Abstract

We present an approach for accurate glucose sensing in turbid media using a spectrally resolved reflectance setup. Our proposed reflectance setup uses specialized source-detector separations (SDSs) to enable an effective separation of diffusion and absorption signals. Additionally, we adjust the selected SDSs to their optimal values to acquire maximum sensitivity to glucose in the two signals. The separation can help to enhance the sensitivity to glucose both for the diffusion and absorption signals, as they always suppress each other by causing opposite effects on the reflected diffuse light intensity. Monte Carlo simulations and experiments for glucose sensing are used to test the method. The acquired optimal SDSs could provide a reference for noninvasive blood glucose sensing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Near-infrared diffuse reflectance spectroscopy (NIR-DRS) has been considered a promising human blood glucose sensing method, as it can potentially provide noninvasive and convenient real-time measurements on human subjects. However, acquiring enough glucose sensitivity could be a critical task for detecting such weak signals of glucose in skin using NIR-DRS. The scattering or absorption change in skin caused by glucose can only result in a weak change in the diffuse reflection light. Just as previous studies have sought optimal optical path lengths (OPLs) to improve the sensitivity to absorption changes for transparent absorption media [14], we believe that optimal light source-detector separations (SDSs) also exist to improve the sensing of scattering and absorption in turbid media. Here, we discuss how to adjust SDSs to obtain the best sensitivity for the measurement of scattering- and absorption-related signals. First, we propose an approach to effectively separate the two kinds of signals since their sensitivity to glucose change is different in measurements with selected SDSs; these measurements vary with SDSs in different ways. These separations can help to analyze the signals as well as their sensitivity to glucose; we realize these separations by using specialized multi-SDSs, in which the SDSs can be adjusted to their optimal values to obtain the best sensitivity for the signals.

For glucose sensing, signal separation could be necessary to obtain enough desired scattering or absorption glucose signals because the two signals with certain SDSs and wavebands may be opposite and thus suppress each other. For some special SDSs, the two signals can be just equal and result in an absolute insensitivity to glucose [5,6]. Therefore, we believe that separation can help both scattering- and absorption-based glucose measurements. Moreover, the separation may also lead to a comprehensive glucose measurement by innovatively fusing the scattering and absorption information. The separation can also be used generally for separating signals from other analytes.

A variety of reports have focused on the analysis of the scattering and absorption sensitivities of diffuse reflectance with different SDSs [713]. These studies showed that the characteristics of the scattering sensitivity vary with the chosen SDS. First, for small SDSs, the optimal sensitivity of scattering can be obtained where the largest reflected light intensity is observed. As the SDS increases, the scattering sensitivity continues to decrease until a particular value of the SDS is reached, at which the scattering sensitivity becomes zero [7,8,10,12,13]. This SDS is called the scattering variation-independent SDS (SVI-SDS) [12] and denotes the point at which the diffuse light intensity no longer varies with the scattering variation of the medium. As the SDS increases beyond the SVI-SDS, the scattering sensitivity increases again, opposite to its prior behavior. The absorption behavior typically exhibits a continuously increasing trend with SDSs [12,13]. Some components cause scattering and absorption changes simultaneously, and their measurement shows a combined result. Even a special analyte-independent SDS can be obtained, where the combined effect of absorption and scattering is zero due to their opposite effects on diffuse light; glucose is such a typical analyte [1416]. The glucose-independent SDSs (also called the floating reference positions of glucose measurement) was therefore proposed for noninvasive glucose measurement to monitor other variables except glucose during long-term blood glucose tests [5,6].

Based on the characteristics of the scattering and absorption sensitivity, using different SDSs is required to sense scattering or absorption for glucose. Small SDSs can be used for sensing glucose scattering, as scattering is dominant in the diffuse light measured here. For sensing absorption, it is difficult to choose SDSs since absorption is always accompanied by scattering. We previously proposed an approach for separating them by using SVI-SDS [13]. At the SVI-SDS, there is not scattering change information; however, the absorption signals can be obtained [12]. Nevertheless, for glucose sensing, the reserved glucose absorption signals appeared weak and were inadequate for desired accurate sensing for some used wavebands. It is because the glucose absorption signal may be suppressed by a negative effect from the scattering of glucose itself. This absorption signal may only be detected by using high signal-to-noise ratio (SNR) detector and fixing the skin in a constant state. Despite of these disadvantages, we believe the SVI-SDS-based method would benefit to calibrate the skin drift caused by other variables with strong absorption during long-term monitoring.

In this paper, we present a method to separate the diffuse reflection light signals into two parts, diffusion signals and absorption signals, according to the diffusion equation as there the equation can clearly distinguish the two signals related to diffusion and absorption. It should be mentioned that this does not result in precise separation of scattering and absorption signals but a simple division of the diffusion and absorption from the measured diffuse light signals. The division can be easily realized by using at least two SDSs.

The separated signals are related to clear physics concepts. The diffusion signals are related to light diffusion and are primarily determined by the diffusion coefficient, whereas the absorption signals are determined by the effective attenuation coefficient (EAC), which is a comprehensive absorption property. We identify the signals as the diffusion-dominated signal (D-signal) and the effective absorption-dominated signal (EA-signal), respectively.

This separation presents a clear explanation for the effects of scattering and absorption on the diffuse reflection light, including the suppression of scattering on absorption. For scattering, it alters the diffusion coefficient, resulting in changes in the D-signal; meanwhile, it can cause a negative change in the EA-signal due to the absorption change in the changed diffusion path. At the SVI-SDS, the change in D-signal and EA-signal caused by scattering is equal, resulting in an absolute invariability of the diffuse light. Similarly, the absorption can also have opposite effects on D-signals and EA-signals because the absorption can also affect the diffusion coefficient and the consequent absorption change which is part of the total absorption. Obviously, diffusion negatively influences the absorption accompanied by both as scattering and absorption varies. By using the separation, both signals could be enhanced.

The separated signals can be used for different purposes. Specifically, the D-signal can be used to sense changes in scattering within the medium, whereas the EA-signal is helpful when attempting to sense components with weak absorption properties, such as blood glucose in the skin. Both the scattering and absorption of glucose are relevant to blood glucose-sensing applications. For example, Maruo, Bruulsema and Heinemann used glucose scattering properties to predict variations in blood glucose [1719]. Goodarzi, Sharma, Wülfert and Xu preferred to collect measurements based on glucose absorption properties because absorption can provide specific features for different variables during testing, which can then be used for analysis of spectral data using chemometric approaches [2023]. In this sense, glucose sensing based on the EA-signal might be a promising method for practical applications due to the enhanced sensitivity of the absorption features by using our approach. However, the scattering signals still should not be ignored. In future, effectively fusing the separated scattering and absorption signals might be a new way to lead a robust glucose sensing.

Taking 5∼15% Intralipid solutions as biological tissues phantoms, we verified the optimal SDSs for glucose measurement. The scattering coefficients of these solutions range from 30-120 cm-1 with a corresponding waveband of 1000∼1600 nm. Theoretical analysis, MC simulations and experiments were also performed and reported here. We analyzed the sensitivity of the two measurements based on the D-signal and EA-signal. The optimal SDSs were then acquired by setting the respective sensitivity of each to maximum.

2. Methods

2.1 Separation based on the diffusion equation

According to Eq. (1), the diffuse reflection light intensity of a medium $I(\rho )$ can be separated into two parts, based on the diffusion equation [24,25]. The D-signal is denoted as ${I_\textrm{D}}(\rho )$ and is related to the diffusion coefficient, D. The EA-signal, is denoted as ${I_\textrm{A}}(\rho )$ and is related to the effective attenuation coefficient, ${\mu _{\textrm{eff}}}$. ${I_0}$ is the incident light intensity, and $\rho$ is the SDS.

$$I(\rho ) = {I_0} \cdot {I_\textrm{D}}(\rho ) \cdot {I_\textrm{A}}(\rho )$$
${A_{\rho }}$ denotes the attenuance of the diffuse light, expressed as:
$${A_{\rho }} = - \ln (I(\rho )/{I_0}) = - \ln {I_\textrm{D}}(\rho ) - \ln {I_\textrm{A}}(\rho )$$
The attenuance, ${A_{\rho }}$, can also be separated into two parts, where ${A_{{\rho ,\textrm{D}}}} = - \ln {I_\textrm{D}}(\rho )$ corresponds to the D-signal, and ${A_{{\rho ,\textrm{A}}}} = - \ln {I_\textrm{A}}(\rho )$ corresponds to the EA-signal.
$${A_{\rho }} = {A_{{\rho ,\textrm{D}}}} + {A_{{\rho ,\textrm{A}}}}$$
The decomposition of the diffusion equation is summarized in Table 1, according to the details reported in Appendixes A and B. It can be applied to infinite and semi-infinite media.

Tables Icon

Table 1. The decomposition of the parameters in the diffusion equation (${I_0}$ is the incident light intensity; $\rho$ is the SDS; D is the diffusion coefficient;$K$ is defined as $1/(2/\mu _\textrm{S}^{\prime} + 4D)$; ${\mu _{\textrm{eff}}}$ is the effective attenuation coefficient.)

It should be mentioned that our derivation is based on the diffusion equation, in which the SDS should be limited to be greater than a minimum value, which can be one or several mean free paths of photons. For the 600∼900 nm, which is commonly used in the measurement of blood oxygen and hemoglobin of tissues, the available SDSs can start with 3∼5 mm; for the 1000∼1600 nm commonly used for blood glucose sensing, the available SDSs can start with 0.5∼1 mm because there the absorption of tissue enhances very much. Also, we should note that there exists an inherent deviation between the equations and real measurement results. For a real application, the calibration is necessary.

From Table 1, we can see the following:

  • (1) The equations for ${\Delta }{A_{\rho }}$, which are caused by absorption or scattering change, indicate that the ${\Delta }{A_{\rho }}$ as a function of $\rho$ is linear, with the decomposition shown in Fig. 1.
  • (2) The ${\Delta }{A_{{\rho ,\textrm{D}}}}$ of the D-signal presents as SDS-independent;
  • (3) The ${A_{{\rho ,\textrm{A}}}}$ of the EA-signal indicates that a distinct linear relationship exists with $\rho$, where the slope is simply ${\mu _{\textrm{eff}}}$. In addition, the slope of ${\Delta }{A_{{\rho ,\textrm{A}}}}$ is simply ${\Delta }{\mu _{\textrm{eff}}}$, which can be obtained from the differential expression using two arbitrary SDSs ${\rho _\textrm{A}}$ and ${\rho _\textrm{B}}$.
    $${\Delta }{\mu _{\textrm{eff}}} = \frac{{{\Delta }{A_{{{\rho }_\textrm{B}}}} - {\Delta }{A_{{{\rho }_\textrm{A}}}}}}{{{\rho _\textrm{B}} - {\rho _\textrm{A}}}}$$
  • (4) For infinite media, the ${\Delta }{A_{{\rho ,\textrm{D}}}}$ and ${\Delta }{A_{{\rho ,\textrm{A}}}}$ seem always opposite when scattering or absorption independently varies, since their partial differential coefficients are opposite. Thus, they are enhanced after separation. However, if scattering and absorption vary at the same time, the ${\Delta }{A_{{{\rho }^\ast }\textrm{,A}}}$ and ${\Delta }{A_{{{\rho }^\ast }\textrm{,D}}}$ might be affected by the signs of ${\Delta }{\mu _\textrm{a}}$ and ${\Delta }{\mu _\textrm{s}}$: if ${\Delta }{\mu _\textrm{a}} \cdot {\Delta }{\mu _\textrm{s}} < 0$, the scattering may furtherly reduce the signals. For infinite media, we can analyze the signals by a specific calculation using the equations.
  • (5) A special SDS may exist for ${\rho ^\ast }$ where ${\Delta }{A_{{{\rho }^{\ast }}}}$=0 or ${\Delta }{I_{{{\rho }^{\ast }}}}$=0, if ${\Delta }D \cdot {\Delta }{\mu _{\textrm{eff}}} < 0$ or $({{\Delta }K/K - {\Delta }{\mu_{\textrm{eff}}}/{\mu_{\textrm{eff}}}} )/{\Delta }{\mu _{\textrm{eff}}} < 0$. if the variable is a component concentration, then ${\rho ^\ast }$ could be an analyte-independent SDS. For glucose, ${\rho ^\ast }$ is a glucose-independent SDS [5,6]. If only the scattering varies, then ${\rho ^\ast }$ could be a scattering variation-independent SDS (SVI-SDS) [12].

 figure: Fig. 1.

Fig. 1. The schematic diagrams. (A) The decomposition of attenuance; (B) The sensitivity of diffuse light intensity.

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2.2 The SDSs for D-signal and EA-signal measurements

From Table 1, the change in attenuance, ${\Delta }{A_{\rho }}$, can easily be separated into two parts, the D-signal and EA-signal. The D-signal can be obtained directly at a small SDS, where $\rho \approx 0$, such that the effect of absorption is negligible.

$$\textrm{D} - \textrm{signal}:\quad \Delta {A_{{\rho ,\textrm{D}}}} \approx \Delta {A_{{\rho = \textrm{small}\ \textrm{value}}}}$$
In contrast, the EA-signal can be obtained by using two SDSs. The slope of ${\Delta }{A_{\rho }}$, i.e., ${\Delta }{\mu _{\textrm{eff}}}$, can be calculated by using Eq. (4), and then, ${\Delta }{A_{{\rho ,\textrm{A}}}}$ can be acquired using Eq. (6). This differential approach can sufficiently reduce any drift in the incident light that might otherwise be caused by the varying incident angle of the light or differences in the reflective properties of the skin, as demonstrated previously in in vivo measurements.
$$\textrm{EA} - \textrm{signal}:\quad {\Delta }{A_{{\rho ,\textrm{A}}}} = {\Delta }{\mu _{\textrm{eff}}} \cdot \rho \quad {\Delta }{\mu _{\textrm{eff}}} = \frac{{{\Delta }{A_{{{\rho }_\textrm{B}}}} - {\Delta }{A_{{{\rho }_\textrm{A}}}}}}{{{\rho _\textrm{B}} - {\rho _\textrm{A}}}}$$

2.3 The concentration determination based on D-signals or EA-signals

Using the D-signals or EA-signals, we can determine the glucose concentration by using the multivariate regression methods [20], such as partial least squares regression (PLSR). A glucose prediction model can be established by using the signals with given glucose concentrations. Then, the model can be used to predict the glucose concentrations for a new spectral signal. If there are multiple variables in the test, the EA-signals are preferred because they can be different due to the different absorption characteristics of the variables; if there only one variable varies with the scattering, the D-signal can be used to sense it. Moreover, our method also needs a calibration by using a chemical test, to acquire the true glucose value for the spectrum recorded at first time, since we always take it as baseline.

2.4 The optimal SDSs for maximum sensitivity

According to Appendixes C, D and F, sensitivity is defined as the change in diffuse light intensity due to variations in analyte concentration, as shown in Eq. (7) and (8).

$${S_{\rho }} = \frac{{\textrm{d}{I_{\rho }}}}{{\textrm{d}C}} = - {I_{\rho }}\frac{{\textrm{d}{A_{\rho }}}}{{\textrm{d}C}}$$
$${S_{{{\rho }_\textrm{A}}\textrm{,}{{\rho }_\textrm{B}}}} = \frac{{\textrm{d}{I_{{{\rho }_\textrm{B}}}}}}{{\textrm{d}C}} - \eta \frac{{\textrm{d}{I_{{{\rho }_\textrm{A}}}}}}{{\textrm{d}C}} \approx - {I_{{{\rho }_\textrm{B}}}}\frac{{\textrm{d}{A_{{\textrm{I}_\textrm{e}}}}}}{{\textrm{d}C}}$$
${S_{\rho }}$ is the sensitivity determined for one SDS $\rho$, whereas ${S_{{{\rho }_\textrm{A}}\textrm{,}{{\rho }_\textrm{B}}}}$ is the sensitivity for the differential test conducted with two SDSs ${\rho _\textrm{A}}$ and ${\rho _\textrm{B}}$. $\eta$ is the differential factor, given as $\eta = {I_{{{\rho }_\textrm{B}}}}/{I_{{{\rho }_\textrm{A}}}}$, and ${A_{{\textrm{I}_\textrm{e}}}}$ is the differential attenuance, defined as ${A_{{\textrm{I}_\textrm{e}}}} = {A_{{{\rho }_\textrm{B}}}} - {A_{{{\rho }_\textrm{A}}}}$. Appendixes C and F present the derivations of the equations, and Table 2 shows the equations.

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Table 2. The optimal SDSs for D-signal-based and EA-signal-based spectroscopy (according to Appendixes C, D&F) (${I_0}$ is the incident light intensity; $\rho$ is the SDS; D is the diffusion coefficient; $K = 1/(2/\mu _\textrm{S}^{\prime} + 4D)$ ; ${\mu _{\textrm{eff}}}$ is the effective attenuation coefficient; $\sigma I$ is the noise of the detector.)

A small SDS is needed when collecting measurements for the D-signal as there is less absorption. The sensitivity is equal to the product of diffuse light intensity ${I_{\rho }}$ and $\textrm{d}{A_{{\rho ,\textrm{D}}}}/\textrm{d}C$. It is possible for $\textrm{d}{A_{{\rho ,\textrm{D}}}}/\textrm{d}C$ to be independent of the SDS $\rho$. Thus, at small SDSs, the maximum sensitivity can be obtained because ${I_{\rho }}$ is at its largest.

Two SDSs should be optimized for the EA-signal-based method. According to the equations in Table 2, the first SDS should be a small SDS such as the optimal SDS of the D-signal-based method, while the second SDS should be at an optimal distance relative to the first SDS.

When the SDS is determined for use, we can evaluate the resolvable glucose concentration limit of the measurement, ${C_{\textrm{limit}}}$, using Eq. (9).

$${C_{\textrm{limit}}} = \frac{{\varDelta {I_{\textrm{noise}}}}}{{\textrm{d}{I_{\rho }}/\textrm{d}C}} = \frac{{3\sigma I}}{{|{{S_{\rho }}} |}}$$
where the $\varDelta {I_{\textrm{noise}}}$ is the noise level of the measured diffuse light ${I_{\rho }}$ at a given SDS $\rho$, as it can be estimated using $\varDelta {I_{\textrm{noise}}} = 3\sigma I$. The $\sigma I$ is the standard derivation of ${I_{\rho }}$. For the measurement using two SDSs, the ${S_{\rho }}$ should be changed to ${S_{{{\rho }_\textrm{A}}\textrm{,}{{\rho }_\textrm{B}}}}$.

3. Application example: glucose measurement

3.1 Monte Carlo simulation and results

Glucose measurements were collected in Intralipid aqueous solutions (5∼15% Intralipid). Only the condition of a semi-infinite medium is considered here. Monte Carlo (MC) simulation [26] is used to get the diffuse reflection light intensity $I(\rho )$ for the scattering media of 5%, 10% and 15% Intralipid solutions with different glucose concentrations over the waveband of 1000∼1600 nm. The glucose concentrations vary from 0 to 900 mg/dL in increments of 180 mg/dL. The optical parameters of these media and glucose can be found in [2730], shown in Fig. 2. The SDSs span from 0.1 mm to 5 mm with interval of 0.1 mm. Based on these data, we computed out the D-signals and EA-signals. Using Eq. (2), the changes in attenuance of diffuse light, ${\Delta }{A_{\rho }}$ caused by glucose are acquired, is shown in Fig. 3(A). The D-signal can be obtained by using a small SDSs, such as 0.5 mm, based on Eq. (5), as shown in Fig. 3(B). The EA-signal can be obtained using Eq. (6), where the two SDSs of ${\rho _\textrm{A}}$ and ${\rho _\textrm{B}}$ should be selected within the SDSs for which the ${\Delta }{A_{\rho }}$ linearly varies, as shown in Fig. 3(C). We also calculated the sensitivity of D-signal and EA-signal to determine the optimal SDSs, using Eq. (7) and Eq. (8), as shown in Fig. 3(D) and Fig. 3(E).

 figure: Fig. 2.

Fig. 2. Optical parameters. (A) for three Intralipid solutions; (B) for glucose of 1 mmol/L(18 mg/dL).

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 figure: Fig. 3.

Fig. 3. Monte Carlo simulation results for 5% Intralipid. (A) Change in attenuance, ${\Delta }{A_{\rho }}$, caused by 1 mg/dL glucose; (B) D-signals, spectra of ${\Delta }{A_{{\rho ,\textrm{D}}}}$ at a SDS of 0.5 mm; (C) EA-signals, spectra of ${\Delta }{A_{{\rho ,\textrm{A}}}}$, calculated by using the data of two SDSs of 0.5 mm and 0.8 mm; (D) Sensitivity using one SDS, ${S_{\rho }}$, caused by 1 mg/dL glucose (1300 nm); (E) Sensitivity using two SDSs, ${S_{{{\rho }_\textrm{A}}\textrm{,\ }{{\rho }_\textrm{B}}}}$, caused by 1 mg/dL glucose (1300 nm); (F) Sensitivity ${S_{{{\rho }_\textrm{A}}\textrm{,\ }{{\rho }_\textrm{B}}}}$ with ${\rho _\textrm{A}}$=0.5 mm, caused by 1 mg/dL glucose, for three Intralipid solutions (1300 nm).

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The following findings can be seen in Fig. 3:

  • (1) Fig. 3(A) shows the ${\Delta }{A_{\rho }}$ caused by glucose, which varies approximately linearly with SDS. The same linear behavior was also observed for the 10% and 15% Intralipid solutions. When the SDS is too small, this linear behavior is not obvious. However, when the SDS is greater than approximately 0.2, 0.3 and 0.5 mm for 5%, 10% and 15% Intralipid solutions, respectively, the linearity of ${\Delta }{A_{\rho }}$ becomes obvious. The slope of ${\Delta }{A_{\rho }}$ can be obtained based on the difference between two SDSs in the linear region. The slope is also an estimation of ${\Delta }{\mu _{\textrm{eff}}}$, based on Eq. (4). We prefer to use SDS values that are greater than 0.5 mm for all three solutions because SDSs that are too small are not convenient for arranging the detectors.
  • (2) Fig. 3(B) shows the ${\Delta }{A_{\rho }}$ at a small SDS of 0.5 mm. Based on the classification scheme, this SDS is considered to belong to the D-signal attenuance, ${\Delta }{A_{{\rho ,\textrm{D}}}}$. This ${\Delta }{A_{{\rho ,\textrm{D}}}}$ displays ambiguous spectral characteristics, except at around 1450 nm.
  • (3) Fig. 3(C) shows the EA-signal, the $\Delta {A_{{\rho ,\textrm{A}}}}$, which is calculated by using the data if the SDSs of 0.5 mm and 0.8 mm.
  • (4) Fig. 3(D) shows the sensitivity using one SDS, ${S_{\rho }}$, at 1300 nm. The ${S_{\rho }}$ at this small SDS exhibits a large sensitivity. Therefore, the D-signal based measurement can be set to use a small SDS. For example, the SDS of 0.5 mm could be an appropriate choice.
  • (5) Fig. 3(E) shows the sensitivity using two SDSs, ${S_{{{\rho }_\textrm{A}}\textrm{,}{{\rho }_\textrm{B}}}}$, at 1300 nm. When ${S_{{{\rho }_\textrm{A}}\textrm{,}{{\rho }_\textrm{B}}}}$ uses a small ${\rho _\textrm{A}}$, its sensitivity is enhanced. Therefore, ${\rho _\textrm{A}}$ should be used at a small SDS first, and then, the optimal ${\rho _\textrm{B}}$ can be obtained, based on the maximum of ${S_{{{\rho }_\textrm{A}}\textrm{,}{{\rho }_\textrm{B}}}}$. For example, ${\rho _\textrm{A}}$ can be set at 0.5 mm, and then, ${\rho _\textrm{B}}$ can be determined by the curve of ${S_{{{\rho }_\textrm{A}}\textrm{,}{{\rho }_\textrm{B}}}}$. In real use, ${\rho _\textrm{B}}$ can be set within a certain range, since at the SDSs in the vicinity of the optimal ${\rho _\textrm{B}}$, a good sensitivity can be obtained. We determined ${\rho _\textrm{B}}$ based on when the sensitivity varies by ± 20% of the maximum sensitivity. Table 3 summarizes the optimal SDSs for the Intralipid solutions.
  • (6) Fig. 3(F) shows a comparison of the three Intralipid solutions. From this comparison, it is clear that the 15% Intralipid solution shows better sensitivity than the 5% Intralipid solution because the scattering in the medium and its diffuse reflection light intensity are larger.
  • (7) From Fig. 3(B) and 3(C), we can see the D-signal is about 3∼4 times the EA-signal when they both use optimal SDSs. Therefore, the D-signals could be easily detected by detectors.

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Table 3. The optimal SDSs for the Intralipid solutions at 1000-1600 nm (according to the MC results)

Fig. 4 shows how ${\rho _\textrm{e}}$ varies with ${\rho _\textrm{A}}$. The value ${\rho _\textrm{e}}$ is the distance between ${\rho _\textrm{A}}$ and ${\rho _\textrm{B}}$. We can see that ${\rho _\textrm{e}}$ is a constant when ${\rho _\textrm{A}}$ is established based on a larger SDS. If ${\rho _\textrm{A}}$ is small, the value of ${\rho _\textrm{e}}$ is also smaller, which means that ${\rho _\textrm{B}}$ will fall close to ${\rho _\textrm{A}}$. In our work, we prefer ${\rho _\textrm{A}}$ to be set at approximately 0.5 mm. The optimal ${\rho _\textrm{B}}$ is listed in Table 3.

 figure: Fig. 4.

Fig. 4. The optimal SDSs for the EA-signal-based measurement of three kinds of Intralipid solutions at 1300 nm. (A) The results according to Eq. (F-8); (B) The results according to the MC simulation.

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3.2 Experiment and results

An experiment to measure glucose in scattering media is used to test the optimal SDSs. The test media is a 5% Intralipid solution with glucose concentrations from 0 to 100 mg/dL in increments of 20 mg/dL (1.1 mmol/L). The samples were prepared by dissolving glucose powder (Type: D-(+)-, Meryer Chemical Technology Co., Ltd., China) into the solutions. A custom designed optical system with 8 detection channels for 8 SDSs is used in the experiment. The optical measurement system is briefly illustrated in Fig. 5. The system consists of six super luminescent emitting diodes as light sources (central wavelengths are 1050 nm, 1219 nm, 1314 nm, 1409 nm, 1550 nm and 1610 nm, InPhenix, USA), an optical switch for 8 channels, and eight InGaAs photoelectric detectors (Hamamatsu Photonics, G5851-21). The fibers are arranged in a probe according to 8 SDSs, which are 0.48 mm, 0.76 mm, 1.04 mm, 1.32 mm, 1.60 mm, 1.88 mm, 2.16 mm and 2.44 mm. Figure 4(b) shows the details of the fibers in the probe end that are used to contact the samples. The optical fibers (NA = 0.27, 200 µm, Nanjing Chunhui Science and Technology Industrial Co. Ltd, China) are used both for illumination and collection. The system was controlled by LabVIEW software. The dark noise of the system was approximately 4e-5 V, and for the commonly used 2 V signals, the noise is approximately 1e-4 V. For an hour test, the system drift is approximately 1e-2 V. It takes approximately 20 minutes to test all samples.

 figure: Fig. 5.

Fig. 5. The schematic diagram of the system. (a) System overview; (b) The probe end

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According to Table 3, we use ${\rho _\textrm{A}}$=0.48 mm and ${\rho _\textrm{B}}$=0.76 mm for the EA-signal-based measurement and ${\rho _\textrm{A}}$=0.48 mm for the D-signal-based measurement.

Fig. 6 shows the experimental results. Figure 6(A) shows the change in attenuance of the diffuse light, $\textrm{d}{A_{\rho }}/\textrm{d}C$, which varies approximately linearly with SDS. We can see the glucose-independent SDSs for the wavelengths used. For 1050 nm and 1219 nm with weak absorption of glucose, the glucose-independent SDSs can be predominantly determined by the SVI-SDSs. Figure 6(B) shows the $\textrm{d}{A_{\rho }}/\textrm{d}C$ at SVI-SDS and the $\textrm{d}{A_{{\textrm{I}_\textrm{e}}}}/\textrm{d}C = ({\textrm{d}{A_{{{\rho }_\textrm{B}}}} - \textrm{d}{A_{{{\rho }_\textrm{A}}}}} )/\textrm{d}C$ using two SDSs of ${\rho _\textrm{A}}$ and ${\rho _\textrm{B}}$. The $\textrm{d}{A_{\rho }}/\textrm{d}C$ at SVI-SDS appears lower. Figure 6(C) shows the D-signals, ${\Delta }{A_{{\rho ,\textrm{D}}}}$ at the SDS of 0.48 mm. Figure 6(D) shows the EA-signals, ${\Delta }{A_{{\rho ,\textrm{A}}}}$ at the SDS of 0.48 mm, when the ${\Delta }{\mu _{\textrm{eff}}}$ calculated by using SDSs of 0.48 and 0.76 mm. We can see that the D-signal is about 2∼3 times EA-signals. We also evaluated the ${C_{\textrm{limit}}}$ using Eq. (10). Figure 6(E) shows the SNR at a SDS of ${\rho _\textrm{B}}$, $SN{R_{{{\rho }_\textrm{B}}}}$. Figure 6(F) shows the evaluated ${C_{\textrm{limit}}}$; we can see that the ${C_{\textrm{limit}}}$ is under 20 mg/dL for most wavelengths, so we test the glucose in 20 mg/dL increments.

 figure: Fig. 6.

Fig. 6. The experiment results. (A) The attenuance change, ${\Delta }{A_{\rho }}$, caused by a glucose change of 1 mg/dL at six wavelengths; (B) Attenuance change with using different SDSs; (C) D-signals, spectra of ${\Delta }{A_{{\rho ,\textrm{D}}}}$ at a SDS of 0.48 mm; (D) EA-signals, spectra of ${\Delta }{A_{{\rho ,\textrm{A}}}}$ calculated by the differential of two SDSs; (E) The signal-to-noise ratio of the detection system at ${\rho _\textrm{B}}$=0.76 mm; (F) Evaluated measurement limit using ${\rho _\textrm{A}}$=0.48 mm and ${\rho _\textrm{B}}$=0.76 mm.

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Table 4. presents the glucose concentration prediction results by the established models using the partial least squares regression (PLSR) method. The spectral data used were the ${\Delta }{A_{{\rho ,\textrm{A}}}}$, ${\Delta }{A_{{\rho ,\textrm{D}}}}$ and ${\Delta }{A_{\rho }}$. The measurement using the ${\Delta }{A_{\rho }}$ at a small SDS of 0.48 mm could be seen based on the D-signal method, which has a better precision due to the lower value of RMSECV (root mean square error of cross validation). The measurement using the ${\Delta }{A_{\rho }}$ at a larger SDS of 0.76 mm yields a lower precision compared to the measurement with 0.48 mm because of the decreasing sensitivity with increasing SDS. The measurement using the ${\Delta }{A_{{\rho ,\textrm{A}}}}$ data at a small SDS of 0.48 mm, which were acquired by performing a differential on the data of 0.48 mm and 0.76 mm, appears to be an average RMSECV level.

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Table 4. The glucose concertation prediction results in 5% Intralipid solutions

Fig. 7 shows a comparison of experimental results and MC simulation results for 5% Intralipid solution. From Fig. 7, we can see that their D-signals or EA-signals present similar spectral characteristics but in different values. Specifically, the experiment results are approximately 10 times the results of MC simulation. It is maybe because that our experiment is more like an infinite medium measurement as in it the test optical fiber was located around 10 mm below the solution surface; however, for the MC simulations, they are in semi-infinite conditions so that the diffuse light can be reduced due to the boundary.

 figure: Fig. 7.

Fig. 7. Comparison of experiment results and MC results for 5% Intralipid solutions. The attenuance change was caused by 1 mmol/L glucose change. (A) MC results. The D-signal uses the SDS of 0.5 mm, and the EA-signal uses 0.5 mm and 0.8 mm; (B) Experiment results. The D-signal uses 0.48 mm, and EA-signal uses 0.48 mm and 0.76 mm.

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4. Discussion

From the equations derived in Table 1 and the MC simulation results, we found out the linearity of diffuse light attenuace change of a variable with SDSs. Based on this finding, we present a method for separating diffuse light attenuance into two parts, a diffuse signal and an absorption signal. We identified them as the D-signal and EA-signal, and then, we determined the optimal SDSs for each of their respective measurements. The optimal SDSs were determined based on a method that relies on the optimal optical pathlength in transparent absorption media.

It should be mentioned that the available SDSs should start with a minimum value, for which linear diffuse light attenuance is observed. The minimum value of SDSs can be determined using MC method or a pre-experiment. For the 1000∼1600 nm, the SDSs should start with 0.5∼1 mm; for shorter wavelengths, like 600∼900 nm, the minimum SDSs could reach to several millimeters.

For the D-signal-based measurement, the optimal SDS should be a small value. In addition, the optimal SDS can be the smallest of the SDSs for which linear diffuse light attenuance is observed. From the results, it is clear that media with strong scattering properties have smaller optimal SDSs. These values are 0.2, 0.3 and 0.5 mm for 5%, 10% and 15% Intralipid solutions, respectively. We prefer 0.5 mm for the Intralipid solutions, because the fibers used for detection can easily be set at this value.

For the EA-signal-based measurement, two SDSs should be used. The first can be the same as the optimal value used for the D-signal-based method. Then, the second SDS can be set near the value of the first one. Table 3 gives the optimal SDSs for the waveband of 1000∼1600 nm. These results are relevant to most biological tissues since the optical properties of 5∼15% Intralipid solutions span a wide range. The results show that the first SDS can be near 0.5 mm, while the second SDS can be at 0.6∼1.0 mm for different scattering media. In practice, we prefer 0.8 mm for the second SDS when using media with stronger scattering tendencies and 1.0 mm for media with weaker scattering. It should be mentioned that the second SDS can be set to values other than the optimal value if the sensitivity is enough for measurement with that SDS.

The results mentioned above are only relevant to single layers and uniform media. For multilayered media, the optimal SDSs could be different from the results presented here. However, the method for their determination would be the same as that proposed in this paper.

Moreover, the two separated signals have strong relationships with relevant physics concepts. According to the derivation for the diffuse equation on infinite media, the D-signal is the relative change of the diffusion coefficient, i.e., ${\Delta }{A_{{\rho ,\textrm{D}}}} = {\Delta }D/D$; the EA-signal is an effective attenuation coefficient change, i.e., ${\Delta }{A_{{\rho ,\textrm{A}}}} = {\Delta }{\mu _{\textrm{eff}}} \cdot \rho$. Since there is a systematic deviation between the equation and real-world conditions, our method can also be taken as an approach for approximating the values of ${\Delta }D/D$ and ${\Delta }{\mu _{\textrm{eff}}}$. Such deviations in the system would not affect the measurements after calibration.

The D-signal-based measurement can be used to measure changes in scattering, because ${\Delta }D/D$ could, alternatively, be determined as ${\ -\Delta }\mu _\textrm{S}^{\prime}/({\mu _\textrm{a}} + \mu _\textrm{S}^{\prime})$. If the media possess properties of strong scattering and low absorption, this expression would otherwise be ${\ -\Delta }\mu _\textrm{S}^{\prime}/\mu _\textrm{S}^{\prime}$, which is the relative change of the reduced scattering coefficient. Therefore, we can easily obtain ${\Delta }\mu _\textrm{S}^{\prime}$ for a medium with a known value of $\mu _\textrm{S}^{\prime}$.

The EA-signal-based measurement can be used to measure absorption, as the specific feature ${\Delta }{\mu _{\textrm{eff}}}$ is related to the analyte and thus can be determined by the analyte’s absorption ${\Delta }{\mu _\textrm{a}}$. However, the same is not true for ${\Delta }\mu _\textrm{S}^{\prime}$ because the feature ${\Delta }\mu _\textrm{S}^{\prime}$ is too similar for various reasons. Furthermore, the EA-signal-based method is useful for component measurements that may be complicated by many factors, thereby allowing a multivariable analysis based on chemometrics to be performed. For example, to perform noninvasive glucose sensing on skin tissue, the EA-signal-based method can be used to test the glucose level or monitor other variables during the test, such as tissue temperature variation or changes in test position.

For long-term component monitoring, we prefer the EA-signal-based method since it demonstrates better stability when used over long testing periods. This is because the EA-signal-based method uses a differential process that is based on two SDSs, which allows for drift in the incident light to be greatly reduced. This differential approach can also reduce the effects of variations in skin properties. Measurement of blood oxygen saturation levels is one example for which the differential method could be aptly applied [31]; the differential approach was applied to photoplethysmography (PPG) signals to obtain differential optical signals between the systolic and diastolic periods. The differential signal reduces the drift in the incident light and reduces the influence of background changes in various tissues, such as muscle and bone.

For the glucose sensing on skin, we believe both the EA-signals and D-signals are useful. The EA-signals could be feasible to monitor the skin conditions, including skin temperature, skin moisture and position change. etc. However, when the skin is in a stable state, their D-signals also can be well applied due to good sensitivity. We think an intelligent fusing the separated EA-signals and D-signals might be a promising work. For example, the EA-signals can be used for calibrating the skin conditions and the D-signals can be used for glucose sensing.

5. Conclusion

We present a comprehensive investigation of near-infrared (NIR) diffuse reflectance spectroscopy and the determination of optimal SDS values. We separated the diffuse light attenuance into a diffusion signal and an absorption signal, and then, we acquired the optimal SDSs for both signals. To sense both signals, two practical approaches for SDS usage were proposed, including the use of one SDS and, alternatively, two SDSs. The test using one SDS was useful for monitoring changes in diffuse light intensity, which are influenced by both light diffusion and absorption in media. However, at small SDSs, only the diffusion signal can be sensed. In contrast, the differential test with two SDSs can be used to obtain the absorption signal and has the ability to greatly reduce any drift in the incident light for a long-term monitoring application. The absorption signals integrated the scattering and absorption of the analyte and shows a more enhanced measurement sensitivity than that using only pure absorption.

In addition to our theoretical analysis, we employed MC simulations and acquired the optimal SDSs at 1000∼1600 nm for 5∼15% Intralipid solutions, which are commonly used to mimic the behavior of most biological tissues. The results are applicable to many types of tests that might be performed on tissues using NIR spectroscopy, such as the in vivo measurement of blood glucose, hemoglobin and blood oxygen levels.

We believe the separation method proposed here is especially suitable for the measurements of in vivo tissues, which always randomly vary during tests. Using our method, we can continuously monitor the in vivo tissues by analyzing the separated scattering and absorption signals. Then, based on the signals we can calibrate the measurements of different tissue conditions or different tissues, since the difference among the tissues can be investigated. Not only calibrating the difference between tissues, the method also can sense a variable of a given tissue with a stable state, like blood glucose or blood oxygen etc. We believe an intelligent use of the scattering and absorption signals during tests might conduct a more accurate result.

In conclusion, we believe that this research will be useful in the area of diffuse reflectance spectroscopy when target information, such as scattering or absorption information, is desired. In addition, the acquired optimal SDSs can be applied to other types of measurements that involve near-infrared reflectance spectroscopy. Additionally, we firmly believe that our proposed separation method for NIR-DRS may further combine the use of scattering and absorption information to perform intelligent analysis of human skin or other turbid media.

Appendix A: Separation of diffuse reflection light in an infinite medium

When continuous monochrome light enters biological tissues, the steady-state light distribution can be described by the diffusion approximation equation. For a medium with strong scattering and weak absorption, the solution of the diffuse reflection light intensity $I(\rho )$ in an infinite medium is [24]:

$$I(\rho ) = {I_0}\frac{1}{{4\pi \rho D}}\exp ( - {\mu _{\textrm{eff}}} \cdot \rho )$$
where ${I_0}$ is the incident light intensity; ${\mu _\textrm{a}}$ and ${\mu _\textrm{s}}$ are the absorption coefficient and scattering coefficient, respectively; g is the anisotropic factor; $\mu _\textrm{s}^{\prime}$ is the reduced scattering coefficient, defined as $\mu _\textrm{s}^{\prime} = (1 - g){\mu _\textrm{s}}$; the diffusion coefficient $D = {[{3({{\mu_\textrm{a}} + \mu_\textrm{s}^{\prime}} )} ]^{ - 1}}$; the effective attenuation coefficient ${\mu _{\textrm{eff}}} = {[{3{\mu_\textrm{a}}({{\mu_\textrm{a}} + \mu_\textrm{s}^{\prime}} )} ]^{1/2}}$; and $\rho$ is the light source detector separation (SDS).

The diffuse reflectance can be separated into two parts, the D-signal ${I_D}(\rho )$ and the EA-signal ${I_A}(\rho )$, as:

$$I(\rho )= {I_0} \cdot {I_\textrm{D}}(\rho )\cdot {I_\textrm{A}}(\rho )\; \;$$
$${I_\textrm{D}}(\rho )= \frac{1}{{4\pi \rho D}}; \quad{I_\textrm{A}}(\rho )= \textrm{exp}( - {\mu _{\textrm{eff}}} \cdot \rho )$$
${A_{\rho }}$ denotes the attenuance of light, expressed as ${A_{\rho }} = - \textrm{ln}\frac{{I(\rho )}}{{{I_0}}}$, and can be decomposed into ${A_{{\rho },\textrm{D}}}$ and ${A_{{\rho },\textrm{A}}}$ as:
$${A_{{\rho },\textrm{D}}} = - \textrm{ln}{I_\textrm{D}}(\rho )= \textrm{ln}4\pi \rho D; \quad{A_{{\rho },\textrm{A}}} = - \textrm{ln}{I_\textrm{A}}(\rho )= {\mu _{\textrm{eff}}} \cdot \rho$$

When a component’s concentration varies, the change in attenuance can be expressed as:

$$\begin{array}{l} \Delta {A_{\rho }} = \frac{1}{D} \cdot \Delta D + \Delta {\mu _{\textrm{eff}}} \cdot \rho \; \\ = - (3D) \cdot {\Delta }{\mu _\textrm{a}} + (\frac{3}{2}D{\mu _{\textrm{eff}}} + \frac{1}{2}{D^{ - 1}}{\mu _{\textrm{eff}}}^{ - 1}) \cdot \rho \cdot {\Delta }{\mu _\textrm{a}}\\ - (3D) \cdot {\Delta }\mu _\textrm{S}^{\prime} + (\frac{3}{2}D{\mu _{\textrm{eff}}}) \cdot \rho \cdot {\Delta }\mu _\textrm{S}^{\prime} \end{array}$$

The change in attenuance for the D-signal and the EA-signal could be:

$$\begin{array}{l} \Delta {A_{{\rho },\textrm{D}}} = \frac{1}{D} \cdot \Delta D = - (3D) \cdot {\Delta }{\mu _\textrm{a}} - (3D) \cdot {\Delta }\mu _\textrm{S}^{\prime};{\ }\\ \Delta {A_{{\rho },\textrm{A}}} = \Delta {\mu _{\textrm{eff}}} \cdot \rho = (\frac{3}{2}D{\mu _{\textrm{eff}}} + \frac{1}{2}{D^{ - 1}}{\mu _{\textrm{eff}}}^{ - 1}) \cdot \rho \cdot {\Delta }{\mu _\textrm{a}} + (\frac{3}{2}D{\mu _{\textrm{eff}}}) \cdot \rho \cdot {\Delta }\mu _\textrm{S}^{\prime} \end{array}$$

Then we derived the scattering-variation-insensitive SDS (SVI-SDS) denoted by ${\rho ^\ast }$, setting $\partial {A_{\rho }}/\partial \mu _\textrm{s}^{\prime} = 0\;$ or $\partial {I_{\rho }}/\partial \mu _\textrm{s}^{\prime} = 0\;$[12].

$${\rho ^\ast } = \frac{2}{{{\mu _{\textrm{eff}}}}},\quad \textrm{let}\quad \frac{{\partial {A_{\rho }}}}{{\partial \mu _\textrm{S}^{\prime}}} = - (3D) + (\frac{3}{2}D{\mu _{\textrm{eff}}}) \cdot \rho = 0$$

And attenuance change ${\Delta }{A_{{{\rho }^\ast }}}$ with SVI-SDS could be:

$$\begin{array}{l} {\Delta }{A_{{{\rho }^{\ast }}\textrm{,D}}} = \frac{{{\Delta }D}}{D} = - 3D({\Delta }{\mu _\textrm{a}} + {\Delta }\mu _\textrm{s}^{\prime}) = - \frac{{{\Delta }{\mu _\textrm{a}} + {\Delta }\mu _\textrm{s}^{\prime}}}{{{\mu _\textrm{a}} + \mu _\textrm{s}^{\prime}}}\\ {\Delta }{A_{{{\rho }^{\ast }}\textrm{,A}}} = \Delta {\mu _{\textrm{eff}}} \cdot {\rho ^\ast }\; = \frac{{{\Delta }{\mu _\textrm{a}}}}{{{\mu _\textrm{a}}}} + 3D({\Delta }{\mu _\textrm{a}} + {\Delta }\mu _\textrm{s}^{\prime}) = \frac{{{\Delta }{\mu _\textrm{a}}}}{{{\mu _\textrm{a}}}} + \frac{{{\Delta }{\mu _\textrm{a}} + {\Delta }\mu _\textrm{s}^{\prime}}}{{{\mu _\textrm{a}} + \mu _\textrm{s}^{\prime}}}\\ {\Delta }{A_{{{\rho }^{\ast }}}} = {\Delta }{A_{{{\rho }^{\ast }}\textrm{,D}}} + {\Delta }{A_{{{\rho }^{\ast }}\textrm{,A}}} = \frac{{{\Delta }{\mu _\textrm{a}}}}{{{\mu _\textrm{a}}}} \end{array}$$

We see that the ${\Delta }{A_{{{\rho }^\ast }}}$ would be only related to the absorption change. The EA-signal with SVI-SDS, i.e. ${\Delta }{A_{{{\rho }^{\ast }}\textrm{,A}}}$, could include richer information than ${\Delta }{A_{{{\rho }^\ast }}}$ due to the contribution of scattering.

Appendix B: Separation of diffuse reflection light in a semi-infinite medium

Under the extrapolated boundary conditions, the solution for the diffuse reflection light intensity $I(\rho )$ in a semi-infinite medium is [5, 9]:

$$I(\rho )= {I_0}\frac{1}{{4\pi }}\left[ {{z_0}\left( {{\mu_{\textrm{eff}}} + \frac{1}{{{r_1}}}} \right)\frac{{{\textrm{e}^{ - {\mu_{\textrm{eff}}}{r_1}}}}}{{r_1^2}} + ({z_0} + 2{z_\textrm{b}})\left( {{\mu_{\textrm{eff}}} + \frac{1}{{{r_2}}}} \right)\frac{{{\textrm{e}^{ - {\mu_{\textrm{eff}}}{r_2}}}}}{{r_2^2}}} \right]$$
where ${r_1} = \sqrt {{\rho ^\textrm{2}}{\ +\ }{\textrm{z}_0}^2} ;{\ }{r_2} = \sqrt {{\rho ^{2}} + {{({\textrm{z}_0} + 2{z_b})}^2};} {\ }{\textrm{z}_0} = 1/\mu _\textrm{s}^{\prime}$; and ${z_b} = 2D$.

Assuming that ${r_1} \approx {r_2} \approx \sqrt {{\rho ^2} + {{({z_0} + 2{z_\textrm{b}})}^2}} \approx \rho$, Eq. (B-1) can be simplified to Eq. (B-2).

$$I(\rho )= {I_0}\frac{1}{{4\pi }}\left[ {\frac{1}{K}\left( {{\mu_{\textrm{eff}}} + \frac{1}{\rho }} \right)\frac{{{\textrm{e}^{ - {\mu_{\textrm{eff}}} \cdot \rho }}}}{{{\rho^2}}}} \right]$$
where K is termed the diffusion coefficient in a semi-infinite situation, expressed as $K = 1/(2{z_0} + 2{z_\textrm{b}})$. In Eq. (B-2), when the SDS is small, we have $\rho \ll \frac{1}{{{\mu _{\textrm{eff}}}}}$, ${\mu _{\textrm{eff}}} + \frac{1}{\rho } \approx \frac{1}{\rho }$; conversely, when the SDS is larger, we have $\rho \gg \frac{1}{{{\mu _{\textrm{eff}}}}}$, ${\mu _{\textrm{eff}}} + \frac{1}{\rho } \approx {\mu _{\textrm{eff}}}$. After these simplifications, Eq. (B-2) can be simplified as Eq. (B-3).
$$\textrm{For}\ \textrm{small}\ \textrm{SDS}:\quad I(\rho )= {I_0}\frac{1}{{4\pi }}\left[ {\frac{1}{K}\frac{{{\textrm{e}^{ - {\mu_{\textrm{eff}}} \cdot \rho }}}}{{{\rho^3}}}} \right]$$
$$\textrm{For}{\ }\textrm{larger}{\ }\textrm{SDS}:\quad I(\rho )= {I_0}\frac{1}{{4\pi }}\left[ {\frac{1}{K}({{\mu_{\textrm{eff}}}} )\frac{{{\textrm{e}^{ - {\mu_{\textrm{eff}}} \cdot \rho }}}}{{{\rho^2}}}} \right]$$

The separated parts, i.e., the D-signal and EA-signal, would be:

$$\textrm{For}{\ }\textrm{small}{\ }\textrm{SDS}:\quad {I_\textrm{D}}(\rho )= \frac{1}{{4\pi K{\rho ^3}}};{\ }{I_\textrm{A}}(\rho )= \textrm{exp}( - {\mu _{\textrm{eff}}} \cdot \rho )$$
$$\textrm{For}{\ }\textrm{larger}{\ }\textrm{SDS}:\quad {I_\textrm{D}}(\rho )= \frac{1}{{4\pi \frac{K}{{{\mu _{\textrm{eff}}}}}{\rho ^2}}};{\ }{I_\textrm{A}}(\rho )= \textrm{exp}( - {\mu _{\textrm{eff}}} \cdot \rho )$$

The separated attenuance ${A_{\rho }}$ could be:

$$\textrm{For}{\ }\textrm{small}{\ }\textrm{SDS}:\quad {A_{{\rho },\textrm{D}}} = \; \textrm{ln}4\pi K{\rho ^3};{\ }{A_{{\rho },\textrm{A}}} = {\mu _{\textrm{eff}}} \cdot \rho$$
$$\textrm{For}{\ }\textrm{larger}{\ }\textrm{SDS}:\quad {A_{{\rho },\textrm{D}}} = \; \textrm{ln}4\pi \frac{K}{{{\mu _{\textrm{eff}}}}}{\rho ^2};{\ }{A_{{\rho },\textrm{A}}} = {\mu _{\textrm{eff}}} \cdot \rho$$

When the component concentrations change, the change in absorbance can be expressed as:

$$\textrm{For}{\ }\textrm{small}{\ }\textrm{SDS}:\quad \Delta {A_{{\rho },\textrm{D}}} = \frac{{\Delta K}}{K};{\ }\Delta {A_{{\rho },\textrm{A}\;\ \;\ }} = \Delta {\mu _{\textrm{eff}}} \cdot \rho ;{\ }\Delta {A_{\rho }} = \frac{{\Delta K}}{K} + \Delta {\mu _{\textrm{eff}}} \cdot \rho$$
$$\textrm{For}{\ }\textrm{larger}{\ }\textrm{SDS}:\quad \Delta {A_{{\rho },\textrm{D}}} = \frac{{\Delta K}}{K} - \frac{{\Delta {\mu _{\textrm{eff}}}}}{{{\mu _{\textrm{eff}}}}};{\ }\Delta {A_{{\rho },\textrm{A}}} = \Delta {\mu _{\textrm{eff}}} \cdot \rho ;{\ }\Delta {A_{\rho }} = \frac{{\Delta K}}{K} - \frac{{\Delta {\mu _{\textrm{eff}}}}}{{{\mu _{\textrm{eff}}}}} + \Delta {\mu _{\textrm{eff}}} \cdot \rho$$

Appendix C: Sensitivity of using single optical path-length or SDS

The single-position measurement sensitivity S can be expressed as Eq. (C-1), where ${I_\textrm{L}}$ is the measurement light intensity; L is the optical path-length (OPL) or SDS between the light source and detector; and $C$ is the analyte concentration.

$$S = \frac{{\textrm{d}{I_\textrm{L}}}}{{\textrm{d}C}}$$
$S$ can also be described as Eq. (C-2).
$$S = - {I_\textrm{L}} \cdot \frac{{\textrm{d}{A_\textrm{L}}}}{{\textrm{d}C}}$$
where ${A_\textrm{L}}$ is the attenuance, given as ${A_\textrm{L}} = - \textrm{ln(}{I_\textrm{L}}/{I_0})$.

For pure absorbing media, Beer-Lambert’s law can be used to express the ${I_\textrm{L}} = {I_0}\textrm{exp}({ - {\mu_\textrm{a}}L} )$. The S can also be expressed as:

$$S = \frac{{\textrm{d}{I_\textrm{L}}}}{{\textrm{d}C}} = - {I_\textrm{L}}L\frac{{\textrm{d}{\mu _\textrm{a}}}}{{\textrm{d}C}} = - {I_0}\textrm{exp}({ - {\mu_\textrm{a}}L} )L\frac{{\textrm{d}{\mu _\textrm{a}}}}{{\textrm{d}C}}$$
where ${I_\textrm{L}}$ is the transmitted light intensity; ${\mu _\textrm{a}}$ is the absorption coefficient of the medium; and L is the optical path-length. The sensitivity S can reach a maximum at the optimal path length ${L_{\textrm{opt}}} = \frac{1}{{{\mu _\textrm{a}}}}$ when setting $\frac{{\textrm{d}S}}{{\textrm{d}L}} = 0$.

For scattering media, the sensitivity can be denoted as ${S_{\rho }}$. For infinite media and semi-infinite media, the sensitivity could be:

$$\textrm{For}{\ }\textrm{infinite}{\ }\textrm{media}:\quad {S_{\rho }} = \frac{{\textrm{d}{I_\rho }}}{{\textrm{d}C}} = - {I_{\rho }}\left( {\rho \frac{{\textrm{d}{\mu_{\textrm{eff}}}}}{{\textrm{d}C}} + \frac{1}{D}\frac{{\textrm{d}D}}{{\textrm{d}C}}} \right)$$
$${\textrm{For}}{\ }{\textrm{semi-infinite}}{\ }\textrm{media}{\ }\textrm{at}{\ }\textrm{small}{\ }\textrm{SDSs}:\quad {S_{\rho }} = \frac{{\textrm{d}{I_\rho }}}{{\textrm{d}C}} = - {I_{\rho }}\left( {\rho \frac{{\textrm{d}{\mu_{\textrm{eff}}}}}{{\textrm{d}C}} + \frac{1}{K}\frac{{\textrm{d}K}}{{\textrm{d}C}}} \right)$$
$$\textrm{For}{\ }\textrm{semi-infinite}{\ }\textrm{media}{\ }\textrm{at}{\ }\textrm{larger}{\ }\textrm{SDSs}:\quad {S_{\rho }} = \frac{{\textrm{d}{I_\rho }}}{{\textrm{d}C}} = - {I_{\rho }}\left( {\rho \frac{{\textrm{d}{\mu_{\textrm{eff}}}}}{{\textrm{d}C}} + \frac{1}{K}\frac{{\textrm{d}K}}{{\textrm{d}C}} - \frac{1}{{{\mu_{\textrm{eff}}}}}\frac{{\textrm{d}{\mu_{\textrm{eff}}}}}{{\textrm{d}C}}} \right)$$
where $K = 1/(2{z_0} + 2{z_\textrm{b}})$.

Appendix D: Sensitivity of using two optical path-lengths or SDSs

${L_\textrm{A}}$ and ${L_\textrm{B}}$ are the two optical path-lengths or SDSs for the EA-signal-based measurement that utilizes a differential processing approach for the signals obtained from two SDSs. ${I_e}$ denotes the equivalent light intensity of the differential, defined as ${I_e} = {I_{{\textrm{L}_\textrm{B}}}}/{I_{{\textrm{L}_\textrm{A}}}}$. Assuming that ${L_\textrm{B}} > {L_\textrm{A}}$, then ${L_\textrm{e}} = {L_\textrm{B}} - {L_\textrm{A}}$, where ${L_\textrm{e}}$ is the equivalent optical path-length or SDS. In addition, ${A_{{\textrm{I}_\textrm{e}}}}$ is the equivalent absorbance or attenuance, given as ${A_{{\textrm{I}_\textrm{e}}}} = {A_{{\textrm{L}_\textrm{B}}}} - {A_{{\textrm{L}_\textrm{A}}}} = - \textrm{ln(}{I_{{\textrm{L}_\textrm{B}}}}/{I_{{\textrm{L}_\textrm{A}}}})$, where ${A_{{\textrm{L}_\textrm{A}}}}$ and ${A_{{\textrm{L}_\textrm{B}}}}$ are the absorbance or attenuance at ${L_\textrm{A}}$ and ${L_\textrm{B}}$, respectively.

The differential measurement sensitivity ${S_{{\textrm{I}_\textrm{e}}}}$ can be determined as in Eq. (D-1).

$${S_{{I_e}}} = \frac{{\textrm{d}{I_e}}}{{\textrm{d}C}} = \frac{{{I_{{\textrm{L}_\textrm{B}}}}}}{{{I_{{\textrm{L}_\textrm{A}}}}}}\left( {\frac{1}{{{I_{{\textrm{L}_\textrm{B}}}}}}\frac{{\textrm{d}{I_{{\textrm{L}_\textrm{B}}}}}}{{\textrm{d}C}} - \frac{1}{{{I_{{\textrm{L}_\textrm{B}}}}}}\frac{{\textrm{d}{I_{{\textrm{L}_\textrm{A}}}}}}{{\textrm{d}C}}} \right) = - {I_e} \cdot \frac{{\textrm{d}{A_{{\textrm{I}_\textrm{e}}}}}}{{\textrm{d}C}}$$

The noise for a differential measurement is defined as ${\sigma }{I_\textrm{e}}$, which could be determined by the noise at both ${L_\textrm{A}}$ and ${L_\textrm{B}}$.

$${\sigma }{I_\textrm{e}} = \frac{{{I_{{\textrm{L}_\textrm{B}}}}}}{{{I_{{\textrm{L}_\textrm{A}}}}}}\sqrt {{{\left( {\frac{1}{{{I_{{\textrm{L}_\textrm{B}}}}}}} \right)}^2} \cdot {\sigma }I_{{\textrm{L}_\textrm{B}}}^2 + {{\left( {\frac{1}{{{I_{{\textrm{L}_\textrm{A}}}}}}} \right)}^2} \cdot {\sigma }I_{{\textrm{L}_\textrm{A}}}^2}$$

It is possible for ${I_{{\textrm{L}_\textrm{A}}}} \gg {I_{{\textrm{L}_\textrm{B}}}}$ when ${L_\textrm{B}} > {L_\textrm{A}}$, since the diffuse light intensity sharply decreases as the path length or SDS increases. Meanwhile, ${\sigma }{I_{{\textrm{L}_\textrm{A}}}} \approx {\sigma }{I_{{\textrm{L}_\textrm{B}}}}$ is true for the same detectors, denoted as ${\sigma }I$. Eq. (D-2) can be simplified as:

$${\sigma }{I_\textrm{e}} \approx \frac{1}{{{I_{{\textrm{L}_\textrm{A}}}}}}{\sigma }I = \beta \cdot {\sigma }I$$
where $\beta = 1/{I_{{\textrm{L}_\textrm{A}}}}$. In Eq. (D-3), the integrated noise ${\sigma }{I_\textrm{e}}$ is $\beta$ times the measurable noise ${\sigma }I$. Therefore, to compare this value with the original noise ${\sigma }I$, we define the sensitivity for the differential measurement as the $1/\beta$ times of ${S_{{I_e}}}$, denoted as ${S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}}$.
$${S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}} = {S_{{\textrm{I}_\textrm{e}}}}/\beta \approx - {I_{{\textrm{L}_\textrm{B}}}} \cdot \frac{{\textrm{d}{A_{{\textrm{I}_\textrm{e}}}}}}{{\textrm{d}C}}$$

Thus, the concentration limit for measurement ${C_{\textrm{limit}}}$ for a noise that is three times a great can be expressed as Eq. (D-5).

$${C_{\textrm{limit}}} = \frac{{\Delta {I_{\textrm{noise}}}}}{{dI/dC}} = \frac{{3{\sigma }{I_\textrm{e}}}}{{|{{S_{{\textrm{I}_\textrm{e}}}}} |}} = \frac{{3{\sigma }I}}{{|{{S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}}} |}}$$

If the noise level of the instrument is given, then the ${C_{\textrm{limit}}}$ can be estimated as:

$${C_{\textrm{limit}}} = \frac{{3{\sigma }I}}{{|{{S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}}} |}} \approx {\;\ }\frac{1}{{\left|{\frac{{{I_{{\textrm{L}_\textrm{B}}}}}}{{3{\sigma }I}} \cdot \frac{{\textrm{d}{A_{{\textrm{I}_\textrm{e}}}}}}{{\textrm{d}C}}} \right|}} = \frac{1}{{\left|{SN{R_{{L_\textrm{B}}}} \cdot \frac{{\textrm{d}{A_{{\textrm{I}_\textrm{e}}}}}}{{\textrm{d}C}}} \right|}}$$
where $SN{R_{{L_\textrm{B}}}} = {I_{{\textrm{L}_\textrm{B}}}}/3{\sigma }I$ is the SNR at ${L_\textrm{B}}$.

Appendix E: Specific equations when two optical path-lengths are used for a pure absorbing medium

Using Beer-Lambert’s law, the equivalent light intensity ${I_e}$ and absorbance ${A_{{\textrm{I}_\textrm{e}}}}$ for pure absorbing media and two path-lengths is given as:

$${I_e} = \frac{{{I_{{\textrm{L}_\textrm{B}}}}}}{{{I_{{\textrm{L}_\textrm{A}}}}}} = \exp ({ - {\mu_\textrm{a}}({L_\textrm{B}} - {L_\textrm{A}})} )$$
where ${\mu _\textrm{a}}$ the is absorption coefficient. The equivalent path-length is ${L_\textrm{e}}$, given as ${L_\textrm{e}} = {L_\textrm{B}} - {L_\textrm{A}}$.

Further, the differential sensitivity ${S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}}$ can be expressed as:

$${S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}} = - {I_{{\textrm{L}_\textrm{B}}}}({{L_\textrm{B}} - {L_\textrm{A}}} )\frac{{\textrm{d}{\mu _\textrm{a}}}}{{\textrm{d}C}} = - {I_{{\textrm{L}_\textrm{A}}}}({\textrm{exp}({ - {\mu_\textrm{a}}{L_\textrm{e}}} )} )({{L_\textrm{e}}} )\frac{{\textrm{d}{\mu _\textrm{a}}}}{{\textrm{d}C}}$$

In addition, the partial differential of ${S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}}$ on ${L_\textrm{A}}$ and ${L_\textrm{e}}$ can be obtained as:

$$\frac{{\partial {S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}}}}{{\partial {L_\textrm{A}}}} = {I_{{\textrm{L}_\textrm{A}}}}{\mu _\textrm{a}}({\textrm{exp}({ - {\mu_\textrm{a}}{L_\textrm{e}}} )} )({{L_\textrm{e}}} )\frac{{\textrm{d}{\mu _\textrm{a}}}}{{\textrm{d}C}} > 0$$
$$\frac{{\partial {S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}}}}{{\partial {L_\textrm{e}}}} = {I_{{\textrm{L}_\textrm{A}}}}({\textrm{exp}({ - {\mu_\textrm{a}}{L_\textrm{e}}} )} )\frac{{\textrm{d}{\mu _\textrm{a}}}}{{\textrm{d}C}}({1 - {L_\textrm{e}}{\mu_\textrm{a}}} )$$

Assuming that $\frac{{\textrm{d}{\mu _\textrm{a}}}}{{\textrm{d}C}} > 0$, we could have $\frac{{\partial {S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}}}}{{\partial {L_\textrm{A}}}} > 0$ and ${S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}} < 0$. If so, then ${S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}}$ would increase with the increase in ${L_\textrm{A}}$; however, this absolute value of this value would decrease with increasing ${L_\textrm{A}}$. Therefore, a smaller value of ${L_\textrm{A}}$ could achieve a larger absolute sensitivity. When ${L_\textrm{A}}$ is given, and setting $\frac{{\partial {S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}}}}{{\partial {L_\textrm{e}}}} = 0$, then the optimal ${L_\textrm{e}}$ can be derived as ${L_\textrm{e}} = \frac{1}{{{\mu _\textrm{a}}}}$, where the ${S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}}$ reaches a maximum. Thus, the optimal two path-lengths for the maximum ${S_{{\textrm{L}_\textrm{A}},{\textrm{L}_\textrm{B}}}}$ should be determined as:

$$\left\{ \begin{array}{l} {L_\textrm{A}}{\ }\textrm{is}\ \textrm{as}\ \textrm{small}\ \textrm{as}\ \textrm{possible}\\ {L_\textrm{B}} - {L_\textrm{A}} = \frac{1}{{{\mu_\textrm{a}}}} \end{array} \right.$$

For the special case of ${L_\textrm{A}} = 0$, then only one path-length ${L_\textrm{B}}$ is used, and the optimal ${L_\textrm{B}}$ can be determined as ${L_\textrm{B}} = \frac{1}{{{\mu _\textrm{a}}}}$, which has been reported previously [4].

Appendix F: Specific equations when using two SDSs for a scattering medium

Similarly, using the diffusion equation, the equivalent light intensity ${I_e}$ and attenuance ${A_{{\textrm{I}_\textrm{e}}}}$ for scattering media when using two SDSs of ${\rho _\textrm{A}}$ and ${\rho _\textrm{B}}$ can be acquired.

For infinite media, using the diffusion equation of Eq. (A-1), the ${I_e}$ and attenuance ${A_{{\textrm{I}_\textrm{e}}}}$ are:

$${I_e} = \frac{{{I_{{{\rho }_\textrm{B}}}}}}{{{I_{{{\rho }_\textrm{A}}}}}}$$
$${A_{{\textrm{I}_\textrm{e}}}} = - \textrm{ln}{I_e}$$

The sensitivity ${S_{{{\rho }_\textrm{A}},{{\rho }_\textrm{B}}}}$ is given as:

$${S_{{{\rho }_\textrm{A}},{{\rho }_\textrm{B}}}} = - {I_{{{\rho }_\textrm{B}}}}\left( {\frac{{\textrm{d}{A_{{\textrm{L}_\textrm{B}}}}}}{{\textrm{d}C}} - \frac{{\textrm{d}{A_{{\textrm{L}_\textrm{A}}}}}}{{\textrm{d}C}}} \right) = - {I_{{{\rho }_\textrm{B}}}}({{\rho_\textrm{B}} - {\rho_\textrm{A}}} )\frac{{\textrm{d}{\mu _{\textrm{eff}}}}}{{\textrm{d}C}}$$
$$= - m\frac{{\exp ({ - {\mu_{\textrm{eff}}}{\rho_\textrm{A}}} )}}{{({{\rho_\textrm{A}} + {\rho_\textrm{e}}} )}}\exp ({ - {\mu_{\textrm{eff}}}{\rho_\textrm{e}}} )({{\rho_\textrm{e}}} )\frac{{\textrm{d}{\mu _{\textrm{eff}}}}}{{\textrm{d}C}}$$
where $m = {I_0}\frac{1}{{4\pi D}}$, and ${\rho _\textrm{e}} = {\rho _\textrm{B}} - {\rho _\textrm{A}}$ denotes the equivalent SDS.

Similarly, the partial differential of ${\rho _\textrm{A}}$ and ${\rho _\textrm{e}}$ is expressed as:

$$\frac{{\partial {S_{{{\rho }_\textrm{A}},{{\rho }_\textrm{B}}}}}}{{\partial {\rho _\textrm{A}}}} = m(\frac{{1 + {\mu _{\textrm{eff}}}({{\rho_\textrm{A}} + {\rho_\textrm{e}}} )}}{{{{({{\rho_\textrm{A}} + {\rho_\textrm{e}}} )}^2}}})\exp ({ - {\mu_{\textrm{eff}}}{\rho_\textrm{A}}} )\textrm{exp}( - {\mu _{\textrm{eff}}}{\rho _\textrm{e}})({{\rho_\textrm{e}}} )\frac{{\textrm{d}{\mu _{\textrm{eff}}}}}{{\textrm{d}C}} > 0$$
$$\frac{{\partial {S_{{{\rho }_\textrm{A}},{{\rho }_\textrm{B}}}}}}{{\partial {\rho _\textrm{e}}}} = - m \cdot \textrm{exp}( - {\mu _{\textrm{eff}}}({\rho _\textrm{A}} + {\rho _\textrm{e}}))\frac{1}{{{{({{\rho_\textrm{A}} + {\rho_\textrm{e}}} )}^2}}}\frac{{\textrm{d}{\mu _{\textrm{eff}}}}}{{\textrm{d}C}}({\mu _{\textrm{eff}}}{\rho _\textrm{e}}^2 + {\mu _{\textrm{eff}}}{\rho _\textrm{A}}{\rho _\textrm{e}} - {\rho _\textrm{A}})$$

When we have $\frac{{\partial {S_{{{\rho }_\textrm{A}},{{\rho }_\textrm{B}}}}}}{{\partial {\rho _\textrm{A}}}} > 0$ & ${S_{{{\rho }_\textrm{A}},{{\rho }_\textrm{B}}}} < 0$, then ${S_{{{\rho }_\textrm{A}},{{\rho }_\textrm{B}}}}$ can yield larger values at for smaller values of ${\rho _\textrm{A}}$. When ${\rho _\textrm{A}}$ is given, and setting $\frac{{\partial {S_{{{\rho }_\textrm{A}},{{\rho }_\textrm{B}}}}}}{{\partial {\rho _\textrm{e}}}} = 0$, then we obtain the optimal ${\rho _\textrm{e}}\;$. Thus, the optimal two SDSs for the maximum ${S_{{{\rho }_\textrm{A}},{{\rho }_\textrm{B}}}}$ should be determined as:

$$\left\{ \begin{array}{l} {\rho_\textrm{A}}{\ }\textrm{is}\ \textrm{as}\ \textrm{small}\ \textrm{as}\ \textrm{possible}\\ {\rho_\textrm{B}} - {\rho_\textrm{A}} = \frac{{\sqrt {{\rho_\textrm{A}}^2 + \frac{{4{\rho_\textrm{A}}}}{{{\mu_{\textrm{eff}}}}}} - {\rho_\textrm{A}}}}{2} \end{array} \right.$$

For semi-infinite media, when using Eq. (B-3) for larger SDSs, the absorbance ${A_{{\textrm{I}_\textrm{e}}}}$ and the sensitivity ${S_{{{\rho }_\textrm{A}},{{\rho }_\textrm{B}}}}$ can be determined as:

$${A_{{\textrm{I}_\textrm{e}}}} = - \textrm{ln}\frac{{{\rho _\textrm{A}}^2}}{{{\rho _\textrm{B}}^2}} + {\mu _{\textrm{eff}}}({{\rho_\textrm{B}} - {\rho_\textrm{A}}} )$$
$${S_{{{\rho }_\textrm{A}},{{\rho }_\textrm{B}}}} = - {I_{{{\rho }_\textrm{B}}}}\left( {\frac{{\textrm{d}{A_{{\textrm{L}_\textrm{B}}}}}}{{\textrm{d}C}} - \frac{{\textrm{d}{A_{{\textrm{L}_\textrm{A}}}}}}{{\textrm{d}C}}} \right) = - {I_{{{\rho }_\textrm{B}}}}({{\rho_\textrm{B}} - {\rho_\textrm{A}}} )\frac{{\textrm{d}{\mu _{\textrm{eff}}}}}{{\textrm{d}C}}$$
$$= - m\frac{{\exp ({ - {\mu_{\textrm{eff}}}{\rho_\textrm{A}}} )}}{{{{({{\rho_\textrm{A}} + {\rho_\textrm{e}}} )}^2}}}\exp ({ - {\mu_{\textrm{eff}}}{\rho_\textrm{e}}} )({{\rho_\textrm{e}}} )\frac{{\textrm{d}{\mu _{\textrm{eff}}}}}{{\textrm{d}C}}$$
where $m = {I_0}\frac{1}{{4\pi }}[{({2{z_0} + 2{z_\textrm{b}}} )({{\mu_{\textrm{eff}}}} )} ]$.

Likewise, the optimal two SDSs for the maximum ${S_{{{\rho }_\textrm{A}},{{\rho }_\textrm{B}}}}$ can be derived as:

$$\left\{ \begin{array}{l} {\rho_\textrm{A}}{\ }\textrm{is}\ \textrm{as}\ \textrm{small}\ as\ \textrm{possible}\\ {\rho_\textrm{B}} - {\rho_\textrm{A}} = \frac{{\sqrt {{\rho_\textrm{A}}^2 + \frac{1}{{{\mu_{\textrm{eff}}}^2}} + \frac{{6{\rho_\textrm{A}}}}{{{\mu_{\textrm{eff}}}}}} - \left( {{\rho_\textrm{A}} + \frac{1}{{{\mu_{\textrm{eff}}}}}} \right)}}{2} \end{array} \right.$$

Funding

National High-tech Research and Development Program (2012AA022602); Project of “111” (B07014).

Acknowledgments

The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Disclosures

The authors declare that there are no known conflicts of interest related to this article.

References

1. R. Cole, “Optimum Optical Density in Spectrophotometry,” J. Opt. Soc. Am. 41(1), 38–40 (1951). [CrossRef]  

2. D. E. Honigs, G. M. Hieftje, and T. Hirschfeld, “An Optimization of Detector-Noise-Limited Infrared Multiwavelength Determinations,” Appl. Spectrosc. 39(2), 253–256 (1985). [CrossRef]  

3. P. S. Jensen and J. Bak, “Near-Infrared Transmission Spectroscopy of Aqueous Solutions: Influence of Optical Pathlength on Signal-to-Noise Ratio,” Appl. Spectrosc. 56(12), 1600–1606 (2002). [CrossRef]  

4. Z. Zhang, Z. Cai, G. Han, W. Sheng, and J. Liu, “Estimation of glucose absorption spectrum at its optimum pathlength for every wavelength over a wide range,” Spectrosc. Lett. 49(9), 588–595 (2016). [CrossRef]  

5. X. Min, R. Liu, B. Fu, and K. Xu, “A preliminary verification of the floating reference measurement method for non-invasive blood glucose sensing,” Opt. Laser Technol. 91, 7–12 (2017). [CrossRef]  

6. G. Han, T. Han, K. Xu, and J. Liu, “Floating reference position-based correction method for near-infrared spectroscopy in long-term glucose concentration monitoring,” J. Biomed. Opt. 22(7), 077001 (2017). [CrossRef]  

7. G. Kumar and J. M. Schmitt, “Optimal probe geometry for near-infrared spectroscopy of biological tissue,” Appl. Opt. 36(10), 2286–2293 (1997). [CrossRef]  

8. J. R. Mourant, I. J. Bigio, D. A. Jack, T. M. Johnson, and H. D. Miller, “Measuring absorption coefficients in small volumes of highly scattering media: source-detector separations for which path lengths do not depend on scattering properties,” Appl. Opt. 36(22), 5655–5661 (1997). [CrossRef]  

9. H. Liu, “Unified analysis of the sensitivities of reflectance and path length to scattering variations in a diffusive medium,” Appl. Opt. 40(10), 1742–1746 (2001). [CrossRef]  

10. S. C. Kanick, H. J. Sterenborg, and A. Amelink, “Empirical model description of photon path length for differential path length spectroscopy: combined effect of scattering and absorption,” J. Biomed. Opt. 13(6), 064042 (2008). [CrossRef]  

11. M. Mehrabi, S. Setayeshi, M. G. Maragheh, S. H. Ardehali, and H. Arabalibeik, “Design of a new reflectance pulse oximeter by obtaining the optimal source-detector space,” Optik 168, 34–45 (2018). [CrossRef]  

12. J. Liu, C. Zhu, J. Jiang, and K. Xu, “Scattering-independent glucose absorption measurement using a spectrally resolved reflectance setup with specialized variable source-detector separations,” Biomed. Opt. Express 9(12), 5903–5914 (2018). [CrossRef]  

13. K. Xu and J. Liu, “Absorption and scattering effects separation from diffuse spectrum data and model establishment and concentration prediction based thereon,” U.S. Patent 9,551,656 (2017).

14. M. Kohl, M. Cope, M. Essenpreis, and D. Böcker, “Influence of glucose concentration on light scattering in tissue-simulating phantoms,” Opt. Lett. 19(24), 2170 (1994). [CrossRef]  

15. A. K. Amerov, J. Chen, and M. A. Arnold, “Molar absorptivities of glucose and other biological molecules in aqueous solutions over the first overtone and combination regions of the near-infrared spectrum,” Appl. Spectrosc. 58(10), 1195–1204 (2004). [CrossRef]  

16. A. K. Amerov, J. Chen, G. W. Small, and M. A. Arnold, “Scattering and absorption effects in the determination of glucose in whole blood by near-infrared spectroscopy,” Anal. Chem. 77(14), 4587–4594 (2005). [CrossRef]  

17. K. Maruo and Y. Yamada, “Near-infrared noninvasive blood glucose prediction without using multivariate analyses: introduction of imaginary spectra due to scattering change in the skin,” J. Biomed. Opt. 20(4), 047003 (2015). [CrossRef]  

18. J. T. Bruulsema, J. E. Hayward, T. J. Farrell, M. S. Patterson, L. Heinemann, M. Berger, T. Koschinsky, J. Sandahl-Christiansen, H. Orskov, and M. Essenpreis, “Correlation between blood glucose concentration in diabetics and noninvasively measured tissue optical scattering coefficient,” Opt. Lett. 22(3), 190–192 (1997). [CrossRef]  

19. L. Heinemann and G. Schmelzeisen-Redeker, “Non-invasive continuous glucose monitoring in Type I diabetic patients with optical glucose sensors,” Diabetologia 41(7), 848–854 (1998). [CrossRef]  

20. M. Goodarzi, S. Sharma, H. Ramon, and W. Saeys, “Multivariate calibration of NIR spectroscopic sensors for continuous glucose monitoring,” TrAC, Trends Anal. Chem. 67, 147–158 (2015). [CrossRef]  

21. S. Sharma, M. Goodarzi, H. Ramon, and W. Saeys, “Performance evaluation of preprocessing techniques utilizing expert information in multivariate calibration,” Talanta 121, 105–112 (2014). [CrossRef]  

22. F. Wülfert, W. T. Kok, O. E. D. Noord, and A. K. Smilde, “Linear techniques to correct for temperature-induced spectral variation in multivariate calibration,” Chemom. Intell. Lab. Syst. 51(2), 189–200 (2000). [CrossRef]  

23. J. Liu, R. Liu, and K. Xu, “Accuracy of Noninvasive Glucose Sensing Based on Near-Infrared Spectroscopy,” Appl. Spectrosc. 69(11), 1313–1318 (2015). [CrossRef]  

24. T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992). [CrossRef]  

25. J. R. Mourant, T. Fuselier, J. Boyer, T. M. Johnson, and I. J. Bigio, “Predictions and measurements of scattering and absorption over broad wavelength ranges in tissue phantoms,” Appl. Opt. 36(4), 949–957 (1997). [CrossRef]  

26. L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo Modeling of Photon Transport in Multi-layered Tissues,” Comput. Meth. Prog. Bio. 47(2), 131–146 (1995). [CrossRef]  

27. T. L. Troy and S. N. Thennadil, “Optical properties of human skin in the near infrared wavelength range of 1000 to 2200 nm,” J. Biomed. Opt. 6(2), 167–176 (2001). [CrossRef]  

28. M. Kohl, M. Essenpreis, and M. Cope, “The influence of glucose concentration upon the transport of light in tissue-simulating phantoms,” Phys. Med. Biol. 40(7), 1267–1287 (1995). [CrossRef]  

29. C. Chen, J. Q. Lu, H. Ding, K. M. Jacobs, Y. Du, and X. H. Hu, “A primary method for determination of optical parameters of turbid samples and application to intralipid between 550 and 1630 nm,” Opt. Express 14(16), 7420–7435 (2006). [CrossRef]  

30. C. Li, H. Zhao, Q. Wang, and K. Xu, “Artificial neural network method for determining optical properties from double-integrating-spheres measurements,” Chin. Opt. Lett. 8, 173–176 (2010). [CrossRef]  

31. Y. Mendelson, “Pulse Oximetery: Theory and Application for Noninvasive Monitoring,” Clin. Chem. 38(9), 1601–1607 (1992).

References

  • View by:

  1. R. Cole, “Optimum Optical Density in Spectrophotometry,” J. Opt. Soc. Am. 41(1), 38–40 (1951).
    [Crossref]
  2. D. E. Honigs, G. M. Hieftje, and T. Hirschfeld, “An Optimization of Detector-Noise-Limited Infrared Multiwavelength Determinations,” Appl. Spectrosc. 39(2), 253–256 (1985).
    [Crossref]
  3. P. S. Jensen and J. Bak, “Near-Infrared Transmission Spectroscopy of Aqueous Solutions: Influence of Optical Pathlength on Signal-to-Noise Ratio,” Appl. Spectrosc. 56(12), 1600–1606 (2002).
    [Crossref]
  4. Z. Zhang, Z. Cai, G. Han, W. Sheng, and J. Liu, “Estimation of glucose absorption spectrum at its optimum pathlength for every wavelength over a wide range,” Spectrosc. Lett. 49(9), 588–595 (2016).
    [Crossref]
  5. X. Min, R. Liu, B. Fu, and K. Xu, “A preliminary verification of the floating reference measurement method for non-invasive blood glucose sensing,” Opt. Laser Technol. 91, 7–12 (2017).
    [Crossref]
  6. G. Han, T. Han, K. Xu, and J. Liu, “Floating reference position-based correction method for near-infrared spectroscopy in long-term glucose concentration monitoring,” J. Biomed. Opt. 22(7), 077001 (2017).
    [Crossref]
  7. G. Kumar and J. M. Schmitt, “Optimal probe geometry for near-infrared spectroscopy of biological tissue,” Appl. Opt. 36(10), 2286–2293 (1997).
    [Crossref]
  8. J. R. Mourant, I. J. Bigio, D. A. Jack, T. M. Johnson, and H. D. Miller, “Measuring absorption coefficients in small volumes of highly scattering media: source-detector separations for which path lengths do not depend on scattering properties,” Appl. Opt. 36(22), 5655–5661 (1997).
    [Crossref]
  9. H. Liu, “Unified analysis of the sensitivities of reflectance and path length to scattering variations in a diffusive medium,” Appl. Opt. 40(10), 1742–1746 (2001).
    [Crossref]
  10. S. C. Kanick, H. J. Sterenborg, and A. Amelink, “Empirical model description of photon path length for differential path length spectroscopy: combined effect of scattering and absorption,” J. Biomed. Opt. 13(6), 064042 (2008).
    [Crossref]
  11. M. Mehrabi, S. Setayeshi, M. G. Maragheh, S. H. Ardehali, and H. Arabalibeik, “Design of a new reflectance pulse oximeter by obtaining the optimal source-detector space,” Optik 168, 34–45 (2018).
    [Crossref]
  12. J. Liu, C. Zhu, J. Jiang, and K. Xu, “Scattering-independent glucose absorption measurement using a spectrally resolved reflectance setup with specialized variable source-detector separations,” Biomed. Opt. Express 9(12), 5903–5914 (2018).
    [Crossref]
  13. K. Xu and J. Liu, “Absorption and scattering effects separation from diffuse spectrum data and model establishment and concentration prediction based thereon,” U.S. Patent 9,551,656 (2017).
  14. M. Kohl, M. Cope, M. Essenpreis, and D. Böcker, “Influence of glucose concentration on light scattering in tissue-simulating phantoms,” Opt. Lett. 19(24), 2170 (1994).
    [Crossref]
  15. A. K. Amerov, J. Chen, and M. A. Arnold, “Molar absorptivities of glucose and other biological molecules in aqueous solutions over the first overtone and combination regions of the near-infrared spectrum,” Appl. Spectrosc. 58(10), 1195–1204 (2004).
    [Crossref]
  16. A. K. Amerov, J. Chen, G. W. Small, and M. A. Arnold, “Scattering and absorption effects in the determination of glucose in whole blood by near-infrared spectroscopy,” Anal. Chem. 77(14), 4587–4594 (2005).
    [Crossref]
  17. K. Maruo and Y. Yamada, “Near-infrared noninvasive blood glucose prediction without using multivariate analyses: introduction of imaginary spectra due to scattering change in the skin,” J. Biomed. Opt. 20(4), 047003 (2015).
    [Crossref]
  18. J. T. Bruulsema, J. E. Hayward, T. J. Farrell, M. S. Patterson, L. Heinemann, M. Berger, T. Koschinsky, J. Sandahl-Christiansen, H. Orskov, and M. Essenpreis, “Correlation between blood glucose concentration in diabetics and noninvasively measured tissue optical scattering coefficient,” Opt. Lett. 22(3), 190–192 (1997).
    [Crossref]
  19. L. Heinemann and G. Schmelzeisen-Redeker, “Non-invasive continuous glucose monitoring in Type I diabetic patients with optical glucose sensors,” Diabetologia 41(7), 848–854 (1998).
    [Crossref]
  20. M. Goodarzi, S. Sharma, H. Ramon, and W. Saeys, “Multivariate calibration of NIR spectroscopic sensors for continuous glucose monitoring,” TrAC, Trends Anal. Chem. 67, 147–158 (2015).
    [Crossref]
  21. S. Sharma, M. Goodarzi, H. Ramon, and W. Saeys, “Performance evaluation of preprocessing techniques utilizing expert information in multivariate calibration,” Talanta 121, 105–112 (2014).
    [Crossref]
  22. F. Wülfert, W. T. Kok, O. E. D. Noord, and A. K. Smilde, “Linear techniques to correct for temperature-induced spectral variation in multivariate calibration,” Chemom. Intell. Lab. Syst. 51(2), 189–200 (2000).
    [Crossref]
  23. J. Liu, R. Liu, and K. Xu, “Accuracy of Noninvasive Glucose Sensing Based on Near-Infrared Spectroscopy,” Appl. Spectrosc. 69(11), 1313–1318 (2015).
    [Crossref]
  24. T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
    [Crossref]
  25. J. R. Mourant, T. Fuselier, J. Boyer, T. M. Johnson, and I. J. Bigio, “Predictions and measurements of scattering and absorption over broad wavelength ranges in tissue phantoms,” Appl. Opt. 36(4), 949–957 (1997).
    [Crossref]
  26. L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo Modeling of Photon Transport in Multi-layered Tissues,” Comput. Meth. Prog. Bio. 47(2), 131–146 (1995).
    [Crossref]
  27. T. L. Troy and S. N. Thennadil, “Optical properties of human skin in the near infrared wavelength range of 1000 to 2200 nm,” J. Biomed. Opt. 6(2), 167–176 (2001).
    [Crossref]
  28. M. Kohl, M. Essenpreis, and M. Cope, “The influence of glucose concentration upon the transport of light in tissue-simulating phantoms,” Phys. Med. Biol. 40(7), 1267–1287 (1995).
    [Crossref]
  29. C. Chen, J. Q. Lu, H. Ding, K. M. Jacobs, Y. Du, and X. H. Hu, “A primary method for determination of optical parameters of turbid samples and application to intralipid between 550 and 1630 nm,” Opt. Express 14(16), 7420–7435 (2006).
    [Crossref]
  30. C. Li, H. Zhao, Q. Wang, and K. Xu, “Artificial neural network method for determining optical properties from double-integrating-spheres measurements,” Chin. Opt. Lett. 8, 173–176 (2010).
    [Crossref]
  31. Y. Mendelson, “Pulse Oximetery: Theory and Application for Noninvasive Monitoring,” Clin. Chem. 38(9), 1601–1607 (1992).

2018 (2)

M. Mehrabi, S. Setayeshi, M. G. Maragheh, S. H. Ardehali, and H. Arabalibeik, “Design of a new reflectance pulse oximeter by obtaining the optimal source-detector space,” Optik 168, 34–45 (2018).
[Crossref]

J. Liu, C. Zhu, J. Jiang, and K. Xu, “Scattering-independent glucose absorption measurement using a spectrally resolved reflectance setup with specialized variable source-detector separations,” Biomed. Opt. Express 9(12), 5903–5914 (2018).
[Crossref]

2017 (2)

X. Min, R. Liu, B. Fu, and K. Xu, “A preliminary verification of the floating reference measurement method for non-invasive blood glucose sensing,” Opt. Laser Technol. 91, 7–12 (2017).
[Crossref]

G. Han, T. Han, K. Xu, and J. Liu, “Floating reference position-based correction method for near-infrared spectroscopy in long-term glucose concentration monitoring,” J. Biomed. Opt. 22(7), 077001 (2017).
[Crossref]

2016 (1)

Z. Zhang, Z. Cai, G. Han, W. Sheng, and J. Liu, “Estimation of glucose absorption spectrum at its optimum pathlength for every wavelength over a wide range,” Spectrosc. Lett. 49(9), 588–595 (2016).
[Crossref]

2015 (3)

K. Maruo and Y. Yamada, “Near-infrared noninvasive blood glucose prediction without using multivariate analyses: introduction of imaginary spectra due to scattering change in the skin,” J. Biomed. Opt. 20(4), 047003 (2015).
[Crossref]

M. Goodarzi, S. Sharma, H. Ramon, and W. Saeys, “Multivariate calibration of NIR spectroscopic sensors for continuous glucose monitoring,” TrAC, Trends Anal. Chem. 67, 147–158 (2015).
[Crossref]

J. Liu, R. Liu, and K. Xu, “Accuracy of Noninvasive Glucose Sensing Based on Near-Infrared Spectroscopy,” Appl. Spectrosc. 69(11), 1313–1318 (2015).
[Crossref]

2014 (1)

S. Sharma, M. Goodarzi, H. Ramon, and W. Saeys, “Performance evaluation of preprocessing techniques utilizing expert information in multivariate calibration,” Talanta 121, 105–112 (2014).
[Crossref]

2010 (1)

2008 (1)

S. C. Kanick, H. J. Sterenborg, and A. Amelink, “Empirical model description of photon path length for differential path length spectroscopy: combined effect of scattering and absorption,” J. Biomed. Opt. 13(6), 064042 (2008).
[Crossref]

2006 (1)

2005 (1)

A. K. Amerov, J. Chen, G. W. Small, and M. A. Arnold, “Scattering and absorption effects in the determination of glucose in whole blood by near-infrared spectroscopy,” Anal. Chem. 77(14), 4587–4594 (2005).
[Crossref]

2004 (1)

2002 (1)

2001 (2)

H. Liu, “Unified analysis of the sensitivities of reflectance and path length to scattering variations in a diffusive medium,” Appl. Opt. 40(10), 1742–1746 (2001).
[Crossref]

T. L. Troy and S. N. Thennadil, “Optical properties of human skin in the near infrared wavelength range of 1000 to 2200 nm,” J. Biomed. Opt. 6(2), 167–176 (2001).
[Crossref]

2000 (1)

F. Wülfert, W. T. Kok, O. E. D. Noord, and A. K. Smilde, “Linear techniques to correct for temperature-induced spectral variation in multivariate calibration,” Chemom. Intell. Lab. Syst. 51(2), 189–200 (2000).
[Crossref]

1998 (1)

L. Heinemann and G. Schmelzeisen-Redeker, “Non-invasive continuous glucose monitoring in Type I diabetic patients with optical glucose sensors,” Diabetologia 41(7), 848–854 (1998).
[Crossref]

1997 (4)

1995 (2)

L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo Modeling of Photon Transport in Multi-layered Tissues,” Comput. Meth. Prog. Bio. 47(2), 131–146 (1995).
[Crossref]

M. Kohl, M. Essenpreis, and M. Cope, “The influence of glucose concentration upon the transport of light in tissue-simulating phantoms,” Phys. Med. Biol. 40(7), 1267–1287 (1995).
[Crossref]

1994 (1)

1992 (2)

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref]

Y. Mendelson, “Pulse Oximetery: Theory and Application for Noninvasive Monitoring,” Clin. Chem. 38(9), 1601–1607 (1992).

1985 (1)

1951 (1)

Amelink, A.

S. C. Kanick, H. J. Sterenborg, and A. Amelink, “Empirical model description of photon path length for differential path length spectroscopy: combined effect of scattering and absorption,” J. Biomed. Opt. 13(6), 064042 (2008).
[Crossref]

Amerov, A. K.

A. K. Amerov, J. Chen, G. W. Small, and M. A. Arnold, “Scattering and absorption effects in the determination of glucose in whole blood by near-infrared spectroscopy,” Anal. Chem. 77(14), 4587–4594 (2005).
[Crossref]

A. K. Amerov, J. Chen, and M. A. Arnold, “Molar absorptivities of glucose and other biological molecules in aqueous solutions over the first overtone and combination regions of the near-infrared spectrum,” Appl. Spectrosc. 58(10), 1195–1204 (2004).
[Crossref]

Arabalibeik, H.

M. Mehrabi, S. Setayeshi, M. G. Maragheh, S. H. Ardehali, and H. Arabalibeik, “Design of a new reflectance pulse oximeter by obtaining the optimal source-detector space,” Optik 168, 34–45 (2018).
[Crossref]

Ardehali, S. H.

M. Mehrabi, S. Setayeshi, M. G. Maragheh, S. H. Ardehali, and H. Arabalibeik, “Design of a new reflectance pulse oximeter by obtaining the optimal source-detector space,” Optik 168, 34–45 (2018).
[Crossref]

Arnold, M. A.

A. K. Amerov, J. Chen, G. W. Small, and M. A. Arnold, “Scattering and absorption effects in the determination of glucose in whole blood by near-infrared spectroscopy,” Anal. Chem. 77(14), 4587–4594 (2005).
[Crossref]

A. K. Amerov, J. Chen, and M. A. Arnold, “Molar absorptivities of glucose and other biological molecules in aqueous solutions over the first overtone and combination regions of the near-infrared spectrum,” Appl. Spectrosc. 58(10), 1195–1204 (2004).
[Crossref]

Bak, J.

Berger, M.

Bigio, I. J.

Böcker, D.

Boyer, J.

Bruulsema, J. T.

Cai, Z.

Z. Zhang, Z. Cai, G. Han, W. Sheng, and J. Liu, “Estimation of glucose absorption spectrum at its optimum pathlength for every wavelength over a wide range,” Spectrosc. Lett. 49(9), 588–595 (2016).
[Crossref]

Chen, C.

Chen, J.

A. K. Amerov, J. Chen, G. W. Small, and M. A. Arnold, “Scattering and absorption effects in the determination of glucose in whole blood by near-infrared spectroscopy,” Anal. Chem. 77(14), 4587–4594 (2005).
[Crossref]

A. K. Amerov, J. Chen, and M. A. Arnold, “Molar absorptivities of glucose and other biological molecules in aqueous solutions over the first overtone and combination regions of the near-infrared spectrum,” Appl. Spectrosc. 58(10), 1195–1204 (2004).
[Crossref]

Cole, R.

Cope, M.

M. Kohl, M. Essenpreis, and M. Cope, “The influence of glucose concentration upon the transport of light in tissue-simulating phantoms,” Phys. Med. Biol. 40(7), 1267–1287 (1995).
[Crossref]

M. Kohl, M. Cope, M. Essenpreis, and D. Böcker, “Influence of glucose concentration on light scattering in tissue-simulating phantoms,” Opt. Lett. 19(24), 2170 (1994).
[Crossref]

Ding, H.

Du, Y.

Essenpreis, M.

Farrell, T. J.

J. T. Bruulsema, J. E. Hayward, T. J. Farrell, M. S. Patterson, L. Heinemann, M. Berger, T. Koschinsky, J. Sandahl-Christiansen, H. Orskov, and M. Essenpreis, “Correlation between blood glucose concentration in diabetics and noninvasively measured tissue optical scattering coefficient,” Opt. Lett. 22(3), 190–192 (1997).
[Crossref]

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref]

Fu, B.

X. Min, R. Liu, B. Fu, and K. Xu, “A preliminary verification of the floating reference measurement method for non-invasive blood glucose sensing,” Opt. Laser Technol. 91, 7–12 (2017).
[Crossref]

Fuselier, T.

Goodarzi, M.

M. Goodarzi, S. Sharma, H. Ramon, and W. Saeys, “Multivariate calibration of NIR spectroscopic sensors for continuous glucose monitoring,” TrAC, Trends Anal. Chem. 67, 147–158 (2015).
[Crossref]

S. Sharma, M. Goodarzi, H. Ramon, and W. Saeys, “Performance evaluation of preprocessing techniques utilizing expert information in multivariate calibration,” Talanta 121, 105–112 (2014).
[Crossref]

Han, G.

G. Han, T. Han, K. Xu, and J. Liu, “Floating reference position-based correction method for near-infrared spectroscopy in long-term glucose concentration monitoring,” J. Biomed. Opt. 22(7), 077001 (2017).
[Crossref]

Z. Zhang, Z. Cai, G. Han, W. Sheng, and J. Liu, “Estimation of glucose absorption spectrum at its optimum pathlength for every wavelength over a wide range,” Spectrosc. Lett. 49(9), 588–595 (2016).
[Crossref]

Han, T.

G. Han, T. Han, K. Xu, and J. Liu, “Floating reference position-based correction method for near-infrared spectroscopy in long-term glucose concentration monitoring,” J. Biomed. Opt. 22(7), 077001 (2017).
[Crossref]

Hayward, J. E.

Heinemann, L.

Hieftje, G. M.

Hirschfeld, T.

Honigs, D. E.

Hu, X. H.

Jack, D. A.

Jacobs, K. M.

Jacques, S. L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo Modeling of Photon Transport in Multi-layered Tissues,” Comput. Meth. Prog. Bio. 47(2), 131–146 (1995).
[Crossref]

Jensen, P. S.

Jiang, J.

Johnson, T. M.

Kanick, S. C.

S. C. Kanick, H. J. Sterenborg, and A. Amelink, “Empirical model description of photon path length for differential path length spectroscopy: combined effect of scattering and absorption,” J. Biomed. Opt. 13(6), 064042 (2008).
[Crossref]

Kohl, M.

M. Kohl, M. Essenpreis, and M. Cope, “The influence of glucose concentration upon the transport of light in tissue-simulating phantoms,” Phys. Med. Biol. 40(7), 1267–1287 (1995).
[Crossref]

M. Kohl, M. Cope, M. Essenpreis, and D. Böcker, “Influence of glucose concentration on light scattering in tissue-simulating phantoms,” Opt. Lett. 19(24), 2170 (1994).
[Crossref]

Kok, W. T.

F. Wülfert, W. T. Kok, O. E. D. Noord, and A. K. Smilde, “Linear techniques to correct for temperature-induced spectral variation in multivariate calibration,” Chemom. Intell. Lab. Syst. 51(2), 189–200 (2000).
[Crossref]

Koschinsky, T.

Kumar, G.

Li, C.

Liu, H.

Liu, J.

J. Liu, C. Zhu, J. Jiang, and K. Xu, “Scattering-independent glucose absorption measurement using a spectrally resolved reflectance setup with specialized variable source-detector separations,” Biomed. Opt. Express 9(12), 5903–5914 (2018).
[Crossref]

G. Han, T. Han, K. Xu, and J. Liu, “Floating reference position-based correction method for near-infrared spectroscopy in long-term glucose concentration monitoring,” J. Biomed. Opt. 22(7), 077001 (2017).
[Crossref]

Z. Zhang, Z. Cai, G. Han, W. Sheng, and J. Liu, “Estimation of glucose absorption spectrum at its optimum pathlength for every wavelength over a wide range,” Spectrosc. Lett. 49(9), 588–595 (2016).
[Crossref]

J. Liu, R. Liu, and K. Xu, “Accuracy of Noninvasive Glucose Sensing Based on Near-Infrared Spectroscopy,” Appl. Spectrosc. 69(11), 1313–1318 (2015).
[Crossref]

K. Xu and J. Liu, “Absorption and scattering effects separation from diffuse spectrum data and model establishment and concentration prediction based thereon,” U.S. Patent 9,551,656 (2017).

Liu, R.

X. Min, R. Liu, B. Fu, and K. Xu, “A preliminary verification of the floating reference measurement method for non-invasive blood glucose sensing,” Opt. Laser Technol. 91, 7–12 (2017).
[Crossref]

J. Liu, R. Liu, and K. Xu, “Accuracy of Noninvasive Glucose Sensing Based on Near-Infrared Spectroscopy,” Appl. Spectrosc. 69(11), 1313–1318 (2015).
[Crossref]

Lu, J. Q.

Maragheh, M. G.

M. Mehrabi, S. Setayeshi, M. G. Maragheh, S. H. Ardehali, and H. Arabalibeik, “Design of a new reflectance pulse oximeter by obtaining the optimal source-detector space,” Optik 168, 34–45 (2018).
[Crossref]

Maruo, K.

K. Maruo and Y. Yamada, “Near-infrared noninvasive blood glucose prediction without using multivariate analyses: introduction of imaginary spectra due to scattering change in the skin,” J. Biomed. Opt. 20(4), 047003 (2015).
[Crossref]

Mehrabi, M.

M. Mehrabi, S. Setayeshi, M. G. Maragheh, S. H. Ardehali, and H. Arabalibeik, “Design of a new reflectance pulse oximeter by obtaining the optimal source-detector space,” Optik 168, 34–45 (2018).
[Crossref]

Mendelson, Y.

Y. Mendelson, “Pulse Oximetery: Theory and Application for Noninvasive Monitoring,” Clin. Chem. 38(9), 1601–1607 (1992).

Miller, H. D.

Min, X.

X. Min, R. Liu, B. Fu, and K. Xu, “A preliminary verification of the floating reference measurement method for non-invasive blood glucose sensing,” Opt. Laser Technol. 91, 7–12 (2017).
[Crossref]

Mourant, J. R.

Noord, O. E. D.

F. Wülfert, W. T. Kok, O. E. D. Noord, and A. K. Smilde, “Linear techniques to correct for temperature-induced spectral variation in multivariate calibration,” Chemom. Intell. Lab. Syst. 51(2), 189–200 (2000).
[Crossref]

Orskov, H.

Patterson, M. S.

J. T. Bruulsema, J. E. Hayward, T. J. Farrell, M. S. Patterson, L. Heinemann, M. Berger, T. Koschinsky, J. Sandahl-Christiansen, H. Orskov, and M. Essenpreis, “Correlation between blood glucose concentration in diabetics and noninvasively measured tissue optical scattering coefficient,” Opt. Lett. 22(3), 190–192 (1997).
[Crossref]

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref]

Ramon, H.

M. Goodarzi, S. Sharma, H. Ramon, and W. Saeys, “Multivariate calibration of NIR spectroscopic sensors for continuous glucose monitoring,” TrAC, Trends Anal. Chem. 67, 147–158 (2015).
[Crossref]

S. Sharma, M. Goodarzi, H. Ramon, and W. Saeys, “Performance evaluation of preprocessing techniques utilizing expert information in multivariate calibration,” Talanta 121, 105–112 (2014).
[Crossref]

Saeys, W.

M. Goodarzi, S. Sharma, H. Ramon, and W. Saeys, “Multivariate calibration of NIR spectroscopic sensors for continuous glucose monitoring,” TrAC, Trends Anal. Chem. 67, 147–158 (2015).
[Crossref]

S. Sharma, M. Goodarzi, H. Ramon, and W. Saeys, “Performance evaluation of preprocessing techniques utilizing expert information in multivariate calibration,” Talanta 121, 105–112 (2014).
[Crossref]

Sandahl-Christiansen, J.

Schmelzeisen-Redeker, G.

L. Heinemann and G. Schmelzeisen-Redeker, “Non-invasive continuous glucose monitoring in Type I diabetic patients with optical glucose sensors,” Diabetologia 41(7), 848–854 (1998).
[Crossref]

Schmitt, J. M.

Setayeshi, S.

M. Mehrabi, S. Setayeshi, M. G. Maragheh, S. H. Ardehali, and H. Arabalibeik, “Design of a new reflectance pulse oximeter by obtaining the optimal source-detector space,” Optik 168, 34–45 (2018).
[Crossref]

Sharma, S.

M. Goodarzi, S. Sharma, H. Ramon, and W. Saeys, “Multivariate calibration of NIR spectroscopic sensors for continuous glucose monitoring,” TrAC, Trends Anal. Chem. 67, 147–158 (2015).
[Crossref]

S. Sharma, M. Goodarzi, H. Ramon, and W. Saeys, “Performance evaluation of preprocessing techniques utilizing expert information in multivariate calibration,” Talanta 121, 105–112 (2014).
[Crossref]

Sheng, W.

Z. Zhang, Z. Cai, G. Han, W. Sheng, and J. Liu, “Estimation of glucose absorption spectrum at its optimum pathlength for every wavelength over a wide range,” Spectrosc. Lett. 49(9), 588–595 (2016).
[Crossref]

Small, G. W.

A. K. Amerov, J. Chen, G. W. Small, and M. A. Arnold, “Scattering and absorption effects in the determination of glucose in whole blood by near-infrared spectroscopy,” Anal. Chem. 77(14), 4587–4594 (2005).
[Crossref]

Smilde, A. K.

F. Wülfert, W. T. Kok, O. E. D. Noord, and A. K. Smilde, “Linear techniques to correct for temperature-induced spectral variation in multivariate calibration,” Chemom. Intell. Lab. Syst. 51(2), 189–200 (2000).
[Crossref]

Sterenborg, H. J.

S. C. Kanick, H. J. Sterenborg, and A. Amelink, “Empirical model description of photon path length for differential path length spectroscopy: combined effect of scattering and absorption,” J. Biomed. Opt. 13(6), 064042 (2008).
[Crossref]

Thennadil, S. N.

T. L. Troy and S. N. Thennadil, “Optical properties of human skin in the near infrared wavelength range of 1000 to 2200 nm,” J. Biomed. Opt. 6(2), 167–176 (2001).
[Crossref]

Troy, T. L.

T. L. Troy and S. N. Thennadil, “Optical properties of human skin in the near infrared wavelength range of 1000 to 2200 nm,” J. Biomed. Opt. 6(2), 167–176 (2001).
[Crossref]

Wang, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo Modeling of Photon Transport in Multi-layered Tissues,” Comput. Meth. Prog. Bio. 47(2), 131–146 (1995).
[Crossref]

Wang, Q.

Wilson, B.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref]

Wülfert, F.

F. Wülfert, W. T. Kok, O. E. D. Noord, and A. K. Smilde, “Linear techniques to correct for temperature-induced spectral variation in multivariate calibration,” Chemom. Intell. Lab. Syst. 51(2), 189–200 (2000).
[Crossref]

Xu, K.

J. Liu, C. Zhu, J. Jiang, and K. Xu, “Scattering-independent glucose absorption measurement using a spectrally resolved reflectance setup with specialized variable source-detector separations,” Biomed. Opt. Express 9(12), 5903–5914 (2018).
[Crossref]

X. Min, R. Liu, B. Fu, and K. Xu, “A preliminary verification of the floating reference measurement method for non-invasive blood glucose sensing,” Opt. Laser Technol. 91, 7–12 (2017).
[Crossref]

G. Han, T. Han, K. Xu, and J. Liu, “Floating reference position-based correction method for near-infrared spectroscopy in long-term glucose concentration monitoring,” J. Biomed. Opt. 22(7), 077001 (2017).
[Crossref]

J. Liu, R. Liu, and K. Xu, “Accuracy of Noninvasive Glucose Sensing Based on Near-Infrared Spectroscopy,” Appl. Spectrosc. 69(11), 1313–1318 (2015).
[Crossref]

C. Li, H. Zhao, Q. Wang, and K. Xu, “Artificial neural network method for determining optical properties from double-integrating-spheres measurements,” Chin. Opt. Lett. 8, 173–176 (2010).
[Crossref]

K. Xu and J. Liu, “Absorption and scattering effects separation from diffuse spectrum data and model establishment and concentration prediction based thereon,” U.S. Patent 9,551,656 (2017).

Yamada, Y.

K. Maruo and Y. Yamada, “Near-infrared noninvasive blood glucose prediction without using multivariate analyses: introduction of imaginary spectra due to scattering change in the skin,” J. Biomed. Opt. 20(4), 047003 (2015).
[Crossref]

Zhang, Z.

Z. Zhang, Z. Cai, G. Han, W. Sheng, and J. Liu, “Estimation of glucose absorption spectrum at its optimum pathlength for every wavelength over a wide range,” Spectrosc. Lett. 49(9), 588–595 (2016).
[Crossref]

Zhao, H.

Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo Modeling of Photon Transport in Multi-layered Tissues,” Comput. Meth. Prog. Bio. 47(2), 131–146 (1995).
[Crossref]

Zhu, C.

Anal. Chem. (1)

A. K. Amerov, J. Chen, G. W. Small, and M. A. Arnold, “Scattering and absorption effects in the determination of glucose in whole blood by near-infrared spectroscopy,” Anal. Chem. 77(14), 4587–4594 (2005).
[Crossref]

Appl. Opt. (4)

Appl. Spectrosc. (4)

Biomed. Opt. Express (1)

Chemom. Intell. Lab. Syst. (1)

F. Wülfert, W. T. Kok, O. E. D. Noord, and A. K. Smilde, “Linear techniques to correct for temperature-induced spectral variation in multivariate calibration,” Chemom. Intell. Lab. Syst. 51(2), 189–200 (2000).
[Crossref]

Chin. Opt. Lett. (1)

Clin. Chem. (1)

Y. Mendelson, “Pulse Oximetery: Theory and Application for Noninvasive Monitoring,” Clin. Chem. 38(9), 1601–1607 (1992).

Comput. Meth. Prog. Bio. (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo Modeling of Photon Transport in Multi-layered Tissues,” Comput. Meth. Prog. Bio. 47(2), 131–146 (1995).
[Crossref]

Diabetologia (1)

L. Heinemann and G. Schmelzeisen-Redeker, “Non-invasive continuous glucose monitoring in Type I diabetic patients with optical glucose sensors,” Diabetologia 41(7), 848–854 (1998).
[Crossref]

J. Biomed. Opt. (4)

K. Maruo and Y. Yamada, “Near-infrared noninvasive blood glucose prediction without using multivariate analyses: introduction of imaginary spectra due to scattering change in the skin,” J. Biomed. Opt. 20(4), 047003 (2015).
[Crossref]

G. Han, T. Han, K. Xu, and J. Liu, “Floating reference position-based correction method for near-infrared spectroscopy in long-term glucose concentration monitoring,” J. Biomed. Opt. 22(7), 077001 (2017).
[Crossref]

S. C. Kanick, H. J. Sterenborg, and A. Amelink, “Empirical model description of photon path length for differential path length spectroscopy: combined effect of scattering and absorption,” J. Biomed. Opt. 13(6), 064042 (2008).
[Crossref]

T. L. Troy and S. N. Thennadil, “Optical properties of human skin in the near infrared wavelength range of 1000 to 2200 nm,” J. Biomed. Opt. 6(2), 167–176 (2001).
[Crossref]

J. Opt. Soc. Am. (1)

Med. Phys. (1)

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref]

Opt. Express (1)

Opt. Laser Technol. (1)

X. Min, R. Liu, B. Fu, and K. Xu, “A preliminary verification of the floating reference measurement method for non-invasive blood glucose sensing,” Opt. Laser Technol. 91, 7–12 (2017).
[Crossref]

Opt. Lett. (2)

Optik (1)

M. Mehrabi, S. Setayeshi, M. G. Maragheh, S. H. Ardehali, and H. Arabalibeik, “Design of a new reflectance pulse oximeter by obtaining the optimal source-detector space,” Optik 168, 34–45 (2018).
[Crossref]

Phys. Med. Biol. (1)

M. Kohl, M. Essenpreis, and M. Cope, “The influence of glucose concentration upon the transport of light in tissue-simulating phantoms,” Phys. Med. Biol. 40(7), 1267–1287 (1995).
[Crossref]

Spectrosc. Lett. (1)

Z. Zhang, Z. Cai, G. Han, W. Sheng, and J. Liu, “Estimation of glucose absorption spectrum at its optimum pathlength for every wavelength over a wide range,” Spectrosc. Lett. 49(9), 588–595 (2016).
[Crossref]

Talanta (1)

S. Sharma, M. Goodarzi, H. Ramon, and W. Saeys, “Performance evaluation of preprocessing techniques utilizing expert information in multivariate calibration,” Talanta 121, 105–112 (2014).
[Crossref]

TrAC, Trends Anal. Chem. (1)

M. Goodarzi, S. Sharma, H. Ramon, and W. Saeys, “Multivariate calibration of NIR spectroscopic sensors for continuous glucose monitoring,” TrAC, Trends Anal. Chem. 67, 147–158 (2015).
[Crossref]

Other (1)

K. Xu and J. Liu, “Absorption and scattering effects separation from diffuse spectrum data and model establishment and concentration prediction based thereon,” U.S. Patent 9,551,656 (2017).

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Figures (7)

Fig. 1.
Fig. 1. The schematic diagrams. (A) The decomposition of attenuance; (B) The sensitivity of diffuse light intensity.
Fig. 2.
Fig. 2. Optical parameters. (A) for three Intralipid solutions; (B) for glucose of 1 mmol/L(18 mg/dL).
Fig. 3.
Fig. 3. Monte Carlo simulation results for 5% Intralipid. (A) Change in attenuance, ${\Delta }{A_{\rho }}$ , caused by 1 mg/dL glucose; (B) D-signals, spectra of ${\Delta }{A_{{\rho ,\textrm{D}}}}$ at a SDS of 0.5 mm; (C) EA-signals, spectra of ${\Delta }{A_{{\rho ,\textrm{A}}}}$ , calculated by using the data of two SDSs of 0.5 mm and 0.8 mm; (D) Sensitivity using one SDS, ${S_{\rho }}$ , caused by 1 mg/dL glucose (1300 nm); (E) Sensitivity using two SDSs, ${S_{{{\rho }_\textrm{A}}\textrm{,\ }{{\rho }_\textrm{B}}}}$ , caused by 1 mg/dL glucose (1300 nm); (F) Sensitivity ${S_{{{\rho }_\textrm{A}}\textrm{,\ }{{\rho }_\textrm{B}}}}$ with ${\rho _\textrm{A}}$ =0.5 mm, caused by 1 mg/dL glucose, for three Intralipid solutions (1300 nm).
Fig. 4.
Fig. 4. The optimal SDSs for the EA-signal-based measurement of three kinds of Intralipid solutions at 1300 nm. (A) The results according to Eq. (F-8); (B) The results according to the MC simulation.
Fig. 5.
Fig. 5. The schematic diagram of the system. (a) System overview; (b) The probe end
Fig. 6.
Fig. 6. The experiment results. (A) The attenuance change, ${\Delta }{A_{\rho }}$ , caused by a glucose change of 1 mg/dL at six wavelengths; (B) Attenuance change with using different SDSs; (C) D-signals, spectra of ${\Delta }{A_{{\rho ,\textrm{D}}}}$ at a SDS of 0.48 mm; (D) EA-signals, spectra of ${\Delta }{A_{{\rho ,\textrm{A}}}}$ calculated by the differential of two SDSs; (E) The signal-to-noise ratio of the detection system at ${\rho _\textrm{B}}$ =0.76 mm; (F) Evaluated measurement limit using ${\rho _\textrm{A}}$ =0.48 mm and ${\rho _\textrm{B}}$ =0.76 mm.
Fig. 7.
Fig. 7. Comparison of experiment results and MC results for 5% Intralipid solutions. The attenuance change was caused by 1 mmol/L glucose change. (A) MC results. The D-signal uses the SDS of 0.5 mm, and the EA-signal uses 0.5 mm and 0.8 mm; (B) Experiment results. The D-signal uses 0.48 mm, and EA-signal uses 0.48 mm and 0.76 mm.

Tables (4)

Tables Icon

Table 1. The decomposition of the parameters in the diffusion equation ( I 0 is the incident light intensity; ρ is the SDS; D is the diffusion coefficient; K is defined as 1 / ( 2 / μ S + 4 D ) ; μ eff is the effective attenuation coefficient.)

Tables Icon

Table 2. The optimal SDSs for D-signal-based and EA-signal-based spectroscopy (according to Appendixes C, D&F) ( I 0 is the incident light intensity; ρ is the SDS; D is the diffusion coefficient; K = 1 / ( 2 / μ S + 4 D ) ; μ eff is the effective attenuation coefficient; σ I is the noise of the detector.)

Tables Icon

Table 3. The optimal SDSs for the Intralipid solutions at 1000-1600 nm (according to the MC results)

Tables Icon

Table 4. The glucose concertation prediction results in 5% Intralipid solutions

Equations (55)

Equations on this page are rendered with MathJax. Learn more.

I ( ρ ) = I 0 I D ( ρ ) I A ( ρ )
A ρ = ln ( I ( ρ ) / I 0 ) = ln I D ( ρ ) ln I A ( ρ )
A ρ = A ρ , D + A ρ , A
Δ μ eff = Δ A ρ B Δ A ρ A ρ B ρ A
D signal : Δ A ρ , D Δ A ρ = small   value
EA signal : Δ A ρ , A = Δ μ eff ρ Δ μ eff = Δ A ρ B Δ A ρ A ρ B ρ A
S ρ = d I ρ d C = I ρ d A ρ d C
S ρ A , ρ B = d I ρ B d C η d I ρ A d C I ρ B d A I e d C
C limit = Δ I noise d I ρ / d C = 3 σ I | S ρ |
I ( ρ ) = I 0 1 4 π ρ D exp ( μ eff ρ )
I ( ρ ) = I 0 I D ( ρ ) I A ( ρ )
I D ( ρ ) = 1 4 π ρ D ; I A ( ρ ) = exp ( μ eff ρ )
A ρ , D = ln I D ( ρ ) = ln 4 π ρ D ; A ρ , A = ln I A ( ρ ) = μ eff ρ
Δ A ρ = 1 D Δ D + Δ μ eff ρ = ( 3 D ) Δ μ a + ( 3 2 D μ eff + 1 2 D 1 μ eff 1 ) ρ Δ μ a ( 3 D ) Δ μ S + ( 3 2 D μ eff ) ρ Δ μ S
Δ A ρ , D = 1 D Δ D = ( 3 D ) Δ μ a ( 3 D ) Δ μ S ;   Δ A ρ , A = Δ μ eff ρ = ( 3 2 D μ eff + 1 2 D 1 μ eff 1 ) ρ Δ μ a + ( 3 2 D μ eff ) ρ Δ μ S
ρ = 2 μ eff , let A ρ μ S = ( 3 D ) + ( 3 2 D μ eff ) ρ = 0
Δ A ρ ,D = Δ D D = 3 D ( Δ μ a + Δ μ s ) = Δ μ a + Δ μ s μ a + μ s Δ A ρ ,A = Δ μ eff ρ = Δ μ a μ a + 3 D ( Δ μ a + Δ μ s ) = Δ μ a μ a + Δ μ a + Δ μ s μ a + μ s Δ A ρ = Δ A ρ ,D + Δ A ρ ,A = Δ μ a μ a
I ( ρ ) = I 0 1 4 π [ z 0 ( μ eff + 1 r 1 ) e μ eff r 1 r 1 2 + ( z 0 + 2 z b ) ( μ eff + 1 r 2 ) e μ eff r 2 r 2 2 ]
I ( ρ ) = I 0 1 4 π [ 1 K ( μ eff + 1 ρ ) e μ eff ρ ρ 2 ]
For   small   SDS : I ( ρ ) = I 0 1 4 π [ 1 K e μ eff ρ ρ 3 ]
For   larger   SDS : I ( ρ ) = I 0 1 4 π [ 1 K ( μ eff ) e μ eff ρ ρ 2 ]
For   small   SDS : I D ( ρ ) = 1 4 π K ρ 3 ;   I A ( ρ ) = exp ( μ eff ρ )
For   larger   SDS : I D ( ρ ) = 1 4 π K μ eff ρ 2 ;   I A ( ρ ) = exp ( μ eff ρ )
For   small   SDS : A ρ , D = ln 4 π K ρ 3 ;   A ρ , A = μ eff ρ
For   larger   SDS : A ρ , D = ln 4 π K μ eff ρ 2 ;   A ρ , A = μ eff ρ
For   small   SDS : Δ A ρ , D = Δ K K ;   Δ A ρ , A     = Δ μ eff ρ ;   Δ A ρ = Δ K K + Δ μ eff ρ
For   larger   SDS : Δ A ρ , D = Δ K K Δ μ eff μ eff ;   Δ A ρ , A = Δ μ eff ρ ;   Δ A ρ = Δ K K Δ μ eff μ eff + Δ μ eff ρ
S = d I L d C
S = I L d A L d C
S = d I L d C = I L L d μ a d C = I 0 exp ( μ a L ) L d μ a d C
For   infinite   media : S ρ = d I ρ d C = I ρ ( ρ d μ eff d C + 1 D d D d C )
For   semi-infinite   media   at   small   SDSs : S ρ = d I ρ d C = I ρ ( ρ d μ eff d C + 1 K d K d C )
For   semi-infinite   media   at   larger   SDSs : S ρ = d I ρ d C = I ρ ( ρ d μ eff d C + 1 K d K d C 1 μ eff d μ eff d C )
S I e = d I e d C = I L B I L A ( 1 I L B d I L B d C 1 I L B d I L A d C ) = I e d A I e d C
σ I e = I L B I L A ( 1 I L B ) 2 σ I L B 2 + ( 1 I L A ) 2 σ I L A 2
σ I e 1 I L A σ I = β σ I
S L A , L B = S I e / β I L B d A I e d C
C limit = Δ I noise d I / d C = 3 σ I e | S I e | = 3 σ I | S L A , L B |
C limit = 3 σ I | S L A , L B |   1 | I L B 3 σ I d A I e d C | = 1 | S N R L B d A I e d C |
I e = I L B I L A = exp ( μ a ( L B L A ) )
S L A , L B = I L B ( L B L A ) d μ a d C = I L A ( exp ( μ a L e ) ) ( L e ) d μ a d C
S L A , L B L A = I L A μ a ( exp ( μ a L e ) ) ( L e ) d μ a d C > 0
S L A , L B L e = I L A ( exp ( μ a L e ) ) d μ a d C ( 1 L e μ a )
{ L A   is   as   small   as   possible L B L A = 1 μ a
I e = I ρ B I ρ A
A I e = ln I e
S ρ A , ρ B = I ρ B ( d A L B d C d A L A d C ) = I ρ B ( ρ B ρ A ) d μ eff d C
= m exp ( μ eff ρ A ) ( ρ A + ρ e ) exp ( μ eff ρ e ) ( ρ e ) d μ eff d C
S ρ A , ρ B ρ A = m ( 1 + μ eff ( ρ A + ρ e ) ( ρ A + ρ e ) 2 ) exp ( μ eff ρ A ) exp ( μ eff ρ e ) ( ρ e ) d μ eff d C > 0
S ρ A , ρ B ρ e = m exp ( μ eff ( ρ A + ρ e ) ) 1 ( ρ A + ρ e ) 2 d μ eff d C ( μ eff ρ e 2 + μ eff ρ A ρ e ρ A )
{ ρ A   is   as   small   as   possible ρ B ρ A = ρ A 2 + 4 ρ A μ eff ρ A 2
A I e = ln ρ A 2 ρ B 2 + μ eff ( ρ B ρ A )
S ρ A , ρ B = I ρ B ( d A L B d C d A L A d C ) = I ρ B ( ρ B ρ A ) d μ eff d C
= m exp ( μ eff ρ A ) ( ρ A + ρ e ) 2 exp ( μ eff ρ e ) ( ρ e ) d μ eff d C
{ ρ A   is   as   small   a s   possible ρ B ρ A = ρ A 2 + 1 μ eff 2 + 6 ρ A μ eff ( ρ A + 1 μ eff ) 2

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