Reference tissue input model for k₂

Starting from SRTM

Simplified reference tissue model (SRTM) was introduced by Lammertsma and Hume (1996) to estimate binding potential (BP_ND) of binding between receptors and PET radioligand, in case of such fast dynamics that a compartmental model with single tissue compartment could describe the kinetic data even in the receptor containing region (Figure 1).

In SRTM (and other reference tissue models for receptor binding) an assumption is that K₁/k₂ is similar in all studied regions. If K₁' and k₂' are used to represent these rate constants in the reference tissue, we can write:

$\frac{K_1}{k_2} = \frac{K^\prime_1}{k^\prime_2}$

, and introduce R₁ (or R_influx):

$R_1 = \frac{K_1}{K^\prime_1} = \frac{k_2}{k^\prime_2}$

Differential equation for SRTM is:

$\frac{dC_{T}(t)}{dt} = R_{1} \frac{dC_{R}(t)}{dt} + k_2 C_{R}(t) - \frac{k_2}{1+BP_{ND}} C_{T}(t)$

or, replacing k₂ with k₂ = R₁×k₂',

$\frac{dC_{T}(t)}{dt} = R_{1} \frac{dC_{R}(t)}{dt} + R_{1} k^\prime_2 C_{R}(t) - \frac{R_{1} k^\prime_2}{1+BP_{ND}} C_{T}(t)$

This formulation enables constraining the reference tissue k₂' to global mean or median.

If there is no specific binding or uptake of the radiopharmaceutical in the tissue, BP_ND=0, and the equations can be written as

$\frac{dC_{T}(t)}{dt} = R_{1} \frac{dC_{R}(t)}{dt} + R_{1} k^\prime_2 \left[ C_{R}(t) - C_{T}(t) \right]$

Integrating the equation, assuming that at time zero all concentrations are zero, gives

$C_{T}(T) = R_{1} C_{R}(t)dt + R_{1} k^\prime_2 \left[ \int_{0}^{T} C_{R}(t) - \int_{0}^{T} C_{T}(t) \right]$

The two parameters of the model (R₁, and k₂' or k₂), can be solved, naturally using nonlinear fitting, but the equation is already in a multilinear form that can be solved using linear methods (Blomqvist, 1984), spectral analysis, or with basis function approach (Gunn et al. 1997). If the shapes of tissue and reference tissue TACs are similar, then only R₁ can be solved, but even that parameter may be of interest, for example in preclinical radiowater PET perfusion studies, instead of using perfusion ratio.

In addition, the equation can be reorganized into linear forms that could allow estimation of the parameters by fitting a line to the linear phase of the plot, similar to multiple-time graphical analysis (Patlak and Logan plots):

$\frac{C_{T}(T)}{\int_{0}^{T} C_{R}(t) - \int_{0}^{T} C_{T}(t)} = R_{1} \times \frac{C_{R}(t)dt}{\int_{0}^{T} C_{R}(t) - \int_{0}^{T} C_{T}(t)} + R_{1} k^\prime_2$

In this representation, the slope of the fitted line would equal the R₁, and y axis intercept the R₁×k₂' or k₂, or directly, k₂' would equal the x axis intercept. Alternatively,

$\frac{C_{T}(T)}{C_{R}(T)} = R_{1} + R_{1} k^\prime_2 \times \frac{\int_{0}^{T} C_{R}(t) - \int_{0}^{T} C_{T}(t)}{C_{R}(t)dt}$

this plot would provide R₁ as the y axis intercept, and k₂ as the slope.

Literature

Blomqvist G. On the construction of functional maps in positron emission tomography. J Cereb Blood Flow Metab. 1984; 4:629-632. doi: 10.1038/jcbfm.1984.89.

Lammertsma AA, Hume SP. Simplified reference tissue model for PET receptor studies. NeuroImage 1996; 4:153-158. doi: 10.1006/nimg.1996.0066.

Yokoi T, Iida H, Itoh H, Kanno I. A new graphic plot analysis for cerebral blood flow and partition coefficient with iodine-123-iodoamphetamine and dynamic SPECT validation studies using oxygen-15-water and PET. J Nucl Med. 1993; 34(3): 498-505. PMID: 8441045.

Tags: Reference tissue

Updated at: 2019-03-24
Created at: 2019-03-24
Written by: Vesa Oikonen

Reference tissue input model for k2

Starting from SRTM

See also:

Literature

Reference tissue input model for k₂