Appendix ATheoretical validation

In the Modified Beer-Lambert Law (MBLL) [33], commonly used in near-infrared spectroscopy and biomedical optics, the Differential Pathlength Factor (DPF) accounts for the increased effective optical path length caused by scattering in a turbid medium (like biological tissue). The modified Beer-Lambert Law is commonly expressed by the equation

$$A=_\left( \frac \right)=\varepsilon .c.L.DPF+G$$

(36)

where A is the measured absorbance (unit less), Io and I are incident and transmitted intensity, ε is the molar extinction coefficient (cm−1.M−1), c is the concentration of the absorber, L is the physical source detector separation (geometric path length in cm), DPF is the differential path length factor (unit less) and G is a geometry dependent factor which is often ignored in differential measurements. The DPF is typically not directly measured, and estimated empirically through models depending on wavelength, tissue type and scattering and absorption properties. A common approach for determination of DPF is.

For homogeneous tissue models, the DPF can be estimated from time-resolved or frequency domain spectroscopy using diffusion theory. An approximate empirical expression for human skin is:

$$DPF\left(\lambda \right)\approx a+b.(\lambda -700)$$

(38)

where λ is the wavelength in nm and a and b are empirical constants for skin determined from experimental calibration [34, 35]. For human skin, typical near infrared wavelengths used in spectroscopy and optical imaging are 700 nm, 800 nm, 850 nm and 900 nm. Typical constants for human skin are a = 4.99 and b = 0.007 [34, 35].

So, DPF at those wavelengths are:

DPF (700 nm) = 4.99 + 0.007 (700–700) = 4.99.

DPF (800 nm) = 4.99 + 0.007 (800–700) = 5.69.

DPF (850 nm) = 4.99 + 0.007 (850–700) = 6.04.

DPF (900 nm) = 4.99 + 0.007 (900–700) = 6.39.

These values assume normal adult human skin, under typical measurement condition. For optical properties like absorption and reduced scattering coefficients, DPF can be calculated using diffusion approximation. In real tissue, due to photon scattering, the actual mean optical path length is 2–6 times the source-detector separation depending on wavelength and tissue types and the practical DPF is computed as:

$$DPF\approx \sqrt_^}_})}$$

(39)

where, \(_^\) and \(_\) have their usual meaning [36]. The DPF values have been calculated for the dataset by taking \(_^\) as per Eq. 17 and Eq. 18 and \(_\) as per Eq. 12. The calculated values are shown in the Table 6 along with values as per equation 38.

Table 6 Comparison of calculated wavelength dependent DPFs with existing model values

It is observed from the Table 6 , that the calculated DPFs are comparable to the literature. Also, approximation of wavelength dependent DPFs by taking \(_^\) as per Eq. 18 are more closer to literature value compared to taking \(_^\) as per Eq. 17. So the calculation of \(_^\) as per Eq. 17 and Eq. 18 and, \(_\) as per Eq. 12 where photons average effective pathlength 0.128 cm has been taken as the average human forearm skin thickness, are valid up to some extent.

Validation by simulationSimulation by adding-doubling technique

Calculated value of \(_\) as per Eq. 13&14 and value of \(_\) as per Eq. 12 and value of anisotropy factor g (= 0.765) as per Eq. 20, average refractive index n = 1.4 and thickness d = 0.128 cm has been taken for the simulation to predict the skin diffuse reflectance spectra by, Adding-Doubling (AD) technique by Scott Prahl [37]. Python scripts has been written for so to get the diffuse reflectance spectra and attempt was made to see if the simulated spectra matches the original value and pattern of diffuse reflectance spectra of the data set. Figure 16 shows obtained spectra from the simulation. It can be seen that the obtained spectra mimics the pattern of the original diffuse reflectance spectra of the data set (Fig. 3). Though the changes in values are less, the overall spectra shows slightly greater value compared to original data set spectra. It has been observed that Scott Prahl’s Adding-Doubling techniques gives inaccurate result in UV–Visible region and it may be due to violation of its assumption or limitations in real-world measurements. In UV–Visible region absorption is very high for biological molecules hemoglobin and melanin, and adding-doubling method assumes multiple scattering dominates, and light becomes diffuse within the sample. Due to high absorption in this region, photons get absorbed before sufficient scattering happens. This leads to non-diffuse transport, and the RTE-based adding-doubling method does not accurately represent the real behavior in that region. Adding-Doubling technique assumes homogeneous, plane-parallel layers, and the actual skin is inhomogeneous in composition and not planner and may be for that the overall predicted diffuse reflectance value by the technique is higher compared to measured diffuse reflectance value by the spectrophotometer for the data set.

Fig. 16

Predicted simulated spectra by A-D technique

Simulation by Kubelka–Munk theory and comparison with original data set

The Kubelka–Munk function for infinitely thick sample (Transmittance, T ≈ 0) is given by

where R is the diffuse reflectance of the optically thick (infinitely thick) sample, and K and S are absorption and scattering coefficients, respectively [38]. Python scripts were written to get the reflectance spectra from Kubelka–Munk (K-M) function (equation 40) for turbid media by taking \(_\) as per Eq. 13&14 and \(_\) as per Eq. 12. Figure 17 shows the diffuse reflected spectra predicted using Kubelka–Munk function along with reflected spectra predicted using the adding-doubling technique, and compares them to original diffuse reflectance spectra from the data set. It is clear from the figure that the reflectance spectra predicted using the K-M function are close to the original spectra, compared to adding-doubling technique. Also, by taking \(_\) as per Eq. 13, the spectra is close to original data set compared to taking \(_\) as per Eq. 14. The spectra obtained from adding doubling simulation mimics the reflectance spectra of original data set (as shown in Fig. 16), however due to scaling factor, when those are compared with spectra obtained from K-M function and original data set, it looks like flattened. It is observed that, the Kubelka–Munk function also gives inaccurate results in the UV region and some portion of the visible region. Kubleka-Munk model is derived from a two flux approximation and assumes, light is fully diffused inside the sample and scattering is isotropic and homogeneous. In biological tissue like skin, light is strongly absorbed due to hemoglobin and melanin in that region, angular scattering is complex, multidimensional and anisotropic, and the assumptions often breakdown. Due to these, K-M function fails to accurately approximate the reflection spectra in UV–Vis region. In K-M function, the K and S, hence the ratio K/S does not accurately represent ratio of absorption and scattering coefficient \(_\)/\(_\). The overall slightly higher predicted reflection spectra from the K-M function may be attributed to that, as well as models simple two flux approximation.

As the diffuse reflected spectra by simulation and using Kubelka–Munk function has been estimated by using of \(_\) as per Eq. 13&14 and \(_\) as per Eq. 12, those equations can be used for the assessment of human forearm skin optics related parameters in the wavelength range of 250–2500 nm.

Fig. 17

Comparison of predicted simulated spectra by A-D technique and K-M function with actual dataset mean

Experimental validation

Agar based skin phantoms for Fitzpatric types I-VI were prepared and absorption coefficients (\(_\)) were determined by Eq. 12 and normal Beer-Lambert law. The materials and their taken amounts for fabrication of the phantoms are given in Table 7. Agar powder has been used as base material. Titanium dioxide (TiO2) has been used as scattering component and coffee powder as epidermal melanin as it has melanoidin which is of brown color and mimics light absorption of melanin especially in 400 nm -700 nm range. Hemoglobin has been used for dermal absorption of light. To prepare the phantoms, firstly 1.6 g agar powder was dissolved in 100 mL deionized water kept in six separate Erlenmeyer flask and the solutions were stirred for 15 min at 90 °C. Then 0.1 g TiO2 was added to each solution and stirred for another 10 min at the same temperature. After that different amount of coffee powder (1 mg/ mL to 12 mg/mL) were added to different solutions and additional 15 min stirring were done. The solutions were then cooled to 45 °C, and 0.75 g bovine hemoglobin (5% blood volume in dermis) was added to each solution and stirred for 15 min at that temperature. Throughout the process, mild stirring speed of 280 rpm was maintained to avoid air bubble formation. The solutions were then poured to petri dishes and kept at 4 °C for overnight. Figure 18 shows the prepared Fitzpatrick types I-VI phantoms.

Table 7 Materials and their amounts for Fitzpatrick Type I-VI skin phantomFig. 18

Fitzpatrick types I-VI phantoms

Vis–NIR spectra of each phantom has been taken by Metrohm NIRS DS2500 analyzer in 400 nm to 2500 nm wavelength range in diffuse reflectance mode. Figure 19a and b shows absorption spectra and absorption coefficient of the phantoms which mimics spectral absorption and absorption coefficient pattern of human skin (Fig. 4a and b). The individual phantoms absorption coefficients are shown in background of mean absorption coefficient in cyan color with thick line. For absorption spectra acquired by the Vis-NIRS instrument, the absorption coefficient of each phantom has been calculated by Eq. 12. Further, an experimental set up (Fig. 20) was built to evaluate the absorption coefficient of the Fitzpatrick Type I-VI skin phantoms. A 100 W mercury short arc lamp (OmniCure SERIES 1000) with tunable optical power output has been used as light source. Different filters has been used to get the light of 400 nm, 420 nm, 440 nm, 460 nm, 480 nm, 500 nm, 520 nm, 540 nm, 560 nm, 580 nm, 600 nm and 620 nm from the light source. Digital handheld laser power meter (COHERENT PowerMax) with detector aperature area 30 mm was placed beneath the samples to measure intensity of the transmitted light. The distance between light source and optical filter, optical filter and phantom sample, and, the phantom sample and detector were kept fixed throughout the experiment. The transmitted power of filtered light through blank petri dish was taken as reference power Po. Petri dish having different Fitzpatrick skin phantoms of thickness 0.128 cm (approx.) was kept in the light path and the transmitted power P was measured for respective samples. Beer-Lambert law relationship Eq. 41 was used to calculate the absorption coefficients of the phantom samples at the chosen wavelengths.

$$_=-\frac\text\left[ \frac_}\right]$$

(41)

Fig. 19

a Absorption spectra of Fitzpatrick skin phantoms b The absorption coefficient of Fitzpatrick skin phantoms versus wavelength

Fig. 20

Schematic of the experimental setup

Table 8 shows absorption coefficients of the prepared Fitzpatrick type I skin phantom, calculated using equation 41. Calculated absorption coefficients of Fitzpatric skin phantom II-VI are shown in Table S1-S5 (supplementary material). Table 9 shows mean absorption coefficients for the phantoms at the selected wavelengths, using both techniques and, compares and contrasts them.

Table 8 The power intensities of different wavelength light through Fitzpatrick type I phantom and calculated absorption coefficientTable 9 Comparison of absorption coefficient for visible spectra acquired by commercial Vis-NIRS instrument and the experimental setup

It is observed that the absorption coefficients at the wavelengths, with both techniques, are nearly close to each other. The errors are may be due to the approximation of the phantoms thickness during assessment by experimental set up and measurements artifacts.

Appendix B

The absorption coefficients and reduced scattering coefficients calculated in current study are compared with previous studies, as shown in below Table 10.

Table 10 Comparison of absorption coefficient (cm−1) and reduced scattering coefficient (cm−1) for human skin at selected wavelengths between the present study and existing literature

The small variations in values are may be due to differences in anatomical site, melanin and hemoglobin concentration and, water and fat fraction or experimental setup. Nevertheless, the consistency supports the reliability of the present findings.