1.

Introduction

When photons propagate through a turbid medium they interact with both static and moving scatterers. Interaction with a moving scatterer, such as a red blood cell (RBC), causes a frequency shift according to the Doppler principle. For a given RBC velocity, the Doppler shift depends on the statistical variations in the scattering angle. Using coherent light, the backscattered photons will form a time-varying speckle pattern that generates a photodetector current that beats with the Doppler frequency.¹ The microcirculatory perfusion, defined as the product of the mean velocity and concentration of moving RBCs, can in theory be assessed from the integrated frequency-weighted power spectrum of the photocurrent, i.e., the first moment of the laser Doppler spectrum. ^{2, 3, 4, 5} In traditional laser Doppler flowmetry (LDF), the perfusion is measured in relative units. Therefore, no information on the absolute RBC velocity or velocity distribution is obtained.

Due to the complex relationship between the RBC velocity and the scattering angle, Dörschel and Müller⁶ developed a method for correcting the laser Doppler spectrum to obtain a more one-to-one relationship between the Doppler spectrum and the corresponding velocity. They assumed an isotropically distributed angle between the velocity and the scattering vector, and a constant scattering angle; the latter assumption being an oversimplification. Furthermore, they integrated the Doppler spectrum over selected frequency intervals (higher frequencies giving an emphasis on higher RBC velocities) to achieve a velocity-resolved tissue perfusion measure. Others^{7, 8, 9} have suggested LDF algorithms that estimate the average RBC velocity by comparing measured and simulated Doppler spectra. These algorithms are only valid/validated for large source detector separations $(>15\mathrm{mm})$ , and thus, they are not suitable for implementation in standard LDF systems, where small source detector separations $(<2\mathrm{mm})$ are used. In addition, these algorithms are capable of estimating only the average RBC velocity and not the RBC velocity distribution.

The method presented in this paper is based on the single Doppler scattering event but with fewer simplifications than the Dörschel and Müller approach. We assume an isotropic distribution of the Doppler-scattering vectors compared to the direction of the moving scatterer, and a negligible probability for multiple Doppler-scattering events. The theory is developed and tested for moving scatterers being either microspheres of a known size or RBCs. As for the scattering angle distribution, the analytical Mie theory is used for the microspheres, and the RBC scattering data are fitted to a Gegenbauer kernel phase function. The theory relates the power spectrum of the detected photocurrent, i.e., the Doppler spectrum, to the distribution of Doppler frequencies, which can be simulated using the Monte Carlo technique. A set of base functions, consisting of simulated Doppler spectra arising from known velocities (in absolute units; millimeters per second) are fitted to a measured Doppler spectrum. The traditional LDF perfusion measure can then be replaced by the relative magnitude of the base spectra, i.e., a velocity-resolved perfusion measure is achieved. The maximum number of base functions is limited to avoid overfitting.

The aim of the study reported here was to propose a method for characterizing the microvascular blood flow in a number of velocity components, rather than a single mean value, as obtained in traditional LDF. The method assesses the relative number of scatterers moving with physiologically relevant, absolute velocities (in millimeters per second).

2.

Material and Methods

2.1.

LDF Theory

When coherent laser light illuminates a turbid medium, a fraction of the injected light will be diffusely backscattered. If a detector is placed above the medium, an interference pattern, or speckle pattern, will be formed on the detector surface. By introducing moving scatterers in the medium, Doppler shifts will occur, and a portion of the backscattered photons will be shifted slightly in frequency. The size of an individual Doppler shift is given by¹⁰

Eq. 1

$${\omega }_d=\frac{4\pi }{{\lambda }_t}v\sin (\theta ∕2)\cos \left(\varphi \right),$$

where ${\lambda }_t$ is the wavelength in the medium, $v$ is the velocity of the moving scatterer, $\theta$ is the scattering angle, and $\varphi$ is the angle between the scattering vector $\left(\mathit{q}\right)$ and the direction of the moving scatterer (Fig. 1 ). In a turbid medium, $v$ , $\theta$ , and $\varphi$ are described by physiological and statistical distributions. Consequently, backscattered light will contain photons with diverse frequency shifts, resulting in a total $E$ field containing a distribution of frequencies rather than a single Doppler frequency.

Fig. 1

Doppler scattering by a single scatterer moving with the velocity $v_s$ ; $\theta$ denotes the scattering angle, $\varphi$ is the angle between the scattering vector $\mathit{q}$ and the direction of the moving scatterer.

The frequency shifts, produced by the moving scatterers, will cause the speckle pattern to fluctuate, generating a time-varying detector current with a frequency content that resembles the Doppler histogram of the detected $E$ field. It has previously been shown^{1, 11} that the power spectrum $P\left(\omega \right)$ of the detected photocurrent, also referred to as the Doppler spectrum, is linked to the distribution of Doppler frequencies, i.e., the Doppler histogram $H\left(\omega \right)$ , by

Eq. 2

$$P\left(\omega \right)={\mathit{q}}_{\mathrm{ac}}H\left(\omega \right)\star H{\left(\omega \right)}^{*},$$

where $\star$ denotes cross-correlation, and ${\mathit{q}}_{\mathrm{ac}}$ is an instrumental constant. The Doppler histogram (optical spectrum), consisting of a nonshifted static part $[H_0=H({\omega }_0\left)\right]$ and a shifted part $[H_d=H(\omega \ne {\omega }_0\left)\right]$ , can be expressed as

Eq. 3

$$H\left(\omega \right)=H{\left(\omega \right)}^{*}=H_0\delta (\omega -{\omega }_0)+H_d\left(\omega \right),$$

where $\delta$ is the dirac function, and ${\omega }_0$ is the frequency of the injected laser light. If the degree of Doppler-shifted photons is low, i.e., $H_0⪢{\Sigma }_{\omega }{H}_d\left(\omega \right)$ , $P\left(\omega \right)$ can be approximated by

Eq. 4

$$P\left(\omega \right)={\mathit{q}}_{\mathrm{ac}}[H_0^2\delta \left(\omega \right)+2H_0{H}_d(\omega -{\omega }_0)+H_d\left(\omega \right)\star H_d\left(\omega \right)]\approx {\mathit{q}}_{\mathrm{ac}}[{\mathrm{dc}}^2\delta \left(\omega \right)+2dc{H}_d(\omega -{\omega }_0)],$$

where $\mathrm{dc}={\Sigma }_{\omega }H\left(\omega \right)$ is the theoretical total light intensity. In the classic LDF measurement setup, the stationary part of Eq. 4 is removed by differentiation and highpass filters, i.e., ${\mathrm{dc}}^2\delta \left(\omega \right)=0$ . Further, the amplitude of the Doppler histogram scales linear against the total light intensity that is injected into the sample. By normalizing the Doppler histogram with the total light intensity, variations due to light intensity perturbations are removed.

In this paper, only cases with a low degree of Doppler-shifted photons were considered. Hence, the approximation

Eq. 5

$$P_n\left(\omega \right)=\frac{P\left(\omega \right)}{{\mathrm{dc}}_m^2}\approx \frac{2{\mathit{q}}_{\mathrm{ac}}}{{\mathit{q}}_{\mathrm{dc}}^2}\frac{H_d\left({\omega }^{\prime }\right)}{\mathrm{dc}}=\frac{{\mathit{q}}_{\mathrm{ac}}}{{\mathit{q}}_{\mathrm{dc}}^2}{H}_{\mathrm{nd}}\left({\omega }^{\prime }\right),$$

can be used for studying the true Doppler frequency distribution, where ${\mathrm{dc}}_m={\mathit{q}}_{\mathrm{dc}}$ dc is the measured total light intensity and ${\mathit{q}}_{\mathrm{dc}}$ an instrumental constant. The intensity-normalized Doppler spectrum and histogram are denoted as $P_n$ and $H_{\mathrm{nd}}$ , respectively.

According to Eq. 1 there is a direct link between the width of the Doppler histogram and the velocity of the moving scatterers. Hence, it is logical to assume that velocity information can be extracted from an LDF measurement by studying the laser Doppler spectrum.

3.

Results

The optimal Gegenbauer kernel phase functions, derived as an average over all RBC measurements (Sec. 2.4.1), was found at ${\alpha }_{\mathrm{gk}}=1.05$ and $g_{\mathrm{gk}}=0.93$ , yielding an anisotropy factor of $⟨\cos \theta ⟩=0.987$ . In Fig. 2 , the optimized Gegenbauer kernel RBC phase function is shown along the anisotropy-equivalent Henyey-Greenstein phase function $(⟨\cos \theta ⟩=0.987)$ and the microsphere Mie phase function $(⟨\cos \theta ⟩=0.494)$ .

Fig. 2

Phase functions of polystyrene microspheres ( $p_{\mathrm{mie}}$ : Mie theory with $⟨\cos \theta ⟩=0.494$ ) and RBCs ( $p_{\mathrm{gk}}$ : optimized Gegenbauer kernel distribution with ${\alpha }_{\mathrm{gk}}=1.05$ , $g_{\mathrm{gk}}=0.93$ and $⟨\cos \theta ⟩=0.987$ ; $p_{\mathrm{hg}}$ : Anisotropy equivalent Henyey-Greenstein distribution with $⟨\cos \theta ⟩=0.987$ ).

A good agreement between simulated and measured single-tube flow Doppler spectra was found for both the polystyrene microspheres (setup 1) and the RBCs, as demonstrated in Fig. 3 . In a few exceptions, a small deviation in the lower frequencies (below $200\mathrm{Hz}$ ) was found.

Fig. 3

Measured (solid lines) and simulated (dashed lines) single-tube flow Doppler spectra for the polystyrene microsphere (setup 1) and RBC solutions. The simulated spectra were derived using a phase function based on Mie theory (microsphere) and a Gegenbauer kernel phase function (RBC), in combination with a parabolic flow profile with an average flow velocity equal to the observed flow velocity (polystyrene microsphere: $2.51\mathrm{mm}∕\mathrm{s}$ ; RBC: $2.38\mathrm{mm}∕\mathrm{s}$ ).

The velocity component estimation is based on the assumption that the detected power spectrum consists of superimposed, single-velocity Doppler histograms [Eq. 6]. The validity of this assumption is strengthened by the results from the combined single and double-tube flow measurements (setup 4), as indicated in Fig. 4 , where the additivity is shown.

Fig. 4

Example showing the additivity of Doppler spectra. The two single-flow spectra, measured at a velocity of $1.5\mathrm{mm}∕\mathrm{s}$ $\left(P_1\right)$ and $7.7\mathrm{mm}∕\mathrm{s}$ $\left(P_2\right)$ , respectively (setup 4), add up to a spectrum $(P_1+P_2)$ similar to a measured double-flow spectrum $\left(P_{12}\right)$ .

A single-velocity component was estimated from the LDF data derived in experimental setup 1, using the observed flow velocity and a parabolic flow profile for simulating each reference spectrum. The result, displayed in Fig. 5 , showed a close similarity between the VC and CMBC behavior. The results also displayed a slowly decreasing VC and CMBC with increasing velocities, even though a constant microsphere concentration was used.

Fig. 5

Estimations of CMBC and a single VC, derived from measurements at a single-tube flow of polystyrene microspheres with a velocity ranging from 0 to $10\mathrm{mm}∕\mathrm{s}$ (setup 1). Each VC was estimated using a single reference spectrum, based on the observed flow velocity and a parabolic flow profile.

The results from the single-tube flow measurements (setup 2) showed a linear relationship between VC and the microsphere concentration (Fig. 6 ). The velocity component was estimated using a single reference spectrum, based on the observed flow velocity $0.91\mathrm{mm}∕\mathrm{s}$ and a parabolic flow profile.

Fig. 6

Estimations of CMBC and a single VC derived from measurements at a single-tube flow of polystyrene microspheres with different concentrations (setup 2). VC reference spectra were simulated using the observed flow velocity $0.91\mathrm{mm}∕\mathrm{s}$ and a parabolic flow profile.

Previously presented VC results are calculated from spectra that were simulated using the observed flow velocity, an approach that is only applicable for model verification. In a real measurement setup, a few fixed velocity components must be selected. The results from VC estimations (setup 1), using reference spectra calculated from the three fixed velocities 0.5, 2, and $8\mathrm{mm}∕\mathrm{s}$ , are presented in Fig. 7 . The three reference spectra were derived using a parabolic flow profile.

Fig. 7

Estimations of CMBC and three fixed VC ( $c_{v_1}$ , $c_{v_2}$ , and $c_{v_3}$ ) derived from measurements at a single-tube flow of polystyrene microspheres with a velocity ranging from 0 to $10\mathrm{mm}∕\mathrm{s}$ (setup 1). The three reference spectra were simulated using a parabolic flow profile and the fixed velocities: $v_1=0.5\mathrm{mm}∕\mathrm{s}$ , $v_2=2\mathrm{mm}∕\mathrm{s}$ , and $v_3=8\mathrm{mm}∕\mathrm{s}$ .

An example of the combined single and double-tube flow experiments (setup 3), using a polystyrene microsphere solution, is presented in Fig. 8 . The displayed result was derived using simulated spectra that were fitted to the measured double-tube flow Doppler spectrum, i.e., no fitting against the single-tube flow measurements was conducted. Two reference spectra ( $P_{v_1}$ and $P_{v_2}$ ) were simulated using the observed flow velocities $v_1=1.5$ (tube 1) and $v_2=7.7\mathrm{mm}∕\mathrm{s}$ (tube 2), and a parabolic flow profile. Both the fitted spectrum and its two components ( $P_{v_1}{c}_{v_1}$ and $P_{v_2}{c}_{v_2}$ ) showed a good agreement with the measured double- and single-tube flow spectra. The results from the other measurements that were conducted using setup 3 displayed a similar agreement when the relative difference between the velocities of the two tube flows was large.

Fig. 8

Decomposed measured double-tube flow spectrum (measured, $P_{12}$ ; component 1, $P_{{\mathrm{VC}}_1}=c_{v_1}{P}_{v_1}$ ; component 2, $P_{{\mathrm{VC}}_2}=c_{v_2}{P}_{v_2}$ ) as compared to measured single-tube flow spectra ( $P_1$ and $P_2$ ), using setup 3. The reference spectra ( $P_{v_1}$ and $P_{v_2}$ ) were simulated using the observed flow velocities $v_1=1.5$ and $v_2=7.7\mathrm{mm}∕\mathrm{s}$ , and a parabolic flow profile. (solid: measured spectrum; dotted: simulated spectrum).

The results from the double-tube measurements, where a fixed microsphere concentration in tube 1 and an alternating microsphere concentration in tube 2 was used (setup 4), showed that it is possible to differentiate the two flows if the flow velocity differs between the tubes. The VC estimation algorithm, using the observed velocity of the two tubes (tube 1: $0.91\mathrm{mm}∕\mathrm{s}$ ; tube 2: $2.90\mathrm{mm}∕\mathrm{s}$ ) and a parabolic flow profile for simulating the two reference spectra, showed a linearly increasing concentration for the tube 2 component $\left(c_{v_2}\right)$ . No such increase was found in the tube 1 component $\left(c_{v_1}\right)$ , which displayed a constant behavior. The variations $\left[sd\right(c)∕\overline{c}]$ in component $c_{v_1}$ ranged 11 to 23%. An example of the results is demonstrated in Fig. 9 .

Fig. 9

Two VC derived from a double-tube flow measurement series where the microsphere concentration in tube 1 was fixed $({\mu }_s=0.12{\mathrm{mm}}^{-1})$ , while the microsphere concentration in tube 2 was altered (setup 4). The two velocity components ( $c_{v_1}$ and $c_{v_2}$ ) were estimated using two reference spectra based on the observed flow velocities $v_1=0.91$ (tube 1) and $v_2=2.90\mathrm{mm}∕\mathrm{s}$ (tube 2), and a parabolic flow profile.

4.

Discussion

In this paper, we proposed a method for separating different flow velocities in LDF. Typically, three concentrations, each associated with a predefined, physiologically relevant, absolute velocity (in millimeters per second), can be estimated simultaneously. The method is based on an analysis of the LDF Doppler spectrum, where a linear combination of predefined simulated single-flow spectra are fitted to a measured spectrum. Numerous measurements were conducted on both single- and double-tube flow phantoms, using a polystyrene microsphere solution to validate the algorithm. The presented theory was also applied to RBC flow measurements to find an RBC phase function that produced simulation results that were consistent with the results from the measurements. The method was developed for low concentrations of scatterers (single Doppler scattering), but can theoretically be extended to the multiple Doppler scattering situation, as discussed in the following.

The simulation model includes an assumption of isotropic direction of the light. According to diffusion theory, this assumption is only valid if the photons have¹⁷ traveled at least $1{\mathrm{mfp}}^{\prime }$ . Based on the geometry of the flow phantoms, a reduced scattering coefficient of at least $0.4{\mathrm{mm}}^{-1}$ is required, assuming negligible absorption. Preliminary oblique-angle measurements¹⁸ indicate that the reduced scattering coefficient of the flow phantoms is about $1{\mathrm{mm}}^{-1}$ , i.e., the isotropic assumption holds. The simulation of the single-tube flow velocity spectrum also includes an assumption of a parabolic flow profile inside the tube. A maximal Reynolds number of 11 indicates that this assumption is valid. However, the fluid contains particles that might cause perturbations in the flow profile, even though the particle sizes are much smaller than the size of the tubes.

In the in vivo case, the RBCs tend to aggregate in the center of the microcirculatory vessels.^{19, 20} As a consequence, the RBC velocity distribution will be nonparabolic with an average RBC velocity greater than the flow velocity of the plasma. Therefore, in an in vivo setup, it is more appropriate to use a fixed velocity rather than the assumed parabolic profile. However, the resulting velocity components will not directly reflect the activity in either the capillaries or the arterioles, but rather the amount of RBC that flows with a specific velocity. Nevertheless, such a measure will contain valuable information about the microcirculation that cannot be found in the traditional integrated LDF perfusion measure.

The phase function of moving scatterers is difficult to characterize experimentally. Therefore, by using polystyrene microspheres, with a known radius and refractive index, we were able to validate our model with an analytical phase function derived using Mie theory. The results showed a close match between the single-tube flow measurements and the simulations, with a few exceptions where a mismatch occurred in frequencies below $200\mathrm{Hz}$ . The origin of this mismatch is unclear to us, but possible explanations are disturbances in the laminar flow and/or air bubbles in the fluid. Nevertheless, our results indicate that the parabolic flow profile and isotropic assumptions are legitimate and that our model is valid in a low-concentration setup.

The results in this paper show that the Gegenbauer kernel phase function is suitable for modeling Doppler-scattering by red blood cells. By freely fitting the two Gegenbauer kernel parameters to measured LDF data, resulting in ${\alpha }_{\mathrm{gk}}=1.05$ and $g_{\mathrm{gk}}=0.93$ at $786\mathrm{nm}$ , we were able to achieve a good agreement between simulated and measured Doppler spectra. Comparing the LDF perfusion measure⁵ (the first moment of the Doppler spectrum) generated by a Gegenbauer kernel phase function and a Heneye-Greenstein phase function with an identical anisotropy resulted in 3.6 times higher values for the Gegenbauer kernel phase function. This indicates the importance of using an accurate RBC phase function when simulating LDF perfusion and LDF Doppler spectra.

It was previously reported that Doppler shifts generated by moving polystyrene microspheres can be used for a precise estimation of the anisotropy factor.¹⁶ Kienle found an anisotropy factor of $0.993\pm 0.001$ at $820\mathrm{nm}$ for diluted blood by using the two-parameter Reynolds-McCormick phase function.²¹ Others have reported that the two-parameter Gegenbauer kernel phase function is suitable for representing single scattering^{12, 13} by RBCs. Double sphere measurements, conducted by Roggan, ¹² showed that an ${\alpha }_{\mathrm{gk}}$ of 1 yielded the closest match between measurements and theory. They found an anisotropy factor of $0.992⩽g⩽0.994$ at $633\mathrm{nm}$ for hematocrit values below 10. We found an ${\alpha }_{\mathrm{gk}}$ of 1.05, which is in good agreement with their results. However, we found an anisotropy factor of 0.987 at $786\mathrm{nm}$ , which is lower than that of both Kienle and Roggan, but in good agreement with the anisotropy of 0.985 at $633\mathrm{nm}$ found by Steinke and Shepherd.²² Some of these deviations are, of course, explained by the differences in wavelength. There are also methodologically induced errors due to the use of saline solution, which has a slightly different refractive index than blood plasma.¹³ In addition, the phase function for whole blood, where the separations between red blood cells are small, is affected by interference between neighboring erythrocytes.²³ Hence, to find an appropriate phase function for modeling Doppler shifts by RBCs in real tissue, further studies are needed.

The proposed velocity component estimation method is capable of accurately estimating the relative concentration of moving scatterers in a single-tube flow phantom (Fig. 6). However, a slight dependency between the estimated concentration and the flow velocity was found (Fig. 5). The origin of this dependency is unclear, but a similar dependency was also found for the classical CMBC measure. Possibly, this might be explained by the extremely low degree of Doppler-shifted photons, a feature that results in an overall poor SNR. This ratio also decreases with increasing flow velocity. By increasing the concentration of moving scatterers, the degree of Doppler-shifted photon can be raised. However, such an approach can cause unwanted nonlinear spectral effects due to multiple Doppler shifts and homodyne detection. The degree of Doppler-shifted photons²⁴ was estimated to be less than 0.4% in our setup.

Results from the double-tube flow measurements, using two velocity components, show that the LDF algorithm is capable of distinguishing two different tube flow velocities (Fig. 8) and to accurately estimate a microsphere concentration increase in one of the tubes (Fig. 9). Further, an estimate of at least three concentration measures, each reflecting the number of moving scatterers having a velocity close to a predefined flow velocity, can be achieved. This is demonstrated in Fig. 7, where the three velocity components 0.5, 2, and $8\mathrm{mm}∕\mathrm{s}$ are estimated from measurements on a single-tube flow model with alternating flow velocity. The true flow velocities, where the velocity component peak values are found, coincide well with their respective predefined velocities. In addition, the three peak values are almost equal in amplitude. Hence, the proposed LDF algorithm produces velocity-accurate concentration measures.

Our method was developed for low degrees of Doppler-shifted photons. For higher degrees of Doppler-shifted photons, nonlinearities are introduced due to homodyne mixing (shifted photons that are mixed with shifted photons) and multiple Doppler shifts. To make the velocity component algorithm valid in an in vivo measurement setup, where a high RBC concentration is expected, this limitation must be overcome. The homodyne mixing effect can be removed by deconvolution of the detected Doppler spectrum. A possible solution to the nonlinearities caused by the multiple Doppler shift is to adjust the simulated single-tube flow velocity spectra used in the fitting procedure. Serov ²⁴ previously presented a way of estimating the degree of Doppler-shifted photons. By combining their results with a Poisson distribution assumption regarding the distribution of multiple Doppler shifts, simulations of multiple Doppler-shifted reference spectra might be achieved.

The classical LDF perfusion measure claims to estimate the RBC concentration times the RBC average velocity. It is a nonabsolute measure that often is calibrated against a motility standard solution consisting of polystyrene microspheres diluted in water. Thus, the perfusion measure is related to the phase function of a microsphere rather than an RBC. The polystyrene microspheres used in this paper resulted in an almost 15 times higher LDF perfusion estimate compared to the RBCs at a similar velocity. Instead, our approach takes into account the phase function of the RBC, which potentially enables the measurement of absolute RBC velocities. However, it is still a nonabsolute measure when it comes to estimating the concentration of moving scatterers at a certain velocity. To achieve an absolute RBC concentration measure, the photon path length in tissue is required.^{5, 25} Nilsson ²⁶ and Larsson ²⁷ showed that it is possible to estimate the photon path length by studying the spatially resolved diffuse reflectance profile, using a multichannel LDF probe. Hence, in theory, it is possible to measure microcirculatory velocities and concentrations in absolute units using LDF, by combining these techniques. However, in real tissue, where the blood flow is localized to the microcirculatory blood vessels, the applicability can be questioned. The robustness of the proposed LDF algorithm must be evaluated further in a tissue-like heterogeneous model, as can be achieved through Monte Carlo simulations.

An attempt to create a velocity-resolved perfusion measure was previously conducted by Dörschel and Müller.⁶ Their approach was based on the differential quotient of the LDF Doppler spectrum to correct for the isotropic nature of the scattering vector. Hence, they only partially solved the problem, as they did not account for the phase function of the moving scatterers. Their approach can be questioned since it is only theoretically valid for a fixed relation between the scattering vector and the direction of the moving scatterer. Yet, they present convincing results from phantom measurements, using a piece of static scattering thin ${\mathrm{Teflon}}^{\circledR }$ foil placed above a moving piece of paper. The dimension and optical properties of their ${\mathrm{Teflon}}^{\circledR }$ foil is, however, not specified, and thus, it cannot be excluded that their results originate from a nonphysiologically relevant phantom setup, where the scattering vector and the direction of the moving scatterer has a fixed nonisotropic relationship.

Several authors previously suggested algorithms that estimate the blood flow root mean square (rms) velocity $v_{\mathrm{rms}}=⟨v_{\mathrm{RBC}}^2{⟩}^{1∕2}$ in absolute terms. Lohwasser and Soelker⁷ matched measured Doppler spectra with Monte Carlo simulations and diffusing wave spectroscopy results. They found that by using a large emitting-receiving fiber separation (25 to $60\mathrm{mm}$ ) it was possible to estimate $v_{\mathrm{rms}}$ . However, they did not consider that the Doppler spectrum depends on the optical properties ( ${\mu }_a$ and ${\mu }_s^{\prime }$ ) of the turbid medium, as shown by Kienle.⁸ Kienle proposes that by matching simulated spectra, derived using the correlation diffusion equation, with measured spectra, and simultaneously determining the optical properties, an accurate $v_{\mathrm{rms}}$ estimate can be achieved. This approach is, however, valid only for large emitting-receiving fiber separations, i.e., a high degree of multiple Doppler shifts, where the Doppler spectrum only depends on ${\mu }_s^{\prime }$ and not the anisotropy.^{8, 28} Binzoni ⁹ suggested an alternative approach that takes multiple Doppler shifts into account. Their approach, based on a theoretical framework by Bonner and Nossal³ assumes a Gaussian RBC velocity distribution in the tissue and that the RBC phase function is described by the Rayleigh-Gans approximation. However, the Rayleigh-Gans approximation is accurate only in describing RBC scattering in the 0- to 4-deg range,¹³ and thus, this approach might not be suitable for small source detector separations (low degrees of multiple Doppler shifts) where the phase function has a profound influence on the Doppler spectrum. It has also been shown that $v_{\mathrm{rms}}$ is estimated by the ratio between the first moment of the Doppler spectrum and the zeroth moment.²⁹ However, this is valid only if the degree of multiple Doppler shifts is negligible, since the zeroth-moment concentration estimate rapidly becomes nonlinear if multiple Doppler shifts are present. By applying a linearization algorithm^{4, 24} to the LDF concentration measure, it might be possible to use the technique in LDF probe setups similar to ours. Still, this approach is capable of estimating only the average RBC velocity, and not multiple velocity components.

What are the possible clinical implications of measuring microcirculatory blood flow in absolute units? Clinicians are used to measuring blood flow in larger vessels in absolute units. By measuring the microcirculatory flow velocity in absolute units, a better acceptance and integration of the method with other blood flow measuring methods can be assumed. In addition, the proposed algorithm might be used for monitoring the tissue perfusion associated with a specific blood flow velocity interval. As the blood flow velocity depends on the dimension of the blood vessel, it might be possible to differentiate between microcirculatory compartments. If successful, this would enable the estimation of the true nutritive capillary blood flow and the detection of shunted blood flow, as associated with diseases and conditions affecting the microcirculation, such as diabetes and leg ulcers.

5.

Conclusions

We proposed an LDF algorithm for velocity-resolved perfusion measurements. The proposed method yields three concentration measures, each associated with a predefined, physiologically relevant, absolute velocity in millimeters per second. Validation measurements, using both single- and double-tube flow phantoms and a microsphere solution, showed that it is possible to track velocity and concentration changes, and to differentiate between flows with different velocities. In vivo, this might enable the differentiation between capillary and arterial blood flow, as the velocity depends on the dimension of the blood vessel. However, further studies are needed to validate the applicability in real tissue. The presented theory was also applied to RBC in vitro flow measurements. A Gegenbauer kernel phase function ( ${\alpha }_{\mathrm{gk}}=1.05$ ; $g_{\mathrm{gk}}=0.93$ ), with an anisotropy factor of 0.987 at $786\mathrm{nm}$ , was found suitable for modeling Doppler scattering by RBCs diluted in saline solution. The method was developed for low concentrations of RBCs, but can in theory be extended to cover multiple Doppler scattering. The current LDF algorithm yields only concentration measures that are relative, but by adding a path length estimation technique, a velocity-resolved absolute LDF perfusion measure might be achievable.