ENAS 915 Instructor: Morris Yale Spring 2013 Homework #6 (and final project in lieu of final) YOU are the professor!
Help Evan create a good homework for next year’s class…
Let’s test some important ideas about irreversible, reversible tracers, and graphical analysis. I would like to have homework problems for next year that test these ideas:
A. If the dephosphorylation of deoxyglucose-6-P is slow enough, then uptake of that tracer (FDG) can be modeled as irreversible (K1, k2, k3 only).
B. There is a t > t* at which the Patlak plot will be linear provided that the eigenvalues of the tissue response function are all slower than some exponential component of the plasma curve. (And if they are not, then it should not be linear.) This principle goes back to the Patlak papers (1983, 95) and Appendix A in Carson, 1993.
C. There is a negative bias (i.e., underestimation of the slope, Vt) of the Logan plot with increasing noise level. This goes back to the Slifstein and Laruelle paper, 2000.
D. The approach to linearity of the Logan plot is governed by the off rate, k4. This phenomenon is described in the Logan review paper from 2003.
To test these concepts, we need to be able to simulate PET time-activity curves. To do so, I am hoping that the all of you can work together using a simulation package that we have available in my lab.
The package is called PETKAT and was developed by my group at IUPUI. It uses a library of modeling and numerical analysis routines, COMKAT, developed in Matlab by Ray Muzic of Case Western Reserve University in Cleveland. The PETKAT package has a decent GUI, so you should not need to do any programming to use it.
Some of the problems require the creation of NOISY time-activity curves. We know that the noise in PET data follow Poisson Statistics. So, a reasonable model for the noise in the tissue curves is the following.
C(t) is the tissue concentration (after decay correction)
RawPET(t) is the radioactivity signal, C(t)*e-at where a is the radioactive decay constant of the isotope. Be sure to set the isotope correctly in PETKAT to 18F for FDG and 11C for raclopride.
The noise is based on the number of counts, which is based on the radioactivity signal, so
SD(i) = *(RawPET(i))0.5/T(i)
1 where
is a scale factor
T(i) is the time frame. (Longer timeframes less noise).
Notice that the SD is different for each time point.
All of this is done for you inside PETKAT. You just choose the scale factor, , to increase the noise level in the data.
Assuming you can get PETKAT to run (try on Yan’s computer in LMP85 first, otherwise, you will need to install an old version of Matlab as well as register and download COMKAT), please do the following:
A. Simulate FDG data (with K1, k2, k3, k4=0.07min-1). Use other values that you find in the literature or make up – make sure they are much faster than k4 – WHY? for 1 hr of dynamic scanning. This will require that you create an input function, Cp. Use a sum of three decaying exponentials. Make sure the slowest decaying exponential is SLOW.
a. Use COMKAT to add noise to the simulated tissue curve. Make 10 different curves.
b. Save the data, reload the data that you created, estimate K1, k2, k3. Are they close to true? How close?
c. Use COMKAT to simulate 4 hrs of dynamic data.
d. Use COMKAT to estimate K1, k2, k3 again. What happened? Did we demonstrate the principle we wanted to? What is it?
B. Now create two different plasma FDG input functions, Cp1(t) and Cp2(t). For good measure, lets make each input function the sum of 3 exponentials (i.e., Ae-at + Be-bt + Ce-ct). The trick here is that one input Cp1(t) should have only fast exponentials, a,b,c, whereas Cp2(t) should have at least one slow exponential to satisfy the theoretical requirement that at some time t>t*, the slow exponential of the plasma will be slower than all the exponential terms in the tissue transfer function. Of course, you are not specifying the transfer function, you are going to select the rate constants. SO some degree of trial and error is required to find the right fast and slow plasma input curves to make the desired point here.
a. simulate irreversible (k4 = 0) FDG uptake curves for the two respective plasma curves Cpi(t) that you have created. Save out the two input curves in a format that can be loaded back into COMKAT.
b. make the Patlak plots for each of the two curves. (you can do this in a program or in Excel if you like). One should be linear and one should not. How can you tell if they are linear or not?
2 Did it work out? What is t*?
C. For this question, we will need to simulate tissue curves for uptake of 11C-raclopride. Because raclopride is a reversibly bound tracer, we need values for K1, k2, k3, k4. Get the parameter values from the Logan 2003 paper that is posted on the class handout wiki page.
a. For a single plasma curve (you can use the slower one of the Cp(t) curves that you created in B – WHY can’t you use the faster one!?) Just as you did in problem A, make noisy tissue curves. In this case, you will need 5 tissue curves at each of three noise levels. They should all use the same input curve and tissue parameters but they should have their own noise realizations (in other words, make them each separately. make sure the noise is NOT identical from curve to curve.)
b. Do Logan plots on all noisy curves.
c. Plot the estimated slope ( = VT) versus the noise (scale factor). Do you replicate the Slifstein finding?
D. Create 3 noiseless curves for 90 min of data with parameters from part C (I am trying to replicate figures 3, 4 of Logan 2003, here.) Make sure to record the curves for Free and Bound (what Logan calls C1(t) and C2(t) in her 2003 paper).
a. Plot Logan plots on all three curves.
b. Incrementally increase t* on each Logan plot – say in increments of 10 min. For each t*, refit the remaining data to a line. Record the slope.
c. Plot slope of the fitted line vs t* for each of three curves (different k4 values).
d. Plot C2/(C1+C2) vs time for each curve (B/(B+F)) i.e., for each k4. Does the approach to steady state for each k4 reflect the approach to linearity of the Logan plot?
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