Table 2.

Error propagation from uncertainty of ${}^{23}\text{Na}$ relaxation times. In the left section the mean TSC and ${\text{C}}_{\text{IC}}$ in an ROI containing the lesion (Patient 1 in Fig. 5) is reported for different values of ${}^{23}\text{Na}$ ${\text{T}}_1$. In the central section mean TSC and ${\text{C}}_{\text{IC}}$ in the lesion are reported for different values of ${\text{T}}_{2,s}$. in the rightmost section mean ${\text{C}}_{\text{IC}}$ and TSC of the tumor are reported for a range of ${\text{T}}_{2,l}$. In the last row the mean uncertainty, defined as mean percentage variation relative to the central values, is reported. The relaxation times that we set in our method and the corresponding mean quantification results in the lesion are highlighted in bold. Due to the scarcity of literature references for ²³Na relaxation times in breast cancer, an abundant range of relaxation properties (± 50%) was simulated. Notations are: ${\text{C}}_{\text{IC}}$ intracellular sodium concentration, TSC total sodium concentration.

	Input	Output		Input	Output		Input	Output
	$T_1$ (ms)	Lesion TSC (mM)	Lesion $C_{\mathit{IC}}$ (mM)	$T_{2,s}$ (ms)	Lesion TSC (mM)	Lesion $C_{\mathit{IC}}$ (mM)	$T_{2,l}$ (ms)	Lesion TSC (mM)	Lesion $C_{\mathit{IC}}$ (mM)
	16	41.2	18.5	0.25	57.2	43.8	8	47.3	30.2
	24	43.6	22.3	0.38	50.3	33.5	12	47.1	29.8
	32	47.0	29.6	0.5	47.0	29.6	16	47.0	29.6
	40	50.9	34.6	0.63	44.9	24.4	20	47.0	29.6
	48	55.1	41.7	1	41.9	22.2	24	47.0	29.6
Mean uncertainty	± 50%	± 14%	± 39%	± 50%	± 15%	± 33%	± 50%	± 0.03%	± 1%