I (Q.IC1.005)](quantities.md#IRF){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [$F_p$ (Q.PH1.002)](quantities.md#Fp){:target="_blank"}, </br> [PS (Q.PH1.004)](quantities.md#PS){:target="_blank"}, </br> [$v_{e}$ (Q.PH1.001.e)](quantities.md#v){:target="_blank"}, </br> [$v_{p}$ (Q.PH1.001.p)](quantities.md#v){:target="_blank"} | (Brix et al. 2004), (Sourbron et al. 2009), (Donaldson et al. 2010) |
 | M.IC1.010 <button class="md-button md-button--hyperlink">COPY LINK</button> | <a id="DPM"></a> Distributed parameter model | -- | DPM | The DP model allows bi-directional exchange of indicator between vascular and extravascular extracellular compartments. The capillary space is modeled as a plug flow system. The EES is modeled as a series of infinitesimal compartments which only exchange indicator with nearby locations in the capillary bed.This forward model is given by the following differential equations: </br> $v_{p}\frac{∂C_{c,p}}{∂t}(x_{ax},t) = F_{p}L_{ax}\frac{∂C_{cp}}{∂x_{ax}}(x_{ax},t) - PSC_{c,p}(x_{ax},t) + PSC_{e}(x_{ax},t)$ </br></br> $v_{e}\frac{∂C_{e}}{∂t}(x_{ax},t) = PSC_{c,p}(x_{ax},t) - PSC_{e}(x_{ax},t)$ </br> The impulse response function is given by  </br> $I(t) = F_{p}(1-u(t{-}\frac{v_{p}}{F_{p}})) e^\frac{-PS}{F_{p}} (1-\int_{0}^{t{-} \frac{v_{p}}{F_{p}}} x(τ)dτ)$ </br> where </br> $x(τ) = u(t) e^\frac{-t\cdot PS}{v_{e}} \sqrt{\frac{PS^2}{t\cdot v_{e} \cdot F_{p}}} I_{1}\Bigl(2\sqrt{\frac{PS^2 \cdot t}{v_{e}\cdot F_{p}}}\Bigl)$ </br></br> Where $I_{1}$ is the first order bessel function of the first kind with </br> [[$C_{c,p}$ ((Q.IC1.001.c,p))](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}],[[$C_{e}$ (Q.IC1.001.e)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [[I (Q.IC1.005)](quantities.md#IRF){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [[u (M.DM1.001)](quantities.md#u){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [$F_p$ (Q.PH1.002)](quantities.md#Fp){:target="_blank"}, </br> [PS (Q.PH1.004)](quantities.md#PS){:target="_blank"}, </br> [$v_{e}$ (Q.PH1.001.e)](quantities.md#v){:target="_blank"}, </br> [$v_{p}$ (Q.PH1.001.p)](quantities.md#v){:target="_blank"}, <br/> [$L_{ax}$ (Q.GE1.007)](quantities.md#Lax){:target="_blank"}, <br/> [$x_{ax}$ (Q.GE1.008)](quantities.md#xax){:target="_blank"} | (Sangren and Sheppard 1953) (Sourbron 2011) |
 | M.IC1.011 <button class="md-button md-button--hyperlink">COPY LINK</button> | <a id="THM"></a> Tissue homogeneity model | Johnson-Wilson model | THM |The TH model allows bi-directional exchange of indicator between vascular and extravascular extracellular compartments. The capillary space is modeled as a plug flow system and the EES as a well mixed compartment. This forward model is given by the following differential equations: </br> $v_{p}\frac{∂C_{e}}{∂t}(x_{ax},t) = -F_{p}L_{ax}\frac{∂C_{c,p}}{∂x_{ax}}(x_{ax},t) - PSC_{c,p}(x_{ax},t) + PSC_{e}$</br></br> $v_{e}\frac{∂C_{e}}{∂t} = \frac{PS}{L_{ax}} \int_{0}^{L_{ax}} C_{c,p}(x_{ax},t)dx - PSC_{e}(t)$</br> The impulse response function is given by:</br></br> $I(t) = u(t) - u(t-\frac{v_{p}}{F_{p}})(1-E)\left\{ 1 + \int_{0}^{t - \frac{v_{p}}{F_{p}}} \sqrt{\frac{F_{p}}{v_{e} τ}} ln(1-E)I_{1}(2ln(1-E)\sqrt{\frac{F_{p}}{v_{e}τ}} dτ)\right\}$ </br></br> Where $I_{1}$ is the first order bessel function of the first kind with </br> [[$C_{c,p}$ ((Q.IC1.001.c,p))](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}],[[$C_{e}$ (Q.IC1.001.e)](quantities.md#C){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [[I (Q.IC1.005)](quantities.md#IRF){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [[u (M.DM1.001)](quantities.md#u){:target="_blank"}, [t (Q.GE1.004)](quantities.md#time){:target="_blank"}], </br> [$F_p$ (Q.PH1.002)](quantities.md#Fp){:target="_blank"}, </br> [PS (Q.PH1.004)](quantities.md#PS){:target="_blank"}, </br> [$v_{e}$ (Q.PH1.001.e)](quantities.md#v){:target="_blank"}, </br> [$v_{p}$ (Q.PH1.001.p)](quantities.md#v){:target="_blank"}, <br/> [$L_{ax}$ (Q.GE1.007)](quantities.md#Lax){:target="_blank"}, <br/> [$x_{ax}$ (Q.GE1.008)](quantities.md#xax){:target="_blank"} | (Johnson and Wilson 1966) (Lawrence and Lee 1998) (Kershaw 2010) (Koh et al. 2003)|