A literature review and novel theoretical approach on the optical properties of whole blood

All spectra were resampled to a 1-nm increment wavelength axis. Depending on the description of sample concentration in hct or total haemoglobin (tHb), the μ_a spectra were rescaled to a hct of 45 % or an equivalent tHb concentration of 150 g/L. Linear rescaling of μ_a spectra with respect to hct = X%, i.e. μ_a,hct = 45 % = (45/X) and μ_a,hct = X%, yields incorrect results if the absorption by the medium (water or plasma) cannot be neglected. This leads to an overestimation of the μ_a spectra at wavelengths where water absorption is substantial (λ > 1,100 nm). We therefore perform a correction for the water absorption on the linearly rescaled μ_a spectra:

Here, μ_{a,hct = 45 %} is the rescaled μ_a spectrum to 45 % hct, μ_{a,hct = X%} is the literature μ_a spectrum at X% hct and μ_a,H2O is the absorption coefficient of pure water, for which we used the spectrum from Hale et al. [31]. The water volume fraction [f_blood] in blood with hct = X% is obtained using:where f_plasma and f_RBC are the water volume fractions in blood plasma and red blood cells, respectively. In our analysis, we used f_plasma = 0.90 and f_RBC = 0.66, which correspond to normal physiological water concentrations in plasma and red blood cells [7]. Equations 1 and 2 show that the correct scaling between haematocrits at a given wavelength depends on the absorption coefficient of water at that wavelength.

For the absorption spectra that were measured on non-scattering homogeneous haemoglobin solutions, also the absorption flattening effect should be taken into account when rescaling the μ_a to that of whole blood. Citing Friebel et al. [8], the absorption flattening effect can be described as: ‘when light passes through a suspension of absorbing particles, such as blood, photons that do not encounter red blood cells pass unattenuated by absorption. As a consequence, the transmitted light intensity is higher than it would be if all the haemoglobin were uniformly dispersed in the solution’, Duysens [32] quantitatively described the reduction of the absorption coefficient obtained from a suspension of particles, with respect to that of a solution in which in the same amounts of absorbing molecules are homogeneously distributed. Following the method of Duysens, adapting only the terminology, we arrive at:

Where μ_a,blood and μ_a,Hb are the absorption coefficient of a blood sample and haemoglobin solution, respectively, and μ_a,RBC is the absorption coefficient of the hemoglobin solution inside the red blood cell. The last two absorption coefficients are related through the haematocrit, μ_a,Hb = Hct∙μ_a,RBC. 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 [33] 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.

The available literature on optical property measurements within our inclusion criteria is summarized in Table 1, with the relevant information (based on the factors of influence that have been listed in the ‘Factors influencing the optical properties of blood’ section) that was available on measurement method and sample preparation. All spectra from samples with intact red blood cells 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 [6], the Gegenbauer–Kernel [7] and the Reynolds–McCormick phase function [8‐10]; details can be found in the respective references. The Gegenbauer–Kernel and the Reynolds–McCormick phase function cited in these publications are the same [34]. Compiled spectra of the absorption coefficient of Hb and HbO₂ solutions are available from Zijlstra [4] and Prahl [5].

Figure 1a, b displays the rescaled μ_a spectra to hct = 45 % for oxygenized blood (SO₂ > 98 %) and deoxygenized blood (SO₂ = 0 %), respectively. For both oxygenized and deoxygenized blood, the rescaled μ_a spectra of Roggan et al. [7] consistently overestimate the other μ_a spectra for nearly all wavelengths up to one order of magnitude. Roggan et al. obtained this overestimation with respect to pure haemoglobin and water solutions also for the original sample haematocrit of 5 %, and ascribed the difference to an increased probability of absorption due to elongated photon paths, resulting from internal photon reflections inside the red blood cells. When rescaling these values to hct = 45 %, the overestimation is magnified to unrealistically high values for μ_a, in spite of the applied correction for the water absorption. We therefore excluded the μ_a spectra of Roggan et al. from the compiled spectra.

The spectra of haemolysed blood from Zijlstra [4] and Prahl [5] 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 [5]. This difference has also been measured by Friebel et al. [8] 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. [9] 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.

Causality dictates a functional relationship between the real and imaginary parts of the complex refractive index. This relation is expressed by the Kramers–Kronig integral dispersion equations. The imaginary part κ(ω) of the complex refractive index m(ω) = n(ω) + iκ(ω) is related to the absorption coefficient μ_a through:where c is the speed of light and ω is the angular frequency of the light. We use a subtractive KK equation [17, 35], so that:where n(ω₀) is the refractive index at some reference frequency ω₀, providing scaling of the calculated spectra. P denotes the Cauchy principle value of the integral. Thus, knowledge of the absorption spectrum of the haemoglobin solution inside an RBC, in combination with a reference value for the refractive index, allows determination of the complex refractive index of the solution at any given frequency ω (or wavelength λ = c/ω).

The scattering properties of a single red blood cell (cross section and anisotropy) are calculated from the angularly resolved scattered intensity I_S(θ), per unit input intensity [36]. The scattering cross section [in square metre] is given by:where k is the wave number k = 2π/λ. By normalization of I_S(θ) on its 4π solid-angle domain, the phase function p_P(θ) is obtained, which is parameterized by the expectation value of the cosine of the scattering angle, the scattering anisotropy g [−]:

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 [37] 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.

We model light scattering of a blood medium by the angular resolved scattered intensity of a collection of N randomly distributed, identical particles:where E* denotes the complex conjugate of E. The ensemble average runs over all possible arrangements of the particles in volume V_T (that contains all particle contributing to the signal). E_s,_n denotes the scattered field amplitude of the nth particle, located at r_n. The scattering vector q has magnitude |q| = 2ksin(θ/2).

The terms m = n in the double sum define the light distribution when no interference between the scattered fields from different particles occurs, e.g. in a dilute medium. This condition is called ‘independent scattering’, and the total scattering cross section is simply N times the scattering cross section of a single particle. The scattering coefficient (or density of the scattering cross section, [in meter]) follows from μ_s = σ_s,TOTAL/V_T or:with hct the particle volume fraction and V_p the particle volume.

If the particles are closely spaced, or when correlations between the particle positions are present, the interference effects cannot be ignored. This condition, usually called dependent scattering in the biomedical optics literature, takes into account the m ≠ n terms as well. Their contribution depends on the ordering in the arrangement of the particles, characterized by the radial distribution function G(r) which describes the probability of finding two particles spaced a difference r apart. We may write [38, 39]:where |q| = 2ksin(θ/2). The term S(θ,hct) is called the structure factor, which thus allows to describe the angular scattering pattern from an ensemble of particles in terms of the scattered intensity pattern of a single particle, by applying a hct-dependent angular weighting of the scattered light. Combining Eq. 10 with Eq. 6, the scattering cross section for dependent scattering is found as:where γ(hct) is the haematocrit-dependent scaling factor between the scattering cross section for dependent scattering and independent scattering, and p_P(θ) is the single-particle phase function. The scattering coefficient follows as:

Expressions for the phase function and scattering anisotropy for dependent scattering can also be derived using the same methods.

Thus, the scattering properties of the blood medium can be calculated, provided a description for the radial distribution function G(r) is available (such as the Percus–Yevick model for non-deformable spheres used in this work). Scaling of the scattering coefficient between haematocrit values takes the following form for a blood medium:

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.

The best known is Twersky’s formula, which starts with the structure factor of Eq. 10 and employs a ‘small particle’ assumption replacing S(θ,hct) with S(0,hct) and uses p_P(θ) = (4π)⁻¹ [40‐42] so that the integral over the radial distribution function G(r) is evaluated at q = 0 (or θ = 0). Assuming scatterers of radius r_p that do not attract or repulse each other (‘gas model’), we have G(r) = 0 (r ≤ r_p); G(r) = 1 (r > r_p). This leads to the simple expression:

Following the same procedure using the radial distribution function of a collection of non-deformable small spheres gives:

Both relations have been tested and have been found to be only in moderate agreement with experimental results on blood [11]. Based on their experiments, Steinke et al. therefore provide the following empirical relation:

We compute γ(hct) at each wavelength using Eqs. 10 and 11 without restrictions on particle size, using the Mie phase function and the Percus–Yevick radial distribution function for non-deformable spheres. Averaging γ(hct, λ) over all wavelengths (250–2,000 nm) and both oxygenated forms and using a Levenberg–Marquardt non-linear least squares curve fitting procedure of γ(hct) versus hct yields the following approximation:

For completeness, we give the equation relating the scattering coefficient of a blood sample (assuming dependent scattering) of given haematocrit to the scattering cross section of a single RBC as:

Equations 17 and 18 are thus one of the main practical results of our work.

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 [17]. 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 [36] 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. [43] at 633 nm and by Meinke et al. [9] 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 [31] with an additional offset to achieve a value of 1.345 at 633 nm [43]. This agrees well with the values of Meinke et al. in the visible wavelength range.

We use the Percus–Yevick approximation [44], solved analytically by Wertheim [45], 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. [35]; the Mie code is benchmarked against the results from Prahl’s web-based Mie calculator [46].

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 [6], the Gegenbauer–Kernel [7] and the Reynolds–McCormick phase function [8‐10]; details can be found in the respective references. The Gegenbauer–Kernel and the Reynolds–McCormick phase functions cited in these publications are the same [34].

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).

The scattering coefficient of most biological tissues exhibits a power law dependency on wavelength in the wavelength regions where μ_a is low with respect to μ_s. This power dependency can be described by:with scaling factor a (in millimetre), scatter power b (dimensionless), scattering coefficient μ_s (in millimetre), wavelength λ and reference wavelength λ₀. To summarize the obtained μ_s spectra of blood, we determined the parameters a and b for both the calculated and compiled spectra. Hereto, we fitted Eq. 19 to the spectra beyond 700 nm with a least squares algorithm, using λ₀ = 700 nm. Error estimations were obtained from the 95 % confidence intervals of the fits.

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.

Figure 2a shows the results of the Kramers–Kronig analysis to obtain the real part of the complex refractive index and for reference the experimental values obtained by Friebel et al. [30]. Also shown is the refractive index of plasma. Subpanels b and c of Fig. 2, respectively, show the calculated spectra of the scattering coefficient and anisotropy for both oxygenated and deoxygenated blood. Figure 2d shows the reduced scattering coefficient μ_s′, obtained from the calculated spectra of μ_s and g. Results of the same calculations using the refractive index from Friebel et al. as input are also shown.

Figure 3a shows the unscaled literature spectra of μ_s at their original haematocrit values (all measured at SO₂ > 98 %). To rescale these spectra to hct = 45 %, we apply our Mie/Percus–Yevick scaling relation (Eqs. 13 and 17 combined), which has been displayed in Fig. 3b. For comparison, also a linear scaling relation (hct/X) and the scaling relations of Twersky (Eqs. 13 and 14 combined and Eqs. 13 and 15 combined) [40, 41] and Steinke et al. (Eqs. 13 and 16 combined) [11] have been displayed. Figure 3c shows that the rescaled literature spectra using Eqs. 13 and 17 are much closer together in magnitude, compared to the unscaled spectra in Fig. 3a. The compiled average μ_s spectrum from the rescaled spectra (N = 8) agrees well in shape and magnitude with the calculated μ_s spectra for those wavelengths where scattering dominates absorption (beyond 700 nm, Fig. 3d). Comparable to the compiled spectrum of μ_a, the jump in μ_s around 1,200 nm is an artifact of our compilation method, caused by the transition of the average of multiple spectra to fewer spectra.

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 λ₀ = 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 %).

Although a non-linear dependency of g on haematocrit has been reported (‘Background’ section), we do not perceive this effect in the literature spectra of g at the original haematocrit values (Fig. 4a). The compiled spectrum for g (Fig. 4b) is therefore obtained using all available, unscaled spectra (N = 9). Note that the data from Roggan et al. [7] are the only data contributing to the compiled spectrum beyond 2,000 nm (indicated by the dashed line in Fig. 4b). The large oscillations in g in this wavelength region are ascribed by Roggan et al. to ‘the small values of the measured quantities’, indicating a low signal-to-noise ratio for these wavelengths. Comparable to the compiled spectra of μ_a and μ_s, the jump in g around 1,200 nm is an artifact of our compilation method, caused by the transition of the average of multiple spectra to fewer spectra. The compiled literature spectrum of g agrees well in magnitude with the calculated spectra of g for those wavelengths where scattering dominates absorption (beyond 700 nm).

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 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. [7‐10]), 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 [34]) 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 [49‐51] and in vivo [52] 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.

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 [32] 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 [33]). The analysis of Duysens assumes random placement of the absorbing particles, with no correlations between their positions (Poisson distribution; so that the spatial variance σ² 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, σ² 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, σ² ultimately determines the flattening effect. In a whole blood medium, possible correlations between the particle positions lead to an increase in σ². This causes a further reduction in the measured absorption coefficient [53]. Interestingly, the increased σ² is determined by the volume integral of the radial distribution function G(r) [54], 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. [55] propose a correction factor introduced by Svaasand [56] 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.

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 [57] 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 [40]. 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. [11] is excellent.

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. [10] 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. [30] (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. [30] 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).

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 [30] 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.