# Support:Documents:Examples:Simplified reference tissue method

This example demonstrates a COMKAT implementation of the simplified reference tissue model (SRTM) described by Lammertsma and Hume [Neuroimage. 4(3 Pt 1), 153-158 (1996)]. By simultaneously analyzing time-activity curves from areas with and without receptors, the SRTM provides a means to estimate the binding potential which is an index of receptor concentration and ligand affinity.

## Background

Input functions are a fundamental component of pharmacokinetic modeling. They describe the time-course of (radio)pharmaceutical which is accessible to the extravascular space. In PET receptor studies the arterial plasma concentration of a ligand is taken as the input function. As there is desire to obviate blood sampling, the SRTM is a popular method used to analyze ligand-receptor data.

The typical model equations used for a region containing receptors include state equations for free (F) and bound (B) ligand:

dF/dt = k1 Cp – (k2+k3)F + k4 B | Eq 1 |

dB/dt = k3 F – k4 B | Eq 2 |

where Cp is the arterial plasma input function and tracer conditions are assumed (B << Bmax) so that k3 = kon (Bmax – B) ≈ kon Bmax.

In a region devoid of receptors, k3 (and therefore B) is zero so Eq 1 may be simplified. Denoting this region as the reference region, or rr, the equation is

dF_{rr}/dt = k1 Cp – k2_{rr} F_{rr} |
Eq 3 |

This may be rearranged to obtain an expression for the plasma input

Cp = Cpn / k1_{rr} |
Eq 4 |

where

Cpn = (dF_{rr}/dt + k2_{rr} F_{rr}) |
Eq 5 |

is the normalized plasma input function.

This expression (Eq 4) for Cp may be substituted into the equations for regions that have receptors, aka, target tissue which we denote as tt

dF_{tt}/dt = (k1_{tt}/k1_{rr}) Cpn – (k2_{tt}+k3_{tt})F_{tt} + k4_{tt} B_{tt} |
Eq 6 |

dB_{tt}/dt = k3_{tt} F_{tt} – k4_{tt} B_{tt} |
Eq 7 |

Data from target tissue and from reference regions could be, in theory, simultaneously fit using the model embodied in Eq. 4 to 6. In practice there is often an issue of identifiability. Hence, assumptions are made to reduce the number of parameters to estimate. One assumption is that the equilibrium concentration ratio between plasma and tissue free space is the same in the reference region and target tissue:

k1_{tt}/ k2_{tt}= k1_{rr} /k2_{rr} |
Eq 8 |

this implies

k1_{tt}/k1_{rr} = k2_{tt}/k2_{rr} |
Eq 9 |

and this ratio may be denoted by R1

R1 = k1_{tt}/k1_{rr} = k2_{tt}/k2_{rr} |
Eq 10 |

Making the substitution into Eq 6 we obtain

dF_{tt}/dt = R1 Cpn – (k2_{tt}+k3_{tt})F_{tt} + k4_{tt} B_{tt} |
Eq 11 |

This equation could be solved simultaneously with Eq. 7 to obtain time-courses of free and bound radioligand. Analytic solutions derived for more general conditions (Eq 1 and 2) are given in the appendix in the document thad describes validation of the COMKAT implementation of the FDG model. Solutions for Ftt and B tt would then be summed to obtain the total concentration in the target tissue which is the quantity measured in an imaging study

C_{t}_{} = F_{tt}+B_{tt} |
Eq 12 |

In general, the solution Ctt is a weighted sum of two exponential terms. For ligands in which the association and dissociation of ligand from the receptor is much faster than the transport of ligand between the vascular and extravascular space, two exponentials are not evident from the data. That is, a single exponential model adequaltey describes the data because the two compartments exchange so quickly that they look like a single compartment. (Mathematically one of the eigenvalues or exponential coefficients is very large in magnitude.) The differential equation for such a single-exponential model is

dC_{tt} /dt = k1_{tt }Cp –k2^{a}_{tt} C _{tt} |
Eq 13 |

where k2att is the the apparent k2tt and is related to the the other parameters as

k2^{a}_{tt} = k2_{tt }/(1+BP)_{} |
Eq 14 |

where BP = k3tt/k4tt is the binding potential. Expressing the input in terms of Cpn, this is equivalent to the one-tissue compartment model depicted below

## COMKAT Implementation

comkat\examples\dasbSRTM.m

Step 1 To set this up in MATLAB, create an input function that implements Eq. 5 [Cpn = (dFrr/dt + k2rr Frr)]. In this k2rr is a parameter to estimate amd Frris taken as experimentally measured reference region concentration. To obtain Frr at arbitrary times requires interpolation. For this a smoothing spline is used. Its piecewise-polynomial coefficients are easily obtained

ppCref = csaps(tref, Cref, 0.01);

where tref and Cref are the sample times and concentrations from the reference region. With Frr expressed in this form, the polynomials may be analytically differentiated once to obtain the piecewise polynomial coefficients of dFrr/dt

ppdCrefdt = fnder(ppCref, 1);

The input Cpn may be evaluated as Cpn(t) = k2rr x ppval(ppCref. t) + ppval(ppdCrefdt.t) with the complete function given in comkat\examples\refInput.m

function [Cpn, dCpndk2ref] = refInput(k2ref, t, pxtra) ppCref = pxtra{1}; ppdCrefdt = pxtra{2}; if (nargout > 0), Cpn = k2ref * ppval(ppCref, t) + ppval(ppdCrefdt,t); idx = find(Cpn < 0); Cpn(idx) = 0; if (nargout > 1), dCpndk2ref = ppval(ppCref, t); dCpndk2ref(idx) = 0; end end return

For the sake of estimating the value of k2rr, this function must provide the derivative of Cpn with respect to k2rr when two output arguments are requested.

This input function is included in the compartment model by using these commands

sa = 1; % specific activity dk = 0; % decay constant (=0 since data are decay corrected) xparm = { ppCref, ppdCrefdt }; cm = addInput(cm, 'Cpn', sa, dk, 'refInput', 'k2ref', xparm);

Step 2 Create a model. To simultaneously fit three regions – midrbain, amygdala, and thalamus – create a model that has the compartments and links depicted as shown. Three Region Model

The commands are

% specify compartments cm = addCompartment(cm, 'Cmid'); cm = addCompartment(cm, 'Camy'); cm = addCompartment(cm, 'Ctha'); cm = addCompartment(cm, 'Sink'); % comkat requires closed system % specify links cm = addLink(cm, 'L', 'Cpn', 'Cmid', 'R1mid'); % connect input to cm = addLink(cm, 'K', 'Cmid', 'Sink', 'k2ppmid'); cm = addLink(cm, 'L', 'Cpn', 'Camy', 'R1amy'); cm = addLink(cm, 'K', 'Camy', 'Sink', 'k2ppamy'); cm = addLink(cm, 'L', 'Cpn', 'Ctha', 'R1tha'); cm = addLink(cm, 'K', 'Ctha', 'Sink', 'k2pptha'); % specify outputs (neglect vascular contrib) % for simultaneously fitting target tissues cm = addOutput(cm, 'Mid', {'Cmid', 'PV'}, { } ); cm = addOutput(cm, 'Amy', {'Camy', 'PV'}, { } ); cm = addOutput(cm, 'Tha', {'Ctha', 'PV'}, { } ); % specify parameters and values cm = addParameter(cm, 'PV', 1); cm = addParameter(cm, 'k2ref', 0.05); cm = addParameter(cm, 'R1mid', 0.8); cm = addParameter(cm, 'R1amy', 1.0); cm = addParameter(cm, 'R1tha', 1.2); cm = addParameter(cm, 'BPmid', 0.5); cm = addParameter(cm, 'BPamy', 1.0); cm = addParameter(cm, 'BPtha', 1.5); cm = addParameter(cm, 'k2ppmid', 'k2ref*R1mid/(1+BPmid)'); cm = addParameter(cm, 'k2ppamy', 'k2ref*R1amy/(1+BPamy)'); cm = addParameter(cm, 'k2pptha', 'k2ref*R1tha/(1+BPtha)');

The rest of the details are quite straightforward and are all contained in the example file comkat\examples\dasbSRTM.m. The only thing slightly unusual is that multiple time-activity curves must be specified so they can be simultaneously fit. To do this, just use

cm = set(cm, 'ExperimentalData', PETdata(:, [1 2 3]));

where columns [1 2 3] contain 'Mid', 'Amy', and 'Tha' data [the same order as the adddOutput() commands] and the rows of PETdata correspond to different frames.