2CXM model: two variants were implemented - NLLS and LLS
With LLS variant a double integration is performed and the corresponding 4x4 linear system is solved using SVD. Resulting solution is converted to physiological parameters {vp,ve,PS,Fp} and substituted to an analytical solution containing a convolution of two different exponents with the AIF.
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Sample ROI for the test
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Tests were performed on the GRASP-4D with 76 timepoints [0,411] sec.
Small ROI (yellow was chosen)
First we performed the fitting with "ROI as a single TAC".
A good fit was observed RMSE=25. But corresponding parameters contain the negative value, which is not physiological.
Next we performed voxel-by-voxel fitting. This model #45 2CXM-LLS has a single hyperparameter "Exclude invalid", which optionally excludes the unphysiological voxels from the map (defaullt=0 - Include all).
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| LLS: Residual distribution [0,100] (include all mode) |
If we turn "Exclude Invalid =1", this would exclude the voxels from the result where one of the key fittted parameters are negarive: this includes transit times {T,Te,Tp} and {vp,ve,PS}
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| Fitting with unphysiological (invalid) voxels removed. |
When such invalid voxels are removed, we can see that a minor portion of the ROI resulted in non-negative parameter values.
NLLS variant: gridded Simplex algorithm was used.
Below is the fitting residual comparison for LLS and NLLS.
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| LLS: fit residual distribution [0,100] |
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NLLS: fit residual distribution [0,100]
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Timing: for the whole slice ROI (24,000 voxels):
Processing times are LLS: 0.5 sec
NLLS: 55 sec