This is a standard receptor-ligand model with two tissue compartments defined by

C_{1} represents free and non- specifically bound tracer in tissue, and C_{2} tracer bound to the receptors. The model accounts for the saturation of receptor sites due to a low specific activity S_{act} of the injected tracer. In this situation, k_{3} is given by k_{on}(B_{max}-C_{2}/S_{act}), and hence is time-dependent. B_{max} is the total receptor concentration, k_{on} the bi-molecular association rate, and k_{4} the dissociation rate constant k_{off}.

System of differential equations [10]:

As described above the 2-tissue compartment model can be described by the linear arrangement of compartments

and the system of differential equations:

Given the input curve C_{plasma}(t) and a set of model parameters K_{1}, .. , k_{4}, the tissue concentrations C_{1}(t) and C_{2}(t) can be calculated by integration of the equation system. However, alternative solutions of the system are possible. With linearized solutions, the equations are integrated twice on both sides, substitutions performed and finally rearranged. This can be done in different ways. The **Linear Least Squares** method implemented in PKIN uses the derivation of Cai et al. [57], equation (6):

Note that the tissue curve C_{PET}(t) which is the target function appears on both sides of the equation. It also includes whole-blood activity C_{Blood}(t) for spillover correction as well as the input curve C_{P}(t). This multi-linear expression can be solved in a least squares sense in one step without iterations. The present implementation uses a singular value decomposition method. As a result the 5 parameters P_{1}, .. , P_{5} are obtained from which the target parameters can be calculated as follows: [57], equation (9)

Recommended Use of the Linear Least Squares Method

aThe advantage of the linearized approach is the fast calculation, a well-defined solution, and no danger to get stuck in a local minimum such as the iterative methods. However, it is well known that it is susceptible to bias, and that small TAC perturbations can cause large changes of the parameter estimates. Therefore it is recommended that the **Linear Least Squares** method is only used for getting a quick solution which is further refined by the iterative methods. In fact, if **Parameters Initialization** is configured in **Extras**, the initial starting parameters of the compartment models are obtained by the **Linear Least Squares** method.

Implementation Notes

The **Linear Least Squares** method can be used for different model configurations:

- Irreversible model: if k
_{4}is disabled from fitting, the program automatically sets k_{4}= 0. and uses a linearized equation which is modified accordingly. - 1-tissue compartment model: if k
_{3}is disabled from fitting, the program automatically sets k_{3}= k_{4}= 0 and uses a modified equation. - In all configurations the vB parameter can optionally be disabled from fitting and fixed at a specific user-defined value.