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SRTM: Simplified Reference Tissue Model

Contents

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 ' , k 2 ' , 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.

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The model relies on the following assumptions:

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

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

The three unknowns, R 1 , k 2 ' and k 2 a , 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 ( t ) = θ 1 C R ( t ) + θ 2 C R ( t ) - θ 3 t ( 2 )

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

B i ( t ) = C R ( t ) - θ 3 , i t ( 3 )

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

C T ( t ) = θ 1 C R ( t ) + θ 2 B i ( t ) ( 4 )

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 θ 1 , θ 2 and θ 3 are obtained. Values for BP , R 1 and k 2 are then easily deduced.

The user can select a logarithmic range of values of θ 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 [ k 2 a min , k 2 a max ] 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

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 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.

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