Inverse problem in the hyperthermia therapy of cancer with laser heating and plasmonic nanoparticles

Figures & data

Figure 1. Sketches of the cases examined in this work: (a) soft tissue phantom; (b) skin model.

Table 1. Auxiliary sampling importance resampling (ASIR) algorithm.[28,29]

Step 1

For i = 1, …, N draw new particles ${\mathbf{x}}_k^i$ from the prior density π (xk|${\mathbf{x}}_{k-1}^i$, θ) by solving the forward problem for ${\mathbf{x}}_{k-1}^i$ and θ. Then, calculate some characterization ${\mathbf{\mu }}_k^i$ of xk, given ${\mathbf{x}}_{k-1}^i$ and use the likelihood density to calculate the corresponding weights $w_k^i$ = π(${\mathbf{z}}_k^{\text{meas}}$|${\mathbf{\mu }}_k^i$,θ)$w_{k-1}^i$.

Step 2

Calculate the total weight t = Σi$w_k^i$ and then normalize the particle weights, that is, for i = 1, …, N let $w_k^i$ = $t^{-1}{w}_k^i$.

Step 3

Resample the particles as follows:

Construct the cumulative sum of weights (CSW) by computing ci = ci−1 + $w_k^i$ for i = 1, …, N, with c0 = 0

Let i = 1 and draw a starting point u1 from the uniform distribution U[0,N^−1]

For j = 1, …, N

Move along the CSW by making uj = u1 + N^−1(j − 1)

While uj  > ci make i = i + 1

Assign samples ${\mathbf{x}}_{k-1}^j={\mathbf{x}}_{k-1}^i$

Assign sample weights $w_k^j$ = N^−1

Assign parent i^j = i

Step 4

For j =1, …, N draw particles x^jk from the prior density π (xk|${\mathbf{x}}_{k-1}^{ij}$, θ), by solving the forward problem using the parent i^j, and then use the likelihood density to calculate the correspondent weights $w_k^j$ = π${\mathbf{z}}_k^{\text{meas}}$|(${\mathbf{x}}_k^j$, θ)/π (${\mathbf{z}}_k^{\text{meas}}$|${\mathbf{\mu }}_k^{ij}$, θ).

Step 5

Calculate the total weight t = Σj $w_k^j$ and then normalize the particle weights; that is, for j = 1, …, N let $w_k^j$ = $t^{-1}{w}_k^j$ .

Table 2. Thermophysical and optical properties of the PVCP phantom.[32,44,45]

Density ρ (kg/m^3)	Specific heat cp (kJ/kgK)	Thermal conductivity k (W/mK)	Absorption coefficient κ (m^−1)	Scattering coefficient σs(m^−1)	Scattering anisotropy factor g
995.1	1.79	0.15	2	12,000	0.9

Table 3. Optical properties of region loaded with gold nanorods.

Concentration of nanoparticles (m^−3)	8 × 10^15
κ (m^−1)	179.02
σs (m^−1)	12,013.83

Figure 2. Exact and estimated temperature distribution at selected times with N = 100 particles and with N = 250 particles.

Figure 3. Estimated and exact temperature distribution at t = 180 s: (Left) along the radius for a line at z = 6 mm of the inclusion with N = 100 particles (a) and N = 250 particles (b); (Right) along the centerline with N = 100 particles (c) and N = 250 particles (d).

Figure 4. Comparison of the estimated and exact transient temperature variations with the temperature measurements at the sensor position (r = 2 mm, z = 6 mm): (a) N = 100 particles; (b) N = 250 particles.

Figure 5. Estimated and exact transient temperature variations: (Left) at (r = 15 mm, z = 6 mm) with N = 100 particles (a) and N = 250 particles (b); (Right) at (r = 0 mm, z = 10 mm) with N = 100 particles (c) and N = 250 particles (d).

Figure 6. (a) Exact fluence rate distribution; (b) Fluence rate distribution estimated with N = 250 particles.

Table 4. Thermophysical and optical properties of tissues.[33,34,47]

Tissue	Epidermis	Tumour	Dermis	Fat	Muscle
Thickness, (mm)	0.1	0.75	1.5	2	8
ρ (kg/m^3)	1200	1030	1200	1000	1085
cp (J/kg K)	3589	3852	3300	2674	3800
k (W/m K)	0.235	0.558	0.445	0.185	0.51
Qmet (W/m^3)	0	3680	368.1	368.3	684.2
ωb (s^−1)	0	63 × 10^−4	2 × 10^−4	10^−4	27 × 10^−4
κ (m^−1)	35	122	122	108	54
σs (m^−1)	21,270	22,500	22,500	20,200	6670

Figure 7. Initial temperature distribution.

Table 5. Optical properties of the tumour-containing gold nanorods.

Concentration of nanoparticles (m^−3)	3 × 10^15
κ (m^−1)	188.38
σs (m^−1)	22,505.19

Figure 8. Exact and estimated temperature distribution at selected times with N = 100 particles and N = 250 particles.

Figure 9. Estimated and exact temperature distribution at t = 20 s: (Left) along the radius for a line at z = 0.7 mm with N = 100 particles (a) and N = 250 particles (b); (Right) along the centerline with N = 100 particles (c) and N = 250 particles (d).

Figure 10. Comparison of the estimated and exact transient temperature variations with the temperature measurements at the sensor position (r = 0.5 mm, z = 0.7 mm): N = 100 particles (a); N = 250 particles (b).

Figure 11. Estimated and exact transient temperature variations: (Left) at (r = 6 mm, z = 0.7 mm) with N = 100 particles (a) and N = 250 particles (b); (Right) at (r = 0 mm, z = 3 mm) with N = 100 particles (c) and N = 250 particles (d).

Figure 12. (a) Exact fluence rate distribution; (b) Fluence rate distribution estimated with N = 250 particles.

Figure 13. (a) Exact thermal damage distribution; (b) Estimated thermal damage distribution with N = 250 particles.

Figure 14. Comparison of the transient variation of the thermal damage calculated with the exact and estimated temperatures at the position (r = 0.5 mm, z = 0.7 mm).