2.1.

Preparation of Tissue Specimens

The mini-pigs were euthanized $48\mathrm{h}$ after irradiation, at which time tissue specimens of each lesion were taken and fixed in 10% neutral buffered formalin, embedded with paraffin, sectioned, and stained with hematoxylin and eosin (HE). To determine the maximum histological diameter of the lesion in the skin, serial consecutive sections were cut through the sample blocks to locate the center of the thermal lesion at the microscopic level.

2.2.

Beam-Profile Measurement

The laser beam spatial profile, which was critical for temperature modeling, was obtained by two methods: knife-edge measurement and spatial IR imaging of the skin by a $30\text{-}\mathrm{ms}$ laser irradiation. The 1/e light penetration depth into the skin at $2000\mathrm{nm}$ was approximately $200\mu \mathrm{m}$ , and the associated characteristic thermal diffusion time for a large spot diameter was about $300\mathrm{ms}$ .¹⁹ Compared to the characteristic diffusion time for skin, the heat conduction within the $30\text{-}\mathrm{ms}$ pulse duration was insignificantly small. The temperature rise was directly proportional to the irradiance $[\mathrm{W}∕{\mathrm{cm}}^2]$ for the case of no heat transfer during the laser pulse. Therefore, the measured surface temperature distribution (Fig. 1 ) obtained from the thermal camera represented the spatial intensity profile of the nominally Gaussian-shaped laser beam. The spatial profile was elliptical rather than circular, with the major axis about 10% longer than the minor axis. The radial profiles along the two axes of the elliptic laser beam (see Fig. 2 ) were essentially Gaussian in shape with $1∕{\mathrm{e}}^2$ radii of 2.55 and $2.33\mathrm{mm}$ . The arithmetic average of the two radii was $2.44\mathrm{mm}$ , which was close to the knife-edge measurement value of $2.42\mathrm{mm}$ for this example. To simplify calculations, the model assumed that the spatial profile was a circular, symmetric, Gaussian beam with a $1∕{\mathrm{e}}^2$ radius of $2.44\mathrm{mm}$ .

Fig. 1

Skin surface temperature distribution after $30\text{-}\mathrm{ms}$ laser irradiation. Laser power is $3.23\mathrm{W}$ and beam radius is $2.44\mathrm{mm}$ .

Fig. 2

Surface temperature distribution along major and minor axes after $30\text{-}\mathrm{ms}$ laser irradiation.

The spatial profiles of two larger laser beams created with various telescopes were measured by using the spatial IR imaging as well as the knife-edge method. For the radii, the model used arithmetical averages of 5.04 and $6.92\mathrm{mm}$ derived from IR imaging. These two radii values were close to the knife-edge measurement values of 4.83 and $7.33\mathrm{mm}$ , respectively, values deviating by about 5% from the averaged radii. The differences in the values from these two methods were due to the elliptical, rather than circular, spatial profile assumed by the knife-edge measurement and the slight curvature of the mini-pig skin surface.

2.3.

Optical-Thermal-Damage Model

An optical-thermal-damage model was developed by using FEMLAB finite-element modeling software (FEMLAB 3.1, Comsol Incorporated, Burlington, Massachusetts) to simulate the transient temperatures and thermal damage produced in skin by a radially symmetric laser beam at a normal angle of incidence. The skin was represented by two homogeneous regions (epidermis and dermis) with a nonlinear air-tissue boundary condition. Because the radially symmetric beam was perpendicular to the skin surface, the skin model was simplified to 2-D (radial $r$ and axial $z$ ) and deployed using cylindrical coordinates. The FEMLAB model provided a time-dependent simulation that was capable of incorporating nonlinear boundary conditions and heterogeneous thermal and optical properties in tissue.

2.4.

Optical Property Measurements

A method that used a transient temperature measurement was developed for measuring the in-vivo optical properties of the skin. Under conditions of insignificant heat conduction (exposure time, $t_0⪡\text{characteristic}$ thermal diffusion time $\tau$ ), the slope of peak temperature response to very short pulses versus time provided a measure of the absorption coefficient ${\mu }_a$ :

The thermal and optical properties of the epidermis and dermis used in the optical-thermal-damage model are summarized in Table 1 .

Table 1

Summary of thermal and optical properties used in the model.

3.1.

ED50 Thresholds for $\mathit{1}\mathit{\text{-}}\mathit{\text{Minute}}$ and $\mathit{48}\mathit{\text{-}}\mathit{\text{Hour}}$ Observations

A probit analysis²⁵ was conducted to estimate the ED50 thresholds according to two different end points of the thermal lesions: 1. instant redness observed on the skin within $1\min$ after laser irradiation, and 2. persistent redness at the site $48\mathrm{h}$ after irradiation.

Most of the lesions appeared immediately, that is, within $1\min$ of the onset of laser irradiation, and remained on the skin after $48\mathrm{h}$ . At some irradiation levels, which are close to the instant redness thresholds radiant exposure, however, several instant red spots recorded immediately after irradiations were not observed at the $48\text{-}\mathrm{h}$ postexposure reading. At persistent redness thresholds, the redness at most of the irradiated spots had appeared within $1\min$ after irradiation. However, at some specific radiant exposure levels near the estimated persistent redness threshold, the redness developed on the skin several minutes after the laser irradiation ceased and persisted past the $48\text{-}\mathrm{h}$ postexposure reading.

The average radiant exposure for the ED50 thresholds at the $1\text{-}\min$ and $48\text{-}\mathrm{h}$ postexposure readings are compared in Table 2 . The radii used in the radiant exposure calculation were derived from the thermal image instead of the knife-edge measurement; therefore, the average radiant exposures listed in Table 2 are slightly different from the values presented in our previous work.³ The standard deviation $\left(\sigma \right)$ was derived from the probit fit curve by the following definition:

Table 2

The ED50 average radiant exposure and standard deviation at damage thresholds defined as instant redness within one minute or persistent redness after 48h .  * are the thresholds of instant redness by observation within 1min of irradiation.  ** are thresholds of apparent persistent redness of the skin visible at 48h after irradiation. # means estimated without using probit fit. ☆ means it did not fit trend.

3.2.

Microscopic Observations of the Skin

Figure 3 shows representative gross images of the mini-pig skin surface $48\mathrm{h}$ after various irradiations and their corresponding microscopic HE stained biopsies. Figures 3a, 3b, 3c, 3d, 3e, 3f correspond to mini-pig skins without irradiation, irradiation at the radiant exposure level near the instant redness threshold, and radiant exposure at the persistent redness threshold, respectively. At the instant redness threshold, no redness was observed after $48\mathrm{h}$ [Fig. 3c] although redness had appeared immediately after irradiation. Mild perivascular edema and focal hyperkeratosis was found at some suspicious sites [Fig. 3d]; however, it was doubtful whether the lesion was caused by laser irradiation or mechanical trauma (scratching).

Fig. 3

The gross pictures of mini-pig skin surface $48\mathrm{h}$ after various irradiation and their corresponding microscopic biopsies (HE stain, original magnification $200\times$ ). (a) and (b): no irradiation. (c) and (d): irradiation at the radiant exposure level close to the instant redness threshold (exposure time $2.5\mathrm{s}$ , beam radius $2.44\mathrm{mm}$ , and laser power $0.29\mathrm{W}$ ). (e) and (f): irradiation at persistent redness threshold (exposure time $2.5\mathrm{s}$ , beam radius $2.44\mathrm{mm}$ , and laser power $0.41\mathrm{W}$ ). Note in (b): normal dermal blood vessels (arrow 1), edema and inflammation (arrow 2). Note in (d): edema and inflammation (arrow 2), focal hyperkeratosis (arrow 3). Note in (f): vascular dilation and thrombosis in dermal blood vessels (arrow 4), regenerated epidermal cells growing under necrotic epidermis (arrow 5), transmural necrosis of epidermis (arrow 6), and perivascular inflammation (arrow 7). In conclusion, all blood vessels show edema and inflammation; however, thrombosis, transmural epidermal necrosis, and dermal vascular dilation were not observed in (b) and (d).

Figure 3e shows a gross image of the persistent redness. The thermal lesion formed flat red papules concomitantly with the shrinking of the epidermis at the center of the irradiation sites. The microscopic image of the injury is illustrated in Fig. 3f. There was coagulative necrosis of varying depths at the burn site, with a loss of epidermis over some of the more severe burns. The pattern of necrosis had roughly the shape of a flattened cone. The necrotic epidermis cells had pyknotic or shrunken dense nuclei or occasionally fragmented nuclei. Regenerated epidermis cells formed underneath the dead epidermis cells at the lesion boundary. Vascular dilation and thrombosis in the dermal blood vessels were observed, and perivascular inflammation appeared in deeper blood vessels.

3.3.

Absorption Coefficient of the Epidermis and the Dermis

A dark mini-pig skin spotted with white areas was used to measure the in-vivo absorption coefficient. The peak temperature responses during the first $100\mathrm{ms}$ of laser irradiation were recorded with an IR camera with an imaging rate of 800 frames per second. The 1/e light penetration depth at $2000\mathrm{nm}$ was reported to be approximately $200\mu \mathrm{m}$ , and the associated characteristic thermal diffusion time for a large spot diameter was about $300\mathrm{ms}$ .¹⁹

Measurements $(n=10)$ were conducted at two different locations on the mini-pig flank and at five different power levels. The absorption coefficient of the epidermis $\left({\mu }_{a̱\mathrm{epi}}\right)$ for dark skin was $21.76\pm 0.99{\mathrm{cm}}^{-1}$ , which was derived from the slope of the linear fit lines of the peak temperature response curve using Eq. 11 (linear correlation coefficient $R\approx 0.99$ ) (Fig. 4 ).

Fig. 4

Linear fits of peak temperature response curve during the period of no heat conduction.

Because water was the primary absorber at the $2000\text{-}\mathrm{nm}$ wavelength, another way to estimate the absorption coefficient of the epidermis was based on the product of absorption coefficient of water and the water content in the epidermis.

Absorption coefficients were also measured at two white areas on the spotted mini-pig skin. Those areas had less melanin density than the dark skin used more commonly in damage experiments. The measured absorption coefficient of the white skin was $21.03{\mathrm{cm}}^{-1}$ , which was close to dark-skin absorption coefficient of $21.76{\mathrm{cm}}^{-1}$ . This similarity indicated that melanin granules played an insignificant role in absorbing the far-IR $2000\text{-}\mathrm{nm}$ wavelength.

From those results, a good assumption is that the absorption coefficient for the $2000\text{-}\mathrm{nm}$ wavelength was directly proportional to the water content in the tissue. Consequently, the absorption coefficient of the dermis was estimated to be $58.02{\mathrm{cm}}^{-1}$ , according to the water content ratio of the dermis and epidermis as well as the measured absorption coefficient of the epidermis $\left({\mu }_{a̱\mathrm{epi}}\right)$ .

3.4.

Temperature Measurements Versus the Model Predictions

To validate the optical-thermal-damage model, model predictions of the temperature response at the skin surface were compared to the experimental results. The peak temperature was defined as the maximum temperature at the irradiation site on the skin. Because of the Gaussian shape of the laser beam, the peak temperature represented the temperature at the irradiation center.

In Figs. 5a, 5b, 5c , the predicted peak temperatures after laser irradiation at various laser power levels are compared with the experimental results measured by the thermal camera. The predicted values for all exposure-duration and spot-size conditions agreed well with the experimental measurements within a 10% error.

Fig. 5

The predicted and experimental peak temperatures after laser irradiation for various exposure durations and (a) 2.44-, (b) 5.04-, and (c) $6.92\text{-}\mathrm{mm}$ spot radii.

Figure 6 is a comparison of the experimental and predicted peak transient temperature curves for the laser radiant exposures close to the persistent redness thresholds for various exposure durations. The predicted peak temperature rise followed the measured heating (with the laser on) curve quite well. After the laser was turned off, the predicted and experimental temperatures were in good agreement for the first $1\text{to}2\mathrm{s}$ . After that period, the computed curves deviated from the experimental results. We believe this deviation was due to our overpredicted cooling rate at a temperature below $50°\mathrm{C}$ .

Fig. 6

The experimental and predicted peak temperature responses by laser irradiation of various durations. Beam radius is $2.44\mathrm{mm}$ .

3.5.

Damage Predictions by the Model

In the standard rate process model, threshold damage was associated with a second-degree burn (irreversible partial-thickness injury), and parameters were selected so that $\Omega =1$ at the boundary of damage and normal tissue. In our experiments, this definition corresponded to the threshold occurrence of a persistent redness lesion on the skin. Therefore, the model calculated $\Omega$ values and defined a contour line of $\Omega =1$ as the persistent redness lesion boundary. Within this boundary, $\Omega$ was greater than 1. Thus, there was a region of super threshold damage according to the damage integral. Experimentally determined threshold energy levels do not correspond to $\Omega =1$ at the center of the laser spot $(r=0)$ . In practice, a finite area and a boundary of damage are necessary for an observer to determine that damage has occurred. Therefore, threshold damage becomes a function of the method of observation and the observer’s ability to see a thermally induced change in natural skin. The following results were based on that definition.

The model predicted thermal lesion boundaries that were bowl shaped. A comparison of the computed surface radii with experimental measurements is given in Table 3 , and the computed depths are listed in Table 4 . Because the ED50 damage threshold power levels usually did not correspond exactly to power levels used in the experiments, a corresponding experimental radius did not exist. The experimental damage radii were estimated using the measurements at the first lower and first higher experimental power levels near the thresholds. A sample of the measured radii of redness versus power levels is shown in Fig. 7 . An example of a probit fit analysis for damage/no damage as a function of power is illustrated in Fig. 7 as well. The predicted temperatures at the thermal lesion boundaries after laser irradiation were compared for various spot sizes and exposure durations [Figs. 8a and 8b ].

Fig. 7

Radii of redness lesions and the probabilities of thermal damage versus power levels. The triangles are the radii of the redness lesions that appeared on the skin after $48\mathrm{h}$ . The circles are the experimental data (the probabilities to find damage after irradiations) and the solid curve is the probit fit curve of the lesion/no-lesion observation as a function of laser power. Zero represents no damage and one represents damage. Some circles are not zero or one due to the variation of multiple measurements at the same power. Laser condition: exposure time $2.5\mathrm{s}$ and beam radius $2.44\mathrm{mm}$ .

Fig. 8

(a) Predicted surface temperature at the end of the laser irradiation at the lesion/no-lesion boundary $(\Omega =1)$ . Lesion radii are given in Table 3. (b) Predicted temperature at the end of the laser irradiation at $r=0$ , $z=\max$ predicted lesion depth at $\Omega =1$ . Maximum lesion depths are given in Table 4.

Table 3

Radii of persistent redness on surface by experimental observations and model predictions using the threshold radiant exposure for persistent damage give in Table 2.  * means the computed radii from threshold radiant exposure did not follow the expected trend.

Table 4

Predicted maximum depths of thermal lesions at threshold radiant exposure computed for Ω=1 .

Another end point of the damage threshold, namely, the instant redness at the $1\text{-}\min$ observation point, was analyzed using the optical-thermal-damage model. The maximum computed $\Omega$ values at both $r$ and $z$ equal to zero for the radiant exposure at the instant-redness thresholds are given in Table 5 .

Table 5

Predicted maximum Ω values at instant damage threshold radiant exposure (at r=0 , z=0 ).  ☆ means the experimental threshold radiant exposure did not fit the trend.

In the far-IR wavelength range of $1.800\text{to}2.600\mu \mathrm{m}$ , the ANSI Z136.1-2000 defined MPE as follows:

MPE values have usually been assumed to be a factor of 10 below the radiant exposures at damage thresholds. Inversely, the threshold radiant exposures for various exposure durations were estimated to be ten times the current ANSI MPE values. Using the optical-thermal-damage model, this study examined the damage for those radiant exposures. The radii and depths of damage boundaries $(\Omega =1)$ that were simulated are listed in Table 6 .

Table 6

Predicted radii and depths of thermal lesions (Ω=1) with radiant exposures (ten times ANSI MPE values Hmax=0.56t0.25 ) and 3.5-mm laser spot diameter.

The optical-thermal-damage model was modified to predict $\mathrm{C}{\mathrm{O}}_2$ laser ( $10.6\text{-}\mu \mathrm{m}$ wavelength) damage thresholds. The absorption coefficients of the epidermis and the dermis were estimated as 256.44 and $683.84{\mathrm{cm}}^{-1}$ , respectively, based on the product of the water absorption coefficient ( $854.8{\mathrm{cm}}^{-1}$ at $10.6\mu \mathrm{m}$ wavelength²⁶) and water content in the epidermis and the dermis. Instead of using Gaussian beam irradiance, a $1.9\text{-}\mathrm{cm}$ flat-top laser beam was employed in the model to simulate the laser profile used in the published pigskin laser threshold damage experiments.²⁷ Simulation determined the threshold radiant exposure, which produced $\Omega =1$ within $9\mathrm{mm}$ of the beam center. Figure 9 is a comparison of the simulated damage thresholds and the pigskin damage threshold data.²⁷ The predicted damage thresholds for $10.6\text{-}\mu \mathrm{m}$ ( $\mathrm{C}{\mathrm{O}}_2$ laser) and $2.0\text{-}\mu \mathrm{m}$ laser irradiation as a function of exposure duration are compared in Fig. 10 . The corresponding ANSI MPE limits for these wavelengths are drawn in Fig. 10 as well.

Fig. 9

The simulated damage threshold radiant exposures to $\mathrm{C}{\mathrm{O}}_2$ laser irradiation compared to published pigskin damage thresholds data as a function of exposure duration. ${}^{*}$ refers to Ref. 30.

Fig. 10

The predicted damage threshold radiant exposures to $10.6\mu \mathrm{m}$ ( $\mathrm{C}{\mathrm{O}}_2$ laser) and $2.0\text{-}\mu \mathrm{m}$ laser irradiation and their corresponding ANSI MPE limits as a function of exposure duration. (Prediction of Gaussian shape irradiance, spot $\text{diameter}=3.5\mathrm{mm}$ .)

4.1.

Histological Analysis of Instant and Persistent Redness in Skin

The average radiant exposures for the ED50 thresholds for instant and persistent redness at the sites of irradiation are compared in Table 2. Overall, the average radiant exposures for the instant redness thresholds were lower than for the persistent redness thresholds. Gross observations also found that at some of the sites irradiated with radiant exposures near the instant redness thresholds, redness developed on the skin immediately after the irradiation but gradually disappeared after several hours.

Histologically, the instant and persistent redness observed on the skin suggests different damage mechanisms are at work. The laser-induced temperature rise in the skin causes a dilation of the blood vessels and an increase in the number of open vessels in the dermis. The resulting increase in blood perfusion transfers more heat out of the high-temperature region to cool the skin to normal temperature.

Instant redness from hyperhemia is a reversible injury, and in this study the skin with instant redness reverted to its normal state after only a few hours without any persistent damage. The HE stained biopsies of the skin sites where redness appeared immediately after irradiation but not after $48\mathrm{h}$ did not present evidence of specific persistent injury [Figs. 3c and 3d]. At some higher input of radiant exposure, the temperature reached a critical point where irreversible redness damage was generated. In other words, a site where a persistent redness was observed $48\mathrm{h}$ after the irradiation represented a more serious thermal injury to the skin. At persistent redness thresholds, the $2000\text{-}\mathrm{nm}$ wavelength laser irradiation produced death and necrosis of the epidermal cells and lethal thermal damage to superficial blood vessels. These results were the outcomes of a complex series of physiological vascular responses to heat, including the following: 1. hemostasis (blood flow stasis), 2. thrombosis, and 3. vascular dilation [see Figs. 3e and 3f]. In the dermis, some cellular elements were more sensitive to injury than others, and they were necrotic at a greater depth than the resistant tissues. Thus, there were no sharp edges to the burn lesions. The endothelia of the blood vessels and supporting tissues were more sensitive than other tissues. The collagen bundles below the epidermis were swollen but there was no change in birefringence image intensity. When heated, the complex type 1 collagen macromolecules and fibrils undergo several configuration changes, depending on the tissue temperature and time at temperature. Thermally associated swelling of the collagen fibers seen at the light microscopic level is associated with an expansion of the collagen fibril diameters due to radial dissociation of the collagen macromolecules detected in transmission electron micrographs. This swelling seen at lower temperatures, as shown in in-vitro experiments of rat skin heated at $50°\mathrm{C}$ for $1000\mathrm{s}$ , is not associated with birefringence image intensity loss. Total birefringence loss occurs at $60\text{to}65°\mathrm{C}$ under the same experimental conditions.²⁸

4.2.

Predicted Thermal Damage

Good agreement between the results from the optical-thermal-damage model and the experimental results indicated that the model had included the major parameters contributing to the thermal response of the skin to $2000\text{-}\mathrm{nm}$ laser irradiation. The discrepancy between the predicted and measured temperatures a few seconds after the laser was turned off is hypothesized to be attributable to surface drying, which reduces the evaporative cooling effect (Fig. 6). The difference between measured and computed temperature appeared several seconds into the relaxation phase, typically, when temperatures were less than $50°\mathrm{C}$ , at which point the damage term $\exp [E_0∕RT\left(t\right)]$ of Eq. 9 did not significantly impact the damage integral. In other words, for exposure duration of less than $2.5\mathrm{s}$ , the predicted temperature transient and the rate process algorithm were sufficient to predict thermal damage with negligible error. However, for longer exposure durations, the boundary condition must be adjusted to reflect the reduced heat loss as the skin surface dries.

The predicted radial temperature distribution was compared to the experimental temperature distributions along the major and minor axes of the experimental irradiance profile at the end of laser irradiation exposure durations of 0.25, 0.5, 1.0, and $2.5\mathrm{s}$ (Fig. 11 ). The circular beam profile simplification did not cause the simulated temperature distributions to significantly deviate from experimental temperature distributions. Near the center of the beam where temperatures are maximum, a couple of degrees difference between computed and measured temperatures would produce a significant error in the value of the damage integral $\Omega$ . However, at the boundary of threshold damage (typically $1\text{-}\mathrm{mm}$ radius in our experiments), the difference between computed and measured temperatures is smaller; moreover, at this lower temperature range, the contribution to the damage integral is more time dependent.

Fig. 11

The predicted radial temperature and experimental temperature distributions along major and minor axes after laser irradiation. Beam radius $2.44\mathrm{mm}$ .

The predicted radii of surface lesions and their experimental counterparts were compared in Table 3. The radii of surface spots with persistent redness were measured by an experienced clinical pathologist $48\mathrm{h}$ after irradiation. Because of the uncertainty of the redness determination and the diversity of mini-pig skins, the measured radii near thresholds varied over a relatively large range. (A sample of measured radii of redness versus power levels is shown in Fig. 7). Typically, predictions were within the range of experimental results. The experimental data indicated that the lesion sizes at thresholds had approximately $1\text{-}\mathrm{mm}$ radii but with a rather large range of values. Overall, the trend of the computed radii in Table 3 showed that damage radius (at $\Omega =1$ ) was directly proportional to spot size and inversely proportional to exposure duration. Exceptions to this trend are marked by ${}^{*}$ in Table 3. At those points, the threshold radiant exposures may have been overestimated; that is, the computed lesion radius using the experimental threshold radiant exposures did not conform to the expected trends. The probit analysis of the experimental data indicated that the relative errors (the ratio of standard deviation and mean value given in Table 2) of these three threshold radiant exposures were over 20%. For our largest spot size $(\text{radius}=6.92\mathrm{mm})$ , the range of radii for experimental lesions was the largest. As spot size increased, the radial gradient of the temperature decreased. Thus, measurement uncertainty was amplified by the nonuniformity of the native tissue; that is, the tissue was not homogeneous. Owing to this nonhomogeneous nature of the skin, we did not have circularly symmetrical lesions. even though the Yucatan mini-pig skin best approximates the properties of human skin, the dark pigmentation of the skin hindered the gross visual determination of threshold damage, and therefore may have contributed to an uncertainty in the threshold value because of observational threshold differences, especially for large laser spots and short exposure durations.

The predicted depths of the lesions at thresholds are listed in Table 4. The predicted values indicated that damage from $2000\text{-}\mathrm{nm}$ wavelength laser irradiation would be confined to a very thin layer with a thickness less than $360\mu \mathrm{m}$ . The HE stained skin biopsies (Fig. 3) revealed coagulative necrosis in the epidermis and blood vessel damage with thrombosis and stasis of blood in dilated blood vessels at the burn sites. The pattern of necrosis was roughly the shape of a flattened cone. Epidermis cells had pyknotic or shrunken dense nuclei or occasionally fragmented nuclei, which were morphological evidence of cell death and necrosis.

The predicted temperatures at the end of the laser irradiation at the predicted damage boundary (Table 3) were computed for the threshold radiant exposure of each spot size-exposure duration condition [Figs. 8a and 8b]. As expected, a smaller temperature rise was needed to generate thermal damage for longer exposure durations. Temperatures after irradiation at the damage boundary decreased concavely from $64.3\text{to}61.4°\mathrm{C}$ , whereas exposure duration increased from $0.25\text{to}2.5\mathrm{s}$ , respectively. Only a 3° change in temperature occurred for a change in irradiation time by a factor of 10. For each exposure duration, however, there was little change in the associated threshold temperature. For our large spot sizes of irradiation, $4.88\text{to}13.84\mathrm{mm}$ diam, most of the heat was transferred along the axial direction rather than the radial direction.¹⁹ Therefore, the shape of the temperature response curve depended less on the surface irradiance profile than axial light distribution, which was governed by Beer’s law of absorption. This trend is not true for small spot sizes relative to optical penetration where radial heat conduction becomes significant.

The other end point of the damage threshold was defined as instant redness, that is, redness within $1\min$ of observation. Radiant exposures at the instant redness thresholds are compared with radiant exposures at the persistent redness thresholds in Table 2. The standard rate process model was used to compute $\Omega$ for the instant redness thresholds. The maximum $\Omega$ values at the center of the laser beam simulated by the model are presented in Table 5. When the exponential dependence of the damage integral on temperature [Eq. 9] is considered, the values of $\Omega$ in Table 5 are not unreasonable. The point marked with a ☆ represents a large threshold for instant damage that exceeded the threshold for persistent damage. Nevertheless, the analysis for instant damage did not predict a finite damage radius. The standard rate process model described the irreversible damage process associated with the thermal denaturation of cells in the tissue. Furthermore, two parameters $A$ and $E_0$ were obtained from experiments that studied the persistent damage to skin. The primary mechanism of instant redness to the skin was the increase of blood flow in the dermis, which was a reversible process without any cell denaturation. In conclusion, the standard rate process was not a good model for estimating the instant redness lesion on the skin, because the damage mechanisms for instant and persistent redness were different.

The ANSI Z136.1-2000 gives MPE limits for far-IR wavelengths between 1.8 and $2.6\mu \mathrm{m}$ as a function of laser exposure duration for a spot diameter of $3.5\mathrm{mm}$ and exposure durations between ${10}^{-3}$ and $10\mathrm{s}$ . Experimental results³ revealed that the ANSI MPE standard was reasonable for the original $3.5\text{-}\mathrm{mm}$ spot diameter and exposure durations from $0.25\text{to}2.5\mathrm{s}$ . Our model, however, was used to evaluate the ANSI MPE standard for a broader range of exposure durations. Table 6 shows the predicted radii and depths of thermal lesions $(\Omega =1)$ with estimated threshold radiant exposures (ten times the ANSI MPE value $H_{\max }=0.56t^{0.25}$ ) and a $3.5\text{-}\mathrm{mm}$ laser spot diameter. For exposure durations from $1\text{to}{10}^{-2}\mathrm{s}$ , thermal lesions were less than $1\mathrm{mm}$ in radius, which was comparable to the values for the experimental sizes of skin thermal lesions at threshold (listed in Table 3). Most computed lesion depths at ten times the standard were less than the predicted depths at the measured persistent redness thresholds (listed in Table 4). Computed maximum $\Omega \mathrm{s}$ were less than 1 for 10- and ${10}^{-3}\text{-}\mathrm{s}$ exposure durations, indicating that no thermal damage occurred with the estimated threshold radiant exposures at those exposure conditions.

It should be noted that the value of the safety factor was based on considerations of the overall level of uncertainty in the data, the experimental detail, the sources of potential error, the differences between animal and humans, and the state of knowledge of the injury mechanism and the biological sequelae. Although a safety factor of less than ten has been used at times when experimental uncertainty was small, the value of ten for the safety factor has been adequate and is most frequently considered.²⁹ Therefore, the safety factor was chosen to be ten in this work.

Furthermore, we used the model to obtain a rough estimate of the MPE values. Based on the discussion before and the simulated and measured radii of the persistent redness lesions at threshold (see in Table 3), we assumed the following: 1. the second-degree burn threshold occurred when the radius of the surface lesion $(\Omega =1)$ was equal to $1.0\mathrm{mm}$ , and 2. the estimation of MPE value was a factor of 10, which is called the safety factor below the radiant exposure at the second-degree burn threshold. The $1.0\text{-}\mathrm{mm}$ damage radius was selected as a typical value from the range of experimental values given in Table 3. In Fig. 12 , the simulated threshold radiant exposure values were compared with $10\times \mathrm{ANSI}$ MPE limits as a function of time for a laser spot of $3.5\mathrm{mm}$ diam. The comparison showed that the ANSI MPE values were close to or less than the corresponding estimations of MPE produced by the model.

Fig. 12

The simulated threshold radiant exposure values compared with $10\times \mathrm{ANSI}$ MPEs as a function of laser exposure time. (Spot $\text{diameter}=3.5\mathrm{mm}$ .)

In conclusion, as shown by the data in Table 6 and Fig. 12, the current ANSI MPE standard matches the model predictions for exposure durations from ${10}^{-1}\text{to}1\mathrm{s}$ ; however, the differences for exposure duration less than $0.1\mathrm{s}$ or longer than $10\mathrm{s}$ need to be examined. For exposure durations less than $0.1\mathrm{s}$ , the model predicts a nearly constant MPE. Generally, the damage that occurs at a point in the tissue is not only a function of the temperature increase, but it also depends on the rise time and decay characteristics of temperature. The characteristic thermal diffusion time in the skin is about $300\mathrm{ms}$ for $2000\text{-}\mathrm{nm}$ laser irradiation and a large spot diameter. Therefore, for exposure durations less than $0.1\mathrm{s}$ , heat conduction during laser irradiation is negligible, and the threshold damage according to the damage integral is a function only of peak temperature at a point and its decay transient at the end of the short exposure.³⁰ In other words, the threshold radiant exposure, which is directly proportional to the temperature increase, is not a function of exposure duration in this range. Consequently, Fig. 12 shows that the ANSI MPE limits may be underestimations for laser pulses less than $0.1\mathrm{s}$ . Furthermore, Fig. 12 predicts a constant radiant exposure level for times between $0.001\text{to}0.1\mathrm{s}$ , and potentially for even shorter times. Currently, the ANSI MPE value is constant for exposure durations from ${10}^{-3}\text{to}{10}^{-9}\mathrm{s}$ . Our modeling results indicate that the range of a constant MPE could be extended to ${10}^{-1}\mathrm{s}$ . The ANSI plateau level for exposure durations less than $0.001\mathrm{s}$ is $0.1\mathrm{J}∕{\mathrm{cm}}^2$ , whereas the model predicts a plateau level of about $0.34\mathrm{J}∕{\mathrm{cm}}^2$ starting at $0.1\mathrm{s}$ . Although it has been suggested that the ANSI MPE level may be too low for exposure durations less than $0.1\mathrm{s}$ , caution should be exercised until further pigskin experimental data are available for this exposure duration range. For exposure time larger than $4\mathrm{s}$ (see Fig. 6), the model may underestimate temperatures due to its overestimation of evaporative cooling rate. Thus the temperature rise from a $10\text{-}\mathrm{s}$ laser irradiation may be lower than the actual temperature and overestimate the MPE values.

The spot-size-dependent threshold radiant exposure values were simulated by the model, and in Fig. 13 they are compared with the experimental results. The experimental study of the in-vivo pigskin damage threshold for beam diameters larger than $3.5\mathrm{mm}$ and exposure durations of $0.25\mathrm{s}$ and longer revealed that it may be necessary to revise the ANSI MPE standard to a lower value.³ Both experimental values (threshold average radiant exposure) and the model simulation for a lesion radius of $1.0\mathrm{mm}$ imply that the predicted MPE values are less than the ANSI standard for the large beam diameters. Our data support the position that the MPE should be decreased as the beam diameter becomes larger than $3.5\mathrm{mm}$ . In summary, both experiment and simulation support the need to revise the MPE standard for far-IR laser beams with large diameters $(>3.5\mathrm{mm})$ . Clearly, the model provides a system for studying the relationships between MPE value and spot size, exposure duration, and wavelength.

Fig. 13

The simulated threshold radiant exposure values (damage $\text{radius}=1.0\mathrm{mm}$ ) and experimental values (threshold average radiant exposure) compared to $10\times \mathrm{ANSI}$ MPE as a function of spot size. (Exposure $\text{duration}=0.5\mathrm{s}$ .)

In Fig. 9, the simulated damage threshold radiant exposures for $\mathrm{C}{\mathrm{O}}_2$ $\left(10.6\mu \mathrm{m}\right)$ laser irradiation are compared with the published pigskin damage threshold data.²⁷ The higher prediction of threshold radiant exposure at $20\mathrm{s}$ was due to the overestimation of evaporative cooling rate. Excluding that point, good agreement was observed between the optical-thermal-damage model and experimental results. By adjustment of optical parameters as a function of wavelength, predictions are possible as a function of exposure duration and spot size at exposure duration less than $10\mathrm{s}$ and at laser wavelengths where water is the primary absorber and scattering is insignificant.

The predicted damage thresholds to $10.6\mu \mathrm{m}$ ( $\mathrm{C}{\mathrm{O}}_2$ laser) and $2.0\text{-}\mu \mathrm{m}$ laser irradiation as a function of exposure duration are compared in Fig. 10. For $10.6\text{-}\mu \mathrm{m}$ wavelength irradiation, Fig. 10 predicts a constant radiant exposure level for time between $0.0001\text{to}0.001\mathrm{s}$ , assuming linear heat conduction. Because the characteristic thermal diffusion time in the skin for $10.6\text{-}\mu \mathrm{m}$ wavelength irradiation is approximately 100 times less than that for $2.0\text{-}\mu \mathrm{m}$ wavelength,³⁰ the time range of constant radiant exposure level for $10.6\text{-}\mu \mathrm{m}$ wavelength ends at $0.001\mathrm{s}$ instead of $0.1\mathrm{s}$ for $2.0\text{-}\mu \mathrm{m}$ wavelength. At $10.6\text{-}\mu \mathrm{m}$ wavelength, the safety factors of the predicted threshold radiant exposure with respect to ANSI MPE level are less than 10 at exposure time from ${10}^{-4}\text{to}3\mathrm{s}$ . Further damage experiments may be necessary to examine the suitability of the ANSI MPE limits for the $10.6\text{-}\mu \mathrm{m}$ wavelength.

Figure 14 shows the experimental threshold irradiance for both 2.0- and $10.6\text{-}\mu \mathrm{m}$ wavelengths laser irradiation and their corresponding model predictions of threshold irradiance. Both experimental and simulated data indicate that a simple power law could be used to describe the relation between threshold irradiance and exposure time:

Fig. 14

The experimental threshold irradiance levels for both 2.0- and $10.6\text{-}\mu \mathrm{m}$ wavelengths laser irradiation and their corresponding model predictions. ${}^{*}$ refers to Ref. 30.

We were aware of the limitations of our model in that several physical processes were excluded from the model analysis. For instance, the model used for the $2000\text{-}\mathrm{nm}$ wavelength did not incorporate photon scattering in skin, nor did it address changes in the optical and thermal properties of the tissue due to the temperature increase. Furthermore, the temperature gradients in skin induced water transport in the tissue from deeper layers to the surface, which may in turn have had a significant effect on the heat and mass transfer and light absorption coefficients; in other words, the heat deposition rate and surface evaporation kinetics may need more consideration in respect to the temperature change. Another important source of uncertainty will always be human judgment in estimating lesion radii for large spot sizes (see Table 3).

Perhaps new technology will provide a method for replacing our qualitative visualization of damage with a more quantitative way. For example, polarization-sensitive optical coherence tomography measurements may provide an in-vivo method for measuring birefringence loss owing to thermal denaturation in tissue. Moreover, if it is possible to quantitatively and separately record thermal damage of each constituent in tissue, a multiple damage process model that includes several standard Arrhenius integrals could be developed. Each integral would predict damage of one specific constituent in tissue.