Delay and Dispersion Correction

Since the blood samples used to estimate C_a(t) are taken from the subject's arm, the activity data gathered from them must be corrected for delay and dispersion within the body. In order to perform the delay correction, we used the technique of Iida, by performing a least squares fitting of the equation:

$$A^{*}(t) = \frac{\int\limits_{T_1}^{T_2} \left( \alpha \left[... ... e^{-\beta t} \right] + \gamma C_a(t) \right) dt} {T_2 - T_1}$$ (11)

where A^*(t) in this case is the average activity over grey matter, $\int_{T_1}^{T_2}dt$ represents integration over frames, $\alpha$, $\beta$, and $\gamma$ are the fitting parameters, and C_a(t) is the blood data. The delay itself enters this equation by generating C_a(t) from:

$$C_a(t) = \bar{g}(t+\delta)$$ (12)

The function $\bar{g}$ is the result of performing dispersion correction on the blood sample data. This is represented by the equation:

$$\bar{g}(t) = g(t) + \tau \frac{dg}{dt}$$ (13)

which is an implicit deconvolution of the equation:

$$g(t) = \bar{g}(t) \otimes \left[ \frac{1}{\tau} e^{\frac{-t}{\tau}} \right]$$ (14)