Abstract
Plasmonic metal nanoparticles are efficient absorbers of optical energy and thus can serve as a heat source for photothermal therapy. In this study, time-dependent solutions to the heat-flow equation are derived for the temperature elevation of tissue arising from the absorption of light by a suspension of plasmonic nanoparticles. Analytical solutions for the temperature are obtained assuming the diffusion approximation for light transport. Two types of heat sources are considered: heat production arising from a point source of light and from planewave illumination. The results will provide a theoretical basis for the design and optimization of experimental systems for photothermal treatment and photo immunotherapy.
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This work was supported by the National Institutes of Health (1R01EB028078-01A1)
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Appendices
Appendix 1: Derivation of the Green’s Functions
For a spherically-symmetric source, the Green’s function, g(r, t|r′, t′), for the diffusion equation is defined as the solution to
To find g(r, t|r′, t′), we employ the completeness relation for the spherical Bessel function j 0(rρ):
We now let [52]
and substitute into (37). Noting the relation,
and using (38) for \(\delta (r\!-\!r')/r^2\) on the right of (37), we obtain the following differential equation for C ρ(t, t′):
The solution to this equation is
where u(t) is the unit step function. Substituting into (39), we obtain (10).
For a planewave source, the Green’s function, g(z, t|z′, t′), for the diffusion equation is defined as the solution to
Using the same method as above, the free-space green’s function is found to be
To enforce boundary conditions 1 and 2 we employ linear combinations of the free-space Green’s functions as follows. The Green’s functions for boundary conditions 1 abd 2 are, respectively, \(g_1(z,t|z',t') = g(z,t|z',t') - g(z,t|-\!z',t')\) and \(g_2(z,t|z',t') = g(z,t|z',t') + g(z,t|-\!z',t')\), which result in (22) and (23).
Appendix 2: Temperature Elevation for Boundary Condition 3
We can derive the temperature elevation for boundary condition 3 using the same approach as above once we have the Green’s function g 3(z, t|z′, t′) that enforces this boundary condition. One can show that
where g 2(z, t|z′, t′) is given by (23) and
The most straightforward method of deriving this result is to use Laplace transforms; the general procedure is outlined in [53]. The Green’s function g 3 reduces to g 2 when h = 0. One can also show that g 3 → g 1 in the limit h →∞.
The temperature elevation T 3(z, t) is now obtained by substituting g 3(z, t|z′, t′) into (24) and integrating with respect to z′ and t′. This results in
Here T 2(z, t) is given by (28) and
where
When h = 0, we have T 3(z, t) = T 2(z, t) and when h →∞, T 3(z, t) → T 1(z, t).
The steady-state temperature (t →∞) may be written
where \(T_2^{(ss)}(z)\) is given by (34) and
This can be readily integrated after expanding the integrand in partial fractions. After some algebra, we obtain
\(T_3^{(ss)}(z)\) will have a peak temperature at some distance z max > 0 that can be found by differentiating \(T_3^{(ss)}(z)\) with respect to z and setting the derivative to zero. This results in
This reduces to (35) when h →∞ and z max = 0 when h = 0. Note that no finite z max exists in the absence of perfusion.
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Norton, S.J., Vo-Dinh, T. (2021). Nanoparticle-Mediated Heating: A Theoretical Study for Photothermal Treatment and Photo Immunotherapy. In: Vo-Dinh, T. (eds) Nanoparticle-Mediated Immunotherapy. Bioanalysis, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-030-78338-9_5
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