Generalized multi-energy model for computing radiation treatment parameters on image domain and automated model-selection framework

Table 1 Modeling and testing root mean square error (%) for stopping power ratio

SPR modeling RMSE	Total	Lung	Soft	Bone	Testing RMSE	Total	Soft	Bone
TSPR,2,1; Tμmr	0.85	0.00	1.03	0.40		2.69	2.72	2.65
TSPRsoft,2,1 bone; Tμmr	0.73	0.00	0.87	0.40		2.63	2.62	2.65
TSPR,1,2; Tμm	0.76	0.13	0.90	0.42		2.66	2.66	2.65
TSPR,1,2; TμL	0.83	0.02	0.96	0.53		7.26	3.42	10.49
Stoichiometric	0.93	-0.01	1.10	0.56		3.40	3.38	3.42
Taasti	0.15	0.01	0.15	0.15		0.67	0.76	0.60
H-S	0.39	0.28	0.41	0.35		0.85	0.91	0.77
TSPR,2,1; Tμmr 4 g	0.69	0.00	0.81	0.40		2.37	2.14	2.65

RMSE: Root mean square error; SPR: Stopping power ratio; H-S: Hunemohr-Saito.

Table 2 Modeling and testing mean error (%) for stopping power ratio

SPR modeling ME	Total	Lung	Soft	Bone	Testing ME	Total	Soft	Bone
TSPR,2,1; Tμmr	-0.01	0.00	-0.01	0.00		-0.59	-0.32	-0.98
TSPRsoft,2,1 bone; Tμmr	0.00	0.00	0.00	0.00		-0.70	-0.50	-0.98
TSPR,1,2; Tμm	-0.01	-0.13	-0.01	-0.01		-0.74	-0.56	-1.00
TSPR,1,2; TμL	0.04	-0.02	0.07	-0.02		1.02	-0.73	3.46
Stoichiometric	-0.01	-0.01	-0.01	0.00		-1.19	-1.00	-1.45
Taasti	0.02	-0.01	0.00	0.08		0.34	0.28	0.43
H-S	0.29	0.28	0.30	0.29		0.74	0.76	0.71
TSPR,2,1; Tμmr 4 g	0.01	0.00	0.02	0.00		-0.51	-0.18	-0.98

ME: Mean error; ρ_e: Relative electron density; SPR: Stopping power ratio; H-S: Hunemohr-Saito.

Table 3 Modeling and testing median absolute error (%) for stopping power ratio

SPR modeling MedAE	Total	Lung	Soft	Bone	Testing MedAE	Total	Soft	Bone
TSPR,2,1; Tμmr	0.37	0.00	0.58	0.20		2.25	1.63	2.87
TSPRsoft,2,1 bone; Tμmr	0.30	0.00	0.31	0.20		2.48	2.26	2.87
TSPR,1,2; Tμm	0.29	0.13	0.29	0.18		2.20	1.81	2.91
TSPR,1,2; TμL	0.30	0.02	0.43	0.27		3.22	2.93	4.22
Stoichiometric	0.37	0.01	0.37	0.35		3.08	2.81	3.35
Taasti	0.08	0.01	0.08	0.08		0.67	0.79	0.54
H-S	0.22	0.28	0.22	0.25		0.62	0.64	0.53
TSPR,2,1; Tμmr 4 g	0.27	0.00	0.29	0.20		1.47	1.22	2.87

MedAE: Median absolute error; SPR: Stopping power ratio; H-S: Hunemohr-Saito.

Table 4 Modeling and testing maximum absolute error (%) for stopping power ratio

SPR modeling MaxAE	Total	Lung	Soft	Bone	Testing MaxAE	Total	Soft	Bone
TSPR,2,1; Tμmr	2.57	0.00	2.57	1.04		4.76	4.76	3.74
TSPRsoft,2,1 bone; Tμmr	2.51	0.00	2.51	1.04		4.83	4.83	3.74
TSPR,1,2; Tμm	2.44	0.13	2.44	1.16		5.05	5.05	3.67
TSPR,1,2; TμL	2.91	0.02	2.91	1.39		22.35	5.87	22.35
Stoichiometric	2.93	0.01	2.91	1.38		5.94	5.94	4.88
Taasti	0.40	0.01	0.40	0.28		1.04	1.04	1.02
H-S	1.21	0.28	1.21	0.67		1.31	1.31	1.13
TSPR,2,1; Tμmr 4 g	2.26	0.00	2.26	1.04		4.49	4.49	3.74

MaxAE: Maximum absolute error; SPR: Stopping power ratio; H-S: Hunemohr-Saito.

Table 5 Modeling and testing root mean square error (%) for relative electron density

ρe modeling RMSE	Total	Lung	Soft	Bone	Testing RMSE	Total	Soft	Bone
Tρe,2,1; Tμmr	0.82	0.99	0.96	0.44		2.48	2.55	2.37
Tρe,1,2; Tμm	0.70	0.16	0.81	0.47		2.32	2.25	2.41
Tρe,1,2; TμL	0.89	0.04	1.02	0.59		3.03	3.00	3.08
H-S	0.04	0.01	0.04	0.04		0.11	0.13	0.08
Tρe,2,1; Tμmr 4 g	0.69	0.99	0.77	0.44		2.31	2.26	2.37

RMSE: Root mean square error; ρ_e: Relative electron density; H-S: Hunemohr-Saito.

Table 6 Modeling and testing mean error (%) for relative electron density

ρe modeling ME	Total	Lung	Soft	Bone	Testing ME	Total	Soft	Bone
Tρe,2,1; Tμmr	-0.02	-0.99	0.01	0.00		0.73	0.41	1.18
Tρe,1,2; Tμm	0.01	0.16	0.00	0.02		0.89	0.65	1.22
Tρe,1,2; TμL	0.01	0.04	0.01	0.00		1.29	1.09	1.56
H-S	0.02	0.01	0.02	0.02		0.10	0.11	0.07
Tρe,2,1; Tμmr 4 g	-0.03	-0.99	0.00	0.00		0.89	0.69	1.18

ME: Mean error; ρ_e: Relative electron density; H-S: Hunemohr-Saito.

Table 7 Modeling and testing median absolute error (%) for relative electron density

ρe modeling MedAE	Total	Lung	Soft	Bone	Testing MedAE	Total	Soft	Bone
Tρe,2,1; Tμmr	0.37	0.99	0.46	0.27		2.33	2.40	2.25
Tρe,1,2; Tμm	0.26	0.16	0.29	0.23		1.90	1.71	2.31
Tρe,1,2; TμL	0.35	0.04	0.37	0.35		2.41	2.40	2.92
H-S	0.02	0.01	0.02	0.03		0.11	0.12	0.05
Tρe,2,1; Tμmr 4 g	0.24	0.99	0.20	0.27		2.09	2.04	2.25

MedAE: Median absolute error; ρ_e: Relative electron density; H-S: Hunemohr-Saito.

Table 8 Modeling and testing maximum absolute error (%) for relative electron density

ρe modeling MaxAE	Total	Lung	Soft	Bone	Testing MaxAE	Total	Soft	Bone
Tρe,2,1; Tμmr	2.27	0.99	2.27	1.11		3.69	3.69	3.35
Tρe,1,2; Tμm	2.15	0.16	2.15	1.24		3.97	3.97	3.29
Tρe,1,2; TμL	2.72	0.04	2.72	1.49		5.07	5.07	4.28
H-S	0.10	0.01	0.10	0.06		0.16	0.16	0.11
Tρe,2,1; Tμmr 4 g	2.27	0.99	2.27	1.11		3.62	3.62	3.35

MaxAE: Maximum absolute error; ρ_e: Relative electron density; H-S: Hunemohr-Saito.

Table 9 Modeling and testing root mean square error (%) for effective atomic number

Zef modeling RMSE	Total	Lung	Soft	Bone	Testing RMSE	Total	Soft	Bone
TZ,2,0; Tμmr	9.34	0.24	11.71	0.45		10.63	13.77	2.45
TZ,1,1; Tμm	4.33	0.06	5.42	0.45		6.28	7.88	2.79
TB,0,1	0.32	0.00	0.40	0.01		0.39	0.49	0.08
H-S	5.27	0.00	0.53	9.11		5.63	1.00	8.64
TZ,2,0; Tμr 4 g	1.61	0.24	1.99	0.45		5.55	6.97	2.45

RMSE: Root mean square error, Z_ef: Effective atomic number; H-S: Hunemohr-Saito.

Table 10 Modeling and testing mean error (%) for effective atomic number

Zef modeling ME	Total	Lung	Soft	Bone	Testing ME	Total	Soft	Bone
TZ,2,0; Tμmr	7.23	0.24	11.34	0.01		6.15	11.42	-1.23
TZ,1,1; Tμm	3.12	-0.06	4.91	0.00		1.25	3.20	-1.49
TB,0,1	0.00	0.00	0.00	0.00		-0.17	-0.29	0.04
H-S	3.10	0.00	0.10	9.10		3.25	-0.55	8.57
TZ,2,0; Tμr 4 g	0.01	0.24	0.01	0.01		-0.79	-0.48	-1.23

ME: Mean error; Z_ef: Effective atomic number; H-S: Hunemohr-Saito.

Table 11 Modeling and testing median absolute error (%) for effective atomic number

Zef modeling MedAE	Total	Lung	Soft	Bone	Testing MedAE	Total	Soft	Bone
TZ,2,0; Tμmr	10.19	0.24	10.74	0.19		5.60	9.04	2.24
TZ,1,1; Tμm	3.93	0.06	4.95	0.33		1.78	1.00	2.22
TB,0,1	0.10	0.00	0.15	0.00		0.27	0.27	0.05
H-S	0.65	0.00	0.25	8.98		1.83	0.20	7.85
TZ,2,0; Tμr 4 g	0.47	0.24	0.53	0.19		3.50	7.20	2.24

MedAE: Median absolute error; Z_ef: Effective atomic number; H-S: Hunemohr-Saito.

Table 12 Modeling and testing maximum absolute error (%) for effective atomic number

Zef modeling MaxAE	Total	Lung	Soft	Bone	Testing MaxAE	Total	Soft	Bone
TZ,2,0; Tμmr	18.39	0.24	18.39	1.19		29.28	29.28	3.53
TZ,1,1; Tμm	10.29	0.06	10.29	0.97		19.97	19.97	4.17
TB,0,1	1.17	0.00	1.17	0.01		0.97	0.97	0.12
H-S	9.64	0.00	1.10	9.64		10.54	1.99	10.54
TZ,2,0; Tμmr 4 g	6.32	0.24	6.32	1.19		11.04	11.04	3.53

MaxAE: Maximum absolute error; Z_ef: Effective atomic number; H-S: Hunemohr-Saito.

Table 13 Modeling and testing root mean square error (%) for mean excitation energy

I modeling RMSE	Total	Lung	Soft	Bone	Testing RMSE	Total	Soft	Bone
TI,2,0; Tμmr	3.94	0.19	4.93	0.49		5.93	6.93	4.14
H-S	1.95	0.00	2.23	1.41		4.71	5.13	4.05
TI,2,0; Tμmr 4 g	0.85	0.19	1.00	0.49		4.92	5.40	4.14

RMSE: Root mean square error; I: Mean excitation energy; H-S: Hunemohr-Saito.

Table 14 Modeling and testing mean error (%) for mean excitation energy

I modeling ME	Total	Lung	Soft	Bone	Testing ME	Total	Soft	Bone
TI,2,0; Tμmr	-0.71	0.19	-1.13	0.01		3.46	4.56	1.91
H-S	-0.04	0.00	-0.06	0.00		-3.14	-3.26	-2.97
TI,2,0; Tμmr 4 g	0.02	0.19	0.02	0.01		1.19	0.68	1.91

ME: Mean error; I: Mean excitation energy; H-S: Hunemohr-Saito.

Table 15 Modeling and testing median absolute error (%) for mean excitation energy

I modeling MedAE	Total	Lung	Soft	Bone	Testing MedAE	Total	Soft	Bone
TI,2,0; Tμmr	2.54	0.19	3.22	0.39		4.59	4.45	4.73
H-S	0.86	0.00	1.01	0.61		2.18	2.29	1.31
TI,2,0; Tμmr 4 g	0.57	0.19	0.73	0.39		4.75	6.13	4.73

MedAE: Median absolute error; I: Mean excitation energy; H-S: Hunemohr-Saito.

Table 16 Modeling and testing maximum absolute error (%) for mean excitation energy

I modeling MaxAE	Total	Lung	Soft	Bone	Testing MaxAE	Total	Soft	Bone
TI,2,0; Tμmr	12.73	0.19	12.73	0.97		13.15	13.15	5.22
H-S	7.52	0.00	7.52	3.09		7.67	7.67	7.03
TI,2,0; Tμmr 4 g	2.01	0.19	2.01	0.97		7.09	7.09	5.22

MaxAE: Maximum absolute error; I: Mean excitation energy; H-S: Hunemohr-Saito.