Table 1.
List of mathematical symbols used for the kinetic model and their meaning.
| Symbol | Description |
|---|---|
| r | Position vector of the voxel of interest |
| t | Time since the start of the VEPCASL pulse train |
| c(r, t) | Normalised concentration of labeled blood |
| d3r | Elemental volume |
| δt(r) | Blood transit time from the labeling plane to the voxel at position r |
| τ | VEPCASL labeling duration |
| κ(r) | Vessel interior flag (1 inside vessels, 0 outside) |
| S(ri, t) | ASL signal intensity in voxel i |
| S0 | Calibration factor giving the ASL signal intensity per unit blood volume |
| td | Time delay caused by dispersion |
| D(r, td) | Dispersion kernel |
| p(r) | Dispersion kernel time to peak |
| s(r) | Dispersion kernel sharpness |
| Г | Gamma function which normalises the dispersion kernel to unit area |
| T(δt, td) | Signal attenuation due to T1 decay |
| T1b | Longitudinal relaxation time of arterial blood |
| R(t, δt, td) | Signal attenuation due to imaging RF pulses |
| TR | Repetition time (i.e. separation of imaging RF pulses) |
| α | Imaging RF pulse flip angle |
| N (r, t, td) | Previous number of RF excitation pulses experienced by the blood |
| t0 | Time at which the first imaging RF pulse is played out |
| δt,min | Time at which labeled blood first arrives in the imaging region |
| v(ri) | Volume of blood within vessels in voxel i |
| A(ri) | Scaling factor proportional to blood volume (A = S0v(ri)) |