The following must be completed in order to implement a two-compartment
rCBF model in Matlab:
1) Get the most recent articles on performing two-compartment analysis
that includes solving for V0.
COMPLETED ON: July 5, 1993
STATUS: Solving for V0 as well as K1 and k2 seems quite straight
forward, and should be no more difficult than solving for
K1 and k2. We will use the triple weighted integration
method, either applied directly, or by using Hiroto's
more elegant modification.
Although the desired final product is a package that solves for K1, k2 and
V0, it is easier to start by neglecting V0, delay, and dispersion. In order
to perform the analysis while ignoring V0, the following must be completed:
2) Create a Matlab routine that generates the rR lookup table. This lookup
table is used to find a value for k2 at each pixel, and should be 100x1
(100 values of rR corresponding to values of k2 from 0.01 minutes to
1 minute).
COMPLETED ON: June 25, 1993
STATUS: June 29, 1993: Overhauled the function since we realized that
the convolution was not being performed correctly.
The function now creates evenly sampled data sets
for the purpose of convolution, convolutes, and then
re-samples back to the original sample spacing. We
discovered that unevenly sampled data sets did not
convolute correctly.
June 25, 1993: Completed.
June 21, 1993: Created an rR lookup table with Matlab, but not
an actual .m file. However, all of the commands
used were saved in a matlab diary.
3) Create a Matlab routine that calculates a value for rL. Since there is a
different value of rL for each pixel in the image, rL should be 16384x1 for
every slice.
COMPLETED ON: June 21, 1993
STATUS: Completed.
4) Create a Matlab routine that integrates the above two routines by
using the look-up table to find an approximation for k2. This
routine can probably also calculate K1 since this is straight
forward once k2 is known. This routine should probably just take
a handle to an input data set, and a handle to an output data set.
COMPLETED ON: July 2, 1993
STATUS: July 5, 1993: Worked out the correct units for K1 and k2.
K1 was being expressed as:
(nCi * sec * (g blood)) / ((mL tissue) * counts * sec)
It should be expressed as:
(mL blood) / ((g tissue) * minute)
k2 was being expressed as s^-1, and should be
expressed as min^-1.
June 30, 1993: Routine now calculates a K1 image as well as
a k2 image. However, while this image appears
qualitatively similar to a K1 image produced
by Jin's K1_image_auto.for, the values themselves
are off by a little more than four orders of
magnitude.
June 25, 1993: A .m file exists, but it is very rudimentary.
Data for a k2 image is generated, but no
K1 image is generated.
June 21, 1993: Played around with getting values for k2 by
using the above two functions. No actual
.m file exists yet, but once again all of
the commands used were saved in a matlab
diary.
The above three routines will provide an rCBF package that does not take
V0, delay or dispersion into account.
We are currently adding delay and dispersion correction to the
above routines, and from there we will move on to performing
an analysis that also solves for V0.
The steps for finding the delay constant delta are:
1) Find A(t) by:
i) integrating a slice across its frames, and then selecting
all pixels that have a value greater than 1.8 times the mean of
the integrated image.
ii) allow the user to interactively adjust the mask by selecting a
new scale factor.
iii) use pixels selected by the above procedure as a mask applied to
every frame in the slice, and then calculate the mean of each
image. The resulting values are used as A(t).
COMPLETED ON: July 16, 1993
STATUS: July 16,1993: Finished
July 9, 1993: Wrote a Matlab script that implements the above
three steps and uses getmask.m.
July 7, 1993: Created getmask.m. This function handles the second
part of the above by displaying the image, and
allowing the user to interactively adjust the scale
factor. getmask returns the resulting mask as
0's and 1's.
2) Use A(t) to do a least squares fit to the blood delay and dispersion
equation.
COMPLETED ON: August 1, 1993
STATUS: Aug. 1, 1993: Completed. The dispersion correction problem
was resolved by converting the routine currently
used by the FORTRAN K1 image generation program
to Matlab. The delay correction was not changed,
except by using a better (trapezoidal) integration
routine. Unfortunately, the curve fitting procedure
is now very slow, and this must be solved.
July 16,1993: After satisfying ourselves that the stepwise
fitting method described below gave reasonable
results, we decided to add dispersion
correction. Implementation was quite easy,
but unfortunately adding dispersion correction
made things worse. The blood curve no longer
followed the brain curve closely.
July 15,1993: In order to try and solve the overdetermination
problem, we have taken a stepwise incrementation
approach to finding delta. We choose a value
of delta, and peform the curve fit while holding
delta constant. We then choose a new value of
delta, and perform the three term curve fit again
(for alpha, beta, and gamma). After having
performed the fit for several values of delta, we
choose the fit that produced the least amount of
error. At the moment, we are stepping through
delta with increments of 1 second.
July 12,1993: Found that the curve fitting problem seems to be
overdetermined. We can find several values of
alpha, beta, gamma, and delta that give good curve
fits.
July 9, 1993: Performed the least squares fit without taking
dispersion into account. The fit along the rising
slope was very good, but at the top of the slope
the fit function exploded. We do have some ideas
for solving this problem, however.
July 7, 1993: Took a first pass look at least squares fitting
in general, and the matlab functions that support
it in particular.
In order to complete two-compartment rCBF analysis, we must:
1) Calculate the delay and dispersion constants, and apply these as
corrections to the blood data used in our analysis. Please see the
previous section for details on this step.
2) Perform a full two-compartment analysis that returns K1, k2, and V0
using the triple-weighted integration method.
COMPLETED ON:
STATUS: Aug. 1, 1993: Our implementation is currently undergoing
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