Optical parameters estimation in inhomogeneous turbid media using backscattered light: for transcutaneous scattering measurement of intravascular blood

Shiyang Liang; Takeo Miyake; Koichi Shimizu

doi:10.1364/BOE.510245

1. Introduction

Noninvasive measurements of subcutaneous physiological conditions hold important value for biomedical applications [1–3]. The optical properties of body fluids, including blood, lymph, saliva, and cerebrospinal fluid can provide valuable insights into various physiological parameters [4–7]. Among these, the assessment of blood lipid levels has gained rapid importance globally, given the increasing population of overweight people. Temporal changes in serum lipid levels serve as crucially important markers for evaluating overall health in terms of lipid metabolism. Future risks of cardiovascular and diabetes diseases can be predicted based on the lipid metabolism [8,9]. Under healthy conditions, serum lipid levels rise after a meal but revert to normal levels within a few hours. By contrast, abnormal blood metabolism such as postprandial hyperlipidemia engenders sustained high lipid levels until the subsequent meal, consequently increasing the risk of lifestyle-related diseases [10–12]. Currently, diagnosing postprandial hyperlipidemia requires that patients remain in a medical institution for more than several hours for frequent blood sampling [12]. To overcome this limitation, noninvasive optical measurement of blood lipid levels can facilitate continuous real-time evaluation of blood lipid metabolism from the comfort of one’s home.

Blood lipids encompass triglycerides (TG), cholesterol, phospholipids, and free fatty acids, with serum TG levels as essential parameters measured in general blood tests. Particularly, TG constitutes a major component of lipoprotein particles such as chylomicrons (CM) and very low-density lipoproteins (VLDL). Because of their larger size than those of other serum lipoproteins, such as low-density lipoproteins (LDL) and high-density lipoproteins (HDL) [13–15], CM and VLDL contribute more to optical scattering from blood. Therefore, serum TG levels can be evaluated using measurements of blood turbidity based on its scattering characteristics [16–18].

For practical measurements of the animal body surface, backscattering measurements are more useful than transmission measurements [19,20]. Many techniques have been developed for measuring the absorption coefficient (µ_a) through backscattering in biomedical applications [4–7,21], but techniques for measuring the scattering coefficient (μ_s) or the reduced scattering coefficient (μ_s’) have been used less commonly in medical practice. For an earlier study [22], we proposed an analytical solution to obtain the effective attenuation coefficient (µ_eff) of blood, extending the applicability of diffusion approximation to human blood that does not satisfy the scattering-dominant condition. This solution allows the estimation of µ_eff from the spatially resolved reflectance observed along the object’s surface. When µ_a is known a priori, µ_s’ is obtainable from the estimated value of µ_eff. However, in real-world scenarios, µ_a might not be known or might vary in time and space.

Furthermore, for our earlier study [22], we developed a technique to suppress the strong light scattering effect in peripheral tissues surrounding the target blood vessel. This technique relies on assumption of the known a priori contribution ratios of the tissue and the blood to the backscattered reflectance. These two assumptions about µ_a and the contribution ratios pose difficult challenges when applying inter-person or in-person comparisons of lipid levels. As described herein, our objective is to advance the noninvasive optical technique for estimating µ_s’ and µ_a in a blood vessel while suppressing the scattering effect from peripheral tissues without requiring a priori spatial information about blood vessels.

2. Methods

2.1 Principle of µ_eff estimation from backscattering measurements

Light propagation in mammalian tissue is commonly described as energy flow in the equation of transfer. Figure 1 portrays the principle for estimating optical parameters from a reflectance measurement. A vertical narrow beam of light is directed onto the horizontal surface of a scattering object. The backscattered light in the normal direction is measured using a light detector. The reflectance R(ρ) is defined as the intensity ratio between the measured and incident light related to the source–detector distance ρ. In the diffusion approximation to the transfer equation, the reflectance is expressed as shown below [23–25].

(1)$$R(\rho )= \frac{{{z_0}{A_d}}}{{2\pi }}\left( {{\mu_{eff}} + \frac{1}{{\sqrt {{\rho^2} + z_0^2} }}} \right)\frac{{\exp \left( { - {\mu_{eff}}\sqrt {{\rho^2} + z_0^2} } \right)}}{{{\rho ^2} + z_0^2}}$$

Fig. 1. Model of reflectance measurement.

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Therein, z₀= 1/µ_s’ and µ_s’=(1-g)µ_s’. In Eq. (1), µ_s’, g, µ_s, A_d and µ_eff respectively represent the reduced scattering coefficient, the anisotropy factor, the scattering coefficient of the medium, the detection area and the effective attenuation coefficient. For practical purposes, we can approximate z₀²$+ $ρ² ≈ρ² [22]. Consequently, the equation can be transformed as

(2)$${ln} [{{{l \rho }^2}{ R}({ \rho } )} ]={-} {{ \mu }_{{{eff}}}}{ \rho } + l{ln} \left( {{{ \mu }_{{{eff}}}} + \frac{1}{{ \rho }}} \right) + {{ln}}\frac{{{{ z}_0}{{ A}_{ d}}}}{{2{ \pi }}}$$

Therefore, one can estimate µ_eff from the measured reflectance R(ρ) using the slope of the curve $\ln [{{\rho^2}R(\rho )} ]$ with respect to distance ρ [22,23].

2.2 Principle of µ_s’ and µ_a estimation with two wavelengths

In diffusion approximation, the diffusion coefficient D is given as

(3)$$D = \frac{{{\mu _a}}}{{{\mu _{eff}}^2}} = {[{3({{\mu_s}^{\prime} + a{\mu_a}} )} ]^{ - 1}}, $$

where ${\mu _a}$ and a respectively denote the absorption coefficient and the adjusting numerical factor which depends on the form of the diffusion equation [26,27]. When a = 1 and 0, Eq. (3) reduces to the classical ($D = {[{3({{\mu_s}^{\prime} + {\mu_a}} )} ]^{ - 1}}$) and newly proposed ($D = {({3{\mu_s}^{\prime}} )^{ - 1}}$) [28] definitions of the diffusion coefficient.

In scattering-dominant cases such as those related to human tissue, the applicability of the diffusion approximation is widely known. However, in the case of whole blood, where the absorption coefficient becomes comparable to the scattering coefficient, the accuracy of the diffusion approximation is not guaranteed. Nevertheless, earlier reports have described that the diffusion approximation still holds in cases where absorption is comparable to scattering if one uses the following factor a [29].

(4)$$a = 1 - \frac{4}{5}\frac{{{\mu _s}^{\prime} + {\mu _a}}}{{{\mu _s}^{\prime}({1 + g} )+ {\mu _a}}}$$

Using Eqs. (3) and (4), the equation for µ_s’ and µ_a can be derived as

(5)$$3{\mu _a}\frac{{({5G{\mu_s}^{\prime} + {\mu_a}} )({{\mu_s}^{\prime} + {\mu_a}} )}}{{5({G{\mu_s}^{\prime} + {\mu_a}} )}} = {\mu _{eff}}^2, $$

where G = 1 + g, and µ_eff is the value obtained in the reflectance R(ρ) measurement using Eq. (2). In a previous study [22], we obtained µ_s’ by solving Eq. (5) as

(6)$${\mu _s}^{\prime} = \frac{{ - B + \sqrt {{B^2} - 4AC} }}{{2A}}, $$

where A = 15G${\mu _a}$, B = 5 G(3${\mu _a}$²-${\mu _{eff}}^2$) + 3${\mu _a}$² and C = (3${\mu _a}$²-5${\mu _{eff}}$²)${\mu _a}$ . In this case, the absorption coefficient µ_a was assumed to be known a priori.

However, in practical applications, µ_a is not always known. It might vary with time such as oxygenation changes in hemoglobin. Therefore, the accuracy of µ_s’ estimation can be improved if we can estimate µ_a in the same measurement. To estimate µ_a during the µ_s’ calculation, we have developed the following new technique.

By conducting the same measurement presented in Fig. 1 using light with two wavelengths λ₁ and λ₂, one can obtain two equations in the form of Eq. (5), which consists of simultaneous equations for unknowns µ_s’ and µ_a. However, the number of unknowns also increases as µ_s’(λ₁), µ_s’(λ₂), µ_a(λ₁), and µ_a(λ₂). Often, spectral information about the target material exists, such as blood or body tissue. In such cases, the number of unknowns can be reduced using spectral ratios.

(7)$$\mu _s^{\prime}\left( {{\lambda _2}} \right) = {k_s}\mu _s^{\prime}\left( {{\lambda _1}} \right)$$

(8)$${\mu _a}\left( {{\lambda _2}} \right) = {k_a}\mu \left( {{\lambda _1}} \right)$$

In these equations, k_s and k_a respectively stand for the spectral ratios of molar scattering and absorption coefficients ε_s(λ₁)/ε_s(λ₂) and ε_a(λ₁)/ε_a(λ₂). Consequently, we obtain the simultaneous equations for two unknowns, µ_s’ and µ_a of wavelength λ₁, as shown below.

(9)$$\left\{ {\begin{array}{{c}} {3{\mu_a}\frac{{({5G{\mu_s}^{\prime} + {\mu_a}} )({{\mu_s}^{\prime} + {\mu_a}} )}}{{5({G{\mu_s}^{\prime} + {\mu_a}} )}} = {\mu_{eff}}^2({{\lambda_1}} )}\\ {3{k_a}{\mu_a}\frac{{({5G{k_s}{\mu_s}^{\prime} + {k_a}{\mu_a}} )({{k_s}{\mu_s}^{\prime} + {k_a}{\mu_a}} )}}{{5({G{k_s}{\mu_s}^{\prime} + {k_a}{\mu_a}} )}} = {\mu_{eff}}^2({{\lambda_2}} )} \end{array}} \right.$$

Therein, µ_s’=µ_s’(λ₁) and µ_a= µ_a (λ₁).

2.3 Convenient method for µs’ and µa estimation

Various approaches exist for solving Eq. (9). A simple technique is numerical two-parameter estimation. Among them, we chose a simple technique by which we can observe the behaviors of estimated µ_s’ and µ_a directly. Because each equation in Eq. (9) is quadratic with respect to µ_s’, Eq. (9) can be transformed to the expression shown below.

(10)$$\left\{ \begin{array}{l} \mu_s^{\prime} = \frac{{ - 5G(3\mu_s^2 - {\mu_e}ff{1^2}) - 3\mu_a^2 + \sqrt {{{[5G(3\mu_a^2 - {\mu}_{eff1}{^2}) + 3\mu_a^2]}^2} - 60G\mu_a^2(3\mu_a^2 - 5\mu_{eff1}{^2})} }}{{30G{\mu_a}}}\\ \mu_s^{\prime}\frac{{ - 5G(3k_a^2\mu_a^2 - {\mu}_{eff2}{^2}) - 3k_a^2\mu_a^2 + \sqrt {{{[5G(3k_a^2\mu_a^2 - {\mu}_{eff2}{^2}) + 3k_a^2\mu_a^2]}^2} - 60Gk_a^2\mu_a^2(3k_a^2\mu_a^2 - 5{\mu}_{eff2}{^2})} }}{{30G{k_s}{k_a}{\mu_a}}} \end{array} \right.$$

Therein, µ_eff1= µ_eff(λ₁) and µ_eff2= µ_eff(λ₂). Each equation in Eq. (10) yields µ_s’ as a function of known G, k_s, k_a, measured µ_eff1, µ_eff2 and one unknown µ_a. If unknown µ_a is the correct value, then both equations will result in the same µ_s’. Therefore, we can ascertain the correct µ_a in the search for the minimum difference in the two µ_s’ values calculated in Eq. (10). To evaluate this difference, we use the Michelson contrast, defined as

(11)$${C_M} = \frac{{{{ {{\mu_s}^{\prime}} |}_f} - {{ {{\mu_s}^{\prime}} |}_l}}}{{{{ {{\mu_s}^{\prime}} |}_f} + {{ {{\mu_s}^{\prime}} |}_l}}}, $$

where C_M, µ_s’|_f and µ_s’|_l respectively represent the Michelson contrast, with µ_s’ obtained in the former and the latter equations in Eq. (10). An example of the dependence of these parameters on µ_a is presented in Fig. 2. µ_s’|_f and µ_s’|_l change at different rates with µ_a crossing at a certain point. The contrast becomes minimum at the specific value of µ_a. This µ_a and the corresponding µ_s’ are the solutions of the simultaneous equation Eq. (9) or Eq. (10).

Fig. 2. Dependence of reduced scattering coefficient µ_s’ in two solutions and its contrast on absorption coefficient µ_a: µ_eff1= 1.046 /mm_, µ_eff2= 2.250 /mm, g = 0.99, k_s= 0.647, and k_a= 3.80 (assuming practical wavelengths λ₁= 660 nm and λ₂= 940 nm).

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Based on the principle presented above, we can estimate µ_s’ and µ_a using the following process. Figure 3 presents an example of the process in a flow chart. Initially, we calculate both equations in Eq. (10) with an assumed initial value of µ_a. After validating the calculated µ_s’, we compare them using contrast C_M. If any µ_s’ falls outside the valid range or if the contrast is insufficiently small, then we update the value of µ_a. By repeating this process until the contrast value becomes smaller than the preset threshold, we can estimate µ_s’ and µ_a using the same procedure.

Fig. 3. Process for µ_s’ and µ_a separated estimation: µ_eff is estimated from the ln[ρ²R(ρ)] curve with respect to ρ using Eq. (2). µ_a is updated to achieve convergence in the contrast C_M.

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In this process, it is necessary to update the µ_a value to minimize contrast C_M. The Newton–Raphson method is one candidate for achieving convergence. The stability of this convergence process has been confirmed when starting with reasonable initial values of µ_a reported in the literature [4–7,25]. An important feature of this process is the direct testability of the estimated µ_s’ and µ_a values during the convergence process. During the process of µ_s’ calculation, we examine the validity of µ_s’ to ensure that the value falls within an appropriate range. We can also confirm the validity of µ_a during its updating process. This feature supports calculation efficiency by eliminating unnecessary calculations of erroneous cases.

2.4 Application of the differential method

The primary objective of this study is the noninvasive measurement of blood in a subcutaneous blood vessel using the reflectance R(ρ) measured at the body surface. Figure 4 illustrates the reflectance measurement at the body surface with a subcutaneous blood vessel.

Fig. 4. Reflectance measurement with subcutaneous blood vessel.

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The probing volume intersects not only the blood vessel but also the surrounding tissue. Consequently, the reflectance R(ρ) is inevitably influenced more by the optical properties of the surrounding tissue than by those within the blood vessel. To evaluate µ_s’ and µ_a within the blood vessel, considering the surrounding tissue’s strong effect, we developed a technique using a differential principle between the reflectances with and without the blood vessel [22]. Figure 5 portrays the measurement setup used for this technique. Reflectances were measured on the body surface along lines just above the blood vessel axis and off the axis. The contribution of the blood vessel component to the reflectance can be differentiated from that of the body tissue using the following equations.

(12)$$\begin{array}{l} \; \; {R_{\textrm{on}}}\left( \rho \right) = \alpha \left( \rho \right){R_{blood}}\left( \rho \right) + \left[ {1 - \alpha \left( \rho \right)} \right]{R_{tissue}}\left( \rho \right)\\ {R_{\textrm{off}}}\left( \rho \right) = \beta \left( \rho \right){R_{blood}}\left( \rho \right) + \left[ {1 - \beta \left( \rho \right)} \right]{R_{tissue}}\left( \rho \right) \end{array}$$

Fig. 5. Measurement of on-axis and off-axis reflectances for differential technique to suppress scattering effect from surrounding tissue: I₀ and s respectively denote the incident light intensity and the shift distance.

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In those equations, R_on(ρ) and R_off(ρ) respectively represent the on-axis and off-axis reflectances. Therein, α(ρ) and β(ρ) respectively denote the contribution factors of the blood vessel component to the reflectances R_on(ρ) and R_off(ρ). This technique can also be employed for the multi-wavelength estimation method proposed in this study. The reflectance of blood in the blood vessel is given as presented below.

(13)$${R_{blood}}({\rho ,\lambda } )= \frac{{[{1 - \beta ({\rho ,\lambda } )} ]{R_{\textrm{on}}}({\rho ,\lambda } )- [{1 - \alpha ({\rho ,\lambda } )} ]{R_{\textrm{off}}}({\rho ,\lambda } )}}{{\alpha ({\rho ,\lambda } )- \beta ({\rho ,\lambda } )}}$$

The effectiveness of this technique depends on the appropriate choice of the contribution factors. Here, α(ρ, λ) and β(ρ, λ) are determined uniquely in Monte Carlo simulations using depth d and diameter ϕ of the blood vessel in the body tissue. In our earlier study, d and ϕ were assumed to be known a priori [22]. However, in real measurements with human subjects, they are often unknown. Additionally, the vessel diameter can vary depending on the physiological and psychological conditions of the subject. To find appropriate contribution factors, we developed a new technique to estimate d and ϕ during the µ_s’ and µ_a estimation process.

The principle of this choice is based on a database search. First, we prepare a database of contribution factors α(ρ) and β(ρ) for different combinations of optical and geometrical parameters, i.e., µ_s’, µ_a, d, and ϕ. The database provides output (α, β) for given input of (µ_s’, µ_a, d, ϕ). When some input parameters are unknown, we scan the unknown parameters within a practically reasonable range, resulting in multiple candidate sets of (α, β) outputs. We then select the appropriate contribution factors from these candidates. Optical parameters µ_s’ and µ_a can be estimated using the different methods described in earlier sections 2.2 and 2.3. Therefore, we typically scan the geometrical parameters d and ϕ.

Among the candidate sets of (α, β) outputs, an appropriate contribution factor set is identifiable by choosing the one which results in the steepest slope of R_blood(ρ), leading to the maximum µ_eff. This principle is derived from large difference in light absorption between blood and body tissue. In the NIR wavelength range, the values of (µ_s’, µ_a) are typically in the orders of (0.5/mm, 0.5/mm) and (1.0/mm, 0.01/mm), respectively, for blood and body tissue. Considering Eq. (3), or µ_eff= [3µ_a(µ_s’+aµ_a)]^1/2, the value of µ_eff for blood is much larger than that of body tissue. Therefore, we can select the appropriate contribution factors as the set that results in the maximum µ_eff. Figure 6 presents µ_eff estimated from the reflectance R_blood(ρ) calculated using R_on(ρ), R_off(ρ), α(ρ), and β(ρ) for assumed d and ϕ. It shows a typical distribution with a single peak at the correct d and ϕ. This unimodal change enables us to obtain the required appropriate contribution factors.

Fig. 6. Characteristics of µ_eff for variation of assumed depth and diameter of blood vessel in body tissue. Vertical axis is the estimated µ_eff value normalized by the value for true geometry. (Tissue: µ_s’=1.0/mm, µ_a= 0.01/m. Blood: µ_s’=0.5/mm, µ_a= 0.5/mm. Detector array shift: s = 5.0 mm, True depth d = 1 mm. True diameter ϕ=5 mm)

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This database approach is akin to the neural network trained with Monte-Carlo simulation data [21]. While the neural network technique offers greater flexibility and versatility for general applications, the proposed technique is straightforward and can be easily implemented in the system described in the following section.

2.5 Integration into a system

The proposed techniques were integrated into a system consisting of a database and data-processing algorithms. Figure 7 depicts the process of creating the database, which can output the contribution factors α(ρ, λ) and β(ρ, λ) for given optical and geometrical parameters, i.e., µ_s’, µ_a, d and ϕ. In this database generation, α(ρ, λ) and β(ρ, λ) are calculated using reflectances of four kinds obtained from Monte Carlo simulations of photon migration in a turbid medium [22,30] as

(14)$$\begin{array}{l} \alpha \left( {\rho ,\lambda } \right) = \left[ {{R_{\textrm{tissue}}}\left( {\rho ,\lambda } \right) - {R_{\textrm{on}}}\left( {\rho ,\lambda } \right)} \right]/\left[ {{R_{\textrm{tissue}}}\left( {\rho ,\lambda } \right) - {R_{\textrm{blood}}}\left( {\rho ,\lambda } \right)} \right]\\ \beta \left( {\rho ,\lambda } \right) = \left[ {{R_{\textrm{tissue}}}\left( {\rho ,\lambda } \right) - {R_{\textrm{off}}}\left( {\rho ,\lambda } \right)} \right]/\left[ {{R_{\textrm{tissue}}}\left( {\rho ,\lambda } \right) - {R_{\textrm{blood}}}\left( {\rho ,\lambda } \right)} \right] \end{array}$$

where R_on(ρ,λ) and R_off(ρ,λ) represent the reflectances measured at the body surface along and off the axis of the subcutaneous blood vessel, respectively. R_tissue(ρ,λ) and R_blood(ρ,λ) correspond to the reflectances of tissue and blood in a homogeneous structure.

Fig. 7. Process of creating database to obtain contribution factors from given optical and geometrical conditions of turbid medium: R_tissue(ρ), R_blood(ρ), R_on(ρ), and R_off(ρ) are obtained from Monte Carlo simulation of photon migration as the ratio of received photon energy at the source–detector distance ρ. The contribution factors α(ρ, λ) and β(ρ, λ) are calculated using Eq. (14). This database provides output (α, β) for the input (µ_s’, µ_a, d, ϕ).

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Figure 8 shows the data-processing algorithm used to estimate µ_s’ and µ_a of the blood in the subcutaneous blood vessel with unknown depth and diameter using the reflectances R_on(ρ) and R_off(ρ) measured on the body surface.

Fig. 8. Overall process to estimate µ_s’ and µ_a of the blood in the blood vessel with unknown depth and diameter in body tissue: Contribution factors α and β are obtained from the database (DB) prepared using the process depicted in Fig. 7. µ_eff is estimated from the ln[ρ²R(ρ)] curve with respect to ρ using Eq. (2). Here, µ_s’ and µ_a are estimated in the process depicted in Fig. 3.

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3. Validity check of the proposed technique

3.1 Conditions of simulation

Figure 9 depicts the model used for the Monte Carlo simulation [22,30–32] to assess the validity of the proposed technique. To simulate the case with a subsurface vein in the forearm of an adult human, the depth and diameter of the blood vessel were set at d = 1.0 mm and ϕ = 5.0 mm, respectively, based on fundamental conditions. Light detecting areas were arranged as small squares aligned in the x-direction along the horizontal surface of a turbid medium. These squares were 1 mm × 1 mm, with centers at ρ = 1, 2, 3…mm. The reflectance R(ρ) was obtained as the integrated photon energy backscattered from the medium and detected in each area. All photons that reached the detection-square from any angle in the half-space were counted as detected photons.

Fig. 9. Model for Monte Carlo simulation: Photons are injected at the center of the upper surface. After propagation in the medium, backscattered photons are received as reflectance R(ρ) as a function of the radial distance ρ from the light incidence point. A blood vessel with a diameter ϕ is located at a depth d below the surface.

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For a homogeneous medium, the optical parameters of the blood vessel part were set to be the same as those of the surrounding tissues. With practical application to blood turbidity measurements in mind, the wavelengths were set as λ₁= 660 nm and λ₂= 940 nm. The optical parameters of blood were varied: µ_s’(λ₁)= 0.1–2.0 mm^-1, µ_a (λ₁)= 0.1–0.9 mm^-1 with k_s= 0.647, k_a= 3.80, and g = 0.99 at both wavelengths. Meanwhile, the parameters of tissue were µ_s’= 1.0 mm^-1, µ_a= 0.01 mm^-1 and g = 0.95 at both wavelengths [33–37]. Reflectances R_blood(ρ, λ) and R_tissue(ρ, λ) were obtained, respectively, from the simulation for homogeneous media with optical parameters corresponding to blood and body tissue. Also, R_on(ρ, λ) and R_off(ρ, λ) were obtained along the on-axis and off-axis lines above the blood vessel with a shift distance s = 5.0 mm. The validity of this model was confirmed by comparing it with experimental data [22].

All the Monte-Carlo simulations were executed on a computer equipped with Intel Xeon Gold 6348 CPU (2.60 GHz, 256GB). The typical runtime for a single set of reflectance calculation was about 260000 seconds.

3.2 Accuracy in a homogeneous medium

The absorption coefficient µ_a of the blood is of the same order as the reduced scattering coefficient, which does not satisfy the conditions of the diffusion approximation. Because Eq. (2) was derived from diffusion approximation, its accuracy in estimating µ_eff is not guaranteed, especially for blood. From simulation and experimentation, we verified that this technique became applicable by adjusting factor a in Eq. (4), even in the case of blood [22]. However, the effectiveness of Eq. (6) for scattering estimation was only examined with a known absorption coefficient µ_a. Therefore, for the current study, we examined the validity of the newly proposed technique to estimate both the reduced scattering coefficient µ_s’ and the absorption coefficient µ_a simultaneously in a homogeneous medium including non-scattering-dominant conditions.

Reflectances R(ρ, λ₁) and R(ρ, λ₂) were obtained in the Monte Carlo simulation of photon migration in a scattering medium. Using Eq. (2), we estimated µ_eff(λ₁) and µ_eff(λ₂) respectively from R(ρ, λ₁) and R(ρ, λ₂). Also, µ_s’ and µ_a for λ₁ were estimated using Eq. (10) in the process depicted in Fig. 3. The estimation error was evaluated as the percentage error, i.e., the absolute value of the difference between the estimated and given values normalized by the given value.

The estimation errors for different combinations of given µ_s’ and µ_a are shown in Fig. 10. Although some estimation errors are unavoidable because of reasons such as the reflectance at discrete detection points and the randomness in Monte Carlo simulation, most of them remained below a few percent for µ_eff, and below several percent for µ_s’ and µ_a. These results verify the validity of the proposed technique for µ_s’ and µ_a estimation in homogeneous media with a non-scattering-dominant condition, such as blood.

Fig. 10. Estimation error in a homogeneous medium: (a) effective attenuation coefficient, (b) reduced scattering coefficient, and (c) absorption coefficient.

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3.3 Effectiveness of the differential technique

In the newly proposed technique, the estimation of µ_eff from R(ρ) serves as the basis for estimating µ_s’ and µ_a in the blood vessel. Reflectance R(ρ) should represent the blood rather than the surrounding tissue. However, as depicted in Fig. 4, the probing volume intersects not only the blood vessel but also the surrounding tissue. Therefore, the reflectance measured on the body surface is inevitably masked by the overwhelming effect of tissue reflectance. The effectiveness of the proposed differential technique was examined in the simulation to estimate µ_eff from the reflectance R(ρ) obtained in the process of Section 3.1.

Figure 11 presents results of µ_eff estimation obtained when both µ_s’ and µ_a in the subcutaneous blood vessel are varied. The estimated µ_eff exhibits a linear dependence on the given µ_eff, even without the differential technique. This finding verifies the fundamental feasibility of noninvasive measurement to detect changes in optical properties inside the human body. However, the low slope of the regression line represents very low sensitivity. The change in the blood vessel was masked by the constant optical parameters of the surrounding tissue. This masking effect was suppressed using the differential technique. In fact, using the differential technique, the sensitivity increased by about 10 times. The estimated values became more reliable. Based on these results, the differential technique was used for the following analyses.

Fig. 11. Effectiveness of differential technique in estimating µ_eff of blood in blood vessel by eliminating the influence of surrounding tissues. (Tissue: µ_s’=1.0/mm, µ_a= 0.01/m. Blood: µ_s’=0.1–2.0/mm, µ_a= 0.3–1.0/mm. Detector array shift: s = 5.0 mm. Blood vessel depth d = 1.0 mm. Blood vessel diameter ϕ=5.0 mm)

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3.4 Accuracy in the blood vessel

The on-axis and off-axis reflectances R_on(ρ, λ) and R_off(ρ, λ) were obtained for two wavelengths λ₁= 660 nm and λ₂= 940 nm in the Monte Carlo simulation with the model presented in Fig. 9. Because the wavelength variations of µ_s’ and µ_a of surrounding tissue are markedly less than those in blood vessel, the optical parameters for surrounding tissues were set as µ_s’= 1.0 mm^-1, µ_a= 0.01 mm^-1 with g = 0.95 for both wavelengths λ₁, and λ₂. Parameters within the blood vessel were varied in the ranges of µ_s’= 0.1–2.0 mm^-1, and µ_a= 0.1–0.9 mm^-1 with g = 0.99. These values were selected based on reported values [4–7,33–37]. Using the process portrayed in Fig. 8, µ_s’ and µ_a were estimated.

Figure 12 presents the accuracy of the proposed technique. The average values over the variation of another parameter are shown as the mean values with error bars representing the standard deviations, i.e., m ± σ. Good agreement can be observed between the given and estimated values, with small variations in both µ_s’ and µ_a estimation. These results also verify the validity of the proposed technique for the separate estimation of µ_s’ and µ_a in a localized part of a turbid medium, such as the blood vessel in the surrounding tissue.

We can expect higher accuracy for a larger vessel at a shallower depth, as it closely approximates the conditions of a homogeneous medium. However, as the vessel becomes smaller and deeper, the accuracy becomes lower because the cross-sectional area between the probing volume and the vessel becomes smaller.

Fig. 12. Results of separate estimation of µ_s’ and µ_a: (a) estimated µ_s’ over µ_a variation (N = 5), (b) estimated µ_a over µ_s’ variation (N = 10); The optical parameters of tissue and blood are respectively (µ_s’ = 1.0/mm, µ_a= 0.01/mm) and (µ_s’ = 0.1–2.0/mm, µ_a= 0.1–0.9/mm). The detector-array shift, the blood vessel depth, and the blood vessel diameter were, respectively, 5.0 mm, 1.0 mm and 5.0 mm.

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4. Practical considerations for application

4.1 Vulnerability and tolerance against measurement noise

The proposed technique can be regarded as an inverse problem aimed at inferring the cause (µ_s’ and µ_a) from the result (R(ρ)). In inverse problems, even slight noise in the measured R(ρ) can engender non-negligible error in the estimation of µ_s’ and µ_a. To investigate this possibility, random noise of various types was added to the reflectance R(ρ). The resulting estimation errors were analyzed.

The reflectance values were varied using the following equations to simulate random noise and bias noise.

(15)$${R_\textrm{r}}(\rho)\textrm{ } = \textrm{ }[{1\textrm{ } + \textrm{ }({RN{-}0.5} )W} ]{R_{\textrm{MC}}}(\rho)$$

(16)$${R_\textrm{b}}(\rho) = {R_{\textrm{MC}}}(\rho)\textrm{ } + W{R_{\textrm{max}}}.$$

In those equations, R_r(ρ), R_b(ρ), and R_MC(ρ) respectively represent reflectances with random noise, with bias noise, and the noise-free reflectance obtained from Monte Carlo simulation. Also, R_max is the maximum reflectance value in the measurement range of ρ. RN and W respectively denote a random number between 0.000 and 1.000 and the weight for noise amplitude. Figure 13 presents the results of the error analysis in the form of mean values with error bars representing the standard deviations, i.e., m ± σ. N corresponds to the µ_a and µ_s’ variations in µ_s’ and µ_a estimation, respectively.

Fig. 13. Results of error-analysis for separated estimation of µ_s’ and µ_a: (a) estimated µ_s’ over µ_a variation with 20% random noise (N = 5, W = 0.2) in reflectance measurement, (b) estimated µ_a over µ_s’ variation with 20% random noise (N = 10, W = 0.2), (c) estimated µ_s’ over µ_a variation with 20% bias noise (N = 5, W = 0.2), and (d) estimated µ_a over µ_s’ variation with 20% bias noise (N = 10, W = 0.2).

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As shown in Figs. 13(a) and 13(c), the µ_s’ estimation error remains in the same order as the noise in reflectance measurement. This robustness apparently arises from the principle of µ_eff estimation. Because µ_eff is estimated from the slope of reflectance R(ρ), the fluctuation attributable to noise can be averaged out in the slope detection. This averaging effect is more effective for random noise than for bias noise. The bias noise raises the reflectance value more in a larger ρ range, where the reflectance value is smaller, resulting in reducing the slope of the reflectance. The slope reduction engenders lower µ_eff estimation. This finding explains the slight underestimation observed in Fig. 13(c).

Regarding µ_a estimation in Figs. 13(b) and 13(d), a non-negligible error was observed in the range µ_a > 0.7. This result suggests the noise-susceptibility of this technique in the measurement of high absorption values. However, the mean estimated values showed strong correlation with the given values, and the error remained in the same order as that of the noise in reflectance measurements. These results verify the applicability of the proposed technique for µ_s’ and µ_a estimation in practical applications.

4.2 Accuracy dependence on the detection range

When a narrow beam of light is directed onto a turbid medium, the incident light disperses extensively after propagating a mean-free-path length, typically around 1 mm for NIR light in human body tissue [4–7,38–41]. The backscattered light is detected by a detector. Even when the detector’s receiving angle is small, its detection volume also significantly expands in the medium due to intense light scattering. The probing volume of the backscattered light with the detector is obtained as the superposition of these two spreads [42,43]. Figure 14(a) portrays the variation of this probing volume with the detector position.

Fig. 14. Measurement of reflectance by scanning the medium’s surface with a photodetector: (a) variation in probing volume, (b) measurement with a variable detection range.

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Using the proposed technique, we estimate µ_s’ and µ_a from reflectance R(ρ) measured as a function of the source–detector distance ρ. The diffusion approximation is applicable when ρ >>1/µ_s’. The backscattered light intensity is less than that of the diffusion approximation in the range of less than a few millimeters [22]. If ρ is set as too small, then estimation based on the diffusion approximation becomes erroneous. By contrast, large ρ guarantees the accuracy of the diffusion approximation. However, as ρ becomes large, the backscattered light intensity drops rapidly below the detection limit. Moreover, for accurate measurement of the slope of ln(ρ²R), the reflectance R(ρ) should be measured over widely various ρ. Nevertheless, this wide range will inevitably include inappropriate positions at either side of the range, leading to potential inaccuracies. Therefore, we analyzed the dependence of the estimated µ_eff accuracy on the ρ-range to identify appropriate conditions for the estimation of µ_s’ and µ_a of the blood.

Figure 15 presents the result of the error analysis for a homogeneous blood case, where µ_s’ =µ_a= 0.5 /mm. The two axes in the horizontal plane correspond to the center coordinate and the width of the detector scan (ρ-range), as portrayed in Fig. 14(b). The ranges of these two axes were chosen considering the practical measurement conditions. The error became smaller with the center position of the ρ-range, indicating an increase in the accuracy of the diffusion approximation. Regarding the ρ-range width, the error remained low in 1–10 mm width range and increased gradually in the 10–20 mm width range. This increase was attributed to the inclusion of inappropriate detector positions. Moreover, this tendency was apparent when the center position was less than 6 mm. Results obtained from this analysis suggest that the position and width of the light detection range should be chosen carefully to ensure accurate measurements.

Fig. 15. Error analysis for appropriate detection range of reflectance measurement: µ_s’ =µ_a= 0.5 /mm. Vertical axis is the percent error in µ_eff estimation.

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5. Discussion

We have developed a noninvasive technique to acquire optical properties of a localized part in a turbid medium, such as blood turbidity in blood vessels. One crucially important issue encountered during the measurement of subcutaneous blood vessels was the strong scattering effect from surrounding body tissue. We can suppress this effect using the differential principle between reflectances with and without the blood vessel. However, to apply this principle, geometrical information of the blood vessel must be obtained. To address this challenge, we introduced a new technique based on the differential principle to suppress the scattering effect without prior knowledge of the blood vessel geometry. By leveraging prior learning or database preparation, we were able to reduce the strong scattering effect from tissues surrounding the target blood vessel. To streamline the estimation processes, we integrated the µ_s’ and µ_a estimation and the differential operation into a single algorithm.

Through extensive computer simulations, we examined the feasibility and characteristics of the proposed technique. We validated the accuracy of µ_s’ and µ_a estimation specifically for non-scattering-dominant media like the blood in subcutaneous blood vessels. Although it was possible to detect changes in µ_eff without employing the differential technique, we observed limited sensitivity resulting in estimated values that were markedly smaller than the true values. However, with application of the differential technique, sensitivity improved approximately ten-fold, leading to estimated values that closely approached the true ones. Our error analysis confirmed that the estimation error in µ_s’ and µ_a estimation remained below 10% within the practical range of human blood measurement.

Because µ_s’ and µ_a estimation from the measured reflectance constitutes an inverse problem, the estimation error attributable to measurement noise might reach unrealistic magnitudes. To assess the susceptibility of the proposed technique against measurement noise, random noise and bias noise were added to the measured reflectance. Then µ_s’ and µ_a were estimated. Because of the averaging function in the process of µ_eff estimation, the estimation errors did not escalate beyond the magnitude of measurement noise. This robustness against measurement noise highlights an important benefit of our proposed technique, particularly for practical applications.

Additionally, we analyzed the dependence of accuracy on the radial range of reflectance measurement to maximize the effectiveness of the proposed technique. Our analysis revealed that a suitable radial range (i.e. ρ = 6–16 mm) from the light incident point facilitated accurate estimation of µ_s’ and µ_a in 5 mm diameter blood vessels located 1 mm below the body surface.

6. Conclusions

For noninvasive measurement of blood turbidity from the body surface, a technique was developed to enable measurement of the optical properties of blood within subcutaneous blood vessels. Our approach is based on the diffusion approximation of the radiative transfer equation, thereby enabling estimation of the effective attenuation coefficient µ_eff of a turbid medium through space-resolved reflectance measurements along the medium surface. Based on analytical equations, we obtained the reduced scattering coefficient µ_s’ and the absorption coefficient µ_a from µ_eff. To accommodate non-scattering-dominant media such as blood, we introduced a modified diffusion coefficient in the equations. To determine both µ_s’ and µ_a, we employed a two-wavelength reflectance measurement, thereby enabling the formulation of two simultaneous equations for the unknowns µ_s’ and µ_a. We proposed a convenient process for solving these equations. In this solution process, we can directly validate the reasonability of the solutions, thereby preventing the occurrence of non-significant or erroneous outcomes.

In conclusion, this study presents an innovative technique for estimating the localized optical properties in a turbid medium. It offers a new tool for noninvasive monitoring of optical properties of blood within subcutaneous blood vessels. We have demonstrated its validity and limitations through extensive analyses. However, all these analyses are based on simulations and are not exempt from their limitations. Ultimately, they must be demonstrated through experimentation. Verification of its practical usefulness in real-world applications also remains an area for future research with animal experiments.

Separate estimation of scattering and absorption coefficients from a reflectance measurement will provide a useful tool for the noninvasive monitoring of subsurface structures in various domains. Overall, the proposed technique holds considerable potential in various biomedical applications.

Funding

Japan Society for the Promotion of Science (20K20537).

Acknowledgments

The authors thank Professor Xiaopeng Shao of Xidian University and Dr. Jing Wang of Xi’an Jiaotong University for their valuable support of this work.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Optical parameters estimation in inhomogeneous turbid media using backscattered light: for transcutaneous scattering measurement of intravascular blood

Abstract

Corrections

1. Introduction