Improved patient-specific hyperthermia planning based on parametrized electromagnetic and thermal models for the SIGMA-30 applicator

Figures & data

Figure 1. General workflow of the novel optimization procedure for improved regional hyperthermia (RHT) planning using optimization of the so-called ‘Parametrized Treatment Models’ (PTMs) via a root-mean-square-deviation (RMSD)-based comparison between calculated and measured temperatures along tumor, rectal and bladder catheters. An oval form of boxes refers to measurement parts of the optimization scheme. The core parts of the optimization routine that are bounded by dashed lines are described in more detail in Figure 2.

Figure 2. Details of the two-step optimization consisting of a ‘Coarse Step’ using the ‘homogeneous’ (H) ‘Perfusion Behavior Type’ (PBT), and a ‘Fine Step’ using either ‘non-dynamic’ (ND) or ‘dynamic’ (D) PBTs. The optimum whole-body perfusion level found by the Coarse Step (here m = m_best_H = 10 ml/100g/min) is used as a starting point for the Fine Parametrizations/Optimizations. For clarity, the particular numbers used for the case presented in the section Results are displayed in Figure 3.

Figure 3. Parametrized treatment models (PTMs) for the case presented in the section Results, created as parameter combinations, involving: •two relative SAR distributions (‘central’ and ‘shifted’ patient positioning). •two therapies [‘Higher-Power’-Therapy (HP_THER) and ‘Lower-Power’-Therapy (LP_THER), each of them represented by a power range of absorbed power in patient, approximated by five power levels each thus resulting of a total of 20 absolute SAR distributions]. •three different PBTs [homogeneous (H) with 7 perfusion models, non-dynamic (ND) with 1350 perfusion models, and, dynamic (D) with 1536 perfusion models], resulting in the total number of PTMs of n = 57.860, equal to the total number of BHTE runs.

Table 1. Clinically measured power and temperature values for two investigated therapies of a pediatric pelvic rhabdomyosarcoma: Higher-Power-Therapy (HP_THER) in the middle column, and Lower-Power- Therapy (LP_THER) in the right column.

	HP_THER	LP_THER
Power-on duration	80 min	80 min
Aver. temperature at power-off in tumor catheter	42.02 °C	41.18 °C
Max. temperature at power-off in tumor catheter	42.6 °C	42.4 °C
Aver. temperature at power-off in rectum catheter	41.33 °C	41.04 °C
Max. temperature at power-off in rectum catheter	41.5 °C	41.3 °C
Temperature at power-off in bladder catheter	43.2 °C	42.1 °C
Aver. temperature at power-off in all 3 catheters	41.71 °C	41,16 °C
Average forward power (measured at amplifier outputs)	205 W	183 W
Average reflected power (measured at amplifier outputs)	31 W	22 W
Derived approximate range of absorbed power, P_range	100 W − 120 W	93 W − 113 W

In the last line the power range P_range is given, derived via a combined experimental-numerical power investigation, involving forward and reflected powers at the amplifier outputs, cable losses between amplifier outputs and antenna feedpoints, as well as and stray power losses.

Figure 4. A root-mean-squared-deviation (RMSD)-based minimization in Equation (7) for the best case in Table 3 for the parameter combination [‘Higher-Power’-Therapy (HP_THER), ‘dynamic Perfusion Behavior Type’ (D_PBT), ‘Central Patient Positioning’ (CE_POS), P = 120 W absorbed power in patient]. Displayed are RMSD values for all (i_perf = 1…n_perf_D = 1536) dynamic (D)-type perfusion models, as defined in Equation (13)(13) $$\begin{array}{c}m\_\mathit{tumor}=\mathrm{ }2.5\to 5\mathrm{ }|\mathrm{ }5\to 10\mathrm{ }|\mathrm{ }7.5\to 15\mathrm{ }|\mathrm{ }10\to 20\\ m\_\mathit{rectum}=\mathrm{ }2.5\to 5\mathrm{ }|\mathrm{ }5\to 10\mathrm{ }|\mathrm{ }7.5\to 15\mathrm{ }|\mathrm{ }10\to 20\\ m\_\mathit{bladder}=\mathrm{ }1.25\to 2.5\mathrm{ }|\mathrm{ }2.5\to 5\mathrm{ }|\mathrm{ }5\to 10\\ m\_\mathit{muscle}=\mathrm{ }3.33\to 10\mathrm{ }|\mathrm{ }5\to 15\mathrm{ }|\mathrm{ }10\to 30\mathrm{ }|\mathrm{ }13.33\to 40\\ m\_\mathit{fat}=\mathrm{ }1\to 3\mathrm{ }|\mathrm{ }2\to 6\mathrm{ }|\mathrm{ }2.5\to 7.5\mathrm{ }|\mathrm{ }3.33\to 10\\ m\_\mathit{bone}=\mathrm{ }1.25\to 2.5\mathrm{ }|\mathrm{ }2.5\to 5,\end{array}$$(13) , associated with this case (indicated as ‘MODEL-NUMBER’ on the abscissa). The minimum RMSD was found for the index i_perf = 1001, i.e. for perfusion model D-1001. RMSD values range from 0.3882 °C (for i_perf = 1001) to almost 5 °C. The latter high RMSD values were obtained for low perfusion model indices, i.e. (too) low tumor and rectum perfusion values.

Table 2. Tissue perfusion values in perfusion models exhibiting the non-dynamic (ND) and dynamic (D) Perfusion Behavior Type (PBT) in the optimization procedure described in Figure 1, 2, and 3.

Tissue	ρ	c	κ
	[kg/m^3]	[Ws/kg/°C]	[W/m/°C]
Bladder	1040	3300	0.43
Rectum	1040	3600	0.53
Tumor	1040	3600	0.5
Fat	916	2500	0.25
Muscle	1040	3600	0.5
Bone	1955	1300	0.5

ρ: mass density; c : specific heat capacity; κ: thermal conductivity. The values for the homogeneous (H) PBT were: ρ = 1040 kg/m³, c = 3600 Ws/kg/°C, κ = 0.5 W/m/°C, applied for all regions.

Table 3. Optimization results in terms of minimum root-mean-square-deviation (RMSD) values, as defined in Equation (7)(7) $$\begin{array}{c}\text{for }i\_\mathit{case}=\mathrm{ }1\dots n\_\mathit{cases}:\hfill \\ \mathit{min}\left\{\mathit{RMSD}(i_{\mathit{case}})\right\}=\mathit{min}\left\{\sqrt{\frac{1}{N}\mathrm{ }\sum _{i=1}^N{(T_i-{\theta }_i)}^2\mathrm{ }}\right\}\begin{array}{c}{n}_{\mathit{perf}}\_\mathit{TYPE}\\ i\_\mathit{perf}=1\end{array}\hfill \end{array}$$(7) , for the Higher-Power-Therapy (HP_THER) for both Optimization Steps (Coarse, Fine).

Power (W)	N (number of squared differences)	Perfusion model		RMSD (°C)
		CE_POS	SH_POS	CE_POS	SH_POS
Homogeneous (H) Perfusion Behavior Type (H_PBT)
100	14	H-4	H-4	1.3250	1.6288
105	14	H-4	H-4	1.3063	1.6158
110	14	H-4	H-4	1.3280	1.6373
115	14	H-4	H-4	1.3959	1.703
120	14	H-4	H-4	1.4919	1.7978
All-5	70	H-4	H-4	1.3711	1.6779
Non-Dynamic (ND) Perfusion Behavior Type (ND_PBT)
100	14	ND-938	ND-937	0.6448	0.6071
105	14	ND-992	ND-938	0.5271	0.5339
110	14	ND-992	ND-955	0.4803	0.5312
115	14	ND-1009	ND-955	0.4935	0.6106
120	14	ND-1011	ND-958	0.4751	0.5964
All-5	70	ND-992	ND-955	0.5882	0.6435
Dynamic (D) Perfusion Behavior Type (D_PBT)
100	14	D-521	D-401	0.4392	0.5237
105	14	D-497	D-898	0.4743	0.5196
110	14	D-884	D-906	0.4698	0.4957
115	14	D-1002	D-906	0.4416	0.4512
120	14	D-1001	D-906	0.3882	0.4943
All-5	70	D-994	D-906	0.5417	0.5445

The results of the Coarse Step (H_PBT) are displayed in the upper segment and those of the Fine Step are displayed in the middle segment (ND_PBT) and in the bottom segment (D_PBT). Each segment involves results of 12 cases, i.e. 10 ‘suitable PTMs’ (‘SINGLE-PTM-RMSD-Minimization’) and 2 ALL5 results (‘ALL5-RMSDMinimization’). For every case, using a short notation (‘H-4’, ND-1011, etc.) a respective number of each ‘suitable perfusion model’ is also reported. For each PBT, the optimum PTMs and the optimum ALL5 results are marked in bold. These values represent also the optimum values for both therapies because the values in Table 3 are generally better than the respective ones in Table 4. Region perfusion values for every perfusion model reported in this table can be found in Table 5.

Table 4. Optimization results only for the ‘Lower-Power’-Therapy (LP_THER) in terms of minimum RMSD values, displayed in an analogous way to Table 3.

Power (W)	N (number of squared differences)	Perfusion Model		RMSD (°C)
		CE_POS	SH_POS	CE_POS	SH_POS
Homogeneous (H) Perfusion Behavior Type (H_PBT)
93	14	H-4	H-4	1.4127	1.7126
98	14	H-4	H-4	1.4360	1.7407
103	14	H-4	H-4	1.5031	1.7964
108	14	H-4	H-4	1.6076	1.8794
113	14	H-5	H-5	1.6127	1.8917
All 5	70	H-4	H-4	1.5415	1.8309
Non-Dynamic (ND) Perfusion Behavior Type (ND_PBT)
93	14	ND-955	ND-955	0.7358	0.7592
98	14	ND-1009	ND-955	0.6612	0.7186
103	14	ND-1010	ND-956	0.6644	0.7521
108	14	ND-943	ND-889	0.6308	0.7892
113	14	ND-944	ND-1228	0.7382	0.7819
All 5	70	ND-1009	ND-956	0.7419	0.8260
Dynamic (D) Perfusion Behavior Type (D_PBT)
93	14	D-906	D-905	0.6274	0.6671
98	14	D-906	D-817	0.6353	0.6574
103	14	D-906	D-825	0.6135	0.6719
108	14	D-913	D-825	0.6100	0.6788
113	14	D-913	D-913	0.6351	0.6814
All 5	70	D-913	D-825	0.6358	0.7035

All RMSD values are worse than respective values in Table 3, i.e. HP_THER was better approximated by the optimization procedure than LP_THER. For each PBT, the optimum PTMs and the optimum ALL5 results are marked in bold. Region perfusion values for every perfusion model reported in this table can be found in Table 5.

Table 5. Region perfusion values of all perfusion models reported in Tables 3 and 4.

Perfusion model	Tumor	Rectum	Bladder	Muscle	Fat	Bone	Incidence index ∑N
H-4	10	10	10	10	10	10	532
H-5	15	15	15	15	15	15	28
ND-889	15	5	5	20	2.5	2.5	14
ND-937	15	10	5	10	2.5	2.5	14
ND-938	15	10	5	10	2.5	5	28
ND-943	15	10	5	20	2.5	2.5	14
ND-944	15	10	5	20	2.5	5	14
ND-955	15	10	10	10	2.5	2.5	140
ND-956	15	10	10	10	2.5	5	84
ND-958	15	10	10	10	5	5	14
ND-992	15	15	5	10	2.5	5	98
ND-1009	15	15	10	10	2.5	2.5	98
ND-1010	15	15	10	10	2.5	5	14
ND-1011	15	15	10	10	5	2.5	14
ND-1228	20	10	10	10	5	5	14
D-401	5→10	2.5→5	1.25 →2.5	10→30	1→3	1.25→2.5	14
D-497	5→10	5→10	1.25 →2.5	10→30	1→3	1.25→2.5	14
D-521	5→10	5→10	2.5 →5	5→15	1→3	1.25→2.5	14
D-817	7.5 →15	2.5→5	2.5 →5	10→30	1→3	1.25→2.5	14
D-825	7.5 →15	2.5→5	2.5 →5	13.33→40	1→3	1.25→2.5	98
D-884	7.5 →15	5 →10	1.25→2.5	10→30	2→6	2.5→5	14
D-898	7.5 →15	5 →10	2.5 →5	3.33 → 10	1→3	2.5→5	14
D-905	7.5 →15	5 →10	2.5 →5	5→15	1→3	1.25→2.5	14
D-906	7.5 →15	5 →10	2.5 →5	5→15	1 →3	2.5→5	154
D-908	7.5 →15	5 →10	2.5 →5	5→15	2 →6	2.5→5	140
D-913	7.5→15	5→10	2.5→5	10→30	1→3	1.25→2.5	112
D-994	7.5→15	7.5→15	2.5→5	3.33→10	1→3	2.5→5	70
D-1001	7.5→15	7.5→15	2.5→5	5→15	1→3	1.25→2.5	14
D-1002	7.5 →15	7.5→15	2.5→5	5→15	1→3	2.5→5	14

In the last row, results of ‘Approach 3’ are reported by displaying the corresponding ‘Incidence Index’, ΣN, defined in Equation (9)(9) $$n\_\mathit{cases}\mathrm{ }=\mathrm{ }n\_\mathit{suitable}\_\mathit{ptm}\mathrm{ }+n\_\mathit{therapies}•n\_\mathit{SAR}$$(9) . The H-type, ND-type, and D-type perfusion models with the highest ΣN values, respectively, are marked in bold.

Figure 5. Comparison of measured [for the Higher-Power-Therapy (HP_THER), black dashed curve with cross symbols] and calculated ‘Temperature Sets’ of the pelvic region along the tumor, rectal, and bladder catheters in 14 points: 5 in tumor catheter (0 cm corresponds to the caudal-medial catheter tip), 8 in rectal catheter (0 cm corresponds to anus), and one in bladder catheter, and Calculated results refer to the optimum Parametrized Treatment Models (PTMs) for each ‘Perfusion Behavior Type’ (PBT), respectively. These PTMs are indicated as follows: •‘H-4-centr-105 W’ (red, circle symbols) [homogeneous perfusion model H-4, central patient positioning (CE_POS), absorbed power in the patient model P = 105W]. •‘ND-1011-centr-120W’ (blue, square symbols) (non-dynamic perfusion model ND-1011, CE_POS, P = 120W), and, • ‘D-1001-centr-120W’ (purple, triangle symbols) (dynamic perfusion model D-1001, CE_POS, P = 120W). Region perfusion values are displayed in Table 5.

Figure 6. Comparison of measured [for the Lower-Power-Therapy (LP_THER), black dashed curve with cross symbols] and calculated ‘Temperature Sets’ in points explained in the legend of Figure 6. Calculated results refer to those PTMs, which exhibited the minimum RMSD in Table 4, i.e. those which are the best for LP_THER only, for each perfusion behavior type (PBT), respectively. These PTMs are indicated as follows: •‘H-4-centr-93 W’ (red, circle symbols) (homogeneous PBT, CE_POS, p = 93 W). •‘ND-943-centr-108’ W (blue, square symbols) (non-dynamic PBT, CE_POS, p = 108 W), and •‘D-913-centr-108W’ (purple, triangle symbols) (dynamic PBT, CE_POS, p = 108 W). Region perfusion values are displayed in Table 5.