## Abstract

This paper discusses the accuracy of the optical determination of the oxygenated and deoxygenated hemoglobin content of human skin under the influence of a melanin layer for a multi-wavelengths imager. The relation between the nonlinear results by Monte Carlo simulation (MCS) and the modified *Lambert Beer’s* law (MLB) is also clarified, emphasizing the importance of the absolute values of skin pigments and their influence on the mean path-length used in MLB. The fitting procedure of the MCS data to the actual skin spectra is shown to obtain the absolute values. It is also shown that once the proper mean path-lengths have been determined, MLB can be used fairly well within an accuracy of 80% compared with MCS. Images of oxygenated hemoglobin with a newly-developed four-wavelength camera are presented to demonstrate the advantages of a multi-wavelength system.

© Optical Society of America

## 1. Introduction

Diffuse reflection spectroscopy for skin is expected to have various applications for noninvasive diagnosis. In particular, recent advances in CCD detectors will facilitate the use of imaging spectroscopy for detecting local changes in oxygenated and deoxygenated hemoglobin, which are related to peripheral circulation diseases or injuries of the skin. In the visible wavelength range between 500 nm and 620 nm, optically absorbing pigments are limited to oxygenated hemoglobin and deoxygenated hemoglobin and melanin. Therefore, the development of a method that can extract the contents of three components from the obtained reflection spectrum of the skin is required. However, skin has a complex multi-layered structure and has strong optical scattering; the relation between the pigment concentration and measurable optical absorbance become nonlinear, which is not easy to predict. Many researchers have tried to solve this problem through a variety of approaches.

*San Wan et al*. (1981)^{1)} and *R. Rox Anderson et al*. (1981)^{2)} used the *Kubelka Munk* approach to deal with nonlinear diffuse reflection from the skin. *I.V. Meglinsky et al*.^{3)4)} used the Monte Carlo technique to determine where the skin signal comes from and to simulate human skin spectra assuming seven skin layers. *M. Shimada et al*.^{5)} discussed the use of modified *Lambert-Beer’s* law combined with multiple linear regression analysis. *M. J. C. Van Gemert et al*. (1989) ^{6)} summarized the optical properties of human skin, which can be used as the basic data to understand skin optics.

*K. H. Frank et al*. (1989)^{7)}, *G. B. Hanna et al*. (1995)^{8)} and *Y. Kakihana et al*. (1997)^{9)} reported skin hemoglobin oxygenation and its concentration using a micro lightguide spectrophotometer. *N. Tsumura et al*. discussed the imaging technology of hemoglobin and melanin using independent-component analysis^{10) 11)} or using an inverse optical scattering technique by multi-spectral camera ^{12)}.

The present paper reports on a new fitting method for extracting the contents of three components from the actual skin spectra. The Monte Carlo simulation (MCS) of the three-layered skin model is described to obtain nonlinear relations between the absorbance and concentration to be used for fitting. Then, the relation between the nonlinear MCS and the linear modified *Lambert Beer’s* law (MLB) is discussed. Finally, a series of typical images of oxygenated hemoglobin for the human foot, during and after arterial occlusion, are presented to demonstrate the advantages of the multi-wavelength system for skin analysis.

## 2. Human skin spectra to determine the absolute concentration of oxygenated and deoxygenated hemoglobin and melanin by fitting

The actual skin spectra, to be used in subsequent fitting analyses for finding absolute concentrations of hemoglobin and melanin, were measured using a spectrophotometer. Fig. 1(a) and (b) present the reflection spectra of the skin from the forearms of three Japanese volunteers. The ordinate of the spectra is the absorbance; i.e. log (*R _{o}/R*), where

*R*is the amount of reflected light from a sample and

*R*is that from barium sulfate powder as a white reference. Fig. 1(a) is for volunteer No.1 at rest as well as with varied skin oxygenation, while Fig. 1(b) is for volunteers No. 2 and No. 3 only at rest. In Fig. 1(a), in addition to the rest condition, a heating or occlusion action was employed to give various hemoglobin concentrations and saturations. First, the skin surface was heated to 43.5°C to increase the amount of oxygenated hemoglobin (oxyHb) and to decrease deoxygenated hemoglobin (deoxyHb). Secondly, the upper arm was occluded by a cuff at a pressure of 200 mmHg; closing both veins and arteries to decrease [oxyHb] and to increase [deoxyHb]. Finally, the pressure was released to reverse this change.

_{o}## 3. Monte Carlo simulation and the result of fitting

#### 3.1 Description of Monte Carlo simulation (MCS) and skin model

The purpose of the MCS is to obtain skin data for fitting by calculating the amount of reflected light from the skin as a function of the concentrations of three pigments: oxyHb, deoxyHb and melanin existing in the respective layers. Fig.2 presents a three-layered skin model used in the MCS^{6)}. Layer1 (epidermis) contains melanin, layer2 (dermis) contains oxyHb and deoxyHb, and layer3 (subcutaneous tissue) contains lipid. The depth of the three layers is assumed to be 0.06 mm, 1.00 mm, and 28.94 mm, respectively. Layer3 is sufficiently thick because the MCS photons only migrate down to the shallow part of this layer.

In order to reduce the number of variables, an absorption coefficient *µ _{a}* (mm

^{-1}) of layer2 was used in place of using two variables: concentration of oxyHb or [oxyHb] and the concentration of deoxyHb or [deoxyHb], noting that both the pigments are contained in the same layer. So the

*µ*of layer2 is defined as:

_{a}where *ε _{oxyHb}* and

*ε*are the millimolar absorptivities in the unit of mM

_{deoxyHb}^{-1}·cm

^{-1}, [oxyHb] and [deoxyHb] are in mM. Thus, the diffusely reflected amount of light is calculated as a function of

*µ*and the melanin concentration for six wavelengths. The refractive index of three-layers is assumed to be 1.34. The

_{a}*ε*and

_{oxyHb}*ε*are taken from the report of

_{deoxyHb}*W. G. Zijlstra et al*. (1991)

^{13)}and are shown in Fig.3 after being multiplied four times, because their molar weight is based on one of four subunits of a full hemoglobin molecule. The absorptivities of melanin ((mg/mL)

^{-1}·cm

^{-1}) are determined by our measurement and are also given in Fig.3. The scattering coefficients

*µ*and the anisotropy factor g of layer1 and 2 are the values;

_{s}*µ*: 41–60,

_{s}*g*: 0.768–0.823 depending on the wavelength according to the data reported by

*M. J. C Gemert et al*. (1989)

^{6)}. The MCS program used here is coded based on the method of

*B. C. Wilson and G. Adam*(1987)

^{14)}.

#### 3.2 Result of MCS

Fig.4 shows the nonlinear relation obtained by the MCS as the basic data for fitting. The four graphs correspond to the relation for 512 nm, 557 nm, 581 nm, and 619 nm, respectively. The ordinate (*Z*-axis) of the graph represents the absorbance by MCS. The absorbance is defined as log (*I _{0}/I_{r}*), where

*I*is the intensity of light illuminated and

_{0}*I*is that reflected. The

_{r}*X*-axis represents the

*µ*(mm

_{a}^{-1}) of layer2, which is related to the concentration of oxyHb and deoxyHb by equation (1). The

*Y*-axis directly represents the concentration of melanin (mg/mL). The surface plot of (

*X, Y, Z*) forms a curved surface, therefore, when the concentration of the hemoglobin or melanin varies, the point (

*X, Y, Z*) will move on the curved surface. The four red dots and yellow and blue dots on the graph stand for actual skin data obtained by the fitting procedure.

In order to use the above relation in the fitting, the absorbance (*Z*) ofMCS is presented by a cubic function (2) of *X* and *Y* for each wavelength.

where *X* is *µ _{a}* (mm

^{-1}) as defined in equation (1) and

*Y*is the concentration of melanin: [melanin]. The coefficients

*A*to

*J*are the numerically determined constants except for

*J*, and are shown in table 1. The last term

*J*is the offset that is common to all wavelengths. Then the fitting procedure can be stated to minimize the sum

*S*

by adjusting the four variables [oxyHb], [deoxyHb], [melanin] and *J* (offset). In the equation (3), abs*(i)* means the absorbance of the skin spectra at *i*th wavelength.

#### 3.3 Results of fitting to actual human spectra

The four variables obtained by the fitting are shown in Table 2. This table presents [totalHb] (the sum of [oxyHb] and [deoxyHb]) and oxygen saturation SO_{2} (100[oxyHb]/[totalHb]), in addition to the four variables. “Max error” listed at the bottom of the table, stands for the maximum of absorbance residues after fitting. This is the maximum difference between the solid curves and the six points in Fig.1.

## 4. Derivation of modified Lambert-Beer’s law from the cubic equation

The modified *Lambert-Beer’s* law^{15)} (MLB) was proposed to improve the original *Lambert-Beer’s* law by introducing a concept of *mean path-length*. The mean path-length, *d* is the average path-length of traveling photons due to multiple scattering. The MLB is usually written in the formula

where *ΔZ* is the change in absorbance caused by changes in the respective concentrations of Δ[oxyHb], Δ[deoxyHb], and Δ[melanin] and *d _{oxyHb}, d_{deoxyHb}* and

*d*are the

_{melanin}*d*values for oxyHb, deoxyHb and melanin, respectively. A method of relating the nonlinear relation obtained by MCS with the MLB follows.

Since the absorbance *Z* is the function of [oxyHb], [deoxyHb], [melanin] and *J*, in Eq. (2) the change in absorbance*ΔZ* is written as

as the first order approximation.

The multiplying factors such as *∂Z/∂[oxyHb]* in Eq.(5) are presented by the following formulas, remembering that *X*=*µ _{a}* and

*µ*is presented by Eq.(1),

_{a}By comparing (4) and (5) in combination with (6)(7)(8), we have as the mean path-lengths for the three pigments,

Partial derivatives by *X* and *Y* in (9) are given from Eq.(2) as follows;.

In order to determine ∂*Z*/∂*X* and ∂*Z*/∂*Y* in (10) and (11) we have to specify the values of *X* and *Y* of a proper subject. Thus, the calculated mean path-lengths for volunteer No.1 (at rest) and for volunteer No.3 (at rest) are shown in Table 3.

Using the mean path-length of volunteer No.1 in Table 3, Eq.(4) can be numerically calculated, becoming a simultaneous equation

which is the MLB equation that we intended to derive.

By solving Eq.(12), we have finally Eq.(13), which converts a set of changes in absorbance at four wavelengths to a set of changes in Δ[oxyHb], Δ[deoxyHb] and Δ[melanin].

If the *X* and *Y* values of volunteer No.3 are adopted, Eqs.(12) and (13) change to

## 5. Hemoglobin images for the human foot with a new four-wavelength optical imager

We developed a four-wavelength imaging system to detect changes in the oxygenated and deoxygenated hemoglobin in the skin.^{16)} The detector part consists of a CCD camera and a filter assembly for wavelength selection, and the light source part consists of a halogen or LED lamp. The filter assembly is made up of four interference filters set at a speed of 0.7 sec/filter. The selected wavelengths are 512 nm, 557 nm, 581 nm and 619 nm, which are the same as described in the above simulation. With this system, four images are obtained every 3 seconds and many sets of four images can be saved on a computer during a series of measurements.

In order to demonstrate the system’s advantages, an arterial occlusion was performed on the skin of human foot of a healthy volunteer. A cuff was fixed at the neck of the left foot. After resting for over 10 minutes, the cuff pressure was increased to 200 mmHg and maintained for 5 minutes so as to stop both venous and arterial blood flow, then the occlusion was released to re-start the blood flow. The camera was operated continuously during the above procedure and four wavelength images of the foot were obtained every 3 to 5 seconds. Then, these images were transformed into the image of [oxyHb] using the MLB method equation (13).

The test results are shown in the six images, from A to F in Fig.5, which visualizes rapid change in the local distribution of [oxyHb] starting from the end of a 5 minute occlusion. As a reference, the distribution of [oxyHb] just before occlusion was taken to be zero (green). When the amount of [oxyHb] increases, the color becomes red and it becomes blue when [oxyHb] decreases below zero. Figure A shows the image taken at the end of a five minute occlusion. The image is blue because it is the most deoxygenated situation. Figure B is the situation 6 seconds after the release of the occlusion when the blood had started to flow again. The increase in [oxyHb] was found to start at the position indicated with the arrow at the inner side of the foot. Figures C, D and E are the images at 13, 26, and 40 seconds after release, respectively. In figure C, D and E, the area showing increased [oxyHb] extends wider and wider, finally reaching maximum in figure E. Then it decreases again in figure F from the area indicated by the arrow. Finally, [oxyHb] gradually returns to the original level (green) although the figure is not shown here.

## 6 Discussion

#### 6.1 Discussion on the fitting result

From the fitting result given in table 2, we can see the total skin hemoglobin concentration at rest ranges from 33 to 61 µM and the oxygen saturation ranges from 61% to 73% for the three volunteers. After heating, there is a remarkable increase in total skin hemoglobin up to 56 µM and an increase in oxygen saturation up to 91% for volunteer No.1. This increase is due to arterialization of the skin and is considered to be qualitatively reasonable. After occlusion, a further increase is found in total hemoglobin to 76 µM in addition to a rapid decrease in oxygen saturation down to 48%. After release, a rapid increase in oxygen saturation up to 89%(at the maximum point) is found again due to the re-start of blood circulation. The behavior of the oxyHb and deoxyHb during the occlusion-release procedure was also reasonable.

The amount of melanin ranges from 0.5 to 0.95 mg/mL for the three volunteers. The quantitative values of oxyHb, deoxyHb and melanin obtained by fitting may change depending on the MCS model (assumed thickness of layers) and the values of the scattering coefficient and g factor.

There is another noteworthy feature concerning the change in melanin concentration during the heating or occlusion-release procedure. In the melanin data shown in table 2, there is significant variation in the values of columns 1, 2, 3 and 4, starting from 0.53 to 0.34 mg/mL. Because melanin is thought to be stable against these procedures, such variations are not reasonable. The cause of the variation could be due to the discrepancy in optical properties adopted in MCS or some error in measurement, etc. Further research is required to determine the cause of these variations.

#### 6.2 Comparison of MLB with MCS

In order to obtain *absolute* concentrations, a fitting procedure based on the MCS is required. On the other hand, where the *changes* in concentration from a particular value are concerned, both the MCS and the MLB are applicable, although in the latter case, the importance of absolute values are still emphasized to determine the mean path-length. The following discussion considers the accuracy of MLB compared with MCS, after a set of mean path-lengths has once been determined. As you can see in table 3, the mean path-length in the MLB equation varies depending on the *X* and *Y* of the subject. So, in this discussion the three cases are compared; the MCS and MLB with two different mean path-lengths. The MLB result with the path-length for volunteer No.1 is referred to as MLB(1), and that for volunteer No.3 is referred to as MLB(2).

Fig.6 represents the results of the comparison. The four graphs show the differences among these three methods for the changes in [oxyHb], [deoxyHb], [totalHb], and [melanin]. The three bars in a group stand for the result with MCS, MLB(1), and MLB(2), respectively. The numbers attached on the abscissa, 1 to 6, correspond to the numbers of the columns (1 to 6) in table 2, which describes the type of subject and the situation of the skin.

All the values in Fig.6 are the difference of the data for columns from “2” to “6” and a common data for column “1.” Note here that all the bars at column “1” are zero, since the values of volunteer No.1 (at rest) are regarded to be the reference.

Difference values with MCS are taken directly from table 2, as the values in the columns “2” to “6” subtracted by a common value in the column “1.”

On the other hand, difference values with MLB(1), or MLB(2) are obtained by Eq.(13) or (13-1) using *ΔZ*(512 nm), *ΔZ*(557 nm), *ΔZ*(581 nm) and *ΔZ*(619 nm), which are the difference in original absorbance between the spectra of 2, 3, 4, 5 or 6 and the common spectrum 1 for volunteer No.1 in Fig.1.

There are some differences in MCS and MLB(1) and MLB(2) with respect to the changes in [oxyHb], [deoxyHb] and [totalHb]. However, these differences in the three methods are found in less than 20% of the changes in MCS, so MLB is a useful method as far as the relative changes are concerned.

In addition, an interesting fact is observed in the trends of the MLB(1), MCS and MLB(2). MLB(2) tends to exhibit the largest changes while MCS lies in the middle and MLB(1) shows the smallest changes. This is understood if we note in Fig.4 that point 6 always shows higher absorbance than point 1. Remember, point 1 corresponds to MLB(1), and point 6 corresponds to MLB(2). And, also note that higher absorbance causes a smaller slope on the surface and this smaller slope results in a smaller mean path-length. Comparison of tables 3(a) and 3(b) supports this fact. Finally, a smaller mean path-length causes larger numerical factors in equations (13) and (13-1). Thus, the differences of 20% appearing in Fig. 6 are due to the variation of absorbance of the skin.

## 7. Conclusion

A systematic analysis was conducted to show the relation between the nonlinear results from the Monte Carlo simulation (MCS) and the results from the modified *Lambert Beer’s* law (MLB). The importance of the absolute concentrations of skin pigments are emphasized to determine the mean path-lengths required in MLB. The result of the MCS with a three-layered skin model were presented by a cubic function and the absolute concentrations of skin pigments were obtained by the fitting procedure to the actual skin spectra. Thus, the obtained [totalHb] at rest ranged from 33 to 61 µM and the oxygen saturation ranged from 61% to 73% for the three volunteers. The mean path-lengths of the dermis layer were from 0.5 to 1 mm for wavelengths less than 581 nm and about 3 mm for 619 nm. It is also shown that once the proper mean path-lengths are determined, the MLB could be used fairly well within an accuracy of 80% compared with the MCS. A series of Δ[oxyHb] images was displayed to demonstrate the effectiveness of the four-wavelength camera. They presented local change in skin oxygenation due to blood flow during a blood occlusion and release procedure. This system could be applied to evaluate peripheral vascular diseases, skin injuries and skin grafting.

## 8. Acknowledgements

The authors wish to thank *Dr. Hayashi* Shimadzu Corporation and *Dr. Eda* KARC, CRL for their helpful support. We also wish to thank *Dr. Tsumura* Chiba University for the fruitful discussions concerning skin optics.

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