I. Introduction

Several research groups have been working to develop laser Doppler light scattering instrumentation to measure flow within blood vessels[1]–[4] or vascularized tissue.[5]–[11] Particularly promising for clinical applications are noninvasive schemes,[3],[5]–[11] where a tissue surface is illuminated and information about blood flow is obtained by analyzing spectral components of backscattered light. Our own interests are to develop techniques for examining local blood flow in tissues which are vascularized primarily by small blood vessels, namely, capillaries, venules, and arterioles.[12],[13] Examples are skin, gums, skeletal muscle, and the surfaces of organs and body extremities. These entities typically glow with a diffuse backscattered pattern when irradiated with a narrow incident beam. Because the scattering cross section for particles of the size of blood cells—predominantly erythrocytes (red blood cells)—is sharply peaked in the forward direction (mean scattering angle 5.4°), such radiation mostly contains photons which have been scattered at least once by surrounding tissue elements. Very little (<0.1%) of the observed backscattered luminescence derives from photons scattered only by the flowing blood cells.

The optical structure of the complex tissue matrix which surrounds the blood vessels generally cannot be characterized in detail. However, we shall show that, if the tissue scattering cross section is much greater than that of the moving blood cells, a rather general model can be devised which is applicable to a wide variety of human and animal tissues. This model can guide instrumental design and use, particularly of a specialized electronic processor for extracting flow parameters from detected fluctuations in photocurrent.

Photons generally suffer several collisions with somatic cells, connective tissue, blood vessel walls, etc. before interacting with a blood cell [Fig. 1(A)]. The effect of such interactions is to randomize the direction of the light which is incident upon the moving erythrocytes. These photons can be scattered into a small forward angle by a blood cell and then scattered back toward the detector by the tissue matrix [Fig. 1(B)]. More complicated scenarios can be envisioned, including sequential scattering by the moving particles [Fig. 1(C)]. Here we examine a simplified theoretical model which takes these features into account, particularly the observed diffuse tissue luminescence and expected small angle scattering of the erythrocytes (Secs. II and III). We also discuss the consequences of photons scattering from more than one moving cell before emerging from the tissue (Sec. IV).

The model leads to predictions of the way in which photon autocorrelations (Secs. III and IV) and photocurrent power spectra (Sec. V) depend on variables of the moving particles such as angular scattering cross section, size, number density, and speed distribution. These theoretical implications have been tested experimentally with a tissue analog in which particles flow through a fiber capillary bundle surrounded by a medium which is a diffuse scatterer of light (agarose impregnated with 0.055-μm radius polystyrene microspheres). The results of these studies are given in Sec. VI.

In particular, we show that the normalized first moment of the spectrum, 〈ω〉, is proportional to the rms speed of the moving particles 〈V²〉^1/2 as

II. Theoretical Model

Several investigators have undertaken mathematical descriptions of the propagation of light in whole blood. The intense scattering and strong absorption which occurs at normal hematocrit values implies that photon diffusion theory or angle dependent transport theory must be used for modeling in that instance (see, e.g., the papers cited in [Ref. 14]). The present situation, i.e., the transmission and scattering of light in the microvasculature, is somewhat simpler because the number of red cells in moderately perfused tissue (0.5–4% by volume of tissue) is much lower than in whole blood (40% v/v). However, scattering of photons by tissue elements surrounding the blood vessels must be taken into account. Although the manner in which light diffuses throughout that tissue matrix cannot be specified in detail, if one assumes the latter to be stationary (or moving very slowly), considerable progress can be made in understanding how the observed photon autocorrelation function (or spectrum) relates to the movement of the blood cells.

We shall now show that the time autocorrelation function of detected photons which undergo a multiple step diffusion in tissue (Fig. 1: A,A,…A,B,A,…A) can be expressed in terms of an intermediate scattering function I₁(τ) given as

Equation (2) can be derived from the following argument. We consider each element of the static tissue matrix to be a separate irradiator. If all scattering centers within the tissue were immobile, the relative phase, in time, of the electromagnetic field incident from the laser would be preserved at all locations within the tissue; i.e., although the intensity of these distributed sources would differ at each location and a spatial point-to-point phase relationship could not be specified, the time coherence of each source would be homologous with the incident field. (The scattering would be described by the sequence A,A,…A shown in Fig. 1.) Changes in temporal phase occur because the red cells move [Fig. 1(B)]. These phase changes in effect are detected by other static tissue elements which, if no subsequent scattering by a red cell occurs, act as direct links to the external detector (a series of steps from Fig. 1: A,A,…,A,B,A,…A); that is, static tissue elements scatter to other static tissue elements and, ultimately, to the detector, yet preserving relative temporal phase in the same way that phase information is preserved in an incident pathway. Of course, each time that a photon interacts with another moving red cell, an additional time-dependent phase shift occurs.

In a distributed network of microvessels, succeeding temporal phase shifts [(Q · vt)] will be totally independent of earlier ones. In contrast, when light diffuses in large blood vessels, e.g., retinal vessels[3] or femoral veins,[4] it might be scattered by several red blood cells whose velocities and orientations are strongly correlated $(〈{\mathbf{\text{v}}}_1·{\mathbf{\text{v}}}_2〉\sim {\mathbf{\text{v}}}_1^2)$. However, if the lack of correlation of the scattering vectors Q₁ with Q₂ (i.e., 〈Q₁ · Q₂〉 = 0) from sequential red blood cell scattering is sufficient to randomize the series Doppler shifts in such large vessels, results developed here for uncorrelated scattering by cells in different microvessels nonetheless could be applicable.

The wave front emanating from any given elemental tissue source may be approximated by a plane wave propagating along the vector directed from the illumination center to an erythrocyte. Deviations from planarity are not significant as long as the distance between the tissue source point and the moving cell is large compared with the displacement ΔR(τ_e) = |r_j(τ_e) − r_j(0)|, where τ_e is the phase correlation time of the field (time such that I₁(τ_e) ∼ 0.3). For moving red blood cells, ΔR(τ_e) is ∼2 μm. Let E(ρ) be the magnitude of the field which derives from a static scattering center at location ρ within the tissue; also designate θ(ρ) as the angle between a line drawn from ρ to a moving red cell, and a line drawn from that cell to an arbitrary tissue element located at some point r_d which, by the argument in the preceding paragraph, acts as the primary detector locus [see Fig. 1(B)]. If this tissue element is sufficiently far from the erythrocyte, the scattered field at the physical detector is[15]

Let us consider the erythrocytes to be spherically symmetric. (For our present calculations we only require that the directions of cell movement are uncorrelated with shape and that any significant changes in cell morphology occur slowly compared with τ_e.) In the Rayleigh-Gans limit,[16] A(Q) thus is given as

The field at the tissue point r_d has components representing scattering from sources which are distributed throughout the tissue. Thus, an integration must be performed over the term e_SC given in Eq (4), i.e.,

We measure the normalized photon autocorrelation function [〈n(t)n(t + τ)〉/〈n〉²] of photons scattered by moving red blood cells as well as photons which interact only with static tissue elements. The total intensity of the light falling upon the photodetector is designated as i₀, and the intensity of that portion which arises from photons which have interacted with moving cells is designated as i_SC. Since the detected light arises from a large number of essentially uncorrelated events, we can assume that the field statistics of the light scattered by the moving cells are Gaussian, in which case [〈n(t)n(t + τ)/〈n〉²] can be expressed as[17]

The phase shifts [φ(ρ)] and [φ(r_d)] are uncorrelated with any other terms, so the expectation in Eq. (9) factors into a product, two multiplicands of which are 〈exp[−iφ(ρ′) exp[iφ(ρ)]〉 and $〈exp[-i\phi ({\mathbf{\text{r}}}_{\text{d}}^{\prime })exp[i\phi ({\mathbf{\text{r}}}_{\mathbf{\text{d}}})]〉$; also, because the phase shifts at different points within the tissue are unrelated, these quantities factor as 〈exp[−iφ(ρ′)]〉 〈exp[iφ(ρ)]〉 and $〈exp[-i\phi (r_d^{\prime })]〉$ 〈exp[iφ(r_d)]〉 unless ρ′ = ρ and ${\mathbf{\text{r}}}_{\mathbf{\text{d}}}^{\prime }={\mathbf{\text{r}}}_{\mathbf{\text{d}}}$ However, since all values of φ between 0 and 2π are equally probable, we find 〈exp(iφ)〉 = 0, from which we infer that the expectation in Eq. (9) can be expressed as [see Eq. (5′)]

Because the photon distribution arising from the diffusion of light through the static tissue is imprecisely known, we now assume that a cell is irradiated with equal intensity from all directions, i.e., that we have 4π illumination. Thus, the numerator in Eq. (2) is obtained upon transforming the integration of ρ in Eq. (10) to an integration over the scattering angle θ and taking account of the geometric solid angle which locates an annulus of tissue sources with respect to the detector [from which the sinθ term in Eq. (5) follows]. We note that scattering occurs from many erythrocytes but assume that the cells move independently and are statistically identical. We thus can associate the term 〈exp{iQ · [r_j(t + τ) − r_j(t)]}〉 with 〈exp[iQ · ΔR(τ)]〉, where ΔR(τ) is the center of mass translation of a (typical) cell over a period of time τ. Finally, the expression given for I₁(τ) in Eq. (2) is obtained by normalizing such that lim_τ→0I₁(τ) = 1.

III. Results for a Specific Speed Distribution and Particle Shape

Equation (2) has been derived by assuming that the scattering system consists of identical, spherically symmetric, particles (or particles whose orientation is uncorrelated with direction of movement), illuminated by a light source which has been totally randomized in direction. The particles are considered to move independently, and interparticle optical interference is neglected. We now determine the form of I₁(τ), using specific particle structure factors $\mathcal{S}(Q)$ and speed distributions in Eq. (2).

To compute the expectation 〈exp[iQ · ΔR(τ)]〉, one must first choose a model of blood flow through the vascular bed. It is simplest to consider the blood to be percolating through a randomly directed network of capillaries, with trajectories which are undisturbed for times which are long compared with the relaxation of I₁(τ). The latter assumption can be easily justified, as decay times τ_1/e of photon autocorrelation functions measured for capillary flow typically are of the order of 10⁻⁴–10⁻³ sec (see Sec. VI and [Refs. 5]–[11]), whereas blood cells move steadily and without dramatic shape change for tenths of a second or longer.[18],[19] The expectation 〈exp[iQ · ΔR(τ)]〉 for spherically symmetric particles moving with constant velocities in random directions is[20]

The structure factor $\mathcal{S}(Q)$ for erythrocytes has only been established experimentally for a limited number of scattering angles.[21] These data and a Rayleigh-Gans scattering approximation for randomly oriented red blood cells are well represented by a Rayleigh-Gans approximation for a sphere having 2.75-μm radius (see Fig. 2). We expect that, for the purposes of the present investigation (using red blood cells and polystyrene latex spheres), a suitable analytic expression is that derived for optically homogeneous spheres in the Rayleigh-Gans approximation, i.e.,[16]

Substituting Eq. (11) into Eq. (2), one finds

Numerical curves determined from Eq. (18) are shown in Fig. 3. We note from the equation that when T gets large, I₁(τ) tends to zero as I₁(τ) ∼ 1/τ², whatever the value of L. In Fig. 4 we show the halfwidth of the decay T_1/2 as a function of L. From Eq. (18) we also see that I₁(τ) becomes a function only of T (independent of L) when 2ξL² = 8ξk²a² is large. One consequence is that the half-decay point of I₁(τ) is then proportional to a/〈V²〉^1/2; from Eq. (17) and the asymptotic value T_1/2 = 0.52 shown in Fig. 4, we find in this case that the half-decay point is given as

The value for τ_1/2 given in Eq. (19) would pertain to the measured autocorrelation function for a heterodyne experiment where most of the photons which arrive at the photodetector are not scattered by moving cells. However, as shown below (see Fig. 7), scattering from the microvasculature is found to be nearly a homodyne process (i.e., the detected photons are predominantly Doppler shifted) and can be described by Eq. (7) with i_SC ≈ i₀ and β = 0.5 for a conventional spectrometer (see Sec. V) and β = 0.2 for a fiber optics instrument.[8],[11] The autocorrelation function in this case contains contributions representing the autocorrelation of photons which are multiply scattered from moving erythrocytes. In the next section we show how the intermediate scattering function I(τ) is related to I₁(τ) when such multiple scattering is taken into account.

IV. Effects of Multiple Scattering

The model described in Secs. II and III is a single-shift model, in that we presume that a photon interacts with a moving cell only once while migrating within the tissue; however, under normal circumstances there is a strong likelihood that a photon will be scattered by more than one moving red cell before it leaves the tissue. If an appreciable number of such events occurs, the measured photon autocorrelation function will include components relating to higher-order scattering processes, i.e., those which involve multiply convoluted spectral shifts. Thus, the normalized intermediate scattering function I(τ) in general must be expressed as

Sorenson et al.[22] show that I₂(t), which is the autocorrelation of field components $I_2(\tau )=〈E_2^{*}(t)E_2(t+\tau )〉$ [where E₂(t) represents an electric field of unit amplitude which has been scattered two times], can be written as the product $I_2(\tau )=I_1^2(\tau )$ if there is no correlation between the positions of any two particles. This is a consequence of taking an ensemble average of unrelated phase shifts; similarly, one can show that terms of the form $〈E_i^{*}(t)E_j(t)〉$ are zero when i is not j, because the phase of E_j(t) is completely independent of the phase of E_i(t) (see footnote 8 of [Ref. 22]). In accordance with the arguments given in Sec. III, the time dependence of the fluctuating field associated with an n step photon diffusion process, of which m steps are collisions with moving cells (Fig. 1: A,A,…, A,B,A,…, A,B,A,…, A), can be written as E_SC(t) = E_SC1(t)E_SC2(t) … E_SCm(t). Because these m phase shifts are independent, the resultant term of the intermediate scattering function is

We next note that P_m can be written as P_m = Σ_rP(m|r) · P(r), where P(m|r) is the probability that a photon makes m collisions with the moving erythrocytes before emerging from the tissue, given that it makes r collisions in total [i.e., (r − m) collisions with static elements]. But, P(m|r) is given by the binomial distribution, i.e., the probability of realizing m successes (collisions with moving cells) out of r tries (total collisions), where the probability of success is proportional to the scattering cross section of the blood relative to the total scattering cross section of the tissue, p = σ_blood/(σ_blood + σ_tissue). If the blood volume is but a small fraction of total tissue volume, p simply will be proportional to blood volume (assuming that blood volume changes will not affect the gross scattering structure of the tissue and the mean path length of photon diffusion). In the limit that r is large and p is much less than 1, the binomial distribution can be represented by the Poisson distribution $P(m|r)\to exp(-\overline{m}){(\overline{m})}^j/(j!)$, independent of r, where m is the average number of collisions with a moving cell. (By the preceding argument m is proportional to red blood cell number density and, consequently, to blood volume.) Thus, if an incident photon experiences many collisions with static elements before leaving the tissue, Eq. (22) can be rewritten as

The normalized autocorrelation function g⁽²⁾(τ) can now be obtained from Eqs. (7) and (23), namely,

In Fig. 5 we show g⁽²⁾(τ) − 1 evaluated with I₁ as given by Eq. (20). When $\overline{m}$ is <1, the amplitude associated with the time varying portion of the autocorrelation function decreases approximately linearly with decreasing $\overline{m}$ (nearly heterodyne), with only minor change in shape or decay rate. For large $\overline{m}$, the amplitude of the decaying portion approaches the maximal value β and the shape of the curves becomes Gaussian with a decay rate proportional to ${(\overline{m})}^{1/2}$. This Gaussian form is obtained for a sufficiently high degree of multiple scattering, regardless of the velocity distribution P(V). Consequently, laser Doppler velocimetry cannot provide information concerning details of the velocity distribution but is sensitive only to the mean speed through the scale factor m^1/2〈V²〉/a.

Several simplifying assumptions have been used to derive Eqs. (2), (14), (20), and (23), which were necessitated by our uncertainty about the optical structure of the vascularized tissues which would be examined by laser Doppler scattering techniques. In Sec. VI we present experimental results which nonetheless indicate that many important features of the scattering process indeed are faithfully represented by these calculations.

V. Spectral Analysis

A spectrum analyzer also can be used to resolve the fluctuating intensity of the light which is scattered to the detector. The resulting signal is given as[15]

The spectrum S(ω) which pertains to the model for multiple scattering as described in the last section can be computed from the following expansion, obtained from Eqs. (26) and (24):

There is a general analytic relationship between $\overline{m}$, blood cell size and speeds, and the width of the spectrum. The latter is characterized by various moments of S(ω). For example, let us define the first moment 〈ω〉 as[11]

The function in Eq. (35) can be evaluated from the expressions given in Eqs. (29), (30), and (20). As shown in the Appendix, we find that 〈ω〉 is given as

VI. Experimental Models

In Fig. 7 we show some typical measured photon autocorrelation functions taken with a spectrometer constructed of a Spectra-Physics model 125A He–Ne laser, a fiber optics delivery system,[11] an ITT F-4085 photon counting tube, and a clipped autocorrelator.[24] Figure 7(a) shows the autocorrelation function of photons backscattered from the surface of a warm (∼37°) finger. Figure 7(b) is the photon autocorrelation function obtained from studies on a model system (to be described below) which we used to test predictions of the theory presented in the preceding section.

Several similar tissue models were used. Each essentially consisted of a bundle of hollow fibers surrounded by a strongly scattering medium. The data presented here were obtained from a bundle of seventeen 180-μm diam silicone fibers (BioRad Laboratories, Biofiber 5) set in a 5.0%-w/v agarose gel impregnated with 2.5-mg/ml, 0.055-μm radius latex polystyrene spheres. The latter material acted as an effective diffuser of the light incident from the laser, resulting in a glow which was similar in distributed intensity to that from living tissue. The fittings to the fiber bundle were connected to a precision infusion pump (Harvard Apparatus Co.). Various homogeneous particle suspensions, composed of either polystyrene spheres diluted in water or heparinized whole human blood diluted with 10% dextran in saline, were passed through the bundle at predetermined flow rates. When filled with diluted blood (0.5–5% RBC v/v) the 180-μm diam tubes have a scattering cross section equal to that of tissue microvessels (5–30 μm). The mean speeds of the flowing particles are reproducible. (The speed distributions probably are uniform, as for Newtonian flow in a cylinder.) The results from the model system, therefore, should deviate only slightly from those predicted using the distribution described by Eq. (12) (compare curves A and B in Figs. 5(c) and (d)].

These models were created to provide a system similar to living tissue, in which photons diffusing among static elements occasionally would encounter moving particles having known scattering properties. By varying particle size, concentration, and velocity, the appropriateness of our theory could then be determined. Specifically, we sought to verify the premise that discerned Doppler shifts arise from forward scattering by red blood cells, even when backscattered light from the tissue is analyzed. We also wished to test our conclusions concerning multiple Doppler scattering by moving particles.

As predicted by our theory, power spectra or autocorrelations of light scattered from the model system, as well as from human and animal tissues, were found to be independent of external scattering angle. This is in marked contrast to the dependence on the (difficult to determine) angle between q and v vectors observed when Doppler ultrasound measurements are made on large arteries[25] where tissue and red blood cell scattering cross sections are ∼10⁻⁵ those of light.

Using the model system and the Doppler apparatus described in [Ref. (11)], we could demonstrate that Doppler shifts do in fact vary linearly with the mean velocity of the flowing cells (Fig. 8). A more critical test concerned the relationship between spectral shifts and the scattering cross section (structure factor) of the particles. The number concentration of different sized particles was adjusted to the same total scattering cross section (to eliminate effects of a variable amount of multiple Doppler scattering; see next paragraph), and the resulting suspensions were then passed through the model at identical flow speeds. In accordance with Eq. (36), the observed mean spectral shift varied as the inverse of particle size (Fig. 9).

Our theoretical analysis of multiple scattering was tested by varying the concentration of red blood cells flowing in the model system. At each concentration, the mean spectral shift was determined for several values of mean velocity (0.2–2.0 mm/sec). The degree of multiple scattering $\overline{m}$ was also evaluated. The latter was accomplished by determining the amplitude of the normalized autocorrelation decay [g⁽²⁾(0) − 1]. As indicated in Eq. (24), this quantity is approximately $(2\beta \overline{m})$ for small $\overline{m}$ and approaches β (here, β = 0.17) for $\overline{m}>2$. As shown in Fig. 10, as the concentration of suspended red blood cells flowing in the model system was increased (while keeping the flow velocity constant), the mean frequency 〈ω〉 of the Doppler shift increased almost linearly at low concentrations $(\overline{m}<1)$; at higher particle concentrations 〈ω〉 tended toward a square root dependence on concentration, as predicted by Eq. (36).

The lower curve in Fig. 10 represents values computed from Eq. (36) using best estimates for 〈V²〉^1/2 and a. The values of the fitted upper curve are everywhere 140% of those of the lower curve. The 40% discrepancy in the coefficient ${〈V^2〉}^{1/2}/(\sqrt{12\xi }a)$ could arise from errors in determining the rms velocity of the red blood cells from the bulk flow through the model system and in estimating $\overline{m}$, or from the theoretical approximations used for the speed distribution [Eq. (12)] and structure factor [Eq. (13′)]. Additionally, the model system contains oriented 180-μm diam tubes (much larger than tissue microvessels) in which multiple Doppler scattering of the type indicated in Fig. 1(C) may occur.

Our theory thus provides a good representation of the behavior of flow models which have been constructed to mimic laser Doppler scattering from the microvasculature of living tissues. Those model systems allow systematic variation of particle radius, concentration, and velocity, as well as external scattering geometry, enabling us to test the quantitative predictions of our theory in a manner not possible in living tissue. The ability of the theory to rationalize accurately the data acquired with the model systems suggests: (1) laser Doppler scattering can be used to quantitate instantaneous microvascular blood flow in small volumes (∼1 mm³) of optically accessible tissue; (2) 〈ω〉 varies linearly with blood flow in tissues containing small relative blood volumes; and (3) 〈ω〉 varies linearly with small changes in blood volume associated with the cardiac cycle.

The theory predicts, with reasonable accuracy, measurements of blood volume flow in a model system where flow speed 〈V²〉^1/2 and collision number $\overline{m}$ can be determined. However, amplitudes of actual physiological flow extend over 2 orders of magnitude, with large variations in both 〈V²〉^1/2 and $\overline{m}$. Estimates of $\overline{m}$ are difficult to obtain for tissues for which the mean number of scattering events from moving cells is >2 (i.e., tissue whose vascular volume is >0.06-ml blood per gram of tissue, given the existing probe design). Therefore, in order to determine an absolute value of blood flow from any given tissue, the laser Doppler flowmeter generally would first have to be calibrated by another quantitative technique. However, this instrument is uniquely able to monitor microvascular flow continuously and noninvasively in many animal and human tissues, with a temporal resolution of ∼100 msec and spatial resolution of 1 mm.

Appendix: Derivation of Eq. (36)

Note that Eqs. (29), (30), and (20) can be used to write Eq. (35) as

(A1)$$f(\overline{m})=2exp(-2\overline{m})\text{∑}_{j=1}^{\infty }\frac{{(2\overline{m})}^j{f}_j}{j!}$$

where f_j is

(A2)$$f_j\equiv \frac{1}{\pi }{\mathit{\int }}_0^{\infty }W[{\mathit{\int }}_0^{\infty }cos\mathit{\text{WZ}}{\left(\frac{1}{1+Z^2}\right)}^j\mathit{\text{dZ}}]\mathit{\text{dW}}.$$

The integral in the brackets can be expressed as[23]

$${\mathit{\int }}_0^{\infty }cos\mathit{\text{WZ}}{\left(\frac{1}{1+Z^2}\right)}^j\mathit{\text{dZ}}={\left(\frac{W}{2}\right)}^{j-1/2}\frac{{\pi }^{1/2}}{\mathrm{\Gamma }(j)}{K}_{j-1/2}(W),$$

where K_j is a modified Bessel function, and Γ(j) is a gamma function. The integral over W then is[26]

$${\mathit{\int }}_0^{\infty }W{\left(\frac{W}{2}\right)}^{j-1/2}{K}_{j-1/2}(W)\mathit{\text{dW}}=\mathrm{\Gamma }(j+½).$$

Thus, the term f_j is given as

(A3)$$f_j=\frac{\mathrm{\Gamma }(j+½)}{{\pi }^{1/2}\mathrm{\Gamma }(j)},$$

and, by Eq. (A1), the expression for $f(\overline{m})$ becomes

(A4)$$f(\overline{m})=\frac{2exp(-2\overline{m})}{{\pi }^{1/2}}\text{∑}_{j=1}^{\infty }\frac{{(2\overline{m})}^j\mathrm{\Gamma }(j+½)}{\mathrm{\Gamma }(j+1)\mathrm{\Gamma }(j)}.$$

Using this expression, one obtains from Eq. (34) the relationship given in Eq. (36).

The authors thank John Bechhoefer, Pat Bowen, and Doug McGuire for their cheerful and competent assistance with several technical details relating to the accomplishment of this work.

Figures

 

Fig. 1 Diffusion of photons within the tissue can be represented as a series of scattering steps of types A, B, or C. Step A represents scattering from a static tissue element which does not impart any Doppler shift (the phase shift of this step, φ_A, is constant). The distance between static scattering centers |ρ − r_j| will depend on the tissue structure but typically is ∼100 μm. Step B represents small-angle Doppler scattering from a moving red blood cell with a phase shift φ_β(t), which varies as Q · vt. The probability of step B scattering increases with local blood cell concentration. Step C represents sequential scattering from two moving red blood cells and occurs within larger vessels (>50-μm diam). In this case the velocity vectors of the two cells are highly correlated, $|v_1·v_2|\sim {\upsilon }_1^2$, although the sign of the phase terms Q₁ · r₁(t) and Q₂ · r₂(t) are random.

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Fig. 2 Angular scattering structure factor $\mathcal{S}(Qa)$ vs Qa. Shown are (a) the approximations given by Eq. (13), which is used in our subsequent analysis, and (b) a curve based on empirical data and a Rayleigh-Gans computation of light scattered by randomly oriented biconcave disk red blood cells.[21] Also shown are curves for (c) spherical red blood cells of 2.75-μm radius and polystyrene latex spheres of (d) 0.55- and (e) 0.26-μm radius, derived from Mie theory for particles in water.

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Fig. 3 Field autocorrelation function given by Eq. (18), I₁(T), for light which is singly scattered by moving particles, plotted as a function of the reduced time variable T for different values of the particle size variable L (L = 1,2,4,15). For large particles with L > 4 (a > 0.15 μm), I₁(T) has the asymptotic form given by Eq. (20).

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Fig. 4 Half time, T_1/2, of I₁(T) determined from Eq. (18) and plotted as a function of the size parameter L = 2ka. For large particles with highly anisotropic forward scattering (L > 4), the reduced time variable is given by Eq. (19).

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Fig. 5 Normalized intensity autocorrelation function g⁽²⁾(T) plotted as a function of the reduced time variable T [i.e., scaled by the rms speed divided by the particle radius, see Eq. (17)] for different degrees of multiple scattering and different speed distributions. Parts (a) and (b) show g⁽²⁾(T) for the Gaussian speed distribution [Eq. (12)] when $\overline{m}$ varies from 0.1 to 4. (a) For small values of $\overline{m}$, the predominant effect on g⁽²⁾(T) is an increase in amplitude with increasing $\overline{m}$. (b) For larger $\overline{m}$ the amplitude of g⁽²⁾(0) approaches a maximum, 1 + β, determined by the spatial coherence at the detector, while the decay rate increases and approaches ${\overline{m}}^{1/2}$ scaling. In (c) and (d) the curves labeled A, B, C, and D represent g⁽²⁾(T) for different speed distributions: A, Gaussian [Eq. (12)]; B, uniform (Newtonian flow in a cylinder); C, single speed; D, two speed ( $1/3\overline{\upsilon }$ and $5/3\overline{\upsilon }$). Differences in these curves are noticeable at large T if $\overline{m}\le 1$ but are insignificant for large $\overline{m}$ (where all curves become Gaussian). The relative amplitude of the slow decay (corresponding to low frequency components) is seen to increase with the variance of the speed distribution.

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Fig. 6 Power spectrum S(W) of the detected photocurrent computed as a function of $W{\overline{m}}^{1/2}$ for $\overline{m}=0.5,1,1.5,2,3,4,6,8,\text{and}10$ (the curves from left to right, respectively). For $\overline{m}<1$, the normalized first moment of these spectra increases more rapidly than ${\overline{m}}^{1/2}$, whereas for large $\overline{m}$ the spectra scale with $W{\overline{m}}^{1/2}$.

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Fig. 7 (A) Normalized photon autocorrelation function g⁽²⁾(τ) − 1 of light backscattered from the fingertip of a normal volunteer. The decay amplitude is 90% of the empirical value of β obtained from homodyne scattering from a solution of polystyrene spheres using the same fiber optic probe and corresponds [see Eq. (24)] to $\overline{m}\simeq 1.2$. (B) Autocorrelation functions obtained from light backscattered from the tissue model using diluted human blood with mean velocity of 1 mm/sec and hematocrits H = 0.05, 0.025, 0.006, and 0.003 (a, b, c, and d, respectively). Increased red blood cell number density (and, therefore, $\overline{m}$) results in increased decay rates and amplitudes of g⁽²⁾(τ) − 1. These experimental curves are similar to the theoretical curves shown in Fig. 5. For clarity, smoothed curves have been drawn through the data points, which are only shown for H = 0.025.

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Fig. 8 Mean Doppler shift 〈ω〉, as defined in Eq. (32), plotted vs mean velocity for a variety of particle sizes and volume fractions of fluid moving through the hollow fibers of the model system. In the 0–2-mm/sec range, the mean frequency 〈ω〉 increases linearly with mean particle velocity. A monotonically increasing dependence on particle density and decreasing dependence on particle radius are also observed, a–g, human blood diluted to hematocrits of 0.003, 0.006, 0.012, 0.012, 0.025, 0.035, 0.050, and 0.12, respectively; h–i, 0.55-μm radius PSL spheres at concentrations of 0.5 and 1.0 % v/v; j, 0.264-μm radius PSL spheres, 1 % v/v.

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Fig. 9 Mean Doppler frequency normalized by $f(\overline{m})$ and rms particle velocity $\overline{\nu }/f(\overline{m})=〈\omega 〉/f(\overline{m}){〈V^2〉}^{1/2}$ vs the reciprocal of particle radius. Shown are data obtained when red blood cells or PSL spheres were passed through the tissue flow model. Mean frequencies have been normalized to $\overline{m}$ and mean particle speed. The solid line drawn through the data points represents values of $\overline{\nu }/f(\overline{m})$ predicted by Eq. (36). [For red blood cells, $\overline{m}$ equals 1.43 times the vol % RBC in our model system (see insert in Fig. 10). Corresponding values of $\overline{m}$ for PSL solutions were determined from literature values[14],[16] of particle scattering cross sections and from particle volume ratios. From the values of $\overline{m}$ so obtained, $f(\overline{m})$ was calculated by Eq. (A4).]

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Fig. 10 Mean Doppler shift normalized by the rms particle velocity plotted as a function of the hematocrit of the fluid flowing through the tissue model (H = 0.3–12% v/v RBC). The average degree of multiple scattering $\overline{m}$ was related to RBC concentration by determining the decay amplitudes (see insert) and fitting according to Eq. (24). Empirical values of $\overline{\nu }=〈\omega 〉/{〈V^2〉}^{1/2}$ are plotted (circles) for rms velocities in the 0.2–2-mm/sec range. The lower solid line is the theoretical curve given by Eq. (36) using an effective RBC radius of 2.8 μm and measured values of 〈V²〉^1/2 and $\overline{m}$. The upper curve is obtained from the lower merely by increasing the ordinate scale [〈V²〉^1/2/(12ξ)^1/2a] by 40%. The shape of the curve depends only on $f(\overline{m})$ [see Eq. (A4)].

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