# 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

 dFrr/dt = k1 Cp – k2rr Frr Eq 3

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

 Cp = Cpn / k1rr Eq 4

where

 Cpn = (dFrr/dt + k2rr Frr) 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

 dFtt/dt = (k1tt/k1rr) Cpn – (k2tt+k3tt)Ftt + k4tt Btt Eq 6 dBtt/dt = k3tt Ftt – k4tt Btt 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:

 k1tt/ k2tt= k1rr /k2rr Eq 8

this implies

 k1tt/k1rr = k2tt/k2rr Eq 9

and this ratio may be denoted by R1

 R1 = k1tt/k1rr = k2tt/k2rr Eq 10

Making the substitution into Eq 6 we obtain

 dFtt/dt = R1 Cpn – (k2tt+k3tt)Ftt + k4tt Btt 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

 Ct = Ftt+Btt 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

 dCtt /dt = k1tt Cp –k2att C tt Eq 13

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

 k2att = k2tt /(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

Simplified Model Figure ## 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

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.