Dynamic model for the strain-modulated spectral reflectance of the human skin in vivo

Hyperspectral imaging (HSI) offers a wealth of information regarding human skin. In this study, we established a dynamic skin spectral reflectance model (DSSR) relating the reflectance to skin surface strain, considering multi physiological and physical parameters of the skin. Experimentally, by HSI, we measured the reflectance variance of the forearm skin in vivo caused by the surface strain, and assessed these key parameters. For the human skin in vivo, within the strain range covered in this paper, stretching increases spectral reflectance, while compression decreases it. Our proposed model provides a possibility for non-contact strain measurement and health monitoring on the skin in vivo based on HSI.


Introduction
Skin is the largest organ of the human body, through which people can perceive their surroundings [1][2][3][4].At the same time, skin is also an optical interface, reflecting light shining on it.Light reflected by the skin contains rich information about our body.Skin strain may alter its optical properties so that the reflectance and absorption of the skin are modulated by the applied strain.Notably, the ability to monitor and analyze these changes has implications for early detection of skin diseases [5], personalized skincare [6], and the evaluation of treatment efficacy in dermatological conditions [7,8].Furthermore, understanding the spectral changes induced by skin strain can enhance the accuracy of non-invasive monitoring techniques, crucial in managing chronic diseases [9][10][11] and in applications ranging from wearable health sensors to telemedicine [12,13].
The optical characteristics of the skin depend on its physiological and physical properties [14][15][16][17][18].The utilization of Hyperspectral Imaging (HSI) as a novel approach has significantly broadened the scope of spectral measurements in non-invasive measurements and health monitoring.Some research revealed that physiological parameters can be well analyzed through reflectance spectra in non-invasive measurements and health monitoring [19,20].
Furthermore, physical parameters of the skin such as thickness [21][22][23], microstructures [24] also affect the optical properties of the skin.For example, due to potential applications in the field of dermatology, Yoshida, K., et al. developed a skin reflectance model that takes into account skin topography [24].Guzelsu, N., et al. found that the reflectance of polarized light exhibits a linear dependence on skin surface stretch [25].Some studies noticed the possible impact of skin thickness on reflectance spectra [23,[26][27][28][29].Meanwhile, the skin thickness is determined by the skin surface deformation [30,31].Till now, understanding how the spectral reflectance of human skin is modulated by strain, especially in vivo, is yet a challenging issue.
In this study, we delve into the mechanism governing the strain-modulated spectral reflectance of human skin in vivo.Specifically, we consider the impact of variations in skin thickness and surface roughness induced by strain on spectral reflectance.To unravel this complex relationship, we have developed a dynamic skin spectral reflectance (DSSR) model.This model draws upon a combination of the hyper-elastic model and the Kubelka-Munk model, enabling us to correlate variations in spectral reflectance with the surface strain experienced by human skin.We conducted in vivo skin loading experiments and captured reflectance spectra of the skin in different surface strain states using HSI.By fitting the spectral reflectance experimentally obtained from HSI, the key parameters of the DSSR model (epidermal thickness, dermal thickness, melanin volume fraction, blood volume fraction in the dermis, oxyhemoglobin saturation), which characterize individual skin differences, can be assessed.Physiological parameters are assumed to remain constant in skin during deformation.The experimental results indicate that, the DSSR model can capture the variations of the epidermal thickness and the dermal thickness in different surface strain through analyzing the HSI data.For the human skin in vivo, within the strain range covered in this paper, stretching increases spectral reflectance, while compression decreases it.This response phenomenon can be well explained by the model.This novel mechanism we propose opens up exciting possibilities for non-contact strain measurements and health monitoring applications based on hyperspectral imaging (HSI).

Dynamic skin spectral reflectance model
Figure 1 shows the skin composition and the optical properties.The skin is thought to be a three-layer assembly consisted of epidermis, dermis, and hypodermis.When light shines on the skin, the skin's rough surface reflects a minor quantity of light.Then, the light enters epidermis, dermis and hypodermis in turn.The melanin, hemoglobin, and water absorb the light in a selective manner, while the collagen and tiny fibrils cause it to scatter.After absorption and scattering, the left light is released back into the air.The reflectance spectra of the skin are always indicative of the body's physiological and physical characteristics.
In this section, we will establish the DSSR model to explain the response of the spectral reflectance of the skin, R skin , to the surface mechanical deformation ε x .In the DSSR model, the topmost epidermis is assumed to possess a rough surface, while both the epidermis and the dermis are treated as thin films.The hypodermis is considered a semi-infinite body.The spectral reflectance R skin is calculated as the ratio of the incident light intensity to the reflected light intensity.It is a measure of the skin's ability to reflect light in the different wavelengths.
The light that gets out of the skin includes two parts.The first part is the light that directly reflects off the epidermal top face.It will be referred as the reflected light.The intensity ratio between it and the indecent light is R reflect .The other part is the light that enters the skin, is partly absorbed and scattered by biological tissues, and finally escapes from it.This part will be referred as the scattered light.The reflectance of the scattered light is noted as R scatter .Thus, we have In this work, R reflect is calculated by the classical Torrance-Sparrow model [24,32] where n 2 (λ) is the spectral refractive index of topmost epidermis introduced by the Cauchy dispersion equation [33], σ(ε x ) is the roughness of the epidermis surface depended on the mechanical strain ε x of the skin.The detailed deduction of Eq. ( 2) can be found in SI Section 1. R scatter in Eq. ( 1) can be expressed as follows by the Kubelka-Munk theory and Kubelka theory as where T i and R i are the transmittance and the reflectance of each layer.For a specific wavelength range, R i and T i can be expressed as [34] In Eq. ( 3) and ( 4), the subscript i = e, d, h represents the epidermis, dermis, and hypodermis.h i is the thickness of i-layer, the parameter a i =µ i,s +µ i,a and b i = 2 √︁ a i 2 − 1.Here, µ i,s and µ i,a are the scattering coefficient and absorption coefficient of i-layer, determined by wavelength.µ i,s of each layer is some fixed spectral curves.The absorption coefficient µ i,a is determined by the material compositions of each layer and light wavelength.

Modulation of the surface strain for the skin thickness and the rough surface
The surface strain on the skin induces changes in the thickness of both the epidermis and dermis.Therefore, we introduce a hyper-elastic model to describe the relationship between the surface strain and thickness.Our model represents the skin as an orthotropic hyper-elastic material.
Figure 2(A) shows the coordinate establishment and the applied loading.We choose the Z-axis to be perpendicular to the skin surface (along the direction of skin thickness), the X-axis to be along the direction of the fibers in skin and the Y-axis to be perpendicular to the X-axis.
According to the incompressibility of a hyper elastic material [35], we have where λ 1 , λ 2 and λ 3 are the stretch ratios along the X, Y, and Z axis.In the mechanical deformation experiments conducted in this study, uniaxial deformation along the X-axis is applied, as shown in Fig. 2(A).Figure 2(B) shows stretch ratios in DSSR model.The three stretch ratios can be calculated through the strain along the X-axis ε x , as λ 1 = 1+ε x , λ 2 = 1-ν s ε x , and λ 3 = 1/((1+ε x )(1-ν s ε x )), where ν s is the Poisson's ratio of the skin.In the incompressible materials, the ν s always is 0.5.The rough degree of the epidermis surface σ under strain ε x can be expressed as where r x (ε x ) and r y (ε x ) represent half of the characteristic wavelength of the epidermal surface texture in x and y directions, respectively, and A(ε x ) represents the amplitude of the epidermal surface texture.By substituting Eq. ( 6) into Eq.( 2), we can derive the reflectance, R reflect , of the strained epidermal surface.The detailed deduction of Eq. ( 6) and the variations in the surface reflected reflectance R reflect with strain of skin through simulation are presented can be found in SI Section 2. The epidermal layer is much thinner than the dermal layer.Therefore, the dermis can be considered to have a compliance with strain to the epidermis.In this work, it is assumed that the strain of the underlying dermis and hypodermis is the same as that of the epidermis [36].Thus, the i-th thickness h i under strain ε x can be calculated by where h i,0 is the normal thickness of the i-th layer without the deformation.By substituting Eq. ( 7) into Eq.( 4) and subsequently into Eq.( 3), we can determine the reflectance, R scatter .
The epidermis, as the outermost layer with textural features, initially undergoes deformation and subsequently transmits the force to the dermis.The dermis, serving as the primary structure resistant to mechanical deformation, subsequently experiences deformation.Therefore, we considered the correlation between the epidermal thickness h e , the dermal thickness h d , and the skin surface strain ε x .
The epidermal thickness under deformation can be expressed as and the dermal thickness under deformation can be expressed as Where h 0,e and h 0,d are the initial epidermal thickness and the initial dermal thickness without any applied loading respectively.

Effect of the skin composition
In Eq. ( 4), µ i,a is the absorption coefficient of i-layer.Since the light absorbers in each layer differ, the calculation of µ i,a for each layer varies.
In the epidermis, the absorbers include melanin, water, and its matrix, so that µ e,a =A l F e,mel +A r F e,water +A t (1−F e,mel −F e,water ), (10) where A mel , A water , and A mat are the absorption spectrum of melanin, water, and the epidermis's matrix, and can be found from a collected database [3].In Eq. (10), F e,mel and F e,water are the volume fractions of melanin and water in the epidermis depended on individual.
In the dermis, the absorbers are hemoglobin of the blood, water, and its matrix.Oxygenated hemoglobin and deoxyhemoglobin have distinct absorption spectra.Therefore, the dermal absorption coefficient µ d,a is calculated as follows: where A hbo and A hb represent the absorption spectra of oxygenated hemoglobin and deoxyhemoglobin, respectively.In Eq. ( 11), F d,blood and F d,water represent the volume fractions of blood and water in the dermis, respectively, while S oxy corresponds to the blood oxygen saturation, indicating the percentage of oxygenated hemoglobin.The absorbers of hypodermis are oxygenated hemoglobin, deoxyhemoglobin, water, and fat matrix.To simplify the model, in addition to the blood, we roughly assume that water and fat matrix each account for half.Thus, the absorption coefficient of hypodermis is expressed as where (0.5A fat + 0.5A water ) is used as the combined absorption spectrum of water and fat matrix.
Because the dermis and the hypodermis are very close in space, it is assumed that the blood oxygen saturation of the hypodermis, S y , is the same as that of the dermis.
The scattering coefficient can be determined using empirical formulas [3].Mie scattering can be described by the equation µ mie = 2 × 10 5 ×λ −1.5 cm −1 , while Rayleigh scattering is represented by µ rayleigh = 2 × 10 12 ×λ −4 cm −1 .In the epidermal layer, the model assigns equal ratios to Mie and Rayleigh scattering, each constituting 50% of the total scattering effect.Conversely, in the dermal layer, the model is configured with a dominant Mie scattering contribution of 90%, while Rayleigh scattering accounts for the remaining 10%.
Figure 3 displays the absorption and scattering coefficients of the main chromophores in skin tissue.The absorption coefficients are from the in-vivo published data [3] as shown in Fig. 3(A).The scattering coefficients are from the empirical formulas as shown in Fig. 3(B).In Eqs.(10) to (12), the parameters F e,mel , F e,water , F d,blood , F d,water , S oxy , and F h,blood can vary among individuals.In addition, there are another two unknown parameters, the epidermis thickness h epi and the dermis thickness h der , in Eq. ( 4).
Thus, the DSSR model needs these physiological parameters to characterize the spectral reflectance of a people's skin.Using the DSSR model, we can accomplish two types of work.On the one hand, by using different parameters, the DSSR model can fit the spectral reflectance of the skin under different conditions.On the other hand, these parameters of one person can be assessed by fitting the model to the experimental spectral reflectance.
The physiological and physical parameters in the skin characterize differences among individuals.By fitting the spectral reflectance experimentally obtained from HSI, the key parameters of the DSSR model (epidermal thickness, dermal thickness, melanin volume fraction, blood volume fraction in the dermis, oxyhemoglobin saturation), which characterize individual skin differences, can be assessed.To simplify the calibration process, we set the epidermal moisture F e,water , dermal moisture F d,water , blood volume fraction in the hypodermis F h,blood as constant, with F e,water = 40%, F d,water = 85%, F h,blood = 3%.These parameters variations within the normal range exhibit minimal influence on spectral reflectance within the 400-1000 nm range.Physiological parameters and scattering coefficients are assumed to remain constant in skin during deformation.

Hyperspectral experiment
Hyperspectral experiment is used to investigate the influence of deformation on the spectral reflectance of the skin.The spectral reflectance of the skin in vivo is measured by a commercial hyperspectral camera GaiaField-F-V10 from ShuangLi Hepu.The spatial resolution of the camera is 387 × 696 pixels, with each pixel containing full reflectance spectra from 377.70 to 1024.70 nm with 128 spectral bands.The reflectance R skin is obtained through a white board calibration.Suppose the captured light intensity from a standard white board provided by the manufacturer is I white , and the captured light intensity from the skin is I skin , so as to R skin = I skin /I white .The dark current I dark was subtracted from both the captured light intensity from the standard white board I white and the skin I skin before calculating the reflectance R skin .
Figure 4 presents the experimental setup and loading setup.As shown in Fig. 4(A), we utilize a hyperspectral camera to capture reflectance data from the subject's forearm.The optical axis of the hyperspectral camera is approximately perpendicular to the skin surface, ensuring accurate data collection.To achieve this, we employ two 1000W halogen lamps as our light source, symmetrically positioned around the hyperspectral camera.These halogen lamps are located 80 cm away from the subject's forearm and are equipped with frosted glass in front of them.The angle of incidence, which is crucial for the precision of reflectance measurements, was set at 45°r elative to the skin surface.The distance between the light source, the skin, and the detector was meticulously maintained throughout the experiment to ensure consistency in the measurements.These halogen lamps provide a broad spectrum of light, ranging from 350 nm to 1050 nm, to illuminate the subject's forearm.To maintain data accuracy, we shield against interference from other indoor and outdoor light sources.There is a scale on the chute to measure the distance between the two crossbars.The initial distance between two crossbars is 35 mm.We draw some markers using a carbon pen on the observation area during the experiment.The markers within the observation area are utilized to calculate surface strain by measuring the variations in their distances.
Firstly, in the HSI experiment, the forearm skin is compressed to ε x = -0.392at the beginning.We gradually decreased the compressed load until the distance between two crossbars is 35 mm.The state in which distance between two crossbars reached 35 mm is considered as the undeformed state.Subsequently, the tension loading is applied and increased to ε x = 0.357.Throughout this process, we recorded HSI images of the forearm skin at different loadings to analyze the spectral reflectance.
The experiments are conducted with five different subjects.Five subjects (including 2 females; Asian; mean age, 24.5 y) participated.All subjects had no skin or blood disorders.Before the experiment, subjects were asked to listen to a piece of quiet music to relax.Only experimental data of one subject (male) is presented in this manuscript, and the spectral datum of others is provided in the supporting datasets and SI Section 3. The variation trends of reflectance spectrum from others have a consistent with the experimental data presented in this manuscript.During experimental measurement, the temperature and humidity of the experimental rooms were kept at 24 ± 2°C and 55 ± 5%, respectively.

Results
The spectral reflectance of the skin with different surface strains is measured by HSI. Figure 5(A) shows the pseudo color images of the reflectance in the visible light band.The color map is generated by averaging the intensity across the specified spectral range of 490-590 nm, according to the consistent variation trend in this wavelength band.This means that the stretch deformation makes the skin reflectance up, while the compressive deformation makes the skin spectral reflectance down in this wavelength range.This phenomenon is so obvious that it can be distinguished by our naked eye during the experiment.Figure 5(B) presents the experimental spectral curves.The spectral curves are the averages of observed points in the experimental observation area in special strain state.The skin area of the observed points has no obvious stains or covers the observable venous vessels.These curves demonstrate the dependence of the spectral reflectance on the mechanical strain.Figure 5(C) presents the model-fitting spectral curves.The classic least square algorithm is used to fit the DSSR model to the experimental data so that the skin parameters can be assessed.The fitting results are presented in Table 1 and Table 2.It should be noted that the appropriate initial values and search scope supplied by previous investigation is helpful to make the fitting successes.

Structure parameter Value Reasonable range
Epidermal thickness (Initial) 0.0593(mm) 0.0664 ± 0.074(mm) [37] Dermal thickness (Initial) 1.69(mm) -Melanin volume fraction 9.2% 8-16% [37,38] Blood volume fraction in the dermis 1.8% 1-30% [39] Oxyhemoglobin saturation 86% 30-90% [37,40] Table 1 shows the fitting results of physiological and physical parameters in initial state, that is, the undeformed state.The calibration results of these parameters obtain a reasonable agreement with those in previous researches [37][38][39][40].The reasonable range of the dermal thickness with similar conditions to us has not been found, so we have not provided it.Subsequently, we keep the melanin volume fraction F mel , the blood volume fraction in the dermis F d,blood , and the oxyhemoglobin saturation S oxy constant, and assess the epidermal thickness h epi and the dermal thickness h der by fitting spectral curves in different strain states.As a result, these two thicknesses under different strains can be determined.
Table 2 presents the results of strain and thickness in various strain states obtained.The "Experimental strain" in the table represents the strain determined by the makers.The "Epidermal thickness by fitting model" and "Dermal thickness by fitting model" are the h epi and h der assessed through fitting spectral curves obtained by HSI.The "Strain by fitting epidermal thickness" and "Strain by fitting dermal thickness" are the strain calculated by the fitting results of the h epi and h der , respectively.Here, the experimental strains determined by the markers can be thought to be the actual strain, while the strain by fitting epidermal thickness and dermal thickness be the strain determined by the spectral analysis.To verify the accuracy of our inverted thickness measurements, we utilized Optical Coherence Tomography (OCT) to measure the changes in epidermal layer thickness under load.We used the thickness results obtained from OCT as reference values to assess the error in thickness inversion by fitting model.The error was determined by calculating the ratio of the difference between the epidermal thickness measured by the fitting model and the OCT experiments, to the epidermal thickness measured by the OCT experiments."Epidermal thickness by OCT" and "Error in epidermal thickness" are the h epi assessed by OCT and error in the h epi by fitting model compared to the OCT experiment, respectively.Details of the OCT experimental setup and results are provided in the SI Section 4. In Table 2, these strains and thicknesses by the two methods obtain a good agreement.Figure 5(D) shows the variations of the reflectance with the strains during three visible light bands, 400-490 nm, 490-590 nm and 590-700 nm.In the visible light bands, the reflectance approximately changes linearly with the applied strain.Some similar results were also observed by Guzelsu, N., et al. [16].They believed that the change of skin reflectance is due to the reduction of skin surface roughness caused by stretching.But, according to our analysis, the thickness variations should also contribute to this phenomenon.Figure 5(E) shows the assessed the thicknesses of epidermis and dermis from the spectral curves, and the calculated strains by them."Exp epidermal thickness" and "Exp dermal thickness" represent the calculated results of the epidermal thickness and dermal thickness based on experimental strain and elastic theory respectively."Fitting epidermal thickness" and "Fitting dermal thickness" represent the fitting results of h epi and h der based on the DSSR model, respectively.The straight line is the line fitting to the discrete data points.In Fig. 5(E), these assessed thickness values show a reasonable agreement with the experimental thickness values, which were calculated using elastic theory based on experimental strain data.The results in Table 2 and Fig. 5(E) indicated that HSI can be employed to be an effective technique to obtain the mechanical state of the human skin in vivo, based on the DSSR model proposed in this work.

Discussion
Our study aimed to gain a deeper understanding of the modulation mechanism of skin spectral reflectance under mechanical strain and experimentally validate this mechanism.Although we have studied the effect of deformation on reflectance through both theoretical analysis and experimental investigation, there remain some ambiguities in this study.In the following, we will engage in a brief discussion to address these issues.
Firstly, in our study, based on simplified assumptions, we assumed that physiological parameters and scattering coefficients remain constant during skin deformation.However, in reality, skin deformation may lead to variations in the volume fraction of moisture and scattering properties.Meanwhile, the observation that the epidermis has higher scattering compared to the dermis, which our model does not currently account for, is indeed an important aspect.Not incorporating this difference in scattering properties could potentially affect the accuracy of the retrieved skin parameters.Assuming equal proportions of water and fat in the skin matrix in our model simplifies its complexity.However, water's higher infrared absorption and fat's unique scattering characteristics may impact the precision of skin's optical behavior simulation in the model, especially in the near-infrared spectrum.In future research, there is room to further investigate the variations in these parameters and their impact on spectral reflectance.This will contribute to a more comprehensive understanding of the optical properties of the skin under different deformation conditions.Secondly, the above theoretical and experiment works are confined to the uniaxial loading, but the complicated deformations in skin should be common situations.If this method is to be extended to the measurement at multi-axial strain in skin, it would necessitate the more advancement of models and the further undertaking of the validation experiments.It would be worthwhile to study how stretching will affect the surface reflectance and scatter reflectance respectively when using measurements with polarizers when the mirror component is excluded [11].
Finally, the DSSR model effectively captures the strain-induced variations to skin reflectance spectra.The results obtained through model fitting closely with the experimentally applied values, indicating the model's ability to represent the responding trends.However, there are errors between the fitted strains and the experimentally strains.The Kubelka-Munk theory has the limitations in its unmodified form.Some recent studies, which propose correction factors to enhance its accuracy [41,42], we are considering these modifications for future iterations of our model to improve its applicability and precision in reflecting actual skin spectral behavior.Further investigations should consider the impact of deformation on physiological parameters, scattering coefficients, and the complex mechanical behavior of the skin, to enhance the accuracy of strain and thickness calculations.In this study, we examined both surface reflectance and scatter reflectance together as a combined calculation.However, investigating these two components individually through separate experiments would be valuable.

Conclusion
In this study, we explored how the skin strain influences spectral reflectance in vivo, by considering the strain-induced alterations in skin thickness and surface roughness.We proposed the DSSR model, combining the hyper-elastic model and Kubelka-Munk theory, and validated it through HSI experiment.Our findings demonstrated that the DSSR model accurately captures the spectral variations in response to the skin strain, enabling calibrations of thicknesses and strains under different deformation conditions.We established a clear correlation between spectral reflectance and surface strain within our studied range, offering potential applications in non-contact strain measurement and health monitoring using HSI.This research holds promise for innovations in healthcare and dermatology.

Fig. 1 .
Fig. 1.Schematic representation of the compositions and the optical properties of the skin.

Fig. 2 .
Fig. 2. Schematic diagram of the hyper-elastic model, the coordinate and stretch ratios in the DSSR model.(A) Schematic diagram of the coordinate establishment and the applied loading along the skin fibers.(B) Schematic diagram of stretch ratios in the DSSR model.

Fig. 3 .
Fig. 3. Absorption and scattering coefficients spectra of skin tissue.(A) Absorption coefficients of main absorbers in skin tissue.(B) Scattering coefficients in skin tissue consisted by Mie scattering and Rayleigh scattering.

Fig. 4 .
Fig. 4. The experimental setup and loading setup.(A) Hyperspectral experimental setup.(B) Schematic of the experimental loading setup.

Figure 4 (
Figure 4(B) is the schematic diagram of the loading device.The device is consisted of two chutes and two crossbars.The chutes are used to hold the crossbars.Two crossbars are affixed to the skin using double-sided adhesive tape to induce deformation.The observation area is between the crossbars.The movement of crossbars determines the strain along the X-axis direction ε x .There is a scale on the chute to measure the distance between the two crossbars.The initial distance between two crossbars is 35 mm.We draw some markers using a carbon pen on the observation area during the experiment.The markers within the observation area are utilized to calculate surface strain by measuring the variations in their distances.Firstly, in the HSI experiment, the forearm skin is compressed to ε x = -0.392at the beginning.We gradually decreased the compressed load until the distance between two crossbars is 35 mm.The state in which distance between two crossbars reached 35 mm is considered as the undeformed state.Subsequently, the tension loading is applied and increased to ε x = 0.357.Throughout this process, we recorded HSI images of the forearm skin at different loadings to analyze the spectral reflectance.The experiments are conducted with five different subjects.Five subjects (including 2 females; Asian; mean age, 24.5 y) participated.All subjects had no skin or blood disorders.Before the

Fig. 5 .
Fig. 5.The spectral response of the skin under deformation.(A) The spectral reflectance of the observation area during 490-590 nm under different deformation.(B and C) Experimental and model fitting reflectance spectra of skin under the different strain ε x , strain ε x = -0.392,-0.245, -0.0983, 0.186, 0.357.(D) The reflectance response to strain during the three visible light bands.(E) The epidermal thickness and dermal thickness determined using the applied strain and the spectral data.