# Simplified Reference Tissue Model 2: Carson Model (Simplified Reference Tissue Model with fixed ${k}_{2}^{\text{'}}$)

## Contents

## Introduction

The basis for *SRTM2* is largely the same as that for *SRTM* . Both are based on the *Full Reference Tissue Model* (or *4 Parameter Reference Tissue Model*), with the hypothesis that:

- The distribution volume is the same for the
*tissue of interest*(tissue including specific binding) and the*reference tissue*(tissue with no specific binding): ${K}_{1}/{k}_{2}={K}_{1}^{\text{'}}/{k}_{2}^{\text{'}}$

- 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.

An additional restriction is added in the approach for *SRTM2*, in order to reduce noise and accelerate the generation of parametric images. The *SRTM* operational equation is:

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

*Implementation*

Since ${k}_{2}^{\text{'}}={k}_{2}/{R}_{1}$ is the clearance rate constant from the *reference region*, it has only one true value, and it should not be necessary to estimate it for every pixel, as *SRTM* does. The *SRTM2* implementation [1] uses this property to reduce noise in the calculations, by considering a fixed ${k}_{2}^{\text{'}}$ parameter. In our implementation, the value for ${k}_{2}^{\text{'}}$ can be established in two ways: either it is directly specified by the user, or it is estimated just once, by applying *SRTM* to the mean TAC of a region with high specific uptake, using the appropriate TAC of the reference region as well. From here on, the obtained ${k}_{2}^{\text{'}}$ is considered fixed, and, thus, the parametric images are generated with a fixed ${k}_{2}^{\text{'}}$. The other parameters, ${k}_{2a}$ and ${R}_{1}$, are estimated using the basis functions approach.

The basis functions can be defined as:

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

Thus, there is one basis function for each value of ${k}_{2a}$. Equation **(1)** can now be written as:

${C}_{T}\left(t\right)={R}_{1}{B}_{i}\left(t\right)\text{}\left(3\right)$

Which shows that a least-squares fit allows calculating ${R}_{1}$ for each basis function. After the index $i$, which minimizes the deviation between the TAC and the model curve is determined, the values of ${k}_{2a}$ and ${R}_{1}$ are obtained. Values for $\mathrm{BP}$ and ${k}_{2}$, are then easily deduced.

The user can select a logarithmic range of values for ${k}_{2a}$, to take into account all plausible values for this parameter.

## Preprocessing algorithm

- 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. - Least squares fit of ${R}_{1}$ for each of the basis functions. This results in a set of optimal parameters and an estimated model curve for each basis function.
- 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.

- ${k}_{2}^{\text{'}}$: the clearance rate constant from the reference region.

## 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

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

## Main references

[1] Wu, Y., & Carson, R. E. (2002). Noise reduction in the simplified reference tissue model for neuroreceptor functional imaging. *Journal of Cerebral Blood Flow and Metabolism*, 22(12), 1440–1452