A neural controller for online laser power adjustment during the heat therapy process in the presence of nanoparticles

Abstract

The present research evaluated the efficiency of a control approach to control the temperature of a breast tumor mass in the presence of nanoparticles exposed to laser radiation. However, if the radiation is carried out in open loop manner it may result in excessive temperature rise healthy cells that exist in the vicinity of tumor’s cells. This may lead to the death of healthy cells. So, using closed loop control methods is necessary to guarantee the preservation of healthy cells during the period of radiation. Therefore, in this study, an artificial neural network was trained as a controller. In other words, the trained neural network adjusted the laser power over a period of time in such a way that the temperature in the center of the tumor reached the desired level with an appropriate temporal behavior. The difference between the real temperature of the tumor and the desired temperature of it is the controller input, while the controller output determined the amount of laser power. The simulation studies were carried out using an appropriate physiological model in the presence of nanoparticles. First, Schrödinger equations were solved followed by the effective mass equation. Afterward the optimum number of nanoparticles to be used in the IR field was calculated. Next, the important electro-optical features related to the nanostructure, such as the absorption continuum and reflection continuum had been calculated. The neural network proposed controller was then evaluated through other simulation studies in the tumor mass model. The results showed a promising performance by the trained artificial neural network in adjusting radiated laser power for the desired temperature increase in the center of a tumor mass.

Appendix

For Eigen-value FEM problem solving, the Gälerkin method was applied. The case of torsional vibrations of a uniform circular cross-section was also considered. The differential equations and boundary conditions required for determining shape modes and natural frequencies are shown in further equations. To apply the Gälerkin method:

$$\mathop \int \limits_0^L \left\{\left({r^2}.\frac{{{d^2}R\left( r \right)}}{{d{r^2}}} + 2r.\frac{{dR\left( r \right)}}{{dr}} + \left[ {\left( {E\left( {n,l} \right) - V\left( r \right)} \right).{r^2} + \left( {2.\frac{{r{e^2}}}{{\varepsilon \left( r \right)}}} \right) - l\left( {l + 1} \right)} \right]\right).R\left( r \right) \right\}\varphi dr = 0$$

(7)

In the above equation, L is the length of the FEM element. The continuity condition at all boundaries of heteronanocrystal layers must be satisfied based on the associated wave function solution. In this manner, we have:

$$\psi \left( {R = L} \right) = 0,\frac{{d\psi }}{{dr\left( {R = 0} \right)}} = 0 ,\frac{{d\psi }}{{dr\left( {R = L} \right)}} = 0$$

(8)

Considering the initial and boundary conditions, leads to modified forms of the Gälerkin approach are obtained:

$$- \mathop \int \limits_0^L {r^2}.\left( {\frac{{{{{{\text{d}}\varphi }}}}}{{{\text{d}}\left( {\text{r}} \right)}}} \right).\left( {\frac{{dR}}{{dr}}} \right)dr - \mathop \int \limits_0^L \left( {\frac{{2rd\varphi }}{{dr}}} \right) + {\mathop \int \limits_0^L ([E\left( {n,l} \right) - V\left( r \right).{r^2} + \left( {2.\frac{{r{e^2}}}{{\varepsilon \left( r \right)}}} \right) - l\left( {l + 1} \right)} ].R\left( r \right))\varphi dr = 0 $$

(9)

The above equations should be solved using the FEM method (L is the length of the FEM element). Therefore, in next step FEM approximation should be implement using a set of basic functions Ni:

$$\psi \,\, = \,\,\,\,\sum\limits_{i = 1}^{} {{\psi _i}{N_i}} = \,\,\,\overline \psi N{\text{ and }}\phi \,\, = \,\,\,\,\sum\limits_{i = 1}^{} {{\phi _i}{N_i}} = \,\,\overline \phi N$$

(10)

where the basis operator, N, in one dimension is given as:

$$N = [{{N_1},{N_2}} ]\to {N_1} = {{1 - \xi/2 }},\quad {N_2} = {{1 + \xi/2 }}$$

(11)

The coordinate transformation can be given as:

$$x = \sum\limits_{i = 1}{{x_i}{N_i}}$$

(12)

Moreover, we can write the derivative transformations:

$${{dx} \mathord{\left/ {\vphantom {{dx} {d\xi }}} \right. \kern-\nulldelimiterspace} {d\xi }} = {{{x_2} - {x_1}} \mathord{\left/ {\vphantom {{{x_2} - {x_1}} 2}} \right. \kern-\nulldelimiterspace} 2} \to {{{l_e}} \mathord{\left/ {\vphantom {{{l_e}} 2}} \right. \kern-\nulldelimiterspace} 2}$$

(13)

$${{d\psi } \mathord{\left/ {\vphantom {{d\psi } {d\xi }}} \right. \kern-\nulldelimiterspace} {d\xi }} = \,{{d\psi } \mathord{\left/ {\vphantom {{d\psi } {dx}}} \right. \kern-\nulldelimiterspace} {dx}}.{{dx} \mathord{\left/ {\vphantom {{dx} {d\xi }}} \right. \kern-\nulldelimiterspace} {d\xi }}\to{{d\psi } \mathord{\left/ {\vphantom {{d\psi } {d\xi }}} \right. \kern-\nulldelimiterspace} {d\xi }} = {2 \mathord{\left/ {\vphantom {2 {{l_e}}}} \right. \kern-\nulldelimiterspace} {{l_e}}}\,.\,{{d\xi } \mathord{\left/ {\vphantom {{d\xi } {dx}}} \right. \kern-\nulldelimiterspace} {dx}}$$

(14)

$${d \mathord{\left/ {\vphantom {d {dx}}} \right. \kern-\nulldelimiterspace} {dx}} = {2 \mathord{\left/ {\vphantom {2 {{l_e}}}} \right. \kern-\nulldelimiterspace} {{l_e}}}.\,{d \mathord{\left/ {\vphantom {d {d\xi }}} \right. \kern-\nulldelimiterspace} {d\xi }}$$

(15)

Using the matrix notation, the following can be written:

$${{d\psi } \mathord{\left/ {\vphantom {{d\psi } {d\xi }}} \right. \kern-\nulldelimiterspace} {d\xi }} = \frac{1}{2} \times \left[ { - 1\,\,\,1} \right] \times \left[ \begin{array}{l} {\psi _1}\\ {\psi _2} \end{array} \right]$$

(16)

$${d \mathord{\left/ {\vphantom {d {d\xi }}} \right. \kern-\nulldelimiterspace} {d\xi }} = \frac{1}{2} \times \left[ { - 1\,\,\,1} \right]$$

(17)

When combining Eqs. (15), (16) and (17):

$${d \mathord{\left/ {\vphantom {d {dx}}} \right. \kern-\nulldelimiterspace} {dx}} = \frac{1}{{{l_e}}} \times \left[ { - 1\,\,\,1} \right] = {B}$$

(18)

where B gives the FEM approximation of the function derivative.

$${{d\psi } \mathord{\left/ {\vphantom {{d\psi } {dx}}} \right. \kern-\nulldelimiterspace} {dx}} = \frac{1}{{{l_e}}} \times \left[ { - 1\,\,\,1} \right]\overline \psi = B\overline \psi$$

(19)

where $\overline \psi$ is the FE approximation vector of the continuous function $\psi$. Then using Eqs. (11) and (19) into Eq. (9) the below equations will be obtained:

$$\begin{gathered} - \int\limits_0^L {{r^2}.({{\overline \varphi }^T}{B^T}B\overline R )({{{l_e}} \mathord{\left/ {\vphantom {{{l_e}} 2}} \right. \kern-\nulldelimiterspace} 2})d\xi } - \int\limits_0^L {2r.({{\overline \varphi }^T}{B^T}N\overline R )({{{l_e}} \mathord{\left/ {\vphantom {{{l_e}} 2}} \right. \kern-\nulldelimiterspace} 2})d\xi } \hfill \\ + \int\limits_0^L {(\,\,({{2r.{m^*}.{e^2}} \mathord{\left/ {\vphantom {{2r.{m^*}.{e^2}} {{\hbar ^2}.{\varepsilon _r}}}} \right. \kern-\nulldelimiterspace} {{\hbar ^2}.{\varepsilon _r}}}) - V(r)\,\, - \,\,l(l + 1)\,\,).({{2.{m^*}} \mathord{\left/ {\vphantom {{2.{m^*}} {{\hbar ^2}}}} \right. \kern-\nulldelimiterspace} {{\hbar ^2}}}).(E{{\overline \varphi }^T}{N^T}N\overline R )({{{l_e}} \mathord{\left/ {\vphantom {{{l_e}} 2}} \right. \kern-\nulldelimiterspace} 2})d\xi = 0} \hfill \\ \end{gathered}$$

(20)

With the use of Eqs. (8), (9), and (18) for one element of the approximation mesh:

$$- \int\limits_{ - 1}^1 {({{\overline {\phi (r)} }^T})} \left\{ \begin{array}{l} \,\left[ {({{{l_e}} \mathord{\left/ {\vphantom {{{l_e}} 8}} \right. \kern-\nulldelimiterspace} 8})\,\left( {\begin{array}{*{20}{c}} {{\xi ^2}}&{ - {\xi ^2}}\\ { - {\xi ^2}}&{{\xi ^2}} \end{array}} \right) + ({{{l_e}} \mathord{\left/ {\vphantom {{{l_e}} 2}} \right. \kern-\nulldelimiterspace} 2})\,\left( {\begin{array}{*{20}{c}} { - {N_1}\xi }&{ - {N_2}\xi }\\ {{N_1}\xi }&{{N_2}\xi } \end{array}} \right) - ({{{l_e}^2.2{m^*} \times ({{{e^2}} \mathord{\left/ {\vphantom {{{e^2}} {{\hbar ^2}{\varepsilon _r}}}} \right. \kern-\nulldelimiterspace} {{\hbar ^2}{\varepsilon _r}}})} \mathord{\left/ {\vphantom {{{l_e}^2.2{m^*} \times ({{{e^2}} \mathord{\left/ {\vphantom {{{e^2}} {{\hbar ^2}{\varepsilon _r}}}} \right. \kern-\nulldelimiterspace} {{\hbar ^2}{\varepsilon _r}}})} 4}} \right. \kern-\nulldelimiterspace} 4})\,\left( {\begin{array}{*{20}{c}} {{N_1}^2\xi }&{{N_2}{N_1}\xi }\\ {{N_2}{N_1}\xi }&{{N_2}^2\xi } \end{array}} \right)} \right] \to \,\,\\ \\ + \left[ {\,( - l(l + 1)) \times {{({l_e}} \mathord{\left/ {\vphantom {{({l_e}} 2}} \right. \kern-\nulldelimiterspace} 2})\left( {\begin{array}{*{20}{c}} {{N_1}^2}&{{N_2}{N_1}}\\ {{N_2}{N_1}}&{{N_2}^2} \end{array}} \right) + ({{{l_e}^3.{m^*}} \mathord{\left/ {\vphantom {{{l_e}^3.{m^*}} {4{\hbar ^2}}}} \right. \kern-\nulldelimiterspace} {4{\hbar ^2}}})\,\left( {\begin{array}{*{20}{c}} {{\xi ^2}{N_1}^2V(\xi )}&{{\xi ^2}{N_2}{N_1}V(\xi )}\\ {{\xi ^2}{N_2}{N_1}V(\xi )}&{{\xi ^2}{N_2}^2V(\xi )} \end{array}} \right)\,} \right] \to - \left[ {({{{l_e}^3 \times {E_{nl}}.{m^*}} \mathord{\left/ {\vphantom {{{l_e}^3 \times {E_{nl}}.{m^*}} {4{\hbar ^2}}}} \right. \kern-\nulldelimiterspace} {4{\hbar ^2}}})\left( {\begin{array}{*{20}{c}} {{\xi ^2}{N_1}^2}&{{\xi ^2}{N_2}{N_1}}\\ {{\xi ^2}{N_2}{N_1}}&{{\xi ^2}{N_2}^2} \end{array}} \right)\,} \right] \end{array} \right\}\, \times ({2 \mathord{\left/ {\vphantom {2 {{l_e}\xi }}} \right. \kern-\nulldelimiterspace} {{l_e}\xi }})\,\overline {R(r)} d\xi = 0$$

(21)

Equation (21) is presented in the form of FEM for solving numerically:

$$\int\limits_{ - 1}^1 \left\{{\overline{\phi(r)}}^T\times \left(\sum{A_i}+B.{E_{nl}}\right) \times {\overline{R(r)}}\right\} d\xi = 0$$

(22)

In Eq. (22), “A_i” and “B_i” are the definition matrix.

The Schrodinger equation using FEM approximation can be obtained as below:

$$\sum\limits_i^{{N_{element}}} {\left\{ {\left[ A \right] - \lambda \left[ B \right]} \right\}} \overline \psi = 0$$

(23)

Using the boundary conditions of a defined potential, the nodal approximations at the edges can be found by considering: ${\left. {{\Psi _1} = r \times {\psi _1}\,} \right|_{r = 0}} = \,\,0$ and ${\Psi _{n + 1}}\, = r \times {\psi _{n + 1}}\,\, = \,\,0$ where N is the number of elements. Therefore, to assemble N elements:

$$\left\{ {\left( {\begin{array}{*{20}{l}} {{a_{11}}} &\ldots& {{a_{1n}}} \\ \vdots &\ddots& \vdots \\ {{a_{n1}}}& \cdots& {{a_{nn}}} \end{array}} \right) -\lambda \left( {\begin{array}{*{20}{l}} {{b_{11}}} &\ldots& {{b_{1n}}} \\ \vdots& \ddots& \vdots \\ {{b_{n1}}} &\cdots& {{b_{nn}}} \end{array}} \right)} \right\}\left[ {\begin{array}{*{20}{l}} {{\Psi _2}} \\ \vdots\\ \vdots \\ {{\Psi _N}} \end{array}} \right] = 0$$

(24)

where λ is the FEM approximation Eigen-value. Therefore, using Eqs. (23) and (24) for “N” elements of mesh, Eigen-value and Eigen-state can be attained by solving Eq. (24).