Update perfusionModels.md

edits to format of TKM equations and references

docs/perfusionModels.md

@@ -96,15 +96,15 @@ We provide the differential equations and impulse response functions using the c
 
 | Code | OSIPI name| Alternative names|Notation|Description|Reference|
 | -- | -- | -- | -- | -- | -- |
-| M.IC1.001 |  <a name="LSS"> Linear and stationary system model | -- | LSS model | This forward model is given by the following equations: </br> $C(t)=I(t)\ast C_{a,p}(t)$ </br> with </br> [[I (Q.IC1.005)](quantities.md#IRF), [t (Q.GE1.004)](quantities.md#time)],</br> [[$C_{a,p}$ (Q.IC1.001.[a,p])](quantities.md#C), [$t$ (Q.GE1.004)](quantities.md#time)],</br> [[$C_t$ (Q.IC1.001.[t])](quantities.md#C), [$t$ (Q.GE1.004)](quantities.md#time)] | Rempp et al. (1994) |
-| M.IC1.002 | <a name="1CNEX"> One-compartment, no indicator exchange  model | -- | 1CNEX model | The one compartment no indicator exchange model describes an intravascular model with no vascular to EES indicator exchange. This forward model is given by the following differential equation: </br> $v_{p}\frac{dC_{t}(t)}{dt} = F_{p}C_{a,p} - F_{p}C_{c,p}(t)$</br>The impulse response function is given by: </br> $I(t) = F_{p}e^{{-\frac{F_{p}}{v_{p}}(t)}}$ </br> with ... TO ADD | Tofts et al. (1999) |
-| M.IC1.003 | <a name="1CFEX"> One-compartment, fast indicator exchange  model | -- | 1CFEX model | The one compartment fast exchange model describes infinitely fast bi-directional exchange of indicator between vascular to extravascular extracellular spaces. The capillary and EES effectively act as a single compartment. This forward model is given by the following differential equation: </br> $\frac{dC_{t}(t)}{dt} = F_{p}C_{a,p} - \frac{F_{p}}{v_{p} + v_{e}}C_{t}(t)$ </br> The impulse response function is given by: </br> $I(t) = F_{p}e^{{-\frac{F_{p}}{v_{p} + v_{e}}(t)}}$ </br> with ... TO ADD PARAMETERS | Sourbron et al. (2013) |
-| M.IC1.004 | <a name="TM"> Standard Tofts Model | Kety model, Generalized Kinetic Model | TM | The Tofts model describes bi-directional exchange of indicator between vascular to extravascular extracellular spaces. The capillary compartment is assumed to have negligible volume. The EES is modeled as well-mixed compartment. The forward model is given by the following differential equation: </br> $\frac{dC_{t}(t)}{dt} = K^{trans}C_{c,p} - \frac{K^{trans}}{v_{e}}C_{t}(t)$ </br> The impulse response function is given by: </br> $I(t) = K^{trans}e^{{-\frac{K^{trans}}{v_{e}}(t)}}$ </br> with ... TO ADD PARAMETERS | Tofts and Kermode (1991) |
-| M.IC1.005 | <a name="ETM"> 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 ... TO ADD PARAMETERS | Tofts (1997) |
-| M.IC1.006 | <a name="Patlak"> 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: $$C_{c,p} = C_{a,p}$$ 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 ... TO ADD PARAMETERS | Patlak et al. (1983) |
-| M.IC1.007 | <a name="2CUM"> 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 ... TO ADD PARAMETERS | Pradel et al. (2003), Sourbron (2009) |
-| M.IC1.008 | <a name="PFUM"> 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 AND PARAMETERS | St. Lawrence and Frank (2000) |
-| M.IC1.009 | <a name="2CXM"> 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 ... TO ADD PARAMETERS | Brix et al. (2004), Sourbron et al. (2009), Donaldson et al. (2010) |
+| M.IC1.001 |  <a name="LSS"> Linear and stationary system model | -- | LSS model | This forward model is given by the following equations: </br> $C(t)=I(t)\ast C_{a,p}(t)$ </br> with </br> [[I (Q.IC1.005)](quantities.md#IRF), [t (Q.GE1.004)](quantities.md#time)],</br> [[$C_{a,p}$ (Q.IC1.001.[a,p])](quantities.md#C), [$t$ (Q.GE1.004)](quantities.md#time)],</br> [[$C_t$ (Q.IC1.001.[t])](quantities.md#C), [$t$ (Q.GE1.004)](quantities.md#time)] | (Rempp et al. 1994) |
+| M.IC1.002 | <a name="1CNEX"> One-compartment, no indicator exchange  model | -- | 1CNEX model | The one compartment no indicator exchange model describes an intravascular model with no vascular to EES indicator exchange. This forward model is given by the following differential equation: </br> $v_{p}\frac{dC_{t}(t)}{dt} = F_{p}C_{a,p} - F_{p}C_{c,p}(t)$</br>The impulse response function is given by: </br> $I(t) = F_{p}e^{{-\frac{F_{p}}{v_{p}}t}}$ </br> with ... TO ADD | (Tofts et al. 1999) |
+| M.IC1.003 | <a name="1CFEX"> One-compartment, fast indicator exchange  model | -- | 1CFEX model | The one compartment fast exchange model describes infinitely fast bi-directional exchange of indicator between vascular to extravascular extracellular spaces. The capillary and EES effectively act as a single compartment. This forward model is given by the following differential equation: </br> $\frac{dC_{t}(t)}{dt} = F_{p}C_{a,p} - \frac{F_{p}}{v_{p} + v_{e}}C_{t}(t)$ </br> The impulse response function is given by: </br> $I(t) = F_{p}e^{{-\frac{F_{p}}{v_{p} + v_{e}}t}}$ </br> with ... TO ADD PARAMETERS | (Sourbron et al. 2013) |
+| M.IC1.004 | <a name="TM"> Standard Tofts Model | Kety model, Generalized Kinetic Model | TM | The Tofts model describes bi-directional exchange of indicator between vascular to extravascular extracellular spaces. The capillary compartment is assumed to have negligible volume. The EES is modeled as well-mixed compartment. The forward model is given by the following differential equation: </br> $\frac{dC_{t}(t)}{dt} = K^{trans}C_{c,p} - \frac{K^{trans}}{v_{e}}C_{t}(t)$ </br> The impulse response function is given by: </br> $I(t) = K^{trans}e^{{-\frac{K^{trans}}{v_{e}}t}}$ </br> with ... TO ADD PARAMETERS | (Tofts and Kermode 1991) |
+| M.IC1.005 | <a name="ETM"> 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 ... TO ADD PARAMETERS | (Tofts 1997) |
+| M.IC1.006 | <a name="Patlak"> 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 ... TO ADD PARAMETERS | (Patlak et al. 1983) |
+| M.IC1.007 | <a name="2CUM"> 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 ... TO ADD PARAMETERS | (Pradel et al. 2003), (Sourbron 2009) |
+| M.IC1.008 | <a name="PFUM"> 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 AND PARAMETERS | (St. Lawrence and Frank 2000) |
+| M.IC1.009 | <a name="2CXM"> 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 ... TO ADD PARAMETERS | (Brix et al. 2004), (Sourbron et al. 2009), (Donaldson et al. 2010) |
 | M.IC1.999 | <a name="not listed IC1"></a> Model not listed | -- | -- | This is a custom free-text item, which can be used if a model of interest is not listed. Please state a literature reference and request the item to be added to the lexicon for future usage. | -- |