Introduction

The two-compartment cerebral blood flow model can be characterized by the following three equations:

$$\frac{dM}{dt} = K_{1}C_{a}(t) - k_{2}M(t)$$ (1)

$$M(t) = K_{1}C_{a}(t) \otimes e^{-k_{2}t}$$ (2)

 

A(t) = M(t) + C_a(t)V₀ (3)

In these equations, A(t) is the PET data collected over a set of frames, C_a(t) is the delay and dispersion corrected arterial blood sample data, and M(t) is the radioactive tracer activity present in cerebral tissue. We know both A(t) and C_a(t), but cannot know M(t) without knowing V₀.

One additional point to consider is that these equations are written for continuous functions. However, the PET data that is actually available is average activity across each frame. Therefore, in order to solve the equations, we must average the non PET data across frames. By combining equations (2) and (3), and averaging the non PET data over frames, we get:

$$A^{*}(t) = \frac{\int\limits_{T_1}^{T_2} \left( K_{1} \left[ ... ...e^{-k_{2}t} \right] + C_{a}(t)V_{0} \right) dt}{T_{2} - T_{1}}$$ (4)

where A^*(t) is the PET data that is actually collected (by definition, averaged over frames), and ${\int_{T_1}^{T_2}dt} / {(T_2 - T_1)}$ performs an averaging over frames (T₁ is the frame start time, T₂ is the frame stop time, and the integral is evaluated for each frame).

We wish to solve equation (4) for K₁, k₂, and V₀. Of course, one approach would be to try to perform an explicit least squares curve fitting. However, this approach would be quite computationally intensive since the fitting would need to be performed for every pixel of a $128 \times 128$ pixel image. Fortunately, there is a method of solution that gives good results with reduced computational difficulty.