Abstract
Diffusing wave spectroscopy (DWS) and diffuse correlation spectroscopy (DCS) can assess blood flow index (BFI) of biological tissue with multiply scattered light. Though the main biological function of red blood cells (RBCs) is advection, in DWS/DCS, RBCs are assumed to undergo Brownian motion. To explain this discrepancy, we critically examine the cumulant approximation, a major assumption in DWS/DCS. We present a precise criterion for validity of the cumulant approximation, and in realistic tissue models, identify conditions that invalidate it. We show that, in physiologically relevant scenarios, the first cumulant term for random flow and second cumulant term for Brownian motion alone can cancel each other. In such circumstances, assuming pure Brownian motion of RBCs and the first cumulant approximation, a routine practice in DWS/DCS of BFI, can yield good agreement with data, but only because errors due to two incorrect assumptions cancel out. We conclude that correctly assessing random flow from scattered light dynamics requires going beyond the cumulant approximation and propose a more accurate model to do so.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Diffusing Wave Spectroscopy (DWS) was formulated in the late 1980s to probe dynamics of scatterers in soft matter based on intensity fluctuations of multiply scattered light. DWS models the field autocorrelation as an integral over photon paths, assuming uncorrelated motion at different scattering events [1–3], in the light diffusion approximation. In biological tissues, DWS and its differential formulation of diffuse correlation spectroscopy (DCS) can measure red blood cell (RBC) dynamics, which correlate empirically with blood flow [4–6]. The assumptions underlying DWS theory are powerful and widely used, even in distant fields such as laser speckle flowmetry [7,8]. However, given recent experimental advances in measuring time-of-flight- (TOF-) resolved light dynamics [9–11] in detail for the first time, some assumptions in DWS/DCS of blood flow merit further scrutiny. Here we focus on two important and inter-related assumptions; 1) the nature of RBC motion, and 2) the first cumulant approximation of the average over dynamic scattering angles.
Particle motion can be ordered, as in advective, randomly oriented, flow, or disordered, as in Brownian motion, or a combination of both [7]. DWS/DCS measurements of particle motion in biological tissues such as the rat brain [12,13], piglet brain [14], human brain [6,15–18], breast [19,20], head and neck [21], and skeletal muscle [22] have shown that experimental data is best fitted by ignoring advection of RBCs, and only considering Brownian motion. Thus, the blood flow index (BFI) in DWS/DCS has units of a Brownian diffusion coefficient.
The starting point of dynamic light scattering theories [23,24] is the normalized field autocorrelation of mth order paths: $\langle \textrm{exp}\left[ {i\mathop \sum \nolimits_{j = 1}^m {{\boldsymbol q}_j} \cdot \Delta {{\boldsymbol r}_j}} \right] \rangle $, where 〈 〉 denotes an average over particle displacement vectors $\Delta {{\boldsymbol r}_j}$ and dynamic momentum transfer vectors ${{\boldsymbol q}_j}$, which are assumed to be independent, and $i = \sqrt { - 1} $. In the multiple scattering regime, assuming uncorrelated dynamic scattering events while neglecting both random flow and higher order terms in the cumulant expansion [23], DWS expresses the normalized TOF-resolved field autocorrelation function, g1, as:
It is notable that most studies of blood flow in biological tissues utilize Eq. (1), although this expression was first derived for uniform colloidal suspensions, foams, and gels with unordered (Brownian) motion. A major biological function of RBCs is the transport of gases in the circulation; thus while hydrodynamic diffusion of RBCs is well-documented [25,26], RBCs should also possess characteristics of ballistic motion [24]. Therefore, it seems incongruous that blood flow, in this conventional sense, is typically assumed not be observable through light scattering techniques.
In this work, we present a plausible explanation to help resolve these discrepancies. We simulate field autocorrelations with a custom extended Monte Carlo method that explicitly allows for a mixture of static and dynamic particles with different scattering phase functions. We attempt to accurately model both the high scattering anisotropy and low volume fraction of RBCs in tissue. We assess the results to critically examine the first cumulant approximation, or first cumulant expansion, which is common in DWS. We show that under certain circumstances, errors due to the cumulant approximation, on one hand, and neglecting random flow, on the other, can cancel, enabling the conventional theory in Eq. (1) to describe autocorrelations well, even while ignoring advection. We then propose a more accurate approach that avoids these assumptions and might enable direct quantification of RBC advection in dynamic light scattering measurements.
2. Field autocorrelation models
In this section, we develop candidate models for TOF-resolved field autocorrelations, highlighting key assumptions in each.
2.1 Nature of scatterer motion
In DWS/DCS, the mean-squared displacement at time lag τd, 〈Δr2(τd)〉, of RBCs is modeled as either random flow, given by 〈Δr2(τd)〉 = v2τd2, where v2 is the second moment of the Gaussian velocity distribution, or as Brownian motion, given by 〈Δr2(τd)〉 = 6DBτd, where DB is the Brownian diffusion coefficient. In this work, we allow for RBCs to exhibit a “hybrid” of both random flow and Brownian motion, given by 〈Δr2(τd)〉 = 6DBτd + v2τd2, where the two types of motion are assumed to be independent.
2.2 Adapting Bonner and Nossal’s theory
To assess the validity of the cumulant approximation (or first cumulant expansion), we start from Bonner and Nossal’s theory [24], which implicitly includes all cumulant terms by directly integrating over the dynamic scattering phase function. This theory envisions tissue as a mixture of static (tissue matrix) and dynamic scatterers (i.e. RBCs and other scattering blood components). We extend this theory as follows: 1) the lower limit of number of dynamic scattering events, m, is set to zero to include purely static scattering paths, 2) the Henyey-Greenstein (and later, the more general Gegenbauer kernel) scattering phase function for dynamic scattering is assumed in deriving the normalized autocorrelation for a single dynamic scattering event (g1ss), 3) the average number of dynamic scattering events, $\bar{m}$(τs), is assumed to be $\bar{m}$(τs) = pdynµsτsc/n, where pdyn is the probability of dynamic scattering, µs is the static scattering coefficient, c is speed of light in vacuum, and n is the index of refraction of medium, and 4) as described above, we explicitly allow for RBC Brownian motion, in addition to random flow considered Bonner and Nossal’s original theory.
It is appropriate to remark on pdyn, the probability that a scattering event is dynamic. Note that pdyn is given by pdyn = VBµs,blood/µs, where VB is blood volume fraction and µs,blood is blood scattering coefficient, and does not depend on anisotropy factor. Given µs = 100 cm−1, at a hematocrit of 33%, µs,blood/µs ≈ 7.8 at 850 nm [27], and pdyn = 10% corresponds to VB = 1.3%. On the other hand, the DCS probability of dynamic scattering, α [14,17], is given by α = VBµs’,blood/(VBµs’,blood+µs’) = pdyn (1-gdyn)/[pdyn(1-gdyn)+(1-gstat)], where gdyn and gstat are anisotropy factors of dynamic and static scatterers, respectively. If pdyn(1-gdyn) << (1-gstat), α ≅ pdyn(1-gdyn)/(1-gstat). For example, α ≅ 2% when pdyn = 10%, gdyn = 0.98 (typical for RBCs [27]) and gstat = 0.9.
Bonner and Nossal [24] assumed that multiple static scattering in tissue is sufficient to achieve random illumination of dynamic particles and uncorrelated dynamic scattering events. This enabled them to construct the field autocorrelation in the multiple dynamic scattering regime from the single dynamic scattering field autocorrelation, g1ss. Our extension of their expression is:
2.3 Critical examination of the cumulant approximation
The cumulant approximation conveniently brings the angular average of Eq. (3) into the exponent, providing a very simple and intuitive expression for the autocorrelation function. In this section, we revisit the cumulant expansion, paying special attention to errors incurred by the high scattering anisotropy of RBCs. Commonly, in the derivation of DWS [23], the normalized field autocorrelation is succinctly expressed as:
Ultimately, the functional form, or “shape” of the autocorrelation decay with respect to time lag, τd, depends on which polynomial orders of τd contribute in the exponent. With hybrid motion, different cumulant orders can contribute to the same polynomial order of τd. Thus, another way to simplify Eq. (8) is to group terms according to the order of τd:
2.3.1 Special case 1: Brownian motion dominates random flow (A1)
As Brownian motion is unordered on time scales larger than the collision time, the mean squared displacement increases as τd, whereas for ordered motion, the mean squared displacement increases as τd2. When τd << 6DB/v2, Brownian diffusion dominates random flow (A1), i.e. 〈Δr2(τd)〉 ≈ 6DBτd. This approximation is implicit in Eq. (1). Many studies have empirically found that the Brownian motion model leads to a better fit to experimental data than the random flow model [12–22], but did not explicitly verify that τd << 6DB/v2. For example, for v = 1.25 mm/s and DB = 6 × 10−11 m2.s−1, this condition corresponds to τd << 220 μs. Thus, given A1, we modify Bonner and Nossal’s theory, which implicitly includes all cumulants [Eq. (6)], to describe Brownian diffusion.
2.3.2 Special case 2: first cumulant dominates second cumulant (A2)
If the first cumulant term dominates the second cumulant term, Eqs. (9) and (13) reduce to Eq. (14). This occurs when the mean-squared displacement of the moving particles satisfies 〈Δr2(τd)〉 << 6 / [$\bar{m}$ R〈q2(θ)〉]. For example, with gdyn = 0.98 (i.e. RBCs), v = 1.25 mm/s and DB = 6 × 10−11 m2.s−1, the condition is satisfied when τd << 141 μs. Thus, given A2, we modify conventional DWS theory, which neglects higher order cumulants, to describe hybrid motion.
2.3.3 Special case 3: first order τd term dominates (A1&2)
While special case 1 compares magnitudes of ordered and unordered motion, in special case 3, we examine τd terms which determine the ultimate shape of the autocorrelation. From Eq. (13), the general condition for the first order τd term to dominate the τd2 term is given by:
2.3.4 First cumulant for random flow cancels second cumulant for Brownian diffusion
From Eq. (13), the coefficient of τd2 includes contributions from the first cumulant term for random flow and second cumulant term for Brownian motion. The velocity distribution may be selected such that these two contributions cancel, minimizing deviations from an exponential decay at small τd. The distribution standard deviation, v*, is then given by:
2.4 Monte Carlo simulations
Because experimental systems with Brownian motion and random flow are challenging to contrive, in this study, Monte Carlo is taken as the gold standard. A previous Monte Carlo algorithm simulating the time resolved reflectance of a semi-infinite medium [37] was modified to calculate the normalized field autocorrelation. In the modified version, both dynamic scattering (moving particles, with a customized anisotropy gdyn) and static scattering (background tissue scattering, with a fixed anisotropy, gstat = 0.9) are explicitly simulated, with a probability of dynamic scattering pdyn. Only dynamic scattering events contribute to the recorded total accumulated momentum transfer Ydyn = Σj(1-cosθj) [34,37,38], where θj is the dynamic scattering angle at dynamic scattering site j (Fig. 2). By explicitly distinguishing dynamic and static scattering events, this approach obviates assumptions, including the cumulant approximation and similarity relation that are often invoked in standard Monte Carlo approaches [39]. Even though their momentum transfer is not stored, static scattering events are important in determining the Ydyn distribution. Unless otherwise stated, the Henyey-Greenstein phase function [Eq. (4)] is assumed for dynamic scattering.
Monte Carlo simulations stochastically estimate the probability density P(Ydyn,τs). When a photon reaches the detector, Ydyn associated with that photon is scored in a P(Ydyn,τs) histogram. Then, the overall normalized field correlation is determined as the weighted average of field correlations across all dimensionless momentum transfers experienced by detected photon paths [3,37,40]:
The propagation of a photon in the medium continues until the photon escapes from the medium, or until TOF exceeds a pre-defined 1 ns threshold. Photons that escape the medium within 5 cm from the source are collected. Absorption in the medium is included by multiplying the photon weight by the albedo at every scattering event [37,54]. The TOF bin width for the simulation is 5 ps, and the autocorrelation is assigned to the center point of each time bin. The normalized TOF-resolved field autocorrelation should not depend on absorption for our narrow TOF bins. Each simulation launched 300 million photons and took an average of 35 hours to run on an Intel i9 Model 9900K 3.6-GHz processor.
3. Results
3.1 First cumulant approximation (A2)
First the consequence of ignoring higher order cumulants on the field autocorrelation is examined by comparing g1 derived from Monte Carlo simulations (MC hybrid) to those derived from the first cumulant approximation (A2) [Eq. (14)], with hybrid motion, at two different TOF values, τs = 47.5 ps (Fig. 3(a)) and 197.5 ps (Fig. 3(b)), and various gdyn values. These TOF values correspond to photon path lengths of 1.01 cm and 4.23 cm in the medium, respectively. In general, both methods agree well at small τd whereas the long time lag tail of A2 is lower than that of MC hybrid, with more disagreement at early TOF (Fig. 3(a)) and higher gdyn (Fig. 3(b)-(c)). While inaccuracy of the cumulant approximation at early TOF (few dynamic scattering events) is well-known [23], its dependency on anisotropy of dynamic scatterers has not been rigorously investigated. Surprisingly, discrepancies in the tails of the autocorrelation extend beyond τs ∼ 500 ps for gdyn = 0.98 (Fig. 3(c)). A zoomed-in linear scale of Fig. 3(c) shows that, for high gdyn, neglecting higher order cumulants significantly underestimates the tails even at large TOFs (Fig. 3(d)).
The role of gdyn in lowest order error term in the cumulant approximation is described by R, the ratio of the second cumulant term to the square of the first cumulant term [Eq. (10) and Fig. 1(b)]. Larger R values at early TOF and larger gdyn (Fig. 1(b)) predict larger errors in the first cumulant approximation.
3.2 Neglecting random flow (A1)
Next the consequence of ignoring random flow on the field autocorrelation is examined by comparing Bonner and Nossal’s theory for Brownian motion alone (A1) to MC hybrid (Fig. 4). As described earlier, A1 accounts for all cumulant terms, but ignores random flow. A1 agrees well with MC hybrid at small τd, but decorrelates slower than MC hybrid at large τd. In other words, ignoring random flow increases the autocorrelation tails. This phenomenon is more obvious at shorter τs (Fig. 4(a)) and at higher gdyn (Fig. 4(b)-(d)) because for fixed particle dynamics, the autocorrelation decays slower for high gdyn (e.g. gdyn = 0.98) than for low gdyn (e.g. gdyn = 0.6).
3.3 Conventional DWS (A1&2)
As mentioned earlier, conventional DWS ignores both random flow and higher cumulant terms (A1&2). Even though A1 and A2 are individually incorrect, Eq. (16) is still approximately satisfied as the velocity distribution width was chosen [Eq. (17)] so that it is valid up to O(τd3). Thus A1&2 surprisingly agrees well with the gold standard MC hybrid, especially when gdyn < 0.8 (Fig. 5(a)), or when gdyn = 0.98 and τs > 200 ps (Fig. 5(c)-(d)). The fact that A1&2 performs better than A1 and A2 is surprising because A1 and A2 include only one incorrect assumption each, whereas A1&2 includes two incorrect assumptions. This result can be explained by the fact that the autocorrelation tails can be reduced either by neglecting higher order cumulant terms (observed in A2 in Fig. 3) or by including random flow (suggested in A1 in Fig. 4). Thus, for A1&2, ignoring flow increases the tail whereas ignoring higher order cumulants reduces the tail. Therefore, in this case, two incorrect assumptions (Fig. 5) provide better predictions than just one (Fig. 3 and Fig. 4). Residual minor errors in Fig. 5 are due to O(τd3) terms in Eq. (13).
To illustrate the role of the first cumulant approximation in DWS theory for pure Brownian motion or diffusion, in Fig. 6, Bonner and Nossal’s theory (A1) and Monte Carlo simulations (MC diff.) are compared to DWS theory (A1&2) In this case, all cumulant terms are accounted for in MC diff. and A1, while only the first order cumulant term is accounted for in A1&2. As expected, A1 and MC diff. agree well while the tail of A1&2 is much lower, even at τs = 997.5 ps (Fig. 6(c)-(d)). This further confirms that neglecting higher order cumulants reduces the autocorrelation tails.
3.4 Full cumulant expansion (Bonner and Nossal) with hybrid motion
To avoid assumptions in DWS/DCS theory, Bonner and Nossal’s approach of directly integrating over the scattering phase function can be used. Thus, a B&N hybrid model accounts for all cumulant terms, and for both Brownian diffusion and random flow of dynamic scatterers. As shown in Fig. 7, B&N hybrid is an improvement over A1&2, agreeing well with MC hybrid regardless of anisotropy of dynamic scatterers (Fig. 7(a)-b), and the agreement is observed at short (Fig. 7(a)) and long τs (Fig. 7(c)-(d)). Thus, eliminating incorrect assumptions altogether achieves excellent agreement with the gold standard.
4. Discussion
Using a customized Monte Carlo of light scattering in realistic tissue models as a gold standard, we highlight that errors due to two common, incorrect assumptions, one regarding the nature of RBC motion and the other regarding the form of the field autocorrelation, can cancel, yielding much better predictions of TOF-resolved field autocorrelations than just one incorrect assumption. This suggest that goodness-of-fit alone is a flawed measure by which to assess assumptions underlying a theory. By carefully examining each assumption, we arrive at a more accurate approach to assess random flow and further improve goodness-of-fit.
4.1 Accuracy of the cumulant approximation
The cumulant approximation, or first cumulant expansion, is widely used in light scattering theory, because it yields simple analytic expressions [1,7,8,23,55]. Here we argue that errors in the cumulant approximation require special attention, particularly at late time lags for highly anisotropic dynamic particles such as RBCs and, as discussed further below, in tissues with low probabilities of dynamic scattering. As such, we encourage the view that the cumulant approximation, which is central in the diffusion approximation for correlation transport, be viewed and evaluated essentially independently of the diffusion approximation for radiative transport. While the cumulant approximation can be improved by restricting fitting to early time lags (τd), our results show significant disagreements at late time lags when τs ≈ 200 ps (Fig. 3(b) and Fig. 6(b)). Importantly, as described in the next section, assessing RBC random flow requires analyzing longer time lags where the accuracy of the cumulant approximation becomes questionable.
4.2 Revealing random flow
The appropriate description of RBC motion (ordered vs. unordered) has been the topic of continued debate in the field of laser speckle [8,55]. Certainly, as discussed in Section 2.3.1, at sufficiently short time lags (i.e. τd << 220 μs), Brownian motion dominates random flow. Meanwhile, random flow just impacts the tails of TOF-resolved field autocorrelations (for instance, where A1 is higher than MC hybrid in Fig. 4). Therefore, assessing random flow is thus only feasible at later time lags.
As discussed in Section 2.3.2, the cumulant approximation is invalid if τd ≈ 141 μs or larger, precisely the lag time range needed to assess random flow. This revelation adds a new wrinkle to the debate on the nature of RBC dynamics, as the cumulant approximation is assumed in standard models for ordered and unordered flow in the literature [7]. Given our assumptions (gdyn = 0.98, v = 1.25 mm/s and DB = 6 × 10−11 m2.s−1) we hypothesize that the first cumulant approximation cannot accurately assess random flow.
To test this hypothesis, we treated the MC hybrid model as experimental data and fit either the A2 or B&N hybrid model to this data, with DB and v as free parameters. Figure 8 compares the fitted DB and v values to the expected (actual) values. Two different fitting ranges of the MC field autocorrelation are examined: g1MC > 0.01 (Fig. 8(a)-(c)), and g1MC > 0.5 (Fig. 8(d)-(f)). B&N hybrid accurately recovers DB and v while A2 underestimates DB and v. As A2 ignores higher order cumulants, the fitting procedure reduces DB to approximate the long time lag tail in g1MC. With a restricted fitting range (to exclude the tail of g1MC), A2 more accurately estimates DB (Fig. 8(d)). However, notably, A2 always yields v∼0 mm/s, suggesting that A2 is a poor model for quantifying random flow.
Thus, particular caution is warranted in applying the first cumulant approximation for small numbers of dynamic scattering events. Thus, for laser speckle flowmetry in parenchymal regions with sparse blood vessels, the range of autocorrelation lags (or the range of exposure times) must be chosen judiciously to avoid model errors.
4.3 Dynamic scattering probability dependence
In the above sections, we show that one confound in applying DWS/DCS theory to tissue measurements is the exceptionally high gdyn of RBCs. Here, we highlight another important confound; the low dynamic scattering probability (pdyn) of tissues. As mentioned in Section 2.4, a smaller VB implies a smaller pdyn. To illustrate the effect of pdyn on DWS/DCS accuracy, pdyn = 30% (VB = 3.9%, Fig. 9(b)) was compared to pdyn = 10% (VB = 1.3%, Fig. 9(a)), used heretofore in this work. Obviously, smaller pdyn values result in a slower autocorrelation decay due to the smaller number of dynamic scattering events. For both pdyn, A1&2, representing conventional DWS/DCS, underestimates the autocorrelation tails of MC hybrid, which are well-represented by B&N hybrid. Importantly, disagreement between A1&2 and MC hybrid persists to later TOFs for smaller pdyn, due to the smaller number of dynamic scattering events. For example, A1&2 agrees well with MC hybrid when pdyn = 30% at 497.5 ps (Fig. 9(b)), but still decays faster when pdyn = 10% (Fig. 9(a)).
To quantify agreement between various models as pdyn changes, the coefficient of determination (r-squared) versus τs is shown (Fig. 10). The MC hybrid model is taken as the gold standard, and hence has an r-squared of unity. Different ranges of the field autocorrelation function are examined, corresponding to g1MC > 0.01, 0.2, and 0.5. As expected, B&N hybrid agrees best, with an r-squared approaching unity in all cases, while A1&2 (conventional DWS/DCS) is the second best, followed by A1 (B&N diffusion). Limiting the fitting range greatly improves the r-squared for A1, especially at early TOF. For example, the lowest r-squared value for A1 is well below 0.4 (0.9) if the tail of autocorrelation function is included (Fig. 10(a),(d)), and above 0.4 (0.9) if the tail is excluded (Fig. 10(c),(f)) when pdyn = 10% (30%). These results are expected since the first cumulant term for Brownian motion dominates early time lags.
Increasing pdyn (the number of dynamic scattering events) improves the r-squared for A1 and A1&2. For instance, A1 yields r-squared below 0.6 at early TOFs when pdyn = 10% (Fig. 10(a)), and well above 0.6 at early TOFs when pdyn = 30% (Fig. 10(d)). The improvement in A1 is expected since with more dynamic scattering events, the first cumulant term for Brownian motion dominates more of the autocorrelation decay. The improvement in A1&2 is also expected since higher order cumulants are less important with more dynamic scattering events.
4.4 Proposed improved expression
A common approach (COM) to determine field autocorrelations is a weighted average over total dimensionless momentum transfer Ytot [4,40,56]:
Here, we argue that the most accurate approach is to use a Monte Carlo simulation that includes both static and dynamic scattering events (Section 2.4), as this approach correctly performs angular averaging over the scattering phase function, implicitly including all cumulant orders. To achieve accurate results with a standard Monte Carlo code that does not explicitly include separate dynamic and static scattering events, we propose an improved expression (IMP):
As shown in Fig. 11, the proposed method (IMP), with a more accurate autocorrelation (adapted from Bonner and Nossal), closely approximates our customized Monte Carlo method (MC hybrid) which explicitly simulates dynamic and static scattering, resulting in the best agreement with near unity r-squared both when pdyn = 10% (α = 2%) [Fig. 11(e)-(f)], and when pdyn = 30% (α = 6%) [Fig. 11(g)-(h)]. On the other hand, the common approach (COM hybrid), which assumes the cumulant approximation, underestimates the autocorrelation tails when τs is less than 300 ps. While agreement improves for τs > 300 ps, these discrepancies are significant because the cumulant approximation is widely assumed to be valid in DCS/DWS across a range of source-detector separations.
In summary, our proposed method [Eq. (22)]: 1) improves the underestimation of autocorrelation tails observed in the common method (and DWS/DCS theory), and 2) enables the use of highly optimized existing Monte Carlo tools. Another potential advantage of the proposed method is that it can be extended to multi-layered tissues, by tracking momentum transfer or partial path length in each tissue type, as is possible in MCX.
4.5 Gegenbauer phase function
Bonner and Nossal’s theory and the MC method incorporate all cumulants and depend on the entire scattering phase function, unlike DWS theory (A1&2), which just depends on 〈cosθ〉. In this section, we examine the effect of phase function choice on the autocorrelation. Early goniophotometric measurement of RBC suspensions showed that light scattering by RBCs is well-described by a Gegenbauer (GE) phase function [58–61]. The GE phase function which is given by:
As above, the velocity distribution v* is selected so that the first cumulant term for random flow cancels second cumulant term for Brownian diffusion. Like R, v* depends on phase function [Eq. (17)], and v* = 0.817 mm/s when a = 1.2, gGE = 0.893 and DB = 6 × 10−11 m2.s−1 [26,33,34]. Figure 13 shows that while the autocorrelation tails are greater for the HG phase function (B&N hybrid, HG) than the GE phase function (MC hybrid, GE and B&N hybrid, GE), all have longer tails than DWS (A2), which ignores higher order cumulants. The smaller cumulant approximation error for the GE phase function is explained by smaller R values in Fig. 12(b).
Figure 14 also confirms that ignoring random flow (A1, GE) increases the autocorrelation tail. Thus assuming a GE rather than HG phase function does not change the essential conclusions regarding autocorrelation tail; namely, that neglecting higher order cumulants reduces the tail, neglecting random flow increases the tails, and that both assumptions in tandem lead to better agreement with the gold standard MC simulation. Moreover, Fig. 14 shows that conventional DWS/DCS (A1&2) with two incorrect assumptions agrees well with the more accurate models that use the GE phase function for both pdyn = 10% and pdyn = 30%.
Figure 15 shows r-squared versus τs for different ranges of the field autocorrelation. As before, the MC model is taken as the gold standard. Like Fig. 10, Fig. 15 shows that MC hybrid, GE agrees well with A1&2 (two incorrect assumptions) at all TOFs but disagrees with A1, GE (one incorrect assumption) at earlier TOFs, and that r-squared for A1, GE improves in tissues with higher pdyn, and when the range is limited to early time lags.
4.6 Simplifying assumptions
To arrive at our proposed improved expression for field autocorrelations in biological tissue [Eq. (22)], several assumptions were adapted from the original Bonner and Nossal theory [24]. Specifically, it was assumed that RBCs move independently, that averaging over photon paths and scattering events is equivalent to ensemble averaging, that the distances between static tissue source points and moving cells is large compared to displacements for decorrelation, and that static scattering randomizes the direction of incident light between dynamic scattering events. In incorporating a hybrid model of RBC motion, we have also assumed that advective and Brownian components of the motion are independent.
Importantly, vascular geometry was not explicitly simulated in the current study. RBCs were assumed to be uniformly distributed throughout the tissue volume, parameterized by the probability of dynamic scattering (pdyn). In a previous study with a proscribed parallel vessel geometry, pdyn was empirically found to be proportional to vessel radius-squared and inversely proportional to vessel spacing-squared [34]. While our simplified tissue model enabled us to investigate the cumulant approximation without assuming a specific geometry, future studies should investigate realistic vascular geometries, accounting for the issues highlighted in the present work.
5. Conclusion
We have shown that field autocorrelations in realistic biological tissue models with hybrid motion are accurately predicted by our adaptation of Bonner and Nossal’s original theory for RBC scattering. Importantly, neglecting higher order cumulants reduces the field autocorrelation tails whereas neglecting random flow increases them. Thus, for particular choices of random flow distribution and Brownian diffusion coefficient, DWS theory with pure Brownian motion might agree with experimental data, leading to the incorrect conclusion that random flow is absent. Furthermore, if experimental data are fit using the DWS cumulant approximation, allowing for hybrid motion, dynamics are underestimated. Here, we argue that dynamics can be accurately assessed by analyzing field autocorrelations with a model that includes higher order cumulants. Altogether, this work supports that random flow may be assessable in TOF-resolved field autocorrelations, but higher order cumulants should be incorporated into models to ensure its accurate assessment.
Appendix A
Here, we derive Eq. (6). In Bonner and Nossal’s original theory [24], the normalized field autocorrelation function can be expressed as
Appendix B
Here, we show how Eqs. (1) and (9) are derived by applying cumulant expansion to Bonner and Nossal’s theory [Eq.(6)]. First, we take logarithm both sides of Eq.(6):
Funding
National Institutes of Health (R01NS094681, R03EB023591, R21NS105043).
Acknowledgments
We thank Oybek Kholiqov and Wenjun Zhou for valuable discussions.
Disclosures
The authors declare no conflicts of interest.
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