2.1.

Skin

The cutaneous tissues are usually divided into three distinct layers, namely the stratum corneum, epidermis, and dermis, from the outermost to the innermost.^24,25 The epidermis can be further subdivided into four sublayers: stratum lucidum, stratum granulosum, stratum spinosum, and stratum basale. The dermis, in turn, can be subdivided into two sublayers: papillary dermis and reticular dermis. It is worth noting that the stratum lucidum is a clear layer found only in thick skin regions, such as the palms and soles.²⁴ For this reason, it is usually not considered in skin optics investigations.

The absorption of light traversing the stratum corneum and epidermis is dominated by melanin with a minor contribution from beta-carotene [Fig. 2(b)].²⁵ Melanin is synthesized and preferentially concentrated in the stratum basale.²⁴ Melanin can be classified as either brown-black eumelanin or yellow-red pheomelanin [Fig. 2(a)], the former of which has a significantly higher concentration in skin. Melanin can be dispersed throughout the epidermis in colloidal form or clustered within organelles called melanosomes.²⁶

In the skin, light is mostly scattered by heterogeneous structures, such as cells, organelles, and fibers.²⁵ Goniometric measured data²⁷ and empirical observations available in the literature^4,28 indicate that both the stratum corneum and epidermis are characterized by a strong forward scattering behavior exhibiting slight variations with wavelength. Scattering in the epidermis is dominated by melanosomes,²⁹ which can be described as particles with the shape of a prolate spheroid³⁰ and a melanin concentration of 17.9% to 72.4%.²⁹ In lightly pigmented specimens, melanosomes can occur in groups surrounded by a transparent membrane known as melanosome complexes.^30,31 In addition to empirical observations indicating that a significant amount of Rayleigh scattering does not occur in the stratum corneum and epidermis,^4,5 it has been shown that the consideration of scattering caused by melanosomes and melanosome complexes is sufficient to predictively model the scattering behavior of the stratum corneum and epidermis in the blue region of the light spectrum.¹⁴

The outermost dermal layer, the papillary dermis, is composed of loosely packed collagen and reticulin fibers, which, in turn, are composed of smaller-sized fibrils. Since these fibrils are smaller in magnitude than the wavelength of light within the visible spectrum, any scattering caused by these structures would likely follow the Rayleigh scattering theory.⁸ The innermost dermal layer, the reticular dermis, is composed of dense irregular connective tissue formed by densely packed collagenous fibers arranged in a tight interwoven pattern.²⁴ A network of blood vessels also runs throughout the dermis with wider vessels in the reticular dermis. Blood-borne pigments within these vessels [Figs. 2(b)–2(d)] dominate the dermis’s absorptive behavior in the visible domain.³² Furthermore, the diffusion of red light traversing the reticular dermis³³ results from scattering by the dense collagen fibers (following a Mie scattering behavior⁹) and the wider blood vessels. Since this scattering is significant enough to diffuse the light,³³ it would likely dominate any Rayleigh scattering that may occur in the reticular dermis.

2.2.

Blood

This highly specialized connective tissue is composed of plasma and various types of cells known as formed elements.³⁴ Red blood cells, or erythrocytes, make up 99% of the formed elements in the blood, and each one consists of a thin membrane encapsulating a hemoglobin solution. The attenuation of light by red blood cells exceeds that of other blood components by 2 to 3 orders of magnitude.³⁴ The absorption of light by red blood cells in the visible domain is primarily attributed to the hemoglobins [Figs. 2(c) and 2(d)], and the scattering of light is determined by their shape and orientation.³⁵

The orientation of red blood cells in a flowing blood sample can change depending on its shear rate (i.e., the velocity gradient in the direction normal to the flow).³⁶ At low, intermediate, and high shear rates, the red blood cells are predominantly randomly oriented, rolling, or aligned with the direction of the flow, respectively.³⁶ When the volume fraction of blood occupied by formed elements, known as hematocrit, is above 40%, the alignment of the red blood cells becomes more pronounced.

2.3.

Hypodermis

This subcutaneous tissue is located immediately beneath the skin except for certain regions of the body (e.g., eyelids) in which it is not present.²⁴ It consists mostly of white fat cells that contain lipidic vesicles.³⁷ The size of these spherical droplets of lipids is larger than that of typical scatterers.³⁸ This may explain the strong backscattering behavior of this adipose tissue, comparable with the dermis’s behavior and much stronger than the behavior of other types of tissues like cartilages and muscles, which has been verified in actual experiments.³⁹ The most abundant light absorbers found in the hypodermis, namely water (10% to 30%³⁷) and lipids (70% to 90%³⁷), have a negligible impact in the visible domain.^40–43 Due to the strong absorptive properties of cutaneous melanin in the blue region of the light spectrum, only a negligible amount of light in this region is likely to reach the hypodermis.²⁵ Experiments by Dinish et al.⁴⁴ show that 90% to 95% of light in the red and green regions of the light spectrum are reflected by white adipose tissue. Accordingly, the majority of the visible light that reaches the hypodermis is likely to be scattered by the droplets of lipids and remitted into the reticular dermis.⁴⁵

2.4.

Vein Wall

Vein walls are composed of elastin, collagen, and muscle tissue.⁴⁶ The absorption of visible light within a vein wall is dominated by hemoglobin in the muscle tissue [Figs. 2(c) and 2(d)].⁴⁷ It is also possible that light traversing a vein wall may be scattered by its heterogeneous structures.

2.5.

Detour and Sieve Effects

Skin and blood are both turbid mediums characterized by their heterogeneously distributed, pigment-containing structures. Notably, the attenuating properties of melanosomes and red blood cells play an important role in the optical processes of skin and blood, respectively. When light traverses a turbid medium, scattering by these structures can increase its optical pathlength, which results in an increased probability of absorption. Conversely, light may not encounter any pigment-containing structure and, therefore, has a minimal chance of absorption. These phenomena are, respectively, known as the detour effect⁴⁸ and the sieve effect.⁴⁹ The combined contribution of these effects to the overall absorptive behavior of a turbid medium depends on the properties and distribution of the medium’s pigment-containing structures.

3.1.

Model Background and Setup

In our in silico experiments, we use HyLIoS¹⁴ and CLBlood^15,16 to simulate the interactions of light with skin and flowing venous blood, respectively. Both of these models have been evaluated extensively against measured data in their original publications.^14,15 CLBlood was also recently updated and re-evaluated.¹⁶

HyLIoS and CLBlood are both first-principles models that employ measured optical properties of constituent materials to control the propagation and attenuation of light within human skin and blood. Although this paper is focused on light in the visible domain, it is worth noting that both models consider all constituent materials with a significant contribution to light attenuation in the ultraviolet, visible, and near-infrared domains. Measured optical properties for these constituent materials were obtained from scientific literature and made available online.^50,51 References to this data can also be found in the models’ original publications.^14,15 Additionally, HyLIoS and CLBlood account for sieve and detour effects by individually distributing melanosomes throughout the epidermis and red blood cells throughout the plasma, respectively. This includes the grouping of melanosomes within melanosome complexes. As pointed out by Lister et al.,¹⁷ the size and distribution of these melanin-containing structures have been often overlooked when simulating their effects on skin color.

In our experiments, we computed modeled reflectance data for skin specimens with and without a vein immediately beneath the skin. Henceforth, we will refer to these specimens as skin specimens with and without a subcutaneous vein. The representation of these specimens in our modeling framework is described in the following sections.

3.2.

Skin Specimen without a Subcutaneous Vein

For the skin specimens without a subcutaneous vein, we used the HyLIoS model, as described in its original publication.¹⁴ In particular, there is a layer below the reticular dermis that represents the hypodermis [Fig. 3(a)]. We remark, as outlined in Sec. 2.3, that the amount of light in the blue region reaching this layer is negligible due to the cutaneous pigmentation²⁵ and most of the light in the green and red regions is reflected back.⁴⁴ Thus, by reflecting the visible light that may reach the hypodermis, a strategy also employed by several relevant works on skin color reproduction (e.g., the interested reader is referred to the works by Cotton and Claridge¹⁰ and by Doi and Tominaga⁵²), one can minimize the introduction of undue computational overhead in the simulations without reducing the fidelity of the simulations. Accordingly, we assume that visible light reaching the hypodermis is diffusely reflected back into the reticular dermis. The diffusion distribution in this case accounts for the overall scattering behaviors of both reticular dermis³³ and hypodermis,³⁹ as also outlined earlier (Secs. 2.1 and 2.3). The implications of this simulation choice are further discussed in Sec. 6.5.

Fig. 3

Diagrams depicting the layers considered in our experimental framework for skin specimens: (a) without a subcutaneous vein and (b) with a subcutaneous vein.

3.3.

Skin Specimen with a Subcutaneous Vein

For the skin specimens with a subcutaneous vein, the hypodermis layer is replaced by the vein [Fig. 3(b)]. We use a three layer model to represent the vein [Fig. 3(b)]. The first and third layers represent the vein wall and the second layer employs CLBlood to account for flowing venous blood. When light encounters the boundary of a layer, a Fresnel test²⁵ is performed to determine whether the light is reflected or transmitted.

We do not consider any absorption or scattering of light within the vein wall. In addition, any light that is transmitted below the bottom vein wall layer is discarded. The scientific reasoning behind each of these simulation choices, which were also implemented to avoid undue computational overhead without reducing the fidelity of our simulations, is further discussed in Secs. 6.1 and 6.6, respectively.

3.4.

Vein Precomputation

Since we know that visible light transmitted by the dermis is diffusely distributed when considering a typical physiological thickness,³³ we can assume a uniform distribution for the angle of incidence of all light that encounters the vein. This assumption allows us to precompute the probability that light is reflected by the vein using the representation outlined in Sec. 3.3. To obtain this probability, we first compute the reflectance of the vein considering polar angles in the range [0 deg, 90 deg) at each 5-deg interval and azimuthal angles in the range [0 deg, 360 deg) at each 90-deg interval. For each angle of incidence, the reflectance calculations are performed considering ${10}^5$ sample rays per wavelength. Since we assume that the light entering the vein is uniformly distributed, we average the reflectance values across all samples at each wavelength to get the overall probability of light being reflected by the vein. When performing our experiments, we employ this precomputed probability to determine whether a light ray that encounters the vein will be reflected diffusely or discarded (i.e., absorbed or transmitted).

4.1.

Experimental Sets

We performed three sets of in silico experiments. In experiment set #1 (Figs. 6 and 7), we consider a baseline skin specimen characterized by the parameter values presented in Tables 1 and 2. Accompanying each of these values is a list of references that provide their physiological basis. We remark that the thickness of our skin specimen is within a measured range of thickness for skin on the dorsum of the hand.⁵⁴

Table 1

HyLIoS parameters employed in the general characterization of all skin specimens considered in this investigation. Note that, in the last two rows, the scatterers correspond to the fibrils in the papillary dermis.

Table 2

HyLIoS melanin and blood content parameters that are modified to produce different skin specimen characterizations. The presented values correspond to the baseline skin specimen.

In experiment set #2 (Fig. 8), we multiply the melanin content and dermal blood content parameters in Table 2 by $f_{\mathrm{mel}}\in \{1.0,1.5,2.0,2.5\}$ and $f_{\mathrm{bl}}\in \{1.0,1.5,2.0,2.5\}$, respectively, to obtain additional skin specimens. For each experiment, the mean epidermal melanin content is in the range 1% to 3% and dermal blood content is in the range 2% to 6%, corresponding to the physiological characterization of a lightly pigmented specimen.⁶⁴ Our justification for focusing on lightly pigmented specimens in our investigation is provided in Sec. 6.3.

In experiment set #3 (Figs. 9–13), we varied the radius of the fibrils in the papillary dermis and the thickness of the papillary dermis. We considered fibril radius values in the range 20 to 60 nm and papillary dermis thickness values in the range from 0.0025 to 0.02 cm. These ranges are based on measurements of collagen and reticulin fibril radius^7,74 as well as reported values for papillary dermis thickness.^4,63,74,91

4.2.

Vein-Related Data

In our precomputation (Sec. 3.4) of the probability of light being reflected by the vein [$p_{\mathrm{vein}}^r(\lambda )$], we considered the blood characterization data presented in Table 3. It is worth noting that the thickness of the blood sample, $1270\mu \mathrm{m}$, corresponds to the lumen diameter of a vein on the dorsum of the hand.⁹⁴ The refractive index selected for the vein wall is 1.39, which corresponds to a measured value for the vena femoralis.⁹⁹ The resulting curve for $p_{\mathrm{vein}}^r(\lambda )$ is presented in Fig. 4.

Table 3

CLBlood parameters employed in the characterization of the blood flowing through the subcutaneous vein.

Fig. 4

The precomputed probability of light being reflected by the subcutaneous vein specimen ($p_{\mathrm{vein}}^r$) employed in our experiments. This curve was computed using the procedure outlined in Sec. 3.4, considering the blood characterization data presented in Table 3.

4.3.

Rayleigh Scattering Simulation

Within the HyLIoS’ formulation,¹⁴ the direction of a light ray traversing the papillary dermis can be perturbed considering the Rayleigh scattering distribution.¹⁰⁰ Ideally, one would like to consider the exact geometric shape of the fibrils and directly simulate the interaction of light with each one of them. However, such an approach not only would represent a challenging project in itself, deserving a publication in its own right like the work of Arifler et al.¹⁰¹ on ocular stroma, but also would be severely constrained by data scarcity. Alternatively, one needs to resort to an approximation and take into account its practical implications.

In the work by Saidi et al.,⁸ the authors employed infinitely long cylinders placed parallel to the skin surface to represent fibers eliciting Mie scattering. Incidentally, their algorithm was based on the work by Bohren and Huffmann.¹⁰² Although a similar approximation could be potentially employed for the scattering elicited by fibrils, a few aspects have to be taken into account when assessing its suitability for the problem at hand.

As pointed out by Bohren,¹⁰³ scattering by spheres and by infinitely long circular cylinders illuminated normally to their axes are in some ways similar. For example, spherical Bessel functions employed in the formulation of sphere scattering coefficients correspond to cylindrical Bessel functions employed in the formulation of cylinder scattering coefficients. Different from a sphere, however, an infinitely long cylinder cannot be confined within a finite volume, which impacts the way the scattering varies with the radius of the scatterers.¹⁰³

Moreover, recall that Rayleigh scattering is dependent on the wavelength of light to the power of −4, which corresponds to the Rayleigh limit of the Mie scattering theory,^9,100 and it is characterized by forward and backward lobes leading to an isotropic behavior on average.^8,100 As also pointed out by Bohren,¹⁰³ the use of other type of approximations (e.g., cylinders) in scattering formulations originally proposed considering homogeneous spheres, like the Rayleigh theory, which is included in the Mie theory, would be “risky,” especially for computing scattering toward the backward direction.

If the fibrils present in the papillary dermis were characterized by a preferred orientation, this might represent an argument in favor of a cylindrical approximation. However, the irregular distribution of collagen fibrils suggested by Oettlé⁶ was corroborated by the electron microscope images provided by Arao et al.,⁷ showing no preferred orientation for these structures in the papillary dermis.

Hence, representing the scatterers of interest, namely the fibrils, by spheres, albeit having the shortcomings of any approximation, provides the best fidelity/cost ratio in comparison with other alternatives, like cylinders, since it allows us to avoid undue complexity and the introduction of bias in the simulation of backscattering, as indicated by Bohren.¹⁰³ For these reasons, we have selected the following formula for the scattering coefficient controlling the probability of Rayleigh scattering events:^14,100,104

To deactivate Rayleigh scattering, we simply set ${\eta }_{\mathrm{f}}={\eta }_{\mathrm{pd}}$, which results in ${\mu }_s^R=0$ and, hence, a 0% probability that a Rayleigh scattering event occurs. The values for $v_{\mathrm{f}}$, ${\eta }_{\mathrm{pd}}$, and the baseline ${\eta }_{\mathrm{f}}$ are given in Table 1. The baseline value for $r$ is given in Table 2.

It is worth noting that the thickness of the papillary dermis ($t_{\mathrm{pd}}$) also indirectly affects how much Rayleigh scattering will occur. This can be explained by the fact that the probability of light being attenuated while traversing a given layer tends to be higher for thicker layers.¹⁰⁵

4.4.

Swatch Generation

Swatches are used in this work to approximate the appearance of a given skin specimen under distinct illumination conditions. To generate the color of a swatch, the skin specimen’s modeled reflectance curve is convolved with the selected illuminant’s relative spectral power distribution and the broad spectral response of the human photoreceptors.¹⁰⁶ The last step involves a standard CIE XYZ to sRGB conversion procedure,¹⁰⁷ which includes a white balancing calculation considering the selected illuminant.

In our investigation, we considered four standard CIE (Commission Internationale de l’Eclairage) illuminants, namely D50, D65, A, and F2,¹⁰⁶ whose relative spectral power distributions are presented in Fig. 5. D50 and D65 approximate daylight at different times of day, A corresponds to an incandescent light, and F2 represents a “cool white fluorescent” light typical of an office environment.¹⁰⁶ Since F2 was published independently of the other illuminants and the employed data were provided in relative units, we decided to increase its intensity in the swatch generation process to get a similar qualitative brightness across the swatches. We selected D65 for the experimental instances in which we considered a single illuminant (Figs. 8 and 13).

Fig. 5

Relative spectral power distribution of each CIE standard illuminant¹⁰⁶ employed in the swatch generation process described in Sec. 4.4.

After computing the color of a swatch, we apply a grayscale texture to it to make the synthetic depiction of skin more realistic. In Figs. 8 and 13, each swatch depicts a zoomed-out view of skin with a vein running beneath it. Note that, in these swatches, not only the skin area above the vein but also the area around it are rendered with their corresponding colors. This allows us to implicitly account for color perception phenomena related to the retinex theory.¹⁰⁸ This theory proposes that the spectral remission from a given area alone does not determine its color, i.e., one must also consider the remission from surrounding areas. According to Kienle et al.,¹¹ retinex theory should be taken into account when assessing the perception of a vein’s appearance as seen by a human observer.

4.5.

Online Reproducibility

The experiments in this paper can be reproduced with our online versions of HyLIoS¹⁰⁹ and CLBlood,¹¹⁰ which are deployed using a custom model distribution system.^111,112 This system allows a user to perform their own experiments with a specified skin specimen characterization and experimental setup. The online version of HyLIoS has been updated to facilitate the reproduction of the reflectance of a skin specimen with a subcutaneous vein considering a chosen venous blood oxygenation. Additionally, we have made it possible to reproduce the reflectance of an isolated vein from a single angle of incidence using our online version of CLBlood.¹¹⁰

5.1.

Experiment Set #1

For our first set of experimental results, we explore the effect of disabling Rayleigh scattering in the papillary dermis on the reflectance and appearance of the baseline skin specimen with and without a subcutaneous vein. First, we examine the reflectance results, which are presented in Fig. 6. When Rayleigh scattering is enabled [Fig. 6(a)], there is a smaller difference between the reflectance curves of the skin specimens with and without a subcutaneous vein in the short (blue) wavelength range. In particular, the difference between the two curves is negligible in the region near 400 nm. Given the near zero probability of light being reflected by the vein in this region (Fig. 4), the light must have been scattered out of the skin before it could reach the vein.

Fig. 6

Modelled reflectance curves obtained for the baseline skin specimen with and without a subcutaneous vein: (a) with Rayleigh scattering and (b) without Rayleigh scattering.

Since this behavior could potentially be attributed to Rayleigh scattering, let us examine the reflectance results for the experiments without Rayleigh scattering presented in Fig. 6(b). We can observe that disabling Rayleigh scattering reduces the reflectance significantly for both specimens in the region near 400 nm. Furthermore, the amount of reflected light in this region is lower for the skin specimen with a subcutaneous vein. This indicates that the absence of Rayleigh scattering allowed more light to reach and, hence, be absorbed by the vein.

Now, let us examine Fig. 7 to see how these differences in reflectance translate to appearance. The appearance swatches presented in Fig. 7(a) were generated using the reflectance data from the experiments with Rayleigh scattering [Fig. 6(a)] and considering several illuminants. The colors depicted in these swatches generally match our expectations for the appearance of skin with and without a subcutaneous vein. Accordingly, the RGB (red, green, blue) color triplets used to generate these swatches, presented in Table 4, quantitatively show that the colors elicited by the skin specimen with a subcutaneous vein have a blue component that exceeds in magnitude its red and green counterparts (for all considered illuminants). Conversely, the swatches presented in Fig. 7(b), computed considering the reflectance data from the experiments without Rayleigh scattering [Fig. 6(b)], present a brownish-red appearance for the skin specimen with a subcutaneous vein. For this specimen, we can also observe that the red component of the RGB triplets presented in Table 5 exceeds in magnitude its green and blue counterparts.

Fig. 7

Generated swatches depicting the effect of Rayleigh scattering in the papillary dermis on the appearance of the baseline skin specimen without (top) and with (bottom) a subcutaneous vein. From left to right, the swatches in each subfigure were generated considering CIE standard illuminants D50, D65, A, and F2: (a) with Rayleigh scattering and (b) without Rayleigh scattering.

Table 4

With Rayleigh scattering, the RGB (red, green, blue) triplets employed in the generation of the swatches presented in Fig. 7. Each RGB triplet is computed using the modelled reflectance data for the baseline skin specimen with and without a subcutaneous vein considering CIE standard illuminants D50, D65, A, and F2, as described in Sec. 4.4.

Table 5

Without Rayleigh scattering, the RGB (red, green, blue) triplets employed in the generation of the swatches presented in Fig. 7. Each RGB triplet is computed using the modelled reflectance data for the baseline skin specimen with and without a subcutaneous vein considering CIE standard illuminants D50, D65, A, and F2, as described in Sec. 4.4.

5.2.

Experiment Set #2

Swatches belonging to our second set of experimental results are presented in Fig. 8. They show that the role of Rayleigh scattering in the appearance of a skin specimen with a subcutaneous vein is consistent when considering different levels of melanin content and dermal blood content.

Fig. 8

Generated swatches depicting several skin specimens with a subcutaneous vein running beneath them considering the CIE standard illuminant D65. Distinct skin specimens were obtained by multiplying melanin and blood content (Table 2) by $f_{\mathrm{mel}}$ and $f_{\mathrm{bl}}$, respectively. Left to right: $f_{\mathrm{mel}}=1.0$, 1.5, 2.0, 2.5. Top to bottom: $f_{\mathrm{bl}}=1.0$, 1.5, 2.0, 2.5: (a) with Rayleigh scattering and (b) without Rayleigh scattering. Note that the baseline skin specimen corresponds to $f_{\mathrm{mel}}=1.0$ and $f_{\mathrm{bl}}=1.0$.

5.3.

Experiment Set #3

Since our results indicate that Rayleigh scattering in the papillary dermis contributes to the appearance of veins, let us examine the effect of varying the values of parameters that have an impact on the amount of Rayleigh scattering occurring in the papillary dermis.

The results presented in Fig. 9 show the effect of varying the radius of the fibrils in the papillary dermis on the reflectance of skin specimens with and without a subcutaneous vein. For the skin specimens without a subcutaneous vein [Fig. 9(a)], we can see that a higher fibril radius corresponds to a lower reflectance, except in the region near 400 nm. For the skin specimen with a subcutaneous vein [Fig. 9(b)], a higher fibril radius corresponds to a higher reflectance across the visible spectrum. We can also observe in Fig. 9(b) that the difference between the reflectance of the baseline skin specimen ($r=40\mathrm{nm}$) and the reflectance of a skin specimen with a higher fibril radius ($r=50$ and 60 nm) is the greatest in the middle of the visible spectrum, a region corresponding to what a human observer would perceive as green color. In both subfigures [Figs. 9(a) and 9(b)], we can see that skin specimens with a small fibril radius ($r=20$ and 30 nm) have reflectance curves that resemble the reflectance curves presented in Fig. 6(b), which correspond to the experiments considering the baseline skin specimen without any Rayleigh scattering.

Fig. 9

Modelled reflectance curves for skin specimens considering varying radius values of the fibrils in the papillary dermis ($r$): (a) without a subcutaneous vein and (b) with a subcutaneous vein. Note that the baseline skin specimen has $r=40\mathrm{nm}$.

Now, let us examine Fig. 10 to see how these observations translate to appearance. In particular, when examining the appearance of the skin specimens with a subcutaneous vein, we can see that decreasing the fibril radius below the baseline value darkens the appearance of the vein and provides a brownish-red or purple coloration. Alternatively, increasing the fibril radius above the baseline value results in a more greenish appearance.

Fig. 10

Generated swatches depicting variations in skin appearance when modifying the radius of the fibrils in the papillary dermis ($r$). For comparison purposes, we present swatches for specimens without (top) and with (bottom) a subcutaneous vein in each subfigure. For each subfigure, from left to right: $r=20$, 30, 40, 50, 60 nm. The swatches were generated considering CIE standard illuminants: (a) D50, (b) D65, (c) A, and (d) F2. Note that the baseline skin specimen has $r=40\hspace{0.17em}\mathrm{nm}$.

We also performed experiments in which we varied the thickness of the papillary dermis. In the corresponding reflectance curves, presented in Fig. 11, we can observe similar trends to the experiments in which we varied the fibril radius (Fig. 9). However, the changes in reflectance are lower in magnitude when varying the papillary dermis thickness given the parameter values that we have selected. Consequently, the corresponding appearance swatches in Fig. 12 show variations in coloration similar, albeit less prominent, to the results presented in Fig. 10.

Fig. 11

Modelled reflectance curves for skin specimens considering varying papillary dermis thickness ($t_{\mathrm{pd}}$): (a) without a subcutaneous vein and (b) with a subcutaneous vein. Note that the baseline skin specimen has $t_{\mathrm{pd}}=0.02\mathrm{cm}$.

Fig. 12

Generated swatches depicting variations in skin appearance when modifying the thickness of the papillary dermis ($t_{\mathrm{pd}}$). For comparison purposes, we present swatches for specimens without (top) and with (bottom) a subcutaneous vein in each subfigure. For each subfigure, from left to right: $t_{\mathrm{pd}}=0.0025$, 0.005, 0.01, 0.015, 0.02 cm. The swatches were generated considering CIE standard illuminants (a) D50, (b) D65, (c) A, and (d) F2. Note that the baseline skin specimen has $t_{\mathrm{pd}}=0.02\mathrm{cm}$.

Now, let us examine the combined effect of varying both the radius of the fibrils and the papillary dermis thickness in Fig. 13. When examining the swatches that consider Rayleigh scattering [Fig. 13(a)], we can see that increasing one parameter value while decreasing the other can lead to more subtle changes in the appearance of the specimen, while increasing or decreasing both parameter values simultaneously can lead to more drastic changes in its appearance. On the other hand, when examining the swatches generated with Rayleigh scattering disabled [Fig. 13(b)], we can verify that similar changes do not take place. This demonstrates that any change in the appearance of a skin specimen with a subcutaneous vein resulting from varying the values of fibril radius and papillary dermis thickness is primarily determined by the occurrence of Rayleigh scattering in this tissue.

Fig. 13

Generated swatches depicting several skin specimens with a subcutaneous vein running beneath them considering the CIE standard illuminant D65. Distinct skin specimens were obtained by varying the radius of the fibrils in the papillary dermis ($r$) and the papillary dermis thickness ($t_{\mathrm{pd}}$). Left to right: $r=20$, 30, 40, 50, 60 nm. Top to bottom: $t_{\mathrm{pd}}=0.0025$ 0.0025, 0.005, 0.01, 0.015, 0.02 cm: (a) with Rayleigh scattering and (b) without Rayleigh scattering. Note that the baseline skin specimen has $r=40\mathrm{nm}$ and $t_{\mathrm{pd}}=0.02\mathrm{cm}$.

6.1.

Intermediate Tissues

In our experiments, we do not fully consider the attenuating properties of the vein wall. We also do not consider a hypodermis between the skin and the vein. In the human body, if a vein is deep enough, then scattering in the hypodermis ensures that the vein is not visible. Accordingly, it is possible that a less visually prominent vein may have a thin hypodermal layer between itself and the skin. Although these intermediate tissues may impact the reflectance of skin in general, it is unlikely that they can be the primary source of a vein’s bluish appearance given the optical properties of their main constituents. In the remainder of this subsection, we examine this claim in more detail.

The weak absorption of visible light in these tissues is dominated by hemoglobin.^45–47 Since hemoglobin is abundant in the subcutaneous vein, these tissues should have a minor impact on the overall absorption. However, if we were to consider the effect of hemoglobin in these tissues, it should be similar to increasing the hemoglobin concentration in the reticular dermis. As we can see in Fig. 8, this does not have a significant effect on the bluish appearance of the vein.

There is also a possibility that small heterogeneous structures in these intermediate tissues produce Rayleigh scattering. However, our experiments indicate that Rayleigh scattering in the skin is strong enough to elicit the bluish appearance of veins without the consideration of these tissues. This is corroborated by the observation that cyanotic fingertips would not appear bluish if Rayleigh scattering in the papillary dermis was not sufficient to prevent blue light from reaching the reticular dermis.¹² Since Rayleigh scattering in the skin occurs before light reaches the hypodermis or vein wall, any additional Rayleigh scattering in these subcutaneous tissues could only increase the, already sufficient, remittance of blue light.

6.2.

Vein Appearance

In this paper, we have been focusing on the bluish appearance of veins. As demonstrated by experiment set #3 (Figs. 9–13), a subcutaneous vein may also appear to be subjectively more green, purple, or gray than blue when considering different physiological states and illumination conditions. These findings indicate that Rayleigh scattering occurring in the papillary dermis plays an important role in the appearance of any visible subcutaneous vein, even though its contribution may not always result in a bluish appearance.

6.3.

Pigmentation Level

We have chosen to focus on lightly pigmented specimens in our investigation since melanin absorbs light in the blue region significantly more than in other regions [Fig. 2(a)]. This indicates that additional melanin in the epidermis would increase the absorption of blue light before it reaches the papillary dermis, which would inhibit our ability to examine the effects of Rayleigh scattering caused by the collagen fibrils. We remark, however, that examining the appearance of veins for moderately and highly pigmented specimens may be a topic of interest for future investigation (Sec. 7).

6.4.

Venous Blood Characterization

We performed several additional experiments to assess the effect of varying the venous blood characterization on the appearance of a skin specimen with a subcutaneous vein. When experimenting with various oxygen saturation levels, we observed that variations within a typical physiological range did not have a noticeable impact on coloration. However, arterial oxygenation levels (e.g., 97%) resulted in a skin specimen with a subcutaneous vein reflecting more red light, which, in turn, elicited a purple appearance. Alternatively, extremely low levels of oxygenation levels (e.g., 0%) resulted in less reflected red light, which, in turn, elicited a more dominant blue coloration. Since this behavior is consistent with observations by Kienle et al.,¹¹ we excluded these results for the sake of conciseness. We also explored different distributions of red blood cell orientations in the flowing venous blood, corresponding to different shear rates. This did not noticeably affect the resulting coloration.

6.5.

Hypodermis Reflectance

In Sec. 3.2, we mentioned that in our simulations, visible light reaching the hypodermis is diffusely reflected back to the reticular dermis. We also noted that this simulation choice was based on reported morphological and optical characteristics of white adipose tissues (Sec. 2.3) and it is consistent with simulation strategies employed by related works.^10,52 We remark that most of our observations presented in this work (Secs. 5 and 6) correspond to a skin specimen with a subcutaneous vein beneath it. Moreover, the direct observation made for a skin specimen with a hypodermis beneath it (Sec. 5.1) focuses on a feature of skin reflectance curve (Fig. 6) in the blue region and emphasizes the light that does not reach the hypodermis. Hence, the outcomes of our investigation are not impaired by the simulation choice outlined above.

6.6.

Vein Transmittance

In Sec. 3.3, we mentioned that our simulations discard light rays that are transmitted by the vein. This strategy was motivated by the low probability of light being transmitted by the vein ($p_{\mathrm{vein}}^t$), depicted in the curve presented in Fig. 14(a). This curve was calculated using the same approach employed for precomputing $p_{\mathrm{vein}}^r$ (Fig. 4), described in Sec. 3.4. Now, let us consider the worst-case scenario, where 100% of light transmitted by the vein would be diffusely remitted by the tissue below. In this case, the amount of light transmitted by the vein and then subsequently remitted by it would be given by ${(p_{\mathrm{vein}}^t)}^2$, which is presented in Fig. 14(b). We note that the maximum value on this curve is 0.53% at 700 nm. This would represent a negligible contribution to the overall reflectance of a skin specimen with a subcutaneous vein (Fig. 6), notably when compared with the probability of light being reflected by the vein without considering a remission of light by tissues below the vein (Fig. 4). Hence, the decision of discarding the light rays that might be transmitted by the vein did not affect the observations derived from our in silico experiments, especially considering that ${(p_{\mathrm{vein}}^t)}^2$ values are even smaller for other wavelengths [Fig. 14(b)].

Fig. 14

Computed transmittance data for the subcutaneous vein specimen employed in our experiments. (a) Probability of light being transmitted by the vein ($p_{\mathrm{vein}}^t$). (b) Squared probability of light being transmitted by the vein [${(p_{\mathrm{vein}}^t)}^2$]. The values for $p_{\mathrm{vein}}^t$ were computed using an analogous procedure to the precomputation outlined in Sec. 3.4, considering the blood characterization data presented in Table 3.