Quantitative interpretations of Visible-NIR reflectance spectra of blood

Yulia M. Serebrennikova; Jennifer M. Smith; Debra E. Huffman; German F. Leparc; Luis H. García-Rubio

doi:10.1364/OE.16.018215

1. Introduction

A new method for detection of microorganisms in blood culture was recently proposed [1]. The method is founded on the following principles: (1) metabolic activity of microorganisms induces changes in the physical and chemical properties of blood and (2) due to distinct optical properties of blood these changes can be captured with diffuse reflectance spectroscopy [1]. In this publication we present a new model implemented with the proposed method designed for quantitative extraction of blood properties from the diffuse reflectance measurements of blood culture systems. This new model is complete in terms of robust solutions of diffusion and scattering problems and their numerical implementation.

The use of spectroscopy to measure the state of blood and blood-infused tissues has long been appealing for biomedical applications both in vitro and in vivo due to, firstly, the distinctive optical signature of hemoglobin and, secondly, the simple, non-invasive, and nondestructive nature of the measurements [2–6]. Yet, since blood is a complex fluid, a multitude of blood parameters of both physical (size, shape, and concentration of erythrocytes and other cells (i.e., leukocytes, platelets, etc.)), and chemical (i.e., concentrations of hemoglobin, bilirubin, myoglobin, cytochromes, nucleic acids, etc.) nature and system parameters (media, temperature, pH, etc.) affect the measurements and, thus, quantitative analysis of spectroscopic measurements has been challenging [7–9].

Numerous approaches (statistical, empirical, numerical, and theoretical) have been undertaken for quantitative analysis of measured spectra. Examples of statistical approaches would be multivariate regression models, i.e. partial least squares or principal component analysis, that employ calibration curves relating the measured spectral amplitudes at characteristic wavelengths or some portions of the spectra to the selected blood parameters [2,3,5,7,9]. Empirical models use arbitrarily derived scaling factors, for example, ratios of spectral amplitudes measured at different wavelengths and/or optical configurations [2,10], to fit desired theoretical approximations [6]. Although these techniques are relatively easy and computationally fast, they are limited by the approximation used, typically require extensive calibration and standardization, and often show poor inter-donor consistency.

Numerical approaches to the solution of Maxwell equations for spectroscopic measurements of blood and tissues such as Monte-Carlo simulations, finite boundary element method, discrete dipole approximation, T-matrix, etc. [6,11–14], have been extensively used and have the advantage of providing rigorous solutions that involve no major approximations [6]. However, cumbersome and time-consuming computations are required [5,6,12].

Of the theoretical methods, the photon diffusion models have been known to provide an accurate description of the optical measurements of blood [2,15]. Unfortunately, since in the past the rigorous solution to the photon diffusion problem was considered to be mathematically complex [6], the simplifying empirical approximations were generally used [4,6,16–20].

To successfully apply the theory, accurate macroscopic scattering and absorption optical properties are needed. Since these optical properties are the functions of multiple blood parameters, their direct extraction from the measurements, for example, with inverse Monte- Carlo simulations [12,21], have produced a wide range of values [16,21–23]. For the same reason simplified empirical approximations for scattering [4,6,18–20] and/or neglecting the effect of the particle size on absorption [19–22] may lead to erroneous results [2].

The macroscopic optical properties of blood can be computed with a choice of scattering theory, i.e., Lorentz-Mie, Rayleigh-Gans, Fraunhofer, or anomalous diffraction [23–26], or a numerical technique [11,14,26–30]. The major advantage of the numerical techniques is the capability of rigorous computation of optical properties of realistic non-spherical particle shapes such as the biconcave disk of erythrocytes [28]; yet due to computational complexities they are not yet applicable for the routine analysis of blood samples. Of the scattering theories, Mie theory provides an exact solution for the spherical particle of the size range of blood components [15,24,31]. Since the general solution to Mie theory is only for spheres, its applicability to non-spherical particles, such as erythrocytes, has been criticized [14,28,32–34]. However, it can be successfully used for non-spherical randomly oriented cells, as those in agitated blood cultures, when size rather than shape determines light scattering [11,27].

The approach to the deconvolution, i.e. quantitative analysis, of diffuse reflectance measurements in terms of optically relevant blood parameters presented in this paper is based on the thorough wavelength dependent solutions to the photon diffusion problem and Mie scattering theory. We show that the accurate implementation of these theories succeeds in the extraction of realistic estimates of the parameters from the diffuse reflectance spectra of blood cultures. Computations are nearly instantaneous enabling the possibility of real-time continuous monitoring applications. Additionally, our approach requires only the knowledge of the refractive indices of the basic constituents of the samples and, therefore, no calibration or standardization is needed as compared to the empirical and statistical methods.

2. Method

2.1 Measurement

Measurements of diffuse reflectance were conducted with a USB-4000 spectrometer characterized by linear CCD detector, signal to noise ratio of 300:1, and 50 µm aperture, equipped with either standard or angled reflectance QR400-Vis-NIR probe each consisting of one 0.2 mm radius read fiber surrounded by six 0.2 mm radius illumination fibers. The probes’ source-detector distance was 0.4 mm and numerical aperture was 0.22. LS-1 or HL-2000 Tungsten Halogen lamps with dynamic range of 360-2000 nm were used as isotropic light sources. The spectrometer, probes, and light sources were purchased from Ocean Optics Inc. (Dunedin, FL). The optical readings were taken by placing the surface of a reflectance probe at 45° angle to the surface of a blood culture vial. Spectra were acquired with the integration time of 150 msec. The spectral resolution of the raw data was 0.33 nm and was interpolated to 1 nm for data analysis. For the measurements reported in this paper we used standard plastic BacT/ALERT blood culture vials [35] purchased from bioMerieux Inc. (Hazelwood, MO) containing 40 ml of growth media. Approximately 8–10 ml of blood was added for the experiments. Blood samples from healthy individuals were provided by Florida Blood Services (St. Petersburg, FL). Some measured spectra presented in this paper were recorded from blood cultures contaminated with microorganisms [36]. In all cases blood cultures were incubated at 37°C. The vials were continuously agitated to ensure that the solutions were well mixed. The diffuse reflectance R(λ) was calculated by scaling the measured reflected intensity of a sample I_sample to that of MgO₂ (assuming it to be 100%) and subtracting the dark intensity I_dark measured when there was no light according to Eq. (1).

R (λ) = \frac{I_{sample} - I_{dark}}{I_{{MgO}_{2}} - I_{dark}}

2.2 Photon diffusion theory

The described measurement configuration satisfied the assumptions under which the photon diffusion theory provides a robust mathematical description of measured diffuse reflectance. These assumptions are that, first, light is scattered multiple times, second, the scattering is larger than absorption, and, third, the depth of the measured sample is larger than the mean photon free path [2,15,16,37,38].

The detailed derivation of the solution to photon diffusion problem has been previously published [15,37]. The resulting solution expresses the reflected intensity R(λ) as a non-linear function of the measurement geometry (detector aperture r_x, incident beam radius a, sourcedetector separation distance b, and depth of the sample d) and optical properties (macroscopic reduced scattering µ_st(λ) and absorption µ_a(λ) cross-sections) of a measured sample:

R (λ) = \frac{r_{x}^{2}}{r_{2}^{2} - r_{1}^{2}} (R_{diff} (r_{2}) - R_{diff} (r_{1}))

where diffuse reflectance intensity of arbitrary radius r, R_diff, is defined as

R_{diff} (r) = \frac{2 μ_{st} (λ)}{a^{2}} \sum_{n = 1}^{\infty} \frac{k_{n} z_{n} \cos γ_{n}}{N_{n} ζ_{n}^{2} (k_{n}^{2} + μ_{t}^{2} (λ))} {\begin{matrix} (\frac{a^{2}}{2}) - ra I_{1} (ζ_{n} a) K_{1} (ζ_{n} r); r > a \\ (\frac{r^{2}}{2}) - ra K_{1} (ζ_{n} a) \cdot I_{1} (ζ_{n} r); r < a \end{matrix}}

The variables of the Eqs. (2) and (3) are defined in Table 1.

2.3 Formulation of the optical properties

The macroscopic absorption µ_a(λ) and scattering µ_s(λ) cross-sections describe the combined optical properties of multiple scatters and chromophores present in a given sample. The first term of Eq. (4) defining total macroscopic absorption cross-section of the sample describes its dependence on single-particle absorption cross-sections C_abs and particle number density N(p) of N populations of particles and the second term incorporates the absorption contributions from M chromophoric compounds with [C_i] being concentration of i^th compound dissolved in the media of the sample. In Eq. (5) the total macroscopic scattering cross-section is a function of single-particle scattering cross-sections C_sca, particle number density N(p) and the mean volume V(p) of the J^th particle population [25,39]. We use the multi-scattering approximation suggested by [39] in a form of non-linear dependence of the macroscopic scattering cross-section on the particle concentration. The scattering component enters the reflectance calculations (Eq.(3)) through the reduced macroscopic scattering cross-section µ_st(λ), which results from the correction of the scattering cross-section µ_s(λ) for anisotropy of the scattering <µ> as shown in Eq. (6).

μ_{a} (λ) = \sum_{J = 1}^{N} N_{{(p)}^{J}} C_{absJ} + \sum_{i = 1}^{M} \frac{4 π k_{i} (λ)}{λ} [C_{i}]

μ_{s} (λ) = \sum_{J = 1}^{N} N_{{(p)}^{J}} (1 - N_{{(p)}^{J}} V_{{(p)}^{J}}) C_{scaJ}

μ_{st} (λ) = \sum_{J = 1}^{N} N_{{(p)}^{J}} (1 - N_{{(P)}^{J}} V_{{(P)}^{J}}) C_{scaJ} (1 - < μ_{J} >)

Table 1. Definition of parameters used in Eqs. (2 and 3).

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The single-particle cross-sections, C_abs and C_sca, calculated for each particle population were products of the corresponding absorption Q_abs and scattering Q_sca efficiencies and average scattering area S of a given particle population (Eq. (7)). The efficiencies were computed as shown in Eqs. (8-10) in terms of the particle size parameter α and the particle relative refractive index m(λ) implicit in the coefficients a_n and b_n, the Mie theory formulas for which can be found in [40]. The asymmetry factor <µ> as formulated in Eq. (11) was also a function of particle size and relative refractive index. The scattering area and size parameter of each particle population were calculated under the assumption of homogeneous spherical scatters characterized by mean volume V(p) (Eq. (12)).

C_{abs} = Q_{abs} (m (λ), V_{(p)}) \cdot S C_{sca} = Q_{sca} (m (λ), V_{(p)}) \cdot S

Q_{sca} = \frac{2}{α^{2}} \sum_{n = 1}^{\infty} (2 n + 1) {{∣ a_{n} ∣}^{2} + {∣ b_{n} ∣}^{2}}

Q_{ext} = \frac{2}{α^{2}} \sum_{n = 1}^{\infty} (2 n + 1) {Re (a_{n} + b_{n})}

Q_{abs} = Q_{ext} - Q_{sca}

< μ > = \frac{4}{α^{2} \cdot Q_{sca}} \sum_{n = 2}^{\infty} [\frac{n^{2} - 1}{n} {Re (a_{n - 1} {\bar{a}}_{n} + b_{n - 1} {\bar{b}}_{n})} + \frac{2 n - 1}{n (n - 1)} {Re (a_{n - 1} {\bar{b}}_{n - 1})}]

S = \frac{{(6 π^{2} V_{(p)})}^{\frac{2}{3}}}{4} α = \frac{{(6 π^{2} V_{(p)})}^{\frac{1}{3}}}{λ}

The Mie scattering algorithm used was constructed following the general computer code of [41] and includes the refractive index increment computations [42]. The computer code has been extensively tested and compared against well-tested Mie codes available [43,44] over the relevant wavelength, size, and optical property sets of interest. The algorithm has been evaluated previously through quantitative analysis of transmission measurements of biological particulates such as blood components and a variety of microorganisms [45, 46].

The effect of multi-component chemical composition for each particle population was introduced through the additive properties of the complex refractive index m(λ) [46-48]. For a particle consisting of M components the real n(λ) and imaginary k(λ) parts of the particle complex refractive index were calculated as weighted sums of the contributions ω_i of each component [46–48]:

m (λ) = \frac{n (λ) + ik (λ)}{n_{0} (λ)} n (λ) = \sum_{i = 1}^{M} ω_{i} n_{i} (λ) k (λ) = \sum_{i = 1}^{M} ω_{i} k_{i} (λ)

For example, for erythrocytes three chromophoric components (oxy-, deoxy-, and methemoglobin) were considered (Fig. 1), for each the weighted contribution ω_i was a product of its relative fraction f_i of the total hemoglobin and hemoglobin concentration within erythrocytes [Hb]_(e). Distinct characteristic spectral signatures of these forms of hemoglobin can be appreciated with Fig. 1. The refractive indices of water n₀(λ) and hemoglobin forms were obtained from the published data [49,50].

Fig. 1. The imaginary parts, k(λ), of refractive indices of oxyhemoglobin (HbO₂), deoxyhemoglobin (Hb), and methemoglobin (HbFe(III)).

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2.4 Analysis of data

The combined solution to photon diffusion approximation and Mie theory constituted the fundamental model that was applied to the diffuse reflectance measurements. With this model the diffuse reflectance spectra were deconvoluted in terms of optically relevant blood culture parameters. In the context of this paper deconvolution (or mathematical interpretation) implies an algorithm-based reconstruction of the measured signal with parameterized mathematical functions by which the values of the parameters are estimated. In practical terms the theoretically predicted spectra were computed as functions of blood culture parameters and then fitted to the corresponding measured spectra in the 500–850 nm wavelength range in the least squares sense. A Nelder-Meade downhill simplex optimization algorithm was used for this purpose [51,52].

The optically relevant blood culture parameters included number density N(p) of particle populations present (erythrocytes and other cells), mean particle volume V(p), the products of these parameters being the volume fraction V_f(p) of a given particle population. Also, total amount of hemoglobin [Hb]_T, hemoglobin concentration within erythrocytes [Hb]_(e), concentration of free hemoglobin in media [C_Hb], fractions of three hemoglobin forms (oxyf(HbO₂), deoxy- f(Hb), and met- f(HbFe(III)), and volume fraction of lysed erythrocytes V_f(e)lysed. The mass balance statements relating some of these parameters are shown in Eqs. (14–16). Firstly, since only three forms of hemoglobin were considered, their fractions always added up to one (Eq. (14)). Second, the total amount of hemoglobin [Hb]_T in a given sample was the sum of total hemoglobin within erythrocytes, i.e. a product of the erythrocyte hemoglobin concentration [Hb]_(e) and volume fraction of erythrocyte population V_(e), and total hemoglobin dispersed in the media, i.e. a product of free hemoglobin concentration [C_Hb] and sample volume V (Eq. (15)). Thirdly, concentration of hemoglobin in the media [C_Hb] was a function of the fraction of lysed erythrocytes V_f(e)lysed as shown in Eq. (16). The mass balance constraints were implemented using variable transformation techniques [53].

f ({HbO}_{2}) + f (Hb) + f (HbFe (III)) = 1

{[Hb]}_{T} = {[Hb]}_{(e)} \cdot V_{f (e)} + [C_{Hb}] \cdot V

[C_{Hb}] = \frac{1}{V} {[Hb]}_{(e)} \cdot V_{f (e)} \cdot V_{f (e) lysed}

Given these relationships, the independently estimated parameters were number density N_(p), mean particle volume V_(p), hemoglobin concentration within erythrocytes [Hb]_(e), fractions of hemoglobin forms fi, and media hemoglobin concentration [C_Hb]. The erythrocytes’ and media hemoglobin concentrations and the fractions of hemoglobin forms entered the model through the formulation of the complex refractive index of the erythrocyte population in Eq. (13) and media absorption, i.e. second term of Eq. (4). The mean particle volume V _(p) was included in the formulation of both single-particle and macroscopic cross-sections (Eqs. (5–12)). The number densities N _(p) of particle populations were modeled with Eqs. (4–6). It is evident from Eqs. (2–13) that the model is highly non-linear in terms of the parameters being estimated and as such the uniqueness of the parameters cannot be ensured. Nevertheless, extensive numerical sensitivity analyses demonstrated that the model is indeed sensitive to the parameters listed. Furthermore, by constraining the parameter search to the range of physiologically feasible values, convergence to a unique solution from a wide variety of initial conditions is always achieved. The average errors of the estimation of the blood culture parameters were ±0.1*10⁸ cell ml^-1 for number density N _(p), ±0.5 µm³ for mean particle volume V_(p), ±0.0003 for particle volume fraction V_f(p), ±0.005 for hemoglobin fraction within erythrocytes [Hb]_(e), ±0.0001 g ml^-1 for concentration of free hemoglobin in media [C_Hb], ±0.0015 for the fractions of hemoglobin forms, and ±0.01 for volume fraction of lysed erythrocytes V_f(e)lysed.

3. Results

Typical sets of wavelength-dependent diffuse reflectance spectra of blood cultures are shown in Fig. 2. These spectral sets were recorded over time from two independent blood cultures of different experimental set-ups. In brief, the spectra plotted in Panel A of Fig. 2 were recorded from a blood culture contaminated with E. coli. The spectra were collected for 12 hours with 2 min interval between the measurements. These spectra show a transition from oxyhemoglobin with characteristic reflectance peak at 660 nm to deoxyhemoglobin with characteristic reflectance peaks at 725 nm and 810 nm over the course of the experiment. The spectra in Panel B of Fig. 2 were recorded from a non-contaminated blood culture for over 120 hours with 17 min interval between the measurements. In this example, a transition from oxyhemoglobin to methemoglobin with characteristic peaks at 630 nm and 690 nm can be observed. These changes in the spectral features were results of metabolic activity of contaminating bacteria (Panel A) and aging process of erythrocytes (Panel B) as discussed in detail in [1].

Fig. 2. Panel A: diffuse reflectance spectra collected during a blood culture experiments with E. coli. Panel B: diffuse reflectance spectra collected during a blood culture experiment with no contaminants.

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In blood culture experiments erythrocytes are the primary scattering elements, as they are the dominant blood components, and they typically have volume fractions of <0.1 in blood cultures. Hemoglobin is the dominant chromophoric component of erythrocytes and, therefore, determined the major features of the measured spectral curves. Therefore, the evident variability in the recorded spectral shapes of Fig. 2 largely resulted from the changes in the chemical composition of hemoglobin over the measurement time periods. The signature absorption bands of hemoglobin forms, for example, 542 nm and 577 nm of oxyhemoglobin, 558 nm and 760 nm of deoxyhemoglobin, and 627 nm of methemoglobin (Fig. 1), appear as depressions in the diffuse reflectance spectra (Fig. 2). Notice that the characteristic absorption signatures of the hemoglobin forms are enhanced in diffuse reflectance spectra through non-trivial dependence of the reflectance on both scattering and absorption (Eqs. (2–3)). By means of such dependence the differences in the characteristic spectroscopic features of hemoglobin forms enable accurate quantitative assessment of hemoglobin composition. Furthermore, the physiological parameters of the measured samples such as number density and mean volume of erythrocytes (and other cells) can be derived from the measured spectra. In the following discussion we illustrate these claims through a series of characteristic examples. The contributions of the different physiological parameters of the components of blood cultures to the diffuse reflectance signals will be addressed individually.

The first set of the characteristic examples of the deconvolution of the measured reflectance spectra of blood cultures is shown in Fig. 3. It addresses the model’ ability to estimate the fractions of the multiple hemoglobin forms present in a given blood culture sample. The examples shown in Fig. 3 are individual spectra from three blood culture experiments. The spectra in Panels A and B were recorded after inoculation of different blood samples to aerobic and anaerobic blood culture vials, respectively. The spectrum in Panel C was recorded after long-term incubation of an aerobic blood culture. Each of the examples had distinct hemoglobin composition; yet all three main forms of hemoglobin contributed to the measured reflectance signals of all spectra. The estimates of the hemoglobin fractions obtained via deconvolution of the measured spectra are listed in Table 2. The estimates of other blood parameters such as the volume fraction and mean volume of erythrocytes are also given (Table 2). Each subplot of Fig. 3 displays a comparison between a measured and the corresponding theoretically predicted reflectance spectra along with the residuals of the fit.

The following examples address the model’s capability to quantify the optically relevant parameters of blood cultures such as the volume fraction and mean volume of the scatters. These properties determine the magnitudes of macroscopic cross-sections (Eqs. (4–6)) and thus notably contribute to the reflectance signal. Figure 4 illustrates typically observed differences in the shape and magnitude of the reflectance spectra of blood cultures resulting from differences in the mean volume of the scattering elements. In this example the scatter number density and chemical composition of the compared blood cultures are similar (Table 3). However, since the volume fraction of the scatters is directly proportional to their mean volume, this parameter also varied between the measured samples. Figure 4 shows that the changes in both the mean volume and the volume fraction of scattering elements affect the relative amplitude and character of the reflectance spectra.

Fig. 3. The measured (R_meas) and predicted (R_calc) diffuse reflectance spectra and residuals (σ_res) of three different blood cultures.

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Table 2. The relevant parameters of blood culture components obtained from the deconvolution of the diffuse reflectance spectra presented in Fig. 2.

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Fig. 4. The measured (R_meas) and predicted (R_calc) diffuse reflectance spectra and residuals (σ_res) for blood culture samples characterized by the mean erythrocyte volume of 82 µm³ (A) and the mean erythrocyte volume of 99 µm³ (B).

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Table 3. The relevant parameters of blood culture components obtained from the deconvolution of the diffuse reflectance spectra presented in Fig. 4.

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Figure 5 demonstrates the spectral differences between two samples characterized by similar chemical composition and the equal volume fractions of scatters but by distinct mean volumes of the scatters (Table 4). In particular, spectrum B was recorded of a blood culture with larger number density of erythrocytes than that of spectrum A. Note, that the effects of the volume fraction and the mean volume of scatters on the reflectance signal are complex through non-linear dependency of macroscopic scattering cross-section on the first parameter (Eq. (5)) and complex formulations of single-particle cross-sections as functions of particle size (Eqs. (7–12))

Fig. 5. The measured (R_meas) and predicted (R_calc) diffuse reflectance spectra and residuals (σ_res) for blood culture samples characterized by the mean erythrocyte volume of 92 µm³ (A) and the mean erythrocyte volume of 77 µm³ (B).

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Table 4. The relevant parameters of blood culture components obtained from the deconvolution of the diffuse reflectance spectra presented in Fig. 5.

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The previous examples demonstrated good agreement between the measured and reconstructed reflectance spectra and the corresponding deconvolution estimates of blood culture parameters for non-contaminated samples, i.e. containing only typical blood components in culture media. Since erythrocytes are the predominant particle population (>99.99% of all particles by volume) of normal healthy blood, it was reasonable to approximate that those samples had only one particle population. With erythrocytes having the number density of 108 cells ml-1 and therefore occupying up to 10% of liquid phase of a typical blood culture, for any other population to have a measurable contribution to the reflectance signal, comparable particle number density and/or volume fraction are needed.

Although microbial contaminants growing in blood cultures typically have small size (for example, characteristic average diameter of bacteria is about 0.5-2 µm [45,47] and yeast cells are typically 3–10 µm in diameter [47]), they usually reach the number densities of 10⁷ -10⁹ cells ml^-1 in positive blood cultures [35,36] and thus can alter the reflectance signal. This outcome is illustrated with Fig. 6, where three spectra recorded at different time steps of the same blood culture experiment with two contaminants (bacterium Staphylococcus epidermidis and yeast Candida glabrata) are presented. Of these spectra, A was recorder prior build-up of microbial biomass, B was recorded when S. epidermidis reached number density of >10⁸ cells ml^-1, and C was recorded when the yeast cells reached similar optically relevant concentration. The experiment started with low inocula of both contaminants and because of the significant dissimilarity in the growth rates of these two microorganisms leading to the difference in time needed to reach the optically relevant cell number densities, it was possible to capture the individual contributions of these microorganisms to the reflectance signal. In Fig. 6, the spectral change from spectrum A to spectrum B resulted from S. epidermidis reaching a calculated number density of 0.8·10⁹ cells ml^-1 (Table 5). Since the cells of S. epidermidis typically form clusters [36,47], the bacterial mean cell volume was approximated as 3 µm³ (Table 5). Further, the growth of C. glabrata produced additional spectral changes (i.e., the shift from spectrum B to spectrum C in Fig. 6). With approximated mean cell volume of yeast of 33.5 µm³ cell number density for C. glabrata was estimated to be 7·10⁷ cells ml^-1 for spectrum C (Table 5). These results suggest that physiological properties of blood culture components other than erythrocytes can also be captured and quantitatively resolved from diffuse reflectance measurements.

Fig. 6. The measured (R_meas) and predicted (R_calc) diffuse reflectance spectra and residuals (σ_res) for blood culture samples characterized by one particle population (erythrocytes) (A), two particle populations (erythrocytes and bacterial cells) (B), and three particle populations (erythrocytes, bacterial cells, and yeast cells) (C).

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Table 5. The relevant parameters of blood culture components obtained from the deconvolution of the diffuse reflectance spectra presented in Fig. 6.

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Another important process that may to occur in blood cultures is lysis of erythrocytes [36]. Erythrocyte destruction unavoidably induces the changes in the number density and mean volume of the scatters. Since erythrocytes are the primary particle population in blood cultures, the effect of this process on the reflectance signal is dramatic as shown in Fig. 7. The spectra in Fig. 7 were collected from an anaerobic blood culture contaminated with Clostridium perfringens, an anaerobic hemolytic bacterium, at two time points of the experiment. Spectrum A was recorded prior to hemolysis and spectrum B was recorded later into the experiment when hemolysis occurred as a result of metabolic activity of C. perfringens. The observed spectral differences are the collective outcome of the physical changes in the sample, i.e. in the size, volume fraction, and number density of particulates as erythrocytes rupture into pieces, and chemical changes resulted from the increase in the concentration of free hemoglobin in the media (Table 6). Both physical and chemical perturbations of lysis led to the observed decrease in the magnitude of the reflected signal. Scattering decreased with reduced volume fraction of scattering elements and absorption increased with hemoglobin becoming free in solution. Moreover, the reduction of scattering caused the absorptive spectral features in diffuse reflectance to be less pronounced and produced an apparent flattening of the spectrum (Fig. 7). By including mathematical description of these processes in the formulation of macroscopic cross-sections (Eqs. (4–6)) the measured reflectance spectra of samples containing lysed erythrocytes can be suitably predicted and reasonable quantitative estimates of the properties of sample components can be estimated (Table 6).

Fig. 7. Temporal progression of oxyhemoglobin (A), fraction of deoxyhemoglobin (B), fraction of methemoglobin (C), and the mean cell volume (MCV) of erythrocytes (D) along with the residual sum of squares (E) obtained from deconvolution of diffuse reflectance spectra recorded during an on-line blood culture experiment.

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Table 6. The relevant parameters of blood culture components obtained from the deconvolution of the diffuse reflectance spectra presented in Fig. 7.

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Further, temporal progression of blood culture parameters estimated from continuously measured reflectance spectra can be evaluated. For example, Fig. 8 shows the temporal changes in the fractions of hemoglobin forms and mean volume of erythrocytes estimated from reflectance measurements of non-contaminated blood culture. The measurements were taken every 15 min for over 120 hrs and each measured spectrum was deconvoluted with the model. Tracking the temporal variability in the estimated parameters allows for quantitative assessment of the processes taking place within the blood culture as described in [1,36,54]. The residual sum of squares plotted in panel E of Fig. 8 indicates good agreement between the measured and theoretically predicted spectra and therefore validates the estimated parameter values.

Fig. 8. The measured (R_meas) and predicted (R_calc) diffuse reflectance spectra and residuals (σ_res) for blood culture samples with intact erythrocytes (A) and partially lysed erythrocytes (B).

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4. Discussion

The solutions to photon diffusion approximation and Mie scattering theory are well-known and have been used for over several decades, but surprisingly their applications to the spectroscopic measurements of blood have been restricted to one or several selected wavelengths [2,15]. Overall, the possibilities offered by the wavelength-dependent measurements have been explored only in few studies [21,22]. Furthermore, the number of reported simulations of the wavelength-dependent spectroscopic behavior of blood in terms of reflectance is very limited [30,55,56]. The approach presented in this paper demonstrates the advantages offered by the non-linear dependence on wavelength of both real and imaginary parts of hemoglobin refractive index that can be exploited in the context of diffuse reflectance measurements of blood cultures. Complete employment of the model in terms of these nonlinear dependencies is vital for mathematical description of scattering and absorption properties of blood.

In this paper we briefly reported on the capabilities of the new approach designed to quantitatively extract the physiological and chemical parameters of the components of blood cultures from diffuse reflectance measurements. This interpretation model has been applied effectively for the analysis of a multitude of reflectance spectra from blood cultures. The examples of the recorded spectra with corresponding theoretical deconvolutions presented in this paper were intended to illustrate how the variability in the optically relevant properties of blood culture components alter the diffuse reflectance spectra and how these properties can be estimated. We used the residuals of the fit between the measured and predicted spectra as a measure of the quality of theoretical predictions. The excellent agreements between the measurements and theory confirmed the validity of our approach. As defined by Eqs. (2–13) the reflectance spectra are functions of the number density, size, and composition as well as the absorption and scattering properties of the particle populations present in a given sample. In terms of blood cultures, such important variables as number density and mean volume of erythrocytes, concentration of hemoglobin within the erythrocytes and dissolved in media, and fractions of the main forms of hemoglobin can be assessed. From these variables, the erythrocyte volume fraction (hematocrit) and the amount of erythrocyte lysis can be derived. Further, we have shown that the composite effects of multiple particle populations can be assessed from diffuse reflectance spectra. The additional populations could be, for instance, microbial contaminants; when their volume fractions exceed 0.01% they can produce measurable changes to the diffuse reflectance signals. Therefore, the mean cell size, number density, and chemical composition (if chromophoric) of the supplementary particle populations can be estimated. The interpretation model was shown to produce comprehensive estimates of these variables.

While a number of comprehensive approaches for theoretical and/or numerical description of reflectance or backscattering measurements has been developed [15–21], they have been restricted to single wavelength measurements and even then their computational complexity often require simplifying approximations. From this perspective, the presented approach for spectral deconvolution has an important advantage of being computationally fast and efficient. Our interpretation model is envisioned to be used for on-line and real-time applications, for example, in clinical blood culture analyses as illustrated in Fig. 8.

5. Conclusion

Diffuse reflectance is simple, non-invasive, and non-destructive type of spectroscopic measurements that by its virtues has been extensively exploited for various applications including those in biomedical field. In this paper we report on a new algorithm for accurate quantitative description of the diffuse reflectance spectra from blood cultures in terms of the relevant parameters of the blood culture components. For this method, the rigorous solutions to photon diffusion approximation and Mie scattering theory have been revisited. Adequate mathematical descriptions of multiple scattering, multiple scattering populations, and multiple chromophores present within the scatters as well as in solution were added. In fact, the success of new interpretation model lies in the utilization of non-linear wavelength dependence of the basic optical properties, i.e., refractive indices, of the chromophoric components present in the measured samples. The displayed solution to the diffuse reflectance problem was found to give appropriate estimates of the blood culture parameters as was illustrated with a scope of characteristic examples. Furthermore, the features of this quantitative theoretical deconvolution model eliminate the need for external calibrations (other than scaling for white intensity) and therefore lead to superior interpretation of the measured spectra.

Acknowledgments

The authors would like to acknowledge the support of Claro Scientific, LLC., Florida Blood Services, and of the College of Marine Science, University of South Florida.

References and links

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Parameter	Description
a	Radius of the incident photon beam
b	Source-detector separation distance
d	Depth of the sample
D_f	Photon diffusion constant defined as D_f =(3(µ_st + µ_a ))^-1
I₀ and I₁	Zeroth and first-order modified Bessel functions of first kind
K₀ and K₁	Zeroth and first-order modified Bessel functions of second kind
k_n	The n^th eigenvalue of the Green’s function defined by tan(k_nd)=-2.142 D_fkn
N_n	Eigenvalue normalization constant defined as
	N_n ==∫^d₀sin²(k_n z+γ_n ) dz==(2k_nd+sin 2γn-sin(2k_nd+2γ))/4k_n
r	Radius of observation
r_x	Detector aperture radius
r₁	b-r_x
r₂	b+r_x
R_diff	Diffuse reflectance intensity of arbitrary radius r
R(λ)	Diffuse reflectance intensity of detector aperture radius r_x
z_n	The n^th term of z (distance along the axis of incident light)-dependence
	z_n =k_n cos γ_n +µ_t (λ) sin γ_n -[k_n cos (k_n d+γ)+µt(λ)sin (k_n d+γ)] e-^µt(λ)d
γ_n	Phase function defined as γ_n =arctan (2.142 D_fk_n )
λ	Wavelength
<µ>	Asymmetry parameter
µ_a(λ)	Macroscopic absorption cross-section
µ_st(λ)	Reduced macroscopic scattering cross-section
µt(λ)	Total macroscopic cross-section defined as µ_t(λ)=µ_s(λ)+µ_a(λ)
ζn	Bessel function parameter of the nth order defined as ζ_n=k_n ²+µ_a(λ)/Df

Deconvoluted blood culture parameters	Spectrum A	Spectrum B	Spectrum C
Volume fraction of erythrocytes	0.0564	0.0345	0.0536
Number density of erythrocytes (10⁸ cells ml^-1)	6.2	4.3	5.9
Mean volume of erythrocytes (µm³)	91.0	81.8	90.9
Volume fraction of hemoglobin in erythrocytes	0.34	0.33	0.34
Fraction of lysed erythrocytes	<0.01	<0.01	<0.01
Concentration of hemoglobin in media (g ml-1)	<0.0001	<0.0001	<0.0001
Fraction of oxyhemoglobin of total hemoglobin	0.945	0.042	0.800
Fraction of deoxyhemoglobin of total hemoglobin	0.053	0.957	0.082
Fraction of methemoglobin of total hemoglobin	0.002	0.001	0.118
Residual sum of squares	0.020	0.026	0.025

Deconvoluted blood culture parameters	Spectrum A	Spectrum B
Volume fraction of erythrocytes	0.055	0.067
Number density of erythrocytes (10⁸ cells ml^-1)	6.8	6.8
Mean volume of erythrocytes (µm³)	82.0	99.0
Volume fraction of hemoglobin in erythrocytes	0.30	0.29
Fraction of lysed erythrocytes	<0.01	<0.01
Concentration of hemoglobin in media (g ml^-1)	<0.0001	<0.0001
Fraction of deoxyhemoglobin of total hemoglobin	1.0	1.0
Residual sum of squares	0.020	0.026

Deconvoluted blood culture parameters	Spectrum A	Spectrum B
Volume fraction of erythrocytes in blood culture	0.073	0.073
Number density of erythrocytes in blood culture (10⁸ cells ml^-1)	7.9	9.5
Mean cell volume of erythrocytes (µm³)	92.0	77.0
Volume fraction of hemoglobin in erythrocytes	0.30	0.29
Fraction of lysed erythrocytes	<0.01	<0.01
Concentration of hemoglobin in media (g ml^-1)	<0.0001	<0.0001
Fraction of oxyhemoglobin of total hemoglobin	0.915	0.915
Fraction of deoxyhemoglobin of total hemoglobin	0.084	0.085
Fraction of methemoglobin of total hemoglobin	0.001	0.0001
Residual sum of squares	0.049	0.089

Deconvoluted blood culture parameters	Spectrum A	Spectrum B	Spectrum C
Total volume fraction of scatters	0.0836	0.0860	0.0880
Number density of erythrocytes (10⁸ cells ml^-1)	9.3	9.2	9.2
Mean cell volume of erythrocytes (µm³)	90.0	90.0	90.0
Volume fraction of hemoglobin in erythrocytes	0.32	0.32	0.32
Fraction of lysed erythrocytes	<0.01	<0.01
Concentration of hemoglobin in media (g ml^-1)	<0.0001	<0.0001
Fraction of deoxyhemoglobin of total hemoglobin	0.9999	0.9999	0.9998
Number density of bacterial cells in blood culture (10⁸ cells ml^-1)	-	8.0	4.0
Mean cell volume of bacteria (µm³)	-	3.0	3.0
Number density of yeast cells in blood culture (10⁸ cells ml^-1)	-	-	0.7
Mean cell volume of yeast (µm³)	-	-	33.5
Residual sum of squares	0.053	0.075	0.067

Quantitative interpretations of Visible-NIR reflectance spectra of blood

Abstract

1. Introduction

2. Method

2.1 Measurement

2.2 Photon diffusion theory

2.3 Formulation of the optical properties

2.4 Analysis of data

3. Results

4. Discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Tables (6)

Equations (16)

Optics Express

Deconvoluted blood culture parameters	Spectrum A	Spectrum B
Volume fraction of erythrocytes in blood culture	0.068	0.055
Number density of erythrocytes in blood culture (10⁸ cells ml^-1)	8.5	6.9
Mean cell volume of erythrocytes (µm³)	80.0	80.0
Volume fraction of hemoglobin in erythrocytes	0.32	0.32
Fraction of lysed erythrocytes	<0.01	0.19
Concentration of free hemoglobin in media (g ml^-1)	<0.0001	0.0044
Fraction of deoxyhemoglobin of total hemoglobin	0.989	0.999
Residual sum of squares	0.015	0.030