2.1.

Numerical Analysis and Modeling of Heat Transfer in the Tissues due to the Laser Source

Heat transfer from the laser source to the surrounding tissues was simulated by using the FEA method and bioheat equation. This model includes heat conduction, convection through blood flow, metabolic heat generation in the tissue, and heat generation by the laser source. In this model, the tissues were assumed to be homogeneous and isotropic. The temperature distribution was obtained by solving the bioheat transfer equation³⁸

To solve the bioheat equation, the boundaries of the model, except for the surface of the skin layer exposed to air, were considered to be at body temperature. At the surface in contact with air, it was assumed that heat transfer occurred as a result of free convection into air (heat transfer by radiation into the air would be negligible), which is described by

2.2.

Modeling the Laser Source

To simulate the heat generated by the laser irradiation (the source term $Q_{\mathrm{ext}}$), a spatially Gaussian and temporally continuous laser beam was modeled perpendicular to the surface of the model to simulate an optical fiber positioned orthogonally to the surface.

Light propagation in biological tissue results from the diffusion approximation of the radiative transfer equation and is defined by⁴¹

2.3.

Model

Two quasirealistic six-layer head models in a 3-D Cartesian ($x$, $y$, and $z$) coordinate system were simulated to solve light propagation equations in neonatal and adult head tissues, as shown in Fig. 1. The thickness of the skin, fat, skull, dura, cerebral spinal fluid (CSF), and brain was defined as shown in Table 2 for the adult and neonatal models. The minimum thickness of neonatal skin was extracted from the distribution of skin thicknesses measured in vitro as a function of gestational maturity.⁵⁰ For the sake of simplicity, the curvature of the six layers was not considered. To assume that the temperature at the boundary of the selected region was equal to body temperature (37°C), a sufficiently large thickness (80 mm) of brain tissue was selected. The model size was defined as ($\text{radius}=80\mathrm{mm}$ and $\text{height}=80\mathrm{mm}$), fourfold larger than the area affected by laser radiation in order to conserve the semi-infinite nature of the model.⁵¹ The optical source was simulated in which the laser source emits a Gaussian and temporally continuous laser beam. Different radii of laser spots (0.2, 0.4, 0.6, 1, 3, 5, 7, 11, and 15 mm) for different powers (from 10 to 200 mW) were investigated.

Fig. 1

(a) Geometrical structure of 3-D-CAD composed of six layers (skin, fat, skull, dura, CSF, and brain) (inset shows a zoom up region). The origin was situated immediately under the source. (b) Distributions of the photon-fluence rate in the head calculated in the head model.

Table 2

Tissue thickness (mm) of six-layer adult and neonatal head models.

The adult and neonatal models consisted of 1,564,530 and 1,024,918 tetrahedral elements, respectively. The maximum element size of the refined mesh around the light source was 0.01 mm. The diffusion equation is valid in the case of large scattering compared with absorption and when studying diffuse light propagation, i.e., at sufficient distances from any light sources. These considerations were applied to the physics of the model.

2.4.

Thermal and Optical Properties for Various Tissues

The tissue parameters of the model, such as thermal conductivity, density, heat capacity at constant pressure, metabolic rate, blood perfusion for the six different layers, are shown in Tables 3 and 4.⁵² The brain perfusion rate used in this study was $0.00932{\mathrm{s}}^{-1}\approx 0.56{\min }^{-1}$, which is comparable with the value measured in vivo with ASL-fMRI.^60,61 Gray and white matters were considered to have the same thermal properties.³⁹ The effect of cerebral metabolism on brain temperature was considered to be negligible,^62,63 allowing the metabolic rate for adults and neonates considered to be identical. Brain function and metabolic activity are indicated by oxygen concentration and glucose intake in brain cells.^64,65

Table 3

Thermal properties of the six layers of the head (skin, fat, skull, dura, CSF, and brain).52

Table 4

Heat equation parameters of the six layers of the head (skin, fat, skull, dura, CSF, and brain).52

The NIR laser wavelength in the middle of biological optical windows (600 to 900 nm) was selected for further investigation (isosbestic point: 800 nm).⁶⁶ The scattering coefficients were acquired by the following equation:

Table 5

Optical properties of the tissue components, corresponding to the six layers of the head (skin, fat, skull, dura, CSF, and brain): absorption coefficient,66 scattering coefficient.67–69

Discrepancies were observed concerning the precise value of absorption and scattering coefficients. This difference could be derived from the discrepancy between theoretical and experimental results. In general, in vivo ${\mu }_a$ and ${\mu }_s$ values for human skin were significantly smaller than those obtained in vitro (about 10 and 2 times, respectively⁷¹). For ${\mu }_a$, the discrepancy may be attributed to the low sensitivity of the double-integrating sphere, and goniometric techniques were applied for in vitro measurements at weak absorption combined with strong scattering (${\mu }_a\ll {\mu }_s$) and sample preparation methods.

The absorption coefficients in our study were adopted from the most recent complete reference concerning the optical properties of tissues.⁶⁶ Different percentages of melanin $(\text{Melanosomes per unit volume})\%=0.87\%$, 1.15%, 1.65% corresponding to the skin absorption coefficient ${\mu }_{a(\text{skin})}=0.52$, 0.65, and $0.88{\mathrm{cm}}^{-1}$ (Ref. 67) were applied in our model. A wide range of skin absorption coefficients ($0.5{\mathrm{cm}}^{-1}$ also including 0.1, 0.2, 0.3, and $0.4{\mathrm{cm}}^{-1}$) was investigated. In addition, different ranges (0.01, 0.05, 0.1, and $0.5{\mathrm{cm}}^{-1}$) of adult brain absorption coefficients were considered.⁷¹

3.1.

Effect of Various Laser Radiant Powers on Neonatal and Adult Head Models

Figure 2 shows the various temperature change profiles as a function of the radial distance around an optical fiber tip ($\text{laser spot size}=1\mathrm{mm}$) in six different layers of the head models (skin, fat, skull, dura, CSF, and brain) under steady-state conditions [Fig. 2(a): adult, Fig. 2(b): neonate]. The maximum temperature increase, using NIR for laser powers between 1 and 100 mW (laser irradiance, $I$, between 0.127 and $12.73\mathrm{W}/{\mathrm{cm}}^2$), with 10 mW increments, ranged from 0.16°C to 16.12°C (on the adult head) and 0.13°C to 13.51°C (on the neonatal head) at a skin depth of 0.4 mm. However, the maximum temperature increase on the brain for laser powers between 1 and 100 mW ranged from 0.0025°C to 0.26°C and from 0.03°C to 2.85°C at depths of 15.9 and 4.9 mm in the adult and neonatal brain, respectively. Figure 2(c) demonstrates the temperature changes at the depth of 0.4 mm with various powers: 10 to 100 mW in 10 mW increments over time.

Fig. 2

Temperature changes according to the radial distance from the laser source with various powers: 1 mW, 10 to 100 mW $10\times 10$ ($0.127\mathrm{W}/{\mathrm{cm}}^2<I<12.73\mathrm{W}/{\mathrm{cm}}^2$), for a six-layered head model (skin, fat, skull, dura, CSF, and brain) obtained from numerical simulations (the inset shows a zoom up region). (a) Adult head model, (b) neonatal head model, and (c) temperature changes at the depth of 0.4 mm with various powers: 10 to 100 mW in 10 mW increments over time.

3.2.

Effect on Neonatal Versus Adult Tissue

The intensity and correlative temperature changes in adult and neonatal head tissues according to the depth from the surface under steady-state conditions were evaluated in order to compare temperature distribution profiles in the adult versus the neonatal model (Fig. 3). For instance, Figs. 3 and 4 show the heat distributions for $3.81\mathrm{W}/{\mathrm{cm}}^2$ and the magnitude of laser penetration, respectively. As the skin and skull are much thinner in neonates, more light reaches the neonatal brain than the adult brain, which consequently results in a more marked temperature increase in the neonatal than in the adult brain. In the present study, using similar characteristics for neonatal and adult skin, the temperature increase was obviously the same in the two models. Due to the shorter distance of the other tissues from the surface in the neonatal head, the maximum temperature increase for $3.81\mathrm{W}/{\mathrm{cm}}^2$ was higher in the neonatal brain ($\mathrm{\Delta }T=0.85°\mathrm{C}$) than in the adult brain ($\mathrm{\Delta }T=0.07°\mathrm{C}$) (Fig. 3).

Fig. 3

Temperature changes according to the radial distance from the laser source ($I=3.81\mathrm{W}/{\mathrm{cm}}^2$) for six layers of adult and neonatal head model (skin, fat, skull, dura, CSF, and brain) obtained from numerical simulations.

Fig. 4

Comparative distributions of the normalized intensity of photons ($I/I_0$) according to the radial distance from the laser source ($I=3.81\mathrm{W}/{\mathrm{cm}}^2$) for six layers of adult (lower diagram) and neonatal head models (skin, fat, skull, dura, CSF, and brain) obtained from numerical simulations.

The light-distribution profiles on the brain surface for the respective models are presented in Fig. 4, which shows the comparative distributions of the normalized intensity of photons ($I/I_0$) from the laser source (30 mW) for six layers of adult (lower diagram) and neonatal head models (skin, fat, skull, dura, CSF, and brain). The depths of the brain from the skin surface were 15.9 mm for the adult model and 4.9 mm for the neonatal model. The horizontal axis represents the distance along the surface of the brain from the source. Comparison of the adult and neonatal models showed deeper scattering of light in the neonatal brain.

3.3.

Effect of Blood Perfusion

As blood flow has a major impact on tissue temperature regulation, we simulated two different conditions (presence and absence of blood perfusion) in the adult model to investigate the influence of blood perfusion on local heating induced by laser irradiation [Fig. 5(a)]. The maximum temperature increase in the brain in the presence of blood perfusion ($\mathrm{\Delta }T=0.07°\mathrm{C}$) was lower than in the absence of blood perfusion ($\mathrm{\Delta }T=0.5°\mathrm{C}$), suggesting that, based on the numerical simulation, blood perfusion decreases the temperature rise by 0.43°C for $3.81\mathrm{W}/{\mathrm{cm}}^2$. Figure 5(b) shows the effect of different blood perfusion and metabolic rates on induced heat distribution in neonatal brain. Doubling the blood perfusion rate decreased temperature changes in the brain by about 0.1°C [Fig. 5(b)], whereas the laser-induced heat changes in the brain when the normal adult metabolic rate ($\mathrm{10,000}\mathrm{W}/{\mathrm{m}}^3$) is decreased by one half can be considered to be negligible.

Fig. 5

Comparative temperature changes according to the radial distance from the laser source ($I=3.81\mathrm{W}/{\mathrm{cm}}^2$) between six layers of the head model (skin, fat, skull, dura, CSF, and brain) (a) with blood perfusion and without blood perfusion in adult and (b) the effect of different blood perfusion rates and metabolism on induced heat distribution in neonatal brain.

3.4.

Effect of Different Skin and Brain Absorption Coefficients

As accurate in vivo measurements of optical properties are not available, a wide range of these values had to be modeled. Figure 6 presents the evolution of the temperature profile as function of radial distance for different skin [Fig. 6(a)] and brain [Fig. 6(b)] absorption coefficients for $3.81\mathrm{W}/{\mathrm{cm}}^2$. The curves depict the temperature distribution in the skin and brain, in which a variation of the absorption coefficient [${\mu }_{a(\text{skin})}$: 0.1 to $0.88{\mathrm{cm}}^{-1}$; ${\mu }_{a(\text{brain})}$: 0.01 to $0.5{\mathrm{cm}}^{-1}$) had a significant influence at a depth of 0.4 mm in the skin ($\mathrm{\Delta }T=1.8°\mathrm{C}$ to 6.9°C), but not in brain tissue ($\mathrm{\Delta }T=0.07100°\mathrm{C}$ to 0.7148°C).

Fig. 6

The temperature changes according to the radial distance from the laser source ($I=3.81\mathrm{W}/{\mathrm{cm}}^2$) with various absorption coefficients are shown; the lines correspond to different (a) skin and (b) brain absorption coefficients. (The inset shows a zoomed up region.)

3.5.

Effect of Different Laser Spot Sizes

Laser spot size impacts the temperature profile. The effect of laser spot size on the temperature distribution in the skin and brain tissue of the adult and neonatal head models was investigated in the laser power range of 10 to 200 mW (in order to investigate the effect of higher powers on the temperature profile by changing spot size). Figure 7 shows a more marked temperature decrease as the laser spot size was increased from 0.2 to 15 mm.

Fig. 7

The effect of laser spot size on the temperature distribution in (a) adult skin and (b) adult brain tissue.

Figure 8 shows the temperature changes versus irradiance [Fig. 8(a), skin; Fig. 8(b), brain]. Temperature changes in the brain were 0.6059°C and 0.7182°C for the same irradiance of $0.113\mathrm{W}/{\mathrm{cm}}^2$ with different powers and spot sizes (150 mW, $\text{spot size}=13\mathrm{mm}$) and (200 mW, $\text{spot size}=15\mathrm{mm}$), respectively. Higher powers and larger spot sizes (but with the same irradiance), therefore, induced higher temperatures, implying that temperature changes depend not only on the magnitude of irradiance but also on the photon distribution (which depends on both power and spot size).

Fig. 8

The effect of laser irradiance on the temperature distribution in (a) adult skin and (b) adult brain tissue.

3.6.

Effect of Different Tissue Thicknesses

Different thicknesses were investigated, as tissue thickness is one of the parameters that needs to be accurate, and they are estimated and variable according to different individuals and medical conditions. Figure 9 shows the analysis of temperature changes in different tissues due to variations of the thickness of individual layers. Temperature changes were not largely affected by variations of the thickness of each layer: skin (3, 4, 5, and 6 mm), fat (0.4, 1.4, 2, and 3 mm), skull (5, 6, 7, and 8 mm), and CSF (0.1, 1, 2, and 3 mm). In particular, the effect of thickness variations on deep tissues is negligible.

Fig. 9

Temperature changes in different tissues due to variations in the thickness of individual layers.

3.7.

Validation of Diffusion Theory

The diffusion theory and the Monte Carlo approach have complementary attributes for modeling photon transport in a scattering medium. The Monte Carlo approach is precise but computationally inefficient, whereas the diffusion theory is inaccurate but computationally efficient. A trade-off must, therefore, be reached between the computational accuracy and efficiency of the two models. Using Monte Carlo improves the computational accuracy at the expense of computational efficiency. Although the diffusion theory is acceptable when the isotropic point source is situated far from the surface of the scattering medium, it becomes less accurate as the source approaches the surface (Fig. 10). To demonstrate this point, we compared the results obtained with the Monte Carlo method⁷² and the diffusion theory. In this section, we evaluate each step of the approximation described above using the accurate Monte Carlo method. The optical properties from Table 5 were used. As shown in Fig. 8, the fluence derived from the diffusion theory was only accurate when $r$ was greater than 0.3 cm. Deviations caused by each step of the approximation are illustrated in Fig. 10. Curve $M$ is derived from the Monte Carlo method, whereas curve $D$ is derived from the diffusion theory.

Fig. 10

Comparisons between the Monte Carlo method and the diffusion theory in terms of (a) distribution of fluence rate, (b) the diffuse reflectance, transmittance. (c) Fluence $\mathrm{J}/{\mathrm{cm}}^2$ (the inset shows relative errors between the results of the Monte Carlo method and the diffusion theory).

The error due to the approximation of Fig. 10(a) (right) with Fig. 10(b) (left) is shown in Fig. 10(c). Curves $M$ and $D$ were calculated by the Monte Carlo method and the diffusion theory, respectively; they show relatively small systematic differences. The relative error decreased with increasing $r$; it was $>100\%$ close to $r=0\%$ and 20% close to $r=0.35\mathrm{cm}$.

4.1.

Effect of Various Laser Radiant Powers on Neonatal and Adult Head Models

By ignoring the optical discontinuity and anisotropic properties of the tissues, the temperature distribution showed a higher surface temperature for a highly scattering medium during laser irradiation. Current research into laser-induced tissue damage has focused on damage generated in superficial tissues in adults.²⁶ Ito et al. investigated the heating effect of NIR irradiation at 789 nm. A $0.101°\mathrm{C}/\mathrm{mW}$ temperature elevation was detected at a depth of 0.5 mm in the human forearm in which ${\mu }_{a(\text{forearm})}=0.2{\mathrm{cm}}^{-1}$.⁶⁶

In agreement with the results of this previous study,²⁶ our results show a 2.7°C temperature increase at a depth of 0.4 mm in the skin [using ${\mu }_{a(\text{skin})}=0.2{\mathrm{cm}}^{-1}$, NIR light at 800 nm, and a power of 30 mW, $I=3.81\mathrm{W}/{\mathrm{cm}}^2$] [Fig. 6(a)]. On the other hand, for a power of 1 mW ($I=0.127\mathrm{W}/{\mathrm{cm}}^2$), a 0.16°C temperature increase was detected at a depth of 0.4 mm in the skin [using ${\mu }_{a(\text{skin})}=0.5{\mathrm{cm}}^{-1}$]. In our simulation, the total fluence rate, especially in the superficial area of skin, was overestimated when the diffusion approximation was used.⁷³ Although the main aim of our study was to investigate thermal effects on brain tissue (situated far away from the laser source), we compared the results obtained in superficial tissue in a previous experimental study in order to validate our model.²⁶

In adults, a 10°C temperature increase is considered to be the safety limit to avoid skin injury⁷⁴ (assuming a skin surface temperature of around 31°C). Consequently, our results show a temperature increase of up to 10°C at a power of about 50 mW ($I=6.36\mathrm{W}/{\mathrm{cm}}^2$) and 60 mW ($I=7.63\mathrm{W}/{\mathrm{cm}}^2$) under steady-state conditions on adult and neonatal skin, respectively.

Considering a 1-mm spot size and assuming a brain blood perfusion rate equal to $0.00166{\mathrm{s}}^{-1}$, the temperature increase in the adult brain for powers ranging from 1 to 100 mW ($0.127\mathrm{W}/{\mathrm{cm}}^2<I<12.73\mathrm{W}/{\mathrm{cm}}^2$) was much lower ($\mathrm{\Delta }T=0.002°\mathrm{C}$ to 0.26°C) than the temperature known to induce cell injury when studied in vitro (10°C). In contrast, in the neonatal brain, the temperature increase was lower ($\mathrm{\Delta }T=0.03°\mathrm{C}$ to 0.86°C) than the critical temperature (1°C) for powers ranging from 1 to 30 mW ($0.127\mathrm{W}/{\mathrm{cm}}^2<I<3.81\mathrm{W}/{\mathrm{cm}}^2$) and a 1-mm spot size.

4.2.

Effect on Neonatal Versus Adult Tissue

As expected, with an irradiance of $0.31\mathrm{W}/{\mathrm{cm}}^{-2}$, the temperature increase in the neonatal brain was higher than that in the adult brain (when comparing the $\mathrm{\Delta }T$ of each neonatal head layer with the same layer in the adult model), notably because of the thinner skin and skull in neonates, facilitating penetration of photons into the brain. Despite similar optical properties of neonatal and adult skin at 800 nm,^50,66 additional factors predisposing to a more marked temperature increase are the greater transparency and lower absorption coefficient of neonatal brain tissues.⁷¹

Nevertheless, the temperature increase at $3.81\mathrm{W}/{\mathrm{cm}}^2$ was about 4.5°C versus 4°C at a depth of 0.4 mm in skin (neonate versus adult) and 0.85°C versus 0.07°C on the brain (neonate versus adult), which is still much lower than the previously reported limit for the skin (10°C) and the brain (1°C) under steady-state conditions²⁴ and also well below the safety limits adopted for laser-induced tissue injury. However, if the temperature changes very slowly (e.g., at a rate of less than $0.5°\mathrm{C}/\min$), the subject may be unaware of a 4°C to 5°C change in temperature⁷⁵ provided the skin temperature remains within the neutral thermal range of 30°C to 36°C.

4.3.

Effect of Blood Perfusion

Blood perfusion plays a significant role in the thermal regulation of a living body. In this study, blood perfusion removed heat away from the laser source and appeared to constitute a cooling mechanism for the brain. Local heating of the model was considerably reduced when blood perfusion in the tissues was taken into account. Body fluids, transporting heat throughout the tissues, act as a convection mechanism. The present study did not consider the blood flow increase induced by vasodilatation resulting from the temperature rise,⁷⁶ which would have further reduced laser-induced heating of the skin as a result of the normal physiological thermoregulation processes. In addition, the superficial brain is spontaneously cooled by the environment and is cooler than arterial blood. Under physiological conditions, an increase in CBF may lead to an increase in superficial brain temperature and a simultaneous decrease in deep brain temperature, emphasizing the complexity of the effects of blood perfusion rate.

Our results show that changes in cerebral metabolism had no significant impact on local brain temperature changes congruent with the experimental results⁷⁷ [Fig. 5(b)]. However, changes in brain temperature were linked with cerebral blood perfusion and blood perfusion contributed to maintain a low brain temperature.

4.4.

Effect of Various Skin and Brain Absorption and Scattering Coefficients

Discordant values have been reported for the exact absorption and scattering coefficients. These discordant values could be due to the discrepancies between theoretical and experimental investigations. In general, in vivo ${\mu }_a$ and ${\mu }_s$ values for human skin are significantly lower than those obtained in vitro (about 10-fold and twofold lower, respectively⁷¹). The discordant values for ${\mu }_a$ may be related to the low sensitivity of the double-integrating sphere, the goniometric techniques used for in vitro measurements at low absorption combined with strong scattering (${\mu }_a\ll {\mu }_s$) and sample preparation methods.⁷¹ Consequently, in the absence of accurate absorption coefficients, these values need to be chosen cautiously, and our temperature profiles were probably overestimated as data were presented as the maximum temperature rise resulting from intentional selection of the maximum absorption coefficients available in the literature. As skin absorption is usually dominated by melanin absorption, various melanin levels in the skin have, therefore, been investigated to study the effect of darker skin. Figure 6 presents the temperature distributions with broad ranges of skin [Fig. 6(a)] and brain [Fig. 6(b)] absorption coefficients and shows that changes in brain or skin absorption coefficients had a greater impact on superficial layers closer to the source, and a lesser impact on deeper tissues. Nevertheless, as the brain is situated far from the source, this impact is limited to the brain surface [Fig. 6(b), inset].

Photons are less scattered and penetrate much more deeply for lower values of scattering coefficient, as they are able to travel over a longer distance with a greater step size before they interact with the tissue at a new position. As the value of the scattering coefficient increases, scattering increases and photons undergo frequent scattering with smaller step sizes, resulting in a more circular profile of fluence rate. For lower scattering coefficients, photons penetrate much more deeply with intensity decreasing outward according to the radial distance. While changes in scattering coefficients can impact the photon density distribution, their effect on temperature is negligible in view of the fact that the range of scattering coefficients of these head layers is low (brain scattering coefficient, 8.74 to $12.17{\mathrm{cm}}^{-1}$; skin, 15.09 to $26.75{\mathrm{cm}}^{-1}$; fat, 8.30 to $22.12{\mathrm{cm}}^{-1}$; skull, 8.89 to $19.24{\mathrm{cm}}^{-1}$; and CSF, 0.1 to $3.2{\mathrm{cm}}^{-1}$).^67,70

4.5.

Effect of Various Laser Spot Sizes

The mechanisms involved in the interaction between laser and biological tissue are also intimately related to the laser characteristics (e.g., wavelength, energy density, and spot size). By increasing the beam spot size, laser light can be diffused over larger areas and, in these situations, laser power can be increased while maintaining safe levels of skin irradiation. Figure 7 shows that the temperature decreased as the laser spot size increased from 0.2 to 15 mm. This figure shows that, for a specific laser power, laser spot size should be kept above a specific limit to avoid skin temperatures exceeding safety limits (e.g., in adults, assuming a skin safety threshold of 10°C, if $\text{laser power}=100\mathrm{mW}$, the spot size should be $\ge 11\mathrm{mm}$).

In the brain, a few orders of magnitude of difference in spot size does not induce any change of brain temperature, but smaller differences in laser power cause measurable differences in brain temperature due to smaller spot size leading to higher irradiance but smaller penetration depth. Laser beam spot size determines the depth of penetration. A larger spot size decreases scattering of light and increases the depth of penetration. A larger spot size, therefore, results in deeper penetration, whereas a smaller spot size induces more rapid scatter and more rapid decay of fluence with depth.⁷⁸ Since the ${\mathrm{e}}^{-1}$ depth of scattered light is unclear when scattering dominates absorption (${\mu }_a\ll {\mu }_s^{\prime }$), the exact depth of light penetration in tissue has not been determined.¹⁹ Therefore, by maintaining the same power, using a larger spot size ($\mathrm{sp}\uparrow$) decreases irradiance ($I\downarrow$), but increases the depth of penetration of the light ($\text{depth}\uparrow$), resulting in insignificant temperature changes in deeper tissues (e.g., brain). By maintaining the same spot size, higher power ($P\uparrow$) increases irradiance ($I\uparrow$) and the depth of penetration of the light ($\text{depth}\uparrow$), resulting in significant brain temperature changes. The temperature in the brain thus depends on the laser power more than on the spot size.

Figures 7(a) and 7(b) show that spot size changes at constant power have a lesser intense heating effect on deeper tissue.

Figure 8 illustrates the temperature changes versus irradiance and shows higher temperature due to higher power and higher spot size with the same irradiance. In skin tissue, for small spot sizes, cooling of surrounding nonirradiated tissue is much more effective than for a large spot size. With larger spot sizes, heat transfer from the center of the spot size cannot occur radially (sideways). Consequently, for the same level of skin irradiance, larger spots produce higher temperatures than smaller spots. This effect of higher temperatures for larger spot sizes reduces the exposure limit, since the effect of cooling means that the damage threshold does not simply depend on the skin irradiance. Exposure limits decrease with increasing spot sizes, reflecting the fact that, for the same level of irradiance, larger spots are more hazardous than smaller spots. However, this dependence of the risk threshold on spot size diameter does not apply to very large spot sizes, since the temperature profile in the center of the spot has a more or less flat profile, and this value is not affected by any further increase in the actual spot size, as the edges that are cooled radially are situated too far away from the center. Consequently, for large sources, the risk threshold depends only on the irradiance and no longer on the spot diameter.⁷⁹

For exposure to IR radiation of the skin lasting for several seconds, involuntary body movements and heat conduction disperse the irradiance profile over an area of at least several square millimeters ($\sim 3.5\mathrm{mm}$), even when the irradiated body part is intentionally kept still, even the smaller spot sizes (0.2 to 1 mm) has been considered in our model in order to study the effect of these range of spot size (i.e., which is below the 10°C limit skin temperature at 3 mm but not below the 10°C limit at 1 mm at the same power). For the wavelength of 800 nm used in our study, with the maximum anticipated exposure time (Table 1) and an ANSI standard MPE equal to $0.3\mathrm{W}/{\mathrm{cm}}^{-2}$, a larger power would be allowable with a larger spot size.

4.6.

Systematic Model Errors

Our simulated model remains a mathematical model, meaning that errors could be come from the simplifications. These errors are essentially due to the following causes: