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**Objectives:** Enhanced infrared neural stimulation (EINS) using nanoparticles is a new research hotspot. In this paper, the numerical modeling of the interaction between a light source and brain tissue during EINS is studied.
**Materials and Methods: **This model is built with the finite element method (FEM) to mimic the propagation and absorption of light in brain tissue with EINS. Only the thermal change is considered in this model since the photothermal effect is the main mechanism of EINS. The temperature response of brain irradiation is governed by the extensively used Pennes’ bio-heat equation in a multilayer model.
**Results:** The temperature distribution in the brain under laser irradiation is determined. And the relationships between the brain tissue temperature and the three factors (the laser pulse time, the laser energy and the enhanced absorption coefficient of the tissue caused by the nanoparticles) are analyzed.
**Conclusions:** The results indicate that the brain tissue is easier to warm up with the enhancement of nanoparticles and parameters of the laser can alter the temperature increase of the brain tissue. These findings offer a theoretical basis for future animal experiments.

Neural functional disorder is a serious disease affecting millions of patients worldwide. All the motor and sensory functions of the body are the process of the muscle or nerve cells changing from a resting potential to an action potential (AP) after receiving a variety of stimuli. Thus, a nerve functional disorder can be repaired by applying an artifact neural interface with an external stimulus such as with electrical, optical, chemical and mechanical methods. Among these methods, electrical stimulation, which has already entered the clinical stage, is the most popular method [

Infrared neural stimulation (INS) with a noncontact interface, high spatial selectivity and no stimulation artifacts has advantages in achieving selective stimulation [

Heat injury is a difficult issue of INS that should not be neglected. When the laser energy exceeds two times the required minimum energy, the tissue produces thermal damage [

Many articles have tried to model the temperature changes in biotic tissue under laser irradiation [

There are two popular numerical methods to implement photothermal model, a finite element method (FEM) and a simple explicit one-step finite difference time domain (FDTD) method [

This model simulates the temperature distribution of the human brain tissue under enhanced near-infrared laser irradiation. In this work, a laser with wavelength of 808 nm is chosen as an example and the model is built using the FEM simulation software COMSOL Multiphysics 3.5a.

In biological tissue, heat transfer is a complex process including energy conduction and radiation, air convection and heat evaporation and the metabolism of biological tissue. In addition, the blood flow also affects the heat transfer. Therefore, the Pennes’ bio-heat equation is used to calculate the temperature of the tissue irradiated by laser exposure:

δ ts ρ C δ T δ t + ∇ ⋅ ( − k ∇ T ) = ρ b C b ω b ( T b − T ) + Q met + Q laser (1)

where:

δ_{ts} = time ratio (the default value is 1)

ρ = density of tissue (kg/m^{3})

C = specific heat of tissue (J/(kg∙K))

T = temperature (˚C)

K = thermal conductivity of tissue (W/(m∙K))

ρ_{b} = density of blood (kg/m)

C_{b} = specific heat of blood (J/(kg∙K))

ω_{b} = volumetric perfusion rate (kg/(s∙m))

T_{b} = temperature of arterial blood (˚C)

Q_{met} = metabolic heat source (W/m^{3})

Q_{laser} = laser heat source (W/m^{3})

In Equation (1), Q_{laser} is derived from the absorption of the laser energy by the tissue. Only a portion of the laser propagating in the tissue is absorbed; the remaining portion will be scattered, reflected or refracted. Here, we suppose the light source is Gaussian laser, so Q_{laser} of Equation (2) is defined such as [

S ( r , z ) = μ α ( 1 − R ) ϕ 0 exp [ − 0.5 r 2 ω 0 2 exp ( − μ s z ) − μ t z ] (2)

S(r, z) is the Q_{laser} of Equation (1) and where:

μ_{α} = absorption coefficient (1/m)

R = specular reflectance

μ_{s} = scattering coefficient (1/m)

μ_{t} = μ_{s} + μ_{α}, Attenuation coefficient of tissue (1/m)

φ_{0} = the flux density of the incident light ( φ 0 = 2 P / π ω 0 2 , W/m^{2})

P = laser power (W)

ω_{0} = Gaussian beam radius (m)

When the biological tissue is divided into multiple layers, the optical properties of each layer are different. The vertical coordinates of the end boundary of each layer are defined as Z_{1}, Z_{2}, Z_{3,} so the heat source of the second layer is:

S 2 ( r , z ) = μ α 2 ϕ 0 exp { − 0.5 r 2 ω 0 2 exp [ − μ s 1 z 1 − μ s 2 ( z − z 1 ) ] − μ t 1 z 1 − μ t 2 ( z − z 1 ) } (3)

The heat source of the third layer is:

S 3 ( r , z ) = μ α 3 ϕ 0 exp { − 0.5 r 2 ω 0 2 exp [ − μ s 1 z 1 − μ s 2 ( z 2 − z 1 ) − μ s 3 ( z − z 2 ) ] − μ t 1 z 1 − μ t 2 ( z 2 − z 1 ) − μ t 3 ( z − z 2 ) } (4)

Since the heat source varies with time, the heat source term of each layer should be multiplied by a time factor:

f ( t ) = exp [ − 4 ( t − τ ) 2 τ 2 ] (5)

where t is the computation time and τ is the laser irradiation time.

The geometry of FEM software COMSOL Multiphysics 3.5a is shown in

And the subdomain setting of the geometry model is calculated by Bioheat Equation, Equation (1). The thermal properties of the brain layers for Equation (1) are summarized in _{0} = 2e−4 m. For the 808 nm laser, the optical parameters of the brain layers are summarized in

Structure | Heat Conductivity W/(m∙K) | Density kg/m^{3} | Specific Heat Capacity J/(kg∙K) | Blood Perfusion kg/(s∙m) | Metabolism W/m^{3} |
---|---|---|---|---|---|

CSF | 0.64 | 1000 | 3850 | 0 | 0 |

Gray | 0.5 | 1080 | 3850 | 14.7e-1 | 16700 |

White | 0.5 | 1080 | 3850 | 3.675e-1 | 4175 |

Structure | Absorption Coefficient (1/m) | Scattering Coefficient (1/m) |
---|---|---|

CSF | 1 | 66 |

Gray | 25 | 7700 |

White | 5 | 36,450 |

Q_{laser} of Equation (1) for the CSF is defined as S(r, z) which is calculated by Equation (2). Q_{laser} for Gray is defined as S_{2}(r, z) which is calculated by Equation (3). And Q_{laser} for White is defined as S_{3}(r, z) which is calculated by Equation (4).

In this COMSOL model, four boundary conditions are set: 1) the axial symmetry is set to thermal insulation; 2) the CSF surface is set as a natural convection boundary considering the convection with the surrounding air, q conv = h ( T inf − T ) [_{conv} is the heat flux due to convection and T is the surface CSF temperature; the heat transfer coefficient, h, is 10 W/(m^{2}∙K) and the external temperature, T_{inf}, is 25˚C; 3) the interior boundaries are set to continuity; and 4) other boundaries are set as Dirichlet constant temperature, where the initial temperature of both gray matter and white matter was fixed at 37.2˚C and the CSF temperature was fixed at 37.1˚C [

Gold nanorods are selected as an example in this paper. As shown in

Simulation results on the temperature distribution of the brain model at different laser pulse duration and varying laser power are displayed in

(laser pulse opens at Time = 0 ms). Although the CSF is in front of the gray matter, the temperature of the CSF is quite lower than that of the gray matter. Furthermore, almost all the energy is gathered in the gray matter, and little energy from the laser can propagate into the white matter.

matter (

For laser irradiation at different pulse durations and powers, Figures 4(a)-(d) provides the temperature distribution of the brain model while the temperature rises to the peak value. All of the results from the above simulations are summarized in

According to Fribance’s simulation result, local temperature needs to be rapidly increased by 6.6˚C - 11.2˚C in 0.1 - 2.6 ms to activate the axon [

Recently, researchers have used different materials nanoparticle, such as gold, silica and silver to enhance the absorption of light because of their localized surface plasmons (LSPs) capability [_{α} of the nanorod at concentration of 3e−5 g/l is about 460 1/m, which is 18.4 times of μ_{α} of gray. And this absorption coefficient can be easily adjusted by changing the concentration of nanorod. Then we analyzed the effect of the tissue absorption coefficient (which are 5 times and 10 times of μ_{α} of gray) on the temperature distribution, results are provided in

INS receives more and more attention due to its advantages including high spatial selectivity and no need for contact. The local transient temperature gradient induced by the absorption of the laser is the main mechanism of INS. Therefore, the study and prediction of the temperature distribution in biological tissues in INS is frequently required.

In this work, the FEM numerical algorithm is proposed to solve the Pennes’ bio-heat equation to get the temperature distribution of the brain irradiated by an 808 nm laser. The results obtained from this study indicate that the parameters of the laser that clearly affect the temperature distribution of brain are pulse duration, laser power and the absorption coefficient. The tissue absorption of the near infrared laser increases by adding nanoparticles around the tissue. By changing the absorption coefficient in this model, we found that the tissue temperature could rise above 6˚C with exposure to the near infrared laser, proving that it is possible to excite the nerve by near-infrared laser enhanced by nanoparticles.

The results of the brain photothermal model can provide a suitable parameter reference for laser treatment. It is possible to improve the model accuracy by finding a more accurate absorption and scattering coefficient of brain for the light, as well as more accurate thermal parameters for the tissue.

This work was supported by the National Natural Science Foundation of China (grant number: 31500796).

The authors have stated explicitly that there are no conflicts of interest in connection with this article.

Zhou, R., Chen, H.L. and Mou, Z.X. (2019) FEM Model of the Temperature Distribution in the Brain during Enhanced Infrared Neural Stimulation Using Nanoparticles. Journal of Applied Mathematics and Physics, 7, 381-393. https://doi.org/10.4236/jamp.2019.72029

According to Beer-Lambert Law,

A U = lg ( 1 / T ) = K C l (A1)

where,

T = the transmittance (the ratio of transmitted light intensity to incident light intensity), _{ }

K = the extinction coefficient of the solution (L/g/cm),

C = the concentration of the solution (g/L),

l = the optical path of the solution (cm).

In this paper, μ_{α} = 1/l_{α}

μ_{α} = the absorption coefficient (1/cm),

l_{α} = the absorption length (cm),

l_{α} is also the optical path when T = 1/e,

where A U α = lg ( 1 / T ) = lg e = 0.43429448 . According to Equation (A1), AU_{α} = KCl_{α}, then

μ α = 1 / l α = K C / A U α (A2)

From the measurement results we can see that AU is about 2 at λ = 808 nm, where C = 3e−5 g/l, and l = 1 cm. So according to Equation (A1),

A U = K C l = 2 (A3)

where,

K and C is equal to those in Equation (A2), and l is the equal the thickness of the cuvette (l = 1 cm). Then solve Equation (A2) and Equation (A3), μ_{α} = 4.6 (1/cm) = 460 (1/m).