A literature review and novel theoretical approach on the optical properties of whole blood
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Optical property measurements on blood are influenced by a large variety of factors of both physical and methodological origin. The aim of this review is to list these factors of influence and to provide the reader with optical property spectra (250–2,500 nm) for whole blood that can be used in the practice of biomedical optics (tabulated in the appendix). Hereto, we perform a critical examination and selection of the available optical property spectra of blood in literature, from which we compile average spectra for the absorption coefficient (μ a), scattering coefficient (μ s) and scattering anisotropy (g). From this, we calculate the reduced scattering coefficient (μ s′) and the effective attenuation coefficient (μ eff). In the compilation of μ a and μ s, we incorporate the influences of absorption flattening and dependent scattering (i.e. spatial correlations between positions of red blood cells), respectively. For the influence of dependent scattering on μ s, we present a novel, theoretically derived formula that can be used for practical rescaling of μ s to other haematocrits. Since the measurement of the scattering properties of blood has been proven to be challenging, we apply an alternative, theoretical approach to calculate spectra for μ s and g. Hereto, we combine Kramers–Kronig analysis with analytical scattering theory, extended with Percus–Yevick structure factors that take into account the effect of dependent scattering in whole blood. We argue that our calculated spectra may provide a better estimation for μ s and g (and hence μ s′ and μ eff) than the compiled spectra from literature for wavelengths between 300 and 600 nm.
KeywordsBlood Optical properties Spectroscopy Absorption coefficient Scattering coefficient Scattering anisotropy
The interaction of light with blood plays an important role in optical diagnostics and therapeutics—for instance for the non-invasive assessment of blood composition  and the laser treatment of varicose veins . Predictions on the accuracy and outcome of these optical methods can be obtained through simulation models of the light–blood interaction. The reliability of these models depends foremost on accurate knowledge of the optical properties of blood, which include the absorption coefficient μ a, scattering coefficient μ s and scattering anisotropy g that parameterizes the phase function p(θ). Dating back to as early as 1943 , many studies have focused on the quantitative assessment of these optical properties [4, 5, 6, 7, 8, 9, 10]. These studies demonstrated that optical property measurements on whole blood are challenging, due to the considerable light attenuation in undiluted blood. Although light attenuation is less in diluted samples, rescaling of the optical properties from these samples to whole blood introduces an additional challenge because the scattering properties of blood scale non-linearly as a function of red blood cell concentration (haematocrit) [10, 11, 12]. As a consequence, sample preparation, but also measurement method and conditions (e.g. blood flow [7, 13, 14, 15, 16]), influences the outcome of the optical property assessment considerably. In this review article, we will therefore provide an overview, interpretation and compilation of the available literature on the optical properties of blood in the visible and near-infrared wavelength range (250–2,500 nm). Our inclusion criteria are (1) publication of both quantitative and spectrally resolved data on μ a, μ s and g and (2) the use of human blood from healthy adults for sample preparation.
In part I (‘Methods’ and ‘Results’ sections) of this article, we focus on the absorption coefficient of whole blood. We compile an average μ a spectrum for blood with a haematocrit of 45 % from rescaled spectra that are available in literature, while excluding outlier spectra. We also incorporate the effect of ‘absorption flattening’: the phenomenon that the absorption spectrum of a system of strongly absorbing particles (i.e. red blood cells in whole blood) is reduced compared to that of a suspension containing the same number of absorbing molecules in homogeneous dispersion (i.e. haemolysed blood).
The scattering properties of blood (μ s and g) are considered in part II (‘Theoretical estimation of μ s and g ’, ‘Methods’ and ‘Results’ sections) of this article. Given the difficulty in measuring the scattering properties of red blood cells, and the relative ease of measuring absorption spectra of the red blood cells’ contents, we previously proposed a computational approach based on a Kramers–Kronig analysis of the complex refractive index of haemoglobin . We obtained estimates of red blood cell scattering by combining this approach with analytical scattering theory. Here, we extend this method using Percus–Yevick structure factors that take into account the spatial correlations between the positions of individual red blood cells in a whole blood medium. From this, we obtain calculated spectra of μ s and g for oxygenized and deoxygenized blood. Moreover, we present a novel scaling relation for μ s to different haematocrit values, which we use to theoretically verify a previously published empirical scaling relation . We apply the novel scaling relation to rescale the available literature spectra for μ s to a haematocrit of 45 %. From the rescaled spectra, we compile an average μ s spectrum for whole blood. We also provide a compiled spectrum of the literature spectra of g. To provide the reader with reasonable means to estimate the scattering coefficient, we present an empirical power law for scattering coefficient versus wavelength (>700 nm). In addition, we provide spectra for the reduced scattering coefficient (μ s′) and the effective attenuation coefficient (μ eff), derived from both the compiled and calculated spectra of μ a, μ s and g.
The main results of this article are ready-to-use compiled spectra of μ a, as well as both compiled and calculated spectra of μ s, μ s′, μ eff and g for whole blood with a haematocrit of 45 %. For convenience, these spectra are tabulated in the Appendix of this article. Moreover, methods for scaling between different haematocrits are presented. We argue that our calculated spectra may provide a better estimation of the scattering properties of whole blood than the compiled spectra from literature for wavelengths <600 nm.
Composition of human blood and its optical properties
Normal human blood consists of red blood cells (RBCs or erythrocytes, ±4,500 × 103/μL blood), white blood cells (leukocytes, ±8 × 103/μL blood), platelets (thrombocytes, ±300 × 103/μL blood) and blood plasma (containing water, electrolytes, plasma proteins, carbohydrates, lipids and various extracellular vesicles [18, 19]). The haematocrit (hct) is defined as the volume percentage of red blood cells in blood and on average amounts to 40 % for adult women and 45 % for adult men. Red blood cells are composed mainly of haemoglobin, with a concentration of ±350 g/L in a cell volume of ±90 fL. In healthy human adults, the average haemoglobin concentration in blood accounts for 140 g/L in women and 155 g/L in men .
Accounting for an absorption contribution of two to three orders of magnitude higher than the other blood components, red blood cells are by far the most dominant absorbing element in the blood in the wavelength range of 250–1,100 nm . Practically, all light absorption by the red blood cells is due to haemoglobin, which exhibits specific absorption features for its various derivatives: bound to oxygen (oxyhaemoglobin, HbO2), unbound to oxygen (deoxyhaemoglobin, Hb), bound to carbon monoxide (carboxyhaemoglobin), oxidized (methaemoglobin), fetal and more . From these haemoglobin derivatives, oxyhaemoglobin and deoxyhaemoglobin are the most abundant types in healthy human adult blood. The oxygen saturation of blood is defined as the ratio of the HbO2 concentration to the total haemoglobin concentration, oxygen saturation (SO2) = [HbO2] / ([HbO2] + [Hb]), and amounts to ∼97.5 % in arterial blood and ∼75 % in venous blood . Of all blood particles, red blood cells also predominate the scattering of blood with two to three orders of magnitude, arising from the difference in refractive index between red blood cells and the surrounding blood plasma .
Without the presence of red blood cells, plasma absorption in the 250–1,100-nm region is dominated by various proteins, nutritive compounds and/or pharmaceuticals, while plasma scattering is dominated by proteins and platelets . Under pathological conditions, the absorption contribution of certain plasma proteins can become significant even in the presence of red blood cells, e.g. the absorption of bilirubin around 460 nm for jaundiced patients .
In the wavelength range beyond 1,100 nm, blood absorption is dominated by the absorption of water [7, 9]. Only when water is removed from the blood, several absorption features due to the presence of haemoglobin, albumin and globulin can be identified as small absorption peaks between 1,690 and 2,400 nm .
Factors influencing the optical properties of blood
Since red blood cells are the main contributor to the optical properties of blood, their volume percentage (i.e. haematocrit), haemoglobin concentration and oxygen saturation directly influence the absorption and scattering properties of blood. Whereas the absorption coefficient μ a is proportional to the haematocrit, the scattering coefficient μ s saturates for hct > 10 %, i.e. μ s, is underestimated for high hct values with respect to a linear relationship between the two parameters . Meinke et al. , in our opinion correctly, ascribed this saturation effect to a decrease of the mean distance between red blood cells, because it violates the assumption of independent single scattering. This group also reported non-linear deviations of g for hct > 10 %. See part II section of this paper for further discussion.
The scattering of blood is primarily caused by the complex refractive index mismatch between red blood cells and plasma. Although most measurements on the optical properties of blood are performed on blood samples where plasma has been replaced by saline/phosphate buffer, Meinke et al. [10, 20] measured that this affects the complex refractive index mismatch considerably, resulting in an overestimation of the scattering coefficient of 5.5–9.4 % with respect to red blood cells in plasma.
The principle of causality dictates that the real and imaginary parts of the complex refractive index are connected as expressed by the Kramers–Kronig relations. The imaginary part is proportional to the absorption coefficient, which in turn depends on the SO2. Thus, the real part of the complex refractive index is also SO2 dependent and so are the scattering properties [9, 17]. This influence is most prominent in the visible wavelength region where differences in μ a due to changes in SO2 are high, leading to deviations up to 15 % in μ s and 12 % in g between fully oxygenated and fully deoxygenated blood .
Various sources have reported that the shear rate due to blood flow [7, 13, 14, 15, 16] and aggregate formation (e.g. rouleaux formation) [13, 23, 24] significantly influence the optical properties of blood due to non-Newtonian flow characteristics. Enejder et al.  measured a decrease in the absorption and reduced scattering of bovine blood of ∼3 % when increasing the average shear rate from 0 to 1,600 s−1, as well as a decrease in reduced scattering of 4 % when randomly oriented red blood cells form aggregates.
Other reported factors of influence on the optical properties are osmolarity , temperature [25, 26], inter-person variability  and pathologic disorders such as sickle cell anemia . A special case is that for adults versus fetuses, whose blood is composed of different types of haemoglobin (adult versus fetal haemoglobin) that exhibit slight variations in their absorption features .
Measurement methods in literature
Most measurements on whole or diluted blood with intact red blood cells have been performed using single or double integrating sphere geometries. The resulting wavelength-dependent transmission and/or reflectance from a thin sample slab is analysed by inverse Monte Carlo models [6, 7, 8, 9, 10] or T-matrix computations  to obtain estimates for μ a, μ s and g. As is acknowledged by various sources [6, 7, 8], the assumed scattering phase function of blood in the inverse Monte Carlo analysis highly influences the inferred optical properties—especially μ s and g. Although other measurement methods have been reported for optical property measurements on whole blood [28, 29], we did not encounter any studies that exploit these methods experimentally or the quantitative assessment of spectra of μ a, μ s and g.
The refractive index of oxygenated haemoglobin solutions was determined by Friebel et al.  from measurements of the Fresnel reflection with an integrating sphere spectrometer. Complementing these measurements, Meinke et al.  measured the refractive index of plasma at four wavelengths using an Abbe refractometer, which yielded a Sellmeier equation for the visible wavelength range.
Part I: the absorption coefficient of whole blood
From the available optical property spectra in literature, we compiled the averaged spectra of μ a for whole blood with a haematocrit of 45 %. Criteria for including optical property data were (1) publication of both absolute and spectrally resolved data on the optical properties and (2) the use of human blood from healthy adults for sample preparation. In case tabulated data were unavailable, the program GetData Graph Digitizer (v188.8.131.52) was used to obtain the digitized optical property spectra from the published graphs. The same criteria were applied for the inclusion and tabulation of literature spectra for μ s and g, which will be considered in part II of this article.
Compiled literature spectrum of μ a
Where μ a,blood and μ a,Hb are the absorption coefficient of a blood sample and haemoglobin solution, respectively. The length d RBC is a typical dimension of a red blood cell. In this derivation, it was assumed that the RBCs can be represented by cubes with volume equal to an RBC (d RBC = 3√90 μm). Following the same approach, Finlay and Foster  derived a more complex version of Eq. 3, valid for equivolumetric spherical particles. Since the difference between both forms is neglicable for the present parameters, we adhere to the much simpler form of Eq. 3 throughout this manuscript.
The compiled spectra of μ a were obtained by averaging the rescaled spectra, with the exclusion of one outlier spectrum, as specified in the ‘Results’ section. The μ a spectra for oxygenized (nine averages) and deoxygenized blood (three averages) were compiled separately.
Optical property spectra of human blood in literature
Literature on the optical properties of blood in the visible and near-infrared
Wavelength range (nm)
Zijlstra et al. 
Hb solution from haemolysed RBCs (human); SO2 = 0, 100 %; T = 20–24 °C
Compiled data from Gratzer and Kollias
Hb solution from haemolysed RBCs; SO2 = 0, 100 %
Yaroslavsky et al. 
Double IS with inverse MC (P HG)
Fresh heparinized whole blood (human); hct = 45–46 %, SO2 > 98 %; no flow, γ = 0 s−1
μ a, μ s, g
Roggan et al. 
Double IS with inverse MC (P GK)
Fresh RBCs (human) in phosphate buffer; hct = 5 %; SO2 = 0, 100 %; in flow, γ = 500 s−1; T = 20 °C
μ a, μ s, g
Friebel et al. 
IS with inverse MC (P RC)
Fresh RBCs (human) in phosphate buffer; hct = 0.84, 42.1 %; SO2 > 99 %; in flow, γ = 600 s−1; T = 20 °C
μ a, μ s, g
Friebel et al. 
IS with inverse MC (P RC)
Fresh RBCs (human) in phosphate buffer; hct = 33.2 %; SO2 = 0, 100 %; in flow, γ = 600 s−1; T = 20 °C
μ a, μ s, g
Meinke et al. 
IS with inverse MC (P RC)
Fresh RBCs (human) in phosphate buffer and saline solution/plasma various samples between hct = 0.84 and 42.1 %
μ a, μ s, g
Compiled literature spectrum of μ a
The spectra of haemolysed blood from Zijlstra  and Prahl  in Fig. 1a, b have been rescaled with the absorption flattening factor from Eq. 3. Unscaled, the μ a spectrum of haemolised blood overestimates the absorption of both oxygenized and deoxygenized blood with approximately 10–20 % at the Soret band around 420 nm . This difference has also been measured by Friebel et al.  when they compared their μ a spectra from samples containing intact red blood cells to those containing haemolysed blood at exactly the same concentrations of haemoglobin. After correcting for the absorption flattening, the spectra are in good agreement with the absorption spectra from (whole) blood measurements.
The compiled μ a spectrum of oxygenized blood is composed of the average of N = 9 spectra (Fig. 1c). Due to the difficulty to fully deoxygenize blood (high oxygen affinity of haemoglobin), fewer literature spectra are available for deoxygenized blood—resulting in a compiled μ a spectrum of the average of N = 3 spectra for deoxygenized blood (Fig. 1c). Note that the data from Friebel et al.  are the only data contributing to the compiled spectrum beyond 1,200 nm for oxygenized blood and beyond 1,000 nm for deoxygenized blood (indicated by the dashed lines in Fig. 1c). The sudden jumps in the compiled spectra at 1,200 and 1,000 nm are caused by this transition of the average of multiple spectra to only one spectrum that differs slightly in amplitude (∼0.1 mm−1) from the other spectra. We consider these jumps as artifacts of our compilation method, which can be ignored or smoothed when using these spectra in practice.
Part II: the scattering properties of whole blood
The determination of the scattering properties of whole blood is extremely challenging because assumptions on the applied scattering phase function are of high influence and the scaling of diluted blood measurements to physiological haematocrit values is not straightforward (‘Background’ section). In our previous work, we therefore proposed to use a ‘forward’ approach to estimate the light scattering properties from accurate measurements of the absorption coefficient of haemoglobin solutions, followed by Kramers–Kronig (KK) analysis and application of light scattering theory . We expand on this theoretical approach here to include dependent scattering effects.
In the first step, the complex refractive index is determined from the absorption coefficient of the contents of one red blood cell. This is used as input to scattering theory in the second step, accounting for inter-particle correlations due to high-volume fractions. This way, the theoretical scattering property spectra of blood can be calculated for any haematocrit at any wavelength. We use this theory to obtain calculated spectra for μ s and g for whole blood with a haematocrit of 45 %.
For practical convenience, we proceed to average the scaling factors for μ s over wavelength, which leads to a simple expression depending on haematocrit only. This novel scaling relation is then used to rescale literature spectra of μ s to a haematocrit of 45 %, from which we compile an average spectrum.
Summarizing, in this part II of the article, we provide both calculated and compiled literature spectra for μ s and g. From this, we calculate the reduced scattering coefficient μ s′ and effective attenuation coefficient μ eff for whole blood.
Theoretical estimation of μ s and g
Scattering properties of red blood cells
These scattering properties can be calculated if an appropriate theory is available to calculate I S(θ). A common approach yielding reasonable agreement with experiment  is to describe the RBC as a sphere with an equivalent volume (90 μm3, ‘Composition of human blood and its optical properties’ section) using Mie theory.
Scattering properties of whole blood
Expressions for the phase function and scattering anisotropy for dependent scattering can also be derived using the same methods.
Practical formula for haematocrit dependent scaling of μ s
From the preceding analysis, it is clear that γ(hct)—the factor ultimately for non-linear scaling of the scattering coefficient with hct—can be a complicated function of wavelength because both S(θ,hct) and p P(θ) are wavelength dependent. However, some practical expressions for γ(hct), depending on haematocrit only, have been presented in the literature.
Calculated spectra of μ s and g
To compute the complex refractive index of an RBC’s contents, we model the RBC as a sphere (90 μm3), containing a homogeneous solution of haemoglobin molecules. Hereto, we use the average of the oxygenized and deoxygenized μ a spectra of haemolysed blood from Prahl and Zijlstra only (part I)—rescaled to the appropriate concentration (350 g/L per RBC; ‘Composition of human blood and its optical properties’ 2.1), but not corrected for absorption flattening. Using Eq. 4, the imaginary part of the complex refractive index is obtained. In the Kramers–Kronig analysis (Eq. 5), we use a reference measurement of the real part of the complex refractive index at 800 nm to scale the computed spectra. Details of this procedure can be found in our previous publication . The obtained complex refractive index spectra of oxygenized and deoxygenized blood are then used as input for subsequent calculations.
To implement the theory of Eqs. 6–13, a consistent combination of scattering theory and structure factor is needed. Here, we use the Mie theory  to calculate the scattered intensity and scattering properties by approximating a red blood cell with an equivolumetric sphere (r = 2.78 μm). Mie calculations also require specification of the refractive index of the medium in which the scattering particles are suspended (i.e. plasma). The refractive index of plasma has been determined experimentally by Streekstra et al.  at 633 nm and by Meinke et al.  at 400, 500, 600 and 700 nm. Since no data is available on the entire required wavelength range (including the near-infrared), we approximate the refractive index of plasma by that of water  with an additional offset to achieve a value of 1.345 at 633 nm . This agrees well with the values of Meinke et al. in the visible wavelength range.
We use the Percus–Yevick approximation , solved analytically by Wertheim , to calculate the structure factor of a suspension of non-deformable spheres. The exact descriptions of the Percus–Yevick radial distribution function can be found elsewhere, e.g. in Refs. [39, 45]. All calculations are performed using self-written routines in Labview. The Kramers–Kronig code is benchmarked against the routines available from Ref. ; the Mie code is benchmarked against the results from Prahl’s web-based Mie calculator .
Compiled literature spectra of μ s and g
The available literature on optical property measurements of μ s and g within our inclusion criteria (‘Methods’ section) is summarized in Table 1. All spectra were obtained using integrating sphere measurements in combination with inverse Monte Carlo simulations. Phase functions that were applied in the analysis of these literature spectra included the Henyey–Greenstein , the Gegenbauer–Kernel  and the Reynolds–McCormick phase function [8, 9, 10]; details can be found in the respective references. The Gegenbauer–Kernel and the Reynolds–McCormick phase functions cited in these publications are the same .
We rescaled the μ s spectra from their original haematocrits (hct = X%) to a whole blood haematocrit of 45 % using Eqs. 13 and 17. From the rescaled spectra (N = 8), we compiled an average spectrum. The compiled spectrum of the anisotropy g was obtained from the average of the unscaled literature spectra of g (N = 9).
Scatter power analysis on μ s
Reduced scattering μ s′ and effective attenuation μ eff
In general, studies that rely on the diffuse reflectance or transmittance of whole blood consider the reduced scattering coefficient μ s′ = μ s(1 − g) and the effective attenuation coefficient μ eff = √[3μ a(μ a + μ s′)], rather than the scattering coefficient μ s and absorption coefficient μ a only. Therefore, we present the compiled spectra of μ s′ and μ eff, using the compiled spectra from literature for μ a, μ s and g. We also present theory-derived spectra of μ s′ and μ eff, using the calculated spectra for μ a, μ s and g, with μ a obtained as μ a = μ ext − μ s with μ ext the calculated extinction coefficient from Mie theory.
Calculated spectra of μ s and g
Compiled literature spectrum of μ s
Scatter power analysis on μ s
The scatter power analysis (Eq. 19) resulted in a values of 82.5 ± 0.2 mm−1 (calculated μ s, SO2 > 98 %), 72.2 ± 0.2 mm−1 (calculated μ s, SO2 = 0 %) and 91.8 ± 0.6 mm−1 (compiled μ s, SO2 > 98 %) using reference λ 0 = 700 nm. The scatter power b values were 1.23 ± 0.005 (calculated μ s, SO2 > 98 %), 1.22 ± 0.006 (calculated μ s, SO2 > 0 %) and 1.19±0.012 (compiled μ s, SO2 > 98 %).
Compiled literature spectrum of g
Reduced scattering and effective attenuation
Figure 4c shows the calculated and compiled reduced scattering coefficient spectra μ s′, which were obtained using the calculated and compiled spectra of μ s and g, respectively. Similar to the spectra of μ s and g, the calculated and compiled μ s′ spectra agree well in magnitude for those wavelengths where scattering dominates absorption (beyond 700 nm).
Figure 4d shows the calculated and compiled effective attenuation coefficient spectra μ eff. For the calculated spectrum of μ eff, the absorption coefficient was calculated using Mie theory as the difference between the extinction coefficient and scattering coefficient (for both SO2 = 0 % and SO2 > 98 %). The compiled spectra were obtained using both the compiled spectrum for μ a (part I) and the compiled spectrum for μ s′ (only for SO2 > 98 %). The absorption coefficient dominates μ eff. The excellent correspondence between the calculated and compiled spectra thus demonstrates that scattering theory is capable of including absorption flattening effects. The jumps in the 1,100–1,200 nm region and/or the oscillations beyond 2,000 nm in μ s′ and μ eff are caused by the compilation artifacts in μ a, μ s and g that have been explained above.
In the Appendix of this article, we provide the tabulated data for the compiled spectra of μ a (oxygenated and deoxygenated blood), μ s and g. The table also includes the calculated spectra for μ s and g (Kramers–Kronig/Percus–Yevick analysis for oxygenated and deoxygenated blood). All spectra are scaled to a haematocrit of 45 %. The data are presented with a resolution of 2 nm up to 600 nm and a resolution of 5 nm beyond 600 nm. From this, the calculated and compiled spectra for μ s′ and μ eff can easily be calculated. The full table can also be downloaded at our website www.biomedicalphysics.org.
Compilation of optical property spectra from literature
In this article, we provided an overview of the available literature on the spectra of the optical properties (μ a, μ s and g) of whole blood. Hereto, we included only data that present quantitative spectra of these properties and were measured on human blood or dilutions thereof. These restrictions limit the available data to the seven contributions as listed in Table 1, from which five contributions are obtained from (dilutions of) whole blood (μ a, μ s and g), and two contributions are obtained from haemolysed blood (μ a only). It should also be noted that experimental studies on the optical properties are scarce for wavelengths beyond 1,100 nm, compared to the visible and near-infrared wavelength range (λ < 1,100 nm). Hence, our compiled spectra beyond 1,100 nm are composed of only one (μ a and μ s) or two (g) literature spectra, which makes them more susceptible to experimental or methodological errors than the compiled values for λ < 1,100 nm.
The compiled spectra for μ s and g are largely dominated by the results from one research group (Roggan et al., Friebel et al. and Meinke et al. [7, 8, 9, 10]), with three out of four literature spectra for μ s and eight out of nine literature spectra for g. All spectra were obtained using integrating sphere setups, in combination with inverse Monte Carlo simulations (IS/iMC, to translate the measured diffuse reflectance and/or collimated and diffuse transmittance to values of μ a, μ s and g). The results of the inverse procedure depend highly on the phase function that is used in the Monte Carlo simulations. For the research group of Roggan et al., Friebel et al. and Meinke et al., the preferred phase function is the Reynolds–McCormick (also called Gegenbauer–Kernel ) phase function. The authors argue that this phase function has better correspondence with their measurements than the often used Henyey−Greenstein phase function or the Mie phase function. This result can be understood considering Eq. 10, which shows that the ‘effective phase function’ of a blood medium is given by the single RBC phase function, multiplied with the concentration-dependent structure factor. An additional drawback of inverse Monte Carlo procedures is that all parameters are optimized independently, whereas, following from causality, a correlation exists between all optical properties (i.e. the Kramers−Kronig relations).
It would be beneficial to investigate the possibilities of other assessment techniques that avoid the methodological uncertainties (e.g. assumptions on phase function) that are associated with IS/iMC measurements. With optical coherence tomography (OCT), the non-diffusive component of the scattered light can be analysed, which facilitates quantification of the scattering properties, in addition to the absorption properties. With spectroscopic OCT [47, 48] and the closely related technique low-coherence spectroscopy (LCS), also the spectrally resolved optical properties can be quantified. LCS has been proven to give accurate estimations of μ a and μ s spectra in turbid media with relatively high attenuation (μ a + μ s up to 35 mm−1) both in vitro [49, 50, 51] and in vivo  in the visible wavelength range. Alternatively, methods that rely on the analysis of diffuse scattering from whole blood may be combined with other analysis models than the regular inverse Monte Carlo simulations.
Absorption flattening in whole blood: rescaling μ a
For the haemolysed blood spectra that contribute to the compiled spectrum of μ a for whole blood, we take into account the absorption flattening effect. This effect involves the reduction of the absorption coefficient of a suspension of absorbing particles (i.e. blood containing RBCs), compared to a homogeneous solution containing the same number of absorbing molecules (i.e. haemolysed blood). The first theoretical assessment of absorption flattening originates from Duysens  for cubical-, spherical- and arbitrary-shaped particles. We use the cubical description (Eq. 3), since it only slightly deviates from Duysens’ more comprehensive spherical particle approximation (which was reintroduced by Finlay and Foster ). The analysis of Duysens assumes random placement of the absorbing particles, with no correlations between their positions (Poisson distribution; so that the spatial variance σ 2 of the number of particles equals the mean number μ of particles). Applying Beer’s law to each of the particles (and unit transmission for the ‘holes’) leads to the result of Eq. 3 upon averaging over all possible particle arrangements. If all particles were stacked together, σ 2 would be 0 (without changing μ) and the measured transmission would correspond to that of a homogeneous solution of the absorbing molecules—without absorption flattening. Thus, σ 2 ultimately determines the flattening effect. In a whole blood medium, possible correlations between the particle positions lead to an increase in σ 2. This causes a further reduction in the measured absorption coefficient . Interestingly, the increased σ 2 is determined by the volume integral of the radial distribution function G(r) , describing spatial arrangement that leads to dependent scattering effects (part II). This clearly emphasizes that the organization of a medium/tissue is reflected in all measurable optical properties. From a practical point of view, Duysens’ simple model of ‘cubic absorbers’ excellently scales data from haemoglobin solutions to the compiled absorption spectrum of blood.
The compartmentalization of haemoglobin in red blood cells causes the absorption flattening effect of blood absorption spectra compared to that of pure haemoglobin solutions. For techniques such as diffuse reflection spectroscopy, the same effect occurs on a larger scale because blood is not distributed homogeneously in tissue, but concentrated in vessels. Van Veen et al.  propose a correction factor introduced by Svaasand  that, interestingly, takes exactly the same form as Eq. 3 (but now with the vessel diameter as the length parameter instead of the diameter of the RBC), although it is derived in a completely different manner.
Dependent scattering in whole blood: rescaling μ s
All literature spectra of μ s were rescaled to a haematocrit of 45 % in the compilation of the average spectrum, while taking into account the effect of dependent scattering. Dependent scattering occurs when particles (i.e. RBCs) are closely spaced, or correlations exist between their positions. In that case, the phase relation between the fields scattered from different particles cannot be neglected. Therefore, the scattered fields should be added, rather than the scattered intensities. We choose a numerical, forward approach to assess the effect of dependent scattering, in which we calculate the scattering properties of blood using Mie theory (independent scattering) and Mie theory in combination with the Percus–Yevick radial distribution function G(r) (dependent scattering). This choice of theoretical descriptions essentially models blood as a high-concentration suspension of non-deformable spheres. This approach does not do full justice to the structural and rheological complexity of RBCs and blood. Future work on scattering formalisms, such as discrete dipole approximations  or models for G(r), can thus be employed using the same methodology.
A main practical result of our work is the scaling factor γ(hct) that scales the scattering coefficients of independent scattering to dependent scattering. The most cited form in the literature is γ(hct) = 1 − hct (Eq. 14), proposed by Twersky . However, in the derivation of this approximation, it is assumed that the scatterers are small and no correlations exist between their spatial positions—which is likely invalid for whole blood. Twersky’s scaling factor has been found in better agreement with experiments compared to linear haematocrit scaling (e.g. γ(hct) = 1), but other theoretical and empirical forms have been proposed, most importantly Eqs. 15 and 16. In this work, we propose γ(hct) = (1 − hct)2 as a practical approximation for the exactly calculated values from Mie/Percus–Yevick theory (Eq. 17). The agreement with the empirical form of Steinke et al.  is excellent.
Theoretical estimation of μ s and g
In addition to the compiled spectra from literature, we also calculated the spectra of μ s and g for whole blood, using only the absorption spectrum of blood as an input. The main advantage of this ‘forward approach’ to calculate these optical properties is that complicated measurements on whole blood are replaced by relatively straightforward absorption measurements on non-scattering haemoglobin solutions. However, both our calculations and whole blood measurements with IS/iMC require assumptions in the analysis method (as discussed for IS/iMC in ‘Compilation of optical property spectra from literature’ section). In our method, a choice for scattering theory and radial distribution function must be made.
Our calculated spectra of the scattering coefficient μ s are in reasonable agreement with the compiled spectra from literature (Fig. 3d). The order of magnitude is the same over the whole wavelength range that is considered, and all spectral features occur at the same wavelengths. The largest deviations are found in the wavelength range where the absorption of blood is strong compared to its scattering (250–600 nm). The same discrepancies are found in the spectrum of the scattering anisotropy g (Fig. 4b). Interestingly, these deviations are less prominent in the compounded parameters μ s′ (reduced scattering coefficient, Fig. 4c) and μ eff (effective scattering coefficient, Fig. 4d). Differences between the compiled and calculated spectra of μ s and g may be caused either by false estimations of the phase function in the iMC analysis of the literature spectra and/or assumptions in our theoretical estimations.
In general, the input to Mie theory (or any other scattering formalism) is the complex refractive index m(ω) = n(ω) + iκ(ω) of the particle and of the suspending medium. In our calculations, its real part is obtained via Kramers–Kronig transformation of the imaginary part, which in turn is obtained from the absorption coefficient of haemoglobin (Eqs. 4 and 5). Meinke et al.  also calculated the scattering properties of blood using the Mie theory, using the experimentally determined values of the real part of the refractive index from haemoglobin solutions by Friebel et al.  (Fig. 2a). These measurements suggest that it can be expected that the refractive index of haemoglobin solutions will increase for wavelengths <300 nm, similar to the refractive index of plasma/water. This is not found in our calculations because haemoglobin absorption spectra (and thus of the imaginary part of the refractive index) are only available down to 250 nm. If these spectra would be available down to wavelengths overlapping with the water absorption in the UV, a similar increase in the refractive index is expected to be found. For this reason, we caution the use of our calculated spectra below 300 nm. However, the values for the refractive index of haemoglobin solutions from Friebel et al.  are on average 0.02 higher in magnitude than the values found through our Kramers−Kronig analysis (Fig. 2a), which Friebel et al. ascribed to sample preparation. Applying the experimentally determined refractive index of Friebel et al. in our analysis would therefore result in unrealistically high values for μ s (Fig. 2b).
Choosing between the compiled and calculated spectra
Since the primary aim of this review is to provide the reader with a set of optical property spectra for whole blood that can be used in the practice of biomedical optics, the question remains which spectra the reader should choose from the provided results. For μ a, we present only compiled literature spectra of oxygenated blood and deoxygenated whole blood (Fig. 1c). Hence, our logical advice is to use these compiled spectra. For μ s, g, μ s′ and μ eff however, we present both the compiled and the calculated Kramers–Kronig/Percus–Yevick spectra (Figs. 3d and 4b–d, respectively). The compiled spectra, as well as the calculated spectra rely on individual assumptions in their analysis. At present, we cannot assess which method provides the most reliable results. It is therefore difficult to draw any solid conclusions on the choice between the compiled and calculated spectra for μ s, g, μ s′ and μ eff.
In the wavelength below 600 nm, both the calculated and compiled experimental spectra of all optical properties show strong spectral fluctuations. This is expected, since the optical properties are strong functions of the complex refractive index. The real (n) and imaginary part (κ) of this quantity are interdependent on grounds of causality and as expressed by the Kramers–Kronig relations. The spectrum of κ can be easily obtained from the well-established absorption spectrum of haemoglobin solutions using Eq. 4. The spectrum of n is available through calculations (this work) and has been determined experimentally  as shown in Fig. 2a. Both methods show fluctuations in n(λ) around the large absorption peaks of haemoglobin with a modulation depth of 0.01–0.05 around their respective baseline values. To the best of our knowledge, no scattering theory applied to blood (cells) predicts the magnitude of the fluctuations in the compiled literature spectra of μ s and particularly g using these input values. We hypothesize that this is largely due to the inverse Monte Carlo procedures as discussed in the ‘Compilation of optical property spectra from literature’ section. We therefore argue that our calculated spectra may provide a more consistent estimation of μ s, g, μ s′ and μ eff for the wavelength range of 300–600 nm.
In this article, we provided a critical review, compilation and calculation of the optical properties of whole blood (hct = 45 %). An important conclusion from our review study is that the optical properties of blood are influenced by a large variety of factors of both physical and methodological origin (‘Factors influencing the optical properties of blood’ section). One should always be aware of these factors when relying on literature spectra of μ a, μ s and g or when performing one’s own optical property measurements on blood.
For two important factors of influence—the effects of absorption flattening and dependent scattering—we provided practical formulas for rescaling literature spectra that have been obtained from haemolysed and diluted blood, respectively. Our theoretically derived formula for the influence of dependent scattering on μ s is in good agreement with the previously reported empirical relation by Steinke et al. .
The main results of this article are the compiled spectra for the μ a of oxygenized and deoxygenized whole blood (Fig. 1c) and both the compiled and calculated spectra for μ s (Fig. 3d), g (Fig. 4b), μ s′ (Fig. 4c) and μ eff (Fig. 4d) of whole blood. We argue that our calculated spectra may provide a better estimation of μ s, g, μ s′ and μ eff in the wavelength range of 300–600 nm. The compiled and/or calculated spectra of μ a, μ s and g have been tabulated in the Appendix of this article. From that, the spectra for μ s′ and μ eff can be easily calculated. With that, we hope that we have provided the reader with a set of optical property spectra for whole blood that can be used in the practice of biomedical optics.
N. Bosschaart is supported by the IOP Photonic Devices program managed by the Technology Foundation STW and AgentschapNL (IPD12020).
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