B_CURVE

bcurve computes the right-hand-side of equation 11, that is,

$$\frac{\int\limits_{T_1}^{T_2} \left( \alpha \left[ C_a(t)... ...e^{-\beta t} \right] + \gamma C_a(t) \right) dt} {T_2 - T_1}$$ (18)

In the early stages of the development of RCBF, b_curve was used in the non-linear fitting required for delay correction. Now, the entire fitting procedure for delay correction is encapsulated in the CMEX routine delaycorrect. However, delaycorrect simply performs one fit (optimising $\alpha$, $\beta$, and $\gamma$for a single value of $\delta$); it is called multiple times, and b_curve is used to pick which value of $\delta$ resulted in the best fit (in the least squares sense). In particular, we wish to satisfy equation 11; therefore, the sum of the squares of the differences between its two sides (A^* (t) and equation 18) for a particular value of $\delta$ is computed, and the value of $\delta$ that results in the smallest residual is picked as the delay factor. The finding of the residual is actually done in fit_b_curve, which due to its simplicity (it has only two line of interesting code, one of which is a call to b_curve) is not shown here.

1.

Input and output arguments:

function integral = b_curve ...
    (args, shifted_g_even, ts_even, A, fstart, flengths)

The output argument integral is just the values of equation 18 after frame-by-frame integration, i.e. there is one number for each frame.

The input arguments are = 0=

<879>>=-0.25in 0 =<: <879>>

args

a three element vector containing $\alpha$, $\beta$, and $\gamma$ from equation . Keep in mind in the following code that args(1) is $\alpha$, args(1) is $\beta$, and args(1) is $\gamma$.

g_even

the blood data resampled at an evenly-spaced time domain, and possibly shifted by some delay factor $\delta$. We are not interested here in what that value of $\delta$ is, however.

ts_even

the evenly-spaced time domain at which g_even is sampled.

A

the PET activity, averaged over gray matter, for the current slice. (This isn't actually used in b_curve, but the argument is still here for historical reasons and to make calls to b_curve and fit_b_curve look the same.)

fstart

the frame start times.

flengths

the frame lengths.

2.

Evaluate the first term of the integrand in equation 18.

expthing = exp(-args(2)*ts_even);
c = nconv(shifted_g_even,expthing,ts_even(2)-ts_even(1));
c = c (1:length(ts_even));
i1 = args(1)*c;                   % alpha * (convolution)

3.

Evaluate the second term of the integrand, and the two terms together to get the integrand, i.

i2 = args(3)*shifted_g_even;            % gamma * g(t - delta)

i = i1+i2;

4.

Perform the frame-by-frame integration.

integral = nframeint (ts_even, i, fstart, flengths);

5.

Clean up any NaN's that have cropped up by setting them to zero. Also, truncate the function to whatever length A is.

nuke = find (isnan (integral));
integral (nuke) = zeros (size (nuke));

integral = integral (1:length(A));