Three‐dimensional irradiance and temperature distributions resulting from transdermal application of laser light to human knee—A numerical approach

The use of light for therapeutic applications requires light‐absorption by cellular chromophores at the target tissues and the subsequent photobiomodulation (PBM) of cellular biochemical processes. For transdermal deep tissue light therapy (tDTLT) to be clinically effective, a sufficiently large number of photons must reach and be absorbed at the targeted deep tissue sites. Thus, delivering safe and effective tDTLT requires understanding the physics of light propagation in tissue. This study simulates laser light propagation in an anatomically accurate human knee model to assess the light transmittance and light absorption‐driven thermal changes for eight commonly used laser therapy wavelengths (600–1200 nm) at multiple skin‐applied irradiances (W cm−2) with continuous wave (CW) exposures. It shows that of the simulated parameters, 2.38 W cm−2 (30 W, 20 mm beam radius) of 1064 nm light generated the least tissue heating −4°C at skin surface, after 30 s of CW irradiation, and the highest overall transmission—approximately 3%, to the innermost muscle tissue.


| INTRODUCTION
Photobiomodulation (PBM), the putative mechanism in light therapy, uses nonionizing light sources in the visible (VIS) and near-infrared (NIR) spectrum (600-1200 nm) [1][2][3] to produce photochemical changes inside a cell's structure to modulate its biochemistry [4,5]. One such change extensively studied and elucidated in the literature [6,7], is to restore mitochondrial homeostasis in cells in injured tissues [8]. In this case, modulation of cellular biochemistry is initiated by photon absorption by cytochrome c oxidase (CCO)-a metalorganic protein embedded in the mitochondrial inner membrane [8], leading to the normalization of adenosine triphosphate (ATP) production rates [9], the release of nitric oxide (NO) and other reactive oxygen species (ROS) [10], and eventually to increase cell survival, proliferation [6], stimulation of extracellular matrix deposition [11], and restoration of mitochondrial homeostasis [6,12]. When the biochemistry of a sufficiently large number of injured cells are appropriately modulated, PBM can cause physiological variations with therapeutic benefits.
The ready availability of high-power light sources, for example, lasers, has made the transdermal delivery of clinically effective quantities of photons to deep tissue structures possible, greatly expanding the range of conditions that can be effectively treated with light. For example, high-intensity laser therapy (HILT) [13], can deliver a larger number of photons to deep tissues, stimulating cellular metabolism [14,15], speeding up the healing process, and reducing pain [16,17]. As a result, there has been an increasing research focus on HILT. Abdullah et al. [18] compared the effects of low-level light therapy (LLLT) and HILT on pain relief and functional improvement in patients with knee osteoarthritis in a shamcontrolled randomized clinical trial. With the same technique, Qian Lu et al. [19] investigated the effectiveness of high-power lasers on chronic refractory wounds. Danhong Han et al. [20] devised a source-detector mechanism to demonstrate the penetrating and heating effect of highpower lasers in porcine tissue. Piao Daqing et al. [21] employed a similar method in a dog cadaver model to measure the transmission efficiency from skin to the spinal cord. Nonetheless, there is still very limited information regarding light absorption-driven thermal effect in human tissues at various wavelengths applied to the skin. It being impossible to test the parameter space of all possible combinations of wavelengths, skin applied irradiances, and exposure times to obtain the desired information, PBM therapy researchers increasingly rely on computational methods to estimate the light transmittance in the tissue of interest and subsequent thermal effect on skin surface.
Numerical simulations using finite element methods (FEM) provide a cost-effective, practical means of investigating light therapy dosing. There are several works reported in the literature utilizing FEM to illustrate light propagation and distribution of light irradiance in layered tissues [22,23]. For example, Kwon et al. [24] used FEM to study light-tissue interaction and found that properly chosen critical PBM parameters, like incident beam power and spot size can enhance light penetration depth. Vysakh et al. [25] deployed a multilayered skin model to model light propagation at 660 nm and concluded that the fluctuation of fluence distribution is mostly based on tissues' optical characteristics. Bhattacharya et al. [22] and Kawther et al. [26] employed a bioheat transfer model in brain and skin tissue respectively at three different wavelengths: 630, 700, and 810 nm, to show that temperature changes flatten as tissue depth increases. This study, unlike the previous ones, looks at the effects of different VIS and NIR skin applied irradiances to quantify light transmission through the tissues of the human kneeepidermis, dermis, subcutaneous fat, and muscle, and to estimate increases in tissue temperature induced by light absorption.

| MATERIALS AND METHODS
In this comprehensive parametric study, we use FEM (COMSOL Multiphysics v5.6, Stockholm, Sweden) to simulate the light propagation and heat transfer in an anatomically accurate human knee model by solving the diffusion approximation of radiative transfer equation (Section 2.2) and the bioheat equation (Section 2.3). The solutions describe the irradiance and temperature distributions produced in the knee tissues from the transdermal application-lateral, external right side of the knee-of a continuous wave (CW), uniform irradiance profile laser at several VIS and NIR wavelengths: 632, 660, 808, 850, 905, 980, 1064, and 1200 nm. Each wavelength applied at 36 different irradiances (between 0.4 and 4.24 W cm À2 ). The irradiance variations were achieved through discrete combinations of 6 power levels (5-30 W, 5 W interval) and 6 beam radii (15-20 mm, 1 mm interval). All simulations used 60 s uniform exposures across the cross-sectional area of the beam. The parameters and parameter values used in the simulations are given in Section 2.4.
We first conduct a baseline simulation of the tissue's transmittance and benchmark the simulation against the work of Vasudevan's [25]. Vasudevan simulated light propagation in a multilayer skin model using similar parameters: 100 mW at 660 nm. The models show similar fluence rates, demonstrating similar light transmission- Figure A1 in the Appendix.

| Construction of an anatomically accurate human knee model
To reconstruct an anatomically accurate human knee model, geometrical information of a male subject-specific right knee was retrieved from the axial medical images (in-plane resolution of 512 by 512 pixels and slice thickness of 1 mm) in the VH Dissector (The Visible Human Project, The National Library of Medicine) ( Figure 1A,B). The (3D) geometry of different knee layers, including subcutaneous fat, muscle, and bones, along with cartilage and meniscus (grouped as one tissue for simplicity) were digitally reconstructed using a freely available open-source software package-3D slicer (Slicer) ( Figure 1C). The outermost layer of skin was also reconstructed in a similar manner, with a layer offset of 0.2 mm added inwards to create the epidermis. The innermost epidermis layer was then joined with the outermost subcutaneous fat layer to form the dermis using Hypermesh software package (v2019). To eliminate the erratic nodes or artefacts in the segmentation, manual editing with semiautomatic filters (e.g., region growing) were used in the segmentation editor and evaluated the accuracy of the segmentation based on criteria such as boundary continuity and anatomical consistency. The reconstructed surface contours of the layered tissue structures were subsequently meshed in COMSOL using 4-noded tetrahedral elements with minimum element sizes of: 0.05 mm (epidermis, aspect ratio 1.3), 0.8 mm (dermis, fat, and muscle, aspect ratio 1.5), 0.5 mm (cartilage-meniscus, aspect ratio 1.3), and 0.8 mm (bones, aspect ratio 1.3) ( Figure 1D). A mesh convergency study was performed to obtain the optimal mesh size and improve computational efficiency (Table A1).

| Modelling light propagation in the tissues of the human knee
Light propagation in turbid media (e.g., biological tissues) is described by the radiative transfer equation (RTE).
However, due to the complexity of finding closed form, exact solutions to the RTE for nontrivial tissue geometries, the diffusion approximation to the RTE [23] is commonly used. Appropriate use of the RTE's diffusion approximation requires that the modelled media be predominantly scattering (scattering>>absorption), which is true for the biological tissues [27] that we are studying. In this study, the light's irradiance distribution in the tissues (3D) was estimated using the diffusion approximation to the RTE (Equation 1) is derived as: where, ϕ r, t ð Þ: absorption coefficient (cm À1 ), μ 0 s : reduced scattering coefficient (cm À1 ), c: speed of light in tissue (m s À1 ), S: source (W cm À3 ). The light source (Dirichlet boundary condition) was modelled using the following expression, , r 0 = beam radius of the laser (cm), R = air-tissue reflectance [1].
The tissue optical properties, namely, absorption coefficient (μ a ) and reduced scattering coefficient (μ 0 s ), have been extensively reported in the literature, The tissue optical properties used in this study were collected from published works [28][29][30] and summarized in Table 1.

| Calculating the light-absorption induced temperature increases in the knee tissues
Light-absorption induced thermal changes in the tissue layers were calculated using the time-dependent bio-heat transfer Equation (2).
where, ρ: tissue density (K gm À3 ), C p : heat capacity at constant pressure (J Kg À1 K À1 ), k: tissue thermal conductivity (W m À1 K À1 ). The heat source, due to the light absorption by the tissues, Q light W cm À3 ½ , and the heat transfer, due to the blood perfusion, Q bio W cm À3 ½ , were calculated using Equations (3) and (4), respectively: where ρ b : blood density, K gm À3 ð Þ , C p:b : blood specific heat J Kg À1 K À1 ð Þ , ω b : blood perfusion rate, s À1 ð Þ, T b : arterial blood temperature, ( 0 C).
The blood and tissue thermal properties used in this study were collected from published works [31,32] and summarized in Table 2.

| Parameters and parameter values used in the models/simulations
In order to investigate the light transmission and tissue temperature effects of HILT, this study simulated (modelled) the application of variations commonly used HILT dosing parameters to the exterior lateral side of the right knee. The parameters and parameter values are detailed in Table 3.

| Effect of wavelength on light transmission and tissue temperature
The baseline simulation applying to the exterior lateral side of the right knee 5 W of 980 nm light (15 mm beam radius, 0.7 W cm À2 uniform light irradiance) showed, as expected, that each tissue layer had different "loss T A B L E 1 Optical properties (absorption (μ a ) and reduced scattering coefficient (μ 0 s )) in different wavelengths. T A B L E 2 Thermal properties of different tissues components and blood [31,32]. coefficients"-slope of irradiance losses in the tissue layers (Figure 2), and that the overall transmission losses across the tissue stack had a roughly exponential decrease. In all cases the light's irradiances at each tissue layer interface were calculated using a surface integral of the entire applied or transmitted beam area. The result shows that more than 90% of the incident's light irradiance was absorbed before reaching the muscle tissue. The other simulations conducted to compare the light's transmission losses at the other wavelengths in the study (Table 3) show that increasing the light's wavelength increased the light's transmission. Specifically, the light's irradiance at innermost muscle tissue was 55% greater in the long NIR wavelengths (1064 and 1200 nm) than in the VIS wavelengths (632 and 660 nm). The NIR versus VIS transmittance through each tissue layer showed similar differences: 32% versus 21% to the innermost skin tissue, 6% versus 3% to the outermost muscle tissue, and 2.33% versus 0.96% to the innermost muscle tissue. The results are shown in Figure 2A-D.

Properties
Unlike LLLT, HILT produces a thermic effect on the treated tissues. Therefore, higher light transmission, while impactful of tissue temperatures, cannot be the only consideration for determining a suitable wavelength. Aside from light transmission, it is also crucial to understand how tissue type, irradiance, and wavelength affect the thermal changes in the tissues. Hence this study examined the absorption-driven tissue heating at all the tissue layers. Figures 3A,B illustrate the changes in skin temperature induced during a 60 s CW application of 5 W of laser power applied in a 15 mm radius beam with a top-hat beam profile (uniform irradiance). Because of the higher absorption coefficients, higher temperatures (approximately 45 C) were observed at the skin surface at the wavelengths of 632 and 660 nm, as compared to the longer wavelength of 1064 nm, which produced the F I G U R E 2 Light irradiance: (A) All tissue layers at 632 nm and 1200 nm (B) skin (epidermis + dermis) tissue, (C) subcutaneous fat tissue, and (D) muscle tissue. Applied HILT dose: 5 W, 15 mm beam radius (0.7 W cm À2 ).

Irradiation location
The exterior lateral side of a right knee smallest temperature increase (approximately 3 C) among all wavelengths after 60 s of CW laser irradiation. Interestingly, the overall higher transmission wavelength of 1200 nm, produced a 5 C higher change of temperature at the skin (nearly 44 C) than 1064 nm. Taking the findings of light transmission and thermal changes into consideration, Figure 3C shows that 1064 nm maximized light transmission to deep muscle tissue while minimizing skin heating.

| Effect of beam size on light transmission and tissue temperature
Following the determination of 1064 nm as an optimal tDTLT wavelength-highest transmission through the tissue layers with lowest increase in skin temperaturethe impact of beam area on light transmission and tissue temperature was investigated. Applied beam radii of 15-20 mm (1 mm increments) were modelled. The modelling results show that increasing the beam radii from 15 to 20 mm (7.06-12.56 cm 2 ) resulted in a more than 30% increase in transmission (from 2.27% to 2.98%) to the innermost muscle tissue ( Figure 4A), while reducing the temperature at the skin's surface by around 2 C ( Figure 4B). This is not unexpected as it is well documented in the literature that due to the strong scattering effect in larger beam sizes, photons are re-emitted into the adjacent tissue, resulting in increased transmission [33].

| Effect of skin applied irradiance on skin temperature
Higher laser irradiances at the skin deliver higher irradiances to the targeted tissue, however, higher skin applied irradiances also induce higher skin surface temperatures. This section investigated the changes in skin temperatures induced by light absorption during a 60 s CW application of 1064 nm light at different irradiances (0.4-2.38 W cm À2 ). Light was applied in a fixed 20 mm beam radius at various powers: 5-30 W, 5 W increments. The results are shown in Figure 5A,B. Figure 5A shows that at 30 W the temperature on the skin surface increased by approximately 4 C after 30 s (41 C, a temperature range that falls within the tolerable limits for skin tissue [34]) and by 8 C after 60 s (45 C, which exceeds the tolerable temperature range for skin tissue as per the safety standards set by the International Electrotechnical Commission; IEC). Figure 5B shows that there were no significant increases in deep tissue temperatures (less than 0.3 C). Higher irradiances at the target tissues are advantageous in that they not only substantially reduce treatment time over LLLT, but also treat a large volume. Therefore, we conducted a comparative analysis of the treated volume, absorbed dose at the targeted deep tissue, and skin temperature for LLLT and HILT doses, both having similar skin-applied energy density. To elucidate, we modelled a HILT dose with a skin-applied energy density of 23.8 J cm À2 , consisting of a power density of 2.38 W cm À2 (30 W, 20 mm beam radius) at 1064 nm, and an LLLT dose with a similar skin-applied energy density (23.8 J cm À2 ) consisting of a power density of 2.35 W cm À2 (0.5 W, 2.6 mm beam radius) at the same wavelength, with a treatment duration of 10 s for both cases. The models show that the LLLT and HILT doses produced similar changes in skin temperature, while the HILT dose treated a significantly larger volume of tissue and delivered a higher absorbed dose at the target tissue (innermost muscle tissue) compared to the LLLT dose (Table 4).

| DISCUSSION AND CONCLUSION
We performed a multiphysics simulation of light propagation and light-absorption driven tissue heating using the finite element approach. Tissue transmittance and temperature changes at the epidermal, dermal, fat, and muscle tissue layers were examined in an anatomically accurate human knee model using a fixed set of opto-thermal parameter values-Tables 1 and 2. A comprehensive parametric study was then conducted to investigate the effect that wavelength, power, and beam area had on transmission and tissue temperature.
The study showed that regardless of wavelength light energy attenuates exponentially as it propagates through the tissues of the skin, subcutaneous fat, and muscle, with less than 3% of the light reaching the innermost muscle tissue. Slightly higher transmissions (3%-4%) have been reported in other published works [20,35]. The study found that of the modelled parameters and parameter values, and due to a beneficial tissue scatter to absorption ratio-experimentally demonstrated by Zhao and Fairchild [36]-1064 nm light had near maximum light transmission to the innermost muscle tissue (3%) while minimizing skin surface temperature increases (4 C, allowable temperature rise as per the safety standards established by the American National Standards Institute [ANSI]). Changing wavelength affected tissue transmittance and heating in accordance to the wavelength's reduced scattering and absorption coefficients. The lower temperature rises observed through 1064 nm irradiation, as compared to 1200 nm irradiation, can be attributed to differences in the respective optical properties employed. Specifically, the 1200 nm wavelength is located on the backslope of the water peak, resulting in higher absorption coefficients.
The study also showed that changing the area (radius) of the applied beam while keeping the applied wavelength and power constant influenced transmittance and tissue heating. Increasing the applied beam area decreased the applied irradiance (1/r 2 dependency) reducing the light's power density, lower tissue heating is expected. Due to the substantial scattering effect of larger beam sizes, photons are re-emitted into the adjoining tissue, leading to an increase in transmission [33].
The study found that increasing the power of the applied beam while keeping the applied wavelength and beam area constant resulted in higher irradiance at the skin, which equated to higher irradiance at all tissue layers and increased heating. Increasing the applied power increased the applied irradiance (1:1 dependency) increasing the light's power density, greater tissue heating is expected.
This study assumes that the selected tissues have isotropic scattering, are homogeneous, and have constant thermal properties. However, biological tissues are often heterogeneous, with variations in both optical and thermal properties. As a result, the diffusion approximation used in the study may not accurately describe the light transport, particularly in tissues with high scattering anisotropy such as skin tissue. Moreover, the simplified models used in this study do not account for the complex molecular and cellular mechanisms of tissue heating, which may limit the accuracy of predicting the effects of specific types of radiation or heating on selected tissue types. Nonetheless, despite these limitations, this noninvasive and cost-effective approach provides a reasonable approximation compared to more complex and invasive clinical studies. While we did not consider estimating tissue damage in our study, but we acknowledge the significance of incorporating this factor in evaluating the safety and efficacy of transdermal deep tissue light therapy. Future research could investigate the impact of this on the effectiveness of light therapy for different conditions and explore the potential benefits and limitations of using high intensity laser. This study is also limited in its scope to the unique tissue structure and opto-thermal parameter values used. Different tissues structures and/or opto-thermal parameter values will yield different results. However, the fundamentals of light propagation and lightabsorption driven tissue heating are unchanged, and the work presents a robust methodology and guidance for further investigations of HILT dosing prior to clinical research.