On the optimal choice of the exposure conditions and the nanoparticle features in magnetic nanoparticle hyperthermia

Figures & data

Figure 1. (a) Geometry assumed for the tumour (grey region) and for the irradiated surrounding tissue (white region). The dashed circle, with radius R₂, represents the transition region surrounding the tumour; (b) a sketch of the actual profile expected for the temperature rise after MNPH treatment.

Table I.  Numerical results obtained from the proposed criterion.

wb (kg m^−3 s^−1)	r (cm)	R1 (cm)	R2 (cm)	R(L) (cm)	pm (kW m^−3)	pe (kW m^−3)	Hf (Am^−1s^−1)	Tmax (°C)	Case
0.5	0	1	2	15 (10)	131	3.12	3.14 × 10^8	45.36	1
	0	1	2	10 (7)	131	1.87	3.67 × 10^8	45.32	2
	0	0.75	1.5	7.5 (5)	213	0.36	2.13 × 10^8	45.16	3
	0	0.75	1.5	5 (5)	213	1.02	5.40 × 10^8	45.15	4
1	0	1	2	15 (10)	175	7.40	4.86 × 10^8	46.14	5
	0	1	2	10 (7)	170	7.44	7.28 × 10^8	46	6
	0	0.75	1.5	7.5 (5)	242	9.02	1.07 × 10^9	45.42	7
	0	0.75	1.5	5 (5)	242	6.43	1.35 × 10^9	45.41	8
2	0	1	2	15 (10)	238	11.18	5.95 × 10^8	46.96	9
	0	1	2	10 (7)	235	13.17	9.69 × 10^8	46.82	10
	0	0.75	1.5	7.5 (5)	324	15.31	1.39 × 10^9	46.19	11
	0	0.75	1.5	5 (5)	304	19.62	2.37 × 10^9	45.92	12
4	0	1	2	15 (10)	344	20.83	8.12 × 10^8	47.85	13
	0	1	2	10 (7)	342	23.60	1.30 × 10^9	47.71	14
	0	0.75	1.5	7.5 (5)	447	26.47	1.83 × 10^9	47.02	15
	0	0.75	1.5	5 (5)	433	32.54	3.05 × 10^9	46.85	16
1	0.5	1	2	5 (5)	158	6.82	1.39 × 10^9	45.89	17
	1	1	2	5 (5)	157	6.88	1.40 × 10^9	45.99	18
	1.5	1	2	5 (5)	154	7.31	1.44 × 10^9	45.92	19

Figure 2. Temperature distribution produced over the irradiated tissues for cases 3 (A and D), 16 (B and E) and 19 (C and F) reported in Table I. (A–C) Distribution in the plane x–z; (D–E) radial profile in the plane z = 0 for cases 3 and 16, respectively; (F) distribution in the plane z = 0 for case 19.

Table II.  Optimal values for H, f, d obtained from the value of p_m, p_e reported in Table I (i.e. cases 3, 7, 16 and 19). For MF amplitudes above 20 kA m⁻¹ Equation 11 is assumed no longer valid.

Case	Hmax (kA m^−1)	Ho (kA m^−1)	fo (kHz)	do (Eq. 11)* (nm)	do (Eq. 13) (nm)	cmin (Eq. 11)* (mg mL^−1)	cmin (Eq. 13) (mg mL^−1)	σd
3	10	10	21.3	20.5 (22–17)	9.5	16.9 (13.8–30.1)	28.6	0.15
7	10	10	106.9	17.5 (19.3–14.5)	9.5	6.5 (4.5–11.3)	6.5	0.15
7	10	10	106.9	17 (19.2–14)	10.5	3.3 (2.3–6)	4.4	0.05
7	10	10	106.9	18.5 (20–16.3)	9.5	10.6 (7.7–16.4)	10	0.3
7	15	15	71.3	18 (19.8–15)	11	4 (2.3–6.6)	6.2	0.15
7	20	20	53.5	18 (20.2–15.6)	12	2.8 (2–4.34)	5.3	0.15
7	35	35	30.6	–	28	–	4	0.15
7	40	40	26.7	–	21	–	3.9	0.15
7	45	40	26.7	–	21	–	3.9	0.15
16	15	15	203.1	16.5 (18.7–13.8)	11	2.9 (2–5.3)	3.9	0.15
19	15	15	96.2	17.5 (19.4–14.6)	11	2 (1.4–3.4)	2.9	0.15

*The values in brackets are estimated by assuming k_a = 10–30 kJ/m³.

Figure 3. Contour plot of the behaviour of c_min versus the radius of the tumour, R₁, and the radius of the surrounding irradiated tissue, R, for different value of the blood perfusion rate: (A) w_b = 0.5 kg m⁻³ s⁻¹; (B) w_b = 1 kg m⁻³ s⁻¹; (C) w_b = 2 kg m⁻³ s⁻¹; (D) w_b = 4 kg m⁻³ s⁻¹.

Figure 4. Behaviour of p_m/ac (a = 4µ₀M_RHf) versus μ_d estimated by using Equation 13 as expression of p_m and in the case of polydisperse MNPs. A lognormal distribution, with σ_d = 0.2, has been assumed. Each curve is relative to a different value of H. The inset reports the maximum values assumed by each curve in figure, versus H and the corresponding values of μ_d.