Evaluation of blood oxygen saturation in vivo from diffuse reflectance spectra

Alexander A. Stratonnikov; Victor B. Loschenov

doi:10.1117/1.1411979

1 October 2001 Evaluation of blood oxygen saturation in vivo from diffuse reflectance spectra

Alexander A. Stratonnikov, Victor B. Loschenov

Author Affiliations +

Journal of Biomedical Optics, Vol. 6, Issue 4, (October 2001). https://doi.org/10.1117/1.1411979

1. Introduction

There are many approaches to evaluate the blood oxygen saturation in vivo from tissue diffuse reflectance spectra.¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ¹⁰ ¹¹ ¹² ¹³ ¹⁴ ¹⁵ ¹⁶ ¹⁷ ¹⁸ ¹⁹ ²⁰ ²¹ ²² Hemoglobin contained in erythrocytes is the main tissue absorber in the visible and very near infrared range. As a hemoglobin absorption spectrum is different for oxygenated and deoxygenated forms¹⁸ ²³ (by this reason the blood in an artery and a vein varies in color), this fact can be used for quantification of the blood oxygen saturation in the tissue in vivo. The central problem with the evaluation of blood oxygen saturation for the tissue in vivo is to take into account the influence of the scattering, other tissue absorbers and heterogeneous hemoglobin distribution.

To quantify the relative oxyhemoglobin and deoxyhemoglobin concentrations as well as other tissue absorbers in the tissue using optical spectroscopy, the following two steps are usually performed. First, the total absorption ( μ_a ) and reduced scattering ( μ_s ^′ ) tissue coefficients are extracted from the experimental measurements at several wavelengths. For a uniform semi-infinite medium it can be done by time resolved methods in frequency and time domain¹⁰ ¹¹ ²⁰ ²⁴ ²⁵ or spatially resolved methods.¹⁷ ²⁹ ²⁶ ²⁷ ²⁸ ²⁹ ³⁰ The number of wavelengths whereby the total absorption coefficient is evaluated should be at least equal to the number of the tissue absorbers contributing to absorption in the spectral range under analysis. If the measurement method gives no way of obtaining absorption and reduced scattering coefficients independently, for example, in steady-state measurements without spatial resolution absorption and reduced scattering coefficients are coupled to each other, some assumption should be done about behavior of the reduced scattering coefficient or this value requires independent measurements. Using this information and applying some model for diffuse reflection dependence upon tissue optical properties, the absorption coefficient can be extracted from simple steady-state diffuse reflectance measurements.³¹ On evaluating the total tissue absorption coefficient at several wavelengths, the concentrations of individual absorbers may be obtained as it is done in the usual spectroscopy problem of multicomponent systems. If the absorption spectra of all the species contributing to the tissue absorption are known, it is done by solving normal equations with the least square method. The sophisticated chemometrics methods (partial least square, principle component analysis¹⁹ or singular value decomposition³²) can be also applied if information about absorbing species is not complete.

An original self-consistent approach for evaluation of the blood oxygen saturation and the relative hemoglobin concentration from diffuse reflectance spectrum obtained with fiber probe geometry without spatial resolution is described in this work. The method is applied in the visible wavelength range, where the spectral difference between oxygenated and deoxygenated hemoglobin is rather high. The advantage of the proposed approach is its simplicity and applicability to different fiber probe geometries used to deliver and collect light passed through the tissue. The relation between a diffuse reflectance signal and tissue optical properties in terms of photon path length distribution function is detailed in the next section. The existing theoretical approaches extracting the absorption coefficient are briefly reviewed and discussed therein. Taylor expansion method is presented in Sec. 3. In linear approximation it results in an effective algorithm for blood oxygen saturation calculation. An experimental setup for diffuse reflectance and diffuse transmittance measurements is described in Sec. 4. Finally, the experimental results obtained on human skin during artery occlusion and blood samples during photochemical deoxygenation are presented and discussed in Sec. 5.

2. Diffuse Reflectance and Photon Path Length Distribution Function

To evaluate the spectral dependencies of absorption and scattering coefficients it is convenient to consider the light attenuation A(λ) defined by the following expression:

Eq. (1)

A (λ) = - ln (\frac{I}{I_{0}}),

where I₀ and I are the total photon flows coming out from the delivery fiber and entering into the receiving fiber correspondingly. In the case of integrating sphere configuration the value I is the integrated photon flow measured by the sphere (see Figures 1 and 2). Sometimes the dependence (1) is called the absorption without correction for the scattering.

Figure 1

Integrating sphere geometry for collecting diffuse reflected photons. The sampling depth for this configuration is determined mainly by the light penetration depth δ.

Figure 2

Probe arrangement with distantly placed fibers for the measurement of tissue diffuse reflectance spectra in vivo. The sampling depth is determined by the light penetration depth δ and fiber separation distance d.

Due to the strong tissue scattering the photon path lengths through the tissue contributing the measured signal have a distribution with some probability function P. The intensity of a light signal in the receiving fiber or in the integrating sphere is determined by absorbing and scattering properties of the tissue as well as by measurement configuration and may be expressed through the path length probability function by the following relation:

Eq. (2)

I / I_{0} = \int_{0}^{\infty} P (μ_{s}, g, l) \cdot exp (- μ_{a} l) dl,

where μ_s is the scattering tissue coefficient, μ_a is the absorption coefficient, g is the scattering anisotropy factor, l is the photon path length through the tissue on the trajectory between the delivering and receiving fibers (see Figure 2), and the dimensionless value P(μ_s,g,l)dl is the probability that the photon path length will be in the interval [l,l+dl] between the delivering and receiving fibers in the absence of absorption (path length distribution function). The path length distribution function P depends on the scattering tissue properties ( μ_s and g) as well as on the geometry of the measurements (fiber probe spatial configuration). The dimension of P function is the inverse dimension of the length. The integral of the distribution function over all photon path lengths (normalization of the path length distribution function) determines the attenuation of photon flux between the delivery and receiving fibers due to the scattering alone. It should be also noted here that integral (2) represents Laplace transform of the path length distribution function P(l).

As mentioned above, the path length distribution function is strongly dependent on the spatial configuration of the fiber probe as its geometry defines the trajectories of photons, which passed through the tissue and reached the detector. Two limiting cases of geometrical configurations for the semi-infinite medium are usually discussed. The first is the integrating sphere arrangement when all light reemitted from the tissue is collected as shown in Figure 1. Practically it may be realized only with the use of an integrating sphere for collecting all the diffuse reflected light. However, the use of the integrating sphere is very inconvenient for in vivo measurements. A sampling depth for this configuration is determined mainly by the light penetration depth δ. In the second limiting case the delivery and receiving fibers (or a light emitter and a detector) are placed apart at the distance d which is much greater than the fiber diameter (size of the emitter and the detector) as shown in Figure 2. In this case the photon trajectories through the tissue form a banana shaped region with the ends at the positions of the delivery and receiving fibers. The sampling depth for this configuration is proportional to $\sqrt{2} d / 4$ in the weak absorption limit (d≪δ) and $\sqrt{d δ / 2}$ in the strong absorption limit ( d ≫δ ).³³ ³⁴ Thus, in this case increasing the fiber separation makes it possible to increase the sampling depth. This configuration is applied to cerebral blood oxygen saturation measurements in the near infrared spectral range. Of course, there may be intermediate cases such as closely spaced delivery and receiving fibers which are greater in diameter than the distance between the fibers. Approximate analytical solutions relating tissue optical properties and diffuse reflected signal intensity have been obtained only for these limiting cases of the integrating sphere and the spatially placed fibers. For intermediate cases one may use a collection efficiency factor scaling the measured signal to these analytical expressions.

In the case of a transparent medium for a configuration when delivering and receiving fibers are placed on opposite sides of a sample with thickness L, all the photons have the same path length L, and the path length distribution function is reduced to Dirac delta function: P(l)=δ(l−L) . On substituting this function in Eq. (2) it is reduced to the Beer–Lambert law relating signal attenuation in the nonscattering absorbing medium with the absorption coefficient μ_a and the path length L

Eq. (3)

A = - ln (\int_{0}^{\infty} δ (l - L) \cdot exp (- μ_{a} l) d l) = μ_{a} \cdot L .

In the presence of the scattering the relation between the measured signal attenuation and the absorption coefficient is complicated. Accurate approaches for obtaining the absorption and scattering tissue coefficients are based on the time resolved measurements. The time resolved reemittance signal represents in fact the path length distribution function with change of product speed of light in the tissue and time on path length l multiplied by the absorption term exp(-μ_al). An analytical approximation for the path length distribution function (with zero boundary condition) for the fiber configuration shown in Figure 2 has been obtained from the diffusion theory²⁴

Eq. (4)

\begin{array}{l} P (l) \cdot exp (- μ_{a} l) = z_{0} {(4 π D)}^{- 3 / 2} l^{- 5 / 2} \\ \times exp (- \frac{d^{2} + z_{0}^{2}}{4 D l}) \cdot exp (- μ_{a} l) \cdot d s, \end{array}

where z₀=1/μ_s ^′,μ_s ^′=μ_s(1−g),D=1/3μ_s ^′, d is distance between the source and the detector, ds is the cross-sectional area of the receiving fiber. Fitting the experimental time resolved reemitted signal with function (4) (or more sophisticated one taking into account different boundary conditions²⁵ ³⁵) one can get the values of μ_a and μ_s ^′. ²⁴ But this method requires a complicated technique for its realization and makes application in clinical conditions difficult.

The steady-state experimental methods are easier to apply in practice. In diffuse approximation the formula relating the intensity of the steady-state measured signal with tissue optical properties and the fiber separation distance can be obtained directly from Eq. (2). Substituting relation (4) for the path length distribution function into Eq. (2) and calculating the resulting integral we obtain the following relation for the steady-state case in the diffusion approximation:

Eq. (5)

\begin{array}{l} I / I_{0} & = & \int_{0}^{\infty} P (l) \cdot exp (- μ_{a} l) d l \\ = & \frac{1}{2 π μ_{s}^{'}} \cdot (\frac{1}{δ} + \frac{1}{d}) \cdot \frac{1}{d^{2}} \cdot exp (- d / δ) \cdot d s, \\ δ & = & 1 / \sqrt{3 μ_{a} (μ_{a} + μ_{s}^{'})} \approx 1 / \sqrt{3 μ_{a} μ_{s}^{'}} . \end{array}

Measuring the reemitted signal at different distances d between delivery and receiving fibers and fitting this dependence by relation (5) (or more sophisticated one taking accurately into account the influence of the boundary²⁵ ²⁶), it is possible to get the values of μ_a and μ_s ^′. In Ref. 19 this technique was applied to the evaluation of the blood oxygen saturation in the near-infrared (NIR) spectral range by fitting the obtained wavelength dependence of tissue absorption coefficient with the spectra of oxygenated and deoxygenated hemoglobin as well as background absorption.

In the following consideration we will be restricted to the case when we measure the reemitted signal without spatial resolution. In this case it is not possible to get the values of absorption and scattering coefficients independently. Nevertheless, assuming the wavelength dependence of scattering and knowing the absorption spectra of the species present in the tissue, it is possible to evaluate concentrations of the tissue absorbers. However, the use of Eq. (5) for this purpose is restricted only to the diffuse approximation region and the spatially separated fiber probe configuration (Figure 2). For example, when we use a probe arrangement with closely spaced fibers the signal dependence is rather described by that obtained for integrating sphere configuration.

For the integrating sphere configuration there are two simple relations between the measured reflected signal and the tissue properties. One is based on Kubelka–Munk theory for light transport in tissue,⁹ and another has been obtained by fitting Monte Carlo results with the analytical expression.³¹ According to the Kubelka–Munk theory the diffuse reflectance coefficient R_d is described as follows:

Eq. (6)

R_{d} = I / I_{0} = exp (- A) = 1 - \frac{K}{S} (\sqrt{1 + 2 \frac{S}{K}} - 1)

where K and S are absorption and scattering Kubelka–Munk coefficients, respectively. The Kubelka–Munk absorption coefficient K can be related to the tissue absorption coefficient μ_a. In Ref. 9 this approach was used to evaluate the blood oxygen saturation by modeling tissue absorption as a sum of oxygenated and deoxygenated hemoglobin and approximating the Kubelka–Munk scattering coefficient by a smooth wavelength function. Actually, this approach is very close to ours given bellow. The difference is that we are not restricted to the particular fiber probe geometry and derive the results starting from the exact relation (2) for the diffuse reflectance (or transmittance) signal expanding it in Taylor series and omitting higher orders (see below).

Another relation for the integrating sphere configuration was obtained by Jacques³¹ and tested using Monte Carlo technique. According to it the expression for diffuse reflectance coefficient can be written as

Eq. (7)

R_{d} \approx exp (- 7.8 \cdot μ_{a} \cdot δ),

where δ is the light penetration depth defined in Eq. (5). This relation was used in Ref. 31 to evaluate bilirubin content in skin. Relation (7) clearly demonstrates that the mean photon path length through the tissue in the integrating sphere configuration can be approximated by 7.8δ. Relations (6) and (7) are valid only for the integrating sphere configuration (i.e., when all the light diffusely reflected from the tissue is collected). Fiber probes collect actually only a part of the reflected signal. This should be taken into account by a collection efficiency factor³¹ when applying these relations to data interpretation. It should be noted also that when relations (5 6 7) are applied to evaluate the concentrations of absorbing species in the tissue one must know the wavelength dependence of the scattering coefficient. This information may be obtained from experimental data or modeled. For example, it was shown in Refs. 14 and 36 that for the visible and NIR range the wavelength dependence of the scattering coefficient can be approximated by: μ_s ^′∼1/λ^0.37 , although the Rayleigh contribution, which is proportional to 1/λ⁴ , may also be substantial for short wavelengths.

A very important question, which we have not touched so far, concerns the heterogeneous distribution of the tissue absorbers. Hemoglobin is located in vessels, which form a complex tangled network inside the tissue, while melanin is concentrated in a thin superficial epidermis layer. These nonuniformities result to distortion of diffuse reflectance spectra as compared to that predicted from the models previously described. The heterogeneous distribution of melanin in the tissue and its influence on the diffuse reflectance spectra were considered in Ref. 31.

3. Tayor Series Expansion Model

Our approach given below is based on the assumption that attenuation A is a smooth (analytical) function of μ_a and the changes of μ_a over the spectral range under analysis are not high. We start from the exact relation for A based on Eq. (2) and approximate the attenuation by Taylor series expansion at some value μ_a=μ_a ⁰ as follows:

Eq. (8)

\begin{matrix} A & = & - ln (I / I_{0}) \\ \approx & A (μ_{a}^{0}) + A^{'} (μ_{a} - μ_{a}^{0}) + \frac{A^{″}}{2!} {(μ_{a} - μ_{a}^{0})}^{2} + \dots \end{matrix}

The value of μ_a ⁰ will be determined in a self-consistent way to minimize difference between experimental and modeled data. Intuition suggests that this value lies in the middle of μ_a variance interval in the spectral range under analysis. For the following analysis (linear approximation) we save only two terms (constant and linear) in expansion (8). It should be emphasized that Taylor coefficients are functions of wavelength through the dependence of μ_s(λ) but do not depend on μ_a. It is also possible to allow for the quadratic term in Eq. (8) resulting in a more complicated model with the greater number of fitted parameters. Such a model may occur to be useful for analysis of data acquired with a high signal to noise ratio or for high changes in the absorption coefficient through the analyzed spectral range. For the data obtained in our experimental setup, the linear approximation gives the required accuracy. The applicability of linear approximation [smallness of the quadratic term in Eq. (8)] can be expressed in the following form:

Eq. (9)

\frac{1}{2} | A^{″} | \cdot Δ μ_{a} ≪ A^{'},

where Δ μ_a is the variance of the tissue absorption coefficient in the spectral range under investigation.

The expansion coefficient A^′ (before a linear term) has the sense of the average photon path length through the tissue 〈L〉. It easily can be observed by calculating the derivative of A over μ_a from Eq. (2)

Eq. (10)

〈 L 〉 = {\frac{\partial A}{\partial μ_{a}} |}_{μ_{a} = μ_{a}^{0}} = \frac{{\int_{0}^{\infty} l \cdot P (μ_{s}, g, l) \cdot exp (- μ_{a}^{0} l) d l}^{​}}{{\int_{0}^{\infty} P (μ_{s}, g, l) \cdot exp (- μ_{a}^{0} l) d l}^{​}} .

It is clear from this relation that the 〈L〉 is the averaged photon path length through the tissue for the distribution function P(μ_s,g,l)exp(−μ_al). The angle brackets denote further the averaging with the distribution function given above. It should be noted that with the use of 〈L〉 from Eq. (10) relation (8) in the linear approximation is similar to the modified Beer–Lambert law

Eq. (11)

A = A_{0} + 〈 L 〉 \cdot μ_{a},

which is applied for evaluating the changes in μ_a and calculating the relative concentrations of oxygenated and deoxygenated cerebral hemoglobin from the NIR spectral range. The difference here is that in our approach we consider 〈L〉 as a Taylor coefficient which is not dependent on μ_a, and the following correction is not intended to model the dependence of 〈L〉 from μ_a as it is done in the modified Beer–Lambert law but to take into account the next (quadratic) term in Taylor series expansion (8). The unknown contribution of scattering A₀ in the modified Beer–Lambert law is compensated by analyzing the changes in attenuation ΔA over time relative to an arbitrary initial moment so that one could get the corresponding changes in time of the absorption coefficient Δ μ_a. In our approach we analyze the spectral dependence of complete attenuation A (to within a constant wavelength term), and the absolute blood oxygen saturation may by extracted from a single spectral curve.

It can also be shown that the second derivative of attenuation over μ_a can be expressed as

Eq. (12)

A^{″} = - (〈 L^{2} 〉 - {〈 L 〉}^{2}) = - 〈 (L - {〈 L 〉)}^{2} 〉 .

Thus, it can be said that application of Taylor series expansion means the approximation of the path length distribution function by its moments. The first moment is a mean value and the second moment is a dispersion of a random variable. The condition for applicability of the linear approximation (9) with allowance for Eq. (12) takes the form

Eq. (13)

(〈 L^{2} 〉 - {〈 L 〉}^{2}) \cdot Δ μ_{a} ≪ 〈 L 〉 .

Intuition suggests that the applicability of the liner approximation is restricted by the region, where the changes of absorption coefficient Δμ_a are small as compared to the averaged value μ_a ⁰ . It is also clear that this criterion is dependent on the path length distribution function, the narrower distribution function the more favorable this criterion. In the limiting case of a super narrow path length distribution function which may be approximated by Dirak delta function P(l)=δ(l−L) the conditions of the criterion (13) will be adequate everywhere as its left side equals to zero. If we take a more realistic case of the path length distribution function with narrow but finite dispersion ΔL approximated by Gaussian distribution P(l)=exp{−[(l−〈L〉)/2ΔL]²}/ΔL(2π)^1/2 so that (ΔL/〈L〉)≪1, the criterion (13) is reduced to the following relation:

Eq. (14a)

(Δ μ_{a} \cdot 〈 L 〉) \cdot {(\frac{Δ L}{〈 L 〉})}^{2} ≪ 1 .

The dimensionless term Δ μ_a〈L〉 has the sense of attenuation changes in the spectral range under analysis. This value typically is less or of the order of unity. For example, the estimation of attenuation changes in human skin in vivo in the visible range for our fiber probe geometry (see Figure 4) gives the value of Δ μ_a〈L〉 in the range between 0.4 and 1. Analyzing relation (14a) one can conclude that for narrow path length distribution functions ((ΔL/〈L〉)≪1) the applicability of the linear approximation is rather favorable and may extend even beyond the small variances in the absorption coefficient. In the opposite case of a broad path length distribution function the condition (13) will not be so favorable. For example, just to evaluate the most unfavorable case we can take a general path length distribution function in the form P(l)=α²l⋅exp(−αl) , which has a rather broad maximum (comparable with the averaged path length) at l=α⁻¹ where α is a parameter. The averaged photon path length for this distribution function calculated from Eq. (10) is expressed as 〈L〉=2/(α+μ_a ⁰ ). We can easily obtain from (13) the following very simple criterion for applicability of the linear approximation with this path length distribution function:

Eq. (14b)

\frac{1}{4} (Δ μ_{a} \cdot 〈 L 〉) ≪ 1 .

Analyzing this relation one can conclude that the linear approximation should work fairly well even for cases with the broad path length distribution function if the changes in attenuation are less or the order of unity. The analytical form of the criterion for applicability of the linear approximation can also be calculated for the case of distantly separated fibers when the diffusion theory is valid. Substituting the path length distribution function obtained from the diffusion approximation for the distantly separated fibers [Eq. (4)] when calculating the averaged values in relation (13) we obtain

Eq. (14c)

\frac{Δ μ_{a}}{μ_{a}^{0}} ≪ 4 \cdot (1 + \frac{δ}{d}) .

We observe that for this particular case the applicability of the linear approximation is also valid for high absorption coefficient variances (Δμ_a≈μ_a ⁰ ) through the spectral range under investigation when δ ≫ d (weak absorption limit). Thus, the linear approximation is valid for small variances of the absorption coefficient and may extend even over this range for narrow path length distribution functions.

Figure 4

Attenuation spectra of a finger during occlusion at different times (thick gray) and fitted spectra, A _model, (thin black) in the range between 510 and 590 nm. The corresponding blood oxygen saturation SO ₂ calculated from the model is shown in the figure.

We assume further that the main chromophore in the tissue contributing to the strongly wavelength variable absorption is hemoglobin in oxygenated and deoxygenated form, while the contribution of other absorbers can be approximated by a smooth wavelength function. Thus, the tissue absorption coefficient is modeled as

Eq. (15)

μ_{a} (λ) = μ_{a}^{b} (λ) + {c_{Hb} \cdot ɛ_{Hb} (λ) + c_{{HbO}_{2}} \cdot ε_{{HbO}_{2}} (λ)} \cdot ln (10),

where μ_a ^b(λ) is the contribution of background to the tissue absorption coefficient (melanin, etc.), ɛ and c are the extinction coefficients and concentrations, respectively, of deoxygenated (Hb) and oxygenated ( HbO ₂ ) hemoglobin. The background contribution should be a smooth function of wavelength. It is also possible to take into account other tissue absorbers in Eq. (15) in the way similar to hemoglobin. For example, for analysis in the NIR wavelength range (between 700 and 1100 nm) it is necessary to add in Eq. (15) the extinction coefficients of fat and water²⁰ ³⁷ or the bilirubin extinction for analysis of bilirubin content in skin.³¹ For further analysis we will be restricted only to hemoglobin content calculation. It is acceptable for the wavelength range between 510 and 590 nm which is used for analysis. Taking into account relations (8) and (15) and approximating the background absorption and scattering contribution by a smooth (polynomial) function of wavelength we arrive at the following simple equation for attenuation:

Eq. (16)

\begin{matrix} A_{model} (λ) & = & ln (\frac{I_{0}}{I}) \\ = & c_{0} + c_{1} \cdot λ + c_{2} \cdot λ^{2} + 〈 L 〉 \\ \cdot {c_{Hb} \cdot ɛ_{Hb} (λ) + c_{{HbO}_{2}} \cdot ε_{{HbO}_{2}} (λ)} \cdot ln (10), \end{matrix}

where c₀ , c₁, and c₂ are coefficients taking into account the contributions of background absorption and the scattering as well as the constant value of μ_a ⁰〈L〉 . The c₀ coefficient also takes into account the uncertain constant additive entering into the experimental attenuation curve [see Eq. (20) argumentation to this]. The values of these coefficients as well as the values of 〈L〉c_Hb and 〈L〉c_HbO
₂ are obtained by minimizing the difference between fitted and experimental spectra with the least square method:

Eq. (17)

χ^{2} = \sum_{λ_{i} = λ_{min}}^{λ_{i} = λ_{max}} {[A_{model} (λ_{i}) - A_{exp} (λ_{i})]}^{2} = min,

where λ_i are the N wavelength values at which the attenuation is measured in the range between λ_min and λ_max. The optimal wavelength range between λ_min and λ_max used for analysis is chosen from the following considerations. The spectra of Hb and HbO ₂ should differ substantially in this spectral range, whereas the contribution of other tissue absorbers and the scattering are described by smooth functions of wavelength. Thus, the wider spectral range used for analysis the greater difference attained between Hb and HbO ₂ spectral curves resulting in error decreasing. However, the wavelength dependent contributions of scattering and other tissue absorbers are more substantial in the wider spectral range resulting in error increasing. The applicability of linear approximation [Eqs. (14a)–(14c)] is also less favorable in the wide spectral range. We found out that the spectral range between 510 nm and 590 nm is the “golden mean” for most cases of blood oxygenation calculations. The parameter c₂ in Eq. (16) can be omitted from the fitting in such a narrow spectral range.

The blood oxygen saturation degree SO ₂ and relative hemoglobin concentration HbT are calculated from the above determined coefficients as follows:

Eq. (18)

\begin{array}{c} {SO}_{2} = \frac{〈 L 〉 c_{{HbO}_{2}}}{〈 L 〉 c_{{HbO}_{2}} + 〈 L 〉 c_{Hb}} = \frac{c_{{HbO}_{2}}}{c_{{HbO}_{2}} + c_{Hb}}, \\ HbT = 〈 L 〉 \cdot (c_{Hb} + c_{{HbO}_{2}}) . \end{array}

The relative hemoglobin concentration values depend on measurement geometry (through the 〈L〉 value) and so only the relative changes taken during the measurements at the same geometry are meaningful. It should also be mentioned that for in vivo measurements the movement of the object (change of the distance between the fiber probe and the tissue for noncontact measurements) may result in the small random changes in 〈L〉 and, consequently, the errors in relative hemoglobin concentration monitoring, while the oxygen saturation value is less receptive to these changes.

The accuracy of the model may be estimated by the difference between fitted and experimental attenuation spectra determined by the following relation:

Eq. (19)

Error = \frac{\sqrt{χ^{2} / N}}{\frac{1}{N} \cdot \sum_{λ_{i} = λ min}^{λ_{i} = λ_{max}} 〈 L 〉 \cdot {c_{Hb} \cdot ε_{Hb} (λ_{i}) + c_{HbO 2} \cdot ε_{{HbO}_{2}} (λ_{i})} \cdot ln (10)} \cdot 100 %,

where the χ² is defined by Eq.(17) and N is the number of points in the spectrum range between λ_min and λ_max. When the error value is over 50 the SO ₂ and HbT calculation results should be considered as invalid. It may take place when the blood content is very low in the tissue so that its contribution to the absorption is compared to that of noise. It may also take place in the presence of other tissue absorbers which are not taken into account by Eq. (15) and when the influence of the scattering cannot be approximated by a smooth wavelength function in the spectral range under consideration. The discrepancy between the experiment and the model may result also from inapplicability of linear approximation (13) due to high variations of μ_a in the analyzed spectral range. The allowance for quadratic term in Eq. (8) may improve the match between experimental and fitted data in this case.

4. Experimental Technique

For the measurements of diffuse reflectance spectra of tissue in vivo and blood samples we applied a fiberoptic spectrometer LESA-5 (Biospec, Russia). An experimental setup used for in vivo measurements is shown in Figure 3. Light from a stabilized light source (halogen lamp) was delivered to the tissue by means of a quartz fiber. After passing through the tissue the light was collected by a receiving fiber. The output end of the receiving fiber was the input slit of the spectrometer. We applied the spectrometer with a fixed grating and a charge coupled device (CCD)-type detector. The total spectrum range (between 400 and 1000 nm) can be acquired for 0.1 s. The spectrometer was controlled by a personal computer. A special program operating in the MS Windows environment allows one to acquire and display the spectra and calculate the oxygen hemoglobin saturation and the relative hemoglobin concentration in real time using the above described algorithm. For calculation we exploited only a part of the spectral range between 510 and 590 nm acquired by the spectrometer.

Figure 3

Experimental setup for measurements of diffuse reflectance spectra in vivo.

For in vivo measurements we used a fiber configuration shown in Figure 2. The distance d between delivery and receiving fibers with core diameters of 200 μm and NA 0.22 was set to 2 mm. During the measurements the fibers were fixed at a distance less than half a millimeter from the tissue. This configuration allowed only diffuse reflected photons to enter into the receiving fiber. There was no contact between fibers and tissue to avoid the influence of fiber pressure on blood circulation.

To demonstrate that Taylor expansion algorithm is applicable to different geometries of the measurement resulting in different path length distribution functions, P(l), of photons through a sample, we present here the results of measuring the hemoglobin oxygen saturation in blood samples during photochemical deoxygenation. The fiber configuration used for the blood samples has quite a different geometry and correspondingly another path length distribution function from that of the tissue. To prepare a blood sample some fresh blood with added heparin to avoid clotting was incubated with sulphonated aluminum phthalocyanine as photosensitizer in concentrations of 1–20 μM for 30 min. To saturate the blood with oxygen an oxygen flow passed through the blood during incubation to attain the initial hemoglobin oxygen saturation level in the range of 80–85. A blood drop was placed between two glasses. The glass thickness was 1.2 mm. The thickness of the blood layer was controlled with the use of a special film and was in the range of d=125±5 μm. The diameter of a sample was 1.5 cm. The delivering and receiving fibers were placed at the opposite sides of a blood sample with the receiving fiber positioned at a small angle to the normal φ≈10° to exclude collimated transmitted photons. Thus, in this case we actually measured the spectral diffuse transmittance of the sample. Further, these samples were irradiated with laser light at fluence rates 10–1200 mW/cm ² at 670 nm incident close to normal to the glass surface. The laser irradiation was absorbed by photosensitizer present in blood and induced photochemical oxygen consumption resulting in slow decrease of the hemoglobin oxygen saturation.

The experimentally measured spectral dependence of the light attenuation is evaluated with the use of the following relation:

Eq. (20)

A_{exp} (λ) = ln (\frac{I_{ref} - I_{dark}}{I - I_{dark}}),

where I_ref is the measured reflectance signal from the reference sample (BaSO ₄ ), which has the uniform diffuse reflectance near unity in the spectral range under consideration (400–1000 nm), I_dark is the signal in the absence of any light (dark current of CCD) and I is the diffuse reflectance signal from the tissue or diffuse transmittance in the case of the blood sample. Equation (20) accounts for the spectral nonuniformity of the light source, the fiber transmission and the detector sensitivity. It should be noted that A_exp(λ) as given by Eq. (20) is defined to within some constant value as compared to that of defined by Eq. (1): A_exp(λ)=A(λ) +const. This constant would be equal to zero if we can substitute the signal coming out from the delivering fiber in Eq. (20) instead of I_ref. It should be noted that I_ref is measured with a fiber probe placed at the distance of about 1–2 cm from the reference sample surface as the I_ref signal should approximate the wavelength dependence of photon flux coming out of the delivering fiber and wavelength dependent contribution of the scattering should be minimized. The reference signal I_ref measured as described above differs from I₀ [see Eq. (1)] by some wavelength independent constant factor resulting in appearance of constant additive in the experimental attenuation curve. However, the exact value of the constant in A_exp(λ) is not important for our algorithm as only its wavelength dependence matters. This additive constant is compensated by fitted c₀ coefficient in Eq. (16).

5. Results and Discussion

To demonstrate the applicability of the above described algorithm we present here the results of deoxygenation observed during the artery occlusion. The occlusion was done by tightly wrapping a rubber tape around the base of a finger, while the measurements were done on the outer surface of an upper finger phalanx. We also did the artery occlusion by placing a pneumatic pressure cuff around an upper arm and applying pressure up to 200 mmHg (data are not shown) but found no substantial difference in deoxygenation dynamics in these two methods. Figure 4 demonstrates the attenuation spectra A_exp(λ) taken from the finger of the author (A.A.S.) at different times after starting the occlusion. The spectra are shifted along the y axis on arbitrary values for observation convenience, the amount of the shift is not important for calculations. We notice that the spectral line shapes strongly correlate with the blood oxygen saturation. In the range between 510 and 590 nm the modeled spectra calculated using Eq. (16) with coefficient c₂ responsible for quadratic contribution fixed to zero are shown in Figure 4 by black thin lines and the corresponding blood oxygen saturation values calculated from the model are given. The closeness of the experimental A_exp(λ) and fitted A_model(λ) spectral curves also suggests the applicability of our model in this case. For this measurement configuration the use of the spectral wavelength range between 510 and 590 nm is most suitable as the spectral features of hemoglobin are most pronounced for this range as can be seen from Figure 4. Anyway, the disadvantage of this spectral range is the small light penetration depth. The sampling depth of our measurements is the fraction of millimeter corresponding to the signal originated from skin capillary loops.⁹ By close analysis of spectra shown in Figure 4 one can also notice the difference in the NIR spectral range (the appearance of a small peak at 755 nm for deoxygenated tissue) resulting from different blood oxygen saturation. However, the spectral differences here are not so pronounced, and the error in evaluation of the blood oxygen saturation in this spectral range from the above described model will be much greater. We tried to interpret the data by Taylor expansion method using the spectral range between 650 and 800 nm. As mentioned above, the deoxygenated hemoglobin has pronounced spectral feature (peak at 755 nm), but spectral dependence of the oxygenated hemoglobin is rather smooth resulting in possible cross talking between smooth background contribution, influence of the scattering [see Eq. (16)] and the contribution of the oxygenated hemoglobin. In this spectral range water contribution should also be taken into account. It should be noted that the modeled attenuation curves fit fairly well into the experimental curves for the NIR range, but the blood oxygen saturation evaluated from this fitting was higher than expected. For example, before starting the occlusion SO ₂ values calculated from NIR range alone (from 650 to 800 nm) were as high as 85–100. The reason for this disagreement is that the smooth contribution of the oxygenated hemoglobin in NIR may be partly compensated by a linear function of the wavelength responsible for scattering and background absorption (the above mentioned cross talking). The expansion of the spectral range to include both visible and NIR in relation (17) also resulted in an increase of the error. It is because of poor applicability of the linear approximation when the variances of the absorption coefficient are rather great [Eqs. (13) and (14)] and insufficient approximation of the background absorption and the scattering contribution by the linear function of the wavelength. One can also notice in Figure 4 the slight tendency to deviation between the experimental attenuation and the modeled curves near the interval boundary at 590 nm. If we increase the wavelength range even just to 600 nm when fitting the modeled curve by Eq. (17), this deviation would be greater. Thus, we think that with the use of the Taylor expansion method the spectral range from 510 up to 590 nm is optimal for evaluating the blood oxygen saturation and the hemoglobin relative concentration.

We can easily evaluate from Figure 4 that the measured attenuation changes Δ μ_a 〈L〉 in the spectral range under analysis are 0.4 and 1.0 for deoxygenated and oxygenated case, correspondingly. Substituting these values into criterion (14b) obtained for broad path length distribution function we get evidence that linear approximation is applied here though with small reserve. If we expand the analyzed spectral range even up to 650 nm, the attenuation changes would increase up to about 2.5 (see Figure 4) resulting to poor applicability of criterion (14b) in this case and giving unreasonable values for blood oxygen saturation as was mentioned above. Another estimation may be received if we assume that criterion (14c) obtained with the use of diffusion theory for distantly separated fibers is valid for our fiber probe configuration. Using the data of hemoglobin extinction spectra, one can make sure that its variances in the absorption (Δ μ_a/μ_a ⁰ ) in the spectral range from 510 to 590 nm are 0.4 and 0.9 for deoxygenated and oxygenated hemoglobin, correspondingly. Assuming further that averaged light penetration depth δ in this spectral range is about 1 mm, and substituting it in criterion (14c), we obtain that the linear approximation is valid for Δ μ_a/μ_a ⁰ ≪6 that is also fulfilled in our case.

Figure 5 demonstrates the monitoring of typical time behavior for the blood oxygen saturation and the relative hemoglobin concentration during the artery occlusion calculated from our model. The finger occlusion was started at t=0 s and released at t=700 s. Under normal conditions (before starting occlusion) the irregular oscillations in SO ₂ in the range of 50–70 are observed (time interval between −600 and 0 s). These oscillations are not random errors but real facts mainly due to the spatial micro nonuniformity of SO ₂ in the skin layer and hand movements during the measurements. The value of the blood oxygen saturation in skin at rest (50–70) at the normal ambient temperature (20–22°C) agrees fairly well with that reported by others.³⁸ ³⁹ ⁴⁰ It should be noted that the measurements done at lower ambient temperatures gave the values of the blood oxygen saturation in skin at rest as low as 15–30 (data are not shown). As may be seen from Figure 5, total hemoglobin deoxygenating in a superficial skin layer is attained 10 min after starting the occlusion. The closeness of SO ₂ value to zero at the end of the occlusion also demonstrates the accuracy of the model. After occlusion release we observe a sharp increase for both blood oxygen saturation and relative hemoglobin concentration due to the intensive blood circulation (reactive hyperemia), which after some time are restored to their previous values. The similar results have been also observed by other researchers.³⁸ ³⁹ For further examples employing this algorithm for in vivo measurements the reader is refereed to Ref. 41.

Figure 5

Time dependence of blood oxygen saturation ( SO ₂ ) and total relative hemoglobin concentration (HbT) calculated from the model during finger occlusion. The occlusion was started at t=0 s and released at t=700 s.

The use of Taylor expansion algorithm for calculating oxygenation in blood samples should be more favorable than for tissues because path length distribution function for blood sample geometry is rather narrow. Really, when passing through the sample, most photons change their initial direction just slightly (single scattering) as the thickness of the blood sample is much less than the reduced photon scattering length (inverse of reduced scattering coefficient). It means that for this configuration the photon path length distribution function has a rather sharp peak at the path length value 〈L〉 a little bit higher than the sample thickness d. As the angle φ between receiving fiber and normal to the blood sample was rather small (about 10°), the broadness ΔL of path length distribution function is just about d(sinφ)²≈0.03⋅d as roughly estimated from single scattering assumption. Substituting this value of ΔL into criterion (14a) obtained for narrow path length distribution functions and evaluating the attenuation changes for blood sample Δμ_a〈L〉≈2.5 we can easily calculate that Δ μ_a〈L〉⋅(ΔL/〈L〉)²≈0.002≪1. Thus, the criterion of applicability of linear approximation is fulfilled here with great reserve, resulting in reasonable and reliable values of blood oxygen saturation obtained in this case.

Additional reduction in noise is also achieved due to absence of subject movement during the measurements. Here we present the observation of oxygen saturation changes in a blood sample incubated with a photosensitizer under the influence of laser irradiation [photodynamic therapy (PDT)].⁴¹ ⁴² ⁴³ This model may be very useful for evaluation of oxygen consumption rate during PDT. Oxygen is consumed in photochemical reactions during PDT and steady decrease in blood oxygen saturation should be observed. The attenuation spectra taken from the blood sample during laser irradiation are shown in Figure 6. The corresponding time behavior of the blood oxygen saturation ( SO ₂ ) and the relative hemoglobin concentration (HbT) calculated with the use of the above described model are shown in Figure 7. A smooth decrease of the oxygen saturation to zero value and the constant relative hemoglobin concentration are observed. As expected, the random noise is extremely low for these measurements. We should be reminded that the fiber configurations for the tissue measurements (Figure 5) and the blood sample measurements (Figure 7) were quite different (in the second case the delivery and receiving fibers were positioned at the opposite sides of a blood sample). Nevertheless, the proposed algorithm may be successfully applied to both cases giving accurate results.

Figure 6

Attenuation spectra of a blood sample incubated with photosensitizer in concentration 0.02 mg/ml during photodynamic therapy with laser power density of 25 mW/cm ² at 670 nm. The time interval between spectra is 20 s. The transformation of hemoglobin from oxygenated to deoxygenated form is clearly observed.

Figure 7

Time dependence of blood oxygen saturation ( SO ₂ ) and total relative hemoglobin concentration (HbT) in a blood sample during PDT. The values are calculated from the spectra shown in Figure 6.

6. Conclusion

A simple algorithm for evaluation of blood oxygen saturation and relative hemoglobin concentration was obtained here by the consistent approach of Taylor expansion of exact relation for light attenuation. The advantage of this approach is that it results in a simple, robust, not very sensitive to noise algorithm for the blood oxygen saturation, which can be implemented for calculations in real time. The algorithm may be successfully applied to the visible wavelength range for different measurement configurations as confirmed by experimental data presented in the paper. Work is in progress to apply this algorithm investigating the influence of temperatures and laser irradiation on blood oxygenation in human skin⁴⁴ and adapt this approach to the near infrared range to increase the sampling depth of the measurements.

Acknowledgments

The authors would like to thank Professor G. L. Kiselev, Dr. A. Y. Douplik, and Dr. G. A. Meerovich for fruitful discussions and valuable remarks concerning this work.

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Notes

Address all correspondence to Dr. Alexander A. Stratonnikov. Tel/Fax: 095-324-6363; E-mail: biospec@online.ru

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Alexander A. Stratonnikov and Victor B. Loschenov "Evaluation of blood oxygen saturation in vivo from diffuse reflectance spectra," Journal of Biomedical Optics 6(4), (1 October 2001). https://doi.org/10.1117/1.1411979

Published: 1 October 2001

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KEYWORDS

Absorption

Tissues

Blood oxygen saturation

Scattering

Signal attenuation

Blood

Tissue optics

1.

Introduction

2.

Diffuse Reflectance and Photon Path Length Distribution Function