Section M: Perfusion Models¶
This section is currently work in progress
General forward model¶
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| M.GF1.001 | Forward model | -- | -- | A forward model to be inverted (select from Section M. | -- |
MR signal models¶
This section covers models that describe how the measured MR signal S depends on electromagnetic properties, such as the relaxation rates R1, R1 and R2* or the magnetic susceptibility \(\chi\) , and on MR sequence parameters such as TR and TE. The exception is section “Magnitude models: DCE - R1 in the fast water exchange limit, direct relationship with indicator concentration”, in which the model describes how the MR signal depends directly on the indicator concentration.
Magnitude models: DSC¶
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| M.SM1.001 | Gradient echo model | -- | GE model | This forward model is given by the following equation: \(S=S_0\cdot e^{-TE\cdot R_2^*}\) with TE (Q.MS1.005), S0 (Q.MS1.010), R2* (Q.EL1.007), S (Q.MS1.001). | Jackson et al. 2005 |
| M.SM1.002 | Spin echo model | -- | SE model | This forward model is given by the following equation: \(S=S_0\cdot e^{-TE\cdot R_2}\) with TE (Q.MS1.005), S0 (Q.MS1.010), R2 (Q.EL1.004), S (Q.MS1.001). | Jackson et al. 2005 |
| M.SM1.999 | 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. | -- |
Magnitude models: DCE - R1 in the fast water exchange limit¶
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| M.SM2.001 | Linear model | -- | Linear | This forward model is given by the following equation: \(S=k \cdot R_1\) with k (Q.GE1.009), R1 (Q.EL1.001), S (Q.MS1.001). | -- |
| M.SM2.002 | Spoiled gradient recalled echo model | FLASH model | SPGR model | This forward model is given by the following equation: \(S=S_0 \cdot \frac{sin(\alpha)[1-e^{TR\cdot R_1}]}{1-cos(\alpha)\cdot e^{-TR\cdot R_1}}\) with S0 (Q.MS1.010), R1 (Q.EL1.001), TR (Q.MS1.006), α (Q.MS1.007), S (Q.MS1.001) | -- |
| M.SM2.003 | Single-shot saturation recovery SPGR with centric encoding model | SS-SR-FLASH-c model | SS-SR-SPGR-c model | This forward model is given by the following equation: \(S = S_0\cdot(1-e^{-PD\cdot R_1})\) with S0 (Q.MS1.010), R1 (Q.EL1.001), PD (Q.MS1.008), S (Q.MS1.001) | -- |
| M.SM2.004 | Saturation-recovery SPGR with linear encoding model | SR-turboFLASH-lin model | SR-turboSPGR-lin model | This forward model is given by the following equation: \(S=S_0\cdot sin(\alpha)\cdot[(1-e^{-PD\cdot R_1})a^{n-1}+b\frac{(1-a^{n-1})}{(1-a)}]\) with \(a=cos(\alpha)\cdot e^{-TR\cdot R_1}\), \(b=1-e^{-TR\cdot R_1}\), S0 (Q.MS1.010), R1 (Q.EL1.001), PD (Q.MS1.008), TR (Q.MS1.006), α (Q.MS1.006)= 90°, n (Q.MS1.011), S (Q.MS1.001) | Larson 2001 |
| M.SM2.005 | Single-shot inversion recovery SPGR with centric encoding model | SS-IR-FLASH-c model | SS-IR-SPGR-c model | This forward model is given by the following equation: \(S = S_0\cdot(1-2e^{-PD\cdot R_1})\) with S0 (Q.MS1.010), R1 (Q.EL1.001), PD (Q.MS1.008), S (Q.MS1.001) | Ordidge et al. 1990 |
| M.SM2.006 | Inversion-recovery spoiled gradient recalled echo with linear encoding model | IR-turboFLASH-lin model | IR-turboSPGR-lin model | This forward model is given by the following equation: \(S=S_0\cdot sin(\alpha)\cdot [\frac{(C+bA-\frac{1}{cos(\alpha)}D+1)}{1+BD}\cdot e^{-PD\cdot R_1}a^{n-1}+\)\((1-e^{-PD\cdot R_1})a^{n-1}+b\frac{1-a^{n-1}}{1-a}]\) with \(a=cos(\alpha)e^{-TR\cdot R_1}\), \(b=1-e^{-TR\cdot R_1}\), \(C=a^{N-1}(1-e^{-PD\cdot R_1})\), \(A=\frac{1-a^{N-1}}{1-a}\), \(D=cos(\alpha)e^{-PD\cdot R_1}\), \(B=a^{N-1}e^{-PD\cdot R_1}\), S0 (Q.MS1.010), R1 (Q.EL1.001), PD (Q.MS1.008), α (Q.MS1.007) , TR (Q.MS1.006), n (Q.MS1.011), N (Q.MS1.012), S (Q.MS1.001) | Larson 2001 |
| M.SM2.999 | 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. | -- |
Magnitude models: DCE - R1 in the fast water exchange limit, direct relationship with indicator concentration¶
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| M.MS3.001 | Linear model | -- | Linear | This forward model is given by the following equation: \(S=k\cdot C\) with C (Q.IC1.001), k (Q.GE1.009), S (Q.MS1.001) |
-- |
| M.MS3.002 | Absolute signal enhancement model | -- | ASE | This forward model is given by the following equation: \(| S-S_{BL} |=k\cdot C\) , with C (Q.IC1.001), SBL (Q.MS1.002), k (Q.GE1.009), S (Q.MS1.001) |
Ingrisch and Sourbron 2013 |
| M.MS3.003 | Relative signal enhancement model | -- | RSE | This forward model is given by the following equation: \(| \frac{S}{S_{BL}}-1 |=k\cdot C\) with C (Q.IC1.001), SBL (Q.MS1.002), k (Q.GE1.009), S (Q.MS1.001) |
Ingrisch and Sourbron 2013 |
| M.SM3.999 | 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. | -- |
Magnitude models: DCE - R1 in the presence of restricted water exchange¶
Magnitude models: Combined DCE/DSC - R1/ R2/ R1*¶
Magnitude models: Combined DCE/DSC - R1/R2/R2*¶
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| M.SM6.001 | DSC Multi-echo (GE) model | -- | -- | This forward model is given by the following equation: \(S=S_{DCE,FXL}(R_1)e^{-TE\cdot R_2^*}\) where SDCE,FXL(R1) is a forward model selected from Magnitude models: DCE- R1 in fast water exchange limit, TE (Q.MS1.005), R2* (Q.EL1.007), S (Q.MS1.001) |
-- |
| M.SM6.002 | DSC Multi-echo (SE) model | -- | -- | This forward model is given by the following equation: \(S=S_{DCE,FXL}(R_1)e^{-TE\cdot R_2}\) where SDCE,FXL(R1) is a forward model selected from Magnitude models: DCE- R1 in fast water exchange limit, TE (Q.MS1.005), R2 (Q.EL1.004), S (Q.MS1.001) |
-- |
| M.SM6.999 | 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. | -- |
Phase models¶
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| M.SM7.001 | Linear susceptibility signal model | -- | -- | This forward model is given by the following equation: \(S=k\cdot \chi\) with k (Q.GE1.009), χ (Q.EL1.011), S (Q.MS1.001). |
-- |
| M.SM7.999 | 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. | -- |
Electromagnetic property models¶
This section covers models that describe how electromagnetic properties (EP), such as relaxation rates R1, R2 and R2* or the magnetic susceptibility χ , are realted to indicator concentrations.
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| M.EL1.001 | Transverse relaxation rate (GE), linear with relaxivity model | Effective relaxation rate (GE), linear with relaxivity model | -- | This forward model is given by the following equation: \(R_2^*=R_{20}^*+r_2^*\cdot C\) with R20* (Q.EL1.008), r2* (Q.EL1.017), C (Q.IC1.001), R2* (Q.EL1.007) |
(Rosen et al. 1990) |
| M.EL1.002 | Transverse relaxation rate (SE), linear with relaxivity model | Natural relaxation rate (GE), linear with relaxivity model | -- | This forward model is given by the following equation: \(R_2=R_{20}+r_2\cdot C\) with R20 (Q.EL1.005), r2 (Q.EL1.016), C (Q.IC1.001), R2 (Q.EL1.004) |
(Rosen et al. 1990) |
| M.EL1.003 | Longitudinal relaxation rate, linear with relaxivity model | -- | -- | This forward model is given by the following equation: \(R_1=R_{10}+r_1\cdot C\) with R10 (Q.EL1.002), r1 (Q.EL1.015), C (Q.IC1.001), R1 (Q.EL1.001) |
(Rosen et al. 1990) |
| M.EL1.004 | Transverse relaxation rate (GE) with gradient leakage correction model | -- | -- | This forward model is given by the following equation: \(R_2^*=R_{20}^*+r_{2v}^*| C_p-C_e | +r_{2e}^*C_e,\) with \(R_{20}^*\) (Q.EL1.008), \(C_p\) (Q.IC1.001.[p]), \(C_e\) (Q.IC1.001.[e]), \(r_{2e}^*\) (Q.EL1.017.[e]), \(r_{2v}^*\) (Q.EL1.017.[v]), \(R_2^*\)(Q.EL1.007) |
Sourbron et al 2012 |
| M.EL1.005 | Transverse relaxation rate (GE), quadratic model | -- | -- | This forward model is given by the following equation: \(R_2^*=R_{20}^*+k_1\cdot C_p+k_2C_p^2\) with \(R_{20}^*\) (Q.EL1.008), \(C_p\) (Q.IC1.001.[p]), [\(k_1\),\(k_2\)] (Q.EL1.020), \(R_2^*\) (Q.EL1.007) |
Van Osch 2003 (also see Calamante 2013) |
| M.EL1.006 | Linear susceptibility concentration model | -- | -- | This forward model is given by the following equation: \(\chi=\chi_0+\delta\chi\cdot C\) with \(\chi_0\) (Q.EL1.012), \(\delta\chi\) (Q.EL1.013), \(C\) (Q.IC1.001), \(\chi\) (Q.EL1.011) |
(Conturo et al. 1992) |
| M.EL1.999 | 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. | -- |
Indicator concentration models¶
This section covers models that describe how indicator concentrations in tissue and blood vary with time.
Indicator kinetic models¶
In the current version of the lexicon the list of indicator kinetic models is restricted to linear and stationary tissues, and specific models with two distribution spaces. A summary of common pharmacokinetic models in contrast-agent based perfusion MRI is given in (Sourbron and Buckley 2013). We provide the differential equations and impulse response functions using the consistent parameterizations to enable straightforward comparisons between models.
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| M.IC1.001 | Linear and stationary system model | -- | LSS model | This forward model is given by the following equations: \(C(t)=I(t)\ast C_{a,p}(t)\) with [I (Q.IC1.005), t (Q.GE1.004)], [\(C_{a,p}\) (Q.IC1.001.[a,p]), \(t\) (Q.GE1.004)], [\(C_t\) (Q.IC1.001.[t]), \(t\) (Q.GE1.004)] | (Rempp et al. 1994) |
| M.IC1.002 | 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: \(v_{p}\frac{dC_{t}(t)}{dt} = F_{p}C_{a,p} - F_{p}C_{c,p}(t)\)The impulse response function is given by: \(I(t) = F_{p}e^{{-\frac{F_{p}}{v_{p}}t}}\) with [\(C_{a,p}\) (Q.IC1.001.[a,p]), t (Q.GE1.004)], [\(C_{c,p}\) (Q.IC1.001.[c,p]), t (Q.GE1.004)], [\(C_t\) (Q.IC1.001.[t]), t (Q.GE1.004)], [I (Q.IC1.005), t (Q.GE1.004)], \(F_p\) (Q.PH1.002), \(v_{p}\) (Q.PH1.001.[p]) | (Tofts et al. 1999) |
| M.IC1.003 | 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: \(\frac{dC_{t}(t)}{dt} = F_{p}C_{a,p} - \frac{F_{p}}{v_{p} + v_{e}}C_{t}(t)\) The impulse response function is given by: \(I(t) = F_{p}e^{{-\frac{F_{p}}{v_{p} + v_{e}}t}}\) with ... TO ADD PARAMETERS | (Sourbron et al. 2013) |
| M.IC1.004 | 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: \(\frac{dC_{t}(t)}{dt} = K^{trans}C_{c,p} - \frac{K^{trans}}{v_{e}}C_{t}(t)\) The impulse response function is given by: \(I(t) = K^{trans}e^{{-\frac{K^{trans}}{v_{e}}t}}\) with ... TO ADD PARAMETERS | (Tofts and Kermode 1991) |
| M.IC1.005 | 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: \(C_{c,p} = C_{a,p}\) The forward model is given by the following differential equation: \(v_{e}\frac{dC_{e}(t)}{dt} = PSC_{c,p} - PSC_{e}(t)\) The impulse response function is given by: \(I(t) = v_{p}\delta(t) + K^{trans}e^{{-\frac{K^{trans}}{v_{e}}t}}\) with ... TO ADD PARAMETERS | (Tofts 1997) |
| M.IC1.006 | 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: \(v_{e}\frac{dC_{e}(t)}{dt} = PSC_{c,p}\) The impulse response function is given by: \(I(t) = v_{p}\delta(t) + PS\) with ... TO ADD PARAMETERS | (Patlak et al. 1983) |
| M.IC1.007 | 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: \(v_{p}\frac{dC_{c,p}(t)}{dt} = F_{p}C_{a,p} - F_{p}C_{c,p} - PSC_{a,p}\) \(v_{e}\frac{dC_{e}(t)}{dt} = PSC_{a,p}\) The impulse response function is given by: \(I(t) = F_{p}e^{-({\frac{F_{p} + PS}{v_{p}}})t} + E(1 - e^{-({\frac{F_{p} + PS}{v_{p}}})t})\) with ... TO ADD PARAMETERS | (Pradel et al. 2003), (Sourbron 2009) |
| M.IC1.008 | 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: \(v_{p}\frac{\partial C_{c,p}(x_{ax}, t)}{\partial t} = -F_{p}L_{ax}\frac{\partial C_{a,p}(x_{ax}, t)}{\partial x_{ax}} - PSC_{a,p}(x_{ax}, t)\) \(v_{e}\frac{dC_{e}(t)}{dt} = PS \int_{0}^{L_{ax}} C_{c,p} (x_{ax},t) dx\) The impulse response function is ... TO ADD IRF AND PARAMETERS | (St. Lawrence and Frank 2000) |
| M.IC1.009 | 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: \(v_{p}\frac{dC_{c,p}}{dt} = F_{p}C_{a,p}(t) - F_{p}C_{c,p}(t) - PSC_{c,p}(t) + PSC_{e}(t)\) \(v_{e}\frac{dC_{e}}{dt} = PSC_{c,p}(t) - PSC_{e}(t)\) The impulse response function is given by \(I(t) = F_{p}e^{-K_{+}t} + E_{-}(e^{-K_{+}t} - e^{-K_{-}t})\) \(K_{\pm} = \frac{1}{2}\Biggl(\frac{F_{p} + PS}{v_{p}} + \frac{PS}{v_{p}} \pm \sqrt{(\frac{F_{p} + PS}{v_{p}} + \frac{PS}{v_{e}})^{2} - 4\frac{F_{p}PS}{v_{p}v_{e}}}\Biggl)\) \(E_{-} = \frac{K_{+} + \frac{F_{p}}{v_{p}}}{K_{+} + K_{-}}\) with ... TO ADD PARAMETERS | (Brix et al. 2004), (Sourbron et al. 2009), (Donaldson et al. 2010) |
| M.IC1.999 | 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. | -- |
Arterial input function models¶
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| M.IC2.001 | Parker AIF model | -- | Parker AIF | This forward model is given by the following equation: \(C_{a,b}(t)=\sum_{n=1}^{2}\frac{A_n}{\sigma_n\sqrt{2\pi}}e^{-\frac{(t-T_n)^2}{2\sigma_n^2}}+\frac{\alpha e^{-\beta t}}{1+e^{-s(t-\tau)}},\) where \(A_n\), \(T_n\) and \(\sigma_n\) are the scaling constants, center and widths of the nth Gaussian; \(\alpha\) and \(\beta\) are the amplitude and decay constants of the exponential; and \(s\) and \(\tau\) are the width and center of the sigmoid, and [\(C_{a,b}\) (Q.IC1.001.[a,b]), t (Q.GE1.004)]. If not specified otherwise, the values from the publication are assumed: [\(A_1\), \(A_2\), \(T_1\), \(T_2\), \(\sigma_1\), \(\sigma_2\), \(\alpha\) , \(\beta\) , s, \(\tau\) ] = [48.54 mmol\(\cdot\)s, 19.8 mmol\(\cdot\)s, 10.2276 s, 21.9 s, 3.378 s, 7.92 s, 1.050 mmol, 0.0028 s-1, 0.6346 s-1, 28.98 s]. | (Parker et al. 2006) |
| M.IC2.002 | Georgiou AIF model | -- | Georgiou AIF | The AIF between the start of the nth recirculation and (n+1)th recirculation is given by: \(C_{a,p}(t)=(\sum_{i=1}^{3}A_ie^{-m_it})\cdot(\sum_{j=0}^{n}\gamma((j+1)\alpha+j,\beta,t-j\tau)),\) with \(n\tau<t<(n+1)\tau\) and \(\gamma(\alpha, \beta, \tau)=\frac{t^\alpha e^{-\frac{t}{\beta}}}{\beta^{\alpha+1}\Gamma(\alpha +1)}\) with \(t\geq 0\), where \(A_1\), \(A_2\), \(A_3\) and \(m_1\), \(m_2\), \(m_3\) are the amplitudes and time constants of the exponential terms, the coefficients and reflect the number of theoretical mixing chambers and the ratio of the volume of the mixing chambers to the volume flow rate, and is the recirculation time, [\(C_{a,p}\) (Q.IC1.001.[a,p]), t (Q.GE1.004)]. \(\gamma = 0\) for \(t<0\). If not specified otherwise, the values from the publication are assumed: [ \(A_1\), \(A_2\), \(A_3\), \(m_1\), \(m_2\), \(m_3\), \(\alpha\), \(\beta\) , \(\tau\) ] = [0.37 mM, 0.33 mM, 10.06 mM, 0.002 s-1, 0.02 \(s^{-1}\), 0.267 \(s^{-1}\), 5.26, 1.92 s, 7.74 s]. | (Georgiou et al. 2019) |
| M.IC2.003 | Weinmann AIF model | -- | Weinmann AIF | This forward model is given by the following equation: \(C_{a,p}(t)=D(a_1e^{-m_1t}+a_2e^{-m_2t}),\) where \(D\) is the dose of contrast agent and \(a_1\), \(a_2\), \(m_1\) and \(m_2\) are the amplitudes and time constants of the exponential terms, and [\(C_{a,p}\) (Q.IC1.001.[a,p]), t (Q.GE1.004)]. If the model parameters are not specified, the values from the publication are assumed: [\(D\), \(a_1\), \(m_1\), \(a_2\), \(m_2\)] = [0.25 mmol/kg, 3.99 kg/l, 0.0024 s-1, 4.78 kg/l, 0.0002 \(s^{-1}\)]. | (Weinmann et al. 1984) |
| M.IC2.999 | 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. | -- |
Descriptive models¶
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| M.DM1.001 | Unit step model | Heaviside step model | u | This forward model is given by the following equation: \(f(x-T)=0,\ \ x\leq T\), \(f(x-T)=1,\ \ x\gt T\) , where T is a defined data grid point, at which the step function changes from 0 to 1 and [x, f(x)]= [Data grid (Q.GE1.001), Data (Q.GE1.002)]. | -- |
| M.DM1.002 | Linear-quadratic model | -- | LQM | This forward model is given by the following equation: \(f(x)=f_{BL},\ \ x\leq BAT\), \(f(x)=f_{BL}+\beta_1(x-BAT)+\beta_2(x-BAT)^2,\)\(x\gt BAT\), where BAT is the bolus arrival time (Q.BA1.001), fBL the baseline (Q.BL1.001), β1 the slope after the BAT, β2 a quadratic component and [x, f(x)]= [Data grid (Q.GE1.001), Data (Q.GE1.002)]. | (Cheong et al. 2003) |
| M.DM1.003 | Two step linear model | -- | 2SLM | This forward model is given by the following equation: \(f(x)=f_{BL},\ \ x\leq BAT,\) \(f(x)=f_{BL}+b_1(x-BAT),\ \ x\gt BAT,\) where BAT is the bolus arrival time (Q.BA1.001), fBL is the baseline (Q.BL1.001), b1 the slope after the BAT and [x, f(x)]= [Data grid (Q.GE1.001), Data (Q.GE1.002)]. | Cheong et al. 2003) |
| M.DM1.004 | Three step linear mode | -- | 3SLM | This forward model is given by the following equation: \(f(x)=f_{BL},\ \ x\leq BAT,\) \(f(x)=f_{BL}+b_1(x-BAT),\ \ BAT\leq x\leq \beta,\) \(f(x)=f_{BL}+b_1(x-BAT)+b_2(x-\beta),\ \ x\gt\beta\) , where BAT is the bolus arrival time (Q.BA1.001), fBL is the baseline (Q.BL1.001), β is the point of intersection of the 2nd and 3rd line segment. b1 and b2 are the slopes of the 2nd and 3rd line segments and [x, f(x)]= [Data grid (Q.GE1.001), Data (Q.GE1.002)]. | (Singh et al. 2009) |
| M.DM1.005 | Multi-exponential model | -- | -- | This forward model is given by the following equation: \(f(x)=A_1\cdot e^{-x\cdot a_1}+;...+A_n\cdot e^{-x\cdot a_n}\) <\br> where A1, …, An and a1,...,an are arbitrary coefficients and [x, f(x)]= [Data grid (Q.GE1.001), Data (Q.GE1.002)]. | -- |
| M.DM1.006 | Gamma-variate model | -- | -- | This forward model is given by the following equation: \(f(x)=\frac{1}{\Gamma(\alpha)\beta^{\alpha}}(x-BAT)^{\alpha-1}e^{-(x-BAT)/\beta}\) where BAT is the bolus arrival time (Q.BA1.001), α is a shape parameter, β is a scale parameter, Γ(α) is the gamma function and [x, f(x)]= [Data grid (Q.GE1.001), Data (Q.GE1.002)]. | Mouridsen 2006 |
| M.DM1.007 | Fermi model | -- | -- | This forward model is given by the following equation: \(f(x)=F\cdot\frac{1+b}{1+b\cdot e^{x\cdot a}}\) , where \(F\) is the blood flow, \(a\) und \(b\) are arbitrary fit parameters and [x, f(x)]= [Data grid (Q.GE1.001), Data (Q.GE1.002)]. |
(Brinch et al. 1999) |
| M.DM1.008 | Normal distribution model | Gaussian distribution model | N | This forward model is given by the following equation: \(f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}},\) where \(\mu\) is the mean (xxx), \(\sigma\) is the standard deviation (xxx) and [x, f(x)]= [Data grid (Q.GE1.001), Data (Q.GE1.002)]. | -- |
| M.DM1.999 | 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. | -- |
Leakage correction models¶
This section is concerned with models for DSC leakage correction. They are not descriptive models in the sense that they are defined for very specific physical quantities, but at the same time cannot be derived as a composition of kinetic models, electromagnetic tissue property models or MR signal models.
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| M.LC1.001 | Boxerman-Schmainda-Weisskoff (BSW) leakage correction model | -- | BSW leakage correction model | This forward model is given by the following equation: \(R_2^*(t)=R_{20}^*+K_1\overline{\Delta R_{2,ref}^*(t)}-K_2\int_0^t \overline{\Delta R_{2,ref}^*(t')dt'},\) with [\(\overline{\Delta R_{2,ref}^*}\)(Q.EL1.010), t (Q.GE1.004)] \(R_{20}^*\) (Q.EL1.008), \(K_1\) (Q.LC1.001), \(K_2\) (Q.LC1.002), [\(R_2^*\) (Q.EL1.007), t (Q.GE1.004)] | -- |
| M.LC1.002 | Bidirectional leakage correction model | -- | -- | This forward model is given by the following equation: \(R_2^*(t)=R_{20}^*+K_1\overline{\Delta R_{2,ref}^*(t)}\)-\(K_2\int_0^t \overline{\Delta R_{2,ref}^*(t')}\cdot e^{-k_{e->p}(t-t')}dt',\) with [\(\overline{\Delta R_{2,ref}^*}\)(Q.EL1.010), t (Q.GE1.004)] \(R_{20}^*\) (Q.EL1.008), \(K_1\) (Q.LC1.001), \(K_2\) (Q.LC1.002), \(k_{e->p}\) (Q.PH1.009.[e->p]) [\(R_2^*\) (Q.EL1.007), t (Q.GE1.004)] | -- |
| M.LC1.999 | 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. | -- |
Perfusion identity models¶
This group lists relationships between perfusion quantities that can be used to derive one quantity from another under certain assumptions. This group is divided into the derivation of scalar quantities and the derivation of scalar derived from dynamic curves.
Scalar quantities¶
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| M.ID1.001 | Central volume theorem | -- | CVT | This forward model is given by the following equation: \(v_p=MTT\cdot F_p\) with MTT (Q.PH1.006), \(F_p\) (Q.PH1.002), \(v_p\) (Q.PH1.001.[p]) | -- |
| M.ID1.002 | Total volume of distribution | -- | -- | This forward model is given by the following equation: \(v=v_p+v_e+v_i\) with \(v_p\) (Q.PH1.001.[p]), \(v_e\) (Q.PH1.001.[e]), \(v_i\) (Q.PH1.001.[i]), \(v\) (Q.PH1.001) | -- |
| M.ID1.003 | Blood vs plasma volume fraction | -- | -- | This forward model is given by the following equation: \(v_b=\frac{v_p}{(1-Hct)}\) with \(v_p\) (Q.PH1.001.[p]), \(Hct\) (Q.PH1.012), \(v_b\) (Q.PH1.001.[b]). | -- |
| M.ID1.004 | Blood vs plasma flow | -- | -- | This forward model is given by the following equation: \(F_b=\frac{F_p}{(1-Hct)}\) with \(F_p\) (Q.PH1.002), \(Hct\) (Q.PH1.012), \(F_b\) (Q.PH1.003) | -- |
| M.ID1.005 | Blood vs plasma AIF | -- | -- | This forward model is given by the following equation: \(C_{a,b}(t)=C_{a,p}(t)\cdot(1-Hct_a)\), with [\(C_{a,p}\) (Q.IC1.001.[a,p]), t (Q.GE1.004)], [\(C_{a,b}\) (Q.IC1.001.[a,b]), t (Q.GE1.004)], \(Hct_a\) (Q.PH1.012.[a]) | -- |
| M.ID1.006 | Small vessel hematocrit correction | -- | -- | This forward model is given by the following equation: \(Hct_f=\frac{1-Hct_a}{1-Hct_t}\) with \(Hct_a\) (Q.PH1.012.[a]), \(Hct_t\) (Q.PH1.012[t]), \(Hct_f\) (Q.PH1.013) | -- |
| M.ID1.007 | Compartment extraction fraction | -- | -- | This forward model is given by the following equation: \(E=\frac{PS}{F_p+PS}\) with \(PS\) (Q.PH1.004), \(F_p\) (Q.PH1.002), \(E\) (Q.PH1.005) | -- |
| M.ID1.008 | Plug flow extraction fraction | -- | -- | This forward model is given by the following equation: \(E=1-e^{-\frac{PS}{F_p}}\) with \(PS\) (Q.PH1.004), \(F_p\) (Q.PH1.002), \(E\) (Q.PH1.005) | -- |
| M.ID1.009 | Plasma MTT identity | -- | -- | This forward model is given by the following equation: \(MTT_p=\frac{v_p}{F_p+PS}\) with \(v_p\) (Q.PH1.001.[p]), \(PS\) (Q.PH1.004), \(F_p\) (Q.PH1.002), \(MTT_p\) (Q.PH1.006.[p]) | -- |
| M.ID1.010 | Interstitial MTT identity | -- | -- | This forward model is given by the following equation: \(MTT_e=\frac{v_e}{PS}\) with \(v_e\) (Q.PH1.001.[e]), \(PS\) (Q.PH1.004), \(MTT_e\) (Q.PH1.006.[e]) | -- |
| M.ID1.011 | \(K^{trans}\) identity | -- | -- | This forward model is given by the following equation: \(K^{trans}=E\cdot F_p\), with E (Q.PH1.005), \(F_p\) (Q.PH1.002), \(K^{trans}\) (Q.PH1.008) | -- |
| M.ID1.012 | \(k_{ep}\) identity | -- | -- | This forward model is given by the following equation: \(k_{ep}=\frac{K^{trans}}{v_e}\), \(K^{trans}\) (Q.PH1.008), \(v_e\) (Q.PH1.001.[e]), \(k_{e->p}\) (Q.PH1.009.[e->p]) | -- |
| M.ID1.999 | 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. | -- |
Scalars derived from dynamic curves¶
| Code | OSIPI name | Alternative names | Notation | Description | Reference |
|---|---|---|---|---|---|
| M.ID2.001 | Bolus delay identity | -- | -- | This forward model is given by the following equation: \(MTT_a=\int_{0}^{\infty}h_a(t)dt\) with [ \(h_a\) (Q.IC1.004), t (Q.GE1.004)], \(MTT_a\) (Q.PH1.006.[a]) | -- |
| M.ID2.002 | Tissue mean transit time identity | -- | -- | This forward model is given by the following equation: \(MTT_t=\int_{0}^{\infty}R(t)dt\) with [ \(R\) (Q.IC1.002), \(t\) (Q.GE1.004)], \(MTT_t\) (Q.PH1.006.[t]) | -- |
| M.ID2.003 | Blood plasma flow from maximum | -- | -- | This forward model is given by the following equation: \(F_p=max(I(t))\) with [ \(I\) (Q.IC1.005), \(t\) (Q.GE1.004)], \(F_p\) (Q.PH1.002) | -- |
| M.ID2.004 | Blood plasma flow from first time frame | -- | -- | This forward model is given by the following equation: with | -- |
| M.ID2.005 | Capillary transit time heterogeneity identity | -- | -- | This forward model is given by the following equation: with | -- |
| M.ID2.999 | 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. | -- |