ANSWERS
Medical Imaging Applications, Spring 2005 BME 595
Modeling HW #3 (Morris)
Due Wed Apr 27
1. a. DOPA is a precursor of (chemical that is made into) dopamine. (It is sometimes used as a treatment for Parkinson's disease.) 18F-DOPA is a tracer that is used to monitor the rate of synthesis of new dopamine in the brain. Like most processes in the body, the synthesis of dopamine from DOPA is enzyme-mediated (controlled by a specific enzyme molecule). The enzyme is Amino Acid Decarboxylase.
One complication of using 18F-DOPA as a tracer is that a metabolite (a breakdown product) of 18F-DOPA is created in various organs of the body and returned to the blood and is able to get across the blood-brain-barrier into the brain.
Consider the following accepted compartmental model of uptake of 18F-DOPA into tissue and its incorporation into dopamine in the brain.
1a. Please write mass balance equations for all of the chemical species that contribute to the PET signal with reference to the following model
1b. Then write the 'Output' Equation for the PET signal to be measured.
1c. How may unique kinetic parameters are needed to describe this system?
1d. Aside from dynamic PET measurements, what concentrations would also need to be measured – from what type of samples – in order to solve this model?
Answer 1a.
Balances on free 18F-FDOPA and on free 18F-30MFD:
Balances on bound 18F-FDOPA:
Answer 1b.
Standard output expression gives all the contributors to radioactivity in the voxel or region that is being modeled.
But in this case, the catch is that we have TWO radioactive species: 18F-FDOPA and on free 18F-30MFD and the two species may have different access to the compartments so to be completely general, I would write the answer as:
Fvx is the fluid (i.e., blood) volume fraction for species x (ie., DOPA or OMFD) and
Tv-x is the tissue volume fraction for species x.
Answer 1c. There are (at least) 6 distinct parameters in this model. We cannot assume that the K1 and k2 parameters for 18F-FDOPA and 18F-30MFD are the same. (If you wrote a bimolecular reaction term (rather than a first order one) for metabolism of 18F-FDOPA then you would probably have an additional Bmax-DOPA parameter in your model.)
Answer 1d. To be able to solve the model in 1a requires knowledge of two input functions. One for plasma concentration of 18F-FDOPA and the other for plasma concentration of 18F-30MFD.
To be able to get two distinct input functions, we would need some sort of assay for the native 18F-FDOPA and for the metabolite 18F-30MFD in blood that we could perform on arterial blood samples drawn from a subject during the course of a PET scan.
Review Question
(remember back to the dark ages when all we knew about was the Blood Flow Model)
2. 15O-water and 15O-butanol are two tracers commonly used to measure regional cerebral blood flow (rCBF) in the brain via PET. Below is a comparison of the measured K1 parameter based on studies with the two respective tracers. (Average brain blood flow is typically around 50 ml/min/gm.)
*2.a. Which tracer (15O-water or 15O-butanl) is most reliable at very high (rCBF > 125 ml/min/gm) flows?
Answer a. Butanol is more reliable. Note that at high (CBF>125 ml/min/gm) flow, rCBF measured with 15O-water will top out and not increase with increased flow (the signal is “clipped”).
*2.b. Professors want to identify the brain region that is only involved in attention (so that we can make slides that keep students interested during class).
Experiment:
· Place volunteer students in the PET scanner,
· inject with a good blood flow tracer,
· show them pictures of interesting lecture slides,
· use an eye-tracking device to monitor whether they are paying attention to the slide,
· analyze the blood flow images to determine parts of brain with increased blood flow.
Unfortunately, some of the slides are funnier than others and we do NOT want to find the brain circuits that respond to humor (students might spend the whole lecture laughing). We just want brain regions that control attention.
If 'attention' regions experience rCBF that reaches 150ml/min/gm during the experiment and 'humor' regions experience rCBF that reaches 100 ml/min/gm, which tracer is best for identifying the 'attention' regions in the brain but NOT confusing them with the 'humor' regions based on our experimental protocol? Why?
Answer b. Again, butanol will allow us to differentiate flow in ‘humor’ regions from flow in ‘attention’ regions. 15O-water imaging would identify both regions as higher than normal when subjects are looking at lecture slides. Note: in any case, this would require an input function and quantitative analysis to estimate a blood flow rate (or at least a K1 value).
We could NOT get by with normalizing all regions of the image by average blood flow alone UNLESS we had a “humor” control condition and were going to analyze our images by subtracting the control condition image from the activated condition image – as if often done with fMRI analysis.
*2.c. We want to test the hypothesis that prolonged use of an anti-epilepsy medicine is believed to cause long term changes in permeability in brain vessels. Which tracer is best to test such an hypothesis. Why? Under what conditions must we scan?
Answer c. Here, we want an extraction-limited tracer as opposed to a flow limited tracer. So, if we could scan subjects on and off epilepsy medicine under the condition of very high flow, we could use 15O-water to look for changes in K1. Such changes in K1 (at high enough flow to cause saturation of the 15O-water uptake) could be attributed to changes in E instead of changes in F. Recall that K1 = EF where in most cases we assume that E = 1. In this case we are imaging in a region that assumes that F is high and constant across scans while E differs depending on presence or absence of epilepsy medicine.
One condition that we talked about in class that causes high cerebral blood flow is breathing of high levels of CO2.
(Of course, we would have to make sure that epilepsy medicine did not cause severe drop in blood flow or that combination of breathing CO2 and epilepsy medicine would not harm the subjects.)
3.
Sometimes, a tracer binds to two different, but related, receptor sites (see Figure 1). For example, raclopride binds to D2 and D3 receptors, which are sometimes collectively referred to as D2/D3 sites.
Say we wanted to use PET and raclopride to measure kinetic rate constants and number of receptors for each type of site, separately. Draw a compartmental model for uptake and binding of 11C-raclopride to two distinct RECEPTOR sites (D2 & D3).
3a. Write the differential equation for the Free compartment, only, for the high specific activity case (i.e., Bound species are negligible compared to number of available receptor sites).
Answer a. Start with the model diagram
note that there are two (equivalent?) binding compartments – one for D2 receptors and one for D3 receptors.
Now proceed to the mass balance for the free case. Nonetheless, if you want to describe both binding processes, F->B2 and F->B3, as bimolecular –which they are- you could start with the following:
but since I specified that “Bound species are negligible compared to number of available receptor sites”, that’s the same as saying that the available receptor sites are always in excess and that we are NOT near saturation. I.e.
so the equation can be written more compactly –for these conditions- as,
3b. From looking at the Free compartment balance, do you expect that we will be able to distinguish between the binding rate constants for the D2 sites and D3 sites? Why or why not. Group the terms in your differential equation to support your claim.
Answer b. Intuitively, we should be skeptical of our ability to identify k3 and k5 separately.
Consider that they both occur in the dF/dt equation as one indistinguishable term, i.e.,
-(k2 + k3 + k5)F.
More formally, we note that the sensitivities of the PET output to k3 and k5: would be expressed as:
will be very similar, and practically, they will probably have indistinguishable effects on the model output.
Intuitively, we should be skeptical of our able to identify k3 and k5 separately.
Consider that they both occur in the dF/dt equation as one indistinguishable term, i.e.,
-(k2 + k3 + k5)F. Note: the sensitivities of the PET output to k3 and k5:
will be very identical, and thus will have indistinguishable effects on the model output. So, any changes in k3 or k5 would have large effects on the free concentration of ligand when F is large – the two parameters will almost certainly not be identifiable.
So, we would likely NOT be able to differentiate a forward rate constant for the D2 receptor separate from a forward rate constant for the D3 receptor.
3c. If we are doing this experiment in rats and the specific activity of the tracer is relatively low (how to gauge ‘relatively’) then how must we augment the equation in 3b?
Answer c. OK. now we’re back to the situation where we must account for ALL the species (hot and cold ligand) that might bring us near to saturation of receptor sites, BmaxB2 or BmaxB3:
Since there is only one ratio of hot to total molecules in this SINGLE INJECTION experiments, there is only one specific activity, SA. Using said specific activity, we could re-express the above equation as:
If we have defined SA= hot molecules to total molecules, then the above equation should be in moles of hot tracer per volume (e.g., pmol/ml) and the Bx/SA terms just convert concentration of bound HOT molecules at a particular type of receptor to bound TOTAL molecules at a particular type of receptor.