Numerical assessment of a criterion for the optimal choice of the operative conditions in magnetic nanoparticle hyperthermia on a realistic model of the human head

Figures & data

Figure 1. Coercivity versus MNP size for an ensemble of magnetite nanoparticles.

Table 1. Thermal and electric properties of some human tissues in the band 50–200 kHz.

				ɛr			σ (S/m)
Tissue	k (W/m/°C)	cbwb (W/m^3/°C)	qm (W/m^3)	50 kHz	100 kHz	200 kHz	50 kHz	100 kHz	200 kHz
Skull	0.32	3400	610	264	228	204	0.02	0.021	0.021
CST	0.57	0	0	109	109	109	2	2	2
Grey matter	0.51	40000	7100	5461	3222	2010	0.13	0.13	0.14
Fat	0.20	1700	300	172	93	56	0.024	0.024	0.025
Muscle	0.49	2700	480	10094	8089	6378	0.35	0.36	0.38
Skin (dry)	0.37	9100	1620	1127	1119	1105	3 × 10^−4	4 × 10^−4	1 × 10^−3
White matter	0.48	40000	7100	3548	2108	1287	0.078	0.082	0.087

Figure 2. A view of the model adopted in the simulation in CST software environment. (A) A cut of the numerical phantom, (B) exposure system and numerical phantom.

Table 2. Tumour and transition region positions and sizes considered in the numerical analysis.

	Tumour position
Case	x (cm)	y (cm)	z (cm)	Tumour size (mm)	Transition regionsize (mm)
1	−1.4	−3	−84	20	40
				15	30
2	2.1	−5	−82	20	40
				15	30
3	−1.4	−3	−78.5	20	40
				15	30

Figure 3. Relative error in dB between the normalised (to Hf) electric power distributions dissipated in the numerical phantom at f₁ = 50 kHz and f₂ = 200 kHz. (A) Cut in the xy plane, (B) cut in the zx plane; (C) cut in the yz plane.

Table 3. Optimal values for p_h, p_e and Hf: MNPs uniformly distributed in the tumour.

		Text = 25 °C			Text = 30 °C
Case	Tumour size (mm)	phopt (kW/m^3)	peopt (kW/m^3)	Hf (A/m/s)	phopt (kW/m^3)	peopt (kW/m^3)	Hf (A/m/s)
1	20	569	1.84	1.77 × 10^9	572	1.71	1.71 × 10^9
	15	687	1.81	1.77 × 10^9	691	1.69	1.71 × 10^9
2	20	593	1.45	1.63 × 10^9	594	1.31	1.55 × 10^9
	15	687	1.45	1.62 × 10^9	689	1.31	1.54 × 10^9
3	20	611	1.46	1.13 × 10^9	611	1.36	1.10 × 10^9
	15	739	1.52	1.15 × 10^9	733	1.43	1.11 × 10^9

Table 4. Optimal values for H, f, d and c_min: MNPs uniformly distributed in the tumour.

			Text=25 °C					Text=30 °C
Case	Tumoursize (mm)	Hopt (kA/m)	fopt (kHz)	dopt (nm)Eq. (1)(1) Eq. (2)(2)		cmin (mg/mL) Eq. (1)(1) Eq. (2)(2)		fopt (kHz)	dopt (nm)Eq. (1)(1) Eq. (2)(2)		cmin (mg/mL) Eq. (1)(1) Eq. (2)(2)
1	20	10	177	17	10	8.9	12.6	171	17	10	9.2	13.2
		15	118	17	100	7.5	5.6	114	17	100	7.8	5.8
		20	89	17	70	6.9	5.5	85	17	70	7.2	5.7
	15	10	177	17	10	10.7	15.2	171	17	10	11.2	15.9
		15	118	17	100	9	6.8	114	17	100	9.4	7.1
		20	89	17	70	8.4	6.7	85	17	70	8.7	6.9
2	20	10	163	17	10	10	14.3	155	17	10	10.6	15.1
		15	109	17	100	8.4	6.4	103	17	100	8.9	6.7
		20	81	18	70	7.8	6.3	77	18	70	8.2	6.6
	15	10	162	17	10	11.7	16.7	154	17	10	12.3	17.6
		15	108	17	100	9.8	7.4	102	17	100	10.4	7.8
		20	81	18	70	9.1	7.2	77	18	70	9.6	7.7
3	20	10	113	17	10	14.6	21.2	110	17	10	15.1	21.9
		15	76	18	100	12.2	9.4	73	18	100	12.6	9.7
		20	57	18	70	11	9.2	55	18	70	11.4	9.6
	15	10	115	17	10	17.5	25.4	111	17	10	17.8	25.9
		15	76	18	100	14.7	11.3	74	18	100	14.9	11.5
		20	57	18	70	13.3	11.1	56	18	70	13.5	11.3

Figure 5. Normalised radial profile of the MNP distribution in the tumour.

Table 5. Optimal vales for H, f, d and c_min: MNPs non-uniformly distributed in the tumour (T_ext = 25 °C).

						dopt (nm)		cmin (mg/mL)
Case	Tumour size (mm)	phopt (kW/m^3)	Hf (A/m/s)	Hopt (kA/m)	fopt (kHz)	Eq. 1	Eq. 2	Eq. 1	Eq. 2	Tmax (°C)
1	15	713	1.75 × 10^9	10	175	17	10	11.2	16	48.6
				15	117	17	100	9.4	7.2	48.6
				20	87	18	70	8.6	7	48.6

Figure 6. Temperature distribution induced over the irradiated tissues (MNPs non-uniformly distributed in the tumour), for case 1 and T_ext = 25 °C: (A) cut in the xy plane, (B) cut in the yz plane, (C) cut in the zx plane. The insets show a zoom of the tumour (smallest black circle) and the transition region (largest black circle).

Table 6. Optimal values for H, f, d and c_min by using the experimental SAR in Malik et al. [21] (T_ext = 25 °C).

Case	Tumour size (mm)	Phopt (W/cm^3)	Hf (A/m/s)	H (kA/m)	f (kHz)	dopt (nm)	SAR (W/g)	cmin (mg/mL)	Tmax (°C)
1	15	0.713	1.75 × 10^9	10.5	166	15	62.5	11.4	48.6
				16	110	15	84	8.5	48.6

Figure 7. Required MNPs concentration (A) and MNPs total mass (B) for successfully heating tumors of different sizes, for each of the three positions in Table 2.

Malik V, Goodwill J, Mallapragada S, Prozorov T, Prozorov R. Comparative study of magnetic properties of nanoparticles by high-frequency heat dissipation and conventional magnetometry. IEEE Magn Lett 2014;5:1–4.

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