COMMENTS FROM EVAN

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…

General comments.

I would have to say that this experiment in home-work creation as final project was not an unqualified success. Some of that can be attributed to PETKAT and the frustrations you experienced in trying to use it. I apologize for that. In the past, at least one student in my class has had prior experience with PETKAT and was able to guide his/her classmates.

Nevertheless, I have some trouble with what you submitted. Again, I may not have made the questions clear enough but the general idea was this. What does it take to be the professor? Well, you have to create problems –in this case datasets- that when analyzed produce answers that reinforce the principles presented in class. You’d be surprised how difficult this is. Perhaps you are. If so, good.

Lets consider (A).

What was the principle I want to demonstrate? Simply put: That even if FDG has some minimal reversibility (ie,. k4>0), it can adequately be modeled as irreversible for short-duration studies. How short is short? Well, as long as the time-constant of reversibility is large compared with the duration of the scan. What is the time-constant (half-time) for a linear rate constant (units = min-1)? t1/2 = 0.693/k4

So, first, I was hoping that you would choose k4 sufficiently small (slow) so that 0.693/k4 would be much larger than 1 hr but in the range of 4hr. so maybe k4 = 0.001- 0.002min-1

Then, we need time-activity curves of 1 hr and 4hr duration.

I asked you to make noisy curves to help illustrate the point. (I am not sure how you decided to make 10 curves for the 1 hr case but only 1 curve of the 4 hr case.)

We need a means of comparing analyzed curves. I suggested fitting the data by estimating K1, k2, k3. All other parameters should have been fixed (I know it may not have been clear how to do this.). Finally, we need a means of assessing the bias in the estimates. You calculated rCMRglc = K1k3/(k2+k3) but you didn’t calculate bias (difference between true and estimated) so I don’t know if the fits of 3-parameter model to the data was actually giving me better answers than the fits to the 4 hr data or not.

One more point here. By floating 7 parameters rather than 3, the whole thing was muddied. A clean demonstration of the principle I wanted to demonstrate required that you estimate only those parameters, K1, k2, k3. I was hoping to see that rCMRglc would be reasonably well estimated from 1 hr data using the 3-parameter model, but not as well estimated from 4 hr of data without estimating the additional parameter, k4.

(B) OK, first, I get a C- for the problem statement here. I think if you read it closely and go back to the Appendix in Carson, 1993, you will see that I got it backwards. If we need the intercept term to be constant, we need the tissue to “follow “ the plasma. That means that the slowest, determining rate component of the system must be in the PLASMA. So I should have said

“eigenvalues of the tissue response function are all slower FASTER than some exponential component of the plasma curve”

In any case, my big problem is with your Patlak plots. The slopes of both of them on page with (ugh, I wish you had numbered the pages)??? “Fast Patlak Plot“and “Slow…” shows that they were both ZERO (2e-5, 3e-7). How can this be? I think there must be something wrong with your Patlak plots. I couldn’t find the problem.. but I suspect its there. Afterall, you know what the slope should be, right? Slope = K1k3/(k2+k3). I think this knowledge of the Patlak slope should have helped you debug your plots.

(C)  This one was at least partially a success. For myself going forward, however, it seems clear that I would need to create more simulations at different noise levels to unambiguously demonstrate the principle. Afterall, there must be error bars on your Logan slopes (they were noisy simulations, right). That means there ought to be error bars on your plot of Slope vs SNR. Do you really think that the points are different?

(D)  Here, I am just a bit puzzled. If you used the same parameters as in the Logan review from 2003, then I would have expected you to re-produce her (relevant) figures. But there really is not much effect related to k4 apparent in the figures. certainly it is subtle. This one was a near partial success, I guess.

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) = a*(RawPET(i))0.5/DT(i)

where

a is a scale factor

DT(i) is the time frame. (Longer timeframes à 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, a, 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.

  1. Use COMKAT to add noise to the simulated tissue curve. Make 10 different curves.
  2. Save the data, reload the data that you created, estimate K1, k2, k3. Are they close to true? How close?
  3. Use COMKAT to simulate 4 hrs of dynamic data.
  4. 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.

  1. 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.
  2. 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?

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.

  1. 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.)
  2. Do Logan plots on all noisy curves.
  3. 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).

  1. Plot Logan plots on all three curves.
  2. 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.
  3. Plot slope of the fitted line vs t* for each of three curves (different k4 values).
  4. 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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