Tests using Arterial Spin Labelling model¶
This model implements a basic resting-state ASL kinetic model for PASL and pCASL acquisitions. The model parameters are \(f_{tiss}\), the relative perfusion and \(\delta t\) the transit time of the blood from the labelling plane to the voxel.
Time points are divided into two categories:
During bolus is defined as \(\delta t < t <= \tau + \delta t\)
Post bolus is defined as \(t > \tau + \delta t\)
Here \(\tau\) is the bolus duration. The model output is zero for pre-bolus time points.
The following rate constant is defined:
\(\frac{1}{T_{1app}} = \frac{1}{(1 / T_1 + f_{calib} / \lambda)}\)
\(\lambda\) is the tissue/blood partition coefficient of water which we take to be 0.9. \(f_{calib}\) is the calibrated CBF which typically we do not do not have access to (since we are inferring relative CBF) so we use a typical value of 0.01 \(s^{-1}\).
CASL model¶
During bolus¶
\(M(t) = 2 f_{tiss} T_{1app} \exp{(\frac{-\delta t}{T_{1b}})} (1 - \exp{(-\frac{(t - \delta t)}{T_{1app}})})\)
Post bolus¶
\(M(t) = 2 f_{tiss} T_{1app} \exp{(-\frac{\delta t}{T_{1b}})} \exp{(-\frac{(t - \tau - \delta t)}{T_{1app}})} (1 - \exp{(-\frac{\tau}{T_{1app}})})\)
PASL model¶
\(r = \frac{1}{T_{1app}} - \frac{1}{T_{1b}}\)
\(f = 2\exp{(-\frac{t}{T_{1app}})}\)
During bolus¶
\(M(t) = f_{tiss} \frac{f}{r} (\exp{(rt)} - \exp{(r\delta t)})\)
Post bolus¶
\(M(t) = f_{tiss} \frac{f}{r} (\exp{(r(\delta t + \tau))} - \exp{(r\delta t)})\)
The time points in evaluating an ASL model are the \(T_i\) values, which may be expressed as the sum of the bolus duration \(\tau\) and a post-labelling delay time. For 2D acquisitions they may be further modified by the additional time delay in acquiring each slice.
Test data¶
The test data used is a pCASL acquisition with \(\tau = 1.8s\) and six post-labelling delays of 0.25, 0.5, 0.75, 1.0, 1.25 and 1.5s. The acquisition was 2D with an additional time delay of 0.0452s per slice. 8 repeats of the full set of PLDs was obtained.
The test data was fitted in two ways. One method was to average over the repeats and fit the model to the repeat-free data. The other is to fit the model to the whole data including repeats. Naturally this involves a larger data size and hence a mini-batch approach to the optimization.
Mean data tests¶
For these tests we have only 6 time points and therefore we do not use a mini-batch approach, instead using a fixed batch size of 6 (all data points).
Convergence by learning rate¶
The convergence of mean cost by learning rate is shown below:
The pattern is closely similar to that obtained using a biexpoential model although the convergence here is generally ‘cleaner’. Learning rates between 0.05 and 0.1 attain the lowest cost within the given number of epochs, with 0.1 converging faster. Higher learning rates are less stable and do not appear to be likely to converge, while lower learning rates converge slowly.
The best cost achieved in 500 epochs is shown below, reinforcing the optimum learning rate range 0.1 - 0.05
Full data tests¶
For these tests we have 8 repeats of the 6 PLDs giving 48 data points. This raises the possibility of a mini-batch approach. Intuitively the obvious choice of batch size is 6, arranged so that each optimization iteration considers one repeat of all 6 PLDs. However we experiment with varying the batch size to see if there is any actual advantage in this structure.
The patterns with convergence and batch size are very similar to those obtained from the biexponential model. In particular there is no visible effect of aligning the batch size with the ASL repeats. Again we find a general optimum learning rate of 0.1 - 0.05 associated with a batch size around 10, although it is noticable that the best cost achieved at lower learning rates is a bit better with smaller batch sizes.