Impact of vessel diameter and bandwidth of illumination in sidestream dark-field oximetry

Tomohiro Kurata; Zhenguang Li; Shigeto Oda; Hiroshi Kawahira; Hideaki Haneishi

doi:10.1364/BOE.6.001616

1. Introduction

Microcirculation refers to the blood circulation in those blood vessels having a diameter of about 100 μm. Microcirculation is found within organ tissues and exists throughout the body. The function of microcirculation is to exchange nutrients and oxygen delivered by erythrocytes for waste products of organ tissues. In other words, microcirculation occurs the region where arteries turn into veins.

Sidestream dark-field (SDF) imaging was developed as an imaging system to observe microcurculation directly [1]. The SDF system provides a clinically-applicable, portable, and hand-held imaging to study the microcirculation of tissue surfaces. A schematic diagram of SDF imaging is shown in Fig. 1. SDF imaging employs a light-guide equipped with concentrically placed light-emitting diodes (LEDs) to provide illumination. The lens system in the core of the light-guide is optically isolated from LEDs of the outer ring, preventing microcirculation images from contamination by tissue surface reflections. As for analysis of the microcirculation by the SDF technique, some physical quantities such as vessel length, vessel diameter, and velocity of red blood cells (RBCs) have been estimated by analyzing the SDF images of individual vessels [2].

Fig. 1 Schematic diagram of the SDF imaging.

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Microcirculation imaging can be applied to diagnose the state of an organ. In digestive tract open surgery, surgeons typically diagnose the state of an organ as normal or ischemia on the basis of its visual color in order to decide whether to remove a part of it. The diagnosis relies on the surgeons’ visual impression and previous experiences. Therefore, a quantitative technique for evaluating the state of an organ is desirable. A macro technique with a hyperspectral camera has been suggested [3]. However, it is difficult for any macro techniques to specifically distinguish the boundary of the ischemia part. For a digestive organ, videoscope techniques such as laparoscopic and endoscopic surgery are commonly used recently. In this case, inaccurate color reproduction disturbs the diagnosis of surgeons. In contrast, a micro technique such as an observation of the microcirculation is more accurate to distinguish the boundary of the ischemia part, although it is a local observation. Therefore, the SDF imaging is useful to help in diagnosing the state of organs. Moreover, the SDF imaging can be applied to videoscope surgery because the SDF probe can be fabricated with a small diameter. However, for accurate diagnosis, monitoring of not only the flow of RBCs but also some other physical quantities are required. In particular, the most important item among the physical quantities of blood to evaluate the level of the state of an organ is oxygen saturation (SO₂).

The measurement of SO₂ from images, which is called image-based oximetry, has been actively studied in ophthalmology. The first retinal image-based oximetry experiment was reported by Hickam et al. [4]. They used broadband illumination provided by a dual-wavelength pair of illumination sources (peak wavelength: 510 and 640 nm) with the bandwidth of > 100 nm FWHM (full width at half maximum). Cohen et al. [5] reported results of an image-based experiment using illumination sources with dual-wavelengths of 470 and 515nm with a 20 nm bandwidth. Besides ophthalmology studies, there have been some to estimate SO₂ of individual vessels. Pittman and Duling [6] reported transmission type SO₂ measurements using a microscope. They used illumination sources with 546 and 555 nm with dual-wavelengths of a 10 nm bandwidth and considered the effect of light scattering. Styp-Rekowska et al. [7] also used a microscope and illumination sources from 500 to 598 nm wavelengths at 2 nm intervals and evaluated the influence of the number of wavelengths used, exposure time, and analysis area.

In the present study, we aim at establishing an estimation method of oxygen levels of the individual vessels from spectral images acquired with the SDF imaging system. For this purpose, we perform the Monte Carlo photon propagation simulation as the first approach in which we evaluate dependence of accuracy of the SDF oximetry on vessel diameters and wavelength-bandwidths of the illumination sources.

2. Methods and materials

2.1. Formula for oximetry

2.1.1. Lambert-Beer’s law for biological tissue

In this section, we discuss image-based oximetry by considering incident light of intensity I_in = I_in(λ) at wavelength λ. When the incident light enters into medium with a thickness d, the Lambert-Beer’s law gives the intensity I_out = I_out(λ) of the transmitted light through the medium as:

I_{out} = I_{in} \cdot 10^{- μ_{a} (λ) d},

where μ_a(λ) represents an absorption coefficient that can be written by μ_a(λ) = ε(λ)c, in which ε(λ) is an extinction coefficient and c is the concentration of the absorber in the medium. Using this expression, we define optical density D = D(λ) as:

D = - log (\frac{I_{out}}{I_{in}}) = μ_{a} (λ) d .

We consider that the medium is blood and assume that the incident light is absorbed by only hemoglobin, then μ_a(λ) is written as:

μ_{a} (λ) = [s ε_{{HbO}_{2}} (λ) + (1 - s) ε_{Hb} (λ)] c,

where ε_HbO₂(λ) and ε_Hb(λ) represent the extinction coefficients of the oxygenated and the deoxygenated hemoglobin respectively, c represents the hemoglobin concentration, and s is the saturation. To be exact, the optical density D(λ) for real blood needs to be corrected with an additional term. However, we will use the above simple form in this subsection and modify it in the next subsection. To estimate the saturation s in the case that ε_HbO₂(λ) and ε_Hb(λ) are known, we solve the follow equations for s using different wavelengths λ₁ and λ₂:

\begin{array}{l} D (λ_{1}) = [s ε_{{HbO}_{2}} (λ_{1}) + (1 - s) ε_{Hb} (λ_{1})] c d, \\ D (λ_{2}) = [s ε_{{HbO}_{2}} (λ_{2}) + (1 - s) ε_{Hb} (λ_{2})] c d . \end{array}

Next, we define the ratio of D(λ₁) and D(λ₂) as Φ = Φ(λ₁, λ₂), which is called the saturation parameter in reference [5] by:

Φ (λ_{1}, λ_{2}) = \frac{D (λ_{2})}{D (λ_{1})} .

From Eqs. (4) and (5), we obtain the saturation s as:

s = \frac{Φ ε_{Hb} (λ_{1}) - ε_{Hb} (λ_{2})}{Δ λ_{2} - Φ Δ λ_{1}},

where Δλ_n := ε_HbO₂(λ_n) − ε_Hb(λ_n) for n = 1, 2. It should be noted that we do not have to know c and d to solve for s.

In the discussion above, we do not take into account the reflected light from the medium. In reality, however, there is a portion of light that returns from the surface of the medium in which light enters into. When a medium is highly scattering, the ratio of the reflected light is larger than the transmitted light. Therefore, the Lambert-Beer’s law can be applied to the reflected intensity of light in the case of a highly scattering medium [8]. A biological tissue is a highly scattering medium, hence the saturation s of a tissue can be estimated from Eq. (6) based on the Lambert-Beer’s law in principle, but there are some uncertainties such as light path length and the effect of scattering. Thus, many studies, as reviewed in [8], have suggested an appropriate way to correct them for each modality. In the case of reflected light, the optical density D = D(λ) of Eq. (2) can be written by using the reflectance R = R(λ):

D = - log R .

As with the transmitted light, Eq. (6) can be applied to estimating s by measuring the reflectance.

2.1.2. SDF oximetry

We propose a SDF oximetry method that estimates the SO₂ from SDF images of a vessel. The proposed method is constructed in an analogy to retinal oximetry methods. In this section, we show the SDF oximetry method using two spectral images. When applying retinal oximetry methods to SDF imaging, we must consider differences between the retinal oximetry and the SDF oximetry. As presented in Fig. 2, there is a difference in the light paths. In retinal oximetry (Fig. 2(a)), photons backscattered from the vessel mainly contribute to the detected intensity at the vessel part in a retinal fundus image [9]. In contrast, photons that pass once the vessel (single pass) mainly contribute to that in a SDF image (Fig. 2(b)). In the SDF technique, light enters a tissue from illumination sources configured around the camera and then the camera captures the light scattered by the tissue. Consequently, we apply retinal oxymetry methods to the SDF oximetry keeping in mind the above consideration. We consider the proposed SDF oximetry method. Figure 3 shows a concept of the proposed method. As shown in Figs. 3(a) and 3(b), we define the light that enters the body with intensity I₀(λ) and returns outside without passing through any vessels as back reflection light and define its intensity as I_in(λ) = R(λ)I₀(λ) where R(λ) denotes the total reflectance of layers behind the vessel near a surface, e.g. muscularis, submucosa, and mucosa. Next we assume the same amount of reflecting light illuminates the vessels. Therefore, the intensity of the transmitted light can be expressed in terms of I₀ as in Eq. (1):

I_{out} = I_{in} (λ) \cdot 10^{- μ_{a} (λ) d} = R (λ) I_{0} (λ) \cdot 10^{- μ_{a} (λ) d},

where μ_a(λ) is the absorption coefficient defined in Eq. (3). Here we ignore the effect of photons that pass more than twice through the vessel (double pass) because all photons enter the peripheral region of the target vessel and most of them pass through the vessel once at most. In this case, the optical density D(λ) of blood can be written as:

D (λ) = μ_{a} (λ) d = [s ε_{{HbO}_{2}} (λ) + (1 - s) ε_{Hb} (λ)] c d .

As in the previous section, we define the saturation parameter by: Ψ(λ₁, λ₂) = D(λ₂)/D(λ₁). To determine s, we solve Eq. (9) for s with two wavelengths λ = λ₁, λ₂ and we obtain:

s = \frac{Ψ_{ε_{Hb}} (λ_{1}) - ε_{Hb} (λ_{2})}{Δ λ_{2} - Ψ Δ λ_{1}},

where Ψ = Ψ(λ₁, λ₂) and Δλ_n := ε_HbO₂(λ_n) − ε_Hb(λ_n) for n = 1, 2.

Fig. 2 Light paths for retinal and SDF imaging. (a) For a retinal fundus camera. Three paths can be considered: incident light is backscattered from a blood vessel (backscattering); reflected or scattered light passes once through a vessel (single pass); the light passes twice through a vessel (double pass). (b) For a SDF imaging system. In contrast with a fundus camera, there is no backscattered light. The dashed line triangles denote the illumination area.

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Fig. 3 A conceptual diagram of the SDF oximetry method. (a) Light with the intensity I₀ enters into a tissue. (b) The incident light is reflected and/or scattered in each layer, then the intensity becomes RI₀. The light with I_in = RI₀ enters into a vessel and then goes through. I_out represents the intensity of the transmitted light.

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2.2. Average extinction coefficient

Light-emitting diodes (LEDs) are commonly used as illumination sources in SDF imaging [1, 10]. As discussed in the previous section, monochromatic light is required for the SO₂ estimation. Thus, we have to take into account the effect of bandwidth of the LEDs since the bandwidth cannot be ignored. Moreover, a typical camera detects intensity of light by integrating over all wavelengths. Hence we also have to consider the effect of this integration on the SO₂ estimation. In order to address these issues, we modified the extinction coefficients of hemoglobin by averaging over the wavelength in the SO₂ estimation by referring to [5].

First, we discuss the general case. As we mentioned earlier, the optical density of real blood is written with an additional term by:

D (λ) = μ_{a} (λ) d + B (λ),

where d is the diameter of a vessel, μ_a(λ) represents the absorption coefficient of hemoglobin. The additional term B(λ) represents a correction for the RBC geometry and the detector geometry, and all phenomena other than the absorption of hemoglobin such as scattering and the sieve effect [11]. The sieve effect is a discrepancy between spectral absorption of RBCs and that of hemoglobin itself. Its dominant wavelength range is approximately from 370 to 430 nm. Fortunately, since this wavelength range is far from that of LED used in this study (see the sub-subsection 2.3.2), this effect is negligible.

We consider light with the amplitude I₀ = I(λ₀) at the peak wavelength λ₀ and the relative spectral intensity distribution F₀(λ) of the light I₀, i.e. the intensity of light represents I₀F₀(λ). A camera detects the intensity of the light transmitting through the vessel as:

{\bar{I}}_{out} (λ_{0}) = \int_{Ω} d λ R (λ) S (λ) I_{0} F_{0} (λ) \cdot 10^{- [μ_{a} (λ) d + B (λ)]},

where S(λ) represents the sensitivity of the camera, R(λ) represents the total reflectance of layers behind the vessel, and Ω represents the spectral sensitivity region of the camera. The intensity of light outside a vessel is written by:

{\bar{I}}_{in} (λ_{0}) = \int_{Ω} d λ R (λ) S (λ) I_{0} F_{0} (λ) .

Similarly to Eq. (2), we define the optical density D̄(λ₀) = −log[Ī_out(λ₀)/Ī_in(λ₀)] in consideration of the influence of the integral detection by the camera and the bandwidth of an illumination by:

\begin{array}{l} \bar{D} (λ_{0}) & = & - log [\frac{\int_{Ω} d λ R (λ) S (λ) F_{0} (λ) \cdot 10^{- [μ_{a} (λ) d + B (λ)]}}{\int_{Ω} d λ R (λ) S (λ) F_{0} (λ)}] \\ = : & [s {\bar{ε}}_{{HbO}_{2}} (λ_{0}, d) + (1 - s) {\bar{ε}}_{Hb} (λ_{0}, d)] c d + B (λ_{0}), \end{array}

where ε̄_HbO₂ and ε̄_Hb represent average extinction coefficients. The average extinction coefficients are defined so that the optical density has the same form as Eq. (2). It should be noted that the average extinction coefficients are a function of not only the peak wavelength, but also the diameter of the vessel. Solving Eq. (14) with s = 1 and s = 0, we obtain:

\begin{array}{l} {\bar{ε}}_{{HbO}_{2}} (λ_{0}, d) & = & - \frac{1}{c d} (log [\frac{\int_{Ω} d λ R (λ) S (λ) F_{0} (λ) \cdot 10^{- [ε_{{HbO}_{2}} (λ) c d + B (λ)]}}{\int_{Ω} d λ R (λ) S (λ) F_{0} (λ)}] + B (λ_{0})), \\ {\bar{ε}}_{Hb} (λ_{0}, d) & = & - \frac{1}{c d} (log [\frac{\int_{Ω} d λ R (λ) S (λ) F_{0} (λ) \cdot 10^{- [ε_{Hb} (λ) c d + B (λ)]}}{\int_{Ω} d λ R (λ) S (λ) F_{0} (λ)}] + B (λ_{0})), \end{array}

respectively. Here μ_a(λ) is explicitly expressed as μ_a(λ) = ε_HbO₂ (λ)c and μ_a(λ) = ε_Hb(λ)c for s = 1 and s = 0, respectively. If light is monochromatic, i.e. F₀(λ) = δ(λ − λ₀) which is a Dirac delta function, then Eq. (15) becomes ε̄_HbO₂ (λ₀) = ε_HbO₂ (λ₀) and ε̄_Hb(λ₀) = ε_Hb(λ₀).

Finally, we discuss the approximation made in the discussion above. Cohen and Laing [5] assumed that the dependences of the RBCs’ scattering part of B(λ) and the reflectance R(λ) on wavelength are much smaller than variations of S(λ), F(λ), and μ_a(λ), then they do not contribute at all to the integral. In addition to Cohen’s approximation, we assume that variation of B(λ) other than RBCs’ scattering is also smaller than that of S(λ), F(λ), and μ_a(λ), i.e. B(λ) ≃ B(λ₀) in the wavelength range of LED used. Therefore, Eq. (15) can be rewritten to:

\begin{array}{l} {\bar{ε}}_{{HbO}_{2}} (λ_{0}, d) & = & - \frac{1}{c d} (log [\frac{\int_{Ω} d λ S (λ) F_{0} (λ) 10^{- ε_{{HbO}_{2}} (λ) c d}}{\int_{Ω} d λ S (λ) F_{0} (λ)}]), \\ {\bar{ε}}_{Hb} (λ_{0}, d) & = & - \frac{1}{c d} (log [\frac{\int_{Ω} d λ S (λ) F_{0} (λ) 10^{- ε_{Hb} (λ) c d}}{\int_{Ω} d λ S (λ) F_{0} (λ)}]) . \end{array}

In this study, we assume that S(λ) is set to unity and F(λ) is set to a Gaussian function (see the next subsection) in order to simplify the calculation. The diameter d of a blood vessel is assumed to be consistent with the FWHM of the profile of a vessel image [12, 13]. Before estimating SO₂, we measure d from images, and then evaluate Eq. (16) with known ε_HbO₂ c and ε_Hbc.

2.3. Setups of experiments

We performed a Monte Carlo photon propagation simulation to evaluate the SO₂ estimation method. As a basic study, we investigated the trend of the estimated SO₂ by changing values of SO₂, vessel diameter, and bandwidth of illumination sources. Our procedure was as follows: first, we made a tissue model with a blood vessel and set blood with known SO₂ into the vessel. Next, we simulated the photon propagation in the tissue and acquired virtual SDF images. Finally, we estimate SO₂ from the virtual SDF images.

2.3.1. Tissue model

To construct a three-dimensional tissue model with a vessel, we used an open-source software iso2mesh [14,15] which is coded with Matlab (MathWorks, Natick, USA). The tissue model is constructed by referring to the result of hematoxylin and eosin (H&E) staining of a pig small intestine shown in Fig. 4(a). The simulated tissue model is shown in Figs. 4(b) and 4(c). As shown in Fig. 4(d), each of the layers simulate serosa, muscularis, submucosa, and mucosa from the top surface respectively. Here the model has a blood vessel for which the uppermost surface is located at 100 μm below the top surface in the serosa. We set the blood vessel diameter as 100, 200, and 300 μm. In this study, the optical properties of the blood vessel and blood were approximated by those of hemoglobin. The oxygenated saturation was varied by the amount of the oxygenated hemoglobin. We used these optical properties in the case of the hematocrit value H = 0.4 reported in the literature [16, 17]. The optical properties of blood are shown in Fig. 5(a). In the optical properties of tissue, we assumed that absorption and scattering of each layer were homogeneous, the saturation of tissue was fixed, and the absorption coefficient of each layer other than the serosa was equal. We used the value of mucosa in the study by Bashkatov et al. [18] as the absorption coefficient of tissue. The scattering coefficient is in proportion to the volume fraction of scatterer in a medium [19]. Accordingly, we assumed that the size of scatterer in each layer was equal. Since the submucosa includes an abundant number of blood vessels and tends to scatter the light, the ratio of the volume fraction of scatterer to that of the submucosa was set as 0.7 in the mucosa and 0.3 in the muscularis. The scattering coefficient of the submucosa was set to the value reference to the data of Bashkatov et al. [18]. Absorption coefficient and scattering coefficient of the submucosa are shown in Fig. 5(b). For the serosa, we set this layer as transparent, i.e. μ_a = 0 mm⁻¹ and μ_s = 0 mm⁻¹. The anisotropy factor of tissue was set as 0.9. This is because this value is approximately equal for any tissue.

Fig. 4 The tissue model. (a) The cross section of a small intestine of a pig H&E stained. The layers, from the top surface, serosa, muscularis, submucosa, and mucosa. The inset on the right is an enlarged image that shows blood vessels are present in the serosa (blue arrows). (b) (c) The simulated tissue model which has a layered structure and a straight blood vessel at 0.1 mm below the top surface in the serosa. (b) Side view. (c) Overview. (d) A schematic diagram of the tissue model and the geometry of the simulation.

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Fig. 5 Optical properties used in this study. (a) Optical properties of blood [16, 17]. Open and closed squares represent the absorption coefficient of oxygenated and de-oxygenated hemoglobin respectively, crosses represent the scattering coefficient, and circles represents the anisotropy factor. (b) Optical properties of the tissue [18]. Squares represent the absorption coefficient and crosses represent the scattering coefficient.

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2.3.2. Trial SDF device

We assumed the illumination geometry was similar to that in the SDF imaging device that we have developed. Figure 6 shows our trial device with LED chips (SMLV56RGB1W, ROHM Co., Ltd., Japan) and a charge-coupled device (CCD) camera. Six LED chips were configured around the exterior tip of a probe to provide uniform illumination. These LED chips could be switched from blue to green light. The peak wavelengths of each color were 470 nm and 527 nm and the bandwidths (FWHM) of each intensity distribution were 27.3 nm and 37.2 nm respectively. The reasons why we chose these wavelengths were that light with these wavelengths is well absorbed by hemoglobin and their penetration distance in a tissue is shorter than that of longer-wavelength light [20], so these wavelengths are suitable for observation of vessels near the surface. The incident angle of our device was 16°. Figure 4(d) shows the geometry of the simulation. A LED chip source was assumed as a set of point sources. In this simulation study, the number of point sources per LED chip was 100. We set the initial launch area to be the same as that of a LED chip of the trial device. The area of a LED chip was 2.8 × 3.1 mm². To simulate an area source with a set of point sources, we generated position launching photons with the uniform probability distribution.

Fig. 6 The trial SDF illumination device. (a) A schematic diagram of the trial SDF device. The outer diameter of this device was 13.5 mm. LED chips were used as illumination. (b) This LED chip could be switched from blue light at the peak wavelength of 470 nm to green light at 527 nm.

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We assumed that the spectral intensity distribution of a LED with a peak wavelength λ₀ has a Gaussian function given by:

F_{0} (λ) = N exp [- \frac{{(λ - λ_{0})}^{2}}{2 σ^{2}}],

where N is the normalization coefficient and σ is the coefficient to control the bandwidth. We set bandwidth w as the FWHM of the Gaussian function, namely:

σ = \sqrt{\frac{w^{2}}{8 ln 2}} .

For w = 30 nm, σ is 12.74 nm.

2.3.3. Monte Carlo photon propagation simulation

We used a mesh-based Monte Carlo (MMC) approach [21, 22] developed based on references [23–25] for a three-dimensional Monte Carlo code. The Monte Carlo simulation procedures are explained in detail in [23–25].

For the simulation of light absorption in a tissue, we used the concept of the photon packet. A photon packet enters into a tissue with an initial weight W₀ = 1, and then after s steps, its weight becomes W_s = W_s−1 exp[−μ_aΔℓ], where W_s−1 is the weight at the (s − 1)-th step, μ_a is the absorption coefficient of the tissue and Δℓ is the step distance as it propagates within the tissue. When a photon packet reaches the top surface, its weight is detected with respect to a sub-region in the top surface. We assumed that a virtual SDF image acquired by a CCD camera was the spatial intensity distribution of the average weight of the sub-region.

As setups for the simulation, we set the number of incident photons as 2.0 × 10⁷ per LED chip. The number of a sub-regions was 100 × 100 and the area of a sub-region was 0.02 × 0.02 mm².

3. Results

3.1. Photon propagation simulation

Figure 7(a) shows a virtual SDF image simulated for the light at 470 nm with bandwidth w = 0 and the oxygenated blood vessel with diameter d = 100 μm. We can see the straight vessel in the image. Figure 7(b) shows the x-directional intensity profiles of the single pass, double pass, and no pass of light for the case of Fig. 7(a); there were obtained by averaging the two-dimensional distribution along the y-direction. This result indicates that the double pass light does not contribute to the virtual SDF image unlike for the single pass light. We measured the FWHM of the profiles from virtual SDF images, and then evaluated average extinction coefficients using Eq. (16). Figure 7(c) shows the average extinction coefficients results. Figure 7(d) shows the relation of intensities between incident and transmitted light for d = 100 μm. We used the weighted average of these intensities to evaluate the Lambert-Beer’s law below.

Fig. 7 Results of the photon propagation simulation for a tissue model with the straight vessel. (a) The virtual SDF image (the gray level map of the intensity). (b) The profile of the virtual SDF image (blue line). The single pass, the double pass, and the no pass are depicted by red, magenta, and green, respectively. (c) Absorption coefficients with average extinction coefficients are plotted as a function of bandwidth of the illumination. ε̄(λ₀) for λ₀ = 470 nm and 527 nm are depicted by squares and triangles, respectively. Open and closed symbols represent SO₂ s = 1 and s = 0, respectively. Diameter d values of 100 μm, 200 μm, and 300 μm are depicted by solid lines, dashed lines, and chain lines, respectively. (d) The histogram of the intensity I_in of light directly entering into the blood vessel and I_out of light passing through it when the model is (a).

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3.2. Three SO₂ estimation methods and mean SO₂ estimation error

We estimated SO₂ using three methods: the first one was the Lambert-Beer’s law using weighted average values of the intensities of the incident and transmitted light of the blood vessel (Fig. 7(d)), which is called the pure Lambert-Beer’s law method (PLB); the second one was the image-based measurement with the extinction coefficients at the peak wavelength, which is called a method using the extinction coefficients at the peak wavelength (EPW); the last one was the imaged-based measurement with the average extinction coefficients defined in Eq. (16), which is called a method using the average extinction coefficients (AEC).

We used the SO₂ estimation error (SEE) as an evaluation index of estimation accuracy. We modeled the estimated saturation s by a linear function to parameterize those trends:

s = α t + β,

where t (0 ≤ t ≤ 1) represents the true SO₂ value and α and β are fitting coefficients. Those coefficients are obtained by the Levenberg-Marquadt method. If these coefficients are close to α = 1 and β = 0, the estimation is accurate. When the SEE Δs(t) is defined as an absolute value of the difference between the estimated curve Eq. (19) and the true curve s = t, we define the mean SEE

\bar{Δ s}

as:

\bar{Δ s} : = \frac{1}{M} \sum_{i = 1}^{M} Δ s (t_{i}) = \frac{1}{M} \sum_{i = 1}^{M} | (α - 1) t_{i} + β |,

where M represents the number of sample points over the SO₂ range and t_i represents the true SO₂ value at the i-th sample point. In this study, we had 21 true SO₂ values, that is, M = 21.

Figures 8, 9 and 10 show the results of the estimated value s and the estimated curve by Eq. (19) for d = 100 μm, 200 μm, and 300 μm, respectively. Three methods (PLB, EPW, and AEC) are shown together. We show the $\bar{Δ s}$ and standard deviation (SD) of each measurement in Table 1 and summarize the $\bar{Δ s}$ of these results in Fig. 11. In Fig. 11, the plots are fitted by a surface with bivariate d and w power series up to the second-order of both variables. Since extinction coefficients at the peak wavelength and average extinction coefficients match in w = 0, $\bar{Δ s}$ of EPW and AEC are equal. As a result, the tendency of the SEE of PLB (Fig. 11(a)) and AEC (Fig. 11(c)) are explicitly similar.

Fig. 8 The SO₂ estimation of d = 100 μm vessel. (a) The monochromatic light with no bandwidth (w = 0). (b) w = 10 nm. (c) w = 20 nm. (d) w = 30 nm. (e) w = 40 nm. (f) w = 50 nm.

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Fig. 9 The SO₂ estimation of d = 200 μm vessel. (a) The monochromatic light with no bandwidth (w = 0). (b) w = 10 nm. (c) w = 20 nm. (d) w = 30 nm. (e) w = 40 nm. (f) w = 50 nm.

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Fig. 10 The SO₂ estimation of d = 300 μm vessel. (a) The monochromatic light with no bandwidth (w = 0). (b) w = 10 nm. (c) w = 20 nm. (d) w = 30 nm. (e) w = 40 nm. (f) w = 50 nm.

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Table 1. The mean SEE $\bar{Δ s} \pm SD$ .

View Table

Fig. 11 The mean of SEE $\bar{Δ s}$ surface with the contour interval color. (a) PLB. (b) EPW. (c) AEC.

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We focused on the changes in $\bar{Δ s}$ due to w. In the comparison between image-based methods, AEC is more accurate than EPW for all bandwidths in each d. As shown in Fig. 11(b) and Table 1, $\bar{Δ s}$ using EPW increases with increment of w. The changing rate of $\bar{Δ s}$ due to w decreases with an increase in d. On the other hand, as shown in Fig. 11(c) and Table 1, $\bar{Δ s}$ using AEC decreases with increment of w. Next, we focused on the changes in $\bar{Δ s}$ due to d of the vessel. From the results, AEC is more precise than EPW for all d values in each w. $\bar{Δ s}$ using EPW increases with increment of d. The changing rate of $\bar{Δ s}$ due to d decreases with an increase in w. By contrast, $\bar{Δ s}$ using AEC increases with increment of d.

4. Discussion and conclusion

We have investigated the validity of an SO₂ estimation of individual vessels using two spectral SDF images obtained by a Monte Carlo photon propagation simulation where two image-based methods (EPW and AEC) were compared with the intensity method (PLB). As an experiment, we performed the simulation with illumination sources from w = 0 nm to 50 nm at 10 nm intervals and the tissue model with vessel diameter d = 100, 200, and 300 μm.

There were strong correspondences between PLB and the image-based methods. The SEE of AEC was especially similar to the SEE of PLB. In the EPW, the SEE was an increasing function with respect to d and w. On the other hand, in the PLB and AEC, the errors were not a monotone function. In addition, for all conditions, AEC was more accurate than EPW, hence using the average extinction coefficients of hemoglobin is effective for the SDF oximetry with narrowband illumination sources such as LEDs.

We considered that the SEE was caused by a deviation from Lambert-Beer’s law due to the scattering event in case of using the extinction coefficient of hemoglobin. In our study we assumed that, the scattering in a tissue and a vessel was more dominant than the absorption over the entire wavelength range and more than 440 nm, respectively. Therefore, its effect was not negligible. We considered that the its possible effects were the overestimation of intensity in the vessel and a positional displacement to reach the surface due to multiple scattering in the tissue. The displacement caused a loss or a gain of detected intensity. A correction of these effects is included in the term of B(λ) in Eq. (11). In our study, we assumed that B(λ) was constant compared to extinction coefficients of hemoglobin and the camera sensitivity, and then it vanishes in Eq. (16), i.e. we did not take into account the influence of the scattering and the other factors. This assumption is not very likely to be valid. Thus, we can correct these influence with the average extinction coefficients that include B(λ).

In addition to the above prediction, we considered that there was another cause of the SEE. Smith [26] pointed out that generally the SEE is due to the combination of the selected wavelengths in retinal vessel oximetry. Here this error is called the potential error (PE). The potential error occurs in the difference of the extinction coefficients of hemoglobin and de-oxygenated hemoglobin between selected wavelengths. Since average extinction coefficients vary in d and w, the PE also varies in a combination of d and w. Here, we evaluated the PE with the procedure proposed by Smith [26] with average extinction coefficients. Figure 12 shows the result with a combination of our calculated average extinction coefficients. We used mean FWHM of profiles as the diameter and ΔT₁ = ΔT₂ was set as 1.0 × 10⁻³ (see Eqs. (21) and in [26]). As a result, we verified that the tendency of our SEEs of AEC was consistent with the mean PEs due to changes in average extinction coefficients. From this result, the mean of SEE was greatly affected by the wavelength combination compared with the aforementioned scattering effect. When monochromatic light is irradiated to the model with vessel d = 300 μm, the mean PE takes the maximum value. In the case of w = 0, the mean PE occurs in the change of d rather than the difference of average extinction coefficients and increases as d increases. This is because average extinction coefficients of any d are equal.

Fig. 12 The mean PE evaluated by Smith’s procedure [26]. The surface is a fitting curve with a bivariate power series until the second-order of both variables.

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As a future plan, we will construct a method for correcting the PE and deviation from Lambert-Beer’s law. One candidate correction method is to use a database. In the method, we obtain α and β for many combinations of d and w by a photon propagation simulation and estimate s from real SDF images. Then, we look up α and β in the corresponding condition in the database; finally, we predict the true SO₂ value from t = (s − β)/α. This method can be used without knowing the effects of B(λ). We expect that this method is useful for the SO₂ estimation of a vessel in microcirculation.

Acknowledgments

This research was supported by Technology Foundation, Kakenhi, the Grant-in-Aid for Scientific Research B25282151.

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d [μm]	w [nm]	Using PLB	Image-based measurement

			Using EPW	Using AEC
100	0	0.0512 ± 0.0237	0.0826 ± 0.0532
100	10	0.0388 ± 0.0267	0.1652 ± 0.0643	0.1183 ± 0.0759
100	20	0.0536 ± 0.0316	0.2194 ± 0.0658	0.0879 ± 0.0569
100	30	0.0524 ± 0.0303	0.2496 ± 0.0942	0.0863 ± 0.0497
100	40	0.0369 ± 0.0216	0.2863 ± 0.1241	0.0860 ± 0.0505
100	50	0.0132 ± 0.0076	0.3377 ± 0.1401	0.0666 ± 0.0394

200	0	0.2121 ± 0.0725	0.3217 ± 0.1532
200	10	0.1407 ± 0.0968	0.3331 ± 0.1627	0.2607 ± 0.1705
200	20	0.1117 ± 0.0727	0.3678 ± 0.1485	0.1925 ± 0.1353
200	30	0.0944 ± 0.0569	0.3845 ± 0.1677	0.1766 ± 0.1206
200	40	0.0789 ± 0.0468	0.3936 ± 0.1867	0.1591 ± 0.1036
200	50	0.0972 ± 0.0667	0.4198 ± 0.1994	0.1422 ± 0.0912

300	0	0.2830 ± 0.1510	0.3593 ± 0.1831
300	10	0.2113 ± 0.1486	0.3640 ± 0.1793	0.2663 ± 0.1823
300	20	0.1686 ± 0.1114	0.3838 ± 0.1824	0.2092 ± 0.1435
300	30	0.1440 ± 0.0880	0.4019 ± 0.1869	0.1875 ± 0.1250
300	40	0.1239 ± 0.0728	0.4158 ± 0.2115	0.1769 ± 0.1118
300	50	0.1397 ± 0.0979	0.4199 ± 0.2181	0.1520 ± 0.0904

Impact of vessel diameter and bandwidth of illumination in sidestream dark-field oximetry

Abstract

1. Introduction