# Diffuse Correlation Spectroscopy at Short Source-Detector Separations: Simulations, Experiments and Theoretical Modeling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experiments

#### 2.1.1. Instrumentation

#### 2.1.2. Liquid Phantoms

^{−1}and reduced scattering coefficient ${\mu}_{s}^{\prime}$ = 12 cm

^{−1}. These values for absorption and scattering at 785 nm (the laser wavelength) are within ranges previously reported for different tissue types including bone, abdomen and white matter of the brain [45,46,47].

#### 2.1.3. Flow-Chamber Description

^{−1}; ${\mu}_{s}^{\prime}$ = 12 cm

^{−1}at 785 nm) and the chamber was placed atop a homogenous solid phantom block with similar optical properties as the liquid phantom at 785 nm. The chamber housed a clear plastic tube (inner diameter = 1.6mm, outer diameter = 3.2 mm; 8000–0004; Thermo Fisher Scientific Inc., Miami, OK, USA) that was attached to two opposite sides of the chamber using moldable glue (Sugru; FormFormForm Ltd., London, UK) and ran along the bottom surface of the plastic chamber, parallel to one edge. One end of the plastic tube was connected to a syringe pump (75900–00; Cole Parmer, Vernon Hills, IL, USA) that was controlled via software to produce a given volume flow rate $F$, while the other end of this tube drained into a sink. Assuming laminar flow, the pump flow rate was converted to an average fluid flow speed in the channel as $F={V}_{ch}A$, where $A$ is the cross-sectional area of the channel and ${V}_{ch}$ the flow speed. Since the same liquid phantom added to the chamber was also loaded into the syringe pump, both the actively pumped and surrounding liquid media had identical optical transport coefficients. $F$ was set via software to one of four values: ${F}_{0}$ (pump turned off; 0 mL/h), ${F}_{1}$ (1.44 mL/h), ${F}_{2}$ (3.60 mL/h) or ${F}_{3}$ (5.76 mL/h). Given a circular tube of radius 0.08 cm, the average channel flow speeds were calculated to be 0.02 cm/s, 0.05 cm/s and 0.08 cm/s at ${F}_{1}$, ${F}_{2}$, and ${F}_{3}$, respectively. These flow rates gave speeds in ranges reported for moving red blood cells in capillaries [48].

#### 2.1.4. Phantom Preparation and Measurement Protocol

#### 2.1.5. Data Acquisition

#### 2.2. Monte Carlo Simulations of Correlation Transport

^{6}–10

^{8}) individual photon packets are launched and aggregate quantities of interest (such as the photon flux, fluence, absorbance and fluorescence) are tracked spatio-temporally, for a given tissue model. For the studies here, we modified a time-resolved MC photon transport model (previously described in detail [49,51]) to include simulate of photon field correlations in reflection geometry. We note that MC modelling of photon transport in scattering media are well discussed elsewhere [50,51], and we therefore confine description of aspects pertinent only to tracking of field correlations in the MC model.

#### 2.3. Tissue Models Simulated

#### 2.3.1. Semi-Infinite Models

^{−1}), seven absorption coefficients (${\mu}_{a}$: 0.05 − 0.5 cm

^{−1}) and six Brownian flow coefficients (${D}_{B}$: 5 × 10

^{−9}–3 × 10

^{−8}cm

^{2}/s) following the ranges as reported recently [31]. These MC simulations were compared against theoretical calculations to assess both the predictions of theory (forward model), as well as parametrization of the simulations (inverse model), as described in Section 2.4.

#### 2.3.2. Three-Layer Models

#### 2.4. Theoretical Analyses

#### 2.5. Goodness-of-Fits: Fit Residuals

## 3. Results

#### 3.1. Simulations in Semi-Infinite Phantoms

#### 3.1.1. Forward Theoretical Calculations vs. Simulations

^{−8}cm

^{2}/s. Data in Figure 2c,d) are for media with higher absorption ${\mu}_{a}$ = 0.42/cm, at same SDS as Figure 2a,b respectively. The ${\chi}^{2}$ fit-residuals are shown within legends for each simulated model in Figure 2a and d and are in accordance with known limitations of diffusion theory—i.e., data simulated for long SDS were fit better Figure 2c,d, relative to data from shorter SDS Figure 2a,c and simulated media with lower albedo (where the albedo $a={\mu}_{s}^{\prime}/\left({\mu}_{s}^{\prime}+{\mu}_{a}\right)$) were fit worse relative, to media with higher albedo.

#### 3.1.2. Fitting Simulations Using Theory

#### 3.1.3. Errors from Theoretical Fits

^{−8}cm

^{2}/s, for three different SDS (Figure 4a: SDS = 0.15 cm; Figure 4b: SDS = 0.45 cm; Figure 4c: SDS = 0.75 cm). With decreasing SDS, the retrieved values of ${D}_{B}$ increased from its expected (known) value of ${D}_{B}$ = 2 × 10

^{−8}cm

^{2}/s. Figure 4d–f show the same data as Figure 4a–c, but as percent-errors relative to the true input value (of ${D}_{B}$ = 2 × 10

^{−8}cm

^{2}/s). As seen before, errors associated with retrieved flow coefficients were highest for small SDS and low albedos. These data also indicate that extracted ${D}_{B}$ depended on the optical properties of the media, and the SDS used.

^{−9}cm

^{2}/s; up-triangles: 1 × 10

^{−8}cm

^{2}/s; circles: 1.5 × 10

^{−8}cm

^{2}/s; asterisk: 2 × 10

^{−8}cm

^{2}/s; diamonds: 2.5 × 10

^{−8}cm

^{2}/s; down-triangles: 3 × 10

^{−8}cm

^{2}/s). For each line (i.e., at fixed ${D}_{B}$), the percent errors at each SDS shown in Figure 5 was the mean across all 35 simulated models (spanning seven ${\mu}_{a}$ and five ${\mu}_{s}^{\prime}$ values with given ${D}_{B}$), while error-bars are standard-deviations. Abscissa of each line (for each ${D}_{B}$) are shown staggered in Figure 5 to show six distinct markers, since the computed errors (and distributions) were nearly identical across all ${D}_{B}$ values. In other words, the percent-error in retrieved ${D}_{B}$ was independent of the flow coefficient used and only depended on the optical properties of the medium and the SDS used. These data are in agreement with findings published in a recent report that investigated short separation DCS with MC simulations [31].

#### 3.1.4. Relative Changes in Flow Coefficients: Simulations vs. Theory

^{−9}cm

^{2}/s. The mean value of the relative changes in ${D}_{B}$ thus calculated across each of the 35 simulations (spanning all 7 ${\mu}_{a}$ and 5 ${\mu}_{s}^{\prime}$ coefficients) are shown by three bars, each for different SDS in Figure 6. The red asterisks indicate error bars computed as the standard deviation across each of these 35 simulations and were nearly zero, indicating that predicted relative changes in ${D}_{B}$ values from theoretical analyses, were near exactly as modeled by the simulations, across all SDS.

#### 3.1.5. Scaling Factors: Linearly Correcting Retrieved ${D}_{B}$ Coefficients

^{−8}cm

^{2}/s; Figure 7b,d: ${D}_{B}$ = 3 × 10

^{−8}cm

^{2}/s) for two different SDS (Figure 7a,b: 0.25 cm; Figure 7c,d: 1.05 cm). The similarity of these derived correction factors in Figure 7a–d indicate that they were nearly identical for either flow coefficient. Since a correction factor of unity would indicate that retrieved flow coefficient was the same as that modeled, the correction factors in Figure 7a,b (short SDS) are correspondingly lower, than those for Figure 7c,d (longer SDS).

^{−9}cm

^{2}/s through 3 × 10

^{−8}cm

^{2}/s in intervals of 5 × 10

^{−9}cm

^{2}/s), for four SDS (squares: 0.15 cm; triangles: 0.55 cm; circles: 1.05 cm; asterisks: 1.95 cm). The error bars show the standard deviation in retrieved ${D}_{B}$ values across the 35 simulated media with varying optical properties (the dashed line is $y=x$). Figure 8b shows the average absolute value of the corrected ${D}_{B}$ coefficients, for the same data shown in Figure 8a where each retrieved flow coefficient was corrected by multiplication with correction factors derived. Correction factors for each tissue model and SDS were obtained from the set simulations with input ${D}_{B}$ = 3 × 10

^{−8}cm

^{2}/s. As was seen previously in Figure 6, corrected flow coefficients yielded values that were near exactly identical to the true (input) value for ${D}_{B}$, across all ${D}_{B}$ coefficients.

#### 3.2. Diffusion-Theory Based Analysis of Experiments

#### 3.2.1. Flow Models for Fitting Experimental Data

#### 3.2.2. Absolute vs. Relative Flow-Coefficients in Phantoms

#### 3.3. Simulating Experimental Data Using MC

#### 3.3.1. Modeling Phantoms with Inactive Flow

#### 3.3.2. Modeling Phantoms with Actively Pumped Flow

^{−3}cm/s, produced the lowest ${\chi}^{2}$ residuals relative to experimental data acquired in phantom D0F1. Once ${h}_{0}$ and ${V}_{1}$ were known, required inputs for all 12 tissue models (${D}_{0}{F}_{1}-{D}_{3}{F}_{1}$, ${D}_{0}{F}_{2}-{D}_{3}{F}_{2}$ and ${D}_{0}{F}_{3}-{D}_{3}{F}_{3}$) were generated by scaling them across phantoms. The (scaled) input values of the coefficients (sought as inputs) for all the 3-layered tissue models constructed are shown in in Table 2.

## 4. Discussion

^{−2}cm/s but the MC inputs for flow speed of in layer 2 was ${V}_{1}$ = 5.7 × 10

^{−3}cm/s (data in Figure 12b).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Photograph of the phantom flow chamber (

**a**) and schematic of the flow chamber’s construction and measurement system (

**b**). The gray arrows in (

**b**) show the direction of the fluid flow. The flow rate F was controlled using a motorized syringe pump (see text). h is the distance from of the liquid surface to the top of the flow channel, which was controlled by addition of specific volumes of the liquid phantom to the chamber. The chamber (edges shown as thick black lines in (

**b**) was placed atop a homogeneous tissue phantom block with optical properties matched to the liquid phantom at the laser wavelength. The DCS probe was positioned directly above the flow channel and at center of the flow chamber.

**Figure 2.**MC simulations (symbols) and forward theoretical calculations using Equation (2) (lines) for media with low and high absorption (top and bottom rows) at SDS of 0.15 and 1.5 cm (left and right columns). Each figure shows data for 5 media with varying scattering distributed between ${\mu}_{s}^{\prime}$ = 3/cm (squares) to ${\mu}_{s}^{\prime}$ = 15/cm (diamonds).

**Figure 3.**MC simulations (symbols) with fitted curves using Equation (2) (lines) via optimizing for the Brownian flow coefficient ${D}_{B}$ (inverse model) to minimize ${\chi}^{2}$ residuals. Simulated data shown in (

**a**,

**d**) are identical (in the same order) as Figure 2a,d. Data in (

**a**,

**c**) are for SDS of 0.15 cm (at low and high absorption coefficients respectively), while data in (

**b**,

**d**) are for SDS of 1.5 cm.

**Figure 4.**Retrieved flow coefficients obtained via using Equation (2) to fit simulations (top-row) and corresponding percent-errors (bottom-row) for 35 different MC simulations spanning seven ${\mu}_{a}$ and five ${\mu}_{s}^{\prime}$ values and fixed ${D}_{B}$ = 2 × 10

^{−8}cm

^{2}/s. Data are shown as contour maps for 3 different SDS (left column: SDS = 0.15 cm; middle column: SDS = 0.45 cm; right column: SDS = 0.75 cm). Note that the color-bar scale is fixed for the top row but changes across SDS for the bottom row.

**Figure 5.**Percent-errors in retrieved Brownian flow coefficient ${D}_{B}$ with varying SDS from analysis of simulations. Lines show data for 6 simulated ${D}_{B}$ coefficients (identified in the legend). Symbols represent mean percent errors calculated across all available simulations for given SDS and ${D}_{B}$, while error bars shows standard deviations. Data for each point (in each line) was computed across 35 simulations run (see text) with varying absorption and scattering. The lines were identical to each other and are shown staggered only for visual clarity.

**Figure 6.**Relative change in extracted flow coefficients (relative to values obtained from media simulated with ${D}_{B}$ = 5 × 10

^{−9}cm

^{2}/s) at three different SDS. The true relative change in ${D}_{B}$ (x-axis) was obtained from input values to simulations, while the derived values (y-axis) were obtained from fitted coefficients (see text).

**Figure 7.**Scaling (correction) factors determined using simulations at two SDS (top: 0.25 cm; bottom: 1.05 cm) shown for two different media with two different flow coefficients (left: ${D}_{B}$ = 1 × 10

^{−8}cm

^{2}/s; right: ${D}_{B}$ = 3 × 10

^{−8}cm

^{2}/s). Each panel shows data for 35 MC simulations spanning seven ${\mu}_{a}$ and five ${\mu}_{s}^{\prime}$ values (see text). Note that the color-bar scale changes for data in top and bottom rows.

**Figure 8.**Estimated Brownian flow coefficients vs. simulated values. (

**a**) shows the coefficients obtained directly by fitting simulated data, where each line shows data for different SDS. Symbols were average values obtained from 35 simulated media having fixed ${D}_{B}$ (but changing optical properties); errorbars show standard deviations. (

**b**) shows the same data in (

**a**) after appropriate scaling by correction factors (see text). The errorbars in (

**b**) are smaller than the size of symbols shown and different lines (for different SDS) are shown staggered only for clarity. The dashed line in Figure 8a,b shows $y=x$.

**Figure 9.**Experimental data (symbols) and fits using Equation (2) for three different flow models (dash-dotted: Brownian; solid: Random; dashed: shear) for two different flow phantoms. (

**a**) shows representative data in phantom ${D}_{0}{F}_{0}$ (pump flow turned off) while (

**b**) shows data from phantom ${D}_{0}{F}_{3}$ (highest pump flow).

**Figure 10.**Derived flow coefficients obtained from fitting phantom data using Equation (2). (

**a**,

**b**) show the retrieved Brownian diffusion ${D}_{B}$ and flow speed $V$, for each phantom. Each line shows fixed channel depth (shown in legend), symbols show mean values and error bars are standard deviations (across the five DCS scans obtained for each phantom). (

**c**,

**d**) show relative changes flow vs. relative change in input flow rates, normalized relative to flow phantoms ${F}_{1}$. Dashed line in (

**c**,

**d**) show $y=x$.

**Figure 11.**DCS measurements (symbols) for phantoms ${D}_{0}{F}_{0}-{D}_{3}{F}_{0}$, (

**a**–

**d**, respectively) with theoretical fits using a Brownian-flow model (dashed line) and MC predictions (solid line). Each MC model used input ${D}_{B}$ values obtained from scaling the theoretically fitted values (see text).

**Figure 12.**Comparisons of experimentally measured data (circles), diffusion theory-based fits (dashed line) and MC predictions (solid line) for the 12 phantoms with actively pumped. (

**a**–

**d**) show these data when flow rate was ${F}_{1}$ = 1.44 mL/h for depths ${D}_{0}$–${D}_{3}$. ((

**e**–

**h**) show the same sequence of depths at flow rate ${F}_{2}$ = 3.6 mL/h and (

**i**–

**l**) for flow rate ${F}_{3}$ = 5.76 mL/h).

Layer # | ${\mathit{\mu}}_{\mathit{a}}\text{}(1/\mathbf{cm})$ | ${\mathit{\mu}}_{\mathit{s}}\text{}(1/\mathbf{cm})$ | $\mathit{g}$ | $\mathbf{Refractive}\text{}\mathbf{Index},\text{}\mathit{n}$ | ${\mathit{z}}_{1}\text{}\left(\mathbf{cm}\right)$ | $\langle \mathbf{\Delta}{\mathit{r}}^{2}\left(\mathit{\tau}\right)\rangle $ |
---|---|---|---|---|---|---|

Layer 1 | 0.075 | 120 | 0.9 | 1.35 | $h$ | $6{D}_{B}\tau $ |

Layer 2 | 0.15 | $6{D}_{B}\tau $ or ${V}^{2}{\tau}^{2}$ | ||||

Layer 3 | 5 | $6{D}_{B}\tau $ |

**Table 2.**Input coefficients for MC flow phantom models. $\Delta h$ was constrained to be 0.03 cm experimentally while the relative change in fluid flow speed from flow rate of ${F}_{1}$ to ${F}_{2}$ was 2.5 (change from ${F}_{1}$ to ${F}_{3}$ was 4).

Phantom | $\mathbf{Layer}1\text{}\mathbf{Thickness}\text{}{\mathit{z}}_{1}\text{}\left[\mathbf{cm}\right]$ | ${\mathit{D}}_{\mathit{B}}\text{}[{\mathbf{cm}}^{2}/\mathbf{s}]\text{}\left(\mathbf{Layers}\text{}1\text{}3\right)$ | ${\mathit{D}}_{\mathit{B}}\text{}[{\mathbf{cm}}^{2}/\mathbf{s}]\text{}\left(\mathbf{Layer}\text{}2\right)$ | $\mathit{V}\text{}[{\mathbf{cm}}^{2}/\mathbf{s}]\text{}\left(\mathbf{Layer}\text{}2\right)$ |
---|---|---|---|---|

D0F0 | ${h}_{0}=$ 0.02 | 3.6 × 10^{−9} | 3.6 × 10^{−9} | - |

D0F1 | - | 5.7 × 10^{−3} | ||

D0F2 | - | 1.42 × 10^{−2} | ||

D0F3 | - | 2.28 × 10^{−2} | ||

D1F0 | ${h}_{1}={h}_{0}+\Delta h=$ 0.05 | 7.9 × 10^{−9} | 7.9 × 10^{−9} | - |

D1F1 | - | 5.7 × 10^{−3} | ||

D1F2 | - | 1.42 × 10^{−2} | ||

D1F3 | - | 2.28 × 10^{−2} | ||

D2F0 | ${h}_{2}={h}_{1}+\Delta h=$ 0.08 | 8.9 × 10^{−9} | 8.9 × 10^{−9} | - |

D2F1 | - | 5.7 × 10^{−3} | ||

D2F2 | - | 1.42 × 10^{−2} | ||

D2F3 | - | 2.28 × 10^{−2} | ||

D3F0 | ${h}_{3}={h}_{2}+\Delta h=$ 0.11 | 9.3 × 10^{−9} | 9.3 × 10^{−9} | - |

D3F1 | - | 5.7 × 10^{−3} | ||

D3F2 | - | 1.42 × 10^{−2} | ||

D3F3 | - | 2.28 × 10^{−2} |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vishwanath, K.; Zanfardino, S.
Diffuse Correlation Spectroscopy at Short Source-Detector Separations: Simulations, Experiments and Theoretical Modeling. *Appl. Sci.* **2019**, *9*, 3047.
https://doi.org/10.3390/app9153047

**AMA Style**

Vishwanath K, Zanfardino S.
Diffuse Correlation Spectroscopy at Short Source-Detector Separations: Simulations, Experiments and Theoretical Modeling. *Applied Sciences*. 2019; 9(15):3047.
https://doi.org/10.3390/app9153047

**Chicago/Turabian Style**

Vishwanath, Karthik, and Sara Zanfardino.
2019. "Diffuse Correlation Spectroscopy at Short Source-Detector Separations: Simulations, Experiments and Theoretical Modeling" *Applied Sciences* 9, no. 15: 3047.
https://doi.org/10.3390/app9153047