# SRTM: Simplified Reference Tissue Model

## Introduction

The SRTM method is mostly used for receptor studies using reversibly binding tracers. It was developed by Lammertsma [1], on the basis of the Full Reference Tissue Method (or 4 Parameter Reference Tissue Method) [2].

The formulation involves a reference region devoid of specific binding, modeled with a one-tissue compartment, and a region with specific binding (region of interest), which is represented with a two-tissue compartment model. The rate constants are ${K}_{1}$ and ${k}_{2}$, representing the exchange of tracer between plasma and a free ligand compartment in the region of interest; ${K}_{1}^{\text{'}}$, ${k}_{2}^{\text{'}}$, which are the equivalent for the reference region; and ${k}_{3}$ and ${k}_{4}$, which represent the exchange of tracer between the free compartment and a specifically bound ligand compartment in the region of interest.

The model relies on the following assumptions:

• The distribution volume is the same for the tissue of interest (tissue including specific binding) and the reference tissue: ${K}_{1}/{k}_{2}={K}_{1}^{\text{'}}/{k}_{2}^{\text{'}}$ .
• The kinetics in the tissue with specific binding is such that it is difficult to distinguish between the specific and the free/non-specific compartments. This happens when the exchange between the free and the specifically bound compartments is sufficiently rapid. In these cases the tissue region of interest may be approximated by a single compartment, with efflux constant ${k}_{2a}={k}_{2}/\left(1+\mathrm{BP}\right)$, where $\mathrm{BP}={k}_{3}/{k}_{4}$ is the binding potential.

Defining ${R}_{1}={K}_{1}/{K}_{1}^{\text{'}}$ as the ratio of tracer delivery, the following operational equation can be derived for the measured TAC in the receptor-rich region:

${C}_{T}\left(t\right)={R}_{1}{C}_{R}\left(t\right)+{R}_{1}\left({k}_{2}^{\text{'}}-{k}_{2a}\right){C}_{R}\left(t\right)\otimes \text{}{e}^{-{k}_{2a}t}\left(1\right)$

The three unknowns, ${R}_{1}$, ${k}_{2}^{\text{'}}$ and ${k}_{2a}$, in this equation can be fitted using nonlinear regression techniques.

Implementation

The implementation of the SRTM model developed by Gunn [3], using basis functions, was better suited for a pixel-wise application than the original approach. Equation (1) can be rewritten as:

${C}_{T}\left(t\right)={\theta }_{1}{C}_{R}\left(t\right)+{\theta }_{2}{C}_{R}\left(t\right)\otimes \text{}{e}^{-{\theta }_{3}t}\left(2\right)$

Where ${\theta }_{1}={R}_{1}$, ${\theta }_{2}={R}_{1}\left({k}_{2}^{\text{'}}-{k}_{2a}\right)$ and ${\theta }_{3}={k}_{2a}$. Since this equation is linear on ${\theta }_{1}$ and ${\theta }_{2}$, these values can be estimated using standard linear least squares, when the value of ${\theta }_{3}$ is fixed. To obatin a solution for the nonlinear term, a discrete set of parameter values for ${\theta }_{3}$ can be chosen, to form the following basis functions:

${B}_{i}\left(t\right)={C}_{R}\left(t\right)\otimes \text{}{e}^{-{\theta }_{3,i}t}\left(3\right)$

Equation (2) can then be transformed into a linear equation for each basis function:

${C}_{T}\left(t\right)={\theta }_{1}{C}_{R}\left(t\right)+{\theta }_{2}{B}_{i}\left(t\right)\left(4\right)$

In this approach, which we have implemented in QModeling, equation (4) is solved using linear least squares for each ${B}_{i}$. After the index $i$, which minimizes the deviation between the TAC and the model curve is determined, the values of ${\theta }_{1}$, ${\theta }_{2}$ and ${\theta }_{3}$ are obtained. Values for $\mathrm{BP}$, ${R}_{1}$ and ${k}_{2}$ are then easily deduced.

The user can select a logarithmic range of values of ${\theta }_{3}$, to take into account all plausible values for this parameter.

## Preprocessing algorithm

1. Calculation of the basis functions: convolution of the reference TAC with decaying exponentials in the range $\left[{k}_{2a}\mathrm{min},\text{}{k}_{2a}\mathrm{max}\right]$ and at the resolution (Resampling) selected by the user.
2. Least squares fit for each of the basis functions. This results in a set of optimal parameters and an estimated model curve for each basis function.
3. The fit with minimal deviation between receptor-rich TAC and model curve is regarded as the result. The parameters of interest can be calculated from the fitted values.

## Input parameters

• TAC 1: TAC from a region with specific uptake (typically a receptor rich area).
• TAC 2: TAC from a reference region with no specific uptake (typically, a region devoid of target receptors).
• ${k}_{2a}\mathrm{min}$: Minimal value of ${k}_{2a}$ (slowest decay of exponential).
• ${k}_{2a}\mathrm{max}$: Maximal value of ${k}_{2a}$ (fastest decay of exponential).
• #Basis: Number of basis functions. Each basis funtion is defined by its ${k}_{2a}$ value. The values of ${k}_{2a}$ for the basis functions will be in the range between ${k}_{2a}\mathrm{min}$ and ${k}_{2a}\mathrm{max}$, and there will be a total of #Basis different values of ${k}_{2a}$. Increments will be taken at logarithmic steps. This number is directly proportional to processing time: the bigger #Basis, the longer the processing time. If the number of basis functions is too low, the estimation may lack precision. However, increasing its value does not indefinitely improve the estimation.
• Resampling: It specifies the interval at which the TACs will be resampled, before convolving them with the exponentials to form the basis functions. This interval should be equal or smaller than the shortest frame duration. Notice that the bigger this value, the shorter the processing time. However, smaller intervals lead to more accurate estimations.
• Threshold: Discrimination threshold for background masking. All pixels with energy below the specified percentage of the maximal energy will be masked to zero.

## Output parameters and goodness of fit

• $\mathrm{BP}$: Binding Potential relative to non-displaceable uptake $\left(\mathrm{BP}={k}_{3}/{k}_{4}\right)$
• ${k}_{2}$: Efflux constant for the free/non-specific binding compartment in the region of interest
• ${R}_{1}$: Relative tracer delivery $\left({R}_{1}={K}_{1}/{K}_{1}^{\text{'}}\right)$
• ${k}_{2a}$ : Apparent efflux constant for the tissue of interest when considered as a single compartment. $\left({k}_{2a}={k}_{2}/\left(1+\mathrm{BP}\right)\right)$

- Goodness of fit:

To show the goodness of fit at the preprocessing step, two parameters are given, together with the estimated parameters:

• Normalized Mean Squared Error (NMSE):

$\mathrm{NSME}=\frac{||{C}_{t}-{C}_{{t}_{\mathrm{estimate}}}{||}^{2}}{||{C}_{t}-\mathrm{mean}\left({C}_{t}\right){||}^{2}}$

It measures the quality of the fit for the TACs. Values vary between $-\infty$ (bad fit) to 1 (perfect fit).

• Correlation coefficient (Corr. Coef.): The correlation coefficient between the values of the TAC of interest and the values of the TAC estimated by the model. Values closer to 1 are better.

## Image generation algorithm

1. The basis functions are already calculated (see preprocessing algorithm).
2. Voxel-wise least squares fit for each of the basis functions. This results in a set of optimal parameters for each voxel.
3. The image for each of the selected parameters is written.

## Main references

[1] Lammertsma, A. A., & Hume, S. P. (1996). Simplified Reference Tissue Model for PET Receptor Studies. NeuroImage , 4(3), 153–158

[2] Lammertsma AA, Bech CJ, Hume SP, Osman S, Gunn K, Brooks DJ, Frackowiak RS (1996). Comparison of methods for analysis of clinical [11C]raclopride studies. Journal of Cerebral Blood Flow and Metabolism , 16(1):42-52

[3] Gunn, R. N., Lammertsma, A. A., Hume, S. P., & Cunningham, V. J. (1997). Parametric Imaging of Ligand-Receptor Binding in PET Using a Simplified Reference Region Model. NeuroImage , 6(4), 279–287.