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Simplified Reference Tissue Model 2: Carson Model (Simplified Reference Tissue Model with fixed k 2 ' )

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:

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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 ( t ) = R 1 C R ( t ) + R 1 ( k 2 ' - k 2 a ) C R ( t ) - k 2 a t ( 1 )

Implementation

Since k 2 ' = 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 ' parameter. In our implementation, the value for k 2 ' 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 ' is considered fixed, and, thus, the parametric images are generated with a fixed k 2 ' . The other parameters, k 2 a and R 1 , are estimated using the basis functions approach.

The basis functions can be defined as:

B i ( t ) = C R ( t ) + ( k 2 ' - k 2 a , i ) C R ( t ) - k 2 a , i t ( 2 )

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

C T ( t ) = R 1 B i ( t ) ( 3 )

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 2 a and R 1 are obtained. Values for BP and k 2 , are then easily deduced.

The user can select a logarithmic range of values for k 2 a , 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 [ k 2 a min , k 2 a max ] and at the resolution (Resampling) selected by the user.
  2. 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.
  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

Output parameters and goodness of fit

- Goodness of fit:

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

NSME = || C t - C t estimate || 2 || C t - mean ( C t ) || 2

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

Image generation algorithm

  1. The basis functions are already calculated (see preprocessing algorithm).
  2. 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.
  3. 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

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