Differential Heating of Metal Nanostructures at Radio Frequencies

Nanoparticles with a strong absorption of incident radio frequency (rf) or microwave irradiation are desirable for remote hyperthermia treatments. While controversy has surrounded the absorption properties of spherical metallic nanoparticles, other geometries, such as prolate and oblate spheroids, have not received sufficient attention for application in hyperthermia therapies. Here, we use the electrostatic approximation to calculate the relative-absorption ratio of metallic nanoparticles in various biological tissues. We consider a broad parameter space, sweeping across frequencies from 1 MHz to 10 GHz, while also tuning the nanoparticle dimensions from spheres to high-aspect-ratio spheroids approximating nanowires and nanodisks. We find that, while spherical metallic nanoparticles do not offer differential heating in tissue, large absorption cross sections can be obtained from long prolate spheroids, while thin oblate spheroids offer minor potential for absorption. Our results suggest that metallic nanowires should be considered for rf- and microwave-based wireless hyperthermia treatments in many tissues going forward.

Figure 1

(a) Schematic of differential heating induced by rf or microwave application to cells targeted by metal nanostructures. (b) Orientation of rf or microwave interactions with nanomaterials. Relative-absorption ratio is averaged over the three propagating waves shown (with equal magnitudes) to account for a collection of randomly oriented spheroids. (c) Negative real component of the dielectric function for metals considered herein. Dashed curves include scattering rate corrections for spherical nanoparticles with 10 nm radius (NP corresponds to nanoparticle). (d) Imaginary component of the dielectric function for metals considered herein. Dashed curves include scattering rate corrections for spherical nanoparticles with 10 nm radius. (e) Real component of the dielectric function for tissues considered herein. Data from Ref. [7]. (f) Imaginary component of the dielectric function for tissues considered herein. Data from Ref. [7].

Figure 2

(a) Labeling the tunable variables in relative-absorption ratio calculations for spherical nanoparticles. (b)–(i) Relative-absorption ratio, $F_{\mathrm{abs}}$, for metallic nanospheres in tissue, as a function of radius (R) and incident electromagnetic wave frequency (f). (b) Pancreas and $\mathrm{Au}$. (c) Pancreas and $\mathrm{Ag}$. (d) Pancreas and $\mathrm{Pt}$. (e) Pancreas and $\mathrm{Cu}$. (f) Cervix and $\mathrm{Pt}$. (g) Breast and $\mathrm{Pt}$. (h) Brain and $\mathrm{Pt}$. (i) Breast and $\mathrm{Au}$. Semiaxes (i.e., radii in the case of spheres) that violate the electrostatic approximation are excluded from all plots; thus, these results represent lower bounds of differential heating.

Figure 3

(a) Labeling the tunable variables in relative-absorption-ratio calculations for prolate spheroids. (b)–(h) Relative-absorption ratio, $F_{\mathrm{abs}}$, for metallic prolate spheroids in tissue, as a function of spheroid diameter (D), spheroid length (L), and incident electromagnetic wave frequency (f). (b) Pancreas and $\mathrm{Au}$, D = 10 nm. (c) Pancreas and $\mathrm{Au}$, D = 100 nm. (d) Pancreas and $\mathrm{Pt}$, D = 10 nm. (e) Pancreas and $\mathrm{Pt}$, D = 100 nm. (f) Pancreas and $\mathrm{Cu}$, D = 10 nm. (g) Pancreas and $\mathrm{Cu}$, D = 100 nm. (h) Pancreas and $\mathrm{Ag}$, D = 10 nm. (i)–(p) Relative-absorption ratio, $F_{\mathrm{abs}}$, for metallic prolate spheroids in tissue at $f=1\mathrm{GHz}$, as a function of spheroid diameter (D) and spheroid length (L). (i) Breast and $\mathrm{Au}$. (j) Breast and $\mathrm{Ag}$. (k) Breast and $\mathrm{Pt}$. (l) Breast and $\mathrm{Cu}$. (m) Brain and $\mathrm{Au}$. (n) Brain and $\mathrm{Ag}$. (o) Brain and $\mathrm{Pt}$. (p) Brain and $\mathrm{Cu}$. Semiaxes that violate the electrostatic approximation are excluded from all plots; thus, these results represent lower bounds of differential heating.

Figure 4

(a) Labeling the tunable variables in relative-absorption-ratio calculations for oblate spheroids. (b)–(h) Relative-absorption ratio, $F_{\mathrm{abs}}$, for metallic oblate spheroids in tissue, as a function of spheroid diameter (D), spheroid thickness (T), and incident electromagnetic wave frequency (f). (b) Pancreas and $\mathrm{Au}$, T = 10 nm. (c) Pancreas and $\mathrm{Au}$, T = 100 nm. (d) Pancreas and $\mathrm{Pt}$, T = 10 nm. (e) Pancreas and $\mathrm{Pt}$, T = 100 nm. (f) Pancreas and $\mathrm{Cu}$, T = 10 nm. (g) Pancreas and $\mathrm{Cu}$, T = 100 nm. (h) Pancreas and $\mathrm{Ag}$, T = 10 nm. (i)–(p) Relative-absorption ratio, $F_{\mathrm{abs}}$, for metallic oblate spheroids in tissue at $f=1\mathrm{GHz}$, as a function of spheroid diameter (D) and spheroid thickness (T). (i) Breast and $\mathrm{Au}$. (j) Breast and $\mathrm{Ag}$. (k) Breast and $\mathrm{Pt}$. (l) Breast and $\mathrm{Cu}$. (m) Brain and $\mathrm{Au}$. (n) Brain and $\mathrm{Ag}$. (o) Brain and $\mathrm{Pt}$. (p) Brain and $\mathrm{Cu}$. Semiaxes that violate the electrostatic approximation are excluded from all plots; thus, these results represent lower bounds of differential heating.

Figure 5

Relative-absorption ratio for spherical nanoparticles (a)–(d), prolate spheroids (e)–(h), and oblate spheroids (i)–(l) in breast and brain cancerous tissues at 1 GHz. (a), (e), and (i) Brain cancer and $\mathrm{Au}$. (b),(f),(j) Breast cancer and $\mathrm{Au}$. (c),(g),(k) Brain cancer and $\mathrm{Pt}$. (d),(h),(l) Breast cancer and $\mathrm{Pt}$. Dielectric properties of cancerous tissues are taken from Ref. [45] for breast cancer and Ref. [44] for brain cancer. Dashed lines in (a)–(d) represent the boundary of the electrostatic approximation.

Figure 6

Lowest (minimum) case of differential heating for spherical gold particles in the pancreas (a), prolate gold spheroids in breast tissue (b), and oblate gold spheroids in breast tissue (c), as obtained by calculations with seven pairs of ${\omega }_p$ and ${\gamma }_{\mathrm{bulk}}$ reported in Table 1 of Ref. [63]. Highest (maximum) case of differential heating for spherical gold particles in the pancreas (d), prolate gold spheroids in breast tissue (f), and oblate gold spheroids in breast tissue (f), as obtained by calculations with seven pairs of ${\omega }_p$ and ${\gamma }_{\mathrm{bulk}}$ reported in Table 1 of Ref. [63]. Semiaxes that violate the electrostatic approximation are excluded from all plots.