Double-Weighted Integration Method

 

Since the time required to perform a least squares curve fitting would be prohibitive, a simpler approach to solving the problem is required. One technique is to use a weighted integration method. We initially approached the problem by assuming that V₀ was negligibly small, in which case the C_aV₀ term is eliminated from equation (4). Taking equation (4) with C_aV₀eliminated, and integrating both sides from time 0 to the end of the last frame, we get:

$$\int_{0}^{T} A^{*}(t) dt = K_{1} \int_{0}^{T} \frac{\int_{T_1... ... \left[ C_{a}(u) \otimes e^{-k_{2}u} \right] du}{T_2 - T_1} dt$$ (5)

We can then take this equation, and divide it by a weighted version of itself:

$$\frac{\int_{0}^{T} A^{*}(t) dt}{\int_{0}^{T} A^{*}(t) t dt} =... ...eft[ C_{a}(u) \otimes e^{-k_{2}u} \right] du}{T_2 - T_1} t dt}$$ (6)

K₁ cancels out of this equation, leaving us with an equation that only involves k₂. The left side of equation (6) is easily evaluated by integrating the PET data. The right side of equation (6) is not easily solved for k₂, so a different approach was taken. A look-up table was generated relating values of k₂ to resulting values of the right hand side of equation (6). A linear interpolation was then performed to choose values of k₂ from this look-up table for each point in the left hand side of equation (6).