Experimental and theoretical heat transmission through a skin phantom using a Ruby IR therapeutic lamp.

A skin phantom model was developed and characterized to study the interaction of infrared radiation during photothermal therapy. The phantom consists of a platinum silicone matrix with polyethylene oxide containing encapsulated sodium alginate to mimic the concentration of water in human skin. The specific heat and thermal conductivity of each component were experimentally measured. A theoretical model was built with heat transfer equations to simulate the temperature distribution. The experimental results and numerical simulations show that the interaction phenomenon with infrared radiation is superficial, concentrating mostly in the epidermis and part of the dermis. This successfully replicates the actual thermal behaviour of the skin during photothermal therapy. The developed phantom adequately represents the thermal properties and response to infrared radiation of the skin, allowing its use as a surrogate model in thermal imaging and diagnostic applications.


Introduction
Phantoms, also known as tissue equivalents or physiological models, are a diverse range of materials and devices designed to replicate the complex characteristics of biological tissue.Their widespread applicability in sensitive imaging and diagnostic applications has fuelled their significant growth [1].
Stepping into the realm of hyperthermia, Ali Dabbagh [2] presented a reusable phantom capable of accurately assessing the heated region under various thermal modalities.This phantom incorporated a thermochromic dye, transitioning from transparent blue to colourless upon surpassing a specific temperature threshold.The threshold temperature was figured out through spectrophotometric analysis.Gelatine was employed as the matrix material, and melanin served as the absorber, emulating the optical properties of human skin for in-depth studies.Furthermore, Alexey N. Bashkatov successfully created phantoms using gelatine as a matrix and melanin as an absorber, effectively emulating the optical properties of human skin for in-depth studies [3].
Regarding numerical modelling, [4] supplies a comprehensive review of ongoing efforts to develop correct numerical models capable of quantifying and predicting the interaction between radiation and biological tissues.The radiative transfer equation (RTE) appears as the most precise approach for modelling the phenomenon of radiation-tissue interaction.Additionally, presents an insightful analysis of heat transfer in biological tissues, using a non-equilibrium thermodynamic model to numerically simulate this phenomenon [5].
This work presents the development and characterization of a skin phantom model designed to study the interaction of infrared radiation during photothermal therapy.The results show that the developed phantom adequately represents the optical and thermal properties of the skin, allowing its application in non-invasive thermal studies and diagnoses.

Theoretical model.
The absorptance  is broadly defined as a weighted fraction between absorbed radiation, while thermal emittance  defined as a weighted fraction between the emitted radiation and the Planck black body distribution, and both can be determined in terms of the surface reflectance ().The equations have been reported by A. Amon [6], they have the following expressions: ) Where   represents the Planck black body distribution,   represents the and incoming radiation.
An in-depth exploration of heat theory is essential for comprehending the thermal dynamics and enhancing the efficacy of the physiological model "phantom" developed for simulating the application of infrared light on the skin during therapy sessions with a Ruby IR therapeutic lamp.We provide a detailed examination of the fundamental principles underlying heat transfer, including specific heat, thermal conductivity, and temperature distribution within the phantom [7].By gaining a comprehensive understanding of the intricate mechanisms governing heat transfer, we can effectively analyze and interpret the thermal properties exhibited by the phantom.
Heat flows from one body to another in three main ways: conduction, convection, and radiation; driven by thermal gradients between the bodies.Heat conduction occurs when heat flows from a higher temperature source to a lower temperature body by passing through a mediumpassing on the energy by collisions between neighboring particles, then in accord with Cho et.al. [8]: where q is the heat transfer by conduction, k is the thermal conductivity of the material, A is the surface area of heat transfer, T1 and T2 are the temperatures of the two different bodies (assuming steady state temperatures) and L is the thickness of the medium.
Heat convection occurs by the motion of a mass of fluid carrying the energy away from the source of heat into a cooler part of the fluid.The convection heat transfer at a constant temperature in a fluid has been reported by A. Dabbagh [9] , then it can be expressed as: where q is the heat transfer by convection, ℎ ̅ is the average heat transfer coefficient, Tbody is the temperature of the body and Tc is the temperature of the surrounding fluid.
All objects therefore emit and absorb radiation at the same time, and the net effect is based on their temperature relative to their surroundings, and their emittance (equation 2).The Stefan-Boltzmann law can be re expressed for a hot object in a cooler environment as [9]: where P is the thermal energy emitted per unit area,  is the emittance of the object, A is the radiating area, T is the temperature of the radiator and Tc is the temperature of the cooler surroundings.Stefan's constant, s, is taken as 5.6703 x 10 8 W m 2 K 4 .
The heat transfer equation is a fundamental thermodynamic equation describing a system's heat flow.It provides a mathematical representation of how heat is transferred from one region to another and how the temperature changes with respect to time and space.The general form of the heat transfer equation is reported by Whittaker [7], then we have: where T is the system's temperature, z represents the spatial coordinate, k is the material's thermal conductivity (thermal conductivity quantifies the material's ability to conduct heat and is a crucial parameter for solving the heat transfer equation), which characterizes its ability to conduct heat, and q represents any heat generation or heat source term, which can be present in certain scenarios where external factors contribute to the heat transfer process.
The term represents the second derivative of temperature with respect to the spatial coordinate z.It describes the spatial variation of temperature and accounts for how heat flows within the system.To solve the heat transfer equation, various pieces of information are typically necessary.Boundary conditions play a crucial role as they specify the temperature or heat flux values at the boundaries of the system.These conditions are essential for solving the equation as they define the heat transfer behavior at the system's edges or surfaces.Additionally, initial conditions are required, which specify the temperature distribution within the system at the initial time.These initial conditions provide the starting point for solving the equation and determine the system's initial state.Based on the anatomical characteristics of the skin, this study adopts a conceptualization of tissues as idealized plates that are infinitely long, wide, and possess a uniform thickness, denoted as H.The temperature distribution within the tissue at the initial time is represented by T0(z), where z denotes the spatial coordinate.The governing equation for the heat transfer phenomenon, Penne's equation [5], can be mathematically expressed as follows.
Where , ,  are respectively the density, the specific heat, and the thermal conductivity of the Dragon Skin Polymer.  , denotes the perfusion and   ,   , represents the density and specific heat of the alginate.  is a constant and  is the temperature of the phantom.  represents the heat source.
The boundary conditions when the skin is heated will be: Where H(t) is the Heaviside function.Those conditions represent the time-dependent surface heat flux, the timedependent temperature of the cooling medium and the heat convection coefficient [10], between the medium and the phantom surface.
The temperature field [11] is: Where: 4th Altamira Materials and Devices Encounter Journal of Physics: Conference Series 2699 (2024) 012005 IOP Publishing doi:10.1088/1742-6596/2699/1/0120054 To take account for deviations from the classical approach involving Fourier conduction [12] and to consider the effect of microstructural interactions on the fast transient heat transport process, we use the double-phase-delay DPL equation, and it is expressed as: is the time lag in the establishing of heat flux and associated conduction through a medium, while   represents the diffusion of heat ahead of the sharp wavefronts that would be induced by   y is the time lag in establishing the temperature gradient across the medium during which conduction occurs through its small-scale structures.

Experimental details
The skin phantom was made using the methodology reported by Monsibais [13], it is composed by matrix of platinum silicone and polyethylene dioxide, which contains encapsulated sodium alginate, with which it is possible to imitate the concentration of water present in each layer of the human skin.The epidermis was built by mixing polyethylene oxide through the spin coating technique.The dermis was made from sodium alginate, allowing the concentration of water contained in the layer to be adjusted to adjust it to real values.The fatty tissue was imitated through release wax, which correctly imitates human fat.Finally, the internal muscle was imitated through silicone rubber.
The determination of the specific heat of each of the components was achieved through a traditional cup calorimeter.The temperature was determined from the following equation [14]: Where Q is the heat, m represents the mass, c refers to specific heat and Δ is the temperature difference.100 grams of bidistilled water at 100 °C were used and the room temperature was 25 °C.
In case of thermal conductivity, an array was built, where below the sample there is water vapor at 100 °C and above the sample we have ice at 0 °C.The rate of heat flow is determined by the amount of melt water.For this purpose, the melted water within a certain time is collected and weighed.The determination of the thermal conductivity was made from Fourier's law [15]: Where Q represents the heat flux, Δx is the thickness of the sample, ΔT is the temperature difference and A is the area of the sample.The specific heat of ice is:  = 334   .
To determine the temperature distribution, an experimental set-up like that used during photothermal therapy was performed, placing a 250 W therapeutic lamp (Ruby IR) 40 cm away from the phantom.The internal temperature of the phantom was obtained through 4 type j thermocouples placed inside it, meanwhile, the external temperature was obtained through a thermographic camera.

Results and discussion
There are two steps during photothermal therapy.First, the tissue is heated directly within the optical absorption depth.Second, the heat diffuses to deeper tissue.Generally, in the initial stage of the second step, heat diffusion does not reach very deep into the tissues.After that, the diffusion of heat has enough time to spread deeper into the tissues.The temperature fields in the skin tissues in each test condition are produced differently because of thermal properties and the absorption coefficient of the epidermal layer.Silicon rubber phantom was built to simulate the interaction of IR radiation with a generic muscle, the value of the specific heat and the thermal conductivity of each of its components was obtained experimentally.It is worth mentioning that the phantom is a biphasic porous material with a fluid phase and a solid phase.Heat transfer through a porous medium could be affected by a non-equilibrium local transfer, originating from convection between regions with different temperatures.For the case of specific heat, the values listed in Table 1 were obtained through equation (15).We must highlight the proximity in the value of the specific heat obtained by the release wax and the reported value of the fatty tissue, confirming that it is possible to use it as a substitute material in the thermal study.The 4th Altamira Materials and Devices Encounter Journal of Physics: Conference Series 2699 (2024) 012005 IOP Publishing doi:10.1088/1742-6596/2699/1/0120055 specific heat values of the dermis and epidermis can be reached by adding the value of the specific heats of silicone rubber and polyethylene oxide.The thermal conductivity was determined through equation, obtaining the values listed in Table1-b.The average thermal conductivity obtained for the phantom was 0.3 ±0.01 [W/mK], with an ambient temperature of 22 °C.The value calculated for silicone rubber and epidermis is very close, in the same way, to the value obtained for release wax and fatty tissue.By implementing them within the phantom construct we can expect the thermal conductivity to be very close to the real one, without considering the metabolic activity of living tissue.From the experimental results, where the phenomenon is clearly superficial (Figure 1-a), with a greater presence in the first two layers of the phantom; Numerical analysis was performed with the help of Wolfram-Mathematica software.The priority was to simulate heat distribution across skin layers, building on work by Gruesbeck (2020) and Muhamad Hamdi (2013).The mathematical theory used for the simulations is described in section theoretical model.The code was modified and adjusted to the required parameters for infrared radiation.In Figure 1-b is presented the theoretical results for the temperature distribution of four layers of skin subjected to an external heat source.Human skin is modeled as a structure consisting of the epidermis, dermis, subcutaneous tissue, and an internal tissue.We assume that each layer is homogeneous, and that blood perfusion, thermal conductivity, and heat capacity are constant in each layer.Also, that the layers are perfectly bonded to each other to allow a continuous flow of heat through the interfaces.Equations function as a mathematical model for twodimensional heat transfer, that is, over an area.Simulated conditions of the experiment include: • Generated heat volume: 250 / 3  • Distance from source to phantom: 40 cm • Phantom surface area: 10 cm 2 • Phantom volume: 50 cm 3   The information provided by Figure 1-b agrees with the experimental results, where the greatest amount of heat is concentrated in the first 0.005 m, that is, the dermis and the epidermis.The temperature can reach 50 °C; when reaching 0.030 m, the temperature practically does not change.Figure 1-b represents the temperature field in the initial stage of the photothermal therapy process.The heating effect of the skin depth causes a significant part of the irradiation to be absorbed within the edge of the layered tissue, due to the absorbed energy, the temperature distribution within the tissue decays along the direction of propagation.This is because the epidermal layer has a higher absorption coefficient value than the dermal and subcutaneous fat layers.Therefore, the outer surface, where the epidermal layer is located, can absorb more energy at all wavelengths.The shorter wavelengths always have a higher temperature at the surface of the skin than the tissue deep within the skin.Because the effect of thermal conductivity plays an important role in the conductance of the absorbed energy and the irradiation time must be long enough for the heat to diffuse to the deeper layer.the heating contours, the main objective of a conduction analysis is to determine the temperature distribution in a tissue medium, that is, to know the temperature in the tissue as a function of space in the steady state and as a function of space.time during the transient state.Once this temperature distribution is known, the heat flux at any point in or on the surface of the tissue can be calculated from Fourier's law.For the simulation, the thermal conductivity coefficients shown in table 1-b and the specific heats reported in table 1-a were used.In addition, equations were used as a mathematical model.We can observe how over 10 mm of length, there is a temperature gradient of 25 °C, with a decreasing exponential attenuation profile, consistent with Figure 1-a.It is possible to observe that the contour with the highest temperature measures approximately 0.001m, from there, the wavefront reaches up to 0.005m, that is, the area where the infrared radiation has the greatest incidence measures 0.0005 m 2 .These simulated results show that the temperature rise was highest in the center of the irradiation region and decreased from the irradiation region towards the tissues, the effect arising from the fact that the heat from the hot spot zone in the central region it will diffuse to the cold region in the surrounding tissues and the longitudinal temperature distribution affects the diffusion of heat through the different tissue layers.It is notable that the temperature decreases with increasing depth of penetration.Figure 2-b shows the temperature changes of the irradiation surface with elapsed time, measured using the four thermocouples.It can be seen in all the curves that the temperature increased rapidly in the initial stage and then continued to increase linearly, without reaching the stable state that all heating processes have.It is known that the non-Fourier state will gradually disappear with increasing time and will reduce to the Fourier model.Photothermal therapy occurs during the linear stage of the heating phenomenon.The temperature distribution study and the thermal images obtained (Figure 2-a), confirm that the phenomenon is mostly superficial, obtaining a difference of up to 8 °C between the temperature of the bottom of the phantom and the surface, with most of the heating occurring in the outer layers and the surface of the phantom in 60 seconds, which simulates the epidermis and part of the skin.dermis.This behavior could be attributed to the alginate capsules, which have a higher coefficient of thermal conductivity, however, thermocouples one and two also show a significant increase in temperature, which indicates that heat continues to flow through the phantom, albeit more slowly, behavior that is observed during actual therapy.

Conclusions
The developed skin phantom adequately mimics the optical and thermal properties of human skin, allowing its application as a surrogate model for non-invasive thermal imaging and diagnostic studies.The phenomenon of interaction of infrared radiation with the skin occurs almost entirely in the first layer of the skin and up to 3 mm below, heating of the subsequent layers occurs by secondary mechanisms in the case of biological tissue and by convection of heat in the case of phantoms.Absorption coefficients are much greater ex-vivo than in-vivo and this is probably due to sample preparation techniques and rehydration leading to unrealistic water content and greater absorption by water for ex-vivo samples.
The specific heat and thermal conductivity values experimentally obtained for each phantom component closely match those reported in literature for their respective native human skin tissues.The temperature distribution trends qualitatively concur across experimental thermographic images and simulated theoretical contours.A thermal gradient of nearly 25°C is witnessed over the first 10mm depth, with an exponentially decaying profile downwards.This agrees with the strong optical absorption expected from skin's outer layers.
In summary, the fabricated platinum silicone and polyethylene oxide phantom, containing encapsulated sodium alginate microcapsules, convincingly emulates skin's thermal characteristics and response to infrared irradiation.It serves as an accurate tissue-mimic for conducting non-invasive infrared heating examinations, enabling analyses of heat transfer phenomena, and facilitating further advancements in photothermal diagnostics. .

Figure 1 .
Figure 1.Thermographic images of the phantom (a) and Numerical modeling of the temperature distribution (b)

Figure 2 7 Figure 2 -
Figure 2 Internal temperature distribution of the phantom (a) .Heating contours (b

Table 1 .
Specific heat and Thermal conductivity coefficients of each layer