Table 1.

Illustration of sets of biophysical (BP) parameter values resulting in the same diffusion–kurtosis (DK) parameters

DK parameters		BP parameters		C new invariants
$[{\mathbf{D}}_{\Vert },{\mathbf{D}}_{\perp },{\mathbf{W}}_{\Vert },{\mathbf{W}}_{\perp },\overline{\mathbf{W}}]$	Branch	$[\mathbf{f},{\mathbf{D}}_{\mathrm{a}},{\mathbf{D}}_{\mathrm{e}}^{\Vert },{\mathbf{D}}_{\mathrm{e}}^{\perp },\kappa ]$	ζ 1	ζ 2
[1.503, 0.195, 1.456, 0.291, 0.926]	+	[0.730, 2.000, 1.000, 0.300, 8.000]	−0.006	0.210
	−	[0.607, 1.287, 2.191, 0.318, 11.49]	0.023	0.053
[1.557, 1.048, 0.396, 0.708, 0.330]	+	[0.250, 2.370, 1.300, 1.390, 50.00]	0.349	0.624
	−	—		—
[0.457, 0.408, 2.901, 2.702, 2.770]	+	[0.879, 1.320, 1.401, −0.232, 0.265]	−0.190	0.022
	−	[0.870, 0.950, 2.000, 0.720, 0.360]	−0.023	0.014
	−	[0.549, 0.182, 1.071, 0.766, 1.414]	0.154	−0.002
	−	[0.510, 0.076, 0.931, 0.794, 3.187]	0.161	−0.005
[1.560, 1.256, 0.423, 0.540, 0.506]	+	—		—
	−	[0.240, 1.450, 2.100, 1.400, 2.330]	0.237	0.125
	−	[0.189, 0.668, 1.887, 1.489, 5.442]	0.325	0.057

Each plus or minus branch can correspond to a single, multiple, or none BP parameters. Some sets of BP parameters fall outside the region of plausible parameters, like the + branch solution of the third example. We can observe that the invariants of the not fully symmetric part of C, incorporated by DDE, discriminate between the BP parameter sets having the same exact DK representation. All diffusivities are in $\mathrm{\mu }{\text{m}}^2/\text{ms}$ and the C components in $\mathrm{\mu }{\text{m}}^4/{\text{ms}}^2$.