SRTM Variants using GLM

For a single run, the use of 3 parameters provides a GLM analysis equivalent to the original SRTM (Lammertsma 1996). Other acronyms for linear SRTM include MRTM (Ichise 2003) and LSRRM (Alpert 2003). If k2' is set as a global constant, then this is MRTM2 (Ichise 2003). Files for describing a single scan using each type of analysis are shown below. Note that the default value of the shape is “constant”, so this can be left out of the file. Also, the default during is a single period that spans the run, so this also can be left out of the file to simply the format.

             LSRRM or MRTM                                                                           MRTM2

0.  # tau2’ for reference region                                                        3.5  # tau2’ for reference region

1 R1   # event 1, constant value across run                                    1 R1       # k2=R1/tau2' above

a k2   # required k2, because tau'=0 above                                     A k2a constant

A k2a  # k2a, from which BP is calculated

In some case, one might want a time dependence on k2a; when using 3 constant parameters, this again is LSRRM.  This could be added to either file above.  A time dependence in k2a starting at 30 minutes into a scan can be added as below. In the example, the constant k2a term is applied to the entire scan, and the gamma time dependence is applied starting at 30 minutes and continuing to the end of the run.

LSRRM without variable k2, or MTRM2 plus time dependence

3.5  # tau2’ for reference region

1 R1 constant  # k2=R1/tau2' above

A k2a constant

B k2a gamma    # normalized gamma = t/tau * exp(1-t/tau)

30 25                 # start at time=30 minutes, use tau=25 minutes

Finally, one could choose to implement a user-specified function implemented through the table file associated with the run, or further expand the time dependence using multiple functions (Normandin, 2012). In practice, only one or a very few time-dependent terms will provide optimal chi2/DOF. However, there may be cases when several functions better fit the data than a single function, because simple functions (gamma, sigmoid) do not adequately capture the shape of the response due to effects like receptor saturation at high occupancies. For instance, the graph here shows a co-infusion of D2 antagonist with the second bolus of 11C-raclopride, and the high occupancy achieved by the antagonist is not well modeled by a single gamma-variate function. Using two gamma-variate functions, as used for the figure, does a much better job. Alternatively, one could fit a single gamma-variate function together with a different constant term across the second run, so that the purpose of the gamma function is simply to refine the peak BP reduction.

Joseph B. Mandeville, Athinoula A. Martinos Center for Biomedical Imaging at MGH/MIT/Harvard