Numerical analysis validation

The analysis tool used in this numerical study was COMSOL Multiphysics, a powerful simulation software based on the finite element method. The backward differentiation formula method was applied to discretize the Pennes Bioheat equation, enabling the calculation of transient temperature distributions over time. Additionally, to validate the numerical analysis model propose herein, a grid independence test and convergence test based on the number of iterations were performed under the laser power of 1.0 W, the laser radius of 0.2 mm, and the irradiation time of 500 s, focusing on the temperature results in the area where the laser first contacts the inflammation. As observed in Fig. 3, when the number of grids reached 1,328,831 or more, the temperature change converged to less than \( ^ \)℃, setting the final number of grids at 1,328,831. Moreover, the error value in relation to the number of iterations dropped below \( ^\%\) after 35 iterations, confirming convergence and validating the effectiveness of the numerical analysis model proposed in this numerical study.

Fig. 3

Mesh validation of the numerical model and iteration error in percentage

Laser irradiation time and power behavior analysis in inflammation

In this numerical study, laser irradiation was focused on the central part of the inflammation (x = 12.3 mm, y = 0 mm, z = 6 mm) to observe the effects of PTT depending on the laser intensity, radius, and irradiation time. To reiterate, the essence of PTT is to increase the temperature of inflammation to maximize irreversible damage. Although the temperature field was calculated in the numerical analysis for the entire model, to focus on the temperature rise in the inflammation owing to the laser, a plot was drawn for the area of x = 11 mm to x = 14 mm, y = -2 mm to y = 2 mm, z = 4 mm to z = 8 mm, as shown in Fig. 4. Figure 4a shows a 3D schematic of the numerical analysis model showing the laser irradiation of the implant and inflamed parts. Figure 4b and c are schematics of the cross-sections in the XZ and YZ directions, respectively, at the inflammation center point (x = 12.8 mm, y = 0, z = 6 mm). In Fig. 4b, the line AA’ represents the tangent to the implant surface where the inflammation, abutment, and artificial tooth root meet.

Fig. 4

Implant model in various directions

First, with the laser radius fixed at 0.15 mm, the laser intensity was set to 0.6 W, 1.0 W, and 1.4 W, and the irradiation time was set to 100 s, 300 s, and 500 s. The temperature distribution under these settings and the corresponding thermal damage (irreversible damage) of the inflammation are depicted at the inflammation center point on the XZ plane (Fig. 5) and the YZ plane (Fig. 6).

Fig. 5

Temperature and \( \) distribution with laser irradiation at various laser powers and laser irradiation times (XZ plane direction) for a laer radius of 0.15 mm

Fig. 6

Temperature and \( \) distribution with laser irradiation at various laser powers and laser irradiation times for a laser radius of 0.15 mm (YZ plane direction)

In both Figs. 5 and 6, the black line represents where the value of the Arrhenius damage integral (\( \)) equals 1, indicating that irreversible damage occurs where \( \) exceeds 1. As observed in these figures, under the same laser power (Figs. 5 and 6 (a, b,c), (d, e,f), (g, h,i)), an increase in the laser irradiation time resulted in an increase in the temperature of the inflammation at the same location, and an expansion of the area where irreversible damage occurred.

Examining the temperature distribution within the inflammation, it was observed that the temperature spread in a slightly distorted shape owing to the influence of the laser irradiation angle. Temperature diffusion is a phenomenon that propagates in all directions. In the case of inflamed tissues, which differ from typical solids in having relatively lower light absorption coefficients, laser light penetrates from the surface to the interior of the inflamed tissue. Consequently, the temperature also appeared to spread in the direction of laser irradiation, as calculated accurately. Under the same laser irradiation time (Figs. 5 and 6 (a, d,g), (b, e,h), (c, f,i)), as the laser power increased, the temperature of the inflammation increased and the area suffering thermal damage expanded. This indicated that, as the laser irradiation time and power increased, the extent of irreversible damage in inflammation increased. This can be interpreted as an increase in the total heat applied by the laser with longer irradiation times, allowing more time for heat to spread within the inflammation, thereby increasing the area of inflammation that undergoes irreversible damage. Additionally, as the laser power increased, the amount of heat applied per unit area to the inflammation increased, leading to an increase in the irreversible damage of inflammation. However, while inducing the irreversible damage of the inflammation is essential for the treatment of peri-implantitis, from the perspective of normal tissues, irreversible damage leads to tissue damage, and thus, an excessive temperature increase should be avoided.

The implant surface temperature is directly related to the inflammation’s temperature. While laser energy absorption is localized within the inflammation, thermal conduction from the inflammation to implant can raises the implant surface temperature. The heat energy absorbed from the inflammation is transferred to the implant surface. If the temperature within the inflammation rises excessively due to overly intense laser power or prolonged irradiation times, it may result in the implant surface temperature exceeding the critical threshold of 47 ℃. This scenario poses a risk of thermal damage to the alveolar bone. Consequently, it is necessary to examine the temperature on the implant surface, which is the hottest part of the implant (AA’ line). As shown in Fig. 7, under all the laser radii and irradiation time conditions, the temperature at point AA’, where the implant physically contacts the inflammation and is the hottest, was examined. Below 1.0 W, even after t = 300 s, the implant surface temperature generally did not reach 47 ℃, except in some cases. However, above 1.0 W, the temperature of a significant portion of the implant surface exceeded 47 ℃; thus, caution is warranted during treatment.

Fig. 7

Temperature distribution of line AA’ with various radii and irradiation times of the laser

Laser radius and power behavior analysis in inflammation

In Sect. 3.2, we analyzed the degree of irreversible damage and the temperature distribution of the inflammation according to the laser irradiation time and intensity in various planar directions (XZ direction, YZ direction). In Sect. 3.3, we fixed the laser intensity at 0.8 W and set the laser irradiation radius to 0.1 mm, 0.15 mm, and 0.2 mm, and the irradiation time to 100 s, 300 s, and 500 s. Subsequently, we plotted the temperature distribution and the corresponding results under these conditions.

Fig. 8

Temperature and \( \) distribution with laser irradiation at various laser irradiation radii and laser irradiation times (XZ plane direction) for a laser power of 0.8 W

Fig. 9

Temperature and \( \) distribution with laser irradiation at various laser irradiation radii and laser irradiation times (YZ plane direction) for a laser power of 0.8 W

Figure 8 shows the temperature distribution results and the corresponding thermal damage to the inflammation when the shape in Fig. 4a is cut along the XZ plane at the inflammation center point. Figure 9 shows the temperature distribution results and the thermal damage to the inflammation when the shape in Fig. 4a is cut along the YZ plane at the inflammation center point. As mentioned in Sect. 3.2, the black line represents where the Arrhenius damage integral value (\( \)) is 1, and if \( \) exceeds 1, it indicates irreversible damage. Observing Figs. 8 and 9, when the laser intensity was fixed, a smaller laser radius increased the intensity per unit area, concentrating the temperature increase at the center of the inflammation. Conversely, as the laser radius increased, the area covered by the laser also increased, showing a similar trend in the extent of irreversible damage of inflammation across different laser radii (Figs. 8 and 9 (a, b,c), (d, e,f), (g, h,i)).

However, as will be discussed in Sect. 3.4, with laser powers above 0.8 W, the Arrhenius thermal damage ratio (\(_\)) varied depending on the radius as the laser power increased. Additionally, at the same laser power, an increase in the irradiation time led to an increase in the area of irreversible damage of inflammation for all laser radii, as calculated. As shown in Fig. 7, the temperature difference at AA’ for each radius is negligible, and at 0.8 W, there is no impact on the alveolar bone. However, as previously described, although the \(_\) value increases with increasing laser intensity, caution is required during treatment because of its potential impact on the alveolar bone.

Arrhenius thermal damage ratio (\(_\)) in inflammation

In Sect. 3.2 and 3.3, we examined the extent of reversible damage and temperature distribution in the XZ and YZ planes inside the inflammation. In Sect. 3.4, we assessed the extent of irreversible damage throughout the entire volume of inflammation. As mentioned in Sect. 2.2, to evaluate the extent of damage across the entire inflammation, we used the Arrhenius thermal damage ratio (\(_\)), which, as Eq. (8) shows, represents the ratio of the volume in which the value of the Arrhenius damage integral exceeds 1 to the total volume of the inflammation.

Fig. 10

Arrhenius thermal damage ratio for various laser irradiation times

(a) laser irradiation time: 100 s, (b) laser irradiation time: 200 s, (c) laser irradiation time: 300 s, (d) laser irradiation time: 400 s, and (e) laser irradiation time: 500 s.

Figure 10 shows the values of \(_\) for different laser radii as the laser irradiation time increases, with the red line representing the point at 1.0 W. For all the laser radii, the \(_\) value increased as both the irradiation time and laser intensity increased. Notably, as the laser intensity increased, the variation in \(_\) values depending on the laser radius also became larger. In addition, at t = 100 s, the \(_\) value was approximately 0.75 for the maximum laser intensity (2.0 W) at a laser radius of 0.2 mm. This indicated that approximately 75% of the inflammation was eradicated. At t = 500 s, with the maximum laser intensity (2.0 W), the \(_\) value was close to 1, indicating that almost all the inflammation was eradicated. It is necessary to examine the \(_\) values at laser intensities below 1.0 W, where the temperature at the surface (AA’ line) where the inflammation and the implant meet approaches 47 ℃ (as shown in Fig. 7). Thus, the \(_\) value obtained at a laser irradiation time of 100 s was approximately 0.26, and it reached approximately 0.5 when the irradiation time was extended to 500 s.      

Fig. 11

Arrhenius thermal damage ratio for various laser irradiation radii (a) laser irradiation radius: 0.1 mm, (b) laser irradiation radius: 0.15 mm, and (c) laser irradiation radius: 0.2 mm

In Fig. 11, the effect of the laser radius on \(_\) is depicted. With increasing laser radius, the \(_\) value increased for the same laser irradiation duration. Furthermore, the variation in \(_\) values for each irradiation time became more pronounced at higher laser intensities, and this amplification was notably greater than that caused by changes in the laser radius.

The values of \(_\) are similar to the results shown in Fig. 10, where conditions close to \(_\) being 1 were obtained at the laser intensity of 2.0 W, the irradiation time of 500 s, and the laser radii of 0.15 mm and 0.2 mm. In addition, at a laser intensity less than 1.0 W, a maximum \(_ \)of approximately 0.5 was achieved. Note that, as shown in Fig. 7, the AA’ line (the line where the inflammation contacts the implant) is more than 2 mm away from the alveolar bone. Therefore, the temperature rise along the AA’ line undergoes a conduction process from the artificial tooth root to the alveolar bone, and the temperature at the contact area between the alveolar bone and the artificial tooth root is lower. Further analysis could lead to the discovery of laser intensities higher than that (1.0 W) set in Figs. 10 and 11 to avoid thermal damage to the alveolar bone, potentially achieving higher\(_\) values without causing thermal damage to the alveolar bone.